Hecke algebras, new vectors and new forms on Γ ₀(m)

We characterize the space of new forms for Γ 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= GL ₂(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” of 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.