I am currently studying "A Classical Introduction to Modern Number Theory", a book by Kenneth Ireland and Michael Rosen.
In that, we have the following two theorems:
$(1)$ The Law of Cubic Reciprocity: Let $\pi_1$ and $\pi_2$ be primary, $N\pi_1,N\pi_2 \neq 3,$ and $N\pi_1 \neq N\pi_2$. Then $\chi_{\pi_1}(\pi_2)=\chi_{\pi_2}(\pi_1)$, and
$(2)$ Supplement to the Cubic Reciprocity Law: Suppose that $N\pi \neq 3$. If $\pi=q$ is rational, write $q=3m-1$. If $\pi=a+b\omega$ is a primary complex prime, write $a=3m-1$. Then $\chi_{\pi}(1-\omega)=\omega^{2m}.$
These two theorems cover the reciprocity laws for all primary primes and all possible cases and combinations except one. How do we calculate $\chi_{(1-\omega)}(\pi)$ for a given primary prime $\pi$, such that $N\pi \neq 3$?