Is every $\rho$ modular?

As in Mazur's letter (Section 1.1), consider f,g,p,q with f, g of level N, M, respectively. Suppose that there is no form h at level NM such that $h \equiv f\pmod{p}$ and $h\equiv g\pmod{q}$ (congruence as in Section 1.2). Our choice of p and q implies that there is h₁ and h₂ at level NM such that $h_1 \equiv f\pmod{p}$ and $h_2\equiv g\pmod{q}$. Using the Cebotarev Density Theorem (see e.g., [Lan94, VIII.4]) and Ribet's level raising theorem [Rib90] we see that there are infinitely many primes $\ell$ for which there exists newforms b₁ and b₂ at level $NM\ell$ such that $b_1 \equiv h_1\pmod{p}$ and $b_2\equiv h_2\pmod{q}$. Does b₁=b₂ in one of these infinitely many cases? More generally, begin with two newforms fand g both of some level N and consider the infinitely many primes $\ell$ such that there exists h₁ and h₂of level $N\ell$ with $h_1 \equiv f\pmod{p}$ and $h_2\equiv g\pmod{q}$. Must there always be an $\ell$so that we can take h₁=h₂? If yes, then every mod pqrepresentation is modular; at least, if we admit the classical Serre conjectures.

Answers to the above questions may be viewed as a generalization of Ribet's level raising machinery [Rib90]. The introduction of two primes at once obscures the algebraic interpretation in terms of ideals of the Hecke algebra; making it appear necessary to find new arguments. For this reason, the author suspects that there are non-modular mod pq representations. Even if not all mod pq representations are modular, the question remains of finding the minimal weight and level of those which are.