The database of Bianchi modular forms was mostly computed by John Cremona using modular symbol algorithms developed in his 1981 DPhil thesis (see also J. E. Cremona,
Hyperbolic tessellations, modular symbols, and
elliptic curves over complex quadratic fields
Compositio Mathematica, tome 51, no 3 (1984), p. 275-324) for the five fields $\mathbb{Q}(\sqrt{-1})$, $\mathbb{Q}(\sqrt{-2})$, $\mathbb{Q}(\sqrt{-3})$, $\mathbb{Q}(\sqrt{-7})$, and $\mathbb{Q}(\sqrt{-11})$. This algorithm was extended to handle the other four fields of class number one ($\mathbb{Q}(\sqrt{-19})$, $\mathbb{Q}(\sqrt{-43})$, $\mathbb{Q}(\sqrt{-67})$, $\mathbb{Q}(\sqrt{-163})$) by Elise Whitley in her 1990 PhD thesis, to $\mathbb{Q}(\sqrt{-5})$ by Jeremy Bygott in his 1999 PhD thesis, and to the fields $\mathbb{Q}(\sqrt{-23})$ and $\mathbb{Q}(\sqrt{-31})$ of class number $3$ by Mark Lingham in his 2005 PhD thesis. These extended algorithms were reimplemented by John Cremona in 2021-2022. The open source code `bianchi-progs` implementing the algorithm in C++ is available on GitHub; it depends on the C++ library eclib. The code is currently (as of December 2022) limited to the fields of absolute discriminant less than \(100\) or equal to \(163\), so includes the nine imaginary quadratic fields of class number one, and a number of fields of class number up to $8$. It only computes newforms data for forms of cuspidal weight \(2\) with trivial character and dimension \(1\).

Dimension data for full cuspidal and new spaces for a range of weights and $\mathrm{SL}_2$ levels over $\mathbb{Q}(\sqrt{-d})$ for $d=2,11,19,43,67,163$ were computed by Alexander Rahm using his own code written in PARI/GP.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by John Cremona on 2022-12-02 05:50:29

**Referred to by:**

**History:**(expand/hide all)

- 2024-05-31 18:21:28 by Sam Schiavone
- 2022-12-02 05:50:29 by John Cremona (Reviewed)
- 2022-02-09 19:48:05 by Andrew Sutherland (Reviewed)
- 2022-01-22 08:21:10 by John Cremona
- 2021-10-09 18:14:49 by Andrew Sutherland (Reviewed)
- 2021-09-29 10:06:32 by John Cremona
- 2021-06-17 15:07:29 by John Cremona (Reviewed)
- 2021-06-02 11:32:23 by John Cremona
- 2021-04-17 07:12:23 by Andrew Sutherland (Reviewed)

**Differences**(show/hide)