  The character formulas of Sally and Shalika are an early triumph in $p$-adic
harmonic analysis, but, to date, the calculations underlying the formulas have
not been available. In this paper, which should be viewed as a precursor of the
forthcoming volume by the authors and Alan Roche, we leverage modern technology
(for example, the Moy-Prasad theory) to compute explicit character tables. An
interesting highlight is the computation of the 'exceptional' supercuspidal
characters, i.e., those depth-zero representations not arising by
inflation-induction from a Deligne-Lusztig representation of finite
$\operatorname{SL}_2$; this provides a concrete application for the recent work
of DeBacker and Kazhdan.
