The usual Eisenstein is defined as $$ E_{2n}(\tau) = \sum_{(m, m') \in \mathbb{Z}^2-(0,0)} \frac{1}{(m + m' \tau)^{2n}} \ . $$ It has a generalization to twisted Eisenstein series ($\phi := e^{2\pi i\lambda}$), \begin{align} E_{n \ge 1}\left[\begin{matrix} \phi \\ \theta \end{matrix}\right] \equiv & \ - \frac{B_n(\lambda)}{n!} \\ & \ + \frac{1}{(n-1)!}\sum_{r \ge 0} \frac{(r + \lambda)^{n - 1}\theta^{-1} q^{r + \lambda}}{1 - \theta^{-1}q^{r + \lambda}} + \frac{(-1)^n}{(n-1)!}\sum_{r \ge 1} \frac{(r - \lambda)^{n - 1}\theta q^{r - \lambda}}{1 - \theta q^{r - \lambda}} \ . \end{align}
I wonder if the $n\to \infty$ asymptotic behavior/large $n$ expansion of $E_{2n}(\tau)$ or $E_n [\phi, \theta](\tau)$ is known in the literature?