Congruence conditions for the mod $\lambda$ values of the Fourier coefficients of classical eigenforms

Abstract

We classify all instances of the condition $a_{p}(f) \equiv x \bmod \lambda$ being related to a congruence on the prime $p$, where $a_{p}(f)$ denotes the $p$th Fourier coefficient of a classical normalised cuspidal eigenform $f$ and $\lambda$ 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 $\lambda$ Galois representation associated with $f$ and extends work by Swinnerton-Dyer. We highlight that for $x = 0$, this condition is more often implied by a congruence on the prime $p$ than the general value of $a_{p}(f) \bmod \lambda$. Finally, we illustrate various instances of these congruences through examples from the setting of weight 2 newforms attached to rational elliptic curves.