  It is well-known that the Ap\'ery sequences which arise in the irrationality
proofs for $\zeta(2)$ and $\zeta(3)$ satisfy many intriguing arithmetic
properties and are related to the $p$th Fourier coefficients of modular forms.
In this paper, we prove that the connection to modular forms persists for
sequences associated to Brown's cellular integrals and state a general
conjecture concerning supercongruences.
