Old subspace of cusp forms
= Old subspace of cusp forms
{title2=$S_k(\Gamma_1(N))^{\mathrm{old}}$}
The old subspace is spanned by $h(tz)$ for <cusp forms> $h$ of level $M<N$, where $M\mid N$ and $t\mid N/M$. It is stable under the <Hecke operators>, but need not have a basis of eigenvectors for every bad-prime operator. Its <Petersson inner product> <orthogonal complement> is the <new subspace of cusp forms>.