ON FUNDAMENTAL FOURIER COEFFICIENTS OF SIEGEL MODULAR FORMS

Abstract We prove that if F is a nonzero (possibly noncuspidal) vector-valued Siegel modular form of any degree, then it has infinitely many Fourier coefficients which are indexed by half-integral matrices having odd, square-free (and thus fundamental) discriminant. The proof uses an induction argument in the setting forms. Further, as application variant our result and complementing work A. Pollack, we show how to obtain unconditional functional equation spinor L -function holomorphic cuspidal eigenform degree $3$ level $1$ .