  We extract the neutron electric dipole moment (nEDM) $ \vert \vec{d}_n \vert
$ on configurations produced with $N_f=2+1+1$ twisted mass fermions with
lattice spacing of $a \simeq 0.082$ fm and a light quark mass that corresponds
to $M_{\pi} \simeq 373$ MeV. We do so by evaluating the $CP$-odd form factor
$F_3$ for small values of the $CP$-violation parameter $\theta$ in the limit of
zero momentum transfer. This limit is extracted using the usual parametrization
but in addition position space methods. The topological charge is computed via
cooling and gradient flow using the Wilson, Symanzik tree-level improved and
Iwasaki actions for smoothing. We obtain consistent results for all choices of
smoothing procedures and methods to extract $F_3$ at zero momentum transfer.
For the ensemble analyzed we find a value of nEDM of $\vert \vec{d}_n \vert /
\theta = -0.045(6)(1) {\rm e} \cdot {\rm fm}$.
