Mean value theorems for automorphic $L$-functions

Let $f$ be a holomorphic Hecke eigencuspform of even weight $k\ge 12$ for $\mathrm{SL}(2, \mathbb{Z})$ and let $L(s,\mathrm{sym}^2f)$ be the symmetric square $L$-function of $f$. Let $C(x)$ be the summatory function of the coefficients of $L(s,\mathrm{sym}^2 f)$. The true order is found for
$$ \int_0^x C(y)^2\,dy. $$