In certain rank-1 groups of Lie type, the conductor c(χ) of an irreducible character equals c(χ(g)) for some single element g, and in quasisimple cases the field Q(χ) is generated by one value.
Steinberg , The representations of (3,q) , (4,q) , (3,q) , _4(q) , Canadian J
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Character values and conductors of low-rank groups of Lie type
In certain rank-1 groups of Lie type, the conductor c(χ) of an irreducible character equals c(χ(g)) for some single element g, and in quasisimple cases the field Q(χ) is generated by one value.