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.



Discussion for the sign of a specific sum
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 $S.$


