def lfuncEPhtml(L, fmt):
""" Euler product as a formula and a table of local factors.
"""
ans = ""
texform_gen = "\[L(A,s) = "
texform_gen += "\prod_{p \\text{ prime}} f_p(p^{-s})^{-1} \]\n"
ans += texform_gen + "where, for $p\\nmid " + str(L.level) + "$,\n"
# ans += "\[f_p(T) = 1 - a_p T + b_p T^2 + \chi(p) a_p p T^3 + \chi(p) p^2 T^4 \]"
ans += "\[f_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]"
# ans += "with $b_p = a_{p^2} - a_p^2$ and $\chi$ is the central character of the L-function."
ans += "with $b_p = a_p^2 - a_{p^2}$. "
ans += "If $p \mid " + str(L.level) + "$, then $f_p$ is a polynomial of degree at most 3, "
ans += "with $f_p(0) = 1$."
factN = list(factor(L.level))
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></tr>\n"
)
eptable += "</thead>"
numfactors = len(L.localfactors)
goodorbad = "bad"
for lf in L.bad_lfactors:
try:
eptable += (
"<tr><td>"
+ goodorbad
+ "</td><td>"
+ str(lf[0])
+ "</td><td>"
+ "$"
+ list_to_factored_poly_otherorder(lf[1])
+ "$"
+ "</td></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
eptable += (
"<tr"
+ firsttime
+ "><td>"
+ goodorbad
+ "</td><td>"
+ str(j)
+ "</td><td>"
+ "$"
+ list_to_factored_poly_otherorder(L.localfactors[this_prime_index])
+ "$"
+ "</td></tr>\n"
)
goodorbad = ""
firsttime = ""
firsttime = " id='moreep'"
for j in good_primes2:
this_prime_index = prime_pi(j) - 1
eptable += (
"<tr"
+ firsttime
+ " class='more nodisplay'"
+ "><td>"
+ goodorbad
+ "</td><td>"
+ str(j)
+ "</td><td>"
+ "$"
+ list_to_factored_poly_otherorder(L.localfactors[this_prime_index])
+ "$"
+ "</td></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):
""" Euler product as a formula and a table of local factors.
"""
texform_gen = "\[L(A,s) = "
texform_gen += "\prod_{p \\text{ prime}} F_p(p^{-s})^{-1} \]\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 $p\\nmid " + str(L.level) + "$,\n"
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 $p \mid " + str(L.level) + "$, then $F_p$ is a polynomial of degree at most 3, "
# ans += "If " + pbadset + ", then $F_p$ is a polynomial of degree at most 3, "
ans += "If " + pbadset + ", then $F_p$ is a polynomial of degree at most "
ans += str(L.degree - 1) + ". "
# ans += "with $F_p(0) = 1$."
factN = list(factor(L.level))
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>"
numfactors = len(L.localfactors)
goodorbad = "bad"
C = getDBConnection()
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,'$C_1$')
else:
eptable += group_display_knowl(this_gal_group[0][0],this_gal_group[0][1],C)
for j in range(1,len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],this_gal_group[j][1],C)
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],C)
for j in range(1,len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],this_gal_group[j][1],C)
eptable += "</td>"
eptable += "</tr>\n"
# eptable += "<td>" + group_display_knowl(4,1,C) + "</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],C)
for j in range(1,len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],this_gal_group[j][1],C)
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):
""" Euler product as a formula and a table of local factors.
"""
texform_gen = "\[L(A,s) = "
texform_gen += "\prod_{p \\text{ prime}} F_p(p^{-s})^{-1} \]\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 $p\\nmid " + str(L.level) + "$,\n"
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 $p \mid " + str(L.level) + "$, then $F_p$ is a polynomial of degree at most 3, "
# ans += "If " + pbadset + ", then $F_p$ is a polynomial of degree at most 3, "
ans += "If " + pbadset + ", then $F_p$ is a polynomial of degree at most "
ans += str(L.degree - 1) + ". "
# ans += "with $F_p(0) = 1$."
factN = list(factor(L.level))
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>"
numfactors = len(L.localfactors)
goodorbad = "bad"
C = getDBConnection()
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,
'$C_1$')
else:
eptable += group_display_knowl(this_gal_group[0][0],
this_gal_group[0][1], C)
for j in range(1, len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],
this_gal_group[j][1], C)
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], C)
for j in range(1, len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],
this_gal_group[j][1], C)
eptable += "</td>"
eptable += "</tr>\n"
# eptable += "<td>" + group_display_knowl(4,1,C) + "</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], C)
for j in range(1, len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],
this_gal_group[j][1], C)
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):
""" Euler product as a formula and a table of local factors.
"""
texform_gen = "\[L(A,s) = "
texform_gen += "\prod_{p \\text{ prime}} F_p(p^{-s})^{-1} \]\n"
pfactors = prime_divisors(L.level)
if len(pfactors) == 1: #i.e., the conductor is prime
pgoodset = "$p \\neq " + str(L.level) + "$"
pbadset = "$p = " + str(L.level) + "$"
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 $p\\nmid " + str(L.level) + "$,\n"
ans += texform_gen + "where, for " + pgoodset + ",\n"
ans += "\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]"
ans += "with $b_p = a_p^2 - a_{p^2}$. "
# ans += "If $p \mid " + str(L.level) + "$, then $F_p$ is a polynomial of degree at most 3, "
ans += "If " + pbadset + ", then $F_p$ is a polynomial of degree at most 3, "
ans += "with $F_p(0) = 1$."
factN = list(factor(L.level))
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></tr>\n"
eptable += "</thead>"
numfactors = len(L.localfactors)
goodorbad = "bad"
for lf in L.bad_lfactors:
try:
eptable += ("<tr><td>" + goodorbad + "</td><td>" + str(lf[0]) + "</td><td>" +
"$" + list_to_factored_poly_otherorder(lf[1]) + "$" +
"</td></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
eptable += ("<tr" + firsttime + "><td>" + goodorbad + "</td><td>" + str(j) + "</td><td>" +
"$" + list_to_factored_poly_otherorder(L.localfactors[this_prime_index]) + "$" +
"</td></tr>\n")
goodorbad = ""
firsttime = ""
firsttime = " id='moreep'"
for j in good_primes2:
this_prime_index = prime_pi(j) - 1
eptable += ("<tr" + firsttime + " class='more nodisplay'" + "><td>" + goodorbad + "</td><td>" + str(j) + "</td><td>" +
"$" + list_to_factored_poly_otherorder(L.localfactors[this_prime_index]) + "$" +
"</td></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)
