  We characterize the space of new forms for $\Gamma_0(m)$ as a common
eigenspace of certain Hecke operators which depend on primes $p$ dividing the
level $m$. To do that we find generators and relations for a $p$-adic Hecke
algebra of functions on $K={\rm GL}_2(\mathbb{Z}_p)$. We explicitly find the
$n+1$ irreducible representations of $K$ which contain a vector of level $n$
including the unique representation that contains the "new vector" at level
$n$. After translating the $p$-adic Hecke operators that we obtain into
classical Hecke operators we obtain the results about the new space mentioned
above.
