def _anihilate_pol(k, M):
'''
k: The weight of an element c, where c is a construction
for generators of M_{det^* sym(10)} and an instance of
ConstDivision.
M: an instance of Sym10EvenDiv or Sym10OddDiv.
Return a polynomial pl such that the subspace of M anihilated by pl(T(2))
is equal to the subspace of holomorphic modular forms.
'''
R = PolynomialRing(QQ, names="x")
x = R.gens()[0]
if k % 2 == 0:
# Klingen-Eisenstein series
f = CuspForms(1, k + 10).basis()[0]
return x - f[2] * (1 + QQ(2) ** (k - 2))
elif k == 13:
# Kim-Ramakrishnan-Shahidi lift
f = CuspForms(1, 12).basis()[0]
a = f[2]
return x - f[2] ** 3 + QQ(2) ** 12 * f[2]
else:
chrply = M.hecke_charpoly(2)
dim = hilbert_series_maybe(10)[k]
l = [(a, b) for a, b in chrply.factor() if a.degree() == dim]
if len(l) > 1 or l[0][1] != 1:
raise RuntimeError
else:
return l[0][0]
def dimension(self):
if self.sym_wt == 8 and self.wt == 4:
return 1
elif self.sym_wt <= 10 and self.wt <= 4:
return 0
elif self.wt > 4:
pl = hilbert_series_maybe(self.sym_wt, prec=self.wt + 1)
return pl[self.wt]
else:
raise NotImplementedError(
"The dimensions of small determinant weights" +
" are not known in general.")
def test_ramanujan_conj(self):
'''Test Ramanujan conjectures for eigenforms of determinant weights
less than or equal to 29.
'''
prec = 6
hpl = hilbert_series_maybe(10)
for k in range(22, 30):
if hpl[k] != 0:
N = sym10_space(k, prec, data_directory=data_dir)
self.assertEqual(N.dimension(), len(N.basis()))
_chply = N.hecke_charpoly(2)
for cply, _ in _chply.factor():
K = NumberField(cply, names="a")
a = K.gens()[0]
f = N.eigenform_with_eigenvalue_t2(a)
self.assert_ramanujan_conj_eigenform(f)
def dimension(self):
return hilbert_series_maybe(10)[self.wt]