Abstract We prove that vector-valued Siegel cusp forms for 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.
Sidney Washburn (Tue,) studied this question.