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# -*- coding: utf-8 -*- r""" Overconvergent p-adic modular forms for small primes
This module implements computations of Hecke operators and `U_p`-eigenfunctions on `p`-adic overconvergent modular forms of tame level 1, where `p` is one of the primes `\{2, 3, 5, 7, 13\}`, using the algorithms described in [Loe2007]_.
- [Loe2007]_
AUTHORS:
- David Loeffler (August 2008): initial version - David Loeffler (March 2009): extensively reworked - Lloyd Kilford (May 2009): add :meth:`~sage.modular.overconvergent.genus0.OverconvergentModularFormsSpace.slopes` method - David Loeffler (June 2009): miscellaneous bug fixes and usability improvements
The Theory ~~~~~~~~~~
Let `p` be one of the above primes, so `X_0(p)` has genus 0, and let
.. MATH::
f_p = \sqrt[p-1]{\frac{\Delta(pz)}{\Delta(z)}}
(an `\eta`-product of level `p` -- see module :mod:`sage.modular.etaproducts`). Then one can show that `f_p` gives an isomorphism `X_0(p) \to \mathbb{P}^1`. Furthermore, if we work over `\CC_p`, the `r`-overconvergent locus on `X_0(p)` (or of `X_0(1)`, via the canonical subgroup lifting), corresponds to the `p`-adic disc
.. MATH::
|f_p|_p \le p^{\frac{12r}{p-1}}.
(This is Theorem 1 of [Loe2007]_.)
Hence if we fix an element `c` with `|c| = p^{-\frac{12r}{p-1}}`, the space `S_k^\dag(1, r)` of overconvergent `p`-adic modular forms has an orthonormal basis given by the functions `(cf)^n`. So any element can be written in the form `E_k \times \sum_{n \ge 0} a_n (cf)^n`, where `a_n \to 0` as `N \to \infty`, and any such sequence `a_n` defines a unique overconvergent form.
One can now find the matrix of Hecke operators in this basis, either by calculating `q`-expansions, or (for the special case of `U_p`) using a recurrence formula due to Kolberg.
An Extended Example ~~~~~~~~~~~~~~~~~~~
We create a space of 3-adic modular forms::
sage: M = OverconvergentModularForms(3, 8, 1/6, prec=60)
Creating an element directly as a linear combination of basis vectors.
.. link
::
sage: f1 = M.3 + M.5; f1.q_expansion() 27*q^3 + 1055916/1093*q^4 + 19913121/1093*q^5 + 268430112/1093*q^6 + ... sage: f1.coordinates(8) [0, 0, 0, 1, 0, 1, 0, 0]
We can coerce from elements of classical spaces of modular forms:
.. link
::
sage: f2 = M(CuspForms(3, 8).0); f2 3-adic overconvergent modular form of weight-character 8 with q-expansion q + 6*q^2 - 27*q^3 - 92*q^4 + 390*q^5 - 162*q^6 ...
We express this in a basis, and see that the coefficients go to zero very fast:
.. link
::
sage: [x.valuation(3) for x in f2.coordinates(60)] [+Infinity, -1, 3, 6, 10, 13, 18, 20, 24, 27, 31, 34, 39, 41, 45, 48, 52, 55, 61, 62, 66, 69, 73, 76, 81, 83, 87, 90, 94, 97, 102, 104, 108, 111, 115, 118, 124, 125, 129, 132, 136, 139, 144, 146, 150, 153, 157, 160, 165, 167, 171, 174, 178, 181, 188, 188, 192, 195, 199, 202]
This form has more level at `p`, and hence is less overconvergent:
.. link
::
sage: f3 = M(CuspForms(9, 8).0); [x.valuation(3) for x in f3.coordinates(60)] [+Infinity, -1, -1, 0, -4, -4, -2, -3, 0, 0, -1, -1, 1, 0, 3, 3, 3, 3, 5, 3, 7, 7, 6, 6, 8, 7, 10, 10, 8, 8, 10, 9, 12, 12, 12, 12, 14, 12, 17, 16, 15, 15, 17, 16, 19, 19, 18, 18, 20, 19, 22, 22, 22, 22, 24, 21, 25, 26, 24, 24]
An error will be raised for forms which are not sufficiently overconvergent:
.. link
::
sage: M(CuspForms(27, 8).0) Traceback (most recent call last): ... ValueError: Form is not overconvergent enough (form is only 1/12-overconvergent)
Let's compute some Hecke operators. Note that the coefficients of this matrix are `p`-adically tiny:
.. link
::
sage: M.hecke_matrix(3, 4).change_ring(Qp(3,prec=1)) [ 1 + O(3) 0 0 0] [ 0 2*3^3 + O(3^4) 2*3^3 + O(3^4) 3^2 + O(3^3)] [ 0 2*3^7 + O(3^8) 2*3^8 + O(3^9) 3^6 + O(3^7)] [ 0 2*3^10 + O(3^11) 2*3^10 + O(3^11) 2*3^9 + O(3^10)]
We compute the eigenfunctions of a 4x4 truncation:
.. link
::
sage: efuncs = M.eigenfunctions(4) sage: for i in [1..3]: ....: print(efuncs[i].q_expansion(prec=4).change_ring(Qp(3,prec=20))) (1 + O(3^20))*q + (2*3 + 3^15 + 3^16 + 3^17 + 2*3^19 + 2*3^20 + O(3^21))*q^2 + (2*3^3 + 2*3^4 + 2*3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + 2*3^10 + 2*3^11 + 2*3^12 + 2*3^13 + 2*3^14 + 2*3^15 + 2*3^16 + 3^17 + 2*3^18 + 2*3^19 + 3^21 + 3^22 + O(3^23))*q^3 + O(q^4) (1 + O(3^20))*q + (3 + 2*3^2 + 3^3 + 3^4 + 3^12 + 3^13 + 2*3^14 + 3^15 + 2*3^17 + 3^18 + 3^19 + 3^20 + O(3^21))*q^2 + (3^7 + 3^13 + 2*3^14 + 2*3^15 + 3^16 + 3^17 + 2*3^18 + 3^20 + 2*3^21 + 2*3^22 + 2*3^23 + 2*3^25 + O(3^27))*q^3 + O(q^4) (1 + O(3^20))*q + (2*3 + 3^3 + 2*3^4 + 3^6 + 2*3^8 + 3^9 + 3^10 + 2*3^11 + 2*3^13 + 3^16 + 3^18 + 3^19 + 3^20 + O(3^21))*q^2 + (3^9 + 2*3^12 + 3^15 + 3^17 + 3^18 + 3^19 + 3^20 + 2*3^22 + 2*3^23 + 2*3^27 + 2*3^28 + O(3^29))*q^3 + O(q^4)
The first eigenfunction is a classical cusp form of level 3:
.. link
::
sage: (efuncs[1] - M(CuspForms(3, 8).0)).valuation() 13
The second is an Eisenstein series!
.. link
::
sage: (efuncs[2] - M(EisensteinForms(3, 8).1)).valuation() 10
The third is a genuinely new thing (not a classical modular form at all); the coefficients are almost certainly not algebraic over `\QQ`. Note that the slope is 9, so Coleman's classicality criterion (forms of slope `< k-1` are classical) does not apply.
.. link
::
sage: a3 = efuncs[3].q_expansion()[3]; a3 3^9 + 2*3^12 + 3^15 + 3^17 + 3^18 + 3^19 + 3^20 + 2*3^22 + 2*3^23 + 2*3^27 + 2*3^28 + 3^32 + 3^33 + 2*3^34 + 3^38 + 2*3^39 + 3^40 + 2*3^41 + 3^44 + 3^45 + 3^46 + 2*3^47 + 2*3^48 + 3^49 + 3^50 + 2*3^51 + 2*3^52 + 3^53 + 2*3^54 + 3^55 + 3^56 + 3^57 + 2*3^58 + 2*3^59 + 3^60 + 2*3^61 + 2*3^63 + 2*3^64 + 3^65 + 2*3^67 + 3^68 + 2*3^69 + 2*3^71 + 3^72 + 2*3^74 + 3^75 + 3^76 + 3^79 + 3^80 + 2*3^83 + 2*3^84 + 3^85 + 2*3^87 + 3^88 + 2*3^89 + 2*3^90 + 2*3^91 + 3^92 + O(3^98) sage: efuncs[3].slope() 9
----------- """
#***************************************************************************** # Copyright (C) 2008 William Stein <wstein@gmail.com> # 2008-9 David Loeffler <d.loeffler.01@cantab.net> # # Distributed under the terms of the GNU General Public License (GPL) # http://www.gnu.org/licenses/ #***************************************************************************** from __future__ import print_function, absolute_import from six.moves import range from six import integer_types
from sage.matrix.all import matrix, MatrixSpace, diagonal_matrix from sage.misc.misc import verbose from sage.misc.cachefunc import cached_method from sage.misc.superseded import deprecated_function_alias from sage.modular.all import (DirichletGroup, trivial_character, EtaProduct, j_invariant_qexp, hecke_operator_on_qexp) from sage.modular.arithgroup.all import (Gamma1, is_Gamma0, is_Gamma1) from sage.modular.modform.element import ModularFormElement from sage.modules.all import vector from sage.modules.module import Module from sage.structure.element import Vector, ModuleElement from sage.structure.richcmp import richcmp from sage.plot.plot import plot from sage.rings.all import (O, Infinity, ZZ, QQ, pAdicField, PolynomialRing, PowerSeriesRing, is_pAdicField) import weakref
from .weightspace import WeightSpace_constructor as WeightSpace, WeightCharacter __ocmfdict = {}
#################### # Factory function # ####################
def OverconvergentModularForms(prime, weight, radius, base_ring=QQ, prec = 20, char = None): r""" Create a space of overconvergent `p`-adic modular forms of level `\Gamma_0(p)`, over the given base ring. The base ring need not be a `p`-adic ring (the spaces we compute with typically have bases over `\QQ`).
INPUT:
- ``prime`` - a prime number `p`, which must be one of the primes `\{2, 3, 5, 7, 13\}`, or the congruence subgroup `\Gamma_0(p)` where `p` is one of these primes.
- ``weight`` - an integer (which at present must be 0 or `\ge 2`), the weight.
- ``radius`` - a rational number in the interval `\left( 0, \frac{p}{p+1} \right)`, the radius of overconvergence.
- ``base_ring`` (default: `\QQ`), a ring over which to compute. This need not be a `p`-adic ring.
- ``prec`` - an integer (default: 20), the number of `q`-expansion terms to compute.
- ``char`` - a Dirichlet character modulo `p` or ``None`` (the default). Here ``None`` is interpreted as the trivial character modulo `p`.
The character `\chi` and weight `k` must satisfy `(-1)^k = \chi(-1)`, and the base ring must contain an element `v` such that `{\rm ord}_p(v) = \frac{12 r}{p-1}` where `r` is the radius of overconvergence (and `{\rm ord}_p` is normalised so `{\rm ord}_p(p) = 1`).
EXAMPLES::
sage: OverconvergentModularForms(3, 0, 1/2) Space of 3-adic 1/2-overconvergent modular forms of weight-character 0 over Rational Field sage: OverconvergentModularForms(3, 16, 1/2) Space of 3-adic 1/2-overconvergent modular forms of weight-character 16 over Rational Field sage: OverconvergentModularForms(3, 3, 1/2, char = DirichletGroup(3,QQ).0) Space of 3-adic 1/2-overconvergent modular forms of weight-character (3, 3, [-1]) over Rational Field """ else: raise ValueError("p must be one of {2, 3, 5, 7, 13}")
######################### # Main class definition # #########################
class OverconvergentModularFormsSpace(Module): r""" A space of overconvergent modular forms of level `\Gamma_0(p)`, where `p` is a prime such that `X_0(p)` has genus 0.
Elements are represented as power series, with a formal power series `F` corresponding to the modular form `E_k^\ast \times F(g)` where `E_k^\ast` is the `p`-deprived Eisenstein series of weight-character `k`, and `g` is a uniformiser of `X_0(p)` normalised so that the `r`-overconvergent region `X_0(p)_{\ge r}` corresponds to `|g| \le 1`.
TESTS::
sage: K.<w> = Qp(13).extension(x^2-13); M = OverconvergentModularForms(13, 20, radius=1/2, base_ring=K) sage: M is loads(dumps(M)) True """
############### # Init script # ###############
def __init__(self, prime, weight, radius, base_ring, prec, char): r""" Create a space of overconvergent `p`-adic modular forms of level `\Gamma_0(p)`, over the given base ring. The base ring need not be a `p`-adic ring (the spaces we compute with typically have bases over `\QQ`).
EXAMPLES::
sage: OverconvergentModularForms(3, 0, 1/2) Space of 3-adic 1/2-overconvergent modular forms of weight-character 0 over Rational Field """
raise TypeError("Base ring must be QQ or a p-adic field")
raise TypeError("Residue characteristic of base ring (=%s) must be %s" % (base_ring, self._p))
else:
raise ValueError("Weight-character must be even")
##################################### # Methods called by the init script # #####################################
def _set_radius(self, radius): r""" Set the radius of overconvergence to be `r`, where `r` is a rational number in the interval `0 < r < \frac{p}{p+1}`.
This only makes sense if the base ring contains an element of normalised valuation `\frac{12r}{p-1}`. If this valuation is an integer, we use the appropriate power of `p`. Otherwise, we assume the base ring has a ``uniformiser`` method and take an appropriate power of the uniformiser, raising an error if no such element exists.
EXAMPLES::
sage: M = OverconvergentModularForms(3, 2, 1/2) # indirect doctest sage: M._set_radius(1/3); M Space of 3-adic 1/3-overconvergent modular forms of weight-character 2 over Rational Field
sage: L.<w> = Qp(3).extension(x^5 - 3) sage: OverconvergentModularForms(3, 2, 1/30, base_ring=L).normalising_factor() # indirect doctest w + O(w^101)
sage: OverconvergentModularForms(3, 2, 1/40, base_ring=L) Traceback (most recent call last): ... ValueError: no element of base ring (=Eisenstein Extension ...) has normalised valuation 3/20 """
raise ValueError("radius (=%s) must be between 0 and p/(p+1)" % radius) else: except AttributeError: # base ring isn't a p-adic ring pi = p e = d
############################################## # Boring functions that access internal data # ##############################################
def is_exact(self): r""" True if elements of this space are represented exactly, i.e., there is no precision loss when doing arithmetic. As this is never true for overconvergent modular forms spaces, this returns False.
EXAMPLES::
sage: OverconvergentModularForms(13, 12, 0).is_exact() False """
def change_ring(self, ring): r""" Return the space corresponding to self but over the given base ring.
EXAMPLES::
sage: M = OverconvergentModularForms(2, 0, 1/2) sage: M.change_ring(Qp(2)) Space of 2-adic 1/2-overconvergent modular forms of weight-character 0 over 2-adic Field with ... """
def base_extend(self, ring): r""" Return the base extension of self to the given base ring. There must be a canonical map to this ring from the current base ring, otherwise a TypeError will be raised.
EXAMPLES::
sage: M = OverconvergentModularForms(2, 0, 1/2, base_ring = Qp(2)) sage: M.base_extend(Qp(2).extension(x^2 - 2, names="w")) Space of 2-adic 1/2-overconvergent modular forms of weight-character 0 over Eisenstein Extension ... sage: M.base_extend(QQ) Traceback (most recent call last): ... TypeError: Base extension of self (over '2-adic Field with capped relative precision 20') to ring 'Rational Field' not defined. """ else:
def _an_element_(self): r""" Return an element of this space (used by the coercion machinery).
EXAMPLES::
sage: OverconvergentModularForms(3, 2, 1/3, prec=4).an_element() # indirect doctest 3-adic overconvergent modular form of weight-character 2 with q-expansion 9*q + 216*q^2 + 2430*q^3 + O(q^4) """
def character(self): r""" Return the character of self. For overconvergent forms, the weight and the character are unified into the concept of a weight-character, so this returns exactly the same thing as self.weight().
EXAMPLES::
sage: OverconvergentModularForms(3, 0, 1/2).character() 0 sage: type(OverconvergentModularForms(3, 0, 1/2).character()) <class '...weightspace.AlgebraicWeight'> sage: OverconvergentModularForms(3, 3, 1/2, char=DirichletGroup(3,QQ).0).character() (3, 3, [-1]) """
def weight(self): r""" Return the character of self. For overconvergent forms, the weight and the character are unified into the concept of a weight-character, so this returns exactly the same thing as self.character().
EXAMPLES::
sage: OverconvergentModularForms(3, 0, 1/2).weight() 0 sage: type(OverconvergentModularForms(3, 0, 1/2).weight()) <class '...weightspace.AlgebraicWeight'> sage: OverconvergentModularForms(3, 3, 1/2, char=DirichletGroup(3,QQ).0).weight() (3, 3, [-1]) """
def normalising_factor(self): r""" The normalising factor `c` such that `g = c f` is a parameter for the `r`-overconvergent disc in `X_0(p)`, where `f` is the standard uniformiser.
EXAMPLES::
sage: L.<w> = Qp(7).extension(x^2 - 7) sage: OverconvergentModularForms(7, 0, 1/4, base_ring=L).normalising_factor() w + O(w^41) """
def __eq__(self, other): r""" Check whether ``self`` is equal to ``other``.
EXAMPLES::
sage: OverconvergentModularForms(3, 12, 1/2) == ModularForms(3, 12) False sage: OverconvergentModularForms(3, 0, 1/2) == OverconvergentModularForms(3, 0, 1/3) False sage: OverconvergentModularForms(3, 0, 1/2) == OverconvergentModularForms(3, 0, 1/2, base_ring = Qp(3)) False sage: OverconvergentModularForms(3, 0, 1/2) == OverconvergentModularForms(3, 0, 1/2) True """ else:
def __ne__(self, other): """ Check whether ``self`` is not equal to ``other``.
EXAMPLES::
sage: OverconvergentModularForms(3, 12, 1/2) != ModularForms(3, 12) True sage: OverconvergentModularForms(3, 0, 1/2) != OverconvergentModularForms(3, 0, 1/3) True sage: OverconvergentModularForms(3, 0, 1/2) != OverconvergentModularForms(3, 0, 1/2, base_ring = Qp(3)) True sage: OverconvergentModularForms(3, 0, 1/2) != OverconvergentModularForms(3, 0, 1/2) False """
def _params(self): r""" Return the parameters that define this module uniquely: prime, weight, character, radius of overconvergence and base ring. Mostly used for pickling.
EXAMPLES::
sage: L.<w> = Qp(7).extension(x^2 - 7) sage: OverconvergentModularForms(7, 0, 1/4, base_ring=L)._params() (7, 0, 1/4, Eisenstein Extension ..., 20, Dirichlet character modulo 7 of conductor 1 mapping 3 |--> 1)
"""
def __reduce__(self): r""" Return the function and arguments used to construct self. Used for pickling.
EXAMPLES::
sage: L.<w> = Qp(7).extension(x^2 - 7) sage: OverconvergentModularForms(7, 0, 1/4, base_ring=L).__reduce__() (<function OverconvergentModularForms at ...>, (7, 0, 1/4, Eisenstein Extension ..., 20, Dirichlet character modulo 7 of conductor 1 mapping 3 |--> 1))
"""
def gen(self, i): r""" Return the ith module generator of self.
EXAMPLES::
sage: M = OverconvergentModularForms(3, 2, 1/2, prec=4) sage: M.gen(0) 3-adic overconvergent modular form of weight-character 2 with q-expansion 1 + 12*q + 36*q^2 + 12*q^3 + O(q^4) sage: M.gen(1) 3-adic overconvergent modular form of weight-character 2 with q-expansion 27*q + 648*q^2 + 7290*q^3 + O(q^4) sage: M.gen(30) 3-adic overconvergent modular form of weight-character 2 with q-expansion O(q^4) """
def _repr_(self): r""" Return a string representation of self.
EXAMPLES::
sage: OverconvergentModularForms(3, 0, 1/2)._repr_() 'Space of 3-adic 1/2-overconvergent modular forms of weight-character 0 over Rational Field' """
def prime(self): r""" Return the residue characteristic of self, i.e. the prime `p` such that this is a `p`-adic space.
EXAMPLES::
sage: OverconvergentModularForms(5, 12, 1/3).prime() 5 """
def radius(self): r""" The radius of overconvergence of this space.
EXAMPLES::
sage: OverconvergentModularForms(3, 0, 1/3).radius() 1/3 """
def gens(self): r""" Return a generator object that iterates over the (infinite) set of basis vectors of self.
EXAMPLES::
sage: o = OverconvergentModularForms(3, 12, 1/2) sage: t = o.gens() sage: next(t) 3-adic overconvergent modular form of weight-character 12 with q-expansion 1 - 32760/61203943*q - 67125240/61203943*q^2 - ... sage: next(t) 3-adic overconvergent modular form of weight-character 12 with q-expansion 27*q + 19829193012/61203943*q^2 + 146902585770/61203943*q^3 + ... """
def prec(self): r""" Return the series precision of self. Note that this is different from the `p`-adic precision of the base ring.
EXAMPLES::
sage: OverconvergentModularForms(3, 0, 1/2).prec() 20 sage: OverconvergentModularForms(3, 0, 1/2,prec=40).prec() 40 """
##################################### # Element construction and coercion # #####################################
def _element_constructor_(self, input): r""" Create an element of this space. Allowable inputs are:
- elements of compatible spaces of modular forms or overconvergent modular forms
- arbitrary power series in `q`
- lists of elements of the base ring (interpreted as vectors in the basis given by self.gens()).
Precision may be specified by padding lists at the end with zeros; inputs with a higher precision than the set precision of this space will be rounded.
EXAMPLES:
From a `q`-expansion::
sage: M = OverconvergentModularForms(3, 0, 1/2, prec=5) sage: R.<q> = QQ[[]] sage: f=M(q + q^2 - q^3 + O(q^16)); f 3-adic overconvergent modular form of weight-character 0 with q-expansion q + q^2 - q^3 + O(q^5) sage: M.coordinate_vector(f) (0, 1/27, -11/729, 173/19683, -3172/531441)
From a list or a vector::
sage: M([1,0,1]) 3-adic overconvergent modular form of weight-character 0 with q-expansion 1 + 729*q^2 + O(q^3) sage: M([1,0,1,0,0]) 3-adic overconvergent modular form of weight-character 0 with q-expansion 1 + 729*q^2 + 17496*q^3 + 236196*q^4 + O(q^5) sage: f = M([1,0,1,0,0]); v = M.coordinate_vector(f); v (1, 0, 1, 0, 0) sage: M(v) == f True
From a classical modular form::
sage: f = CuspForms(Gamma0(3), 12).0; f q - 176*q^4 + 2430*q^5 + O(q^6) sage: fdag = OverconvergentModularForms(3, 12, 1/3, prec=8)(f); fdag 3-adic overconvergent modular form of weight-character 12 with q-expansion q - 176*q^4 + 2430*q^5 - 5832*q^6 - 19336*q^7 + O(q^8) sage: fdag.parent().coordinate_vector(f)*(1 + O(3^2)) (0, 3^-2 + O(3^0), 2*3^-3 + 2*3^-2 + O(3^-1), 3^-4 + 3^-3 + O(3^-2), 2 + 3 + O(3^2), 2*3 + 3^2 + O(3^3), 2*3^4 + 2*3^5 + O(3^6), 3^5 + 3^6 + O(3^7)) sage: OverconvergentModularForms(3, 6, 1/3)(f) Traceback (most recent call last): ... TypeError: Cannot create an element of 'Space of 3-adic ...' from element of incompatible space 'Cuspidal subspace ...'
We test that zero elements are handled properly::
sage: M(0) 3-adic overconvergent modular form of weight-character 0 with q-expansion O(q^5) sage: M(O(q^3)) 3-adic overconvergent modular form of weight-character 0 with q-expansion O(q^3)
We test coercion between spaces of different precision::
sage: M10 = OverconvergentModularForms(3, 0, 1/2, prec=10) sage: f = M10.1 sage: M(f) 3-adic overconvergent modular form of weight-character 0 with q-expansion 27*q + 324*q^2 + 2430*q^3 + 13716*q^4 + O(q^5) sage: M10(M(f)) 3-adic overconvergent modular form of weight-character 0 with q-expansion 27*q + 324*q^2 + 2430*q^3 + 13716*q^4 + O(q^5) """
return input
and input.weight() == self.weight().k() and input.character().primitive_character() == self.weight().chi().primitive_character()): else: else:
else: raise TypeError("Don't know how to create an overconvergent modular form from %s" % input)
@cached_method def zero(self): """ Return the zero of this space.
EXAMPLES::
sage: K.<w> = Qp(13).extension(x^2-13); M = OverconvergentModularForms(13, 20, radius=1/2, base_ring=K) sage: K.zero() 0 """
zero_element = deprecated_function_alias(17694, zero)
def _coerce_from_ocmf(self, f): r""" Try to convert the overconvergent modular form `f` into an element of self. An error will be raised if this is obviously nonsense.
EXAMPLES::
sage: M = OverconvergentModularForms(3, 0, 1/2) sage: MM = M.base_extend(Qp(3)) sage: R.<q> = Qp(3)[[]]; f = MM(q + O(q^2)); f 3-adic overconvergent modular form of weight-character 0 with q-expansion (1 + O(3^20))*q + O(q^2) sage: M._coerce_from_ocmf(f) 3-adic overconvergent modular form of weight-character 0 with q-expansion q + O(q^2) sage: f in M # indirect doctest True """ raise TypeError("Cannot create an element of '%s' from element of incompatible space '%s'" % (self, input.parent()))
def _coerce_map_from_(self, other): r""" Canonical coercion of x into self. Here the possibilities for x are more restricted.
TESTS::
sage: M = OverconvergentModularForms(3, 0, 1/2) sage: MM = M.base_extend(Qp(3)) sage: MM.has_coerce_map_from(M) # indirect doctest True sage: MM.coerce(M.1) 3-adic overconvergent modular form of weight-character 0 with q-expansion (3^3 + O(3^23))*q + (3^4 + 3^5 + O(3^24))*q^2 ... sage: M.has_coerce_map_from(MM) False sage: M.coerce(1) 3-adic overconvergent modular form of weight-character 0 with q-expansion 1 + O(q^20) """ self.base_ring().has_coerce_map_from(other.base_ring())): else:
def coordinate_vector(self, x): r""" Write x as a vector with respect to the basis given by self.basis(). Here x must be an element of this space or something that can be converted into one. If x has precision less than the default precision of self, then the returned vector will be shorter.
EXAMPLES::
sage: M = OverconvergentModularForms(Gamma0(3), 0, 1/3, prec=4) sage: M.coordinate_vector(M.gen(2)) (0, 0, 1, 0) sage: q = QQ[['q']].gen(); M.coordinate_vector(q - q^2 + O(q^4)) (0, 1/9, -13/81, 74/243) sage: M.coordinate_vector(q - q^2 + O(q^3)) (0, 1/9, -13/81) """ return self.base_extend(x.base_ring()).coordinate_vector(x)
########################################################## # Pointless routines required by parent class definition # ##########################################################
def ngens(self): r""" The number of generators of self (as a module over its base ring), i.e. infinity.
EXAMPLES::
sage: M = OverconvergentModularForms(2, 4, 1/6) sage: M.ngens() +Infinity """
def gens_dict(self): r""" Return a dictionary mapping the names of generators of this space to their values. (Required by parent class definition.) As this does not make any sense here, this raises a TypeError.
EXAMPLES::
sage: M = OverconvergentModularForms(2, 4, 1/6) sage: M.gens_dict() Traceback (most recent call last): ... TypeError: gens_dict does not make sense as number of generators is infinite """
##################################### # Routines with some actual content # #####################################
def hecke_operator(self, f, m): r""" Given an element `f` and an integer `m`, calculates the Hecke operator `T_m` acting on `f`.
The input may be either a "bare" power series, or an OverconvergentModularFormElement object; the return value will be of the same type.
EXAMPLES::
sage: M = OverconvergentModularForms(3, 0, 1/2) sage: f = M.1 sage: M.hecke_operator(f, 3) 3-adic overconvergent modular form of weight-character 0 with q-expansion 2430*q + 265356*q^2 + 10670373*q^3 + 249948828*q^4 + 4113612864*q^5 + 52494114852*q^6 + O(q^7) sage: M.hecke_operator(f.q_expansion(), 3) 2430*q + 265356*q^2 + 10670373*q^3 + 249948828*q^4 + 4113612864*q^5 + 52494114852*q^6 + O(q^7) """
# This should just be an instance of hecke_operator_on_qexp but that # won't accept arbitrary power series as input, although it's clearly # supposed to, which seems rather to defy the point but never mind...
if f.parent() is self.base_extend(f.parent().base_ring()): return f.parent().hecke_operator(f, m) else: raise TypeError("Not an element of this space") else:
def _convert_to_basis(self, qexp): r""" Given a `q`-expansion, converts it to a vector in the basis of this space, to the maximum possible precision (which is the minimum of the `q`-adic precision of the `q`-expansion and the precision of self).
EXAMPLES::
sage: M = OverconvergentModularForms(2, 0, 1/2) sage: R.<q> = QQ[[]] sage: M._convert_to_basis(q + q^2 + O(q^4)) 1/64*g - 23/4096*g^2 + 201/65536*g^3 + O(g^4) """
def hecke_matrix(self, m, n, use_recurrence = False, exact_arith = False): r""" Calculate the matrix of the `T_m` operator in the basis of this space, truncated to an `n \times n` matrix. Conventions are that operators act on the left on column vectors (this is the opposite of the conventions of the sage.modules.matrix_morphism class!) Uses naive `q`-expansion arguments if use_recurrence=False and uses the Kolberg style recurrences if use_recurrence=True.
The argument "exact_arith" causes the computation to be done with rational arithmetic, even if the base ring is an inexact `p`-adic ring. This is useful as there can be precision loss issues (particularly with use_recurrence=False).
EXAMPLES::
sage: OverconvergentModularForms(2, 0, 1/2).hecke_matrix(2, 4) [ 1 0 0 0] [ 0 24 64 0] [ 0 32 1152 4608] [ 0 0 3072 61440] sage: OverconvergentModularForms(2, 12, 1/2, base_ring=pAdicField(2)).hecke_matrix(2, 3) * (1 + O(2^2)) [ 1 + O(2^2) 0 0] [ 0 2^3 + O(2^5) 2^6 + O(2^8)] [ 0 2^4 + O(2^6) 2^7 + 2^8 + O(2^9)] sage: OverconvergentModularForms(2, 12, 1/2, base_ring=pAdicField(2)).hecke_matrix(2, 3, exact_arith=True) [ 1 0 0] [ 0 33881928/1414477 64] [ 0 -192898739923312/2000745183529 1626332544/1414477] """
raise ValueError("n is too large for current precision") else: raise ValueError("n is too large computing initial conds: can't work out u[%s, %s]" % (i,j)) else: # can only apply recurrence if have i,j both >= p. for i in range(self.prime()): if self.weight() != 0: raise ValueError("n is too large for current precision") else: if j <= self.prime() * i: raise ValueError("n is too large computing initial conds: can't work out u[%s,%s]" % (i,j)) mat[i,j] = 0
else:
else: raise ValueError("n is too large")
def slopes(self, n, use_recurrence=False): r""" Compute the slopes of the `U_p` operator acting on self, using an n x n matrix.
EXAMPLES::
sage: OverconvergentModularForms(5,2,1/3,base_ring=Qp(5),prec=100).slopes(5) [0, 2, 5, 6, 9] sage: OverconvergentModularForms(2,1,1/3,char=DirichletGroup(4,QQ).0).slopes(5) [0, 2, 4, 6, 8] """ else: print("slopes are only defined for base field QQ or a p-adic field")
def eigenfunctions(self, n, F = None, exact_arith=True): """ Calculate approximations to eigenfunctions of self. These are the eigenfunctions of self.hecke_matrix(p, n), which are approximations to the true eigenfunctions. Returns a list of OverconvergentModularFormElement objects, in increasing order of slope.
INPUT:
- ``n`` - integer. The size of the matrix to use.
- ``F`` - None, or a field over which to calculate eigenvalues. If the field is None, the current base ring is used. If the base ring is not a `p`-adic ring, an error will be raised.
- ``exact_arith`` - True or False (default True). If True, use exact rational arithmetic to calculate the matrix of the `U` operator and its characteristic power series, even when the base ring is an inexact `p`-adic ring. This is typically slower, but more numerically stable.
NOTE: Try using ``set_verbose(1, 'sage/modular/overconvergent')`` to get more feedback on what is going on in this algorithm. For even more feedback, use 2 instead of 1.
EXAMPLES::
sage: X = OverconvergentModularForms(2, 2, 1/6).eigenfunctions(8, Qp(2, 100)) sage: X[1] 2-adic overconvergent modular form of weight-character 2 with q-expansion (1 + O(2^74))*q + (2^4 + 2^5 + 2^9 + 2^10 + 2^12 + 2^13 + 2^15 + 2^17 + 2^19 + 2^20 + 2^21 + 2^23 + 2^28 + 2^30 + 2^31 + 2^32 + 2^34 + 2^36 + 2^37 + 2^39 + 2^40 + 2^43 + 2^44 + 2^45 + 2^47 + 2^48 + 2^52 + 2^53 + 2^54 + 2^55 + 2^56 + 2^58 + 2^59 + 2^60 + 2^61 + 2^67 + 2^68 + 2^70 + 2^71 + 2^72 + 2^74 + 2^76 + O(2^78))*q^2 + (2^2 + 2^7 + 2^8 + 2^9 + 2^12 + 2^13 + 2^16 + 2^17 + 2^21 + 2^23 + 2^25 + 2^28 + 2^33 + 2^34 + 2^36 + 2^37 + 2^42 + 2^45 + 2^47 + 2^49 + 2^50 + 2^51 + 2^54 + 2^55 + 2^58 + 2^60 + 2^61 + 2^67 + 2^71 + 2^72 + O(2^76))*q^3 + (2^8 + 2^11 + 2^14 + 2^19 + 2^21 + 2^22 + 2^24 + 2^25 + 2^26 + 2^27 + 2^28 + 2^29 + 2^32 + 2^33 + 2^35 + 2^36 + 2^44 + 2^45 + 2^46 + 2^47 + 2^49 + 2^50 + 2^53 + 2^54 + 2^55 + 2^56 + 2^57 + 2^60 + 2^63 + 2^66 + 2^67 + 2^69 + 2^74 + 2^76 + 2^79 + 2^80 + 2^81 + O(2^82))*q^4 + (2 + 2^2 + 2^9 + 2^13 + 2^15 + 2^17 + 2^19 + 2^21 + 2^23 + 2^26 + 2^27 + 2^28 + 2^30 + 2^33 + 2^34 + 2^35 + 2^36 + 2^37 + 2^38 + 2^39 + 2^41 + 2^42 + 2^43 + 2^45 + 2^58 + 2^59 + 2^60 + 2^61 + 2^62 + 2^63 + 2^65 + 2^66 + 2^68 + 2^69 + 2^71 + 2^72 + O(2^75))*q^5 + (2^6 + 2^7 + 2^15 + 2^16 + 2^21 + 2^24 + 2^25 + 2^28 + 2^29 + 2^33 + 2^34 + 2^37 + 2^44 + 2^45 + 2^48 + 2^50 + 2^51 + 2^54 + 2^55 + 2^57 + 2^58 + 2^59 + 2^60 + 2^64 + 2^69 + 2^71 + 2^73 + 2^75 + 2^78 + O(2^80))*q^6 + (2^3 + 2^8 + 2^9 + 2^10 + 2^11 + 2^12 + 2^14 + 2^15 + 2^17 + 2^19 + 2^20 + 2^21 + 2^23 + 2^25 + 2^26 + 2^34 + 2^37 + 2^38 + 2^39 + 2^40 + 2^41 + 2^45 + 2^47 + 2^49 + 2^51 + 2^53 + 2^54 + 2^55 + 2^57 + 2^58 + 2^59 + 2^60 + 2^61 + 2^66 + 2^69 + 2^70 + 2^71 + 2^74 + 2^76 + O(2^77))*q^7 + O(q^8) sage: [x.slope() for x in X] [0, 4, 8, 14, 16, 18, 26, 30] """
#raise TypeError, "cannot calculate eigenfunctions over exact base fields"
continue
# Annoying thing: r isn't quite as precise as it claims to be # (bug reported to sage-support list) else: mr = None break
verbose("Unable to calculate exact root in slope %s" % r.valuation()) continue
# now calculate the kernel using PARI
verbose("PARI returned empty eigenspace in slope %s" % r.valuation()) continue # Can't happen? Does PARI always return a # nonempty kernel for matrices that have det # indistinguishable from 0?
verbose("PARI returned non-simple eigenspace in slope %s" % r.valuation()) continue
# This is slightly subtle. We want all eigenfunctions to have q-exps in Z_p. # Normalising the q-term to be 1 doesn't work for the Eisenstein series if # we're in the 0 component of weight-character space. But normalising the const term # to 1 works as *none of the small primes we deal with are irregular*! :-) else: raise ValueError("Constant and linear terms both zero!") # if this gets called something is very wrong.
# This sometimes fails if n is too large -- last row of matrix fills # up with garbage. I don't know why. XXX FIX THIS XXX
def recurrence_matrix(self, use_smithline=True): r""" Return the recurrence matrix satisfied by the coefficients of `U`, that is a matrix `R =(r_{rs})_{r,s=1 \dots p}` such that `u_{ij} = \sum_{r,s=1}^p r_{rs} u_{i-r, j-s}`. Uses an elegant construction which I believe is due to Smithline. See [Loe2007]_.
EXAMPLES::
sage: OverconvergentModularForms(2, 0, 0).recurrence_matrix() [ 48 1] [4096 0] sage: OverconvergentModularForms(2, 0, 1/2).recurrence_matrix() [48 64] [64 0] sage: OverconvergentModularForms(3, 0, 0).recurrence_matrix() [ 270 36 1] [ 26244 729 0] [531441 0 0] sage: OverconvergentModularForms(5, 0, 0).recurrence_matrix() [ 1575 1300 315 30 1] [ 162500 39375 3750 125 0] [ 4921875 468750 15625 0 0] [ 58593750 1953125 0 0 0] [244140625 0 0 0 0] sage: OverconvergentModularForms(7, 0, 0).recurrence_matrix() [ 4018 8624 5915 1904 322 28 1] [ 422576 289835 93296 15778 1372 49 0] [ 14201915 4571504 773122 67228 2401 0 0] [ 224003696 37882978 3294172 117649 0 0 0] [ 1856265922 161414428 5764801 0 0 0 0] [ 7909306972 282475249 0 0 0 0 0] [13841287201 0 0 0 0 0 0] sage: OverconvergentModularForms(13, 0, 0).recurrence_matrix() [ 15145 124852 354536 ... """
def _discover_recurrence_matrix(self, use_smithline=True): r""" Does hard work of calculating recurrence matrix, which is cached to avoid doing this every time.
EXAMPLES::
sage: o = OverconvergentModularForms(3,12,0) sage: o._discover_recurrence_matrix() == o.recurrence_matrix() True """
# Compute Smithline's polynomial H_p
# avoid dividing by q so as not to instantiate a Laurent series raise ValueError("Something strange is happening here")
else: # compute from U(f^j) for small j via Newton's identities # to be implemented when I can remember Newton's identities! raise NotImplementedError
def cps_u(self, n, use_recurrence=False): r""" Compute the characteristic power series of `U_p` acting on self, using an n x n matrix.
EXAMPLES::
sage: OverconvergentModularForms(3, 16, 1/2, base_ring=Qp(3)).cps_u(4) 1 + O(3^20) + (2 + 2*3 + 2*3^2 + 2*3^4 + 3^5 + 3^6 + 3^7 + 3^11 + 3^12 + 2*3^14 + 3^16 + 3^18 + O(3^19))*T + (2*3^3 + 3^5 + 3^6 + 3^7 + 2*3^8 + 2*3^9 + 2*3^10 + 3^11 + 3^12 + 2*3^13 + 2*3^16 + 2*3^18 + O(3^19))*T^2 + (2*3^15 + 2*3^16 + 2*3^19 + 2*3^20 + 2*3^21 + O(3^22))*T^3 + (3^17 + 2*3^18 + 3^19 + 3^20 + 3^22 + 2*3^23 + 2*3^25 + 3^26 + O(3^27))*T^4 sage: OverconvergentModularForms(3, 16, 1/2, base_ring=Qp(3), prec=30).cps_u(10) 1 + O(3^20) + (2 + 2*3 + 2*3^2 + 2*3^4 + 3^5 + 3^6 + 3^7 + 2*3^15 + O(3^16))*T + (2*3^3 + 3^5 + 3^6 + 3^7 + 2*3^8 + 2*3^9 + 2*3^10 + 2*3^11 + 2*3^12 + 2*3^13 + 3^14 + 3^15 + O(3^16))*T^2 + (3^14 + 2*3^15 + 2*3^16 + 3^17 + 3^18 + O(3^19))*T^3 + (3^17 + 2*3^18 + 3^19 + 3^20 + 3^21 + O(3^24))*T^4 + (3^29 + 2*3^32 + O(3^33))*T^5 + (2*3^44 + O(3^45))*T^6 + (2*3^59 + O(3^60))*T^7 + (2*3^78 + O(3^79))*T^8
NOTES:
Uses the Hessenberg form of the Hecke matrix to compute the characteristic polynomial. Because of the use of relative precision here this tends to give better precision in the p-adic coefficients. """
# From a conversation with David Loeffler, apparently self.base_ring() # is either the field of rational numbers or some p-adic field. In the # first case we want to use the linbox algorithm, and in the second # case the Hessenberg form algorithm. # else:
class OverconvergentModularFormElement(ModuleElement): r""" A class representing an element of a space of overconvergent modular forms.
EXAMPLES::
sage: K.<w> = Qp(5).extension(x^7 - 5); s = OverconvergentModularForms(5, 6, 1/21, base_ring=K).0 sage: s == loads(dumps(s)) True """
def __init__(self, parent, gexp=None, qexp=None): r""" Create an element of this space.
EXAMPLES::
sage: OverconvergentModularForms(3, 2, 1/6,prec=5).an_element() # indirect doctest 3-adic overconvergent modular form of weight-character 2 with q-expansion 3*q + 72*q^2 + 810*q^3 + 6096*q^4 + O(q^5) """
#self.weight = self.parent().weight raise ValueError("Must supply exactly one of a q-expansion and a g-expansion") else: # qexp is not None
def _add_(self, other): r""" Add self to other (where other has the same parent as self).
EXAMPLES::
sage: M = OverconvergentModularForms(2, 12, 1/6) sage: f = M.0 sage: f + f # indirect doctest 2-adic overconvergent modular form of weight-character 12 with q-expansion 2 - 131040/1414477*q ... """
def _lmul_(self, x): r""" Left multiplication by other.
EXAMPLES::
sage: M = OverconvergentModularForms(2, 12, 1/6) sage: f = M.0 sage: 2*f # indirect doctest 2-adic overconvergent modular form of weight-character 12 with q-expansion 2 - 131040/1414477*q ...
"""
def _rmul_(self, x): r""" Right multiplication by other.
EXAMPLES::
sage: M = OverconvergentModularForms(2, 12, 1/6) sage: f = M.0 sage: f * 3 # indirect doctest 2-adic overconvergent modular form of weight-character 12 with q-expansion 3 - 196560/1414477*q ...
"""
def prec(self): r""" Return the series expansion precision of this overconvergent modular form. (This is not the same as the `p`-adic precision of the coefficients.)
EXAMPLES::
sage: OverconvergentModularForms(5, 6, 1/3,prec=15).gen(1).prec() 15 """
def is_eigenform(self): r""" Return True if this is an eigenform. At present this returns False unless this element was explicitly flagged as an eigenform, using the _notify_eigen function.
EXAMPLES::
sage: M = OverconvergentModularForms(3, 0, 1/2) sage: f = M.eigenfunctions(3)[1] sage: f.is_eigenform() True sage: M.gen(4).is_eigenform() False """
def slope(self): r""" Return the slope of this eigenform, i.e. the valuation of its `U_p`-eigenvalue. Raises an error unless this element was explicitly flagged as an eigenform, using the _notify_eigen function.
EXAMPLES::
sage: M = OverconvergentModularForms(3, 0, 1/2) sage: f = M.eigenfunctions(3)[1] sage: f.slope() 2 sage: M.gen(4).slope() Traceback (most recent call last): ... TypeError: slope only defined for eigenfunctions """
def eigenvalue(self): r""" Return the `U_p`-eigenvalue of this eigenform. Raises an error unless this element was explicitly flagged as an eigenform, using the _notify_eigen function.
EXAMPLES::
sage: M = OverconvergentModularForms(3, 0, 1/2) sage: f = M.eigenfunctions(3)[1] sage: f.eigenvalue() 3^2 + 3^4 + 2*3^6 + 3^7 + 3^8 + 2*3^9 + 2*3^10 + 3^12 + 3^16 + 2*3^17 + 3^18 + 3^20 + 2*3^21 + 3^22 + 2*3^23 + 3^25 + 3^26 + 2*3^27 + 2*3^29 + 3^30 + 3^31 + 3^32 + 3^33 + 3^34 + 3^36 + 3^40 + 2*3^41 + 3^43 + 3^44 + 3^45 + 3^46 + 3^48 + 3^49 + 3^50 + 2*3^51 + 3^52 + 3^54 + 2*3^57 + 2*3^59 + 3^60 + 3^61 + 2*3^63 + 2*3^66 + 2*3^67 + 3^69 + 2*3^72 + 3^74 + 2*3^75 + 3^76 + 2*3^77 + 2*3^78 + 2*3^80 + 3^81 + 2*3^82 + 3^84 + 2*3^85 + 2*3^86 + 3^87 + 3^88 + 2*3^89 + 2*3^91 + 3^93 + 3^94 + 3^95 + 3^96 + 3^98 + 2*3^99 + O(3^100) sage: M.gen(4).eigenvalue() Traceback (most recent call last): ... TypeError: eigenvalue only defined for eigenfunctions """
def q_expansion(self, prec=None): r""" Return the `q`-expansion of self, to as high precision as it is known.
EXAMPLES::
sage: OverconvergentModularForms(3, 4, 1/2).gen(0).q_expansion() 1 - 120/13*q - 1080/13*q^2 - 120/13*q^3 - 8760/13*q^4 - 15120/13*q^5 - 1080/13*q^6 - 41280/13*q^7 - 5400*q^8 - 120/13*q^9 - 136080/13*q^10 - 159840/13*q^11 - 8760/13*q^12 - 263760/13*q^13 - 371520/13*q^14 - 15120/13*q^15 - 561720/13*q^16 - 45360*q^17 - 1080/13*q^18 - 823200/13*q^19 + O(q^20) """ raise ValueError else:
def gexp(self): r""" Return the formal power series in `g` corresponding to this overconvergent modular form (so the result is `F` where this modular form is `E_k^\ast \times F(g)`, where `g` is the appropriately normalised parameter of `X_0(p)`).
EXAMPLES::
sage: M = OverconvergentModularForms(3, 0, 1/2) sage: f = M.eigenfunctions(3)[1] sage: f.gexp() (3^-3 + O(3^95))*g + (3^-1 + 1 + 2*3 + 3^2 + 2*3^3 + 3^5 + 3^7 + 3^10 + 3^11 + 3^14 + 3^15 + 3^16 + 2*3^19 + 3^21 + 3^22 + 2*3^23 + 2*3^24 + 3^26 + 2*3^27 + 3^29 + 3^31 + 3^34 + 2*3^35 + 2*3^36 + 3^38 + 2*3^39 + 3^41 + 2*3^42 + 2*3^43 + 2*3^44 + 2*3^46 + 2*3^47 + 3^48 + 2*3^49 + 2*3^50 + 3^51 + 2*3^54 + 2*3^55 + 2*3^56 + 3^57 + 2*3^58 + 2*3^59 + 2*3^60 + 3^61 + 3^62 + 3^63 + 3^64 + 2*3^65 + 3^67 + 3^68 + 2*3^69 + 3^70 + 2*3^71 + 2*3^74 + 3^76 + 2*3^77 + 3^78 + 2*3^79 + 2*3^80 + 3^84 + 2*3^85 + 2*3^86 + 3^88 + 2*3^89 + 3^91 + 3^92 + 2*3^94 + 3^95 + O(3^97))*g^2 + O(g^3) """
def coordinates(self, prec=None): r""" Return the coordinates of this modular form in terms of the basis of this space.
EXAMPLES::
sage: M = OverconvergentModularForms(3, 0, 1/2, prec=15) sage: f = (M.0 + M.3); f.coordinates() [1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] sage: f.coordinates(6) [1, 0, 0, 1, 0, 0] sage: OverconvergentModularForms(3, 0, 1/6)(f).coordinates(6) [1, 0, 0, 729, 0, 0] sage: f.coordinates(100) Traceback (most recent call last): ... ValueError: Precision too large for space """
def prime(self): r""" If this is a `p`-adic modular form, return `p`.
EXAMPLES::
sage: OverconvergentModularForms(2, 0, 1/2).an_element().prime() 2 """
def _notify_eigen(self, eigenvalue): """ Flags this element as an eigenform. It then remembers some extra data.
EXAMPLES::
sage: OverconvergentModularForms(3, 16, 1/3).eigenfunctions(4) # indirect doctest [...] """
def is_integral(self): r""" Test whether or not this element has `q`-expansion coefficients that are `p`-adically integral. This should always be the case with eigenfunctions, but sometimes if n is very large this breaks down for unknown reasons!
EXAMPLES::
sage: M = OverconvergentModularForms(2, 0, 1/3) sage: q = QQ[['q']].gen() sage: M(q - 17*q^2 + O(q^3)).is_integral() True sage: M(q - q^2/2 + 6*q^7 + O(q^9)).is_integral() False """
def _repr_(self): r""" String representation of self.
EXAMPLES::
sage: o=OverconvergentModularForms(3, 0, 1/2) sage: o([1, 0, 1, 3])._repr_() '3-adic overconvergent modular form of weight-character 0 with q-expansion 1 + 729*q^2 + 76545*q^3 + O(q^4)' """
def _richcmp_(self, other, op): r""" Compare self to other.
EXAMPLES::
sage: o = OverconvergentModularForms(3, 0, 1/2) sage: o([1, 1, 1, 0, 0, 0, 0]) == o([2, 1, 0]) False sage: o([1, 1, 1, 0, 0, 0, 0]) == o([1,1]) True """
def r_ord(self, r): r""" The `p`-adic valuation of the norm of self on the `r`-overconvergent region.
EXAMPLES::
sage: o=OverconvergentModularForms(3, 0, 1/2) sage: t = o([1, 1, 1/3]) sage: t.r_ord(1/2) 1 sage: t.r_ord(2/3) 3 """
def valuation(self): r""" Return the `p`-adic valuation of this form (i.e. the minimum of the `p`-adic valuations of its coordinates).
EXAMPLES::
sage: M = OverconvergentModularForms(3, 0, 1/2) sage: (M.7).valuation() 0 sage: (3^18 * (M.2)).valuation() 18 """ else:
def governing_term(self, r): r""" The degree of the series term with largest norm on the `r`-overconvergent region.
EXAMPLES::
sage: o=OverconvergentModularForms(3, 0, 1/2) sage: f=o.eigenfunctions(10)[1] sage: f.governing_term(1/2) 1 """ F = pAdicField(p)
raise RuntimeError("Can't get here")
def valuation_plot(self, rmax = None): r""" Draw a graph depicting the growth of the norm of this overconvergent modular form as it approaches the boundary of the overconvergent region.
EXAMPLES::
sage: o=OverconvergentModularForms(3, 0, 1/2) sage: f=o.eigenfunctions(4)[1] sage: f.valuation_plot() Graphics object consisting of 1 graphics primitive """
def weight(self): r""" Return the weight of this overconvergent modular form.
EXAMPLES::
sage: M = OverconvergentModularForms(13, 10, 1/2, base_ring = Qp(13).extension(x^2 - 13,names='a')) sage: M.gen(0).weight() 10 """
def additive_order(self): r""" Return the additive order of this element (required attribute for all elements deriving from sage.modules.ModuleElement).
EXAMPLES::
sage: M = OverconvergentModularForms(13, 10, 1/2, base_ring = Qp(13).extension(x^2 - 13,names='a')) sage: M.gen(0).additive_order() +Infinity sage: M(0).additive_order() 1 """
def base_extend(self, R): r""" Return a copy of self but with coefficients in the given ring.
EXAMPLES::
sage: M = OverconvergentModularForms(7, 10, 1/2, prec=5) sage: f = M.1 sage: f.base_extend(Qp(7, 4)) 7-adic overconvergent modular form of weight-character 10 with q-expansion (7 + O(7^5))*q + (6*7 + 4*7^2 + 7^3 + 6*7^4 + O(7^5))*q^2 + (5*7 + 5*7^2 + 7^4 + O(7^5))*q^3 + (7^2 + 4*7^3 + 3*7^4 + 2*7^5 + O(7^6))*q^4 + O(q^5) """
def __pari__(self): r""" Return the Pari object corresponding to self, which is just the `q`-expansion of self as a formal power series.
EXAMPLES::
sage: f = OverconvergentModularForms(3, 0, 1/2).1 sage: pari(f) # indirect doctest 27*q + 324*q^2 + 2430*q^3 + 13716*q^4 + 64557*q^5 + 265356*q^6 + 983556*q^7 + 3353076*q^8 + 10670373*q^9 + 32031288*q^10 + 91455804*q^11 + 249948828*q^12 + 657261999*q^13 + 1669898592*q^14 + 4113612864*q^15 + 9853898292*q^16 + 23010586596*q^17 + 52494114852*q^18 + 117209543940*q^19 + O(q^20) sage: pari(f.base_extend(Qp(3))) # indirect doctest (3^3 + O(3^23))*q + (3^4 + 3^5 + O(3^24))*q^2 + (3^5 + 3^7 + O(3^25))*q^3 + (3^3 + 3^4 + 2*3^5 + 2*3^8 + O(3^23))*q^4 + (2*3^4 + 3^5 + 3^6 + 2*3^7 + 3^10 + O(3^24))*q^5 + (3^6 + 3^7 + 3^8 + 3^9 + 3^10 + 3^11 + O(3^26))*q^6 + (2*3^3 + 3^4 + 2*3^6 + 2*3^7 + 2*3^8 + 3^9 + 3^10 + 2*3^11 + 3^12 + O(3^23))*q^7 + (2*3^4 + 3^5 + 3^8 + 2*3^9 + 2*3^10 + 2*3^13 + O(3^24))*q^8 + (3^7 + 2*3^9 + 2*3^12 + 2*3^14 + O(3^27))*q^9 + (2*3^5 + 3^8 + 3^9 + 2*3^10 + 2*3^13 + 2*3^15 + O(3^25))*q^10 + (3^4 + 2*3^5 + 2*3^6 + 3^8 + 2*3^9 + 3^12 + 3^14 + 2*3^16 + O(3^24))*q^11 + (3^5 + 3^6 + 2*3^8 + 2*3^9 + 2*3^10 + 2*3^12 + 3^14 + 2*3^15 + 2*3^16 + 3^17 + O(3^25))*q^12 + (2*3^3 + 2*3^4 + 2*3^5 + 3^8 + 2*3^9 + 2*3^11 + 3^13 + 2*3^14 + 2*3^17 + 3^18 + O(3^23))*q^13 + (2*3^4 + 2*3^6 + 2*3^7 + 3^8 + 2*3^9 + 3^10 + 3^12 + 3^14 + 2*3^15 + 2*3^16 + 3^18 + 3^19 + O(3^24))*q^14 + (2*3^6 + 3^7 + 3^9 + 3^10 + 3^11 + 2*3^14 + 3^15 + 2*3^16 + 3^17 + 3^18 + 3^20 + O(3^26))*q^15 + (3^3 + 2*3^4 + 2*3^7 + 2*3^8 + 3^9 + 3^10 + 2*3^11 + 3^12 + 2*3^14 + 2*3^15 + 3^17 + 3^18 + 2*3^19 + 2*3^20 + O(3^23))*q^16 + (2*3^5 + 2*3^7 + 2*3^8 + 3^10 + 3^11 + 2*3^12 + 2*3^13 + 3^14 + 3^15 + 3^17 + 2*3^18 + 3^19 + 2*3^21 + O(3^25))*q^17 + (3^8 + 3^9 + 2*3^10 + 2*3^11 + 3^12 + 3^14 + 3^15 + 3^16 + 3^17 + 2*3^21 + 3^22 + O(3^28))*q^18 + (2*3^3 + 3^5 + 2*3^6 + 2*3^8 + 2*3^9 + 3^11 + 2*3^12 + 3^13 + 3^14 + 2*3^15 + 3^16 + 3^17 + 2*3^18 + 3^19 + 2*3^21 + O(3^23))*q^19 + O(q^20) """ |