  The Grothendieck ring of Chow motives admits two natural opposite
$\lambda$-ring structures, one of which is a special structure allowing the
definition of Adams operations on the ring. In this work I present algorithms
which allow an effective simplification of expressions that involve both
$\lambda$-ring structures, as well as Adams operations. In particular, these
algorithms allow the symbolic simplification of algebraic expressions in the
sub-$\lambda$-ring of motives generated by a finite set of curves into
polynomial expressions in a small set of motivic generators. As a consequence,
the explicit computation of motives of some moduli spaces is performed,
allowing the computational verification of some conjectural formulas for these
spaces.
