Are the Satake parameters of an automorphic representation pairwise distinct?

Question

I have a rather naive question whose answer could turn out to be pretty useful to me: given an automorphic representation $\pi$ of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$, are its Satake parameters at an unramified prime $p$ necessarily pairwise distinct?

Answer 1

In this question in contrary to the others you probably meant primitive L-function, ie. irreducible representation, otherwise $\zeta(s)L(s,\chi_4)$ is a counter-example.

Then look at https://www.lmfdb.org/L/ArtinRepresentation/2.23.3t2.b.a/#moreep

($59$ splits completely in the ring of integers of the splitting field of $x^3-x^2+1\in \Bbb{Q}[x]$)