In these lecture notes of Zagier, I read that generalized theta functions are still modular forms. Let $q = e^{2\pi i z}$
$$\theta(z) = \sum_{(x,y) \in \mathbb{Z}} \Big[ x^4 - 6 x^2 y^2 + y^4 \Big] \, q^{x^2 + y^2} \tag{$\ast$}$$ is a modular form over $\Gamma_0(4)$. In the one-variables case we argue using Poisson summation, e.g. the identity:
$$\sum_{n \in \mathbb{Z}} e^{\pi n^2 t} = \frac{1}{\sqrt{t}} \sum_{n \in \mathbb{Z}}e^{\pi n^2 / t} $$
Since this is Poisson summation for $f(x) = e^{-x^2 t} $. Is there any analogous Poisson-summation identity we can use here or this generalized $\theta$ funnction in ($\ast$)?
Also if anyone can help, I am still confused by the jargon.
is this a cusp form ?
is this a Maass waveform ?