# Programs for computing with automorphic forms

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I plan to use this page to make available various useful little scripts and tables of data that I and other people have worked out in connection with the explicit calculation of spaces of automorphic forms. These scripts are written in various combinations of Sage, PARI/GP and Magma. (The first two are open source and free; the latter is not open source and not free.)

## Overconvergent modular forms at small weights and levels

These scripts exist in two versions. The most up-to-date is a Sage version, which will soon be available as a part of Sage. A slightly older Sage version can be downloaded from here: overconvergent.py. Examples of its use can be found in the slides for a talk I gave at the Heilbronn Institute in August 2008.

The original versions, in PARI/GP, are available below. These scripts all run happily in the antiquated GP/PARI version 2.1.6 on crackpipe.ma.ic.ac.uk, and should work on any more recent versions too. To use them, just download the script and save it, and run the GP interpreter on them by typing gp hijikata_padic.gp (for instance). This will load the script into GP and print full documentation of all the functions in each module.

• hijikata_padic.gp, which calculates the characteristic power series of the U_p operator by taking a p-adic limit in the trace formula. Valid for any p and N, and returns a power series over weight space; assumes trivial character at p. Based on hijikata.gp by William Stein.
• hijikata_verbose.gp is a minimally modified version of William Stein's hijikata.gp which returns its results in a symbolic form.
• umatrix_small_p.gp calculates the matrix of the U operator acting on overconvergent forms of tame level 1 and weight 0, when p is 2, 3, 5, 7 or 13. The calculation is "longhand", without using Kolberg's recurrences. Also includes code to calculate the T_l operators for primes l =/= p; this bit has not been very carefully checked.
• umatrix_recurrence.gp calculates the matrix of the U operator using Kolberg's recurrences; this is much faster than the above, but less full-featured, and p = 13 is not implemented yet. It also includes functions to calculate the norm of U regarded as an operator from S0(r1) to S0(r2) for various r1 < r2, which were used in the calculations for my note on the norm of the U operator.

## Automorphic forms on unitary groups

These programs are intended to accompany my paper "Explicit calculations of automorphic forms on unitary groups". They calculate automorphic forms for the unitary group in 3 variables attached to Q(sqrt(-7)).

• Classical forms of level 1: level1.zip. This is a suite of programs, mostly in Sage but using the Sage interface to Magma at one point, for computing the space of automorphic forms of level G(Z-hat). The code included can compute the Hecke operators at any split prime, and there is also experimental code for non-split primes. Users without access to Magma can still compute at 2, 3 and 11 with the pre-calculated data included. (To do: reimplement the Magma-based parts in GAP, which is included with Sage.)
• Classical forms of level 2: level2.zip. Written entirely in Magma. Level 2 here means a certain parahoric subgroup of G(Z_2) (not the Iwahori).

Last updated 2008-10-16