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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 $\bigcup_{N,k} S_k^+(\Gamma_1(N))$?

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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 $2$.

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