We prove that if F is a nonzero (possibly noncuspidal) vector-valued Siegel modular form of any degree, then it has infinitely many nonzero 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. Further, as an application of a variant of our result and complementing 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 $3$ and level $1$ .

中文翻译：

---

关于SIEGEL模形式的基本傅立叶系数

我们证明，如果F是任何次数的非零（可能非尖角）向量值 Siegel 模形式，则它具有无限多个非零傅里叶系数，这些系数由具有奇数、无平方（因此是基本的）判别式的半积分矩阵索引. 该证明在向量值模形式的设置中使用归纳论证。此外，作为我们结果变体的应用和 A. Pollack 工作的补充，我们展示了如何获得阶 $3$ 和水平的全纯尖尖 Siegel 本征形式的旋量L函数的函数方程的无条件证明 $1$ 。