  Bloch's theorem is the centerpiece of topological band theory, which itself
has defined an era of quantum materials research. However, Bloch's theorem is
broken by a perpendicular magnetic field, making it difficult to study
topological systems in strong flux. For the first time, moir\'e materials have
made this problem experimentally relevant, and its solution is the focus of
this work. We construct gauge-invariant irreps of the magnetic translation
group at $2\pi$ flux on infinite boundary conditions, allowing us to give
analytical expressions in terms of the Siegel theta function for the magnetic
Bloch Hamiltonian, non-Abelian Wilson loop, and many-body form factors. We
illustrate our formalism using a simple square lattice model and the
Bistritzer-MacDonald Hamiltonian of twisted bilayer graphene, obtaining
reentrant ground states at $2\pi$ flux under the Coulomb interaction.
