def render_webpage(args={}):
"""
Configure and return a template for the Siegel modular forms pages.
"""
info = dict(args)
# info['learnmore'] = [ ('Siegel modular forms', 'http://en.wikipedia.org/wiki/Siegel_modular_form')]
info['learnmore'] = []
bread = [('Siegel modular forms', url_for('ModularForm_GSp4_Q_top_level'))]
if len(args) == 0:
return render_template("ModularForm_GSp4_Q/ModularForm_GSp4_Q_navigation.html",
title='Siegel Modular Forms',
bread=bread,
**info)
# possible keys for the URL
group = args.get('group')
character = args.get('character')
weight = args.get('weight')
level = args.get('level')
form = args.get('form')
page = args.get('page')
weight_range = args.get('weight_range')
ev_modulus = args.get('ev_modulus')
fc_modulus = args.get('fc_modulus')
# set info
info['group'] = group
info['form'] = form
info['level'] = level
# We check first the key 'group' since it is needed always
tmp_parent_as_tex = '%s'
if args['group']:
if 'Sp4Z' == args['group']:
info['parent_as_tex'] = 'M_{k,j}\\big({\\rm Sp}(4,\\mathbb{Z})\\big)'
# dimension = siegel_core._dimension_Sp4Z
dimension = dimensions.dimension_Sp4Z
info['generators'] = 'smf.Igusa_generators'
elif 'Gamma0_2' == args['group']:
info['parent_as_tex'] = 'M_{k,j}\\big(\\Gamma_0(2)\\big)'
dimension = dimensions.dimension_Gamma0_2
elif 'Gamma1_2' == args['group']:
info['parent_as_tex'] = 'M_{k,j}\\big(\\Gamma_1(2)\\big)'
dimension = dimensions.dimension_Gamma1_2
elif 'Gamma_2' == args['group']:
info['parent_as_tex'] = 'M_{k,j}\\big(\\Gamma(2)\\big)'
dimension = dimensions.dimension_Gamma_2
elif 'Sp4Z_2' == args['group']:
info['parent_as_tex'] = 'M_{k,2}\\big({\\rm Sp}(4,\\mathbb{Z})\\big)'
dimension = siegel_core._dimension_Sp4Z_2
elif 'Sp6Z' == args['group']:
info['parent_as_tex'] = 'M_k\\big({\\rm Sp}(6,\\mathbb{Z})\\big)'
dimension = siegel_core._dimension_Sp6Z
elif 'Sp8Z' == args['group']:
info['parent_as_tex'] = 'M_k\\big({\\rm Sp}(8,\\mathbb{Z})\\big)'
dimension = siegel_core._dimension_Sp8Z
elif 'Kp' == args['group']:
info['parent_as_tex'] = 'M_k\\big(K(p)\\big)'
info['learnmore'] += [('Paramodular forms', 'http://math.lfc.edu/~yuen/paramodular/')]
info['generators'] = 'smf.Kp_generators'
dimension = siegel_core._dimension_Kp
elif 'Gamma0_2' == args['group']:
info['parent_as_tex'] = 'M_k\\big(\\Gamma_0(2)\\big)'
dimension = siegel_core._dimension_Gamma0_2
elif 'Gamma0_3' == args['group']:
info['parent_as_tex'] = 'M_k\\big(\\Gamma_0(3)\\big)'
dimension = siegel_core._dimension_Gamma0_3
elif 'Gamma0_3_psi_3' == args['group']:
info['parent_as_tex'] = 'M_k\\big(\\Gamma_0(3,\\psi_3)\\big)'
dimension = siegel_core._dimension_Gamma0_3_psi_3
elif 'Gamma0_4' == args['group']:
info['parent_as_tex'] = 'M_k\\big(\\Gamma_0(4)\\big)'
dimension = siegel_core._dimension_Gamma0_4
elif 'Gamma0_4_psi_4' == args['group']:
info['parent_as_tex'] = 'M_k\\big(\\Gamma_0(4,\\psi_4)\\big)'
dimension = siegel_core._dimension_Gamma0_4_psi_4
elif 'Gamma0_4_half' == group:
info['parent_as_tex'] = 'M_{k-1/2}\\big(\\Gamma_0(4)\\big)'
dimension = siegel_core._dimension_Gamma0_4_half
else:
info['error'] = 'Request for unavailable type of Siegel modular form'
return render_template("ModularForm_GSp4_Q/None.html", **info)
info['learnmore'] += [('The spaces \(' + info['parent_as_tex'] + '\)', url_for(
'ModularForm_GSp4_Q_top_level', group=group, page='basic'))]
bread += [('\(' + info['parent_as_tex'] + '\)', url_for('ModularForm_GSp4_Q_top_level',
group=group, page='forms'))]
else:
# some nonsense request came in, we answer by nonsense too
return render_template("ModularForm_GSp4_Q/None.html")
# We branch now according to the value of the key 'page'
##########################################################
## FORM COLLECTION REQUEST
##########################################################
if page == 'forms':
try:
f = urllib.urlopen(DATA + group + '/available_eigenforms.p')
go = pickle.load(f)
f.close()
forms_exist = True
except (IOError, EOFError, KeyError):
info['error'] = 'No data access'
forms_exist = False
if True == forms_exist:
info['forms'] = [(k, [(form, go[k][form]) for form in go[k]]) for k in go]
return render_template("ModularForm_GSp4_Q/ModularForm_GSp4_Q_forms.html",
title='Siegel modular forms \(' + info['parent_as_tex'] + '\)',
bread=bread, **info)
if page == 'basic':
bread += [('Basic information', url_for('ModularForm_GSp4_Q_top_level', group=group, page=page))]
return render_template("ModularForm_GSp4_Q/ModularForm_GSp4_Q_basic.html",
title='Siegel modular forms basic information',
bread=bread, **info)
##########################################################
## DIMENSIONS REQUEST
##########################################################
if page == 'dimensions':
# We check whether the weight_range makes sense to us and, if so, dispatch it
info['weight_range'] = weight_range
try:
assert info['weight_range'], 'Please enter a valid argument'
min_wt, max_wt, sym_pow = input_parser.kj_parser( weight_range)
min_wt = Integer( min_wt)
if None == max_wt or max_wt < min_wt:
max_wt = min_wt
if None == sym_pow:
sym_pow = 0
assert min_wt < 1000000 and (max_wt - min_wt + 1) * max_wt < 10000 and sym_pow < 1000, '%d-%d,%d: Input too large: Please enter smaller range or numbers.' % (max_wt, min_wt, sym_pow)
except Exception as e:
info['error'] = str(e)
return render_template( "ModularForm_GSp4_Q/ModularForm_GSp4_Q_dimensions.html",
title='Siegel modular forms dimensions \(' + info['parent_as_tex'] + '\)',
bread=bread, **info)
# A priori the request is reasonable, so we try to get the data for the answer
try:
info['new_method'] = None
if 'Gamma_2' == group or 'Gamma0_2' == group or 'Gamma1_2' == group or 'Sp4Z' == group:
info['sym_pow'] = sym_pow
info['table_headers'], info['dimensions'] = dimension( range( min_wt, max_wt + 1), sym_pow)
####### a hack ########
info['new_method'] = 'new_method'
bread += [('Dimensions',
url_for('ModularForm_GSp4_Q_top_level', group=group, page=page, level=level, weight_range=weight_range))]
elif 'Kp' == group:
info['dimensions'] = [(k, dimension(k, tp=int(level))) for k in range(min_wt, max_wt + 1)]
bread += [('Dimensions',
url_for('ModularForm_GSp4_Q_top_level', group=group, page=page, level=level, weight_range=weight_range))]
else:
info['dimensions'] = [(k, dimension(k)) for k in range(min_wt, max_wt + 1)]
bread += [('Dimensions',
url_for('ModularForm_GSp4_Q_top_level', group=group, page=page, weight_range=weight_range))]
except Exception as e:
info['error'] = 'Functional error: %s' % (str(e)) #(sys.exc_info()[0])
return render_template( "ModularForm_GSp4_Q/ModularForm_GSp4_Q_dimensions.html",
title='Siegel modular forms dimensions \(' + info['parent_as_tex'] + '\)',
bread=bread, **info)
# We provide some headers for the 'old' method and ask for rendering an answer
if info['new_method']:
info['table_headers'] = info['table_headers']
elif 'Sp8Z' == group:
info['table_headers'] = ['Weight', 'Total', 'Ikeda lifts', 'Miyawaki lifts', 'Other']
elif 'Sp6Z' == group:
info['table_headers'] = ['Weight', 'Total', 'Miyawaki lifts I', 'Miyawaki lifts II', 'Other']
elif group == 'Kp':
info['table_headers'] = ["Weight", "Total", "Gritsenko Lifts", "Nonlifts", "Oldforms"]
elif 'Sp4Z_2' == group or 'Gamma0_4_half' == group:
info['table_headers'] = ['Weight', 'Total', 'Non cusp', 'Cusp']
else:
info['table_headers'] = ["Weight", "Total", "Eisenstein", "Klingen", "Maass", "Interesting"]
return render_template("ModularForm_GSp4_Q/ModularForm_GSp4_Q_dimensions.html",
title='Siegel modular forms dimensions \(' + info['parent_as_tex'] + '\)',
bread=bread, **info)
##########################################################
## SPECIFIC FORM REQUEST
##########################################################
if page == 'specimen':
info['weight'] = weight
# try to load data
try:
file_name = weight + '_' + form + '.sobj'
f_url = DATA + group + '/eigenforms/' + file_name
# print 'fafaf %s'%f_url
f = load(f_url)
file_name = weight + '_' + form + '-ev.sobj'
g_url = DATA + group + '/eigenvalues/' + file_name
# print 'gagag %s'%g_url
g = load(g_url)
loaded = True
except:
info['error'] = 'Data not available'
loaded = False
print 'hahahah %s' % loaded
if True == loaded:
# throw out disc = 0 keys for cusp forms
f_keys = f[2].keys()
if 'Sp4Z' == group and 'E' != form and 'Klingen' != form:
f_keys = filter(lambda (a, b, c): b ^ 2 < 4 * a * c, f_keys)
# sort the table of Fourier coefficients by discriminant, forms in increasing lexicographic order
if 'Sp8Z' != group and 'Sp6Z' != group:
__disc = lambda (a, b, c): 4 * a * c - b ** 2
__cmp = lambda (
a, b, c), (A, B, C): cmp((4 * a * c - b ** 2, a, b, c), (4 * A * C - B ** 2, A, B, C))
f_keys.sort(cmp=__cmp)
if 'Sp8Z' == group:
# matrix index is given as [m11 m22 m33 m44 m12 m13 m23 m14 m24 m34]
__mat = lambda (m11, m22, m33, m44, m12, m13, m23, m14, m24, m34): \
matrix(ZZ, 4, 4, [m11, m12, m13, m14, m12, m22, m23, m24,
m13, m23, m33, m34, m14, m24, m34, m44])
__disc = lambda i: __mat(i).det()
__cmp = lambda f1, f2: cmp([__mat(f1).det()] + list(f1), [__mat(f2).det()] + list(f2))
# print 'before: ', f_keys
f_keys.sort(cmp=__cmp)
# print f_keys
if 'Sp6Z' == group:
# matrix index is given as [m11/2 m22/2 m33/2 m12 m13 m23]
__mat = lambda (a, b, c, d, e, f): \
matrix(ZZ, 3, 3, [2 * a, d, e, d, 2 * b, f, e, f, 2 * c])
__disc = lambda i: __mat(i).det()
__cmp = lambda f1, f2: cmp([__mat(f1).det()] + list(f1), [__mat(f2).det()] + list(f2))
# print 'before: ', f_keys
f_keys.sort(cmp=__cmp)
# print f_keys
# make the coefficients of the M_k(Sp(4,Z)) forms integral
if 'Sp4Z' == group: # or 'Sp4Z_2' == group:
d = lcm(map(lambda n: denominator(n), f[1].coefficients()))
f = list(f)
f[1] *= d
for k in f[2]:
f[2][k] *= d
try:
if not ev_modulus:
m = 0
else:
m = int(ev_modulus)
info['ev_modulus'] = m
K = g[0].parent().fraction_field()
if m != 0:
if QQ == K:
for i in g[1]:
g[1][i] = Integer(g[1][i]) % m
else:
I = K.ideal(m)
for i in g[1]:
g[1][i] = I.reduce(g[1][i])
except:
info['fc_modulus'] = 0
pass
try:
if not fc_modulus:
m = 0
else:
m = int(fc_modulus)
info['fc_modulus'] = m
K = g[0].parent().fraction_field()
if m != 0:
if 'Sp4Z_2' == group:
if QQ == K:
for i in f_keys:
f[2][i] = sum((v[0] % m) * v[1] for v in list(f[2][i]))
else:
I = K.ideal(m)
for i in f_keys:
f[2][i] = sum(I.reduce(v[0]) * v[1] for v in list(f[2][i]))
else:
if QQ == K:
for i in f_keys:
f[2][i] = Integer(f[2][i]) % m
else:
I = K.ideal(m)
for i in f_keys:
f[2][i] = I.reduce(f[2][i])
except:
info['fc_modulus'] = 0
pass
info['the_form'] = [f[0].parent(), f[1],
[(l, g[1][l]) for l in g[1]],
[(i, f[2][i], __disc(i)) for i in f_keys],
f_url, g_url]
# info['friends'] = [ ('Spin L-function', url_for('not_yet_implemented'))]#, \
## ('Standard L-function', url_for('not_yet_implemented')), \
## ('First Fourier-Jacobi coefficient', url_for('not_yet_implemented'))]
location = url_for('ModularForm_GSp4_Q_top_level', group=group, page=page, weight=weight, form=form)
info['form_name'] = form
bread += [(weight + '_' + form, location)]
return render_template("ModularForm_GSp4_Q/ModularForm_GSp4_Q_specimen.html",
title='Siegel modular form ' + weight + '_' + form,
bread=bread, **info)
# if a nonexisting page was requested return the homepage of Siegel modular forms
return render_webpage()
def render_webpage(args={}):
"""
Configure and return a template for the Siegel modular forms pages.
"""
info = dict(args)
# info['learnmore'] = [ ('Siegel modular forms', 'http://en.wikipedia.org/wiki/Siegel_modular_form')]
info['learnmore'] = []
bread = [('Siegel modular forms', url_for('ModularForm_GSp4_Q_top_level'))]
if len(args) == 0:
return render_template("ModularForm_GSp4_Q_navigation.html",
title='Siegel Modular Forms',
bread=bread,
**info)
# possible keys for the URL
group = args.get('group')
character = args.get('character')
weight = args.get('weight')
level = args.get('level')
form = args.get('form')
page = args.get('page')
weight_range = args.get('weight_range')
# set info
info['group'] = group
info['form'] = form
info['level'] = level
# We check first the key 'group' since it is needed always
tmp_parent_as_tex = '%s'
if args['group']:
if 'Sp4Z' == args['group']:
info['parent_as_tex'] = 'M_{k}\\big({\\rm Sp}(4,\\mathbb{Z})\\big)'
# dimension = siegel_core._dimension_Sp4Z
dimension = dimensions.dimension_Sp4Z
info['generators'] = 'smf.Igusa_generators'
elif 'Gamma0_2' == args['group']:
info['parent_as_tex'] = 'M_{k}\\big(\\Gamma_0(2)\\big)'
dimension = dimensions.dimension_Gamma0_2
elif 'Gamma1_2' == args['group']:
info['parent_as_tex'] = 'M_{k}\\big(\\Gamma_1(2)\\big)'
dimension = dimensions.dimension_Gamma1_2
elif 'Gamma_2' == args['group']:
info['parent_as_tex'] = 'M_{k}\\big(\\Gamma(2)\\big)'
dimension = dimensions.dimension_Gamma_2
elif 'Sp4Z_2' == args['group']:
info[
'parent_as_tex'] = 'M_{k,2}\\big({\\rm Sp}(4,\\mathbb{Z})\\big)'
dimension = siegel_core._dimension_Sp4Z_2
elif 'Sp6Z' == args['group']:
info['parent_as_tex'] = 'M_k\\big({\\rm Sp}(6,\\mathbb{Z})\\big)'
# dimension = siegel_core._dimension_Sp6Z
dimension = dimensions.dimension_Sp6Z
elif 'Sp8Z' == args['group']:
info['parent_as_tex'] = 'M_k\\big({\\rm Sp}(8,\\mathbb{Z})\\big)'
# dimension = siegel_core._dimension_Sp8Z
dimension = dimensions.dimension_Sp8Z
elif 'Gamma0_4_half' == group:
info['parent_as_tex'] = 'M_{k-1/2}\\big(\\Gamma_0(4)\\big)'
# dimension = siegel_core._dimension_Gamma0_4_half
dimension = dimensions.dimension_Gamma0_4_half
elif 'Kp' == args['group']:
info['parent_as_tex'] = 'M_k\\big(K(p)\\big)'
info['learnmore'] += [('Paramodular forms',
'http://math.lfc.edu/~yuen/paramodular/')]
info['generators'] = 'smf.Kp_generators'
dimension = siegel_core._dimension_Kp
elif 'Gamma0_2' == args['group']:
info['parent_as_tex'] = 'M_k\\big(\\Gamma_0(2)\\big)'
dimension = siegel_core._dimension_Gamma0_2
elif 'Gamma0_3' == args['group']:
info['parent_as_tex'] = 'M_k\\big(\\Gamma_0(3)\\big)'
dimension = siegel_core._dimension_Gamma0_3
elif 'Gamma0_3_psi_3' == args['group']:
info['parent_as_tex'] = 'M_k\\big(\\Gamma_0(3,\\psi_3)\\big)'
dimension = siegel_core._dimension_Gamma0_3_psi_3
elif 'Gamma0_4' == args['group']:
info['parent_as_tex'] = 'M_k\\big(\\Gamma_0(4)\\big)'
dimension = siegel_core._dimension_Gamma0_4
elif 'Gamma0_4_psi_4' == args['group']:
info['parent_as_tex'] = 'M_k\\big(\\Gamma_0(4,\\psi_4)\\big)'
dimension = siegel_core._dimension_Gamma0_4_psi_4
else:
info[
'error'] = 'Request for unavailable type of Siegel modular form'
return render_template("None.html", **info)
info['learnmore'] += [('The spaces \(' + info['parent_as_tex'] + '\)',
url_for('ModularForm_GSp4_Q_top_level',
group=group,
page='basic'))]
bread += [('\(' + info['parent_as_tex'] + '\)',
url_for('ModularForm_GSp4_Q_top_level',
group=group,
page='forms'))]
else:
# some nonsense request came in, we answer by nonsense too
return render_template("None.html")
# We branch now according to the value of the key 'page'
##########################################################
## FORM COLLECTION REQUEST
##########################################################
if page == 'forms':
try:
f = urllib.urlopen(DATA + group + '/available_eigenforms.p')
go = pickle.load(f)
f.close()
forms_exist = True
except (IOError, EOFError, KeyError):
info['error'] = 'No data access'
forms_exist = False
if True == forms_exist:
info['forms'] = [(k, [(form, go[k][form]) for form in go[k]])
for k in go]
return render_template("ModularForm_GSp4_Q_forms.html",
title='Siegel modular forms \(' +
info['parent_as_tex'] + '\)',
bread=bread,
**info)
if page == 'basic':
bread += [('Basic information',
url_for('ModularForm_GSp4_Q_top_level',
group=group,
page=page))]
return render_template("ModularForm_GSp4_Q_basic.html",
title='Siegel modular forms basic information',
bread=bread,
**info)
##########################################################
## DIMENSIONS REQUEST
##########################################################
if page == 'dimensions':
# We check whether the weight_range makes sense to us and, if so, dispatch it
info['weight_range'] = weight_range
try:
assert info['weight_range'], 'Please enter a valid argument'
min_wt, max_wt, sym_pow = input_parser.kj_parser(weight_range)
min_wt = Integer(min_wt)
if None == max_wt or max_wt < min_wt:
max_wt = min_wt
if None == sym_pow:
sym_pow = 0
assert min_wt < 1000000 and (
max_wt - min_wt + 1
) * max_wt < 10000 and sym_pow < 1000, '%d-%d,%d: Input too large: Please enter smaller range or numbers.' % (
max_wt, min_wt, sym_pow)
except Exception as e:
info['error'] = str(e)
return render_template("ModularForm_GSp4_Q_dimensions.html",
title='Siegel modular forms dimensions \(' +
info['parent_as_tex'] + '\)',
bread=bread,
**info)
# A priori the request is reasonable, so we try to get the data for the answer
try:
info['new_method'] = None
if 'Gamma_2' == group or 'Gamma0_2' == group or 'Gamma1_2' == group or 'Sp4Z' == group or 'Sp6Z' == group or 'Sp8Z' == group or 'Gamma0_4_half' == group:
info['sym_pow'] = sym_pow
info['table_headers'], info['dimensions'] = dimension(
range(min_wt, max_wt + 1), sym_pow)
####### a hack ########
info['new_method'] = 'new_method'
bread += [('Dimensions',
url_for('ModularForm_GSp4_Q_top_level',
group=group,
page=page,
level=level,
weight_range=weight_range))]
elif 'Kp' == group:
info['dimensions'] = [(k, dimension(k, tp=int(level)))
for k in range(min_wt, max_wt + 1)]
bread += [('Dimensions',
url_for('ModularForm_GSp4_Q_top_level',
group=group,
page=page,
level=level,
weight_range=weight_range))]
else:
info['dimensions'] = [(k, dimension(k))
for k in range(min_wt, max_wt + 1)]
bread += [('Dimensions',
url_for('ModularForm_GSp4_Q_top_level',
group=group,
page=page,
weight_range=weight_range))]
except Exception as e:
info['error'] = 'Functional error: %s' % (str(e)
) #(sys.exc_info()[0])
return render_template("ModularForm_GSp4_Q_dimensions.html",
title='Siegel modular forms dimensions \(' +
info['parent_as_tex'] + '\)',
bread=bread,
**info)
# We provide some headers for the 'old' method and ask for rendering an answer
if info['new_method']:
info['table_headers'] = info['table_headers']
# elif 'Sp8Z' == group:
# info['table_headers'] = ['Weight', 'Total', 'Ikeda lifts', 'Miyawaki lifts', 'Other']
elif 'Sp6Z' == group:
info['table_headers'] = [
'Weight', 'Total', 'Miyawaki lifts I', 'Miyawaki lifts II',
'Other'
]
elif group == 'Kp':
info['table_headers'] = [
"Weight", "Total", "Gritsenko Lifts", "Nonlifts", "Oldforms"
]
elif 'Sp4Z_2' == group or 'Gamma0_4_half' == group:
info['table_headers'] = ['Weight', 'Total', 'Non cusp', 'Cusp']
else:
info['table_headers'] = [
"Weight", "Total", "Eisenstein", "Klingen", "Maass",
"Interesting"
]
return render_template("ModularForm_GSp4_Q_dimensions.html",
title='Siegel modular forms dimensions \(' +
info['parent_as_tex'] + '\)',
bread=bread,
**info)
##########################################################
## SPECIFIC FORM REQUEST
##########################################################
if page == 'specimen':
info['weight'] = weight
ev_modulus = args.get('emod')
fc_modulus = args.get('fcmod')
erange = args.get('erange')
fcrange = args.get('fcrange')
# try to load data
if 'Kp' == group or 'Sp4Z_2' == group or 'Sp4Z' == group:
# fetch from mongodb
try:
smple = sample.Sample([group], weight + '_' + form)
f = (smple.field()(0), smple.explicit_formula(),
smple.Fourier_coefficients()
if smple.Fourier_coefficients() else {})
g = (smple.field()(0),
smple.eigenvalues() if smple.eigenvalues() else {})
file_name = weight + '_' + form + '.sobj'
f_url = DATA + group + '/eigenforms/' + file_name
file_name = weight + '_' + form + '-ev.sobj'
g_url = DATA + group + '/eigenvalues/' + file_name
loaded = True
except Exception as e:
info['error'] = 'Data not available: %s %s' % (str(e), weight +
'_' + form)
loaded = False
else:
try:
file_name = weight + '_' + form + '.sobj'
f_url = DATA + group + '/eigenforms/' + file_name
# print 'fafaf %s'%f_url
f = load(f_url)
file_name = weight + '_' + form + '-ev.sobj'
g_url = DATA + group + '/eigenvalues/' + file_name
# print 'gagag %s'%g_url
g = load(g_url)
loaded = True
except:
info['error'] = 'Data not available'
loaded = False
if True == loaded:
# define specific methods for computing discriminant and ordering of form
if 'Sp8Z' != group and 'Sp6Z' != group: # with current data this is all degree 2 SMFs
__disc = lambda (a, b, c): 4 * a * c - b**2
__cmp = lambda (a, b, c), (A, B, C): cmp(
(4 * a * c - b**2, a, b, c), (4 * A * C - B**2, A, B, C))
if 'Sp8Z' == group:
# matrix index is given as [m11 m22 m33 m44 m12 m13 m23 m14 m24 m34]
__mat = lambda (m11, m22, m33, m44, m12, m13, m23, m14, m24, m34): \
matrix(ZZ, 4, 4, [m11, m12, m13, m14, m12, m22, m23, m24,
m13, m23, m33, m34, m14, m24, m34, m44])
__disc = lambda i: __mat(i).det()
__cmp = lambda f1, f2: cmp([__mat(f1).det()] + list(f1),
[__mat(f2).det()] + list(f2))
if 'Sp6Z' == group:
# matrix index is given as [m11/2 m22/2 m33/2 m12 m13 m23]
__mat = lambda (a, b, c, d, e, f): \
matrix(ZZ, 3, 3, [2 * a, d, e, d, 2 * b, f, e, f, 2 * c])
__disc = lambda i: __mat(i).det()
__cmp = lambda f1, f2: cmp([__mat(f1).det()] + list(f1),
[__mat(f2).det()] + list(f2))
# make the coefficients of the M_k(Sp(4,Z)) forms integral
# if 'Sp4Z' == group: # or 'Sp4Z_2' == group:
# d = lcm(map(lambda n: denominator(n), f[1].coefficients()))
# f = list(f)
# f[1] *= d
# for k in f[2]:
# f[2][k] *= d
# replace generator with a to make things prettier
if isinstance(f[0].parent(), Field):
if f[0].parent() != QQ:
gen = str(f[0].parent().gen())
info['gen_coeff_field'] = teXify_pol(
str(f[0].parent().gen()).replace(gen, 'a'))
info['poly_coeff_field'] = teXify_pol(
str(f[0].parent().polynomial()).replace(gen, 'a'))
info['poly_in_gens'] = teXify_pol(
str(f[1]).replace(gen, 'a'))
else:
info['poly_in_gens'] = teXify_pol(str(f[1]))
else:
# coefficient field is not a sage field, so just assume its supposed to be rationals
info['poly_in_gens'] = teXify_pol(str(f[1]))
# isolate requested eigenvalue indices
if erange == 'all':
filt_evals = g[1]
eval_index = filt_evals.keys()
info['erangedesc'] = 'all available eigenvalues'
else:
if erange:
spliterange = erange.split('-')
if len(spliterange) > 1 and spliterange[0].isdigit(
) and spliterange[1].isdigit():
elow, ehigh = int(spliterange[0]), int(spliterange[1])
# filter out to have eigenvalues in [elow, ehigh]
filt_evals = {
n: lam
for n, lam in g[1].iteritems()
if int(n) >= elow and int(n) <= ehigh
}
eval_index = filt_evals.keys()
info[
'erangedesc'] = 'eigenvalues with $n$ in [' + ` elow ` + ', ' + ` ehigh ` + ']'
else:
# can't make sense of the range, return a default
info['erange'] = ''
filt_evals = g[1]
eval_index = filt_evals.keys()[0:20]
info['erangedesc'] = 'the first few eigenvalues'
# prepare formatted eigenvalues
ftd_evals = []
try:
if not ev_modulus:
m = 0
else:
m = int(ev_modulus)
info['ev_modulus'] = m
K = g[0].parent().fraction_field()
if m != 0:
if QQ == K:
for i in eval_index:
rdcd_eval = Integer(g[1][i]) % m
ftd_evals.append(
(str(i), teXify_pol(str(rdcd_eval))))
else:
I = K.ideal(m)
for i in eval_index:
rdcd_eval = I.reduce(g[1][i])
ftd_evals.append(
(str(i),
teXify_pol(str(rdcd_eval).replace(gen, 'a'))))
info['emoddesc'] = 'reduced modulo ' + ` m ` + '.'
else:
for i in eval_index:
if QQ == K:
ftd_evals.append(
(str(i), teXify_pol(str(g[1][i]))))
else:
ftd_evals.append(
(str(i),
teXify_pol(str(g[1][i]).replace(gen, 'a'))))
info['emoddesc'] = 'with no reduction.'
except:
pass
if (fcrange == 'all'):
filt_fcs = f[2]
fc_index = filt_fcs.keys()
fc_index.sort(cmp=__cmp)
info['fcrangedesc'] = 'all available Fourier coefficients'
else:
if fcrange:
splitfcrange = fcrange.split('-')
if len(splitfcrange) > 1 and splitfcrange[0].isdigit(
) and splitfcrange[1].isdigit():
fclow, fchigh = int(splitfcrange[0]), int(
splitfcrange[1])
filt_fcs = {
n: fc
for n, fc in f[2].iteritems()
if __disc(n) >= fclow and __disc(n) <= fchigh
}
fc_index = filt_fcs.keys()
fc_index.sort(cmp=__cmp)
info[
'fcrangedesc'] = 'Fourier coefficients with index such that $D$ is in [' + ` fclow ` + ', ' + ` fchigh ` + ']'
else:
# can't make sense of the range, return a default
info['fcrange'] = ''
filt_fcs = f[2]
fc_index = filt_fcs.keys()
fc_index.sort(cmp=__cmp)
fc_index = fc_index[0:20]
info['fcrangedesc'] = 'the first few Fourier coefficients'
# prepare formatted fourier coefficients
ftd_fcs = []
try:
if not fc_modulus:
m = 0
else:
m = int(fc_modulus)
info['fc_modulus'] = m
K = g[0].parent().fraction_field()
if m != 0:
if 'Sp4Z_2' == group:
if QQ == K:
for i in fc_index:
ftd_fc = sum(
(v[0] % m) * v[1] for v in list(f[2][i]))
ftd_fcs.append(
(str(i), teXify_pol(str(ftd_fc)),
str(__disc(i))))
else:
I = K.ideal(m)
for i in fc_index:
ftd_fc = sum(
I.reduce(v[0]) * v[1]
for v in list(f[2][i]))
ftd_fcs.append(
(str(i),
teXify_pol(str(ftd_fc).replace(gen, 'a')),
str(__disc(i))))
else:
if QQ == K:
for i in fc_index:
ftd_fc = Integer(f[2][i]) % m
ftd_fcs.append(
(str(i), teXify_pol(str(ftd_fc)),
str(__disc(i))))
else:
I = K.ideal(m)
for i in fc_index:
ftd_fc = I.reduce(f[2][i])
ftd_fcs.append(
(str(i),
teXify_pol(str(ftd_fc).replace(gen, 'a')),
str(__disc(i))))
info['fcmoddesc'] = 'reduced modulo ' + ` m ` + '.'
else:
for i in fc_index:
ftd_fc = f[2][i]
if QQ == K:
ftd_fcs.append((str(i), teXify_pol(str(ftd_fc)),
str(__disc(i))))
else:
ftd_fcs.append(
(str(i),
teXify_pol(str(ftd_fc).replace(gen, 'a')),
str(__disc(i))))
info['fcmoddesc'] = 'with no reduction.'
except:
pass
location = url_for('ModularForm_GSp4_Q_top_level',
group=group,
page=page,
weight=weight,
form=form)
properties2 = [('Species', '$' + info['parent_as_tex'] + '$'),
('Weight', '%s' % weight)]
# if implemented, add L-function to friends
if 'Sp4Z' == group:
numEmbeddings = f[0].parent().degree()
friends = []
if form.endswith('E'):
# form is a Siegel-Eisenstein series, nothing interesting
# to show here
pass
elif form.endswith('Klingen'):
# form is a Klingen-Eisenstein series, so plausibly could
# link to the elliptic cusp form on the boundary it comes
# from
pass
elif form.endswith('Maass'):
# link the the underlying elliptic modular form. This
# datum is not included with the modular form, so just
# link to the unique Galois orbit of full level and
# weight 2k-2 (this code assumes Maeda).
ellWeight = 2 * int(weight) - 2
friends.append(
('Elliptic modular form',
'/ModularForm/GL2/Q/holomorphic/1/' + str(ellWeight)))
else:
# there are no other lifts to full level, so the L-function
# is primitive and therefore interesting
for embedding in range(0, numEmbeddings):
friends.append(
('Spin L-function for ' + str(weight) + '_' +
form + '.' + str(embedding),
'/L/ModularForm/GSp/Q/Sp4Z/specimen/' +
str(weight) + '/' + form + '/' + str(embedding)))
else:
friends = []
#TODO implement remaining spin L-functions, standard L-functions,
# and first Fourier-Jacobi coefficient
downloads = [('Fourier coefficients', f_url),
('Eigenvalues', g_url)]
location = url_for('ModularForm_GSp4_Q_top_level',
group=group,
page=page,
weight=weight,
form=form)
bread += [(weight + '_' + form, location)]
info['ftd_evals'] = ftd_evals
info['ftd_fcs'] = ftd_fcs
info['location'] = location
info['form_name'] = form
return render_template("ModularForm_GSp4_Q_specimen.html",
title='Siegel modular form ' + weight +
'_' + form,
bread=bread,
properties2=properties2,
friends=friends,
downloads=downloads,
**info)
# if a nonexisting page was requested return the homepage of Siegel modular forms
return render_webpage()
def render_webpage(args={}):
"""
Configure and return a template for the Siegel modular forms pages.
"""
info = dict(args)
# info['learnmore'] = [ ('Siegel modular forms', 'http://en.wikipedia.org/wiki/Siegel_modular_form')]
info['learnmore'] = []
bread = [('Siegel modular forms', url_for('ModularForm_GSp4_Q_top_level'))]
if len(args) == 0:
return render_template("ModularForm_GSp4_Q_navigation.html",
title='Siegel Modular Forms',
bread=bread,
**info)
# possible keys for the URL
group = args.get('group')
character = args.get('character')
weight = args.get('weight')
level = args.get('level')
form = args.get('form')
page = args.get('page')
weight_range = args.get('weight_range')
# set info
info['group'] = group
info['form'] = form
info['level'] = level
# We check first the key 'group' since it is needed always
tmp_parent_as_tex = '%s'
if args['group']:
if 'Sp4Z' == args['group']:
info['parent_as_tex'] = 'M_{k}\\big({\\rm Sp}(4,\\mathbb{Z})\\big)'
# dimension = siegel_core._dimension_Sp4Z
dimension = dimensions.dimension_Sp4Z
info['generators'] = 'smf.Igusa_generators'
elif 'Gamma0_2' == args['group']:
info['parent_as_tex'] = 'M_{k}\\big(\\Gamma_0(2)\\big)'
dimension = dimensions.dimension_Gamma0_2
elif 'Gamma1_2' == args['group']:
info['parent_as_tex'] = 'M_{k}\\big(\\Gamma_1(2)\\big)'
dimension = dimensions.dimension_Gamma1_2
elif 'Gamma_2' == args['group']:
info['parent_as_tex'] = 'M_{k}\\big(\\Gamma(2)\\big)'
dimension = dimensions.dimension_Gamma_2
elif 'Sp4Z_2' == args['group']:
info['parent_as_tex'] = 'M_{k,2}\\big({\\rm Sp}(4,\\mathbb{Z})\\big)'
dimension = siegel_core._dimension_Sp4Z_2
elif 'Sp6Z' == args['group']:
info['parent_as_tex'] = 'M_k\\big({\\rm Sp}(6,\\mathbb{Z})\\big)'
# dimension = siegel_core._dimension_Sp6Z
dimension = dimensions.dimension_Sp6Z
elif 'Sp8Z' == args['group']:
info['parent_as_tex'] = 'M_k\\big({\\rm Sp}(8,\\mathbb{Z})\\big)'
# dimension = siegel_core._dimension_Sp8Z
dimension = dimensions.dimension_Sp8Z
elif 'Gamma0_4_half' == group:
info['parent_as_tex'] = 'M_{k-1/2}\\big(\\Gamma_0(4)\\big)'
# dimension = siegel_core._dimension_Gamma0_4_half
dimension = dimensions.dimension_Gamma0_4_half
elif 'Kp' == args['group']:
info['parent_as_tex'] = 'M_k\\big(K(p)\\big)'
info['learnmore'] += [('Paramodular forms', 'http://math.lfc.edu/~yuen/paramodular/')]
info['generators'] = 'smf.Kp_generators'
dimension = siegel_core._dimension_Kp
elif 'Gamma0_2' == args['group']:
info['parent_as_tex'] = 'M_k\\big(\\Gamma_0(2)\\big)'
dimension = siegel_core._dimension_Gamma0_2
elif 'Gamma0_3' == args['group']:
info['parent_as_tex'] = 'M_k\\big(\\Gamma_0(3)\\big)'
dimension = siegel_core._dimension_Gamma0_3
elif 'Gamma0_3_psi_3' == args['group']:
info['parent_as_tex'] = 'M_k\\big(\\Gamma_0(3,\\psi_3)\\big)'
dimension = siegel_core._dimension_Gamma0_3_psi_3
elif 'Gamma0_4' == args['group']:
info['parent_as_tex'] = 'M_k\\big(\\Gamma_0(4)\\big)'
dimension = siegel_core._dimension_Gamma0_4
elif 'Gamma0_4_psi_4' == args['group']:
info['parent_as_tex'] = 'M_k\\big(\\Gamma_0(4,\\psi_4)\\big)'
dimension = siegel_core._dimension_Gamma0_4_psi_4
else:
info['error'] = 'Request for unavailable type of Siegel modular form'
return render_template("None.html", **info)
info['learnmore'] += [('The spaces \(' + info['parent_as_tex'] + '\)', url_for(
'ModularForm_GSp4_Q_top_level', group=group, page='basic'))]
bread += [('\(' + info['parent_as_tex'] + '\)', url_for('ModularForm_GSp4_Q_top_level',
group=group, page='forms'))]
else:
# some nonsense request came in, we answer by nonsense too
return render_template("None.html")
# We branch now according to the value of the key 'page'
##########################################################
## FORM COLLECTION REQUEST
##########################################################
if page == 'forms':
try:
f = urllib.urlopen(DATA + group + '/available_eigenforms.p')
go = pickle.load(f)
f.close()
forms_exist = True
except (IOError, EOFError, KeyError):
info['error'] = 'No data access'
forms_exist = False
if True == forms_exist:
info['forms'] = [(k, [(form, go[k][form]) for form in go[k]]) for k in go]
return render_template("ModularForm_GSp4_Q_forms.html",
title='Siegel modular forms \(' + info['parent_as_tex'] + '\)',
bread=bread, **info)
if page == 'basic':
bread += [('Basic information', url_for('ModularForm_GSp4_Q_top_level', group=group, page=page))]
return render_template("ModularForm_GSp4_Q_basic.html",
title='Siegel modular forms basic information',
bread=bread, **info)
##########################################################
## DIMENSIONS REQUEST
##########################################################
if page == 'dimensions':
# We check whether the weight_range makes sense to us and, if so, dispatch it
info['weight_range'] = weight_range
try:
assert info['weight_range'], 'Please enter a valid argument'
min_wt, max_wt, sym_pow = input_parser.kj_parser( weight_range)
min_wt = Integer( min_wt)
if None == max_wt or max_wt < min_wt:
max_wt = min_wt
if None == sym_pow:
sym_pow = 0
assert min_wt < 1000000 and (max_wt - min_wt + 1) * max_wt < 10000 and sym_pow < 1000, '%d-%d,%d: Input too large: Please enter smaller range or numbers.' % (max_wt, min_wt, sym_pow)
except Exception as e:
info['error'] = str(e)
return render_template( "ModularForm_GSp4_Q_dimensions.html",
title='Siegel modular forms dimensions \(' + info['parent_as_tex'] + '\)',
bread=bread, **info)
# A priori the request is reasonable, so we try to get the data for the answer
try:
info['new_method'] = None
if 'Gamma_2' == group or 'Gamma0_2' == group or 'Gamma1_2' == group or 'Sp4Z' == group or 'Sp6Z' == group or 'Sp8Z' == group or 'Gamma0_4_half' == group:
info['sym_pow'] = sym_pow
info['table_headers'], info['dimensions'] = dimension( range( min_wt, max_wt + 1), sym_pow)
####### a hack ########
info['new_method'] = 'new_method'
bread += [('Dimensions',
url_for('ModularForm_GSp4_Q_top_level', group=group, page=page, level=level, weight_range=weight_range))]
elif 'Kp' == group:
info['dimensions'] = [(k, dimension(k, tp=int(level))) for k in range(min_wt, max_wt + 1)]
bread += [('Dimensions',
url_for('ModularForm_GSp4_Q_top_level', group=group, page=page, level=level, weight_range=weight_range))]
else:
info['dimensions'] = [(k, dimension(k)) for k in range(min_wt, max_wt + 1)]
bread += [('Dimensions',
url_for('ModularForm_GSp4_Q_top_level', group=group, page=page, weight_range=weight_range))]
except Exception as e:
info['error'] = 'Functional error: %s' % (str(e)) #(sys.exc_info()[0])
return render_template( "ModularForm_GSp4_Q_dimensions.html",
title='Siegel modular forms dimensions \(' + info['parent_as_tex'] + '\)',
bread=bread, **info)
# We provide some headers for the 'old' method and ask for rendering an answer
if info['new_method']:
info['table_headers'] = info['table_headers']
# elif 'Sp8Z' == group:
# info['table_headers'] = ['Weight', 'Total', 'Ikeda lifts', 'Miyawaki lifts', 'Other']
elif 'Sp6Z' == group:
info['table_headers'] = ['Weight', 'Total', 'Miyawaki lifts I', 'Miyawaki lifts II', 'Other']
elif group == 'Kp':
info['table_headers'] = ["Weight", "Total", "Gritsenko Lifts", "Nonlifts", "Oldforms"]
elif 'Sp4Z_2' == group or 'Gamma0_4_half' == group:
info['table_headers'] = ['Weight', 'Total', 'Non cusp', 'Cusp']
else:
info['table_headers'] = ["Weight", "Total", "Eisenstein", "Klingen", "Maass", "Interesting"]
return render_template("ModularForm_GSp4_Q_dimensions.html",
title='Siegel modular forms dimensions \(' + info['parent_as_tex'] + '\)',
bread=bread, **info)
##########################################################
## SPECIFIC FORM REQUEST
##########################################################
if page == 'specimen':
info['weight'] = weight
ev_modulus = args.get('emod')
fc_modulus = args.get('fcmod')
erange = args.get('erange')
fcrange = args.get('fcrange')
# try to load data
if 'Kp' == group or 'Sp4Z_2' == group or 'Sp4Z' == group:
# fetch from mongodb
try:
smple = sample.Sample( [group], weight + '_' + form)
f = (smple.field()(0), smple.explicit_formula(), smple.Fourier_coefficients() if smple.Fourier_coefficients() else {})
g = (smple.field()(0), smple.eigenvalues() if smple.eigenvalues() else {})
file_name = weight + '_' + form + '.sobj'
f_url = DATA + group + '/eigenforms/' + file_name
file_name = weight + '_' + form + '-ev.sobj'
g_url = DATA + group + '/eigenvalues/' + file_name
loaded = True
except Exception as e:
info['error'] = 'Data not available: %s %s' % (str(e), weight + '_' + form)
loaded = False
else:
try:
file_name = weight + '_' + form + '.sobj'
f_url = DATA + group + '/eigenforms/' + file_name
# print 'fafaf %s'%f_url
f = load(f_url)
file_name = weight + '_' + form + '-ev.sobj'
g_url = DATA + group + '/eigenvalues/' + file_name
# print 'gagag %s'%g_url
g = load(g_url)
loaded = True
except:
info['error'] = 'Data not available'
loaded = False
if True == loaded:
# define specific methods for computing discriminant and ordering of form
if 'Sp8Z' != group and 'Sp6Z' != group: # with current data this is all degree 2 SMFs
__disc = lambda (a, b, c): 4 * a * c - b ** 2
__cmp = lambda (
a, b, c), (A, B, C): cmp((4 * a * c - b ** 2, a, b, c), (4 * A * C - B ** 2, A, B, C))
if 'Sp8Z' == group:
# matrix index is given as [m11 m22 m33 m44 m12 m13 m23 m14 m24 m34]
__mat = lambda (m11, m22, m33, m44, m12, m13, m23, m14, m24, m34): \
matrix(ZZ, 4, 4, [m11, m12, m13, m14, m12, m22, m23, m24,
m13, m23, m33, m34, m14, m24, m34, m44])
__disc = lambda i: __mat(i).det()
__cmp = lambda f1, f2: cmp([__mat(f1).det()] + list(f1), [__mat(f2).det()] + list(f2))
if 'Sp6Z' == group:
# matrix index is given as [m11/2 m22/2 m33/2 m12 m13 m23]
__mat = lambda (a, b, c, d, e, f): \
matrix(ZZ, 3, 3, [2 * a, d, e, d, 2 * b, f, e, f, 2 * c])
__disc = lambda i: __mat(i).det()
__cmp = lambda f1, f2: cmp([__mat(f1).det()] + list(f1), [__mat(f2).det()] + list(f2))
# make the coefficients of the M_k(Sp(4,Z)) forms integral
# if 'Sp4Z' == group: # or 'Sp4Z_2' == group:
# d = lcm(map(lambda n: denominator(n), f[1].coefficients()))
# f = list(f)
# f[1] *= d
# for k in f[2]:
# f[2][k] *= d
# replace generator with a to make things prettier
if isinstance(f[0].parent(), Field):
if f[0].parent()!=QQ:
gen = str(f[0].parent().gen())
info['gen_coeff_field'] = teXify_pol(str(f[0].parent().gen()).replace(gen, 'a'))
info['poly_coeff_field'] = teXify_pol(str(f[0].parent().polynomial()).replace(gen, 'a'))
info['poly_in_gens'] = teXify_pol(str(f[1]).replace(gen, 'a'))
else:
info['poly_in_gens'] = teXify_pol(str(f[1]))
else:
# coefficient field is not a sage field, so just assume its supposed to be rationals
info['poly_in_gens'] = teXify_pol(str(f[1]))
# isolate requested eigenvalue indices
if erange=='all':
filt_evals = g[1]
eval_index = filt_evals.keys()
info['erangedesc']= 'all available eigenvalues'
else:
if erange:
spliterange = erange.split('-')
if len(spliterange)>1 and spliterange[0].isdigit() and spliterange[1].isdigit():
elow, ehigh = int(spliterange[0]), int(spliterange[1])
# filter out to have eigenvalues in [elow, ehigh]
filt_evals = {n: lam for n, lam in g[1].iteritems() if int(n)>=elow and int(n)<=ehigh}
eval_index = filt_evals.keys()
info['erangedesc'] = 'eigenvalues with $n$ in [' + `elow` + ', ' + `ehigh` + ']'
else:
# can't make sense of the range, return a default
info['erange'] = ''
filt_evals = g[1]
eval_index = filt_evals.keys()[0:20]
info['erangedesc'] = 'the first few eigenvalues'
# prepare formatted eigenvalues
ftd_evals = []
try:
if not ev_modulus:
m = 0
else:
m = int(ev_modulus)
info['ev_modulus'] = m
K = g[0].parent().fraction_field()
if m != 0:
if QQ == K:
for i in eval_index:
rdcd_eval = Integer(g[1][i]) % m
ftd_evals.append((str(i), teXify_pol(str(rdcd_eval))))
else:
I = K.ideal(m)
for i in eval_index:
rdcd_eval = I.reduce(g[1][i])
ftd_evals.append((str(i), teXify_pol(str(rdcd_eval).replace(gen, 'a'))))
info['emoddesc'] = 'reduced modulo ' + `m` + '.'
else:
for i in eval_index:
if QQ == K:
ftd_evals.append((str(i), teXify_pol(str(g[1][i]))))
else:
ftd_evals.append((str(i), teXify_pol(str(g[1][i]).replace(gen, 'a'))))
info['emoddesc'] = 'with no reduction.'
except:
pass
if (fcrange=='all'):
filt_fcs = f[2]
fc_index = filt_fcs.keys()
fc_index.sort(cmp=__cmp)
info['fcrangedesc'] = 'all available Fourier coefficients'
else:
if fcrange:
splitfcrange = fcrange.split('-')
if len(splitfcrange)>1 and splitfcrange[0].isdigit() and splitfcrange[1].isdigit():
fclow, fchigh = int(splitfcrange[0]), int(splitfcrange[1])
filt_fcs = {n: fc for n, fc in f[2].iteritems() if __disc(n)>=fclow and __disc(n)<=fchigh}
fc_index = filt_fcs.keys()
fc_index.sort(cmp=__cmp)
info['fcrangedesc'] = 'Fourier coefficients with index such that $D$ is in [' + `fclow` + ', ' + `fchigh` + ']'
else:
# can't make sense of the range, return a default
info['fcrange'] = ''
filt_fcs = f[2]
fc_index = filt_fcs.keys()
fc_index.sort(cmp=__cmp)
fc_index = fc_index[0:20]
info['fcrangedesc'] = 'the first few Fourier coefficients'
# prepare formatted fourier coefficients
ftd_fcs = []
try:
if not fc_modulus:
m = 0
else:
m = int(fc_modulus)
info['fc_modulus'] = m
K = g[0].parent().fraction_field()
if m != 0:
if 'Sp4Z_2' == group:
if QQ == K:
for i in fc_index:
ftd_fc = sum((v[0] % m) * v[1] for v in list(f[2][i]))
ftd_fcs.append((str(i),
teXify_pol(str(ftd_fc)),
str(__disc(i))))
else:
I = K.ideal(m)
for i in fc_index:
ftd_fc = sum(I.reduce(v[0]) * v[1] for v in list(f[2][i]))
ftd_fcs.append((str(i),
teXify_pol(str(ftd_fc).replace(gen, 'a')),
str(__disc(i))))
else:
if QQ == K:
for i in fc_index:
ftd_fc = Integer(f[2][i]) % m
ftd_fcs.append((str(i),
teXify_pol(str(ftd_fc)),
str(__disc(i))))
else:
I = K.ideal(m)
for i in fc_index:
ftd_fc = I.reduce(f[2][i])
ftd_fcs.append((str(i),
teXify_pol(str(ftd_fc).replace(gen, 'a')),
str(__disc(i))))
info['fcmoddesc'] = 'reduced modulo ' + `m` + '.'
else:
for i in fc_index:
ftd_fc = f[2][i]
if QQ == K:
ftd_fcs.append((str(i),
teXify_pol(str(ftd_fc)),
str(__disc(i))))
else:
ftd_fcs.append((str(i),
teXify_pol(str(ftd_fc).replace(gen, 'a')),
str(__disc(i))))
info['fcmoddesc'] = 'with no reduction.'
except:
pass
location = url_for('ModularForm_GSp4_Q_top_level', group=group, page=page, weight=weight, form=form)
properties2 = [('Species', '$' + info['parent_as_tex'] + '$'),
('Weight', '%s' % weight)]
# if implemented, add L-function to friends
if 'Sp4Z'== group:
numEmbeddings = f[0].parent().degree()
friends = []
if form.endswith('E'):
# form is a Siegel-Eisenstein series, nothing interesting
# to show here
pass
elif form.endswith('Klingen'):
# form is a Klingen-Eisenstein series, so plausibly could
# link to the elliptic cusp form on the boundary it comes
# from
pass
elif form.endswith('Maass'):
# link the the underlying elliptic modular form. This
# datum is not included with the modular form, so just
# link to the unique Galois orbit of full level and
# weight 2k-2 (this code assumes Maeda).
ellWeight = 2*int(weight) - 2
friends.append(('Elliptic modular form',
'/ModularForm/GL2/Q/holomorphic/1/' + str(ellWeight)))
else:
# there are no other lifts to full level, so the L-function
# is primitive and therefore interesting
for embedding in range(0, numEmbeddings):
friends.append(('Spin L-function for '
+ str(weight) + '_' + form + '.' + str(embedding),
'/L/ModularForm/GSp/Q/Sp4Z/specimen/'
+ str(weight) + '/' + form + '/' + str(embedding)))
else:
friends = []
#TODO implement remaining spin L-functions, standard L-functions,
# and first Fourier-Jacobi coefficient
downloads = [('Fourier coefficients', f_url),
('Eigenvalues', g_url)]
location = url_for('ModularForm_GSp4_Q_top_level', group=group, page=page, weight=weight, form=form)
bread += [(weight + '_' + form, location)]
info['ftd_evals'] = ftd_evals
info['ftd_fcs'] = ftd_fcs
info['location'] = location
info['form_name'] = form
return render_template("ModularForm_GSp4_Q_specimen.html",
title = 'Siegel modular form ' + weight + '_' + form,
bread=bread, properties2=properties2, friends=friends, downloads=downloads, **info)
# if a nonexisting page was requested return the homepage of Siegel modular forms
return render_webpage()
