The squarefree-level degree-count conjecture for weight 2 newforms

Let $d$ be a positive integer, let $X$ tend to infinity, and count weight $2$ newforms of degree $d$ and squarefree level $N\leq X$, up to the counting convention used in the paper. Squarefree-level degree-count conjecture. For every $\varepsilon>0$, the number of such newforms is

$$O\!\left(X^{1-d/6+\varepsilon}\right).$$

In particular, this number is finite if $d\geq 7$. The conjecture refines heuristic and empirical predictions for the distribution of degrees; the paper presents data supporting it, while noting that the actual growth rate may be smaller.