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

Brown has shown that MTM($\mathbb{Z}$) is generated by the motivic fundamental group of $\mathbb{P}^1-\{0,1,\infty\}$, and that periods of mixed Tate motives are $\mathbb{Q}[(2\pi i)^{-1}]$-linear combinations of multi-zeta values $\zeta(n_1, \dots, n_r)$, $n_i \in \mathbb{N}, n_1 \geq 2$. One can even take $n_i \in \{2,3\}$.

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 $\chi_\ell$.

By 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?

It might be made out of idèle class characters corresponding to its constituent pure Tate motives, but I am not able to see how.

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?