  We prove a new upper bound for the dimension of the space of cohomological
automorphic forms of fixed level and growing parallel weight on $\mathrm{GL}_2$
over a number field which is not totally real, improving the one obtained by
Marshall. The main tool of the proof is the mod $p$ representation theory of
$\mathrm{GL}_2(\mathbb{Q}_p)$ as started by Barthel-Livne and Breuil, and
developed by Paskunas.
