The exact ion-electron and electron-electron interaction energies in quantum dots (QDs) with parabolic potentials are calculated and studied in detail. On the basis of calculated results, one- and two-electron spectra in QDs are revealed. It is found that the spectra are dramatically changed with the variation of the size and field. The even-parity-odd-parity and spin-singlet-spin-triplet oscillations with magnetic field are clearly shown for one- and two-electron ground states in QDs. The variations of the spectra with field are quite different for different dot sizes, and represent magnetic 'fingerprints' of the QDs. These phenomena are induced by the negative-ion-electron and electron-electron interactions in QDs.