I've been carrying out some experiments on the computer and I noticed the following congruence phenomenon: fixing a prime $p$, it seems that any modular form over $SL_2(\mathbb{Z})$ and of weight $k \geq p+1$ is congruent $\pmod{p}$ to some modular form over $\Gamma_0(p)$ and of weight $k-(p-1)$.

(Edit: by congruent I mean the formal q-expansions at infinity are congruent).

Of course the case where $k = p+1$ is a result of Serre in his "p-adic zeta functions" paper.

I can prove the statement above is true when $p \equiv 11 \pmod{12}$ and $k \geq p+5$. I'm interested to know if someone knows if this is true in general and possibily a proof or a reference. Thanks!