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We characterize the space of new forms for $\Gamma_0(m)$ as a commoneigenspace of certain Hecke operators which depend on primes $p$ dividing thelevel $m$. To do that we find generators and relations for a $p$-adic Heckealgebra 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 intoclassical Hecke operators we obtain the results about the new space mentionedabove.

Hecke algebras, new vectors and new forms on $\Gamma_0(m)$