Can Eisenstein Series output complex numbers? The 2nd Eisenstein Series, defined by,
$$E_{2}(\tau)=1-24\sum_{n=0}^{\infty} \frac{nq^{2n}}{1-q^{2n}},$$
where $q=e^{i\pi \tau}$ is the nome acts on upper-half plane.  Must it always output real numbers for all $\tau$?  Simple question but I can't find any info on it in the documentation online.  
 A: If the real part of $\tau$ is an integer, then $q^2$ is a positive real.
If the real part of $\tau$ is a half-integer, then $q^2$ is a negative real.
In both cases, $\operatorname{E}_2(\tau)$ must be real.
Due to $\operatorname{E}_2$ being holomorphic,
infinitesimal deviations from those vertical rays $2\Re\tau\in\mathbb{Z}$
result in non-real changes to $\operatorname{E}_2(\tau)$ whenever $\frac{\mathrm{d}\!\operatorname{E}_2(\tau)}{\mathrm{d}\tau}\neq0$.
Nevertheless, there are more curves beside the above vertical rays
where $\operatorname{E}_2(\tau)$ is real.
The plot below, together with a colormap, shows 
$\operatorname{E}_2(\tau)$ for $\Re\tau\in[-\frac{3}{2},+\frac{3}{2}],
\Im\tau\in(0,\frac{15}{8}]$.
The colormap ignores magnitude and quantizes polar angles into $24$ bins.

Accordingly, the curves where $\operatorname{E}_2(\tau)$ is real
are the boundaries between the magenta and fully saturated red regions (for positive values) and the boundaries between brightest cyan and light greenish tints (for negative values). The curves split at points where
$\frac{\mathrm{d}\!\operatorname{E}_2(\tau)}{\mathrm{d}\tau}=0$.
The color vortex points mark simple zeros (in the plane) resp. double poles
(on the real axis).
