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We prove a new upper bound for the dimension of the space of cohomologicalautomorphic 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 byMarshall. 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, anddeveloped by Paskunas.

Multiplicities of cohomological automorphic forms on $\mathrm{GL}_2$ and mod $p$ representations of $\mathrm{GL}_2(\mathbb{Q}_p)$