A First Course in Modular Forms
《A First Course in Modular Forms》 · 第 187–188 页 · Section 5.6「Oldforms and Newforms」，Definition 5.6.1
The subspace of oldforms at level $N$ is $S_k(\Gamma_1(N))^{\mathrm{old}}=\sum_{p\mid N,\ p\text{ prime}}i_p((S_k(\Gamma_1(N/p)))^2)$.