  We classify all modular categories of dimension $4m$, where $m$ is an odd
square-free integer, and all ranks $6$ and $7$ weakly integral modular
categories. This completes the classification of weakly integral modular
categories through rank $7$. Our results imply that all integral modular
categories of rank at most $7$ are pointed (that is, every simple object has
dimension $1$). All strictly weakly integral (weakly integral but non-integral)
modular categories of ranks $6$ and $7$ have dimension $4m$, with $m$ an odd
square free integer, so their classification is an application of our main
result. The classification of rank $7$ integral modular categories is
facilitated by an analysis of two actions on modular categories: the Galois
group of the field generated by the entries of the $S$-matrix and the group of
isomorphism classes of invertible simple objects. The interplay of these two
actions is of independent interest, and we derive some valuable arithmetic
consequences from their actions.
