  We prove that if $F$ is a non-zero (possibly non-cuspidal) vector-valued
Siegel modular form of any degree, then it has infinitely many non-zero 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 of vector-valued modular forms. In an Appendix, as an
application of a variant of our result and building upon the work of A.
Pollack, we show how to obtain an unconditional proof of the functional
equation of the spinor $L$-function of a holomorphic cuspidal Siegel eigenform
of degree $3$.
