Let $K$ be a number field, which can be $\mathbb Q(\zeta)$ with $\zeta$ a root of unity if that helps, and let $\operatorname{SL}_n(\mathbb{Z})$ act on $(K^\times)^n:=K^\times\times\cdots\times K^\times$ by «matrix multplication», so that $M\cdot (\lambda_1,\dots,\lambda_n)=(\lambda_1^{m_{1,1}}\cdots\lambda_n^{m_{1,n}},\dots, \lambda_1^{m_{n,1}}\cdots\lambda_n^{m_{n,n}})$.

Is the cohomology $H^*(\operatorname{SL}_n(\mathbb{Z}),(K^\times)^n)$ known, at least in degree $2$?

If $\mu$ is the group of roots of unity in $K$, then $\mu^n=\mu\times\cdots\times\mu$ is a $\operatorname{SL}_2(\mathbb{Z})$-submodule of $(K^\times)^n$, and one can also ask

Is the cohomology $H^*(\operatorname{SL}_n(\mathbb{Z}),\mu^n)$ known, at least in degree $2$?

The two cohomologies are related by a long exact sequence, of course.

I am really interested in the case where $n=2$ which should be easier (?)...

Let $K$ be a number field, which can be $\mathbb Q(\zeta)$ with $\zeta$ a root of unity if that helps, and let $\operatorname{SL}_n(\mathbb{Z})$ act on $(K^\times)^n:=K^\times\times\cdots\times K^\times$ by «matrix multplication», so that $M\cdot (\lambda_1,\dots,\lambda_n)=(\lambda_1^{m_{1,1}}\cdots\lambda_n^{m_{1,n}},\dots, \lambda_1^{m_{n,1}}\cdots\lambda_n^{m_{n,n}})$.

Is the cohomology $H^*(\operatorname{SL}_n(\mathbb{Z}),(K^\times)^n)$ known, at least in degree $2$?

If $\mu$ is the group of roots of unity in $K$, then $\mu^n=\mu\times\cdots\times\mu$ is a $\operatorname{SL}_n(\mathbb{Z})$-submodule of $(K^\times)^n$, and one can also ask

Is the cohomology $H^*(\operatorname{SL}_n(\mathbb{Z}),\mu^n)$ known, at least in degree $2$?

The two cohomologies are related by a long exact sequence, of course.

I am really interested in the case where $n=2$ which should be easier (?)...