I'm looking at $\theta$ constants with characteristic, defined by $$\theta\left(\begin{array}{c} \epsilon \\ \epsilon' \end{array}\right)(z,\tau) = \sum_{n\in \mathbb{Z}} \exp 2\pi i \left\{ \frac{1}{2}\left(n+\frac{\epsilon}{2}\right)^2\tau + \left( n + \frac{\epsilon}{2}\right)\left(z + \frac{\epsilon'}{2}\right)\right\}$$
I know that they obey a transformation equation $$\theta\left(\begin{array}{c} \epsilon \\ \epsilon' \end{array}\right)(0,\gamma(\tau)) = \kappa (c\tau + d)^{\frac{1}{2}} \theta\left(\begin{array}{c} a\epsilon + c\epsilon' -ac \\ b\epsilon +d\epsilon'+bd\end{array}\right)(0,\tau)$$ for $\gamma \in \mathrm{SL}(2,\mathbb{Z})$. However, I was wondering if one could place some conditions on the characteristic $\chi = \left(\begin{array}{c} \epsilon \\ \epsilon' \end{array}\right)$ so that $\tau \mapsto \theta(\chi)(0,\tau)$ (or maybe $\tau \mapsto \theta(\chi)(0,2\tau)$) would be modular on $\Gamma_0(N)$. If you have some literature I can look at that has these results I would be very grateful. I have the book Theta Constants, Riemann Surfaces and the Modular Group on my lap but I can't seem to find the result I'm looking for.