Character values and conductors of low-rank groups of Lie type

Let $\chi$ be a complex irreducible character of a finite group $G$. The conductor of $\chi$, denoted $c(\chi )$, is the smallest positive integer $n$ such that $\chi (x)\in \mathbb{Q}(\exp (2\pi i\mathrm{/}n))$ for all $x\in G$. We show that for certain rank $1$ finite groups of Lie type, the conductor $c(\chi )$ is realized at a single group element; that is, there exists $g\in G$ such that $c(\chi )=c(\chi (g))$. In some quasisimple cases, we further prove that the field of values $\mathbb{Q}(\chi )$ is generated by a single value. This phenomenon, which is related to a well-known conjecture of W.~Feit, was recently observed by Boltje \emph{et al.} in their reduction of the conjecture to finite simple groups. Our approach uses techniques from algebraic number theory together with the known character tables of these groups.