# Neutrino Masses and Mixing from Double Covering of Finite Modular Groups

## TL;DR

This paper extends the modular forms framework to integral weights using double coverings of finite modular groups, constructing models for lepton masses and mixing with new mathematical structures.

## Contribution

It introduces a novel approach to integral weight modular forms via double covering groups, enabling new models for lepton flavor physics.

## Key findings

- Constructed lowest weight 1 modular forms using Dedekind eta-function.
- Presented modular forms of weights 2 to 6 for level 3.
- Developed a lepton mass and mixing model based on T' modular symmetry.

## Abstract

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 $\Gamma'_N$ which is the double covering of $\Gamma_N$. The lowest weight 1 modular forms of level 3 are constructed in terms of Dedekind eta-function, and they transform as a doublet of $\Gamma'_3 \cong T'$. 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.