Serre invariants (awaiting review)

Let $\bar \rho_{E,\ell}$ be the mod-$\ell$ Galois representation of an elliptic curve $E/\Q$.

The Serre invariants $(k,M)$ of $\bar \rho_{E,\ell}$ consist of the Serre weight $k$ and the Serre conductor $M$ giving the weight and minimal level of a newform $f\in S_{k}^{\textrm new}(\Gamma_1(M))$ whose associated mod-$\ell$ Galois representation is isomorphic to $\bar \rho_{E,\ell}$.

This means that $a_p(E)$ and $a_p(f)$ reduce to the same element of the residue field of a prime above $\ell$ in the coefficient field of $f$ (this residue field need not have degree one, but every $a_p(f)$ must reduce to an element of $\F_\ell$ in order for this condition to hold).

The modular form $f$ is not uniquely determined, but the minimal level $M$ arising among all such $f$ is uniquely determined, and among those with level $M$, the weight is uniquely determined.

For all but finitely many primes $\ell$, including all $\ell>7$ of good reduction for $E$, the Serre invariants are $(2,N)$, where $N$ is the conductor of the elliptic curve. The primes $\ell$ for which this does not hold are exceptional.

In general, the Serre weight $k$ is divisible by $2$ and the Serre conductor $M$ divides $N$.