Find the $L^2$ norm of $\theta_\chi(z)^2$ in $\mathbb{H}/\Gamma_0(4)$ I'd like to learn about $L^2$ norms on Hyperbolic space.  My Automorphic forms textbook says this function is a "cusp form" so it's in $L^2(\mathbb{H}/\Gamma_0(4))$:
$$ \theta_\chi(z) = \sum_{n \in \mathbb{Z}} \chi(n) \, e^{-\pi n^2 \, z} $$
This is a cusp form, since $\chi(0) = 0$ for any Dirichlet character $\chi$.  Can we do the integral over the fundamental region?
$$ L^2(\theta_\chi) =  \int_{\mathbb{H}/\Gamma_0(4)} |\theta_\chi(z)|^2 \, \frac{dx^2 + dy^2 }{y^2} $$
Have I written the norm correctly?  It's very likely that in some cases, my question is not well-posed and that can be discussed in the comments or I'll up-vote an answer.

It seems the original version of my question is problematic. At least 
$\theta_\chi(z)^2$ is a weight 1 (?) cusp form over $\Gamma_0(4)$ and the question $L^2$ makes sense. 
$$ \theta_\chi(z)^2 = \sum_{(m,n) \in \mathbb{Z}^2} \chi(n) \, e^{-2\pi i (m^2+n^2 )\, z} $$
I may have to name a specific Dirichlet character. Perhaps the mod 4 character with $\chi(3)=-1$? And then we are integrating the absolute value squared over hyperbolic space:
$$| \theta_\chi(z)|^4 = \sum_{(m,n) \in \mathbb{Z}^4} \chi(n) \, e^{-2\pi i (a^2+b^2) -(c^2+d^2) )\, z} $$
We obtain a theta series associated to a quadratic form $q(z)=(a^2+b^2) -(c^2+d^2) $. The area form should be corrected as well 
$$ L^2(\theta_\chi^2) =  \int_{\mathbb{H}/\Gamma_0(4)} |\theta_\chi(z)|^4 \, \frac{dx \, dy}{y^2} $$
Is this question well stated?

The standard, but non-obvious, way to compute an $L^2$ norm of a modular form $f$ is as the residue at $s=1$ of the Rankin-Selberg $L$-function (or Dirichlet series) obtained as an integral $\int_{\Gamma\\\mathfrak H} E_s\cdot |f|^2$. This is because the residue at $s=1$ of that Eisenstein series is a constant (basically the multiplicative inverse of the natural volume of $\Gamma\backslash\mathfrak H$). For half-integral weight theta series, after the corrections indicated by @Peter Humphries are made, you can identify this Rankin-Selberg thing as an already-familiar Dirichlet series, whose reside at $1$ is known in other terms.
EDIT: "The" weight-zero Eisenstein series attached to the cusp $i\infty$ for $\Gamma_0(N)$ is $\sum_{c,d} y^2/|cz+d|^{2s}$ where $c,d$ are summed over relatively prime integers congruent to $0,1$ mod $N$, and multiplicatively modulo $\pm 1$ for $N=2$.
