# Discussion for the sign of a specific sum

Given a modular form $f$ of an even weight $k$ for the full modular group. Let $\lambda_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ For a fixed positive integers $a$ and $b,$ I want to discuss the sign of this sum $$S=\frac{\lambda_f(a)\lambda_f(b)}{(ab)^{3/4}} \sum_{\substack{l=1\\\gcd(l,a)=\gcd(l,b)=1}}^{+\infty}\left(\frac{\lambda_f(l)}{l^{3/4}}\right)^3.$$ Is there a method to do this? Many thanks.

Using The Euler product, we can express the sum $$\sum_{\substack{l=1\\\gcd(l,a)=\gcd(l,b)=1}}^{+\infty}\left(\frac{\lambda_f(l)}{l^{3/4}}\right)^3$$ as an infinite product over prime numbers so that the sum $S.$