show that $\sum P(x,y) e^{x^2 + y^2}$ is a modular form over $\Gamma_0(4)$ where $P(x,y) = x^4 - 6 x^2 y^2 + y^4$ 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 ?
 A: This is an example taken from Zagier's "Introduction to Modular Forms", page 246 Section C. If you read Section E beginning on page 249, Zagier introduces the differential operator $F_\nu(f,g)$ of H. Cohen. If $f$ and $g$ are modular forms of a particular subgroup of weights $k$ and $l$ respectively, then 
$F_\nu$ maps to a cuspform of weight $k+l+2\nu$. It is then pointed out on page 250 that
$$\frac{8}{3}F_2(\vartheta,\vartheta)=\theta(z)$$where $\vartheta(z)=\sum_{n\in{Z}}q^{n^2}$. Since $\vartheta(z)$ is of weight 1/2 on $\Gamma(4)$, $\theta(z)$ is indeed a modular form of weight 5 and a cusp form since it's constant term is $0$.
A: The sense of "generalized" (theta series) is that the polynomial coefficients should be harmonic (for exponentials of the form $e^{2\pi i(m^2+n^2)}$ or the $N$-dimensional analogue) and homogeneous of total degree $d$. The resulting modular form is of weight ${N\over 2}+d$. 
Thus, it is easiest to understand for $N$ even.
We can take advantage of the fact that $z \to -1/4z$ and $z\to z+1$ generate $\Gamma_0(4)$, and Poisson summation, to prove this. A new ingredient, in addition to the fact that Gaussians are essentially their own Fourier transforms, is Hecke's identity, namely, that on $\mathbb R^n$, functions $P(x)e^{-\pi |x|^2}$ (with $x\in \mathbb R^n$) with $P$ harmonic homogeneous of degree $d$ are simply multiplied by $i^{-d}$ under Fourier transform.
EDIT: and it is certainly holomorphic, if we express it more clearly in terms of the complex variable $z\in \mathfrak H$. It vanishes at the cusp $i\infty$ visibly because its $0$th Fourier coefficient is indeed $0$, because $P(0,0)=0$. There is a not-completely-trivial computation to be done to show that it vanishes at the other cusps (inequivalent to $i\infty$).
