def get_new_and_oldspace_decomposition(k, N, xi=0):
r"""
Get decomposition of the new and oldspace S_k(N,xi) into submodules.
"""
M = ModularSymbols(N, k, sign=1).cuspidal_submodule()
L = list()
L = [M.new_submodule().dimension()]
check_dim = M.new_submodule().dimension()
for d in divisors(N):
if (d == 1):
continue
O = M.old_submodule(d)
Od = O.dimension()
if (d == N and k == 2 or Od == 0):
continue
S = ModularSymbols(ZZ(N / d), k,
sign=1).cuspidal_submodule().new_submodule()
Sd = S.dimension()
if (Sd == 0):
logger.info("%s, %s" % (O, Od))
logger.info("%s, %s" % (S, Sd))
mult = len(divisors(ZZ(d)))
check_dim = check_dim + mult * Sd
L.append((ZZ(N / d), mult, Sd))
check_dim = check_dim - M.dimension()
if (check_dim <> 0):
raise ArithmeticError, "Something wrong! check_dim=%s" % check_dim
return str(M.dimension(), L)
def get_new_and_oldspace_decomposition(k,N,xi=0):
r"""
Get decomposition of the new and oldspace S_k(N,xi) into submodules.
"""
M=ModularSymbols(N,k,sign=1).cuspidal_submodule()
L=list()
L=[M.new_submodule().dimension()]
check_dim=M.new_submodule().dimension()
for d in divisors(N):
if(d==1):
continue
O=M.old_submodule(d); Od=O.dimension()
if(d==N and k==2 or Od==0):
continue
S=ModularSymbols(ZZ(N/d),k,sign=1).cuspidal_submodule().new_submodule(); Sd=S.dimension()
if(Sd==0):
logger.info("%s, %s" % (O,Od))
logger.info("%s, %s" % (S,Sd))
mult=len(divisors(ZZ(d)))
check_dim=check_dim+mult*Sd
L.append((ZZ(N/d),mult,Sd))
check_dim=check_dim-M.dimension()
if(check_dim<>0):
raise ArithmeticError, "Something wrong! check_dim=%s" % check_dim
return str(M.dimension(),L)
def html_table(tbl):
r""" Takes a dictonary and returns an html-table.
INPUT:
-''tbl'' -- dictionary with the following keys
- headersh // horozontal headers
- headersv // vertical headers
- rows -- dictionary of rows of data
"""
ncols = len(tbl["headersh"])
nrows = len(tbl["headersv"])
data = tbl['data']
if (len(data) <> nrows):
logger.error("wrong number of rows!")
for i in range(nrows):
logger.info("len(%s)=%s" % (i, len(data[i])))
if (len(data[i]) <> ncols):
logger.error("wrong number of cols [=%s]!" % ncols)
if (tbl.has_key('atts')):
s = "<table " + str(tbl['atts']) + ">\n"
else:
s = "<table>\n"
format = dict()
for i in range(ncols):
format[i] = ''
if (tbl.has_key('data_format')):
if isinstance(tbl['data_format'], dict):
if (tbl['data_format'].has_key(i)):
format[i] = tbl['data_format'][i]
elif (isinstance(tbl['data_format'], str)):
format[i] = tbl['data_format']
if (tbl.has_key('header')):
s += "<thead><tr><th><td colspan=\"" + str(
ncols) + "\">" + tbl['header'] + "</td></th></tr></thead>"
s = s + "<tbody>"
#smath="<span class=\"math\">"
# check which type of content we have
h1 = tbl['headersh'][0]
sheaderh = ""
sheaderv = ""
h1 = tbl['headersv'][0]
col_width = dict()
if not tbl.has_key('col_width'):
# use maximum width as default
maxw = 0
for k in range(ncols):
for r in range(nrows):
l = len_as_printed(str(data[r][k]))
if l > maxw:
maxw = l
l = l * 10.0 # use true font size?
for k in range(ncols):
col_width[k] = maxw
else:
for i in range(ncols):
col_width[i] = 0
if tbl.has_key('col_width'):
if tbl['col_width'].has_key(i):
col_width[i] = tbl['col_width'][i]
if (tbl.has_key("corner_label")):
l = len_as_printed(str(tbl["corner_label"])) * 10
row = "<tr><td width=\"%s\">" % l
row += str(tbl["corner_label"]) + "</td>"
else:
row = "<tr><td></td>"
for k in range(ncols):
row = row + "<td>" + sheaderh + str(tbl['headersh'][k]) + "</td> \n"
row = row + "</tr> \n"
s = s + row
for r in range(nrows):
l = len_as_printed(str(tbl["headersv"][r])) * 10
logger.info("l=%s head=%s" % (l, tbl["headersv"]))
row = "<tr><td width=\"%s\">" % l
row += sheaderv + str(tbl['headersv'][r]) + "</td>"
for k in range(ncols):
wid = col_width[k]
if format[k] == 'html' or format[k] == 'text':
row = row + "\t<td halign=\"center\" width=\"" + str(
wid) + "\">"
if isinstance(data[r][k], list):
for ss in data[r][k]:
sss = str(ss)
if (len(sss) > 0):
row += sss
else:
sss = str(data[r][k])
row += sss
row = row + "</td> \n"
else:
row = row + "\t<td width=\"" + str(wid) + "\">"
if isinstance(data[r][k], list):
for ss in data[r][k]:
sss = latex(ss)
if (len(sss) > 0):
row += "\(" + sss + "\)"
else:
sss = latex(data[r][k])
if (len(sss) > 0):
row = row + "\(" + sss + "\)</td> \n"
row += "</td>\n"
# allow for different format in different columns
row = row + "</tr> \n"
s = s + row
s = s + "</tbody></table>"
return s
def find_inverse_images_of_twists(k, N=1, chi=0, fi=0, prec=10, verbose=0):
r"""
Checks if f is minimal and if not, returns the associated
minimal form to precision prec.
INPUT:
- ''k'' -- positive integer : the weight
- ''N'' -- positive integer (default 1) : level
- ''chi'' -- non-neg. integer (default 0) use character nr. chi
- ''fi'' -- non-neg. integer (default 0) We want to use the element nr. fi f=Newforms(N,k)[fi]
- ''prec'' -- integer (the number of coefficients to get)
- ''verbose'' -- integer
OUTPUT:
-''[t,l]'' -- tuple of a Bool t and a list l. The list l contains all tuples of forms which twists to the given form.
The actual minimal one is the first element of this list.
EXAMPLES::
"""
(t, f) = _get_newform(k, N, chi, fi)
if (not t):
return f
if (is_squarefree(ZZ(N))):
return [True, f]
# We need to check all square factors of N
logger.info("investigating: %s" % f)
N_sqfree = squarefree_part(ZZ(N))
Nsq = ZZ(N / N_sqfree)
twist_candidates = list()
KF = f.base_ring()
# check how many Hecke eigenvalues we need to check
max_nump = number_of_hecke_to_check(f)
maxp = max(primes_first_n(max_nump))
for d in divisors(N):
# we look at all d such that d^2 divdes N
if (not ZZ(d**2).divides(ZZ(N))):
continue
D = DirichletGroup(d)
# check possible candidates to twist into f
# g in S_k(M,chi) wit M=N/d^2
M = ZZ(N / d**2)
logger.info("Checking level %s" % M)
for xig in range(euler_phi(M)):
(t, glist) = _get_newform(k, M, xig)
if (not t):
return glist
for g in glist:
logger.debug("Comparing to function %s" % g)
KG = g.base_ring()
# we now see if twisting of g by xi in D gives us f
for xi in D:
try:
for p in primes_first_n(max_nump):
if (ZZ(p).divides(ZZ(N))):
continue
bf = f.q_expansion(maxp + 1)[p]
bg = g.q_expansion(maxp + 1)[p]
if (bf == 0 and bg == 0):
continue
elif (bf == 0 and bg <> 0 or bg == 0 and bf <> 0):
raise StopIteration()
if (ZZ(p).divides(xi.conductor())):
raise ArithmeticError, ""
xip = xi(p)
# make a preliminary check that the base rings match with respect to being real or not
try:
QQ(xip)
XF = QQ
if (KF <> QQ or KG <> QQ):
raise StopIteration
except TypeError:
# we have a non-rational (i.e. complex) value of the character
XF = xip.parent()
if ((KF == QQ or KF.is_totally_real()) and
(KG == QQ or KG.is_totally_real())):
raise StopIteration
## it is diffcult to compare elements from diferent rings in general but we make some checcks
## is it possible to see if there is a larger ring which everything can be coerced into?
ok = False
try:
a = KF(bg / xip)
b = KF(bf)
ok = True
if (a <> b):
raise StopIteration()
except TypeError:
pass
try:
a = KG(bg)
b = KG(xip * bf)
ok = True
if (a <> b):
raise StopIteration()
except TypeError:
pass
if (
not ok
): # we could coerce and the coefficients were equal
return "Could not compare against possible candidates!"
# otherwise if we are here we are ok and found a candidate
twist_candidates.append([dd, g.q_expansion(prec), xi])
except StopIteration:
# they are not equal
pass
#logger.debug("Candidates=%s" % twist_candidates)
if (len(twist_candidates) == 0):
return (True, None)
else:
return (False, twist_candidates)
def html_table(tbl):
r""" Takes a dictonary and returns an html-table.
INPUT:
-''tbl'' -- dictionary with the following keys
- headersh // horozontal headers
- headersv // vertical headers
- rows -- dictionary of rows of data
"""
ncols=len(tbl["headersh"])
nrows=len(tbl["headersv"])
data=tbl['data']
if(len(data)<>nrows):
logger.error("wrong number of rows!")
for i in range(nrows):
logger.info("len(%s)=%s" % (i,len(data[i])))
if(len(data[i])<>ncols):
logger.error("wrong number of cols [=%s]!" % ncols)
if(tbl.has_key('atts')):
s="<table "+str(tbl['atts'])+">\n"
else:
s="<table>\n"
format = dict()
for i in range(ncols):
format[i]=''
if(tbl.has_key('data_format')):
if isinstance(tbl['data_format'],dict):
if(tbl['data_format'].has_key(i)):
format[i]=tbl['data_format'][i]
elif(isinstance(tbl['data_format'],str)):
format[i]=tbl['data_format']
if(tbl.has_key('header')):
s+="<thead><tr><th><td colspan=\""+str(ncols)+"\">"+tbl['header']+"</td></th></tr></thead>"
s=s+"<tbody>"
#smath="<span class=\"math\">"
# check which type of content we have
h1=tbl['headersh'][0]
sheaderh=""; sheaderv=""
h1=tbl['headersv'][0]
col_width=dict()
if not tbl.has_key('col_width'):
# use maximum width as default
maxw = 0
for k in range(ncols):
for r in range(nrows):
l =len_as_printed(str(data[r][k]))
if l>maxw:
maxw = l
l = l*10.0 # use true font size?
for k in range(ncols):
col_width[k]=maxw
else:
for i in range(ncols):
col_width[i]=0
if tbl.has_key('col_width'):
if tbl['col_width'].has_key(i):
col_width[i]=tbl['col_width'][i]
if(tbl.has_key("corner_label")):
l = len_as_printed(str(tbl["corner_label"]))*10
row="<tr><td width=\"%s\">" % l
row+=str(tbl["corner_label"])+"</td>"
else:
row="<tr><td></td>"
for k in range(ncols):
row=row+"<td>"+sheaderh+str(tbl['headersh'][k])+"</td> \n"
row=row+"</tr> \n"
s=s+row
for r in range(nrows):
l = len_as_printed(str(tbl["headersv"][r]))*10
logger.info("l=%s head=%s" % (l,tbl["headersv"]))
row="<tr><td width=\"%s\">" %l
row+=sheaderv+str(tbl['headersv'][r])+"</td>"
for k in range(ncols):
wid = col_width[k]
if format[k]=='html' or format[k]=='text':
row=row+"\t<td halign=\"center\" width=\""+str(wid)+"\">"
if isinstance(data[r][k],list):
for ss in data[r][k]:
sss = str(ss)
if(len(sss)>0):
row+=sss
else:
sss = str(data[r][k])
row+=sss
row=row+"</td> \n"
else:
row=row+"\t<td width=\""+str(wid)+"\">"
if isinstance(data[r][k],list):
for ss in data[r][k]:
sss = latex(ss)
if(len(sss)>0):
row+="\("+sss+"\)"
else:
sss=latex(data[r][k])
if(len(sss)>0):
row=row+"\("+sss+"\)</td> \n"
row+="</td>\n"
# allow for different format in different columns
row=row+"</tr> \n"
s=s+row
s=s+"</tbody></table>"
return s
def find_inverse_images_of_twists(k,N=1,chi=0,fi=0,prec=10,verbose=0):
r"""
Checks if f is minimal and if not, returns the associated
minimal form to precision prec.
INPUT:
- ''k'' -- positive integer : the weight
- ''N'' -- positive integer (default 1) : level
- ''chi'' -- non-neg. integer (default 0) use character nr. chi
- ''fi'' -- non-neg. integer (default 0) We want to use the element nr. fi f=Newforms(N,k)[fi]
- ''prec'' -- integer (the number of coefficients to get)
- ''verbose'' -- integer
OUTPUT:
-''[t,l]'' -- tuple of a Bool t and a list l. The list l contains all tuples of forms which twists to the given form.
The actual minimal one is the first element of this list.
EXAMPLES::
"""
(t,f) = _get_newform(k,N,chi,fi)
if(not t):
return f
if(is_squarefree(ZZ(N))):
return [True,f]
# We need to check all square factors of N
logger.info("investigating: %s" % f)
N_sqfree=squarefree_part(ZZ(N))
Nsq=ZZ(N/N_sqfree)
twist_candidates=list()
KF=f.base_ring()
# check how many Hecke eigenvalues we need to check
max_nump=number_of_hecke_to_check(f)
maxp=max(primes_first_n(max_nump))
for d in divisors(N):
# we look at all d such that d^2 divdes N
if(not ZZ(d**2).divides(ZZ(N))):
continue
D=DirichletGroup(d)
# check possible candidates to twist into f
# g in S_k(M,chi) wit M=N/d^2
M=ZZ(N/d**2)
logger.info("Checking level %s"%M)
for xig in range(euler_phi(M)):
(t,glist) = _get_newform(k,M,xig)
if(not t):
return glist
for g in glist:
logger.debug("Comparing to function %s" %g)
KG=g.base_ring()
# we now see if twisting of g by xi in D gives us f
for xi in D:
try:
for p in primes_first_n(max_nump):
if(ZZ(p).divides(ZZ(N))):
continue
bf=f.q_expansion(maxp+1)[p]
bg=g.q_expansion(maxp+1)[p]
if(bf == 0 and bg == 0):
continue
elif(bf==0 and bg<>0 or bg==0 and bf<>0):
raise StopIteration()
if(ZZ(p).divides(xi.conductor())):
raise ArithmeticError,""
xip=xi(p)
# make a preliminary check that the base rings match with respect to being real or not
try:
QQ(xip)
XF=QQ
if( KF<>QQ or KG<>QQ):
raise StopIteration
except TypeError:
# we have a non-rational (i.e. complex) value of the character
XF=xip.parent()
if( (KF == QQ or KF.is_totally_real()) and (KG == QQ or KG.is_totally_real())):
raise StopIteration
## it is diffcult to compare elements from diferent rings in general but we make some checcks
## is it possible to see if there is a larger ring which everything can be coerced into?
ok=False
try:
a=KF(bg/xip); b=KF(bf)
ok=True
if(a<>b):
raise StopIteration()
except TypeError:
pass
try:
a=KG(bg); b=KG(xip*bf)
ok=True
if(a<>b):
raise StopIteration()
except TypeError:
pass
if(not ok): # we could coerce and the coefficients were equal
return "Could not compare against possible candidates!"
# otherwise if we are here we are ok and found a candidate
twist_candidates.append([dd,g.q_expansion(prec),xi])
except StopIteration:
# they are not equal
pass
#logger.debug("Candidates=%s" % twist_candidates)
if(len(twist_candidates)==0):
return (True,None)
else:
return (False,twist_candidates)