def render_bmf_webpage(field_label, level_label, label_suffix):
label = "-".join([field_label, level_label, label_suffix])
credit = "John Cremona"
info = {}
title = "Bianchi cusp forms"
data = None
properties = []
friends = []
bread = get_bread()
try:
numeigs = request.args['numeigs']
numeigs = int(numeigs)
except:
numeigs = 20
info['numeigs'] = numeigs
try:
data = WebBMF.by_label(label, max_eigs=numeigs)
title = "Bianchi cusp form {} over {}".format(
data.short_label, field_pretty(data.field_label))
bread = get_bread([(field_pretty(data.field_label),
url_for(".render_bmf_field_dim_table_gl2",
field_label=data.field_label)),
(data.level_label,
url_for('.render_bmf_space_webpage',
field_label=data.field_label,
level_label=data.level_label)),
(data.short_label, '')])
properties = data.properties
friends = data.friends
info['downloads'] = [('Modular form to Magma',
url_for(".render_bmf_webpage_download",
field_label=field_label,
label_suffix=label_suffix,
level_label=level_label,
download_type='magma')),
('Eigenvalues to Sage',
url_for(".render_bmf_webpage_download",
field_label=field_label,
label_suffix=label_suffix,
level_label=level_label,
download_type='sage'))]
except ValueError:
flash_error("No Bianchi modular form in the database has label %s",
label)
return redirect(url_for(".index"))
return render_template(
"bmf-newform.html",
downloads=info["downloads"],
title=title,
credit=credit,
bread=bread,
data=data,
properties=properties,
friends=friends,
info=info,
learnmore=learnmore_list(),
KNOWL_ID="mf.bianchi.%s" % label,
)
def render_bmf_webpage(field_label, level_label, label_suffix):
label = "-".join([field_label, level_label, label_suffix])
credit = "John Cremona"
bread = []
info = {}
title = "Bianchi cusp forms"
data = None
properties = []
friends = []
bread = [('Modular forms', url_for('modular_forms')),
('Bianchi', url_for(".index"))]
try:
data = WebBMF.by_label(label)
title = "Bianchi cusp form {} over {}".format(
data.short_label, field_pretty(data.field_label))
bread = [('Modular forms', url_for('modular_forms')),
('Bianchi', url_for(".index")),
(field_pretty(data.field_label),
url_for(".render_bmf_field_dim_table_gl2",
field_label=data.field_label)),
(data.level_label,
url_for('.render_bmf_space_webpage',
field_label=data.field_label,
level_label=data.level_label)),
(data.short_label, '')]
properties = data.properties
friends = data.friends
except ValueError:
raise
info[
'err'] = "No Bianchi modular form in the database has label {}".format(
label)
return render_template("bmf-newform.html",
title=title,
credit=credit,
bread=bread,
data=data,
properties=properties,
friends=friends,
info=info,
learnmore=learnmore_list())
def render_bmf_webpage(field_label, level_label, label_suffix):
label = "-".join([field_label, level_label, label_suffix])
credit = "John Cremona"
bread = []
info = {}
title = "Bianchi cusp forms"
data = None
properties2 = []
friends = []
bread = [('Bianchi modular forms', url_for(".index"))]
try:
data = WebBMF.by_label(label)
title = "Bianchi cusp form {} over {}".format(data.short_label,field_pretty(data.field_label))
bread = [('Bianchi modular forms', url_for(".index")),
(field_pretty(data.field_label), url_for(".render_bmf_field_dim_table_gl2", field_label=data.field_label)),
(data.level_label, url_for('.render_bmf_space_webpage', field_label=data.field_label, level_label=data.level_label)),
(data.short_label, '')]
properties2 = data.properties2
friends = data.friends
except ValueError:
info['err'] = "No Bianchi modular form in the database has label {}".format(label)
return render_template("bmf-newform.html", title=title, credit=credit, bread=bread, data=data, properties2=properties2, friends=friends, info=info)
def download_bmf_sage(**args):
"""Generates the sage code for the user to obtain the BMF eigenvalues.
As in the HMF case, and unlike the website, we export *all* eigenvalues in
the database, not just 50, and not just those away from the level."""
label = "-".join([args['field_label'], args['level_label'], args['label_suffix']])
try:
f = WebBMF.by_label(label)
except ValueError:
return "Bianchi newform not found"
hecke_pol = f.hecke_poly_obj
hecke_eigs = f.hecke_eigs
F = WebNumberField(f.field_label)
K = f.field.K()
primes_in_K = [p for p,_ in zip(primes_iter(K),hecke_eigs)]
prime_gens = [p.gens_reduced() for p in primes_in_K]
outstr = '"""\n This code can be loaded, or copied and paste using cpaste, into Sage.\n'
outstr += ' It will load the data associated to the BMF, including\n'
outstr += ' the field, level, and Hecke and Atkin-Lehner eigenvalue data (if known).\n'
outstr += '"""\n\n'
outstr += 'P = PolynomialRing(QQ, "x")\nx = P.gen()\n'
outstr += 'g = P(' + str(F.coeffs()) + ')\n'
outstr += 'F = NumberField(g, "{}")\n'.format(K.gen())
outstr += '{} = F.gen()\n'.format(K.gen())
outstr += 'ZF = F.ring_of_integers()\n\n'
outstr += 'NN = ZF.ideal({})\n\n'.format(f.level.gens())
outstr += 'primes_array = [\n' + ','.join([str(st).replace(' ', '') for st in prime_gens]).replace('],[',
'],\\\n[') + ']\n'
outstr += 'primes = [ZF.ideal(I) for I in primes_array]\n\n'
Qx = PolynomialRing(QQ,'x')
if hecke_pol != 'x':
outstr += 'heckePol = P({})\n'.format(str((Qx(hecke_pol)).list()))
outstr += 'K = NumberField(heckePol, "z")\nz = K.gen()\n'
else:
outstr += 'heckePol = x\nK = QQ\ne = 1\n'
hecke_eigs_processed = [str(st).replace(' ', '') if st != 'not known' else '"not known"' for st in hecke_eigs]
outstr += '\nhecke_eigenvalues_array = [' + ', '.join(hecke_eigs_processed) + ']'
outstr += '\nhecke_eigenvalues = {}\n'
outstr += 'for i in range(len(hecke_eigenvalues_array)):\n hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]\n\n'
if f.have_AL:
AL_eigs = f.AL_table_data
outstr += 'AL_eigenvalues = {}\n'
for s in AL_eigs:
outstr += 'AL_eigenvalues[ZF.ideal(%s)] = %s\n' % (s[0],s[1])
else:
outstr += 'AL_eigenvalues ="not known"\n'
outstr += '\n# EXAMPLE:\n# pp = ZF.ideal(2).factor()[0][0]\n# hecke_eigenvalues[pp]\n'
return outstr
def download_bmf_magma(**args):
label = "-".join([args['field_label'], args['level_label'], args['label_suffix']])
try:
f = WebBMF.by_label(label)
except ValueError:
return "Bianchi newform not found"
hecke_pol = f.hecke_poly_obj
hecke_eigs = f.hecke_eigs
F = WebNumberField(f.field_label)
K = f.field.K()
primes_in_K = [p for p,_ in zip(primes_iter(K),hecke_eigs)]
prime_gens = [list(p.gens()) for p in primes_in_K]
outstr = '/*\n This code can be loaded, or copied and pasted, into Magma.\n'
outstr += ' It will load the data associated to the BMF, including\n'
outstr += ' the field, level, and Hecke and Atkin-Lehner eigenvalue data.\n'
outstr += ' At the *bottom* of the file, there is code to recreate the\n'
outstr += ' Bianchi modular form in Magma, by creating the BMF space\n'
outstr += ' and cutting out the corresponding Hecke irreducible subspace.\n'
outstr += ' From there, you can ask for more eigenvalues or modify as desired.\n'
outstr += ' It is commented out, as this computation may be lengthy.\n'
outstr += '*/\n\n'
outstr += 'P<x> := PolynomialRing(Rationals());\n'
outstr += 'g := P!' + str(F.coeffs()) + ';\n'
outstr += 'F<{}> := NumberField(g);\n'.format(K.gen())
outstr += 'ZF := Integers(F);\n\n'
outstr += 'NN := ideal<ZF | {}>;\n\n'.format(set(f.level.gens()))
outstr += 'primesArray := [\n' + ','.join([str(st).replace(' ', '') for st in prime_gens]).replace('],[',
'],\n[') + '];\n'
outstr += 'primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];\n\n'
if hecke_pol != 'x':
outstr += 'heckePol := ' + hecke_pol + ';\n'
outstr += 'K<z> := NumberField(heckePol);\n'
else:
outstr += 'heckePol := x;\nK := Rationals(); e := 1;\n'
hecke_eigs_processed = [str(st).replace(' ', '') if st != 'not known' else '"not known"' for st in hecke_eigs]
outstr += '\nheckeEigenvaluesList := [*\n'+ ',\n'.join(hecke_eigs_processed) + '\n*];\n'
outstr += '\nheckeEigenvalues := AssociativeArray();\n'
outstr += 'for i in [1..#heckeEigenvaluesList] do\n heckeEigenvalues[primes[i]] := heckeEigenvaluesList[i];\nend for;\n'
if f.have_AL:
AL_eigs = f.AL_table_data
outstr += '\nALEigenvalues := AssociativeArray();\n'
for s in AL_eigs:
outstr += 'ALEigenvalues[ideal<ZF | {}>] := {};\n'.format(set(s[0]), s[1])
else:
outstr += '\nALEigenvalues := "not known";\n'
outstr += '\n// EXAMPLE:\n// pp := Factorization(2*ZF)[1][1];\n// heckeEigenvalues[pp];\n\n'
outstr += '\n'.join([
'print "To reconstruct the Bianchi newform f, type',
' f, iso := Explode(make_newform());";',
'',
'function make_newform();',
' M := BianchiCuspForms(F, NN);',
' S := NewSubspace(M);',
' // SetVerbose("Bianchi", 1);',
' NFD := NewformDecomposition(S);',
' newforms := [* Eigenform(U) : U in NFD *];',
'',
' if #newforms eq 0 then;',
' print "No Bianchi newforms at this level";',
' return 0;',
' end if;',
'',
' print "Testing ", #newforms, " possible newforms";',
' newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *];',
' print #newforms, " newforms have the correct Hecke field";',
'',
' if #newforms eq 0 then;',
' print "No Bianchi newform found with the correct Hecke field";',
' return 0;',
' end if;',
'',
' autos := Automorphisms(K);',
' xnewforms := [* *];',
' for f in newforms do;',
' if K eq RationalField() then;',
' Append(~xnewforms, [* f, autos[1] *]);',
' else;',
' flag, iso := IsIsomorphic(K,BaseField(f));',
' for a in autos do;',
' Append(~xnewforms, [* f, a*iso *]);',
' end for;',
' end if;',
' end for;',
' newforms := xnewforms;',
'',
' for P in primes do;',
' if Valuation(NN,P) eq 0 then;',
' xnewforms := [* *];',
' for f_iso in newforms do;',
' f, iso := Explode(f_iso);',
' if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then;',
' Append(~xnewforms, f_iso);',
' end if;',
' end for;',
' newforms := xnewforms;',
' if #newforms eq 0 then;',
' print "No Bianchi newform found which matches the Hecke eigenvalues";',
' return 0;',
' else if #newforms eq 1 then;',
' print "success: unique match";',
' return newforms[1];',
' end if;',
' end if;',
' end if;',
' end for;',
' print #newforms, "Bianchi newforms found which match the Hecke eigenvalues";',
' return newforms[1];',
'',
'end function;'])
return outstr