Let $K(\sqrt{D})$ be a real quadratic field with discriminant D and $a$ an integral ideal with its norm $N(a)$ .
I do not know why the following equation holds using the theta transformation formula $$\sum_{\lambda\equiv0(a)}e^{-\pi t(\lambda^2e^v+\overline \lambda^2e^{-v})}=\frac{1}{t\sqrt{D}N(a)}\sum_{v\equiv0 \frac{1}{a\sqrt{D}}} e^{\frac{-\pi v^2e^{-v}+\overline v^2e^v}{t}}$$