Let $\Gamma_n=Sp_{2n}(\mathbb{Z})$ and write $\gamma=\left(\matrix{A & B \\ C & D}\right)\in\Gamma_n$ for its block decomposition. Further let $\Gamma_n^0$ be the subset consisting of symplectic matrices such that $C=0$. Let $\mathcal{H}_n=\{Z\in\mathcal{M}_{n\times n}(\mathbb{C}): Z^t=Z, Y>0\}$, where $Z=X+iY$, be the usual Siegel upper half plane. We define $j(\gamma,Z)=\det(C_\gamma Z+D_\gamma)$.

The non-holomorphic Eisenstein series $E_n^{k,s}$ is defined as $$E_n^{k,s}(Z)=\det(Y)^s\sum_{\Gamma_n^0\backslash\Gamma_n}j(\gamma,Z)^{-k}|j(\gamma,Z)|^{-2s}$$ for $k\in\mathbb{N}$ and $s\in\mathbb{C}$. In order to prove its absolute convergence, we need to estimate $$\sum_{\Gamma_n^0\backslash\Gamma_n}|j(\gamma,Z)|^{\alpha}$$ where $\alpha\in\mathbb{R}$. I know that there are explicit conditions on $\alpha(k,s,n)$ so that the series is convergent, but I could not find any reference for a proof.

If $n=1$, this series turns out to be $\le c_0\sum_{c,d\in\mathbb{Z}}^{(c,d)\not=(0,0)}|cz+d|^{\alpha}$ for some constant $c_0$, and I have seen (again no reference) that this is $\le c_1 y^{\alpha+1}$ on the usual fundamental domain for $\Gamma_1\backslash \mathcal{H}_1$ (which I am happy to believe, as it looks like taking that sum is **more or less** like integrating over one variable, therefore raising the exponent by one).

Is something similar happening in general $n\ge 1$, maybe like $$\sum_{\Gamma_n^0\backslash\Gamma_n}|j(\gamma,Z)|^{\alpha}\le c_0 \det(Y)^{\alpha+n}$$ on a fundamental domain for $\Gamma_n\backslash \mathcal{H}_n$ chosen so that $Y\ge c_1 I$ (like Klingen explains, using the Minkowski reduced set)? Can you suggest any reference on the topic?