**********************************************************************
* The data in -big.db- are
* Copyright (C) 2008 Nils-Peter Skoruppa
* Arnaud Jehanne
*
* Distributed under the terms of the GNU General Public License (GPL)
* http://www.gnu.org/licenses/
**********************************************************************
Author: Arnaud Jehanne
File Type:
plain text file which can be read into PARI/GP using under GP the
command all = read("FormesPoids1_15000.gp").
Remarks:
The PARI/GP Script -printToFiles.gp- can be used to generate the files
-form.*.*.*.txt-.
Description:
The file -FormesPoids1_15000.gp- was submitted to me by Arnaud Jehanne
(University Bordeaux I) March 2007. This file contains 27 icosahedral
forms of weight 1 and real nebentyp. The format of the file
-FormesPoids1_15000.gp- is as follows:
[space, space, ...]
space:
[ n, level, character, form, form, ..., form, 512]
weight:
should always be equal to $1$. However, Arnaud put here the number
of forms.
level:
the $l$ such that the form is a new form for $\Gamma_0(l)$.
character:
the (negative) discriminant $D$ such that the Kronecker symbol
$\big(\frac D*\big)$ is the nebentyp of the forms in this space.
form:
[ name, basis, [pol_5, pol_24], coefficients]
name:
the forms in a given space are called "a", "b", ....
basis:
[1, Mod(1/2*y - 1/2, y^2 - 5), I, Mod(1/2*I*y - 1/2*I, y^2 - 5)]~
pol_5:
polynomial whose decomposition field is the fixed field
of the kernel of the corresponding projective Galois representation
given as list of the coefficients in ascending order.
pol_24:
polynomial whose decomposition field is the fixed field
of the kernel of the corresponding (proper) Galois representation
given as list of the coefficients in ascending order.
coefficients:
5000 x 4 matrix $M$ , such that the vector $M*(1, S, i, iS)^t$
contains the first 15000 Fourier coefficients of the form in
question. Here $S=(-1+sqrt(5))/2$ and $i = \sqrt{-1}$.
When working under PARI/GP you may use M*basis for obtaining
the vector containing these Fourier coefficients.
The number 512 is meaningless (it used to be magic number for a PARI/GP
program).
The levels and characters of the 18 spaces containing these 27 forms
are as follows:
No. level char. n. of forms
-----------------------------------
1 1948 -487 1
2 2083 -2083 1
3 2336 -292 2
4 2707 -2707 1
5 2863 -2863 2
6 3004 -751 1
7 3203 -3203 1
8 3547 -3547 1
9 3548 -887 1
10 3587 -3587 2
11 3676 -919 1
12 3775 -151 2
13 3775 -755 2
14 3875 -31 2
15 3875 -155 2
16 4000 -4 2
17 4000 -20 2
18 4027 -4027 1
Copyright (C) 2007
Arnaud Jehanne (Universit\'e Bordeaux I)
Nils Skoruppa (University of Siegen)