def spread(N):
l = range(1, N + 1)
max_c = count_c(l)
min_c = prime_pi(N)
if N == 1:
min_c = 1
return max_c - min_c
def make_one_euler_factor_db(E, p):
r"""
Returns the Euler factor at p from a database elliptic curve E.
"""
ld = [ld for ld in E['local_data'] if ld['p']==p]
if ld: # p is bad, we get ap from the stored local data:
ap = ld[0]['red']
if ap:
return [1,-ap]
else:
return [1]
# Now p is good and < 100 so we retrieve ap from the stored aplist:
ap = E['aplist'][prime_pi(p)-1] # rebase count from 1 to 0
return [1,-ap,int(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 check_hecke_ring_character_values_and_an(self, rec, verbose=False):
"""
check that hecke_ring_character_values has the correct format, depending on whether hecke_ring_cyclotomic_generator is set or not
check that an has length 100 and that each entry is either a list of integers of length hecke_ring_rank (if hecke_ring_cyclotomic_generator=0) or a list of pairs
check that ap has length pi(maxp) and that each entry is formatted correctly (as for an)
"""
# TIME about 4000s for full table
an = rec['an']
if len(an) != 100:
if verbose:
print("Length an", len(an))
return False
ap = rec['ap']
maxp = rec['maxp']
if len(ap) != prime_pi(maxp):
if verbose:
print("Length ap", len(ap), prime_pi(maxp))
return False
if maxp < 997:
if verbose:
print("Maxp", maxp)
return False
m = rec['hecke_ring_cyclotomic_generator']
d = rec['hecke_ring_rank']
def check_val(val):
if not isinstance(val, list):
return False
if m == 0:
return len(val) == d and all(isinstance(c, integer_types) for c in val)
else:
for pair in val:
if len(pair) != 2:
return False
if not isinstance(pair[0], integer_types):
return False
e = pair[1]
if not (isinstance(e, integer_types) and 0 <= 2*e < m):
return False
return True
if not all(check_val(a) for a in an):
if verbose:
for n, a in enumerate(an, 1):
if not check_val(a):
print("Check an failure (m=%s, d=%s)"%(m, d), n, a)
return False
if not all(check_val(a) for a in ap):
if verbose:
for p, a in zip(prime_range(maxp), ap):
if not check_val(a):
print("Check ap failure (m=%s, d=%s)"%(m, d), p, a)
return False
for p, a in zip(prime_range(100), ap):
if a != an[p-1]:
if verbose:
print("Match failure", p, a, an[p-1])
return False
if rec['char_orbit_index'] != 1:
if rec.get('hecke_ring_character_values') is None:
if verbose:
print("No hecke_ring_character_values")
return False
N = rec['level']
total_order = 1
for g, val in rec['hecke_ring_character_values']:
total_order *= mod(g, N).multiplicative_order()
if not check_val(val):
if verbose:
print("Bad character val (m=%s, d=%s)"%(m, d), g, val)
return False
success = (total_order == euler_phi(N))
if not success and verbose:
print("Generators failed", total_order, euler_phi(N))
return success
return True
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 primes_cc_bound(self):
return prime_pi(self.an_cc_bound)
def rational_euler_factors(traces, euler_factors_cc, level, weight):
dirichlet = [1] * 11
dirichlet[0] = 0
euler_factors = []
bad_lfactors = []
halfdegree = len(euler_factors_cc)
PS = PowerSeriesRing(ZZ, "X")
CCCx = PolynomialRing(CCC, "x")
todo = list(enumerate(primes_first_n(30)))
for p in sorted(ZZ(level).prime_divisors()):
p_index = prime_pi(p) - 1
if p_index >= 30:
todo.append((p_index, p))
for p_index, p in todo:
if p_index > len(euler_factors_cc[0]):
assert level % p == 0
bad_lfactors.append([int(p), [int(1)] + [None] * halfdegree])
continue
#try to guess the rest by multiplying them
roots = []
for lf in euler_factors_cc:
roots += reciprocal_roots(lf[p_index])
root_powers = [None] * (halfdegree + 1)
for j in range(1, halfdegree + 1):
try:
root_powers[j] = RRR(sum(map(lambda z: (z**j).real(),
roots))).unique_integer()
except ValueError:
root_powers = root_powers[:j]
break
partial_efzz = from_power_sums(root_powers)
efzz = map(
int, partial_efzz) + [None] * (halfdegree + 1 - len(partial_efzz))
if len(traces) > p:
if efzz[1] is None:
efzz[1] = int(-traces[p - 1])
else:
assert efzz[1] == -traces[p - 1]
# to check that from_power_sums is correct
ef = prod([CCCx(lf[p_index]) for lf in euler_factors_cc])
for j, elt in enumerate(ef.list()[:len(partial_efzz)]):
try:
efj = RRR(elt.real()).unique_integer()
except ValueError:
break
assert efj == efzz[j]
if level % p != 0:
sign = RRR(ef.list()[-1].real() /
p**((halfdegree) * (weight - 1))).unique_integer()
assert sign in [1, -1], "%s\n%s" % (RRR(
prod(lf[p_index][2]
for lf in euler_factors_cc).real()).unique_integer(), p**(
(halfdegree) * (weight - 1)))
efzz2 = [None] * halfdegree
for i, elt in enumerate(reversed(efzz[:-1])):
if elt is None:
efzz2[i] = None
else:
efzz2[i] = int(sign * p**((i + 1) * (weight - 1)) * elt)
efzz += efzz2
euler_factors.append(efzz)
else:
if None not in efzz:
k = len(efzz)
while efzz[k - 1] == 0 and k >= 1:
k -= 1
efzz = efzz[:k]
bad_lfactors.append([int(p), efzz])
if p_index < 30:
euler_factors.append(efzz)
if p < 11:
if p == 2:
foo = (1 / PS(efzz[:4])).padded_list(4)
for i in range(1, 4):
dirichlet[p**i] = foo[i]
elif p == 3:
foo = (1 / PS(efzz[:3])).padded_list(4)
for i in range(1, 3):
dirichlet[p**i] = foo[i]
else:
dirichlet[p] = -efzz[1] if len(efzz) > 1 else 0
assert dirichlet[p] == traces[
p - 1], "p = %s, ap = %s, tp = %s, efzz = %s" % (
p, dirichlet[p], traces[p - 1], efzz)
extend_multiplicatively(dirichlet)
assert len(euler_factors) == 30
return euler_factors, bad_lfactors, dirichlet
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 rational_euler_factors(euler_factors_cc, level, weight, an_list_bound = 11):
dirichlet = [1]*an_list_bound
dirichlet[0] = 0
euler_factors = []
bad_lfactors = []
halfdegree = len(euler_factors_cc)
PS = PowerSeriesRing(ZZ, "X")
CCCx = PolynomialRing(CCC, "x")
todo = list(enumerate(primes_first_n(30)))
for p in sorted(ZZ(level).prime_divisors()):
p_index = prime_pi(p) - 1
if p_index >= 30:
todo.append((p_index, p))
for p_index, p in todo:
if p_index >= len(euler_factors_cc[0]):
assert level % p == 0, "%s, %s, %s" % (level, weight, len(euler_factors_cc))
bad_lfactors.append([int(p), [int(1)] + [None]*halfdegree])
continue
#try to guess the rest by multiplying them
roots = []
for lf in euler_factors_cc:
roots += reciprocal_roots(lf[p_index])
root_powers = [None] * (halfdegree + 1)
for j in range(1,halfdegree + 1):
try:
root_powers[j] = RRR(sum( map(lambda z: (z**j).real(), roots) )).unique_integer()
except ValueError:
root_powers = root_powers[:j]
break
partial_efzz = from_power_sums(root_powers)
efzz = map(int, partial_efzz) + [None]*(halfdegree +1 - len(partial_efzz))
# to check that from_power_sums is correct
ef = prod([CCCx(lf[p_index]) for lf in euler_factors_cc])
for j, elt in enumerate(ef.list()[:len(partial_efzz)]):
try:
efj = int(RRR(elt.real()).unique_integer())
except ValueError:
#print j
#print RRR(elt.real())
#print p
#print "[%s]" % (", ".join(["[%s]" % (", ".join(map(print_CCC, lf[p_index]))) for ef in euler_factors_cc]))
#assert False
break;
assert efj == efzz[j], "%s, %s, %s, %s != %s" % (level, weight, len(euler_factors_cc), efj, efzz[j])
if (level % p) != 0:
sign = RRR(ef.list()[-1].real()/p**((halfdegree)*(weight - 1))).unique_integer()
assert sign in [1,-1], "%s\n%s" % (RRR(prod( lf[p_index][2] for lf in euler_factors_cc).real()).unique_integer(),p**((halfdegree)*(weight - 1)))
efzz2 = [None] * halfdegree
for i, elt in enumerate(reversed(efzz[:-1])):
if elt is None:
efzz2[i] = None
else:
efzz2[i] = int(sign*p**((i+1)*(weight - 1)) * elt)
efzz += efzz2
euler_factors.append(efzz)
else:
if None not in efzz:
k = len(efzz)
while efzz[k - 1] == 0 and k >= 1:
k -= 1
efzz = efzz[:k]
bad_lfactors.append([int(p), efzz])
if p_index < 30:
euler_factors.append(efzz)
if p < an_list_bound:
k = RR(an_list_bound).log(p).floor()+1
foo = (1/PS(efzz)).padded_list(k)
for i in range(1, k):
dirichlet[p**i] = foo[i]
extend_multiplicatively(dirichlet)
assert len(euler_factors) == 30, "%s, %s, %s, %s != 30" % (level, weight, len(euler_factors_cc), len(euler_factors))
return euler_factors, bad_lfactors, dirichlet
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 check_hecke_ring_character_values_and_an(self, rec, verbose=False):
"""
check that hecke_ring_character_values has the correct format, depending on whether hecke_ring_cyclotomic_generator is set or not
check that an has length 100 and that each entry is either a list of integers of length hecke_ring_rank (if hecke_ring_cyclotomic_generator=0) or a list of pairs
check that ap has length pi(maxp) and that each entry is formatted correctly (as for an)
"""
# TIME about 4000s for full table
an = rec['an']
if len(an) != 100:
if verbose:
print "Length an", len(an)
return False
ap = rec['ap']
maxp = rec['maxp']
if len(ap) != prime_pi(maxp):
if verbose:
print "Length ap", len(ap), prime_pi(maxp)
return False
if maxp < 997:
if verbose:
print "Maxp", maxp
return False
m = rec['hecke_ring_cyclotomic_generator']
d = rec['hecke_ring_rank']
def check_val(val):
if not isinstance(val, list):
return False
if m == 0:
return len(val) == d and all(isinstance(c, integer_types) for c in val)
else:
for pair in val:
if len(pair) != 2:
return False
if not isinstance(pair[0], integer_types):
return False
e = pair[1]
if not (isinstance(e, integer_types) and 0 <= 2*e < m):
return False
return True
if not all(check_val(a) for a in an):
if verbose:
for n, a in enumerate(an, 1):
if not check_val(a):
print "Check an failure (m=%s, d=%s)"%(m, d), n, a
return False
if not all(check_val(a) for a in ap):
if verbose:
for p, a in zip(prime_range(maxp), ap):
if not check_val(a):
print "Check ap failure (m=%s, d=%s)"%(m, d), p, a
return False
for p, a in zip(prime_range(100), ap):
if a != an[p-1]:
if verbose:
print "Match failure", p, a, an[p-1]
return False
if rec['char_orbit_index'] != 1:
if rec.get('hecke_ring_character_values') is None:
if verbose:
print "No hecke_ring_character_values"
return False
N = rec['level']
total_order = 1
for g, val in rec['hecke_ring_character_values']:
total_order *= mod(g, N).multiplicative_order()
if not check_val(val):
if verbose:
print "Bad character val (m=%s, d=%s)"%(m, d), g, val
return False
success = (total_order == euler_phi(N))
if not success and verbose:
print "Generators failed", total_order, euler_phi(N)
return success
return True
