The sign changes of Fourier coefficients of Eisenstein series

Abstract

In this paper we prove a number of theorems that determine the extent to which the signs of the Hecke eigenvalues of an Eisenstein newform determine the newform. We address this problem broadly and provide theorems of both individual and statistical nature. Many of these results are Eisenstein series analogs of well-known theorems for cusp forms. For instance, we determine how often the pth Fourier coefficients of an Eisenstein newform begin with a fixed sequence of signs $$\varepsilon _p = \{\pm 1, 0\}$$ . Moreover, we prove the following variant of the strong multiplicity-one theorem: an Eisenstein newform is uniquely determined by the signs of its Hecke eigenvalues with respect to any set of primes with density greater than $$1/2$$ .