Let $S$ be the (countable) set of holomorphic cuspidal new eigenforms of weight $\geq 2$. Any $f\in S$ has a level $N\_f$ and a canonically normalized Fourier expansion $f(z)=\sum\_{n=1}^{\infty}a\_f(n)e^{2\pi i nz}$ with $a\_f(1)=1$ and $a\_f(n) \in \overline{\mathbf{Z}}$. 


Form a graph $\mathcal{G}$ as follows: Take the vertex set of $\mathcal{G}$ to be the set $S$, and join $f$ and $g$ by an edge if the algebraic integers $(a\_f(n)-a\_g(n))\_{n \nmid N\_f N\_g}$ generate a nontrivial ideal in $\overline{\mathbf{Z}}$; in other words, $f$ and $g$ are joined if their Fourier coefficients are related by a nontrivial congruence.


Is $\mathcal{G}$ connected?



The graph of congruences between modular forms
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
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>


