Products of Hecke eigenforms

Let f(z) and g(z) be Hecke eigenforms for Γ0(p), where p is a prime. If both f(z) and g(z) are non-cuspidal forms and p⩾7, then the product is a Hecke eigenform only if it comes trivially from a level 1 solution. If g(z) is a cuspform and p⩾5, then in addition to the level 1 solutions, there are 8 new cases where the product of Hecke eigenforms is a Hecke eigenform.