 \\                 &=  \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}}                                 }}                      \varphi_0''({\mathbf x}_0'')                     \Gamma_{({\mathbf x}_0'', {\mathbf y})} \big\backslash    {\mathbb D}_{({\mathbf x}_0'', {\mathbf y})}.                 \right)                  \cdot                 \left(                       \sum_{\substack{                                 