Experimental Mathematics

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

Josep González and Jorge Jiménez-Urroz

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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.

Article information

Experiment. Math., Volume 21, Issue 1 (2012), 84-102.

First available in Project Euclid: 31 May 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F30: Fourier coefficients of automorphic forms

Sato–Tate distribution Lang–Trotter conjecture Lehmer’s conjecture


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

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