A modular supercongruence for 6F5: An apéry-like story

We prove a supercongruence modulo p3 between the pth Fourier coefficient of a weight 6 modular form and a truncated 6F5-hypergeometric series. Novel ingredients in the proof are the comparison of two rational approximations to δ(3) to produce non-trivial harmonic sum identities and the reduction of the resulting congruences between harmonic sums via a congruence relating the Apéry numbers to another Apéry-like sequence.