def compute_from_startpoints(self, points, p = None, cut_nonsimple_aniso = True, fast = 1, **kwds):
self._reduction = kwds.get('reduction', self._reduction)
if NCPUS0 == 1:
for a in points:
self._compute_simple_modules_graph_from_startpoint(a, p, cut_nonsimple_aniso, fast)
else:
computations = self._compute_simple_modules_graph_from_startpoint_parallel([(a, p, cut_nonsimple_aniso, fast) for a in points])
for r in computations:
if not isinstance(r[1], SimpleModulesGraph):
logger.error("Got something else... {0}".format(r))
# print r
else:
G = r[1]
logger.debug("simple from G: {0}".format(len(G._simple)))
logger.debug(G._simple)
self.add_vertices(G.vertices())
self.add_edges(G.edges())
self._vertex_colors[
self._nonsimple_color] += G._vertex_colors[G._nonsimple_color]
self._vertex_colors[
self._simple_color] += G._vertex_colors[G._simple_color]
self._simple = uniq(self._simple + G._simple)
for h in G._heights.keys():
if not self._heights.has_key(h):
self._heights[h] = G._heights[h]
else:
self._heights[h] = uniq(self._heights[h] + G._heights[h])
logger.info("Found {0} {1}-simple module{2} so far.".format(
len(self._simple), self._weight, "s" if len(self._simple) != 1 else ""))
logger.info("Found in total {0} {1}-simple module{3} with p-rank <= {2}".format(
len(self._simple), self._weight, self._rank_limit, "s" if len(self._simple) != 1 else ""))
return 0
def aniso_dict_to_latex_table(d, max_per_table=10, include_weight=True):
if include_weight:
beginstr = r"\begin{tabularx}{llX} \toprule $k$ & signature & $k$-simple modules\\\midrule" + \
"\n"
else:
beginstr = r"\begin{tabular}{llX} \toprule $k$ & signature & $k$-simple modules\\\midrule" + \
"\n"
endstr = r"\bottomrule\end{tabularx}" + "\n"
s = beginstr
ks = sorted(d.keys())
for i in range(len(d.keys())):
k = ks[i]
sigs = uniq([_.signature() for _ in d[k]])
for sig in sigs:
if include_weight:
s += "$" + str(k) + "$" + " & "
s += " " + str(sig) + " & "
ms = [m for m in d[k] if m.signature() == sig]
ms = sorted(
ms, lambda x, y: int(-1) if x.order() <= y.order() else int(1))
for m in ms:
s += str(FQM_vertex(m)).strip() + ", "
s = s[:-2]
s += r"\\\midrule" + "\n"
if (i + 1) % max_per_table == 0:
s += endstr + beginstr
s += endstr
return s
def aniso_dict_to_latex_table(d, max_per_table=10, include_weight=True):
if include_weight:
beginstr = r"\begin{tabularx}{llX} \toprule $k$ & signature & $k$-simple modules\\\midrule" + \
"\n"
else:
beginstr = r"\begin{tabular}{llX} \toprule $k$ & signature & $k$-simple modules\\\midrule" + \
"\n"
endstr = r"\bottomrule\end{tabularx}" + "\n"
s = beginstr
ks = sorted(d.keys())
for i in range(len(d.keys())):
k = ks[i]
sigs = uniq([_.signature() for _ in d[k]])
for sig in sigs:
if include_weight:
s += "$" + str(k) + "$" + " & "
s += " " + str(sig) + " & "
ms = [m for m in d[k] if m.signature() == sig]
ms = sorted(
ms, lambda x, y: int(-1) if x.order() <= y.order() else int(1))
for m in ms:
s += str(FQM_vertex(m)).strip() + ", "
s = s[:-2]
s += r"\\\midrule" + "\n"
if (i + 1) % max_per_table == 0:
s += endstr + beginstr
s += endstr
return s
def compute_from_startpoints(self,
points,
p=None,
cut_nonsimple_aniso=True,
fast=1,
**kwds):
self._reduction = kwds.get('reduction', self._reduction)
if NCPUS0 == 1:
for a in points:
self._compute_simple_modules_graph_from_startpoint(
a, p, cut_nonsimple_aniso, fast)
else:
computations = self._compute_simple_modules_graph_from_startpoint_parallel(
[(a, p, cut_nonsimple_aniso, fast) for a in points])
for r in computations:
if not isinstance(r[1], SimpleModulesGraph):
logger.error("Got something else... {0}".format(r))
# print r
else:
G = r[1]
logger.debug("simple from G: {0}".format(len(G._simple)))
logger.debug(G._simple)
self.add_vertices(G.vertices())
self.add_edges(G.edges())
self._vertex_colors[
self._nonsimple_color] += G._vertex_colors[
G._nonsimple_color]
self._vertex_colors[
self._simple_color] += G._vertex_colors[
G._simple_color]
self._simple = uniq(self._simple + G._simple)
for h in G._heights.keys():
if not self._heights.has_key(h):
self._heights[h] = G._heights[h]
else:
self._heights[h] = uniq(self._heights[h] +
G._heights[h])
logger.info(
"Found {0} {1}-simple module{2} so far.".
format(len(self._simple), self._weight,
"s" if len(self._simple) != 1 else ""))
logger.info(
"Found in total {0} {1}-simple module{3} with p-rank <= {2}".
format(len(self._simple), self._weight, self._rank_limit,
"s" if len(self._simple) != 1 else ""))
return 0
def find_simple_anisotropic_parallel(num,
s,
weights=range(2,
Integer(27) / 2),
dynamic=True,
lower=1,
only_2_n=False,
reduction=False):
r"""
Test for anisotropic $k$-simple fqm's for $k$ in weights.
INPUT::
- num: upper bound
- s: number of iterations
- lower: lower bound
"""
simple = dict()
for kk in weights:
simple[kk] = list()
m = round(RR(num) / s)
args = [(m * a + lower, min(m * (a + lower),
lower + num - 1), lower + num - 1, None, None,
weights, True, dynamic, only_2_n, reduction) for a in range(s)]
tests = find_simple_anisotropic_wrapper(args)
done = 0
#starttime = walltime()
for test in tests:
done += 1
simple_part = test[1][0]
if isinstance(simple_part, str):
print "Result: ", test
continue
tt = test[1][1]
for kk in simple_part.keys():
print "Result from process: ", simple_part[kk]
simple[kk] = uniq(simple[kk] + simple_part[kk])
#tt = walltime(starttime)
sl = [len(sp) for sp in simple.values()]
num_simple = sum(sl)
timeest = (s - done) * RR(tt) / RR(60)
if timeest > 1:
if timeest > 60:
print("%g%% done, ETA: %d hour" +
("s" if timeest / 60 >= 2 else "") +
", %d minutes, simple lattices: %d") % (
RR(done) / RR(s) * 100, timeest / 60,
(timeest / 60).frac() * 60, num_simple)
else:
print("%g%% done, ETA: %d minute" +
("s" if timeest >= 2 else "") + ", simple lattices: %d"
) % (RR(done) / RR(s) * 100, timeest, num_simple)
else:
print "%g%% done, ETA: %d seconds, simple lattices: %d" % (
RR(done) / RR(s) * 100, timeest * 60, num_simple)
return simple
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()
emf_logger.debug("info={0}".format(info))
WMFS = None
if level <= 0:
info["error"] = "Got wrong level: %s " % level
return info
try:
WMFS = WebModFormSpace_cached(level=level, weight=weight, cuspidal=True, character=character)
emf_logger.debug("Created WebModFormSpace %s" % WMFS)
if "download" in info and "tempfile" in info:
save(WNF, info["tempfile"])
info["filename"] = str(weight) + "-" + str(level) + "-" + str(character) + "-" + label + ".sobj"
return info
except ValueError as e:
emf_logger.debug(e)
emf_logger.debug(e.message)
# if isinstance(e,IndexError):
info["error"] = e.message
WMFS = None
if WMFS is None:
info["error"] = (
"We are sorry. The sought space can not be found in the database. "
+ "<br> Detailed information: {0}".format(info.get("error", ""))
)
return info
else:
### Somehow the Hecke orbits are sometimes not in the space...
# if WMFS.dimension_new_cusp_forms()!=len(WMFS.hecke_orbits):
# ## Try to add them here...
# for d in range(len(WMFS.dimension_new_cusp_forms())):
# F =
# WMFS.hecke_orbits.append(F)
info = {"space": WMFS}
# info['old_decomposition'] = WMFS.oldspace_decomposition()
## For side-bar
lifts = list()
lifts.append(("Half-Integral Weight Forms", "/ModularForm/Mp2/Q"))
lifts.append(("Siegel Modular Forms", "/ModularForm/GSp4/Q"))
info["lifts"] = lifts
friends = list()
for label in WMFS.hecke_orbits:
f = WMFS.hecke_orbits[label]
friends.append(("Number field " + f.base_field_label(), f.base_field_url()))
if f.coefficient_field_label(check=True):
friends.append(("Number field " + f.coefficient_field_label(), f.coefficient_field_url()))
friends.append(("Dirichlet character \(" + WMFS.character.latex_name + "\)", WMFS.character.url()))
friends = uniq(friends)
info["friends"] = friends
return info
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()
emf_logger.debug("info={0}".format(info))
WMFS = None
if level <= 0:
info['error'] = "Got wrong level: %s " % level
return info
try:
WMFS = WebModFormSpace_cached(level = level, weight = weight, cuspidal=True,character = character)
emf_logger.debug("Created WebModFormSpace %s"%WMFS)
if 'download' in info and 'tempfile' in info:
save(WNF,info['tempfile'])
info['filename'] = str(weight) + '-' + str(level) + '-' + str(character) + '-' + label + '.sobj'
return info
except ValueError as e:
emf_logger.debug(e)
emf_logger.debug(e.message)
#if isinstance(e,IndexError):
info['error'] = e.message
WMFS = None
if WMFS is None:
info['error'] = "We are sorry. The sought space can not be found in the database. "+"<br> Detailed information: {0}".format(info.get('error',''))
return info
else:
### Somehow the Hecke orbits are sometimes not in the space...
#if WMFS.dimension_new_cusp_forms()<>len(WMFS.hecke_orbits):
# ## Try to add them here...
# for d in range(len(WMFS.dimension_new_cusp_forms())):
# F =
# WMFS.hecke_orbits.append(F)
info = {'space':WMFS}
# info['old_decomposition'] = WMFS.oldspace_decomposition()
## For side-bar
lifts = list()
lifts.append(('Half-Integral Weight Forms', '/ModularForm/Mp2/Q'))
lifts.append(('Siegel Modular Forms', '/ModularForm/GSp4/Q'))
info['lifts'] = lifts
friends = list()
for label in WMFS.hecke_orbits:
f = WMFS.hecke_orbits[label]
friends.append(('Number field ' + f.base_field_label(), f.base_field_url()))
if f.coefficient_field_label(check=True):
friends.append(('Number field ' + f.coefficient_field_label(), f.coefficient_field_url()))
friends.append(("Dirichlet character \(" + WMFS.character.latex_name + "\)", WMFS.character.url()))
friends = uniq(friends)
info['friends'] = friends
return info
def set_info_for_web_newform(level=None,
weight=None,
character=None,
label=None,
**kwds):
r"""
Set the info for on modular form.
"""
info = to_dict(kwds)
info['level'] = level
info['weight'] = weight
info['character'] = character
info['label'] = label
if level is None or weight is None or character is None or label is None:
s = "In set info for one form but do not have enough args!"
s += "level={0},weight={1},character={2},label={3}".format(
level, weight, character, label)
emf_logger.critical(s)
emf_logger.debug("In set_info_for_one_mf: info={0}".format(info))
prec = my_get(info, 'prec', default_prec, int)
bprec = my_get(info, 'bprec', default_display_bprec, int)
emf_logger.debug("PREC: {0}".format(prec))
emf_logger.debug("BITPREC: {0}".format(bprec))
try:
WNF = WebNewForm_cached(level=level,
weight=weight,
character=character,
label=label)
emf_logger.critical("defined webnewform for rendering!")
# if info.has_key('download') and info.has_key('tempfile'):
# WNF._save_to_file(info['tempfile'])
# info['filename']=str(weight)+'-'+str(level)+'-'+str(character)+'-'+label+'.sobj'
# return info
except IndexError as e:
WNF = None
info['error'] = e.message
url1 = url_for("emf.render_elliptic_modular_forms")
url2 = url_for("emf.render_elliptic_modular_forms", level=level)
url3 = url_for("emf.render_elliptic_modular_forms",
level=level,
weight=weight)
url4 = url_for("emf.render_elliptic_modular_forms",
level=level,
weight=weight,
character=character)
bread = [(EMF_TOP, url1)]
bread.append(("of level %s" % level, url2))
bread.append(("weight %s" % weight, url3))
if int(character) == 0:
bread.append(("trivial character", url4))
else:
bread.append(("\( %s \)" % (WNF.character.latex_name), url4))
info['bread'] = bread
properties2 = list()
friends = list()
space_url = url_for('emf.render_elliptic_modular_forms',
level=level,
weight=weight,
character=character)
friends.append(
('\( S_{%s}(%s, %s)\)' %
(WNF.weight, WNF.level, WNF.character.latex_name), space_url))
if WNF.coefficient_field_label(check=True):
friends.append(('Number field ' + WNF.coefficient_field_label(),
WNF.coefficient_field_url()))
friends.append(
('Number field ' + WNF.base_field_label(), WNF.base_field_url()))
friends = uniq(friends)
friends.append(("Dirichlet character \(" + WNF.character.latex_name + "\)",
WNF.character.url()))
if WNF.dimension == 0:
info['error'] = "This space is empty!"
# emf_logger.debug("WNF={0}".format(WNF))
#info['name'] = name
info['title'] = 'Modular Form ' + WNF.hecke_orbit_label
if 'error' in info:
return info
# info['name']=WNF._name
## Until we have figured out how to do the embeddings correctly we don't display the Satake
## parameters for non-trivial characters....
cdeg = WNF.coefficient_field.absolute_degree()
bdeg = WNF.base_ring.absolute_degree()
if WNF.coefficient_field.absolute_degree() == 1:
rdeg = 1
else:
rdeg = WNF.coefficient_field.relative_degree()
if cdeg == 1:
info['satake'] = WNF.satake
info['qexp'] = WNF.q_expansion_latex(prec=10, name='a')
info['qexp_display'] = url_for(".get_qexp_latex",
level=level,
weight=weight,
character=character,
label=label)
# info['qexp'] = WNF.q_expansion_latex(prec=prec)
#c_pol_st = str(WNF.absolute_polynomial)
#b_pol_st = str(WNF.polynomial(type='base_ring',format='str'))
#b_pol_ltx = str(WNF.polynomial(type='base_ring',format='latex'))
#print "c=",c_pol_ltx
#print "b=",b_pol_ltx
if cdeg > 1: ## Field is QQ
if bdeg > 1 and rdeg > 1:
p1 = WNF.coefficient_field.relative_polynomial()
c_pol_ltx = latex(p1)
lgc = p1.variables()[0]
c_pol_ltx = c_pol_ltx.replace(lgc, 'a')
z = p1.base_ring().gens()[0]
p2 = z.minpoly()
b_pol_ltx = latex(p2)
b_pol_ltx = b_pol_ltx.replace(latex(p2.variables()[0]), latex(z))
info['polynomial_st'] = 'where \({0}=0\) and \({1}=0\).'.format(
c_pol_ltx, b_pol_ltx)
else:
c_pol_ltx = latex(WNF.coefficient_field.relative_polynomial())
lgc = str(
latex(WNF.coefficient_field.relative_polynomial().variables()
[0]))
c_pol_ltx = c_pol_ltx.replace(lgc, 'a')
info['polynomial_st'] = 'where \({0}=0\)'.format(c_pol_ltx)
else:
info['polynomial_st'] = ''
info['degree'] = int(cdeg)
if cdeg == 1:
info['is_rational'] = 1
else:
info['is_rational'] = 0
# info['q_exp_embeddings'] = WNF.print_q_expansion_embeddings()
# if(int(info['degree'])>1 and WNF.dimension()>1):
# s = 'One can embed it into \( \mathbb{C} \) as:'
# bprec = 26
# print s
# info['embeddings'] = ajax_more2(WNF.print_q_expansion_embeddings,{'prec':[5,10,25,50],'bprec':[26,53,106]},text=['more coeffs.','higher precision'])
# elif(int(info['degree'])>1):
# s = 'There are '+str(info['degree'])+' embeddings into \( \mathbb{C} \):'
# bprec = 26
# print s
# info['embeddings'] = ajax_more2(WNF.print_q_expansion_embeddings,{'prec':[5,10,25,50],'bprec':[26,53,106]},text=['more coeffs.','higher precision'])
# else:
# info['embeddings'] = ''
emf_logger.debug("PREC2: {0}".format(prec))
info['embeddings'] = WNF._embeddings[
'values'] #q_expansion_embeddings(prec, bprec,format='latex')
info['embeddings_len'] = len(info['embeddings'])
properties2 = []
if (ZZ(level)).is_squarefree():
info['twist_info'] = WNF.twist_info
if isinstance(info['twist_info'],
list) and len(info['twist_info']) > 0:
info['is_minimal'] = info['twist_info'][0]
if (info['twist_info'][0]):
s = '- Is minimal<br>'
else:
s = '- Is a twist of lower level<br>'
properties2 = [('Twist info', s)]
else:
info['twist_info'] = 'Twist info currently not available.'
properties2 = [('Twist info', 'not available')]
args = list()
for x in range(5, 200, 10):
args.append({'digits': x})
alev = None
CM = WNF._cm_values
if CM is not None:
if CM.has_key('tau') and len(CM['tau']) != 0:
info['CM_values'] = CM
info['is_cm'] = WNF.is_cm
if WNF.is_cm is None:
s = '- Unknown (insufficient data)<br>'
elif WNF.is_cm is True:
s = '- Is a CM-form<br>'
else:
s = '- Is not a CM-form<br>'
properties2.append(('CM info', s))
alev = WNF.atkin_lehner_eigenvalues()
info['atkinlehner'] = None
if isinstance(alev, dict) and len(alev.keys()) > 0 and level != 1:
s1 = " Atkin-Lehner eigenvalues "
s2 = ""
for Q in alev.keys():
s2 += "\( \omega_{ %s } \) : %s <br>" % (Q, alev[Q])
properties2.append((s1, s2))
emf_logger.debug("properties={0}".format(properties2))
# alev = WNF.atkin_lehner_eigenvalues_for_all_cusps()
# if isinstance(alev,dict) and len(alev.keys())>0:
# emf_logger.debug("alev={0}".format(alev))
# info['atkinlehner'] = list()
# for Q in alev.keys():
# s = "\(" + latex(c) + "\)"
# Q = alev[c][0]
# ev = alev[c][1]
# info['atkinlehner'].append([Q, c, ev])
if (level == 1):
poly = WNF.explicit_formulas.get('as_polynomial_in_E4_and_E6', '')
if poly <> '':
d, monom, coeffs = poly
emf_logger.critical("poly={0}".format(poly))
info['explicit_formulas'] = '\('
for i in range(d):
c = QQ(coeffs[i])
s = ""
if d > 1 and i > 0 and c > 0:
s = "+"
if c < 0:
s = "-"
if c.denominator() > 1:
cc = "\\frac{{ {0} }}{{ {1} }}".format(
abs(c.numerator()), c.denominator())
else:
cc = str(abs(c))
s += "{0} \cdot ".format(cc)
a = monom[i][0]
b = monom[i][1]
if a == 0 and b <> 0:
s += "E_6^{{ {0} }}".format(b)
elif b == 0 and a <> 0:
s += "E_4^{{ {0} }}".format(a)
else:
s += "E_4^{{ {0} }}E_6^{{ {1} }}".format(a, b)
info['explicit_formulas'] += s
info['explicit_formulas'] += " \)"
cur_url = '?&level=' + str(level) + '&weight=' + str(weight) + '&character=' + str(character) + \
'&label=' + str(label)
if len(WNF.parent.hecke_orbits) > 1:
for label_other in WNF.parent.hecke_orbits.keys():
if (label_other != label):
s = 'Modular form '
if character:
s = s + str(level) + '.' + str(weight) + '.' + str(
character) + str(label_other)
else:
s = s + str(level) + '.' + str(weight) + str(label_other)
url = url_for('emf.render_elliptic_modular_forms',
level=level,
weight=weight,
character=character,
label=label_other)
friends.append((s, url))
s = 'L-Function '
if character:
s = s + str(level) + '.' + str(weight) + '.' + str(character) + str(
label)
else:
s = s + str(level) + '.' + str(weight) + str(label)
# url =
# "/L/ModularForm/GL2/Q/holomorphic?level=%s&weight=%s&character=%s&label=%s&number=%s"
# %(level,weight,character,label,0)
url = '/L' + url_for('emf.render_elliptic_modular_forms',
level=level,
weight=weight,
character=character,
label=label)
if WNF.coefficient_field_degree > 1:
for h in range(WNF.coefficient_field_degree):
s0 = s + ".{0}".format(h)
url0 = url + "{0}/".format(h)
friends.append((s0, url0))
else:
friends.append((s, url))
# if there is an elliptic curve over Q associated to self we also list that
if WNF.weight == 2 and WNF.coefficient_field_degree == 1:
llabel = str(level) + '.' + label
s = 'Elliptic curve isogeny class ' + llabel
url = '/EllipticCurve/Q/' + llabel
friends.append((s, url))
info['properties2'] = properties2
info['friends'] = friends
info['max_cn'] = WNF.max_cn()
return info
def set_info_for_web_newform(level=None, weight=None, character=None, label=None, **kwds):
r"""
Set the info for on modular form.
"""
info = to_dict(kwds)
info['level'] = level
info['weight'] = weight
info['character'] = character
info['label'] = label
if level is None or weight is None or character is None or label is None:
s = "In set info for one form but do not have enough args!"
s += "level={0},weight={1},character={2},label={3}".format(level, weight, character, label)
emf_logger.critical(s)
emf_logger.debug("In set_info_for_one_mf: info={0}".format(info))
prec = my_get(info, 'prec', default_prec, int)
bprec = my_get(info, 'bprec', default_display_bprec, int)
emf_logger.debug("PREC: {0}".format(prec))
emf_logger.debug("BITPREC: {0}".format(bprec))
try:
WNF = WebNewForm_cached(level=level, weight=weight, character=character, label=label)
if not WNF.has_updated():
raise IndexError("Unfortunately, we do not have this newform in the database.")
info['character_order'] = WNF.character.order
info['code'] = WNF.code
emf_logger.debug("defined webnewform for rendering!")
except IndexError as e:
info['error'] = e.message
url0 = url_for("mf.modular_form_main_page")
url1 = url_for("emf.render_elliptic_modular_forms")
url2 = url_for("emf.render_elliptic_modular_forms", level=level)
url3 = url_for("emf.render_elliptic_modular_forms", level=level, weight=weight)
url4 = url_for("emf.render_elliptic_modular_forms", level=level, weight=weight, character=character)
bread = [(MF_TOP, url0), (EMF_TOP, url1)]
bread.append(("Level %s" % level, url2))
bread.append(("Weight %s" % weight, url3))
bread.append(("Character \( %s \)" % (WNF.character.latex_name), url4))
bread.append(("Newform %d.%d.%d.%s" % (level, weight, int(character), label),''))
info['bread'] = bread
properties2 = list()
friends = list()
space_url = url_for('emf.render_elliptic_modular_forms',level=level, weight=weight, character=character)
friends.append(('\( S_{%s}(%s, %s)\)'%(WNF.weight, WNF.level, WNF.character.latex_name), space_url))
if hasattr(WNF.base_ring, "lmfdb_url") and WNF.base_ring.lmfdb_url:
friends.append(('Number field ' + WNF.base_ring.lmfdb_pretty, WNF.base_ring.lmfdb_url))
if hasattr(WNF.coefficient_field, "lmfdb_url") and WNF.coefficient_field.lmfdb_label:
friends.append(('Number field ' + WNF.coefficient_field.lmfdb_pretty, WNF.coefficient_field.lmfdb_url))
friends = uniq(friends)
friends.append(("Dirichlet character \(" + WNF.character.latex_name + "\)", WNF.character.url()))
if WNF.dimension==0 and not info.has_key('error'):
info['error'] = "This space is empty!"
info['title'] = 'Newform ' + WNF.hecke_orbit_label
info['learnmore'] = [('History of modular forms', url_for('.holomorphic_mf_history'))]
if 'error' in info:
return info
## Until we have figured out how to do the embeddings correctly we don't display the Satake
## parameters for non-trivial characters....
## Example to illustrate the different cases
## base = CyclotomicField(n) -- of degree phi(n)
## coefficient_field = NumberField( p(x)) for some p in base['x'] of degree m
## we would then have cdeg = m*phi(n) and bdeg = phi(n)
## and rdeg = m
## Unfortunately, for e.g. base = coefficient_field = CyclotomicField(6)
## we get coefficient_field.relative_degree() == 2 although it should be 1
cdeg = WNF.coefficient_field.absolute_degree()
bdeg = WNF.base_ring.absolute_degree()
if cdeg == 1:
rdeg = 1
else:
## just setting rdeg = WNF.coefficient_field.relative_degree() does not give correct result...
##
rdeg = QQ(cdeg)/QQ(bdeg)
cf_is_QQ = (cdeg == 1)
br_is_QQ = (bdeg == 1)
if cf_is_QQ:
info['satake'] = WNF.satake
if WNF.complexity_of_first_nonvanishing_coefficients() > default_max_height:
info['qexp'] = ""
info['qexp_display'] = ''
info['hide_qexp'] = True
n,c = WNF.first_nonvanishing_coefficient()
info['trace_nv'] = latex(WNF.first_nonvanishing_coefficient_trace())
info['norm_nv'] = '\\approx ' + latex(WNF.first_nonvanishing_coefficient_norm().n())
info['index_nv'] = n
else:
if WNF.prec < prec:
#get WNF record at larger prec
WNF.prec = prec
WNF.update_from_db()
info['qexp'] = WNF.q_expansion_latex(prec=10, name='\\alpha ')
info['qexp_display'] = url_for(".get_qexp_latex", level=level, weight=weight, character=character, label=label)
info["hide_qexp"] = False
info['max_cn_qexp'] = WNF.q_expansion.prec()
## All combinations should be tested...
## 13/4/4/a -> base ring = coefficient_field = QQ(zeta_6)
## 13/3/8/a -> base_ring = QQ(zeta_4), coefficient_field has poly x^2+(2\zeta_4+2x-3\zeta_$ over base_ring
## 13/4/3/a -> base_ring = coefficient_field = QQ(zeta_3)
## 13/4/1/a -> all rational
## 13/6/1/a/ -> base_ring = QQ, coefficient_field = Q(sqrt(17))
## These are variables which needs to be set properly below
info['polvars'] = {'base_ring':'x','coefficient_field':'\\alpha'}
if not cf_is_QQ:
if rdeg>1: # not WNF.coefficient_field == WNF.base_ring:
## Here WNF.base_ring should be some cyclotomic field and we have an extension over this.
p1 = WNF.coefficient_field.relative_polynomial()
c_pol_ltx = web_latex_poly(p1, '\\alpha') # make the variable \alpha
c_pol_ltx_x = web_latex_poly(p1, 'x')
zeta = p1.base_ring().gens()[0]
# p2 = zeta.minpoly() #this is not used anymore
# b_pol_ltx = web_latex_poly(p2, latex(zeta)) #this is not used anymore
z1 = zeta.multiplicative_order()
info['coeff_field'] = [ WNF.coefficient_field.absolute_polynomial_latex('x'),c_pol_ltx_x, z1]
if hasattr(WNF.coefficient_field, "lmfdb_url") and WNF.coefficient_field.lmfdb_url:
info['coeff_field_pretty'] = [ WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty, WNF.coefficient_field.lmfdb_label]
if z1==4:
info['polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\) and \(\zeta_4=i\).</div><br/>'.format(c_pol_ltx)
info['polvars']['base_ring']='i'
elif z1<=2:
info['polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\).</div><br/>'.format(c_pol_ltx)
else:
info['polynomial_st'] = '<div class="where">where</div> %s\(\mathstrut=0\) and \(\zeta_{%s}=e^{\\frac{2\\pi i}{%s}}\) '%(c_pol_ltx, z1,z1)
info['polvars']['base_ring']='\zeta_{{ {0} }}'.format(z1)
if z1==3:
info['polynomial_st'] += 'is a primitive cube root of unity.'
else:
info['polynomial_st'] += 'is a primitive {0}-th root of unity.'.format(z1)
elif not br_is_QQ:
## Now we have base and coefficient field being equal, meaning that since the coefficient field is not QQ it is some cyclotomic field
## generated by some \zeta_n
p1 = WNF.coefficient_field.absolute_polynomial()
z1 = WNF.coefficient_field.gens()[0].multiplicative_order()
c_pol_ltx = web_latex_poly(p1, '\\zeta_{{{0}}}'.format(z1))
c_pol_ltx_x = web_latex_poly(p1, 'x')
info['coeff_field'] = [ WNF.coefficient_field.absolute_polynomial_latex('x'), c_pol_ltx_x]
if hasattr(WNF.coefficient_field, "lmfdb_url") and WNF.coefficient_field.lmfdb_url:
info['coeff_field_pretty'] = [ WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty, WNF.coefficient_field.lmfdb_label]
if z1==4:
info['polynomial_st'] = '<div class="where">where \(\zeta_4=e^{{\\frac{{\\pi i}}{{ 2 }} }}=i \).</div>'.format(c_pol_ltx)
info['polvars']['coefficient_field']='i'
elif z1<=2:
info['polynomial_st'] = ''
else:
info['polynomial_st'] = '<div class="where">where \(\zeta_{{{0}}}=e^{{\\frac{{2\\pi i}}{{ {0} }} }}\) '.format(z1)
info['polvars']['coefficient_field']='\zeta_{{{0}}}'.format(z1)
if z1==3:
info['polynomial_st'] += 'is a primitive cube root of unity.</div>'
else:
info['polynomial_st'] += 'is a primitive {0}-th root of unity.</div>'.format(z1)
else:
info['polynomial_st'] = ''
if info["hide_qexp"]:
info['polynomial_st'] = ''
info['degree'] = int(cdeg)
if cdeg==1:
info['is_rational'] = 1
info['coeff_field_pretty'] = [ WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty ]
else:
info['is_rational'] = 0
emf_logger.debug("PREC2: {0}".format(prec))
info['embeddings'] = WNF._embeddings['values'] #q_expansion_embeddings(prec, bprec,format='latex')
info['embeddings_len'] = len(info['embeddings'])
properties2 = [('Level', str(level)),
('Weight', str(weight)),
('Character', '$' + WNF.character.latex_name + '$'),
('Label', WNF.hecke_orbit_label),
('Dimension of Galois orbit', str(WNF.dimension))]
if (ZZ(level)).is_squarefree():
info['twist_info'] = WNF.twist_info
if isinstance(info['twist_info'], list) and len(info['twist_info'])>0:
info['is_minimal'] = info['twist_info'][0]
if(info['twist_info'][0]):
s = 'Is minimal<br>'
else:
s = 'Is a twist of lower level<br>'
properties2 += [('Twist info', s)]
else:
info['twist_info'] = 'Twist info currently not available.'
properties2 += [('Twist info', 'not available')]
args = list()
for x in range(5, 200, 10):
args.append({'digits': x})
alev = None
CM = WNF._cm_values
if CM is not None:
if CM.has_key('tau') and len(CM['tau']) != 0:
info['CM_values'] = CM
info['is_cm'] = WNF.is_cm
if WNF.is_cm == 1:
info['cm_field'] = "2.0.{0}.1".format(-WNF.cm_disc)
info['cm_disc'] = WNF.cm_disc
info['cm_field_knowl'] = nf_display_knowl(info['cm_field'], getDBConnection(), field_pretty(info['cm_field']))
info['cm_field_url'] = url_for("number_fields.by_label", label=info["cm_field"])
if WNF.is_cm is None or WNF.is_cm==-1:
s = '- Unknown (insufficient data)<br>'
elif WNF.is_cm == 1:
s = 'Yes<br>'
else:
s = 'No<br>'
properties2.append(('CM', s))
alev = WNF.atkin_lehner_eigenvalues()
info['atkinlehner'] = None
if isinstance(alev,dict) and len(alev.keys())>0 and level != 1:
s1 = " Atkin-Lehner eigenvalues "
s2 = ""
for Q in alev.keys():
s2 += "\( \omega_{ %s } \) : %s <br>" % (Q, alev[Q])
properties2.append((s1, s2))
emf_logger.debug("properties={0}".format(properties2))
# alev = WNF.atkin_lehner_eigenvalues_for_all_cusps()
# if isinstance(alev,dict) and len(alev.keys())>0:
# emf_logger.debug("alev={0}".format(alev))
# info['atkinlehner'] = list()
# for Q in alev.keys():
# s = "\(" + latex(c) + "\)"
# Q = alev[c][0]
# ev = alev[c][1]
# info['atkinlehner'].append([Q, c, ev])
if(level == 1):
poly = WNF.explicit_formulas.get('as_polynomial_in_E4_and_E6','')
if poly != '':
d,monom,coeffs = poly
emf_logger.critical("poly={0}".format(poly))
info['explicit_formulas'] = '\('
for i in range(len(coeffs)):
c = QQ(coeffs[i])
s = ""
if d>1 and i >0 and c>0:
s="+"
if c<0:
s="-"
if c.denominator()>1:
cc = "\\frac{{ {0} }}{{ {1} }}".format(abs(c.numerator()),c.denominator())
else:
cc = str(abs(c))
s += "{0} \cdot ".format(cc)
a = monom[i][0]; b = monom[i][1]
if a == 0 and b != 0:
s+="E_6^{{ {0} }}".format(b)
elif b ==0 and a != 0:
s+="E_4^{{ {0} }}".format(a)
else:
s+="E_4^{{ {0} }}E_6^{{ {1} }}".format(a,b)
info['explicit_formulas'] += s
info['explicit_formulas'] += " \)"
# cur_url = '?&level=' + str(level) + '&weight=' + str(weight) + '&character=' + str(character) + '&label=' + str(label) # never used
if len(WNF.parent.hecke_orbits) > 1:
for label_other in WNF.parent.hecke_orbits.keys():
if(label_other != label):
s = 'Modular form '
if character:
s += newform_label(level,weight,character,label_other)
else:
s += newform_label(level,weight,1,label_other)
url = url_for('emf.render_elliptic_modular_forms', level=level,
weight=weight, character=character, label=label_other)
friends.append((s, url))
s = 'L-Function '
if character:
s += newform_label(level,weight,character,label)
else:
s += newform_label(level,weight,1,label)
# url =
# "/L/ModularForm/GL2/Q/holomorphic?level=%s&weight=%s&character=%s&label=%s&number=%s"
# %(level,weight,character,label,0)
url = '/L' + url_for(
'emf.render_elliptic_modular_forms', level=level, weight=weight, character=character, label=label)
if WNF.coefficient_field_degree > 1:
for h in range(WNF.coefficient_field_degree):
s0 = s + ".{0}".format(h)
url0 = url + "{0}/".format(h)
friends.append((s0, url0))
else:
friends.append((s, url))
# if there is an elliptic curve over Q associated to self we also list that
if WNF.weight == 2 and WNF.coefficient_field_degree == 1:
llabel = str(level) + '.' + label
s = 'Elliptic curve isogeny class ' + llabel
url = '/EllipticCurve/Q/' + llabel
friends.append((s, url))
info['properties2'] = properties2
info['friends'] = friends
info['max_cn'] = WNF.max_available_prec()
return info
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()
emf_logger.debug("info={0}".format(info))
WMFS = None
if info.has_key('error'):
return info
if level <= 0:
info['error'] = "Got wrong level: %s " % level
return info
try:
WMFS = WebModFormSpace_cached(level = level, weight = weight, cuspidal=True, character = character, update_from_db=True)
WMFS.update_from_db() ## Need to call an extra time for some reason...
if not WMFS.has_updated():
stop = False
orbit = WMFS.character.character.galois_orbit()
while not stop:
if len(orbit) == 0:
stop = True
continue
c = orbit.pop()
if c.number() == WMFS.character.number:
continue
print c.number()
WMFS_rep = WebModFormSpace_cached(level = level, weight = weight, cuspidal=True, character = c.number(), update_from_db=True)
if WMFS_rep.has_updated_from_db():
stop = True
info['wmfs_rep_url'] = url_for('emf.render_elliptic_modular_forms', level=level, weight=weight, character = c.number())
info['wmfs_rep_number'] = c.number()
emf_logger.debug("Created WebModFormSpace %s"%WMFS)
if 'download' in info and 'tempfile' in info:
save(WNF,info['tempfile'])
info['filename'] = str(weight) + '-' + str(level) + '-' + str(character) + '-' + label + '.sobj'
return info
except ValueError as e:
emf_logger.debug(e)
emf_logger.debug(e.message)
#if isinstance(e,IndexError):
info['error'] = e.message
WMFS = None
if WMFS is None:
info['error'] = "We are sorry. The sought space can not be found in the database. "+"<br> Detailed information: {0}".format(info.get('error',''))
return info
else:
### Somehow the Hecke orbits are sometimes not in the space...
#if WMFS.dimension_new_cusp_forms()!=len(WMFS.hecke_orbits):
# ## Try to add them here...
# for d in range(len(WMFS.dimension_new_cusp_forms())):
# F =
# WMFS.hecke_orbits.append(F)
info['space'] = WMFS
# info['old_decomposition'] = WMFS.oldspace_decomposition()
info['oldspace_decomposition']=''
try:
emf_logger.debug("Oldspace = {0}".format(WMFS.oldspace_decomposition))
if WMFS.oldspace_decomposition != []:
emf_logger.debug("oldspace={0}".format(WMFS.oldspace_decomposition))
l = []
for t in WMFS.oldspace_decomposition:
emf_logger.debug("t={0}".format(t))
N,k,chi,mult,d = t
url = url_for('emf.render_elliptic_modular_forms', level=N, weight=k, character=chi)
if chi != 1:
sname = "S^{{ new }}_{{ {k} }}(\\Gamma_0({N}),\\chi_{{ {N} }}({chi},\\cdot))".format(k=k,N=N,chi=chi)
else:
sname = "S^{{ new }}_{{ {k} }}(\\Gamma_0({N}))".format(k=k,N=N)
l.append("\href{{ {url} }}{{ {sname} }}^{{\oplus {mult} }}".format(sname=sname,mult=mult,url=url))
if l != []:
s = "\\oplus ".join(l)
info['oldspace_decomposition']=' $ {0} $'.format(s)
except Exception as e:
emf_logger.critical("Oldspace decomposition failed. Error:{0}".format(e))
## For side-bar
lifts = list()
lifts.append(('Half-Integral Weight Forms', '/ModularForm/Mp2/Q'))
lifts.append(('Siegel Modular Forms', '/ModularForm/GSp4/Q'))
info['lifts'] = lifts
friends = list()
for label in WMFS.hecke_orbits:
f = WMFS.hecke_orbits[label]
# catch the url being None or set to '':
if hasattr(f.base_ring, "lmfdb_url") and f.base_ring.lmfdb_url:
friends.append(('Number field ' + f.base_ring.lmfdb_pretty, f.base_ring.lmfdb_url))
if hasattr(f.coefficient_field, "lmfdb_url") and f.coefficient_field.lmfdb_url:
friends.append(('Number field ' + f.coefficient_field.lmfdb_pretty, f.coefficient_field.lmfdb_url))
friends.append(("Dirichlet character \(" + WMFS.character.latex_name + "\)", WMFS.character.url()))
friends = uniq(friends)
info['friends'] = friends
return info
def set_info_for_web_newform(level=None,
weight=None,
character=None,
label=None,
**kwds):
r"""
Set the info for on modular form.
"""
info = to_dict(kwds)
info['level'] = level
info['weight'] = weight
info['character'] = character
info['label'] = label
if level is None or weight is None or character is None or label is None:
s = "In set info for one form but do not have enough args!"
s += "level={0},weight={1},character={2},label={3}".format(
level, weight, character, label)
emf_logger.critical(s)
emf_logger.debug("In set_info_for_one_mf: info={0}".format(info))
prec = my_get(info, 'prec', default_prec, int)
bprec = my_get(info, 'bprec', default_display_bprec, int)
emf_logger.debug("PREC: {0}".format(prec))
emf_logger.debug("BITPREC: {0}".format(bprec))
try:
WNF = WebNewForm_cached(level=level,
weight=weight,
character=character,
label=label)
if not WNF.has_updated():
raise IndexError(
"Unfortunately, we do not have this newform in the database.")
info['character_order'] = WNF.character.order
info['code'] = WNF.code
emf_logger.debug("defined webnewform for rendering!")
except IndexError as e:
info['error'] = e.message
url0 = url_for("mf.modular_form_main_page")
url1 = url_for("emf.render_elliptic_modular_forms")
url2 = url_for("emf.render_elliptic_modular_forms", level=level)
url3 = url_for("emf.render_elliptic_modular_forms",
level=level,
weight=weight)
url4 = url_for("emf.render_elliptic_modular_forms",
level=level,
weight=weight,
character=character)
bread = [(MF_TOP, url0), (EMF_TOP, url1)]
bread.append(("Level %s" % level, url2))
bread.append(("Weight %s" % weight, url3))
bread.append(("Character \( %s \)" % (WNF.character.latex_name), url4))
bread.append(
("Newform %d.%d.%d.%s" % (level, weight, int(character), label), ''))
info['bread'] = bread
properties2 = list()
friends = list()
space_url = url_for('emf.render_elliptic_modular_forms',
level=level,
weight=weight,
character=character)
friends.append(
('\( S_{%s}(%s, %s)\)' %
(WNF.weight, WNF.level, WNF.character.latex_name), space_url))
if hasattr(WNF.base_ring, "lmfdb_url") and WNF.base_ring.lmfdb_url:
friends.append(('Number field ' + WNF.base_ring.lmfdb_pretty,
WNF.base_ring.lmfdb_url))
if hasattr(WNF.coefficient_field,
"lmfdb_url") and WNF.coefficient_field.lmfdb_label:
friends.append(('Number field ' + WNF.coefficient_field.lmfdb_pretty,
WNF.coefficient_field.lmfdb_url))
friends = uniq(friends)
friends.append(("Dirichlet character \(" + WNF.character.latex_name + "\)",
WNF.character.url()))
if WNF.dimension == 0 and not info.has_key('error'):
info['error'] = "This space is empty!"
info['title'] = 'Newform ' + WNF.hecke_orbit_label
info['learnmore'] = [('History of modular forms',
url_for('.holomorphic_mf_history'))]
if 'error' in info:
return info
## Until we have figured out how to do the embeddings correctly we don't display the Satake
## parameters for non-trivial characters....
## Example to illustrate the different cases
## base = CyclotomicField(n) -- of degree phi(n)
## coefficient_field = NumberField( p(x)) for some p in base['x'] of degree m
## we would then have cdeg = m*phi(n) and bdeg = phi(n)
## and rdeg = m
## Unfortunately, for e.g. base = coefficient_field = CyclotomicField(6)
## we get coefficient_field.relative_degree() == 2 although it should be 1
cdeg = WNF.coefficient_field.absolute_degree()
bdeg = WNF.base_ring.absolute_degree()
if cdeg == 1:
rdeg = 1
else:
## just setting rdeg = WNF.coefficient_field.relative_degree() does not give correct result...
##
rdeg = QQ(cdeg) / QQ(bdeg)
cf_is_QQ = (cdeg == 1)
br_is_QQ = (bdeg == 1)
if cf_is_QQ:
info['satake'] = WNF.satake
if WNF.complexity_of_first_nonvanishing_coefficients(
) > default_max_height:
info['qexp'] = ""
info['qexp_display'] = ''
info['hide_qexp'] = True
n, c = WNF.first_nonvanishing_coefficient()
info['trace_nv'] = latex(WNF.first_nonvanishing_coefficient_trace())
info['norm_nv'] = '\\approx ' + latex(
WNF.first_nonvanishing_coefficient_norm().n())
info['index_nv'] = n
else:
if WNF.prec < prec:
#get WNF record at larger prec
WNF.prec = prec
WNF.update_from_db()
info['qexp'] = WNF.q_expansion_latex(prec=10, name='\\alpha ')
info['qexp_display'] = url_for(".get_qexp_latex",
level=level,
weight=weight,
character=character,
label=label)
info["hide_qexp"] = False
info['max_cn_qexp'] = WNF.q_expansion.prec()
## All combinations should be tested...
## 13/4/4/a -> base ring = coefficient_field = QQ(zeta_6)
## 13/3/8/a -> base_ring = QQ(zeta_4), coefficient_field has poly x^2+(2\zeta_4+2x-3\zeta_$ over base_ring
## 13/4/3/a -> base_ring = coefficient_field = QQ(zeta_3)
## 13/4/1/a -> all rational
## 13/6/1/a/ -> base_ring = QQ, coefficient_field = Q(sqrt(17))
## These are variables which needs to be set properly below
info['polvars'] = {'base_ring': 'x', 'coefficient_field': '\\alpha'}
if not cf_is_QQ:
if rdeg > 1: # not WNF.coefficient_field == WNF.base_ring:
## Here WNF.base_ring should be some cyclotomic field and we have an extension over this.
p1 = WNF.coefficient_field.relative_polynomial()
c_pol_ltx = web_latex_poly(p1,
'\\alpha') # make the variable \alpha
c_pol_ltx_x = web_latex_poly(p1, 'x')
zeta = p1.base_ring().gens()[0]
# p2 = zeta.minpoly() #this is not used anymore
# b_pol_ltx = web_latex_poly(p2, latex(zeta)) #this is not used anymore
z1 = zeta.multiplicative_order()
info['coeff_field'] = [
WNF.coefficient_field.absolute_polynomial_latex('x'),
c_pol_ltx_x, z1
]
if hasattr(WNF.coefficient_field,
"lmfdb_url") and WNF.coefficient_field.lmfdb_url:
info['coeff_field_pretty'] = [
WNF.coefficient_field.lmfdb_url,
WNF.coefficient_field.lmfdb_pretty,
WNF.coefficient_field.lmfdb_label
]
if z1 == 4:
info[
'polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\) and \(\zeta_4=i\).</div><br/>'.format(
c_pol_ltx)
info['polvars']['base_ring'] = 'i'
elif z1 <= 2:
info[
'polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\).</div><br/>'.format(
c_pol_ltx)
else:
info[
'polynomial_st'] = '<div class="where">where</div> %s\(\mathstrut=0\) and \(\zeta_{%s}=e^{\\frac{2\\pi i}{%s}}\) ' % (
c_pol_ltx, z1, z1)
info['polvars']['base_ring'] = '\zeta_{{ {0} }}'.format(z1)
if z1 == 3:
info[
'polynomial_st'] += 'is a primitive cube root of unity.'
else:
info[
'polynomial_st'] += 'is a primitive {0}-th root of unity.'.format(
z1)
elif not br_is_QQ:
## Now we have base and coefficient field being equal, meaning that since the coefficient field is not QQ it is some cyclotomic field
## generated by some \zeta_n
p1 = WNF.coefficient_field.absolute_polynomial()
z1 = WNF.coefficient_field.gens()[0].multiplicative_order()
c_pol_ltx = web_latex_poly(p1, '\\zeta_{{{0}}}'.format(z1))
c_pol_ltx_x = web_latex_poly(p1, 'x')
info['coeff_field'] = [
WNF.coefficient_field.absolute_polynomial_latex('x'),
c_pol_ltx_x
]
if hasattr(WNF.coefficient_field,
"lmfdb_url") and WNF.coefficient_field.lmfdb_url:
info['coeff_field_pretty'] = [
WNF.coefficient_field.lmfdb_url,
WNF.coefficient_field.lmfdb_pretty,
WNF.coefficient_field.lmfdb_label
]
if z1 == 4:
info[
'polynomial_st'] = '<div class="where">where \(\zeta_4=e^{{\\frac{{\\pi i}}{{ 2 }} }}=i \).</div>'.format(
c_pol_ltx)
info['polvars']['coefficient_field'] = 'i'
elif z1 <= 2:
info['polynomial_st'] = ''
else:
info[
'polynomial_st'] = '<div class="where">where \(\zeta_{{{0}}}=e^{{\\frac{{2\\pi i}}{{ {0} }} }}\) '.format(
z1)
info['polvars']['coefficient_field'] = '\zeta_{{{0}}}'.format(
z1)
if z1 == 3:
info[
'polynomial_st'] += 'is a primitive cube root of unity.</div>'
else:
info[
'polynomial_st'] += 'is a primitive {0}-th root of unity.</div>'.format(
z1)
else:
info['polynomial_st'] = ''
if info["hide_qexp"]:
info['polynomial_st'] = ''
info['degree'] = int(cdeg)
if cdeg == 1:
info['is_rational'] = 1
info['coeff_field_pretty'] = [
WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty
]
else:
info['is_rational'] = 0
emf_logger.debug("PREC2: {0}".format(prec))
info['embeddings'] = WNF._embeddings[
'values'] #q_expansion_embeddings(prec, bprec,format='latex')
info['embeddings_len'] = len(info['embeddings'])
properties2 = [('Level', str(level)), ('Weight', str(weight)),
('Character', '$' + WNF.character.latex_name + '$'),
('Label', WNF.hecke_orbit_label),
('Dimension of Galois orbit', str(WNF.dimension))]
if (ZZ(level)).is_squarefree():
info['twist_info'] = WNF.twist_info
if isinstance(info['twist_info'],
list) and len(info['twist_info']) > 0:
info['is_minimal'] = info['twist_info'][0]
if (info['twist_info'][0]):
s = 'Is minimal<br>'
else:
s = 'Is a twist of lower level<br>'
properties2 += [('Twist info', s)]
else:
info['twist_info'] = 'Twist info currently not available.'
properties2 += [('Twist info', 'not available')]
args = list()
for x in range(5, 200, 10):
args.append({'digits': x})
alev = None
CM = WNF._cm_values
if CM is not None:
if CM.has_key('tau') and len(CM['tau']) != 0:
info['CM_values'] = CM
info['is_cm'] = WNF.is_cm
if WNF.is_cm == 1:
info['cm_field'] = "2.0.{0}.1".format(-WNF.cm_disc)
info['cm_disc'] = WNF.cm_disc
info['cm_field_knowl'] = nf_display_knowl(
info['cm_field'], getDBConnection(),
field_pretty(info['cm_field']))
info['cm_field_url'] = url_for("number_fields.by_label",
label=info["cm_field"])
if WNF.is_cm is None or WNF.is_cm == -1:
s = '- Unknown (insufficient data)<br>'
elif WNF.is_cm == 1:
s = 'Yes<br>'
else:
s = 'No<br>'
properties2.append(('CM', s))
alev = WNF.atkin_lehner_eigenvalues()
info['atkinlehner'] = None
if isinstance(alev, dict) and len(alev.keys()) > 0 and level != 1:
s1 = " Atkin-Lehner eigenvalues "
s2 = ""
for Q in alev.keys():
s2 += "\( \omega_{ %s } \) : %s <br>" % (Q, alev[Q])
properties2.append((s1, s2))
emf_logger.debug("properties={0}".format(properties2))
# alev = WNF.atkin_lehner_eigenvalues_for_all_cusps()
# if isinstance(alev,dict) and len(alev.keys())>0:
# emf_logger.debug("alev={0}".format(alev))
# info['atkinlehner'] = list()
# for Q in alev.keys():
# s = "\(" + latex(c) + "\)"
# Q = alev[c][0]
# ev = alev[c][1]
# info['atkinlehner'].append([Q, c, ev])
if (level == 1):
poly = WNF.explicit_formulas.get('as_polynomial_in_E4_and_E6', '')
if poly != '':
d, monom, coeffs = poly
emf_logger.critical("poly={0}".format(poly))
info['explicit_formulas'] = '\('
for i in range(len(coeffs)):
c = QQ(coeffs[i])
s = ""
if d > 1 and i > 0 and c > 0:
s = "+"
if c < 0:
s = "-"
if c.denominator() > 1:
cc = "\\frac{{ {0} }}{{ {1} }}".format(
abs(c.numerator()), c.denominator())
else:
cc = str(abs(c))
s += "{0} \cdot ".format(cc)
a = monom[i][0]
b = monom[i][1]
if a == 1:
a = ""
if b == 1:
b = ""
if a == 0 and b != 0:
s += "E_6^{{ {0} }}(z)".format(b)
elif b == 0 and a != 0:
s += "E_4^{{ {0} }}(z)".format(a)
else:
s += "E_4^{{ {0} }}(z) \cdot E_6^{{ {1} }}(z)".format(a, b)
info['explicit_formulas'] += s
info['explicit_formulas'] += " \)"
# cur_url = '?&level=' + str(level) + '&weight=' + str(weight) + '&character=' + str(character) + '&label=' + str(label) # never used
if len(WNF.parent.hecke_orbits) > 1:
for label_other in WNF.parent.hecke_orbits.keys():
if (label_other != label):
s = 'Modular form '
if character:
s += newform_label(level, weight, character, label_other)
else:
s += newform_label(level, weight, 1, label_other)
url = url_for('emf.render_elliptic_modular_forms',
level=level,
weight=weight,
character=character,
label=label_other)
friends.append((s, url))
s = 'L-Function '
if character:
s += newform_label(level, weight, character, label)
else:
s += newform_label(level, weight, 1, label)
# url =
# "/L/ModularForm/GL2/Q/holomorphic?level=%s&weight=%s&character=%s&label=%s&number=%s"
# %(level,weight,character,label,0)
url = '/L' + url_for('emf.render_elliptic_modular_forms',
level=level,
weight=weight,
character=character,
label=label)
if WNF.coefficient_field_degree > 1:
for h in range(WNF.coefficient_field_degree):
s0 = s + ".{0}".format(h)
url0 = url + "{0}/".format(h)
friends.append((s0, url0))
else:
friends.append((s, url))
# if there is an elliptic curve over Q associated to self we also list that
if WNF.weight == 2 and WNF.coefficient_field_degree == 1:
llabel = str(level) + '.' + label
s = 'Elliptic curve isogeny class ' + llabel
url = '/EllipticCurve/Q/' + llabel
friends.append((s, url))
info['properties2'] = properties2
info['friends'] = friends
info['max_cn'] = WNF.max_available_prec()
return info
def submatrix_configurations(S):
configs = orbit(S)
[m.set_immutable() for m in configs]
return uniq(configs)
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()
emf_logger.debug("info={0}".format(info))
WMFS = None
if info.has_key('error'):
return info
if level <= 0:
info['error'] = "Got wrong level: %s " % level
return info
try:
WMFS = WebModFormSpace_cached(level = level, weight = weight, cuspidal=True, character = character, update_from_db=True)
emf_logger.debug("Created WebModFormSpace %s"%WMFS)
if not WMFS.has_updated():
#get the representative we have in the db for this space (Galois conjugate)
#note that this does not use the web_object infrastructure at all right now
#which should be changed for sure!
dimension_table_name = WebModFormSpace._dimension_table_name
db_dim = getDBConnection()['modularforms2'][dimension_table_name]
rep = db_dim.find_one({'level': level, 'weight': weight, 'character_orbit': {'$in': [character]}})
if not rep is None and not rep['cchi'] == character: # don't link back to myself!
info['wmfs_rep_url'] = url_for('emf.render_elliptic_modular_forms', level=level, weight=weight, character=rep['cchi'])
info['wmfs_rep_number'] = rep['cchi']
# FIXME: the variable WNF is not defined above, so the code below cannot work (I don't think it is ever used)
# if 'download' in info and 'tempfile' in info:
# save(WNF,info['tempfile'])
# info['filename'] = str(weight) + '-' + str(level) + '-' + str(character) + '-' + label + '.sobj'
# return info
except ValueError as e:
emf_logger.debug(e)
emf_logger.debug(e.message)
#if isinstance(e,IndexError):
info['error'] = e.message
WMFS = None
if WMFS is None:
info['error'] = "We are sorry. The sought space can not be found in the database. "+"<br> Detailed information: {0}".format(info.get('error',''))
return info
else:
### Somehow the Hecke orbits are sometimes not in the space...
#if WMFS.dimension_new_cusp_forms()!=len(WMFS.hecke_orbits):
# ## Try to add them here...
# for d in range(len(WMFS.dimension_new_cusp_forms())):
# F =
# WMFS.hecke_orbits.append(F)
info['space'] = WMFS
info['max_height'] = default_max_height
# info['old_decomposition'] = WMFS.oldspace_decomposition()
info['oldspace_decomposition']=''
try:
emf_logger.debug("Oldspace = {0}".format(WMFS.oldspace_decomposition))
if WMFS.oldspace_decomposition != []:
emf_logger.debug("oldspace={0}".format(WMFS.oldspace_decomposition))
l = []
for t in WMFS.oldspace_decomposition:
emf_logger.debug("t={0}".format(t))
N,k,chi,mult,d = t
url = url_for('emf.render_elliptic_modular_forms', level=N, weight=k, character=chi)
if chi != 1:
sname = "S^{{ new }}_{{ {k} }}(\\Gamma_0({N}),\\chi_{{ {N} }}({chi},\\cdot))".format(k=k,N=N,chi=chi)
else:
sname = "S^{{ new }}_{{ {k} }}(\\Gamma_0({N}))".format(k=k,N=N)
l.append("\href{{ {url} }}{{ {sname} }}^{{\oplus {mult} }}".format(sname=sname,mult=mult,url=url))
if l != []:
s = "\\oplus ".join(l)
info['oldspace_decomposition']=' $ {0} $'.format(s)
except Exception as e:
emf_logger.critical("Oldspace decomposition failed. Error:{0}".format(e))
## For side-bar
lifts = list()
lifts.append(('Half-Integral Weight Forms', '/ModularForm/Mp2/Q'))
lifts.append(('Siegel Modular Forms', '/ModularForm/GSp4/Q'))
info['lifts'] = lifts
friends = list()
for label in WMFS.hecke_orbits:
f = WMFS.hecke_orbits[label]
# catch the url being None or set to '':
if hasattr(f.base_ring, "lmfdb_url") and f.base_ring.lmfdb_url:
friends.append(('Number field ' + f.base_ring.lmfdb_pretty, f.base_ring.lmfdb_url))
if hasattr(f.coefficient_field, "lmfdb_url") and f.coefficient_field.lmfdb_url:
friends.append(('Number field ' + f.coefficient_field.lmfdb_pretty, f.coefficient_field.lmfdb_url))
friends.append(("Dirichlet character \(" + WMFS.character.latex_name + "\)", WMFS.character.url()))
friends = uniq(friends)
info['friends'] = friends
info['code'] = WMFS.code
return info
def _compute_simple_modules_graph_from_startpoint(self, s, p=None, cut_nonsimple_aniso=True, fast=1):
# for forking this is necessary
logger = get_logger(s)
# print logger
k = self._weight
###########################################################
# Determine which primes need to be checked
# According to the proof of Proposition XX in [BEF], we
# only need to check primesnot dividing the 6*level(s),
# for which prime_pol(s,p,k) <= 0.
# For those primes, we check if there is any
# k-simple fqm in s.C(p) and if not, we do not have to
# consider p anymore.
###########################################################
if p == None:
p = 2
N = Integer(6) * s.level()
slp = N.prime_factors()
for q in prime_range(next_prime(N) + 1):
if not q in slp:
logger.info(
"Smallest prime not dividing 6*level({0}) = {1} is p = {2}".format(s, Integer(6) * s.level(), q))
p = q
break
while prime_pol(s, p, k) <= 0 or p in slp:
p = next_prime(p)
p = uniq(prime_range(p) + slp)
logger.info("Starting with s = {0} and primes = {1}".format(s, p))
if isinstance(p, list):
primes = p
else:
primes = [p]
simple = s.is_simple(k, reduction = self._reduction, bound = self._bound)
if not simple:
logger.info("{0} is not simple.".format(s))
# print simple
s = FQM_vertex(s)
# print s
self.add_vertex(s)
# print "added vertex ", s
############################################################
# Starting from the list of primes we generated,
# we now compute which primes we actually need to consider.
# That is, the primes such that there is a fqm in s.C(p)
# which is k-simple.
############################################################
np = list()
if cut_nonsimple_aniso and simple:
for i in range(len(primes)):
p = primes[i]
fs = False
for t in s.genus_symbol().C(p, False):
if t.is_simple(k, bound = self._bound):
fs = True
logger.debug("Setting fs = True")
break
if fs:
np.append(p)
primes = np
# print "here", primes
logger.info("primes for graph for {0}: {1}".format(s, primes))
# if len(primes) == 0:
# return
heights = self._heights
h = 0
if not heights.has_key(h):
heights[h] = [s]
else:
if heights[h].count(s) == 0:
heights[h].append(s)
vertex_colors = self._vertex_colors
nonsimple_color = self._nonsimple_color
simple_color = self._simple_color
# Bs contains the modules of the current level (= height = h)
Bs = [s]
# set the correct color for the vertex s
if simple:
if vertex_colors[simple_color].count(s) == 0:
vertex_colors[simple_color].append(s)
elif vertex_colors[nonsimple_color].count(s) == 0:
vertex_colors[nonsimple_color].append(s)
###################################################
# MAIN LOOP
# we loop until we haven't found any simple fqm's
###################################################
while simple:
h = h + 1
if not heights.has_key(h):
heights[h] = list()
# the list Bss will contain the k-simple modules of the next height level
# recall that Bs contains the modules of the current height level
Bss = list()
simple = False
# checklist = modules that we need to check for with .is_simple(k)
# we assemble this list because afterwards
# we check them in parallel
checklist = []
for s1 in Bs:
Bs2 = list()
for p in primes:
# check if we really need to check p for s1
# otherwise none of the fqm's in s1.C(p) are simple
# and we will not consider them.
if prime_pol(s1.genus_symbol(), p, k) <= 0:
Bs2 = Bs2 + s1.genus_symbol().C(p, False)
else:
logger.info(
"Skipping p = {0} for s1 = {1}".format(p, s1))
# print "Skipping p = {0} for s1 = {1}".format(p, s1)
# print "Bs2 = ", Bs2
# now we actually check the symbols in Bs2
for s2 in Bs2:
if s2.max_rank() > self._rank_limit:
# we skip s2 if its minimal number of generators
# is > than the given rank_limit.
continue
s2 = FQM_vertex(s2)
skip = False
for v in self._heights[h]:
# we skip symbols that correspond to isomorphic modules
if v.genus_symbol().defines_isomorphic_module(s2.genus_symbol()):
skip = True
logger.debug(
"skipping {0} b/c isomorphic to {1}".format(s2.genus_symbol(), v.genus_symbol()))
s2 = v
break
if skip:
continue
if not skip:
self.add_vertex(s2)
heights[h].append(s2)
self.update_edges(s2, h, fast=fast)
# before using the actual dimension formula
# we check if there is already a non-k-simple neighbor
# (an incoming edge from a non-k-simple module)
# which would imply that s2 is not k-simple.
has_nonsimple_neighbor = False
for e in self.incoming_edges(s2):
if vertex_colors[nonsimple_color].count(e[0]) > 0:
has_nonsimple_neighbor = True
logger.debug(
"Has nonsimple neighbor: {0}".format(s2.genus_symbol()))
break
if has_nonsimple_neighbor:
#not simple
if not vertex_colors[nonsimple_color].count(s2) > 0:
vertex_colors[nonsimple_color].append(s2)
else:
checklist.append((s2, k, self._reduction, self._bound))
logger.debug("checklist = {0}".format(checklist))
# check the modules in checklist
# for being k-simple
# this is done in parallel
# when a process returns
# we add the vertex and give it its appropriate color
if NCPUS1 == 1:
checks = [([[s[0]]],check_simple(*s)) for s in checklist]
else:
checks = list(check_simple(checklist))
logger.info("checks = {0}".format(checks))
for check in checks:
s2 = check[0][0][0]
if check[1]:
simple = True
logger.info(
"Found simple module: {0}".format(s2.genus_symbol()))
Bss.append(s2)
if not vertex_colors[simple_color].count(s2) > 0:
vertex_colors[simple_color].append(s2)
else:
if not vertex_colors[nonsimple_color].count(s2) > 0:
vertex_colors[nonsimple_color].append(s2)
Bs = Bss
simple = [v.genus_symbol() for v in vertex_colors[simple_color]]
self._simple = uniq(simple)
def _compute_simple_modules_graph_from_startpoint(self,
s,
p=None,
cut_nonsimple_aniso=True,
fast=1):
# for forking this is necessary
logger = get_logger(s)
# print logger
k = self._weight
###########################################################
# Determine which primes need to be checked
# According to the proof of Proposition XX in [BEF], we
# only need to check primesnot dividing the 6*level(s),
# for which prime_pol(s,p,k) <= 0.
# For those primes, we check if there is any
# k-simple fqm in s.C(p) and if not, we do not have to
# consider p anymore.
###########################################################
if p == None:
p = 2
N = Integer(6) * s.level()
slp = N.prime_factors()
for q in prime_range(next_prime(N) + 1):
if not q in slp:
logger.info(
"Smallest prime not dividing 6*level({0}) = {1} is p = {2}"
.format(s,
Integer(6) * s.level(), q))
p = q
break
while prime_pol(s, p, k) <= 0 or p in slp:
p = next_prime(p)
p = uniq(prime_range(p) + slp)
logger.info("Starting with s = {0} and primes = {1}".format(s, p))
if isinstance(p, list):
primes = p
else:
primes = [p]
simple = s.is_simple(k, reduction=self._reduction, bound=self._bound)
if not simple:
logger.info("{0} is not simple.".format(s))
# print simple
s = FQM_vertex(s)
# print s
self.add_vertex(s)
# print "added vertex ", s
############################################################
# Starting from the list of primes we generated,
# we now compute which primes we actually need to consider.
# That is, the primes such that there is a fqm in s.C(p)
# which is k-simple.
############################################################
np = list()
if cut_nonsimple_aniso and simple:
for i in range(len(primes)):
p = primes[i]
fs = False
for t in s.genus_symbol().C(p, False):
if t.is_simple(k, bound=self._bound):
fs = True
logger.debug("Setting fs = True")
break
if fs:
np.append(p)
primes = np
# print "here", primes
logger.info("primes for graph for {0}: {1}".format(s, primes))
# if len(primes) == 0:
# return
heights = self._heights
h = 0
if not heights.has_key(h):
heights[h] = [s]
else:
if heights[h].count(s) == 0:
heights[h].append(s)
vertex_colors = self._vertex_colors
nonsimple_color = self._nonsimple_color
simple_color = self._simple_color
# Bs contains the modules of the current level (= height = h)
Bs = [s]
# set the correct color for the vertex s
if simple:
if vertex_colors[simple_color].count(s) == 0:
vertex_colors[simple_color].append(s)
elif vertex_colors[nonsimple_color].count(s) == 0:
vertex_colors[nonsimple_color].append(s)
###################################################
# MAIN LOOP
# we loop until we haven't found any simple fqm's
###################################################
while simple:
h = h + 1
if not heights.has_key(h):
heights[h] = list()
# the list Bss will contain the k-simple modules of the next height level
# recall that Bs contains the modules of the current height level
Bss = list()
simple = False
# checklist = modules that we need to check for with .is_simple(k)
# we assemble this list because afterwards
# we check them in parallel
checklist = []
for s1 in Bs:
Bs2 = list()
for p in primes:
# check if we really need to check p for s1
# otherwise none of the fqm's in s1.C(p) are simple
# and we will not consider them.
if prime_pol(s1.genus_symbol(), p, k) <= 0:
Bs2 = Bs2 + s1.genus_symbol().C(p, False)
else:
logger.info("Skipping p = {0} for s1 = {1}".format(
p, s1))
# print "Skipping p = {0} for s1 = {1}".format(p, s1)
# print "Bs2 = ", Bs2
# now we actually check the symbols in Bs2
for s2 in Bs2:
if s2.max_rank() > self._rank_limit:
# we skip s2 if its minimal number of generators
# is > than the given rank_limit.
continue
s2 = FQM_vertex(s2)
skip = False
for v in self._heights[h]:
# we skip symbols that correspond to isomorphic modules
if v.genus_symbol().defines_isomorphic_module(
s2.genus_symbol()):
skip = True
logger.debug(
"skipping {0} b/c isomorphic to {1}".format(
s2.genus_symbol(), v.genus_symbol()))
s2 = v
break
if skip:
continue
if not skip:
self.add_vertex(s2)
heights[h].append(s2)
self.update_edges(s2, h, fast=fast)
# before using the actual dimension formula
# we check if there is already a non-k-simple neighbor
# (an incoming edge from a non-k-simple module)
# which would imply that s2 is not k-simple.
has_nonsimple_neighbor = False
for e in self.incoming_edges(s2):
if vertex_colors[nonsimple_color].count(e[0]) > 0:
has_nonsimple_neighbor = True
logger.debug("Has nonsimple neighbor: {0}".format(
s2.genus_symbol()))
break
if has_nonsimple_neighbor:
#not simple
if not vertex_colors[nonsimple_color].count(s2) > 0:
vertex_colors[nonsimple_color].append(s2)
else:
checklist.append((s2, k, self._reduction, self._bound))
logger.debug("checklist = {0}".format(checklist))
# check the modules in checklist
# for being k-simple
# this is done in parallel
# when a process returns
# we add the vertex and give it its appropriate color
if NCPUS1 == 1:
checks = [([[s[0]]], check_simple(*s)) for s in checklist]
else:
checks = list(check_simple(checklist))
logger.info("checks = {0}".format(checks))
for check in checks:
s2 = check[0][0][0]
if check[1]:
simple = True
logger.info("Found simple module: {0}".format(
s2.genus_symbol()))
Bss.append(s2)
if not vertex_colors[simple_color].count(s2) > 0:
vertex_colors[simple_color].append(s2)
else:
if not vertex_colors[nonsimple_color].count(s2) > 0:
vertex_colors[nonsimple_color].append(s2)
Bs = Bss
simple = [v.genus_symbol() for v in vertex_colors[simple_color]]
self._simple = uniq(simple)
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()
emf_logger.debug("info={0}".format(info))
WMFS = None
if info.has_key('error'):
return info
if level <= 0:
info['error'] = "Got wrong level: %s " % level
return info
try:
WMFS = WebModFormSpace_cached(level = level, weight = weight, cuspidal=True, character = character, update_from_db=True)
if not WMFS.has_updated():
stop = False
orbit = WMFS.character.character.galois_orbit()
while not stop:
if len(orbit) == 0:
stop = True
continue
c = orbit.pop()
if c.number() == WMFS.character.number:
continue
print c.number()
WMFS_rep = WebModFormSpace_cached(level = level, weight = weight, cuspidal=True, character = c.number(), update_from_db=True)
if WMFS_rep.has_updated_from_db():
stop = True
info['wmfs_rep_url'] = url_for('emf.render_elliptic_modular_forms', level=level, weight=weight, character = c.number())
info['wmfs_rep_number'] = c.number()
emf_logger.debug("Created WebModFormSpace %s"%WMFS)
if 'download' in info and 'tempfile' in info:
save(WNF,info['tempfile'])
info['filename'] = str(weight) + '-' + str(level) + '-' + str(character) + '-' + label + '.sobj'
return info
except ValueError as e:
emf_logger.debug(e)
emf_logger.debug(e.message)
#if isinstance(e,IndexError):
info['error'] = e.message
WMFS = None
if WMFS is None:
info['error'] = "We are sorry. The sought space can not be found in the database. "+"<br> Detailed information: {0}".format(info.get('error',''))
return info
else:
### Somehow the Hecke orbits are sometimes not in the space...
#if WMFS.dimension_new_cusp_forms()!=len(WMFS.hecke_orbits):
# ## Try to add them here...
# for d in range(len(WMFS.dimension_new_cusp_forms())):
# F =
# WMFS.hecke_orbits.append(F)
info['space'] = WMFS
# info['old_decomposition'] = WMFS.oldspace_decomposition()
## For side-bar
lifts = list()
lifts.append(('Half-Integral Weight Forms', '/ModularForm/Mp2/Q'))
lifts.append(('Siegel Modular Forms', '/ModularForm/GSp4/Q'))
info['lifts'] = lifts
friends = list()
for label in WMFS.hecke_orbits:
f = WMFS.hecke_orbits[label]
# catch the url being None or set to '':
if hasattr(f.base_ring, "lmfdb_url") and f.base_ring.lmfdb_url:
friends.append(('Number field ' + f.base_ring.lmfdb_pretty, f.base_ring.lmfdb_url))
if hasattr(f.coefficient_field, "lmfdb_url") and f.coefficient_field.lmfdb_url:
friends.append(('Number field ' + f.coefficient_field.lmfdb_pretty, f.coefficient_field.lmfdb_url))
friends.append(("Dirichlet character \(" + WMFS.character.latex_name + "\)", WMFS.character.url()))
friends = uniq(friends)
info['friends'] = friends
return info
def set_info_for_web_newform(level=None, weight=None, character=None, label=None, **kwds):
r"""
Set the info for on modular form.
"""
info = to_dict(kwds)
info["level"] = level
info["weight"] = weight
info["character"] = character
info["label"] = label
if level is None or weight is None or character is None or label is None:
s = "In set info for one form but do not have enough args!"
s += "level={0},weight={1},character={2},label={3}".format(level, weight, character, label)
emf_logger.critical(s)
emf_logger.debug("In set_info_for_one_mf: info={0}".format(info))
prec = my_get(info, "prec", default_prec, int)
bprec = my_get(info, "bprec", default_display_bprec, int)
emf_logger.debug("PREC: {0}".format(prec))
emf_logger.debug("BITPREC: {0}".format(bprec))
try:
WNF = WebNewForm_cached(level=level, weight=weight, character=character, label=label)
emf_logger.critical("defined webnewform for rendering!")
# if info.has_key('download') and info.has_key('tempfile'):
# WNF._save_to_file(info['tempfile'])
# info['filename']=str(weight)+'-'+str(level)+'-'+str(character)+'-'+label+'.sobj'
# return info
except IndexError as e:
WNF = None
info["error"] = e.message
url1 = url_for("emf.render_elliptic_modular_forms")
url2 = url_for("emf.render_elliptic_modular_forms", level=level)
url3 = url_for("emf.render_elliptic_modular_forms", level=level, weight=weight)
url4 = url_for("emf.render_elliptic_modular_forms", level=level, weight=weight, character=character)
bread = [(EMF_TOP, url1)]
bread.append(("of level %s" % level, url2))
bread.append(("weight %s" % weight, url3))
if int(character) == 0:
bread.append(("trivial character", url4))
else:
bread.append(("\( %s \)" % (WNF.character.latex_name), url4))
info["bread"] = bread
properties2 = list()
friends = list()
space_url = url_for("emf.render_elliptic_modular_forms", level=level, weight=weight, character=character)
friends.append(("\( S_{%s}(%s, %s)\)" % (WNF.weight, WNF.level, WNF.character.latex_name), space_url))
if WNF.coefficient_field_label(check=True):
friends.append(("Number field " + WNF.coefficient_field_label(), WNF.coefficient_field_url()))
friends.append(("Number field " + WNF.base_field_label(), WNF.base_field_url()))
friends = uniq(friends)
friends.append(("Dirichlet character \(" + WNF.character.latex_name + "\)", WNF.character.url()))
if WNF.dimension == 0:
info["error"] = "This space is empty!"
# emf_logger.debug("WNF={0}".format(WNF))
# info['name'] = name
info["title"] = "Modular Form " + WNF.hecke_orbit_label
if "error" in info:
return info
# info['name']=WNF._name
## Until we have figured out how to do the embeddings correctly we don't display the Satake
## parameters for non-trivial characters....
cdeg = WNF.coefficient_field.absolute_degree()
bdeg = WNF.base_ring.absolute_degree()
if WNF.coefficient_field.absolute_degree() == 1:
rdeg = 1
else:
rdeg = WNF.coefficient_field.relative_degree()
if cdeg == 1:
info["satake"] = WNF.satake
info["qexp"] = WNF.q_expansion_latex(prec=10, name="a")
info["qexp_display"] = url_for(".get_qexp_latex", level=level, weight=weight, character=character, label=label)
# info['qexp'] = WNF.q_expansion_latex(prec=prec)
# c_pol_st = str(WNF.absolute_polynomial)
# b_pol_st = str(WNF.polynomial(type='base_ring',format='str'))
# b_pol_ltx = str(WNF.polynomial(type='base_ring',format='latex'))
# print "c=",c_pol_ltx
# print "b=",b_pol_ltx
if cdeg > 1: ## Field is QQ
if bdeg > 1 and rdeg > 1:
p1 = WNF.coefficient_field.relative_polynomial()
c_pol_ltx = latex(p1)
lgc = p1.variables()[0]
c_pol_ltx = c_pol_ltx.replace(lgc, "a")
z = p1.base_ring().gens()[0]
p2 = z.minpoly()
b_pol_ltx = latex(p2)
b_pol_ltx = b_pol_ltx.replace(latex(p2.variables()[0]), latex(z))
info["polynomial_st"] = "where \({0}=0\) and \({1}=0\).".format(c_pol_ltx, b_pol_ltx)
else:
c_pol_ltx = latex(WNF.coefficient_field.relative_polynomial())
lgc = str(latex(WNF.coefficient_field.relative_polynomial().variables()[0]))
c_pol_ltx = c_pol_ltx.replace(lgc, "a")
info["polynomial_st"] = "where \({0}=0\)".format(c_pol_ltx)
else:
info["polynomial_st"] = ""
info["degree"] = int(cdeg)
if cdeg == 1:
info["is_rational"] = 1
else:
info["is_rational"] = 0
# info['q_exp_embeddings'] = WNF.print_q_expansion_embeddings()
# if(int(info['degree'])>1 and WNF.dimension()>1):
# s = 'One can embed it into \( \mathbb{C} \) as:'
# bprec = 26
# print s
# info['embeddings'] = ajax_more2(WNF.print_q_expansion_embeddings,{'prec':[5,10,25,50],'bprec':[26,53,106]},text=['more coeffs.','higher precision'])
# elif(int(info['degree'])>1):
# s = 'There are '+str(info['degree'])+' embeddings into \( \mathbb{C} \):'
# bprec = 26
# print s
# info['embeddings'] = ajax_more2(WNF.print_q_expansion_embeddings,{'prec':[5,10,25,50],'bprec':[26,53,106]},text=['more coeffs.','higher precision'])
# else:
# info['embeddings'] = ''
emf_logger.debug("PREC2: {0}".format(prec))
info["embeddings"] = WNF._embeddings["values"] # q_expansion_embeddings(prec, bprec,format='latex')
info["embeddings_len"] = len(info["embeddings"])
properties2 = []
if (ZZ(level)).is_squarefree():
info["twist_info"] = WNF.twist_info
if isinstance(info["twist_info"], list) and len(info["twist_info"]) > 0:
info["is_minimal"] = info["twist_info"][0]
if info["twist_info"][0]:
s = "- Is minimal<br>"
else:
s = "- Is a twist of lower level<br>"
properties2 = [("Twist info", s)]
else:
info["twist_info"] = "Twist info currently not available."
properties2 = [("Twist info", "not available")]
args = list()
for x in range(5, 200, 10):
args.append({"digits": x})
alev = None
CM = WNF._cm_values
if CM is not None:
if CM.has_key("tau") and len(CM["tau"]) != 0:
info["CM_values"] = CM
info["is_cm"] = WNF.is_cm
if WNF.is_cm is None:
s = "- Unknown (insufficient data)<br>"
elif WNF.is_cm is True:
s = "- Is a CM-form<br>"
else:
s = "- Is not a CM-form<br>"
properties2.append(("CM info", s))
alev = WNF.atkin_lehner_eigenvalues()
info["atkinlehner"] = None
if isinstance(alev, dict) and len(alev.keys()) > 0 and level != 1:
s1 = " Atkin-Lehner eigenvalues "
s2 = ""
for Q in alev.keys():
s2 += "\( \omega_{ %s } \) : %s <br>" % (Q, alev[Q])
properties2.append((s1, s2))
emf_logger.debug("properties={0}".format(properties2))
# alev = WNF.atkin_lehner_eigenvalues_for_all_cusps()
# if isinstance(alev,dict) and len(alev.keys())>0:
# emf_logger.debug("alev={0}".format(alev))
# info['atkinlehner'] = list()
# for Q in alev.keys():
# s = "\(" + latex(c) + "\)"
# Q = alev[c][0]
# ev = alev[c][1]
# info['atkinlehner'].append([Q, c, ev])
if level == 1:
poly = WNF.explicit_formulas.get("as_polynomial_in_E4_and_E6", "")
if poly != "":
d, monom, coeffs = poly
emf_logger.critical("poly={0}".format(poly))
info["explicit_formulas"] = "\("
for i in range(d):
c = QQ(coeffs[i])
s = ""
if d > 1 and i > 0 and c > 0:
s = "+"
if c < 0:
s = "-"
if c.denominator() > 1:
cc = "\\frac{{ {0} }}{{ {1} }}".format(abs(c.numerator()), c.denominator())
else:
cc = str(abs(c))
s += "{0} \cdot ".format(cc)
a = monom[i][0]
b = monom[i][1]
if a == 0 and b != 0:
s += "E_6^{{ {0} }}".format(b)
elif b == 0 and a != 0:
s += "E_4^{{ {0} }}".format(a)
else:
s += "E_4^{{ {0} }}E_6^{{ {1} }}".format(a, b)
info["explicit_formulas"] += s
info["explicit_formulas"] += " \)"
cur_url = (
"?&level=" + str(level) + "&weight=" + str(weight) + "&character=" + str(character) + "&label=" + str(label)
)
if len(WNF.parent.hecke_orbits) > 1:
for label_other in WNF.parent.hecke_orbits.keys():
if label_other != label:
s = "Modular form "
if character:
s = s + str(level) + "." + str(weight) + "." + str(character) + str(label_other)
else:
s = s + str(level) + "." + str(weight) + str(label_other)
url = url_for(
"emf.render_elliptic_modular_forms",
level=level,
weight=weight,
character=character,
label=label_other,
)
friends.append((s, url))
s = "L-Function "
if character:
s = s + str(level) + "." + str(weight) + "." + str(character) + str(label)
else:
s = s + str(level) + "." + str(weight) + str(label)
# url =
# "/L/ModularForm/GL2/Q/holomorphic?level=%s&weight=%s&character=%s&label=%s&number=%s"
# %(level,weight,character,label,0)
url = "/L" + url_for(
"emf.render_elliptic_modular_forms", level=level, weight=weight, character=character, label=label
)
if WNF.coefficient_field_degree > 1:
for h in range(WNF.coefficient_field_degree):
s0 = s + ".{0}".format(h)
url0 = url + "{0}/".format(h)
friends.append((s0, url0))
else:
friends.append((s, url))
# if there is an elliptic curve over Q associated to self we also list that
if WNF.weight == 2 and WNF.coefficient_field_degree == 1:
llabel = str(level) + "." + label
s = "Elliptic curve isogeny class " + llabel
url = "/EllipticCurve/Q/" + llabel
friends.append((s, url))
info["properties2"] = properties2
info["friends"] = friends
info["max_cn"] = WNF.max_cn()
return info
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()
emf_logger.debug("info={0}".format(info))
WMFS = None
if info.has_key('error'):
return info
if level <= 0:
info['error'] = "Got wrong level: %s " % level
return info
try:
WMFS = WebModFormSpace_cached(level = level, weight = weight, cuspidal=True, character = character, update_from_db=True)
emf_logger.debug("Created WebModFormSpace %s"%WMFS)
if not WMFS.has_updated():
#get the representative we have in the db for this space (Galois conjugate)
#note that this does not use the web_object infrastructure at all right now
#which should be changed for sure!
dimension_table_name = WebModFormSpace._dimension_table_name
db_dim = getDBConnection()['modularforms2'][dimension_table_name]
rep = db_dim.find_one({'level': level, 'weight': weight, 'character_orbit': {'$in': [character]}})
if not rep is None and not rep['cchi'] == character: # don't link back to myself!
info['wmfs_rep_url'] = url_for('emf.render_elliptic_modular_forms', level=level, weight=weight, character=rep['cchi'])
info['wmfs_rep_number'] = rep['cchi']
# FIXME: the variable WNF is not defined above, so the code below cannot work (I don't think it is ever used)
# if 'download' in info and 'tempfile' in info:
# save(WNF,info['tempfile'])
# info['filename'] = str(weight) + '-' + str(level) + '-' + str(character) + '-' + label + '.sobj'
# return info
except ValueError as e:
emf_logger.debug(e)
emf_logger.debug(e.message)
#if isinstance(e,IndexError):
info['error'] = e.message
WMFS = None
if WMFS is None:
info['error'] = "We are sorry. The sought space can not be found in the database. "+"<br> Detailed information: {0}".format(info.get('error',''))
return info
else:
### Somehow the Hecke orbits are sometimes not in the space...
#if WMFS.dimension_new_cusp_forms()!=len(WMFS.hecke_orbits):
# ## Try to add them here...
# for d in range(len(WMFS.dimension_new_cusp_forms())):
# F =
# WMFS.hecke_orbits.append(F)
info['space'] = WMFS
info['max_height'] = default_max_height
# info['old_decomposition'] = WMFS.oldspace_decomposition()
info['oldspace_decomposition']=''
try:
emf_logger.debug("Oldspace = {0}".format(WMFS.oldspace_decomposition))
if WMFS.oldspace_decomposition != []:
emf_logger.debug("oldspace={0}".format(WMFS.oldspace_decomposition))
l = []
for t in WMFS.oldspace_decomposition:
emf_logger.debug("t={0}".format(t))
N,k,chi,mult,d = t
url = url_for('emf.render_elliptic_modular_forms', level=N, weight=k, character=chi)
if chi != 1:
sname = "S^{{ new }}_{{ {k} }}(\\Gamma_0({N}),\\chi_{{ {N} }}({chi},\\cdot))".format(k=k,N=N,chi=chi)
else:
sname = "S^{{ new }}_{{ {k} }}(\\Gamma_0({N}))".format(k=k,N=N)
l.append("\href{{ {url} }}{{ {sname} }}^{{\oplus {mult} }}".format(sname=sname,mult=mult,url=url))
if l != []:
s = "\\oplus ".join(l)
info['oldspace_decomposition']=' $ {0} $'.format(s)
except Exception as e:
emf_logger.critical("Oldspace decomposition failed. Error:{0}".format(e))
## For side-bar
lifts = list()
lifts.append(('Half-Integral Weight Forms', '/ModularForm/Mp2/Q'))
lifts.append(('Siegel Modular Forms', '/ModularForm/GSp4/Q'))
info['lifts'] = lifts
friends = list()
for label in WMFS.hecke_orbits:
f = WMFS.hecke_orbits[label]
# catch the url being None or set to '':
if hasattr(f.base_ring, "lmfdb_url") and f.base_ring.lmfdb_url:
friends.append(('Number field ' + f.base_ring.lmfdb_pretty, f.base_ring.lmfdb_url))
if hasattr(f.coefficient_field, "lmfdb_url") and f.coefficient_field.lmfdb_url:
friends.append(('Number field ' + f.coefficient_field.lmfdb_pretty, f.coefficient_field.lmfdb_url))
friends.append(("Dirichlet character \(" + WMFS.character.latex_name + "\)", WMFS.character.url()))
friends = uniq(friends)
info['friends'] = friends
info['code'] = WMFS.code
return info
def set_info_for_web_newform(level=None, weight=None, character=None, label=None, **kwds):
r"""
Set the info for on modular form.
"""
info = to_dict(kwds)
info['level'] = level
info['weight'] = weight
info['character'] = character
info['label'] = label
if level is None or weight is None or character is None or label is None:
s = "In set info for one form but do not have enough args!"
s += "level={0},weight={1},character={2},label={3}".format(level, weight, character, label)
emf_logger.critical(s)
emf_logger.debug("In set_info_for_one_mf: info={0}".format(info))
prec = my_get(info, 'prec', default_prec, int)
bprec = my_get(info, 'bprec', default_display_bprec, int)
emf_logger.debug("PREC: {0}".format(prec))
emf_logger.debug("BITPREC: {0}".format(bprec))
try:
WNF = WebNewForm_cached(level=level, weight=weight, character=character, label=label)
emf_logger.critical("defined webnewform for rendering!")
# if info.has_key('download') and info.has_key('tempfile'):
# WNF._save_to_file(info['tempfile'])
# info['filename']=str(weight)+'-'+str(level)+'-'+str(character)+'-'+label+'.sobj'
# return info
except IndexError as e:
WNF = None
info['error'] = e.message
url1 = url_for("emf.render_elliptic_modular_forms")
url2 = url_for("emf.render_elliptic_modular_forms", level=level)
url3 = url_for("emf.render_elliptic_modular_forms", level=level, weight=weight)
url4 = url_for("emf.render_elliptic_modular_forms", level=level, weight=weight, character=character)
bread = [(EMF_TOP, url1)]
bread.append(("of level %s" % level, url2))
bread.append(("weight %s" % weight, url3))
if int(character) == 0:
bread.append(("trivial character", url4))
else:
bread.append(("\( %s \)" % (WNF.character.latex_name), url4))
info['bread'] = bread
properties2 = list()
friends = list()
space_url = url_for('emf.render_elliptic_modular_forms',level=level, weight=weight, character=character)
friends.append(('\( S_{%s}(%s, %s)\)'%(WNF.weight, WNF.level, WNF.character.latex_name), space_url))
if hasattr(WNF.base_ring, "lmfdb_url") and WNF.base_ring.lmfdb_url:
friends.append(('Number field ' + WNF.base_ring.lmfdb_pretty, WNF.base_ring.lmfdb_url))
if hasattr(WNF.coefficient_field, "lmfdb_url") and WNF.coefficient_field.lmfdb_label:
friends.append(('Number field ' + WNF.coefficient_field.lmfdb_pretty, WNF.coefficient_field.lmfdb_url))
friends = uniq(friends)
friends.append(("Dirichlet character \(" + WNF.character.latex_name + "\)", WNF.character.url()))
if WNF.dimension==0:
info['error'] = "This space is empty!"
# emf_logger.debug("WNF={0}".format(WNF))
#info['name'] = name
info['title'] = 'Modular Form ' + WNF.hecke_orbit_label
if 'error' in info:
return info
# info['name']=WNF._name
## Until we have figured out how to do the embeddings correctly we don't display the Satake
## parameters for non-trivial characters....
cdeg = WNF.coefficient_field.absolute_degree()
bdeg = WNF.base_ring.absolute_degree()
if cdeg == 1:
rdeg = 1
else:
rdeg = WNF.coefficient_field.relative_degree()
cf_is_QQ = (cdeg == 1)
br_is_QQ = (bdeg == 1)
if cf_is_QQ:
info['satake'] = WNF.satake
info['qexp'] = WNF.q_expansion_latex(prec=10, name='\\alpha ')
info['qexp_display'] = url_for(".get_qexp_latex", level=level, weight=weight, character=character, label=label)
info['max_cn_qexp'] = WNF.q_expansion.prec()
if not cf_is_QQ:
if not br_is_QQ and rdeg>1: # not WNF.coefficient_field == WNF.base_ring:
p1 = WNF.coefficient_field.relative_polynomial()
c_pol_ltx = web_latex_poly(p1, '\\alpha') # make the variable \alpha
c_pol_ltx_x = web_latex_poly(p1, 'x')
zeta = p1.base_ring().gens()[0]
# p2 = zeta.minpoly() #this is not used anymore
# b_pol_ltx = web_latex_poly(p2, latex(zeta)) #this is not used anymore
z1 = zeta.multiplicative_order()
info['coeff_field'] = [ WNF.coefficient_field.absolute_polynomial_latex('x'),c_pol_ltx_x, z1]
if hasattr(WNF.coefficient_field, "lmfdb_url") and WNF.coefficient_field.lmfdb_url:
info['coeff_field_pretty'] = [ WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty, WNF.coefficient_field.lmfdb_label]
if z1==4:
info['polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\) and \(\zeta_4=i\).</div><br/>'.format(c_pol_ltx)
elif z1<=2:
info['polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\).</div><br/>'.format(c_pol_ltx)
else:
info['polynomial_st'] = '<div class="where">where</div> %s\(\mathstrut=0\) and \(\zeta_{%s}=e^{\\frac{2\\pi i}{%s}}\).'%(c_pol_ltx, z1,z1)
else:
p1 = WNF.coefficient_field.relative_polynomial()
c_pol_ltx = web_latex_poly(p1, '\\alpha')
c_pol_ltx_x = web_latex_poly(p1, 'x')
z1 = p1.base_ring().gens()[0].multiplicative_order()
info['coeff_field'] = [ WNF.coefficient_field.absolute_polynomial_latex('x'), c_pol_ltx_x, z1]
if hasattr(WNF.coefficient_field, "lmfdb_url") and WNF.coefficient_field.lmfdb_url:
info['coeff_field_pretty'] = [ WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty, WNF.coefficient_field.lmfdb_label]
if z1==4:
info['polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\) and \(\zeta_4=i\).'.format(c_pol_ltx)
elif z1<=2:
info['polynomial_st'] = '<div class="where">where</div> {0}\(\mathstrut=0\).</div><br/>'.format(c_pol_ltx)
else:
info['polynomial_st'] = '<div class="where">where</div> %s\(\mathstrut=0\) and \(\zeta_{%s}=e^{\\frac{2\\pi i}{%s}}\).'%(c_pol_ltx, z1,z1)
else:
info['polynomial_st'] = ''
info['degree'] = int(cdeg)
if cdeg==1:
info['is_rational'] = 1
info['coeff_field_pretty'] = [ WNF.coefficient_field.lmfdb_url, WNF.coefficient_field.lmfdb_pretty ]
else:
info['is_rational'] = 0
# info['q_exp_embeddings'] = WNF.print_q_expansion_embeddings()
# if(int(info['degree'])>1 and WNF.dimension()>1):
# s = 'One can embed it into \( \mathbb{C} \) as:'
# bprec = 26
# print s
# info['embeddings'] = ajax_more2(WNF.print_q_expansion_embeddings,{'prec':[5,10,25,50],'bprec':[26,53,106]},text=['more coeffs.','higher precision'])
# elif(int(info['degree'])>1):
# s = 'There are '+str(info['degree'])+' embeddings into \( \mathbb{C} \):'
# bprec = 26
# print s
# info['embeddings'] = ajax_more2(WNF.print_q_expansion_embeddings,{'prec':[5,10,25,50],'bprec':[26,53,106]},text=['more coeffs.','higher precision'])
# else:
# info['embeddings'] = ''
emf_logger.debug("PREC2: {0}".format(prec))
info['embeddings'] = WNF._embeddings['values'] #q_expansion_embeddings(prec, bprec,format='latex')
info['embeddings_len'] = len(info['embeddings'])
properties2 = []
if (ZZ(level)).is_squarefree():
info['twist_info'] = WNF.twist_info
if isinstance(info['twist_info'], list) and len(info['twist_info'])>0:
info['is_minimal'] = info['twist_info'][0]
if(info['twist_info'][0]):
s = 'Is minimal<br>'
else:
s = 'Is a twist of lower level<br>'
properties2 = [('Twist info', s)]
else:
info['twist_info'] = 'Twist info currently not available.'
properties2 = [('Twist info', 'not available')]
args = list()
for x in range(5, 200, 10):
args.append({'digits': x})
alev = None
CM = WNF._cm_values
if CM is not None:
if CM.has_key('tau') and len(CM['tau']) != 0:
info['CM_values'] = CM
info['is_cm'] = WNF.is_cm
if WNF.is_cm:
info['cm_field'] = "2.0.{0}.1".format(-WNF.cm_disc)
info['cm_disc'] = WNF.cm_disc
info['cm_field_knowl'] = nf_display_knowl(info['cm_field'], getDBConnection(), field_pretty(info['cm_field']))
info['cm_field_url'] = url_for("number_fields.by_label", label=info["cm_field"])
if WNF.is_cm is None:
s = '- Unknown (insufficient data)<br>'
elif WNF.is_cm:
s = 'Is a CM-form<br>'
else:
s = 'Is not a CM-form<br>'
properties2.append(('CM info', s))
alev = WNF.atkin_lehner_eigenvalues()
info['atkinlehner'] = None
if isinstance(alev,dict) and len(alev.keys())>0 and level != 1:
s1 = " Atkin-Lehner eigenvalues "
s2 = ""
for Q in alev.keys():
s2 += "\( \omega_{ %s } \) : %s <br>" % (Q, alev[Q])
properties2.append((s1, s2))
emf_logger.debug("properties={0}".format(properties2))
# alev = WNF.atkin_lehner_eigenvalues_for_all_cusps()
# if isinstance(alev,dict) and len(alev.keys())>0:
# emf_logger.debug("alev={0}".format(alev))
# info['atkinlehner'] = list()
# for Q in alev.keys():
# s = "\(" + latex(c) + "\)"
# Q = alev[c][0]
# ev = alev[c][1]
# info['atkinlehner'].append([Q, c, ev])
if(level == 1):
poly = WNF.explicit_formulas.get('as_polynomial_in_E4_and_E6','')
if poly != '':
d,monom,coeffs = poly
emf_logger.critical("poly={0}".format(poly))
info['explicit_formulas'] = '\('
for i in range(len(coeffs)):
c = QQ(coeffs[i])
s = ""
if d>1 and i >0 and c>0:
s="+"
if c<0:
s="-"
if c.denominator()>1:
cc = "\\frac{{ {0} }}{{ {1} }}".format(abs(c.numerator()),c.denominator())
else:
cc = str(abs(c))
s += "{0} \cdot ".format(cc)
a = monom[i][0]; b = monom[i][1]
if a == 0 and b != 0:
s+="E_6^{{ {0} }}".format(b)
elif b ==0 and a != 0:
s+="E_4^{{ {0} }}".format(a)
else:
s+="E_4^{{ {0} }}E_6^{{ {1} }}".format(a,b)
info['explicit_formulas'] += s
info['explicit_formulas'] += " \)"
cur_url = '?&level=' + str(level) + '&weight=' + str(weight) + '&character=' + str(character) + \
'&label=' + str(label)
if len(WNF.parent.hecke_orbits) > 1:
for label_other in WNF.parent.hecke_orbits.keys():
if(label_other != label):
s = 'Modular form '
if character:
s = s + str(level) + '.' + str(weight) + '.' + str(character) + str(label_other)
else:
s = s + str(level) + '.' + str(weight) + str(label_other)
url = url_for('emf.render_elliptic_modular_forms', level=level,
weight=weight, character=character, label=label_other)
friends.append((s, url))
s = 'L-Function '
if character:
s = s + str(level) + '.' + str(weight) + '.' + str(character) + str(label)
else:
s = s + str(level) + '.' + str(weight) + str(label)
# url =
# "/L/ModularForm/GL2/Q/holomorphic?level=%s&weight=%s&character=%s&label=%s&number=%s"
# %(level,weight,character,label,0)
url = '/L' + url_for(
'emf.render_elliptic_modular_forms', level=level, weight=weight, character=character, label=label)
if WNF.coefficient_field_degree > 1:
for h in range(WNF.coefficient_field_degree):
s0 = s + ".{0}".format(h)
url0 = url + "{0}/".format(h)
friends.append((s0, url0))
else:
friends.append((s, url))
# if there is an elliptic curve over Q associated to self we also list that
if WNF.weight == 2 and WNF.coefficient_field_degree == 1:
llabel = str(level) + '.' + label
s = 'Elliptic curve isogeny class ' + llabel
url = '/EllipticCurve/Q/' + llabel
friends.append((s, url))
info['properties2'] = properties2
info['friends'] = friends
info['max_cn'] = WNF.max_cn()
return info
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()
emf_logger.debug("info={0}".format(info))
WMFS = None
if info.has_key('error'):
return info
if level <= 0:
info['error'] = "Got wrong level: %s " % level
return info
try:
WMFS = WebModFormSpace_cached(level = level, weight = weight, cuspidal=True, character = character, update_from_db=True)
if not WMFS.has_updated():
WMFS.update_from_db() ## Need to call an extra time for some reason...
if not WMFS.has_updated():
stop = False
orbit = WMFS.character.character.galois_orbit()
while not stop:
if len(orbit) == 0:
stop = True
continue
c = orbit.pop()
if c.number() == WMFS.character.number:
continue
print c.number()
WMFS_rep = WebModFormSpace_cached(level = level, weight = weight, cuspidal=True, character = c.number(), update_from_db=True)
if WMFS_rep.has_updated_from_db():
stop = True
info['wmfs_rep_url'] = url_for('emf.render_elliptic_modular_forms', level=level, weight=weight, character = c.number())
info['wmfs_rep_number'] = c.number()
emf_logger.debug("Created WebModFormSpace %s"%WMFS)
if 'download' in info and 'tempfile' in info:
save(WNF,info['tempfile'])
info['filename'] = str(weight) + '-' + str(level) + '-' + str(character) + '-' + label + '.sobj'
return info
except ValueError as e:
emf_logger.debug(e)
emf_logger.debug(e.message)
#if isinstance(e,IndexError):
info['error'] = e.message
WMFS = None
if WMFS is None:
info['error'] = "We are sorry. The sought space can not be found in the database. "+"<br> Detailed information: {0}".format(info.get('error',''))
return info
else:
### Somehow the Hecke orbits are sometimes not in the space...
#if WMFS.dimension_new_cusp_forms()!=len(WMFS.hecke_orbits):
# ## Try to add them here...
# for d in range(len(WMFS.dimension_new_cusp_forms())):
# F =
# WMFS.hecke_orbits.append(F)
info['space'] = WMFS
info['max_height'] = default_max_height
# info['old_decomposition'] = WMFS.oldspace_decomposition()
info['oldspace_decomposition']=''
try:
emf_logger.debug("Oldspace = {0}".format(WMFS.oldspace_decomposition))
if WMFS.oldspace_decomposition != []:
emf_logger.debug("oldspace={0}".format(WMFS.oldspace_decomposition))
l = []
for t in WMFS.oldspace_decomposition:
emf_logger.debug("t={0}".format(t))
N,k,chi,mult,d = t
url = url_for('emf.render_elliptic_modular_forms', level=N, weight=k, character=chi)
if chi != 1:
sname = "S^{{ new }}_{{ {k} }}(\\Gamma_0({N}),\\chi_{{ {N} }}({chi},\\cdot))".format(k=k,N=N,chi=chi)
else:
sname = "S^{{ new }}_{{ {k} }}(\\Gamma_0({N}))".format(k=k,N=N)
l.append("\href{{ {url} }}{{ {sname} }}^{{\oplus {mult} }}".format(sname=sname,mult=mult,url=url))
if l != []:
s = "\\oplus ".join(l)
info['oldspace_decomposition']=' $ {0} $'.format(s)
except Exception as e:
emf_logger.critical("Oldspace decomposition failed. Error:{0}".format(e))
## For side-bar
lifts = list()
lifts.append(('Half-Integral Weight Forms', '/ModularForm/Mp2/Q'))
lifts.append(('Siegel Modular Forms', '/ModularForm/GSp4/Q'))
info['lifts'] = lifts
friends = list()
for label in WMFS.hecke_orbits:
f = WMFS.hecke_orbits[label]
# catch the url being None or set to '':
if hasattr(f.base_ring, "lmfdb_url") and f.base_ring.lmfdb_url:
friends.append(('Number field ' + f.base_ring.lmfdb_pretty, f.base_ring.lmfdb_url))
if hasattr(f.coefficient_field, "lmfdb_url") and f.coefficient_field.lmfdb_url:
friends.append(('Number field ' + f.coefficient_field.lmfdb_pretty, f.coefficient_field.lmfdb_url))
friends.append(("Dirichlet character \(" + WMFS.character.latex_name + "\)", WMFS.character.url()))
friends = uniq(friends)
info['friends'] = friends
return info
