Enumerating Newforms

Let $H_1(X_0(N),\mathbf{Z})^+$ denote the $+1$-eigenspace for the action of the involution induced by complex conjugation, which we can compute using modular symbols. We list all newforms of a given level $N$ by decomposing the new subspace of $H_1(X_0(N),\mathbf{Q})^+$ under the action of the the Hecke operators and listing the corresponding systems of Hecke eigenvalues (see [Ste02a]). First we compute the characteristic polynomial of $T_2$, and use it to break up the new space. We apply this process recursively with $T_3, T_5, \ldots$ until either we have exceeded the bound coming from [Stu87] (see [AS]), or we have found a Hecke operator $T_n$ whose characteristic polynomial is irreducible.

We order the newforms in a way that extends the ordering in [Cre97]: First sort by dimension, with smallest dimension first; within each dimension, sort in binary by the signs of the Atkin-Lehner involutions, e.g., $+++$, $++-$, $+-+$, $+-$, $-++$, etc. When two forms have the same Atkin-Lehner sign sequence, order by $\vert\Tr(a_p)\vert$ with ties broken by taking the positive trace first. We denote a Galois-conjugacy class of newforms by a bold symbol such as $\mathbf{389E}$, which consists of a level and isogeny class, where $\mathbf{A}$ denotes the first class, $\mathbf{B}$ the second, $\mathbf{E}$ the fifth, $\mathbf{BB}$ the $28$th, etc. As discussed in [Cre97, pg. 5], for certain small levels the above ordering, when restricted to elliptic curves, does not agree with the ordering used in the tables of [Cre97]. For example, our $\mathbf{446B}$ is Cremona's $\mathbf{446D}$.