The real Eisenstein series
$G_s^* = \frac{\Gamma(s)}{\pi^s} \sum'_{m,n}\frac{Im(\tau)}{|m+n \tau|^{2s}}$
admits the following integral representation (their Mellin transform):
$G_s^* = \frac{1}{2} \int_0^\infty (\Theta_\tau(t)-1) t^{s-1} dt$
where $\Theta_\tau(t)$ is the associated theta series $\Theta_\tau(t) = \sum_{m,n}e^{-\pi Q_\tau(m,n)t}$ . In general, we can construct real modular invariant quantities by considering:
$\hat M(\tau) = \frac{1}{2} \int_0^\infty (\Theta_\tau(t)-1) M(t) dt$
Does anyone know if it is possible to invert this relation? I.e. given a real modular form $\hat M(\tau)$, is it possible to find $M(t)$? Are there any references I could look at? Thank you very much!