Computing Zeroes of Modular Forms

A century ago, Hardy and Ramanujan revolutionized the study of partitions by introducing analytic methods to count how many ways a number can be written as a sum of positive integers. A crucial component of their method is an understanding of zeroes of functions with many symmetries, such as modular forms. 

Two distinct phenomena can occur: either the zeroes are distributed randomly, or they cluster along curves. For instance, for a concrete modular form that occurs in the echelon form of a specific linear transformation, its several hundred zeroes cluster along two circles. 

In this computational project, we would like to determine such clusters of zeros when the "level" of the modular form goes beyond 1. The project involves the following:

• Implementing in Sage/Python code to compute zeroes of modular forms.
• Plotting the zeroes in the upper half plane and surmising the types of clusters that occur.
• Determine any consistent behavior as the characteristics of the modular form, such as weight and degree of first Taylor coefficient, vary. For instance, in "level 1," the clusters only depend on the ratio of these two quantities.

This project will, if all goes according to plan, provide a computational basis for a theoretical result.