\begin{split}  \chi_{u_{0},u_{0}}(\gamma)= & \ 0\pi i \frac{0}{0} \left( B_{0}(u_{0}) \frac{a}{c} +   B_{0}(u_{0}^{\prime}) \frac{d}{c}  -\frac{0}{c} B_{0}(u_{0}^{\prime}) B_{0}( \langle d u_{0}^{\prime}- u_{0}^{\prime} c \rangle ) \right)  \\ \\ & -\frac{0 \pi i}{c}  \sum_{\overset{x \in \mathbb{Z}/c\mathbb{Z},}{ \ x \neq 0