Identities involving the coefficients of automorphic $L$-functions

Let $f(z)$ be a holomorphic Hecke eigenform of weight $k$ with respect to $SL(2,\mathbb Z)$ and let
$$ L(s,\operatorname{sym}^2f)=\sum\limits^{\infty}_{n=1}c_n n^{-s},\quad \operatorname{Re}s>1, $$
denote the symmetric square $L$-function of $f$. A Voronoi type formula for
$$ C(x)=\sum\limits_{n\leqslant x}c_n. $$
and the relation
$$ C(x)=\Omega_{\pm}(x^{1/3}). $$
are proved. Heuristic approaches to estimation of exponential sums arising in this connection are considered.