Hecke Relations on Fourier Coeficients for GL(n), n>2

Question

Let $\pi$ be a cuspidal automorphic representation of $GL_2(\mathbb{A}_\mathbb{Q})$ with trivial central character. Then we can attach an L-function to $\pi$ which can be written as a Dirichlet series

$ L(s,\pi) = \sum_{m\in\mathbb{N}} \frac{\lambda_\pi(m)}{m^s}. $

Using Hecke operators one can prove the following relation on the coefficients

$ \lambda_\pi(m)\lambda_\pi(n) = \sum_{a|(m,n)} \lambda_\pi(a^{-2}mn). $

From this one can, in particular, derive the relation $\lambda_\pi(p)^2-\lambda_\pi(p^2) = 1$, where $p$ is prime and $L$ has been properly normalized. I need a similar formula for $\pi$ a representation of $GL(4)$ instead of $GL(2)$. Has this been calculated, either in the adelic setting, or for classical modular forms on $GL_4(\mathbb{R})$? Thanks!

Answer 1

Have you checked Sections 9.3 and 9.4 in Goldfeld: Automorphic forms and L-functions for the group GL(n,R)? It might have what you need.

Note also that the Euler product translates into the Hecke relations, even over GL(n). Here it is good to know that at an unramified prime $p$ the Euler factor is of the form $H_p(p^{-s})^{-1}$, where $H_p(x)$ is a polynomial of degree $n$ and constant term $1$ (while for a ramified prime $p$ it is of degree less than $n$).

You might also find useful Lemma 5.2 in Qu: Linnik-type problems for automorphic L-functions (Journal of Number Theory 130 (2010), 786-802), which is a variant of the Newton-Girard formulae for power sums. Lemma 5.3 is a nice consequence, which might also serve you well.