def modform_cusp_info(calc, S, l, precLimit):
"""
This goes through all the cusps and compares the space given by `(f|R)[S]`
with the space of Elliptic modular forms expansion at those cusps.
"""
assert l == S.det()
assert list(calc.curlS) == [S]
D = calc.D
HermWeight = calc.HermWeight
reducedCurlFSize = calc.matrixColumnCount
herm_modform_fe_expannsion = FreeModule(QQ, reducedCurlFSize)
if not Integer(l).is_squarefree():
# The calculation of the cusp expansion space takes very long here, thus
# we skip them for now.
return None
for cusp in Gamma0(l).cusps():
if cusp == Infinity: continue
M = cusp_matrix(cusp)
try:
gamma, R, tM = solveR(M, S, space=CurlO(D))
except Exception:
print (M, S)
raise
R.set_immutable() # for caching, we need it hashable
herm_modforms = herm_modform_fe_expannsion.echelonized_basis_matrix().transpose()
ell_R_denom, ell_R_order, M_R = calcMatrixTrans(calc, R)
CycloDegree_R = CyclotomicField(ell_R_order).degree()
print "M_R[0] nrows, ell_R_denom, ell_R_order, Cyclo degree:", \
M_R[0].nrows(), ell_R_denom, ell_R_order, CycloDegree_R
# The maximum precision we can use is M_R[0].nrows().
# However, that can be quite huge (e.g. 600).
ce_prec = min(precLimit, M_R[0].nrows())
ce = cuspExpansions(level=l, weight=2*HermWeight, prec=ce_prec)
ell_M_denom, ell_M = ce.expansion_at(SL2Z(M))
print "ell_M_denom, ell_M nrows:", ell_M_denom, ell_M.nrows()
ell_M_order = ell_R_order # not sure here. just try the one from R. toCyclPowerBase would fail if this doesn't work
# CyclotomicField(l / prod(l.prime_divisors())) should also work.
# Transform to same denom.
denom_lcm = int(lcm(ell_R_denom, ell_M_denom))
ell_M = addRows(ell_M, denom_lcm / ell_M_denom)
M_R = [addRows(M_R_i, denom_lcm / ell_R_denom) for M_R_i in M_R]
ell_R_denom = ell_M_denom = denom_lcm
print "new denom:", denom_lcm
assert ell_R_denom == ell_M_denom
# ell_M rows are the elliptic FE. M_R[i] columns are the elliptic FE.
# We expect that M_R gives a higher precision for the ell FE. I'm not sure
# if this is always true but we expect it here (maybe not needed, though).
print "precision of M_R[0], ell_M, wanted:", M_R[0].nrows(), ell_M.ncols(), ce_prec
assert ell_M.ncols() >= ce_prec
prec = min(M_R[0].nrows(), ell_M.ncols())
# cut to have same precision
M_R = [M_R_i[:prec,:] for M_R_i in M_R]
ell_M = ell_M[:,:prec]
assert ell_M.ncols() == M_R[0].nrows() == prec
print "M_R[0] rank, herm rank, mult rank:", \
M_R[0].rank(), herm_modforms.rank(), (M_R[0] * herm_modforms).rank()
ell_R = [M_R_i * herm_modforms for M_R_i in M_R]
# I'm not sure on this. Seems to be true and it simplifies things in the following.
assert ell_M_order <= ell_R_order, "{0}".format((ell_M_order, ell_R_order))
assert ell_R_order % ell_M_order == 0, "{0}".format((ell_M_order, ell_R_order))
# Transform to same Cyclomotic Field in same power base.
ell_M2 = toCyclPowerBase(ell_M, ell_M_order)
ell_R2 = toLowerCyclBase(ell_R, ell_R_order, ell_M_order)
# We must work with the matrix. maybe we should transform hf_M instead to a
# higher order field instead, if this ever fails (I'm not sure).
assert ell_R2 is not None
assert len(ell_M2) == len(ell_R2) # They should have the same power base & same degree now.
print "ell_M2[0], ell_R2[0] rank with order %i:" % ell_M_order, ell_M2[0].rank(), ell_R2[0].rank()
assert len(M_R) == len(ell_M2)
for i in range(len(ell_M2)):
ell_M_space = ell_M2[i].row_space()
ell_R_space = ell_R2[i].column_space()
merged = ell_M_space.intersection(ell_R_space)
herm_modform_fe_expannsion_Ci = M_R[i].solve_right( merged.basis_matrix().transpose() )
herm_modform_fe_expannsion_Ci_module = herm_modform_fe_expannsion_Ci.column_module()
herm_modform_fe_expannsion_Ci_module += M_R[i].right_kernel()
extra_check_on_herm_superspace(
vs=herm_modform_fe_expannsion_Ci_module,
D=D, B_cF=calc.B_cF, HermWeight=HermWeight
)
herm_modform_fe_expannsion = herm_modform_fe_expannsion.intersection( herm_modform_fe_expannsion_Ci_module )
print "power", i, merged.dimension(), herm_modform_fe_expannsion_Ci_module.dimension(), \
herm_modform_fe_expannsion.dimension()
current_dimension = herm_modform_fe_expannsion.dimension()
return herm_modform_fe_expannsion
