  We compute the non-Eisenstein systems of Hecke eigenvalues contributing to
the $p$-arithmetic homology of irreducible smooth mod $p$ representations $\pi$
of $\mathrm{GL}_2(\mathbb{Q}_p)$ and to the cohomology of their duals. We show
that in most cases they are associated to odd irreducible 2-dimensional Galois
representations whose local component at $p$ corresponds under the mod $p$
local Langlands correspondence to a smooth representation that contains $\pi$
as a subrepresentation.
