Let $$ L(s)=\sum_{n=1}^{\infty}a(n)n^{-s} $$ be a modular $L$-function of conductor $N$, and let
$$ F_d(s)=\sum_{n=1}^{\infty} \frac{a(d_0 n^2)}{(d_0 n^2)^s} \prod_{p|4Nnd}\left(1+\frac{1}{p}\right)^{-1}, $$ where $d=d_0d_1^2$, $d_0$ is squarefree. In p.460 of 'Murty, M. Ram ; Murty, V. Kumar, Mean values of derivatives of modular L -series. Ann. of Math. (2) 133 (1991), no. 3, 447–475.', the author says that 'by factoring $F_d(s)$ as an Euler product and using simple estimates, we find that $$ |F_d(s)| \ll c^{v(d)}d_0^{1/2-\sigma}|L(2s,\mathrm{Sym}^2)\zeta(4s-2)^{-1}|, $$ for $\sigma>3/4$. Here $c$ is an absolute positive constant and $v(d)$ is the number of prime factors of $d$.'
Howerver, I have no idea how to get this estimate.