The $pq$ analog of Serre's conjecture (see "Mod pq Galois representations and Serre's Conjecture"- Khare, Kiming) states that if $\bar{\rho}_1:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F}_p)$ is a mod $p$ Galois representation and $\bar{\rho_2}:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F}_q)$ is a mod $q$ Galois representation, then if $\bar{\rho}_1$ and $\bar{\rho}_2$ are odd, irreducible, ramified at only finitely many primes and of the same "weight", then under some further hypotheses, $\bar{\rho}_1$ and $\bar{\rho}_2$ both lift to Galois representations coming from the same eigenform.

Before Serre made his conjecture (for one prime) there was certainly a lot of computational evidence for it, is there some computational evidence for this conjecture?

It seems that the easiest way to come up with examples of $\bar{\rho}_1$ and $\bar{\rho}_2$ as described would be to take the mod $p$ and mod $q$ representations associated a newform, but such representations don't serve the purpose of producing evidence for the conjecture. One would instead seek to contrive to find $\bar{\rho}_1$ and $\bar{\rho}_2$ that do not arise in this way to find computational evidence.

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