Contribution of symmetric power transfers to the cuspidal cohomology of $\mathrm{G}{\mathrm{L}}_{\mathrm{n}}$

Let $\pi$ be a cuspidal automorphic representation of ${\mathrm{G}\mathrm{L}}_2({\mathbb{A}}_{\mathbb{Q}})$. Newton and Thorne have proved that for every $n\ge 1$, the symmetric power lifting ${\mathrm{s}\mathrm{y}\mathrm{m}}^n(\pi )$ is automorphic if $\pi$ is attached to a non-CM Hecke eigenform. In this article, we establish an asymptotic estimate of the number of cuspidal automorphic representations of ${\mathrm{G}\mathrm{L}}_{n+1}({\mathbb{A}}_{\mathbb{Q}})$ which contribute to the cuspidal cohomology of ${\mathrm{G}\mathrm{L}}_{n+1}$ and are obtained by symmetric $n$th transfer of cuspidal representations of ${\mathrm{G}\mathrm{L}}_2({\mathbb{A}}_{\mathbb{Q}})$. Here we fix the weight and vary the level. This generalises the previous works done for ${\mathrm{G}\mathrm{L}}_3$ and ${\mathrm{G}\mathrm{L}}_4$.