On cusp forms associated with binary theta series

Abstract

Let f ∈ Z[x, y] be a primitive positive binary quadratic form with fundamental discriminant and let S(f, z) := ∞ n=1 a(f, n)e(nz) be the associated cusp form, i.e., the projection of the theta series of f onto the subspace of cusp forms. For any real β > 0, the exact order of magnitude of the counting function n x |a(f, n)| 2β is given. For integral β > 0, a meromorphic continuation of |a(f, n)| 2β n −s to the half plane s > 0 is obtained. The number of sign changes of a(f, n) for n x is estimated.

On the Number of Sign Changes of Hecke Eigenvalues of Newforms

“…It is well known that there are infinitely many n ∈ N such that a(n) > 0 as well as infinitely many n with a(n) < 0. For an extension of this result and a discussion of related questions, see [8] (compare also [2] in connection with binary theta functions).…”