We develop the formalism of the finite modular group Gamma'(4) equivalent to S'(4), a double cover of the modular permutation group Gamma(4) similar or equal to 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 e(t) and.(t), tbeing the modulus. We show that these forms furnish a 3D representation of S'(4) not present for S'(4). Having derived the S'4 multiplication rules and Clebsch-Gordan coefficients, we construct multiplets of modular forms of weights up to k=10. These are expressed as polynomials in eand., bypassing the need to search for non-linear constraints. We further show that within S'(4) 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. (C) 2020 The Authors. Published by Elsevier B.V.
