Congruence terminology

In order to work entirely over the integers and determine congruences exactly, we use the following modified notion of congruence. Let N be a positive integer and consider distinct weight 2 newforms f and g whose levels N_f and N_gboth divide N. Denote by $\tilde{A}_f$ the abelian variety generated by the images of A_f in J₀(N)under the degeneracy maps, and similarly define  $\tilde{A}_g$. We say that f and g are congruent at level N modulo a rational prime p, written $f\equiv g\pmod{p}$, if p divides the order of the finite group $\tilde{A}_f\cap\tilde{A}_g$. Deciding whether or not f and g are congruent can be accomplished by working in the integral homology H₁(X₀(N),Z)using the algorithm of Section .