In the space 
of cusp forms of weight 
and modular
form level 
,
the old subspace 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.
A modular newform (or normalized newform) is a modular eigenform in the new subspace, normalized so that its first Fourier coefficient is 1. Thus it has an expansion
 |
and, with the standard Hecke operators at modular form level 
,
. Its coefficients are multiplicative
and determine an Euler product for the associated
L-series. In particular, the Gross-Zagier
formula uses a weight-2 modular newform.
."
Math. Ann. 185, 134-160, 1970.