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Using 20.3 fb^{-1} of e^{+}e^{-} collision data taken at the center-of-mass energy of sqrt[s]=3.773 GeV, we report the first experimental constraint on the scalar current in the D^{0(+)}→K[over ¯]ℓ^{+}ν_{ℓ} transitions, based on a simultaneous fit to the first measured forward-backward asymmetries and precisely determined partial decay rates. The parameters of the scalar current are determined to be Re(c_{S}^{μ})=0.017±0.008_{stat}±0.006_{syst} and Im(c_{S}^{μ})=±(0.077±0.011_{stat}±0.009_{syst}), which deviates from the standard model by a significance of 2.3σ. The branching fractions of D^{0(+)}→K[over ¯]ℓ^{+}ν_{ℓ}, the hadronic form factor f_{+}(0), and the modulus of the c→s Cabibbo-Kobayahsi-Maskawa matrix element |V_{cs}| are also determined with improved precision. In addition, lepton flavor universality is tested via the ratios of the decay rates between semimuonic and semielectronic decays in the full momentum transfer range and in subranges.