In his 2002 University of London PhD thesis, Daniel Jacobs proved results about the 3-adic slopes of the U_3 operator acting on certain spaces of automorphic forms defined over the definite quaternion algebra Q_2. He wrote gp-pari programs that computed these slopes.

His thesis contains one of these programs, and can be obtained from here.
It's a nice introduction to the subject, as is Kevin Buzzard's
paper *On p-adic families of automorphic forms*, which is available
from his
webpage.

I have been working on generalizing these programs to different spaces of automorphic forms and different U_p.

Here are the files for my program.

The generalized program which calculates the matrix of the U_p operator and its slopes

Auxiliary file which implements arithmetic functions on quaternions

Some code that computes the T_l operator (for l!=p)

There is also code to compute these automorphic forms with even smaller level structure at 2:

The generalized program which calculates the matrix of the U_p operator and its slopes

Auxiliary file which implements arithmetic functions on quaternions

Functions to compute T_l for l!=p

This program runs under GP-Pari and works okay with version 2.2.9. It should work with other recent versions of Pari.

Here's an example run:

[Pari header]? \r quaternionic_modular_forms.g

******************************

Program for calculating the action of U_p on quaternionic modular forms

Original written by Dan Jacobs, (c) 2002

Extended by L J P Kilford, (c) 2004-2005

******************************

Version 0.1

Verbose mode on

The prime p is 11

with extra level structure at 2

of the smaller level U_{sq}(11)

The p-adic accuracy is 40

The cosets are [[0, 10; 1, 0], [9, 0; 1, 5], [9, 0; 4, 5]]

Testing the factorisations ...

The factorisations are OK

Functions for testing matrices

Further factorisations

Finding the matrices for the generating functions

Defining the U_p operator

1

0

%27 = [2, 1, 1, 1, 1, 0]

Upmatrix(p_adic_accuracy,wt,dimension) - calculates the (dimension x dimension) matrix of the U_p operator acting on weight wt automorphic forms, calculated to p-adic precision p_adic_accuracy

charpowerseries(p_adic_accuracy,wt,dimension) - calculates the characteristic power series of Upmatrix(p_adic_accuracy,wt,dimension)

findSlopes(p_adic_accuracy,wt,dimension) - calculates the Newton slopes of charpowerseries(p_adic_accuracy,wt,dimension)

Tlmatrix(p_adic_accuracy,wt,dimension) - calculates the (dimension x dimension) matrix of the T_l operator acting on weight wt automorphic forms, calculated to p-adic precision p_adic_accuracy

charpowerseries_l(p_adic_accuracy,wt,dimension) - calculates the characteristic power series of Tlmatrix(p_adic_accuracy,wt,dimension)

findSlopes_l(p_adic_accuracy,wt,dimension) - calculates the Newton slopes of charpowerseries_l(p_adic_accuracy,wt,dimension)