Eigenvalues · Superchallenge problem

Let $C$ be the $6\times 6$ cyclic shift matrix ($C\mathbf e_k = \mathbf e_{k+1}$ for $k < 6$ and $C\mathbf e_6 = \mathbf e_1$), and let $A = I + C + C^2$. Find the sum of the absolute values of the six eigenvalues of $A$ (counted with multiplicity).