On counting twists of a character appearing in its associated Weil representation

Abstract

Consider an irreducible, admissible representation $\pi$ of GL(2,$F$) whose restriction to GL(2,$F)^+$ breaks up as a sum of two irreducible representations $\pi_+ + \pi_-$. If $\pi=r_{\theta}$, the Weil representation of GL(2,$F$) attached to a character $\theta$ of $K^*$ which does not factor through the norm map from $K$ to $F$, then $\chi\in \hat{K^*}$ with $(\chi >. \theta ^{-1})|_{F^{*}}=\omega_{{K/F}}$ occurs in ${r_{\theta}}_+$ if and only if $\epsilon(\theta\chi^{-1},\psi_0)=\epsilon(\bar \theta\chi^{-1},\psi_0)=1$ and in ${r_{\theta}}_-$ if and only if both the epsilon factors are -1. But given a conductor $n$, can we say precisely how many such $\chi$ will appear in $\pi$? We calculate the number of such characters at each given conductor $n$ in this work.