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We are developing a positron emission tomography and field-cycled magnetic resonance imaging (PET/FCMRI) dual modality system. The low magnetic field strength of a FCMRI system makes it possible to implement a PET system based upon novel photomultiplier tubes (PMTs), instead of solid-state photodetectors. In this work, the performance of a fine-mesh PMT (Hamamatsu H6153-70) has been investigated. In the static magnetic field, the response of gain, energy resolution, time resolution and efficiency of the mesh PMT, as well as its angular response and positioning uniformity were studied as a function of field strength (from zero up to 24 mT). In the dynamic magnetic field, the short-term recovery behavior and the long-term performance stability of the mesh PMT were studied under three dB/dt configurations (7.3, 11.0 and 14.5 T/s). The relevance of these results to the possible integration of PET with FCMRI systems is discussed.