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# Atkin/Hecke series for overconvergent modular forms.

This file contains a function `hecke_series()` to compute the characteristic series \(P(t)\) modulo \(p^m\) of the Atkin/Hecke operator \(U_p\) upon the space of p-adic overconvergent modular forms of level \(\Gamma_0(N)\). The input weight `k` can also be a list `klist` of weights which must all be congruent modulo \((p-1)\).

Two optional parameters `modformsring` and `weightbound` can be specified, and in most cases for levels \(N > 1\) they can be used to obtain the output more quickly. When \(m \le k-1\) the output \(P(t)\) is also equal modulo \(p^m\) to the reverse characteristic polynomial of the Atkin operator \(U_p\) on the space of classical modular forms of weight k and level \(\Gamma_0(Np)\). In addition, provided \(m \le (k-2)/2\) the output \(P(t)\) is equal modulo \(p^m\) to the reverse characteristic polynomial of the Hecke operator \(T_p\) on the space of classical modular forms of weight k and level \(\Gamma_0(N)\). The function is based upon the main algorithm in [AGBL], and has linear running time in the logarithm of the weight k.

AUTHORS:

- Alan G.B. Lauder (2011-11-10): original implementation.
- David Loeffler (2011-12): minor optimizations in review stage.

EXAMPLES:

The characteristic series of the U_11 operator modulo 11^10 on the space of 11-adic overconvergent modular forms of level 1 and weight 10000:

    sage: hecke_series(11,1,10000,10)
    10009319650*x^4 + 25618839103*x^3 + 6126165716*x^2 + 10120524732*x + 1

The characteristic series of the U_5 operator modulo 5^5 on the space of 5-adic overconvergent modular forms of level 3 and weight 1000:

    sage: hecke_series(5,3,1000,5)
    1875*x^6 + 1250*x^5 + 1200*x^4 + 1385*x^3 + 1131*x^2 + 2533*x + 1

The characteristic series of the U_7 operator modulo 7^5 on the space of 7-adic overconvergent modular forms of level 5 and weight 1000. Here the optional parameter `modformsring` is set to true:

    sage: hecke_series(7,5,1000,5,modformsring = True)  # long time (21s on sage.math, 2012)
    12005*x^7 + 10633*x^6 + 6321*x^5 + 6216*x^4 + 5412*x^3 + 4927*x^2 + 4906*x + 1

The characteristic series of the U_13 operator modulo 13^5 on the space of 13-adic overconvergent modular forms of level 2 and weight 10000. Here the optional parameter `weightbound` is set to 4:

    sage: hecke_series(13,2,10000,5,weightbound = 4)  # long time (17s on sage.math, 2012)
    325156*x^5 + 109681*x^4 + 188617*x^3 + 220858*x^2 + 269566*x + 1

A list containing the characteristic series of the U_23 operator modulo 23^10 on the spaces of 23-adic overconvergent modular forms of level 1 and weights 1000 and 1022, respectively.

    sage: hecke_series(23,1,[1000,1022],10)
    [7204610645852*x^6 + 2117949463923*x^5 + 24152587827773*x^4 + 31270783576528*x^3 + 30336366679797*x^2
    + 29197235447073*x + 1, 32737396672905*x^4 + 36141830902187*x^3 + 16514246534976*x^2 + 38886059530878*x + 1]

REFERENCES:

[AGBL] | (1, 2, 3, 4, 5, 6, 7, 8) Alan G.B. Lauder, “Computations with classical and p-adic modular forms”, LMS J. of Comput. Math. 14 (2011), 214-231.

`sage.modular.overconvergent.hecke_series.``complementary_spaces`(N, p, k0, n, mdash, elldashp, elldash, modformsring, bound)

  Returns a list `Ws`, each element in which is a list `Wi` of q-expansions modulo \((p^\text{mdash},q^\text{elldashp})\). The list `Wi` is a basis for a choice of complementary space in level \(\Gamma_0(N)\) and weight \(k\) to the image of weight \(k - (p-1)\) forms under multiplication by the Eisenstein series \(E_{p-1}\).

  The lists `Wi` play the same role as \(W_i\) in Step 2 of Algorithm 2 in [AGBL]. (The parameters `k0,n,mdash,elldash,elldashp = elldash*p` are defined as in Step 1 of that algorithm when this function is used in `hecke_series()`.) However, the complementary spaces are computed in a different manner, combining a suggestion of David Loeffler with one of John Voight. That is, one builds these spaces recursively using random products of forms in low weight, first searching for suitable products modulo \((p,q^\text{elldash})\), and then later reconstructing only the required products to the full precision modulo \((p^\text{mdash},q^{elldashp})\). The forms in low weight are chosen from either bases of all forms up to weight `bound` or from a (tentative) generating set for the ring of all modular forms, according to whether `modformsring` is `False` or `True`.

  INPUT:

  - `N` – positive integer at least 2 and not divisible by p (level).
  - `p` – prime at least 5.
  - `k0` – integer in range 0 to `p-1`.
  - `n,mdash,elldashp,elldash` – positive integers.
  - `modformsring` – True or False.
  - `bound` – positive (even) integer (ignored if `modformsring` is True).

  OUTPUT:

  - list of lists of q-expansions modulo `(p^\text{mdash},q^\text{elldashp})`.

  EXAMPLES:

      sage: from sage.modular.overconvergent.hecke_series import complementary_spaces
      sage: complementary_spaces(2,5,0,3,2,5,4,true,6) # random
      [[1],
      [1 + 23*q + 24*q^2 + 19*q^3 + 7*q^4 + O(q^5)],
      [1 + 21*q + 2*q^2 + 17*q^3 + 14*q^4 + O(q^5)],
      [1 + 19*q + 9*q^2 + 11*q^3 + 9*q^4 + O(q^5)]]
      sage: complementary_spaces(2,5,0,3,2,5,4,false,6) # random
      [[1],
      [3 + 4*q + 2*q^2 + 12*q^3 + 11*q^4 + O(q^5)],
      [2 + 2*q + 14*q^2 + 19*q^3 + 18*q^4 + O(q^5)],
      [6 + 8*q + 10*q^2 + 23*q^3 + 4*q^4 + O(q^5)]]

`sage.modular.overconvergent.hecke_series.``complementary_spaces_modp`(N, p, k0, n, elldash, LWBModp, bound)

  Returns a list of lists of lists of lists `[j,a]`. The pairs `[j,a]` encode the choice of the \(a\)-th element in the \(j\)-th list of the input `LWBModp`, i.e., the \(a\)-th element in a particular basis modulo \((p,q^\text{elldash})\) for the space of modular forms of level \(\Gamma_0(N)\) and weight \(2(j+1)\). The list `[[j_1,a_1],...,[j_r,a_r]]` then encodes the product of the r modular forms associated to each `[j_i,a_i]`; this has weight \(k + (p-1)i\) for some \(0 \le i \le n\); here the i is such that this list of lists occurs in the ith list of the output. The ith list of the output thus encodes a choice of basis for the complementary space \(W_i\) which occurs in Step 2 of Algorithm 2 in [AGBL]. The idea is that one searches for this space \(W_i\) first modulo \((p,q^\text{elldash})\) and then, having found the correct products of generating forms, one can reconstruct these spaces modulo \((p^\text{mdash},q^\text{elldashp})\) using the output of this function. (This idea is based upon a suggestion of John Voight.)

  INPUT:

  - `N` – positive integer at least 2 and not divisible by p (level).
  - `p` – prime at least 5.
  - `k0` – integer in range 0 to \(p-1\).
  - `n,elldash` – positive integers.
  - `LWBModp` – list of lists of \(q\)-expansions over \(GF(p)\).
  - `bound` – positive even integer (twice the length of the list `LWBModp`).

  OUTPUT:

  - list of list of list of lists.

  EXAMPLES:

      sage: from sage.modular.overconvergent.hecke_series import random_low_weight_bases, complementary_spaces_modp
      sage: LWB = random_low_weight_bases(2,5,2,4,6)
      sage: LWBModp = [[f.change_ring(Zmod(5)) for f in x] for x in LWB]
      sage: complementary_spaces_modp(2,5,0,3,4,LWBModp,6) # random, indirect doctest
      [[[]], [[[0, 0], [0, 0]]], [[[0, 0], [2, 1]]], [[[0, 0], [0, 0], [0, 0], [2, 1]]]]

`sage.modular.overconvergent.hecke_series.``compute_G`(p, F)

  Given a power series \(F \in R[[q]]^\times\), for some ring \(R\), and an integer \(p\), compute the quotient

  \[\frac{F(q)}{F(q^p)}.\]

  Used by `level1_UpGj()` and by `higher_level_UpGj()`, with \(F\) equal to the Eisenstein series \(E_{p-1}\).

  INPUT:

  - `p` – integer
  - `F` – power series (with invertible constant term)

  OUTPUT:

  the power series \(F(q) / F(q^p)\), to the same precision as \(F\)

  EXAMPLE:

      sage: E = sage.modular.overconvergent.hecke_series.eisenstein_series_qexp(2, 12, Zmod(9),normalization="constant")
      sage: sage.modular.overconvergent.hecke_series.compute_G(3, E)
      1 + 3*q + 3*q^4 + 6*q^7 + O(q^12)

`sage.modular.overconvergent.hecke_series.``compute_Wi`(k, p, h, hj, E4, E6)

  This function computes a list \(W_i\) of q-expansions, together with an auxilliary quantity \(h^j\) (see below) which is to be used on the next call of this function. (The precision is that of input q-expansions.)

  The list \(W_i\) is a certain subset of a basis of the modular forms of weight \(k\) and level 1. Suppose \((a, b)\) is the pair of non-negative integers with \(4a + 6b = k\) and \(a\) minimal among such pairs. Then this space has a basis given by

  \[\begin{split}\{ \Delta^j E_6^{b - 2j} E_4^a : 0 \le j < d\}\end{split}\]

  where \(d\) is the dimension.

  What this function returns is the subset of the above basis corresponding to \(e \le j < d\) where \(e\) is the dimension of the space of modular forms of weight \(k - (p-1)\). This set is a basis for the complement of the image of the weight \(k - (p-1)\) forms under multiplication by \(E_{p-1}\).

  This function is used repeatedly in the construction of the Katz expansion basis. Hence considerable care is taken to reuse steps in the computation wherever possible: we keep track of powers of the form \(h = \Delta / E_6^2\).

  INPUT:

  - `k` – non-negative integer.
  - `p` – prime at least 5.
  - `h` – q-expansion of \(h\) (to some finite precision).
  - `hj` – q-expansion of h^j where j is the dimension of the space of modular forms of level 1 and weight \(k - (p-1)\) (to same finite precision).
  - `E4` – q-expansion of `E_4` (to same finite precision).
  - `E6` – q-expansion of `E_6` (to same finite precision).

  The Eisenstein series q-expansions should be normalized to have constant term 1.

  OUTPUT:

  - list of q-expansions (to same finite precision), and q-expansion.

  EXAMPLES:

      sage: from sage.modular.overconvergent.hecke_series import compute_Wi
      sage: p = 17
      sage: prec = 10
      sage: k = 24
      sage: S = Zmod(17^3)
      sage: E4 = eisenstein_series_qexp(4, prec, K=S, normalization="constant")
      sage: E6 = eisenstein_series_qexp(6, prec, K=S, normalization="constant")
      sage: h = delta_qexp(prec, K=S) / E6^2
      sage: j = dimension_modular_forms(1, k - (p-1))
      sage: hj = h**j
      sage: c = compute_Wi(k,p,h,hj,E4,E6); c
      ([q + 3881*q^2 + 4459*q^3 + 4665*q^4 + 2966*q^5 + 1902*q^6 + 1350*q^7 + 3836*q^8 + 1752*q^9 + O(q^10), q^2 + 4865*q^3 + 1080*q^4 + 4612*q^5 + 1343*q^6 + 1689*q^7 + 3876*q^8 + 1381*q^9 + O(q^10)], q^3 + 2952*q^4 + 1278*q^5 + 3225*q^6 + 1286*q^7 + 589*q^8 + 122*q^9 + O(q^10))
      sage: c == ([delta_qexp(10) * E6^2, delta_qexp(10)^2], h**3)
      True

`sage.modular.overconvergent.hecke_series.``compute_elldash`(p, N, k0, n)

  Returns the “Sturm bound” for the space of modular forms of level \(\Gamma_0(N)\) and weight \(k_0 + n(p-1)\).

  See also

  `sturm_bound()`

  INPUT:

  - `p` – prime.
  - `N` – positive integer (level).
  - `k0`, `n` - non-negative integers not both zero.

  OUTPUT:

  - positive integer.

  EXAMPLES:

      sage: from sage.modular.overconvergent.hecke_series import compute_elldash
      sage: compute_elldash(11,5,4,10)
      53

`sage.modular.overconvergent.hecke_series.``ech_form`(A, p)

  Returns echelon form of matrix `A` over the ring of integers modulo \(p^m\), for some prime \(p\) and \(m \ge 1\).

  Todo

  This should be moved to `sage.matrix.matrix_modn_dense` at some point.

  INPUT:

  - `A` – matrix over `Zmod(p^m)` for some m.
  - `p` - prime p.

  OUTPUT:

  - matrix over `Zmod(p^m)`.

  EXAMPLES:

      sage: from sage.modular.overconvergent.hecke_series import ech_form
      sage: A = MatrixSpace(Zmod(5**3),3)([1,2,3,4,5,6,7,8,9])
      sage: ech_form(A,5)
      [1 2 3]
      [0 1 2]
      [0 0 0]

`sage.modular.overconvergent.hecke_series.``hecke_series`(p, N, klist, m, modformsring=False, weightbound=6)

  Returns the characteristic series modulo \(p^m\) of the Atkin operator \(U_p\) acting upon the space of p-adic overconvergent modular forms of level \(\Gamma_0(N)\) and weight `klist`. The input `klist` may also be a list of weights congruent modulo \((p-1)\), in which case the output is the corresponding list of characteristic series for each \(k\) in `klist`; this is faster than performing the computation separately for each \(k\), since intermediate steps in the computation may be reused.

  If `modformsring` is True, then for \(N > 1\) the algorithm computes at one step `ModularFormsRing(N).generators()`. This will often be faster but the algorithm will default to `modformsring = False` if the generators found are not p-adically integral. Note that `modformsring` is ignored for \(N = 1\) and the ring structure of modular forms is always used in this case.

  When `modformsring` is False and \(N > 1\), \(weightbound\) is a bound set on the weight of generators for a certain subspace of modular forms. The algorithm will often be faster if `weightbound = 4`, but it may fail to terminate for certain exceptional small values of \(N\), when this bound is too small.

  The algorithm is based upon that described in [AGBL].

  INPUT:

  - `p` – a prime greater than or equal to 5.
  - `N` – a positive integer not divisible by \(p\).
  - `klist` – either a list of integers congruent modulo \((p-1)\), or a single integer.
  - `m` – a positive integer.
  - `modformsring` – `True` or `False` (optional, default `False`). Ignored if \(N = 1\).
  - `weightbound` – a positive even integer (optional, default 6). Ignored if \(N = 1\) or `modformsring` is True.

  OUTPUT:

  Either a list of polynomials or a single polynomial over the integers modulo \(p^m\).

  EXAMPLES:

      sage: hecke_series(5,7,10000,5, modformsring = True) # long time (3.4s)
      250*x^6 + 1825*x^5 + 2500*x^4 + 2184*x^3 + 1458*x^2 + 1157*x + 1
      sage: hecke_series(7,3,10000,3, weightbound = 4)
      196*x^4 + 294*x^3 + 197*x^2 + 341*x + 1
      sage: hecke_series(19,1,[10000,10018],5)
      [1694173*x^4 + 2442526*x^3 + 1367943*x^2 + 1923654*x + 1,
      130321*x^4 + 958816*x^3 + 2278233*x^2 + 1584827*x + 1]

  Check that silly weights are handled correctly:

      sage: hecke_series(5, 7, [2, 3], 5)
      Traceback (most recent call last):
      ...
      ValueError: List of weights must be all congruent modulo p-1 = 4, but given list contains 2 and 3 which are not congruent
      sage: hecke_series(5, 7, [3], 5)
      [1]
      sage: hecke_series(5, 7, 3, 5)
      1

`sage.modular.overconvergent.hecke_series.``hecke_series_degree_bound`(p, N, k, m)

  Returns the `Wan bound` on the degree of the characteristic series of the Atkin operator on p-adic overconvergent modular forms of level \(\Gamma_0(N)\) and weight k when reduced modulo \(p^m\). This bound depends only upon p, \(k \pmod{p-1}\), and N. It uses Lemma 3.1 in [DW].

  INPUT:

  - `p` – prime at least 5.
  - `N` – positive integer not divisible by p.
  - `k` – even integer.
  - `m` – positive integer.

  OUTPUT:

  A non-negative integer.

  EXAMPLES:

      sage: from sage.modular.overconvergent.hecke_series import hecke_series_degree_bound
      sage: hecke_series_degree_bound(13,11,100,5)
      39

  REFERENCES:

  [DW] | Daqing Wan, “Dimension variation of classical and p-adic modular forms”, Invent. Math. 133, (1998) 449-463.

`sage.modular.overconvergent.hecke_series.``higher_level_UpGj`(p, N, klist, m, modformsring, bound)

  Returns a list `[A_k]` of square matrices over `IntegerRing(p^m)` parameterised by the weights k in `klist`. The matrix \(A_k\) is the finite square matrix which occurs on input p,k,N and m in Step 6 of Algorithm 2 in [AGBL]. Notational change from paper: In Step 1 following Wan we defined j by \(k = k_0 + j(p-1)\) with \(0 \le k_0 < p-1\). Here we replace j by `kdiv` so that we may use j as a column index for matrices.)

  INPUT:

  - `p` – prime at least 5.
  - `N` – integer at least 2 and not divisible by p (level).
  - `klist` – list of integers all congruent modulo (p-1) (the weights).
  - `m` – positive integer.
  - `modformsring` – True or False.
  - `bound` – (even) positive integer.

  OUTPUT:

  - list of square matrices.

  EXAMPLES:

      sage: from sage.modular.overconvergent.hecke_series import higher_level_UpGj
      sage: higher_level_UpGj(5,3,[4],2,true,6)
      [
      [ 1  0  0  0  0  0]
      [ 0  1  0  0  0  0]
      [ 0  7  0  0  0  0]
      [ 0  5 10 20  0  0]
      [ 0  7 20  0 20  0]
      [ 0  1 24  0 20  0]
      ]

`sage.modular.overconvergent.hecke_series.``higher_level_katz_exp`(p, N, k0, m, mdash, elldash, elldashp, modformsring, bound)

  Returns a matrix \(e\) of size `ell x elldashp` over the integers modulo \(p^\text{mdash}\), and the Eisenstein series \(E_{p-1} = 1 + .\dots \bmod (p^\text{mdash},q^\text{elldashp})\). The matrix e contains the coefficients of the elements \(e_{i,s}\) in the Katz expansions basis in Step 3 of Algorithm 2 in [AGBL] when one takes as input to that algorithm \(p\),`N`,`m` and \(k\) and define `k0`, `mdash`, `n`, `elldash`, `elldashp = ell*dashp` as in Step 1.

  INPUT:

  - `p` – prime at least 5.
  - `N` – positive integer at least 2 and not divisible by p (level).
  - `k0` – integer in xrange 0 to p-1.
  - `m,mdash,elldash,elldashp` – positive integers.
  - `modformsring` – True or False.
  - `bound` – positive (even) integer.

  OUTPUT:

  - matrix and q-expansion.

  EXAMPLES:

      sage: from sage.modular.overconvergent.hecke_series import higher_level_katz_exp
      sage: e,Ep1 = higher_level_katz_exp(5,2,0,1,2,4,20,true,6)
      sage: e
      [ 1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0]
      [ 0  1 18 23 19  6  9  9 17  7  3 17 12  8 22  8 11 19  1  5]
      [ 0  0  1 11 20 16  0  8  4  0 18 15 24  6 15 23  5 18  7 15]
      [ 0  0  0  1  4 16 23 13  6  5 23  5  2 16  4 18 10 23  5 15]
      sage: Ep1
      1 + 15*q + 10*q^2 + 20*q^3 + 20*q^4 + 15*q^5 + 5*q^6 + 10*q^7 +
      5*q^9 + 10*q^10 + 5*q^11 + 10*q^12 + 20*q^13 + 15*q^14 + 20*q^15 + 15*q^16 +
      10*q^17 + 20*q^18 + O(q^20)

`sage.modular.overconvergent.hecke_series.``is_valid_weight_list`(klist, p)

  This function checks that `klist` is a nonempty list of integers all of which are congruent modulo \((p-1)\). Otherwise, it will raise a ValueError.

  INPUT:

  - `klist` – list of integers.
  - `p` – prime.

  EXAMPLES:

      sage: from sage.modular.overconvergent.hecke_series import is_valid_weight_list
      sage: is_valid_weight_list([10,20,30],11)
      sage: is_valid_weight_list([-3, 1], 5)
      sage: is_valid_weight_list([], 3)
      Traceback (most recent call last):
      ...
      ValueError: List of weights must be non-empty
      sage: is_valid_weight_list([-3, 2], 5)
      Traceback (most recent call last):
      ...
      ValueError: List of weights must be all congruent modulo p-1 = 4, but given list contains -3 and 2 which are not congruent

`sage.modular.overconvergent.hecke_series.``katz_expansions`(k0, p, ellp, mdash, n)

  Returns a list e of q-expansions, and the Eisenstein series \(E_{p-1} = 1 + \dots\), all modulo \((p^\text{mdash},q^\text{ellp})\). The list e contains the elements \(e_{i,s}\) in the Katz expansions basis in Step 3 of Algorithm 1 in [AGBL] when one takes as input to that algorithm p,m and k and define `k0`, `mdash`, n, `ellp = ell*p` as in Step 1.

  INPUT:

  - `k0` – integer in range 0 to p-1.
  - `p` – prime at least 5.
  - `ellp,mdash,n` – positive integers.

  OUTPUT:

  - list of q-expansions and the Eisenstein series E_{p-1} modulo \((p^\text{mdash},q^\text{ellp})\).

  EXAMPLES:

      sage: from sage.modular.overconvergent.hecke_series import katz_expansions
      sage: katz_expansions(0,5,10,3,4)
      ([1 + O(q^10), q + 6*q^2 + 27*q^3 + 98*q^4 + 65*q^5 + 37*q^6 + 81*q^7 + 85*q^8 + 62*q^9 + O(q^10)],
      1 + 115*q + 35*q^2 + 95*q^3 + 20*q^4 + 115*q^5 + 105*q^6 + 60*q^7 + 25*q^8 + 55*q^9 + O(q^10))

`sage.modular.overconvergent.hecke_series.``level1_UpGj`(p, klist, m)

  Returns a list \([A_k]\) of square matrices over `IntegerRing(p^m)` parameterised by the weights k in `klist`. The matrix \(A_k\) is the finite square matrix which occurs on input p,k and m in Step 6 of Algorithm 1 in [AGBL]. Notational change from paper: In Step 1 following Wan we defined j by \(k = k_0 + j(p-1)\) with \(0 \le k_0 < p-1\). Here we replace j by `kdiv` so that we may use j as a column index for matrices.

  INPUT:

  - `p` – prime at least 5.
  - `klist` – list of integers congruent modulo \((p-1)\) (the weights).
  - `m` – positive integer.

  OUTPUT:

  - list of square matrices.

  EXAMPLES:

      sage: from sage.modular.overconvergent.hecke_series import level1_UpGj
      sage: level1_UpGj(7,[100],5)
      [
      [    1   980  4802     0     0]
      [    0 13727 14406     0     0]
      [    0 13440  7203     0     0]
      [    0  1995  4802     0     0]
      [    0  9212 14406     0     0]
      ]

`sage.modular.overconvergent.hecke_series.``low_weight_bases`(N, p, m, NN, weightbound)

  Returns a list of integral bases of modular forms of level N and (even) weight at most `weightbound`, as \(q\)-expansions modulo \((p^m,q^{NN})\).

  These forms are obtained by reduction mod \(p^m\) from an integral basis in Hermite normal form (so they are not necessarily in reduced row echelon form mod \(p^m\), but they are not far off).

  INPUT:

  - `N` – positive integer (level).
  - `p` – prime.
  - `m`, `NN` – positive integers.
  - `weightbound` – (even) positive integer.

  OUTPUT:

  - list of lists of \(q\)-expansions modulo \((p^m,q^{NN})\).

  EXAMPLES:

      sage: from sage.modular.overconvergent.hecke_series import low_weight_bases
      sage: low_weight_bases(2,5,3,5,6)
      [[1 + 24*q + 24*q^2 + 96*q^3 + 24*q^4 + O(q^5)],
      [1 + 115*q^2 + 35*q^4 + O(q^5), q + 8*q^2 + 28*q^3 + 64*q^4 + O(q^5)],
      [1 + 121*q^2 + 118*q^4 + O(q^5), q + 32*q^2 + 119*q^3 + 24*q^4 + O(q^5)]]

`sage.modular.overconvergent.hecke_series.``low_weight_generators`(N, p, m, NN)

  Returns a list of lists of modular forms, and an even natural number. The first output is a list of lists of modular forms reduced modulo \((p^m,q^{NN})\) which generate the \((\ZZ / p^m \ZZ)\)-algebra of mod \(p^m\) modular forms of weight at most 8, and the second output is the largest weight among the forms in the generating set.

  We (Alan Lauder and David Loeffler, the author and reviewer of this patch) conjecture that forms of weight at most 8 are always sufficient to generate the algebra of mod \(p^m\) modular forms of all weights. (We believe 6 to be sufficient, and we can prove that 4 is sufficient when there are no elliptic points, but using weights up to 8 acts as a consistency check.)

  INPUT:

  - `N` – positive integer (level).
  - `p` – prime.
  - `m`, `NN` – positive integers.

  OUTPUT:

  a tuple consisting of:

  - a list of lists of \(q\)-expansions modulo \((p^m,q^{NN})\),
  - an even natural number (twice the length of the list).

  EXAMPLES:

      sage: from sage.modular.overconvergent.hecke_series import low_weight_generators
      sage: low_weight_generators(3,7,3,10)
      ([[1 + 12*q + 36*q^2 + 12*q^3 + 84*q^4 + 72*q^5 + 36*q^6 + 96*q^7 + 180*q^8 + 12*q^9 + O(q^10)],
      [1 + 240*q^3 + 102*q^6 + 203*q^9 + O(q^10)],
      [1 + 182*q^3 + 175*q^6 + 161*q^9 + O(q^10)]], 6)
      sage: low_weight_generators(11,5,3,10)
      ([[1 + 12*q^2 + 12*q^3 + 12*q^4 + 12*q^5 + 24*q^6 + 24*q^7 + 36*q^8 + 36*q^9 + O(q^10),
      q + 123*q^2 + 124*q^3 + 2*q^4 + q^5 + 2*q^6 + 123*q^7 + 123*q^9 + O(q^10)],
      [q + 116*q^4 + 115*q^5 + 102*q^6 + 121*q^7 + 96*q^8 + 106*q^9 + O(q^10)]], 4)

`sage.modular.overconvergent.hecke_series.``random_low_weight_bases`(N, p, m, NN, weightbound)

  Returns list of random integral bases of modular forms of level \(N\) and (even) weight at most weightbound with coefficients reduced modulo \((p^m,q^{NN})\).

  INPUT:

  - `N` – positive integer (level).
  - `p` – prime.
  - `m`, `NN` – positive integers.
  - `weightbound` – (even) positive integer.

  OUTPUT:

  - list of lists of \(q\)-expansions modulo \((p^m,q^{NN})\).

  EXAMPLES:

      sage: from sage.modular.overconvergent.hecke_series import random_low_weight_bases
      sage: S = random_low_weight_bases(3,7,2,5,6); S # random
      [[4 + 48*q + 46*q^2 + 48*q^3 + 42*q^4 + O(q^5)],
      [3 + 5*q + 45*q^2 + 22*q^3 + 22*q^4 + O(q^5),
      1 + 3*q + 27*q^2 + 27*q^3 + 23*q^4 + O(q^5)],
      [2*q + 4*q^2 + 16*q^3 + 48*q^4 + O(q^5),
      2 + 6*q + q^2 + 3*q^3 + 43*q^4 + O(q^5),
      1 + 2*q + 6*q^2 + 14*q^3 + 4*q^4 + O(q^5)]]
      sage: S[0][0].parent()
      Power Series Ring in q over Ring of integers modulo 49
      sage: S[0][0].prec()
      5

`sage.modular.overconvergent.hecke_series.``random_new_basis_modp`(N, p, k, LWBModp, TotalBasisModp, elldash, bound)

  Returns `NewBasisCode`. Here \(NewBasisCode\) is a list of lists of lists `[j,a]`. This encodes a choice of basis for the ith complementary space \(W_i\), as explained in the documentation for the function `complementary_spaces_modp()`.

  INPUT:

  - `N` – positive integer at least 2 and not divisible by p (level).
  - `p` – prime at least 5.
  - `k` – non-negative integer.
  - `LWBModp` – list of list of q-expansions modulo \((p,q^\text{elldash})\).
  - `TotalBasisModp` – matrix over GF(p).
  - `elldash` - positive integer.
  - `bound` – positive even integer (twice the length of the list `LWBModp`).

  OUTPUT:

  - A list of lists of lists `[j,a]`.

  Note

  As well as having a non-trivial return value, this function also modifies the input matrix `TotalBasisModp`.

  EXAMPLES:

      sage: from sage.modular.overconvergent.hecke_series import random_low_weight_bases, complementary_spaces_modp
      sage: LWB = random_low_weight_bases(2,5,2,4,6)
      sage: LWBModp = [ [f.change_ring(GF(5)) for f in x] for x in LWB]
      sage: complementary_spaces_modp(2,5,2,3,4,LWBModp,4) # random, indirect doctest
      [[[[0, 0]]], [[[0, 0], [1, 1]]], [[[0, 0], [1, 0], [1, 1]]], [[[0, 0], [1, 0], [1, 1], [1, 1]]]]

`sage.modular.overconvergent.hecke_series.``random_solution`(B, K)

  Returns a random solution in non-negative integers to the equation \(a_1 + 2 a_2 + 3 a_3 + ... + B a_B = K\), using a greedy algorithm.

  Note that this is much faster than using `WeightedIntegerVectors.random_element()`.

  INPUT:

  - `B`, `K` – non-negative integers.

  OUTPUT:

  - list.

  EXAMPLES:

      sage: from sage.modular.overconvergent.hecke_series import random_solution
      sage: random_solution(5,10)
      [1, 1, 1, 1, 0]

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