Geodesic
On the connection of the eigenvalues of Hecke operators and the Fourier coefficients of eigenfunctions for Siegel's modular forms of genus $n$
Sbornik. Mathematics, Tome 25 (1975) no. 4, pp. 549-557 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $f(z)=\sum_{N\geqslant0}a(N)\exp2\pi i\sigma(NZ)$ be Siegel's modular form of genus $n$ which is an eigenfunction for all operators in the $p$-component of a Hecke ring; in particular, $T_{p^\delta}f(Z)=\lambda_f(p^\delta)f(Z)$. This paper examines the series $\sum_{\delta=0}^\infty a(p^\delta N)t^\delta$ ($p$ does not divide $N$). It is proved that each such series is a rational function, where the degree of the numerator of this function does not exceed $2^n-2$ and the denominator coincides with the denominator of the series $\sum_{\delta=0}^\infty \lambda_f(p^\delta)t^\delta$. Bibliography: 6 titles. 
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     author = {N. A. Zharkovskaya},
     title = {On~the connection of the eigenvalues {of~Hecke} operators and the {Fourier} coefficients of eigenfunctions for {Siegel's} modular forms of genus~$n$},
     journal = {Sbornik. Mathematics},
     pages = {549--557},
     year = {1975},
     volume = {25},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1975_25_4_a5/}
}
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N. A. Zharkovskaya. On the connection of the eigenvalues of Hecke operators and the Fourier coefficients of eigenfunctions for Siegel's modular forms of genus $n$. Sbornik. Mathematics, Tome 25 (1975) no. 4, pp. 549-557. http://geodesic.mathdoc.fr/item/SM_1975_25_4_a5/

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[4] H. Maass, “Die Primzahlen in der Theorie der Siegelschen Modulfunktionen”, Math., Ann., 124:1 (1951), 87–122 | DOI | MR | Zbl

[5] I. Satake, “Theory of spherical functions on reductive algebraic groups over $p$-adic fields”, Publ. Math. IHES, 18 (1963), 229–293 | MR | Zbl

[6] G. Shimura, “On modular correspondences for $Sp (n,\mathbf{Z})$ and their congruence relations”, Proc. Nat. Acad. Sci. USA, 49:6 (1963), 824–828 | DOI | MR | Zbl