I'm going to be completely honest about this: I need the solution of this to get permitted to the exam in complex analysis. The topic is not even relevant for the exam and I am absolutely not able to solve this.
Let $\Gamma = SL_2(\mathbb{Z})$, let $p$ be an uneven prime number.
Let $\Gamma _0(N) = \left\{\left (\begin{matrix}a&b\\c&d\end{matrix}\right )\in SL(2,\mathbb{Z}) \mid c\equiv0\pmod N\right \}$ a subgroup of $\;\Gamma$.
I need to show:
$\Gamma _0(N)$ has excactly two cuscs that can be represented by $0$ and $\infty$.
A hint is given:
"Let $(A_k)_{k=0,...,p}$ be a system of distinct representatives for $\Gamma _0(p)\backslash \Gamma$ of the Form $\{ E_2\} \cup\left\{\left(\begin{matrix}0&-1\\1&j\end{matrix}\right)
| j \in \{-\frac{p-1}{2},...,\frac{p-1}{2}\}\right\}$.
One can show, that every cusc has a (not neccessarily unique) representative in the set $\{A_k\langle\infty\rangle|k=0,...,p\}$."
I am looking forward to your answers!