Solve the following

Let $\omega$ be a complex number such that $2 \omega+1=z$ where $z=\sqrt{-3}$. If $\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & -\omega^{2}-1 & \omega^{2} \\ 1 & \omega^{2} & \omega^{7}\end{array}\right|=3 k$, then $\mathrm{k}$ is equal to :-