I want to know about low-weight Galois representations in $H^i(\overline{\mathcal M}_{g,n}, \overline{\mathbb Q}_\ell)$ that aren't cyclotomic. This should be equivalent to finding $p,q$ such that $H^{p,q}( \overline{\mathcal M}_{g,n})\neq 0$, $p\neq q$, and $p+q$ is small, or to finding non-algebraic cohomology classes in $H^i(\overline{\mathcal M}_{g,n},\mathbb Q)$ for small $i$.
My question is: What's the smallest weight / degree that can occur?
For $g=0$, this never happens. For $g=1$, the cohomology comes from modular forms of level $1$, and the lowest weight one is weight $11$, arising from the modular form $\Delta(q)$. For $g\geq 2$, I don't know if you can do better than $11$.
I'd also be intersted in knowing about especially low weight Galois representations that don't arise from classical modular forms, even if their weight is larger than $11$. I think the lowest one I know about is weight $19$ on $\overline{\mathcal M}_{2,16}$ from a Siegel modular form. Should I expect the weights to keep increasing as $g$ goes to infinity, so that these are the smallest ones?
I became interested in these because they are natural examples of everywhere unramified Galois representations arising from geometry, and the lowest weight ones seem like the simplest in some sense. I'd also like to know the answer purely to get a better intuition about the cohomology of these spaces.