Do I have this equivalence? $$ \sum_{\beta \equiv 1 \mod \sqrt{-7}} \mathrm{Re}(\beta) e^{\beta \overline{\beta}/7} \stackrel{?}{=} \sum_{m, n } m e^{m^2 + 7 n^2}$$
Let $\beta \in \mathbb{Z}[\frac{1 + \sqrt{-7}}{2}]$ be the ring of integers of $\mathbb{Q}(\sqrt{-7}) = \mathbb{Q}[x]/(x^2 + 7)$. $$ \beta \equiv 1 \mod \sqrt{-7}$$ If I let $\beta = m \cdot 1 + n \cdot \frac{1 + \sqrt{-7}}{2} $ what does this condition mean in terms of $m, n, \in \mathbb{Z}$ ? The actual theta function equation should be something quite normal looking.