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It follows from the representation of the central extension of the covering group of the classical Galilei group how the wave function transforms. I have worked out once the representation theory for the Galilei group as lecture notes for a QM2 lecture. It's, however, in German (the quantum theoretical part starts with Sect. 2.5):
https://itp.uni-frankfurt.de/~hees/publ/hqm.pdf
https://itp.uni-frankfurt.de/~hees/publ/hqm.pdf