Modular invariant quark and lepton models in double covering of S4 modular group

We perform a comprehensive analysis of the homogeneous finite modular group Gamma(4)' S-4', which is the double covering of S-4 group. The weight 1 modular forms of level 4 arc constructed in terms of the Dedekind eta function, and they transform as a triplet (3) over cap' of S-4'. The integral weight modular forms until weight 6 are built from the tensor products of weight 1 modular forms. We perform a systematical classification of S-4' modular models for lepton masses and mixing with and without generalized CP, where the left-handed leptons are assigned to a triplet of S-4' and right-handed charged leptons transform as singlets under S-4', and we consider both scenarios where the neutrino masses arise from a Weinberg operator or type-I seesaw mechanism. The phenomenological implications of the minimal models for lepton masses, mixing angles, CP violation phases, and neutrinoless double decay are discussed. The S-4' modular symmetry is extended to the quark sector, and we present several predictive models which use nine or ten free parameters including real and imaginary parts of tau to describe quark masses and the Cabibbo-Kobayashi-Maskawa mixing matrix. We give a quark-lepton unified model which can explain the flavor structure of quarks and leptons simultaneously for a common value of tau.