Locally induced Galois representations with exceptional residual images

In this paper, we classify all continuous Galois representations $\rho:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to \mathrm{GL}_2(\overline{\mathbf{Q}}_p)$ which are unramified outside $\{p,\infty\}$ and locally induced at $p$, under the assumption that $\overline{\rho}$ is exceptional, that is, has image of order prime to $p$. We prove two results. If $f$ is a level one cuspidal eigenform and one of the $p$-adic Galois representations $\rho_f$ associated to $f$ has exceptional residual image, then $\rho_f$ is not locally induced and $a_p(f)\neq 0$. If $\rho$ is locally induced at $p$ and with exceptional residual image, and furthermore certain subfields of the fixed field of the kernel of $\overline{\rho}$ are assumed to have class numbers prime to $p$, then $\rho$ has finite image up to a twist.