  The planned DUNE experiment will have excellent sensitivity to the vector and
axial couplings of the electron to the $Z$-boson via precision measurements of
neutrino--electron scattering. We investigate the sensitivity of DUNE-PRISM, a
movable near detector in the direction perpendicular to the beam line, and find
that it will qualitatively impact our ability to constrain the weak couplings
of the electron. We translate these neutrino--electron scattering measurements
into a determination of the weak mixing angle at low scales and estimate that,
with seven years of data taking, the DUNE near-detector can be used to measure
$\sin^2\theta_W$ with about 2\% precision. We also discuss the impact of
combining neutrino--electron scattering data with neutrino trident production
at DUNE-PRISM.
