New subspace of cusp forms
= New subspace of cusp forms
{title2=$S_k(\Gamma_1(N))^{\mathrm{new}}$}
The new subspace is the <orthogonal complement> of the <old subspace of cusp forms> for the <Petersson inner product>. A form belongs to it precisely when the adjoints of all lower-level <oldform by argument dilation> maps annihilate it.