We discuss two related problems in conformal field theory. The first is the construction of the modular transformation matrix S for integer spin modular invariants in which some characters appear with multiplicity larger than 1. The second problem is the relation between the characters and the branching functions in coset theories in which the field identification identifies some fields with themselves ("fixed points"). We find that these problems are closely related, and that the solution is remarkably interesting. The fixed points of any conformal field theory seem always to define a new (not necessarily unitary) conformal field theory whose primary fields are in one-to-one correspondence with the fixed points. The characters of this conformal field theory are needed to modify the coset branching functions.