The spatiotemporal mobility patterns and next location prediction of fake base stations (FBS) provide important technical support for the police to prevent spam messages from FBS. However, due to the difficulty in locating their real-time locations, our understanding of the mobility patterns and predictability of FBS is still limited. Based on the crowdsourced spam data, we extract the time and potential locations of FBS and propose a Tucker-MMC method that combines Tucker decomposition with a Mobility Markov Chain (MMC) model to investigate the mobility patterns and predictability of FBS sending spam messages. First, we utilize Tucker decomposition to reflect the spatial and temporal preferences during the movement of the corresponding FBS. Then the mobility regularity and the theoretical maximum predictability of the FBS trajectories with similar mobility preferences are analyzed by entropy and Fano's inequality. A Tucker-MMC is also established for the next location prediction. The results using the spam dataset in Beijing show that the accuracy of Tucker-MMC is more than double that of the MMC. The accuracy of the actual location prediction model is more likely to approach the theoretical maximum predictability when FBS send spam messages in a shorter time, shorter transfer distance, and smaller access range.