Let $\Gamma$ be a concruence subgroup of $SL_2(\mathbb{Z})$, $k$ an integer and $f, g \in M_k(\Gamma)$.
We want to show that for $M \in GL_2(\mathbb{Q})⁺$ and $M \Gamma M^{-1} \subset SL_2(\mathbb{Z}) $:
$ [\bar{SL_2(\mathbb{Z})}: \bar{\Gamma}] = [\bar{SL_2(\mathbb{Z})}:M \Gamma M^{-1}]$
So we want to show that the index for those two subgroups is the same. I think it is clear to me why $M \Gamma M^{-1}$ is a subgroup. But how does one compare these indices? By comparing the cosets and showing two inequalities or is there another trick.