def friends(self):
f = []
cglink = url_character(type=self.type,number_field=self.nflabel,modulus=self.modlabel)
f.append( ("Character group", cglink) )
if self.nflabel:
f.append( ('Number Field', '/NumberField/' + self.nflabel) )
if self.type == 'Dirichlet':
f.append( ('L function', '/L'+ url_character(type=self.type,
number_field=self.nflabel,
modulus=self.modlabel,
number=self.numlabel) ) )
f.append( ("Value Field", '/NumberField/' + self.vflabel) )
return f
def character_search(**args):
info = to_dict(args)
for field in ['modulus', 'conductor', 'order']:
info[field] = info.get(field, '')
query = {}
if 'natural' in args:
label = info.get('natural', '')
try:
modulus = int(str(label).partition('.')[0])
number = int(str(label).partition('.')[2])
except ValueError:
return "<span style='color:red;'>ERROR: bad query</span>"
return redirect(url_character(type='Dirichlet',modulus=modulus, number=number))
else:
for field in ['modulus', 'conductor', 'order']:
if info.get(field):
query[field] = parse_range(info[field])
info['bread'] = [('Characters','/Character'), ('search results', ' ') ]
info['credit'] = 'Sage'
if (len(query) != 0):
from sage.modular.dirichlet import DirichletGroup
info['contents'] = charactertable(query)
info['title'] = 'Dirichlet Characters'
return render_template("character_search.html", **info)
else:
return "<span style='color:red;'>ERROR: bad query</span>"
def friends(self):
f = []
cglink = url_character(type=self.type,
number_field=self.nflabel,
modulus=self.modlabel)
f.append(("Character group", cglink))
if self.nflabel:
f.append(('Number Field', '/NumberField/' + self.nflabel))
if self.type == 'Dirichlet':
f.append(
('L function', '/L' + url_character(type=self.type,
number_field=self.nflabel,
modulus=self.modlabel,
number=self.numlabel)))
f.append(("Value Field", '/NumberField/' + self.vflabel))
return f
def character_search(**args):
info = to_dict(args)
for field in ["modulus", "conductor", "order"]:
info[field] = info.get(field, "")
query = {}
if "natural" in args:
label = info.get("natural", "")
try:
modulus = int(str(label).partition(".")[0])
number = int(str(label).partition(".")[2])
except ValueError:
return "<span style='color:red;'>ERROR: bad query</span>"
return redirect(url_character(type="Dirichlet", modulus=modulus, number=number))
else:
for field in ["modulus", "conductor", "order"]:
if info.get(field):
query[field] = parse_range(info[field])
info["bread"] = [("Characters", "/Character"), ("search results", " ")]
info["credit"] = "Sage"
if len(query) != 0:
from sage.modular.dirichlet import DirichletGroup
info["contents"] = charactertable(query)
info["title"] = "Dirichlet Characters"
return render_template("character_search.html", **info)
else:
return "<span style='color:red;'>ERROR: bad query</span>"
def character_search(**args):
info = to_dict(args)
for field in ['modulus', 'conductor', 'order']:
info[field] = info.get(field, '')
query = {}
if 'natural' in args:
label = info.get('natural', '')
try:
modulus = int(str(label).partition('.')[0])
number = int(str(label).partition('.')[2])
except ValueError:
return "<span style='color:red;'>ERROR: bad query</span>"
return redirect(
url_character(type='Dirichlet', modulus=modulus, number=number))
else:
for field in ['modulus', 'conductor', 'order']:
if info.get(field):
query[field] = parse_range(info[field])
info['bread'] = [('Characters', '/Character'), ('search results', ' ')]
info['credit'] = 'Sage'
if (len(query) != 0):
from sage.modular.dirichlet import DirichletGroup
info['contents'] = charactertable(query)
info['title'] = 'Dirichlet Characters'
return render_template("character_search.html", **info)
else:
return "<span style='color:red;'>ERROR: bad query</span>"
def set_info_for_modular_form_space(level=None, weight=None, character=None, label=None, **kwds):
r"""
Set information about a space of modular forms.
"""
info = dict()
info['level'] = level
info['weight'] = weight
info['character'] = character
emf_logger.debug("info={0}".format(info))
if(level > N_max_db or weight > k_max_db):
info['error'] = "Currently not available"
WMFS = None
if level <= 0:
info['error'] = "Got wrong level: %s " % level
return info
try:
if use_db:
WMFS = WebModFormSpace(weight, level, character, use_db=True)
else:
WMFS = WebModFormSpace(weight, level, character)
if 'download' in info and 'tempfile' in info:
WNF._save_to_file(info['tempfile'])
info['filename'] = str(weight) + '-' + str(level) + '-' + str(character) + '-' + label + '.sobj'
return info
except RuntimeError:
info['error'] = "Sage error: Could not construct the desired space!"
if WMFS.level() == 1:
info['group'] = "\( \mathrm{SL}_{2}(\mathbb{Z})\)"
else:
info['group'] = "\( \Gamma_{{0}}( {0} ) \)".format(WMFS.level())
if character == 0:
info['name_new'] = "\(S_{ %s }^{new}(%s) \)" % (WMFS.weight(), WMFS.level())
info['name_old'] = "\(S_{ %s }^{old}(%s) \)" % (WMFS.weight(), WMFS.level())
else:
conrey_char = WMFS.conrey_character()
conrey_char_name = WMFS.conrey_character_name()
info['conrey_character_name'] = '\( ' + conrey_char_name + '\)'
info['character_url'] = url_character(type='Dirichlet', modulus=WMFS.level(), number=conrey_char.number())
info['name_new'] = "\(S_{ %s }^{new}(%s,%s) \)" % (WMFS.weight(), WMFS.level(), conrey_char_name)
info['name_old'] = "\(S_{ %s }^{old}(%s,%s) \)" % (WMFS.weight(), WMFS.level(), conrey_char_name)
info['dimension_cusp_forms'] = WMFS.dimension_cusp_forms()
info['dimension_mod_forms'] = WMFS.dimension_modular_forms()
info['dimension_new_cusp_forms'] = WMFS.dimension_new_cusp_forms()
info['dimension_newspace'] = WMFS.dimension_newspace()
info['dimension_oldspace'] = WMFS.dimension_oldspace()
info['dimension'] = WMFS.dimension()
info['galois_orbits'] = WMFS.get_all_galois_orbit_info()
lifts = list()
if WMFS.dimension() == 0: # we don't need to work with an empty space
info['sturm_bound'] = 0
info['new_decomposition'] = ''
info['is_empty'] = 1
lifts.append(('Half-Integral Weight Forms', '/ModularForm/Mp2/Q'))
lifts.append(('Siegel Modular Forms', '/ModularForm/GSp4/Q'))
info['lifts'] = lifts
return info
info['sturm_bound'] = WMFS.sturm_bound()
info['new_decomposition'] = WMFS.print_galois_orbits()
emf_logger.debug("new_decomp={0}".format(info['new_decomposition']))
info['nontrivial_new'] = len(info['new_decomposition'])
## we try to catch well-known bugs...
info['old_decomposition'] = "n/a"
if level < N_max_comp:
try:
O = WMFS.print_oldspace_decomposition()
info['old_decomposition'] = O
except:
emf_logger.critical("Error in computing oldspace decomposition")
O = []
info['old_decomposition'] = "n/a"
(A, B, C) = sys.exc_info()
# build an error message...
errtype = A.__name__
errmsg = B
s = "%s: %s at:" % (errtype, errmsg)
next = C.tb_next
while(next):
ln = next.tb_lineno
filen = next.tb_frame.f_code.co_filename
s += "\n line no. %s in file %s" % (ln, filen)
next = next.tb_next
info['error_note'] = "Could not construct oldspace!\n" + s
# properties for the sidebar
prop = []
if WMFS._cuspidal == 1:
prop = [('Dimension newforms', [info['dimension_newspace']])]
prop.append(('Dimension oldforms', [info['dimension_oldspace']]))
else:
prop = [('Dimension modular forms', [info['dimension_mod_forms']])]
prop.append(('Dimension cusp forms', [info['dimension_cusp_forms']]))
prop.append(('Sturm bound', [WMFS.sturm_bound()]))
info['properties2'] = prop
## Make parent spaces of S_k(N,chi) for the sidebar
par_lbl = '\( S_{*} (\Gamma_0(' + str(level) + '),\cdot )\)'
par_url = '?level=' + str(level)
parents = [[par_lbl, par_url]]
par_lbl = '\( S_{k} (\Gamma_0(' + str(level) + '),\cdot )\)'
par_url = '?level=' + str(level) + '&weight=' + str(weight)
parents.append((par_lbl, par_url))
info['parents'] = parents
if 'character' in info:
info['character_order'] = WMFS.character_order()
info['character_conductor'] = WMFS.character_conductor()
friends = list()
lifts = list()
if(('label' not in info) and info['old_decomposition'] != 'n/a'):
O = WMFS.oldspace_decomposition()
try:
for (old_level, chi, mult, d) in O:
if chi != 0:
s = "\(S_{%s}(\Gamma_0(%s),\chi_{%s}) \) " % (weight, old_level, chi)
friends.append((
s, '?weight=' + str(weight) + '&level=' + str(old_level) + '&character=' + str(chi)))
else:
s = "\(S_{%s}(\Gamma_0(%s)) \) " % (weight, old_level)
friends.append(
(s, '?weight=' + str(weight) + '&level=' + str(old_level) + '&character=' + str(0)))
except:
pass
info['friends'] = friends
lifts.append(('Half-Integral Weight Forms', '/ModularForm/Mp2/Q'))
lifts.append(('Siegel Modular Forms', '/ModularForm/GSp4/Q'))
info['lifts'] = lifts
return info
def evs_table2(search, twodarray=False):
DB = connect_db()
table = []
nrows = 0
fs = DB.get_Maass_forms(search)
mwf_logger.debug("numrec:{0}".format(len(fs)))
for f in fs: # indices:
row = {}
R = f.get('Eigenvalue', None)
N = f.get('Level', None)
k = f.get('Weight', None)
if R is None or N is None or k is None:
continue
row['R'] = R
row['N'] = N
if k == 0 or k == 1:
row['k'] = int(k)
else:
row['k'] = k
##
chi = f.get('Character', 0)
conrey = f.get('Conrey', 0)
if conrey == 0: # we need to change to conrey's notation
chi = DB.getDircharConrey(N, chi)
## Now get the COnrey number.
## First the character
if k == 0:
url = url_character(type='Dirichlet', modulus=N, number=chi)
s = "<a href={0}>{1}</a>".format(url, chi)
row['ch'] = s
else:
row['ch'] = "eta"
st = f.get('Symmetry', -1)
if st == 1:
st = "odd"
elif st == 0:
st = "even"
else:
st = "n/a"
row['symmetry'] = st
er = f.get('Error', 0)
if er > 0:
er = "{0:1.0e}".format(er)
else:
er = "unknown"
row['err'] = er
dim = f.get('Dim', 0)
if dim is None:
dim = 1 # "undefined"
row['dim'] = dim
numc = f.get('Numc', 0)
row['numc'] = numc
cev = f.get('Cusp_evs', [])
row['fricke'] = 'n/a'
row['cuspevs'] = 'n/a'
if row['k'] == 0 and isinstance(cev, list):
if len(cev) > 1:
fricke = cev[1]
row['fricke'] = fricke
s = '{0}'.format(cev[0])
for j in range(1, len(cev)):
s += ",{0}".format(cev[j])
elif len(cev) == 1:
s = str(cev[0])
elif len(cev) == 0:
s = 'n/a'
row['cuspevs'] = s
url = url_for('mwf.render_one_maass_waveform', maass_id=f.get('_id'))
row['url'] = url
nrows += 1
if twodarray:
s = '<a href="{0}">{1}</a>'.format(row['url'], row['R'])
rowr = [row['N'], row['k'], row['ch'], s,
row['symmetry'], row['err'], row['dim'], row['numc'],
row['fricke'], row['cuspevs']]
table.append(rowr)
else:
# row=row.values()
table.append(row)
mwf_logger.debug("nrows:".format(nrows))
evs = {'table': {}}
evs['table']['data'] = table
evs['table']['nrows'] = nrows
evs['table']['ncols'] = 10
evs['table']['colheads'] = []
knowls = ['mf.maass.mwf.level', 'mf.maass.mwf.weight', 'mf.maass.mwf.character',
'mf.maass.mwf.eigenvalue', 'mf.maass.mwf.symmetry',
'mf.maass.mwf.precision', 'mf.maass.mwf.dimension',
'mf.maass.mwf.ncoefficients', 'mf.maass.mwf.fricke',
'mf.maass.mwf.atkinlehner']
titles = ['Level', 'Weight', 'Char',
'Eigenvalue', 'Symmetry',
'Precision', 'Mult.',
'Coeff.', 'Fricke', 'Atkin-Lehner']
for i in range(10):
evs['table']['colheads'].append((knowls[i], titles[i]))
if 'limit' in search:
search.pop('limit')
if 'skip' in search:
search.pop('skip')
evs['totalrecords'] = DB.count(search, filtered=False)
evs['totalrecords_filtered'] = DB.count(search, filtered=True)
return evs
def dirichlet_table():
args = to_dict(request.args)
mod = args.get("modulus", 1)
return redirect(url_character(type="Dirichlet", modulus=mod))
def navi(L):
r = [(l, url_character(**args)) for l, args in L if l]
return r
def render_field_webpage(args):
data = None
C = base.getDBConnection()
info = {}
bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]
if 'label' in args:
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(('Search results', ' '))
label2 = re.sub(r'[<>]', '', args['label'])
if 'You need to enter a field' in label2:
info['err'] = label2
else:
info['err'] = 'No such field: %s in the database' % label2
info['label'] = args['label_orig'] if 'label_orig' in args else args['label']
return search_input_error(info, bread)
info['wnf'] = nf
from lmfdb.WebNumberField import nf_display_knowl
data['degree'] = nf.degree()
data['class_number'] = nf.class_number()
t = nf.galois_t()
n = nf.degree()
data['is_galois'] = nf.is_galois()
data['is_abelian'] = nf.is_abelian()
if nf.is_abelian():
conductor = nf.conductor()
data['conductor'] = conductor
dirichlet_chars = nf.dirichlet_group()
if len(dirichlet_chars)>0:
data['dirichlet_group'] = ['<a href = "%s">$\chi_{%s}(%s,·)$</a>' % (url_character(type='Dirichlet',modulus=data['conductor'], number=j), data['conductor'], j) for j in dirichlet_chars]
data['dirichlet_group'] = r'$\lbrace$' + ', '.join(data['dirichlet_group']) + r'$\rbrace$'
if data['conductor'].is_prime() or data['conductor'] == 1:
data['conductor'] = "\(%s\)" % str(data['conductor'])
else:
data['conductor'] = "\(%s=%s\)" % (str(data['conductor']), latex(data['conductor'].factor()))
data['galois_group'] = group_display_knowl(n, t, C)
data['cclasses'] = cclasses_display_knowl(n, t, C)
data['character_table'] = character_table_display_knowl(n, t, C)
data['class_group'] = nf.class_group()
data['class_group_invs'] = nf.class_group_invariants()
data['signature'] = nf.signature()
data['coefficients'] = nf.coeffs()
D = nf.disc()
ram_primes = D.prime_factors()
data['disc_factor'] = nf.disc_factored_latex()
if D.abs().is_prime() or D == 1:
data['discriminant'] = "\(%s\)" % str(D)
else:
data['discriminant'] = "\(%s=%s\)" % (str(D), data['disc_factor'])
npr = len(ram_primes)
ram_primes = str(ram_primes)[1:-1]
if ram_primes == '':
ram_primes = r'\textrm{None}'
data['frob_data'], data['seeram'] = frobs(nf.K())
data['phrase'] = group_phrase(n, t, C)
zk = pari(nf.K()).nf_subst('a')
zk = list(zk.nf_get_zk())
Ra = PolynomialRing(QQ, 'a')
zk = [latex(Ra(x)) for x in zk]
zk = ['$%s$' % x for x in zk]
zk = ', '.join(zk)
grh_label = '<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh() else ''
# Short version for properties
grh_lab = nf.short_grh_string()
pretty_label = field_pretty(label)
if label != pretty_label:
pretty_label = "%s: %s" % (label, pretty_label)
info.update(data)
info.update({
'label': pretty_label,
'label_raw': label,
'polynomial': web_latex_split_on_pm(nf.K().defining_polynomial()),
'ram_primes': ram_primes,
'integral_basis': zk,
'regulator': web_latex(nf.regulator()),
'unit_rank': nf.unit_rank(),
'root_of_unity': web_latex(nf.K().primitive_root_of_unity()),
'fund_units': nf.units(),
'grh_label': grh_label
})
bread.append(('%s' % info['label_raw'], ' '))
info['downloads_visible'] = True
info['downloads'] = [('worksheet', '/')]
info['friends'] = []
if nf.can_class_number():
info['friends'].append(('L-function', "/L/NumberField/%s" % label))
info['friends'].append(('Galois group', "/GaloisGroup/%dT%d" % (n, t)))
if 'dirichlet_group' in info:
info['friends'].append(('Dirichlet group', url_for("characters.dirichlet_group_table",
modulus=int(conductor),
char_number_list=','.join(
[str(a) for a in dirichlet_chars]),
poly=info['polynomial'])))
info['learnmore'] = [('Global number field labels', url_for(
".render_labels_page")), (Completename, url_for(".render_discriminants_page"))]
# With Galois group labels, probably not needed here
# info['learnmore'] = [('Global number field labels',
# url_for(".render_labels_page")), ('Galois group
# labels',url_for(".render_groups_page")),
# (Completename,url_for(".render_discriminants_page"))]
title = "Global Number Field %s" % info['label']
if npr == 1:
primes = 'prime'
else:
primes = 'primes'
properties2 = [('Degree:', '%s' % data['degree']),
('Signature:', '$%s$' % data['signature']),
('Discriminant:', '$%s$' % data['disc_factor']),
('Ramified ' + primes + ':', '$%s$' % ram_primes),
('Class number:', '%s %s' % (data['class_number'], grh_lab)),
('Class group:', '%s %s' % (data['class_group_invs'], grh_lab)),
('Galois Group:', group_display_short(data['degree'], t, C))
]
from lmfdb.math_classes import NumberFieldGaloisGroup
try:
info["tim_number_field"] = NumberFieldGaloisGroup.find_one({"label": label})
v = nf.factor_perm_repn(info["tim_number_field"])
info["mydecomp"] = ['*' if x>0 else '' for x in v]
except AttributeError:
pass
# del info['_id']
return render_template("number_field.html", properties2=properties2, credit=NF_credit, title=title, bread=bread, friends=info.pop('friends'), learnmore=info.pop('learnmore'), info=info)
def render_field_webpage(args):
data = None
C = base.getDBConnection()
info = {}
bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]
if 'label' in args:
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(('Search results', ' '))
label2 = re.sub(r'[<>]', '', args['label'])
if 'You need to enter a field' in label2:
info['err'] = label2
else:
info['err'] = 'No such field: %s in the database' % label2
info['label'] = args['label_orig'] if 'label_orig' in args else args['label']
return search_input_error(info, bread)
info['wnf'] = nf
data['degree'] = nf.degree()
data['class_number'] = nf.class_number()
t = nf.galois_t()
n = nf.degree()
data['is_galois'] = nf.is_galois()
data['is_abelian'] = nf.is_abelian()
if nf.is_abelian():
conductor = nf.conductor()
data['conductor'] = conductor
dirichlet_chars = nf.dirichlet_group()
if len(dirichlet_chars)>0:
data['dirichlet_group'] = ['<a href = "%s">$\chi_{%s}(%s,·)$</a>' % (url_character(type='Dirichlet',modulus=data['conductor'], number=j), data['conductor'], j) for j in dirichlet_chars]
data['dirichlet_group'] = r'$\lbrace$' + ', '.join(data['dirichlet_group']) + r'$\rbrace$'
if data['conductor'].is_prime() or data['conductor'] == 1:
data['conductor'] = "\(%s\)" % str(data['conductor'])
else:
data['conductor'] = "\(%s=%s\)" % (str(data['conductor']), latex(data['conductor'].factor()))
data['galois_group'] = group_display_knowl(n, t, C)
data['cclasses'] = cclasses_display_knowl(n, t, C)
data['character_table'] = character_table_display_knowl(n, t, C)
data['class_group'] = nf.class_group()
data['class_group_invs'] = nf.class_group_invariants()
data['signature'] = nf.signature()
data['coefficients'] = nf.coeffs()
D = nf.disc()
ram_primes = D.prime_factors()
data['disc_factor'] = nf.disc_factored_latex()
if D.abs().is_prime() or D == 1:
data['discriminant'] = "\(%s\)" % str(D)
else:
data['discriminant'] = "\(%s=%s\)" % (str(D), data['disc_factor'])
npr = len(ram_primes)
ram_primes = str(ram_primes)[1:-1]
if ram_primes == '':
ram_primes = r'\textrm{None}'
data['frob_data'], data['seeram'] = frobs(nf.K())
data['phrase'] = group_phrase(n, t, C)
zk = pari(nf.K()).nf_subst('a')
zk = list(zk.nf_get_zk())
Ra = PolynomialRing(QQ, 'a')
zk = [latex(Ra(x)) for x in zk]
zk = ['$%s$' % x for x in zk]
zk = ', '.join(zk)
grh_label = '<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh() else ''
# Short version for properties
grh_lab = nf.short_grh_string()
pretty_label = field_pretty(label)
if label != pretty_label:
pretty_label = "%s: %s" % (label, pretty_label)
info.update(data)
info.update({
'label': pretty_label,
'label_raw': label,
'polynomial': web_latex_split_on_pm(nf.K().defining_polynomial()),
'ram_primes': ram_primes,
'integral_basis': zk,
'regulator': web_latex(nf.regulator()),
'unit_rank': nf.unit_rank(),
'root_of_unity': web_latex(nf.K().primitive_root_of_unity()),
'fund_units': nf.units(),
'grh_label': grh_label
})
bread.append(('%s' % info['label_raw'], ' '))
info['downloads_visible'] = True
info['downloads'] = [('worksheet', '/')]
info['friends'] = []
if nf.can_class_number():
info['friends'].append(('L-function', "/L/NumberField/%s" % label))
info['friends'].append(('Galois group', "/GaloisGroup/%dT%d" % (n, t)))
if 'dirichlet_group' in info:
info['friends'].append(('Dirichlet group', url_for("characters.dirichlet_group_table",
modulus=int(conductor),
char_number_list=','.join(
[str(a) for a in dirichlet_chars]),
poly=info['polynomial'])))
info['learnmore'] = [('Global number field labels', url_for(
".render_labels_page")), (Completename, url_for(".render_discriminants_page"))]
# With Galois group labels, probably not needed here
# info['learnmore'] = [('Global number field labels',
# url_for(".render_labels_page")), ('Galois group
# labels',url_for(".render_groups_page")),
# (Completename,url_for(".render_discriminants_page"))]
title = "Global Number Field %s" % info['label']
if npr == 1:
primes = 'prime'
else:
primes = 'primes'
properties2 = [('Degree:', '%s' % data['degree']),
('Signature:', '$%s$' % data['signature']),
('Discriminant:', '$%s$' % data['disc_factor']),
('Ramified ' + primes + ':', '$%s$' % ram_primes),
('Class number:', '%s %s' % (data['class_number'], grh_lab)),
('Class group:', '%s %s' % (data['class_group_invs'], grh_lab)),
('Galois Group:', group_display_short(data['degree'], t, C))
]
from lmfdb.math_classes import NumberFieldGaloisGroup
try:
info["tim_number_field"] = NumberFieldGaloisGroup.find_one({"label": label})
v = nf.factor_perm_repn(info["tim_number_field"])
info["mydecomp"] = ['*' if x>0 else '' for x in v]
except AttributeError:
pass
# del info['_id']
return render_template("number_field.html", properties2=properties2, credit=NF_credit, title=title, bread=bread, friends=info.pop('friends'), learnmore=info.pop('learnmore'), info=info)
def initLfunction(L, args, request):
''' Sets the properties to show on the homepage of an L-function page.
'''
info = {'title': L.title}
try:
info['citation'] = L.citation
except AttributeError:
info['citation'] = ""
try:
info['support'] = L.support
except AttributeError:
info['support'] = ""
info['Ltype'] = L.Ltype()
# Here we should decide which values are indeed special values
# According to Brian, odd degree has special value at 1, and even
# degree has special value at 1/2.
# (however, I'm not sure this is true if L is not primitive -- GT)
# Now we usually display both
if L.Ltype() != "artin" or (L.Ltype() == "artin" and L.sign != 0):
# if is_even(L.degree) :
# info['sv12'] = specialValueString(L, 0.5, '1/2')
# if is_odd(L.degree):
# info['sv1'] = specialValueString(L, 1, '1')
info['sv1'] = specialValueString(L, 1, '1')
info['sv12'] = specialValueString(L, 0.5, '1/2')
info['args'] = args
info['credit'] = L.credit
# info['citation'] = L.citation
try:
info['factorization'] = L.factorization
except:
pass
try:
info['url'] = L.url
except:
info['url'] = ''
info['degree'] = int(L.degree)
info['zeroeslink'] = (request.url.replace('/L/', '/L/Zeros/').replace(
'/Lfunction/', '/L/Zeros/').replace('/L-function/', '/L/Zeros/')
) # url_for('zeroesLfunction', **args)
info['plotlink'] = (
request.url.replace('/L/', '/L/Plot/').replace(
'/Lfunction/', '/L/Plot/').replace('/L-function/', '/L/Plot/')
) # info['plotlink'] = url_for('plotLfunction', **args)
info['bread'] = []
info['properties2'] = set_gaga_properties(L)
# Create friendlink by removing 'L/' and ending '/'
friendlink = request.url.replace('/L/',
'/').replace('/L-function/', '/').replace(
'/Lfunction/', '/')
splitlink = friendlink.rpartition('/')
friendlink = splitlink[0] + splitlink[2]
logger.debug(L.Ltype())
if L.Ltype() == 'maass':
if L.group == 'GL2':
minNumberOfCoefficients = 100 # TODO: Fix this to take level into account
if len(L.dirichlet_coefficients) < minNumberOfCoefficients:
info['zeroeslink'] = ''
info['plotlink'] = ''
info['bread'] = get_bread(
2, [('Maass Form', url_for('.l_function_maass_browse_page')),
('\(' + L.texname + '\)', request.url)])
info['friends'] = [('Maass Form ', friendlink)]
else:
info['bread'] = get_bread(
L.degree, [('Maass Form',
url_for('.l_function_maass_gln_browse_page',
degree='degree' + str(L.degree))),
(L.dbid, request.url)])
elif L.Ltype() == 'riemann':
info['bread'] = get_bread(1, [('Riemann Zeta', request.url)])
info['friends'] = [('\(\mathbb Q\)',
url_for('number_fields.by_label',
label='1.1.1.1')),
('Dirichlet Character \(\\chi_{1}(1,\\cdot)\)',
url_character(type='Dirichlet',
modulus=1,
number=1))]
elif L.Ltype() == 'dirichlet':
mod, num = L.charactermodulus, L.characternumber
Lpattern = r"\(L(s,\chi_{%s}(%s,·))\)"
if mod > 1:
pmod, pnum = WebDirichlet.prevprimchar(mod, num)
Lprev = (Lpattern % (pmod, pnum),
url_for('.l_function_dirichlet_page',
modulus=pmod,
number=pnum))
else:
Lprev = ('', '')
nmod, nnum = WebDirichlet.nextprimchar(mod, num)
Lnext = (Lpattern % (nmod, nnum),
url_for('.l_function_dirichlet_page',
modulus=nmod,
number=nnum))
info['navi'] = (Lprev, Lnext)
snum = str(L.characternumber)
smod = str(L.charactermodulus)
charname = WebDirichlet.char2tex(smod, snum)
info['bread'] = get_bread(1, [(charname, request.url)])
info['friends'] = [('Dirichlet Character ' + str(charname), friendlink)
]
elif L.Ltype() == 'ellipticcurveQ':
label = L.label
while friendlink[len(friendlink) - 1].isdigit(
): # Remove any number at the end to get isogeny class url
friendlink = friendlink[0:len(friendlink) - 1]
info['friends'] = [('Isogeny class ' + label, friendlink)]
for i in range(1, L.nr_of_curves_in_class + 1):
info['friends'].append(
('Elliptic curve ' + label + str(i), friendlink + str(i)))
if L.modform:
info['friends'].append(
('Modular form ' + label.replace('.', '.2'),
url_for("emf.render_elliptic_modular_forms",
level=L.modform['level'],
weight=2,
character=0,
label=L.modform['iso'])))
info['friends'].append(('L-function ' + label.replace('.', '.2'),
url_for('.l_function_emf_page',
level=L.modform['level'],
weight=2,
character=0,
label=L.modform['iso'],
number=0)))
info['friends'].append(('Symmetric square L-function',
url_for(".l_function_ec_sym_page",
power='2',
label=label)))
info['friends'].append(('Symmetric cube L-function',
url_for(".l_function_ec_sym_page",
power='3',
label=label)))
info['bread'] = get_bread(
2, [('Elliptic curve', url_for('.l_function_ec_browse_page')),
(label, url_for('.l_function_ec_page', label=label))])
elif L.Ltype() == 'ellipticmodularform':
friendlink = friendlink + L.addToLink # Strips off the embedding
friendlink = friendlink.rpartition('/')[0] # number for the L-function
if L.character:
info['friends'] = [
('Modular form ' + str(L.level) + '.' + str(L.weight) + '.' +
str(L.character) + str(L.label), friendlink)
]
else:
info['friends'] = [('Modular form ' + str(L.level) + '.' +
str(L.weight) + str(L.label), friendlink)]
if L.ellipticcurve:
info['friends'].append(('EC isogeny class ' + L.ellipticcurve,
url_for("by_ec_label",
label=L.ellipticcurve)))
info['friends'].append(
('L-function ' + str(L.level) + '.' + str(L.label),
url_for('.l_function_ec_page', label=L.ellipticcurve)))
for i in range(1, L.nr_of_curves_in_class + 1):
info['friends'].append(
('Elliptic curve ' + L.ellipticcurve + str(i),
url_for("by_ec_label", label=L.ellipticcurve + str(i))))
info['friends'].append(('Symmetric square L-function',
url_for(".l_function_ec_sym_page",
power='2',
label=label)))
info['friends'].append(('Symmetric cube L-function',
url_for(".l_function_ec_sym_page",
power='3',
label=label)))
elif L.Ltype() == 'hilbertmodularform':
friendlink = '/'.join(friendlink.split('/')[:-1])
info['friends'] = [('Hilbert Modular Form',
friendlink.rpartition('/')[0])]
elif L.Ltype() == 'dedekindzeta':
info['friends'] = [('Number Field', friendlink)]
elif L.Ltype() in ['lcalcurl', 'lcalcfile']:
info['bread'] = [('L-functions', url_for('.l_function_top_page'))]
elif L.Ltype() == 'SymmetricPower':
def ordinal(n):
if n == 2:
return "Square"
elif n == 3:
return "Cube"
elif 10 <= n % 100 < 20:
return str(n) + "th Power"
else:
return str(n) + {
1: 'st',
2: 'nd',
3: 'rd'
}.get(n % 10, "th") + " Power"
if L.m == 2:
info['bread'] = get_bread(3, [
("Symmetric square of Elliptic curve",
url_for('.l_function_ec_sym2_browse_page')),
(L.label,
url_for('.l_function_ec_sym_page', label=L.label, power=L.m))
])
elif L.m == 3:
info['bread'] = get_bread(4, [
("Symmetric cube of Elliptic curve",
url_for('.l_function_ec_sym3_browse_page')),
(L.label,
url_for('.l_function_ec_sym_page', label=L.label, power=L.m))
])
else:
info['bread'] = [
('L-functions', url_for('.l_function_top_page')),
('Symmetric %s of Elliptic curve ' % ordinal(L.m) +
str(L.label),
url_for('.l_function_ec_sym_page', label=L.label, power=L.m))
]
friendlink = request.url.replace('/L/SymmetricPower/%d/' % L.m, '/')
splitlink = friendlink.rpartition('/')
friendlink = splitlink[0] + splitlink[2]
friendlink2 = request.url.replace('/L/SymmetricPower/%d/' % L.m, '/L/')
splitlink = friendlink2.rpartition('/')
friendlink2 = splitlink[0] + splitlink[2]
info['friends'] = [('Isogeny class ' + L.label, friendlink),
('Symmetric 1st Power', friendlink2)]
for j in range(2, L.m + 2):
if j != L.m:
friendlink3 = request.url.replace(
'/L/SymmetricPower/%d/' % L.m, '/L/SymmetricPower/%d/' % j)
info['friends'].append(
('Symmetric %s' % ordinal(j), friendlink3))
elif L.Ltype() == 'siegelnonlift' or L.Ltype(
) == 'siegeleisenstein' or L.Ltype(
) == 'siegelklingeneisenstein' or L.Ltype() == 'siegelmaasslift':
weight = str(L.weight)
number = str(L.number)
info['friends'] = [('Siegel Modular Form', friendlink)]
elif L.Ltype() == "artin":
# info['zeroeslink'] = ''
# info['plotlink'] = ''
info['friends'] = [('Artin representation', L.artin.url_for())]
if L.sign == 0: # The root number is now unknown
info['zeroeslink'] = ''
info['plotlink'] = ''
info['dirichlet'] = lfuncDStex(L, "analytic")
info['eulerproduct'] = lfuncEPtex(L, "abstract")
info['functionalequation'] = lfuncFEtex(L, "analytic")
info['functionalequationSelberg'] = lfuncFEtex(L, "selberg")
if len(request.args) == 0:
lcalcUrl = request.url + '?download=lcalcfile'
else:
lcalcUrl = request.url + '&download=lcalcfile'
info['downloads'] = [('Lcalcfile', lcalcUrl)]
return info
def dirichlet_table():
args = to_dict(request.args)
mod = args.get('modulus', 1)
return redirect(url_character(type='Dirichlet', modulus=mod))
def navi(L):
r = [(l, url_character(**args)) for l, args in L if l]
return r
def evs_table2(search, twodarray=False):
DB = connect_db()
table = []
nrows = 0
fs = DB.get_Maass_forms(search)
mwf_logger.debug("numrec:{0}".format(len(fs)))
for f in fs: # indices:
row = {}
R = f.get('Eigenvalue', None)
N = f.get('Level', None)
k = f.get('Weight', None)
if R is None or N is None or k is None:
continue
row['R'] = R
row['N'] = N
if k == 0 or k == 1:
row['k'] = int(k)
else:
row['k'] = k
##
chi = f.get('Character', 0)
conrey = f.get('Conrey', 0)
if conrey == 0: # we need to change to conrey's notation
chi = DB.getDircharConrey(N, chi)
## Now get the COnrey number.
## First the character
if k == 0:
url = url_character(type='Dirichlet', modulus=N, number=chi)
s = "<a href={0}>{1}</a>".format(url, chi)
row['ch'] = s
else:
row['ch'] = "eta"
st = f.get('Symmetry', -1)
if st == 1:
st = "odd"
elif st == 0:
st = "even"
else:
st = "n/a"
row['symmetry'] = st
er = f.get('Error', 0)
if er > 0:
er = "{0:1.0e}".format(er)
else:
er = "unknown"
row['err'] = er
dim = f.get('Dim', 0)
if dim is None:
dim = 1 # "undefined"
row['dim'] = dim
numc = f.get('Numc', 0)
row['numc'] = numc
cev = f.get('Cusp_evs', [])
row['fricke'] = 'n/a'
row['cuspevs'] = 'n/a'
if row['k'] == 0 and isinstance(cev, list):
if len(cev) > 1:
fricke = cev[1]
row['fricke'] = fricke
s = '{0}'.format(cev[0])
for j in range(1, len(cev)):
s += ",{0}".format(cev[j])
elif len(cev) == 1:
s = str(cev[0])
elif len(cev) == 0:
s = 'n/a'
row['cuspevs'] = s
url = url_for('mwf.render_one_maass_waveform', maass_id=f.get('_id'))
row['url'] = url
nrows += 1
if twodarray:
s = '<a href="{0}">{1}</a>'.format(row['url'], row['R'])
rowr = [
row['N'], row['k'], row['ch'], s, row['symmetry'], row['err'],
row['dim'], row['numc'], row['fricke'], row['cuspevs']
]
table.append(rowr)
else:
# row=row.values()
table.append(row)
mwf_logger.debug("nrows:".format(nrows))
evs = {'table': {}}
evs['table']['data'] = table
evs['table']['nrows'] = nrows
evs['table']['ncols'] = 10
evs['table']['colheads'] = []
knowls = [
'mf.maass.mwf.level', 'mf.maass.mwf.weight', 'mf.maass.mwf.character',
'mf.maass.mwf.eigenvalue', 'mf.maass.mwf.symmetry',
'mf.maass.mwf.precision', 'mf.maass.mwf.dimension',
'mf.maass.mwf.ncoefficients', 'mf.maass.mwf.fricke',
'mf.maass.mwf.atkinlehner'
]
titles = [
'Level', 'Weight', 'Char', 'Eigenvalue', 'Symmetry', 'Precision',
'Mult.', 'Coeff.', 'Fricke', 'Atkin-Lehner'
]
for i in range(10):
evs['table']['colheads'].append((knowls[i], titles[i]))
if 'limit' in search:
search.pop('limit')
if 'skip' in search:
search.pop('skip')
evs['totalrecords'] = DB.count(search, filtered=False)
evs['totalrecords_filtered'] = DB.count(search, filtered=True)
return evs
def add_row(self, nflabel):
nf = WebNumberField(nflabel)
#nflink = (nflabel, url_for('number_fields.by_label',label=nflabel))
nflink = (nflabel, url_character(type='Hecke',number_field=nflabel))
F = WebHeckeFamily(number_field=nflabel)
self._contents.append( (nflink, nf.signature(), nf.web_poly() ) )
def initLfunction(L, args, request):
''' Sets the properties to show on the homepage of an L-function page.
'''
info = {'title': L.title}
try:
info['citation'] = L.citation
except AttributeError:
info['citation'] = ""
try:
info['support'] = L.support
except AttributeError:
info['support'] = ""
info['Ltype'] = L.Ltype()
# Here we should decide which values are indeed special values
# According to Brian, odd degree has special value at 1, and even
# degree has special value at 1/2.
# (however, I'm not sure this is true if L is not primitive -- GT)
# Now we usually display both
if L.Ltype() != "artin" or (L.Ltype() == "artin" and L.sign != 0):
# if is_even(L.degree) :
# info['sv12'] = specialValueString(L, 0.5, '1/2')
# if is_odd(L.degree):
# info['sv1'] = specialValueString(L, 1, '1')
info['sv1'] = specialValueString(L, 1, '1')
info['sv12'] = specialValueString(L, 0.5, '1/2')
info['args'] = args
info['credit'] = L.credit
# info['citation'] = L.citation
try:
info['factorization'] = L.factorization
except:
pass
try:
info['url'] = L.url
except:
info['url'] = ''
info['degree'] = int(L.degree)
info['zeroeslink'] = (request.url.replace('/L/', '/L/Zeros/').
replace('/Lfunction/', '/L/Zeros/').
replace('/L-function/', '/L/Zeros/')) # url_for('zeroesLfunction', **args)
info['plotlink'] = (request.url.replace('/L/', '/L/Plot/').
replace('/Lfunction/', '/L/Plot/').
replace('/L-function/', '/L/Plot/')) # info['plotlink'] = url_for('plotLfunction', **args)
info['bread'] = []
info['properties2'] = set_gaga_properties(L)
# Create friendlink by removing 'L/' and ending '/'
friendlink = request.url.replace('/L/', '/').replace('/L-function/', '/').replace('/Lfunction/', '/')
splitlink = friendlink.rpartition('/')
friendlink = splitlink[0] + splitlink[2]
logger.debug(L.Ltype())
if L.Ltype() == 'maass':
if L.group == 'GL2':
minNumberOfCoefficients = 100 # TODO: Fix this to take level into account
if len(L.dirichlet_coefficients) < minNumberOfCoefficients:
info['zeroeslink'] = ''
info['plotlink'] = ''
info['bread'] = get_bread(2, [('Maass Form',
url_for('.l_function_maass_browse_page')),
('\(' + L.texname + '\)', request.url)])
info['friends'] = [('Maass Form ', friendlink)]
else:
info['bread'] = get_bread(L.degree,
[('Maass Form', url_for('.l_function_maass_gln_browse_page',
degree='degree' + str(L.degree))),
(L.dbid, request.url)])
elif L.Ltype() == 'riemann':
info['bread'] = get_bread(1, [('Riemann Zeta', request.url)])
info['friends'] = [('\(\mathbb Q\)', url_for('number_fields.by_label', label='1.1.1.1')), ('Dirichlet Character \(\\chi_{1}(1,\\cdot)\)',
url_character(type='Dirichlet', modulus=1, number=1))]
elif L.Ltype() == 'dirichlet':
mod, num = L.charactermodulus, L.characternumber
Lpattern = r"\(L(s,\chi_{%s}(%s,·))\)"
if mod > 1:
pmod,pnum = WebDirichlet.prevprimchar(mod, num)
Lprev = (Lpattern%(pmod,pnum),url_for('.l_function_dirichlet_page',modulus=pmod,number=pnum))
else:
Lprev = ('','')
nmod,nnum = WebDirichlet.nextprimchar(mod, num)
Lnext = (Lpattern%(nmod,nnum),url_for('.l_function_dirichlet_page',modulus=nmod,number=nnum))
info['navi'] = (Lprev,Lnext)
snum = str(L.characternumber)
smod = str(L.charactermodulus)
charname = WebDirichlet.char2tex(smod, snum)
info['bread'] = get_bread(1, [(charname, request.url)])
info['friends'] = [('Dirichlet Character ' + str(charname), friendlink)]
elif L.Ltype() == 'ellipticcurveQ':
label = L.label
while friendlink[len(friendlink) - 1].isdigit(): # Remove any number at the end to get isogeny class url
friendlink = friendlink[0:len(friendlink) - 1]
info['friends'] = [('Isogeny class ' + label, friendlink)]
for i in range(1, L.nr_of_curves_in_class + 1):
info['friends'].append(('Elliptic curve ' + label + str(i), friendlink + str(i)))
if L.modform:
info['friends'].append(('Modular form ' + label.replace('.', '.2'), url_for("emf.render_elliptic_modular_forms",
level=L.modform['level'], weight=2, character=0, label=L.modform['iso'])))
info['friends'].append(('L-function ' + label.replace('.', '.2'),
url_for('.l_function_emf_page', level=L.modform['level'],
weight=2, character=0, label=L.modform['iso'], number=0)))
info['friends'].append(
('Symmetric square L-function', url_for(".l_function_ec_sym_page", power='2', label=label)))
info['friends'].append(
('Symmetric cube L-function', url_for(".l_function_ec_sym_page", power='3', label=label)))
info['bread'] = get_bread(2, [('Elliptic curve', url_for('.l_function_ec_browse_page')),
(label, url_for('.l_function_ec_page', label=label))])
elif L.Ltype() == 'ellipticmodularform':
friendlink = friendlink + L.addToLink # Strips off the embedding
friendlink = friendlink.rpartition('/')[0] # number for the L-function
if L.character:
info['friends'] = [('Modular form ' + str(
L.level) + '.' + str(L.weight) + '.' + str(L.character) + str(L.label), friendlink)]
else:
info['friends'] = [('Modular form ' + str(L.level) + '.' + str(L.weight) + str(
L.label), friendlink)]
if L.ellipticcurve:
info['friends'].append(
('EC isogeny class ' + L.ellipticcurve, url_for("by_ec_label", label=L.ellipticcurve)))
info['friends'].append(('L-function ' + str(L.level) + '.' + str(L.label),
url_for('.l_function_ec_page', label=L.ellipticcurve)))
for i in range(1, L.nr_of_curves_in_class + 1):
info['friends'].append(('Elliptic curve ' + L.ellipticcurve + str(i),
url_for("by_ec_label", label=L.ellipticcurve + str(i))))
info['friends'].append(
('Symmetric square L-function', url_for(".l_function_ec_sym_page", power='2', label=label)))
info['friends'].append(
('Symmetric cube L-function', url_for(".l_function_ec_sym_page", power='3', label=label)))
elif L.Ltype() == 'hilbertmodularform':
friendlink = '/'.join(friendlink.split('/')[:-1])
info['friends'] = [('Hilbert Modular Form', friendlink.rpartition('/')[0])]
elif L.Ltype() == 'dedekindzeta':
info['friends'] = [('Number Field', friendlink)]
elif L.Ltype() in ['lcalcurl', 'lcalcfile']:
info['bread'] = [('L-functions', url_for('.l_function_top_page'))]
elif L.Ltype() == 'SymmetricPower':
def ordinal(n):
if n == 2:
return "Square"
elif n == 3:
return "Cube"
elif 10 <= n % 100 < 20:
return str(n) + "th Power"
else:
return str(n) + {1: 'st', 2: 'nd', 3: 'rd'}.get(n % 10, "th") + " Power"
if L.m == 2:
info['bread'] = get_bread(3, [("Symmetric square of Elliptic curve",
url_for('.l_function_ec_sym2_browse_page')),
(L.label, url_for('.l_function_ec_sym_page',
label=L.label,power=L.m))])
elif L.m == 3:
info['bread'] = get_bread(4, [("Symmetric cube of Elliptic curve",
url_for('.l_function_ec_sym3_browse_page')),
(L.label, url_for('.l_function_ec_sym_page',
label=L.label,power=L.m))])
else:
info['bread'] = [('L-functions', url_for('.l_function_top_page')),
('Symmetric %s of Elliptic curve ' % ordinal(L.m)
+ str(L.label),
url_for('.l_function_ec_sym_page',
label=L.label,power=L.m))]
friendlink = request.url.replace('/L/SymmetricPower/%d/' % L.m, '/')
splitlink = friendlink.rpartition('/')
friendlink = splitlink[0] + splitlink[2]
friendlink2 = request.url.replace('/L/SymmetricPower/%d/' % L.m, '/L/')
splitlink = friendlink2.rpartition('/')
friendlink2 = splitlink[0] + splitlink[2]
info['friends'] = [('Isogeny class ' + L.label, friendlink), ('Symmetric 1st Power', friendlink2)]
for j in range(2, L.m + 2):
if j != L.m:
friendlink3 = request.url.replace('/L/SymmetricPower/%d/' % L.m, '/L/SymmetricPower/%d/' % j)
info['friends'].append(('Symmetric %s' % ordinal(j), friendlink3))
elif L.Ltype() == 'siegelnonlift' or L.Ltype() == 'siegeleisenstein' or L.Ltype() == 'siegelklingeneisenstein' or L.Ltype() == 'siegelmaasslift':
weight = str(L.weight)
number = str(L.number)
info['friends'] = [('Siegel Modular Form', friendlink)]
elif L.Ltype() == "artin":
# info['zeroeslink'] = ''
# info['plotlink'] = ''
info['friends'] = [('Artin representation', L.artin.url_for())]
if L.sign == 0: # The root number is now unknown
info['zeroeslink'] = ''
info['plotlink'] = ''
info['dirichlet'] = lfuncDStex(L, "analytic")
info['eulerproduct'] = lfuncEPtex(L, "abstract")
info['functionalequation'] = lfuncFEtex(L, "analytic")
info['functionalequationSelberg'] = lfuncFEtex(L, "selberg")
if len(request.args) == 0:
lcalcUrl = request.url + '?download=lcalcfile'
else:
lcalcUrl = request.url + '&download=lcalcfile'
info['downloads'] = [('Lcalcfile', lcalcUrl)]
return info
def add_row(self, nflabel):
nf = WebNumberField(nflabel)
#nflink = (nflabel, url_for('number_fields.by_label',label=nflabel))
nflink = (nflabel, url_character(type='Hecke',number_field=nflabel))
F = WebHeckeFamily(number_field=nflabel)
self._contents.append( (nflink, nf.signature(), nf.web_poly() ) )
def set_info_for_modular_form_space(level=None,
weight=None,
character=None,
label=None,
**kwds):
r"""
Set information about a space of modular forms.
"""
info = dict()
info['level'] = level
info['weight'] = weight
info['character'] = character
emf_logger.debug("info={0}".format(info))
if (level > N_max_db or weight > k_max_db):
info['error'] = "Currently not available"
WMFS = None
if level <= 0:
info['error'] = "Got wrong level: %s " % level
return info
try:
if use_db:
WMFS = WebModFormSpace(weight, level, character, use_db=True)
else:
WMFS = WebModFormSpace(weight, level, character)
if 'download' in info and 'tempfile' in info:
WNF._save_to_file(info['tempfile'])
info['filename'] = str(weight) + '-' + str(level) + '-' + str(
character) + '-' + label + '.sobj'
return info
except RuntimeError:
info['error'] = "Sage error: Could not construct the desired space!"
if WMFS.level() == 1:
info['group'] = "\( \mathrm{SL}_{2}(\mathbb{Z})\)"
else:
info['group'] = "\( \Gamma_{{0}}( {0} ) \)".format(WMFS.level())
if character == 0:
info['name_new'] = "\(S_{ %s }^{new}(%s) \)" % (WMFS.weight(),
WMFS.level())
info['name_old'] = "\(S_{ %s }^{old}(%s) \)" % (WMFS.weight(),
WMFS.level())
else:
conrey_char = WMFS.conrey_character()
conrey_char_name = WMFS.conrey_character_name()
info['conrey_character_name'] = '\( ' + conrey_char_name + '\)'
info['character_url'] = url_character(type='Dirichlet',
modulus=WMFS.level(),
number=conrey_char.number())
info['name_new'] = "\(S_{ %s }^{new}(%s,%s) \)" % (
WMFS.weight(), WMFS.level(), conrey_char_name)
info['name_old'] = "\(S_{ %s }^{old}(%s,%s) \)" % (
WMFS.weight(), WMFS.level(), conrey_char_name)
info['dimension_cusp_forms'] = WMFS.dimension_cusp_forms()
info['dimension_mod_forms'] = WMFS.dimension_modular_forms()
info['dimension_new_cusp_forms'] = WMFS.dimension_new_cusp_forms()
info['dimension_newspace'] = WMFS.dimension_newspace()
info['dimension_oldspace'] = WMFS.dimension_oldspace()
info['dimension'] = WMFS.dimension()
info['galois_orbits'] = WMFS.get_all_galois_orbit_info()
lifts = list()
if WMFS.dimension() == 0: # we don't need to work with an empty space
info['sturm_bound'] = 0
info['new_decomposition'] = ''
info['is_empty'] = 1
lifts.append(('Half-Integral Weight Forms', '/ModularForm/Mp2/Q'))
lifts.append(('Siegel Modular Forms', '/ModularForm/GSp4/Q'))
info['lifts'] = lifts
return info
info['sturm_bound'] = WMFS.sturm_bound()
info['new_decomposition'] = WMFS.print_galois_orbits()
emf_logger.debug("new_decomp={0}".format(info['new_decomposition']))
info['nontrivial_new'] = len(info['new_decomposition'])
## we try to catch well-known bugs...
info['old_decomposition'] = "n/a"
if level < N_max_comp:
try:
O = WMFS.print_oldspace_decomposition()
info['old_decomposition'] = O
except:
emf_logger.critical("Error in computing oldspace decomposition")
O = []
info['old_decomposition'] = "n/a"
(A, B, C) = sys.exc_info()
# build an error message...
errtype = A.__name__
errmsg = B
s = "%s: %s at:" % (errtype, errmsg)
next = C.tb_next
while (next):
ln = next.tb_lineno
filen = next.tb_frame.f_code.co_filename
s += "\n line no. %s in file %s" % (ln, filen)
next = next.tb_next
info['error_note'] = "Could not construct oldspace!\n" + s
# properties for the sidebar
prop = []
if WMFS._cuspidal == 1:
prop = [('Dimension newforms', [info['dimension_newspace']])]
prop.append(('Dimension oldforms', [info['dimension_oldspace']]))
else:
prop = [('Dimension modular forms', [info['dimension_mod_forms']])]
prop.append(('Dimension cusp forms', [info['dimension_cusp_forms']]))
prop.append(('Sturm bound', [WMFS.sturm_bound()]))
info['properties2'] = prop
## Make parent spaces of S_k(N,chi) for the sidebar
par_lbl = '\( S_{*} (\Gamma_0(' + str(level) + '),\cdot )\)'
par_url = '?level=' + str(level)
parents = [[par_lbl, par_url]]
par_lbl = '\( S_{k} (\Gamma_0(' + str(level) + '),\cdot )\)'
par_url = '?level=' + str(level) + '&weight=' + str(weight)
parents.append((par_lbl, par_url))
info['parents'] = parents
if 'character' in info:
info['character_order'] = WMFS.character_order()
info['character_conductor'] = WMFS.character_conductor()
friends = list()
lifts = list()
if (('label' not in info) and info['old_decomposition'] != 'n/a'):
O = WMFS.oldspace_decomposition()
try:
for (old_level, chi, mult, d) in O:
if chi != 0:
s = "\(S_{%s}(\Gamma_0(%s),\chi_{%s}) \) " % (
weight, old_level, chi)
friends.append((s, '?weight=' + str(weight) + '&level=' +
str(old_level) + '&character=' + str(chi)))
else:
s = "\(S_{%s}(\Gamma_0(%s)) \) " % (weight, old_level)
friends.append((s, '?weight=' + str(weight) + '&level=' +
str(old_level) + '&character=' + str(0)))
except:
pass
info['friends'] = friends
lifts.append(('Half-Integral Weight Forms', '/ModularForm/Mp2/Q'))
lifts.append(('Siegel Modular Forms', '/ModularForm/GSp4/Q'))
info['lifts'] = lifts
return info
def render_field_webpage(args):
data = None
C = base.getDBConnection()
info = {}
bread = [("Global Number Fields", url_for(".number_field_render_webpage"))]
if "label" in args:
label = clean_input(args["label"])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(("Search results", " "))
label2 = re.sub(r"[<>]", "", args["label"])
if "You need to enter a field" in label2:
info["err"] = label2
else:
info["err"] = "No such field: %s in the database" % label2
info["label"] = args["label_orig"] if "label_orig" in args else args["label"]
return search_input_error(info, bread)
info["wnf"] = nf
data["degree"] = nf.degree()
data["class_number"] = nf.class_number()
t = nf.galois_t()
n = nf.degree()
data["is_galois"] = nf.is_galois()
data["is_abelian"] = nf.is_abelian()
if nf.is_abelian():
conductor = nf.conductor()
data["conductor"] = conductor
dirichlet_chars = nf.dirichlet_group()
if len(dirichlet_chars) > 0:
data["dirichlet_group"] = [
'<a href = "%s">$\chi_{%s}(%s,·)$</a>'
% (url_character(type="Dirichlet", modulus=data["conductor"], number=j), data["conductor"], j)
for j in dirichlet_chars
]
data["dirichlet_group"] = r"$\lbrace$" + ", ".join(data["dirichlet_group"]) + r"$\rbrace$"
if data["conductor"].is_prime() or data["conductor"] == 1:
data["conductor"] = "\(%s\)" % str(data["conductor"])
else:
data["conductor"] = "\(%s=%s\)" % (str(data["conductor"]), latex(data["conductor"].factor()))
data["galois_group"] = group_display_knowl(n, t, C)
data["cclasses"] = cclasses_display_knowl(n, t, C)
data["character_table"] = character_table_display_knowl(n, t, C)
data["class_group"] = nf.class_group()
data["class_group_invs"] = nf.class_group_invariants()
data["signature"] = nf.signature()
data["coefficients"] = nf.coeffs()
D = nf.disc()
ram_primes = D.prime_factors()
data["disc_factor"] = nf.disc_factored_latex()
if D.abs().is_prime() or D == 1:
data["discriminant"] = "\(%s\)" % str(D)
else:
data["discriminant"] = "\(%s=%s\)" % (str(D), data["disc_factor"])
npr = len(ram_primes)
ram_primes = str(ram_primes)[1:-1]
if ram_primes == "":
ram_primes = r"\textrm{None}"
data["frob_data"], data["seeram"] = frobs(nf.K())
data["phrase"] = group_phrase(n, t, C)
zk = pari(nf.K()).nf_subst("a")
zk = list(zk.nf_get_zk())
Ra = PolynomialRing(QQ, "a")
zk = [latex(Ra(x)) for x in zk]
zk = ["$%s$" % x for x in zk]
zk = ", ".join(zk)
grh_label = (
'<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh() else ""
)
# Short version for properties
grh_lab = nf.short_grh_string()
pretty_label = field_pretty(label)
if label != pretty_label:
pretty_label = "%s: %s" % (label, pretty_label)
info.update(data)
info.update(
{
"label": pretty_label,
"label_raw": label,
"polynomial": web_latex_split_on_pm(nf.K().defining_polynomial()),
"ram_primes": ram_primes,
"integral_basis": zk,
"regulator": web_latex(nf.regulator()),
"unit_rank": nf.unit_rank(),
"root_of_unity": web_latex(nf.K().primitive_root_of_unity()),
"fund_units": nf.units(),
"grh_label": grh_label,
}
)
bread.append(("%s" % info["label_raw"], " "))
info["downloads_visible"] = True
info["downloads"] = [("worksheet", "/")]
info["friends"] = []
if nf.can_class_number():
info["friends"].append(("L-function", "/L/NumberField/%s" % label))
info["friends"].append(("Galois group", "/GaloisGroup/%dT%d" % (n, t)))
if "dirichlet_group" in info:
info["friends"].append(
(
"Dirichlet group",
url_for(
"characters.dirichlet_group_table",
modulus=int(conductor),
char_number_list=",".join([str(a) for a in dirichlet_chars]),
poly=info["polynomial"],
),
)
)
info["learnmore"] = [
("Global number field labels", url_for(".render_labels_page")),
(Completename, url_for(".render_discriminants_page")),
]
# With Galois group labels, probably not needed here
# info['learnmore'] = [('Global number field labels',
# url_for(".render_labels_page")), ('Galois group
# labels',url_for(".render_groups_page")),
# (Completename,url_for(".render_discriminants_page"))]
title = "Global Number Field %s" % info["label"]
if npr == 1:
primes = "prime"
else:
primes = "primes"
properties2 = [
("Degree:", "%s" % data["degree"]),
("Signature:", "$%s$" % data["signature"]),
("Discriminant:", "$%s$" % data["disc_factor"]),
("Ramified " + primes + ":", "$%s$" % ram_primes),
("Class number:", "%s %s" % (data["class_number"], grh_lab)),
("Class group:", "%s %s" % (data["class_group_invs"], grh_lab)),
("Galois Group:", group_display_short(data["degree"], t, C)),
]
from lmfdb.math_classes import NumberFieldGaloisGroup
try:
info["tim_number_field"] = NumberFieldGaloisGroup.find_one({"label": label})
v = nf.factor_perm_repn(info["tim_number_field"])
info["mydecomp"] = ["*" if x > 0 else "" for x in v]
except AttributeError:
pass
# del info['_id']
return render_template(
"number_field.html",
properties2=properties2,
credit=NF_credit,
title=title,
bread=bread,
friends=info.pop("friends"),
learnmore=info.pop("learnmore"),
info=info,
)