Fix a prime $p\geqslant 5$ and weight $k\leqslant p+1$, and let $f\in S_k(N,\overline{\mathbb{Q}})$ be an eigenform.

Due to the congruence $E_{p-1} \equiv 1 \mod p$, we know that $\overline{E_{p-1}f}\in S_{k+p-1}(N,\overline{\mathbb{F_p}})$ is congruent to $f$ mod $p$, and it is a result (of Deligne-Serre?) that $E_{p-1}$ can be lifted to an eigenform in $S_{k+p-1}(N,\overline{\mathbb{Q}})$.

From some limited calculations I've done it seems to be that there are two such lifts if $f$ is $p$-non-ordinary, and there is almost always only one if $f$ is $p$-ordinary. Does anyone know if this is true in general? It seems to have a bit of a "theta-cycles" flavour to it, although I'm not sure if you can use that theory to get this kind of "characteristic-zero information"?

Fix a prime $p\geqslant 5$ and weight $k\leqslant p+1$, and let $f\in S_k(N,\overline{\mathbb{Q}})$ be an eigenform.

Due to the congruence $E_{p-1} \equiv 1 \mod p$, we know that $\overline{E_{p-1}f}\in S_{k+p-1}(N,\overline{\mathbb{F_p}})$ is congruent to $f$ mod $p$, and it is a result (of Deligne-Serre?) that $E_{p-1}$ can be lifted to an eigenform in $S_{k+p-1}(N,\overline{\mathbb{Q}})$.

If $f$ is defined over $\mathbb{Z}$, from some limited calculations I've done it seems to be that there are two such lifts if $f$ is $p$-non-ordinary, and there is almost always only one if $f$ is $p$-ordinary. Does anyone know if this is true in general? It seems to have a bit of a "theta-cycles" flavour to it, although I'm not sure if you can use that theory to get this kind of "characteristic-zero information"?