Yesterday I gave a talk at the University of Oregon Number Theory seminar on Visualizing Modular Forms. This is a spiritual successor to my paper on Visualizing modular forms that is to appear in Simons Symposia volume Arithmetic Geometry, Number Theory, and Computation.
I've worked with modular forms for almost 10 years now, but I've only known what a modular form looks like for about 2 years. In this talk, I explored visual representations of modular forms, with lots of examples.
The slides are available here.
I'll share one visualization here that I liked a lot: a visualization of a particular Maass form on $\mathrm{SL}(2, \mathbb{Z})$.
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