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 
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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     title = {On~the connection of the eigenvalues {of~Hecke} operators and the {Fourier} coefficients of eigenfunctions for {Siegel's} modular forms of genus~$n$},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] A. N. Andrianov, “Teoremy ratsionalnosti dlya ryadov Gekke i dzeta-funktsii grupp $GL_n$ i $Sp_n$ nad lokalnymi polyami”, Izv. AN SSSR, seriya matem., 33 (1969), 466–505 | MR | Zbl

[2] A. N. Andrianov, “Sfericheskie funktsii dlya $GL_n$ nad lokalnymi polyami i summirovanie ryadov Gekke”, Matem. sb., 83 (125) (1970), 429–451 | MR | Zbl

[3] A. N. Andrianov, “Eilerovy proizvedeniya, otvechayuschie modulyarnym formam Zigelya roda 2”, Uspekhi matem. nauk, XXIX:3 (117) (1974), 43–110 | MR

[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