  In this paper, we call strongly modular those reducible semi-simple odd mod
$l$ Galois representations for which the conclusion of the strongest form of
Serre's original modularity conjecture holds. Under the assumption that the
Serre weight $k$ satisfies $l\textgreater{}k+1$, we give a precise
characterization of strongly modular representations, hence generalizing a
classical theorem of Ribet pertaining to the case of conductor $1$.When the
representation $\rho$ is not strongly modular, we give a necessary and
sufficient condition on the primes $p$ not dividing $Nl$ for which it arises in
level $Np$, where $N$ denotes the conductor of $\rho$. This generalizes a
result of Mazur on the case $(N,k)=(1,2)$.
