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On arbitrary products of eigenforms

Arvind Kumar, Jaban Meher (2016)

Acta Arithmetica

We characterize all the cases in which products of arbitrary numbers of nearly holomorphic eigenforms and products of arbitrary numbers of quasimodular eigenforms for the full modular group SL₂(ℤ) are again eigenforms.

On higher moments of Hecke eigenvalues attached to cusp forms

Guodong Hua (2022)

Czechoslovak Mathematical Journal

Let f , g and h be three distinct primitive holomorphic cusp forms of even integral weights k 1 , k 2 and k 3 for the full modular group Γ = SL ( 2 , ) , respectively, and let λ f ( n ) , λ g ( n ) and λ h ( n ) denote the n th normalized Fourier coefficients of f , g and h , respectively. We consider the cancellations of sums related to arithmetic functions λ g ( n ) , λ h ( n ) twisted by λ f ( n ) and establish the following results: n x λ f ( n ) λ g ( n ) i λ h ( n ) j f , g , h , ε x 1 - 1 / 2 i + j + ε for any ε > 0 , where 1 i 2 , j 5 are any fixed positive integers.

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