Double cover of modular S4 for flavour model building

We develop the formalism of the finite modular group Γ4′≡S4′, a double cover of the modular permutation group Γ4≃S4, 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 ε(τ) and θ(τ), τ being the modulus. We show that these forms furnish a 3D representation of S4′ not present for S4. Having derived the S4′ 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 S4′ 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.