On Hecke eigenforms of degree \(n\) (Q5954689)

Let \(F\), \(G\) be two Siegel cusp forms of degree \(n\) and weight \(k\), which are simultaneous Hecke eigenforms. Suppose that their spinor zeta functions possess a meromorphic continuation to \(\mathbb{C}\) and satisfy the conjectured functional equation. The authors show that the assumption on their Fourier coefficients \(a_F(mT)= Ha_G(mT)\) for all primitive \(T\) and \(m\in \mathbb{N}\) with \(\nu_p(m)\leq 2^n-2\) for all primes \(p\) imply \(F=G\). This generalizes their earlier result [Nagoya Math. J. 155, 153-160 (1999; Zbl 0936.11030)] for the case \(n=2\).