def sample_apoly_points_via_giac_rur(manifold, n):
import giac_rur
I = extended_ptolemy_equations(manifold)
R = I.ring()
p = cyclotomic_polynomial(n, var=R('M'))
I = I + [p]
return giac_rur.rational_univariate_representation(I)
def is_semisimple_modular(M, m, nprimes=5):
"""M is a pari matrix over Q(zeta_m). Check if M mod p has squarefree
char poly for nprimes primes p=1 (mod m). If True for any p,
return True since then the char poly of M itself must be
square-free. If False for all p, return False since the M
probably has non-squarefree char poly. There may therefore be
false negatives.
"""
pol = cyclotomic_polynomial(m)
pt = pari("t")
np = 0
for p in prime_range(1000000):
if m > 1 and not p % m == 1:
continue
np += 1
#print("testing modulo p={}".format(p))
if np > nprimes:
#print("op not semisimple modulo {} primes, so returning False".format(nprimes))
return False
zmodp = pari(pol.roots(GF(p))[0][0])
#print("zmodp = {}".format(zmodp))
try:
Mmodp = M.lift() * pari(mod(1, p))
#print("Lifted matrix = {}".format(Mmodp))
Mmodp = Mmodp.subst(pt, zmodp)
#print("matrix (mod {}) = {}".format(p,Mmodp))
modpcharpoly = Mmodp.charpoly()
#print("char poly (mod {}) = {}".format(p,modpcharpoly))
if modpcharpoly.issquarefree():
#print("op is semisimple mod {}".format(p))
return True
else:
#print("op is not semisimple mod {}".format(p))
pass
except PariError: ## happens if M not integral mod p
np -= 1
def Newforms_v2(N, k, chi_number, dmax=20, nan=100, Detail=0):
t0 = time.time()
G = pari(N).znstar(1)
chi_dc = char_orbit_index_to_DC_number(N, chi_number)
chi_gp = G.znconreylog(chi_dc)
chi_order = ZZ(G.charorder(chi_gp))
if Detail:
print("Decomposing space {}:{}:{}".format(N, k, chi_number))
NK = [N, k, [G, chi_gp]]
pNK = pari(NK)
if Detail > 1:
print("NK = {} (gp character = {})".format(NK, chi_gp))
SturmBound = pNK.mfsturm()
Snew = pNK.mfinit(0)
total_dim = Snew.mfdim(
) # this is the relative dimension i.e. degree over Q(chi)
# Get the character polynomial
# Note that if the character order is 2*m with m odd then Pari uses the
# m'th cyclotomic polynomial and not the 2m'th (e.g. for a
# character of order 6 it uses t^2+t+1 and not t^2-t+1).
chi_order_2 = chi_order // 2 if chi_order % 4 == 2 else chi_order
chipoly = cyclotomic_polynomial(chi_order_2, 't')
chi_degree = chipoly.degree()
assert chi_degree == euler_phi(chi_order) == euler_phi(chi_order_2)
t05 = time.time()
if Detail:
print(
"Computed newspace {}:{}:{} in {:0.3f}, dimension={}*{}={}, now splitting into irreducible subspaces"
.format(N, k, chi_number, t05 - t0, chi_degree, total_dim,
chi_degree * total_dim))
if Detail > 1:
print("Sturm bound = {}".format(SturmBound))
print("character order = {}".format(chi_order))
if total_dim == 0:
if Detail:
print("The space {}:{}:{} is empty".format(N, k, chi_number))
return []
# First just compute Hecke matrices one at a time, to find a splitting operator
def Hecke_op_iter():
p = ZZ(1)
while True:
p = p.next_prime()
# while p.divides(N):
# p=p.next_prime()
#print("Computing T_{}".format(p))
yield p, Snew.mfheckemat(p)
Tp_iter = Hecke_op_iter()
p, op = Tp_iter.next()
s1 = time.time()
if Detail:
print("testing T_{}".format(p))
ok = is_semisimple_modular(op, chi_order_2)
# f = op.charpoly()
# ok = f.issquarefree()
if ok:
if Detail:
print("Lucky first time: semisimple. Finding char poly")
f = op.charpoly()
ops = [(p, op)]
while not ok:
pi, opi = Tp_iter.next()
if Detail:
print("testing T_{}".format(pi))
ok = is_semisimple_modular(op, chi_order_2)
# f = opi.charpoly()
# ok = f.issquarefree()
if ok:
if Detail:
print("success using T_{}. Finding char poly".format(pi))
op = opi
f = op.charpoly()
break
else:
#ops.append((pi,opi))
ops += [(pi, opi)]
if Detail:
print("T_{} not semisimple".format(pi))
print("testing combinations...")
for j in range(5):
co = [ZZ.random_element(-5, 5) for _ in ops]
while not co:
co = [ZZ.random_element(-5, 5) for _ in ops]
if Detail:
print("Testing lin comb of {} ops with coeffs {}".format(
len(co), co))
op = sum([ci * opj[1] for ci, opj in zip(co, ops)])
ok = is_semisimple_modular(op, chi_order_2)
# f=op.charpoly()
# ok = f.issquarefree()
if ok:
if Detail:
print(
"success using {}-combo of T_p for p in {}. Finding char poly"
.format(co, [opj[0] for opj in ops]))
f = op.charpoly()
break
if not ok:
raise RuntimeError(
"failed to find a 0,1-combination of Tp which is semisimple")
ffac = f.factor()
nnf = ffac.matsize()[0]
gp_pols = pari_col1(ffac)
pols = [pol for pol in gp_pols]
reldims = [pol.poldegree() for pol in pols]
dims = [d * chi_degree for d in reldims]
# We'll need the coefficients an, if any newforms have dimension >1 and <=dmax.
an_needed = [
i for i, d in enumerate(dims) if d > 1 and (dmax == 0 or d <= dmax)
]
if Detail:
print("Need to compute a_n for {} newforms: {}".format(
len(an_needed), an_needed))
s2 = time.time()
if Detail:
print("Computed splitting in {:0.3f}, # newforms = {}".format(
s2 - s1, nnf))
print("relative dims = {}, absolute dims = {}".format(reldims, dims))
# Compute AL-matrices if character is trivial:
if chi_order == 1:
Qlist = [(pr, pr**e) for pr, e in ZZ(N).factor()]
ALs = [Snew.mfatkininit(Q[1])[1] for Q in Qlist]
if Detail:
print("AL-matrices:")
for Q, AL in zip(Qlist, ALs):
print("W_{}={}".format(Q[1], AL))
if nnf == 1 and dims[0] > dmax and dmax > 0:
if Detail:
print(
"Only one newform and dim={}, so use traceform to get traces".
format(dims[0]))
traces = pNK.mftraceform().mfcoefs(nan)
if Detail > 1:
print("raw traces: {}".format(traces))
if chi_degree > 1:
# This is faster than the more simple
# traces = [c.trace() for c in traces]
gptrace = pari_trace(chi_degree)
traces = pari.apply(gptrace, traces)
if Detail > 1:
print("traces to QQ: {}".format(traces))
traces = gen_to_sage(traces)[1:]
traces[0] = dims[0]
if Detail > 1:
print("final traces: {}".format(traces))
traces = [traces]
else: # >1 newform, or just one but its absolute dim is <=dmax
hs = [f / fi for fi in pols]
if Detail > 1:
print("fs: {}".format(pols))
print("hs: {}".format(hs))
print(" with degrees {}".format([h.poldegree() for h in hs]))
if Detail > 1:
print("Starting to compute gcds")
As = [(hi * (fi.gcdext(hi)[2])).subst(pari_x, op)
for fi, hi in zip(pols, hs)]
if Detail:
print("Computed idempotent matrix decomposition")
ims = [A.matimage() for A in As]
U = pari.matconcat(ims)
Uinv = U**(-1)
if Detail:
print("Computed U and U^-1")
starts = [1 + sum(d for d in reldims[:i]) for i in range(len(reldims))]
stops = [sum(d for d in reldims[:i + 1]) for i in range(len(reldims))]
slicers = [pari_row_slice(r1, r2) for r1, r2 in zip(starts, stops)]
ums = [slice(Uinv) for slice in slicers]
imums = [imA * umA for imA, umA in zip(ims, ums)]
s3 = time.time()
if Detail:
print("Computed projectors in {:0.3f}".format(s3 - s2))
print("Starting to compute {} Hecke matrices T_n".format(nan))
heckemats = Snew.mfheckemat(pari(range(1, nan + 1)))
s4 = time.time()
if Detail:
print("Computed {} Hecke matrices in {:0.3f}s".format(
nan, s4 - s3))
# If we are going to compute any a_n then we now compute
# umA*T*imA for all Hecke matrices T, whose traces give the
# traces and whose first columns (or any row or column) give
# the coefficients of the a_n with respect to some
# Q(chi)-basis for the Hecke field.
# But if we only need the traces then it is faster to
# precompute imA*umA=imAumA and then the traces are
# trace(imAumA*T). NB trace(UMV)=trace(VUM)!
if Detail:
print("Computing traces")
# Note that computing the trace of a matrix product is faster
# than first computing the product and then the trace:
gptrace = pari(
'c->if(type(c)=="t_POLMOD",trace(c),c*{})'.format(chi_degree))
traces = [[
gen_to_sage(gptrace(pari_trace_product(T, imum)))
for T in heckemats
] for imum in imums]
s4 = time.time()
if Detail:
print("Computed traces to Z in {:0.3f}".format(s4 - s3))
for tr in traces:
print(tr[:20])
ans = [None for _ in range(nnf)]
bases = [None for _ in range(nnf)]
if an_needed:
if Detail:
print("...computing a_n...")
for i in an_needed:
dr = reldims[i]
if Detail:
print("newform #{}/{}, relative dimension {}".format(
i, nnf, dr))
# method: for each irreducible component we have matrices
# um and im (sizes dr x n and n x dr where n is the
# relative dimension of the whole space) such that for
# each Hecke matrix T, its restriction to the component is
# um*T*im. To get the eigenvalue in a suitable basis all
# we need do is take any one column (or row): we choose
# the first column. So the computation can be done as
# um*(T*im[,1]) (NB the placing of parentheses).
imsicol1 = pari_col1(ims[i])
umsi = ums[i]
ans[i] = [(umsi * (T * imsicol1)).Vec() for T in heckemats]
if Detail:
print("ans done")
if Detail > 1:
print("an: {}...".format(ans[i]))
# Now compute the bases (of the relative Hecke field over
# Q(chi) w.r.t which these coefficients are given. Here
# we use e1 because we picked the first column just now.
B = ums[i] * op * ims[i]
e1 = pari_e1(dr)
cols = [e1]
while len(cols) < dr:
cols.append(B * cols[-1])
W = pari.matconcat(cols)
bases[i] = W.mattranspose()**(-1)
if Detail > 1:
print("basis = {}".format(bases[i].lift()))
# Compute AL-eigenvalues if character is trivial:
if chi_order == 1:
ALeigs = [[[Q[0],
gen_to_sage((umA * (AL * (pari_col1(imA))))[0])]
for Q, AL in zip(Qlist, ALs)] for umA, imA in zip(ums, ims)]
if Detail > 1: print("ALeigs: {}".format(ALeigs))
else:
ALeigs = [[] for _ in range(nnf)]
Nko = (N, k, chi_number)
#print("len(traces) = {}".format(len(traces)))
#print("len(newforms) = {}".format(len(newforms)))
#print("len(pols) = {}".format(len(pols)))
#print("len(ans) = {}".format(len(ans)))
#print("len(ALeigs) = {}".format(len(ALeigs)))
pari_nfs = [{
'Nko': Nko,
'SB': SturmBound,
'chipoly': chipoly,
'poly': pols[i],
'ans': ans[i],
'basis': bases[i],
'ALeigs': ALeigs[i],
'traces': traces[i],
} for i in range(nnf)]
# We could return these as they are but the next processing step
# will fail if the underlying gp process has quit, so we do the
# processing here.
# This processing returns full data but the polynomials have not
# yet been polredbested and the an coefficients have not been
# optimized (or even made integral):
#return pari_nfs
t1 = time.time()
if Detail:
print("{}: finished constructing pari newforms (time {:0.3f})".format(
Nko, t1 - t0))
nfs = [process_pari_nf(nf, dmax, Detail) for nf in pari_nfs]
if len(nfs) > 1:
nfs.sort(key=lambda f: f['traces'])
t2 = time.time()
if Detail:
print(
"{}: finished first processing of newforms (time {:0.3f})".format(
Nko, t2 - t1))
if Detail > 2:
for nf in nfs:
if 'eigdata' in nf:
print(nf['eigdata']['ancs'])
nfs = [bestify_newform(nf, dmax, Detail) for nf in nfs]
t3 = time.time()
if Detail:
print("{}: finished bestifying newforms (time {:0.3f})".format(
Nko, t3 - t2))
if Detail > 2:
for nf in nfs:
if 'eigdata' in nf:
print(nf['eigdata']['ancs'])
nfs = [integralify_newform(nf, dmax, Detail) for nf in nfs]
t4 = time.time()
if Detail:
print("{}: finished integralifying newforms (time {:0.3f})".format(
Nko, t4 - t3))
if Detail > 2:
for nf in nfs:
if 'eigdata' in nf:
print(nf['eigdata']['ancs'])
print("Total time for space {}: {:0.3f}".format(Nko, t4 - t0))
return nfs
def Newforms_v1(N, k, chi_number, dmax=20, nan=100, Detail=0):
t0 = time.time()
G = pari(N).znstar(1)
chi_dc = char_orbit_index_to_DC_number(N, chi_number)
chi_gp = G.znconreylog(chi_dc)
chi_order = ZZ(G.charorder(chi_gp))
if Detail:
print("Decomposing space {}:{}:{}".format(N, k, chi_number))
NK = [N, k, [G, chi_gp]]
pNK = pari(NK)
if Detail > 1:
print("NK = {} (gp character = {})".format(NK, chi_gp))
SturmBound = pNK.mfsturm()
Snew = pNK.mfinit(0)
total_dim = Snew.mfdim()
# Get the character polynomial
# Note that if the character order is 2*m with m odd then Pari uses the
# m'th cyclotomic polynomial and not the 2m'th (e.g. for a
# character of order 6 it uses t^2+t+1 and not t^2-t+1).
chi_order_2 = chi_order // 2 if chi_order % 4 == 2 else chi_order
chipoly = cyclotomic_polynomial(chi_order_2, 't')
chi_degree = chipoly.degree()
assert chi_degree == euler_phi(chi_order) == euler_phi(chi_order_2)
if Detail:
print(
"Computed newspace {}:{}:{}, dimension={}*{}={}, now splitting into irreducible subspaces"
.format(N, k, chi_number, chi_degree, total_dim,
chi_degree * total_dim))
if Detail > 1:
print("Sturm bound = {}".format(SturmBound))
print("character order = {}".format(chi_order))
# Get the relative polynomials: these are polynomials in y with coefficients either integers or polmods with modulus chipoly
pols = Snew.mfsplit(0, 1)[1]
if Detail > 2: print("pols[GP] = {}".format(pols))
nnf = len(pols)
dims = [chi_degree * f.poldegree() for f in pols]
if nnf == 0:
if Detail:
print("The space {}:{}:{} is empty".format(N, k, chi_number))
return []
if Detail:
print("The space {}:{}:{} has {} newforms, dimensions {}".format(
N, k, chi_number, nnf, dims))
# Get the traces. NB (1) mftraceform will only give the trace
# form on the whole space so we only use this when nnf==1,
# i.e. the space is irreducible. Otherwise we'll need to compute
# traces from the ans. (2) these are only traces down to Q(chi)
# so when that has degree>1 (and only then) we need to take an
# additional trace.
traces = [None for _ in range(nnf)]
if nnf == 1:
d = ZZ(chi_degree * (pols[0]).poldegree())
if Detail:
print("Only one newform so use traceform to get traces")
traces = pNK.mftraceform().mfcoefs(nan)
if Detail > 1:
print("raw traces: {}".format(traces))
if chi_degree > 1:
# This is faster than the more simple
# traces = [c.trace() for c in traces]
gptrace = pari_trace(chi_degree)
traces = pari.apply(gptrace, traces)
if Detail > 1:
print("traces to QQ: {}".format(traces))
traces = gen_to_sage(traces)[1:]
traces[0] = d
if Detail > 1:
print("final traces: {}".format(traces))
traces = [traces]
# Get the coefficients an. We'll need these for a newform f if
# either (1) its dimension is >1 and <= dmax, when we want to
# store them, or (2) there is more than one newform, so we can
# later compute the traces from them. So we don't need them if
# nnf==1 and the dimension>dmax.
if dmax == 0 or nnf > 1 or ((chi_degree * (pols[0]).poldegree()) <= dmax):
if Detail > 1:
print("...computing mfeigenbasis...")
newforms = Snew.mfeigenbasis()
if Detail > 1:
print("...computing {} mfcoefs...".format(nan))
coeffs = Snew.mfcoefs(nan)
ans = [coeffs * Snew.mftobasis(nf) for nf in newforms]
if Detail > 2:
print("ans[GP] = {}".format(ans))
else:
# there is only one newform (so we have the traces) and its
# dimension is >dmax (so we will not need the a_n):
ans = [None for _ in range(nnf)]
newforms = [None for _ in range(nnf)]
# Compute AL-eigenvalues if character is trivial:
if chi_order == 1:
Qlist = [(p, p**e) for p, e in ZZ(N).factor()]
ALs = [gen_to_sage(Snew.mfatkineigenvalues(Q[1])) for Q in Qlist]
if Detail: print("ALs: {}".format(ALs))
# "transpose" this list of lists:
ALeigs = [[[Q[0], ALs[i][j][0]] for i, Q in enumerate(Qlist)]
for j in range(nnf)]
if Detail: print("ALeigs: {}".format(ALeigs))
else:
ALeigs = [[] for _ in range(nnf)]
Nko = (N, k, chi_number)
# print("len(traces) = {}".format(len(traces)))
# print("len(newforms) = {}".format(len(newforms)))
# print("len(pols) = {}".format(len(pols)))
# print("len(ans) = {}".format(len(ans)))
# print("len(ALeigs) = {}".format(len(ALeigs)))
pari_nfs = [{
'Nko': Nko,
'SB': SturmBound,
'chipoly': chipoly,
'pari_newform': newforms[i],
'poly': pols[i],
'ans': ans[i],
'ALeigs': ALeigs[i],
'traces': traces[i],
} for i in range(nnf)]
# This processing returns full data but the polynomials have not
# yet been polredbested and the an coefficients have not been
# optimized (or even made integral):
#return pari_nfs
t1 = time.time()
if Detail:
print("{}: finished constructing GP newforms (time {:0.3f})".format(
Nko, t1 - t0))
nfs = [process_pari_nf_v1(nf, dmax, Detail) for nf in pari_nfs]
if len(nfs) > 1:
nfs.sort(key=lambda f: f['traces'])
t2 = time.time()
if Detail:
print(
"{}: finished first processing of newforms (time {:0.3f})".format(
Nko, t2 - t1))
if Detail > 2:
for nf in nfs:
if 'eigdata' in nf:
print(nf['eigdata']['ancs'])
nfs = [bestify_newform(nf, dmax, Detail) for nf in nfs]
t3 = time.time()
if Detail:
print("{}: finished bestifying newforms (time {:0.3f})".format(
Nko, t3 - t2))
if Detail > 2:
for nf in nfs:
if 'eigdata' in nf:
print(nf['eigdata']['ancs'])
nfs = [integralify_newform(nf, dmax, Detail) for nf in nfs]
t4 = time.time()
if Detail:
print("{}: finished integralifying newforms (time {:0.3f})".format(
Nko, t4 - t3))
if Detail > 2:
for nf in nfs:
if 'eigdata' in nf:
print(nf['eigdata']['ancs'])
print("Total time for space {}: {:0.3f}".format(Nko, t4 - t0))
return nfs
def Newforms_v1(N, k, chi_number, dmax=20, nan=100, Detail=0, dims_only=False):
t0=time.time()
G = pari(N).znstar(1)
Qx = PolynomialRing(QQ,'x')
chi_dc = char_orbit_index_to_DC_number(N,chi_number)
chi_gp = G.znconreylog(chi_dc)
chi_order = ZZ(G.charorder(chi_gp))
if Detail:
print("Decomposing space {}:{}:{}".format(N,k,chi_number))
NK = [N,k,[G,chi_gp]]
pNK = pari(NK)
if Detail>1:
print("NK = {} (gp character = {})".format(NK,chi_gp))
SturmBound = pNK.mfsturm()
Snew = pNK.mfinit(0)
reldim = Snew.mfdim()
if reldim==0:
if Detail:
print("The space {}:{}:{} is empty".format(N,k,chi_number))
return []
# Get the character polynomial
# Note that if the character order is 2*m with m odd then Pari uses the
# m'th cyclotomic polynomial and not the 2m'th (e.g. for a
# character of order 6 it uses t^2+t+1 and not t^2-t+1).
chi_order_2 = chi_order//2 if chi_order%4==2 else chi_order
chipoly = cyclotomic_polynomial(chi_order_2,'t')
chi_degree = chipoly.degree()
totdim = reldim * chi_degree
assert chi_degree==euler_phi(chi_order)==euler_phi(chi_order_2)
if Detail:
print("Computed newspace {}:{}:{}, dimension={}*{}={}, now splitting into irreducible subspaces".format(N,k,chi_number, chi_degree, reldim, totdim))
if Detail>1:
print("Sturm bound = {}".format(SturmBound))
print("character order = {}".format(chi_order))
# Get the relative polynomials: these are polynomials in y with coefficients either integers or polmods with modulus chipoly
# But we only need the poly for the largest space if its absolute dimension is less than 20
d = max(reldim // 2, dmax // chi_degree) if dmax else 0
if dims_only:
if Detail: t1=time.time(); print("...calling mfsplit({},{})...".format(d,1))
forms, pols = Snew.mfsplit(d,1)
if Detail: print("Call to mfsplit took {:.3f} secs".format(time.time()-t1))
if Detail>2: print("mfsplit({},1) returned pols[GP] = {}".format(d,pols))
dims = [chi_degree*f.poldegree() for f in pols]
dims += [] if sum(dims+[0]) == totdim else [totdim-sum(dims+[0])] # add dimension of largest newform if needed
nnf = len(dims)
if Detail:
print("The space {}:{}:{} has {} newforms, dimensions {}".format(N,k,chi_number,nnf,dims))
pari_nfs = [
{ 'Nko': (N,k,chi_number),
'SB': SturmBound,
'chipoly': chipoly,
'dim': dims[i],
'traces': [],
'ALeigs': [],
'ans': [],
'poly': Qx.gen() if dims[i] == 1 else (pols[i] if dims[i] <= dmax else None),
'pari_newform':None,
'best_poly':None
} for i in range(nnf)]
return pari_nfs
if Detail: t1=time.time(); print("...calling mfsplit({},{})...".format(d,d))
forms, pols = Snew.mfsplit(d,d)
if Detail: print("Call to mfsplit took {:.3f} secs".format(time.time()-t1))
if Detail>2: print("mfsplit({},1) returned pols[GP] = {}".format(d,pols))
dims = [chi_degree*f.poldegree() for f in pols]
dims += [] if sum(dims+[0]) == totdim else [totdim-sum(dims+[0])] # add dimension of largest newform if needed
nnf = len(dims)
if Detail:
print("The space {}:{}:{} has {} newforms, dimensions {}".format(N,k,chi_number,nnf,dims))
# Compute trace forms using a combination of mftraceform and mfsplit (this avoids the need to compute a complete eigenbasis)
# When the newspace contains a large irreducible subspace (and zero or more small ones) this saves a huge amount of time (e.g. 1000x faster)
traces = [None for _ in range(nnf)]
if Detail: t1=time.time(); print("...calling mftraceform...")
straces = pNK.mftraceform().mfcoefs(nan)
if Detail: print("Call to mftraceform took {:.3f} secs".format(time.time()-t1))
straces = gen_to_sage(pari.apply(pari_trace(chi_degree),straces))
if Detail>1: print("Newspace traceform: {}".format(straces))
# Compute coefficients here (but note that we don't need them if there is only one newform and its dimension is > dmax)
if nnf>1 or dmax==0 or dims[0] <= dmax:
if Detail: t1=time.time(); print("...computing {} mfcoefs...".format(nan))
coeffs = Snew.mfcoefs(nan)
if Detail: print("Call to mfcoefs took {:.3f} secs".format(time.time()-t1))
if nnf==1:
traces[0] = straces[1:]
else:
if Detail: s0=time.time()
tforms = [pari.apply("trace",forms[i]) if pols[i].poldegree() > 1 else forms[i] for i in range(nnf-1)]
ttraces = [pari.apply(pari_trace(chi_degree),coeffs*nf) for nf in tforms]
ltraces = [straces[i] - sum([ttraces[j][i] for j in range(len(ttraces))]) for i in range(nan+1)]
traces = [list(t)[1:] for t in ttraces] + [ltraces[1:]]
if Detail>1: print("Traceforms: {}".format(traces))
if Detail: print("Spent {:.3f} secs computing traceforms".format(time.time()-s0))
# Get coefficients an for all newforms f of dim <= dmax (if dmax is set)
# Note that even if there are only dimension 1 forms we want to compute an so that pari puts the AL-eigenvalues in the right order
# m = max([d for d in dims if (dmax==0 or d<=dmax)] + [0])
d1 = len([d for d in dims if d == 1])
dm = len([i for i in range(nnf) if dmax==0 or dims[i] <= dmax])
ans = [traces[i] for i in range(d1)] + [coeffs*forms[i] for i in range(d1,dm)] + [None for i in range(dm,nnf)]
if Detail>2: print("ans[GP] = {}".format(ans))
newforms = [None for i in range(d1)] + [forms[i] for i in range(d1,dm)] + [None for i in range(dm,nnf)]
# Compute AL-eigenvalues if character is trivial:
if chi_order==1:
Qlist = [(p,p**e) for p,e in ZZ(N).factor()]
ALs = [gen_to_sage(Snew.mfatkineigenvalues(Q[1])) for Q in Qlist]
if Detail: print("ALs: {}".format(ALs))
# "transpose" this list of lists:
ALeigs = [[[Q[0],ALs[i][j][0]] for i,Q in enumerate(Qlist)] for j in range(nnf)]
if Detail: print("ALeigs: {}".format(ALeigs))
else:
ALeigs = [[] for _ in range(nnf)]
Nko = (N,k,chi_number)
if Detail: print("traces set = {}".format([1 if t else 0 for t in traces]))
if Detail: print("ans set = {}".format([1 if a else 0 for a in ans]))
pari_nfs = [
{ 'Nko': Nko,
'SB': SturmBound,
'chipoly': chipoly,
'dim': dims[i],
'pari_newform': newforms[i],
'poly': Qx.gen() if dims[i] == 1 else (pols[i] if dims[i] <= dmax else None),
'best_poly': Qx.gen() if dims[i] == 1 else None,
'ans': ans[i],
'ALeigs': ALeigs[i],
'traces': traces[i],
} for i in range(nnf)]
if len(pari_nfs)>1:
pari_nfs.sort(key=lambda f: f['traces'])
t1=time.time()
if Detail:
print("{}: finished constructing GP newforms (time {:0.3f})".format(Nko,t1-t0))
if d1 == dm:
return pari_nfs
# At this point we have everything we need, but the ans have not been optimized (or even made integral)
nfs = [process_pari_nf_v1(pari_nfs[i], dmax, Detail) for i in range(d1,dm)]
t2=time.time()
if Detail:
print("{}: finished first processing of newforms (time {:0.3f})".format(Nko,t2-t1))
if Detail>2:
for nf in nfs:
if 'eigdata' in nf:
print(nf['eigdata']['ancs'])
nfs = [bestify_newform(nf,dmax,Detail) for nf in nfs]
t3=time.time()
if Detail:
print("{}: finished bestifying newforms (time {:0.3f})".format(Nko,t3-t2))
if Detail>2:
for nf in nfs:
if 'eigdata' in nf:
print(nf['eigdata']['ancs'])
nfs = [integralify_newform(nf,dmax,Detail) for nf in nfs]
t4=time.time()
if Detail:
print("{}: finished integralifying newforms (time {:0.3f})".format(Nko,t4-t3))
if Detail>2:
for nf in nfs:
if 'eigdata' in nf:
print(nf['eigdata']['ancs'])
print("Total time for space {}: {:0.3f}".format(Nko,t4-t0))
return [pari_nfs[i] for i in range(d1)] + nfs + [pari_nfs[i] for i in range(dm,nnf)]
def cyclotomic_polynomial_bound_4():
return [cyclotomic_polynomial(k) for k in [1, 2, 3, 4, 5, 6, 8, 10, 12] ]