Hecke Eigenforms as Products of Eigenforms

TL;DR

This paper classifies when the product of two normalized Hecke eigenforms for (N) is itself an eigenform, providing a complete list of 61 such cases, extending previous results for special cases.

Contribution

It offers a complete classification of all instances where the product of two normalized eigenforms is again an eigenform for (N), including a new, elementary proof approach.

Findings

Identified exactly 61 cases where the product of two eigenforms is an eigenform.

Extended previous results to more general levels and forms.

Provided an elementary, effective proof method without Rankin-Selberg convolutions.

Abstract

We investigate when the product of two Hecke eigenforms for {\Gamma}_1(N) is again a Hecke eigenform. In this paper we prove that the product of two normalized eigenforms for {\Gamma}_1(N), of weight greater than 1, is an eigenform only 61 times, and give a complete list. Duke [Duk99] and Ghate [Gha00] independently proved that with eigenforms for SL_2(Z), there are only 16 such product identities. Ghate [Gha02] also proved related results for {\Gamma}_1(N), with N square free. Emmons [Emm05] proved results for eigenforms away from the level, for prime level. The methods we use are elementary and effective, and do not rely on the Rankin-Selberg convolution method used by both Duke and Ghate.