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Let $f$ be an automorphism of a complex Enriques surface $Y$ and let $p_f$denote the characteristic polynomial of the isometry $f^*$ of the numericalN\'eron-Severi lattice of $Y$ induced by $f$. We apply a modification ofMcMullen's method to prove that the modulo-$2$ reduction $(p_f(x) \bmod 2)$ isa product of modulo-$2$ reductions of (some of) the five cyclotomic polynomials$\Phi_m$, where $m \leq 9$ and $m$ is odd. We study Enriques surfaces thatrealize modulo-$2$ reductions of $\Phi_7$, $\Phi_9$ and show that each of thefive polynomials $(\Phi_m(x) \bmod 2)$ is a factor of the modulo-$2$ reduction$(p_f(x) \bmod 2)$ for a complex Enriques surface.

On characteristic polynomials of automorphisms of Enriques surfaces