Let $q = e^{2\pi i\tau}$, $\operatorname{Im}\tau > 0$ and let $G(\tau) = 1 - 24\sum_{n = 1}^{\infty}\frac{nq^{n}}{1 - q^{n}}$ be the weight 2 Eisenstein series for $\Gamma(1)$. Consider the inequivalent cusps of $\Gamma_{1}(N)$. For each cusp, there is an Eisenstein series of weight 2 associated to this cusp. How does one relate this Eisenstein series to a linear combination of $G(k\tau)$'s (for some integers $k$)?
In particular, the case I am considering is when $N = 7$. The inequivalent cusps are $0$, $2/7$, $1/3$, $3/7$, $1/2$, and $\infty$. Consider the cusp $2/7$ and denote by $E_{2/7}$ the associated Eisenstein series. I would like to write $E_{2/7}$ as a finite sum of $G(k\tau)$ for some integers $k$. One of the first issues I'm having is how would I know the $q$-expansions of Eisenstein series of $E_{2/7}$ (which is related to the issue that I can't seem to find a good reference for the definition of a weight 2 Eisenstein series associated to a cusp)? Second, would finding the desired linear combination of $G(k\tau)$'s be just an exercise in matching up finitely many coefficients in the $q$-expansion?