Finite-level and nonexistence conjecture for rational newforms

For each even weight $k$ and level $N$, let $Q_k(N)$ be the set of quadratic-twist classes of classical newforms with rational coefficients and no complex multiplication, represented at minimal level $N$. Finite-level and nonexistence conjecture. For $6\leq k\leq 50$, there is a largest $N_k$ such that $|Q_k(N_k)|\geq 1$; for $18\leq k\leq 50$, this $N_k$ is one of $30$, $10$, $6$, or $2$, as reported in the paper's table; and for $k\geq 52$, there are no non-CM newforms with rational coefficients at all. This is an explicitly computationally motivated expectation based on the displayed table, and the supplied text gives no proof or resolution status.