Compute the index $[Γ( 1 ) ′ : Γ_0 ( N ) ′ ]$
where $Γ(1)' := SL(2,\mathbb{Z})$
$Γ_0(N)':= \{ \begin{pmatrix} a & b\\ c & d\\ \end{pmatrix} \in Γ(1)' : c \equiv 0 \mod{N} \} $
I'm basically stuck on how to get started. Any help would be greatly appreciated!
First assume that $N=p^m$ is a prime power.
Consider $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL(2,\mathbb Z)$. If $p\not\mid d$, then $dx+c\equiv 0\pmod {p^m}$ has a unique solution mod $p^m$ (for example with $x\in\{0,\ldots, p^m-1\}$). With such an $x$ we have $$\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}1&0\\x&1\end{pmatrix}\in\Gamma_0'(p^m)$$ i.e. there are (at most, but by uniqueness of $x$ exactly) $p^m$ cosets in $\Gamma_0'(1)/\Gamma_0'(p^m)$. However these cosets do not cover the case $p\mid d$ yet. But if $p\mid d$ then necessarily $p\not\mid c$ and we can apply the reasoning above after multiplication with $\begin{pmatrix}0&1\\-1&0\end{pmatrix}$. This gives us $2p^m$ cosets in total.
You can combine this using the Chinese Remainder Theorem to find that the index in the general case is $2^rN$ if $N$ has $r$ prime divisors.