\[\theta\genfrac{[}{]}{0.0pt}{}{\tfrac{1}{2}[\mathbf{c}_{1}+\mathbf{c}_{2}+%
\mathbf{c}_{3}+\mathbf{c}_{4}]}{\tfrac{1}{2}[\mathbf{d}_{1}+\mathbf{d}_{2}+%
\mathbf{d}_{3}+\mathbf{d}_{4}]}\left(\frac{\mathbf{x}+\mathbf{y}+\mathbf{u}+%
\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{%
}{\tfrac{1}{2}[\mathbf{c}_{1}+\mathbf{c}_{2}-\mathbf{c}_{3}-\mathbf{c}_{4}]}{%
\tfrac{1}{2}[\mathbf{d}_{1}+\mathbf{d}_{2}-\mathbf{d}_{3}-\mathbf{d}_{4}]}%
\left(\frac{\mathbf{x}+\mathbf{y}-\mathbf{u}-\mathbf{v}}{2}\middle|\boldsymbol%
{{\Omega}}\right)\*\theta\genfrac{[}{]}{0.0pt}{}{\tfrac{1}{2}[\mathbf{c}_{1}-%
\mathbf{c}_{2}+\mathbf{c}_{3}-\mathbf{c}_{4}]}{\tfrac{1}{2}[\mathbf{d}_{1}-%
\mathbf{d}_{2}+\mathbf{d}_{3}-\mathbf{d}_{4}]}\left(\frac{\mathbf{x}-\mathbf{y%
}+\mathbf{u}-\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{%
[}{]}{0.0pt}{}{\tfrac{1}{2}[\mathbf{c}_{1}-\mathbf{c}_{2}-\mathbf{c}_{3}+%
\mathbf{c}_{4}]}{\tfrac{1}{2}[\mathbf{d}_{1}-\mathbf{d}_{2}-\mathbf{d}_{3}+%
\mathbf{d}_{4}]}\left(\frac{\mathbf{x}-\mathbf{y}-\mathbf{u}+\mathbf{v}}{2}%
\middle|\boldsymbol{{\Omega}}\right)\\
=\frac{1}{2^{g}}\sum_{\boldsymbol{{\alpha}}\in\frac{1}{2}{\mathbb{Z}}^{g}/{%
\mathbb{Z}}^{g}}\,\sum_{\boldsymbol{{\beta}}\in\frac{1}{2}{\mathbb{Z}}^{g}/{%
\mathbb{Z}}^{g}}e^{-2\pi i\boldsymbol{{\beta}}\cdot[\mathbf{c}_{1}+\mathbf{c}_%
{2}+\mathbf{c}_{3}+\mathbf{c}_{4}]}\*\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_%
{1}+\boldsymbol{{\alpha}}}{\mathbf{d}_{1}+\boldsymbol{{\beta}}}\left(\mathbf{x%
}\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_%
{2}+\boldsymbol{{\alpha}}}{\mathbf{d}_{2}+\boldsymbol{{\beta}}}\left(\mathbf{y%
}\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_%
{3}+\boldsymbol{{\alpha}}}{\mathbf{d}_{3}+\boldsymbol{{\beta}}}\left(\mathbf{u%
}\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_%
{4}+\boldsymbol{{\alpha}}}{\mathbf{d}_{4}+\boldsymbol{{\beta}}}\left(\mathbf{v%
}\middle|\boldsymbol{{\Omega}}\right).\]