Congruence conditions for the mod λ values of the Fourier coefficients of classical eigenforms

We classify all instances of the condition aₚ(f) ≡ x λ a p ( f ) ≡ x mod λ being related to a congruence on the prime p , where aₚ(f) a p ( f ) denotes the p th Fourier coefficient of a classical normalised cuspidal eigenform f and λ λ is a prime in the number field generated by the Fourier coefficients of f . This classification is done in terms of the (projective) image of the mod λ λ Galois representation associated with f and extends work by Swinnerton-Dyer. We highlight that for $$x = 0$$ x = 0 , this condition is more often implied by a congruence on the prime p than the general value of aₚ(f) λ a p ( f ) mod λ . Finally, we illustrate various instances of these congruences through examples from the setting of weight 2 newforms attached to rational elliptic curves.