I am interested in the claim from F.Diamond & J.Shurman, "A First Course in Modular Forms" that if $f$ is a modular form w.r.t. a congruence subgroup, then
in the Fourier expansion $f(τ) = \sum_{n=0}^{\infty} a_nq_N^n$, the coefficients satisfy the condition $|a_n|≤Cn^r$ for some positive constants $C$ and $r$.
For cusp forms, we can utilize Petersson inner product to show this. On the other hand, for Eisenstein series, the authors prove this claim first for Eisenstein series for $\Gamma_1(N)$. Because they have a explicit basis of the space of such series, they directly confirm that members of the basis satisfy the estimate in the claim to get the desired result. Then, for Eisenstein series for a general congruence subgroup, or equivalently for those for $\Gamma(N)$, they only say that the evaluation for the series for $\Gamma_1(N)$ readily extends to the general case.
Luckily, a basis of the space of Eisenstein series for $\Gamma(N)$ is on the book, for weight more than one. For weight one, the authors construct some series for $\Gamma(N)$, i.e. $g_1^{v}(\tau) = \frac{1}{N}Z_{\Lambda} \left(\frac{c\tau + d}{N}\right)-\frac{c\eta _1(\Lambda) + d\eta _2(\Lambda)}{N^2}$, where $Z_{\Lambda}$ is Weierstress Zeta Function w.r.t. the lattice $\Lambda$ generated by $1$ and $\tau$, $v = (c,d)\in (\Bbb Z/N\Bbb Z)^2$ is of order $N$, and $\eta _1$, etc is the constants appearing in Legendre relation. However, they do not mention if they make up a basis of the space. (See http://people.reed.edu/~jerry/MF/errata4.pdf for those who have the book.)
My question is:
is there any way to easily show that the series above span the space of Eisenstein series for $\Gamma(N)$ of weight one? Or, is there another way to show the first polynomial evaluation?
P.S.
I am also aware of https://link.springer.com/book/10.1007/BFb0065293, which include an article by J-P.Serre, and H.M. Stark determining the basis of the space of weight one modular forms for $\Gamma(N)$. So, I would read it if there are no other ways.