For an integer $n$, let $$ \Gamma_0(n) = \left\{ \left( \begin{matrix}a & b \\ c & d \end{matrix} \right) \in \operatorname{SL}_2 (\mathbb{Z})\bigg | c \equiv 0 \mod n\right\}.$$
Let $N, l$ be different primes. Then what is a representation system of $\Gamma_0(Nl) \subseteq \Gamma_0(N)$?
It seems that $\beta_i = \left( \begin{matrix}1 & 0 \\ Ni & 1 \end{matrix} \right)$ for $i = 0, \dots l-1$, and $\beta_\infty = \left( \begin{matrix}a & b \\ N & l \end{matrix} \right)$, where $al -Nb = 1$, represent it.
But I can't show.
Let $B(l)$ be the image of $\Gamma_0(l)$ in $SL_2(\Bbb{Z}/l\Bbb{Z})$. Take some $r_j,s_i$ such that $$SL_2(\Bbb{Z}/l\Bbb{Z})=\bigcup_j r_j B(l)=\bigcup_i B(l)s_i\qquad (disjoint\ union)$$
The reduction $\bmod l$ is an homomorphism $$f:\Gamma_0(N)\to SL_2(\Bbb{Z}/l\Bbb{Z})$$ Since $N,l$ are coprime then it is surjective. The kernel is contained in $\Gamma_0(Nl)$, and the image of $\Gamma_0(Nl)$ is $B(l)$. This implies that
$$\Gamma_0(N)/\Gamma_0(Nl)\cong f(\Gamma_0(N))/f(\Gamma_0(Nl))=SL_2(\Bbb{Z}/l\Bbb{Z})/B(l)$$ and similarly for the quotient on the left.
Take some representatives $R_j,S_i\in \Gamma_0(N),f(R_j)=r_j,f(S_i)=s_i$. Then $$\Gamma_0(N)=\bigcup_j R_j \Gamma_0(Nl)= \bigcup_i \Gamma_0(Nl) S_i\qquad (disjoint\ union)$$ Finally, show that $\{r_j\}$ can be taken to be $ \pmatrix{1&0\\*&1}\cup \pmatrix{0&-1\\1&0}$ equivalently $\pmatrix{1&0\\*&1}\cup \pmatrix{0&b\\N&0}$ where $-Nb\equiv 1\bmod l$.
What about the $s_i$ ?