Square root of Eisenstein series

Question

I am interested in square roots of Eisenstein series, such as $\sqrt{E_4}$ (see also this question). If we define the square root through its Taylor series, $$\sqrt{E_4}(\tau)=1 + 120 q - 6120 q^2 + 737760 q^3+\mathcal O(q^4), \quad q=e^{2\pi i\tau}$$ is an integer $q$-series with integral coefficients. Using the root test or the quotient test I find that it converges for $|q|<0.004$ or $\text{Im}(\tau)>0.878$. Using the $S$-transformation $$E_4(-1/\tau)=\tau^4E_4(\tau)$$ one could define $\sqrt{E_4}$ also near $\tau=0$ (or $q=1$). Can $\sqrt{E_4}$ in this way be turned into a holomorphic function on $\mathbb H$, using analytic continuation...? Is there a fundamental domain for $\sqrt{E_4}$ and would it just be given by two copies of a fundamental domain for $SL(2,\mathbb Z)$?

Answer 1

The zeros of the $E_4$ modular form occur at the points ${SL}_2(\Bbb Z)$-equivalent to $\pm\frac12+\frac12\sqrt{-3}$. They are simple, so the largest half-plane that $\sqrt{E_4}$ can be defined as a holomorphic function is $\{x+yi:y>\sqrt 3/2\}$. This is equivalent to $|q|<\exp(-\sqrt3\pi)$. Does this agree with your numerics?