  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.
