#  USTC-ICTS/PCFT-21-41 Modular flavor symmetry and vector-valued modular
forms

Xiang-Gan Liu111E-mail<EMAIL_ADDRESS>Gui-Jun Ding222E-mail:
<EMAIL_ADDRESS>
1Interdisciplinary Center for Theoretical Study and Department of Modern
Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
2Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China

###### Abstract

We revisit the modular flavor symmetry from a more general perspective. The
scalar modular forms of principal congruence subgroups are extended to the
vector-valued modular forms, then we have more possible finite modular groups
including $\Gamma_{N}$ and $\Gamma^{\prime}_{N}$ as the flavor symmetry. The
theory of vector-valued modular forms provide a method of differential
equation to construct the modular multiplets, and it also reveals the simple
structure of the modular invariant mass models. We review the theory of
vector-valued modular forms and give general results for the lower dimensional
vector-valued modular forms. The general finite modular groups are listed up
to order 72. We apply the formalism to construct two new lepton mass models
based on the finite modular groups $A_{4}\times Z_{2}$ and $GL(2,3)$.

###### Contents

  1. 1 Introduction
  2. 2 The theory of vector-valued modular forms
    1. 2.1 Vector-valued modular forms and free-module theorem
    2. 2.2 Modular linear differential equation
  3. 3 Some results on low dimensional VVMFs
    1. 3.1 One-dimensional irreducible VVMFs
    2. 3.2 Two-dimensional irreducible VVMFs
    3. 3.3 Three-dimensional irreducible VVMFs
  4. 4 The general finite modular groups and modular invariant theory
  5. 5 New modular invariant lepton models
    1. 5.1 Model 1 based on finite modular group $A_{4}\times Z_{2}$
      1. 5.1.1 Structure of the model
    2. 5.2 Model 2 based on finite modular group $GL(2,3)$
      1. 5.2.1 Structure of the model
  6. 6 Summary and outlook
  7. A All 2-d irreducible representations of $SL(2,\mathbb{Z})$ with finite image
  8. B VVMF modules of the finite modular group $A_{4}\times Z_{2}$
  9. C Finite number of the modular invariant models
  10. D $GL(2,3)$ group

##  1 Introduction

The Standard Model (SM) of particle physics is an extremely successful theory.
It describes the interactions between fundamental particles including quarks
and leptons. The SM can explain almost all experimental results and precisely
predict a wide variety of phenomena. However, it does not predict neither the
masses nor the mixing parameters of the fundamental particles . The origin of
flavor mixing parameters and hierarchical fermion mass is of the greatest
puzzle of SM, see Ref. [1] for a overview of the flavor structure of SM. The
flavor symmetry acting on flavour space is one of the few guiding principles
to address this problem. The flavor group $G_{f}$ can be choosed as continuous
Lie groups or discrete non-abelian finite groups to associate the fermions
between different family. In order to get a realistic mass spectrum and flavor
mixing pattern, flavor group $G_{f}$ must be broken down to different
subgroups in different flavor sectors, which relies on a carefully designed
potential and often introduces extra symmetries and the so-called flavons. A
latest review can be seen in [2].

Recently modular invariance has been proposed from a bottom-up perspective as
a promising framework for understanding the flavor puzzle [3], in which
modular symmetry based on the infinite discrete group $SL(2,\mathbb{Z})$ plays
the role of flavor symmetry, but in a nonlinear way. In addition, the
traditional flavons are replaced by modulus field $\tau$, and the modular
invariance restricts the Yukawa couplings to be the modular forms, which are
holomorphic functions of $\tau$. If the integer weight modular forms of level
$N$ are used as the building block, the finite modular group is
$\Gamma_{N}\equiv\bar{\Gamma}/\bar{\Gamma}(N)$ or its double covering group
$\Gamma^{\prime}_{N}\equiv\Gamma/\Gamma(N)$ [4]. Modular flavor symmetry has
been exploited to explain the hierarchical masses and mixing patterns in the
lepton and quark sectors, and many models have been built by using the groups
$\Gamma_{2}\cong S_{3}$ [5, 6], $\Gamma_{3}\cong A_{4}$ [3, 7, 5, 8, 9, 6, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], $\Gamma_{4}\cong S_{4}$ [23,
24, 25, 26, 27, 28, 29, 14, 30, 31], $\Gamma_{5}\cong A_{5}$ [32, 33, 28],
$\Gamma_{7}\cong\mathrm{PSL}(2,\mathbb{Z}_{7})$ [34],
$\Gamma^{\prime}_{3}\cong T^{\prime}$ [4, 35], $\Gamma^{\prime}_{4}\cong
S^{\prime}_{4}$ [36, 37], $\tilde{\Gamma}_{4}\cong\tilde{S}_{4}$ [38],
$\Gamma^{\prime}_{5}\cong A^{\prime}_{5}$ [39, 40], $\Gamma^{\prime}_{6}\cong
S_{3}\times T^{\prime}$ [41] and $\tilde{\Gamma}_{5}\cong A^{\prime}_{5}\times
Z_{5}$ [40]. The possible role of fractional weight modular forms has been
explored, and the finite modular group should be extend to the metaplectic
cover $\widetilde{\Gamma}_{N}\equiv\widetilde{\Gamma}/\Gamma(N)$ should be
considered [38]. The modular forms of weights $k/2$ with finite metaplectic
modular group $\tilde{\Gamma}_{4}\cong\tilde{S}_{4}$ [38] and the weight $k/5$
modular forms with finite metaplectic modular group $\tilde{\Gamma}_{5}\cong
A^{\prime}_{5}\times Z_{5}$ [40] have been studied. Furthermore, the modular
invariance framework is extended to allow for multiple moduli, and the
classical modular forms are replaced by more extensive automorphic forms [42].
Moreover, as the first nontrivial example, the symplectic modular symmetry is
presented [42], which is based on Siegel modular group $Sp(2g,\mathbb{Z})$ and
is a natural extension of modular group $SL(2,\mathbb{Z})$. The generalized CP
symmetry can be consistently combined with symplectic modular symmetries for
both single modulus with $g=1$ [43, 44, 45] and multiple moduli with $g\geq 2$
[46], so that all the coupling coefficients become real under the canonical CP
basis and real Clebsch-Gordan coefficients. There have also been attempts to
embed modular symmetry into Grand Unified Theories [47, 48, 49, 50, 51, 52,
53, 54]. Some semi-quantitative characteristics and formalism of fermion mass
matrices near fixed points of $\tau$ have also been developed [55, 56, 57].
The modular flavor symmetry has been explored from top-down perspective, and
some finite modular groups can be naturally realized in top-down constructions
based on string theory [44, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68].

In the modular flavor symmetry [3], it is assumed that the Yukawa couplings
are modular forms of the principal congruence subgroup $\Gamma(N)$ and the
matter fields transform in representations of the homogeneous (inhomogeneous)
finite modular group $\Gamma^{\prime}_{N}$ ($\Gamma_{N}$). The purpose of this
paper is to revisit the modular invariance framework from a more general
perspective. We extend the modular form of $\Gamma(N)$ to vector-valued
modular form in irreducible representation of $SL(2,\mathbb{Z})$, and we focus
on the representations with finite image. The matter fields are characterized
by their modular weights and transformation properties under
$SL(2,\mathbb{Z})$. Then the finite modular groups acting as flavor symmetry
emerge as the images of the involved representations. As a result, we have
more candidates for the finite modular groups other than $\Gamma^{\prime}_{N}$
and $\Gamma_{N}$, which are generally the quotients of $SL(2,\mathbb{Z})$
modulo its normal subgroups of finite index. Moreover, from the view of
vector-valued modular form, the structure of the modular form space becomes
much simpler and we can see there are only finite inequivalent modular
invariant models with a given finite modular group.

This paper is organized as follows. In section 2, the theory of vector-valued
modular forms is reviewed. In section 3, we give the general forms of the low
dimensional vector-valued modular forms by solving the modular linear
differential equation. We list the general finite modular groups up to order
72 in section 4, and the modular flavor symmetry is extended by considering
the vector-valued modular forms of $SL(2,\mathbb{Z})$ instead of the scalar
modular forms of $\Gamma(N)$. In section 5, we present two example models for
lepton masses and mixing based on the finite modular groups $A_{4}\times
Z_{2}$ and $GL(2,3)$. Finally, we draw the conclusions in section 6. In
Appendix A, we list all two-dimensional irreducible representations of
$SL(2,\mathbb{Z})$ with finite image. The vector-valued modular form modules
of $A_{4}\times Z_{2}$ are presented in Appendix B. We explain why there are
finite possible lepton models based on a finite modular group in Appendix C.
The finite modular groups $GL(2,3)$, the tensor product decomposition and the
Clebsch-Gordan Coefficients are given in Appendix D.

##  2 The theory of vector-valued modular forms

The special linear group $SL(2,\mathbb{Z})\equiv\Gamma$ called the (full)
modular group consists of matrices with integer entries and determinant 1:

$SL(2,\mathbb{Z})=\left\\{\begin{pmatrix}a&~{}b\\\
c&~{}d\end{pmatrix}\Big{|}ad-bc=1\,,\quad a,b,c,d\in\mathbb{Z}\,\right\\}\,.$
(2.1)

It is an infinite discrete group and can be generated by two generators $S$
and $T$ obeying the following relations

$S^{4}=(ST)^{3}=1\,,\quad S^{2}T=TS^{2}\,,$ (2.2)

with

$S=\begin{pmatrix}0&~{}1\\\ -1&~{}0\end{pmatrix}\,,\qquad
T=\begin{pmatrix}1&~{}1\\\ 0&~{}1\end{pmatrix}\,.$ (2.3)

Note that $S^{2}=-\mathbb{1}_{2}$, where $\mathbb{1}_{2}$ denotes the two-
dimensional identity matrix. The modular group $\Gamma$ acts on the complex
upper half-plane $\mathcal{H}=\\{\tau\in\mathbb{C}~{}|~{}\Im\tau>0\\}$ by
linear fractional transformation

$\gamma\tau\equiv\frac{a\tau+b}{c\tau+d}\,,\qquad\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma\,.$ (2.4)

It is easy to find that $\gamma$ and $-\gamma$ give the same action on $\tau$
and thus the faithful action of the linear fractional transformation is given
by the projective special linear group $PSL(2,\mathbb{Z})$ which is the
quotient group $PSL(2,\mathbb{Z})\cong
SL(2,\mathbb{Z})/\\{\mathbb{1}_{2},-\mathbb{1}_{2}\\}$. Obviously
$S^{2}=-\mathbb{1}_{2}$ commutes with all element of $SL(2,\mathbb{Z})$,
consequently $\\{\mathbb{1}_{2},-\mathbb{1}_{2}\\}$ is exactly the center of
$SL(2,\mathbb{Z})$ and by Schur’s Lemma $\rho(S^{2})$ is proportional to an
identity matrix for any representation $\rho$ of $SL(2,\mathbb{Z})$.
Furthermore we have $\rho(S^{2})=\pm\mathbb{1}_{d}$ because of $S^{4}=1$,
where $d$ denotes the dimension of the representation $\rho$. We call the
representations with $\rho(S^{2})=\mathbb{1}_{d}$ and
$\rho(S^{2})=-\mathbb{1}_{d}$ as even and odd representations respectively.
Any irreducible representation is either even or odd.

###  2.1 Vector-valued modular forms and free-module theorem

The theory of vector-valued modular form (VVMF) first appeared in the work of
Selberg in 1960s [69], but it is not until the last two decades that the
general theory of VVMF has been systematically developed thanks to the work of
Mason, Gannon, Franc and et al [70, 71, 72, 73, 74, 75]. In physics, VVMFs
appear as characters in rational conformal field theory [76, 77].

Let $\rho$ denote a $d$-dimensional representation of $SL(2,\mathbb{Z})$. A
vector-valued modular form $Y(\tau)=(Y_{1}(\tau),\dots,Y_{d}(\tau))^{T}$ of
weight $k$ in representation $\rho$ is a holomorphic vector-valued function in
the upper half-plane $\mathcal{H}$, and transforms as333In addition, $Y(\tau)$
is generally required to satisfy the so-called moderate growth condition,
i.e., there is an integer $N$ such that for each components $Y_{j}(\tau)$ we
have $|Y_{j}(\tau)|<\Im(\tau)^{N}$ for any fixed $\tau\in\mathcal{H}$ and
$\Im(\tau)\gg 0$. In other word, $Y_{j}(\tau)$ is bounded as
$\Im(\tau)\rightarrow+\infty$.

$Y(\gamma\tau)=(c\tau+d)^{k}\rho(\gamma)Y(\tau)\,,$ (2.5)

where $k$ is called the modular weight and $\rho(\gamma)$ is the irreducible
representation matrix of $SL(2,\mathbb{Z})$. In the present work, we will
focus on the case that $k$ is non-negative integer and $\rho$ has finite
image. The benefits are as follows:

  * (i)

$\rho$ can be taken to be unitary since the image of $\rho$ is finite, and it
is also completely reducible by Maschke theorem [78], i.e., it can be
decomposed into a direct sum of irreducible representations. Therefore, we can
further restrict $\rho$ to the unitary irreducible representations without
loss of generality.

  * (ii)

The order of $\rho(\gamma)$ is finite for any $\gamma\in SL(2,\mathbb{Z})$
such that we can always work in the basis where $\rho(T)$ is diagonal, and the
eigenvalues of $\rho(T)$ are roots of unity.

  * (iii)

Finiteness of the image of $\rho$ implies that the kernel $\ker(\rho)$ is the
finite index normal subgroup of $SL(2,\mathbb{Z})$.

Notice that the automorphy factor $j^{k}(\gamma,\tau)\equiv(c\tau+d)^{k}$ is a
monodrome function for integral $k$. If the modular weight $k$ is a rational
number, the automorphy factor would be a multi-valued function and $\rho$ is a
projective representation. Then $SL(2,\mathbb{Z})$ should be extended to its
metaplectic cover group $Mp(2,\mathbb{Z})$ in order to avoid the multi-valued
ambiguity [38]. By taking $\gamma=S^{2}$ in Eq. (2.5) and noticing
$S^{2}\tau=\tau$, we have

$Y(\tau)=(-1)^{k}\rho(S^{2})Y(\tau)\,,$ (2.6)

which leads to

$\rho(S^{2})=(-1)^{k}\mathbb{1}_{d}\,.$ (2.7)

Therefore the odd/even weight VVMFs are in odd/even irreps. It is remarkable
that each component of VVMF $Y(\tau)$ in the representation $\rho$ is scalar
modular form of $\ker(\rho)$, this is obvious when $\gamma$ in Eq. (2.5) is
limited to $\ker(\rho)$ rather than $SL(2,\mathbb{Z})$. The converse is also
true, i.e., scalar modular forms for normal subgroups of $SL(2,\mathbb{Z})$
can be organized into the VVMFs for $SL(2,\mathbb{Z})$, as shown in Ref. [4].
Notice that VVMFs in representation $\rho$ are referred to as modular
multiplets of the finite modular group $\Gamma/\ker(\rho)$ in the modular
flavor symmetry [3].

As mentioned above, it is convenient to work in the $T$ diagonal basis and the
representation matrix $\rho(T)$ can be written as

$\rho(T)=\texttt{diag}(e^{2\pi ir_{1}},\dots,e^{2\pi ir_{d}})\,,\qquad 0\leq
r_{i}<1\,,$ (2.8)

where $r_{i}$ are rational numbers. In this $T$-diagonal basis, the moderate
growth condition implies that every nonzero VVMF
$Y(\tau)\in\mathcal{M}_{k}(\rho)$ has the following holomorphic $q$-expansion
[79]

$Y(\tau)=\begin{pmatrix}Y_{1}(\tau)\\\ \vdots\\\
Y_{d}(\tau)\end{pmatrix}=\begin{pmatrix}q^{r_{1}}\sum_{n\geq
0}a_{1}(n)q^{n}\\\ \vdots\\\ q^{r_{d}}\sum_{n\geq
0}a_{d}(n)q^{n}\end{pmatrix},~{}~{}~{}~{}q=e^{2\pi i\tau}\,,$ (2.9)

where the coefficients $a_{j}(n)$ are generally complex.

As the simplest examples, it is easy to observe that VVMFs are reduced to the
(scalar) modular forms of $SL(2,\mathbb{Z})$ for trivial representation
$\rho=\mathbf{1}$. As a well-known result, the modular forms space
$\mathcal{M}(\mathbf{1})=\bigoplus^{\infty}_{k=0}\mathcal{M}_{k}(\mathbf{1})$
is just a graded ring generated by the Eisenstein series $E_{4}(\tau)$ and
$E_{6}(\tau)$ [80]:

$E_{4}(\tau)=1+240\sum_{n=1}^{\infty}\sigma_{3}(n)q^{n}\,,~{}~{}~{}E_{6}(\tau)=1-504\sum_{n=1}^{\infty}\sigma_{5}(n)q^{n}\,,$
(2.10)

where $\sigma_{k}(n)=\sum_{d|n}d^{k}$ is the sum of the $k$th power of the
divisors of $n$. We will denote by $\mathcal{M}_{k}(\rho)$ the linear space of
holomorphic VVMFs of weight $k$ in irrep $\rho$, and the direct sum
$\mathcal{M}(\rho)=\bigoplus_{k=0}^{\infty}\mathcal{M}_{k}(\rho)$ stands for
all the VVMFs in the irrep $\rho$. Obviuosly multiplying a VVMF
$Y(\tau)\in\mathcal{M}_{k}(\rho)$ with a scalar modular form
$f(\tau)\in\mathcal{M}_{2m}(\mathbf{1})$ results in a new VVMF
$f(\tau)Y(\tau)\in\mathcal{M}_{k+2m}(\rho)$. Hence the direct sums
$\mathcal{M}(\rho)=\bigoplus^{\infty}_{k=0}\mathcal{M}_{k}(\rho)$ is a graded
module over the ring $\mathcal{M}(\mathbf{1})=\mathbb{C}[E_{4},E_{6}]$ 444A
module is a generalization of the notion of vector space, where scalars are on
a ring and no longer has to be on a field. If a module $\mathcal{M}$ can be
generated by a linearly independent set $X$, we say $\mathcal{M}$ is a free
module and $X$ is a basis of $\mathcal{M}$. The cardinality of the basis $X$
is called the rank of the free module $\mathcal{M}$ [81].. It can be proved
that $\mathcal{M}(\rho)$ is a free module whose rank is equal to the dimension
of the representation $\rho$ [72]. This is the famous free-module theorem
about VVMFs, and it is important to explicitly construct VVMFs, as shown in
the following section.

###  2.2 Modular linear differential equation

Let us consider the modular differential operators acting on the VVMFs of
weight $k$ [82],

$D_{k}\equiv\theta-\frac{kE_{2}}{12}\,,\quad\theta\equiv
q\frac{d}{dq}=\frac{1}{2\pi i}\frac{d}{d\tau},\quad k\in\mathbb{Z}\,,$ (2.11)

where $E_{2}(\tau)$ is the well-known quasi-modular Eisenstein series [80]

$E_{2}(\tau)=1-24\sum_{n=1}^{\infty}\sigma_{1}(n)q^{n}\,.$ (2.12)

Notice that $E_{2}(\tau)$ is not a modular form of weight $2$, and it fulfills
the following transformation formula

$E_{2}(\gamma\tau)=(c\tau+d)^{2}E_{2}(\tau)-\frac{6i}{\pi}c(c\tau+d)\,,$
(2.13)

where the last term violates the modularity. It is not hard to verify, $D_{k}$
preserves modularity of modular forms and increasing its weight by 2, i.e.

$(D_{k}f)(\gamma\tau)=(c\tau+d)^{k+2}(D_{k}f)(\tau)\,\qquad\text{for}\quad
f\in\mathcal{M}_{k}(\mathbf{1})\,,$ (2.14)

and

$(D_{k}Y)(\gamma\tau)=(c\tau+d)^{k+2}\rho(\gamma)(D_{k}Y)(\tau)\,\qquad\text{for}\quad
Y\in\mathcal{M}_{k}(\rho)\,.$ (2.15)

A very common and important example first observed by Ramanujan is [82]

$D_{2}(E_{2})=-\frac{E_{4}}{12}\,,\qquad D_{4}(E_{4})=-\frac{E_{6}}{3}\,,\quad
D_{6}(E_{6})=-\frac{E_{4}^{2}}{2}\,.$ (2.16)

For any two scalar modular forms $f$ and $g$ which are of weight $k$ and $m$
respectively, the modular derivative satisfies the Leibniz law as follows,

$D_{k+m}(fg)=(D_{k}f)g+f(D_{m}g)\,.$ (2.17)

For $n\geq 0$ , the $n$th iteration of $D_{k}$ is denoted by

$D^{n}_{k}\equiv D_{k+2(n-1)}\circ D_{k+2(n-2)}\circ\dots\circ D_{k}\,,$
(2.18)

For a generic $d-$dimensional VVMF
$F=\left(f_{1},\ldots,f_{d}\right)^{T}\in\mathcal{M}_{k}(\rho)$, then the
$d+1$ VVMFs $\\{F,D_{k}F,\dots,D_{k}^{d}F\\}$ are linearly dependent over
$\mathcal{M}(\mathbf{1})$ since the rank of the module $\mathcal{M}(\rho)$ is
equal to $d$. Hence there must exist $M_{j}\in\mathcal{M}_{j}(\mathbf{1})$
such that the following equation on the module $\mathcal{M}(\rho)$ is
satisfied

$(D^{d}_{k}+M_{2}D_{k}^{d-1}+M_{4}D_{k}^{d-2}+\dots+M_{2(d-1)}D_{k}+M_{2d})f_{i}=0\,,$
(2.19)

The above equation is actually the so-called $d$th order modular linear
differential equation (MLDE) in weight $k$. From the basic theory of linear
differential equations, we know that a homogeneous linear differential
equation of order $d$ has $d$ linearly independent solutions. Hence the
solution system forms $d$ linearly independent components of $F$.

On the other hand, the set VVMFs $\\{F,D_{k}F,\dots,D_{k}^{d-1}F\\}$ are
linearly independent over $\mathcal{M}(\mathbf{1})$, otherwise they would
satisfy a MLDE of order at most $d-1$, for which each of the $d$ linearly
independent components of $F$ is a solution, this contradicts the
irreducibility of $F$. It is worth noting that linear independence of
$\\{F,D_{k}F,\dots,D_{k}^{d-1}F\\}$ does not mean they can be used as a basis
of module $\mathcal{M}(\rho)$. In fact, they generally span rank $d$ free
submodule of $\mathcal{M}(\rho)$. Unless the module $\mathcal{M}(\rho)$ is
cyclic555A module $\mathcal{M}(\rho)$ is called cyclic if and only if there is
$F$ of minimal weight $k_{0}$ such that
$\\{F,D_{k_{0}}F,\dots,D^{d-1}_{k_{0}}F\\}$ generates $\mathcal{M}(\rho)$. We
simply call $\rho$ is cyclic type in this case. and $F$ is the VVMF of minimal
weight. Interestingly, the module $\mathcal{M}(\rho)$ of irrep $\rho$ with
$\dim\rho\leq 3$ is always cyclic, while there is only non-cyclic module
$\mathcal{M}(\rho)$ for $\dim\rho>6$ [75].

Dimension of $\rho$ | Weight profile | Minimal weight $k_{0}$
---|---|---
1 | $(k_{0})$ | $12\text{Tr}(L)$
2 | $(k_{0},k_{0}+2)$ | $6\text{Tr}(L)-1$
3 | $(k_{0},k_{0}+2,k_{0}+4)$ | $4\text{Tr}(L)-2$
4 | $(k_{0},k_{0}+2,k_{0}+4,k_{0}+6)$ | $3\text{Tr}(L)-3$
$(k_{0},k_{0}+2,k_{0}+2,k_{0}+4)$ | $3\text{Tr}(L)-2$
5 | $(k_{0},k_{0}+2,k_{0}+4,k_{0}+6,k_{0}+8)$ | $(12\text{Tr}(L)-20)/5$
$(k_{0},k_{0}+2,k_{0}+4,k_{0}+4,k_{0}+6)$ | $(12\text{Tr}(L)-16)/5$
$(k_{0},k_{0}+2,k_{0}+2,k_{0}+4,k_{0}+4)$ | $(12\text{Tr}(L)-12)/5$
$(k_{0},k_{0},k_{0}+2,k_{0}+2,k_{0}+4)$ | $(12\text{Tr}(L)-8)/5$
$(k_{0},k_{0}+2,k_{0}+2,k_{0}+4,k_{0}+6)$ | $(12\text{Tr}(L)-14)/5$
Table 1:  The weight profile and minimal weight of the module
$\mathcal{M}(\rho)$ with $\dim\rho<6$, and the parameter $L$ is defined by
$\rho(T)\equiv e^{2\pi iL}$ which implies
$L=\text{diag}(r_{1},r_{2},\ldots,r_{d})$. There are two types of irreps for
dimension $4$, namely cyclic and non-cyclic, and corresponding to
$\rho(S^{2})=-(-1)^{3\text{Tr}(L)\,\text{mod}\,2}$ and
$\rho(S^{2})=(-1)^{3\text{Tr}(L)\,\text{mod}\,2}$ respectively [83]. There are
five types of irreps for dimension $5$, corresponding to
$\text{Tr}(L)=\frac{5n+i}{12}$ with $i=0,1,2,3,4$ and $n\in\mathbb{Z}$.

The structure of the module $\mathcal{M}(\rho)$ is characterized by the
weights of a basis of $\mathcal{M}(\rho)$, they are denoted as a tuple
$(k_{0},\dots,k_{d-1})$ which is called the weight profile of $\rho$. Here
$k_{0}$ is the minimal weight and it is the minimal integer such that
$\mathcal{M}_{k_{0}}(\rho)\neq 0$. The minimal weights and the weight profiles
of the module $\mathcal{M}(\rho)$ for $\dim\rho<6$ are summarized in the table
1, which is adopted from Ref. [75]. With these weight profiles, we can get a
convenient dimension formula by using the Hilbert-Poincaré series which is the
formal power series for module $\mathcal{M}(\rho)$ [84]:

$P(\mathcal{M}(\rho))(t)=\sum_{k\geq
k_{0}}t^{k}\dim(\mathcal{M}_{k}(\rho))\,,$ (2.20)

since $\mathcal{M}(\rho)$ is a free module of rank $d$ over
$\mathbb{C}[E_{4},E_{6}]$, its Hilbert-Poincaré series has the following
general form:

$P(\mathcal{M}(\rho))(t)=\frac{t^{k_{0}}+\dots+t^{k_{d-1}}}{(1-t^{4})(1-t^{6})}\,,$
(2.21)

where $(k_{0},\dots,k_{d-1})$ is the weight profile of $\rho$. From table 1,
we can easily write the generating function (2.21) of each module
$\mathcal{M}(\rho)$, and then read the corresponding dimension of
$\mathcal{M}_{k}(\rho)$ from the power series expansion666The more general
dimension formula of the linear space $\mathcal{M}_{k}(\rho)$ for $k\geq 2$
[85]
$\dim\mathcal{M}_{k}(\rho)=\frac{(5+k)\dim\rho}{12}+\frac{i^{k}\text{Tr}(\rho(S^{3}))}{4}+\frac{(-\omega^{2})^{k}\text{Tr}(\rho(S^{3}T))}{3(1-\omega)}+\frac{\omega^{k}\text{Tr}(\rho((ST)^{2}))}{3(1-\omega^{2})}-\text{Tr}(L)$
(2.22) with $\omega=e^{2\pi i/3}$. There is still no general dimension formula
for $k=1$, although Eq. (2.22) is still applicable in some cases.. Taking the
three-dimensional case as an example, the Hilbert-Poincaré series of
$\mathcal{M}(\rho)$ is

$\frac{t^{k_{0}}(1+t^{2}+t^{4})}{(1-t^{4})(1-t^{6})}=t^{k_{0}}+t^{k_{0}+2}+2t^{k_{0}+4}+2t^{k_{0}+6}+\dots\,,$
(2.23)

with $k_{0}=4\text{Tr}(L)-2$. Therefore we know that there is only one modular
multiplet at the weights $k_{0}$ and $k_{0}+2$, two modular multiplets at the
weights $k_{0}+4$ and $k_{0}+6$, and so on.

For these low-dimensional VVMFs, the free base of module $\mathcal{M}(\rho)$
can be inferred from the weight profile and the free module theorem. For
instance, at lower dimensions $d=1,2,3,4$ [84], we have

$\displaystyle\dim\rho=1:~{}~{}\mathcal{M}(\rho)=\langle F\rangle\,,$
$\displaystyle\dim\rho=2:~{}~{}\mathcal{M}(\rho)=\langle
F,~{}D_{k_{0}}F\rangle\,,$
$\displaystyle\dim\rho=3:~{}~{}\mathcal{M}(\rho)=\langle
F,~{}D_{k_{0}}F,~{}D^{2}_{k_{0}}F\rangle\,,$
$\displaystyle\dim\rho=4:~{}~{}\mathcal{M}(\rho)=\begin{cases}\langle
F,~{}D_{k_{0}}F,~{}D^{2}_{k_{0}}F,~{}D^{3}_{k_{0}}F\rangle~{}~{}\text{for
cyclic type}~{}~{}\rho\\\ \langle
F,~{}D_{k_{0}}F,~{}G,~{}D^{2}_{k_{0}}F\rangle~{}~{}~{}~{}~{}~{}\,\text{for
non-cyclic type}~{}~{}\rho\end{cases}\,,$ (2.24)

where $F$ in each case is the corresponding minimal weight VVMF, and $G$
denotes the another independent VVMF at weight $k_{0}+2$ in addition to
$D_{k_{0}}F$. With these bases, we can write any high-weight VVMF as a linear
combination of them over the module $\mathcal{M}(\mathbf{1})$. For
$\dim\rho=3$ as an example, the VVMF $D^{3}_{k_{0}}F$ of weight $k_{0}+6$ can
be written as

$D^{3}_{k_{0}}F=aE_{4}D_{k_{0}}F+bE_{6}F\,,$ (2.25)

where $a$ and $b$ is the free complex numbers. This happens to be a $3$rd
order MLDE as shown in Eq. (2.19), and it can be used to solve the concrete
form of $F$, as demonstrated in section 3.

##  3 Some results on low dimensional VVMFs

In this section, we report the explicit form of one-dimensional, two-
dimensional and three-dimensional VVMFs by solving the MLDE in Eq. (2.19). It
is worth mentioning that the MLDE of Eq. (2.19) satisfied by 2-d and 3-d VVMF
can be transformed into the generalized hypergeometric equation of the
following form [74]:

$\left[(\theta_{K}+\beta_{1}-1)\cdots(\theta_{K}+\beta_{n}-1)-K(\theta_{K}+\alpha_{1})\cdots(\theta_{K}+\alpha_{n})\right]f_{i}=0\,.$
(3.1)

with

$\theta_{K}=K\frac{d}{dK}\,,\quad K(\tau)=1728/j(\tau)\,.$ (3.2)

Here $j(\tau)$ is the modular $j$-invariant [80] and it is a modular function
of weight 0,

$j(\tau)=\frac{E^{3}_{4}(\tau)}{\Delta(\tau)}\,,\quad\Delta(\tau)=\frac{E_{4}^{3}(\tau)-E_{6}^{2}(\tau)}{1728}\,,$
(3.3)

The function $j(\tau)$ has a simple pole at $i\infty$, and its $q$-expansion
is

$j(\tau)=q^{-1}+744+196884q+21493760q^{2}+\dots\,.$ (3.4)

When the numbers $\beta_{1},\dots,\beta_{n}$ are distinct mod $\mathbb{Z}$,
the equation (3.1) has $n$ independent solutions which are given by the
generalized hypergeometric series

$f_{i}=K^{1-\beta_{i}}~{}{}_{n}F_{n-1}(1+\alpha_{1}-\beta_{i},\dots,1+\alpha_{n}-\beta_{i};1+\beta_{1}-\beta_{i},\check{\dots},1+\beta_{n}-\beta_{i};K)\,,$
(3.5)

where $\check{}$ denotes omission of $1+\beta_{i}-\beta_{i}$ and the
generalized hypergeometric series ${}_{n}F_{n-1}$ is defined by the formula

${}_{n}F_{n-1}(\alpha_{1},\dots,\alpha_{n};\beta_{1},\dots,\beta_{n-1};z)=\sum_{m\geq
0}^{\infty}\dfrac{\prod_{j=1}^{n}(\alpha_{j})_{m}}{\prod_{j=1}^{n-1}(\beta_{k})_{m}}\dfrac{z^{m}}{m!}\,,$
(3.6)

with $m\in\mathbb{N}$, $\alpha,\beta\in\mathbb{C}$ and $(a)_{m}$ is the
Pochhammer symbol defined by

$(a)_{m}=\begin{cases}1\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\qquad
m=0\,,\\\ a(a+1)\dots(a+m-1)\,,\ \qquad m\geq 1\,.\end{cases}$ (3.7)

For the cases of $d>3$, MLDE can only be solved recursively to obtain the
$q$-expansion of the corresponding VVMFs.

###  3.1 One-dimensional irreducible VVMFs

From the relations $S^{4}=(ST)^{3}=1$ satisfied by the modular generators $S$
and $T$, we know that the $SL(2,\mathbb{Z})$ group has 12 irreps denoted as
$\mathbf{1}_{p}$ with

$\mathbf{1}_{p}:~{}~{}~{}~{}~{}\rho_{\mathbf{1}_{p}}(S)=i^{p}\,,\qquad\rho_{\mathbf{1}_{p}}(T)=e^{\frac{i\pi}{6}p}\,,$
(3.8)

where $p=0,1,\dots,11$. Then one can read off the exponent matrix $L=p/12$ for
$\mathbf{1}_{p}$.

From the previous general result, the 1-d VVMFs space $\mathcal{M}(\rho)$ is
the free $\mathcal{M}(\mathbf{1})$-module of rank 1. Let $F$ denote the basis
of $\mathcal{M}(\rho)$ with the minimal weight $k_{0}$, thus $D_{k_{0}}F$ must
be a linearly dependent on $F$, i.e. $D_{k_{0}}F=gF$ with coefficients
$g\in\mathcal{M}(\mathbf{1})$. Since there are no holomorphic modular forms of
weight 2 in $\mathcal{M}(\mathbf{1})$, it follows that $g=0$. Therefore, the
1-d VVMFs satisfies the following 1st order modular linear differential
equation:

$D_{k_{0}}F=0\,.$ (3.9)

From the Eq. (2.16) and Eq. (3.3), we can know that $D_{12}\Delta=0$. Because
$\Delta=\eta^{24}$ where $\eta(\tau)$ is the Dedekind eta function777It is
defined by $\eta(\tau)=q^{1/24}\prod_{n=1}^{\infty}\left(1-q^{n}\right)$ with
$q=e^{2\pi i\tau}$. and using the Leibniz law (2.17), we can obtain the
important result $D_{1}\eta^{2}=0$. This means that the eta products
$\eta^{2k_{0}}$ is a solution of Eq. (3.9):

$F=\eta^{2k_{0}}\,.$ (3.10)

Furthermore, one can check that $\eta^{2k_{0}}(\tau)$ satisfies the following
transformation form

$\displaystyle\eta^{2k_{0}}(S\tau)=(-\tau)^{k_{0}}i^{k_{0}}\eta^{2k_{0}}(\tau)=(-\tau)^{k_{0}}\rho_{\mathbf{1}_{k_{0}}}(S)\eta^{2k_{0}}(\tau),$
$\displaystyle\eta^{2k_{0}}(T\tau)=e^{\frac{i\pi}{6}k_{0}}\eta^{2k_{0}}(\tau)=\rho_{\mathbf{1}_{k_{0}}}(T)\eta^{2k_{0}}(\tau)\,.$
(3.11)

Therefore $\eta^{2k_{0}}(\tau)$ is the weight $k_{0}$ one-dimensional VVMFs of
$SL(2,\mathbb{Z})$ in the irrep $\rho_{\mathbf{1}_{k_{0}}}$. As a result, the
one-dimensional VVMFs module is

$\mathcal{M}(\mathbf{1}_{p})=\mathbb{C}[E_{4},E_{6}]\eta^{2p}\,,\qquad
p=0,1,\dots,11\,.$ (3.12)

###  3.2 Two-dimensional irreducible VVMFs

All the 2-d irreducible unitary representations of $SL(2,\mathbb{Z})$ with
finite image have been fully obtained in Ref. [86]. In the $T$-diagonal basis,
the representation matrix of $T$ can be generally parameterized as

$\rho(T)=\begin{pmatrix}e^{2\pi ir_{1}}&0\\\ 0&e^{2\pi
ir_{2}}\end{pmatrix}\,,\qquad 0\leq r_{1},r_{2}<1\,.$ (3.13)

A striking feature is that these 2-d irreps are completely specified by the
parameters $r_{1}$ and $r_{2}$. The representation matrix $\rho(S)$ is
uniquely determined by $r_{1,2}$ up to a possible similar transformation, and
it can be parameterized as [86]

$\rho(S)=\frac{1}{\lambda_{1}\lambda_{2}(\lambda_{1}-\lambda_{2})}\begin{pmatrix}1&\sqrt{-(\lambda_{1}\lambda_{2})^{5}(\lambda_{1}-\lambda_{2})^{2}-1}\\\
\sqrt{-(\lambda_{1}\lambda_{2})^{5}(\lambda_{1}-\lambda_{2})^{2}-1}&-1\end{pmatrix}\,,$
(3.14)

with $\lambda_{1,2}=e^{2\pi ir_{1,2}}$. The representation $\rho$ is
irreducible and its determinant $\det\rho$ is an even 1-d representation of
$SL(2,\mathbb{Z})$, thus the parameters $r_{1,2}$ should satisfy [86]

$r_{1}+r_{2}\in\frac{1}{6}\mathbb{Z}\,,\quad
r_{1}-r_{2}\notin\mathbb{Z}\,,~{}\mathbb{Z}\pm\frac{1}{6}\,.$ (3.15)

We will denote the 2-d irreps as $\mathbf{2}_{(r_{1},r_{2})}$ whose exponent
matrix $L$ reads as $L=\texttt{diag}(r_{1},r_{2})$. The modular group
$SL(2,\mathbb{Z})$ has 54 inequivalent irreps with finite image, and the
corresponding values of the unordered pair $(r_{1},r_{2})$ are summarized in
the table 3.

From the general theory of VVMFs in section 2, we know that the 2-d VVMFs
space $\mathcal{M}(\rho)$ is a free $\mathcal{M}(\mathbf{1})$-module of rank
2, and it has the free basis $\\{F,D_{k_{0}}F\\}$ with the minimal weight
$k_{0}=6\text{Tr}(L)-1=6(r_{1}+r_{2})-1$. Because $D^{2}_{k_{0}}F$ is a weight
$k_{0}+4$ VVMF, it can be written as the linear combination of the free basis
over $\mathcal{M}(\mathbf{1})$. Thus, we have

$(D^{2}_{k_{0}}+aE_{4})F=0\,,$ (3.16)

with $a$ is a complex number. This is just a second order MLDE.

Since $D_{k_{0}}(\eta^{2k_{0}})=0$ as shown in section 3.1, we can always
reduce the weight $k_{0}$ MLDE to weight zero MLDE by rescaling $F$ to
$\widetilde{F}=\eta^{-2k_{0}}F$:

$(D^{2}_{0}+aE_{4})\widetilde{F}=0\,,$ (3.17)

As shown in Ref. [74], by introducing the notations

$\displaystyle A=E_{6}/E_{4}\,,\qquad K(\tau)=1728/j(\tau)\,,$
$\displaystyle\theta=q\frac{d}{dq}\,,~{}~{}~{}~{}\qquad\theta_{K}=K\frac{d}{dK}\,,$
(3.18)

where $K(\tau)=1728q(1-744q+\dots)$ is a local parameter, the MLDE (3.17) can
be written into a hypergeometric differential equation:

$\left(\theta_{K}^{2}-\frac{2K+1}{6(1-K)}\theta_{K}+\frac{a}{1-K}\right)\widetilde{F}=0\,.$
(3.19)

From the general $q$-expansion of Eq. (2.9), we know $\widetilde{F}$ can be
expanded in a $K$-series,

$\widetilde{F}=\eta^{-2k_{0}}F=\begin{pmatrix}K^{r_{1}-\frac{k_{0}}{12}}\sum_{n\geq
0}a_{1}(n)K^{n}\\\ K^{r_{2}-\frac{k_{0}}{12}}\sum_{n\geq
0}a_{2}(n)K^{n}\end{pmatrix}\,.$ (3.20)

Considering the leading term of Eq. (3.19) in $K$, we can fix the values of
$k_{0}$ and $a$,

$k_{0}=6(r_{1}+r_{2})-1\,,\qquad
a=\left(r_{1}-\frac{k_{0}}{12}\right)\left(r_{2}-\frac{k_{0}}{12}\right)\,.$
(3.21)

Then the differential equation (3.19) can be written as the generalized
hypergeometric equations of Eq. (3.1) with

$\beta_{1}=\frac{r_{2}-r_{1}}{2}+\frac{11}{12}\,,\quad\beta_{2}=\frac{r_{1}-r_{2}}{2}+\frac{11}{12}\,,\quad\alpha_{1}=0\,,\quad\alpha_{2}=\frac{1}{3}\,.$
(3.22)

Therefore the components of $\widetilde{F}$ as the solutions of Eq. (3.17) are
hypergeometric series of type ${}_{2}F_{1}$, as shown in Eq. (3.5), and the
2-d VVMF $F$ is of the following form

$F(\tau)=\begin{pmatrix}~{}~{}\eta^{12(r_{1}+r_{2})-2}K^{\frac{6(r_{1}-r_{2})+1}{12}}~{}{}_{2}F_{1}(\frac{6(r_{1}-r_{2})+1}{12},\frac{6(r_{1}-r_{2})+5}{12};r_{1}-r_{2}+1;K)\\\
C\eta^{12(r_{1}+r_{2})-2}K^{\frac{6(r_{2}-r_{1})+1}{12}}~{}{}_{2}F_{1}(\frac{6(r_{2}-r_{1})+1}{12},\frac{6(r_{2}-r_{1})+5}{12};r_{2}-r_{1}+1;K)\\\
\end{pmatrix}$ (3.23)

up to an overall irrelevant constant. The coefficient $C$ depends on the
explicit form of the representation matrix $\rho(S)$, and its value can be
fixed to satisfy the condition Eq. (2.5) of VVMF. Hence the module of two-
dimensional VVMFs is given by

$\mathcal{M}(\mathbf{2}_{(r_{1},r_{2})})=\mathbb{C}[E_{4},E_{6}]F\oplus\mathbb{C}[E_{4},E_{6}]D_{k_{0}}F\,.$
(3.24)

###  3.3 Three-dimensional irreducible VVMFs

The 3-d irreducible unitary representations of $SL(2,\mathbb{Z})$ with finite
image have not been fully studied as far as we know, a complete classification
is not available in the literature at present. In the $T$ diagonal basis, the
3-d representation matrix $\rho(T)$ can be parameterized as

$\rho(T)=\begin{pmatrix}e^{2\pi ir_{1}}&0&0\\\ 0&e^{2\pi ir_{2}}&0\\\
0&0&e^{2\pi ir_{3}}\end{pmatrix}\,,\qquad 0\leq r_{1},r_{2},r_{3}<1\,,$ (3.25)

which implies the exponent matrix $L=\texttt{diag}(r_{1},r_{2},r_{3})$. See
Refs. [87, 88] for alternative parameterizations for both $\rho(S)$ and
$\rho(T)$. Similar to the 2-d case, these 3-d irreps are uniquely determined
by the parameters $r_{1},r_{2}$ and $r_{3}$ [87]. Consequently we denote these
3-d irreps by $\mathbf{3}_{(r_{1},r_{2},r_{3})}$. The determinant of $\rho$
forms a 1-d irrep of $SL(2,\mathbb{Z})$ in Eq. (3.8), therefore the parameters
$r_{1,2,3}$ have to fulfill the following constraint:

$r_{1}+r_{2}+r_{3}\in\frac{1}{12}\mathbb{Z}\,.$ (3.26)

The space $\mathcal{M}(\rho)$ of 3-d VVMFs is a rank 3 module over
$\mathcal{M}(\mathbf{1})$ with the free basis
$\\{F,D_{k_{0}}F,D^{2}_{k_{0}}F\\}$ and the minimal weight
$k_{0}=4\text{Tr}(L)-2=4(r_{1}+r_{2}+r_{3})-2$. As shown in Eq. (2.25),
$D^{3}_{k_{0}}F$ can be expressed by the free basis and consequently the
minimal weight VVMF $F$ satisfies the following 3rd order MLDE

$(D^{3}_{k_{0}}+aE_{4}D_{k_{0}}+bE_{6})F=0\,,$ (3.27)

where $a$ and $b$ are two complex numbers. Reparameterizing this equation via
the local parameter $K$, one obtains

$\left(\theta_{K}^{3}-\frac{2K+1}{2(1-K)}\theta^{2}_{K}+\frac{18a+1-4K}{18(1-K)}\theta_{K}+\frac{b}{1-K}\right)\widetilde{F}=0\,.$
(3.28)

where $\widetilde{F}=\eta^{-2k_{0}}F$. In the same fashion as section 3.2, we
find the values of $k_{0}$, $a$ and $b$ are

$\displaystyle k_{0}=4(r_{1}+r_{2}+r_{3})-2\,,$ $\displaystyle
a=\left(r_{1}-\frac{k_{0}}{12}\right)\left(r_{2}-\frac{k_{0}}{12}\right)+\left(r_{1}-\frac{k_{0}}{12}\right)\left(r_{3}-\frac{k_{0}}{12}\right)+\left(r_{2}-\frac{k_{0}}{12}\right)\left(r_{3}-\frac{k_{0}}{12}\right)-\frac{1}{18}\,,$
$\displaystyle
b=-\left(r_{1}-\frac{k_{0}}{12}\right)\left(r_{2}-\frac{k_{0}}{12}\right)\left(r_{3}-\frac{k_{0}}{12}\right)\,.$
(3.29)

The differential equation (3.28) can immediately be expressed in the form of
Eq. (3.1) with

$\displaystyle\beta_{1}=\frac{5}{6}-\frac{1}{6}(4r_{1}-2r_{2}-2r_{3})\,,$
$\displaystyle\beta_{2}=\frac{5}{6}-\frac{1}{6}(4r_{2}-2r_{3}-2r_{1})\,,$
$\displaystyle\beta_{3}=\frac{5}{6}-\frac{1}{6}(4r_{3}-2r_{1}-2r_{2})\,,$
$\displaystyle\alpha_{1}=0\,,\quad\alpha_{2}=\frac{1}{3}\,,\quad\alpha_{2}=\frac{2}{3}\,.$
(3.30)

Hence the VVMF $F$ is determined to be

$F(\tau)=\begin{pmatrix}~{}~{}\eta^{8(r_{1}+r_{2}+r_{3})-4}K^{\frac{a_{1}+1}{6}}~{}{}_{3}F_{2}(\frac{a_{1}+1}{6},\frac{a_{1}+3}{6},\frac{a_{1}+5}{6};r_{1}-r_{2}+1,r_{1}-r_{3}+1;K)\\\
C_{1}\eta^{8(r_{1}+r_{2}+r_{3})-4}K^{\frac{a_{2}+1}{6}}~{}{}_{3}F_{2}(\frac{a_{2}+1}{6},\frac{a_{2}+3}{6},\frac{a_{2}+5}{6};r_{2}-r_{3}+1,r_{2}-r_{1}+1;K)\\\
C_{2}\eta^{8(r_{1}+r_{2}+r_{3})-4}K^{\frac{a_{3}+1}{6}}~{}{}_{3}F_{2}(\frac{a_{3}+1}{6},\frac{a_{3}+3}{6},\frac{a_{3}+5}{6};r_{3}-r_{1}+1,r_{3}-r_{2}+1;K)\end{pmatrix}\,,$
(3.31)

with $a_{1}=4r_{1}-2r_{2}-2r_{3},a_{2}=4r_{2}-2r_{1}-2r_{3}$ and
$a_{3}=4r_{3}-2r_{1}-2r_{2}$. The constants $C_{1,2}$ are fixed by the
representation matrix of $\rho(S)$ in the $T$-diagonal basis. The 3-d VVMFs
module is of the form

$\mathcal{M}(\mathbf{3}_{(r_{1},r_{2},r_{3})})=\mathbb{C}[E_{4},E_{6}]F\oplus\mathbb{C}[E_{4},E_{6}]D_{k_{0}}F\oplus\mathbb{C}[E_{4},E_{6}]D^{2}_{k_{0}}F\,.$
(3.32)

For high dimensional irreducible VVMFs, the MLDE cannot be converted to
generalize hypergeometric equation, and can only be solved recursively. Some
results of 4-d irreducible VVMFs can be found in Ref. [83], we shall not
discuss these cases further in the present work.

##  4 The general finite modular groups and modular invariant theory

The fundamental theorem of homomorphism implies that $\ker(\rho)$ is a normal
subgroup of $SL(2,\mathbb{Z})$, and the image
$\text{Im}(\rho)\cong\Gamma/\ker(\rho)$ forms a discrete finite group since
the irrep $\rho$ has finite image. For the modular transformation
$\gamma\in\ker(\rho)$ in Eq. (2.5), we have $\rho(\gamma)=\mathbb{1}_{d}$ and
consequently each component of VVMF $Y(\tau)$ in the representation $\rho$ is
scalar modular form of $\ker(\rho)$. In the paradigm of modular flavor
symmetry [3], $\ker(\rho)$ was only restricted to principle congruence
subgroups $\Gamma(N)$. The image $\text{Im}(\rho)$ is the homogeneous finite
modular group $\Gamma^{\prime}_{N}=\Gamma/\Gamma(N)$ and it is taken as the
flavor symmetry to address the flavor structure of quarks and leptons. From
the view of VVMF, the kernel $\ker(\rho)$ can be a general normal subgroup of
$SL(2,\mathbb{Z})$ instead of $\Gamma(N)$, and accordingly the finite modular
group does not necessarily have to be $\Gamma^{\prime}_{N}$. Thus the
framework of modular invariance can be extended significantly by considering
VVMFs, and we have more possible choices for the finite modular groups to
construct modular invariant models. Obviously the irreps of the finite modular
groups can be lifted to the irreps of $SL(2,\mathbb{Z})$. The finite image
irreps of $SL(2,\mathbb{Z})$ can be obtained from these (infinite many)
general finite modular groups.

The classification of normal subgroups of $PSL(2,\mathbb{Z})$ was first
studied in the 1960s [89, 90], and the normal subgroups of index up to $1500$
are given in Ref. [91]. The normal subgroups of $SL(2,\mathbb{Z})$ have not
been fully discussed as those of $PSL(2,\mathbb{Z})$. Some methods for finding
normal subgroups of small index in finitely-presented groups have been
developed in computational group theory [92], and an algorithm has been
implemented as the command LowIndexNormalSubgroups in the package MAGMA [93].
The normal subgroups of $SL(2,\mathbb{Z})$ with small index can be easily
obtained with the help of MAGMA. In table 2 we list the normal subgroups of
$SL(2,\mathbb{Z})$ with index $\leq 72$ as well as the corresponding finite
modular group, where we omit the few cases for which the finite modular group
is abelian. Besides the known principal congruence subgroups $\Gamma(N)$ as
well as the inhomogeneous and homogeneous finite modular groups, we see other
normal subgroups of $SL(2,\mathbb{Z})$. Additional relators are elements of
$\ker(\rho)$, when they are added to the $SL(2,\mathbb{Z})$ presentation
relations $S^{4}=(ST)^{3}=1$, $S^{2}T=TS^{2}$ in Eq. (2.2), the finite modular
groups $\text{Im}(\rho)\cong\Gamma/\ker(\rho)$ are produced. Moreover, if the
element $S^{2}\in\ker(\rho)$, we have $\rho(S^{2})=\mathbb{1}_{d}$. Then the
corresponding finite modular group has only even representations, and the
VVMFs in these irreps must be of even weights.

Normal subgroups $\ker(\rho)$ | Finite modular groups $\Gamma/\ker(\rho)\cong\text{Im}(\rho)$
---|---
Index | Label | Additional relators | Group structure |  GAP Id
6 | $\Gamma(2)$ | $T^{2}$ | $S_{3}$ | $[6,1]$
12 | $-$ | $S^{2}T^{2}$ | $Z_{3}\rtimes Z_{4}\cong 2D_{3}$ | $[12,1]$
$\pm\Gamma(3)$ | $S^{2},T^{3}$ | $A_{4}$ | $[12,3]$
18 | $-$ | $ST^{-2}ST^{2}$ | $S_{3}\times Z_{3}$ | $[18,3]$
24 | $\Gamma(3)$ | $T^{3}$ | $T^{\prime}$ | $[24,3]$
$-$ | $S^{2}T^{3}$
$\pm\Gamma(4)$ | $S^{2},T^{4}$ | $S_{4}$ | $[24,12]$
$-$ | $S^{2},(ST^{-1}ST)^{2}$ | $A_{4}\times Z_{2}$ | $[24,13]$
36 | $-$ | $S^{3}T^{-2}ST^{2}$ | $(Z_{3}\rtimes Z_{4})\times Z_{3}$ | $[36,6]$
42 | $-$ | $T^{6},(ST^{-1}S)^{2}TST^{-1}ST^{2}$ | $Z_{7}\rtimes Z_{6}$ | $[42,1]$
$-$ | $T^{6},ST^{-1}ST(ST^{-1}S)^{2}T^{2}$
48 | $-$ | $S^{2}T^{4}$ | $2O$ | $[48,28]$
$-$ | $T^{8},ST^{4}ST^{-4}$ | $GL(2,3)$ | $[48,29]$
$\Gamma(4)$ | $T^{4}$ | $A_{4}\rtimes Z_{4}\cong S^{\prime}_{4}$ | $[48,30]$
$-$ | $(ST^{-1}ST)^{2}$ | $A_{4}\times Z_{4}$ | $[48,31]$
$-$ | $S^{2}(ST^{-1}ST)^{2}$ | $T^{\prime}\times Z_{2}$ | $[48,32]$
$-$ | $T^{12},ST^{3}ST^{-3}$ | $((Z_{4}\times Z_{2})\rtimes Z_{2})\rtimes Z_{3}$ | $[48,33]$
54 | $-$ | $T^{6},(ST^{-1}ST)^{3}$ | $(Z_{3}\times Z_{3})\rtimes Z_{6}$ | $[54,5]$
60 | $\pm\Gamma(5)$ | $S^{2},T^{5}$ | $A_{5}$ | $[60,5]$
72 | $-$ | $T^{12},ST^{4}ST^{-4}$ | $S_{4}\times Z_{3}$ | $[72,42]$
$\pm\Gamma(6)$ | $S^{2},T^{6},(ST^{-1}STST^{-1}S)^{2}T^{2}$ | $A_{4}\times S_{3}$ | $[72,44]$
Table 2:  The normal subgroups of $SL(2,\mathbb{Z})$ with index less than $78$
and the corresponding finite modular groups. It is defined
$\pm\Gamma(N)=\left\\{\pm\gamma,\gamma\in\Gamma(N)\right\\}$. Note that
$2D_{3}$ is the binary dihedral group of order 12 and $2O$ is the binary
octahedral group which is the Schur cover of permutation group $S_{4}$ of type
“$-$”. Additional relators are some elements of $\ker(\rho)$, the quotient
group $\Gamma/\ker(\rho)$ is produced when the relations “Additional relators
=1” together with $S^{4}=(ST)^{3}=1$ and $S^{2}T=TS^{2}$ in Eq. (2.2) are
imposed.

In the following, we generalize the modular invariant theory [3] by
considering the general finite modular groups denoted as $\mathcal{G}_{f}$
rather than $\Gamma^{\prime}_{N}$ and $\Gamma_{N}$. The theory depends on a
set of chiral supermultiplets $\Phi_{I}$ and the modulus superfield $\tau$
which transform under the modular group $\Gamma$ as

$\begin{cases}\tau\to\gamma\tau=\dfrac{a\tau+b}{c\tau+d}\,,\\\\[7.22743pt]
\Phi_{I}\to(c\tau+d)^{-k_{I}}\rho_{I}(\gamma)\Phi_{I}\,,\end{cases}\qquad\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma\,,$ (4.1)

where $-k_{I}$ is the modular weight of matter field $\Phi_{I}$, and
$\rho_{I}(\gamma)$ is the unitary irreducible representation of
$\mathcal{G}_{f}$ which is the quotient between $\Gamma$ and its certain
normal subgroups. $\mathcal{G}_{f}$ acts as a flavor symmetry group and it is
kept fixed in the construction. We choose a minimal form of the Kahler
potential [3]

$\mathcal{K}(\Phi_{I},\bar{\Phi}_{I};\tau,\bar{\tau})=-h\Lambda^{2}\log(-i\tau+i\bar{\tau})+\sum_{I}(-i\tau+i\bar{\tau})^{-k_{I}}|\Phi_{I}|^{2}\,,$
(4.2)

where $h$ is a positive constant and $\Lambda$ is the cut-off scale of the
theory. After the modulus $\tau$ acquires a nonzero vacuum expectation value,
this Kahler potential gives rise to kinetic terms for both the matter fields
and the modulus field $\tau$. Similar to the original modular flavor symmetry,
the most general Kähler potential consistent with modular symmetry contains
additional terms [94]. The superpotential $\mathcal{W}(\Phi_{I},\tau)$ can be
expanded in power series of the matter supermultiplets $\Phi_{I}$,

$\mathcal{W}(\Phi_{I},\tau)=\sum_{n}Y_{I_{1}...I_{n}}(\tau)~{}\Phi_{I_{1}}...\Phi_{I_{n}}\,.$
(4.3)

Clearly modular invariance of the superpotential requires the function
$Y_{I_{1}...I_{n}}(\tau)$ should transform in the following way:

$Y_{I_{1}...I_{n}}(\tau)\stackrel{{\scriptstyle\gamma}}{{\longrightarrow}}Y_{I_{1}...I_{n}}(\gamma\tau)=(c\tau+d)^{k_{Y}}\rho_{Y}(\gamma)Y_{I_{1}...I_{n}}(\tau),~{}~{}~{}\gamma=\begin{pmatrix}a&~{}b\\\
c&~{}d\end{pmatrix}\in\Gamma\,,$ (4.4)

where $\rho_{Y}$ is an irrep of $\mathcal{G}_{f}$, and $k_{Y}$ and $\rho_{Y}$
satisfy the conditions

$\displaystyle k_{Y}=k_{I_{1}}+...+k_{I_{n}}\,,$
$\displaystyle\rho_{Y}\otimes\rho_{I_{1}}\otimes\cdots\otimes\rho_{I_{n}}\ni\bm{1}\,.$
(4.5)

Hence $Y_{I_{1}...I_{n}}(\tau)$ is a VVMF of weight $k_{Y}$ and in the
representation $\rho_{Y}$ of the finite modular group $\mathcal{G}_{f}$, and
its explicit forms can be obtained by solving the MLDE, as shown in section 3.
The number of VVMFs in a given irrep $\rho_{Y}$ generally increases with the
modular weight $k_{Y}$. Consequently the larger is $k_{I_{1}}+...+k_{I_{n}}$,
more modular invariant terms are involved in $\mathcal{W}$. However, the
module $\mathcal{M}(\rho_{Y})$ is finitely generated by $d=\dim\rho_{Y}$
independent bases denoted by $\\{Y_{1},\dots,Y_{d}\\}$, thus
$Y_{I_{1}...I_{n}}(\tau)$ can be written uniformly in the form of
$Y^{(k_{Y})}_{\rho_{Y}}=\alpha_{1}Y_{1}+\dots+\alpha_{d}Y_{d}$ with
$\alpha_{i}\in\mathbb{C}[E_{4},E_{6}]$, note that $\alpha_{i}$ are polynomials
of $E_{4}$ and $E_{6}$. Some $\alpha_{i}$ for vanishing for small value of
$k_{Y}$, nevertheless there are at most $d$ independent terms. This implies
that the superpotential is strongly constrained by the modular symmetry, and
it can only take a finite number of possible forms for a given
$\mathcal{G}_{f}$, although there are infinite possible weight and
representation assignments for the matter fields. See Appendix C for
explanation.

##  5 New modular invariant lepton models

In this section, we shall apply the formalism of section 4 to construct models
of lepton masses and mixings. We shall be concerned with the finite modular
groups beyond $\Gamma_{N}$ and $\Gamma^{\prime}_{N}$. For illustration, we
will consider $A_{4}\times Z_{2}$ and $GL(2,3)$ as the finite flavor groups
and two benchmark models will be constructed without introducing any flavon.
Notice that these two finite groups as flavor symmetry have not been discussed
in the literature. As shown in table 2, it is remarkable that there are some
other interesting finite modular groups which could be helpful to understand
the flavor puzzle, such as the binary dihedral group $2D_{3}$ and the binary
octahedral group $2O$ which are left for future work888The group $2D_{3}$ has
four 1-d irreps and two 2-d irreps. The group $2O$ is of order 48 and it is
the Schur cover of the permutation group $S_{4}$ of type “$-$”. It has two 1-d
irreps, three 2-d irreps, two 3-d irreps and one 4-d irrep, the three
generations of lepton doublets could be embedded into a triplet of $2O$ as
usual..

###  5.1 Model 1 based on finite modular group $A_{4}\times Z_{2}$

The group $A_{4}\times Z_{2}$ is the direct product of the tetrahedral group
$A_{4}$ and the cyclic group $Z_{2}$, and it can be generated by two elements
$S$ and $T$ obeying the relations

$S^{2}=(ST)^{3}=(ST^{-1}ST)^{2}=1\,.$ (5.1)

The 24 group elements can be expressed in terms of $S$ and $T$ as : $1$, $S$,
$T^{-1}$, $T$, $ST^{-1}$, $ST$, $T^{-1}S$, $T^{-2}$, $TS$, $T^{2}$,
$ST^{-1}S$, $ST^{-2}$, $STS$, $ST^{2}$, $T^{-1}ST$, $T^{-2}S$, $TST^{-1}$,
$T^{2}S$, $T^{3}$, $ST^{-2}S$, $ST^{2}S$, $T^{-1}STS$, $TST^{-1}S$, $T^{3}S$.
This group has eight inequivalent irreps: six 1-d irreducible representations
denoted by
$\bm{1}_{0},\bm{1}_{0}^{\prime},\bm{1}_{0}^{\prime\prime},\bm{1}_{1},\bm{1}_{1}^{\prime},\bm{1}_{1}^{\prime\prime}$
and two 3-d irreducible representations denoted by $\bm{3}_{0},\bm{3}_{1}$. It
is immediate to see that 1-d unitary representations are given by

$\displaystyle\bm{1}_{i}~{}:~{}~{}S=(-1)^{i}\,,~{}~{}~{}~{}~{}\quad
T=(-1)^{i}\,,$
$\displaystyle\bm{1}^{\prime}_{i}~{}:~{}~{}S=(-1)^{i}\,,~{}~{}~{}~{}~{}\quad
T=(-1)^{i}\omega\,,$
$\displaystyle\bm{1}^{\prime\prime}_{i}~{}:~{}~{}S=(-1)^{i}\,,~{}~{}~{}~{}~{}\quad
T=(-1)^{i}\omega^{2}\,,$ (5.2)

where $i=0,1$ and $\omega=e^{2\pi i/3}$. We see that these six 1-d
representations coincide with the $SL(2,\mathbb{Z})$ singlet representations
$\mathbf{1}_{0}$, $\mathbf{1}_{2}$, $\mathbf{1}_{4}$, $\mathbf{1}_{6}$,
$\mathbf{1}_{8}$, $\mathbf{1}_{10}$ and $\mathbf{1}_{12}$ in Eq. (3.8). The
3-d irreps $\bm{3}_{0}$ and $\bm{3}_{1}$ in the $T$ diagonal basis is given
by,

$\bm{3}_{i}~{}:~{}~{}S=\frac{(-1)^{i}}{3}\begin{pmatrix}-1&2&2\\\ 2&-1&2\\\
2&2&-1\end{pmatrix}\,,\,\qquad T=(-1)^{i}\begin{pmatrix}1&0&0\\\ 0&\omega&0\\\
0&0&\omega^{2}\end{pmatrix}\,,$ (5.3)

which are exactly the triplet representation
$\mathbf{3}_{(0,\frac{1}{3},\frac{2}{3})}$ and
$\mathbf{3}_{(\frac{1}{2},\frac{5}{6},\frac{1}{6})}$ of $SL(2,\mathbb{Z})$.
The relevant multiplication rules are

$\displaystyle\bm{1}^{\prime}_{i}\otimes\bm{1}^{\prime}_{j}=\bm{1}^{\prime\prime}_{\langle
i+j\rangle},~{}~{}~{}\bm{1}^{\prime\prime}_{i}\otimes\bm{1}^{\prime\prime}_{j}=\bm{1}^{\prime}_{\langle
i+j\rangle},~{}~{}~{}\bm{1}^{\prime}_{i}\otimes\bm{1}^{\prime\prime}_{j}=\bm{1}_{\langle
i+j\rangle}\,,$ $\displaystyle\bm{3}_{i}\otimes\bm{3}_{j}=\bm{1}_{\langle
i+j\rangle}\oplus\bm{1^{\prime}}_{\langle
i+j\rangle}\oplus\bm{1^{\prime\prime}}_{\langle
i+j\rangle}\oplus\bm{3}_{\langle i+j\rangle,S}\oplus\bm{3}_{\langle
i+j\rangle,A}$ (5.4)

where $\langle i+j\rangle\equiv i+j~{}\texttt{mod}\;2$, and $\bm{3}_{\langle
i+j\rangle,S}$ and $\bm{3}_{\langle i+j\rangle,A}$ stand for the symmetric and
the antisymmetric triplet combinations respectively. The decomposition of the
tensor product of two triplets
$\bm{\alpha}=(\alpha_{1},\alpha_{2},\alpha_{3})^{T}\sim\mathbf{3}_{i}$ and
$\bm{\beta}=(\beta_{1},\beta_{2},\beta_{3})^{T}\sim\mathbf{3}_{j}$ is

$\displaystyle\left(\bm{\alpha}\otimes\bm{\beta}\right)_{\bm{1}_{\langle
i+j\rangle}}=\alpha_{1}\beta_{1}+\alpha_{2}\beta_{3}+\alpha_{3}\beta_{2}\,,$
$\displaystyle\left(\bm{\alpha}\otimes\bm{\beta}\right)_{\bm{1^{\prime}}_{\langle
i+j\rangle}}=\alpha_{3}\beta_{3}+\alpha_{1}\beta_{2}+\alpha_{2}\beta_{1}\,,$
$\displaystyle\left(\bm{\alpha}\otimes\bm{\beta}\right)_{\bm{1^{\prime\prime}}_{\langle
i+j\rangle}}=\alpha_{2}\beta_{2}+\alpha_{1}\beta_{3}+\alpha_{3}\beta_{1}\,,$
$\displaystyle\left(\bm{\alpha}\otimes\bm{\beta}\right)_{\bm{3}_{\langle
i+j\rangle,S}}=\begin{pmatrix}2\alpha_{1}\beta_{1}-\alpha_{2}\beta_{3}-\alpha_{3}\beta_{2}\\\
2\alpha_{3}\beta_{3}-\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\\\
2\alpha_{2}\beta_{2}-\alpha_{1}\beta_{3}-\alpha_{3}\beta_{1}\\\
\end{pmatrix}\,,$
$\displaystyle\left(\bm{\alpha}\otimes\bm{\beta}\right)_{\bm{3}_{\langle
i+j\rangle,A}}=\begin{pmatrix}\alpha_{3}\beta_{2}-\alpha_{2}\beta_{3}\\\
\alpha_{2}\beta_{1}-\alpha_{1}\beta_{2}\\\
\alpha_{1}\beta_{3}-\alpha_{3}\beta_{1}\\\ \end{pmatrix}\,.$ (5.5)

There are no odd weight modular forms in the representation of $A_{4}\times
Z_{2}$, because $S^{2}=1$ shown in Eq. (5.1) and consequently
$\rho(S^{2})=\mathbb{1}_{d}$. From the table 1 or the dimension formula of
VVMF in Eq. (2.22), we can deduce that there are two modular multiplets in the
representation $\bm{1}^{\prime\prime}_{1}$ and $\bm{3}_{0}$ at weight 2. Their
concrete forms can be directly obtained from the general results of Eqs.
(3.10, 3.31) by considering the $S$ transformations, i.e.,

$Y^{(2)}_{\bm{1}_{1}^{\prime\prime}}(\tau)=\eta^{4}(\tau),\quad~{}~{}~{}Y^{(2)}_{\bm{3}_{0}}(\tau)=\begin{pmatrix}\eta^{4}(\tau)(\frac{K(\tau)}{1728})^{-\frac{1}{6}}~{}{}_{3}F_{2}(-\frac{1}{6},\frac{1}{6},\frac{1}{2};\frac{2}{3},\frac{1}{3};K(\tau))\\\
-6\eta^{4}(\tau)(\frac{K(\tau)}{1728})^{\frac{1}{6}}~{}{}_{3}F_{2}(\frac{1}{6},\frac{1}{2},\frac{5}{6};\frac{4}{3},\frac{2}{3};K(\tau))\\\
-18\eta^{4}(\tau)(\frac{K(\tau)}{1728})^{\frac{1}{2}}~{}{}_{3}F_{2}(\frac{1}{2},\frac{5}{6},\frac{7}{6};\frac{5}{3},\frac{4}{3};K(\tau))\end{pmatrix}\,,$
(5.6)

Their $q$-expansions can be read out as follows,

$\displaystyle
Y^{(2)}_{\bm{1}_{1}^{\prime\prime}}(\tau)=q^{1/6}(1-4q+2q^{2}+8q^{3}-5q^{4}-4q^{5}-10q^{6}+\dots)\,,$
$\displaystyle
Y^{(2)}_{\bm{3}_{0}}(\tau)=\begin{pmatrix}1+12q+36q^{2}+12q^{3}+84q^{4}+72q^{5}+36q^{6}+\dots\\\
-6q^{1/3}(1+7q+8q^{2}+18q^{3}+14q^{4}+31q^{5}+20q^{6}+\dots)\\\
-18q^{2/3}(1+2q+5q^{2}+4q^{3}+8q^{4}+6q^{5}+14q^{6}+\dots)\end{pmatrix}\,.$
(5.7)

We see $Y^{(2)}_{\bm{3}_{0}}(\tau)$ is exactly the triplet modular form of
$\Gamma(3)$ [3], and the over factor $q^{1/6}$ in
$Y^{(2)}_{\bm{1}_{1}^{\prime\prime}}$ provides new ingredient to explain the
mass hierarchy of charged leptons and quarks. The higher weight modular
multiplets can be built from the tensor product of
$Y^{(2)}_{\bm{3}_{0}}(\tau)$ and $Y^{(2)}_{\bm{1}^{\prime\prime}_{1}}(\tau)$
in eq. (5.6). For example, at weight 4, we have four modular multiplets

$\displaystyle
Y^{(4)}_{\bm{1}_{0}}=(Y^{(2)}_{\bm{3}_{0}}Y^{(2)}_{\bm{3}_{0}})_{\bm{1}_{0}}=\big{(}Y^{(2)}_{\bm{3}_{0},1}\big{)}^{2}+2Y^{(2)}_{\bm{3}_{0},2}Y^{(2)}_{\bm{3}_{0},3}\,,$
$\displaystyle
Y^{(4)}_{\bm{1}^{\prime}_{0}}=(Y^{(2)}_{\bm{3}_{0}}Y^{(2)}_{\bm{3}_{0}})_{\bm{1}^{\prime}_{0}}=\big{(}Y^{(2)}_{\bm{3}_{0},3}\big{)}^{2}+2Y^{(2)}_{\bm{3}_{0},1}Y^{(2)}_{\bm{3}_{0},2}\,,$
$\displaystyle
Y^{(4)}_{\bm{3}_{0}}=\frac{1}{2}(Y^{(2)}_{\bm{3}_{0}}Y^{(2)}_{\bm{3}_{0}})_{\bm{3}_{0}}=\begin{pmatrix}\big{(}Y^{(2)}_{\bm{3}_{0},1}\big{)}^{2}-Y^{(2)}_{\bm{3}_{0},2}Y^{(2)}_{\bm{3}_{0},3}\\\
\big{(}Y^{(2)}_{\bm{3}_{0},3}\big{)}^{2}-Y^{(2)}_{\bm{3}_{0},1}Y^{(2)}_{\bm{3}_{0},2}\\\
\big{(}Y^{(2)}_{\bm{3}_{0},2}\big{)}^{2}-Y^{(2)}_{\bm{3}_{0},1}Y^{(2)}_{\bm{3}_{0},3}\end{pmatrix}\,,$
$\displaystyle
Y^{(4)}_{\bm{3}_{1}}=Y^{(2)}_{\bm{3}_{0}}Y^{(2)}_{\bm{1}^{\prime\prime}_{1}}=\begin{pmatrix}Y^{(2)}_{\bm{3}_{0},2}Y^{(2)}_{\bm{1}^{\prime\prime}_{1}}\\\
Y^{(2)}_{\bm{3}_{0},3}Y^{(2)}_{\bm{1}^{\prime\prime}_{1}}\\\
Y^{(2)}_{\bm{3}_{0},1}Y^{(2)}_{\bm{1}^{\prime\prime}_{1}}\end{pmatrix}\,,$
(5.8)

where $Y^{(k)}_{\mathbf{r},i}$ refers to the $i$th component of the VVMF
$Y^{(k)}_{\mathbf{r}}$. The concrete forms of the higher weight modular
multiplets in the representations of $A_{4}\times Z_{2}$ can be obtained in a
similar way.

####  5.1.1 Structure of the model

The neutrino masses are described by type-I seesaw mechanism in this model. We
introduce three right-handed neutrinos
$N^{c}=(N^{c}_{1},N^{c}_{2},N^{c}_{3})^{T}$ which furnish a triplet
$\bm{3}_{1}$ of $A_{4}\times Z_{2}$. The three generations of the left-handed
leptons $L=(L_{1},L_{2},L_{3})^{T}$ transform as a triplet $\bm{3}_{0}$ under
$A_{4}\times Z_{2}$, and the right-handed charged leptons $E^{c}_{1,2,3}$ are
assigned to singlet representations
$\bm{1}_{0},\bm{1}^{\prime\prime}_{0},\bm{1}^{\prime}_{0}$ respectively. The
representation and weight assignments of the matter fields are summarized as
follows:

$\displaystyle\rho_{E^{c}}=\mathbf{1}^{\prime\prime}_{0}\oplus\mathbf{1}^{\prime}_{0}\oplus\mathbf{1}^{\prime\prime}_{1},~{}\rho_{L}=\mathbf{3}_{0},~{}\quad\rho_{N^{c}}=\mathbf{3}_{1},~{}\rho_{H_{u}}=\rho_{H_{d}}=\mathbf{1}_{0}\,,$
$\displaystyle k_{E_{1,2,3}^{c}}=-1,1,1\,,~{}\quad k_{L}=3\,,~{}\quad
k_{N^{c}}=1,~{}\quad k_{H_{u}}=k_{H_{d}}=0\,.$ (5.9)

The modular invariant superpotential for the charged lepton and neutrino
masses is given by

$\displaystyle\hskip-7.22743pt\mathcal{W}_{e}=\alpha\left(E_{1}^{c}LY^{(2)}_{\bm{3}_{0}}\right)_{\bm{1}_{0}}H_{d}+\beta\left(E_{2}^{c}LY^{(4)}_{\bm{3}_{0}}\right)_{\bm{1}_{0}}H_{d}+\gamma\left(E_{3}^{c}LY^{(4)}_{\bm{3}_{1}}\right)_{\bm{1}_{0}}H_{d}\,,$
$\displaystyle\hskip-7.22743pt\mathcal{W}_{\nu}=g_{1}\left((N^{c}L)_{\bm{3}_{1},S}Y^{(4)}_{\bm{3}_{1}}\right)_{\bm{1}_{0}}H_{u}+g_{2}\left((N^{c}L)_{\bm{3}_{1},A}Y^{(4)}_{\bm{3}_{1}}\right)_{\bm{1}_{0}}H_{u}+\Lambda\left((N^{c}N^{c})_{\bm{3}_{0},S}Y^{(2)}_{\bm{3}_{0}}\right)_{\bm{1}_{0}}\,.$
(5.10)

Using the decomposition rules in Eq. (5.5), we can read out the charged lepton
and neutrino mass matrices as follows

$\displaystyle M_{e}=\begin{pmatrix}\alpha Y^{(2)}_{\bm{3}_{0},2}&\alpha
Y^{(2)}_{\bm{3}_{0},1}&\alpha Y^{(2)}_{\bm{3}_{0},3}\\\ \beta
Y^{(4)}_{\bm{3}_{0},3}&\beta Y^{(4)}_{\bm{3}_{0},2}&\beta
Y^{(4)}_{\bm{3}_{0},1}\\\ \gamma Y^{(4)}_{\bm{3}_{1},2}&\gamma
Y^{(4)}_{\bm{3}_{1},1}&\gamma Y^{(4)}_{\bm{3}_{1},3}\end{pmatrix}v_{d}\,,$
$\displaystyle
M_{D}=\begin{pmatrix}2g_{1}Y^{(4)}_{\bm{3}_{1},1}&(-g_{1}+g_{2})Y^{(4)}_{\bm{3}_{1},3}&-(g_{1}+g_{2})Y^{(4)}_{\bm{3}_{1},2}\\\
-(g_{1}+g_{2})Y^{(4)}_{\bm{3}_{1},3}&2g_{1}Y^{(4)}_{\bm{3}_{1},2}&(-g_{1}+g_{2})Y^{(4)}_{\bm{3}_{1},1}\\\
(-g_{1}+g_{2})Y^{(4)}_{\bm{3}_{1},2}&-(g_{1}+g_{2})Y^{(4)}_{\bm{3}_{1},1}&2g_{1}Y^{(4)}_{\bm{3}_{1},3}\end{pmatrix}v_{u}\,,$
$\displaystyle
M_{N}=\Lambda\begin{pmatrix}2Y^{(2)}_{\bm{3}_{0},1}&-Y^{(2)}_{\bm{3}_{0},3}&-Y^{(2)}_{\bm{3}_{0},2}\\\
-Y^{(2)}_{\bm{3}_{0},3}&2Y^{(2)}_{\bm{3}_{0},2}&-Y^{(2)}_{\bm{3}_{0},1}\\\
-Y^{(2)}_{\bm{3}_{0},2}&-Y^{(2)}_{\bm{3}_{0},1}&2Y^{(2)}_{\bm{3}_{0},3}\end{pmatrix}\,.$
(5.11)

the light neutrino Majorana mass matrix $m_{\nu}$ is given by the see-saw
relation

$M_{\nu}=-M_{D}^{T}M_{N}^{-1}M_{D}\,.$ (5.12)

Without loss of generality, the parameters $\alpha$, $\beta$, $\gamma$ and
$g_{1}$ can be taken real, since their phases can be removed by field
redefinition. Nevertheless the coupling $g_{2}$ is complex without imposing CP
symmetry. As a consequence, all the lepton masses and mixing parameters only
depend on six real parameters besides the complex modulus $\tau$. By scanning
over the parameter space, if neutrino masses are normal hierarchy, we find
that a good agreement between the model predictions and the experimental data
can be achieved for the following choice of parameter values:

$\displaystyle\tau=0.103786+1.34097i\,,\quad\beta/\alpha=2321.27\,,\quad\gamma/\alpha=798.326\,,$
$\displaystyle g_{2}/g_{1}=13.8646-4.23100i\,,\quad\alpha
v_{d}=0.536619~{}\text{MeV}\,,\quad
g_{1}^{2}v^{2}_{u}/\Lambda=5.84234~{}\text{meV}\,.$ (5.13)

The lepton masses and mixing parameters at this point are determined to be:

$\displaystyle\sin^{2}\theta_{12}=0.315878\,,\quad\sin^{2}\theta_{13}=0.021913\,,\quad\sin^{2}\theta_{23}=0.531526\,,\quad\delta_{CP}=1.13711\pi\,,$
$\displaystyle\alpha_{21}=1.03704\pi\,,~{}~{}\alpha_{31}=1.15945\pi\,,~{}~{}m_{e}/m_{\mu}=0.00480007,~{}~{}m_{\mu}/m_{\tau}=0.0566796\,,$
$\displaystyle m_{1}=6.47366~{}\text{meV}\,,\quad
m_{2}=10.7754~{}\text{meV}\,,\quad m_{3}=49.9964~{}\text{meV}\,,$ (5.14)

which are within the $3\sigma$ intervals of the experimental data without SK
atmospheric data [95]. The neutrino masses are of normal hierarchy type, and
the neutrino mass sum is fixed to be

$m_{1}+m_{2}+m_{3}=67.2454~{}\mathrm{meV}\,,$ (5.15)

which is well compatible with the latest result $\sum_{i}m_{i}<120$ meV
obtained from the combination of Planck TT, TE, EE+lowE+lensing+BAO data [96].
From the predicted values of mixing angles and neutrino masses in Eq. (5.14),
one can determine the effective electron anti-neutrino mass
$m_{\beta}=\sqrt{\sum_{i}|U_{ei}|^{2}m^{2}_{i}}$ measured in $\beta$ decay
experiments and the effective Majorana neutrino mass
$m_{\beta\beta}=|\sum_{i}U^{2}_{ei}m_{i}|$ characterizing the amplitude of
neutrinoless double decay,

$m_{\beta}=10.8945~{}\text{meV}\,,\quad m_{\beta\beta}=2.63536\times
10^{-6}~{}\text{meV}\,.$ (5.16)

The quantity $m_{\beta}$ is much smaller than the KATRIN tritium beta decay
upper limit $m_{\beta}<0.8$ eV at $90\%$ confidence level [97, 98]. A strong
cancellation in $m_{\beta\beta}$ occurs such that $m_{\beta\beta}$ almost
vanishes. On the other hand, our model would be disfavoured if a positive
signal is found.

This model can also accommodate inverted ordering neutrino masses in some
region of parameter space. The agreement between predictions and data without
SK [95] is optimized by the following values of free parameters:

$\displaystyle\tau=-0.497747+1.25636i\,,\quad\beta/\alpha=0.056594\,,\quad\gamma/\alpha=0.0017391\,,$
$\displaystyle g_{2}/g_{1}=-0.494173+0.23773i\,,\quad\alpha
v_{d}=1227.75~{}\text{MeV}\,,\quad
g_{1}^{2}v^{2}_{u}/\Lambda=538.424~{}\text{meV}\,.$ (5.17)

The corresponding predictions for lepton masses, mixing angles, and CP
violating phases as well as the effective neutrino masses $m_{\beta}$ and
$m_{\beta\beta}$ are

$\displaystyle\sin^{2}\theta_{12}=0.303001\,,\quad\sin^{2}\theta_{13}=0.0224254\,,\quad\sin^{2}\theta_{23}=0.520345\,,$
$\displaystyle\delta_{CP}=1.77550\pi\,,\quad\alpha_{21}=1.23377\pi\,,~{}~{}\alpha_{31}=0.525454\pi\,,$
$\displaystyle m_{e}/m_{\mu}=0.00480000,~{}~{}m_{\mu}/m_{\tau}=0.0565946\,,$
$\displaystyle m_{1}=57.4753~{}\text{meV}\,,\quad
m_{2}=58.1172~{}\text{meV}\,,\quad m_{3}=29.8200~{}\text{meV}\,,$
$\displaystyle m_{\beta}=57.1948~{}\text{meV}\,,\quad
m_{\beta\beta}=28.1651\text{meV}\,.$ (5.18)

The most stringent upper limit on $m_{\beta\beta}$ at present is provided by
the KamLAND-Zen collaboration [99],

$m_{\beta\beta}<(61-165)\;\text{meV}~{}\mathrm{at}~{}90\%\;\mathrm{C.L.}\,,$
(5.19)

where the uncertainty comes from the calculations of the numerical matrix
elements. Future tonne-scale experiments such as nEXO [100] and LEGEND [101]
aim to probe the full region of the inverted neutrino mass ordering and a
significant portion of the normal ordering can be explored. They intend to
improve their sensitivity by a factor of approximately ten,

$\displaystyle
m_{\beta\beta}<\left\\{\begin{array}[]{ll}(4.7-20.3)\;\text{meV},&~{}~{}~{}~{}\text{nEXO}\,,\\\
(9-21)\;\text{meV},&~{}~{}~{}~{}\text{LEGEND}-1000\,.\end{array}\right.$
(5.22)

We see that our prediction of $m_{\beta\beta}$ for the inverted ordering is
within the reach of nEXO and LEGEND sensitivities. Moreover, the neutrino mass
sum is $m_{1}+m_{2}+m_{3}=145.413$ meV in this case, it is slightly larger
than the aggressive upper bound of $\sum_{i}m_{i}<120$meV although still
compatible with the conservative bound $m_{1}+m_{2}+m_{3}<600$ meV extracted
from the measurement of Planck lensing+BAO+$\theta_{\text{MC}}$ [96]. Notice
that the neutrino mass limits depend not only the data sets analyzed but also
on the cosmological model. The limits also become weaker when one departs from
the framework of $\Lambda$CDM plus neutrino mass to frameworks with more
cosmological parameters.

###  5.2 Model 2 based on finite modular group $GL(2,3)$

In this section, we proceed to consider another finite modular group $GL(2,3)$
and model construction for lepton masses and mixing. The group $GL(2,3)$ is
the Schur covering of $S_{4}$ of “+” type, and its representations are those
of $S_{4}$ plus two doublets $\bm{2}^{\prime},\bm{2}^{\prime\prime}$ and one
quartet $\bm{4}$. We summarize the group theory of $GL(2,3)$ in Appendix D. It
is remarkable that $GL(2,3)$ can produce interesting flavor structures which
are unavailable in both $S_{4}$ and $S^{\prime}_{4}$ [36, 37] which is the
double cover of $S_{4}$.

Since the representation matrix $\rho(S^{2})=\mathbb{1}_{d}$ for all irreps of
$GL(2,3)$, the VVMFs transforming in the irreps of $GL(2,3)$ must be of even
weight. From the table 1 or the dimension formula of VVMF in Eq. (2.22), we
can find that there are three modular multiplets at weight 2 transforming as
$\bm{2}$, $\bm{2^{\prime}}$, $\bm{3}$ under $GL(2,3)$. They can be constructed
in terms of the generalized hypergeometric series, as shown in section 3.
Using the general results of Eqs. (3.23, 3.31), we obtain

$\displaystyle
Y^{(2)}_{\bm{2}}(\tau)=\begin{pmatrix}\eta^{4}(\tau)(\frac{K(\tau)}{1728})^{-\frac{1}{6}}~{}{}_{2}F_{1}(-\frac{1}{6},\frac{1}{6};\frac{1}{2};K(\tau))\\\
8\sqrt{3}\eta^{4}(\tau)(\frac{K(\tau)}{1728})^{\frac{1}{3}}~{}{}_{2}F_{1}(\frac{1}{3},\frac{2}{3};\frac{3}{2};K(\tau))\\\
\end{pmatrix}\,,$ (5.23) $\displaystyle
Y^{(2)}_{\bm{2}^{\prime}}(\tau)=\begin{pmatrix}-2\eta^{4}(\tau)(\frac{K(\tau)}{1728})^{\frac{5}{24}}~{}{}_{2}F_{1}(\frac{5}{24},\frac{13}{24};\frac{5}{4};K(\tau))\\\
\eta^{4}(\tau)(\frac{K(\tau)}{1728})^{-\frac{1}{24}}~{}{}_{2}F_{1}(-\frac{1}{24},\frac{7}{24};\frac{3}{4};K(\tau))\\\
\end{pmatrix}\,,$ (5.24) $\displaystyle
Y^{(2)}_{\bm{3}}(\tau)=\begin{pmatrix}\eta^{4}(\tau)(\frac{K(\tau)}{1728})^{-\frac{1}{6}}~{}{}_{3}F_{2}(-\frac{1}{6},\frac{1}{6},\frac{1}{2};\frac{3}{4},\frac{1}{4};K(\tau))\\\
-4\sqrt{2}\eta^{4}(\tau)(\frac{K(\tau)}{1728})^{\frac{1}{12}}~{}{}_{3}F_{2}(\frac{1}{12},\frac{5}{12},\frac{3}{4};\frac{1}{2},\frac{5}{4};K(\tau))\\\
-16\sqrt{2}\eta^{4}(\tau)(\frac{K(\tau)}{1728})^{\frac{7}{12}}~{}{}_{3}F_{2}(\frac{7}{12},\frac{11}{12},\frac{5}{4};\frac{7}{4},\frac{3}{2};K(\tau))\end{pmatrix}\,.$
(5.25)

The $q$-expansions for these modular forms are given by

$\displaystyle
Y^{(2)}_{\bm{2}}(\tau)=\begin{pmatrix}1+24q+24q^{2}+96q^{3}+24q^{4}+144q^{5}+96q^{6}+\dots\\\
8\sqrt{3}q^{1/2}(1+4q+6q^{2}+8q^{3}+13q^{4}+12q^{5}+14q^{6}+\dots)\\\
\end{pmatrix}\,,$ $\displaystyle
Y^{(2)}_{\bm{2}^{\prime}}(\tau)=\begin{pmatrix}-2q^{3/8}(1-3q+q^{2}+2q^{3}+5q^{5}-6q^{6}+\dots)\\\
q^{1/8}(1-q-6q^{2}+5q^{3}+12q^{4}-6q^{5}-7q^{6}+\dots)\\\ \end{pmatrix}\,,$
$\displaystyle
Y^{(2)}_{\bm{3}}(\tau)=\begin{pmatrix}1-8q+24q^{2}-32q^{3}+24q^{4}-48q^{5}+96q^{6}+\dots\\\
-4\sqrt{2}q^{1/4}(1+6q+13q^{2}+14q^{3}+18q^{4}+32q^{5}+31q^{6}+\dots)\\\
-16\sqrt{2}q^{3/4}(1+2q+3q^{2}+6q^{3}+5q^{4}+6q^{5}+10q^{6}+\dots)\\\
\end{pmatrix}\,,$ (5.26)

Note that the components of $Y^{(2)}_{\bm{2}}(\tau)$ and
$Y^{(2)}_{\bm{3}}(\tau)$ are scalar valued modular forms of $\Gamma(4)$. Based
on the lowest modular forms $Y^{(2)}_{\bm{2}}$, $Y^{(2)}_{\bm{2}^{\prime}}$
and $Y^{(2)}_{\bm{3}}$, one can construct the modular forms of higher weights
by using the decomposition rules of $GL(2,3)$ listed in tables 5 and 6. At
weight 4, there are totally six independent VVMFs given by

$\displaystyle
Y^{(4)}_{\bm{1}}=(Y^{(2)}_{\bm{2}}Y^{(2)}_{\bm{2}})_{\bm{1}}=(Y^{(2)}_{\bm{2},1})^{2}+(Y^{(2)}_{\bm{2},2})^{2}\,,$
$\displaystyle
Y^{(4)}_{\bm{2}}=(Y^{(2)}_{\bm{2}}Y^{(2)}_{\bm{2}})_{\bm{2}}=\begin{pmatrix}(Y^{(2)}_{\bm{2},2})^{2}-(Y^{(2)}_{\bm{2},1})^{2}\\\
2Y^{(2)}_{\bm{2},1}Y^{(2)}_{\bm{2},2}\\\ \end{pmatrix}\,,$ $\displaystyle
Y^{(4)}_{\bm{2}^{\prime}}=(Y^{(2)}_{\bm{2}^{\prime}}Y^{(2)}_{\bm{3}})_{\bm{2}^{\prime}}=\begin{pmatrix}\sqrt{2}Y^{(2)}_{\bm{2}^{\prime},2}Y^{(2)}_{\bm{3},2}+Y^{(2)}_{\bm{2}^{\prime},1}Y^{(2)}_{\bm{3},1}\\\
\sqrt{2}Y^{(2)}_{\bm{2}^{\prime},1}Y^{(2)}_{\bm{3},3}-Y^{(2)}_{\bm{2}^{\prime},2}Y^{(2)}_{\bm{3},1}\\\
\end{pmatrix}\,,$ $\displaystyle
Y^{(4)}_{\bm{3}}=(Y^{(2)}_{\bm{2}}Y^{(2)}_{\bm{3}})_{\bm{3}}=\begin{pmatrix}-2Y^{(2)}_{\bm{2},1}Y^{(2)}_{\bm{3},1}\\\
Y^{(2)}_{\bm{2},1}Y^{(2)}_{\bm{3},2}+\sqrt{3}Y^{(2)}_{\bm{2},2}Y^{(2)}_{\bm{3},3}\\\
Y^{(2)}_{\bm{2},1}Y^{(2)}_{\bm{3},3}+\sqrt{3}Y^{(2)}_{\bm{2},2}Y^{(2)}_{\bm{3},2}\\\
\end{pmatrix}\,,$ $\displaystyle
Y^{(4)}_{\bm{3}^{\prime}}=\frac{1}{\sqrt{2}}(Y^{(2)}_{\bm{2}^{\prime}}Y^{(2)}_{\bm{2}^{\prime}})_{\bm{3}^{\prime}}=\begin{pmatrix}\sqrt{2}Y^{(2)}_{\bm{2}^{\prime},1}Y^{(2)}_{\bm{2}^{\prime},2}\\\
-(Y^{(2)}_{\bm{2}^{\prime},1})^{2}\\\ (Y^{(2)}_{\bm{2}^{\prime},2})^{2}\\\
\end{pmatrix}\,,$ $\displaystyle
Y^{(4)}_{\bm{4}}=(Y^{(2)}_{\bm{2}}Y^{(2)}_{\bm{2}^{\prime}})_{\bm{4}}=\begin{pmatrix}Y^{(2)}_{\bm{2},2}Y^{(2)}_{\bm{2}^{\prime},2}\\\
-Y^{(2)}_{\bm{2},1}Y^{(2)}_{\bm{2}^{\prime},1}\\\
Y^{(2)}_{\bm{2},2}Y^{(2)}_{\bm{2}^{\prime},1}\\\
Y^{(2)}_{\bm{2},1}Y^{(2)}_{\bm{2}^{\prime},2}\\\ \end{pmatrix}\,.$ (5.27)

Notice that
$Y^{(4)}_{\bm{1}}=E_{4},Y^{(4)}_{\bm{2}}=6D_{2}Y^{(2)}_{\bm{2}},Y^{(4)}_{\bm{2}^{\prime}}=24D_{2}Y^{(2)}_{\bm{2}^{\prime}}$
and $Y^{(4)}_{\bm{3}}=12D_{2}Y^{(2)}_{\bm{3}}$.

####  5.2.1 Structure of the model

In the following, we present a viable lepton model based on the finite modular
group $GL(2,3)$. The neutrino masses arise from the seesaw mechanism. Both the
left-handed lepton doublets $L$ and right-handed neutrinos $N^{c}$ are
organized into two triplet $\mathbf{3}$ of $GL(2,3)$ with the modular weights
$k_{L}=2$ and $k_{N^{c}}=0$. The first two generations of the right-handed
charged leptons $E_{D}^{c}\equiv(E_{1}^{c},E_{2}^{c})^{T}$ are assigned to a
doublet $\bm{2^{\prime\prime}}$ with weight $k_{E_{D}^{c}}=2$, and the third
right-handed charged lepton $E_{3}^{c}$ transform as $\mathbf{1}$ with weight
$k_{E_{3}^{c}}=0$. The Higgs doublets $H_{u}$ and $H_{d}$ are assumed to be
$GL(2,3)$ trivial singlets of zero modular weight. Then the superpotentials
for the charged lepton and neutrino masses take the form

$\displaystyle\mathcal{W}_{e}$
$\displaystyle=\alpha\left(E_{D}^{c}LY^{(4)}_{\bm{2}^{\prime}}\right)_{\bm{1}}H_{d}+\beta\left(E_{D}^{c}LY^{(4)}_{\bm{4}}\right)_{\bm{1}}H_{d}+\gamma\left(E_{3}^{c}LY^{(2)}_{\bm{3}}\right)_{\bm{1}}H_{d}\,,$
$\displaystyle\mathcal{W}_{\nu}$
$\displaystyle=g_{1}\left(N^{c}LY^{(2)}_{\bm{2}}\right)_{\bm{1}}H_{u}+g_{2}\left(N^{c}LY^{(2)}_{\bm{3}}\right)_{\bm{1}}H_{u}+\Lambda\left(N^{c}N^{c}\right)_{\bm{1}}\,.$
(5.28)

With the Clebsch-Gordon coefficients of $GL(2,3)$ in Appendix D, we can
straightforwardly read out the charged lepton and neutrino mass matrices as
follows,

$\displaystyle M_{e}=\begin{pmatrix}-\alpha
Y^{(4)}_{\bm{2}^{\prime},2}+\sqrt{2}\beta Y^{(4)}_{\bm{4},4}&\sqrt{3}\beta
Y^{(4)}_{\bm{4},3}&\sqrt{2}\alpha Y^{(4)}_{\bm{2}^{\prime},1}-\beta
Y^{(4)}_{\bm{4},2}\\\ \alpha Y^{(4)}_{\bm{2}^{\prime},1}+\sqrt{2}\beta
Y^{(4)}_{\bm{4},2}&\sqrt{2}\alpha Y^{(4)}_{\bm{2}^{\prime},2}+\beta
Y^{(4)}_{\bm{4},4}&\sqrt{3}\beta Y^{(4)}_{\bm{4},1}\\\ \gamma
Y^{(2)}_{\bm{3},1}&\gamma Y^{(2)}_{\bm{3},3}&\gamma
Y^{(2)}_{\bm{3},2}\end{pmatrix}v_{d}\,,$ $\displaystyle
M_{D}=\begin{pmatrix}-2g_{1}Y^{(2)}_{\bm{2},1}&-g_{2}Y^{(2)}_{\bm{3},3}&g_{2}Y^{(2)}_{\bm{3},2}\\\
g_{2}Y^{(2)}_{\bm{3},3}&\sqrt{3}g_{1}Y^{(2)}_{\bm{2},2}&g_{1}Y^{(2)}_{\bm{2},1}-g_{2}Y^{(2)}_{\bm{3},1}\\\
-g_{2}Y^{(2)}_{\bm{3},2}&g_{1}Y^{(2)}_{\bm{2},1}+g_{2}Y^{(2)}_{\bm{3},1}&\sqrt{3}g_{1}Y^{(2)}_{\bm{2},2}\end{pmatrix}v_{u}\,,$
$\displaystyle M_{N}=\Lambda\begin{pmatrix}1~{}&0~{}&0\\\ 0~{}&0~{}&1\\\
0~{}&1~{}&0\end{pmatrix}\,.$ (5.29)

We can set the parameters $\alpha$, $\gamma$ and $g_{1}$ to be real, since
their phases can be absorbed into matter fields. The other two couplings
$\beta$ and $g_{2}$ are complex, Through numerical search, we can find the
best fit values of the parameters compatible with normal ordering data [95]
are given by:

$\displaystyle\tau=-0.499623+0.903453i\,,\quad\beta/\alpha=-2.90228-0.156823i\,,\quad\gamma/\alpha=0.000752696\,,$
$\displaystyle g_{2}/g_{1}=0.204168-0.909287i\,,\quad\alpha
v_{d}=327.660~{}\text{MeV}\,,\quad
g_{1}^{2}v^{2}_{u}/\Lambda=18.4597~{}\text{meV}\,.$ (5.30)

The value of $\tau$ is very close to the left boundary of the fundamental
domain. The predictions for the mass eigenvalues, mixing angles and CP
violation phases are given by

$\displaystyle\sin^{2}\theta_{12}=0.303987\,,\quad\sin^{2}\theta_{13}=0.0222024\,,\quad\sin^{2}\theta_{23}=0.578472\,,$
$\displaystyle\delta_{CP}=1.59938\pi\,,\quad\alpha_{21}=1.44951\pi\,,\quad\alpha_{31}=0.518406\pi\,,$
$\displaystyle
m_{e}/m_{\mu}=0.00480015\,,~{}~{}~{}m_{\mu}/m_{\tau}=0.0566693\,,$
$\displaystyle m_{1}=76.4654~{}\text{meV}\,,\quad
m_{2}=76.9491~{}\text{meV}\,,\quad m_{3}=91.4362~{}\text{meV}\,,$
$\displaystyle m_{\beta}=76.9729~{}\text{meV}\,,\quad
m_{\beta\beta}=53.2091~{}\text{meV}\,.$ (5.31)

All the experimental bounds in neutrino oscillation, tritium beta decays and
neutrinoless double beta decay are satisfied. The neutrino mass sum is
$m_{1}+m_{2}+m_{2}=235.45$ meV, this is compatible with the conservative bound
$\sum_{i}m_{i}<600$ meV from cosmology although larger than the most
aggressive upper bound $\sum_{i}m_{i}<120$ meV. This model can also fit the
data of inverted neutrino mass spectrum. As an example, we give a point in the
parameter space:

$\displaystyle\tau=-0.474696+0.880646i\,,\quad\beta/\alpha=-2.99553-0.191581i\,,\quad\gamma/\alpha=0.000992565\,,$
$\displaystyle g_{2}/g_{1}=0.813157+1.55181i\,,\quad\alpha
v_{d}=305.208~{}\text{MeV}\,,\quad
g_{1}^{2}v^{2}_{u}/\Lambda=13.0640~{}\text{meV}\,,$ (5.32)

The corresponding flavor observables are determined as: which leads to the
following values for the flavor observables

$\displaystyle\sin^{2}\theta_{12}=0.304346\,,\quad\sin^{2}\theta_{13}=0.0224044\,,\quad\sin^{2}\theta_{23}=0.611139\,,$
$\displaystyle\delta_{CP}=1.58351\pi\,,\quad\alpha_{21}=0.745402\pi\,,\quad\alpha_{31}=1.56947\pi\,,$
$\displaystyle
m_{e}/m_{\mu}=0.00480415\,,~{}\quad~{}m_{\mu}/m_{\tau}=0.0565167\,,$
$\displaystyle m_{1}=101.897~{}\text{meV}\,,\quad
m_{2}=102.260~{}\text{meV}\,,\quad m_{3}=89.2362~{}\text{meV}\,,$
$\displaystyle m_{\beta}=101.739~{}\text{meV}\,,\quad
m_{\beta\beta}=54.1723~{}\text{meV}\,.$ (5.33)

Similarly to previous case, all the current experimental bounds from neutrino
oscillation, endpoint $\beta$ decay, neutrinoless double beta decay as well as
conservative bound from cosmology are fulfilled.

##  6 Summary and outlook

The modular flavor symmetry is an important theoretical advance to address the
flavor puzzle of the SM, it is assumed that the Yukawa couplings are modular
forms of the principal congruence subgroup $\Gamma(N)$ and the matter fields
transform in representations of the homogeneous (inhomogeneous) finite modular
group $\Gamma^{\prime}_{N}$ ($\Gamma_{N}$). In the present work, we show that
the requirement of modular forms of $\Gamma(N)$ is not a necessity. One can
start from the vector-valued modular form which is a holomorphic function of
$\tau$ and satisfies $Y(\gamma\tau)=(c\tau+d)^{k}\rho(\gamma)Y(\tau)$. Here
$k$ is the modular weight, $\rho$ is an irrep of the modular group
$SL(2,\mathbb{Z})$ and it is assumed to have finite image. The vector-valued
modular form $Y(\tau)$ is the scalar modular form of the kernel $\ker(\rho)$,
and the image $\text{Im}(\rho)\cong\Gamma/\ker(\rho)$ plays the role of
discrete finite flavor symmetry. The case of $\ker(\rho)=\Gamma(N)$ is
considered in the original framework of modular flavor symmetry. The VVMFs can
considerably extend the modular flavor symmetry, and we have more possible
choices for the finite modular groups to construct modular invariant models.
The general finite modular groups up to order 72 are listed in table 2, they
provide new opportunity for future study.

The theory of vector-valued modular form is reviewed in section 2. The low
dimensional vector-valued modular forms at the minimal weight can be
determined by solving the modular linear differential equation. The one-
dimensional vector-valued modular forms are powers of the Dedekind eta
function. The two-dimensional and three-dimensional vector-valued modular
forms can be expressed in terms of the generalized hypergeometric series. The
space of vector-valued modular forms $\mathcal{M}(\rho)$ in representation
$\rho$ is a module over the ring $\mathbb{C}[E_{4},E_{6}]$, and it can be
generated from a set of basis by multiplying polynomials of $E_{4},E_{6}$.
Hence the modular multiplets in any representation $\rho$ have a finite number
of $\tau$ dependent alignments, then only a finite number of models could be
constructed with a finite modular groups, although there are infinite possible
weight and representation assignments for the matter fields. Guided by the
theory of vector-valued modular form, the construction of high weight modular
multiplets is much easier, it is not necessary to search for non-linear
constraints as in the approach of tensor products.

We have provided examples of applying our formalism in section 5, and
constructed two lepton flavor models based on the finite modular groups
$A_{4}\times Z_{2}$ and $GL(2,3)$. The neutrino masses are generated by the
type I seesaw mechanism and no flavon field other than the modulus $\tau$ is
used. The neutrino masses can be with either normal ordering or inverted
ordering in both models, and the experimental data from neutrino oscillation,
tritium beta decay, neutrinoless double beta decay and cosmology can be
accommodated in certain parameter space. It is known that CP symmetry can be
consistently combined with modular symmetrys [43, 44], and the modulus
transforms as $\tau\rightarrow-\tau^{*}$ under the action of CP. The CP
symmetry can constrain the phases of the coupling constants, and thus make the
modular models more predictive. It is interesting to discuss the implications
of CP symmetry in the context of modular flavor symmetry extended by vector-
valued modular forms in future.

Although we have focused on the modular group $SL(2,\mathbb{Z})$, the modular
invariance framework can be extended to more general Fuchsian group of genus
zero, such as $\Gamma(2)$[41] and $\Gamma_{0}(2)$ [102], even the symplectic
modular group $Sp(2g,\mathbb{Z})$ [42] and so on. The well-defined vector-
valued modular forms exist for these group. It would be interesting to explore
the application of their vector-valued modular forms to the flavor puzzle.
Finally we mention that the matter fields and Yukawa couplings transform as
irreducible representations of $SL(2,\mathbb{Z})$ in top-down constructions,
and the flavor symmetry group could be read from the image of the
representations. We expect that the bottom-up analyses in view of vector-
valued modular forms could correspond to some top-down realizations based on
ultraviolet-complete theories.

### Acknowledgements

This work is supported by the National Natural Science Foundation of China
under Grant Nos 11975224, 11835013 and 11947301 and the Key Research Program
of the Chinese Academy of Sciences under Grant NO. XDPB15.

## Appendix

##  Appendix A All 2-d irreducible representations of $SL(2,\mathbb{Z})$ with
finite image

There are only four finite subgroups of $SU(2)$: the binary dihedral group
$2D_{3}$, the binary tetrahedral group $T^{\prime}$, the binary octahedral
group $2O$ and the binary isosahedral group $A^{\prime}_{5}$, which are
homomorphic to $SL(2,\mathbb{Z})$ [86]. All the 2-d irreps of
$SL(2,\mathbb{Z})$ with finite image are the direct product of the 1-d irreps
of $SL(2,\mathbb{Z})$ with the 2-d irreps of the finite groups $2D_{3}$,
$T^{\prime}$, $2O$ and $A^{\prime}_{5}$ [86]. It is found that there are
totally 54 inequivalent irreps with finite image, and they are listed in table
3 in terms of the parameters $r_{1}$ and $r_{2}$ [86].

Binary dihedral type
---
|  |  |
$r_{1}$ |  $r_{2}$ |  $N$ |  $k_{0}$
$3/4$ | $1/4$ | $4$ | $5$
$5/6$ | $1/3$ | $6$ | $6$
$11/12$ | $5/12$ | $12$ | $7$
$0$ | $1/2$ | $2$ | $2$
$1/12$ | $7/12$ | $12$ | $3$
$1/6$ | $2/3$ | $6$ | $4$

Binary tetrahedral type
---
|  |  |
$r_{1}$ |  $r_{2}$ |  $N$ |  $k_{0}$
$2/3$ | $1/3$ | $3$ | $5$
$3/4$ | $5/12$ | $12$ | $6$
$5/6$ | $1/2$ | $6$ | $7$
$11/12$ | $7/12$ | $12$ | $8$
$0$ | $2/3$ | $3$ | $3$
$1/12$ | $3/4$ | $12$ | $4$
$1/6$ | $5/6$ | $6$ | $5$
$1/4$ | $11/12$ | $12$ | $6$
$1/3$ | $0$ | $3$ | $1$
$5/12$ | $1/12$ | $12$ | $2$
$1/2$ | $1/6$ | $6$ | $3$
$7/12$ | $1/4$ | $12$ | $4$

Binary octahedral type
---
|  |  |
$r_{1}$ |  $r_{2}$ |  $N$ |  $k_{0}$
$7/8$ | $1/8$ | $8$ | $5$
$23/24$ | $5/24$ | $24$ | $6$
$1/24$ | $7/24$ | $24$ | $1$
$1/8$ | $3/8$ | $8$ | $2$
$5/24$ | $11/24$ | $24$ | $3$
$7/24$ | $13/24$ | $24$ | $4$
$3/8$ | $5/8$ | $8$ | $5$
$11/24$ | $17/24$ | $24$ | $6$
$13/24$ | $19/24$ | $24$ | $7$
$5/8$ | $7/8$ | $8$ | $8$
$17/24$ | $23/24$ | $24$ | $9$
$19/24$ | $1/24$ | $24$ | $4$

Binary Icosahedral type I
---
|  |  |
$r_{1}$ |  $r_{2}$ |  $N$ |  $k_{0}$
$4/5$ | $1/5$ | $5$ | $5$
$53/60$ | $17/60$ | $60$ | $6$
$29/30$ | $11/30$ | $30$ | $7$
$9/20$ | $1/20$ | $20$ | $2$
$2/15$ | $8/15$ | $15$ | $3$
$13/60$ | $37/60$ | $60$ | $4$
$3/10$ | $7/10$ | $10$ | $5$
$23/60$ | $47/60$ | $60$ | $6$
$7/15$ | $13/15$ | $15$ | $7$
$11/20$ | $19/20$ | $20$ | $8$
$19/30$ | $1/30$ | $30$ | $3$
$43/60$ | $7/60$ | $60$ | $4$

Binary Icosahedral type II
---
|  |  |
$r_{1}$ |  $r_{2}$ |  $N$ |  $k_{0}$
$3/5$ | $2/5$ | $5$ | $5$
$41/60$ | $29/60$ | $60$ | $6$
$23/30$ | $17/30$ | $30$ | $7$
$17/20$ | $13/20$ | $20$ | $8$
$14/15$ | $11/15$ | $15$ | $9$
$1/60$ | $49/60$ | $60$ | $4$
$1/10$ | $9/10$ | $10$ | $5$
$11/60$ | $59/60$ | $60$ | $4$
$4/15$ | $1/15$ | $15$ | $1$
$7/20$ | $3/20$ | $20$ | $2$
$13/30$ | $7/30$ | $30$ | $3$
$31/60$ | $19/60$ | $60$ | $4$

Table 3: Summary of the 54 irreducible 2-d representations of
$SL(2,\mathbb{Z})$ with finite image, where $N$ denotes the order of
$\rho(T)$. The unordered pair $(r_{1},r_{2})$ uniquely determines the 2-d
irreps up to a similar transformation, and the representation matrices
$\rho(S)$ and $\rho(T)$ are given in Eqs. (3.13,3.14).

##  Appendix B VVMF modules of the finite modular group $A_{4}\times Z_{2}$

Traditionally the higher weight modular multiplets are built from the tensor
product of the lower weight modular multiplets. For instance, at weight 6, we
have the following six linearly independent modular multiplets of $A_{4}\times
Z_{2}$,

$\displaystyle
Y^{(6)}_{\bm{1}_{0}}=(Y^{(2)}_{\bm{3}_{0}}Y^{(4)}_{\bm{3}_{0}})_{\bm{1}_{0}}=(Y^{(2)}_{\bm{3}_{0},1})^{3}+(Y^{(2)}_{\bm{3}_{0},2})^{3}+(Y^{(2)}_{\bm{3}_{0},3})^{3}-3Y^{(2)}_{\bm{3}_{0},1}Y^{(2)}_{\bm{3}_{0},2}Y^{(2)}_{\bm{3}_{0},3}\,,$
$\displaystyle
Y^{(6)}_{\bm{1}_{1}}=Y^{(2)}_{\bm{1}^{\prime\prime}_{1}}Y^{(4)}_{\bm{1}^{\prime}}=(Y^{(2)}_{\bm{3}_{0},3})^{2}Y^{(2)}_{\bm{1}^{\prime\prime}_{1}}+2Y^{(2)}_{\bm{3}_{0},1}Y^{(2)}_{\bm{3}_{0},2}Y^{(2)}_{\bm{1}^{\prime\prime}_{1}}\,,$
$\displaystyle
Y^{(6)}_{\bm{1}^{\prime\prime}_{1}}=Y^{(2)}_{\bm{1}^{\prime\prime}_{1}}Y^{(4)}_{\bm{1}}=(Y^{(2)}_{\bm{3}_{0},1})^{2}Y^{(2)}_{\bm{1}^{\prime\prime}_{1}}+2Y^{(2)}_{\bm{3}_{0},2}Y^{(2)}_{\bm{3}_{0},3}Y^{(2)}_{\bm{1}^{\prime\prime}_{1}}\,,$
$\displaystyle
Y^{(6)}_{\bm{3}_{0}I}=Y^{(2)}_{\bm{3}_{0}}Y^{(4)}_{\bm{1}_{0}}=\begin{pmatrix}(Y^{(2)}_{\bm{3}_{0},1})^{3}+2Y^{(2)}_{\bm{3}_{0},1}Y^{(2)}_{\bm{3}_{0},2}Y^{(2)}_{\bm{3}_{0},3}\\\
(Y^{(2)}_{\bm{3}_{0},1})^{2}Y^{(2)}_{\bm{3}_{0},2}+2(Y^{(2)}_{\bm{3}_{0},2})^{2}Y^{(2)}_{\bm{3}_{0},3}\\\
(Y^{(2)}_{\bm{3}_{0},1})^{2}Y^{(2)}_{\bm{3}_{0},3}+2(Y^{(2)}_{\bm{3}_{0},3})^{2}Y^{(2)}_{\bm{3}_{0},2}\end{pmatrix}\,,$
$\displaystyle
Y^{(6)}_{\bm{3}_{0}II}=Y^{(2)}_{\bm{3}_{0}}Y^{(4)}_{\bm{1}^{\prime}_{0}}=\begin{pmatrix}(Y^{(2)}_{\bm{3}_{0},3})^{3}+2Y^{(2)}_{\bm{3}_{0},1}Y^{(2)}_{\bm{3}_{0},2}Y^{(2)}_{\bm{3}_{0},3}\\\
(Y^{(2)}_{\bm{3}_{0},3})^{2}Y^{(2)}_{\bm{3}_{0},1}+2(Y^{(2)}_{\bm{3}_{0},1})^{2}Y^{(2)}_{\bm{3}_{0},2}\\\
(Y^{(2)}_{\bm{3}_{0},3})^{2}Y^{(2)}_{\bm{3}_{0},2}+2(Y^{(2)}_{\bm{3}_{0},2})^{2}Y^{(2)}_{\bm{3}_{0},1}\end{pmatrix}\,,$
$\displaystyle
Y^{(6)}_{\bm{3}_{1}}=Y^{(2)}_{\bm{1}^{\prime\prime}_{1}}Y^{(4)}_{\bm{3}_{0}}=\begin{pmatrix}Y^{(2)}_{\bm{3}_{0},3}(Y^{(2)}_{\bm{1}^{\prime\prime}_{1}})^{2}\\\
Y^{(2)}_{\bm{3}_{0},1}(Y^{(2)}_{\bm{1}^{\prime\prime}_{1}})^{2}\\\
Y^{(2)}_{\bm{3}_{0},2}(Y^{(2)}_{\bm{1}^{\prime\prime}_{1}})^{2}\\\
\end{pmatrix}\,,$ (B.1)

where $Y^{(6)}_{\bm{3}_{0}I}$ and $Y^{(6)}_{\bm{3}_{0}II}$ stand for the two
independent weight 6 VVMFs transforming as $\bm{3}_{0}$ under $A_{4}\times
Z_{2}$. Other higher weight VVMFs can also be obtained in a similar way. The
theory of VVMF gives us new insight to the structure of the space of the
modular multiplets. As shown in section 2, all the modular multiplets
$\mathcal{M}(\rho)=\bigoplus^{\infty}_{k=0}\mathcal{M}_{k}(\rho)$ in a
representation $\rho$ is a graded module over the ring
$\mathcal{M}(\mathbf{1})=\mathbb{C}[E_{4},E_{6}]$, and a set of basis of the
module $\mathcal{M}(\rho)$ is sufficient to generate the whole space of
modular forms. Hence the VVMFs $\mathcal{M}(A_{4}\times Z_{2})$ in the
representations of the finite modular group $A_{4}\times Z_{2}$ can be
organized by representations and it comprises eight VVMF modules

$\mathcal{M}(A_{4}\times
Z_{2})=\mathcal{M}(\mathbf{1}_{0})\oplus\mathcal{M}(\mathbf{1}_{1})\oplus\mathcal{M}(\mathbf{1}^{\prime}_{0})\oplus\mathcal{M}(\mathbf{1}^{\prime}_{1})\oplus\mathcal{M}(\mathbf{1}^{\prime\prime}_{0})\oplus\mathcal{M}(\mathbf{1}^{\prime\prime}_{1})\oplus\mathcal{M}(\mathbf{3}_{0})\oplus\mathcal{M}(\mathbf{3}_{1})\,.$
(B.2)

From table 1 and the representation matrix of the modular generator $T$, we
can read off the minimal weight and the basis of each module, and the basis
can be generated from the minimal weight VVMF by using the modular
differential operator $D^{n}_{k}$ for the triplet modules
$\mathcal{M}(\mathbf{3}_{0})$ and $\mathcal{M}(\mathbf{3}_{1})$. Namely, we
have

$\displaystyle\mathcal{M}(\mathbf{1}_{0})=\langle 1\rangle\,,$
$\displaystyle\mathcal{M}(\mathbf{1}_{1})=\langle\widetilde{Y}^{(6)}_{\mathbf{1}_{1}}\rangle\,,$
$\displaystyle\mathcal{M}(\mathbf{1}^{\prime}_{0})=\langle\widetilde{Y}^{(4)}_{\mathbf{1}^{\prime}_{0}}\rangle\,,$
$\displaystyle\mathcal{M}(\mathbf{1}^{\prime}_{1})=\langle\widetilde{Y}^{(10)}_{\mathbf{1}^{\prime}_{1}}\rangle\,,$
$\displaystyle\mathcal{M}(\mathbf{1}^{\prime\prime}_{0})=\langle\widetilde{Y}^{(8)}_{\mathbf{1}^{\prime\prime}_{0}}\rangle\,,$
$\displaystyle\mathcal{M}(\mathbf{1}^{\prime\prime}_{1})=\langle\widetilde{Y}^{(2)}_{\mathbf{1}^{\prime\prime}_{1}}\rangle\,,$
$\displaystyle\mathcal{M}(\mathbf{3}_{0})=\langle\widetilde{Y}^{(2)}_{\mathbf{3}_{0}},D_{2}\widetilde{Y}^{(2)}_{\mathbf{3}_{0}},D_{2}^{2}\widetilde{Y}^{(2)}_{\mathbf{3}_{0}}\rangle\,,$
$\displaystyle\mathcal{M}(\mathbf{3}_{1})=\langle\widetilde{Y}^{(4)}_{\mathbf{3}_{1}},D_{4}\widetilde{Y}^{(4)}_{\mathbf{3}_{1}},D_{4}^{2}\widetilde{Y}^{(4)}_{\mathbf{3}_{1}}\rangle\,.$
(B.3)

Furthermore the minimal weight VVMFs can be extracted from the solutions of
corresponding MLDEs shown in section 3, i.e.

$\displaystyle\widetilde{Y}^{(6)}_{\mathbf{1}_{1}}(\tau)=\eta^{12}(\tau)\,,~{}\widetilde{Y}^{(4)}_{\mathbf{1}^{\prime}_{0}}(\tau)=\eta^{8}(\tau)\,,~{}\widetilde{Y}^{(10)}_{\mathbf{1}^{\prime}_{1}}(\tau)=\eta^{20}(\tau)\,,$
$\displaystyle\widetilde{Y}^{(8)}_{\mathbf{1}^{\prime\prime}_{0}}(\tau)=\eta^{16}(\tau)\,,~{}~{}~{}~{}\widetilde{Y}^{(2)}_{\mathbf{1}^{\prime\prime}_{1}}(\tau)=\eta^{4}(\tau)\,,$
$\displaystyle\widetilde{Y}^{(2)}_{\mathbf{3}_{0}}(\tau)=\begin{pmatrix}\eta^{4}(\tau)(\frac{K(\tau)}{1728})^{-\frac{1}{6}}~{}{}_{3}F_{2}(-\frac{1}{6},\frac{1}{6},\frac{1}{2};\frac{2}{3},\frac{1}{3};K(\tau))\\\
-6\eta^{4}(\tau)(\frac{K(\tau)}{1728})^{\frac{1}{6}}~{}{}_{3}F_{2}(\frac{1}{6},\frac{1}{2},\frac{5}{6};\frac{2}{3},\frac{4}{3};K(\tau))\\\
-18\eta^{4}(\tau)(\frac{K(\tau)}{1728})^{\frac{1}{2}}~{}{}_{3}F_{2}(\frac{1}{2},\frac{5}{6},\frac{7}{6};\frac{5}{3},\frac{4}{3};K(\tau))\end{pmatrix}\,,$
$\displaystyle\widetilde{Y}^{(4)}_{\mathbf{3}_{1}}(\tau)=\begin{pmatrix}-6\eta^{8}(\tau)(\frac{K(\tau)}{1728})^{\frac{1}{6}}~{}{}_{3}F_{2}(\frac{1}{6},\frac{1}{2},\frac{5}{6};\frac{2}{3},\frac{4}{3};K(\tau))\\\
-18\eta^{8}(\tau)(\frac{K(\tau)}{1728})^{\frac{1}{2}}~{}{}_{3}F_{2}(\frac{1}{2},\frac{5}{6},\frac{7}{6};\frac{5}{3},\frac{4}{3};K(\tau))\\\
\eta^{8}(\tau)(\frac{K(\tau)}{1728})^{-\frac{1}{6}}~{}{}_{3}F_{2}(-\frac{1}{6},\frac{1}{6},\frac{1}{2};\frac{2}{3},\frac{1}{3};K(\tau))\end{pmatrix}\,.$
(B.4)

The linearly independent modular multiplets of $A_{4}\times Z_{2}$ at each
weight can be straightforwardly obtained by multiplying the polynomial of
$E_{4},E_{6}$ with the bases of modules in Eq. (B.3)

$\displaystyle
k=2:\widetilde{Y}^{(2)}_{\mathbf{3}_{0}}\,,\widetilde{Y}^{(2)}_{\mathbf{1}^{\prime\prime}_{1}}\,,$
$\displaystyle
k=4:E_{4}\,,\widetilde{Y}^{(4)}_{\mathbf{1}^{\prime}_{0}}\,,D_{2}\widetilde{Y}^{(2)}_{\mathbf{3}_{0}}\,,\widetilde{Y}^{(4)}_{\mathbf{3}_{1}}\,,$
$\displaystyle
k=6:E_{6}\,,\widetilde{Y}^{(6)}_{\mathbf{1}_{1}}\,,E_{4}\widetilde{Y}^{(2)}_{\mathbf{1}^{\prime\prime}_{1}}\,,D^{2}_{2}\widetilde{Y}^{(2)}_{\mathbf{3}_{0}}\,,E_{4}\widetilde{Y}^{(2)}_{\mathbf{3}_{0}}\,,D_{4}\widetilde{Y}^{(4)}_{\mathbf{3}_{1}}\,,$
$\displaystyle
k=8:E_{4}^{2}\,,E_{4}\widetilde{Y}^{(4)}_{\mathbf{1}^{\prime}_{0}}\,,\widetilde{Y}^{(8)}_{\mathbf{1}^{\prime\prime}_{0}}\,,E_{6}\widetilde{Y}^{(2)}_{\mathbf{1}^{\prime\prime}_{1}}\,,E_{6}\widetilde{Y}^{(2)}_{\mathbf{3}_{0}}\,,E_{4}D_{2}\widetilde{Y}^{(2)}_{\mathbf{3}_{0}}\,,E_{4}\widetilde{Y}^{(4)}_{\mathbf{3}_{1}}\,,D_{4}^{2}\widetilde{Y}^{(4)}_{\mathbf{3}_{1}}\,,$
(B.5)

Comparing the $q$-expressions, one can see that the modular multiplets in Eq.
(B.5) match those built from tensor products in Eqs. (5.8, B.1),

$\displaystyle Y^{(4)}_{\mathbf{1}_{0}}=E_{4}\,,\quad
Y^{(4)}_{\mathbf{1}^{\prime}_{0}}=-12\widetilde{Y}^{(4)}_{\mathbf{1}^{\prime}_{0}}\,,\quad
Y^{(4)}_{\mathbf{3}_{0}}=-6D_{2}\widetilde{Y}^{(2)}_{\mathbf{3}_{0}}\,,\quad
Y^{(4)}_{\mathbf{3}_{1}}=\widetilde{Y}^{(4)}_{\mathbf{3}_{1}}\,,$
$\displaystyle Y^{(6)}_{\mathbf{1}_{0}}=E_{6}\,,\quad
Y^{(6)}_{\mathbf{1}_{1}}=-12\widetilde{Y}^{(6)}_{\mathbf{1}_{1}}\,,\quad
Y^{(6)}_{\mathbf{1}^{\prime\prime}_{1}}=E_{4}\widetilde{Y}^{(2)}_{\mathbf{1}^{\prime\prime}_{1}}\,,\quad
Y^{(6)}_{\mathbf{3}_{0}I}=E_{4}\widetilde{Y}^{(2)}_{\mathbf{3}_{0}}\,,$
$\displaystyle
Y^{(6)}_{\mathbf{3}_{0}II}=2E_{4}\widetilde{Y}^{(2)}_{\mathbf{3}_{0}}-36D_{2}^{2}\widetilde{Y}^{(2)}_{\mathbf{3}_{0}}\,,\quad
Y^{(6)}_{\mathbf{3}_{1}}=-6D_{4}\widetilde{Y}^{(4)}_{\mathbf{3}_{1}}\,.$ (B.6)

Therefore the module structure of VVMFs makes the construction of modular
multiplets much easier, and one doesn’t need to search for the non-linear
constraints which relate dependent multiplets coming from tensor products.
Since multiplication of the polynomial of $E_{4}$, $E_{6}$ doesn’t change the
alignment, the modular multiplets in any given representation only have finite
possible $\tau$ dependent alignments. These two approaches of constructing
modular forms are complementary. For example, the MLDEs for some minimal
weight VVMFs in high dimensional representations are difficult to be solved
analytically, but they may be obtained from the tensor products of lower
weight VVMFs in low dimensional representations.

##  Appendix C Finite number of the modular invariant models

There are no restrictions on the representation and weight assignments for the
matter fields, and the dimension of space of modualar forms generally
increases with the modular weight. Therefore it seems that one can construct
infinite possible models for neutrino masses and mixing based on a finite
modular group. In this section, we show that only a finite number of models
could be constructed because the modular multiplets $\mathcal{M}(\rho)$ in any
representation $\rho$ have a few possible alignments, as explained in section
B. In the following, we use a concrete example to explain the finiteness of
modular invariant models. The neutrino masses are generated from the Weinberg
operator, we assign the lepton doublets $L$ and right-handed charged leptons
$E^{c}$ to triplets of $A_{4}\times Z_{2}$, both Higgs doublets $H_{u}$ and
$H_{d}$ are trivial singlets and their modular weights are assumed to be
vanishing without loss of generality,

$\displaystyle\rho_{E^{c}}=\mathbf{3}_{1},~{}~{}\rho_{L}=\mathbf{3}_{0},~{}~{}\rho_{H_{u}}=\rho_{H_{d}}=\mathbf{1}_{0}\,.$
(C.1)

The superpotentials for lepton masses can be written as:

$\displaystyle\mathcal{W}_{e}$
$\displaystyle=\alpha_{1}\left(Y^{(k_{e})}_{\bm{3}_{1}}(E^{c}L)_{\bm{3}_{1},S}\right)_{\bm{1}_{0}}H_{d}+\alpha_{2}\left(Y^{(k_{e})}_{\bm{3}_{1}}(E^{c}L)_{\bm{3}_{1},A}\right)_{\bm{1}_{0}}H_{d}+\alpha_{3}Y^{(k_{e})}_{\bm{1}_{1}}(E^{c}L)_{\bm{1}_{1}}H_{d}$
$\displaystyle~{}~{}+\alpha_{4}Y^{(k_{e})}_{\bm{1}^{\prime\prime}_{1}}(E^{c}L)_{\bm{1}^{\prime}_{1}}H_{d}+\alpha_{5}Y^{(k_{e})}_{\bm{1}^{\prime}_{1}}(E^{c}L)_{\bm{1}^{\prime\prime}_{1}}H_{d}\,,$
$\displaystyle\mathcal{W}_{\nu}$
$\displaystyle=\frac{g_{1}}{\Lambda}\left(Y^{(k_{\nu})}_{\bm{3}_{0}}(LL)_{\bm{3}_{0},S}\right)_{\bm{1}_{0}}H_{u}H_{u}+\frac{g_{2}}{\Lambda}Y^{(k_{\nu})}_{\bm{1}_{0}}(LL)_{\bm{1}_{0}}H_{u}H_{u}$
$\displaystyle~{}~{}+\frac{g_{3}}{\Lambda}Y^{(k_{\nu})}_{\bm{1}^{\prime\prime}_{0}}(LL)_{\bm{1}^{\prime}_{0}}H_{u}H_{u}+\frac{g_{4}}{\Lambda}Y^{(k_{\nu})}_{\bm{1}^{\prime}_{0}}(LL)_{\bm{1}^{\prime\prime}_{0}}H_{u}H_{u}\,.$
(C.2)

The modular form singlets can be absorbed into the free couplings
$\alpha_{3,4,5}$ and $g_{2,3,4}$. It can occur that there are several
independent triplet modular forms $Y^{(k_{e})}_{\bm{3}_{1}}$ and
$Y^{(k_{\nu})}_{\bm{3}_{0}}$. Then all terms with arbitrary coefficients
should be included and the superpotential $\mathcal{W}_{e,\nu}$ contains more
parameters than those explicitly indicated. Modular invariance of the
superpotential requires the following relations for the weights

$k_{e}=k_{L}+k_{E^{c}},~{}~{}~{}~{}k_{\nu}=2k_{L}\,,$ (C.3)

where $k_{L}$ and $k_{E^{c}}$ are the modular weights of $L$ and $E^{c}$
respectively. By choosing the values of $k_{L}$ and $k_{E^{c}}$, we can
achieve any integer values for the modular weights $k_{e}$ and $k_{\nu}$. Thus
it seems there are infinite possible models differing in the values of
$k_{e,\nu}$, but different choices of $k_{e,\nu}$ can effectively lead to the
same model so that one has a finite number of possibilities. The key point is
that, the modular multiplets $Y^{(k_{e})}_{\bm{3}_{1}}$ and
$Y^{(k_{\nu})}_{\bm{3}_{0}}$ for any values of $k_{e,\nu}$ can be expressed in
terms of the three independent bases of the modules
$\mathcal{M}(\mathbf{3}_{0})$ and $\mathcal{M}(\mathbf{3}_{1})$ over the ring
$\mathbb{C}[E_{4},E_{6}]$. Namely, they can always be written as

$\displaystyle
Y^{(k_{e})}_{\bm{3}_{1}}=f_{1}(\tau)Y^{(4)}_{\bm{3}_{1}}+f_{2}(\tau)D_{4}Y^{(4)}_{\bm{3}_{1}}+f_{3}(\tau)D_{4}^{2}Y^{(4)}_{\bm{3}_{1}}\,,$
$\displaystyle
Y^{(k_{\nu})}_{\bm{3}_{0}}=h_{1}(\tau)Y^{(2)}_{\bm{3}_{0}}+h_{2}(\tau)D_{2}Y^{(2)}_{\bm{3}_{0}}+h_{3}(\tau)D_{2}^{2}Y^{(2)}_{\bm{3}_{0}}\,,$
(C.4)

where $f_{1}(\tau)$, $f_{2}(\tau)$ and $f_{3}(\tau)$ are polynomials of
$E_{4}$, $E_{6}$, and they are scalar modular forms of $SL(2,\mathbb{Z})$ with
weights $k_{e}-4$, $k_{e}-6$ and $k_{e}-8$ respectively. Analogously
$h_{1}(\tau)$, $h_{2}(\tau)$ and $h_{3}(\tau)$ are modular forms of
$SL(2,\mathbb{Z})$ with weights $k_{\nu}-2$, $k_{\nu}-4$ and $k_{\nu}-6$
respectively. The corresponding functions are vanishing if the modular forms
at some of these weights don’t exist. As a consequence, we see that the
superpotential $\mathcal{W}_{e,\nu}$ has at most 15 independent couplings for
$L\sim\mathbf{3}_{0}$ and $E^{c}\sim\mathbf{3}_{1}$. Even after considering
other possible representation assignments for the matter fields, one can only
construct a finite number of modular invariant superpotential based on
$A_{4}\times Z_{2}$. Hence the charged lepton and neutrino mass matrices can
only take a finite number of possible forms as well. This conclusion also
applies to all other finite modular groups.

We proceed to investigate the Kähler potential before closing this section.
The Kähler potential of the lepton doublets $L$ is considered for
illustration, and its most general form is given by

$\displaystyle\mathcal{K}_{L}=(-i\tau+i\bar{\tau})^{-k_{L}}(L^{\dagger}L)_{\bm{1}}+\sum_{k_{Y},\mathbf{r}_{1},\mathbf{r}_{2}}(-i\tau+i\bar{\tau})^{-k_{L}+k_{1}}(L^{\dagger}Y^{(k_{Y})\dagger}_{\mathbf{r}_{1}}LY^{(k_{Y})}_{\mathbf{r}_{2}})_{\mathbf{1}_{0}}+\mathrm{h.c.}\,,$
(C.5)

where we suppress all coupling constants and each operator could have several
independent $A_{4}\times Z_{2}$ contractions. Because the modular multiplets
in any representation of $A_{4}\times Z_{2}$ have a few alignments, it is
sufficient to only consider the modular multiplets
$Y^{(k_{Y})}_{\mathbf{r}_{1}}$ and $Y^{(k_{Y})}_{\mathbf{r}_{2}}$ which are
the bases of the modules of $A_{4}\times Z_{2}$ shown in Eq. (B.3). Therefore
the modular symmetry constrains the Kähler potential to depend on a finite
(not infinite) number of parameters. The presence of the additional terms
besides the minimal Kähler potential reduces the predictive power of modular
flavor symmetry [94].

##  Appendix D $GL(2,3)$ group

The group $GL(2,3)$ consists of $2\times 2$ invertible matrices whose elements
are added and multiplied as integers modulo 3. It is the Schur cover of the
permutation group $S_{4}$ of “+” type. The group $GL(2,3)$ can be generated by
the modular generators $S$ and $T$ satisfying the relations:

$S^{2}=(ST)^{3}=T^{8}=ST^{4}ST^{-4}=1\,.$ (D.1)

It has 48 elements and the group ID is $[48,29]$ in GAP [103]. The element
$T^{4}$ commutes with all group elements, and the cyclic group
$Z^{T^{4}}_{2}\equiv\left\\{1,T^{4}\right\\}$ is the center, and the quotient
group $GL(2,3)/Z^{T^{4}}_{2}$ is isomorphic to $S_{4}$.

In addition to the representations of $S_{4}$: $\bm{1}$, $\bm{1}^{\prime}$,
$\bm{2}$, $\bm{3}$ and $\bm{3}^{\prime}$, $GL(2,3)$ possesses two doublet
representations $\bm{2}^{\prime},\bm{2}^{\prime\prime}$ and one quartet
representation $\bm{4}$. The explicit forms of the representation matrices
$\rho(S)$ and $\rho(T)$ in each of the irreducible representations are given
in the following,

$\displaystyle\hskip-28.90755pt\mathbf{1}:~{}S=1\,,~{}~{}T=1\,,$
$\displaystyle\hskip-28.90755pt\mathbf{1}^{\prime}:~{}S=-1\,,~{}~{}T=-1\,,$
$\displaystyle\hskip-28.90755pt\mathbf{2}:~{}S=\dfrac{1}{2}\left(\begin{array}[]{cc}-1&-\sqrt{3}\\\
-\sqrt{3}&1\\\ \end{array}\right)\,,~{}~{}T=\left(\begin{array}[]{cc}1&0\\\
0&-1\\\ \end{array}\right)\,,$ (D.6)
$\displaystyle\hskip-28.90755pt\mathbf{2}^{\prime}:~{}S=\dfrac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&1\\\
1&-1\\\ \end{array}\right)\,,~{}~{}T=\left(\begin{array}[]{cc}e^{3\pi
i/4}&0\\\ 0&e^{\pi i/4}\\\ \end{array}\right)\,,$ (D.11)
$\displaystyle\hskip-28.90755pt\mathbf{2}^{\prime\prime}:~{}S=\dfrac{1}{\sqrt{2}}\left(\begin{array}[]{cc}-1&1\\\
1&1\\\ \end{array}\right)\,,~{}~{}T=\left(\begin{array}[]{cc}e^{-\pi i/4}&0\\\
0&e^{-3\pi i/4}\\\ \end{array}\right)\,,$ (D.16)
$\displaystyle\hskip-28.90755pt\mathbf{3}:~{}S=\dfrac{1}{2}\left(\begin{array}[]{ccc}0&\sqrt{2}&\sqrt{2}\\\
\sqrt{2}&-1&1\\\ \sqrt{2}&1&-1\\\
\end{array}\right)\,,~{}~{}T=\left(\begin{array}[]{ccc}1&0&0\\\ 0&i&0\\\
0&0&-i\\\ \end{array}\right)\,,$ (D.23)
$\displaystyle\hskip-28.90755pt\mathbf{3^{\prime}}:~{}S=-\dfrac{1}{2}\left(\begin{array}[]{ccc}0&\sqrt{2}&\sqrt{2}\\\
\sqrt{2}&-1&1\\\ \sqrt{2}&1&-1\\\
\end{array}\right)\,,~{}~{}T=-\left(\begin{array}[]{ccc}1&0&0\\\ 0&i&0\\\
0&0&-i\\\ \end{array}\right)$ (D.30)
$\displaystyle\hskip-28.90755pt\mathbf{4}:~{}S=\dfrac{1}{2\sqrt{2}}\left(\begin{array}[]{cccc}-1&\sqrt{3}&1&\sqrt{3}\\\
\sqrt{3}&-1&\sqrt{3}&1\\\ 1&\sqrt{3}&1&-\sqrt{3}\\\ \sqrt{3}&1&-\sqrt{3}&1\\\
\end{array}\right)\,,~{}~{}T=\left(\begin{array}[]{cccc}e^{-3\pi i/4}&0&0&0\\\
0&e^{3\pi i/4}&0&0\\\ 0&0&e^{-\pi i/4}&0\\\ 0&0&0&e^{\pi i/4}\\\
\end{array}\right)\,.$ (D.39)

Notice that the two 1-d irreps $\bm{1}$ and $\bm{1}^{\prime}$ correspond to
$\mathbf{1}_{0}$ and $\mathbf{1}_{6}$ of $SL(2,\mathbb{Z})$, these three 2-d
irreps $\mathbf{2}$, $\mathbf{2}^{\prime}$ and $\mathbf{2}^{\prime\prime}$ are
the doublet representations $\mathbf{2}_{(0,\frac{1}{2})}$,
$\mathbf{2}_{(\frac{3}{8},\frac{1}{8})}$ and
$\mathbf{2}_{(\frac{7}{8},\frac{5}{8})}$ of $SL(2,\mathbb{Z})$, and the two
3-d irreps $\mathbf{3}$ and $\mathbf{3}^{\prime}$ correspond to
$\mathbf{3}_{(0,\frac{1}{4},\frac{3}{4})}$ and
$\mathbf{3}_{(\frac{1}{2},\frac{3}{4},\frac{1}{4})}$ of $SL(2,\mathbb{Z})$. We
see the representation matrix $\rho(T^{4})=1$ for the irreps $\bm{1}$,
$\bm{1}^{\prime}$, $\bm{2}$, $\bm{3},\bm{3}^{\prime}$ and $\rho(T^{4})=-1$ for
the irreps $\bm{2}^{\prime}$, $\bm{2}^{\prime\prime}$, $\bm{4}$. Hence there
is no way to distinguish the group $GL(2,3)$ from $S_{4}$ when working with
the irreps $\bm{1}$, $\bm{1}^{\prime}$, $\bm{2}$, $\bm{3}$ and
$\bm{3}^{\prime}$. The 48 elements of $GL(2,3)$ can be divided into 8
conjugacy classes, and the character table is shown in table 4, it is obtain
by taking trace of the explicit representation matrices.

Classes | $1C_{1}$ | $1C_{2}$ | $6C_{8}$ | $6C^{\prime}_{8}$ | $6C_{4}$ | $8C_{6}$ | $8C_{3}$ | $12C^{\prime}_{2}$
---|---|---|---|---|---|---|---|---
Sample element | $1$ | $T^{4}$ | $T$ | $ST^{2}$ | $(T^{2}S)^{2}$ | $TST^{3}S$ | $TST^{4}$ | $TST^{3}$
Order of class | 1 | 1 | 6 | 6 | 6 | 8 | 8 | 12
Order of element | 1 | 2 | 8 | 8 | 4 | 6 | 3 | 2
$\mathbf{1}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
$\mathbf{1}^{\prime}$ | $1$ | $1$ | $-1$ | $-1$ | $1$ | $1$ | $1$ | $-1$
$\mathbf{2}$ | $2$ | $2$ | $0$ | $0$ | $2$ | $-1$ | $-1$ | $0$
$\mathbf{2}^{\prime}$ | $2$ | $-2$ | $\sqrt{2}\,i$ | $-\sqrt{2}\,i$ | $0$ | $-1$ | $1$ | $0$
$\mathbf{2}^{\prime\prime}$ | $2$ | $-2$ | $-\sqrt{2}\,i$ | $\sqrt{2}\,i$ | $0$ | $-1$ | $1$ | $0$
$\mathbf{3}$ | $3$ | $3$ | $1$ | $1$ | $-1$ | $0$ | $0$ | $-1$
$\mathbf{3}^{\prime}$ | $3$ | $3$ | $-1$ | $-1$ | $-1$ | $0$ | $0$ | $1$
$\mathbf{4}$ | $4$ | $-4$ | $0$ | $0$ | $0$ | $1$ | $-1$ | $0$
Table 4:  The character table of the group $GL(2,3)$.

We present the decompositions of tensor products of different $GL(2,3)$ irreps
and the corresponding Clebsch-Gordan coefficients in our basis. We use
$\alpha_{i}$ to indicate the elements of the first representation of the
product and $\beta_{i}$ to indicate those of the second representation. The
results are collected in table 5 and table 6.

$\bm{1}^{\prime}\otimes\bm{2}=\bm{2}$ | $\bm{1}^{\prime}\otimes\bm{2}^{\prime}=\bm{2}^{\prime\prime}$ |  $\bm{1}^{\prime}\otimes\bm{2}^{\prime\prime}=\bm{2}^{\prime}$ | $\bm{1}^{\prime}\otimes\bm{4}=\bm{4}$
---|---|---|---
$\mathbf{2}\sim\begin{pmatrix}\alpha\beta_{2}\\\ -\alpha\beta_{1}\\\ \end{pmatrix}$ | $\mathbf{2}^{\prime\prime}\sim\begin{pmatrix}-\alpha\beta_{1}\\\ \alpha\beta_{2}\\\ \end{pmatrix}$ | $\mathbf{2}^{\prime}\sim\begin{pmatrix}-\alpha\beta_{1}\\\ \alpha\beta_{2}\\\ \end{pmatrix}$ | $\mathbf{4}\sim\begin{pmatrix}\alpha\beta_{4}\\\ \alpha\beta_{3}\\\ -\alpha\beta_{2}\\\ -\alpha\beta_{1}\\\ \end{pmatrix}$
$\mathbf{1}\otimes\mathbf{3}=\mathbf{1}^{\prime}\otimes\mathbf{3}^{\prime}=\mathbf{3}$ | $\mathbf{1}\otimes\mathbf{3}^{\prime}=\mathbf{1}^{\prime}\otimes\mathbf{3}=\mathbf{3}^{\prime}$
$\mathbf{3}\sim\begin{pmatrix}\alpha\beta_{1}\\\ \alpha\beta_{2}\\\ \alpha\beta_{3}\\\ \end{pmatrix}$ | $\mathbf{3^{\prime}}\sim\begin{pmatrix}\alpha\beta_{1}\\\ \alpha\beta_{2}\\\ \alpha\beta_{3}\\\ \end{pmatrix}$
$\mathbf{2}\otimes\mathbf{2}=\mathbf{1_{s}}\oplus\mathbf{1^{\prime}_{a}}\oplus\mathbf{2_{s}}$ | $\mathbf{2}\otimes\mathbf{2}^{\prime}=\mathbf{4}$ | $\mathbf{2}\otimes\mathbf{2}^{\prime\prime}=\mathbf{4}$
$\begin{array}[]{l}\mathbf{1_{s}}\sim\alpha_{1}\beta_{1}+\alpha_{2}\beta_{2}\\\ \mathbf{1^{\prime}_{a}}\sim\alpha_{2}\beta_{1}-\alpha_{1}\beta_{2}\\\ \mathbf{2_{s}}\sim\begin{pmatrix}\alpha_{2}\beta_{2}-\alpha_{1}\beta_{1}\\\ \alpha_{2}\beta_{1}+\alpha_{1}\beta_{2}\\\ \end{pmatrix}\\\ \end{array}$ | $\mathbf{4}\sim\begin{pmatrix}\alpha_{2}\beta_{2}\\\ -\alpha_{1}\beta_{1}\\\ \alpha_{2}\beta_{1}\\\ \alpha_{1}\beta_{2}\\\ \end{pmatrix}$ | $\mathbf{4}\sim\begin{pmatrix}-\alpha_{1}\beta_{2}\\\ \alpha_{2}\beta_{1}\\\ \alpha_{1}\beta_{1}\\\ \alpha_{2}\beta_{2}\\\ \end{pmatrix}$
$\mathbf{2^{\prime}}\otimes\mathbf{2^{\prime}}=\mathbf{1^{\prime}_{a}}\oplus\mathbf{3^{\prime}_{s}}$ | $\mathbf{2^{\prime}}\otimes\mathbf{2^{\prime\prime}}=\mathbf{1}\oplus\mathbf{3}$ | $\mathbf{2^{\prime\prime}}\otimes\mathbf{2^{\prime\prime}}=\mathbf{1^{\prime}_{a}}\oplus\mathbf{3^{\prime}_{s}}$
$\begin{array}[]{l}\mathbf{1^{\prime}_{a}}\sim\alpha_{2}\beta_{1}-\alpha_{1}\beta_{2}\\\ \mathbf{3^{\prime}_{s}}\sim\begin{pmatrix}\alpha_{2}\beta_{1}+\alpha_{1}\beta_{2}\\\ -\sqrt{2}\alpha_{1}\beta_{1}\\\ \sqrt{2}\alpha_{2}\beta_{2}\\\ \end{pmatrix}\\\ \end{array}$ | $\begin{array}[]{l}\mathbf{1}\sim\alpha_{2}\beta_{1}+\alpha_{1}\beta_{2}\\\ \mathbf{3}\sim\begin{pmatrix}\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\\\ \sqrt{2}\alpha_{1}\beta_{1}\\\ \sqrt{2}\alpha_{2}\beta_{2}\\\ \end{pmatrix}\\\ \end{array}$ | $\begin{array}[]{l}\mathbf{1^{\prime}_{a}}\sim\alpha_{2}\beta_{1}-\alpha_{1}\beta_{2}\\\ \mathbf{3^{\prime}_{s}}\sim\begin{pmatrix}\alpha_{2}\beta_{1}+\alpha_{1}\beta_{2}\\\ \sqrt{2}\alpha_{1}\beta_{1}\\\ -\sqrt{2}\alpha_{2}\beta_{2}\\\ \end{pmatrix}\\\ \end{array}$
$\mathbf{2}\otimes\mathbf{3}=\mathbf{3}\oplus\mathbf{3^{\prime}}$ | $\mathbf{2}\otimes\mathbf{3^{\prime}}=\mathbf{3}\oplus\mathbf{3^{\prime}}$ | $\mathbf{2^{\prime}}\otimes\mathbf{3}=\mathbf{2^{\prime}}\oplus\mathbf{4}$
$\begin{array}[]{l}\mathbf{3}\sim\begin{pmatrix}-2\alpha_{1}\beta_{1}\\\ \alpha_{1}\beta_{2}+\sqrt{3}\alpha_{2}\beta_{3}\\\ \sqrt{3}\alpha_{2}\beta_{2}+\alpha_{1}\beta_{3}\\\ \end{pmatrix}\\\ \mathbf{3^{\prime}}\sim\begin{pmatrix}-2\alpha_{2}\beta_{1}\\\ \alpha_{2}\beta_{2}-\sqrt{3}\alpha_{1}\beta_{3}\\\ \alpha_{2}\beta_{3}-\sqrt{3}\alpha_{1}\beta_{2}\\\ \end{pmatrix}\\\ \end{array}$ | $\begin{array}[]{l}\mathbf{3}\sim\begin{pmatrix}-2\alpha_{2}\beta_{1}\\\ \alpha_{2}\beta_{2}-\sqrt{3}\alpha_{1}\beta_{3}\\\ \alpha_{2}\beta_{3}-\sqrt{3}\alpha_{1}\beta_{2}\\\ \end{pmatrix}\\\ \mathbf{3^{\prime}}\sim\begin{pmatrix}-2\alpha_{1}\beta_{1}\\\ \alpha_{1}\beta_{2}+\sqrt{3}\alpha_{2}\beta_{3}\\\ \sqrt{3}\alpha_{2}\beta_{2}+\alpha_{1}\beta_{3}\\\ \end{pmatrix}\\\ \end{array}$ | $\begin{array}[]{l}\mathbf{2^{\prime}}\sim\begin{pmatrix}\alpha_{1}\beta_{1}+\sqrt{2}\alpha_{2}\beta_{2}\\\ \sqrt{2}\alpha_{1}\beta_{3}-\alpha_{2}\beta_{1}\\\ \end{pmatrix}\\\ \mathbf{4}\sim\begin{pmatrix}\sqrt{3}\alpha_{1}\beta_{2}\\\ \sqrt{2}\alpha_{1}\beta_{1}-\alpha_{2}\beta_{2}\\\ \sqrt{3}\alpha_{2}\beta_{3}\\\ \sqrt{2}\alpha_{2}\beta_{1}+\alpha_{1}\beta_{3}\\\ \end{pmatrix}\\\ \end{array}$
$\mathbf{2^{\prime}}\otimes\mathbf{3^{\prime}}=\mathbf{2^{\prime\prime}}\oplus\mathbf{4}$ | $\mathbf{2^{\prime\prime}}\otimes\mathbf{3}=\mathbf{2^{\prime\prime}}\oplus\mathbf{4}$ | $\mathbf{2^{\prime\prime}}\otimes\mathbf{3^{\prime}}=\mathbf{2^{\prime}}\oplus\mathbf{4}$
$\begin{array}[]{l}\mathbf{2^{\prime\prime}}\sim\begin{pmatrix}\alpha_{1}\beta_{1}+\sqrt{2}\alpha_{2}\beta_{2}\\\ \alpha_{2}\beta_{1}-\sqrt{2}\alpha_{1}\beta_{3}\\\ \end{pmatrix}\\\ \mathbf{4}\sim\begin{pmatrix}\sqrt{2}\alpha_{2}\beta_{1}+\alpha_{1}\beta_{3}\\\ \sqrt{3}\alpha_{2}\beta_{3}\\\ \alpha_{2}\beta_{2}-\sqrt{2}\alpha_{1}\beta_{1}\\\ -\sqrt{3}\alpha_{1}\beta_{2}\\\ \end{pmatrix}\\\ \end{array}$ | $\begin{array}[]{l}\mathbf{2^{\prime\prime}}\sim\begin{pmatrix}\sqrt{2}\alpha_{2}\beta_{2}-\alpha_{1}\beta_{1}\\\ \alpha_{2}\beta_{1}+\sqrt{2}\alpha_{1}\beta_{3}\\\ \end{pmatrix}\\\ \mathbf{4}\sim\begin{pmatrix}\sqrt{2}\alpha_{2}\beta_{1}-\alpha_{1}\beta_{3}\\\ \sqrt{3}\alpha_{2}\beta_{3}\\\ \sqrt{2}\alpha_{1}\beta_{1}+\alpha_{2}\beta_{2}\\\ \sqrt{3}\alpha_{1}\beta_{2}\\\ \end{pmatrix}\\\ \end{array}$ | $\begin{array}[]{l}\mathbf{2^{\prime}}\sim\begin{pmatrix}\alpha_{1}\beta_{1}-\sqrt{2}\alpha_{2}\beta_{2}\\\ \alpha_{2}\beta_{1}+\sqrt{2}\alpha_{1}\beta_{3}\\\ \end{pmatrix}\\\ \mathbf{4}\sim\begin{pmatrix}\sqrt{3}\alpha_{1}\beta_{2}\\\ \sqrt{2}\alpha_{1}\beta_{1}+\alpha_{2}\beta_{2}\\\ -\sqrt{3}\alpha_{2}\beta_{3}\\\ \alpha_{1}\beta_{3}-\sqrt{2}\alpha_{2}\beta_{1}\\\ \end{pmatrix}\\\ \end{array}$
$\mathbf{2}\otimes\mathbf{4}=\mathbf{2^{\prime}}\oplus\mathbf{2^{\prime\prime}}\oplus\mathbf{4}$ | $\mathbf{2^{\prime}}\otimes\mathbf{4}=\mathbf{2}\oplus\mathbf{3}\oplus\mathbf{3^{\prime}}$ | $\mathbf{2^{\prime\prime}}\otimes\mathbf{4}=\mathbf{2}\oplus\mathbf{3}\oplus\mathbf{3^{\prime}}$
$\begin{array}[]{l}\mathbf{2^{\prime}}\sim\begin{pmatrix}\alpha_{2}\beta_{3}-\alpha_{1}\beta_{2}\\\ \alpha_{2}\beta_{1}+\alpha_{1}\beta_{4}\\\ \end{pmatrix}\\\ \mathbf{2^{\prime\prime}}=\begin{pmatrix}\alpha_{2}\beta_{2}+\alpha_{1}\beta_{3}\\\ \alpha_{2}\beta_{4}-\alpha_{1}\beta_{1}\\\ \end{pmatrix}\\\ \mathbf{4}\sim\begin{pmatrix}\alpha_{1}\beta_{1}+\alpha_{2}\beta_{4}\\\ -\alpha_{1}\beta_{2}-\alpha_{2}\beta_{3}\\\ \alpha_{1}\beta_{3}-\alpha_{2}\beta_{2}\\\ \alpha_{2}\beta_{1}-\alpha_{1}\beta_{4}\\\ \end{pmatrix}\\\ \end{array}$ | $\begin{array}[]{l}\mathbf{2}\sim\begin{pmatrix}\alpha_{2}\beta_{3}-\alpha_{1}\beta_{1}\\\ \alpha_{2}\beta_{2}+\alpha_{1}\beta_{4}\\\ \end{pmatrix}\\\ \mathbf{3}\sim\begin{pmatrix}\sqrt{2}\alpha_{1}\beta_{1}+\sqrt{2}\alpha_{2}\beta_{3}\\\ \alpha_{1}\beta_{3}+\sqrt{3}\alpha_{2}\beta_{4}\\\ \sqrt{3}\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\\\ \end{pmatrix}\\\ \mathbf{3^{\prime}}\sim\begin{pmatrix}\sqrt{2}\alpha_{2}\beta_{2}-\sqrt{2}\alpha_{1}\beta_{4}\\\ \sqrt{3}\alpha_{2}\beta_{1}+\alpha_{1}\beta_{2}\\\ \alpha_{2}\beta_{4}-\sqrt{3}\alpha_{1}\beta_{3}\\\ \end{pmatrix}\\\ \end{array}$ | $\begin{array}[]{l}\mathbf{2}\sim\begin{pmatrix}\alpha_{1}\beta_{4}-\alpha_{2}\beta_{2}\\\ \alpha_{1}\beta_{1}+\alpha_{2}\beta_{3}\\\ \end{pmatrix}\\\ \mathbf{3}\sim\begin{pmatrix}\sqrt{2}\alpha_{2}\beta_{2}+\sqrt{2}\alpha_{1}\beta_{4}\\\ \sqrt{3}\alpha_{2}\beta_{1}-\alpha_{1}\beta_{2}\\\ \sqrt{3}\alpha_{1}\beta_{3}+\alpha_{2}\beta_{4}\\\ \end{pmatrix}\\\ \mathbf{3^{\prime}}\sim\begin{pmatrix}\sqrt{2}\alpha_{2}\beta_{3}-\sqrt{2}\alpha_{1}\beta_{1}\\\ \sqrt{3}\alpha_{2}\beta_{4}-\alpha_{1}\beta_{3}\\\ -\alpha_{2}\beta_{1}-\sqrt{3}\alpha_{1}\beta_{2}\\\ \end{pmatrix}\\\ \end{array}$

Table 5: Multiplication rules of different irreducible representations of
$GL(2,3)$ and the corresponding Clebsch-Gordan coefficients.

$\mathbf{3}\otimes\mathbf{3}=\mathbf{3^{\prime}}\otimes\mathbf{3^{\prime}}=\mathbf{1_{s}}\oplus\mathbf{2_{s}}\oplus\mathbf{3_{a}}\oplus\mathbf{3^{\prime}_{s}}$ | $\mathbf{3}\otimes\mathbf{3^{\prime}}=\mathbf{1^{\prime}}\oplus\mathbf{2}\oplus\mathbf{3}\oplus\mathbf{3^{\prime}}$
---|---
$\begin{array}[]{l}\mathbf{1_{s}}\sim\alpha_{1}\beta_{1}+\alpha_{3}\beta_{2}+\alpha_{2}\beta_{3}\\\ \mathbf{2_{s}}\sim\begin{pmatrix}\alpha_{3}\beta_{2}+\alpha_{2}\beta_{3}-2\alpha_{1}\beta_{1}\\\ \sqrt{3}\alpha_{2}\beta_{2}+\sqrt{3}\alpha_{3}\beta_{3}\\\ \end{pmatrix}\\\ \mathbf{3_{a}}\sim\begin{pmatrix}\alpha_{3}\beta_{2}-\alpha_{2}\beta_{3}\\\ \alpha_{2}\beta_{1}-\alpha_{1}\beta_{2}\\\ \alpha_{1}\beta_{3}-\alpha_{3}\beta_{1}\\\ \end{pmatrix}\\\ \mathbf{3^{\prime}_{s}}\sim\begin{pmatrix}\alpha_{3}\beta_{3}-\alpha_{2}\beta_{2}\\\ \alpha_{3}\beta_{1}+\alpha_{1}\beta_{3}\\\ -\alpha_{2}\beta_{1}-\alpha_{1}\beta_{2}\\\ \end{pmatrix}\\\ \end{array}$ | $\begin{array}[]{l}\mathbf{1^{\prime}}\sim\alpha_{1}\beta_{1}+\alpha_{3}\beta_{2}+\alpha_{2}\beta_{3}\\\ \mathbf{2}\sim\begin{pmatrix}\sqrt{3}\alpha_{2}\beta_{2}+\sqrt{3}\alpha_{3}\beta_{3}\\\ 2\alpha_{1}\beta_{1}-\alpha_{3}\beta_{2}-\alpha_{2}\beta_{3}\\\ \end{pmatrix}\\\ \mathbf{3}\sim\begin{pmatrix}\alpha_{3}\beta_{3}-\alpha_{2}\beta_{2}\\\ \alpha_{3}\beta_{1}+\alpha_{1}\beta_{3}\\\ -\alpha_{2}\beta_{1}-\alpha_{1}\beta_{2}\\\ \end{pmatrix}\\\ \mathbf{3^{\prime}}\sim\begin{pmatrix}\alpha_{3}\beta_{2}-\alpha_{2}\beta_{3}\\\ \alpha_{2}\beta_{1}-\alpha_{1}\beta_{2}\\\ \alpha_{1}\beta_{3}-\alpha_{3}\beta_{1}\\\ \end{pmatrix}\\\ \end{array}$
$\mathbf{3}\otimes\mathbf{4}=\mathbf{2^{\prime}}\oplus\mathbf{2^{\prime\prime}}\oplus\mathbf{4_{1}}\oplus\mathbf{4_{2}}$ | $\mathbf{3^{\prime}}\otimes\mathbf{4}=\mathbf{2^{\prime}}\oplus\mathbf{2^{\prime\prime}}\oplus\mathbf{4_{1}}\oplus\mathbf{4_{2}}$
$\begin{array}[]{l}\mathbf{2^{\prime}}\sim\begin{pmatrix}-\sqrt{3}\alpha_{3}\beta_{1}-\sqrt{2}\alpha_{1}\beta_{2}-\alpha_{2}\beta_{4}\\\ \alpha_{3}\beta_{2}-\sqrt{3}\alpha_{2}\beta_{3}-\sqrt{2}\alpha_{1}\beta_{4}\\\ \end{pmatrix}\\\ \mathbf{2^{\prime\prime}}\sim\begin{pmatrix}\sqrt{2}\alpha_{1}\beta_{3}+\sqrt{3}\alpha_{3}\beta_{4}-\alpha_{2}\beta_{1}\\\ \sqrt{2}\alpha_{1}\beta_{1}+\sqrt{3}\alpha_{2}\beta_{2}+\alpha_{3}\beta_{3}\\\ \end{pmatrix}\\\ \mathbf{4_{1}}\sim\begin{pmatrix}\sqrt{2}\alpha_{3}\beta_{3}-\alpha_{1}\beta_{1}\\\ \alpha_{1}\beta_{2}-\sqrt{2}\alpha_{2}\beta_{4}\\\ \sqrt{2}\alpha_{2}\beta_{1}+\alpha_{1}\beta_{3}\\\ -\sqrt{2}\alpha_{3}\beta_{2}-\alpha_{1}\beta_{4}\\\ \end{pmatrix}\\\ \mathbf{4_{2}}\sim\begin{pmatrix}3\alpha_{1}\beta_{1}-\sqrt{6}\alpha_{2}\beta_{2}\\\ \alpha_{1}\beta_{2}+2\sqrt{2}\alpha_{2}\beta_{4}-\sqrt{6}\alpha_{3}\beta_{1}\\\ \sqrt{6}\alpha_{3}\beta_{4}-3\alpha_{1}\beta_{3}\\\ 2\sqrt{2}\alpha_{3}\beta_{2}+\sqrt{6}\alpha_{2}\beta_{3}-\alpha_{1}\beta_{4}\\\ \end{pmatrix}\\\ \end{array}$ | $\begin{array}[]{l}\mathbf{2^{\prime}}\sim\begin{pmatrix}\sqrt{2}\alpha_{1}\beta_{3}+\sqrt{3}\alpha_{3}\beta_{4}-\alpha_{2}\beta_{1}\\\ -\sqrt{2}\alpha_{1}\beta_{1}-\sqrt{3}\alpha_{2}\beta_{2}-\alpha_{3}\beta_{3}\\\ \end{pmatrix}\\\ \mathbf{2^{\prime\prime}}\sim\begin{pmatrix}\sqrt{3}\alpha_{3}\beta_{1}+\sqrt{2}\alpha_{1}\beta_{2}+\alpha_{2}\beta_{4}\\\ \alpha_{3}\beta_{2}-\sqrt{3}\alpha_{2}\beta_{3}-\sqrt{2}\alpha_{1}\beta_{4}\\\ \end{pmatrix}\\\ \mathbf{4_{1}}\sim\begin{pmatrix}2\sqrt{2}\alpha_{3}\beta_{2}+\sqrt{6}\alpha_{2}\beta_{3}-\alpha_{1}\beta_{4}\\\ \sqrt{6}\alpha_{3}\beta_{4}-3\alpha_{1}\beta_{3}\\\ \sqrt{6}\alpha_{3}\beta_{1}-\alpha_{1}\beta_{2}-2\sqrt{2}\alpha_{2}\beta_{4}\\\ \sqrt{6}\alpha_{2}\beta_{2}-3\alpha_{1}\beta_{1}\\\ \end{pmatrix}\\\ \mathbf{4_{2}}\sim\begin{pmatrix}\sqrt{2}\alpha_{3}\beta_{2}+\alpha_{1}\beta_{4}\\\ -\sqrt{2}\alpha_{2}\beta_{1}-\alpha_{1}\beta_{3}\\\ \alpha_{1}\beta_{2}-\sqrt{2}\alpha_{2}\beta_{4}\\\ \sqrt{2}\alpha_{3}\beta_{3}-\alpha_{1}\beta_{1}\\\ \end{pmatrix}\\\ \end{array}$
$\mathbf{4}\otimes\mathbf{4}=\mathbf{1_{s}}\oplus\mathbf{1^{\prime}_{a}}\oplus\mathbf{2_{a}}\oplus\mathbf{3_{1,a}}\oplus\mathbf{3_{2,s}}\oplus\mathbf{3^{\prime}_{1,s}}\oplus\mathbf{3^{\prime}_{2,s}}$
$\begin{array}[]{l}\mathbf{1_{s}}\sim\alpha_{2}\beta_{1}+\alpha_{1}\beta_{2}+\alpha_{4}\beta_{3}+\alpha_{3}\beta_{4}\\\
\mathbf{1^{\prime}_{a}}\sim\alpha_{3}\beta_{1}-\alpha_{1}\beta_{3}+\alpha_{4}\beta_{2}-\alpha_{2}\beta_{4}\\\
\mathbf{2_{a}}\sim\begin{pmatrix}\alpha_{2}\beta_{1}-\alpha_{1}\beta_{2}+\alpha_{4}\beta_{3}-\alpha_{3}\beta_{4}\\\
\alpha_{3}\beta_{1}-\alpha_{1}\beta_{3}+\alpha_{2}\beta_{4}-\alpha_{4}\beta_{2}\\\
\end{pmatrix}\\\
\mathbf{3_{1,a}}\sim\begin{pmatrix}\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}+\alpha_{4}\beta_{3}-\alpha_{3}\beta_{4}\\\
\sqrt{2}\alpha_{2}\beta_{3}-\sqrt{2}\alpha_{3}\beta_{2}\\\
\sqrt{2}\alpha_{4}\beta_{1}-\sqrt{2}\alpha_{1}\beta_{4}\\\ \end{pmatrix}\\\
\mathbf{3_{2,s}}\sim\begin{pmatrix}\sqrt{2}\alpha_{2}\beta_{1}+\sqrt{2}\alpha_{1}\beta_{2}-\sqrt{2}\
\alpha_{4}\beta_{3}-\sqrt{2}\alpha_{3}\beta_{4}\\\
\sqrt{3}\alpha_{4}\beta_{4}+\alpha_{3}\beta_{2}+\alpha_{2}\beta_{3}-\sqrt{3}\
\alpha_{1}\beta_{1}\\\
\sqrt{3}\alpha_{3}\beta_{3}+\alpha_{4}\beta_{1}+\alpha_{1}\beta_{4}-\sqrt{3}\
\alpha_{2}\beta_{2}\\\ \end{pmatrix}\\\
\mathbf{3^{\prime}_{1,s}}=\begin{pmatrix}\alpha_{3}\beta_{1}+\alpha_{1}\beta_{3}+3\alpha_{4}\beta_{2}+3\alpha_{2}\beta_{4}\\\
2\sqrt{2}\alpha_{3}\beta_{3}-\sqrt{6}\alpha_{4}\beta_{1}-\sqrt{6}\alpha_{1}\beta_{4}\\\
-2\sqrt{2}\alpha_{1}\beta_{1}-\sqrt{6}\alpha_{3}\beta_{2}-\sqrt{6}\alpha_{2}\beta_{3}\\\
\end{pmatrix}\\\
\mathbf{3^{\prime}_{2,s}}=\begin{pmatrix}2\sqrt{2}\alpha_{3}\beta_{1}+2\sqrt{2}\alpha_{1}\beta_{3}\\\
-3\alpha_{2}\beta_{2}-\alpha_{3}\beta_{3}-\sqrt{3}\alpha_{4}\beta_{1}-\sqrt{3}\alpha_{1}\beta_{4}\\\
\alpha_{1}\beta_{1}+3\alpha_{4}\beta_{4}-\sqrt{3}\alpha_{3}\beta_{2}-\sqrt{3}\alpha_{2}\beta_{3}\\\
\end{pmatrix}\\\ \end{array}$

Table 6: Continuum of table 5.

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