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 $\bigcup\_{N,k} S\_k^+(\Gamma\_1(N))$? 



Density of p-ordinary modular forms
An ordinary modular form needs to be $p$-stable (by definition). If you have a modular form of level $N$ co-prime to $p$, you need to associate a $p$-stabilization to this latter to get an ordinary modular form with level $Np$.


If you take $T$ the Hecke algebra acting on Katz p-adic modular forms, it is known that the classical points are dense in $\operatorname{Spec}T$, and $T$ has krull dimension at least $4$, but the ordinary Hecke algebra has a dimension equal to $2$.


