Modular transformation of $\theta_3(\tau)$ like function

Question

For the Jacobi $\theta_3(\tau)$ function we can find its $\tau \mapsto -1/\tau$ transformation using Poisson resummation. It is quite straight forward using its Fourier series $$ \theta_3(\tau) = \sum_n e^{\pi i \tau n^2} $$ Then we simply compute its Fourier transform. Similarly we can prove that the Fourier transform of $e^{-\pi i x^2}$ is $e^{-\pi i x^2}$. But what can we do for a function that has indefinite signature like the following? $$ \theta(\tau) = \sum_n e^{-\pi i \tau n_+^2 - \pi i \bar{\tau}n_-^2} $$ Above $\tau$ is a coordinate on the upper-half-plane (complex numbers with positive imaginary part). Also, $n$ belongs to some lattice that can be split into positive subspace with elements $n_+$ and negative subspace with elements $n_-$ such that $n= n_+ + n_-$. How can we find the transformation properties of this function?