Certain Siegel Cusp Forms with Level are Determined by their Fundamental Fourier Coefficients

We prove that vector-valued Siegel cusp forms for Γ ₀ⁿ(N) Γ 0 n ( N ) with certain nebentypus are determined by their fundamental Fourier coefficients with discriminants coprime to N , assuming N is odd and square-free. In the case of genus 3, we strengthen this to Fourier coefficients corresponding to maximal orders in quaternion algebras. We also prove that Jacobi forms with odd, square-free level N and odd, square-free index with discriminant coprime to N are determined by their primitive theta components.