  Let $G(q)$ be a Chevalley group over a finite field $F_q$. By Lusztig's and
Shoji's work, the problem of computing the values of the unipotent characters
of $G(q)$ is solved, in principle, by the theory of character sheaves; one
issue in this solution is the determination of certain scalars relating two
types of class functions on $G(q)$. We show that this issue can be reduced to
the case where $q$ is a prime, which opens the way to use computer algebra
methods. Here, and in a sequel to this article, we use this approach to solve a
number of cases in groups of exceptional type which seemed hitherto out of
reach.
