A $2k\times 2k$ circulant matrix $\ C$ takes the form

\begin{align} C= \begin{bmatrix} c_0 & c_{2k-1} & \dots & c_{2} & c_{1} \\ c_{1} & c_0 & c_{2k-1} & & c_{2} \\ \vdots & c_{1}& c_0 & \ddots & \vdots \\ c_{2k-2} & & \ddots & \ddots & c_{2k-1} \\ c_{2k-1} & c_{2k-2} & \dots & c_{1} & c_0 \\ \end{bmatrix}. \end{align}

Now we consider a $2k\times 2k$ generalized circulant matrix $\ C_{1}$

\begin{align} C_{1}= \begin{bmatrix} c_0 & -c_{2k-1} & \dots & c_{2} & -c_{1} \\ c_{1} & -c_0 & c_{2k-1} & & -c_{2} \\ \vdots & -c_{1}& c_0 & \ddots & \vdots \\ c_{2k-2} & & \ddots & \ddots & -c_{2k-1} \\ c_{2k-1} & -c_{2k-2} & \dots & c_{1} & -c_0 \\ \end{bmatrix}. \end{align}

That is the first column shifts with a multiplication of $-1$.

Then what are the eigenvalues of $\ C_{1}$? Thank you!