It is known that Eisenstein series has a Fourier expansion $$ G_{k}(\tau)=-\frac{B_{k}}{2k}+\sum_{n=1}^{\infty}\sigma_{k-1}(n)q^{n} $$ where $\sigma_{k-1}(n)=\sum_{d|n}d^{k-1}$ if $k\geq 4$ is even. For $k=2$, this is not a modular form, but it is a quasi-modular form in some sense.
Question : For odd $k\geq 5$, does $G_{k}$ defined as above has any interesting automorphic property? (In this case, $B_{k}=0$ and it might be something like cusp form.)