Congruences between modular forms are certainly a big topic in number theory, maybe with $$E_{p-1}\equiv 1 \mod p \qquad \text{for a prime }p\geq 5$$ as the easiest example. Sometimes, $p$ might be replaced by a more general natural number. But one could equally well replace $p$ also by another modular form like the discriminant $\Delta$ or an Eisenstein series $E_n$ or consider congruences modulo an ideal like $(p, E_n)$.
Edit: In concrete terms, two modular forms $f$ and $g$ for some congruence group $\Gamma$ are congruent to each other modulo $E_n$ if $f-g = cE_n$ for some modular form $c$ (for the same congruence subgroup).
1) Is there any source where such congruences are considered?
2) If not, is there any reason why these are not interesting?
