  We investigate whether it is possible to extract the quark mass anomalous
dimension and its scale dependence from the spectrum of the twisted mass Dirac
operator in Lattice QCD. The answer to this question appears to be positive,
provided that one goes to large enough eigenvalues, sufficiently above the
non-perturbative regime. The obtained results are compared to continuum
perturbation theory. By analyzing possible sources of systematic effects, we
find the domain of applicability of the approach, extending from an energy
scale of around 1.5 to 4 GeV. The lower limit is dictated by physics
(non-perturbative effects at low energies), while the upper bound is set by the
ultraviolet cut-off of present-day lattice simulations. The information about
the scale dependence of the anomalous dimension allows also to extract the
value of the $\Lambda_{\overline{\textrm{MS}}}$-parameter of 2-flavour QCD,
yielding the value $303(13)(25)$ MeV, where the first error is statistical and
the second one systematic. We use gauge field configuration ensembles generated
by the European Twisted Mass Collaboration (ETMC) with 2 flavours of dynamical
twisted mass quarks, at 4 lattice spacings in the range between around 0.04 and
0.08 fm.
