def decomposition(self, B, verbose=False):
"""
Return Hecke decomposition of self using Hecke operators T_p
coprime to the level with norm(p) <= B.
"""
# TODO: rewrite to use primes_of_bounded_norm so that we
# iterate through primes ordered by *norm*, which is
# potentially vastly faster. Delete these functions
# involving characteristic!
p = next_prime_of_characteristic_coprime_to(F.ideal(1), self.level())
T = self.hecke_matrix(p)
D = T.decomposition()
while len([X for X in D if not X[1]]) > 0:
p = next_prime_of_characteristic_coprime_to(p, self.level())
if p.norm() > B:
break
if verbose: print p.norm()
T = self.hecke_matrix(p)
D2 = []
for X in D:
if X[1]:
D2.append(X)
else:
if verbose: print T.restrict(X[0]).fcp()
for Z in T.decomposition_of_subspace(X[0]):
D2.append(Z)
D = D2
D = [self.subspace(X[0]) for X in D]
D.sort()
S = Sequence(D, immutable=True, cr=True, universe=int, check=False)
return S
def new_decomposition(self, verbose=False):
"""
Return complete irreducible Hecke decomposition of new subspace of self.
"""
V = self.degeneracy_matrix().kernel()
p = next_prime_of_characteristic_coprime_to(F.ideal(1), self.level())
T = self.hecke_matrix(p)
D = T.decomposition_of_subspace(V)
while len([X for X in D if not X[1]]) > 0:
p = next_prime_of_characteristic_coprime_to(p, self.level())
if verbose: print p.norm()
T = self.hecke_matrix(p)
D2 = []
for X in D:
if X[1]:
D2.append(X)
else:
if verbose: print T.restrict(X[0]).fcp()
for Z in T.decomposition_of_subspace(X[0]):
D2.append(Z)
D = D2
D = [self.subspace(X[0]) for X in D]
D.sort()
S = Sequence(D, immutable=True, cr=True, universe=int, check=False)
return S
def new_decomposition(self, verbose=False):
"""
Return complete irreducible Hecke decomposition of new subspace of self.
"""
V = self.degeneracy_matrix().kernel()
p = next_prime_of_characteristic_coprime_to(F.ideal(1), self.level())
T = self.hecke_matrix(p)
D = T.decomposition_of_subspace(V)
while len([X for X in D if not X[1]]) > 0:
p = next_prime_of_characteristic_coprime_to(p, self.level())
if verbose: print p.norm()
T = self.hecke_matrix(p)
D2 = []
for X in D:
if X[1]:
D2.append(X)
else:
if verbose: print T.restrict(X[0]).fcp()
for Z in T.decomposition_of_subspace(X[0]):
D2.append(Z)
D = D2
D = [self.subspace(X[0]) for X in D]
D.sort()
S = Sequence(D, immutable=True, cr=True, universe=int, check=False)
return S
def decomposition(self, B, verbose=False):
"""
Return Hecke decomposition of self using Hecke operators T_p
coprime to the level with norm(p) <= B.
"""
# TODO: rewrite to use primes_of_bounded_norm so that we
# iterate through primes ordered by *norm*, which is
# potentially vastly faster. Delete these functions
# involving characteristic!
p = next_prime_of_characteristic_coprime_to(F.ideal(1), self.level())
T = self.hecke_matrix(p)
D = T.decomposition()
while len([X for X in D if not X[1]]) > 0:
p = next_prime_of_characteristic_coprime_to(p, self.level())
if p.norm() > B:
break
if verbose: print p.norm()
T = self.hecke_matrix(p)
D2 = []
for X in D:
if X[1]:
D2.append(X)
else:
if verbose: print T.restrict(X[0]).fcp()
for Z in T.decomposition_of_subspace(X[0]):
D2.append(Z)
D = D2
D = [self.subspace(X[0]) for X in D]
D.sort()
S = Sequence(D, immutable=True, cr=True, universe=int, check=False)
return S
def ideals_of_norm(v):
"""
INPUT:
- `v` -- integer >= 2, or list of integers >= 2
OUTPUT:
- list of ideals of all ideals that have norm in v that is >= 2.
EXAMPLES::
sage: sage.modular.hilbert.sqrt5_tables.ideals_of_norm(4)
[Fractional ideal (2)]
sage: sage.modular.hilbert.sqrt5_tables.ideals_of_norm([9,11])
[Fractional ideal (3), Fractional ideal (-3*a + 1), Fractional ideal (-3*a + 2)]
sage: sage.modular.hilbert.sqrt5_tables.ideals_of_norm([4*5])
[Fractional ideal (-4*a + 2)]
sage: sage.modular.hilbert.sqrt5_tables.ideals_of_norm(4*5)
[Fractional ideal (-4*a + 2)]
"""
try:
v = list(v)
except TypeError:
v = [Integer(v)]
z = F.ideals_of_bdd_norm(max(v))
return sum([z[n] for n in v if n > 1], [])
def find_mod2pow_splitting(i):
P = F.primes_above(2)[0]
if i == 1:
R = ResidueRing(P, 1)
M = MatrixSpace(R, 2)
return [M(matrix_lift(a).list()) for a in find_mod2_splitting()]
R = ResidueRing(P, i)
M = MatrixSpace(R, 2)
# arbitrary lift
wbar = [M(matrix_lift(a).list()) for a in find_mod2pow_splitting(i - 1)]
# Find lifts of wbar[1] and wbar[2] that have square -1
k = P.residue_field()
Mk = MatrixSpace(k, 2)
t = 2**(i - 1)
s = M(t)
L = []
for j in [1, 2]:
C = Mk(matrix_lift(wbar[j]**2 + M(1)) / t)
A = Mk(matrix_lift(wbar[j]))
# Find all matrices B in Mk such that AB+BA=C.
L.append([
wbar[j] + s * M(matrix_lift(B)) for B in Mk if A * B + B * A == C
])
g = M(F.gen())
t = M(t)
two = M(2)
ginv = M(F.gen()**(-1))
for w1 in L[0]:
for w2 in L[1]:
w0 = ginv * (two * g * wbar[3] - w1 - w2 - w1 * w2)
w3 = g * w0 + w2 * w1
if w0 * w0 != w0 - M(1):
continue
if w3 * w3 != M(-1):
continue
return w0, w1, w2, w3
raise ValueError
def next_prime_not_dividing(P, I):
while True:
p = P.smallest_integer()
if p == 1:
Q = F.ideal(2)
elif p % 5 in [2,3]: # inert
Q = F.primes_above(next_prime(p))[0]
elif p == 5:
Q = F.ideal(7)
else: # p split
A = F.primes_above(p)
if A[0] == P:
Q = A[1]
else:
Q = F.primes_above(next_prime(p))[0]
if not Q.divides(I):
return Q
else:
P = Q # try again
def next_prime_not_dividing(P, I):
while True:
p = P.smallest_integer()
if p == 1:
Q = F.ideal(2)
elif p % 5 in [2,3]: # inert
Q = F.primes_above(next_prime(p))[0]
elif p == 5:
Q = F.ideal(7)
else: # p split
A = F.primes_above(p)
if A[0] == P:
Q = A[1]
else:
Q = F.primes_above(next_prime(p))[0]
if not Q.divides(I):
return Q
else:
P = Q # try again
def find_mod2pow_splitting(i):
P = F.primes_above(2)[0]
if i == 1:
R = ResidueRing(P, 1)
M = MatrixSpace(R, 2)
return [M(matrix_lift(a).list()) for a in find_mod2_splitting()]
R = ResidueRing(P, i)
M = MatrixSpace(R, 2)
# arbitrary lift
wbar = [M(matrix_lift(a).list()) for a in find_mod2pow_splitting(i - 1)]
# Find lifts of wbar[1] and wbar[2] that have square -1
k = P.residue_field()
Mk = MatrixSpace(k, 2)
t = 2 ** (i - 1)
s = M(t)
L = []
for j in [1, 2]:
C = Mk(matrix_lift(wbar[j] ** 2 + M(1)) / t)
A = Mk(matrix_lift(wbar[j]))
# Find all matrices B in Mk such that AB+BA=C.
L.append([wbar[j] + s * M(matrix_lift(B)) for B in Mk if A * B + B * A == C])
g = M(F.gen())
t = M(t)
two = M(2)
ginv = M(F.gen() ** (-1))
for w1 in L[0]:
for w2 in L[1]:
w0 = ginv * (two * g * wbar[3] - w1 - w2 - w1 * w2)
w3 = g * w0 + w2 * w1
if w0 * w0 != w0 - M(1):
continue
if w3 * w3 != M(-1):
continue
return w0, w1, w2, w3
raise ValueError
def elliptic_curve_factors(self):
D = [X for X in self.new_decomposition() if X.dimension() == 1]
# Have to get rid of the Eisenstein factor
p = next_prime_of_characteristic_coprime_to(F.ideal(1), self.level())
while True:
q = p.residue_field().cardinality() + 1
E = [A for A in D if A.hecke_matrix(p)[0,0] == q]
if len(E) == 0:
break
elif len(E) == 1:
D = [A for A in D if A != E[0]]
break
else:
p = next_prime_of_characteristic_coprime_to(p, self.level())
return Sequence([EllipticCurveFactor(X, number) for number, X in enumerate(D)],
immutable=True, cr=True, universe=int, check=False)
def elliptic_curve_factors(self):
D = [X for X in self.new_decomposition() if X.dimension() == 1]
# Have to get rid of the Eisenstein factor
p = next_prime_of_characteristic_coprime_to(F.ideal(1), self.level())
while True:
q = p.residue_field().cardinality() + 1
E = [A for A in D if A.hecke_matrix(p)[0,0] == q]
if len(E) == 0:
break
elif len(E) == 1:
D = [A for A in D if A != E[0]]
break
else:
p = next_prime_of_characteristic_coprime_to(p, self.level())
return Sequence([EllipticCurveFactor(X, number) for number, X in enumerate(D)],
immutable=True, cr=True, universe=int, check=False)
def find_mod2_splitting():
"""
Return elements w0, w1, w2, w3 that determine a mod-2 splitting
of the Hamilton quaternion algebra over Q(sqrt(5)).
OUTPUT:
- four 2x2 matrices over the residue field of Q(sqrt(5)) of order 4.
EXAMPLES::
sage: import sage.modular.hilbert.sqrt5_fast_python
sage: w = sage.modular.hilbert.sqrt5_fast_python.find_mod2_splitting(); w
(
[ 1 abar] [ 0 abar + 1] [abar + 1 abar] [1 1]
[abar + 1 0], [ abar 0], [ abar abar + 1], [0 1]
)
sage: w[1]^2, w[2]^2
(
[1 0] [1 0]
[0 1], [0 1]
)
sage: span([vector(z.list()) for z in w]).dimension()
4
"""
P = F.primes_above(2)[0]
k = P.residue_field()
M = MatrixSpace(k,2)
V = k**4
g = k.gen() # image of golden ratio
m1 = M(-1)
sqrt_minus_1 = [(w, V(w.list())) for w in M if w*w == m1]
one = M(1)
v_one = V(one.list())
for w1, v1 in sqrt_minus_1:
for w2, v2 in sqrt_minus_1:
w0 = (g-1)*(w1+w2 - w1*w2)
w3 = g*w0 + w2*w1
if w0*w0 != w0 - 1:
continue
if w3*w3 != -1:
continue
if V.span([w0.list(),v1,v2,w3.list()]).dimension() == 4:
return w0, w1, w2, w3
def test_reduced_rep(B=50):
"""
Unit test: run a simple consistency check that reduced_rep is
working sensibly.
INPUT:
- `B` -- positive integer
OUTPUT:
- None (assertion raised if something detected wrong)
EXAMPLES::
sage: sage.modular.hilbert.sqrt5_tables.test_reduced_rep()
"""
a = F.gen()
z = -45 * a + 28
v = [reduced_rep(a ** i * z) for i in range(-B, B)]
assert len(set(v)) == 1
def find_mod2_splitting():
P = F.primes_above(2)[0]
k = P.residue_field()
M = MatrixSpace(k, 2)
V = k ** 4
g = k.gen() # image of golden ratio
m1 = M(-1)
sqrt_minus_1 = [(w, V(w.list())) for w in M if w * w == m1]
one = M(1)
v_one = V(one.list())
for w1, v1 in sqrt_minus_1:
for w2, v2 in sqrt_minus_1:
w0 = (g - 1) * (w1 + w2 - w1 * w2)
w3 = g * w0 + w2 * w1
if w0 * w0 != w0 - 1:
continue
if w3 * w3 != -1:
continue
if V.span([w0.list(), v1, v2, w3.list()]).dimension() == 4:
return w0, w1, w2, w3
def ideals_of_bounded_norm(B):
r"""
Return all ideals in the ring of integers of Q(sqrt(5)) with norm
bigger than 1 and <= B.
INPUT:
- `B` -- positive integer
OUTPUT:
- list of ideals
EXAMPLES::
sage: v = sage.modular.hilbert.sqrt5_tables.ideals_of_bounded_norm(11); v
[Fractional ideal (2), Fractional ideal (-2*a + 1), Fractional ideal (3), Fractional ideal (-3*a + 1), Fractional ideal (-3*a + 2)]
sage: [I.norm() for I in v]
[4, 5, 9, 11, 11]
"""
return sum([v for n, v in F.ideals_of_bdd_norm(B).iteritems() if n != 1], [])
def find_mod2_splitting():
P = F.primes_above(2)[0]
k = P.residue_field()
M = MatrixSpace(k, 2)
V = k**4
g = k.gen() # image of golden ratio
m1 = M(-1)
sqrt_minus_1 = [(w, V(w.list())) for w in M if w * w == m1]
one = M(1)
v_one = V(one.list())
for w1, v1 in sqrt_minus_1:
for w2, v2 in sqrt_minus_1:
w0 = (g - 1) * (w1 + w2 - w1 * w2)
w3 = g * w0 + w2 * w1
if w0 * w0 != w0 - 1:
continue
if w3 * w3 != -1:
continue
if V.span([w0.list(), v1, v2, w3.list()]).dimension() == 4:
return w0, w1, w2, w3
def next_prime_of_characteristic_coprime_to(P, I):
p = next_prime(P.smallest_integer())
N = ZZ(I.norm())
while N%p == 0:
p = next_prime(p)
return F.primes_above(p)[0]
def find_mod2pow_splitting(i):
"""
Return elements w0, w1, w2, w3 that determine a mod-`2^i` splitting
of the Hamilton quaternion algebra over Q(sqrt(5)).
INPUT:
- `i` -- positive integer
OUTPUT:
- four 2x2 matrices over a residue ring of Q(sqrt(5)) of characteristic a power of 2
EXAMPLES::
sage: import sage.modular.hilbert.sqrt5_fast_python
sage: w = sage.modular.hilbert.sqrt5_fast_python.find_mod2pow_splitting(1); w
(
[ 1 g] [ 0 1 + g] [1 + g g] [1 1]
[1 + g 0], [ g 0], [ g 1 + g], [0 1]
)
sage: w = sage.modular.hilbert.sqrt5_fast_python.find_mod2pow_splitting(2); w
(
[ 3 2 + 3*g] [ 0 1 + 3*g] [3 + 3*g 3*g]
[ 1 + g 2], [ g 0], [ 3*g 1 + g],
[3 + 2*g 3 + 2*g]
[ 2 1 + 2*g]
)
sage: w[1]^2
[3 0]
[0 3]
sage: w[2]^2
[3 0]
[0 3]
sage: w = sage.modular.hilbert.sqrt5_fast_python.find_mod2pow_splitting(4)
sage: w[1]^2
[15 0]
[ 0 15]
sage: w[2]^2
[15 0]
[ 0 15]
sage: w[1], w[2]
(
[ 0 1 + 15*g] [15 + 15*g 12 + 7*g]
[ g 0], [ 4 + 11*g 1 + g]
)
TESTS:
The power must be positive::
sage: sage.modular.hilbert.sqrt5_fast_python.find_mod2pow_splitting(0)
Traceback (most recent call last):
...
ValueError: i must be positive
"""
P = F.primes_above(2)[0]
i = ZZ(i)
if i <= 0:
raise ValueError, "i must be positive"
if i == 1:
R = ResidueRing(P, 1)
M = MatrixSpace(R,2)
return tuple([M(matrix_lift(a).list()) for a in find_mod2_splitting()])
R = ResidueRing(P, i)
M = MatrixSpace(R,2)
# arbitrary lift
wbar = [M(matrix_lift(a).list()) for a in find_mod2pow_splitting(i-1)]
# Find lifts of wbar[1] and wbar[2] that have square -1
k = P.residue_field()
Mk = MatrixSpace(k, 2)
t = 2**(i-1)
s = M(t)
L = []
for j in [1,2]:
C = Mk(matrix_lift(wbar[j]**2 + M(1)) / t)
A = Mk(matrix_lift(wbar[j]))
# Find all matrices B in Mk such that AB+BA=C.
L.append([wbar[j]+s*M(matrix_lift(B)) for B in Mk if A*B + B*A == C])
g = M(F.gen())
t = M(t)
two = M(2)
ginv = M(F.gen()**(-1))
for w1 in L[0]:
for w2 in L[1]:
w0 = ginv*(two*g*wbar[3] -w1 -w2 - w1*w2)
w3 = g*w0 + w2*w1
if w0*w0 != w0 - M(1):
continue
if w3*w3 != M(-1):
continue
return w0, w1, w2, w3
raise ValueError
def next_prime_of_characteristic_coprime_to(P, I):
p = next_prime(P.smallest_integer())
N = ZZ(I.norm())
while N%p == 0:
p = next_prime(p)
return F.primes_above(p)[0]
def dimensions(v, filename=None):
"""
Compute dimensions of spaces of Hilbert modular forms for all the levels in v.
The format is:
Norm dimension generator time
INPUT:
- `v` -- list of positive integers
- ``filename`` -- optional string; if given, output is also written
to that file (in addition to stdout).
OUTPUT:
- appends to table with above format and rows corresponding to the
ideals of Q(sqrt(5)) with norm in v, and optionally creates a
file
EXAMPLES::
sage: from sage.modular.hilbert.sqrt5_tables import dimensions
sage: out = dimensions([1..40])
4 1 2 ...
5 1 -2*a+1 ...
9 1 3 ...
11 1 -3*a+1 ...
11 1 -3*a+2 ...
16 1 4 ...
19 1 -4*a+1 ...
19 1 -4*a+3 ...
20 1 -4*a+2 ...
25 1 5 ...
29 1 a-6 ...
29 1 -a-5 ...
31 2 5*a-3 ...
31 2 5*a-2 ...
36 2 6 ...
sage: out = dimensions([36, 4])
36 2 6 ...
4 1 2 ...
Test writing to a file::
sage: if os.path.exists('tmp_table.txt'): os.unlink('tmp_table.txt')
sage: out = dimensions([36, 4], 'tmp_table.txt')
36 2 6 ...
4 1 2 ...
sage: '36 2 6' in open('tmp_table.txt').read()
True
sage: open('tmp_table.txt').read().count('\n')
2
sage: os.unlink('tmp_table.txt')
"""
if len(v) == 0:
return ""
F = open(filename, "a") if filename else None
out = ""
for N in ideals_of_norm(v):
t = cputime()
H = IcosiansModP1ModN(N)
tm = "%.2f" % cputime(t)
s = "%s %s %s %s" % (N.norm(), H.cardinality(), no_space(reduced_gen(N)), tm)
print s
out += s + "\n"
if F:
F.write(s + "\n")
F.flush()
return out
def charpolys(v, B, filename=None):
"""
Compute characteristic polynomials of T_P for primes P with norm
<= B coprime to the level, for all spaces of Hilbert modular forms
for all the levels in v.
INPUT:
- `v` -- list of positive integers
- `B` -- positive integer
- ``filename`` -- optional string; if given, output is also written
to that file (in addition to stdout).
OUTPUT:
- outputs a table with rows corresponding to the ideals
of Q(sqrt(5)) with norm in v, and optionally creates a file
EXAMPLES::
sage: from sage.modular.hilbert.sqrt5_tables import charpolys
sage: out = charpolys([1..20], 10)
4 2 ... [(5,x-6),(3,x-10)]
5 -2*a+1 ... [(2,x-5),(3,x-10)]
9 3 ... [(2,x-5),(5,x-6)]
11 -3*a+1 ... [(2,x-5),(5,x-6),(3,x-10)]
11 -3*a+2 ... [(2,x-5),(5,x-6),(3,x-10)]
16 4 ... [(5,x-6),(3,x-10)]
19 -4*a+1 ... [(2,x-5),(5,x-6),(3,x-10)]
19 -4*a+3 ... [(2,x-5),(5,x-6),(3,x-10)]
20 -4*a+2 ... [(3,x-10)]
sage: out = charpolys([20, 11], 10)
20 -4*a+2 ... [(3,x-10)]
11 -3*a+1 ... [(2,x-5),(5,x-6),(3,x-10)]
11 -3*a+2 ... [(2,x-5),(5,x-6),(3,x-10)]
Test writing to a file::
sage: if os.path.exists('tmp_table.txt'): os.unlink('tmp_table.txt')
sage: out = charpolys([20, 11], 10, 'tmp_table.txt')
20 -4*a+2 ... [(3,x-10)]
11 -3*a+1 ... [(2,x-5),(5,x-6),(3,x-10)]
11 -3*a+2 ... [(2,x-5),(5,x-6),(3,x-10)]
sage: r = open('tmp_table.txt').read()
sage: 'x-10' in r
True
sage: r.count('\n')
3
sage: os.unlink('tmp_table.txt')
"""
if len(v) == 0:
return ""
out = ""
F = open(filename, "a") if filename else None
P = [p for p in ideals_of_bounded_norm(B) if p.is_prime()]
for N in ideals_of_norm(v):
t = cputime()
H = IcosiansModP1ModN(N)
T = [
(p.smallest_integer(), H.hecke_matrix(p).fcp()) for p in P if gcd(Integer(p.norm()), Integer(N.norm())) == 1
]
tm = "%.2f" % cputime(t)
s = "%s %s %s %s" % (N.norm(), no_space(reduced_gen(N)), tm, no_space(T))
print s
out += s + "\n"
if F:
F.write(s + "\n")
F.flush()
return out
def rational_newforms(v, B=100, filename=None, ncpu=1):
"""
Return system of Hecke eigenvalues corresponding to rational
newforms of level whose norm is in v. Compute the Hecke
eigenvalues a_P for all good primes P with norm < B.
INPUT:
- `v` -- list of integers
- `B` -- positive integer
- ``filename`` -- optional filename
- ``ncpu`` -- positive integer (default: 1); if > 1 then use ncpu
simultaneous processes. Note that that displayed output during
the computation and to the file may be out of order.
OUTPUT:
- outputs a table with rows corresponding to the ideals
of Q(sqrt(5)) with norm in v, and optionally creates a file
Table columns:
norm_of_level generator_of_level number time_for_level a_P a_P ...
EXAMPLES::
sage: from sage.modular.hilbert.sqrt5_tables import rational_newforms
sage: out = rational_newforms([1..76], B=20)
31 5*a-3 0 ... -3 -2 2 4 -4 -4 4
31 5*a-2 0 ... -3 -2 2 -4 4 4 -4
36 6 0 ... ? -4 ? 2 2 0 0
41 a-7 0 ... -2 -1 -4 -2 5 -1 6
41 a+6 0 ... -2 -1 -4 5 -2 6 -1
45 -6*a+3 0 ... -3 ? ? -4 -4 4 4
49 7 0 ... 0 -4 5 -3 -3 0 0
55 a+7 0 ... -1 ? -2 ? 0 8 -4
55 -a+8 0 ... -1 ? -2 0 ? -4 8
64 8 0 ... 0 -2 2 -4 -4 4 4
71 a-9 0 ... -1 0 -2 0 0 -4 2
71 a+8 0 ... -1 0 -2 0 0 2 -4
76 -8*a+2 0 ... ? -3 1 3 -6 ? -7
76 -8*a+2 1 ... ? 1 -5 -3 2 ? 5
76 -8*a+6 0 ... ? -3 1 -6 3 -7 ?
76 -8*a+6 1 ... ? 1 -5 2 -3 5 ?
Test writing to a file::
sage: if os.path.exists('tmp_table.txt'): os.unlink('tmp_table.txt')
sage: out = rational_newforms([1..36], 20,'tmp_table.txt')
31 5*a-3 0 ... -3 -2 2 4 -4 -4 4
31 5*a-2 0 ... -3 -2 2 -4 4 4 -4
36 6 0 ... ? -4 ? 2 2 0 0
sage: r = open('tmp_table.txt').read()
sage: r.count('\n')
3
sage: os.unlink('tmp_table.txt')
"""
if len(v) == 0:
return ""
if ncpu < 1:
raise ValueError, "ncpu must be >= 1"
F = open(filename, "a") if filename else None
if ncpu > 1:
from sage.all import parallel
@parallel(ncpu)
def f(N):
return rational_newforms([N], B, filename=None, ncpu=1)
d = {}
for X in f(v):
N = X[0][0]
ans = X[1].strip()
if ans:
d[N] = ans
if F:
F.write(ans + "\n")
return "\n".join(d[N] for N in sorted(d.keys()))
out = ""
from sqrt5_hmf import QuaternionicModule
for N in ideals_of_norm(v):
t = cputime()
H = QuaternionicModule(N)
EC = H.rational_newforms()
tm = "%.2f" % cputime(t)
for i, E in enumerate(EC):
v = E.aplist(B)
data = [N.norm(), no_space(reduced_gen(N)), i, tm, " ".join([no_space(x) for x in v])]
s = " ".join([str(x) for x in data])
print s
out += s + "\n"
if F:
F.write(s + "\n")
F.flush()
return out
