I want to know how to prove that the function $$ E(z) = 1 + 6\sum_{n\geq 1} \sum_{d|n} \chi(d) e^{2\pi i n z} $$ is a modular form of level $3$ with character $\chi$, where $\chi$ is a mod 3 odd Dirichlet character such that
$$ \chi(d) = \begin{cases} 0 & d \equiv 0 \mod{3} \\ 1 & d \equiv 1 \mod{3} \\ -1 & d \equiv -1 \mod {3}\end{cases}. $$
I heard that this can be proved using Weil's converse theorem, but I think there would be a more direct and easy proof without using it. When we prove modular property of Eisenstein series, we usually write it as $$\sum_{\gamma\in \Gamma_{\infty} \backslash \Gamma} 1 |_{\gamma}$$ where $\Gamma$ is a group that a given Eisenstein series has a transformation property, and $\Gamma_{\infty}$ is a stabilizer of the constant function $1$. However, the summation absolutely converges only when the weight $k$ is larger than 2, and when the weight $\leq 2$ we need to treat more carefully. I think the above Eisenstein series would be a form of $$ \sum_{(c, d) = 1, 3|c} \frac{\chi(d)}{cz + d} $$ which does not converge absolutely. How to show this function is actually a modular form of weight 1? What is the "right" way to make this sum converges to the our function?