Trace identities of twisted Hecke operators on the spaces of cusp forms of half-integral weight

Abstract

Let Rψ be a twisting operator for a quadratic primitive character ψ and ˜ T (n 2 ) the n 2 -th Hecke operator of half-integral weight. When ψ has an odd conductor, we already found trace identities between twisted Hecke operators Rψ ˜ T (n 2 ) of half-integral weight and certain Hecke operators of integral weight for almost all cases (cf. (U1-3)). In this paper, the restriction is removed and we give similar trace identities for every quadratic primitive character ψ, including the case that ψ has an even conductor.