  We introduce the notion of (nondegenerate) strongly-modular fusion algebras.
Here strongly-modular means that the fusion algebra is induced via Verlinde's
formula by a representation of the modular group whose kernel contains a
congruence subgroup. Furthermore, nondegenerate means that the conformal
dimensions of possibly underlying rational conformal field theories do not
differ by integers. Our first main result is the classification of all
strongly-modular fusion algebras of dimension two, three and four and the
classification of all nondegenerate strongly-modular fusion algebras of
dimension less than 24. Secondly, we show that the conformal characters of
various rational models of W-algebras can be determined from the mere knowledge
of the central charge and the set of conformal dimensions. We also describe how
to actually construct conformal characters by using theta series associated to
certain lattices. On our way we develop several tools for studying
representations of the modular group on spaces of modular functions. These
methods, applied here only to certain rational conformal field theories, are in
general useful for the analysis rational models.
