Twisting of Siegel paramodular forms

Abstract

Let Sk(Γpara(N)) be the space of Siegel paramodular forms of level N and weight k. Fix an odd prime p ∤ N and let χ be a nontrivial quadratic Dirichlet character mod p. Based on [Twisting of paramodular vectors, Int. J. Number Theory 10 (2014) 1043–1065], we define a linear twisting map 𝒯χ : Sk(Γpara(N)) → Sk(Γpara(Np4)). We calculate an explicit expression for this twist, give the commutation relations of this map with the Hecke operators and Atkin–Lehner involution for primes ℓ ≠p, and prove that the L-function of the twist has the expected form.