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Recently, Juyal, Maji, and Sathyanarayana have studied a Lambert seriesassociated with a cusp form over the full modular group and the M\"{o}biusfunction. In this paper, we investigate the Lambert series$\sum_{n=1}^{\infty}[a_f(n)\psi(n)*\mu(n)\psi'(n)]\exp(-ny),$ where $a_f(n)$ isthe $n$th Fourier coefficient of a cusp form $f$ over any congruence subgroup,and $\psi$ and $ \psi'$ are primitive Dirichlet characters. This extends theearlier work to the case of higher level subgroups and also gives a characteranalogue.

An asymptotic expansion for a twisted Lambert series associated to a cusp form and the M\"{o}bius function: level aspect