In his 1975 Inventiones paper "On $\ell$-adic representations attached to modular forms", Ken Ribet shows that if $f_1, f_2$ are any two cuspidal modular eigenforms for $\operatorname{SL}_2(\mathbb{Z})$, not Galois-conjugate to each other, then the image of the Galois representation on $\rho_{f_1, \ell} \times \rho_{f_2, \ell}$ is "as large as possible" for all but finitely many $\ell$; i.e the image is $$ \{ (u, v) \in \operatorname{GL}_2(\mathcal{O}_{f_1, \ell}) \times \operatorname{GL}_2(\mathcal{O}_{f_2, \ell}) : \exists x \in \mathbb{Z}_\ell^\times \text{ such that } \det(u) = x^{k_1 - 1}, \det(v) = x^{k_2 - 1}\} $$ where $\mathcal{O}_{f_1, \ell}$ and $\mathcal{O}_{f_2, \ell}$ are the rings of integers of the relevant completions of the coefficient fields of $f_1$ and $f_2$, and $k_1, k_2$ are the weights. Moreover, for any prime $\ell$ the image is open in the above group.

Has this theorem been generalized to modular forms of higher levels and non-trivial characters? If so, what is a good reference for this? (I know lots of references for "big image" statements for individual modular forms, but for pairs of modular forms I don't know where to look.)



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