We define Hecke operators on vector-valued modular forms of the type that appear as characters of rational conformal field theories (RCFTs). We apply our results to derive a number of relations between characters of known RCFTs with different central charges, and extend the previously stud- ied Galois symmetry of modular representations and fusion algebras. We show that the conductor N of a RCFT and the quadratic residues modulo N play an important role in the computation and classification of Galois permutations. We establish a field correspondence in different theories through the picture of effective central charge, which combines Galois inner automorphisms and the structure of simple currents. We then make a first attempt to extend Hecke operators to the full data of modular tensor categories. The Galois symmetry encountered in the modular data trans- forms the fusion and the braiding matrices as well, and yields isomorphic structures in theories related by Hecke operators.