def formtest_2(minD, maxD):
s = 0
for D in range(minD, maxD):
if is_fundamental_discriminant(-D):
for d in divisors(D):
if is_fundamental_discriminant(-d):
sd = RR(
RR(1) / RR(6) * (RR(d) + old_div(RR(D), RR(d))) -
RR(sqrt(D)) / RR(pi) * log(d))
print(D, d, sd)
s += sd
if s <= 0:
print("s= {0} D={1}".format(s, D))
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 discrinimant * 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 neg_fdd_div(D):
s = 0
for d in divisors(D):
if is_fundamental_discriminant(-d):
if Zmod(d)(old_div(D, d)).is_square():
s += 1
return s
def orbittest(minD, maxD, eps=-1):
for D in range(minD, maxD + 1):
if is_fundamental_discriminant(eps * D) and is_odd(D):
print(D)
print(
CMorbits(eps * D) -
len(Newforms(kronecker_character(eps * D), 3, names='a')))
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 compare_formulas_2a(D, k):
d1 = dimension_new_cusp_forms(kronecker_character(D), k)
if D < 0:
D = -D
d2 = RR(1 / pi * sqrt(D) * sum([
log(d) * sigma(old_div(D, d), 0) for d in divisors(D) if Zmod(d)
(old_div(D, d)).is_square() and is_fundamental_discriminant(-d)
]))
return d1 - d2
def sigma_rep(Delta, print_divisors=False):
s = 0
for DD in divisors(Delta):
if is_fundamental_discriminant(-DD):
D = -DD
for d in divisors(old_div(Delta, D)):
s += kronecker(D, d)
if print_divisors:
print(D, d, kronecker(D, d))
return s
def compare_formulas_2(D, k):
d1 = old_div(RR(abs(D)), RR(6))
if D < 0:
D = -D
s1 = RR(
sqrt(abs(D)) * sum([
log(d) for d in divisors(D) if is_fundamental_discriminant(-d)
and kronecker(-d, old_div(D, d)) == 1
]))
d2 = RR((old_div(2, (sqrt(3) * pi))) * s1)
return d1 - d2, d2, RR(2 * sqrt(D) * log(D) / pi)
def quadratic_twist_zeros(D, n, algorithm='clib'):
"""
Return imaginary parts of the first n zeros of all the Dirichlet
character corresponding to the quadratic field of discriminant D.
INPUT:
- D -- fundamental discriminant
- n -- number of zeros to find
- algorithm -- 'clib' (use C library) or 'subprocess' (open another process)
"""
if algorithm == 'clib':
L = Lfunction_from_character(kronecker_character(D), type="int")
return L.find_zeros_via_N(n)
elif algorithm == 'subprocess':
assert is_fundamental_discriminant(D)
cmd = "lcalc -z %s --twist-quadratic --start %s --finish %s"%(n, D, D)
out = os.popen(cmd).read().split()
return [float(out[i]) for i in range(len(out)) if i%2!=0]
else:
raise ValueError, "unknown algorithm '%s'"%algorithm
def quadratic_twist_zeros(D, n, algorithm='clib'):
"""
Return imaginary parts of the first n zeros of all the Dirichlet
character corresponding to the quadratic field of discriminant D.
INPUT:
- D -- fundamental discriminant
- n -- number of zeros to find
- algorithm -- 'clib' (use C library) or 'subprocess' (open another process)
"""
if algorithm == 'clib':
L = Lfunction_from_character(kronecker_character(D), type="int")
return L.find_zeros_via_N(n)
elif algorithm == 'subprocess':
assert is_fundamental_discriminant(D)
cmd = "lcalc -z %s --twist-quadratic --start %s --finish %s" % (n, D,
D)
out = os.popen(cmd).read().split()
return [float(out[i]) for i in range(len(out)) if i % 2 != 0]
else:
raise ValueError, "unknown algorithm '%s'" % algorithm
def test_sigma_rep(Dmin=1, Dmax=1000, print_divisors=False):
n = 0
sa = 0
sw = [1, 0]
sm = [1, 0]
for D in range(Dmin, Dmax):
if is_fundamental_discriminant(-D):
n += 1
sr = sigma_rep(D, True)
sa += sr
sm[1] = max(sr, sm[1])
swt = old_div(sr, D)
sw[1] = max(swt, RR(sw[1]))
if sm[1] == sr:
sm[0] = -D
if sw[1] == swt:
sw[0] = -D
ct = RR(100) * (RR(D)**(old_div(1.0, 25)))
#if ct<=RR(sr):
print(-D, sr, ct, sigma(D, 0))
sa = old_div(sa, n)
print("Max: {0}".format(sm))
print("Avg: {0}, {1}".format(sa, RR(sa)))
print("Worst case: {0}".format(sw))
def compare_formulas_1(D, k):
DG = DirichletGroup(abs(D))
chi = DG(kronecker_character(D))
d1 = dimension_new_cusp_forms(chi, k)
#if D>0:
# lvals=sage.lfunctions.all.lcalc.twist_values(1,2,D)
#else:
# lvals=sage.lfunctions.all.lcalc.twist_values(1,D,0)
#s1=RR(sum([sqrt(abs(lv[0]))*lv[1]*2**len(prime_factors(D/lv[0])) for lv in lvals if lv[0].divides(D) and Zmod(lv[0])(abs(D/lv[0])).is_square()]))
#d2=RR(1/pi*s1)
d2 = 0
for d in divisors(D):
if is_fundamental_discriminant(-d):
K = QuadraticField(-d)
DD = old_div(ZZ(D), ZZ(d))
ep = euler_phi((chi * DG(kronecker_character(-d))).conductor())
#ep=euler_phi(squarefree_part(abs(D*d)))
print("ep=", ep, D, d)
ids = [a for a in K.ideals_of_bdd_norm(-DD)[-DD]]
eulers1 = []
for a in ids:
e = a.euler_phi()
if e != 1 and ep == 1:
if K(-1).mod(a) != K(1).mod(a):
e = old_div(e, (2 * ep))
else:
e = old_div(e, ep)
eulers1.append(e)
print(eulers1, ep)
s = sum(eulers1)
if ep == 1 and not (d.divides(DD) or abs(DD) == 1):
continue
print(d, s)
if len(eulers1) > 0:
d2 += s * K.class_number()
return d1 - d2
def fundamental_discriminants(A, B):
"""Return the fundamental discriminants between A and B (inclusive), as Sage integers,
ordered by absolute value, with negatives first when abs same."""
v = [ZZ(D) for D in range(A, B + 1) if is_fundamental_discriminant(D)]
v.sort(lambda x, y: cmp((abs(x), sgn(x)), (abs(y), sgn(y))))
return v
def test_neg_fdd_div(minD, maxD):
for D in range(minD, maxD):
if is_fundamental_discriminant(-D):
print("{0}, {1}, {2}".format(D, neg_fdd_div(D),
old_div(sigma(D, 0), 2)))
def fundamental_discriminants(A, B):
"""Return the fundamental discriminants between A and B (inclusive), as Sage integers,
ordered by absolute value, with negatives first when abs same."""
v = [ZZ(D) for D in range(A, B+1) if is_fundamental_discriminant(D)]
v.sort(lambda x,y: cmp((abs(x),sgn(x)),(abs(y),sgn(y))))
return v