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I've got a new criterion for comparing Galois reps which are four dimensional if we know the kernel of the residual representation mod $5$ (or any large odd prime) and the Sato-Tate groups (should be same for each one) are connected. Note that we don't need to have a large residual image, so this applies to a different set of representations than Faltings-Serre. It also doesn't use a large tower, making it stronger and more efficient than existing variants of the Livne method.

What I don't have are some good examples: the one I was working on when I discovered this method really requires a criterion that works for $p=2$, which is a lot more complicated (and I don't have a handle on yet). So does anyone have some Galois representations to compare lying around meeting my critera? The kernel of the residual representation may be hard to determine on the automorphic side: I'm working on ways to fix that.

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