In Jochnowitz's paper "Congruences between Systems of Eigenvalues of Modular Forms," she appeals to Katz's paper "A result on Modular Forms in Characteristic p" to prove the equality of mod $p$ filtrations $w(f^p) = p w(f)$ (Fact 1.7 in the Jochnowitz). Here, for $f$ a modular form mod $p$, $w(f)$ denotes the smallest weight $k$ for which $f$ lifts to a modular form with $\mathbb{Q}$-coefficients in $M_k(\Gamma_0(N))$.
However, Jochnowitz is working with modular forms on $\Gamma_0(N)$, and Katz with modular forms on $\Gamma_1(N)$. Moreover, the fact is not true for $p = 2,3$, a hypothesis that Jochnowitz assumes, but does not explicitly mention in the proof. Why is it that Katz's $\Gamma_1(N)$ results can be readily extended to $\Gamma_0(N)$ for $p \ge 5$? I suspect that the $p \ge 5$ dependence has something to do with the Hasse invariant in characteristic $p$ lifting to a characteristic $0$ modular form, but I have not yet figured out the exact relationship.
So far, I have tried showing that if $f$ is a modular form with $\mathbb{Q}$-coefficients on $\Gamma_1(N)$ congruent mod $p$ to a modular form on $\Gamma_0(N)$, then $f$ is itself on $\Gamma_0(N)$. However, this appears not to be true.