Fundamental Fourier coefficients of Siegel modular forms of higher degrees and levels

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.