Fourier expansion of half-integral weight Eisenstein series associated with Kohnen's plus space

Question

The Eisenstein series associated with Kohnen's plus space in $\Gamma_{0}(4)$ is expressed as follows, \begin{align} \begin{split} E_{k + \tfrac{1}{2}}^{\infty}(\tau) =& \sum\limits_{\substack{c>0,\ \text{odd}\\ d}}\left(\frac{c}{d}\right)\left(\frac{-4}{d}\right)^{-\left(k + \tfrac{1}{2}\right)}\left(c\tau + d\right)^{-\left(k + \tfrac{1}{2}\right)},\ k\geq 2\\ E_{k + \tfrac{1}{2}}^{0}(\tau) =& \left(-1\right)^{-k}i\tau^{-\left(k + \tfrac{1}{2}\right)}E_{k + \tfrac{1}{2}}^{\infty}\left(-\frac{1}{4\tau}\right),\ k\geq 2 \end{split} \end{align} where $\left(\tfrac{c}{d}\right)$ and $\left(\tfrac{-4}{d}\right)$ are the Legendre symbols, and $E^{\infty}_{k + 1/2}$ and $E^{0}_{k + 1/2}$ are half-integral weight versions of the Eisenstein series associated with cusps $\infty$ and $0$. How do I obtain the $q$-series expansion of these half-integral weight Eisenstein series? I found the following potentially useful formula from Shimura, \begin{align} \begin{split} E_{k + \tfrac{1}{2}}^{\infty}(\tau) =& (2\pi)^{k + \tfrac{1}{2}}e^{-\tfrac{i\pi}{2}\left(k + \tfrac{1}{2}\right)}\Gamma\left(k + \frac{1}{2}\right)^{-1}\sum\limits_{n=1}^{\infty}a_{n}q^{n},\\ a_{n} =& n^{k - \tfrac{1}{2}}\sum\limits_{m>0,\ \text{odd}}\frac{\varepsilon_{m}^{2k + 1}}{m^{k + \tfrac{1}{2}}}\left(\sum\limits_{0<r<m}\left(\frac{-r}{m}\right)e^{\tfrac{2\pi inr}{m}}\right),\\ \varepsilon_{d} =& \begin{cases} 1,\ \text{if}\ d\equiv 1\ (\text{mod}\ 4),\\ i,\ \text{if}\ d\equiv 3\ (\text{mod}\ 4). \end{cases} \end{split} \end{align} From the definition, it seems that the $q$-series expansion will contain $\text{Log}(q)$ terms, not terms that go as a polynomial in $q$. Can an explicit $q$-series expansion of these half-integral weight forms be obtained?

Answer 1

In addition to the original papers (by yours truly, Joris, and others from the 1970's), you can look at Theorem 15.1.5 of my book with F. Str"omberg, GSM 179, AMS.