 \\                 &=  \sum_{\substack{ {\mathbf y}\in \Omega(T_0) \\ \text{mod } {\Gamma}}} \varphi_0({\mathbf y}) \, \sum_{m \in F}                  \left(                   \sum_{\substack{                          {\mathbf x}_0'' \in    U_{{\mathbf y}}^{\perp} \\                         \langle {\mathbf x}_0'', {\mathbf x}_0'' \rangle = m \\                         \text{mod } \Gamma_{{\mathbf y}}                 