In Zagier's chapter of The 1-2-3 of Modular Forms, he defines the Theta series of a quadratic form $Q$ as $$\Theta_Q(z) = \sum_{n=0}^\infty r(Q,n) q^n$$ where $q = e^{2\pi i z}$. He identifies orbits of quadratic forms under an action of $SL_2(\mathbb{Z})$ with ideal classes in $\mathbb{Q}(\sqrt{D})$, and then takes a homomorphism $\chi$ from the ideal class group of $\mathbb{Q}(\sqrt{D})$ to $\mathbb{C}^*$, and sets $$f_\chi = \frac{1}{w} \sum_{\mathcal{A}} \chi(\mathcal{A}) \Theta_\mathcal{A}(z)$$ where $w$ is a constant, the sum is over ideal classes in $\mathbb{Q}(\sqrt{D})$, and $\Theta_\mathcal{A}(z)$ is $\Theta_Q(z)$ where $\mathcal{A}$ and the orbit of $Q$ under the action of $SL_2(\mathbb{Z})$ are identified.
He then claims that $f_\chi$ is a modular form of weight $1$, but I can't tell why that is the case. I'm guessing that it must be of weight $1$ and character $\chi$, because I don't see how it makes sense otherwise, but the definition I'm familiar with for modular forms of weight $k$ and character $\chi$ is that they are modular forms $f$ of weight $k$ on $\Gamma_1(N)$ such that $$f\left(\frac{az+b}{cz+d}\right) = \chi(d) (cz+d)^k f(z)$$ for all $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(N)$. Is this the right definition?
Here is the relevant section of the text.
Also, at the end, he claims that the Euler product for the $L$-series of $f_\chi$ means that $f_\chi$ is a normalized eigenform, but I thought that the Euler product must be of the form $$\prod_{p} \frac{1}{1 - a_p p^{-s} + \chi(p) p^{k-1-2s}}$$ where the product is over primes $p$ and $a_p$ are the Fourier coefficients of $f_\chi$. I can see why the $L$-series has a product over prime ideals of the ring of integers of $\mathbb{Q}(\sqrt{D})$, but not why it has this form.