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Cross-posted to MO

Let $F$ be a Siegel modular form for $\text{Sp}_4(\mathbb{Z})$ of genus two. Let it also be an eigenform for all the Hecke operators $T_n$, $n\geq1$ (genus two).

Are the eigenvalues corresponding to these Hecke operators algebraic?

I know that they are real owing to the fact that the Hecke operators are self adjoint. I'm aware of this analogous question in the genus 1 case regarding elliptic modular form, where the heart of the proof is the fact that we can find a basis with integer Fourier coefficients. In the genus two case, I know that (read in text) the Fourier coefficients are algebraic, but we do not have nice relation (as in genus 1 case) between them and the Hecke eigenvalues.

Any help, reference or related information?

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    I don’t really understand the story beyond $g>1$, but doesn’t the same argument applies – eigenforms correspond of systems of eigenvalues of Hecke operators act on a certain finitely generated abelian group (for $g=1$, the singular homology of modular curves). – Aphelli Nov 24 '24 at 13:09

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