Let $\Gamma_n = \operatorname{Sp}_n(\Bbb Z)$ and $\Gamma_{n,0} \subset \Gamma_n$ be a subgroup. We write $$ M = \begin{pmatrix}A & B\\ C & D \end{pmatrix} \in \operatorname{Sp}_n(\mathbb{Z})$$ where $M$ is a $(2n)\times(2n)$ matrix and $A,B,C,D$ are $n\times n$ matrices.
I am reading some function theory script and I encountered the writing (definition of the Eisensteinseries for Siegel modular forms)
$$ E_k(Z) := \sum_{M : \Gamma_{n,0} \backslash \Gamma_n} \det(CZ + D)^{-k} . $$
What does $M : \Gamma_{n,0} \backslash \Gamma_n$ mean? What is $M$?
I know "$\backslash$" as a set-substraction but that doesn't seem to make sense here.
It is probably meant as some form of Quotient space but I'm not sure what exactly. Also, I only know "$/$" for the notation of Quotient spaces.
Also, if $\Gamma_{n,0} \backslash \Gamma_n$ means something (a set), why does the author write "$M:$" and not "$M \in$"? (The same author uses "$\in$" everywhere else.)