Sorry if my question is stupid but it comes to my mind whenever I read about the theory of symmetric power $L$ functions. Out of curiosity, I did a web search and found only the explicit expression of these coefficients in the case of modular forms (established by Emmanuel Royer and Jie Wu) and for the case of functions of symmetric square $L$-functions (established by Lau, Liu and Wu).
What is the interest of finding the explicit expression of the Fourier coefficients of symmetric powers $L$-functions $\lambda_{Sym^rf}(n)$ and what problems they may be used? Thank you.
