Modular Newform

A modular newform (or normalized newform) is a modular eigenform in the new subspace of , normalized so that its first Fourier coefficient is 1. The old subspace of this space of cusp forms of weight and modular form level is spanned by forms obtained from lower modular-form levels properly dividing , with dividing . Its orthogonal complement with respect to the Petersson inner product is the new subspace.

In terms of , a modular newform has an expansion

With the standard Hecke operators at modular form level , it satisfies . Its coefficients are multiplicative and determine an Euler product for the associated -series. In particular, the Gross-Zagier formula uses a weight-2 modular newform.