  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.
