Siegel modular form with character (awaiting review)

Given a group $\Gamma\subset \Sp(2g,\R)$,

a character $\psi$ is a homomorphism $\psi:\Gamma\to\C^\times$.

A function $f$ is a Siegel modular form of weight $k$ with respect to subgroup $\Gamma$ and character $\psi$ is such that for all $\gamma=\begin{pmatrix}A&B\\C&D \end{pmatrix} \in\Gamma$, we have $$f((A\Omega+B)(C\Omega+D)) =\psi(\gamma) \det(C\Omega+D)^k f(\Omega),$$ and we write $M_k(\Gamma,\psi)$ for the space of such forms.