Scanning tunneling microscopy is used to image the additional quasi-particle states generated by quantized vortices in the high critical temperature superconductor Bi2Sr2CaCu2O8+delta. They exhibit a copper-oxygen bond-oriented "checkerboard" pattern, with four unit cell (4a0) periodicity and a approximately 30 angstrom decay length. These electronic modulations may be related to the magnetic field-induced, 8a0 periodic, spin density modulations with decay length of approximately 70 angstroms recently discovered in La1.84Sr0.16CuO4. The proposed explanation is a spin density wave localized surrounding each vortex core. General theoretical principles predict that, in the cuprates, a localized spin modulation of wavelength lambda should be associated with a corresponding electronic modulation of wavelength lambda/2, in good agreement with our observations.