We prove the following statement about any Siegel modular form F of degree n and arbitrary odd level N on the group Γ₁^ (n) (N). Let A (F, T) denote the Fourier coefficients of F and write T= (T (i, j) ). Suppose that F has a non-zero Fourier coefficient A (F, T₀) such that (T₀ (n, n), N) =1. Then there exist infinitely many odd and square-free (and thus fundamental) integers m such that m=discriminant (T) and A (F, T) 0. In the case of odd degrees, we prove a stronger result by replacing odd and square-free with odd and prime. We also prove quantitative results towards this. As a consequence, we can show in particular that the statement of the main result in arXiv: 2408. 03442 about the algebraicity of certain critical values of the spinor L-functions of holomorphic newforms (in the ambit of Deligne's conjectures) on congruence subgroups of GSp (3) is unconditional.
Anamby et al. (Mon,) studied this question.