We report on the first Q^{2}-dependent measurement of the beam-normal single spin asymmetry A_{n} in the elastic scattering of 570 MeV vertically polarized electrons off ^{12}C. We cover the Q^{2} range between 0.02 and 0.05 GeV^{2}/c^{2} and determine A_{n} at four different Q^{2} values. The experimental results are compared to a theoretical calculation that relates A_{n} to the imaginary part of the two-photon exchange amplitude. The result emphasizes that the Q^{2} behavior of A_{n} given by the ratio of the Compton to charge form factors cannot be treated independently of the target nucleus.