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: $$L(s) = \sum_{k=0}^\infty \frac{a_n}{n^s} = \prod_{p} \left( 1 - \alpha(p)p^{-s} \right)^{-1} \left( 1 - \beta(p)p^{-s} \right)^{-1} \left( 1 - \gamma(p)p^{-s} \right)^{-1}$$

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: $$a_{p^k} = \frac{ \left| \begin{array}{ccc} \alpha(p)^{k+2} & \beta(p)^{k+2} & \gamma(p)^{k+2} \\ \alpha(p) & \beta(p) & \gamma(p) \\ 1 & 1 & 1 \end{array} \right| }{ \left| \begin{array}{ccc} \alpha(p)^{2} & \beta(p)^{2} & \gamma(p)^{2} \\ \alpha(p) & \beta(p) & \gamma(p) \\ 1 & 1 & 1 \end{array} \right| }$$

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.