Abstract We present a set of diagonal matrices which index enough Fourier coefficients for a complete characterization of all Siegel cusp forms of degree 2, weight k, level N and character χ, where k is an even integer ≥4, N is an odd, square-free positive integer, and χ has conductor equal to N. As an application, we show that the Koecher-Maass series of any F ∈ S k 2 twisted by the set of Maass waveforms whose eigenvalues are in the continuum spectrum of the hyperbolic Laplacian determines F. We also generalize a result due to Skogman about the non-vanishing of all theta components of a Jacobi cusp form of even weight and prime index, which may have some independent interest.

中文翻译：

---

关于 2 次 Siegel 尖峰形式及其傅立叶系数

摘要 我们提出了一组对角矩阵，它们索引足够的傅立叶系数，以完整表征所有 2 次、权重 k、级别 N 和字符 χ 的 Siegel 尖峰形式，其中 k 是一个 ≥4 的偶数，N 是奇数平方-free 正整数，并且 χ 的导体等于 N。 作为一个应用，我们证明了任何 F ∈ S k 2 的 Koecher-Maass 级数被一组 Maass 波形扭曲，其特征值在双曲拉普拉斯算子的连续谱中确定 F。我们还概括了由于 Skogman 关于偶数权重和质数指数的雅可比尖峰形式的所有 theta 分量不消失的结果，这可能具有一些独立的兴趣。