Deligne, Goncharov and Levine have constructed a Tannakian category of mixed Tate motives, MTM($\mathcal{O}_{K,S}$), over the ring of integers of a number field $K$ unramified outside a finite set of places $S$. 

In particular there is a category MTM($\mathbb{Z}$) of mixed Tate motives unramified over $\mathbb{Z}$. Its objects are extensions of pure Tate motives $\mathbb{Q}(n)$ where $n$ is an integer. They arise from motivic sheaves on the moduli spaces of marked genus 0 curves.

The $\ell$-adic realization of the pure Tate motive $\mathbb{Q}(n)$ is the $Gal (\overline{\mathbb{Q}}/\mathbb{Q})$-module $\mathbb{Q}_\ell$, the Galois action being given by the $n$-th power of the $\ell$-adic cyclotomic character $\chi_\ell$.

In Langlands philosophy every motive should correspond to an automorphic form. For pure Tate motives $\mathbb{Q}(n)$ class field theory shows how: they correspond to powers of idèle class characters.

> Which automorphic form/representation does a *truly* mixed Tate motive correspond to?

For example we know that $M^1_n := Ext^1(\mathbb{Q}(0), \mathbb{Q}(n)) = K_{2n-1}(\mathbb{Z})\otimes \mathbb{Q}$ for $n > 1$ where $K_.$ are the Milnor K-groups. What makes $M^1_n$ automorphic?