Euler expansions of theta-transforms of Siegel modular forms of degree $n$

Let $F(Z)$ be a Siegel modular form of degree $n$, weight $k$ and character $\chi$ for the congruence subgroup $\Gamma_0^n(q)$ of the Siegel modular group $\Gamma^n$. Suppose that $F$ is an eigenfunction for all Hecke operators with index relatively prime to $q$. It is proven that for each fixed, symmetric, semi-integral, positive definite matrix $N$ of order $n$ and for each Dirichlet character $\psi$, equal to zero on all prime divisors of $q\operatorname{det}2N$, the Dirichlet series
$$ \sum_{M\in\operatorname{SL}_n(\mathbf Z)\setminus M_n^+(\mathbf Z)}\frac{\psi(\operatorname{det}M)f(MN^tM)}{(\operatorname{det}M)^s}, $$
where $f(N')$ are the Fourier coefficients of $F$ and $M_n^+(\mathbf Z)$ is the set of integral matrices of order $n$ with positive determinant, has an expansion as an Euler product which can be explicitly calculated.
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