Double cover of modular S4 for flavour model building

We develop the formalism of finite modular group Γ4′≡S4′, a double cover permutation Γ4≃S4, for theories flavour. The integer weight k>0 level 4 forms indispensable can be even or odd. explicitly construct lowest-weight (k=1) in terms two Jacobi theta constants, denoted as ε(τ) and θ(τ), τ being modulus. show that these furnish 3D representation S4′ not present S4. Having derived multiplication rules Clebsch-Gordan coefficients, we multiplets weights up to k=10. These are expressed polynomials ε θ, bypassing need search non-linear constraints. further within there options define (generalised) CP transformation discuss possible residual symmetries based on invariance. Finally, provide examples application our results, constructing phenomenologically viable lepton flavour models.