  To every $h + \mathbb{N}$-graded module $M$ over an $\mathbb{N}$-graded
conformal vertex algebra $V$, we associate an increasing filtration $(G^pM)_{p
\in \mathbb{Z}}$ which is compatible with the filtrations introduced by
Haisheng Li. The associated graded vector space $\mathrm{gr}^G(M)$ is naturally
a module over the vertex Poisson algebra $\mathrm{gr}^G(V)$. We study
$\mathrm{gr}^G(M)$ for the three irreducible modules of the Ising model
$\mathrm{Vir}_{3, 4}$, namely $\mathrm{Vir}_{3,4} = L(1/2, 0)$, $L(1/2, 1/2)$
and $L(1/2, 1/16)$. We obtain an explicit monomial basis of each of these
modules and a formula for their refined characters which are related to Nahm
sums for the matrix $\left(\begin{smallmatrix} 8 & 3 \\ 3 & 2
\end{smallmatrix}\right)$.
