PyCox: computing with (finite) Coxeter groups and Iwahori-Hecke algebras. We introduce the computer algebra package PyCox, written entirely in the Python language. It implements a set of algorithms, in a spirit similar to the older 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 ⩽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.