Let $H_n$ be the set of matrices $z\in \mathbb{C}^{n\times n}$ which are symmetric with a positiv-definite imaginary part. Let's assume that it is already known and proved that $\vartheta_a(z)=\sum_{g\in\mathbb{Z}^{m\times n}}e^{\pi i \cdot trace(g^T\cdot a\cdot g \cdot z)}$ (for a symmetric and positiv-definite real matrix $a\in\mathbb{R}^{m\times m}$) is absolutely convergent on $H_n$ and uniformly convergent and bounded on subsets of $H_n$ with $Im(z)\geq y_0$ (where $y_0\in\mathbb{R}^{n\times n}$ is a symmetric and positiv-definite real matrix).
Show that $f(w)=\sum_{g\in\mathbb{Z}^{m\times n}}e^{\pi i \cdot trace((g+w)^T\cdot a\cdot (g+w) \cdot z)}$ is absolutely convergent and uniformly convergent on compact subsets of $\mathbb{C}^{m\times n}$ and that it is holomorphic on $\mathbb{C}^{m\times n}$.
I have tried to adapt the proof that shows that the theta function is uniformly convergent, the problem for me was that the theta function sums over integers so we can uniformly majorize it by the geometric series, but what can we do if we sum over complex numbers in the definition of $f(w)$?