Let $f$ be a primitive form of an even weight $k$ and level $N\geq 1.$ By the theory of Hecke operators, $$\lambda_f(n)=\frac{\hat{f}(n)}{n^{\frac{k-1}{2}}}$$ is a real number. Studying the distribution of signs of Hecke eigenvalues has become an active area of research in recent past. Many authors investigated the problem of sign changes of the sequence $(\lambda_f(n))$ like Landau, Murty and Kohnen....

My question is: What's the reason of studying the sign changes problem, I mean are there any algebraic or geometric purposes behind this study?

Many thanks