Hecke eigenforms as products of eigenforms

We investigate when the product of two Hecke eigenforms for Γ1(N) is again an eigenform. In this paper we prove that among all levels N, the product of two eigenforms for Γ1(N) of weight 2 or greater is an eigenform only 61 times, and give a complete list. Duke and Ghate independently addressed this topic for eigenforms at level 1, proving there are only 16 such identities. Ghate subsequently proved related results for almost everywhere eigenforms of weight 3 or greater for Γ1(N), with N squarefree. Our work extends the results of Ghate to eigenforms of weight 2 or greater, with no restrictions on the level. The methods we use are elementary and effective, and make no use of the Rankin–Selberg convolution.