A Test for Identifying Fourier Coefficients of Automorphic Forms and Application to Kloosterman Sums

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 ±S(l, 1; p)/√p are not the coefficients of a Maass form with small level and eigenvalue. Source code and thecalculated Kloosterman sums are available electron ically.