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Hafner and Stopple proved a conjecture of Zagier that the inverse Mellin transform of the symmetric square [Formula: see text]-function associated to the Ramanujan tau function has an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function [Formula: see text]. Later, Chakraborty et al. extended this phenomenon for any Hecke eigenform over the full modular group. In this paper, we study an asymptotic expansion of the Lambert series [Formula: see text] where [Formula: see text] is the [Formula: see text]th Fourier coefficient of a Hecke eigenform [Formula: see text] of weight [Formula: see text] over the full modular group.

Hafner and Stopple proved a conjecture of Zagier, that the inverse Mellin transform of the symmetric square L-function associated to the Ramanujan tau function has an asymptotic expansion in terms of the non-trivial zeros of the Riemann zeta function ζ(s). Later, Chakraborty, Kanemitsu and the second author extended this phenomenon for any Hecke eigenform over the full modular group. In this paper, we study an asymptotic expansion of the Lambert serieswhere λ f (n) is the nth Fourier coefficient of a Hecke eigen form f (z) of weight k over the full modular group.

Recently, Juyal, Maji and Sathyanarayana have studied a Lambert series associated with a cusp form over the full modular group and the Möbius function. In this paper, we investigate the Lambert series, where a f (n) is the nth Fourier coefficient of a cusp form f over any congruence subgroup, and ψ and ψ ′ are primitive Dirichlet characters. This extends the earlier work to the case of higher level subgroups and also gives a character analogue.