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
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
