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# 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] | (1, 2, 3) David Loeffler, Spectral expansions of overconvergent modular functions, Int. Math. Res. Not 2007 (050). Arxiv preprint.

AUTHORS:

- David Loeffler (August 2008): initial version
- David Loeffler (March 2009): extensively reworked
- Lloyd Kilford (May 2009): add `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

\[f_p = \sqrt[p-1]{\frac{\Delta(pz)}{\Delta(z)}}\]

(an \(\eta\)-product of level \(p\) – see module `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

\[|f_p|_p \le p^{\frac{12r}{p-1}}.\]

(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.

    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:

    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:

    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:

    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:

    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:

    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:

    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:

    sage: (efuncs[1] - M(CuspForms(3, 8).0)).valuation()
    13

The second is an Eisenstein series!

    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.

    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

class `sage.modular.overconvergent.genus0.``OverconvergentModularFormElement`(parent, gexp=None, qexp=None)

  Bases: `sage.structure.element.ModuleElement`

  A class representing an element of a space of overconvergent modular forms.

  EXAMPLE:

      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

  `additive_order`()

    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

  `base_extend`(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)

  `coordinates`(prec=None)

    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

  `eigenvalue`()

    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.

    EXAMPLE:

        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

  `gexp`()

    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 \(X_0(p)\)).

    EXAMPLE:

        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)

  `governing_term`(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

  `is_eigenform`()

    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.

    EXAMPLE:

        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

  `is_integral`()

    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!

    EXAMPLE:

        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

  `prec`()

    Return the series expansion precision of this overconvergent modular form. (This is not the same as the \(p\)-adic precision of the coefficients.)

    EXAMPLE:

        sage: OverconvergentModularForms(5, 6, 1/3,prec=15).gen(1).prec()
        15

  `prime`()

    If this is a \(p\)-adic modular form, return \(p\).

    EXAMPLE:

        sage: OverconvergentModularForms(2, 0, 1/2).an_element().prime()
        2

  `q_expansion`(prec=None)

    Return the \(q\)-expansion of self, to as high precision as it is known.

    EXAMPLE:

        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)

  `r_ord`(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

  `slope`()

    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.

    EXAMPLE:

        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

  `valuation`()

    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

  `valuation_plot`(rmax=None)

    Draw a graph depicting the growth of the norm of this overconvergent modular form as it approaches the boundary of the overconvergent region.

    EXAMPLE:

        sage: o=OverconvergentModularForms(3, 0, 1/2)
        sage: f=o.eigenfunctions(4)[1]
        sage: f.valuation_plot()
        Graphics object consisting of 1 graphics primitive

  `weight`()

    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

`sage.modular.overconvergent.genus0.``OverconvergentModularForms`(prime, weight, radius, base_ring=Rational Field, prec=20, char=None)

  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

class `sage.modular.overconvergent.genus0.``OverconvergentModularFormsSpace`(prime, weight, radius, base_ring, prec, char)

  Bases: `sage.modules.module.Module_old`

  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

  `base_extend`(ring)

    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 of 2-adic Field ...
        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.

  `change_ring`(ring)

    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 ...

  `character`()

    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().

    EXAMPLE:

        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])

  `coordinate_vector`(x)

    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)

  `cps_u`(n, use_recurrence=False)

    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.

  `eigenfunctions`(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]

  `gen`(i)

    Return the ith module generator of self.

    EXAMPLE:

        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)

  `gens`()

    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: t.next()
        3-adic overconvergent modular form of weight-character 12 with q-expansion 1 - 32760/61203943*q - 67125240/61203943*q^2 - ...
        sage: t.next()
        3-adic overconvergent modular form of weight-character 12 with q-expansion 27*q + 19829193012/61203943*q^2 + 146902585770/61203943*q^3 + ...

  `gens_dict`()

    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

  `hecke_matrix`(m, n, use_recurrence=False, exact_arith=False)

    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]

  `hecke_operator`(f, m)

    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.

    EXAMPLE:

        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)

  `is_exact`()

    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

  `ngens`()

    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

  `normalising_factor`()

    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.

    EXAMPLE:

        sage: L.<w> = Qp(7).extension(x^2 - 7)
        sage: OverconvergentModularForms(7, 0, 1/4, base_ring=L).normalising_factor()
        w + O(w^41)

  `prec`()

    Return the series precision of self. Note that this is different from the \(p\)-adic precision of the base ring.

    EXAMPLE:

        sage: OverconvergentModularForms(3, 0, 1/2).prec()
        20
        sage: OverconvergentModularForms(3, 0, 1/2,prec=40).prec()
        40

  `prime`()

    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

  `radius`()

    The radius of overconvergence of this space.

    EXAMPLE:

        sage: OverconvergentModularForms(3, 0, 1/3).radius()
        1/3

  `recurrence_matrix`(use_smithline=True)

    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 ...

  `slopes`(n, use_recurrence=False)

    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]

  `weight`()

    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().

    EXAMPLE:

        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])

  `zero_element`()

    Return the zero of this space.

    EXAMPLE:

        sage: K.<w> = Qp(13).extension(x^2-13); M = OverconvergentModularForms(13, 20, radius=1/2, base_ring=K)
        sage: K.zero_element()
        0

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