We evaluate the neutron electric dipole
moment (nEDM) on the lattice using configurations produced with
$N_f=2+1+1$ twisted mass fermions at lattice spacing of
$a=0.082\,\mathrm{fm}$ and a light quark mass that corresponds to
$M_{\pi}=370 \, \mathrm{MeV}$. We do so by extracting the CP-odd form
factor $F_3$ at the limit of zero momentum transfer. This limit is taken
using a parametrization of the momentum dependence by a dipole fit as
well as using the position space methods of "Application of the
derivative to the ratio technique" and "elimination of the momentum in
the plateau region technique". The evaluation of $F_3$ requires the
calculation of the topological charge. We measure the topological charge
via cooling and the gradient flow using the Wilson-Symanzik tree-level
improved and Iwasaki actions for smoothing. We obtain consistent results
for all choices of smoothing actions, smoothing procedures and momentum
dependence treating techniques. We report an nEDM in units of the
$\theta$ vacuum angle of $|d_n|/ \theta = −0.045(6)(1) e \cdot
\mathrm{fm}$.