I've noticed that there is some extremely intriguing self-similar/fractal branch cut structure on certain Eisenstein series. In particular, the following holomorphic, quasi-modular function on the upper-half complex plane:
$$\frac{1}{144}\big(E_{2}^{2}(\tau) - E_{4}(\tau)\big).$$
The plot I've included here is of the argument of this function. Clearly, it's invariant under $\tau \to \tau +1$, but most certainly not invariant under $\tau \to -1/\tau$.
I believe those black lines represent branch cuts. As you can see, this function exhibits what appears to be very rich branch cut structure, and it even appears to become fractal-like, at small imaginary parts of $\tau$.
Now I know that with functions on $\mathbb{C}$, branch cuts imply that the functions aren't really holomorphic on $\mathbb{C}$, but rather are holomorphic on some Riemann surface. Does anyone have any thoughts on an elegant geometrical way to think of the above function? As in, do we know really what space this thing is a function on? From this picture, it really doesn't look like it's natural to think of this as a function on the upper-half plane! Believe it or not, I actually have a physical reason why I don't think this is naturally considered as defined on the upper-half plane.
A couple of notes that might jog someone's thoughts:
(I) Those black "dots" at the top of the black lines are where the function vanishes identically,
(II) Those points that look like "saddle-points" are where the derivative of the function vanishes identically.
Thanks for any thoughts!