Let $f$ be a newform of weight $k \geq 2$ and level $N \geq 1$ without complex multiplication. A prime $p$ is said to be ordinary for $f$ if the $p$-th Fourier coefficient $a\_p(f)$ is a $p$-adic unit (to make sense of this in general, one needs to choose a prime ideal above $p$ in the field $K\_f$ of Fourier coefficients of $f$, equivalently an embedding of $K\_f$ into $\overline{\mathbf{Q}\_p}$).


Non-ordinary primes seem to be rare. For example, if $f$ has coefficients in $\mathbf{Z}$, then a very rough guess is that $a\_p(f)$ modulo $p$ is randomly distributed, so we may expect that


\begin{equation\*}
\# \{p \leq x : a\_p \equiv 0 \textrm{ mod } p \} \stackrel{?}{\approx} \sum\_{p \leq x} \frac{1}{p} \sim \log \log x.
\end{equation\*}
On the other hand, for the modular form $\Delta = \sum\_{n \geq 1} \tau(n) q^n$, it seems not to be known that there are infinitely many primes $p$ such that $\tau(p) \not\equiv 0$ mod $p$, so estimating the number of ordinary or non-ordinary primes is difficult in general.


If we consider elliptic curves, we may ask whether every prime $p$ is non-ordinary for some elliptic curve $E$ over $ \mathbf{ Q } $, which means that the reduction of $E$ mod $p$ is supersingular. In fact, Deuring has shown that given any integer $a$ such that $|a|<2\sqrt{p}$, there exists an elliptic curve $E\_p$ over the finite field $\mathbf{F}\_p$ with exactly $p+1-a$ points over $\mathbf{F}\_p$. Now take any elliptic curve $E$ over $\mathbf{Q}$ whose reduction mod $p$ is $E\_p$ and use the modularity theorem. We get a modular form $f$ of weight $2$ with integral coefficients satisfying $a\_p(f)=a$, and we may choose it to be non-CM (in fact, we get infinitely many such modular forms).


My question is whether this kind of result is known or even expected in higher weight. Certainly, if you have a finite set of newforms without CM, the above heuristics suggest that there exist infinitely many primes which are ordinary for all these newforms. But I don't know what happens for an infinite set of newforms, even if this set is "thin" in some sense.


Here is another example of question which arises.



> 
> Fix a weight $k \geq 3$. Is every prime $p$ non-ordinary for some non-CM newform of weight $k$ and level not dividing $p$? At least, are there infinitely many such primes?
> 
> 
> 


Hida's theory implies that if $p \geq 5$ and $k$ is equal to $\{3,4,5,\ldots,10,14\}$ modulo $p-1$, then all newforms of weight $k$ and level $1$ are non-ordinary at $p$. This gives a positive answer to the question for some primes $p$ which are small with respect to $k$ (in particular, these primes satisfy $p \leq k-2$).


One may try to use the theory of congruences between modular forms, like the theory of Hida families, but usually one starts with a modular form which is ordinary at $p$, and I don't know to which extent the theory has been generalized to the non-ordinary case. 



Existence of newforms which are non-ordinary at a given prime
Given p and k, it's clear we can find a CM-type newform of weight k and some level which is supersingular at p (just choose an imaginary quadratic field in which p is inert, and some sufficiently large conductor away from p). 


So it suffices to find a second newform that is congruent to the first one mod p and is *not* of CM type. This can always be done (after adding some auxiliary primes to the level) by the answer to this old question of mine: [Congruences between CM and non-CM modular forms](https://mathoverflow.net/questions/132605/congruences-between-cm-and-non-cm-modular-forms).


