Primitive central idempotents of a finite group over a field

We give a method to compute the primitive central idempotents of a semi-simple finite group algebra associated with an irreducible character $\chi$ such that for every $g\in G$, either $\chi (g)=0$ or all eigenvalues of $\rho (g)$ have the same order. We then show several consequences which highlight the significance and possible applications of our approach.