def CMorbits(D):
s = sum([
2**(len(prime_divisors(old_div(D, d))))
for d in divisors(D) if is_fundamental_discriminant(-d) and Zmod(d)
(old_div(D, d)).is_square()
]) + 1
return s
def class_nr_pos_def_qf(D):
r"""
Compute the class number of positive definite quadratic forms.
For fundamental discriminants this is the class number of Q(sqrt(D)),
otherwise it is computed using: Cohen 'A course in Computational Algebraic Number Theory', p. 233
"""
if D>0:
return 0
D4 = D % 4
if D4 == 3 or D4==2:
return 0
K = QuadraticField(D)
if is_fundamental_discriminant(D):
return K.class_number()
else:
D0 = K.discriminant()
Df = ZZ(D).divide_knowing_divisible_by(D0)
if not is_square(Df):
raise ArithmeticError("Did not get a discrimimant * square! D={0} disc(D)={1}".format(D,D0))
D2 = sqrt(Df)
h0 = QuadraticField(D0).class_number()
w0 = _get_w(D0)
w = _get_w(D)
#print "w,w0=",w,w0
#print "h0=",h0
h = 1
for p in prime_divisors(D2):
h = QQ(h)*(1-kronecker(D0,p)/QQ(p))
#print "h=",h
#print "fak=",
h=QQ(h*h0*D2*w)/QQ(w0)
return h
def get_atkin_lehner_eigenvalues(self):
r"""
Get the Atkin-Lehner eigenvalues from database if they exist.
"""
if not ((self.character().is_trivial() or self.character().order() == 2) and not self._atkin_lehner_eigenvalues is None):
return None
C = connect_to_modularforms_db('Atkin-Lehner.files')
fs = get_files_from_gridfs('Atkin-Lehner')
s = {'N':int(self.level()),'k':int(self.weight()),
'cchi':int(self.chi()),'newform':int(self.newform_number())}
res = C.find_one(s)
self._atkin_lehner_eigenvalues = {}
if not res is None:
alid = res['_id']
al = loads(fs.get(alid).read())
for d in prime_divisors(self.level()):
self._atkin_lehner_eigenvalues[d] = 1 if al[d]=='+' else -1
else: # We compute them
A = self.as_factor()
for p in prime_divisors(self.level()):
self._atkin_lehner_eigenvalues[p]= int(A.atkin_lehner_operator(p).matrix()[0,0])
def bad_formal_immersion_data(d):
"""
This is the Oesterlé for type 1 primes with modular symbols main routine.
The computation is actually a two step rocket. First Proposition 6.8 of
Derickx-Kamienny-Stein-Stoll is used to replace Parents polynomial of
degree 6 bound by something reasonable, and then Proposition 6.3 is used
to go from something reasonable to the exact list.
"""
assert d > 0
p_bad = prime_range(11)
p_done = {}
q_to_bad_p = {}
M = construct_M(d)[0]
for p in prime_range(11, 2 * M * d):
# first do a relatively cheap test
if is_formal_immersion_fast(d, p):
continue
# this is less cheap but still quite fast
if is_formal_immersion_medium(d, p) == 1:
continue
# this is the most expensive but give the correct answer
is_formal = is_formal_immersion(d, p)
if is_formal:
if is_formal > 1:
p_done[p] = is_formal
else:
p_bad.append(p)
for p, q_prod in p_done.items():
for q in prime_divisors(q_prod):
q_to_bad_p[q] = q_to_bad_p.get(q, 1) * p
return p_bad, q_to_bad_p
def lfuncEPhtml(L,fmt):
""" Euler product as a formula and a table of local factors.
"""
texform_gen = "\[L(s) = " # "\[L(A,s) = "
texform_gen += "\prod_{p \\text{ prime}} F_p(p^{-s})^{-1} \]\n"
pfactors = prime_divisors(L.level)
if len(pfactors) == 1: #i.e., the conductor is prime
pgoodset = "$p \\neq " + str(pfactors[0]) + "$"
pbadset = "$p = " + str(pfactors[0]) + "$"
else:
badset = "\\{" + str(pfactors[0])
for j in range(1,len(pfactors)):
badset += ",\\;"
badset += str(pfactors[j])
badset += "\\}"
pgoodset = "$p \\notin " + badset + "$"
pbadset = "$p \\in " + badset + "$"
ans = ""
ans += texform_gen + "where, for " + pgoodset + ",\n"
if L.degree == 4 and L.motivic_weight == 1:
ans += "\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]"
ans += "with $b_p = a_p^2 - a_{p^2}$. "
elif L.degree == 2 and L.motivic_weight == 1:
ans += "\[F_p(T) = 1 - a_p T + p T^2 .\]"
else:
ans += "\(F_p\) is a polynomial of degree " + str(L.degree) + ". "
ans += "If " + pbadset + ", then $F_p$ is a polynomial of degree at most "
ans += str(L.degree - 1) + ". "
bad_primes = []
for lf in L.bad_lfactors:
bad_primes.append(lf[0])
eulerlim = 25
good_primes = []
for j in range(0, eulerlim):
this_prime = Primes().unrank(j)
if this_prime not in bad_primes:
good_primes.append(this_prime)
eptable = "<table id='eptable' class='ntdata euler'>\n"
eptable += "<thead>"
eptable += "<tr class='space'><th class='weight'></th><th class='weight'>$p$</th><th class='weight'>$F_p$</th>"
if L.degree > 2:
eptable += "<th class='weight galois'>$\Gal(F_p)$</th>"
eptable += "</tr>\n"
eptable += "</thead>"
goodorbad = "bad"
for lf in L.bad_lfactors:
try:
thispolygal = list_to_factored_poly_otherorder(lf[1], galois=True)
eptable += ("<tr><td>" + goodorbad + "</td><td>" + str(lf[0]) + "</td><td>" +
"$" + thispolygal[0] + "$" +
"</td>")
if L.degree > 2:
eptable += "<td class='galois'>"
this_gal_group = thispolygal[1]
if this_gal_group[0]==[0,0]:
pass # do nothing, because the local faco is 1
elif this_gal_group[0]==[1,1]:
eptable += group_display_knowl(this_gal_group[0][0],this_gal_group[0][1],'$C_1$')
else:
eptable += group_display_knowl(this_gal_group[0][0],this_gal_group[0][1])
for j in range(1,len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],this_gal_group[j][1])
eptable += "</td>"
eptable += "</tr>\n"
except IndexError:
eptable += "<tr><td></td><td>" + str(j) + "</td><td>" + "not available" + "</td></tr>\n"
goodorbad = ""
goodorbad = "good"
firsttime = " class='first'"
good_primes1 = good_primes[:9]
good_primes2 = good_primes[9:]
for j in good_primes1:
this_prime_index = prime_pi(j) - 1
thispolygal = list_to_factored_poly_otherorder(L.localfactors[this_prime_index],galois=True)
eptable += ("<tr" + firsttime + "><td>" + goodorbad + "</td><td>" + str(j) + "</td><td>" +
"$" + thispolygal[0] + "$" +
"</td>")
if L.degree > 2:
eptable += "<td class='galois'>"
this_gal_group = thispolygal[1]
eptable += group_display_knowl(this_gal_group[0][0],this_gal_group[0][1])
for j in range(1,len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],this_gal_group[j][1])
eptable += "</td>"
eptable += "</tr>\n"
# eptable += "<td>" + group_display_knowl(4,1) + "</td>"
# eptable += "</tr>\n"
goodorbad = ""
firsttime = ""
firsttime = " id='moreep'"
for j in good_primes2:
this_prime_index = prime_pi(j) - 1
thispolygal = list_to_factored_poly_otherorder(L.localfactors[this_prime_index],galois=True)
eptable += ("<tr" + firsttime + " class='more nodisplay'" + "><td>" + goodorbad + "</td><td>" + str(j) + "</td><td>" +
"$" + list_to_factored_poly_otherorder(L.localfactors[this_prime_index], galois=True)[0] + "$" +
"</td>")
if L.degree > 2:
this_gal_group = thispolygal[1]
eptable += "<td class='galois'>"
eptable += group_display_knowl(this_gal_group[0][0],this_gal_group[0][1])
for j in range(1,len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],this_gal_group[j][1])
eptable += "</td>"
eptable += "</tr>\n"
firsttime = ""
eptable += "<tr class='less toggle'><td></td><td></td><td> <a onclick='"
eptable += 'show_moreless("more"); return true' + "'"
eptable += ' href="#moreep" '
eptable += ">show more</a></td></tr>\n"
eptable += "<tr class='more toggle nodisplay'><td></td><td></td><td> <a onclick='"
eptable += 'show_moreless("less"); return true' + "'"
eptable += ' href="#eptable" '
eptable += ">show less</a></td></tr>\n"
eptable += "</table>\n"
ans += "\n" + eptable
return(ans)
def lfuncEPhtml(L,fmt, prec = None):
"""
Euler product as a formula and a table of local factors.
"""
# Formula
texform_gen = "\[L(s) = " # "\[L(A,s) = "
texform_gen += "\prod_{p \\text{ prime}} F_p(p^{-s})^{-1} \]\n"
pfactors = prime_divisors(L.level)
if len(pfactors) == 0:
pgoodset = None
pbadset = None
elif len(pfactors) == 1: #i.e., the conductor is prime
pgoodset = "$p \\neq " + str(pfactors[0]) + "$"
pbadset = "$p = " + str(pfactors[0]) + "$"
else:
badset = "\\{" + str(pfactors[0])
for j in range(1,len(pfactors)):
badset += ",\\;"
badset += str(pfactors[j])
badset += "\\}"
pgoodset = "$p \\notin " + badset + "$"
pbadset = "$p \\in " + badset + "$"
ans = ""
ans += texform_gen + "where"
if pgoodset is not None:
ans += ", for " + pgoodset
ans += ",\n"
if L.motivic_weight == 1 and L.characternumber == 1 and L.degree in [2,4]:
if L.degree == 4:
ans += "\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]"
ans += "with $b_p = a_p^2 - a_{p^2}$. "
elif L.degree == 2:
ans += "\[F_p(T) = 1 - a_p T + p T^2 .\]"
else:
ans += "\(F_p\) is a polynomial of degree " + str(L.degree) + ". "
if pbadset is not None:
ans += "If " + pbadset + ", then $F_p$ is a polynomial of degree at most "
ans += str(L.degree - 1) + ". "
# Figuring out good and bad primes
bad_primes = []
for lf in L.bad_lfactors:
bad_primes.append(lf[0])
eulerlim = 25
good_primes = []
for p in primes_first_n(eulerlim):
if p not in bad_primes:
good_primes.append(p)
#decide if we display galois
display_galois = True
if L.degree <= 2 or L.degree >= 12:
display_galois = False
if L.coefficient_field == "CDF":
display_galois = False
def pretty_poly(poly, prec = None):
out = "1"
for i,elt in enumerate(poly):
if elt is None or (i == prec and prec != len(poly) - 1):
out += "O(%s)" % (seriesvar(i, "polynomial"),)
break;
elif i > 0:
out += seriescoeff(elt, i, "series", "polynomial", 3)
return out
eptable = r"""<div style="max-width: 100%; overflow-x: auto;">"""
eptable += "<table class='ntdata euler'>\n"
eptable += "<thead>"
eptable += "<tr class='space'><th class='weight'></th><th class='weight'>$p$</th>"
if L.degree > 2 and L.degree < 12:
display_galois = True
eptable += "<th class='weight galois'>$\Gal(F_p)$</th>"
else:
display_galois = False
eptable += r"""<th class='weight' style="text-align: left;">$F_p$</th>"""
eptable += "</tr>\n"
eptable += "</thead>"
def row(trclass, goodorbad, p, poly):
out = ""
try:
if L.coefficient_field == "CDF" or None in poly:
factors = str(pretty_poly(poly, prec = prec))
elif not display_galois:
factors = list_to_factored_poly_otherorder(poly, galois=display_galois, prec = prec, p = p)
else:
factors, gal_groups = list_to_factored_poly_otherorder(poly, galois=display_galois, p = p)
out += "<tr" + trclass + "><td>" + goodorbad + "</td><td>" + str(p) + "</td>";
if display_galois:
out += "<td class='galois'>"
if gal_groups[0]==[0,0]:
pass # do nothing, because the local faco is 1
elif gal_groups[0]==[1,1]:
out += group_display_knowl(gal_groups[0][0], gal_groups[0][1],'$C_1$')
else:
out += group_display_knowl(gal_groups[0][0], gal_groups[0][1])
for n, k in gal_groups[1:]:
out += "$\\times$"
out += group_display_knowl(n, k)
out += "</td>"
out += "<td>" +"$" + factors + "$" + "</td>"
out += "</tr>\n"
except IndexError:
out += "<tr><td></td><td>" + str(j) + "</td><td>" + "not available" + "</td></tr>\n"
return out
goodorbad = "bad"
trclass = ""
for lf in L.bad_lfactors:
eptable += row(trclass, goodorbad, lf[0], lf[1])
goodorbad = ""
trclass = ""
goodorbad = "good"
trclass = " class='first'"
good_primes1 = good_primes[:9]
good_primes2 = good_primes[9:]
for j in good_primes1:
this_prime_index = prime_pi(j) - 1
eptable += row(trclass, goodorbad, j, L.localfactors[this_prime_index])
goodorbad = ""
trclass = ""
trclass = " id='moreep' class='more nodisplay'"
for j in good_primes2:
this_prime_index = prime_pi(j) - 1
eptable += row(trclass, goodorbad, j, L.localfactors[this_prime_index])
trclass = " class='more nodisplay'"
eptable += r"""<tr class="less toggle"><td colspan="2"> <a onclick="show_moreless("more"); return true" href="#moreep">show more</a></td>"""
last_entry = ""
if display_galois:
last_entry +="<td></td>"
last_entry += "<td></td>"
eptable += last_entry
eptable += "</tr>"
eptable += r"""<tr class="more toggle nodisplay"><td colspan="2"><a onclick="show_moreless("less"); return true" href="#eptable">show less</a></td>"""
eptable += last_entry
eptable += "</tr>\n</table>\n</div>\n"
ans += "\n" + eptable
return(ans)
def lfuncEPhtml(L, fmt):
"""
Euler product as a formula and a table of local factors.
"""
# Formula
texform_gen = r"\[L(s) = " # r"\[L(A,s) = "
texform_gen += r"\prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]" + "\n"
pfactors = prime_divisors(L.level)
if len(pfactors) == 0:
pgoodset = None
pbadset = None
elif len(pfactors) == 1: #i.e., the conductor is prime
pgoodset = r"$p \neq " + str(pfactors[0]) + "$"
pbadset = "$p = " + str(pfactors[0]) + "$"
else:
badset = r"\{" + str(pfactors[0])
for j in range(1,len(pfactors)):
badset += r",\;"
badset += str(pfactors[j])
badset += r"\}"
pgoodset = r"$p \notin " + badset + "$"
pbadset = r"$p \in " + badset + "$"
ans = ""
ans += texform_gen + "where"
if pgoodset is not None:
ans += ", for " + pgoodset
ans += ",\n"
if L.motivic_weight == 1 and L.characternumber == 1 and L.degree in [2,4]:
if L.degree == 4:
ans += r"\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]"
ans += "with $b_p = a_p^2 - a_{p^2}$. "
elif L.degree == 2:
ans += r"\[F_p(T) = 1 - a_p T + p T^2 .\]"
else:
ans += r"\(F_p(T)\) is a polynomial of degree " + str(L.degree) + ". "
if pbadset is not None:
ans += "If " + pbadset + ", then $F_p(T)$ is a polynomial of degree at most "
ans += str(L.degree - 1) + ". "
# Figuring out good and bad primes
bad_primes = [p for p, _ in L.bad_lfactors]
good_primes = [p for p in prime_range(100) if p not in bad_primes]
p_index = {p: i for i, p in enumerate(prime_range(100))}
#decide if we display galois
display_galois = True
if L.degree <= 2 or L.degree >= 12:
display_galois = False
elif L.coefficient_field == "CDF":
display_galois = False
elif all(None in elt for elt in (L.localfactors + L.bad_lfactors)):
display_galois = False
def pretty_poly(poly, prec=None):
out = "1"
for i, elt in enumerate(poly):
if elt is None or (i == prec and prec != len(poly) - 1):
out += "+O(%s)" % (seriesvar(i, "polynomial"),)
break
elif i > 0:
out += seriescoeff(elt, i, "series", "polynomial", 3)
return out
eptable = r"""<div style="max-width: 100%; overflow-x: auto;">"""
eptable += "<table class='ntdata euler'>\n"
eptable += "<thead>"
eptable += "<tr class='space'><th class='weight'></th><th class='weight'>$p$</th>"
if display_galois:
eptable += r"<th class='weight galois'>$\Gal(F_p)$</th>"
eptable += r"""<th class='weight' style="text-align: left;">$F_p(T)$</th>"""
eptable += "</tr>\n"
eptable += "</thead>"
def row(trclass, goodorbad, p, poly):
if isinstance(poly[0], list):
galois_pretty_factors = list_factored_to_factored_poly_otherorder
else:
galois_pretty_factors = list_to_factored_poly_otherorder
out = ""
try:
if L.coefficient_field == "CDF" or None in poly:
factors = r'\( %s \)' % pretty_poly(poly)
gal_groups = [[0,0]]
elif not display_galois:
factors = galois_pretty_factors(poly, galois=display_galois, p = p)
factors = make_bigint(r'\( %s \)' % factors)
else:
factors, gal_groups = galois_pretty_factors(poly, galois=display_galois, p = p)
factors = make_bigint(r'\( %s \)' % factors)
out += "<tr" + trclass + "><td>" + goodorbad + "</td><td>" + str(p) + "</td>"
if display_galois:
out += "<td class='galois'>"
if gal_groups[0]==[0,0]:
pass # do nothing, because the local factor is 1
else:
out += r"$\times$".join( [group_display_knowl_C1_as_trivial(n,k) for n, k in gal_groups] )
out += "</td>"
out += "<td> %s </td>" % factors
out += "</tr>\n"
except IndexError:
out += "<tr><td></td><td>" + str(j) + "</td><td>" + "not available" + "</td></tr>\n"
return out
goodorbad = "bad"
trclass = ""
for p, lf in L.bad_lfactors:
if p in L.localfactors_factored_dict:
lf = L.localfactors_factored_dict[p]
eptable += row(trclass, goodorbad, p, lf)
goodorbad = ""
trclass = ""
goodorbad = "good"
trclass = " class='first'"
for j, good_primes in enumerate([good_primes[:9], good_primes[9:]]):
for p in good_primes:
if p in L.localfactors_factored_dict:
lf = L.localfactors_factored_dict[p]
else:
lf = L.localfactors[p_index[p]]
eptable += row(trclass, goodorbad, p, lf)
goodorbad = ""
if j == 0:
trclass = ""
elif j == 1:
trclass = " class='more nodisplay'"
else:
if j == 0:
trclass = " id='moreep' class='more nodisplay'"
eptable += r"""<tr class="less toggle"><td colspan="2"> <a onclick="show_moreless("more"); return true" href="#moreep">show more</a></td>"""
last_entry = ""
if display_galois:
last_entry +="<td></td>"
last_entry += "<td></td>"
eptable += last_entry
eptable += "</tr>"
eptable += r"""<tr class="more toggle nodisplay"><td colspan="2"><a onclick="show_moreless("less"); return true" href="#eptable">show less</a></td>"""
eptable += last_entry
eptable += "</tr>\n</table>\n</div>\n"
ans += "\n" + eptable
return(ans)
def lfuncEPhtml(L, fmt):
""" Euler product as a formula and a table of local factors.
"""
texform_gen = "\[L(s) = " # "\[L(A,s) = "
texform_gen += "\prod_{p \\text{ prime}} F_p(p^{-s})^{-1} \]\n"
pfactors = prime_divisors(L.level)
if len(pfactors) == 1: #i.e., the conductor is prime
pgoodset = "$p \\neq " + str(pfactors[0]) + "$"
pbadset = "$p = " + str(pfactors[0]) + "$"
else:
badset = "\\{" + str(pfactors[0])
for j in range(1, len(pfactors)):
badset += ",\\;"
badset += str(pfactors[j])
badset += "\\}"
pgoodset = "$p \\notin " + badset + "$"
pbadset = "$p \\in " + badset + "$"
ans = ""
ans += texform_gen + "where, for " + pgoodset + ",\n"
if L.degree == 4 and L.motivic_weight == 1:
ans += "\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]"
ans += "with $b_p = a_p^2 - a_{p^2}$. "
elif L.degree == 2 and L.motivic_weight == 1:
ans += "\[F_p(T) = 1 - a_p T + p T^2 .\]"
else:
ans += "\(F_p\) is a polynomial of degree " + str(L.degree) + ". "
ans += "If " + pbadset + ", then $F_p$ is a polynomial of degree at most "
ans += str(L.degree - 1) + ". "
bad_primes = []
for lf in L.bad_lfactors:
bad_primes.append(lf[0])
eulerlim = 25
good_primes = []
for j in range(0, eulerlim):
this_prime = Primes().unrank(j)
if this_prime not in bad_primes:
good_primes.append(this_prime)
eptable = "<table id='eptable' class='ntdata euler'>\n"
eptable += "<thead>"
eptable += "<tr class='space'><th class='weight'></th><th class='weight'>$p$</th><th class='weight'>$F_p$</th>"
if L.degree > 2:
eptable += "<th class='weight galois'>$\Gal(F_p)$</th>"
eptable += "</tr>\n"
eptable += "</thead>"
goodorbad = "bad"
for lf in L.bad_lfactors:
try:
thispolygal = list_to_factored_poly_otherorder(lf[1], galois=True)
eptable += ("<tr><td>" + goodorbad + "</td><td>" + str(lf[0]) +
"</td><td>" + "$" + thispolygal[0] + "$" + "</td>")
if L.degree > 2:
eptable += "<td class='galois'>"
this_gal_group = thispolygal[1]
if this_gal_group[0] == [0, 0]:
pass # do nothing, because the local faco is 1
elif this_gal_group[0] == [1, 1]:
eptable += group_display_knowl(this_gal_group[0][0],
this_gal_group[0][1],
'$C_1$')
else:
eptable += group_display_knowl(this_gal_group[0][0],
this_gal_group[0][1])
for j in range(1, len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],
this_gal_group[j][1])
eptable += "</td>"
eptable += "</tr>\n"
except IndexError:
eptable += "<tr><td></td><td>" + str(
j) + "</td><td>" + "not available" + "</td></tr>\n"
goodorbad = ""
goodorbad = "good"
firsttime = " class='first'"
good_primes1 = good_primes[:9]
good_primes2 = good_primes[9:]
for j in good_primes1:
this_prime_index = prime_pi(j) - 1
thispolygal = list_to_factored_poly_otherorder(
L.localfactors[this_prime_index], galois=True)
eptable += ("<tr" + firsttime + "><td>" + goodorbad + "</td><td>" +
str(j) + "</td><td>" + "$" + thispolygal[0] + "$" +
"</td>")
if L.degree > 2:
eptable += "<td class='galois'>"
this_gal_group = thispolygal[1]
eptable += group_display_knowl(this_gal_group[0][0],
this_gal_group[0][1])
for j in range(1, len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],
this_gal_group[j][1])
eptable += "</td>"
eptable += "</tr>\n"
# eptable += "<td>" + group_display_knowl(4,1) + "</td>"
# eptable += "</tr>\n"
goodorbad = ""
firsttime = ""
firsttime = " id='moreep'"
for j in good_primes2:
this_prime_index = prime_pi(j) - 1
thispolygal = list_to_factored_poly_otherorder(
L.localfactors[this_prime_index], galois=True)
eptable += ("<tr" + firsttime + " class='more nodisplay'" + "><td>" +
goodorbad + "</td><td>" + str(j) + "</td><td>" + "$" +
list_to_factored_poly_otherorder(
L.localfactors[this_prime_index], galois=True)[0] +
"$" + "</td>")
if L.degree > 2:
this_gal_group = thispolygal[1]
eptable += "<td class='galois'>"
eptable += group_display_knowl(this_gal_group[0][0],
this_gal_group[0][1])
for j in range(1, len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],
this_gal_group[j][1])
eptable += "</td>"
eptable += "</tr>\n"
firsttime = ""
eptable += "<tr class='less toggle'><td></td><td></td><td> <a onclick='"
eptable += 'show_moreless("more"); return true' + "'"
eptable += ' href="#moreep" '
eptable += ">show more</a></td></tr>\n"
eptable += "<tr class='more toggle nodisplay'><td></td><td></td><td> <a onclick='"
eptable += 'show_moreless("less"); return true' + "'"
eptable += ' href="#eptable" '
eptable += ">show less</a></td></tr>\n"
eptable += "</table>\n"
ans += "\n" + eptable
return (ans)
def read_all(filename):
# label -> [p, Lp]
euler_factors_cc = {}
# label -> labels
orbits = {}
# label -> postgres row as list
rows = {}
instances = {}
base_dir = os.path.dirname(os.path.abspath(filename))
lfun_dir = os.path.join(base_dir, 'lfun')
linecount = line_count(filename)
check_all_files(filename, linecount)
k = 0
with open(filename, 'r') as F:
for line in F:
linesplit = line[:-1].split(':')
hoc, label, conrey_label_tmp, embedding_index_tmp, embedding_m, ap_txt = linesplit
level, weight, char_orbit, hecke_orbit, conrey_label, embedding_index = label.split(".")
assert conrey_label_tmp == conrey_label
assert embedding_index_tmp == embedding_index
mf_orbit_label = ".".join([level, weight, char_orbit, hecke_orbit])
if mf_orbit_label not in orbits:
orbits[mf_orbit_label] = []
orbits[mf_orbit_label].append(label)
ap_list = [ toCCC(*elt.split(',')) for elt in ap_txt[2:-2].split('],[')]
ap_list = zip(primes_first_n(len(ap_list)),ap_list)
lpfilename = os.path.join(lfun_dir, label + ".lpdata")
lffilename = os.path.join(lfun_dir, label + ".lpdata.lfunction")
if not all(map(os.path.exists, [lpfilename, lffilename])):
continue
level, weight, conrey_label, embedding_index = map(int, [level, weight, conrey_label, embedding_index])
G = DirichletGroup_conrey(level, CCC)
char = DirichletCharacter_conrey(G, conrey_label)
# the basis
row = constant_lf(level, weight, 2)
row['origin'] = "ModularForm/GL2/Q/holomorphic/%d/%d/%s/%s/%d/%d" % (level, weight, char_orbit, hecke_orbit, conrey_label, embedding_index)
row['self_dual'] = self_dual(char, ap_list)
row['central_character'] = "%s.%s" % (level, conrey_label)
# sets accuracy, root_number, sign_arg, leading_term, order_of_vanishing, positive_zeros, plot_delta, plot_values
for key, val in read_lfunction_file(lffilename).iteritems():
row[key] = val
if row['self_dual']:
row['conjugate'] = '\N'
else:
lfconjfilename= os.path.join(lfun_dir, label + ".lpdata.conj.lfunction")
assert os.path.exists(lfconjfilename)
row['conjugate'] = read_lfunction_file(lfconjfilename)['Lhash']
def euler_factor(p, ap):
if p.divides(level):
return [1, -ap]
charval = CCC(2*char.logvalue(p)).exppii()
if charval.contains_exact(ZZ(1)):
charval = 1
elif charval.contains_exact(ZZ(-1)):
charval = -1
return [1, -ap, (p**(weight-1))*charval]
cut = 30 if level == 1 else max(30, prime_pi(max(prime_divisors(level))))
euler_factors = [ euler_factor(*elt) for elt in ap_list[:cut] ]
bad_euler_factors = [ [elt[0], euler_factor(*elt)] for elt in ap_list if elt[0].divides(level)]
euler_factors_cc[label] = euler_factors
row['euler_factors'] = map( lambda x : map(CBF_to_pair, x), euler_factors)
row['bad_lfactors'] = map( lambda x: [int(x[0]), map(CBF_to_pair, x[1])], bad_euler_factors)
row['coefficient_field'] = 'CDF'
for i, ai in enumerate(dirichlet(CCC, euler_factors[:11])[2:12]):
if i + 2 <= 10:
row['a' + str(i+2)] = CBF_to_pair(ai)
for key in schema_lf:
assert key in row, "%s not in row = %s" % (key, row)
assert len(row) == len(schema_lf), "%s != %s" % (len(row) , len(schema_lf))
rows[label] = [row[key] for key in schema_lf]
instances[label] = (row['origin'], row['Lhash'], 'CMF')
k += 1
if linecount > 10:
if (k % (linecount//10)) == 0:
print "read_all %.2f%% done" % (k*100./linecount)
print "read_all Done"
rational_rows = populate_rational_rows(orbits, euler_factors_cc, rows, instances)
positive_zeros = schema_lf_dict['positive_zeros']
for elt, val in rows.iteritems():
assert isinstance(val[positive_zeros], str), "%s, %s, %s" % (elt, type(val[positive_zeros]), val[positive_zeros])
lfun_filename = filename + ".lfunctions"
instances_filename = filename + ".instances"
export_lfunctions(rows, rational_rows, instances, lfun_filename, instances_filename)
return 0
def lfuncEPhtml(L, fmt, prec=None):
"""
Euler product as a formula and a table of local factors.
"""
# Formula
texform_gen = "\[L(s) = " # "\[L(A,s) = "
texform_gen += "\prod_{p \\text{ prime}} F_p(p^{-s})^{-1} \]\n"
pfactors = prime_divisors(L.level)
if len(pfactors) == 0:
pgoodset = None
pbadset = None
elif len(pfactors) == 1: #i.e., the conductor is prime
pgoodset = "$p \\neq " + str(pfactors[0]) + "$"
pbadset = "$p = " + str(pfactors[0]) + "$"
else:
badset = "\\{" + str(pfactors[0])
for j in range(1, len(pfactors)):
badset += ",\\;"
badset += str(pfactors[j])
badset += "\\}"
pgoodset = "$p \\notin " + badset + "$"
pbadset = "$p \\in " + badset + "$"
ans = ""
ans += texform_gen + "where"
if pgoodset is not None:
ans += ", for " + pgoodset
ans += ",\n"
if L.motivic_weight == 1 and L.characternumber == 1 and L.degree in [2, 4]:
if L.degree == 4:
ans += "\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]"
ans += "with $b_p = a_p^2 - a_{p^2}$. "
elif L.degree == 2:
ans += "\[F_p(T) = 1 - a_p T + p T^2 .\]"
else:
ans += "\(F_p\) is a polynomial of degree " + str(L.degree) + ". "
if pbadset is not None:
ans += "If " + pbadset + ", then $F_p$ is a polynomial of degree at most "
ans += str(L.degree - 1) + ". "
# Figuring out good and bad primes
bad_primes = []
for lf in L.bad_lfactors:
bad_primes.append(lf[0])
eulerlim = 25
good_primes = []
for p in primes_first_n(eulerlim):
if p not in bad_primes:
good_primes.append(p)
#decide if we display galois
display_galois = True
if L.degree <= 2 or L.degree >= 12:
display_galois = False
if L.coefficient_field == "CDF":
display_galois = False
def pretty_poly(poly, prec=None):
out = "1"
for i, elt in enumerate(poly):
if elt is None or (i == prec and prec != len(poly) - 1):
out += "O(%s)" % (seriesvar(i, "polynomial"), )
break
elif i > 0:
out += seriescoeff(elt, i, "series", "polynomial", 3)
return out
eptable = r"""<div style="max-width: 100%; overflow-x: auto;">"""
eptable += "<table class='ntdata euler'>\n"
eptable += "<thead>"
eptable += "<tr class='space'><th class='weight'></th><th class='weight'>$p$</th>"
if L.degree > 2 and L.degree < 12:
display_galois = True
eptable += "<th class='weight galois'>$\Gal(F_p)$</th>"
else:
display_galois = False
eptable += r"""<th class='weight' style="text-align: left;">$F_p$</th>"""
eptable += "</tr>\n"
eptable += "</thead>"
def row(trclass, goodorbad, p, poly):
out = ""
try:
if L.coefficient_field == "CDF" or None in poly:
factors = str(pretty_poly(poly, prec=prec))
elif not display_galois:
factors = list_to_factored_poly_otherorder(
poly, galois=display_galois, prec=prec, p=p)
else:
factors, gal_groups = list_to_factored_poly_otherorder(
poly, galois=display_galois, p=p)
out += "<tr" + trclass + "><td>" + goodorbad + "</td><td>" + str(
p) + "</td>"
if display_galois:
out += "<td class='galois'>"
if gal_groups[0] == [0, 0]:
pass # do nothing, because the local faco is 1
elif gal_groups[0] == [1, 1]:
out += group_display_knowl(gal_groups[0][0],
gal_groups[0][1], '$C_1$')
else:
out += group_display_knowl(gal_groups[0][0],
gal_groups[0][1])
for n, k in gal_groups[1:]:
out += "$\\times$"
out += group_display_knowl(n, k)
out += "</td>"
out += "<td>" + "$" + factors + "$" + "</td>"
out += "</tr>\n"
except IndexError:
out += "<tr><td></td><td>" + str(
j) + "</td><td>" + "not available" + "</td></tr>\n"
return out
goodorbad = "bad"
trclass = ""
for lf in L.bad_lfactors:
eptable += row(trclass, goodorbad, lf[0], lf[1])
goodorbad = ""
trclass = ""
goodorbad = "good"
trclass = " class='first'"
good_primes1 = good_primes[:9]
good_primes2 = good_primes[9:]
for j in good_primes1:
this_prime_index = prime_pi(j) - 1
eptable += row(trclass, goodorbad, j, L.localfactors[this_prime_index])
goodorbad = ""
trclass = ""
trclass = " id='moreep' class='more nodisplay'"
for j in good_primes2:
this_prime_index = prime_pi(j) - 1
eptable += row(trclass, goodorbad, j, L.localfactors[this_prime_index])
trclass = " class='more nodisplay'"
eptable += r"""<tr class="less toggle"><td colspan="2"> <a onclick="show_moreless("more"); return true" href="#moreep">show more</a></td>"""
last_entry = ""
if display_galois:
last_entry += "<td></td>"
last_entry += "<td></td>"
eptable += last_entry
eptable += "</tr>"
eptable += r"""<tr class="more toggle nodisplay"><td colspan="2"><a onclick="show_moreless("less"); return true" href="#eptable">show less</a></td>"""
eptable += last_entry
eptable += "</tr>\n</table>\n</div>\n"
ans += "\n" + eptable
return (ans)
def atkin_lehner_signs(A):
N = A.level()
return [A.atkin_lehner_operator(p).matrix()[0,0] for p in prime_divisors(N)]
