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 $sin^2theta_W$ with about 2% precision. We also discuss the impact of combining neutrino--electron scattering data with neutrino trident production at DUNE-PRISM.
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