db-hmfs.md

Database hmfs

Description	Hilbert modular forms
Status	production
Contact	John Voight
Code	hilbert_modular_forms
Collections	fields, forms, forms.search, hecke

Notes: In good shape, but there are many remaining feature requests. The original collection 'forms' has been split: 'forms.search' contains all data fields except 'hecke_polynomial', 'hecke_eigenvalues', 'AL_eigenvalues', which makes it very much smaller and faster for operations such as searching. The collection 'hecke' contains these three data fields as well as the (full) label.

Todo: See issues

Collection fields

• Content: Totally real fields, for Hilbert modular forms
• Contributors: John Cremona, Lassina Dembele, Steve Donnelly, Aurel Page, and John Voight
• Origin: see arxiv:1605.02637
• Extent: The database contains 400 totally real fields.

Field	Description	Type of stored data	Mathematical type	Example of stored data	Remarks
_id	Mongo id	ObjectId	-		assigned by Mongo; contains creation timestamp
label	field label	string	-	'3.3.49.1'
degree	field degree	int	N	2
discriminant	field discriminant	int	Z	49
narrow_class_no	narrow class number	int	N	2
primes	fixed ordered list of primes	list of strings representing prime ideals [norm, integer generator, extra generator]	list of prime ideals	['[7, 7, 2*w^2 - w - 3]', '[8, 2, 2]', ...]
ideals	fixed ordered list of ideals	list of strings representing ideals [norm, integer generator, extra generator]	list of ideals	['[1, 1, 1]', '[7, 7, 2*w^2 - w - 3]', ...]

Index information on collection fields:

• {'_id': 1} (created by mongo)
• {'label': 1} (for retrieving data)

Collection forms

• Content: Hilbert modular forms over totally real fields
• Contributors: John Cremona, Lassina Dembele, Steve Donnelly, Aurel Page, and John Voight
• Origin: see arxiv:1605.02637
• Extent: The database contains 368,356 Hilbert newforms over 400 totally real number fields of degree up to 6.

Field	Description	Type of stored data	Mathematical type	Example of stored data	Remarks
_id	Mongo id	ObjectId	-		assigned by Mongo; contains creation timestamp
field_label	base field label	string	-	'3.3.49.1'
deg	base field degree	int	N	3
disc	base field discriminant	int	Z	49
label	form label	string	-	'3.3.49.1-27.1-a'
short_label	form short label	string	-	'27.1-a'
label_nsuffix	label numerical suffix	int	N	0
label_suffix	label (alpha) suffix	string	-	'a'
level_ideal	level ideal definition	string	ideal	'[27, 3, 3]'
level_label	level ideal label	string	-	'27.1'
level_norm	level ideal norm	int	N	27
weight	weight	string	-	'[2, 2, 2]'
parallel_weight	parallel weight	int	N	2
dimension	dimension of Hecke constituent space	int	N	1
is_CM	is a CM form?	string	bool	'no'	in {'yes','no'}
is_base_change	is a base change form?	string	bool	'yes'	in {'yes','no'}
hecke_polynomial	defining polynomial for Hecke field	string representing a polynomial in x	Z[x]	'x^2 - 5'
hecke_eigenvalues	Hecke eigenvalues	list of strings representing polynomials in x	list from Z[x]	['-5', '-4', '1', '1', ...]
AL_eigenvalues	Atkin-Lehner eigenvalues	list of lists of strings, a pair [ideal definition, eigenvalue] for each Atkin-Lehner prime	-	[['[27, 3, 3]', '-1']]

Index information on collection forms:

• {'_id': 1} (created by mongo)
• {'label': 1} (for data retrieval; use collection forms.search for searching)

Collection forms.search

• Content: Hilbert modular forms over totally real fields (data excluding Hecke polynomial and eigenvalues and Atkin-Lehner eigenvalues)
• Contributors: John Cremona, Lassina Dembele, Steve Donnelly, Aurel Page, and John Voight
• Origin: see arxiv:1605.02637
• Extent: The database contains 368,356 Hilbert newforms over 400 totally real number fields of degree up to 6.

Field	Description	Type of stored data	Mathematical type	Example of stored data	Remarks
_id	Mongo id	ObjectId	-		assigned by Mongo; contains creation timestamp
field_label	base field label	string	-	'3.3.49.1'
deg	base field degree	int	N	3
disc	base field discriminant	int	Z	49
label	form label	string	-	'3.3.49.1-27.1-a'
short_label	form short label	string	-	'27.1-a'
label_nsuffix	label numerical suffix	int	N	0
label_suffix	label (alpha) suffix	string	-	'a'
level_ideal	level ideal definition	string	ideal	'[27, 3, 3]'
level_label	level ideal label	string	-	'27.1'
level_norm	level ideal norm	int	N	27
weight	weight	string	-	'[2, 2, 2]'
parallel_weight	parallel weight	int	N	2
dimension	dimension of Hecke constituent space	int	N	1
is_CM	is a CM form?	string	bool	'no'	in {'yes','no'}
is_base_change	is a base change form?	string	bool	'yes'	in {'yes','no'}

Index information on collection forms.search:

• [('_id', 1)] (created by mongo)
• [('deg', 1), ('disc', 1), ('level_norm', 1), ('level_label', 1), ('label_nsuffix', 1)]
• [('field_label', 1), ('deg', 1), ('disc', 1), ('dimension', 1), ('level_norm', 1), ('level_label', 1), ('label_nsuffix', 1), ('is_CM', 1), ('is_base_change', 1)]
• [('label', 1)]
• [('field_label', 1), ('level_ideal', 1)]]

Collection hecke

• Content: Hilbert modular forms over totally real fields (Hecke polynomial and eigenvalues and Atkin-Lehner eigenvalues)
• Contributors: John Cremona, Lassina Dembele, Steve Donnelly, Aurel Page, and John Voight
• Origin: see arxiv:1605.02637
• Extent: The database contains 368,356 Hilbert newforms over 400 totally real number fields of degree up to 6.

Field	Description	Type of stored data	Mathematical type	Example of stored data	Remarks
_id	Mongo id	ObjectId	-		assigned by Mongo; contains creation timestamp
label	form label	string	-	'3.3.49.1-27.1-a'
hecke_polynomial	defining polynomial for Hecke field	string representing a polynomial in x	Z[x]	'x^2 - 5'
hecke_eigenvalues	Hecke eigenvalues	list of strings representing polynomials in x	list from Z[x]	['-5', '-4', '1', '1', ...]
AL_eigenvalues	Atkin-Lehner eigenvalues	list of lists of strings, a pair [ideal definition, eigenvalue] for each Atkin-Lehner prime	-	[['[27, 3, 3]', '-1']]

Index information on collection hecke:

• {'_id': 1} (created by mongo)
• {'label': 1} (for data retrieval)