Let $M$ be a modular curve and $\pi:E\to M$ the universal elliptic curve. For a prime $\ell$, let $\mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$. I am wondering whether there are any explicit constructions of elements of $H^1_{\mathrm{et}}(M,\mathrm{Sym}^k\mathcal{H}(1))$. Any related reference would be helpful.

1$\begingroup$ Do you want classes in the étale cohomology of $M$ over $\mathbb Q$? or over the algebraic closure? (If the latter a weight $k+2$ cuspform $f$ yields such an element via the holomorphic oneform $f(q)dq(q,1)^k$ attached to $f$). $\endgroup$ – Olivier May 7 '18 at 14:27

$\begingroup$ @Olivier Maybe, a better description of my cohomology would be $H^1_{\mathrm{et}}(M_{\overline{\mathbb{Q}}},\mathrm{Sym}^k\mathcal{H}(1))^{\mathrm{Gal}_\mathbb{Q}}$ $\endgroup$ – User0829 May 8 '18 at 1:59