In Cornell, Silverman, Stevens, "Modular Forms and Fermat's Last Theorem," Edixhoven, the author of the chapter on Serre's Conjecture, asserts a few times, beginning on page 234 of the text, that if $E/\mathbb Q$ is an elliptic curve such that the residual mod $3$ representation
$$\rho_3: G_{\mathbb Q} \to \operatorname{GL}_2(\mathbb F_3)$$
attached to $E$ is irreducible, where $G_{\mathbb Q}$ is the absolute Galois group of $\mathbb Q$, then this representation is not induced by $G(\overline{\mathbb Q}/\mathbb Q(\sqrt{-3}))$. I cannot for the life of me find where the text has proven or given reference for anything resembling this result. What we do know is that this representation is absolutely irreducible.
Can someone provide me with either a reference for this fact or a proof?