Computing GL(3) automorphic forms

Abstract

We present a method to find the Fourier coefficients of a GL(3) automorphic form given its Casimir eigenvalues. The method uses a linear system derived from twisted functional equations. To demonstrate the correctness of our results, we test various conjectures, such as the multiplicativity of Fourier coefficients, an analogue of the Sato–Tate conjecture on the distribution of prime Fourier coefficients and the Riemann hypothesis. Similar work has been done by Miller (‘A method for computing general automorphic forms on general groups’, Preprint, 2008, http://arxiv.org/abs/0801.3299v2) and Farmer et al. (‘A direct serach for degree 3 L-functions’). As far as we are aware, our numerical results are the first examples of GL(3) cusp forms that do not arise from special cases of the Langlands functoriality conjectures.

Locations

• Bulletin of the London Mathematical Society - View - PDF