A theorem of Deligne asserts that to cusp forms with Euler product, there is for each prime $\ell$ a Galois representation of $G(K_{\ell}/\mathbb{Q})$, where $K_{\ell}$ is the maximal abelian extension of $\mathbb{Q}$ ramified only at $\ell$, satisfying
$ \det \circ \rho_{\ell} = \chi_{\ell}^{k-1} $
where $\chi_{\ell}$ is the natural map to $Gal(K_{\ell}^{ab}/\mathbb{Q}) \simeq \mathbb{Z}_{\ell}^*$. This Galois representation is "exceptional at $\ell$" if it is not surjective.
If a Galois representation is exceptional, then its image in $GL_2(\mathbb{F}_{\ell})$ must either be (i) contained in a Borel subgroup, (ii) contained in the normalizer of a Cartan but not the Cartan itself, or (iii) isomorphic to $S_4$ under the map to $PGL_2(\mathbb{F}_{\ell})$.
I understand how one might prove that a Galois representation is not exceptional for a prime $\ell$. Exceptionality imposes conditions, e.g. on the coefficients, and there are results that can immediately reduce one to checking a finite set of primes for a given modular form.
In addition, showing that a Galois representation is exceptional of type (i) or (ii) amounts of an equality of two modular forms, which can be checked with a finite computation.
How can one prove that a Galois representation is exceptional of type (iii)?