Computation of Harmonic Weak Maass Forms

Abstract

Harmonic weak Maass forms of half-integral weight have been the subject of much recent work. They are closely related to Ramanujan's mock theta functions, and their theta lifts give rise to Arakelov Green functions, and their coefficients are often related to central values and derivatives of Hecke L-functions. We present an algorithm to compute harmonic weak Maass forms numerically, based on the automorphy method due to Hejhal and Stark. As explicit examples we consider harmonic weak Maass forms of weight 1/2 associated to the elliptic curves 11a1, 37a1, 37b1. We have made extensive numerical computations, and the data we obtained are presented in this paper. We expect that experiments based on our data will lead to a better understanding of the arithmetic properties of the Fourier coefficients of harmonic weak Maass forms of half-integral weight.

Keywords:

• modular form
• harmonic Maass form
• elliptic curve
• L-function
• Weil representation
• 11Y35
• 11Y40
• 11F30
• 11G05

Notes

¹An implementation of this algorithm can be found online at http://code.google.com/r/fredrik314-psage/ .

²Current versions of the algorithm can be obtained at http://code.google.com/r/fredrik314-psage/ .

³The interested reader is encouraged to use http://code.google.com/r/fredrik314-psage/ to compute this function.

⁴ lcalc, a library for computing zeros and values of L-functions. Available at www.math.uwaterloo.ca/~mrubinst.