# A computation of modular forms

of weight one and small level

This page contains data on classical (holomorphic) modular forms of weight one.

## Fourier expansions of cuspidal eigenforms

*Magma readable files.* Please load the instructions file magma-instructions.m
and the file of your choice below. Each such file contains q-expansions of all cuspidal
eigenforms of weight one which are new at the levels within the given bounds. (Each form is given to sufficient
precision to enable one to recover it to arbitrary precision using a magma function in
the instructions file: see the preamble in the instructions file. One Dirichlet character from
each Galois conjugacy class has been chosen: see characters.m for further details.)

Eigenforms in levels:

- from 1 to 200: qexps-1-200.m (0.1 MB)
- from 201 to 400: qexps-201-400.m (0.4 MB)
- from 401 to 600: qexps-401-600.m (0.9 MB)
- from 601 to 800: qexps-601-800.m (1.6 MB)
- from 801 to 1000: qexps-801-1000.m (2.6 MB)
- from 1001 to 1100: qexps-1001-1100.m (4 MB)
- from 1101 to 1200: qexps-1101-1200.m (3.8 MB)
- from 1201 to 1300: qexps-1201-1300.m (2.3 MB)
- from 1301 to 1400: qexps-1301-1400.m (8.1 MB)
- from 1401 to 1500: qexps-1401-1500.m (5.9 MB)

The following *very* large files give the q-expansions of the cuspidal eigenforms in levels up to 1000 to precision q^{10000}:
long-qexps-1-200.m (7 MB),
long-qexps-201-400.m (30 MB), long-qexps-401-600.m (47 MB),
long-qexps-601-800.m (70 MB),
long-qexps-801-1000.m (91 MB).

*Sage readable files.* Exactly as above for magma, but please read and load the instruction file sage-instructions.sage
and then load the file of your choice below.

Eigenforms in levels:

- from 1 to 200: qexps_1-200.sage (0.1 MB)
- from 201 to 400: qexps_201-400.sage (0.6 MB)
- from 401 to 600: qexps_401-600.sage (1.7 MB)
- from 601 to 800: qexps_601-800.sage (3.6 MB)
- from 801 to 1000: qexps_801-1000.sage (5.8 MB)

The eigenforms in level above 1000 and the *very* large files are not available in sage format.

## Eigenforms sorted by projective image

*Magma readable files.* Please load the instructions file magma-instructions.m
and the file of your choice below. Each such file contains q-expansions of all cuspidal
eigenforms of weight one with level at most 1000, or between 1001 and 1500, and whose associated Galois representation has projective
image of the given type. (As above, each form is given to sufficient
precision to enable one to recover it to arbitrary precision using a magma function in
the instructions file, and one Dirichlet character from each Galois conjugacy class has been chosen.)

- Dihedral projective image level up to 1000: dihedral.m (5 MB)
- Dihedral projective image level 1001 to 1500: dihedral_1001_1500.m (20.2 MB)
- Projective image A
_{4}level up to 1000: A4.m (0.15 MB) - Projective image A
_{4}level 1001 to 1500: A4_1001_1500.m (0.23 MB) - Projective image S
_{4}level up to 1000: S4.m (0.24 MB) - Projective image S
_{4}level 1001 to 1500: S4_1001_1500.m (3.2 MB) - Projective image A
_{5}level up to 1000: A5.m (0.05 MB) - Projective image A
_{5}level 1001 to 1500: A5_1001_1500.m (0.13 MB)

This exhausts the possible projective images. The Galois representations
with dihedral image are induced from characters of absolute Galois groups of quadratic fields. The "smallest"
dihedral form is in level 23. The smallest dihedral form whose Galois representation is induced from a character of
a real quadratic field but no imaginary quadratic field is in level 145 (there are several in that level). When the associated projective image is not
dihedral one calls the form "exotic". The smallest level in which exotic forms occur with projective image A_{4}, S_{4}, and
A_{5} are 124, 148 and 633, respectively.

*Sage readable files. * Exactly as above for magma, though only for levels up to 1000. Please read and load the instruction file sage-instructions.sage
and then load the file of your choice below.

- Dihedral projective image: dihedral_forms.sage (10.9 MB)
- Projective image A
_{4}: A4_forms.sage (0.3 MB) - Projective image S
_{4}: S4_forms.sage (0.57 MB) - Projective image A
_{5}: A5_forms.sage (0.12 MB)

## Dimensions of spaces of cuspidal eigenforms

*Magma readable files.*
The next file gives the dimensions of the old and new spaces in level N for levels N from 1 to 1000:
see the preamble and load the file.

- Dimensions in levels 1 to 1000: gamma1-dims-1-1000.m
- Dimensions in levels 1001 to 1500: gamma1-dims-1001-1500.m

The dimensions in each level are given below, broken down according to choosing a representative for each Galois conjugacy class of Dirichlet characters. (The choice of representative is as for the q-expansions above.) Please load the instructions file magma-dims-instructions.m and the file of your choice.

Dimensions by character in levels:

- from 1 to 200: dims-1-200.m
- from 201 to 400: dims-201-400.m
- from 401 to 600: dims-401-600.m
- from 601 to 800: dims-601-800.m
- from 801 to 1000: dims-801-1000.m
- from 1001 to 1100: dims-1001-1100.m
- from 1101 to 1200: dims-1101-1200.m
- from 1201 to 1300: dims-1201-1300.m
- from 1301 to 1400: dims-1301-1400.m
- from 1401 to 1500: dims-1401-1500.m

*Sage readable files.*
The dimensions of the old and new spaces in level N for levels N from 1 to 1000:
see the preamble and load the file.

- Dimensions in levels 1 to 1000: gamma1-dims_1-1000.sage
- Dimensions in levels 1001 to 1500: gamma1-dims_1001-1500.sage

## Bases of old and new cuspidal spaces

*Magma readable files only.* Please load the instructions file magma-bases-instructions.m
and the file of your choice below. Each such file contains q-expansions of bases for the old and new cuspidal spaces for all
characters with modulus in the given range. (As above, each form is given to sufficient
precision to enable one to recover it to arbitrary precision using a magma function in
the instructions file.)

Bases in levels:

- from 1 to 200: basis-1-200.m (0.3 MB)
- from 201 to 400: basis-201-400.m (0.8 MB)
- from 401 to 600: basis-401-600.m (1.3 MB)
- from 601 to 800: basis-601-800.m (2 MB)
- from 801 to 1000: basis-801-1000.m (2.7 MB)