In an assignment, I have to give an example of a 2-dimensional $\ell$-adic representation of the absolute Galois group of $\mathbb{Q}$, bu I am faced with the problem that I do not a lot of these. Or not enough to find the example I am looking for.
More precisely, let $G$ be the absolute Galois group in question, and let $K$ be a (fixed) finite extension of $\mathbb{Q}_\ell$, for some prime $\ell$. I am looking for a representation $$ \rho : G \to GL_2(\bar{K}).$$
The examples I do know are the trivial representation and the ones arising by taking the direct sum or the tensor product of characters (that is, 1-dimensional representations) of $G$. The problem is that they do not seem to give what I am looking for.
Added: Most $\ell$-adic representations that arise from geometry have image lying in $GL_2(\mathcal{O}_K)$. Although these are very interesting representations, and that (in some sense) you can always reduce it to this case (see BR comments below), what I am looking for are representations that cannot be conjugated (by any element of $GL_2(\bar{K})$) in such a way that the image lies in $GL_2(\mathcal{O}_K)$.
The reason the original phrasing was ambiguous and unclear is that I was hoping to build a better repertoire of Galois representations, in the hope of eventually finding an example with the desired properties.
The question remains unclear. As BR and Keenan Kidwell have pointed out, any continuous Galois representations with values in $GL_2(K)$ (with $K$ a finite extension of $\mathbb Q_{\ell}$) can always be conjugated by an element of $GL_2(K)$ into $GL_2(\mathcal O_K)$.
What exactly is is that you want?
Added in response to the comment below: If a continuous Galois rep. takes values in $GL_2(\overline{K})$, then it lies in $GL_2(L)$ for some finite extension $L$ of $K$, and so lies in $GL_2(\mathcal O_L)$ after conjugating by an element of $GL_2(L)$. So this changes nothing, except for replacing the label $K$ by the label $L$ (and since $K$ was arbitrary, this is not a real change).