This image shows the modulus |j| of the J-invariant as a function of the square of the nome on the unit disk |q| < 1. That is, runs from 0 to along the edge of the disk. Black indicates regions where the modulus is near zero, green where the modulus is about one, and red where the modulus is greater than ten. The color scale is logarithmic. Inside of every black region is a zero; note the zeros are cubic.
The small black dot on the far right, in the large red cardioid, is a numerical artifact.
The fractal self-similarity of this function is that of the modular group; note that this function is a modular form. Every modular function will have this general kind of self-similarity. In this sense, this particular image clearly illustrates the tesselation of the q-disk by the modular group.