The Sato–Tate Distribution and the Values of Fourier Coefficients of Modular Newforms

Abstract

The Sato–Tate conjecture has been recently settled in great generality. One natural question now concerns the rate of convergence of the distribution of the Fourier coefficients of modular newforms to the Sato–Tate distribution. In this paper, we address this issue, imposing congruence conditions on the primes and on the Fourier coefficients as well. Assuming a proper error term in the convergence to a conjectural limiting distribution, supported by experimental data, we prove the Lang–Trotter conjecture, and in the direction of Lehmer's conjecture, we prove that τ(p)=0 has at most finitely many solutions. In fact, we propose a conjecture, much more general than Lehmer's, about the vanishing of Fourier coefficients of any modular newform.

2000 AMS Subject Classification:

• 11F30

Keywords:

• Sato–Tate distribution
• Lang–Trotter conjecture
• Lehmer's conjecture

ACKNOWLEDGMENTS

We would like to thank Carles Barcelo for his help in the statistical part of this article, and also the referee for the suggestions that made the paper more complete.

The first author is partially supported by DGICYT grant MTM2009-13060-C02-02, the second by DGICYT grant MTM2009-11068.