## Modular Form Level 1, Weight 22 (1.22.a.a) Visualised

Modular forms are highly self-symmetric functions studied in number theory. They have surprising connections to several different fields of mathematics. Modular forms are rarely visualised, especially not in 3D. David Lowry-Duda reached out to me after seeing some of my other graphics on YouTube and we set to work attempting to create some new visualisations. These visualisations were first shared by David at the 2021 Bridges conference. (Paper). (Slides)

Thanks to David Lowry-Duda for his work coding the Modular Form calculation. I've used his procedure to calculate only on the fundamental domain, which improves accuracy.

LMFDB Entry

Due to the very large values, we have applied a damping function to control the height of the 3D outputs.

The colour represents the argument (phase) of the function. Where hue=arg(z). Red is the positive real axis, cyan is the negative real axis.

## Poincaré disk

The Poincaré disk is a conformal mapping of the upper-half plane onto the unit circle.

Height damping function: atan(log(sqrt(z)+1)), then scaled for visual appeal.

The fractal like nature of the modular form shape becomes clearer when it is rendered with only a single colour. If you compare this image to plot 1 above, you will see that the colours meet at each of the holes, these are the zeros of the modular form.