Submodules of Hecke modules¶
- class sage.modular.hecke.submodule.HeckeSubmodule(ambient, submodule, dual_free_module=None, check=True)[source]¶
- Bases: - HeckeModule_free_module- Submodule of a Hecke module. - ambient()[source]¶
- Synonym for ambient_hecke_module. - EXAMPLES: - sage: CuspForms(2, 12).ambient() Modular Forms space of dimension 4 for Congruence Subgroup Gamma0(2) of weight 12 over Rational Field - >>> from sage.all import * >>> CuspForms(Integer(2), Integer(12)).ambient() Modular Forms space of dimension 4 for Congruence Subgroup Gamma0(2) of weight 12 over Rational Field 
 - ambient_hecke_module()[source]¶
- Return the ambient Hecke module of which this is a submodule. - EXAMPLES: - sage: CuspForms(2, 12).ambient_hecke_module() Modular Forms space of dimension 4 for Congruence Subgroup Gamma0(2) of weight 12 over Rational Field - >>> from sage.all import * >>> CuspForms(Integer(2), Integer(12)).ambient_hecke_module() Modular Forms space of dimension 4 for Congruence Subgroup Gamma0(2) of weight 12 over Rational Field 
 - complement(bound=None)[source]¶
- Return the largest Hecke-stable complement of this space. - EXAMPLES: - sage: M = ModularSymbols(15, 6).cuspidal_subspace() sage: M.complement() Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 20 for Gamma_0(15) of weight 6 with sign 0 over Rational Field sage: E = EllipticCurve("128a") sage: ME = E.modular_symbol_space() sage: ME.complement() Modular Symbols subspace of dimension 17 of Modular Symbols space of dimension 18 for Gamma_0(128) of weight 2 with sign 1 over Rational Field - >>> from sage.all import * >>> M = ModularSymbols(Integer(15), Integer(6)).cuspidal_subspace() >>> M.complement() Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 20 for Gamma_0(15) of weight 6 with sign 0 over Rational Field >>> E = EllipticCurve("128a") >>> ME = E.modular_symbol_space() >>> ME.complement() Modular Symbols subspace of dimension 17 of Modular Symbols space of dimension 18 for Gamma_0(128) of weight 2 with sign 1 over Rational Field 
 - degeneracy_map(level, t=1)[source]¶
- The \(t\)-th degeneracy map from - selfto the space of ambient modular symbols of the given level. The level of- selfmust be a divisor or multiple of level, and \(t\) must be a divisor of the quotient.- INPUT: - level– positive integer; the level of the codomain of the map
- t– integer; the parameter of the degeneracy map, i.e., the map is related to \(f(q)\) - \(f(q^t)\)
 - OUTPUT: - A linear function from - selfto the space of modular symbols of given level with the same weight, character, sign, etc., as this space.- EXAMPLES: - sage: D = ModularSymbols(10,4).cuspidal_submodule().decomposition(); D [Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 10 for Gamma_0(10) of weight 4 with sign 0 over Rational Field, Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 10 for Gamma_0(10) of weight 4 with sign 0 over Rational Field] sage: d = D[1].degeneracy_map(5); d Hecke module morphism defined by the matrix [ 0 0 -1 1] [ 0 1/2 3/2 -2] [ 0 -1 1 0] [ 0 -3/4 -1/4 1] Domain: Modular Symbols subspace of dimension 4 of Modular Symbols space ... Codomain: Modular Symbols space of dimension 4 for Gamma_0(5) of weight ... - >>> from sage.all import * >>> D = ModularSymbols(Integer(10),Integer(4)).cuspidal_submodule().decomposition(); D [Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 10 for Gamma_0(10) of weight 4 with sign 0 over Rational Field, Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 10 for Gamma_0(10) of weight 4 with sign 0 over Rational Field] >>> d = D[Integer(1)].degeneracy_map(Integer(5)); d Hecke module morphism defined by the matrix [ 0 0 -1 1] [ 0 1/2 3/2 -2] [ 0 -1 1 0] [ 0 -3/4 -1/4 1] Domain: Modular Symbols subspace of dimension 4 of Modular Symbols space ... Codomain: Modular Symbols space of dimension 4 for Gamma_0(5) of weight ... - sage: d.rank() 2 sage: d.kernel() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 10 for Gamma_0(10) of weight 4 with sign 0 over Rational Field sage: d.image() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 4 for Gamma_0(5) of weight 4 with sign 0 over Rational Field - >>> from sage.all import * >>> d.rank() 2 >>> d.kernel() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 10 for Gamma_0(10) of weight 4 with sign 0 over Rational Field >>> d.image() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 4 for Gamma_0(5) of weight 4 with sign 0 over Rational Field 
 - dual_free_module(bound=None, anemic=True, use_star=True)[source]¶
- Compute embedded dual free module if possible. - In general this will not be possible, e.g., if this space is not Hecke equivariant, possibly if it is not cuspidal, or if the characteristic is not 0. In all these cases we raise a - RuntimeErrorexception.- If - use_staris- True(which is the default), we also use the +/- eigenspaces for the star operator to find the dual free module of- self. If- selfdoes not have a star involution,- use_starwill automatically be set to- False.- EXAMPLES: - sage: M = ModularSymbols(11, 2) sage: M.dual_free_module() Vector space of dimension 3 over Rational Field sage: Mpc = M.plus_submodule().cuspidal_submodule() sage: Mcp = M.cuspidal_submodule().plus_submodule() sage: Mcp.dual_free_module() == Mpc.dual_free_module() True sage: Mpc.dual_free_module() Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [ 1 5/2 5] sage: M = ModularSymbols(35,2).cuspidal_submodule() sage: M.dual_free_module(use_star=False) Vector space of degree 9 and dimension 6 over Rational Field Basis matrix: [ 1 0 0 0 -1 0 0 4 -2] [ 0 1 0 0 0 0 0 -1/2 1/2] [ 0 0 1 0 0 0 0 -1/2 1/2] [ 0 0 0 1 -1 0 0 1 0] [ 0 0 0 0 0 1 0 -2 1] [ 0 0 0 0 0 0 1 -2 1] sage: M = ModularSymbols(40,2) sage: Mmc = M.minus_submodule().cuspidal_submodule() sage: Mcm = M.cuspidal_submodule().minus_submodule() sage: Mcm.dual_free_module() == Mmc.dual_free_module() True sage: Mcm.dual_free_module() Vector space of degree 13 and dimension 3 over Rational Field Basis matrix: [ 0 1 0 0 0 0 1 0 -1 -1 1 -1 0] [ 0 0 1 0 -1 0 -1 0 1 0 0 0 0] [ 0 0 0 0 0 1 1 0 -1 0 0 0 0] sage: M = ModularSymbols(43).cuspidal_submodule() sage: S = M[0].plus_submodule() + M[1].minus_submodule() sage: S.dual_free_module(use_star=False) Traceback (most recent call last): ... RuntimeError: Computation of complementary space failed (cut down to rank 7, but should have cut down to rank 4). sage: S.dual_free_module().dimension() == S.dimension() True - >>> from sage.all import * >>> M = ModularSymbols(Integer(11), Integer(2)) >>> M.dual_free_module() Vector space of dimension 3 over Rational Field >>> Mpc = M.plus_submodule().cuspidal_submodule() >>> Mcp = M.cuspidal_submodule().plus_submodule() >>> Mcp.dual_free_module() == Mpc.dual_free_module() True >>> Mpc.dual_free_module() Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [ 1 5/2 5] >>> M = ModularSymbols(Integer(35),Integer(2)).cuspidal_submodule() >>> M.dual_free_module(use_star=False) Vector space of degree 9 and dimension 6 over Rational Field Basis matrix: [ 1 0 0 0 -1 0 0 4 -2] [ 0 1 0 0 0 0 0 -1/2 1/2] [ 0 0 1 0 0 0 0 -1/2 1/2] [ 0 0 0 1 -1 0 0 1 0] [ 0 0 0 0 0 1 0 -2 1] [ 0 0 0 0 0 0 1 -2 1] >>> M = ModularSymbols(Integer(40),Integer(2)) >>> Mmc = M.minus_submodule().cuspidal_submodule() >>> Mcm = M.cuspidal_submodule().minus_submodule() >>> Mcm.dual_free_module() == Mmc.dual_free_module() True >>> Mcm.dual_free_module() Vector space of degree 13 and dimension 3 over Rational Field Basis matrix: [ 0 1 0 0 0 0 1 0 -1 -1 1 -1 0] [ 0 0 1 0 -1 0 -1 0 1 0 0 0 0] [ 0 0 0 0 0 1 1 0 -1 0 0 0 0] >>> M = ModularSymbols(Integer(43)).cuspidal_submodule() >>> S = M[Integer(0)].plus_submodule() + M[Integer(1)].minus_submodule() >>> S.dual_free_module(use_star=False) Traceback (most recent call last): ... RuntimeError: Computation of complementary space failed (cut down to rank 7, but should have cut down to rank 4). >>> S.dual_free_module().dimension() == S.dimension() True - We test that Issue #5080 is fixed: - sage: EllipticCurve('128a').congruence_number() 32 - >>> from sage.all import * >>> EllipticCurve('128a').congruence_number() 32 
 - free_module()[source]¶
- Return the free module corresponding to - self.- EXAMPLES: - sage: M = ModularSymbols(33,2).cuspidal_subspace() ; M Modular Symbols subspace of dimension 6 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field sage: M.free_module() Vector space of degree 9 and dimension 6 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 -1 1] [ 0 0 1 0 0 0 0 -1 1] [ 0 0 0 1 0 0 0 -1 1] [ 0 0 0 0 1 0 0 -1 1] [ 0 0 0 0 0 1 0 -1 1] [ 0 0 0 0 0 0 1 -1 0] - >>> from sage.all import * >>> M = ModularSymbols(Integer(33),Integer(2)).cuspidal_subspace() ; M Modular Symbols subspace of dimension 6 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field >>> M.free_module() Vector space of degree 9 and dimension 6 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 -1 1] [ 0 0 1 0 0 0 0 -1 1] [ 0 0 0 1 0 0 0 -1 1] [ 0 0 0 0 1 0 0 -1 1] [ 0 0 0 0 0 1 0 -1 1] [ 0 0 0 0 0 0 1 -1 0] 
 - hecke_bound()[source]¶
- Compute the Hecke bound for - self.- This is a number \(n\) such that the \(T_m\) for \(m \leq n\) generate the Hecke algebra. - EXAMPLES: - sage: M = ModularSymbols(24,8) sage: M.hecke_bound() 53 sage: M.cuspidal_submodule().hecke_bound() 32 sage: M.eisenstein_submodule().hecke_bound() 53 - >>> from sage.all import * >>> M = ModularSymbols(Integer(24),Integer(8)) >>> M.hecke_bound() 53 >>> M.cuspidal_submodule().hecke_bound() 32 >>> M.eisenstein_submodule().hecke_bound() 53 
 - intersection(other)[source]¶
- Return the intersection of - selfand- other, which must both lie in a common ambient space of modular symbols.- EXAMPLES: - sage: M = ModularSymbols(43, sign=1) sage: A = M[0] + M[1] sage: B = M[1] + M[2] sage: A.dimension(), B.dimension() (2, 3) sage: C = A.intersection(B); C.dimension() 1 - >>> from sage.all import * >>> M = ModularSymbols(Integer(43), sign=Integer(1)) >>> A = M[Integer(0)] + M[Integer(1)] >>> B = M[Integer(1)] + M[Integer(2)] >>> A.dimension(), B.dimension() (2, 3) >>> C = A.intersection(B); C.dimension() 1 
 - is_ambient()[source]¶
- Return - Trueif- selfis an ambient space of modular symbols.- EXAMPLES: - sage: M = ModularSymbols(17,4) sage: M.cuspidal_subspace().is_ambient() False sage: A = M.ambient_hecke_module() sage: S = A.submodule(A.basis()) sage: sage.modular.hecke.submodule.HeckeSubmodule.is_ambient(S) True - >>> from sage.all import * >>> M = ModularSymbols(Integer(17),Integer(4)) >>> M.cuspidal_subspace().is_ambient() False >>> A = M.ambient_hecke_module() >>> S = A.submodule(A.basis()) >>> sage.modular.hecke.submodule.HeckeSubmodule.is_ambient(S) True 
 - is_new(p=None)[source]¶
- Return - Trueif this Hecke module is \(p\)-new. If \(p\) is None, returns- Trueif it is new.- EXAMPLES: - sage: M = ModularSymbols(1,16) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) sage: S.is_new() True - >>> from sage.all import * >>> M = ModularSymbols(Integer(1),Integer(16)) >>> S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) >>> S.is_new() True 
 - is_old(p=None)[source]¶
- Return - Trueif this Hecke module is \(p\)-old. If \(p\) is- None, returns- Trueif it is old.- EXAMPLES: - sage: M = ModularSymbols(50,2) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.old_submodule().free_module()) sage: S.is_old() True sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.new_submodule().free_module()) sage: S.is_old() False - >>> from sage.all import * >>> M = ModularSymbols(Integer(50),Integer(2)) >>> S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.old_submodule().free_module()) >>> S.is_old() True >>> S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.new_submodule().free_module()) >>> S.is_old() False 
 - is_submodule(V)[source]¶
- Return - Trueif and only if- selfis a submodule of V.- EXAMPLES: - sage: M = ModularSymbols(30,4) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) sage: S.is_submodule(M) True sage: SS = sage.modular.hecke.submodule.HeckeSubmodule(M, M.old_submodule().free_module()) sage: S.is_submodule(SS) False - >>> from sage.all import * >>> M = ModularSymbols(Integer(30),Integer(4)) >>> S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) >>> S.is_submodule(M) True >>> SS = sage.modular.hecke.submodule.HeckeSubmodule(M, M.old_submodule().free_module()) >>> S.is_submodule(SS) False 
 - linear_combination_of_basis(v)[source]¶
- Return the linear combination of the basis of - selfgiven by the entries of \(v\).- The result can be of different types, and is printed accordingly, depending on the type of submodule. - EXAMPLES: - sage: M = ModularForms(Gamma0(2),12) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) sage: S.basis() ((1, 0, 0, 0), (0, 1, 0, 0)) sage: S.linear_combination_of_basis([3, 10]) (3, 10, 0, 0) sage: S = M.cuspidal_submodule() sage: S.basis() [q + 252*q^3 - 2048*q^4 + 4830*q^5 + O(q^6), q^2 - 24*q^4 + O(q^6)] sage: S.linear_combination_of_basis([3, 10]) 3*q + 10*q^2 + 756*q^3 - 6384*q^4 + 14490*q^5 + O(q^6) - >>> from sage.all import * >>> M = ModularForms(Gamma0(Integer(2)),Integer(12)) >>> S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) >>> S.basis() ((1, 0, 0, 0), (0, 1, 0, 0)) >>> S.linear_combination_of_basis([Integer(3), Integer(10)]) (3, 10, 0, 0) >>> S = M.cuspidal_submodule() >>> S.basis() [q + 252*q^3 - 2048*q^4 + 4830*q^5 + O(q^6), q^2 - 24*q^4 + O(q^6)] >>> S.linear_combination_of_basis([Integer(3), Integer(10)]) 3*q + 10*q^2 + 756*q^3 - 6384*q^4 + 14490*q^5 + O(q^6) 
 - module()[source]¶
- Alias for code{self.free_module()}. - EXAMPLES: - sage: M = ModularSymbols(17,4).cuspidal_subspace() sage: M.free_module() is M.module() True - >>> from sage.all import * >>> M = ModularSymbols(Integer(17),Integer(4)).cuspidal_subspace() >>> M.free_module() is M.module() True 
 - new_submodule(p=None)[source]¶
- Return the new or \(p\)-new submodule of this space of modular symbols. - EXAMPLES: - sage: M = ModularSymbols(20,4) sage: M.new_submodule() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_0(20) of weight 4 with sign 0 over Rational Field sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) sage: S Rank 12 submodule of a Hecke module of level 20 sage: S.new_submodule() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_0(20) of weight 4 with sign 0 over Rational Field - >>> from sage.all import * >>> M = ModularSymbols(Integer(20),Integer(4)) >>> M.new_submodule() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_0(20) of weight 4 with sign 0 over Rational Field >>> S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) >>> S Rank 12 submodule of a Hecke module of level 20 >>> S.new_submodule() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_0(20) of weight 4 with sign 0 over Rational Field 
 - nonembedded_free_module()[source]¶
- Return the free module corresponding to - selfas an abstract free module, i.e. not as an embedded vector space.- EXAMPLES: - sage: M = ModularSymbols(12,6) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) sage: S Rank 14 submodule of a Hecke module of level 12 sage: S.nonembedded_free_module() Vector space of dimension 14 over Rational Field - >>> from sage.all import * >>> M = ModularSymbols(Integer(12),Integer(6)) >>> S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) >>> S Rank 14 submodule of a Hecke module of level 12 >>> S.nonembedded_free_module() Vector space of dimension 14 over Rational Field 
 - old_submodule(p=None)[source]¶
- Return the old or \(p\)-old submodule of this space of modular symbols. - EXAMPLES: We compute the old and new submodules of \(\mathbf{S}_2(\Gamma_0(33))\). - sage: M = ModularSymbols(33); S = M.cuspidal_submodule(); S Modular Symbols subspace of dimension 6 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field sage: S.old_submodule() Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field sage: S.new_submodule() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field - >>> from sage.all import * >>> M = ModularSymbols(Integer(33)); S = M.cuspidal_submodule(); S Modular Symbols subspace of dimension 6 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field >>> S.old_submodule() Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field >>> S.new_submodule() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field 
 - rank()[source]¶
- Return the rank of - selfas a free module over the base ring.- EXAMPLES: - sage: ModularSymbols(6, 4).cuspidal_subspace().rank() 2 sage: ModularSymbols(6, 4).cuspidal_subspace().dimension() 2 - >>> from sage.all import * >>> ModularSymbols(Integer(6), Integer(4)).cuspidal_subspace().rank() 2 >>> ModularSymbols(Integer(6), Integer(4)).cuspidal_subspace().dimension() 2 
 - submodule(M, Mdual=None, check=True)[source]¶
- Construct a submodule of - selffrom the free module M, which must be a subspace of- self.- EXAMPLES: - sage: M = ModularSymbols(18,4) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) sage: S[0] Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_0(18) of weight 4 with sign 0 over Rational Field sage: S.submodule(S[0].free_module()) Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_0(18) of weight 4 with sign 0 over Rational Field - >>> from sage.all import * >>> M = ModularSymbols(Integer(18),Integer(4)) >>> S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) >>> S[Integer(0)] Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_0(18) of weight 4 with sign 0 over Rational Field >>> S.submodule(S[Integer(0)].free_module()) Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_0(18) of weight 4 with sign 0 over Rational Field 
 - submodule_from_nonembedded_module(V, Vdual=None, check=True)[source]¶
- Construct a submodule of - selffrom V. Here V should be a subspace of a vector space whose dimension is the same as that of- self.- INPUT: - V– submodule of ambient free module of the same rank as the rank of- self
- check– whether to check that V is Hecke equivariant
 - OUTPUT: Hecke submodule of self - EXAMPLES: - sage: M = ModularSymbols(37,2) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) sage: V = (QQ**4).subspace([[1,-1,0,1/2],[0,0,1,-1/2]]) sage: S.submodule_from_nonembedded_module(V) Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 5 for Gamma_0(37) of weight 2 with sign 0 over Rational Field - >>> from sage.all import * >>> M = ModularSymbols(Integer(37),Integer(2)) >>> S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) >>> V = (QQ**Integer(4)).subspace([[Integer(1),-Integer(1),Integer(0),Integer(1)/Integer(2)],[Integer(0),Integer(0),Integer(1),-Integer(1)/Integer(2)]]) >>> S.submodule_from_nonembedded_module(V) Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 5 for Gamma_0(37) of weight 2 with sign 0 over Rational Field 
 
- sage.modular.hecke.submodule.is_HeckeSubmodule(x)[source]¶
- Return - Trueif x is of type HeckeSubmodule.- EXAMPLES: - sage: sage.modular.hecke.submodule.is_HeckeSubmodule(ModularForms(1, 12)) doctest:warning... DeprecationWarning: the function is_HeckeSubmodule is deprecated; use 'isinstance(..., HeckeSubmodule)' instead See https://github.com/sagemath/sage/issues/37895 for details. False sage: sage.modular.hecke.submodule.is_HeckeSubmodule(CuspForms(1, 12)) True - >>> from sage.all import * >>> sage.modular.hecke.submodule.is_HeckeSubmodule(ModularForms(Integer(1), Integer(12))) doctest:warning... DeprecationWarning: the function is_HeckeSubmodule is deprecated; use 'isinstance(..., HeckeSubmodule)' instead See https://github.com/sagemath/sage/issues/37895 for details. False >>> sage.modular.hecke.submodule.is_HeckeSubmodule(CuspForms(Integer(1), Integer(12))) True