Many arithmetically interesting operators act on spaces of modular forms. One of the these operators is the Atkin-Lehner involution, which splits a space of modular forms into a direct sum of a plus-eigenspace and a minus-eigenspace. I will first say a bit about the classical split in the dimensions between these two eigenspaces, and then refine this story to account for congruences between modular forms. This is an application of a new technique for counting mod-p modular forms recursively in the weight by establishing deep congruences between traces of certain Hecke operators. Joint with Samuele Anni and Alexandru Ghitza.