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""" 

Hecke modules 

""" 

 

#***************************************************************************** 

# Copyright (C) 2004,2005,2006 William Stein <wstein@gmail.com> 

# 

# This program is free software: you can redistribute it and/or modify 

# it under the terms of the GNU General Public License as published by 

# the Free Software Foundation, either version 2 of the License, or 

# (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from __future__ import print_function 

from __future__ import absolute_import 

 

import sage.rings.all 

import sage.arith.all as arith 

import sage.misc.misc as misc 

import sage.modules.module 

from sage.structure.all import Sequence 

import sage.matrix.matrix_space as matrix_space 

from sage.structure.parent import Parent 

 

import sage.misc.prandom as random 

 

from . import algebra 

from . import element 

from . import hecke_operator 

 

from sage.modules.all import FreeModule 

 

def is_HeckeModule(x): 

r""" 

Return True if x is a Hecke module. 

 

EXAMPLES:: 

 

sage: from sage.modular.hecke.module import is_HeckeModule 

sage: is_HeckeModule(ModularForms(Gamma0(7), 4)) 

True 

sage: is_HeckeModule(QQ^3) 

False 

sage: is_HeckeModule(J0(37).homology()) 

True 

""" 

return isinstance(x, HeckeModule_generic) 

 

class HeckeModule_generic(sage.modules.module.Module): 

r""" 

A very general base class for Hecke modules. 

 

We define a Hecke module of weight `k` to be a module over a commutative 

ring equipped with an action of operators `T_m` for all positive integers `m` 

coprime to some integer `n`(the level), which satisfy `T_r T_s = T_{rs}` for 

`r,s` coprime, and for powers of a prime `p`, `T_{p^r} = T_{p} T_{p^{r-1}} - 

\varepsilon(p) p^{k-1} T_{p^{r-2}}`, where `\varepsilon(p)` is some 

endomorphism of the module which commutes with the `T_m`. 

 

We distinguish between *full* Hecke modules, which also have an action of 

operators `T_m` for `m` not assumed to be coprime to the level, and 

*anemic* Hecke modules, for which this does not hold. 

""" 

 

Element = element.HeckeModuleElement 

 

def __init__(self, base_ring, level, category=None): 

r""" 

Create a Hecke module. Not intended to be called directly. 

 

EXAMPLES:: 

 

sage: CuspForms(Gamma0(17),2) # indirect doctest 

Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(17) of weight 2 over Rational Field 

sage: ModularForms(3, 3).category() 

Category of Hecke modules over Rational Field 

""" 

if not isinstance(base_ring, sage.rings.all.CommutativeRing): 

raise TypeError("base_ring must be commutative ring") 

 

from sage.categories.hecke_modules import HeckeModules 

default_category = HeckeModules(base_ring) 

if category is None: 

category = default_category 

else: 

assert category.is_subcategory(default_category), "%s is not a subcategory of %s"%(category, default_category) 

 

sage.modules.module.Module.__init__(self, base_ring, category=category) 

 

level = sage.rings.all.ZZ(level) 

if level <= 0: 

raise ValueError("level (=%s) must be positive"%level) 

self.__level = level 

self._hecke_matrices = {} 

self._diamond_matrices = {} 

 

def __setstate__(self, state): 

r""" 

Ensure that the category is initialized correctly on unpickling. 

 

EXAMPLES:: 

 

sage: loads(dumps(ModularSymbols(11))).category() # indirect doctest 

Category of Hecke modules over Rational Field 

""" 

if not self._is_category_initialized(): 

from sage.categories.hecke_modules import HeckeModules 

self._init_category_(HeckeModules(state['_base'])) 

sage.modules.module.Module.__setstate__(self, state) 

 

def __hash__(self): 

r""" 

The hash is determined by the base ring and the level. 

 

EXAMPLES:: 

 

sage: MS = sage.modular.hecke.module.HeckeModule_generic(QQ,1) 

sage: hash(MS) == hash((MS.base_ring(), MS.level())) 

True 

 

""" 

return hash((self.base_ring(), self.__level)) 

 

def _compute_hecke_matrix_prime_power(self, p, r, **kwds): 

r""" 

Compute the Hecke matrix T_{p^r}, where `p` is prime and `r \ge 2`, assuming that 

`T_p` is known. This is carried out by recursion. 

 

All derived classes must override either this function or ``self.character()``. 

 

EXAMPLES:: 

 

sage: M = ModularForms(SL2Z, 24) 

sage: M._compute_hecke_matrix_prime_power(3, 3) 

[ -4112503986561480 53074162446443642880 0] 

[ 2592937954080 -1312130996155080 0] 

[ 0 0 834385168339943471891603972970040] 

""" 

# convert input arguments to int's. 

(p,r) = (int(p), int(r)) 

if not arith.is_prime(p): 

raise ArithmeticError("p must be a prime") 

# T_{p^r} := T_p * T_{p^{r-1}} - eps(p)p^{k-1} T_{p^{r-2}}. 

pow = p**(r-1) 

if pow not in self._hecke_matrices: 

# The following will force computation of T_{p^s} 

# for all s<=r-1, except possibly s=0. 

self._hecke_matrices[pow] = self._compute_hecke_matrix(pow) 

if 1 not in self._hecke_matrices: 

self._hecke_matrices[1] = self._compute_hecke_matrix(1) 

Tp = self._hecke_matrices[p] 

Tpr1 = self._hecke_matrices[pow] 

eps = self.character() 

if eps is None: 

raise NotImplementedError("either character or _compute_hecke_matrix_prime_power must be overloaded in a derived class") 

k = self.weight() 

Tpr2 = self._hecke_matrices[pow // p] 

return Tp*Tpr1 - eps(p)*(p**(k-1)) * Tpr2 

 

def _compute_hecke_matrix_general_product(self, F, **kwds): 

r""" 

Compute the matrix of a general Hecke operator acting on this space, by 

factorising n into prime powers and multiplying together the Hecke 

operators for each of these. 

 

EXAMPLES:: 

 

sage: M = ModularSymbols(Gamma0(3), 4) 

sage: M._compute_hecke_matrix_general_product(factor(10)) 

[1134 0] 

[ 0 1134] 

""" 

prod = None 

for p, r in F: 

pow = int(p**r) 

if pow not in self._hecke_matrices: 

self._hecke_matrices[pow] = self._compute_hecke_matrix(pow) 

if prod is None: 

prod = self._hecke_matrices[pow] 

else: 

prod *= self._hecke_matrices[pow] 

return prod 

 

def _compute_dual_hecke_matrix(self, n): 

r""" 

Compute the matrix of the Hecke operator `T_n` acting on the dual of self. 

 

EXAMPLES:: 

 

sage: M = ModularSymbols(Gamma0(3), 4) 

sage: M._compute_dual_hecke_matrix(10) 

[1134 0] 

[ 0 1134] 

""" 

return self.hecke_matrix(n).transpose() 

 

def _compute_hecke_matrix(self, n, **kwds): 

r""" 

Compute the matrix of the Hecke operator `T_n` acting on self. 

 

EXAMPLES:: 

 

sage: M = EisensteinForms(DirichletGroup(3).0, 3) 

sage: M._compute_hecke_matrix(16) 

[205 0] 

[ 0 205] 

""" 

n = int(n) 

if n<1: 

raise ValueError("Hecke operator T_%s is not defined."%n) 

if n==1: 

Mat = matrix_space.MatrixSpace(self.base_ring(),self.rank()) 

return Mat(1) 

 

if arith.is_prime(n): 

return self._compute_hecke_matrix_prime(n, **kwds) 

 

F = arith.factor(n) 

if len(F) == 1: # nontrivial prime power case 

return self._compute_hecke_matrix_prime_power(F[0][0],F[0][1], **kwds) 

 

else: 

return self._compute_hecke_matrix_general_product(F, **kwds) 

 

def _compute_hecke_matrix_prime(self, p, **kwds): 

""" 

Compute and return the matrix of the p-th Hecke operator for p prime. 

Derived classes should overload this function, and they will inherit 

the machinery for calculating general Hecke operators. 

 

EXAMPLES:: 

 

sage: M = EisensteinForms(DirichletGroup(3).0, 3) 

sage: sage.modular.hecke.module.HeckeModule_generic._compute_hecke_matrix_prime(M, 3) 

Traceback (most recent call last): 

... 

NotImplementedError: All subclasses must implement _compute_hecke_matrix_prime 

""" 

raise NotImplementedError("All subclasses must implement _compute_hecke_matrix_prime") 

 

def _compute_diamond_matrix(self, d): 

r""" 

Compute the matrix of the diamond bracket operator `\langle d \rangle` on this space, 

in cases where this isn't self-evident (i.e. when this is not a space 

with fixed character). 

 

EXAMPLES:: 

 

sage: M = EisensteinForms(Gamma1(5), 3) 

sage: sage.modular.hecke.module.HeckeModule_generic._compute_diamond_matrix(M, 2) 

Traceback (most recent call last): 

... 

NotImplementedError: All subclasses without fixed character must implement _compute_diamond_matrix 

""" 

raise NotImplementedError("All subclasses without fixed character must implement _compute_diamond_matrix") 

 

def _hecke_operator_class(self): 

""" 

Return the class to be used for instantiating Hecke operators 

acting on self. 

 

EXAMPLES:: 

 

sage: sage.modular.hecke.module.HeckeModule_generic(QQ,1)._hecke_operator_class() 

<class 'sage.modular.hecke.hecke_operator.HeckeOperator'> 

sage: ModularSymbols(1,12)._hecke_operator_class() 

<class 'sage.modular.modsym.hecke_operator.HeckeOperator'> 

""" 

return hecke_operator.HeckeOperator 

 

def _diamond_operator_class(self): 

r""" 

Return the class to be used for instantiating diamond bracket operators 

acting on self. 

 

EXAMPLES:: 

 

sage: sage.modular.hecke.module.HeckeModule_generic(QQ,1)._diamond_operator_class() 

<class 'sage.modular.hecke.hecke_operator.DiamondBracketOperator'> 

sage: ModularSymbols(1,12)._diamond_operator_class() 

<class 'sage.modular.hecke.hecke_operator.DiamondBracketOperator'> 

""" 

return hecke_operator.DiamondBracketOperator 

 

def anemic_hecke_algebra(self): 

""" 

Return the Hecke algebra associated to this Hecke module. 

 

EXAMPLES:: 

 

sage: T = ModularSymbols(1,12).hecke_algebra() 

sage: A = ModularSymbols(1,12).anemic_hecke_algebra() 

sage: T == A 

False 

sage: A 

Anemic Hecke algebra acting on Modular Symbols space of dimension 3 for Gamma_0(1) of weight 12 with sign 0 over Rational Field 

sage: A.is_anemic() 

True 

""" 

return algebra.AnemicHeckeAlgebra(self) 

 

def character(self): 

r""" 

The character of this space. As this is an abstract base class, return None. 

 

EXAMPLES:: 

 

sage: sage.modular.hecke.module.HeckeModule_generic(QQ, 10).character() is None 

True 

""" 

return None 

 

def dimension(self): 

r""" 

Synonym for rank. 

 

EXAMPLES:: 

 

sage: M = sage.modular.hecke.module.HeckeModule_generic(QQ, 10).dimension() 

Traceback (most recent call last): 

... 

NotImplementedError: Derived subclasses must implement rank 

""" 

return self.rank() 

 

def hecke_algebra(self): 

""" 

Return the Hecke algebra associated to this Hecke module. 

 

EXAMPLES:: 

 

sage: T = ModularSymbols(Gamma1(5),3).hecke_algebra() 

sage: T 

Full Hecke algebra acting on Modular Symbols space of dimension 4 for Gamma_1(5) of weight 3 with sign 0 and over Rational Field 

sage: T.is_anemic() 

False 

 

:: 

 

sage: M = ModularSymbols(37,sign=1) 

sage: E, A, B = M.decomposition() 

sage: A.hecke_algebra() == B.hecke_algebra() 

False 

""" 

return algebra.HeckeAlgebra(self) 

 

def is_zero(self): 

""" 

Return True if this Hecke module has dimension 0. 

 

EXAMPLES:: 

 

sage: ModularSymbols(11).is_zero() 

False 

sage: ModularSymbols(11).old_submodule().is_zero() 

True 

sage: CuspForms(10).is_zero() 

True 

sage: CuspForms(1,12).is_zero() 

False 

""" 

return self.dimension() == 0 

 

def is_full_hecke_module(self): 

""" 

Return True if this space is invariant under all Hecke operators. 

 

Since self is guaranteed to be an anemic Hecke module, the significance 

of this function is that it also ensures invariance under Hecke 

operators of index that divide the level. 

 

EXAMPLES:: 

 

sage: M = ModularSymbols(22); M.is_full_hecke_module() 

True 

sage: M.submodule(M.free_module().span([M.0.list()]), check=False).is_full_hecke_module() 

False 

""" 

try: 

return self._is_full_hecke_module 

except AttributeError: 

pass 

 

# now compute whether invariant under Hecke operators of index 

# dividing the level 

misc.verbose("Determining if Hecke module is full.") 

N = self.level() 

for p in arith.prime_divisors(N): 

if not self.is_hecke_invariant(p): 

self._is_full_hecke_module = False 

return False 

self._is_full_hecke_module = True 

return True 

 

def is_hecke_invariant(self, n): 

""" 

Return True if self is invariant under the Hecke operator 

`T_n`. 

 

Since self is guaranteed to be an anemic Hecke module it is only 

interesting to call this function when `n` is not coprime 

to the level. 

 

EXAMPLES:: 

 

sage: M = ModularSymbols(22).cuspidal_subspace() 

sage: M.is_hecke_invariant(2) 

True 

 

We use check=False to create a nasty "module" that is not invariant 

under `T_2`:: 

 

sage: S = M.submodule(M.free_module().span([M.0.list()]), check=False); S 

Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 7 for Gamma_0(22) of weight 2 with sign 0 over Rational Field 

sage: S.is_hecke_invariant(2) 

False 

sage: [n for n in range(1,12) if S.is_hecke_invariant(n)] 

[1, 3, 5, 7, 9, 11] 

""" 

if arith.gcd(n, self.level()) == 1: 

return True 

if self.is_ambient(): 

return True 

try: 

self.hecke_operator(n).matrix() 

except ArithmeticError: 

return False 

return True 

 

def level(self): 

""" 

Returns the level of this modular symbols space. 

 

INPUT: 

 

 

- ``ModularSymbols self`` - an arbitrary space of 

modular symbols 

 

 

OUTPUT: 

 

 

- ``int`` - the level 

 

 

EXAMPLES:: 

 

sage: m = ModularSymbols(20) 

sage: m.level() 

20 

""" 

return self.__level 

 

def rank(self): 

r""" 

Return the rank of this module over its base ring. Returns 

NotImplementedError, since this is an abstract base class. 

 

EXAMPLES:: 

 

sage: sage.modular.hecke.module.HeckeModule_generic(QQ, 10).rank() 

Traceback (most recent call last): 

... 

NotImplementedError: Derived subclasses must implement rank 

""" 

raise NotImplementedError("Derived subclasses must implement rank") 

 

def submodule(self, X): 

r""" 

Return the submodule of self corresponding to X. As this is an abstract 

base class, this raises a NotImplementedError. 

 

EXAMPLES:: 

 

sage: sage.modular.hecke.module.HeckeModule_generic(QQ, 10).submodule(0) 

Traceback (most recent call last): 

... 

NotImplementedError: Derived subclasses should implement submodule 

""" 

raise NotImplementedError("Derived subclasses should implement submodule") 

 

 

class HeckeModule_free_module(HeckeModule_generic): 

""" 

A Hecke module modeled on a free module over a commutative ring. 

""" 

def __init__(self, base_ring, level, weight, category=None): 

r""" 

Initialise a module. 

 

EXAMPLES:: 

 

sage: M = sage.modular.hecke.module.HeckeModule_free_module(QQ, 12, -4); M 

<class 'sage.modular.hecke.module.HeckeModule_free_module_with_category'> 

sage: TestSuite(M).run(skip = ["_test_additive_associativity",\ 

"_test_an_element",\ 

"_test_elements",\ 

"_test_elements_eq_reflexive",\ 

"_test_elements_eq_symmetric",\ 

"_test_elements_eq_transitive",\ 

"_test_elements_neq",\ 

"_test_pickling",\ 

"_test_some_elements",\ 

"_test_zero",\ 

"_test_eq"]) # is this supposed to be an abstract parent without elements? 

""" 

HeckeModule_generic.__init__(self, base_ring, level, category=category) 

self.__weight = weight 

 

# def __contains__(self, x): 

# r""" 

# Return True if x is an element of self. 

# 

# This shouldn't be getting called, ever (?) 

# """ 

# if not element.is_HeckeModuleElement(x): 

# return False 

# if x.parent() == self: # easy case 

# return True 

# return x.element() in self.free_module() 

 

def _repr_(self): 

r""" 

 

EXAMPLES:: 

 

sage: M = sage.modular.hecke.module.HeckeModule_free_module(QQ, 12, -4); M 

<class 'sage.modular.hecke.module.HeckeModule_free_module_with_category'> 

 

.. TODO:: 

 

Implement a nicer repr, or implement the methods required 

by :class:`ModulesWithBasis` to benefit from 

:meth:`ModulesWithBasis.ParentMethods._repr_`. 

""" 

return repr(type(self)) 

 

def __getitem__(self, n): 

r""" 

Return the nth term in the decomposition of self. See the docstring for 

``decomposition`` for further information. 

 

EXAMPLES:: 

 

sage: ModularSymbols(22)[0] 

Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 7 for Gamma_0(22) of weight 2 with sign 0 over Rational Field 

""" 

n = int(n) 

D = self.decomposition() 

if n < 0 or n >= len(D): 

raise IndexError("index (=%s) must be between 0 and %s"%(n, len(D)-1)) 

return D[n] 

 

def __hash__(self): 

r""" 

The hash is determined by the weight, the level and the base ring. 

 

EXAMPLES:: 

 

sage: MS = ModularSymbols(22) 

sage: hash(MS) == hash((MS.weight(), MS.level(), MS.base_ring())) 

True 

 

""" 

return hash((self.__weight, self.level(), self.base_ring())) 

 

def __len__(self): 

r""" 

The number of factors in the decomposition of self. 

 

EXAMPLES:: 

 

sage: len(ModularSymbols(22)) 

2 

""" 

return len(self.decomposition()) 

 

def _eigen_nonzero(self): 

""" 

Return smallest integer i such that the i-th entries of the entries 

of a basis for the dual vector space are not all 0. 

 

EXAMPLES:: 

 

sage: M = ModularSymbols(31,2) 

sage: M._eigen_nonzero() 

0 

sage: M.dual_free_module().basis() 

[ 

(1, 0, 0, 0, 0), 

(0, 1, 0, 0, 0), 

(0, 0, 1, 0, 0), 

(0, 0, 0, 1, 0), 

(0, 0, 0, 0, 1) 

] 

sage: M.cuspidal_submodule().minus_submodule()._eigen_nonzero() 

1 

sage: M.cuspidal_submodule().minus_submodule().dual_free_module().basis() 

[ 

(0, 1, 0, 0, 0), 

(0, 0, 1, 0, 0) 

] 

""" 

try: 

return self.__eigen_nonzero 

except AttributeError: 

pass 

A = self.ambient_hecke_module() 

V = self.dual_free_module() 

B = V.basis() 

for i in range(V.degree()): 

for b in B: 

if b[i] != 0: 

self.__eigen_nonzero = i 

return i 

assert False, 'bug in _eigen_nonzero' 

 

def _eigen_nonzero_element(self, n=1): 

r""" 

Return `T_n(x)` where `x` is a sparse modular 

symbol such that the image of `x` is nonzero under the dual 

projection map associated to this space, and `T_n` is the 

`n^{th}` Hecke operator. 

 

Used in the dual_eigenvector and eigenvalue methods. 

 

EXAMPLES:: 

 

sage: ModularSymbols(22)._eigen_nonzero_element(3) 

4*(1,0) + (2,21) - (11,1) + (11,2) 

""" 

if self.rank() == 0: 

raise ArithmeticError("the rank of self must be positive") 

A = self.ambient_hecke_module() 

i = self._eigen_nonzero() 

return A._hecke_image_of_ith_basis_vector(n, i) 

 

def _hecke_image_of_ith_basis_vector(self, n, i): 

r""" 

Return `T_n(e_i)`, where `e_i` is the 

`i`th basis vector of the ambient space. 

 

EXAMPLES:: 

 

sage: ModularSymbols(Gamma0(3))._hecke_image_of_ith_basis_vector(4, 0) 

7*(1,0) 

sage: ModularForms(Gamma0(3))._hecke_image_of_ith_basis_vector(4, 0) 

7 + 84*q + 252*q^2 + 84*q^3 + 588*q^4 + 504*q^5 + O(q^6) 

""" 

T = self.hecke_operator(n) 

return T.apply_sparse(self.gen(i)) 

 

def _element_eigenvalue(self, x, name='alpha'): 

r""" 

Return the dot product of self with the eigenvector returned by dual_eigenvector. 

 

EXAMPLES:: 

 

sage: M = ModularSymbols(11)[0] 

sage: M._element_eigenvalue(M.0) 

1 

""" 

if not element.is_HeckeModuleElement(x): 

raise TypeError("x must be a Hecke module element.") 

if not x in self.ambient_hecke_module(): 

raise ArithmeticError("x must be in the ambient Hecke module.") 

v = self.dual_eigenvector(names=name) 

return v.dot_product(x.element()) 

 

def _is_hecke_equivariant_free_module(self, submodule): 

""" 

Returns True if the given free submodule of the ambient free module 

is invariant under all Hecke operators. 

 

EXAMPLES:: 

 

sage: M = ModularSymbols(11); V = M.free_module() 

sage: M._is_hecke_equivariant_free_module(V.span([V.0])) 

False 

sage: M._is_hecke_equivariant_free_module(V) 

True 

sage: M._is_hecke_equivariant_free_module(M.cuspidal_submodule().free_module()) 

True 

 

We do the same as above, but with a modular forms space:: 

 

sage: M = ModularForms(11); V = M.free_module() 

sage: M._is_hecke_equivariant_free_module(V.span([V.0 + V.1])) 

False 

sage: M._is_hecke_equivariant_free_module(V) 

True 

sage: M._is_hecke_equivariant_free_module(M.cuspidal_submodule().free_module()) 

True 

""" 

misc.verbose("Determining if free module is Hecke equivariant.") 

bound = self.hecke_bound() 

for p in arith.primes(bound+1): 

try: 

self.T(p).matrix().restrict(submodule, check=True) 

except ArithmeticError: 

return False 

return True 

 

def _set_factor_number(self, i): 

r""" 

For internal use. If this Hecke module was computed via a decomposition of another 

Hecke module, this method stores the index of this space in that decomposition. 

 

EXAMPLES:: 

 

sage: ModularSymbols(Gamma0(3))[0].factor_number() # indirect doctest 

0 

""" 

self.__factor_number = i 

 

def ambient(self): 

r""" 

Synonym for ambient_hecke_module. Return the ambient module associated to this module. 

 

EXAMPLES:: 

 

sage: CuspForms(1, 12).ambient() 

Modular Forms space of dimension 2 for Modular Group SL(2,Z) of weight 12 over Rational Field 

""" 

return self.ambient_hecke_module() 

 

def ambient_module(self): 

r""" 

Synonym for ambient_hecke_module. Return the ambient module associated to this module. 

 

EXAMPLES:: 

 

sage: CuspForms(1, 12).ambient_module() 

Modular Forms space of dimension 2 for Modular Group SL(2,Z) of weight 12 over Rational Field 

sage: sage.modular.hecke.module.HeckeModule_free_module(QQ, 10, 3).ambient_module() 

Traceback (most recent call last): 

... 

NotImplementedError 

""" 

return self.ambient_hecke_module() 

 

def ambient_hecke_module(self): 

r""" 

Return the ambient module associated to this module. As this is 

an abstract base class, raise NotImplementedError. 

 

EXAMPLES:: 

 

sage: sage.modular.hecke.module.HeckeModule_free_module(QQ, 10, 3).ambient_hecke_module() 

Traceback (most recent call last): 

... 

NotImplementedError 

""" 

raise NotImplementedError 

 

def atkin_lehner_operator(self, d=None): 

""" 

Return the Atkin-Lehner operator `W_d` on this space, if 

defined, where `d` is a divisor of the level `N` 

such that `N/d` and `d` are coprime. 

 

EXAMPLES:: 

 

sage: M = ModularSymbols(11) 

sage: w = M.atkin_lehner_operator() 

sage: w 

Hecke module morphism Atkin-Lehner operator W_11 defined by the matrix 

[-1 0 0] 

[ 0 -1 0] 

[ 0 0 -1] 

Domain: Modular Symbols space of dimension 3 for Gamma_0(11) of weight ... 

Codomain: Modular Symbols space of dimension 3 for Gamma_0(11) of weight ... 

sage: M = ModularSymbols(Gamma1(13)) 

sage: w = M.atkin_lehner_operator() 

sage: w.fcp('x') 

(x - 1)^7 * (x + 1)^8 

 

:: 

 

sage: M = ModularSymbols(33) 

sage: S = M.cuspidal_submodule() 

sage: S.atkin_lehner_operator() 

Hecke module morphism Atkin-Lehner operator W_33 defined by the matrix 

[ 0 -1 0 1 -1 0] 

[ 0 -1 0 0 0 0] 

[ 0 -1 0 0 -1 1] 

[ 1 -1 0 0 -1 0] 

[ 0 0 0 0 -1 0] 

[ 0 -1 1 0 -1 0] 

Domain: Modular Symbols subspace of dimension 6 of Modular Symbols space ... 

Codomain: Modular Symbols subspace of dimension 6 of Modular Symbols space ... 

 

:: 

 

sage: S.atkin_lehner_operator(3) 

Hecke module morphism Atkin-Lehner operator W_3 defined by the matrix 

[ 0 1 0 -1 1 0] 

[ 0 1 0 0 0 0] 

[ 0 1 0 0 1 -1] 

[-1 1 0 0 1 0] 

[ 0 0 0 0 1 0] 

[ 0 1 -1 0 1 0] 

Domain: Modular Symbols subspace of dimension 6 of Modular Symbols space ... 

Codomain: Modular Symbols subspace of dimension 6 of Modular Symbols space ... 

 

:: 

 

sage: N = M.new_submodule() 

sage: N.atkin_lehner_operator() 

Hecke module morphism Atkin-Lehner operator W_33 defined by the matrix 

[ 1 2/5 4/5] 

[ 0 -1 0] 

[ 0 0 -1] 

Domain: Modular Symbols subspace of dimension 3 of Modular Symbols space ... 

Codomain: Modular Symbols subspace of dimension 3 of Modular Symbols space ... 

""" 

if d is None: 

d = self.level() 

d = int(d) 

if self.level() % d != 0: 

raise ArithmeticError("d (=%s) must be a divisor of the level (=%s)"%(d,self.level())) 

 

N = self.level() 

for p, e in arith.factor(d): 

v = arith.valuation(N, p) 

if e < v: 

d *= p**(v-e) 

d = int(d) 

try: 

return self.__atkin_lehner_operator[d] 

except AttributeError: 

self.__atkin_lehner_operator = {} 

except KeyError: 

pass 

Wmat = self._compute_atkin_lehner_matrix(d) 

H = self.endomorphism_ring() 

W = H(Wmat, "Atkin-Lehner operator W_%s"%d) 

self.__atkin_lehner_operator[d] = W 

return W 

 

def basis(self): 

""" 

Returns a basis for self. 

 

EXAMPLES:: 

 

sage: m = ModularSymbols(43) 

sage: m.basis() 

((1,0), (1,31), (1,32), (1,38), (1,39), (1,40), (1,41)) 

""" 

try: 

return self.__basis 

except AttributeError: 

self.__basis = self.gens() 

return self.__basis 

 

def basis_matrix(self): 

r""" 

Return the matrix of the basis vectors of self (as vectors in some 

ambient module) 

 

EXAMPLES:: 

 

sage: CuspForms(1, 12).basis_matrix() 

[1 0] 

""" 

return self.free_module().basis_matrix() 

 

def coordinate_vector(self, x): 

""" 

Write x as a vector with respect to the basis given by 

self.basis(). 

 

EXAMPLES:: 

 

sage: S = ModularSymbols(11,2).cuspidal_submodule() 

sage: S.0 

(1,8) 

sage: S.basis() 

((1,8), (1,9)) 

sage: S.coordinate_vector(S.0) 

(1, 0) 

""" 

return self.free_module().coordinate_vector(x.element()) 

 

def decomposition(self, bound=None, anemic=True, height_guess=1, sort_by_basis = False, 

proof=None): 

""" 

Returns the maximal decomposition of this Hecke module under the 

action of Hecke operators of index coprime to the level. This is 

the finest decomposition of self that we can obtain using factors 

obtained by taking kernels of Hecke operators. 

 

Each factor in the decomposition is a Hecke submodule obtained as 

the kernel of `f(T_n)^r` acting on self, where n is 

coprime to the level and `r=1`. If anemic is False, instead 

choose `r` so that `f(X)^r` exactly divides the 

characteristic polynomial. 

 

INPUT: 

 

 

- ``anemic`` - bool (default: True), if True, use only 

Hecke operators of index coprime to the level. 

 

- ``bound`` - int or None, (default: None). If None, 

use all Hecke operators up to the Sturm bound, and hence obtain the 

same result as one would obtain by using every element of the Hecke 

ring. If a fixed integer, decompose using only Hecke operators 

`T_p`, with `p` prime, up to bound. 

- ``sort_by_basis`` - bool (default: ``False``); If True the resulting 

decomposition will be sorted as if it was free modules, ignoring the 

Hecke module structure. This will save a lot of time. 

 

 

OUTPUT: 

 

 

- ``list`` - a list of subspaces of self. 

 

 

EXAMPLES:: 

 

sage: ModularSymbols(17,2).decomposition() 

[ 

Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(17) of weight 2 with sign 0 over Rational Field, 

Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(17) of weight 2 with sign 0 over Rational Field 

] 

sage: ModularSymbols(Gamma1(10),4).decomposition() 

[ 

Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_1(10) of weight 4 with sign 0 and over Rational Field, 

Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_1(10) of weight 4 with sign 0 and over Rational Field, 

Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_1(10) of weight 4 with sign 0 and over Rational Field, 

Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 18 for Gamma_1(10) of weight 4 with sign 0 and over Rational Field, 

Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 18 for Gamma_1(10) of weight 4 with sign 0 and over Rational Field, 

Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 18 for Gamma_1(10) of weight 4 with sign 0 and over Rational Field 

] 

sage: ModularSymbols(GammaH(12, [11])).decomposition() 

[ 

Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 9 for Congruence Subgroup Gamma_H(12) with H generated by [11] of weight 2 with sign 0 and over Rational Field, 

Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 9 for Congruence Subgroup Gamma_H(12) with H generated by [11] of weight 2 with sign 0 and over Rational Field, 

Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 9 for Congruence Subgroup Gamma_H(12) with H generated by [11] of weight 2 with sign 0 and over Rational Field, 

Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 9 for Congruence Subgroup Gamma_H(12) with H generated by [11] of weight 2 with sign 0 and over Rational Field, 

Modular Symbols subspace of dimension 5 of Modular Symbols space of dimension 9 for Congruence Subgroup Gamma_H(12) with H generated by [11] of weight 2 with sign 0 and over Rational Field 

] 

 

TESTS:: 

 

sage: M = ModularSymbols(1000,2,sign=1).new_subspace().cuspidal_subspace() 

sage: M.decomposition(3, sort_by_basis = True) 

[ 

Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 154 for Gamma_0(1000) of weight 2 with sign 1 over Rational Field, 

Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 154 for Gamma_0(1000) of weight 2 with sign 1 over Rational Field, 

Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 154 for Gamma_0(1000) of weight 2 with sign 1 over Rational Field, 

Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 154 for Gamma_0(1000) of weight 2 with sign 1 over Rational Field, 

Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 154 for Gamma_0(1000) of weight 2 with sign 1 over Rational Field, 

Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 154 for Gamma_0(1000) of weight 2 with sign 1 over Rational Field 

] 

""" 

if not isinstance(anemic, bool): 

raise TypeError("anemic must be of type bool.") 

 

key = (bound, anemic) 

 

try: 

if self.__decomposition[key] is not None: 

return self.__decomposition[key] 

except AttributeError: 

self.__decomposition = {} 

except KeyError: 

pass 

if self.rank() == 0: 

self.__decomposition[key] = Sequence([], immutable=True, cr=True) 

return self.__decomposition[key] 

 

is_rational = self.base_ring() == sage.rings.all.QQ 

 

time = misc.verbose("Decomposing %s"%self) 

T = self.ambient_hecke_module().hecke_algebra() 

if bound is None: 

bound = self.ambient_hecke_module().hecke_bound() 

D = Sequence([], cr=True) 

U = [self.free_module()] 

p = 2 

while len(U) > 0 and p <= bound: 

misc.verbose(mesg="p=%s"%p,t=time) 

if anemic: 

while arith.GCD(p, self.level()) != 1: 

p = arith.next_prime(p) 

misc.verbose("Decomposition using p=%s"%p) 

t = T.hecke_operator(p).matrix() 

Uprime = [] 

for i in range(len(U)): 

if self.base_ring().characteristic() == 0 and self.level()%p != 0: 

is_diagonalizable = True 

else: 

is_diagonalizable = False 

if is_rational: 

X = t.decomposition_of_subspace(U[i], check_restrict = False, 

algorithm='multimodular', 

height_guess=height_guess, proof=proof) 

else: 

X = t.decomposition_of_subspace(U[i], check_restrict = False, 

is_diagonalizable=is_diagonalizable) 

for i in range(len(X)): 

W, is_irred = X[i] 

if is_irred: 

A = self.submodule(W, check=False) 

D.append(A) 

else: 

Uprime.append(W) 

# end for 

p = arith.next_prime(p) 

U = Uprime 

#end while 

for i in range(len(U)): 

A = self.submodule(U[i], check=False) 

D.append(A) 

for A in D: 

if anemic: 

A.__is_splittable_anemic = False 

A.__is_splittable = False 

else: 

A.__is_splittable = False 

self.__is_splittable = len(D) > 1 

if anemic: 

self.__is_splittable_anemic = len(D) > 1 

from sage.modules.free_module import EchelonMatrixKey 

D.sort(key=None if not sort_by_basis 

else lambda ss: EchelonMatrixKey(ss.free_module())) 

D.set_immutable() 

self.__decomposition[key] = D 

for i in range(len(D)): 

self.__decomposition[key][i]._set_factor_number(i) 

return self.__decomposition[key] 

 

def degree(self): 

r""" 

The degree of this Hecke module (i.e. the rank of the ambient free 

module) 

 

EXAMPLES:: 

 

sage: CuspForms(1, 12).degree() 

2 

""" 

return self.free_module().degree() 

 

def dual_eigenvector(self, names='alpha', lift=True, nz=None): 

""" 

Return an eigenvector for the Hecke operators acting on the linear 

dual of this space. This eigenvector will have entries in an 

extension of the base ring of degree equal to the dimension of this 

space. 

 

.. warning: 

 

The input space must be simple. 

 

INPUT: 

 

 

- ``name`` - print name of generator for eigenvalue 

field. 

 

- ``lift`` - bool (default: True) 

 

- ``nz`` - if not None, then normalize vector so dot 

product with this basis vector of ambient space is 1. 

 

 

OUTPUT: A vector with entries possibly in an extension of the base 

ring. This vector is an eigenvector for all Hecke operators acting 

via their transpose. 

 

If lift = False, instead return an eigenvector in the subspace for 

the Hecke operators on the dual space. I.e., this is an eigenvector 

for the restrictions of Hecke operators to the dual space. 

 

.. note:: 

 

#. The answer is cached so subsequent calls always return 

the same vector. However, the algorithm is randomized, 

so calls during another session may yield a different 

eigenvector. This function is used mainly for computing 

systems of Hecke eigenvalues. 

 

#. One can also view a dual eigenvector as defining (via 

dot product) a functional phi from the ambient space of 

modular symbols to a field. This functional phi is an 

eigenvector for the dual action of Hecke operators on 

functionals. 

 

EXAMPLES:: 

 

sage: SF = ModularSymbols(14).cuspidal_subspace().simple_factors() 

sage: sorted([u.dual_eigenvector() for u in SF]) 

[(0, 1, 0, 0, 0), (1, 0, -3, 2, -1)] 

""" 

# TODO -- optimize by computing the answer for i not None in terms 

# of the answer for a given i if known !! 

try: 

w, w_lift = self.__dual_eigenvector[(names,nz)] 

if lift: 

return w_lift 

else: 

return w 

except KeyError: 

pass 

except AttributeError: 

self.__dual_eigenvector = {} 

 

if not self.is_simple(): 

raise ArithmeticError("self must be simple") 

 

# Find a Hecke operator that acts irreducibly on this space: 

p = 2 

t = self.dual_hecke_matrix(p) 

while True: 

f = t.charpoly('x') 

if f.is_irreducible(): 

break 

p = arith.next_prime(p) 

t += random.choice([-2,-1,1,2]) * self.dual_hecke_matrix(p) 

 

# Write down the eigenvector. 

# Write f(x) = (x-alpha)*g(x), where alpha is a root 

# of f(x). 

n = f.degree() 

if n > 1: 

R = f.parent() 

K = R.base_ring().extension(f, names=names) 

alpha = K.gen() 

beta = ~alpha # multiplicative inverse of alpha 

c = [-f[0]*beta] 

for i in range(1,n-1): 

c.append((c[i-1] - f[i])*beta) 

c.append( K(1) ) 

else: 

K = self.base_ring() 

c = [1] 

 

# The entries of c are the coefficients of g (as stated in 

# William Stein's Ph.D. thesis, Section 3.5.3). We compute 

# g(t)v for a some vector v, and get an eigenvector. 

V = FreeModule(K, n) 

t = t.change_ring(K) # coerce t to be over K. 

for j in range(n): 

v = V.gen(j) 

I = t.iterates(v, n) # iterates v, v*t, v*t^2, ... 

w = V(0) 

for i in range(n): 

w += c[i]*V(I.row(i).list()) 

if w != 0: 

break 

 

# Now w is an eigenvector for the action of the Hecke 

# operators on the subspace. We need an eigenvector 

# in the original space, so we take the linear combination 

# of the basis for the embedded dual vector space given 

# by the entries of w. 

Vdual = self.dual_free_module().change_ring(K) 

w_lift = Vdual.linear_combination_of_basis(w) 

 

# Finally rescale so the dot product of this vector and 

# the _eigen_nonzero_element is 1. 

if nz is not None: 

x = self.ambient().gen(nz) 

else: 

x = self._eigen_nonzero_element() 

alpha = w_lift.dot_product(x.element()) 

beta = ~alpha 

w_lift = w_lift * beta 

w = w * beta 

 

self.__dual_eigenvector[(names,nz)] = (w, w_lift) 

if lift: 

return w_lift 

else: 

return w 

 

def dual_hecke_matrix(self, n): 

""" 

The matrix of the `n^{th}` Hecke operator acting on the dual 

embedded representation of self. 

 

EXAMPLES:: 

 

sage: CuspForms(1, 24).dual_hecke_matrix(5) 

[ 44656110 -15040] 

[-307849789440 28412910] 

""" 

n = int(n) 

try: 

self._dual_hecke_matrices 

except AttributeError: 

self._dual_hecke_matrices = {} 

if n not in self._dual_hecke_matrices: 

T = self._compute_dual_hecke_matrix(n) 

self._dual_hecke_matrices[n] = T 

return self._dual_hecke_matrices[n] 

 

def eigenvalue(self, n, name='alpha'): 

""" 

Assuming that self is a simple space, return the eigenvalue of the 

`n^{th}` Hecke operator on self. 

 

INPUT: 

 

 

- ``n`` - index of Hecke operator 

 

- ``name`` - print representation of generator of 

eigenvalue field 

 

 

EXAMPLES:: 

 

sage: A = ModularSymbols(125,sign=1).new_subspace()[0] 

sage: A.eigenvalue(7) 

-3 

sage: A.eigenvalue(3) 

-alpha - 2 

sage: A.eigenvalue(3,'w') 

-w - 2 

sage: A.eigenvalue(3,'z').charpoly('x') 

x^2 + 3*x + 1 

sage: A.hecke_polynomial(3) 

x^2 + 3*x + 1 

 

:: 

 

sage: M = ModularSymbols(Gamma1(17)).decomposition()[8].plus_submodule() 

sage: M.eigenvalue(2,'a') 

a 

sage: M.eigenvalue(4,'a') 

4/3*a^3 + 17/3*a^2 + 28/3*a + 8/3 

 

.. note:: 

 

#. In fact there are `d` systems of eigenvalues 

associated to self, where `d` is the rank of 

self. Each of the systems of eigenvalues is conjugate 

over the base field. This function chooses one of the 

systems and consistently returns eigenvalues from that 

system. Thus these are the coefficients `a_n` for 

`n\geq 1` of a modular eigenform attached to self. 

 

#. This function works even for Eisenstein subspaces, 

though it will not give the constant coefficient of one 

of the corresponding Eisenstein series (i.e., the 

generalized Bernoulli number). 

 

TESTS: 

 

This checks that :trac:`15201` is fixed:: 

 

sage: M = ModularSymbols(5, 6, sign=1) 

sage: f = M.decomposition()[0] 

sage: f.eigenvalue(10) 

50 

""" 

if not self.is_simple(): 

raise ArithmeticError("self must be simple") 

n = int(n) 

try: 

return self.__eigenvalues[n][name] 

except AttributeError: 

self.__eigenvalues = {} 

except KeyError: 

pass 

if n <= 0: 

raise IndexError("n must be a positive integer") 

 

ev = self.__eigenvalues 

 

if (arith.is_prime(n) or n==1): 

Tn_e = self._eigen_nonzero_element(n) 

an = self._element_eigenvalue(Tn_e, name=name) 

_dict_set(ev, n, name, an) 

return an 

 

# Now use the Hecke eigenvalue recurrence, since arithmetic in 

# a field is faster than computing Heilbronn matrices for 

# non-prime n and doing some big sum (i.e., computing T_n(e)). 

# Also by computing using the recurrence on eigenvalues 

# we use information about divisors. 

F = arith.factor(n) 

prod = None 

for p, r in F: 

(p, r) = (int(p), int(r)) 

pow = p**r 

if not (pow in ev and name in ev[pow]): 

# TODO: Optimization -- do something much more 

# intelligent in case character is not defined. For 

# example, compute it using the diamond operators <d> 

eps = self.character() 

if eps is None: 

Tn_e = self._eigen_nonzero_element(pow) 

_dict_set(ev, pow, name, self._element_eigenvalue(Tn_e, name=name)) 

else: 

# a_{p^r} := a_p * a_{p^{r-1}} - eps(p)p^{k-1} a_{p^{r-2}} 

ap = self.eigenvalue(p, name=name) 

if r == 1: 

apow = ap 

else: 

apr1 = self.eigenvalue(pow//p, name=name) 

k = self.weight() 

apr2 = self.eigenvalue(pow//(p*p), name=name) 

apow = ap*apr1 - eps(p)*(p**(k-1)) * apr2 

_dict_set(ev, pow, name, apow) 

if prod is None: 

prod = ev[pow][name] 

else: 

prod *= ev[pow][name] 

_dict_set(ev, n, name, prod) 

return prod 

 

def factor_number(self): 

""" 

If this Hecke module was computed via a decomposition of another 

Hecke module, this is the corresponding number. Otherwise return 

-1. 

 

EXAMPLES:: 

 

sage: ModularSymbols(23)[0].factor_number() 

0 

sage: ModularSymbols(23).factor_number() 

-1 

""" 

try: 

return self.__factor_number 

except AttributeError: 

return -1 

 

def gens(self): 

""" 

Return a tuple of basis elements of ``self``. 

 

EXAMPLES:: 

 

sage: ModularSymbols(23).gens() 

((1,0), (1,17), (1,19), (1,20), (1,21)) 

""" 

return tuple(self(x) for x in self.free_module().gens()) 

 

def gen(self, n): 

r""" 

Return the nth basis vector of the space. 

 

EXAMPLES:: 

 

sage: ModularSymbols(23).gen(1) 

(1,17) 

""" 

return self(self.free_module().gen(n)) 

 

def hecke_matrix(self, n): 

""" 

The matrix of the `n^{th}` Hecke operator acting on given 

basis. 

 

EXAMPLES:: 

 

sage: C = CuspForms(1, 16) 

sage: C.hecke_matrix(3) 

[-3348] 

""" 

n = int(n) 

if n <= 0: 

raise IndexError("n must be positive.") 

if n not in self._hecke_matrices: 

T = self._compute_hecke_matrix(n) 

T.set_immutable() 

self._hecke_matrices[n] = T 

return self._hecke_matrices[n] 

 

def hecke_operator(self, n): 

""" 

Returns the `n`-th Hecke operator `T_n`. 

 

INPUT: 

 

 

- ``ModularSymbols self`` - Hecke equivariant space of 

modular symbols 

 

- ``int n`` - an integer at least 1. 

 

 

EXAMPLES:: 

 

sage: M = ModularSymbols(11,2) 

sage: T = M.hecke_operator(3) ; T 

Hecke operator T_3 on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field 

sage: T.matrix() 

[ 4 0 -1] 

[ 0 -1 0] 

[ 0 0 -1] 

sage: T(M.0) 

4*(1,0) - (1,9) 

sage: S = M.cuspidal_submodule() 

sage: T = S.hecke_operator(3) ; T 

Hecke operator T_3 on Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field 

sage: T.matrix() 

[-1 0] 

[ 0 -1] 

sage: T(S.0) 

-(1,8) 

""" 

return self.hecke_algebra().hecke_operator(n) 

 

def diamond_bracket_matrix(self, d): 

r""" 

Return the matrix of the diamond bracket operator `\langle d \rangle` on self. 

 

EXAMPLES:: 

 

sage: M = ModularSymbols(DirichletGroup(5).0, 3) 

sage: M.diamond_bracket_matrix(3) 

[-zeta4 0] 

[ 0 -zeta4] 

sage: ModularSymbols(Gamma1(5), 3).diamond_bracket_matrix(3) 

[ 0 -1 0 0] 

[ 1 0 0 0] 

[ 0 0 0 1] 

[ 0 0 -1 0] 

""" 

d = int(d) % self.level() 

if d not in self._diamond_matrices: 

if self.character() is not None: 

D = matrix_space.MatrixSpace(self.base_ring(),self.rank())(self.character()(d)) 

else: 

D = self._compute_diamond_matrix(d) 

D.set_immutable() 

self._diamond_matrices[d] = D 

return self._diamond_matrices[d] 

 

def diamond_bracket_operator(self, d): 

r""" 

Return the diamond bracket operator `\langle d \rangle` on self. 

 

EXAMPLES:: 

 

sage: M = ModularSymbols(DirichletGroup(5).0, 3) 

sage: M.diamond_bracket_operator(3) 

Diamond bracket operator <3> on Modular Symbols space of dimension 2 and level 5, weight 3, character [zeta4], sign 0, over Cyclotomic Field of order 4 and degree 2 

""" 

return self.hecke_algebra().diamond_bracket_operator(d) 

 

def T(self, n): 

r""" 

Returns the `n^{th}` Hecke operator `T_n`. This 

function is a synonym for :meth:`.hecke_operator`. 

 

EXAMPLES:: 

 

sage: M = ModularSymbols(11,2) 

sage: M.T(3) 

Hecke operator T_3 on Modular Symbols ... 

""" 

return self.hecke_operator(n) 

 

def hecke_polynomial(self, n, var='x'): 

""" 

Return the characteristic polynomial of the `n^{th}` Hecke operator 

acting on this space. 

 

INPUT: 

 

 

- ``n`` - integer 

 

 

OUTPUT: a polynomial 

 

EXAMPLES:: 

 

sage: ModularSymbols(11,2).hecke_polynomial(3) 

x^3 - 2*x^2 - 7*x - 4 

""" 

return self.hecke_operator(n).charpoly(var) 

 

def is_simple(self): 

r""" 

Return True if this space is simple as a module for the corresponding 

Hecke algebra. Raises NotImplementedError, as this is an abstract base 

class. 

 

EXAMPLES:: 

 

sage: sage.modular.hecke.module.HeckeModule_free_module(QQ, 10, 3).is_simple() 

Traceback (most recent call last): 

... 

NotImplementedError 

""" 

raise NotImplementedError 

 

def is_splittable(self): 

""" 

Returns True if and only if only it is possible to split off a 

nontrivial generalized eigenspace of self as the kernel of some Hecke 

operator (not necessarily prime to the level). Note that the direct sum 

of several copies of the same simple module is not splittable in this 

sense. 

 

EXAMPLES:: 

 

sage: M = ModularSymbols(Gamma0(64)).cuspidal_subspace() 

sage: M.is_splittable() 

True 

sage: M.simple_factors()[0].is_splittable() 

False 

""" 

if not hasattr(self, "__is_splittable"): 

self.decomposition(anemic=False) 

return self.__is_splittable 

 

def is_submodule(self, other): 

r""" 

Return True if self is a submodule of other. 

 

EXAMPLES:: 

 

sage: M = ModularSymbols(Gamma0(64)) 

sage: M[0].is_submodule(M) 

True 

sage: CuspForms(1, 24).is_submodule(ModularForms(1, 24)) 

True 

sage: CuspForms(1, 12).is_submodule(CuspForms(3, 12)) 

False 

""" 

if not isinstance(other, HeckeModule_free_module): 

return False 

return self.ambient_free_module() == other.ambient_free_module() and \ 

self.free_module().is_submodule(other.free_module()) 

 

def is_splittable_anemic(self): 

""" 

Returns true if and only if only it is possible to split off a 

nontrivial generalized eigenspace of self as the kernel of some 

Hecke operator of index coprime to the level. Note that the direct sum 

of several copies of the same simple module is not splittable in this 

sense. 

 

EXAMPLES:: 

 

sage: M = ModularSymbols(Gamma0(64)).cuspidal_subspace() 

sage: M.is_splittable_anemic() 

True 

sage: M.simple_factors()[0].is_splittable_anemic() 

False 

""" 

if not hasattr(self,"__is_splittable_anemic"): 

self.decomposition(anemic=True) 

return self.__is_splittable_anemic 

 

def ngens(self): 

r""" 

Number of generators of self (equal to the rank). 

 

EXAMPLES:: 

 

sage: ModularForms(1, 12).ngens() 

2 

""" 

return self.rank() 

 

def projection(self): 

r""" 

Return the projection map from the ambient space to self. 

 

ALGORITHM: Let `B` be the matrix whose columns are obtained 

by concatenating together a basis for the factors of the ambient 

space. Then the projection matrix onto self is the submatrix of 

`B^{-1}` obtained from the rows corresponding to self, 

i.e., if the basis vectors for self appear as columns `n` 

through `m` of `B`, then the projection matrix is 

got from rows `n` through `m` of `B^{-1}`. 

This is because projection with respect to the B basis is just 

given by an `m-n+1` row slice `P` of a diagonal 

matrix D with 1's in the `n` through `m` positions, 

so projection with respect to the standard basis is given by 

`P\cdot B^{-1}`, which is just rows `n` 

through `m` of `B^{-1}`. 

 

EXAMPLES:: 

 

sage: e = EllipticCurve('34a') 

sage: m = ModularSymbols(34); s = m.cuspidal_submodule() 

sage: d = s.decomposition(7) 

sage: d 

[ 

Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 9 for Gamma_0(34) of weight 2 with sign 0 over Rational Field, 

Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 9 for Gamma_0(34) of weight 2 with sign 0 over Rational Field 

] 

sage: a = d[0]; a 

Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 9 for Gamma_0(34) of weight 2 with sign 0 over Rational Field 

sage: pi = a.projection() 

sage: pi(m([0,oo])) 

-1/6*(2,7) + 1/6*(2,13) - 1/6*(2,31) + 1/6*(2,33) 

sage: M = ModularSymbols(53,sign=1) 

sage: S = M.cuspidal_subspace()[1] ; S 

Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 5 for Gamma_0(53) of weight 2 with sign 1 over Rational Field 

sage: p = S.projection() 

sage: S.basis() 

((1,33) - (1,37), (1,35), (1,49)) 

sage: [ p(x) for x in S.basis() ] 

[(1,33) - (1,37), (1,35), (1,49)] 

""" 

 

# Compute the Hecke-stable projection map pi from the ambient 

# space of M to M. Computing the projection map is the same 

# as writing the ambient space as a direct sum of M and its 

# Hecke-stable complement, which is the old subspace plus the 

# other new factors, then *inverting*. With the projection 

# map in hand, we can compute Hecke operators directly on M 

# fairly quickly without having to compute them on the whole 

# ambient space. Of course, computing this inverse is way too 

# much work to be useful in general (!). (I sort of learned 

# this trick from Joe Wetherell, or at least he was aware of 

# it when I mentioned it to him in an airport once. It's also 

# sort of like a trick Cremona uses in his book for elliptic 

# curves.) It's not a very good trick though. 

 

try: 

return self.__projection 

except AttributeError: 

i = self.factor_number() 

if i == -1: 

raise NotImplementedError("Computation of projection only implemented "+\ 

"for decomposition factors.") 

A = self.ambient_hecke_module() 

B = A.decomposition_matrix_inverse() 

i = (A.decomposition()).index(self) 

n = sum([A[j].rank() for j in range(i)]) 

C = B.matrix_from_columns(range(n,n+self.rank())) 

H = A.Hom(self) 

pi = H(C, "Projection"%self) 

self.__projection = pi 

return self.__projection 

 

 

 

def system_of_eigenvalues(self, n, name='alpha'): 

r""" 

Assuming that self is a simple space of modular symbols, return the 

eigenvalues `[a_1, \ldots, a_nmax]` of the Hecke 

operators on self. See ``self.eigenvalue(n)`` for more 

details. 

 

INPUT: 

 

 

- ``n`` - number of eigenvalues 

 

- ``alpha`` - name of generate for eigenvalue field 

 

 

EXAMPLES: 

 

The outputs of the following tests are very unstable. The algorithms 

are randomized and depend on cached results. A slight change in the 

sequence of pseudo-random numbers or a modification in caching is 

likely to modify the results. We reset the random number generator and 

clear some caches for reproducibility:: 

 

sage: set_random_seed(0) 

sage: ModularSymbols_clear_cache() 

 

We compute eigenvalues for newforms of level 62:: 

 

sage: M = ModularSymbols(62,2,sign=-1) 

sage: S = M.cuspidal_submodule().new_submodule() 

sage: [[o.minpoly() for o in A.system_of_eigenvalues(3)] for A in S.decomposition()] 

[[x - 1, x - 1, x], [x - 1, x + 1, x^2 - 2*x - 2]] 

 

Next we define a function that does the above:: 

 

sage: def b(N,k=2): 

....: t=cputime() 

....: S = ModularSymbols(N,k,sign=-1).cuspidal_submodule().new_submodule() 

....: for A in S.decomposition(): 

....: print("{} {}".format(N, A.system_of_eigenvalues(5))) 

 

:: 

 

sage: b(63) 

63 [1, 1, 0, -1, 2] 

63 [1, alpha, 0, 1, -2*alpha] 

 

This example illustrates finding field over which the eigenvalues 

are defined:: 

 

sage: M = ModularSymbols(23,2,sign=1).cuspidal_submodule().new_submodule() 

sage: v = M.system_of_eigenvalues(10); v 

[1, alpha, -2*alpha - 1, -alpha - 1, 2*alpha, alpha - 2, 2*alpha + 2, -2*alpha - 1, 2, -2*alpha + 2] 

sage: v[0].parent() 

Number Field in alpha with defining polynomial x^2 + x - 1 

 

This example illustrates setting the print name of the eigenvalue 

field. 

 

:: 

 

sage: A = ModularSymbols(125,sign=1).new_subspace()[0] 

sage: A.system_of_eigenvalues(10) 

[1, alpha, -alpha - 2, -alpha - 1, 0, -alpha - 1, -3, -2*alpha - 1, 3*alpha + 2, 0] 

sage: A.system_of_eigenvalues(10,'x') 

[1, x, -x - 2, -x - 1, 0, -x - 1, -3, -2*x - 1, 3*x + 2, 0] 

""" 

return [self.eigenvalue(m, name=name) for m in range(1,n+1)] 

 

def weight(self): 

""" 

Returns the weight of this Hecke module. 

 

INPUT: 

 

 

- ``self`` - an arbitrary Hecke module 

 

 

OUTPUT: 

 

 

- ``int`` - the weight 

 

 

EXAMPLES:: 

 

sage: m = ModularSymbols(20, weight=2) 

sage: m.weight() 

2 

""" 

return self.__weight 

 

def zero_submodule(self): 

""" 

Return the zero submodule of self. 

 

EXAMPLES:: 

 

sage: ModularSymbols(11,4).zero_submodule() 

Modular Symbols subspace of dimension 0 of Modular Symbols space of dimension 6 for Gamma_0(11) of weight 4 with sign 0 over Rational Field 

sage: CuspForms(11,4).zero_submodule() 

Modular Forms subspace of dimension 0 of Modular Forms space of dimension 4 for Congruence Subgroup Gamma0(11) of weight 4 over Rational Field 

""" 

return self.submodule(self.free_module().zero_submodule(), check=False) 

 

def _dict_set(v, n, key, val): 

r""" 

Rough-and-ready implementation of a two-layer-deep dictionary. 

 

EXAMPLES:: 

 

sage: from sage.modular.hecke.module import _dict_set 

sage: v = {} 

sage: _dict_set(v, 1, 2, 3) 

sage: v 

{1: {2: 3}} 

sage: _dict_set(v, 1, 3, 4); v 

{1: {2: 3, 3: 4}} 

""" 

if n in v: 

v[n][key] = val 

else: 

v[n] = {key:val}