Rankin-Cohen brackets of eigenforms and modular forms

Abstract

We use Maeda's Conjecture to prove that the Rankin-Cohen bracket of an eigenform and any modular form is only an eigenform when forced to be because of the dimensions of the underlying spaces. This occurs, for example, when the Rankin-Cohen bracket covers the entirety of Sn. We further determine when the Rankin-Cohen bracket of an eigenform and modular form is not forced to produce an eigenform and when it is determined by the injectivity of the operator itself. This can also be interpreted as using the Rankin-Cohen bracket operator of eigenforms to create evidence for Maeda's Conjecture.