def download_hmf_magma(**args):
C = getDBConnection()
data = None
label = str(args['label'])
f = C.hmfs.forms.find_one({'label': label})
if f is None:
return "No such form"
F = WebNumberField(f['field_label'])
F_hmf = C.hmfs.fields.find_one({'label': f['field_label']})
outstr = 'P<x> := PolynomialRing(Rationals());\n'
outstr += 'g := P!' + str(F.coeffs()) + ';\n'
outstr += 'F<w> := NumberField(g);\n'
outstr += 'ZF := Integers(F);\n\n'
# outstr += 'ideals_str := [' + ','.join([st for st in F_hmf["ideals"]]) + '];\n'
# outstr += 'ideals := [ideal<ZF | {F!x : x in I}> : I in ideals_str];\n\n'
outstr += 'NN := ideal<ZF | {' + f["level_ideal"][1:-1] + '}>;\n\n'
outstr += 'primesArray := [\n' + ','.join(
[st for st in F_hmf["primes"]]).replace('],[', '],\n[') + '];\n'
outstr += 'primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];\n\n'
if f["hecke_polynomial"] != 'x':
outstr += 'heckePol := ' + f["hecke_polynomial"] + ';\n'
outstr += 'K<e> := NumberField(heckePol);\n'
else:
outstr += 'heckePol := x;\nK := Rationals(); e := 1;\n'
outstr += '\nheckeEigenvaluesArray := [' + ', '.join(
[st for st in f["hecke_eigenvalues"]]) + '];'
outstr += '\nheckeEigenvalues := AssociativeArray();\n'
outstr += 'for i := 1 to #heckeEigenvaluesArray do\n heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];\nend for;\n\n'
outstr += 'ALEigenvalues := AssociativeArray();\n'
for s in f["AL_eigenvalues"]:
outstr += 'ALEigenvalues[ideal<ZF | {' + s[0][1:-1] + '}>] := ' + s[
1] + ';\n'
outstr += '\n// EXAMPLE:\n// pp := Factorization(2*ZF)[1][1];\n// heckeEigenvalues[pp];\n\n'
outstr += '/* EXTRA CODE: recompute eigenform (warning, may take a few minutes or longer!):\n'
outstr += 'M := HilbertCuspForms(F, NN);\n'
outstr += 'S := NewSubspace(M);\n'
outstr += '// SetVerbose("ModFrmHil", 1);\n'
outstr += 'newspaces := NewformDecomposition(S);\n'
outstr += 'newforms := [Eigenform(U) : U in newspaces];\n'
outstr += 'ppind := 0;\n'
outstr += 'while #newforms gt 1 do\n'
outstr += ' pp := primes[ppind];\n'
outstr += ' newforms := [f : f in newforms | HeckeEigenvalue(f,pp) eq heckeEigenvalues[pp]];\n'
outstr += 'end while;\n'
outstr += 'f := newforms[1];\n'
outstr += '// [HeckeEigenvalue(f,pp) : pp in primes] eq heckeEigenvaluesArray;\n'
outstr += '*/\n'
return outstr
def download_hmf_magma(**args):
C = getDBConnection()
data = None
label = str(args['label'])
f = C.hmfs.forms.find_one({'label': label})
if f is None:
return "No such form"
F = WebNumberField(f['field_label'])
F_hmf = C.hmfs.fields.find_one({'label': f['field_label']})
outstr = 'P<x> := PolynomialRing(Rationals());\n'
outstr += 'g := P!' + str(F.coeffs()) + ';\n'
outstr += 'F<w> := NumberField(g);\n'
outstr += 'ZF := Integers(F);\n\n'
# outstr += 'ideals_str := [' + ','.join([st for st in F_hmf["ideals"]]) + '];\n'
# outstr += 'ideals := [ideal<ZF | {F!x : x in I}> : I in ideals_str];\n\n'
outstr += 'NN := ideal<ZF | {' + f["level_ideal"][1:-1] + '}>;\n\n'
outstr += 'primesArray := [\n' + ','.join([st for st in F_hmf["primes"]]).replace('],[', '],\n[') + '];\n'
outstr += 'primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];\n\n'
if f["hecke_polynomial"] != 'x':
outstr += 'heckePol := ' + f["hecke_polynomial"] + ';\n'
outstr += 'K<e> := NumberField(heckePol);\n'
else:
outstr += 'heckePol := x;\nK := Rationals(); e := 1;\n'
outstr += '\nheckeEigenvaluesArray := [' + ', '.join([st for st in f["hecke_eigenvalues"]]) + '];'
outstr += '\nheckeEigenvalues := AssociativeArray();\n'
outstr += 'for i := 1 to #heckeEigenvaluesArray do\n heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];\nend for;\n\n'
outstr += 'ALEigenvalues := AssociativeArray();\n'
for s in f["AL_eigenvalues"]:
outstr += 'ALEigenvalues[ideal<ZF | {' + s[0][1:-1] + '}>] := ' + s[1] + ';\n'
outstr += '\n// EXAMPLE:\n// pp := Factorization(2*ZF)[1][1];\n// heckeEigenvalues[pp];\n\n'
outstr += '/* EXTRA CODE: recompute eigenform (warning, may take a few minutes or longer!):\n'
outstr += 'M := HilbertCuspForms(F, NN);\n'
outstr += 'S := NewSubspace(M);\n'
outstr += '// SetVerbose("ModFrmHil", 1);\n'
outstr += 'newspaces := NewformDecomposition(S);\n'
outstr += 'newforms := [Eigenform(U) : U in newspaces];\n'
outstr += 'ppind := 0;\n'
outstr += 'while #newforms gt 1 do\n'
outstr += ' pp := primes[ppind];\n'
outstr += ' newforms := [f : f in newforms | HeckeEigenvalue(f,pp) eq heckeEigenvalues[pp]];\n'
outstr += 'end while;\n'
outstr += 'f := newforms[1];\n'
outstr += '// [HeckeEigenvalue(f,pp) : pp in primes] eq heckeEigenvaluesArray;\n'
outstr += '*/\n'
return outstr
def download_hmf_sage(**args):
C = getDBConnection()
data = None
label = str(args['label'])
f = C.hmfs.forms.find_one({'label': label})
if f is None:
return "No such form"
F = WebNumberField(f['field_label'])
F_hmf = C.hmfs.fields.find_one({'label': f['field_label']})
outstr = 'P.<x> = PolynomialRing(QQ)\n'
outstr += 'g = P(' + str(F.coeffs()) + ')\n'
outstr += 'F.<w> = NumberField(g)\n'
outstr += 'ZF = F.ring_of_integers()\n\n'
outstr += 'NN = ZF.ideal(' + f["level_ideal"] + ')\n\n'
outstr += 'primes_array = [\n' + ','.join(
[st for st in F_hmf["primes"]]).replace('],[', '],\\\n[') + ']\n'
outstr += 'primes = [ZF.ideal(I) for I in primes_array]\n\n'
if f["hecke_polynomial"] != 'x':
outstr += 'hecke_pol = ' + f["hecke_polynomial"] + '\n'
outstr += 'K.<e> = NumberField(heckePol)\n'
else:
outstr += 'heckePol = x\nK = QQ\ne = 1\n'
outstr += '\nhecke_eigenvalues_array = [' + ', '.join(
[st for st in f["hecke_eigenvalues"]]) + ']'
outstr += '\nhecke_eigenvalues = {}\n'
outstr += 'for i in range(len(hecke_eigenvalues_array)):\n hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]\n\n'
outstr += 'AL_eigenvalues = {}\n'
for s in f["AL_eigenvalues"]:
outstr += 'ALEigenvalues[ZF.ideal(s[0])] = s[1]\n'
outstr += '\n# EXAMPLE:\n# pp = ZF.ideal(2).factor()[0][0]\n# hecke_eigenvalues[pp]\n'
return outstr
def download_hmf_sage(**args):
C = getDBConnection()
data = None
label = str(args['label'])
f = C.hmfs.forms.find_one({'label': label})
if f is None:
return "No such form"
F = WebNumberField(f['field_label'])
F_hmf = C.hmfs.fields.find_one({'label': f['field_label']})
outstr = 'P.<x> = PolynomialRing(QQ)\n'
outstr += 'g = P(' + str(F.coeffs()) + ')\n'
outstr += 'F.<w> = NumberField(g)\n'
outstr += 'ZF = F.ring_of_integers()\n\n'
outstr += 'NN = ZF.ideal(' + f["level_ideal"] + ')\n\n'
outstr += 'primes_array = [\n' + ','.join([st for st in F_hmf["primes"]]).replace('],[',
'],\\\n[') + ']\n'
outstr += 'primes = [ZF.ideal(I) for I in primes_array]\n\n'
if f["hecke_polynomial"] != 'x':
outstr += 'hecke_pol = ' + f["hecke_polynomial"] + '\n'
outstr += 'K.<e> = NumberField(heckePol)\n'
else:
outstr += 'heckePol = x\nK = QQ\ne = 1\n'
outstr += '\nhecke_eigenvalues_array = [' + ', '.join([st for st in f["hecke_eigenvalues"]]) + ']'
outstr += '\nhecke_eigenvalues = {}\n'
outstr += 'for i in range(len(hecke_eigenvalues_array)):\n hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]\n\n'
outstr += 'AL_eigenvalues = {}\n'
for s in f["AL_eigenvalues"]:
outstr += 'ALEigenvalues[ZF.ideal(s[0])] = s[1]\n'
outstr += '\n# EXAMPLE:\n# pp = ZF.ideal(2).factor()[0][0]\n# hecke_eigenvalues[pp]\n'
return outstr
def render_hmf_webpage(**args):
C = getDBConnection()
data = None
if 'label' in args:
label = str(args['label'])
data = C.hmfs.forms.find_one({'label': label})
if data is None:
return "No such field"
info = {}
try:
info['count'] = args['count']
except KeyError:
info['count'] = 10
try:
numeigs = request.args['numeigs']
numeigs = int(numeigs)
except:
numeigs = 20
hmf_field = C.hmfs.fields.find_one({'label': data['field_label']})
nf = WebNumberField(data['field_label'])
info['base_galois_group'] = nf.galois_string()
info['field_degree'] = nf.degree()
info['field_disc'] = str(nf.disc())
info['field_poly'] = teXify_pol(str(nf.poly()))
info.update(data)
downloadslabel = '/ModularForm/GL2/' + info[
'field_label'] + '/holomorphic/' + info['label']
info['downloads'] = [
('Download to Magma', downloadslabel + '/download/magma'),
('Download to Sage', downloadslabel + '/download/sage')
]
info['friends'] = []
info['friends'] = [
('L-function', '/L/ModularForm/GL2/' + data['field_label'] +
'/holomorphic/' + info['label'] + '/0/0')
]
# info['learnmore'] = [('Number Field labels',
# url_for("render_labels_page")), ('Galois group
# labels',url_for("render_groups_page")), ('Discriminant
# ranges',url_for("render_discriminants_page"))]
bread = [('Hilbert Modular Forms',
url_for("hilbert_modular_form_render_webpage")),
('%s' % data['label'], ' ')]
t = "Hilbert Cusp Form %s" % info['label']
credit = 'Lassina Dembele, Steve Donnelly and <A HREF="http://www.cems.uvm.edu/~voight/">John Voight</A>'
forms_space = C.hmfs.forms.find({
'field_label': data['field_label'],
'level_ideal': data['level_ideal']
})
dim_space = 0
for v in forms_space:
dim_space += v['dimension']
info['newspace_dimension'] = dim_space
w = polygen(QQ, 'w')
e = polygen(QQ, 'e')
eigs = data['hecke_eigenvalues']
eigs = eigs[:min(len(eigs), numeigs)]
primes = hmf_field['primes']
n = min(len(eigs), len(primes))
info['eigs'] = [{
'eigenvalue': teXify_pol(eigs[i]),
'prime_ideal': teXify_pol(primes[i]),
'prime_norm': primes[i][1:primes[i].index(',')]
} for i in range(n)]
try:
display_eigs = request.args['display_eigs']
if display_eigs in ['True', 'true', '1', 'yes']:
display_eigs = True
else:
display_eigs = False
except KeyError:
display_eigs = False
if 'numeigs' in request.args:
display_eigs = True
info['hecke_polynomial'] = teXify_pol(info['hecke_polynomial'])
if 'AL_eigenvalues_fixed' in data:
if data['AL_eigenvalues_fixed'] == 'done':
info['AL_eigs'] = [{
'eigenvalue': teXify_pol(al[1]),
'prime_ideal': teXify_pol(al[0]),
'prime_norm': al[0][1:al[0].index(',')]
} for al in data['AL_eigenvalues']]
else:
info['AL_eigs'] = [{'eigenvalue': '?', 'prime_ideal': '?'}]
else:
info['AL_eigs'] = [{'eigenvalue': '?', 'prime_ideal': '?'}]
info['AL_eigs_count'] = len(info['AL_eigs']) != 0
if not display_eigs:
for eig in info['eigs']:
if len(eig['eigenvalue']) > 300:
eig['eigenvalue'] = '...'
for eig in info['AL_eigs']:
if len(eig['eigenvalue']) > 300:
eig['eigenvalue'] = '...'
info['level_ideal'] = teXify_pol(info['level_ideal'])
if 'is_CM' in data:
is_CM = data['is_CM']
else:
is_CM = '?'
info['is_CM'] = is_CM
if 'is_base_change' in data:
is_base_change = data['is_base_change']
else:
is_base_change = '?'
info['is_base_change'] = is_base_change
if 'q_expansions' in data:
info['q_expansions'] = data['q_expansions']
properties2 = [('Field', '%s' % data['field_label']),
('Weight', '%s' % data['weight']),
('Level Norm', '%s' % data['level_norm']),
('Level', '$' + teXify_pol(data['level_ideal']) + '$'),
('Label', '%s' % data['label_suffix']),
('Dimension', '%s' % data['dimension']), ('CM?', is_CM),
('Base Change?', is_base_change)]
return render_template("hilbert_modular_form/hilbert_modular_form.html",
downloads=info["downloads"],
info=info,
properties2=properties2,
credit=credit,
title=t,
bread=bread,
friends=info['friends'])
def render_hmf_webpage(**args):
C = getDBConnection()
data = None
if 'label' in args:
label = str(args['label'])
data = C.hmfs.forms.find_one({'label': label})
if data is None:
return "No such field"
info = {}
try:
info['count'] = args['count']
except KeyError:
info['count'] = 10
try:
numeigs = request.args['numeigs']
numeigs = int(numeigs)
except:
numeigs = 20
hmf_field = C.hmfs.fields.find_one({'label': data['field_label']})
nf = WebNumberField(data['field_label'])
info['base_galois_group'] = nf.galois_string()
info['field_degree'] = nf.degree()
info['field_disc'] = str(nf.disc())
info['field_poly'] = teXify_pol(str(nf.poly()))
info.update(data)
downloadslabel = '/ModularForm/GL2/' + info['field_label'] + '/holomorphic/' + info['label']
info['downloads'] = [('Download to Magma', downloadslabel + '/download/magma'), (
'Download to Sage', downloadslabel + '/download/sage')]
info['friends'] = []
info['friends'] = [('L-function', '/L/ModularForm/GL2/' + data['field_label'] + '/holomorphic/' +
info['label'] + '/0/0')]
# info['learnmore'] = [('Number Field labels',
# url_for("render_labels_page")), ('Galois group
# labels',url_for("render_groups_page")), ('Discriminant
# ranges',url_for("render_discriminants_page"))]
bread = [('Hilbert Modular Forms', url_for("hilbert_modular_form_render_webpage")), ('%s' % data[
'label'], ' ')]
t = "Hilbert Cusp Form %s" % info['label']
credit = 'Lassina Dembele, Steve Donnelly and <A HREF="http://www.cems.uvm.edu/~voight/">John Voight</A>'
forms_space = C.hmfs.forms.find(
{'field_label': data['field_label'], 'level_ideal': data['level_ideal']})
dim_space = 0
for v in forms_space:
dim_space += v['dimension']
info['newspace_dimension'] = dim_space
w = polygen(QQ, 'w')
e = polygen(QQ, 'e')
eigs = data['hecke_eigenvalues']
eigs = eigs[:min(len(eigs), numeigs)]
primes = hmf_field['primes']
n = min(len(eigs), len(primes))
info['eigs'] = [{'eigenvalue': teXify_pol(eigs[i]),
'prime_ideal': teXify_pol(primes[i]),
'prime_norm': primes[i][1:primes[i].index(',')]} for i in range(n)]
try:
display_eigs = request.args['display_eigs']
if display_eigs in ['True', 'true', '1', 'yes']:
display_eigs = True
else:
display_eigs = False
except KeyError:
display_eigs = False
if 'numeigs' in request.args:
display_eigs = True
info['hecke_polynomial'] = teXify_pol(info['hecke_polynomial'])
if 'AL_eigenvalues_fixed' in data:
if data['AL_eigenvalues_fixed'] == 'done':
info['AL_eigs'] = [{'eigenvalue': teXify_pol(al[1]),
'prime_ideal': teXify_pol(al[0]),
'prime_norm': al[0][1:al[0].index(',')]} for al in data['AL_eigenvalues']]
else:
info['AL_eigs'] = [{'eigenvalue': '?', 'prime_ideal': '?'}]
else:
info['AL_eigs'] = [{'eigenvalue': '?', 'prime_ideal': '?'}]
info['AL_eigs_count'] = len(info['AL_eigs']) != 0
if not display_eigs:
for eig in info['eigs']:
if len(eig['eigenvalue']) > 300:
eig['eigenvalue'] = '...'
for eig in info['AL_eigs']:
if len(eig['eigenvalue']) > 300:
eig['eigenvalue'] = '...'
info['level_ideal'] = teXify_pol(info['level_ideal'])
if 'is_CM' in data:
is_CM = data['is_CM']
else:
is_CM = '?'
info['is_CM'] = is_CM
if 'is_base_change' in data:
is_base_change = data['is_base_change']
else:
is_base_change = '?'
info['is_base_change'] = is_base_change
if 'q_expansions' in data:
info['q_expansions'] = data['q_expansions']
properties2 = [('Field', '%s' % data['field_label']),
('Weight', '%s' % data['weight']),
('Level Norm', '%s' % data['level_norm']),
('Level', '$' + teXify_pol(data['level_ideal']) + '$'),
('Label', '%s' % data['label_suffix']),
('Dimension', '%s' % data['dimension']),
('CM?', is_CM),
('Base Change?', is_base_change)
]
return render_template("hilbert_modular_form/hilbert_modular_form.html", downloads=info["downloads"], info=info, properties2=properties2, credit=credit, title=t, bread=bread, friends=info['friends'])
