Modular symmetry at level 6 and a new route towards finite modular groups

We propose to construct the finite modular groups from the quotient of two principal congruence subgroups as $Gamma(N)/Gamma(N)$, and the modular group $SL(2,mathbb{Z})$ is extended to a principal congruence subgroup $Gamma(N)$. The original modular invariant theory is reproduced when $N=1$. We perform a comprehensive study of $Gamma_6$ modular symmetry corresponding to $N=1$ and $N=6$, five types of models for lepton masses and mixing with $Gamma_6$ modular symmetry are discussed and some example models are studied numerically. The case of $N=2$ and $N=6$ is considered, the finite modular group is $Gamma(2)/Gamma(6)cong T$, and a benchmark model is constructed.