def get_new_and_oldspace_decomposition(k, N, xi=0):
r"""
Get decomposition of the new and oldspace S_k(N,xi) into submodules.
"""
M = ModularSymbols(N, k, sign=1).cuspidal_submodule()
L = list()
L = [M.new_submodule().dimension()]
check_dim = M.new_submodule().dimension()
for d in divisors(N):
if(d == 1):
continue
O = M.old_submodule(d)
Od = O.dimension()
if(d == N and k == 2 or Od == 0):
continue
S = ModularSymbols(ZZ(N / d), k, sign=1).cuspidal_submodule().new_submodule()
Sd = S.dimension()
if(Sd == 0):
logger.info("%s, %s" % (O, Od))
logger.info("%s, %s" % (S, Sd))
mult = len(divisors(ZZ(d)))
check_dim = check_dim + mult * Sd
L.append((ZZ(N / d), mult, Sd))
check_dim = check_dim - M.dimension()
if(check_dim != 0):
raise ArithmeticError("Something wrong! check_dim=%s" % check_dim)
return str(M.dimension(), L)
def render_elliptic_modular_forms(level=0, weight=0, character=None, label='', **kwds):
r"""
Default input of same type as required. Note that for holomorphic modular forms: level=0 or weight=0 are non-existent.
"""
if character is None and level == 0 and weight == 0:
character = 0
elif character is None:
character = -1
emf_logger.debug(
"In render: level={0},weight={1},character={2},label={3}".format(level, weight, character, label))
emf_logger.debug("args={0}".format(request.args))
emf_logger.debug("args={0}".format(request.form))
emf_logger.debug("met={0}".format(request.method))
keys = ['download', 'jump_to']
info = get_args(request, level, weight, character, label, keys=keys)
level = info['level']
weight = info['weight']
character = info['character']
label = info['label']
emf_logger.debug("info={0}".format(info))
emf_logger.debug("level=%s, %s" % (level, type(level)))
emf_logger.debug("label=%s, %s" % (label, type(label)))
emf_logger.debug("wt=%s, %s" % (weight, type(weight)))
emf_logger.debug("character=%s, %s" % (character, type(character)))
if 'download' in info:
return get_downloads(**info)
emf_logger.debug("info=%s" % info)
## Consistency of arguments>
# if level<=0: level=None
# if weight<=0: weight=None
if 'jump_to' in info: # try to find out which form we want to jump
s = my_get(info, 'jump_to', '', str)
emf_logger.info("info.keys1={0}".format(info.keys()))
info.pop('jump_to')
emf_logger.info("info.keys2={0}".format(info.keys()))
args = extract_data_from_jump_to(s)
emf_logger.debug("args=%s" % args)
return redirect(url_for("emf.render_elliptic_modular_forms", **args), code=301)
# return render_elliptic_modular_forms(**args)
if level > 0 and weight > 0 and character > -1 and label != '':
emf_logger.debug("info=%s" % info)
return render_one_elliptic_modular_form(**info)
if level > 0 and weight > 0 and character > -1:
return render_elliptic_modular_form_space(**info)
if level > 0 and weight > 0:
return browse_elliptic_modular_forms(**info)
if (level > 0 and weight == 0) or (weight > 0 and level == 0):
emf_logger.debug("Have level or weight only!")
return browse_elliptic_modular_forms(**info)
# return render_elliptic_modular_form_navigation_wp(**info)
# Otherwise we go to the main navigation page
return render_elliptic_modular_form_navigation_wp(**info)
def browse_elliptic_modular_forms(level=0, weight=0, character=-1, label='', limits=None, **kwds):
r"""
Renders the webpage for browsing modular forms of given level and/or weight.
"""
emf_logger.debug("In browse_elliptic_modular_forms kwds: {0}".format(kwds))
emf_logger.debug(
"Input: level={0},weight={1},character={2},label={3}".format(level, weight, character, label))
bread = [(EMF_TOP, url_for('emf.render_elliptic_modular_forms'))]
# if level <0:
# level=None
# if weight<0:
# weight=None
info = dict()
drawdomain = False
if character == 0:
info['grouptype'] = 0
info['groupother'] = 1
dimtbl = DimensionTable(0)
if level <= N_max_Gamma0_fdraw:
drawdomain = True
else:
info['grouptype'] = 1
info['groupother'] = 0
if level <= N_max_Gamma1_fdraw:
drawdomain = True
dimtbl = DimensionTable(1)
emf_logger.info("level=%s, %s" % (level, type(level)))
emf_logger.info("wt=%s, %s" % (weight, type(weight)))
if level > 0:
info['geometric'] = get_geometric_data(level, info['grouptype'])
# if info.has_key('plot'):
if drawdomain:
info['fd_plot_url'] = url_for('emf.render_plot', level=level, grouptype=info['grouptype'])
emf_logger.info("PLOT: %s" % info['fd_plot_url'])
if level > 0 and weight == 0:
# print "here1!"
title = "Newforms for \(\Gamma_{0}({1})\)".format(info['grouptype'], level)
level = int(level)
info['level_min'] = level
info['level_max'] = level
info['weight_min'] = 1
info['weight_max'] = 20
# largs = [ {'level':level,'character':character,'weight_block':k} for k in range(100)]
disp = ClassicalMFDisplay('modularforms')
disp.set_table_browsing(limit=[(2, 12), (level, level)], keys=['Weight', 'Level'],
character=character, dimension_table=dimtbl, title='Dimension of cusp forms')
tbl = disp._table
if tbl is not None:
info['browse_table'] = tbl
# info['list_spaces']=ajax_more(make_table_of_spaces_fixed_level,*largs,text='more')
bread.append(("Level %s" % level, url_for("emf.render_elliptic_modular_forms", level=level)))
info['browse_type'] = " of level %s " % level
info['title'] = title
info['bread'] = bread
info['level'] = level
return render_template("emf_browse_fixed_level.html", **info)
elif level == 0 and weight > 0:
title = "Newforms of weight %s " % weight
bread.append(("Weight %s" % level, url_for("emf.render_elliptic_modular_forms", weight=weight)))
level = int(weight)
info['level_min'] = 1
info['level_max'] = 20
info['weight_min'] = weight
info['weight_max'] = weight
# largs = [ {'level':level,'character':character,'weight_block':k} for k in range(100)]
# info['show_all_characters']=1
disp = ClassicalMFDisplay('modularforms')
disp.set_table_browsing(
limit=[(weight, weight), (info['level_min'], info['level_max'])], keys=['Weight', 'Level'],
character=character, dimension_table=dimtbl, title='Dimension of cusp forms')
tbl = disp._table
if tbl is not None:
info['browse_table'] = tbl
# info['list_spaces']=ajax_more(make_table_of_spaces_fixed_level,*largs,text='more')
info['title'] = title
info['bread'] = bread
info['level'] = level
return render_template("emf_navigation.html", info=info, title=title, bread=bread)
emf_logger.debug("here2!")
info['level_min'] = level
info['level_max'] = level
info['weight_min'] = weight
info['weight_max'] = weight
return render_elliptic_modular_form_space_list_chars(level, weight)
def render_elliptic_modular_forms(level=None, weight=None, character=None, label=None,group=None, **kwds):
r"""
Default input of same type as required. Note that for holomorphic modular forms: level=0 or weight=0 are non-existent.
"""
emf_logger.debug(
"In render: level={0},weight={1},character={2},group={3},label={4}".format(level, weight, character, group, label))
emf_logger.debug("args={0}".format(request.args))
emf_logger.debug("args={0}".format(request.form))
emf_logger.debug("met={0}".format(request.method))
keys = ['download', 'jump_to']
info = get_args(request, level, weight, character, group, label, keys=keys)
valid = validate_parameters(level,weight,character,label,info)
if isinstance(valid,basestring):
return redirect(valid,code=301)
level = info['level']; weight = info['weight']; character = info['character']
#if info.has_key('error'):
# return render_elliptic_modular_form_navigation_wp(error=info['error'])
emf_logger.debug("info={0}".format(info))
emf_logger.debug("level=%s, %s" % (level, type(level)))
emf_logger.debug("label=%s, %s" % (label, type(label)))
emf_logger.debug("wt=%s, %s" % (weight, type(weight)))
group = info.get('group',None)
emf_logger.debug("group=%s, %s" % (group, type(group)))
if group == 0:
info['character'] = character = 1 # only trivial character for Gamma_0(N)
try:
if 'download' in info:
return get_downloads(**info)
emf_logger.debug("info=%s" % info)
## Consistency of arguments>
# if level<=0: level=None
# if weight<=0: weight=None
if 'jump_to' in info: # try to find out which form we want to jump
s = my_get(info, 'jump_to', '', str)
emf_logger.info("info.keys1={0}".format(info.keys()))
info.pop('jump_to')
emf_logger.info("info.keys2={0}".format(info.keys()))
args = extract_data_from_jump_to(s)
emf_logger.debug("args=%s" % args)
return redirect(url_for("emf.render_elliptic_modular_forms", **args), code=301)
# return render_elliptic_modular_forms(**args)
emf_logger.debug("HERE! weight={0} level={1} char={2}".format(weight,level,character))
if level > 0 and weight > 0 and character > 0:
if label != '' and not label is None:
return render_web_newform(**info)
else:
return render_web_modform_space(**info)
if level > 0 and weight > 0 and (group != 0 or character == None):
return render_web_modform_space_gamma1(**info)
return render_elliptic_modular_form_navigation_wp(**info)
# Otherwise we go to the main navigation page
except IndexError as e: # catch everything here except KeyError below...
emf_logger.debug("catching exceptions. info={0}".format(info))
errst = str(e)
## Try to customise some of the error messages:
if 'Character' and 'not exist' in errst:
errst += " Please choose a character from the table below!"
flash(errst,'error')
return render_elliptic_modular_form_navigation_wp(**info)
if 'WebNewForm_computing' in errst:
errst = "The space {0}.{1}.{2} is not in the database!".format(level,weight,character)
flash(errst)
return render_elliptic_modular_form_navigation_wp()
except KeyError as e:
emf_logger.debug("catching exceptions. info={0}".format(info))
errst = "The space {0}.{1}.{2} is not in the database!".format(level,weight,character)
flash(errst)
return render_elliptic_modular_form_navigation_wp()
def html_table(tbl):
r""" Takes a dictonary and returns an html-table.
INPUT:
-''tbl'' -- dictionary with the following keys
- headersh // horozontal headers
- headersv // vertical headers
- rows -- dictionary of rows of data
"""
ncols = len(tbl["headersh"])
nrows = len(tbl["headersv"])
data = tbl['data']
if(len(data) != nrows):
logger.error("wrong number of rows!")
for i in range(nrows):
logger.info("len(%s)=%s" % (i, len(data[i])))
if(len(data[i]) != ncols):
logger.error("wrong number of cols [=%s]!" % ncols)
if('atts' in tbl):
s = "<table " + str(tbl['atts']) + ">\n"
else:
s = "<table>\n"
format = dict()
for i in range(ncols):
format[i] = ''
if('data_format' in tbl):
if isinstance(tbl['data_format'], dict):
if(i in tbl['data_format']):
format[i] = tbl['data_format'][i]
elif(isinstance(tbl['data_format'], str)):
format[i] = tbl['data_format']
if('header' in tbl):
s += "<thead><tr><th><td colspan=\"" + str(ncols) + "\">" + tbl['header'] + "</td></th></tr></thead>"
s = s + "<tbody>"
# smath="<span class=\"math\">"
# check which type of content we have
h1 = tbl['headersh'][0]
sheaderh = ""
sheaderv = ""
h1 = tbl['headersv'][0]
col_width = dict()
if 'col_width' not in tbl:
# use maximum width as default
maxw = 0
for k in range(ncols):
for r in range(nrows):
l = len_as_printed(str(data[r][k]))
if l > maxw:
maxw = l
l = l * 10.0 # use true font size?
for k in range(ncols):
col_width[k] = maxw
else:
for i in range(ncols):
col_width[i] = 0
if 'col_width' in tbl:
if i in tbl['col_width']:
col_width[i] = tbl['col_width'][i]
if("corner_label" in tbl):
l = len_as_printed(str(tbl["corner_label"])) * 10
row = "<tr><td width=\"%s\">" % l
row += str(tbl["corner_label"]) + "</td>"
else:
row = "<tr><td></td>"
for k in range(ncols):
row = row + "<td>" + sheaderh + str(tbl['headersh'][k]) + "</td> \n"
row = row + "</tr> \n"
s = s + row
for r in range(nrows):
l = len_as_printed(str(tbl["headersv"][r])) * 10
logger.info("l=%s head=%s" % (l, tbl["headersv"]))
row = "<tr><td width=\"%s\">" % l
row += sheaderv + str(tbl['headersv'][r]) + "</td>"
for k in range(ncols):
wid = col_width[k]
if format[k] == 'html' or format[k] == 'text':
row = row + "\t<td halign=\"center\" width=\"" + str(wid) + "\">"
if isinstance(data[r][k], list):
for ss in data[r][k]:
sss = str(ss)
if(len(sss) > 0):
row += sss
else:
sss = str(data[r][k])
row += sss
row = row + "</td> \n"
else:
row = row + "\t<td width=\"" + str(wid) + "\">"
if isinstance(data[r][k], list):
for ss in data[r][k]:
sss = latex(ss)
if(len(sss) > 0):
row += "\(" + sss + "\)"
else:
sss = latex(data[r][k])
if(len(sss) > 0):
row = row + "\(" + sss + "\)</td> \n"
row += "</td>\n"
# allow for different format in different columns
row = row + "</tr> \n"
s = s + row
s = s + "</tbody></table>"
return s
def find_inverse_images_of_twists(k, N=1, chi=0, fi=0, prec=10, verbose=0):
r"""
Checks if f is minimal and if not, returns the associated
minimal form to precision prec.
INPUT:
- ''k'' -- positive integer : the weight
- ''N'' -- positive integer (default 1) : level
- ''chi'' -- non-neg. integer (default 0) use character nr. chi
- ''fi'' -- non-neg. integer (default 0) We want to use the element nr. fi f=Newforms(N,k)[fi]
- ''prec'' -- integer (the number of coefficients to get)
- ''verbose'' -- integer
OUTPUT:
-''[t,l]'' -- tuple of a Bool t and a list l. The list l contains all tuples of forms which twists to the given form.
The actual minimal one is the first element of this list.
EXAMPLES::
"""
(t, f) = _get_newform(k, N, chi, fi)
if(not t):
return f
if(is_squarefree(ZZ(N))):
return [True, f]
# We need to check all square factors of N
logger.info("investigating: %s" % f)
N_sqfree = squarefree_part(ZZ(N))
Nsq = ZZ(N / N_sqfree)
twist_candidates = list()
KF = f.base_ring()
# check how many Hecke eigenvalues we need to check
max_nump = number_of_hecke_to_check(f)
maxp = max(primes_first_n(max_nump))
for d in divisors(N):
# we look at all d such that d^2 divdes N
if(not ZZ(d ** 2).divides(ZZ(N))):
continue
D = DirichletGroup(d)
# check possible candidates to twist into f
# g in S_k(M,chi) wit M=N/d^2
M = ZZ(N / d ** 2)
logger.info("Checking level %s" % M)
for xig in range(euler_phi(M)):
(t, glist) = _get_newform(k, M, xig)
if(not t):
return glist
for g in glist:
logger.debug("Comparing to function %s" % g)
KG = g.base_ring()
# we now see if twisting of g by xi in D gives us f
for xi in D:
try:
for p in primes_first_n(max_nump):
if(ZZ(p).divides(ZZ(N))):
continue
bf = f.q_expansion(maxp + 1)[p]
bg = g.q_expansion(maxp + 1)[p]
if(bf == 0 and bg == 0):
continue
elif(bf == 0 and bg != 0 or bg == 0 and bf != 0):
raise StopIteration()
if(ZZ(p).divides(xi.conductor())):
raise ArithmeticError("")
xip = xi(p)
# make a preliminary check that the base rings match with respect to being
# real or not
try:
QQ(xip)
XF = QQ
if(KF != QQ or KG != QQ):
raise StopIteration
except TypeError:
# we have a non-rational (i.e. complex) value of the character
XF = xip.parent()
if((KF == QQ or KF.is_totally_real()) and (KG == QQ or KG.is_totally_real())):
raise StopIteration
## it is diffcult to compare elements from diferent rings in general but we make some checcks
# is it possible to see if there is a larger ring which everything can be
# coerced into?
ok = False
try:
a = KF(bg / xip)
b = KF(bf)
ok = True
if(a != b):
raise StopIteration()
except TypeError:
pass
try:
a = KG(bg)
b = KG(xip * bf)
ok = True
if(a != b):
raise StopIteration()
except TypeError:
pass
if(not ok): # we could coerce and the coefficients were equal
return "Could not compare against possible candidates!"
# otherwise if we are here we are ok and found a candidate
twist_candidates.append([dd, g.q_expansion(prec), xi])
except StopIteration:
# they are not equal
pass
# logger.debug("Candidates=%s" % twist_candidates)
if(len(twist_candidates) == 0):
return (True, None)
else:
return (False, twist_candidates)
def render_elliptic_modular_forms(level=None, weight=None, character=None, label=None,group=None, **kwds):
r"""
Default input of same type as required. Note that for holomorphic modular forms: level=0 or weight=0 are non-existent.
"""
emf_logger.debug(
"In render: level={0},weight={1},character={2},group={3},label={4}".format(level, weight, character, group, label))
emf_logger.debug("args={0}".format(request.args))
emf_logger.debug("args={0}".format(request.form))
emf_logger.debug("met={0}".format(request.method))
keys = ['download', 'jump_to']
info = get_args(request, level, weight, character, group, label, keys=keys)
valid = validate_parameters(level,weight,character,label,info)
if isinstance(valid,basestring):
return redirect(valid,code=301)
level = info['level']; weight = info['weight']; character = info['character']
#if info.has_key('error'):
# return render_elliptic_modular_form_navigation_wp(error=info['error'])
emf_logger.debug("info={0}".format(info))
emf_logger.debug("level=%s, %s" % (level, type(level)))
emf_logger.debug("label=%s, %s" % (label, type(label)))
emf_logger.debug("wt=%s, %s" % (weight, type(weight)))
group = info.get('group',None)
emf_logger.debug("group=%s, %s" % (group, type(group)))
if group == 0:
info['character'] = character = 1 # only trivial character for Gamma_0(N)
try:
if 'download' in info:
return get_downloads(**info)
emf_logger.debug("info=%s" % info)
## Consistency of arguments>
# if level<=0: level=None
# if weight<=0: weight=None
if 'jump_to' in info: # try to find out which form we want to jump
s = my_get(info, 'jump_to', '', str)
emf_logger.info("info.keys1={0}".format(info.keys()))
info.pop('jump_to')
emf_logger.info("info.keys2={0}".format(info.keys()))
args = extract_data_from_jump_to(s)
emf_logger.debug("args=%s" % args)
return redirect(url_for("emf.render_elliptic_modular_forms", **args), code=301)
# return render_elliptic_modular_forms(**args)
emf_logger.debug("HERE! weight={0} level={1} char={2}".format(weight,level,character))
if level > 0 and weight > 0 and character > 0:
if label != '' and not label is None:
return render_web_newform(**info)
else:
return render_web_modform_space(**info)
if level > 0 and weight > 0 and (group != 0 or character == None):
return render_web_modform_space_gamma1(**info)
return render_elliptic_modular_form_navigation_wp(**info)
# Otherwise we go to the main navigation page
except IndexError as e: # catch everything here except KeyError below...
emf_logger.debug("catching exceptions. info={0} e={1}".format(info,e))
errst = str(e)
## Try to customise some of the error messages:
if 'Character' and 'not exist' in errst:
errst += " Please choose a character from the table below!"
flash(errst,'error')
return render_elliptic_modular_form_navigation_wp(**info)
if 'WebNewForm_computing' in errst:
errst = "The space {0}.{1}.{2} is not in the database!".format(level,weight,character)
flash(errst)
return render_elliptic_modular_form_navigation_wp()
except KeyError as e:
emf_logger.debug("catching exceptions. info={0} e={1}".format(info,e))
errst = "The orbit {0} is not in the database!".format(newform_label(level,weight,character,label))
flash(errst)
return render_elliptic_modular_form_navigation_wp()
def browse_elliptic_modular_forms(level=0,
weight=0,
character=-1,
label='',
limits=None,
**kwds):
r"""
Renders the webpage for browsing modular forms of given level and/or weight.
"""
emf_logger.debug("In browse_elliptic_modular_forms kwds: {0}".format(kwds))
emf_logger.debug(
"Input: level={0},weight={1},character={2},label={3}".format(
level, weight, character, label))
bread = [(EMF_TOP, url_for('emf.render_elliptic_modular_forms'))]
# if level <0:
# level=None
# if weight<0:
# weight=None
info = dict()
drawdomain = False
if character == 0:
info['grouptype'] = 0
info['groupother'] = 1
dimtbl = DimensionTable(0)
if level <= N_max_Gamma0_fdraw:
drawdomain = True
else:
info['grouptype'] = 1
info['groupother'] = 0
if level <= N_max_Gamma1_fdraw:
drawdomain = True
dimtbl = DimensionTable(1)
emf_logger.info("level=%s, %s" % (level, type(level)))
emf_logger.info("wt=%s, %s" % (weight, type(weight)))
if level > 0:
info['geometric'] = get_geometric_data(level, info['grouptype'])
# if info.has_key('plot'):
if drawdomain:
info['fd_plot_url'] = url_for('emf.render_plot',
level=level,
grouptype=info['grouptype'])
emf_logger.info("PLOT: %s" % info['fd_plot_url'])
if level > 0 and weight == 0:
# print "here1!"
title = "Newforms for \(\Gamma_{0}({1})\)".format(
info['grouptype'], level)
level = int(level)
info['level_min'] = level
info['level_max'] = level
info['weight_min'] = 1
info['weight_max'] = 20
# largs = [ {'level':level,'character':character,'weight_block':k} for k in range(100)]
disp = ClassicalMFDisplay('modularforms')
disp.set_table_browsing(limit=[(2, 20), (level, level)],
keys=['Weight', 'Level'],
character=character,
dimension_table=dimtbl,
title='Dimension of cusp forms')
tbl = disp._table
if tbl is not None:
info['browse_table'] = tbl
# info['list_spaces']=ajax_more(make_table_of_spaces_fixed_level,*largs,text='more')
bread.append(("Level %s" % level,
url_for("emf.render_elliptic_modular_forms",
level=level)))
info['browse_type'] = " of level %s " % level
info['title'] = title
info['bread'] = bread
info['level'] = level
return render_template("emf_browse_fixed_level.html", **info)
elif level == 0 and weight > 0:
title = "Newforms of weight %s " % weight
bread.append(("Weight %s" % level,
url_for("emf.render_elliptic_modular_forms",
weight=weight)))
level = int(weight)
info['level_min'] = 1
info['level_max'] = 20
info['weight_min'] = weight
info['weight_max'] = weight
# largs = [ {'level':level,'character':character,'weight_block':k} for k in range(100)]
# info['show_all_characters']=1
disp = ClassicalMFDisplay('modularforms')
disp.set_table_browsing(limit=[(weight, weight),
(info['level_min'], info['level_max'])],
keys=['Weight', 'Level'],
character=character,
dimension_table=dimtbl,
title='Dimension of cusp forms')
tbl = disp._table
if tbl is not None:
info['browse_table'] = tbl
# info['list_spaces']=ajax_more(make_table_of_spaces_fixed_level,*largs,text='more')
info['title'] = title
info['bread'] = bread
info['level'] = level
return render_template("emf_navigation.html",
info=info,
title=title,
bread=bread)
emf_logger.debug("here2!")
info['level_min'] = level
info['level_max'] = level
info['weight_min'] = weight
info['weight_max'] = weight
return render_elliptic_modular_form_space_list_chars(level, weight)