Let $G$ be the absolute Galois group of a number field $K$. Let $\ell$ be a prime number. There are representations $\mathbb{Z}_\ell(n)$ of $G$ on the group of $\ell$-adic integers given by the formula $g.x=\chi(g)^nx$ where $\chi:G\to \mathbb{Z}_\ell^{\times}$ is the cyclotomic character.

Question: Is it true that the ext groups $\mathrm{Ext}^*(\mathbb{Z}_\ell(0),\mathbb{Z}_\ell(n))$ vanish for $n$ negative ?

The reason I am asking this is that this would imply that the subtriangulated category of Galois representations spanned by the objects $\mathbb{Z}_\ell(n)$ has a weight filtration constructed for instance in Lemma 1.2. of this paper of Marc Levine:

https://www.uni-due.de/~bm0032/publ/TateMotives.pdf

Let $G$ be the absolute Galois group of a number field $K$. Let $\ell$ be a prime number. There are representations $\mathbb{Z}_\ell(n)$ of $G$ on the group of $\ell$-adic integers given by the formula $g.x=\chi(g)^nx$ where $\chi:G\to \mathbb{Z}_\ell^{\times}$ is the cyclotomic character.

Question: Is it true that the ext groups $\mathrm{Ext}^*(\mathbb{Z}_\ell(0),\mathbb{Z}_\ell(n))$ vanish for $n$ negative ?

The reason I am asking this is that this would imply that the triangulated subcategory of the derived category of Galois representations spanned by the objects $\mathbb{Z}_\ell(n)$ has a weight filtration, as constructed for instance in Lemma 1.2. of this paper of Marc Levine:

https://www.uni-due.de/~bm0032/publ/TateMotives.pdf