Computing isolated coefficients of the $j$-function

TL;DR

This paper presents an efficient hybrid numerical-modular method for computing specific Fourier coefficients of the elliptic modular function j(τ), successfully locating prime coefficients and demonstrating practical efficiency.

Contribution

It introduces a new hybrid computational approach with near-linear complexity for isolated coefficients of the j-function, enabling prime coefficient discovery.

Findings

Method achieves complexity n^{1+o(1)}

First prime coefficient found at n=457871

Method is practical for large n

Abstract

We consider the problem of efficiently computing isolated coefficients $c_n$ in the Fourier series of the elliptic modular function $j(\tau)$. We show that a hybrid numerical-modular method with complexity $n^{1+o(1)}$ is efficient in practice. As an application, we locate the first few values of $c_n$ that are prime, the first occurring at $n = 457871$.