The definition of "newform with CM" I'm interested in is given on p. 415 of this paper. You start by specifying your imaginary quadratic field $K = \mathbb{Q}(\sqrt{D})$ for $D < 0$, an integer $k \geq 2$, and a Hecke character $c: I(\mathfrak{f}_{c}) \rightarrow \mathbb{C}^{\times}$ of conductor $\mathfrak{f}_{c} \subset \mathcal{O}_{K}$. Here,
$$c(\alpha \mathcal{O}_{K}) = \alpha^{k-1}, \quad \text{for } \alpha \equiv 1 \hspace{-2ex} \mod \mathfrak{f}_{c}.$$
We get a Dirichlet character $\omega_{c}$ given by
$$\omega_{c}(n) := \frac{c(n \mathcal{O}_{K})}{n^{k-1}}, \quad \text{for } n \in \mathbb{Z} \text{ coprime to } \mathfrak{f}_{c}.$$
We also have the usual character $\chi_{K} := \left( \frac{D}{*} \right)$. Now our CM newform is
$$\Phi_{K, c}(z) := \sum_{\mathfrak{a}} c(\mathfrak{a})q^{N(\mathfrak{a})} \in S_{k}\left(|D|N(\mathfrak{f}_{c}), \chi_{K}\cdot\omega_{c}\right),$$
where the sum runs over integral ideals $\mathfrak{a}$ of $K$ prime to $\mathfrak{f}_{c}$, and where $N$ is the usual ideal norm.
Example: Let $K = \mathbb{Q}(\sqrt{-3})$, $k = 2$. Note that $\omega_{c}$ is trivial. I choose $\mathfrak{f}_{c} := (3) \subset \mathcal{O}_{K}$. I can only guess what my Hecke character $c$ should be... that's what I need help with. Here, $\mathcal{O}_{K}$ is a PID. My newform with CM should then look something like
$$\sum_{a \not\equiv b \hspace{0.5ex}(3)} \left(a + \frac{b}{2}(1 + \sqrt{-3})\right)q^{a^{2} + ab + b^{2}},$$
or at least I think it should. Note that it lies in $S_{2}(27)$. The problem I now have is that I'd get cancellation with $a + \frac{b}{2}(1 + \sqrt{-3})$ and $-a - \frac{b}{2}(1 + \sqrt{-3})$ in my sum, making the whole thing zero. Fine -- I did something wrong with the coefficients. It seems to come down to choosing the representative that $c$ spits out when inputing the ideal $\left(a + \frac{b}{2}(1 + \sqrt{-3})\right)$. I know that this form should look like
$$q - 2q^{4} - q^{7} + 5q^{13} + \cdots.$$