If $\omega_0, \omega_1, \ldots, \omega_{n-1}$ are the $n$-th roots of unity,then $(1+2 \omega_0)(1+2 \omega_1)(1+2 \omega_2) \ldots (1+2 \omega_{n-1})=$

If $\omega$ is the complex cube root of unity and $\left(\frac{a+b \omega+c \omega^2}{c+a \omega+b \omega^2}\right)^k+\left(\frac{a+b \omega+c \omega^2}{b+a \omega^2+c \omega}\right)^l=2$,then $2k+l$ is always

If $\sin \theta + 2\sin \phi + 3\sin \psi = 0$ and $\cos \theta + 2\cos \phi + 3\cos \psi = 0$,then the value of $\cos 3\theta + 8\cos 3\phi + 27\cos 3\psi = $

If $z^2+z+1=0$,then find the value of $\left(z^3+\frac{1}{z^3}\right)^2+\left(z^4+\frac{1}{z^4}\right)^2$,where $z$ is a complex cube root of unity.

If $\omega$ is a non-real cube root of unity,then the value of $\cos \left[ \left\{ (1-\omega)(1-\omega^2) + (2-\omega)(2-\omega^2) + \dots + (2017-\omega)(2017-\omega^2) \right\} \cdot \frac{\pi}{2017} \right]$ is -

The two numbers such that each one is the square of the other are: