It is standard notation but I define them anyway to avoid ambiguity: $$\Gamma_0(N) := \left\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma : c \equiv 0 \text{(mod $N$)}\right\}$$ and $$\Gamma:=SL_2(\mathbb{Z})$$
With this notation I'm trying to figure out what the subgroups of finite index of $\Gamma_0(4)$ and $\Gamma$ are. In particular, given $\Gamma'$ a subgroup of $\Gamma_0(4)$ of finite index I have to prove that $\Gamma' \cap \left\{ \pm \begin{pmatrix} 1 & j \\ 0 & 1 \end{pmatrix} , j\in \mathbb{Z} \right\}$ is of the form $\left\{\pm\begin{pmatrix} 1 & h \\ 0 & 1 \end{pmatrix}^j , j\in \mathbb{Z}\right\}$ for $h>0$
- Is there a direct way to see this?
- Are finite index subgroups simply subgroups $\Gamma_0(4N)$ or are they something more general?