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 $S\_4$. 



Exceptional primes
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 $\ell$.)


