The characters of a rational conformal field theory are the solutions of a modular invariant differential equation. We cast this equation into a form in which the independent variable is not the modular parameter τ, but rather a modular function of τ, successively the Picard λ-function and the modular invariant function j. This enables us to determine the modular transformations of the characters from the monodromy of the solutions of the equation. The requirement that the characters form a unitary representation of the modular group results in restrictions on h and c. Knowledge of the modular transformations also allows us to find the relative normalization of the characters in the one-loop partition function, which specifies the absolute multiplicities of states in the conformal field theory. We carry this out explicitly for a certain class of second order equations in which the characters can be written in terms of hypergeometric functions.