Completeness of classical modular form data

The classical modular form database currently contains information for all newforms $f\in S_k^\mathrm{new}(N,\chi)$ of level $N\ge 1$, weight $k\ge 1$, character $\chi\colon (\Z/N\Z)^\times\to \C$, for which any of the following hold:

• $Nk^2\le 4000$;
• $\chi$ trivial and $Nk^2\le 40\,000$;
• $N \le 24$ and $Nk^2 \le 40\,000$;
• $N \le 10$ and $Nk^2 \le 100\,000$;
• $N \le 100$ and $k \le 12$;
• $k > 1$ and $\dim S_k^{\rm new}(N,\chi) \le 100$ and $Nk^2 \le 40\,000$.
• $k = 2$, $\chi$ is trivial, and $N\le 50\,000$ or $N \le 10^6$ is prime.

In addition to the newspaces identified above, there are 131 newspaces that are present because they contain the minimal twist of a newform in one of the newspaces above.

For each newform with q-expansion $f=\sum a_nq^n$ the database contains the integers $\mathrm{tr}(a_n)$ for $1\le n \le 1000$, and when the level is at most $10\,000$ and the dimension of $f$ is at most $20$, the algebraic integers $a_n$ for $1\le n \le 1000$ expressed in terms of an explicit basis for the coefficient ring $\Q(f)$. For $1000 < N \le 4000$ this data is available for all $n\le 2000$, and for $4000 < N\le 10\,000$, for all $n\le 3000$ (these values exceed both the Sturm bound and $30\sqrt{N}$ in every case). When the level is at most $10\,000$ and the dimension of $f$ is at most $20$ Hecke characteristic polynomials for primes $p\le 100$ are also available.

For each newform $f$ of level $N\le 10\,000$ and each embedding $\rho\colon \Q(f)\to \C$ the complex numbers $\rho(a_n)$ are available as floating point numbers with a precision of at least $52$ bits (separately, for both real and imaginary parts); this information is available for all newforms, regardless of their dimension, even when algebraic $a_n$ are not available (for the same ranges of $n$ as above).

Dimension tables are available for all newspaces $S_k^\mathrm{new}(N,\chi)$ with $Nk^2 \le 4000$, and also for those with $k>1$ and $Nk^2\le 40\,000$, and those with $N\le 10$ and $Nk^2\le 100\,000$. For newspaces in these ranges with $N\le 4000$ we have also computed the first $1000$ coefficients of the trace form of the newspace.

Not every invariant of every newform has been computed (this is computationally infeasible). Below is completeness information for some specific invariants:

• The following information is available for every newform of level $N\le 10\,000$: level, weight, character, dimension, analytic conductor, Sato-Tate group, trace form, complex embedding data, all self twists, all inner twists, all twists to other newforms in the database, the minimal twist, and an upper bound on the analytic rank.

• The following information is available for every newform of dimension at most $20$ and level $N\le 10000$, and also for every weight one newform: coefficient field, exact algebraic coefficient data, generators for the coefficient ring, and a lower bound on the index of the coefficient ring in the ring of integers of the coefficient field.

• The following information is available for every weight one newform: projective image and projective field.

• In cases where the odd 2-dimensional Artin representation corresponding to a weight one newform is present in the LMFDB, it is listed as a related object and the Artin image and Artin field of the weight one newform are available, and this information is also available in some cases where a twist of a weight one newform has a corresponding Artin representation in the LMFDB, even though the twisted Artin representation is not in the LMFDB. As of January 2020 this applies to about 30% of the weight one newforms in the database.

• The database includes all of the 43 newforms that can be expressed as eta quotients [10.1090/S0002-9947-96-01743-6, MR:1376550].

• Stark units for $b=1$ are available for weight 1 newforms of dimension 1.