Products of nearly holomorphic eigenforms

Abstract

We prove that the product of two nearly holomorphic Hecke eigenforms is again a Hecke eigenform for only finitely many choices of factors.

Some Remarks on Rankin–cohen Brackets of Eigenforms

“…By comparing the constant coefficients of both sides of the equality given in Proposition 2.3 of [1], we get similar identies for the operator D. We now state two results which follow the same way as was done in [1].…”

“…A quasimodular form is said to be an eigenform if it is an eigenvector for all of the Hecke operators T n for n ∈ N. It is known that the differential operator D = 1 2πi d dz takes M k to M k+2 . We have the following proposition which follows by a similar argument as done in Proposition 2.4 and 2.5 of [1].…”

“…They also proved that except for finitely many cases, the Rankin-Cohen brackets of two eigenforms is not an eigenform. Recently, Beyerl, James, Trentacoste, Xue [1] have proved that this phenomenon extends to a certain class of nearly holomorphic modular forms. More explicitly, they have proved that there is only one more case apart from the cases listed in [3] and [4] for which the product of two nearly holomorphic eigenforms of certain type is a nearly holomorphic eigenform.…”

“…Using the above propositions and following the method as in [1], we have a result analogous to Theorem 3.1 of [1].…”

Computing Power Series Expansions of Modular Forms

“…For an eigenform f ∈ M k (Γ ), we can further use the Hecke operators in conjunction with the Shimura-Maass derivatives. Proposition 3.8 (Beyerl, James, Trentacose, Xue [7]). Let f ∈ M k (Γ ).…”