Experimental Mathematics

A test for identifying Fourier coefficients of automorphic forms and application to Kloosterman sums

Andrew R. Booker

Abstract

We present a numerical test for determining whether a given set of numbers is the set of Fourier coefficients of a Maass form, without knowing its eigenvalue. Our method extends directly to consideration of holomorphic newforms. The test is applied to show that the Kloosterman sums $\pm S(1,1;p)\big/\hskip-1pt\sqrt p$ are not the coefficients of a Maass form with small level and eigenvalue. Source code and the calculated Kloosterman sums are available electronically.

Article information

Source
Experiment. Math., Volume 9, Issue 4 (2000), 571-581.

Dates
First available in Project Euclid: 20 February 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1045759522

Mathematical Reviews number (MathSciNet)
MR1806292

Zentralblatt MATH identifier
0966.11018

Subjects
Primary: 11F30: Fourier coefficients of automorphic forms
Secondary: 11L05: Gauss and Kloosterman sums; generalizations 11Y35: Analytic computations

Citation

Booker, Andrew R. A test for identifying Fourier coefficients of automorphic forms and application to Kloosterman sums. Experiment. Math. 9 (2000), no. 4, 571--581. https://projecteuclid.org/euclid.em/1045759522


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