I would like to point out that both Tiberius and Claudius expelled the Jews from Rome because of the troublesome nature of their newest sect— the followers of Christus. The followers of Christus were still considered Jews by Rome and other Jews at this point.
Caligula (after Clavdivs) began officiating about them specifically as Christians.
Tiberius first note of the troubles the execution by Pilate of Christus had instigated in these followers of Christus, was only 7 years after the execution.