  We derive an explicit formula for the trace of an arbitrary Hecke operator on
spaces of twist-minimal holomorphic cusp forms with arbitrary level and
character, and weight at least 2. We show that this formula provides an
efficient way of computing basis elements for newform or cusp form spaces. This
work was motivated by the development of a twist-minimal trace formula in the
non-holomorphic case by Booker, Lee and Str\"ombergsson, as well as the
presentation of a fully generalised trace formula for the holomorphic case by
Cohen and Str\"omberg.

Number Theory