- Experiment. Math.
- Volume 21, Issue 1 (2012), 84-102.
The Sato–Tate Distribution and the Values of Fourier Coefficients of Modular Newforms
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 $\tau (p) = 0$ 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.
Experiment. Math., Volume 21, Issue 1 (2012), 84-102.
First available in Project Euclid: 31 May 2012
Permanent link to this document
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11F30: Fourier coefficients of automorphic forms
González, Josep; Jiménez-Urroz, Jorge. The Sato–Tate Distribution and the Values of Fourier Coefficients of Modular Newforms. Experiment. Math. 21 (2012), no. 1, 84--102. https://projecteuclid.org/euclid.em/1338430816