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Package sage :: Package modular :: Package modform :: Module modform :: Class ModularFormsSpace |
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HeckeModule
--+
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ModularFormsSpace
ModularForms
,
ModularFormsSubspace
Method Summary | |
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__init__(self,
group,
weight,
character,
base_field)
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__add__(self,
right)
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__and__(self,
right)
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__call__(self,
x,
check)
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__cmp__(self,
x)
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True if x is an element or subspace of self. | |
ambient_space(self)
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base_field(self)
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basis(self)
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change_base(self)
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character(self)
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cuspidal_subspace(self)
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This function returns a list of subspaces V(f_i,t) corresponding to newforms f_i of some level dividing the level of self, such that the direct sum of the subspaces equals self, if possible. | |
eisenstein_subspace(self)
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embedded_subspace(self)
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group(self)
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has_character(self)
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hecke_matrix(self,
n)
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intersect(self,
right)
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is_ambient(self)
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key(self)
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level(self)
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modular_symbols(self)
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This function returns a list of subspaces S(M,t) and E(M,t), corresponding to levels M dividing N and integers t dividing N/M, such that self is the direct sum of these spaces, if possible. | |
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. | |
weight(self)
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Inherited from HeckeModule | |
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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. | |
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. | |
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Method Details |
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__contains__(self,
x)
True if x is an element or subspace of self.
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decompose(self)This function returns a list of subspaces V(f_i,t) corresponding to newforms f_i of some level dividing the level of self, such that the direct sum of the subspaces 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. |
newspaces(self)This function returns a list of subspaces S(M,t) and E(M,t), corresponding to levels M dividing N and integers t dividing N/M, such that self is the direct sum of these spaces, if possible. Here S(M,t) is by definition the image under f(q) maps to f(q^t) of the new subspace of cusp forms of level M, and similarly E(M,t) is the image of Eisenstein series. Notes: (1) the subspaces S(M,t) need not be stable under Hecke operators of index dividing N/M. (2) Since self can be an arbitrary subspace, there's no guarantee any S(M,t) or E(M,t) is in self, so the return list could be empty. |
sturm_bound(self, 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. If M is none, then M is set equal to self. NOTES: 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: 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 = 0 (mod p) for first s coefficients, then f^r = 0 (mod p) for first s*r coefficents. Since weight of f^r is r*weight(f), it follows that if s >= sturm bound for Gamma_0 at weight(f), then f^r has valuation large enough to be forced to be 0 at r*weight(f) by Sturm bound (which is valid if we choose r right). Thus f = 0 (mod 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 Z[eps] generators for the Hecke algebra here, not Z-generators. So if one wants generators for the Hecke algebra over Z, this bound is wrong. This bound works over any base, even a finite field. There might be much better bounds over Q, or for comparing two eigenforms. |
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