Questions
I am having trouble understanding some details in the computation of the q-expansion of weight-1 Eisenstein series, as done in Section 4.8 of Diamond and Shurman's "A First Course in Modular Forms". My questions are the following, which are in fact Exercises 4.8.4(a) and 4.8.5.
Why is $F_1^{v}$ well defined? It is not clear to me that this is true. Since $Z_{\Lambda}$ is not periodic on the lattice $\Lambda$, this function actually depends on the representatives of $\omega_1/N + \Lambda$ and $\omega_2/N + \Lambda$. I understand however, that as a function on $\tau=\omega_1/\omega_2$ (with $\tau\in\mathcal{H}$) it is well defined and weight-1 invariant under $\Gamma(N)$ (essentially because it is the function $g_1^{v}$).
What happens with the term $\delta(c_v)(\pi i /N)$ in the q-expansion? I am able to follow all the steps until equation (4.34), but it seems to me that this term is missing in the presented formula. I am wondering if I computed $\zeta^{d_v}(1)$ correctly, but this should be precisely $(\pi/N) \cot(\pi d_v/N)$ right?
The second question bothers me more as some calculations done afterwards to compute basis for $\mathcal{E}_1(\Gamma_1(N))$ and $\mathcal{E}_1(N,\chi)$ depend on the explicit formula for the q-expansion of $g_1^{v}$.
Formulas
For those who don't have the book, here are some of the formulas I am referring to. These are the functions:
\begin{equation*} Z_\Lambda(z) = \frac{1}{z} + \sum_{\omega \in \Lambda}(\frac{1}{z-\omega}+\frac{1}{\omega}+\frac{z}{\omega^2}), \end{equation*}
\begin{equation*} Z_{\Lambda_\tau}(z) = \eta_2(\Lambda_\tau)z - \pi i \frac{1+e^{2\pi i z}}{1-e^{2\pi i z}} - 2\pi i \sum_{n=1}^\infty\left(\frac{e^{2\pi i z}·q^n}{1-e^{2\pi i z}·q^n} - \frac{e^{-2\pi i z}·q^n}{1-e^{-2\pi i z}·q^n}\right), \end{equation*}
\begin{equation*} F_1^{v}(\mathbb{C}/\Lambda, (\omega_1/N + \Lambda, \omega_2/N+\Lambda))=Z_\Lambda(\frac{c_v \omega_1 + d_v \omega_2}{N}) - \frac{c_v·\eta_1(\Lambda)+d_v·\eta_2(\Lambda)}{N}, \end{equation*}
\begin{equation*} g_1^{v}(\tau)=\frac{1}{N} Z_{\Lambda_\tau}(\frac{c_v \tau + d_v}{N}) - \frac{c_v·\eta_1(\Lambda)+d_v·\eta_2(\Lambda)}{N}. \end{equation*}
These are some intermediate steps in computing the q-expansion of $g_1^{v}$ (here is where the term $\delta(c_v)(\pi i/N)$ appears).
\begin{equation*} \frac{\eta_2(\Lambda_\tau)}{N} - \frac{c_v·\eta_1(\Lambda_\tau)+d_v·\eta_2(\Lambda_\tau)}{N^2}=\frac{2\pi i c_v}{N^2}, \end{equation*}
\begin{equation*} -\pi i \frac{1+e^{2\pi i z}}{1-e^{2\pi i z}} = \delta(c_v)\frac{\pi}{N}\cot(\frac{\pi d_v}{N}) + (1-\delta(c_v))\left(-\frac{\pi i}{N}+\frac{C_1}{N}\sum_{n=1}^\infty \mu_N^{d_v m} q^{c_v m}\right), \end{equation*}
\begin{equation*} -2\pi i \sum_{n=1}^\infty\frac{e^{2\pi i z}·q^n}{1-e^{2\pi i z}·q^n} = \frac{C_1}{N} \sum_{n=1}^\infty(\sum_{m}\text{sgn}(m)\mu_N^{d_vm})·q_N^n-(1-\delta(c_v))·\frac{C_1}{N}·\sum_{m=1}^\infty\mu_N^{d_vm}·q_N^{c_vm}, \end{equation*}
\begin{equation*} 2\pi i \sum_{n=1}^\infty\frac{e^{-2\pi i z}·q^n}{1-e^{-2\pi i z}·q^n} = \frac{C_1}{N} \sum_{n=1}^\infty(\sum_{m}\text{sgn}(m)\mu_N^{d_vm})·q_N^n, \end{equation*}
where in the third equation $m$ runs through positive divisors of $n$ such that $n/m \equiv c_v$ modulo $N$, and in the fourth equation through negative divisors of $n$ satisfying the same condition. Lastly, we have equation (4.34), which is,
\begin{equation*} g_1^{v}(\tau) = G_1^v(\tau) - \frac{C_1}{N}(\frac{c_v}{N}-\frac{1}{2}), \end{equation*}
where
\begin{equation*} G_1^v(\tau) = \delta(c_v)\zeta^{d_v}(1)+\frac{C_1}{N}\sum_{n=1}^\infty\sigma_0^v(n)q_N^n, \end{equation*}
\begin{equation*} \zeta^{d_v}(k)=\sum_{d\equiv d_v(N)}{d_v^{-k}}, \end{equation*}
\begin{equation*} \sigma_{k}^{v}(n)=\sum_{m}\text{sgn}(d)d^{k}·\mu_N^{d_v m}, \end{equation*}
$d$ being non-zero zero in the second equation, and $m$ running through all divisors of $n$ (positive or negative) such that $n/m\equiv c_v$ modulo $N$. Also, $C_1=-2\pi i$.