Non-vanishing of Fundamental Fourier Coefficients of Paramodular Forms

of fundamental Fourier coefficients, which plays an important role in the theory of Bessel models and L-functions. For instance, in certain cases, non-vanishing of a fundamental Fourier coefficient of a cuspidal Siegel modular form F is equivalent to existence of a global Bessel model of fundamental type (cf. [16, Lemma 4.1]) and is used to show analytic properties and special value results for L-functions for GSp4 × GL2 associated to various twists of F (e.g. [8], [12], [17], [18]). It is also known [19] that fundamental Fourier coefficients determine cuspidal Siegel modular forms of degree 2 of full level. Our result extends previous work by Saha [16, Theorem 3.4], [19, Theorem 1] and Saha, Schmidt [20, Theorem 2] in case of the levels Sp4(Z) and Γ (2) 0 (N). Theorem. Let F ∈ Sk(Γpara(N)) be a non-zero paramodular cusp form of an arbitrary integer weight k and odd square-free level N which is an eigenfunction of the operators T (p) + T (p) for primes p ∤ N , U(p) for p | N and μN . Then F has infinitely many non-zero fundamental Fourier coefficients.