Let $L$ be a CM field of degree $4$ over the rationals, and let $p$ be a prime number. If $q$ is a power of $p$, I would like to know if it is possible to characterize (in some way) all Weil ${\bf F}_q$-numbers inside $L$.

I was inspired by the corresponding question when $L$ has absolute degree $2$ (i.e., it is an imaginary quadratic field), which has a simple answer: Weil $q$-numbers $\pi$, up to roots of unity in $L$, correspond to principal ideals of norm $q$. The answer in this case is especially simple to get because $L$ has only one archimedean place.

How harder is the problem when $L$ is quartic? Thanks.