We extend the even weight modular forms of modular invariant approach to general integral weight modular forms. We find that the modular forms of integral weights and level N can be arranged into irreducible representations of the homogeneous finite modular group $$ {\Gamma}_N^{\prime } $$ Γ N ′ which is the double covering of Γ _N . The lowest weight 1 modular forms of level 3 are constructed in terms of Dedekind eta-function, and they transform as a doublet of $$ {\Gamma}_3^{\prime } $$ Γ 3 ′ ≅ T′. The modular forms of weights 2, 3, 4, 5 and 6 are presented. We build a model of lepton masses and mixing based on T′ modular symmetry.

In the modular symmetry approach to neutrino models, the flavour symmetry emerges as a finite subgroup Γ _N of the modular symmetry, broken by the vacuum expec- tation value (VEV) of a modulus field τ. If the VEV of the modulus τ takes some special value, a residual subgroup of Γ _N would be preserved. We derive the fixed points τ _S = i, τ _ST = (−1 + i $$ \sqrt{3} $$ 3 )/2, τ _TS = (1 + i $$ \sqrt{3} $$ 3 )/2, τ _T = i∞ in the fundamental domain which are in-variant under the modular transformations indicated. We then generalise these fixed points to τ _f = γτ _S , γτ _ST , γτ _TS and γτ _T in the upper half complex plane, and show that it is suffi-cient to consider γ ∈ ΓN. Focussing on level N = 4, corresponding to the flavour group S ₄, we consider all the resulting triplet modular forms at these fixed points up to weight 6. We then apply the results to lepton mixing, with different residual subgroups in the charged lepton sector and each of the right-handed neutrinos sectors. In the minimal case of two right-handed neutrinos, we find three phenomenologically viable cases in which the light neutrino mass matrix only depends on three free parameters, and the lepton mixing takes the trimaximal TM1 pattern for two examples. One of these cases corresponds to a new Littlest Modular Seesaw based on CSD(n) with n = 1 + $$ \sqrt{6} $$ 6 ≈ 3.45, intermediate between CSD(3) and CSD(4). Finally, we generalize the results to examples with three right-handed neutrinos, also considering the level N = 3 case, corresponding to A ₄ flavour symmetry.

We present a comprehensive analysis of neutrino mass and lepton mixing in theories with $A_5$ modular symmetry. We construct the weight 2, weight 4, and weight 6 modular forms of level 5 in terms of Dedekind eta-functions and Klein forms, and their decomposition into irreducible representation of $A_5$. We construct all the simplest models based on $A_5$ modular symmetry, including scenarios of models with and without flavons in the charged lepton sectors. For each case, the neutrino masses can be generated through either the Weinberg operator or the type I seesaw mechanism. We perform an exhaustive numerical analysis, organizing our results in an extensive set of figures and tables.

We present a comprehensive analysis of neutrino mass and lepton mixing in theories with A ₄ modular symmetry, where the only flavon field is the single modulus field τ, and all masses and Yukawa couplings are modular forms. Similar to previous analyses, we discuss all the simplest neutrino sectors arising from both the Weinberg operator and the type I seesaw mechanism, with lepton doublets and right-handed neutrinos assumed to be triplets of A ₄. Unlike previous analyses, we allow right-handed charged leptons to transform as all combinations of 1, 1′ and 1′′ representations of A ₄, using the simplest different modular weights to break the degeneracy, leading to ten different charged lepton Yukawa matrices, instead of the usual one. This implies ten different Weinberg models and thirty different type I seesaw models, which we analyse in detail. We find that fourteen models for both NO and IO neutrino mass ordering can accommodate the data, as compared to one in previous analyses, providing many new possibilities.