How to Compute the Coefficients of the Elliptic Modular Function j(z)

Abstract

We discuss various methods to compute the Fourier coefficients of the elliptic modular function j(z). We present run times to compute the coefficients in practice. If possible, we discuss the theoretical complexity of the corresponding method, too. We conclude that, in practice, an approach due to Kaneko and Zagier turns out to be most efficient.

Keywords:

• Algorithmic number theory
• elliptic modular function j
• Fourier coefficients
• Fourier series