Trace formula of twisting operators of half-integral weight in the case of even conductors

Abstract

Let $S(k + 1/2, N, \chi)$ denote the space of cusp forms of weight $k+1/2$, level $N$, and character $\chi$. Let $R_{\psi}$ be a twisting operator for a quadratic primitive character $\psi$ of even conductor and $\tilde{T}(n^2)$ the $n^2$-th Hecke operator. We give an explicit trace formula of $R_{\psi} \tilde{T}(n^2)$ on $S(k + 1/2, N, \chi)$.