Idempotent Nilpotent

Some circulants that pop up in M Theory have very nice properties. For example, consider the idempotent operator $\frac{1}{3} C$ in If we were doing arithmetic modulo 3 this would look like the equation making the democratic matrix into a nilpotent operator. For a phase of $\delta = \pi$ the eigenvalues of $C$ are $(0, \sqrt{3}^{-1}, \sqrt{3}^{-1})$, which may be normalised to $(0,1,1)$. Note that, modulo 3, $C$ is the same as the modular operator $S$, which squares to unity and represents inversion in the unit circle. Modulo 2, the operator $C$ is the complement of the identity $S$.