MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

Post Made Community Wiki by Lavender Honey
2 deleted 1 characters in body

Suppose that $f$ has level $N$, and suppose that $N$ is divisible by $p$. Then it is well known that $f$ is congruent modulo (some prime above) $p$ to a form $g$ of level $M$ dividing $N$ (and high weight), where $M$ is prime to $p$. In particular, by induction, all forms $f$ are connected to a form $g$ of level $1$ in at most $d$ steps, where $d$ is the number of distinct prime divisors of $N$. Yet all weight level one forms are congruent to $\Delta$ modulo $2$. This is actually related to ideas behind the proof of Serre's conjecture:

http://en.wikipedia.org/wiki/Serre%27s_modularity_conjecture

1

Suppose that $f$ has level $N$, and suppose that $N$ is divisible by $p$. Then it is well known that $f$ is congruent modulo (some prime above) $p$ to a form $g$ of level $M$ dividing $N$ (and high weight), where $M$ is prime to $p$. In particular, by induction, all forms $f$ are connected to a form $g$ of level $1$ in at most $d$ steps, where $d$ is the number of distinct prime divisors of $N$. Yet all weight one forms are congruent to $\Delta$ modulo $2$. This is actually related to ideas behind the proof of Serre's conjecture:

http://en.wikipedia.org/wiki/Serre%27s_modularity_conjecture