  Magnetically levitated superconducting microparticles offer a promising path
to quantum experiments with picogram to microgram objects. In this work, we
levitate a 700ng $\sim 10^{17}$amu superconducting microsphere in a magnetic
chip trap in which detection is integrated. We measure the particle's
center-of-mass motion using a DC-SQUID magnetometer. The trap frequencies are
continuously tunable between 30 and 160 Hz and the particle remains stably
trapped over days in a dilution refrigerator environment. We characterize
motional-amplitude-dependent frequency shifts, which arise from trap
anharmonicities, namely Duffing nonlinearities and mode couplings. We explain
this nonlinear behavior using finite element modelling of the chip-based trap
potential. This work constitutes a first step towards quantum experiments and
ultrasensitive inertial sensors with magnetically levitated superconducting
microparticles.
