mjhjmtu

Let $L(s)$ be an automorphic L-function (attached to a self contragredient automorphic representation on $GL(3)$), according to the following notations for $s$ of sufficiently large real part:
$$L(s) = sum_k=0^infty fraca_nn^s = prod_p left( 1 - alpha(p)p^-s right)^-1 left( 1 - beta(p)p^-s right)^-1 left( 1 - gamma(p)p^-s right)^-1$$

Straightforwardly developing the Euler product provides expressions of the Fourier coefficients $a_n$'s in terms of the Satake parameters $alpha(p)$, $beta(p)$ and $gamma(p)$. I am not particularly aware of others standard useful relations between them. I bumped into the following one:
$$a_p^k = frac
left
left$$

I guess this can be verified, but even the case $k=1$ seems obscure to me. I do not want to believe such a formula to be a (verifiable) accident. Despite it works computationally, am I missing something lying behind? How strongly is the self-contragredience assumption necessary?

Any insight is welcome, as well as other ways to embrace the relations between spectral parameters and coefficients.

There is no coincidence, this is the Weyl character formula for the representation $operatornameSym^k$ of $GL_3$. The reason that the Langlands dual group comes up is, unsurprisingly, the Satake isomorphism.

The general statement is: For an automorphic representation associated to a group $G$ with dual group $hatG$, the coefficient of $p^k$ in the $L$-function associated to a representation $rho$ of $hatG$ is equal to the trace of the Satake parameter (a conjugacy class on $hatG$) acting on $Sym^k rho$.

No assumption beyond unramifiedness should be necessary.