Gaussian periods for seventh roots of unity ($\alpha ,\beta =(-1\pm i\sqrt{7})/2$)

With $\zeta =e^{2\pi i/7}$, put $\alpha =\zeta +{\zeta }^2+{\zeta }^4$ and $\beta ={\zeta }^3+{\zeta }^5+{\zeta }^6$. These Gaussian periods obey $\beta =\overline{\alpha }$, $\alpha +\beta =-1$, and ${\alpha }^2=\alpha +2\beta$. Consequently ${\alpha }^2+\alpha +2=0$. The positive imaginary part of $\alpha$ fixes $\alpha =(-1+i\sqrt{7})/2$ and $\beta =(-1-i\sqrt{7})/2$. They are the nonidentity kernel values of the two degree-three irreducible characters of the nonabelian group of order 21.