has $13$ irreducible characters, which are labelled by certain pairs of partitions. By [@C2 p. 471], we have $$X(W)=\{x_\epsilon\mid \epsilon\in{{\operatorname{Irr}}}(W)\} \cup\{x_0\}.$$ By the explicit description of the Fourier matrices in [@C2 p. 472], we find that

$R_{x_0}=\frac{1}{2}(\rho_{(21,1)}-\rho_{(22,\varnothing)}-
\rho_{(2,11)}+ \rho_{x_0})$;

here, $(1,21)$, $(\varnothing,22)$, $(11,2)$ indicate irreducible characters $\epsilon\in{{\operatorname{Irr}}}(W)$ (as in [@C2 p. 449]). If $q$ is odd, then the values of $R_{x_0}$ are explicitly computed in [@GP1 Prop. 4.5] (and this provides an identification of $R_{x_0}$ with a characteristic function of a cuspidal character sheaf). Now assume that $q=2^f$ where $f{\geqslant}1$. Then, by [@L2d 22.3], there is a unique cuspidal character sheaf $A_0$ on $G$, and it is contained in $\hat{G}^{\text{un}}$. Again, by the explicit description in [@LuSp1 3.3], we have $A_0=A_{x_0}=\mbox{IC}(\overline{C},{{\mathcal{E}}})
[\dim C]$ where $C$ is the class of regular unipotent elements and ${{\mathcal{E}}}\not\cong \overline{{{\mathbb{Q}}}}_\ell$; we have $\dim G=28$ and $\dim C=24$. Let us fix $$g_1=x_a(1)x_b(1)x_c(1)x_d(1) \in C^F$$ where $\{a,b,c,d\}$ is a set of simple roots in the root system of type $D_4$. One checks that $g_1$ has order $8$ and that, if $a',b',c',d'$ is any permutation of $a,b,c,d$, then $g_1$ is conjugate in $G^F$ to $x_{a'}(1)
x_{b'}(1)x_{c'}(1)x_{d'}(1)$. Furthermore, $A(g_1)\cong {{\mathbb{Z}}}/2{{\mathbb{Z}}}$ is abelian and $F$ acts trivially on $A(g_1)$. Let 