Serre's conjecture modulo pq

Let p and q be primes, and consider a continuous representation $\rho:G_\mathbf{Q}\rightarrow\mbox{\rm GL}(2,\mathbf{Z}/pq\mathbf{Z})$that is irreducible in the sense that its reductions modulo p and modulo q are both irreducible. Call $\rho$ modular if there is a modular form f such that a mod p representation attached to fis the mod p reduction of $\rho$, and ditto for q. I have carried out specific computations suggested by Mazur in hopes of determining when one should expect that such mod pq representations are modular; the computation suggests that the right conjectures are elusive. Ribet's theorem (see [22]) produces infinitely many levels $pq\ell$ at which there is a form giving rise to $\rho$ mod p and another giving rise to $\rho$ mod q; we hope to determine if for some $\ell$there is a single form giving rise to both reductions.