Consider the group $$\Gamma_0(2)=\left\lbrace \begin{pmatrix} a & b \\ c & d \end{pmatrix}\in SL_2(\mathbb{Z})\mid c\equiv 0 \pmod2\right\rbrace$$ and $ I=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $ , $S=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} $ , $T^{-1}S=\begin{pmatrix} -1 & -1 \\ 1 & 0 \end{pmatrix}$ , with $T=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$
Now I want to show that these matrices are coset representatives of $SL_2(\mathbb{Z}) /\Gamma_0(2)$
I know that there are three cosets since $|SL_2(\mathbb{Z}) /\Gamma_0(2)|=3$
$SL_2(\mathbb{Z}) /\Gamma_0(2)=\lbrace I,S,T^{-1}S \rbrace$
I think $"\supset"$ is clear because these matrices are in $SL_2(\mathbb{Z})$ and $I \in \Gamma_0(2)$ .
For the other inclusion, I have no idea.
Thanks for the help.