Exceptional primes

Question

Let $f$ be a cuspidal Hecke Eigenform of weight $k \geq 2$ and let $\rho_{f, \lambda}:G_{\mathbb{Q}} \rightarrow GL_2(E_{\lambda})$ be the corresponding Galois representation with $2 \mid \lambda$ constructed by Deligne. Assume now that $\pi_f={\otimes}' \pi_p$ be the corresponding automorphic representation.

My question is the following: If we assume $\pi_2$ non-monomial (it is not induced by character), can you say that projective image $\tilde{\rho_{f, \lambda}}:G_{\mathbb{Q}} \rightarrow PGL_2(E_{\lambda})$ is an exceptional group (S_4). I am aware the image of the corresponding Weil-Deligne representation is $S_4$.

Answer 1

No, this does not work: for any modular form of weight $k \ge 2$, the image of the projective representation $\tilde\rho_{f, \lambda}$ is infinite for every prime $\lambda$.

Proof: if $\tilde{\rho}_{f, \lambda}$ has finite image, then so does the adjoint representation $\operatorname{Ad}^0 \tilde{\rho}_{f, \lambda}$, since the adjoint representation of $GL_2$ factors through $PGL_2$. In particular, the adjoint representation has all Hodge--Tate weights 0, which is a contradiction since $\rho_{f, \lambda}$ has Hodge--Tate weights $0$ and $1-k$ and thus $\operatorname{Ad}^0 \tilde{\rho}_{f, \lambda}$ has weights $\{0, k-1, 1-k\}$.

(Note that this even shows that $\tilde\rho_{f, \lambda}(D_\ell)$ is infinite, where $\lambda \mid \ell$ and $D_\ell$ is a decomposition group at $\ell$.)