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Does any expert here know something about the summation formula of the Voronoi type for the sum $$\sum_{n\le X} a_{f}(n)\chi(n) e\left(\frac{an}{c}\right)?$$

Here $f$ is a newform of level $N$, $\chi$ is a primitive Dirichlet character modulo $N$, and $a,c$ are co-prime integers satisfying $(c,N)=1$.

Much obliged if anyone could give some comments or guide the references. Thanks in advance.

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    $\begingroup$ Try Appendix A of "Rankin-Selberg $L$-functions in the level aspect" by Kowalski, Michel, and Vanderkam. (Note that $a_f(n) \chi(n)$ is basically equal to $a_{f \otimes \chi}(n)$.) $\endgroup$
    – Peter Humphries
    Commented Jul 18, 2018 at 7:31
  • $\begingroup$ @ Peter Humphries Thanks for comments. $\endgroup$
    – FeiHou
    Commented Jul 18, 2018 at 8:51

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