  A sufficiently high magnetic field applied to a superconductor will act on
both the charge and the spin of the individual electrons breaking up the Cooper
pairs. If the spin effect dominates, the superconducting state can develop a
texture before eventually entering the normal state with increasing magnetic
field. This spatially varying superconducting state is a periodic array of
magnetic walls separated by superconducting regions. Known as the
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, it was predicted in 1964. We
report heat capacity measurements on the heavy fermion superconductor, CeCoIn5,
which reveal a second phase transition within the superconducting state, clear
evidence of the FFLO phase. We also report magnetization measurements that
display a cascade of first order phase transitions within the FFLO region. Each
transition indicates an increase in orbital momentum of the superconducting
order parameter and corresponds to a specific Landau level vortex state
comprised of multiquanta vortices. The experimental realization of the FFLO
state provides a new opportunity to study the symbiosis of magnetism and
superconductivity, two states of matter once thought to be mutually exclusive.
