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

Abstract We prove that vector-valued Siegel cusp forms for $$\Gamma _0^n(N)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>Γ</mml:mi> <mml:mn>0</mml:mn> <mml:mi>n</mml:mi> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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.