I'm interested in knowing to what extent is level lowering known to hold in weight 1. Specifically, let's say I have an eigenform $f$ in $S_1(N,\chi)$ and a prime $p$ which doesn't divide $N$. Let's assume that the mod $p$ Galois representation attached to $f$ is unramified at some prime $q$ dividing $N$. Is there then an eigenform $g$ in $S_1(N/q,\chi)$ with the same mod $p$ Galois representation as $f$?

I think I can show this if $p>5$, but the argument relies on Artin's conjecture and feels like overkill. Let me sketch the argument, and then ask some more specific questions at the end.

Here's the argument: if $f$ is CM, check directly that no such $q$ can exist. So the projective image of $\overline{\rho}_f$ must be $A_4$, $S_4$, or $A_5$. In particular, if $p>5$ then one can lift $\overline{\rho}_f$ to a $p$-adic representation with the same image. In particular, the ramification properties are unchanged by the lift. Take this $p$-adic representation and view it as a complex representation (i.e. an Artin representation). By Artin's conjecture (which I guess is now a theorem in this 2-dimensional case by Serre's conjecture), there is a weight 1 eigenform $g$ giving rise to this representation. But since the representation is unramified at $q$, we can take $g$ to have level $N/q$, and we're done.

So questions:

0) Does this argument look okay?

1) Is there a more direct argument that doesn't rely on Artin's conjecture to achieve this. (I'm ultimately interested in the Hilbert modular case, and so I don't want to be using relying on Artin's conjecture).

2) Is there a way to handle $p=2,3,5$ even assuming Artin? Is this even true with $p=2$??