New Ramanujan-type congruences for overpartitions modulo $11$ and $13$

In this paper, we establish two new Ramanujan-type congruences for the overpartition function: $\bar{p}(11\times (8n+5))\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}11)$ and $\bar{p}(13\times 2^6(8n+7))\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}13)$. The proofs rely on the theory of modular forms. We conjecture potential Ramanujan-type congruences for overpartitions modulo 7, 17, 19 and 23.