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 ±S(1, 1; p)/\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.