Theta constant (awaiting review)

Given a degree $g$ and a vector $m = (\mu,\nu)\in {\Bbb Z}^{g}\times {\Bbb Z}^{g}$, define the theta constant of characteristic $m$ by $$\theta_m(\Omega) = \sum_{n\in{\Bbb Z}^{g}} \exp\left( \frac12(n+\frac\mu2)'\Omega(n+\frac\mu2) +(n+\frac\mu2)\frac\nu2\right). $$

Then $\theta_m$ is a Siegel modular form of weight $\frac12$ with respect to some subgroup of the integral symplectic group with some character.