We provide an easy method for the construction of characteristic polynomials of simple ordinary abelian varieties ${{\mathcal A}}$ of dimension g over a finite field ${{\mathbb F}}_q$ , when $q\ge 4$ and $2g=\rho ^{b-1}(\rho -1)$ , for some prime $\rho \ge 5$ with $b\ge 1$ . Moreover, we show that ${{\mathcal A}}$ is absolutely simple if $b=1$ and g is prime, but ${{\mathcal A}}$ is not absolutely simple for any prime $\rho \ge 5$ with $b>1$ .