Fix an odd prime $p$.

For concreteness, let $N$ be coprime to $p$, and let $2 \leq k \leq p$. Let $S^+(N,k)$ be the newforms in $S_k(\Gamma_1(N))$.

Let $f = \sum a_n q^n \in S^+(N,k)$. We say that $f$ is $p$-ordinary if $v_p(a_p)=0$. (If $p$ splits in the field of coefficients of $f$, we require that $a_p$ not be divisible by any prime $\mathfrak{p}$ above $p$.)

For a "random" $N$ and $k$ satisfying the conditions I set out at the beginning, what can be said about the proportion of $p$-ordinary forms in $S^+(N,k)$?

If we let $N$ and $k$ vary, can we say anything about the density of $p$-ordinary forms in the union $\bigcup_{N,k} S_k^+(\Gamma_1(N))$?