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 $|d_n|/ \theta = −0.045(6)(1) e \cdot \mathrm{fm}$.