I am learning about theta functions. Let $q = e^{2\pi i \, z}$ and $\theta(z) = \sum q^{n^2}$. How does it behave under $\mathrm{SL}_2(\mathbb{Z})$ ? In general we have:
$$ \theta\left( - \frac{1}{4z} \right) = \sqrt{- 2\pi i \, z} \; \theta(z) $$
Unfortunately, the maps $z \mapsto z + 1$ and $z \mapsto - \frac{1}{4z}$ do not generate all of $\mathrm{SL}_2(\mathbb{Z})$ but only $\Gamma_0(4)$ which is:
$$ \left\{ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right): ad-bc=1 \text{ and } 4 \, \big| \,c \right\} $$
What is happening then under the map $z \mapsto - \frac{1}{z}$? My guess it is some other theta function, but which kind?