Geodesic
Euler expansions of theta-transforms of Siegel modular forms of degree $n$
Sbornik. Mathematics, Tome 34 (1978) no. 3, pp. 259-300 
A. N. Andrianov. Euler expansions of theta-transforms of Siegel modular forms of degree $n$. Sbornik. Mathematics, Tome 34 (1978) no. 3, pp. 259-300. http://geodesic.mathdoc.fr/item/SM_1978_34_3_a0/
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     title = {Euler expansions of theta-transforms of {Siegel} modular forms of degree~$n$},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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. Bibliography: 13 titles.

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