Definite orthogonal modular forms: computations, excursions, and discoveries

We consider spaces of modular forms attached to definite orthogonal groups of low even rank and nontrivial level, equipped with Hecke operators defined by Kneser neighbours. After reviewing algorithms to compute with these spaces, we investigate endoscopy using theta series and a theorem of Rallis. Along the way, we exhibit many examples and pose several conjectures. As a first application, we express counts of Kneser neighbours in terms of coefficients of classical or Siegel modular forms, complementing work of Chenevier–Lannes. As a second application, we prove new instances of Eisenstein congruences of Ramanujan and Kurokawa–Mizumoto type.