Generic spaces of modular forms

EXAMPLES (computation of base ring): Return the base ring of this space of modular forms.

EXAMPLES: For spaces of modular forms for \Gamma_0(N) or \Gamma_1(N), the default base ring is \QQ:

sage: ModularForms(11,2).base_ring()
Rational Field
sage: ModularForms(1,12).base_ring()
Rational Field
sage: CuspForms(Gamma1(13),3).base_ring()
Rational Field            

The base ring can be explicitly specified in the constructor function.

sage: ModularForms(11,2,base_ring=GF(13)).base_ring()
Finite Field of size 13

For modular forms with character the default base ring is the field generated by the image of the character.

sage: ModularForms(DirichletGroup(13).0,3).base_ring()
Cyclotomic Field of order 12 and degree 4

For example, if the character is quadratic then the field is \QQ (if the characteristic is 0).

sage: ModularForms(DirichletGroup(13).0^6,3).base_ring()
Rational Field

An example in characteristic 7:

sage: ModularForms(13,3,base_ring=GF(7)).base_ring()
Finite Field of size 7
class sage.modular.modform.space.ModularFormsSpace(group, weight, character, base_ring)

A generic space of modular forms.

_ModularFormsSpace__normalize_prec(prec)

If prec=None, return self.prec(). Otherwise, make sure prec is a sensible choice of precision and return it.

EXAMPLES:

sage: N = ModularForms(6,4) 
sage: N._ModularFormsSpace__normalize_prec(int(3))
3
sage: type(N._ModularFormsSpace__normalize_prec(int(3)))
<type 'sage.rings.integer.Integer'>
_ModularFormsSpace__submodule_from_subset_of_basis(x)

Return the submodule of self generated by the elements of x.

EXAMPLES:

sage: N = ModularForms(6,4)
sage: N._ModularFormsSpace__submodule_from_subset_of_basis( [0,2] )
Vector space of degree 5 and dimension 2 over Rational Field
Basis matrix:
[1 0 0 0 0]
[0 0 1 0 0]
__add__(right)

If self and right live inside the same ambient module, return the sum of the two spaces (as modules).

EXAMPLES:

sage: N = CuspForms(44,2) ; ls = [N.submodule([N(u) for u in x.q_expansion_basis(20)]) for x in N.modular_symbols().decomposition()]; ls
[Modular Forms subspace of dimension 1 of Modular Forms space of dimension 9 for Congruence Subgroup Gamma0(44) of weight 2 over Rational Field,
Modular Forms subspace of dimension 3 of Modular Forms space of dimension 9 for Congruence Subgroup Gamma0(44) of weight 2 over Rational Field]
sage: N1 = ls[0] ; N2 = ls[1]
sage: N1 + N2 # indirect doctest
Modular Forms subspace of dimension 4 of Modular Forms space of dimension 9 for Congruence Subgroup Gamma0(44) of weight 2 over Rational Field
__call__(x, check=True)

Try to coerce x into self. If x is a vector of length self.dimension(), interpret it as a list of coefficients for self.basis() and return that linear combination. If x is a power series, it tries to determine whether or not x lives in self. If so, it returns x as an element of M, and throws an error if not.

EXAMPLES:

sage: M = ModularForms(13,4)
sage: M.dimension()
5
sage: M([1,2,3,4,5])
4 + 6*q + 47*q^2 + 143*q^3 + 358*q^4 + 630*q^5 + O(q^6)
sage: M([1,3])
...
TypeError: entries must be a list of length 5
sage: R = M._q_expansion_ring()
sage: M(R([0,1,0,0,0,-2,-4,-2,-12]).add_bigoh(9))
q - 2*q^5 + O(q^6)
sage: M.set_precision(9)         
sage: M(R([0,1,0,0,0,-2,-4,-2,-12]).add_bigoh(9))
q - 2*q^5 - 4*q^6 - 2*q^7 - 12*q^8 + O(q^9)

Note that one only needs coefficients up to self.sturm_bound() to determine the form:

sage: M(R([0,1,0,0,0,-2,-4,-2,-12]).add_bigoh(8))
q - 2*q^5 - 4*q^6 - 2*q^7 - 12*q^8 + O(q^9)
sage: M(R([0,1,1,0,0,0,-4,-2,-12]).add_bigoh(9))
...
ValueError: q-expansion does not correspond to a form in self
sage: S = CuspForms(1,12) ; R = PowerSeriesRing(QQ,'q') ; q = R.0
sage: f = q+O(q^2) ; S(f)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
sage: f = q+2*q^2+O(q^3) ; S(f)
...
ValueError: q-expansion does not correspond to a form in self
sage: f = q-24*q^2+O(q^3) ; S(f)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
__cmp__(x)

Compare self and x.

For spaces of modular forms, we order first by signature, then by dimension, and then by the ordering on the underlying free modules.

EXAMPLES:

sage: N = ModularForms(6,4) ; S = N.cuspidal_subspace()
sage: S.__cmp__(N)
-1
sage: N.__cmp__(S)
1
sage: N.__cmp__(N)
0
sage: M = ModularForms(11,2)
sage: N.__cmp__(M)
-1
sage: M.__cmp__(N)
-1
__init__(group, weight, character, base_ring)

Generic spaces of modular forms. For spaces of modular forms for \Gamma_0(N) or \Gamma_1(N), the default base ring is \QQ.

EXAMPLES:

sage: ModularForms(11,2)
Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field
sage: ModularForms(11,2,base_ring=GF(13))
Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Finite Field of size 13
sage: ModularForms(DirichletGroup(13).0,3)
Modular Forms space of dimension 3, character [zeta12] and weight 3 over Cyclotomic Field of order 12 and degree 4
sage: M = ModularForms(11,2)
sage: M == loads(dumps(M))
True
_coerce_impl(x)

Code to coerce an element into self.

EXAMPLES:

sage: M = ModularForms(22,2) ; S = CuspForms(22,2)
sage: sum(S.basis()) 
q + q^2 - q^3 - 4*q^4 + q^5 + O(q^6)
sage: sum(S.basis() + M.basis())
1 + 3*q + 3*q^2 + 2*q^3 - 7*q^4 + 8*q^5 + O(q^6)
sage: M._coerce_impl(S.basis()[0])
q - q^3 - 2*q^4 + q^5 + O(q^6)
sage: M = ModularForms(Gamma0(22)) ; N = ModularForms(Gamma0(44))
sage: M.basis()[0]
q - q^3 - 2*q^4 + q^5 + O(q^6)
sage: N(M.basis()[0])
q - q^3 - 2*q^4 + q^5 + O(q^6)
_compute_hecke_matrix(n)

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

EXAMPLES:

sage: M = ModularForms(11,2)
sage: M._compute_hecke_matrix(6)
[ 2  0]
[ 0 12]            
sage: M = ModularForms(11,2)
sage: M2 = M.span([M.0 + M.1])
sage: M2.hecke_matrix(2)
...
ArithmeticError: vector is not in free module
_compute_hecke_matrix_prime(p, prec=None)

Compute the matrix of the Hecke operator T_p acting on self.

EXAMPLES:

sage: M = ModularForms(11,2)
sage: M._compute_hecke_matrix_prime(2)
[-2  0]
[ 0  3]            
sage: M = ModularForms(11,2)
sage: M2 = M.span([M.0 + M.1])
sage: M2.hecke_matrix(2)
...
ArithmeticError: vector is not in free module
_compute_q_expansion_basis(prec)

EXAMPLES:

sage: sage.modular.modform.space.ModularFormsSpace(Gamma0(11),2,DirichletGroup(1).0,QQ)._compute_q_expansion_basis(5)
...
NotImplementedError: this must be implemented in the derived class
_has_natural_inclusion_map_to(right)

Return true if there is a natural inclusion map from modular forms in self to modular forms in right.

INPUT:

  • self, right - spaces of modular forms

OUTPUT: True if self embeds in right, and False otherwise.

TODO: Barring a few trivial cases, this only works in the case that right.is_ambient() returns True.

EXAMPLES:

sage: N = ModularForms(6,4) ; S = N.cuspidal_subspace()
sage: N._has_natural_inclusion_map_to(S)
...
NotImplementedError
sage: S._has_natural_inclusion_map_to(N)
True
sage: M = ModularForms(11,2)
sage: N._has_natural_inclusion_map_to(M)
False
_q_expansion(element, prec)

Take an element of self (specified as a list, tuple, or vector), and return the corresponding q-expansion.

EXAMPLES:

sage: m = ModularForms(Gamma0(23),2); m
Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(23) of weight 2 over Rational Field
sage: m.basis()
[
q - q^3 - q^4 + O(q^6),
q^2 - 2*q^3 - q^4 + 2*q^5 + O(q^6),
1 + 12/11*q + 36/11*q^2 + 48/11*q^3 + 84/11*q^4 + 72/11*q^5 + O(q^6)
]
sage: m._q_expansion([1,2,0], 5)
q + 2*q^2 - 5*q^3 - 3*q^4 + O(q^5)
_q_expansion_module()

Return module spanned by coefficients of q-expansions to sufficient precision to determine elements of this space.

EXAMPLES:

sage: M = ModularForms(11,2)
sage: M._q_expansion_module()
Vector space of degree 3 and dimension 2 over Rational Field
User basis matrix:
[   0    1   -2]
[   1 12/5 36/5]
sage: CuspForms(1,12)._q_expansion_module()
Vector space of degree 2 and dimension 1 over Rational Field
User basis matrix:
[0 1]
_q_expansion_ring()

Returns the parent for q-expansions of modular forms in self.

EXAMPLES:

sage: M = ModularForms(11,2)
sage: M._q_expansion_ring()
Power Series Ring in q over Rational Field
_q_expansion_zero()

Returns the q-expansion of the modular form 0.

EXAMPLES:

sage: M = ModularForms(11,2)
sage: M._q_expansion_zero()
0
sage: M._q_expansion_zero() == M._q_expansion_ring()(0)
True
base_extend(base_ring)

Return the base extension of self to base_ring.

EXAMPLES:

sage: M = ModularForms(11,2) ; M
Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field
sage: M.base_extend(CyclotomicField(5))
Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Cyclotomic Field of order 5 and degree 4
basis()

Return a basis for self.

EXAMPLES:

sage: MM = ModularForms(11,2)
sage: MM.basis()
[
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6),
1 + 12/5*q + 36/5*q^2 + 48/5*q^3 + 84/5*q^4 + 72/5*q^5 + O(q^6)
]
change_ring(R)

Change the base ring of this space of modular forms.

TODO: Write this function.

EXAMPLES:

sage: sage.modular.modform.space.ModularFormsSpace(Gamma0(11),2,DirichletGroup(1).0,QQ).change_ring(GF(7))
...
NotImplementedError: This function has not yet been implemented.
character()

Return the Dirichlet character of this space.

EXAMPLES:

sage: M = ModularForms(DirichletGroup(11).0, 3)
sage: M.character()
[zeta10]
sage: s = M.cuspidal_submodule()
sage: s.character()
[zeta10]
sage: CuspForms(DirichletGroup(11).0,3).character()
[zeta10]
cuspidal_submodule()

Return the cuspidal submodule of self.

EXAMPLES:

sage: N = ModularForms(6,4) ; N
Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
sage: N.eisenstein_subspace().dimension()
4
sage: N.cuspidal_submodule()
Cuspidal subspace of dimension 1 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
sage: N.cuspidal_submodule().dimension()
1
cuspidal_subspace()

Synonym for cuspidal_submodule.

EXAMPLES:

sage: N = ModularForms(6,4) ; N
Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
sage: N.eisenstein_subspace().dimension()
4
sage: N.cuspidal_subspace()
Cuspidal subspace of dimension 1 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
sage: N.cuspidal_submodule().dimension()
1
decomposition()

This function returns a list of submodules V(f_i,t) corresponding to newforms f_i of some level dividing the level of self, such that the direct sum of the submodules equals self, if possible. The space V(f_i,t) is the image under g(q) maps to g(q^t) of the intersection with R[[q]] of the space spanned by the conjugates of f_i, where R is the base ring of self.

TODO: Implement this function.

EXAMPLES:

sage: M = ModularForms(11,2); M.decomposition()
...
NotImplementedError
echelon_basis()

Return a basis for self in reduced echelon form. This means that if we view the q-expansions of the basis as defining rows of a matrix (with infinitely many columns), then this matrix is in reduced echelon form.

EXAMPLES:

sage: M = ModularForms(Gamma0(11),4)
sage: M.echelon_basis()
[
1 + O(q^6),
q - 9*q^4 - 10*q^5 + O(q^6),
q^2 + 6*q^4 + 12*q^5 + O(q^6),
q^3 + q^4 + q^5 + O(q^6)
]
sage: M.cuspidal_subspace().echelon_basis()
[
q + 3*q^3 - 6*q^4 - 7*q^5 + O(q^6),
q^2 - 4*q^3 + 2*q^4 + 8*q^5 + O(q^6)
]
sage: M = ModularForms(SL2Z, 12)
sage: M.echelon_basis()
[
1 + 196560*q^2 + 16773120*q^3 + 398034000*q^4 + 4629381120*q^5 + O(q^6),
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
]
sage: M = CuspForms(Gamma0(17),4, prec=10)
sage: M.echelon_basis()
[
q + 2*q^5 - 8*q^7 - 8*q^8 + 7*q^9 + O(q^10),
q^2 - 3/2*q^5 - 7/2*q^6 + 9/2*q^7 + q^8 - 4*q^9 + O(q^10),
q^3 - 2*q^6 + q^7 - 4*q^8 - 2*q^9 + O(q^10),
q^4 - 1/2*q^5 - 5/2*q^6 + 3/2*q^7 + 2*q^9 + O(q^10)
]
echelon_form()

Return a space of modular forms isomorphic to self but with basis of q-expansions in reduced echelon form.

This is useful, e.g., the default basis for spaces of modular forms is rarely in echelon form, but echelon form is useful for quickly recognizing whether a q-expansion is in the space.

EXAMPLES: We first illustrate two ambient spaces and their echelon forms.

sage: M = ModularForms(11)
sage: M.basis()
[
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6),
1 + 12/5*q + 36/5*q^2 + 48/5*q^3 + 84/5*q^4 + 72/5*q^5 + O(q^6)
]
sage: M.echelon_form().basis()
[
1 + 12*q^2 + 12*q^3 + 12*q^4 + 12*q^5 + O(q^6),
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)
]
sage: M = ModularForms(Gamma1(6),4)
sage: M.basis()
[
q - 2*q^2 - 3*q^3 + 4*q^4 + 6*q^5 + O(q^6),
1 + O(q^6),
q - 8*q^4 + 126*q^5 + O(q^6),
q^2 + 9*q^4 + O(q^6),
q^3 + O(q^6)
]
sage: M.echelon_form().basis()
[
1 + O(q^6),
q + 94*q^5 + O(q^6),
q^2 + 36*q^5 + O(q^6),
q^3 + O(q^6),
q^4 - 4*q^5 + O(q^6)
]

We create a space with a funny basis then compute the corresponding echelon form.

sage: M = ModularForms(11,4)
sage: M.basis()
[
q + 3*q^3 - 6*q^4 - 7*q^5 + O(q^6),
q^2 - 4*q^3 + 2*q^4 + 8*q^5 + O(q^6),
1 + O(q^6),
q + 9*q^2 + 28*q^3 + 73*q^4 + 126*q^5 + O(q^6)
]
sage: F = M.span_of_basis([M.0 + 1/3*M.1, M.2 + M.3]); F.basis()
[
q + 1/3*q^2 + 5/3*q^3 - 16/3*q^4 - 13/3*q^5 + O(q^6),
1 + q + 9*q^2 + 28*q^3 + 73*q^4 + 126*q^5 + O(q^6)
]
sage: E = F.echelon_form(); E.basis()
[
1 + 26/3*q^2 + 79/3*q^3 + 235/3*q^4 + 391/3*q^5 + O(q^6),
q + 1/3*q^2 + 5/3*q^3 - 16/3*q^4 - 13/3*q^5 + O(q^6)
]
eisenstein_series()

Compute the Eisenstein series associated to this space.

Note

This function should be overridden by all derived classes.

EXAMPLES:

sage: M = sage.modular.modform.space.ModularFormsSpace(Gamma0(11),2,DirichletGroup(1).0,base_ring=QQ) ; M.eisenstein_series()
...
NotImplementedError: computation of Eisenstein series in this space not yet implemented
eisenstein_submodule()

Return the Eisenstein submodule for this space of modular forms.

EXAMPLES:

sage: M = ModularForms(11,2)
sage: M.eisenstein_submodule()
Eisenstein subspace of dimension 1 of Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field
eisenstein_subspace()

Synonym for eisenstein_submodule.

EXAMPLES:

sage: M = ModularForms(11,2)
sage: M.eisenstein_subspace()
Eisenstein subspace of dimension 1 of Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field
embedded_submodule()

Return the underlying module of self.

EXAMPLES:

sage: N = ModularForms(6,4)
sage: N.dimension()
5
sage: N.embedded_submodule()
Vector space of dimension 5 over Rational Field
find_in_space(f, forms=None, prec=None, indep=True)

INPUT:

  • f - a modular form or power series
  • forms - (default: None) a specific list of modular forms or q-expansions.
  • prec - if forms are given, compute with them to the given precision
  • indep - (default: True) whether the given list of forms are assumed to form a basis.

OUTPUT: A list of numbers that give f as a linear combination of the basis for this space or of the given forms if independent=True.

Note

If the list of forms is given, they do not have to be in self.

EXAMPLES:

sage: M = ModularForms(11,2)
sage: N = ModularForms(10,2)
sage: M.find_in_space( M.basis()[0] )
[1, 0]
sage: M.find_in_space( N.basis()[0], forms=N.basis() )
[1, 0, 0]
sage: M.find_in_space( N.basis()[0] )
...
ArithmeticError: vector is not in free module
gen(n)

Return the nth generator of self.

EXAMPLES:

sage: N = ModularForms(6,4)
sage: N.basis()
[
q - 2*q^2 - 3*q^3 + 4*q^4 + 6*q^5 + O(q^6),
1 + O(q^6),
q - 8*q^4 + 126*q^5 + O(q^6),
q^2 + 9*q^4 + O(q^6),
q^3 + O(q^6)
]
sage: N.gen(0)
q - 2*q^2 - 3*q^3 + 4*q^4 + 6*q^5 + O(q^6)
sage: N.gen(4)
q^3 + O(q^6)
sage: N.gen(5)
...
ValueError: Generator 5 not defined
gens()

Return a complete set of generators for self.

EXAMPLES:

sage: N = ModularForms(6,4)
sage: N.gens()
[
q - 2*q^2 - 3*q^3 + 4*q^4 + 6*q^5 + O(q^6),
1 + O(q^6),
q - 8*q^4 + 126*q^5 + O(q^6),
q^2 + 9*q^4 + O(q^6),
q^3 + O(q^6)
]
group()

Return the congruence subgroup associated to this space of modular forms.

EXAMPLES:

sage: ModularForms(Gamma0(12),4).group()
Congruence Subgroup Gamma0(12)
sage: CuspForms(Gamma1(113),2).group()
Congruence Subgroup Gamma1(113)

Note that \Gamma_1(1) and \Gamma_0(1) are replaced by \mathrm{SL}_2(\ZZ).

sage: CuspForms(Gamma1(1),12).group()
Modular Group SL(2,Z)
sage: CuspForms(SL2Z,12).group()
Modular Group SL(2,Z)
has_character()

Return True if this space of modular forms has a specific character.

This is True exactly when the character() function does not return None.

EXAMPLES: A space for \Gamma_0(N) has trivial character, hence has a character.

sage: CuspForms(Gamma0(11),2).has_character()
True

A space for \Gamma_1(N) (for N\geq 2) never has a specific character.

sage: CuspForms(Gamma1(11),2).has_character()
False
sage: CuspForms(DirichletGroup(11).0,3).has_character()
True
has_coerce_map_from_impl(from_par)

Code to make ModularFormsSpace work well with coercion framework.

EXAMPLES:

sage: M = ModularForms(22,2)
sage: M.has_coerce_map_from_impl(M.cuspidal_subspace())
True
sage: M.has_coerce_map_from(ModularForms(22,4))
False
integral_basis()

Return an integral basis for this space of modular forms.

EXAMPLES: In this example the integral and echelon bases are different.

sage: m = ModularForms(97,2,prec=10)
sage: s = m.cuspidal_subspace()
sage: s.integral_basis()
[
q + 2*q^7 + 4*q^8 - 2*q^9 + O(q^10),
q^2 + q^4 + q^7 + 3*q^8 - 3*q^9 + O(q^10),
q^3 + q^4 - 3*q^8 + q^9 + O(q^10),
2*q^4 - 2*q^8 + O(q^10),
q^5 - 2*q^8 + 2*q^9 + O(q^10),
q^6 + 2*q^7 + 5*q^8 - 5*q^9 + O(q^10),
3*q^7 + 6*q^8 - 4*q^9 + O(q^10)
]
sage: s.echelon_basis()
[
q + 2/3*q^9 + O(q^10),
q^2 + 2*q^8 - 5/3*q^9 + O(q^10),
q^3 - 2*q^8 + q^9 + O(q^10),
q^4 - q^8 + O(q^10),
q^5 - 2*q^8 + 2*q^9 + O(q^10),
q^6 + q^8 - 7/3*q^9 + O(q^10),
q^7 + 2*q^8 - 4/3*q^9 + O(q^10)
]

Here’s another example where there is a big gap in the valuations:

sage: m = CuspForms(64,2)
sage: m.integral_basis()
[
q + O(q^6),
q^2 + O(q^6),
q^5 + O(q^6)
]        

TESTS:

sage: m = CuspForms(11*2^4,2, prec=13); m
Cuspidal subspace of dimension 19 of Modular Forms space of dimension 30 for Congruence Subgroup Gamma0(176) of weight 2 over Rational Field
sage: m.integral_basis()          # takes a long time (3 or 4 seconds)
[
q + O(q^13),
q^2 + O(q^13),
q^3 + O(q^13),
q^4 + O(q^13),
q^5 + O(q^13),
q^6 + O(q^13),
q^7 + O(q^13),
q^8 + O(q^13),
q^9 + O(q^13),
q^10 + O(q^13),
q^11 + O(q^13),
q^12 + O(q^13),
O(q^13),
O(q^13),
O(q^13),
O(q^13),
O(q^13),
O(q^13),
O(q^13)
]
is_ambient()

Return True if this an ambient space of modular forms.

EXAMPLES:

sage: M = ModularForms(Gamma1(4),4)
sage: M.is_ambient()
True
sage: E = M.eisenstein_subspace()
sage: E.is_ambient()
False
level()

Return the level of self.

EXAMPLES:

sage: M = ModularForms(47,3)
sage: M.level()
47
modular_symbols(sign=0)

Return the space of modular symbols corresponding to self with the given sign.

EXAMPLES:

sage: M = sage.modular.modform.space.ModularFormsSpace(Gamma0(11),2,DirichletGroup(1).0,base_ring=QQ) ; M.modular_symbols()
...
NotImplementedError: computation of associated modular symbols space not yet implemented
new_submodule(p=None)

Return the new submodule of self. If p is specified, return the p-new submodule of self.

Note

This function should be overridden by all derived classes.

EXAMPLES:

sage: M = sage.modular.modform.space.ModularFormsSpace(Gamma0(11),2,DirichletGroup(1).0,base_ring=QQ) ; M.new_submodule()
...
NotImplementedError: computation of new submodule not yet implemented
new_subspace(p=None)

Synonym for new_submodule.

EXAMPLES:

sage: M = sage.modular.modform.space.ModularFormsSpace(Gamma0(11),2,DirichletGroup(1).0,base_ring=QQ) ; M.new_subspace()
...
NotImplementedError: computation of new submodule not yet implemented
newforms(names=None)

Return all newforms in the cuspidal subspace of self.

EXAMPLES:

sage: CuspForms(18,4).newforms()
[q + 2*q^2 + 4*q^4 - 6*q^5 + O(q^6)]
sage: CuspForms(32,4).newforms()
[q - 8*q^3 - 10*q^5 + O(q^6), q + 22*q^5 + O(q^6), q + 8*q^3 - 10*q^5 + O(q^6)]
sage: CuspForms(23).newforms('b') 
[q + b0*q^2 + (-2*b0 - 1)*q^3 + (-b0 - 1)*q^4 + 2*b0*q^5 + O(q^6)] 
sage: CuspForms(23).newforms() 
... 
ValueError: Please specify a name to be used when generating names for generators of Hecke eigenvalue fields corresponding to the newforms.
prec(new_prec=None)

Return or set the default precision used for displaying q-expansions of elements of this space.

INPUT:

  • new_prec - positive integer (default: None)

OUTPUT: if new_prec is None, returns the current precision.

EXAMPLES:

sage: M = ModularForms(1,12)
sage: S = M.cuspidal_subspace()
sage: S.prec()
6
sage: S.basis()
[
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
]
sage: S.prec(8)
8
sage: S.basis()
[
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + O(q^8)
]
q_echelon_basis(prec=None)

Return the echelon form of the basis of q-expansions of self up to precision prec.

The q-expansions are power series (not actual modular forms). The number of q-expansions returned equals the dimension.

EXAMPLES:

sage: M = ModularForms(11,2)
sage: M.q_expansion_basis()
[
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6),
1 + 12/5*q + 36/5*q^2 + 48/5*q^3 + 84/5*q^4 + 72/5*q^5 + O(q^6)
]
sage: M.q_echelon_basis()
[
1 + 12*q^2 + 12*q^3 + 12*q^4 + 12*q^5 + O(q^6),
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)
]
q_expansion_basis(prec=None)

Return a sequence of q-expansions for the basis of this space computed to the given input precision.

INPUT:

  • prec - integer (=0) or None

If prec is None, the prec is computed to be at least large enough so that each q-expansion determines the form as an element of this space.

Note

In fact, the q-expansion basis is always computed to at least self.prec().

EXAMPLES:

sage: S = ModularForms(11,2).cuspidal_submodule()
sage: S.q_expansion_basis()
[
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)
]            
sage: S.q_expansion_basis(5)
[
q - 2*q^2 - q^3 + 2*q^4 + O(q^5)
]
sage: S = ModularForms(1,24).cuspidal_submodule()
sage: S.q_expansion_basis(8)
[
q + 195660*q^3 + 12080128*q^4 + 44656110*q^5 - 982499328*q^6 - 147247240*q^7 + O(q^8),
q^2 - 48*q^3 + 1080*q^4 - 15040*q^5 + 143820*q^6 - 985824*q^7 + O(q^8)
]
q_integral_basis(prec=None)

Return a \ZZ-reduced echelon basis of q-expansions for self.

The q-expansions are power series with coefficients in \ZZ; they are not actual modular forms.

The base ring of self must be \QQ. The number of q-expansions returned equals the dimension.

EXAMPLES:

sage: S = CuspForms(11,2)
sage: S.q_integral_basis(5)
[
q - 2*q^2 - q^3 + 2*q^4 + O(q^5)
]
set_precision(new_prec)

Set the default precision used for displaying q-expansions.

INPUT:

  • new_prec - positive integer

EXAMPLES:

sage: M = ModularForms(Gamma0(37),2)
sage: M.set_precision(10)
sage: S = M.cuspidal_subspace()
sage: S.basis()
[
q + q^3 - 2*q^4 - q^7 - 2*q^9 + O(q^10),
q^2 + 2*q^3 - 2*q^4 + q^5 - 3*q^6 - 4*q^9 + O(q^10)
]
sage: S.set_precision(0)
sage: S.basis()
[
O(q^0),
O(q^0)
]

The precision of subspaces is the same as the precision of the ambient space.

sage: S.set_precision(2)
sage: M.basis()
[
q + O(q^2),
O(q^2),
1 + 2/3*q + O(q^2)
]

The precision must be nonnegative:

sage: S.set_precision(-1)
...
ValueError: n (=-1) must be >= 0 

We do another example with nontrivial character.

sage: M = ModularForms(DirichletGroup(13).0^2)
sage: M.set_precision(10)
sage: M.cuspidal_subspace().0
q + (-zeta6 - 1)*q^2 + (2*zeta6 - 2)*q^3 + zeta6*q^4 + (-2*zeta6 + 1)*q^5 + (-2*zeta6 + 4)*q^6 + (2*zeta6 - 1)*q^8 - zeta6*q^9 + O(q^10)
span(B)

Take a set B of forms, and return the subspace of self with B as a basis.

EXAMPLES:

sage: N = ModularForms(6,4) ; N
Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
sage: N.span_of_basis([N.basis()[0]])
Modular Forms subspace of dimension 1 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
sage: N.span_of_basis([N.basis()[0], N.basis()[1]])
Modular Forms subspace of dimension 2 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
sage: N.span_of_basis( N.basis() )
Modular Forms subspace of dimension 5 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
span_of_basis(B)

Take a set B of forms, and return the subspace of self with B as a basis.

EXAMPLES:

sage: N = ModularForms(6,4) ; N
Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
sage: N.span_of_basis([N.basis()[0]])
Modular Forms subspace of dimension 1 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
sage: N.span_of_basis([N.basis()[0], N.basis()[1]])
Modular Forms subspace of dimension 2 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
sage: N.span_of_basis( N.basis() )
Modular Forms subspace of dimension 5 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(6) of weight 4 over Rational Field
sturm_bound(M=None)

For a space M of modular forms, this function returns an integer B such that two modular forms in either self or M are equal if and only if their q-expansions are equal to precision B (note that this is 1+ the usual Sturm bound, since O(q^\mathrm{prec}) has precision prec). If M is none, then M is set equal to self.

EXAMPLES:

sage: S37=CuspForms(37,2)
sage: S37.sturm_bound()
8
sage: M = ModularForms(11,2)
sage: M.sturm_bound()
3
sage: ModularForms(Gamma1(15),2).sturm_bound()
33

Note

Reference for the Sturm bound that we use in the definition of of this function:

J. Sturm, On the congruence of modular forms, Number theory (New York, 1984-1985), Springer, Berlin, 1987, pp. 275-280.

Useful Remark:

Kevin Buzzard pointed out to me (William Stein) in Fall 2002 that the above bound is fine for Gamma1 with character, as one sees by taking a power of f. More precisely, if f\cong 0\pmod{p} for first s coefficients, then f^r = 0 \pmod{p} for first s r coefficients. Since the weight of f^r is r \text{weight}(f), it follows that if s \geq the sturm bound for \Gamma_0 at weight(f), then f^r has valuation large enough to be forced to be 0 at r\cdot weight(f) by Sturm bound (which is valid if we choose r right). Thus f \cong 0 \pmod{p}. Conclusion: For \Gamma_1 with fixed character, the Sturm bound is exactly the same as for \Gamma_0. A key point is that we are finding \ZZ[\varepsilon] generators for the Hecke algebra here, not \ZZ-generators. So if one wants generators for the Hecke algebra over \ZZ, this bound is wrong.

This bound works over any base, even a finite field. There might be much better bounds over \QQ, or for comparing two eigenforms.

weight()

Return the weight of this space of modular forms.

EXAMPLES:

sage: M = ModularForms(Gamma1(13),11)
sage: M.weight()
11
sage: M = ModularForms(Gamma0(997),100)
sage: M.weight()
100
sage: M = ModularForms(Gamma0(97),4)
sage: M.weight()
4
sage: M.eisenstein_submodule().weight()
4
sage.modular.modform.space.contains_each(V, B)

Determine whether or not V contains every element of B. Used here for linear algebra, but works very generally.

EXAMPLES:

sage: contains_each = sage.modular.modform.space.contains_each
sage: contains_each( range(20), prime_range(20) )
True
sage: contains_each( range(20), range(30) )
False
sage.modular.modform.space.is_ModularFormsSpace(x)

Return True if x is a `ModularFormsSpace`.

EXAMPLES:

sage: from sage.modular.modform.space import is_ModularFormsSpace
sage: is_ModularFormsSpace(ModularForms(11,2))
True
sage: is_ModularFormsSpace(CuspForms(11,2))
True
sage: is_ModularFormsSpace(3)
False

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