Sorry to disturb. Does any experts here know something upon the Voronoi type for the symmetric $L$-functions$$\sum_{n\le X} A_F(1,n)\left ( \frac{an}{c}\right)=?$$ Here $F$ is a symmetric-lift of a $GL_2$ cusp form of square-free level $N$ (i.e., $F=\text{sym}^2 f$), and $A_F(1,n)$ is the $n$-th coefficient of the symmetric $L$-function $L(s,\text{sym}^2f)$. The denominator $c$ is co-prime with $N$ (i.e., $(c,N)=1$).

I checked F. Zhou's paper "The Voronoi formula on $GL(3)$ with ramification" at https://arxiv.org/abs/1806.10786. But I am not sure if his definition of the $GL_3$ cusp form on $\Gamma_{N}$ covers my case.

Note that in the paper of Buttcane Jack and Rizwanur Khan's “$L^4$-norms of Hecke newforms of large level'', the authors said it seems challengeable to give a Voronoi type for the symmetric $L$-functions with $(c,N)=1$ as in the present case.

If any expert here leans something upon this type of Voronoi formula, please guide some reference or share some your very valuable comments here. Very grateful for your time.

Much obliged, and thanks in advance.

Sorry to disturb. Does any experts here know something upon the Voronoi type for the symmetric $L$-functions$$\sum_{n\le X} A_F(1,n)e\left ( \frac{an}{c}\right)=?$$ Here $F$ is a symmetric-lift of a $GL_2$ cusp form of square-free level $N$ (i.e., $F=\text{sym}^2 f$), and $A_F(1,n)$ is the $n$-th coefficient of the symmetric $L$-function $L(s,\text{sym}^2f)$. The denominator $c$ is co-prime with $N$ (i.e., $(c,N)=1$).

I checked F. Zhou's paper "The Voronoi formula on $GL(3)$ with ramification" at https://arxiv.org/abs/1806.10786. But I am not sure if his definition of the $GL_3$ cusp form on $\Gamma_{N}$ covers my case.

Note that in the paper of Buttcane Jack and Rizwanur Khan's “$L^4$-norms of Hecke newforms of large level'', the authors said it seems challengeable to give a Voronoi type for the symmetric $L$-functions with $(c,N)=1$ as in the present case.

If any expert here leans something upon this type of Voronoi formula, please guide some reference or share some your very valuable comments here. Very grateful for your time.

Much obliged, and thanks in advance.