\(\Gamma(1) \coloneqq{\operatorname{SL}}_2({\mathbf{Z}})\), and principal congruence subgroups of level \(N\) for \(\Gamma(1)\) are defined as \begin{align*} \Gamma(N) \coloneqq\ker\qty{\Gamma(1) \twoheadrightarrow{\operatorname{SL}}_2({\mathbf{Z}}/N)} = \left\{{M\in \Gamma(1) {~\mathrel{\Big\vert}~}M\cong I \operatorname{mod}N}\right\} ,\end{align*} so the kernels of reduction mod \(N\). Unsorted/songruence subgroups are any subgroups \(H\) such that \(\Gamma(N) \subseteq H \leq \Gamma(1)\) for some \(N\).