  Using an effective field theory approach, we study coherent neutrino
scattering on nuclei, in the setup pertinent to the COHERENT experiment. We
include non-standard effects both in neutrino production and detection, with an
arbitrary flavor structure, with all leading Wilson coefficients simultaneously
present, and without assuming factorization in flux times cross section. A
concise description of the COHERENT event rate is obtained by introducing three
generalized weak charges, which can be associated (in a certain sense) to the
production and scattering of $\nu_e$, $\nu_\mu$ and $\bar{\nu}_\mu$ on the
nuclear target. Our results are presented in a convenient form that can be
trivially applied to specific New Physics scenarios. In particular, we find
that existing COHERENT measurements provide percent level constraints on two
combinations of Wilson coefficients. These constraints have a visible impact on
the global SMEFT fit, even in the constrained flavor-blind setup. The
improvement, which affects certain 4-fermion LLQQ operators, is significantly
more important in a flavor-general SMEFT. Our work shows that COHERENT data
should be included in electroweak precision studies from now on.
