def DirichletCharacterGaloisReps(N):
global DCGR_cache
if not N in DCGR_cache:
Chars = [ch[0] for ch in DG(N).galois_orbits()]
vv = [[[DC.multiplicative_order(chi)] +
character_traces(DC.sage_character(chi)), i]
for i, chi in enumerate(Chars)]
vv.sort()
DCGR_cache[N] = [Chars[v[1]] for v in vv]
return DCGR_cache[N]
class WebChar(WebObject, CachedRepresentation):
r"""
Class which should/might be replaced with
WebDirichletCharcter once this is ok.
"""
_key = ['modulus', 'number', 'version']
_file_key = ['modulus', 'number', 'version']
_collection_name = 'webchar'
def __init__(self,
modulus=1,
number=1,
update_from_db=True,
compute_values=False,
init_dynamic_properties=True):
r"""
Init self.
"""
emf_logger.debug("In WebChar {0}".format(
(modulus, number, update_from_db, compute_values)))
if isinstance(modulus, basestring):
try:
m, n = modulus.split('.')
modulus = int(m)
number = int(n)
except:
raise ValueError, "{0} does not correspond to the label of a WebChar".format(
modulus)
if not gcd(number, modulus) == 1:
raise ValueError, "Character number {0} of modulus {1} does not exist!".format(
number, modulus)
if number > modulus:
number = number % modulus
self._properties = WebProperties(
WebInt('conductor'), WebInt('modulus', value=modulus),
WebInt('number', value=number), WebInt('modulus_euler_phi'),
WebInt('order'), WebStr('latex_name'),
WebStr('label', value="{0}.{1}".format(modulus, number)),
WebNoStoreObject('sage_character', type(trivial_character(1))),
WebDict('_values_algebraic'), WebDict('_values_float'),
WebDict('_embeddings'),
WebFloat('version', value=float(emf_version)))
emf_logger.debug('Set properties in WebChar!')
super(WebChar,
self).__init__(update_from_db=update_from_db,
init_dynamic_properties=init_dynamic_properties)
#if not self.has_updated_from_db():
# self.init_dynamic_properties() # this was not done if we exited early
# compute = True
if compute_values:
self.compute_values()
#emf_logger.debug('In WebChar, self.__dict__ = {0}'.format(self.__dict__))
emf_logger.debug('In WebChar, self.number = {0}'.format(self.number))
def compute_values(self, save=False):
emf_logger.debug(
'in compute_values for WebChar number {0} of modulus {1}'.format(
self.number, self.modulus))
if self._values_algebraic == {} or self._values_float == {}:
changed = True
for i in range(self.modulus):
self.value(i, value_format='float')
self.value(i, value_format='algebraic')
if changed and save:
self.save_to_db()
else:
emf_logger.debug('Not saving.')
def init_dynamic_properties(self, embeddings=False):
if self.number is not None:
emf_logger.debug('number: {0}'.format(self.number))
self.character = DirichletCharacter_conrey(
DirichletGroup_conrey(self.modulus), self.number)
if not self.number == 1:
self.sage_character = self.character.sage_character()
else:
self.sage_character = trivial_character(self.modulus)
self.name = "Character nr. {0} of modulus {1}".format(
self.number, self.modulus)
if embeddings:
from lmfdb.modular_forms.elliptic_modular_forms.backend.emf_utils import dirichlet_character_conrey_galois_orbit_embeddings
emb = dirichlet_character_conrey_galois_orbit_embeddings(
self.modulus, self.number)
self.set_embeddings(emb)
c = self.character
if self.conductor == 0:
self.conductor = c.conductor()
if self.order == 0:
self.order = c.multiplicative_order()
if self.modulus_euler_phi == 0:
self.modulus_euler_phi = euler_phi(self.modulus)
if self.latex_name == '':
self.latex_name = "\chi_{" + str(self.modulus) + "}(" + str(
self.number) + ", \cdot)"
def is_trivial(self):
r"""
Check if self is trivial.
"""
return self.character.is_trivial()
def embeddings(self):
r"""
Returns a dictionary that maps the Conrey numbers
of the Dirichlet characters in the Galois orbit of ```self```
to the powers of $\zeta_{\phi(N)}$ so that the corresponding
embeddings map the labels.
Let $\zeta_{\phi(N)}$ be the generator of the cyclotomic field
of $N$-th roots of unity which is the base field
for the coefficients of a modular form contained in the database.
Considering the space $S_k(N,\chi)$, where $\chi = \chi_N(m, \cdot)$,
if embeddings()[m] = n, then $\zeta_{\phi(N)}$ is mapped to
$\mathrm{exp}(2\pi i n /\phi(N))$.
"""
return self._embeddings
def set_embeddings(self, d):
self._embeddings = d
def embedding(self):
r"""
Let $\zeta_{\phi(N)}$ be the generator of the cyclotomic field
of $N$-th roots of unity which is the base field
for the coefficients of a modular form contained in the database.
If ```self``` is given as $\chi = \chi_N(m, \cdot)$ in the Conrey naming scheme,
then we have to apply the map
\[
\zeta_{\phi(N)} \mapsto \mathrm{exp}(2\pi i n /\phi(N))
\]
to the coefficients of normalized newforms in $S_k(N,\chi)$
as in the database in order to obtain the coefficients corresponding to ```self```
(that is to elements in $S_k(N,\chi)$).
"""
return self._embeddings[self.number]
def __repr__(self):
r"""
Return the string representation of the character of self.
"""
return self.name
def value(self, x, value_format='algebraic'):
r"""
Return the value of self as an algebraic integer or float.
"""
x = int(x)
if value_format == 'algebraic':
if self._values_algebraic is None:
self._values_algebraic = {}
y = self._values_algebraic.get(x)
if y is None:
y = self._values_algebraic[x] = self.sage_character(x)
else:
self._values_algebraic[x] = y
return self._values_algebraic[x]
elif value_format == 'float': ## floating point
if self._values_float is None:
self._values_float = {}
y = self._values_float.get(x)
if y is None:
y = self._values_float[x] = self.character(x)
else:
self._values_float[x] = y
return self._values_float[x]
else:
raise ValueError, "Format {0} is not known!".format(value_format)
def url(self):
r"""
Return the url of self.
"""
if not hasattr(self, '_url') or self._url is None:
self._url = url_for('characters.render_Dirichletwebpage',
modulus=self.modulus,
number=self.number)
return self._url
class WebChar(WebObject, CachedRepresentation):
r"""
Class which should/might be replaced with
WebDirichletCharcter once this is ok.
"""
_key = ['modulus', 'number','version']
_file_key = ['modulus', 'number','version']
_collection_name = 'webchar'
def __init__(self, modulus=1, number=1, update_from_db=True, compute_values=False, init_dynamic_properties=True):
r"""
Init self.
"""
emf_logger.debug("In WebChar {0}".format((modulus,number,update_from_db,compute_values)))
if isinstance(modulus,basestring):
try:
m,n=modulus.split('.')
modulus = int(m); number=int(n)
except:
raise ValueError,"{0} does not correspond to the label of a WebChar".format(modulus)
if not gcd(number,modulus)==1:
raise ValueError,"Character number {0} of modulus {1} does not exist!".format(number,modulus)
if number > modulus:
number = number % modulus
self._properties = WebProperties(
WebInt('conductor'),
WebInt('modulus', value=modulus),
WebInt('number', value=number),
WebInt('modulus_euler_phi'),
WebInt('order'),
WebStr('latex_name'),
WebStr('label',value="{0}.{1}".format(modulus,number)),
WebNoStoreObject('sage_character', type(trivial_character(1))),
WebDict('_values_algebraic'),
WebDict('_values_float'),
WebDict('_embeddings'),
WebFloat('version', value=float(emf_version))
)
emf_logger.debug('Set properties in WebChar!')
super(WebChar, self).__init__(
update_from_db=update_from_db,
init_dynamic_properties=init_dynamic_properties
)
#if not self.has_updated_from_db():
# self.init_dynamic_properties() # this was not done if we exited early
# compute = True
if compute_values:
self.compute_values()
#emf_logger.debug('In WebChar, self.__dict__ = {0}'.format(self.__dict__))
emf_logger.debug('In WebChar, self.number = {0}'.format(self.number))
def compute_values(self, save=False):
emf_logger.debug('in compute_values for WebChar number {0} of modulus {1}'.format(self.number, self.modulus))
if self._values_algebraic == {} or self._values_float == {}:
changed = True
for i in range(self.modulus):
self.value(i,value_format='float')
self.value(i,value_format='algebraic')
if changed and save:
self.save_to_db()
else:
emf_logger.debug('Not saving.')
def init_dynamic_properties(self, embeddings=False):
if self.number is not None:
emf_logger.debug('number: {0}'.format(self.number))
self.character = DirichletCharacter_conrey(DirichletGroup_conrey(self.modulus),self.number)
if not self.number == 1:
self.sage_character = self.character.sage_character()
else:
self.sage_character = trivial_character(self.modulus)
self.name = "Character nr. {0} of modulus {1}".format(self.number,self.modulus)
if embeddings:
from lmfdb.modular_forms.elliptic_modular_forms.backend.emf_utils import dirichlet_character_conrey_galois_orbit_embeddings
emb = dirichlet_character_conrey_galois_orbit_embeddings(self.modulus,self.number)
self.set_embeddings(emb)
c = self.character
if self.conductor == 0:
self.conductor = c.conductor()
if self.order == 0:
self.order = c.multiplicative_order()
if self.modulus_euler_phi == 0:
self.modulus_euler_phi = euler_phi(self.modulus)
if self.latex_name == '':
self.latex_name = "\chi_{" + str(self.modulus) + "}(" + str(self.number) + ", \cdot)"
def is_trivial(self):
r"""
Check if self is trivial.
"""
return self.character.is_trivial()
def embeddings(self):
r"""
Returns a dictionary that maps the Conrey numbers
of the Dirichlet characters in the Galois orbit of ```self```
to the powers of $\zeta_{\phi(N)}$ so that the corresponding
embeddings map the labels.
Let $\zeta_{\phi(N)}$ be the generator of the cyclotomic field
of $N$-th roots of unity which is the base field
for the coefficients of a modular form contained in the database.
Considering the space $S_k(N,\chi)$, where $\chi = \chi_N(m, \cdot)$,
if embeddings()[m] = n, then $\zeta_{\phi(N)}$ is mapped to
$\mathrm{exp}(2\pi i n /\phi(N))$.
"""
return self._embeddings
def set_embeddings(self, d):
self._embeddings = d
def embedding(self):
r"""
Let $\zeta_{\phi(N)}$ be the generator of the cyclotomic field
of $N$-th roots of unity which is the base field
for the coefficients of a modular form contained in the database.
If ```self``` is given as $\chi = \chi_N(m, \cdot)$ in the Conrey naming scheme,
then we have to apply the map
\[
\zeta_{\phi(N)} \mapsto \mathrm{exp}(2\pi i n /\phi(N))
\]
to the coefficients of normalized newforms in $S_k(N,\chi)$
as in the database in order to obtain the coefficients corresponding to ```self```
(that is to elements in $S_k(N,\chi)$).
"""
return self._embeddings[self.number]
def __repr__(self):
r"""
Return the string representation of the character of self.
"""
return self.name
def value(self, x, value_format='algebraic'):
r"""
Return the value of self as an algebraic integer or float.
"""
x = int(x)
if value_format =='algebraic':
if self._values_algebraic is None:
self._values_algebraic = {}
y = self._values_algebraic.get(x)
if y is None:
y = self._values_algebraic[x]=self.sage_character(x)
else:
self._values_algebraic[x]=y
return self._values_algebraic[x]
elif value_format=='float': ## floating point
if self._values_float is None:
self._values_float = {}
y = self._values_float.get(x)
if y is None:
y = self._values_float[x]=self.character(x)
else:
self._values_float[x]=y
return self._values_float[x]
else:
raise ValueError,"Format {0} is not known!".format(value_format)
def url(self):
r"""
Return the url of self.
"""
if not hasattr(self, '_url') or self._url is None:
self._url = url_for('characters.render_Dirichletwebpage',modulus=self.modulus, number=self.number)
return self._url
