Dimensions of Atkin-Lehner subspaces (reviewed)

For each prime dividing the level \(N\), the Atkin-Lehner operator scales each newform by either \(1\) or \(-1\). Let \(j\) be the number of primes dividing \(N\). Ordering the \(2^j\) vectors of \(\pm 1\) lexicographically (with \(1\) before \(-1\)), we show the dimensions of the subspaces of \(S_k^{\mathrm{new}}(\Gamma_0(N),\chi)\) with prescribed Atkin-Lehner eigenvalues.