  We describe the image of general families of two-dimensional representations
over compact semi-local rings. Applying this description to the family carried
by the universal Hecke algebra acting on the space of modular forms of level
$N$ modulo a prime $p$, we prove new results about the coefficients of modular
forms mod $p$. If $f=\sum_{n=0}^\infty a_n q^n$ is such a form, for which we
can assume without loss of generality that $a_n=0$ if $(n,Np)>1$, calling
$\delta(f)$ the density of the set of primes $\ell$ such that $a_\ell \neq 0$,
we prove that $\delta(f)>0$ provided that $f$ is not zero (and if $p=2$, not a
multiple of $\Delta$). More importantly, we prove, when $p>2$, a {\it uniform}
version of this result, namely that there exists a constant $c>0$ depending
only on $N$ and $p$ such that $\delta(f)>c$ for all forms $f$ except for those
in an explicit subspace of infinite codimension of the space of all modular
forms mod $p$ of level $N$. Forms in this subspace, called {\it special}
modular forms mod $p$, are proved to be closely related to certain classes of
modular forms mod $p$ previously studied by the author, Nicolas and Serre,
called cyclotomic and CM modular forms mod $p$.
