analysis - Complex equation simplification

Let $k$ be a positive integer and $c_0$ a positive constant. Consider the following expression:
\begin{equation} \left(2 i c_0-i+\sqrt{3}\right)^2 \left(-2 c_0+i \sqrt{3}+1\right)^k+\left(-2 i c_0+i+\sqrt{3}\right)^2 \left(-2 c_0-i \sqrt{3}+1\right)^k
\end{equation}
I would like to find a simple expression for the above in which only real numbers appear. It is clear that the above expression is always a real number since

\begin{equation}
\overline{\left(2 i c_0-i+\sqrt{3}\right)^2 \left(-2 c_0+i \sqrt{3}+1\right)^k}= \left(-2 i c_0+i+\sqrt{3}\right)^2 \left(-2 c_0-i \sqrt{3}+1\right)^k.
\end{equation}
But I am not able to simplify it. I am pretty sure I once saw how to do this in a complex analysis course but I cannot recall the necessary tools. Help is much appreciated.