Summing Hecke eigenvalues over polynomials

In this paper we estimate sums of the form $$\sum _{n\le X}|a_{{\text {Sym}}^m \pi }(|f(n)|)|$$ , for symmetric power lifts automorphic representations $$\pi $$ attached to holomorphic forms and polynomials $$f(x)\in {\mathbb {Z}}[x]$$ arbitrary degree. We give new upper bounds these under certain natural assumptions on f. Our results are unconditional when $$\deg (f)\le 4$$ . Moreover, study analogous sum over in several variables. obtain an all cubic two variables that define elliptic curves.