  We use the method of optimal distillation profiles to compute the low-lying
charmonium spectrum in an $N_f = 3+1$ ensemble at the $SU(3)$ light flavor
symmetric point ($m_{\pi} \approx 420$ MeV), physical charm quark mass and
lattice spacing $a\approx 0.0429$ fm. The spectrum and mass splittings display
good agreement with their values in nature and the statistical errors are
comparable, if not smaller, than those of state-of-the-art lattice
calculations. We also present first results on the mixing of charmonium with
glueballs and light hadrons obtained in a similar $N_f = 3+1$ ensemble but at
larger pion mass.
