  The electronic states of lateral many electron quantum dots in high magnetic
fields are analyzed in terms of energy and spin. In a regime with two Landau
levels in the dot, several Coulomb blockade peaks are measured. A zig-zag
pattern is found as it is known from the Fock-Darwin spectrum. However, only
data from Landau level 0 show the typical spin-induced bimodality, whereas
features from Landau level 1 cannot be explained with the Fock-Darwin picture.
Instead, by including the interaction effects within spin-density-functional
theory a good agreement between experiment and theory is obtained. The absence
of bimodality on Landau level 1 is found to be due to strong spin polarization.
