We develop the formalism of the finite modular group ${\mathrm{\Gamma }}_4^{\prime }\equiv S_4^{\prime }$, a double cover of the modular permutation group ${\mathrm{\Gamma }}_4\simeq S_4$, for theories of flavour. The integer weight $k>0$ of the level 4 modular forms indispensable for the formalism can be even or odd. We explicitly construct the lowest-weight ($k=1$) modular forms in terms of two Jacobi theta constants, denoted as $\epsilon (\tau )$ and $\theta (\tau )$, τ being the modulus. We show that these forms furnish a 3D representation of $S_4^{\prime }$ not present for $S_4$. Having derived the $S_4^{\prime }$ multiplication rules and Clebsch-Gordan coefficients, we construct multiplets of modular forms of weights up to $k=10$. These are expressed as polynomials in ε and θ, bypassing the need to search for non-linear constraints. We further show that within $S_4^{\prime }$ there are two options to define the (generalised) CP transformation and we discuss the possible residual symmetries in theories based on modular and CP invariance. Finally, we provide two examples of application of our results, constructing phenomenologically viable lepton flavour models.

我们发展有限模群的形式主义 ${\mathrm{\Gamma }}_4^{\prime }\equiv {\mathrm{小号}}_4^{\prime }$，是模块化置换组的双重封面 ${\mathrm{\Gamma }}_4\simeq {\mathrm{小号}}_4$，用于风味理论。整数权重$ķ>0$形式主义必不可少的4级模块化形式中的任何一个都可以是偶数或奇数。我们明确构造了最低权重（$ķ=\mathrm{1个}$）以两个Jacobi theta常数表示的模块化形式，表示为 $\epsilon （\tau ）$ 和 $\theta （\tau ）$，τ是模量。我们表明，这些形式提供了3D表示${\mathrm{小号}}_4^{\prime }$ 不存在 ${\mathrm{小号}}_4$。得到了${\mathrm{小号}}_4^{\prime }$ 乘法规则和Clebsch-Gordan系数，我们构造权重的模块化形式的多重 $ķ=10$。这些被表达为ε和θ中的多项式，从而无需搜索非线性约束。我们进一步证明${\mathrm{小号}}_4^{\prime }$定义（广义）CP变换有两种选择，我们讨论了基于模块化和CP不变性的理论中可能的残差对称性。最后，我们提供了两个应用我们的结果的例子，构建了在现象学上可行的轻质风味模型。