If $\alpha$ is a non-real root of $x^7=1$,then $\alpha(1+\alpha)(1+\alpha^2+\alpha^4) = $

The real part of $\frac{(\cos a+i \sin a)^6}{(\sin b+i \cos b)^8}$ is

If $1, \omega, \omega^2$ are the cube roots of unity,$k$ is a positive integer and $(1-\omega+\omega^2)^{3k} + (1-\omega^2+\omega)^{3k} = (1-\omega+\omega^2)^{3k+1} + (1+\omega-\omega^2)^{3k+1}$,then $k=$

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2x+4=0$,then $\alpha^9+\beta^9$ is equal to

If $\alpha$ and $\beta$ are the roots of the equation $x^2 + x + 1 = 0$, then $\alpha^{2026} + \beta^{2026} = $

Let $\omega \neq 1$ be a cube root of unity. Then the minimum value of the set $\{|a + b\omega + c\omega^2|^2 : a, b, c \text{ are distinct non-zero integers}\}$ is equal to: