  We present an algorithm to compute the Hecke operators on the equivariant
cohomology of an arithmetic subgroup $\Gamma$ of the general linear group
$\mathrm{GL}_n$. This includes $\mathrm{GL}_n$ over a number field or a
finite-dimensional division algebra. As coefficients, we may use any
finite-dimensional local coefficient system. Unlike earlier methods, the
algorithm works for the cohomology $H^i$ in all degrees $i$. It starts from the
well-rounded retract $\tilde{W}$, a $\Gamma$-invariant cell complex which
computes the cohomology. It extends $\tilde{W}$ to a new well-tempered complex
$\tilde{W}^+$ of one higher real dimension, using a real parameter called the
temperament. The algorithm has been coded up for $\mathrm{SL}_n(\mathbb{Z})$
for $n=2,3,4$; we present some results for congruence subgroups of
$\mathrm{SL}_3(\mathbb{Z})$.
