the character table of $G({{\mathbb{F}}}_p)$ is explicitly known; namely, $F_4$, $E_6$ and $p=2$. For type $F_4$, our results complete earlier results of Marcelo–Shinoda [@MaSh]. See also [@GeHe] for the discussion of further examples, where the complete character table of $G({{\mathbb{F}}}_p)$ is not known.

We assume some familiarity with the character theory of finite groups of Lie type; see, e.g., [@C2], [@first]. The basic reference for the theory of character sheaves are Lusztig’s papers [@L2a]–[@L2e]. In Section \[sec1\], we review the classification of unipotent characters of $G(q)$ and the analogous classification of the unipotent character sheaves on $G$. These two classifications are known to be the same for $G(q)$ of split type (a fact which has only recently found a conceptual explanation; see Lusztig [@L11]). In Section \[sec2\], we can then formulate in precise terms the problem of equating class functions $R_x=\xi\chi_A$ as above, and establish Proposition \[p1\]. Finally, Sections \[sec3\], \[secF4\], \[secE6\] contain a number of examples where we show how the scalars $\xi$ can be determined using standard functions and algorithms in [GAP]{}.

[**Notation.**]{} \[abs11\] Let $\ell$ be a prime such that $\ell\nmid q$. If $\Gamma$ is a finite group, we denote by $\mbox{CF}
(\Gamma)$ the vector space of 