How can I draw a function of the nome $q$ on the unit disk?

Question

This figure is the real part of the discriminant as a function of the nome $q$ on the unit disk. It is taken from the wiki link:

https://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions

I know how to compute the real part of the discriminant. How can I draw this figure? In particular, what means "the nome $q$ on the unit disk"?

Answer 1

The Mathematica code

colarg[z_] := {Mod[3 Pi/2+Arg[z], 2 Pi]/(2 Pi), If[z == 0, 0, 1], (1 + Log[Abs[z]]^2)^(-1/6)};
Image[
 Table[
  colarg[If[Abs[x + I y] > .999999, 0, DedekindEta[Log[x + I y]/(2 Pi I)]^24]],
  {x, -1 + 10^-6 RandomReal[], 1, .0035}, {y, -1 + 10^-6 RandomReal[], 1, .0035}
 ],
 ColorSpace -> Hue, Magnification -> 1
]

produces

Answer 2

The nome is the quantity $q=\exp(i \pi \tau)$, which lies in the unit disk $|q|<1$ when $\tau$ lies in the upper half plane.

So you can view the function as a function of $\tau$, and draw the picture by coloring the points of the upper half plane, or (as in this case) you can view it as a function of $q$, and draw the picture by coloring the points of unit disk.