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

We prove that vector-valued Siegel cusp forms for ${\mathrm{\Gamma }}_0^n(N)$ with certain nebentypus are determined by their fundamental Fourier coefficients with discriminants coprime to the level $N$, 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 of fundamental index with discriminant coprime to the odd level $N$ are determined by their primitive theta components.