  We introduce the computer algebra package {\sf PyCox}, written entirely in
the {\sf Python} language. It implements a set of algorithms - in a spirit
similar to the older {\sf CHEVIE} system - for working with Coxeter groups and
Hecke algebras. This includes a new variation of the traditional algorithm for
computing Kazhdan--Lusztig cells and $W$-graphs, which works efficiently for
all finite groups of rank $\leq 8$ (except $E_8$). We also discuss the
computation of Lusztig's leading coefficients of character values and
distinguished involutions (which works for $E_8$ as well). Our experiments
suggest a re-definition of Lusztig's "special" representations which,
conjecturally, should also apply to the unequal parameter case.
