Implement basis_of_weight for rings of quasimodular forms

[DavidAyotte]

📚 Description

This PR implements the method QuasiModularForms.basis_of_weight(k). It returns a list of quasimodular forms which generate the space of weight k. Some examples:

sage: QM = QuasiModularForms(1)
sage: QM.basis_of_weight(12)
[q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6),
 1 + 65520/691*q + 134250480/691*q^2 + 11606736960/691*q^3 + 274945048560/691*q^4 + 3199218815520/691*q^5 + O(q^6),
 1 - 288*q - 129168*q^2 - 1927296*q^3 + 65152656*q^4 + 1535768640*q^5 + O(q^6),
 1 + 432*q + 39312*q^2 - 1711296*q^3 - 14159664*q^4 + 317412000*q^5 + O(q^6),
 1 - 576*q + 21168*q^2 + 308736*q^3 - 15034608*q^4 - 39208320*q^5 + O(q^6),
 1 + 144*q - 17712*q^2 + 524736*q^3 - 2279088*q^4 - 79760160*q^5 + O(q^6),
 1 - 144*q + 8208*q^2 - 225216*q^3 + 2634192*q^4 + 1488672*q^5 + O(q^6)]
sage: QM = QuasiModularForms(Gamma1(3))
sage: QM.basis_of_weight(3)
[1 + 54*q^2 + 72*q^3 + 432*q^5 + O(q^6),
 q + 3*q^2 + 9*q^3 + 13*q^4 + 24*q^5 + O(q^6)]
sage: QM.basis_of_weight(5)
[1 - 90*q^2 - 240*q^3 - 3744*q^5 + O(q^6),
 q + 15*q^2 + 81*q^3 + 241*q^4 + 624*q^5 + O(q^6),
 1 - 24*q - 18*q^2 - 1320*q^3 - 5784*q^4 - 10080*q^5 + O(q^6),
 q - 21*q^2 - 135*q^3 - 515*q^4 - 1392*q^5 + O(q^6)]

CC: @videlec

📝 Checklist

• I have made sure that the title is self-explanatory and the description concisely explains the PR.
• I have linked an issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation accordingly.

⌛ Dependencies

[videlec]

This is a really cool feature.

It would be nice if this was used to refactor a bit _homogeneous_to_polynomial.

[codecov-commenter]

Codecov Report

Base: 88.59% // Head: 88.58% // Decreases project coverage by -0.01% ⚠️

Coverage data is based on head (6185e8c) compared to base (104dde9).
Patch coverage: 100.00% of modified lines in pull request are covered.

Additional details and impacted files

@@             Coverage Diff             @@
##           develop   #35029      +/-   ##
===========================================
- Coverage    88.59%   88.58%   -0.01%     
===========================================
  Files         2136     2136              
  Lines       396142   396150       +8     
===========================================
- Hits        350948   350946       -2     
- Misses       45194    45204      +10     

Impacted Files	Coverage Δ
src/sage/modular/quasimodform/ring.py	95.65% <100.00%> (+0.26%)	⬆️
src/sage/crypto/util.py	91.37% <0.00%> (-5.18%)	⬇️
src/sage/homology/matrix_utils.py	84.40% <0.00%> (-3.67%)	⬇️
src/sage/interfaces/qsieve.py	71.30% <0.00%> (-2.61%)	⬇️
src/sage/rings/polynomial/pbori/parallel.py	59.82% <0.00%> (-1.79%)	⬇️
src/sage/categories/weyl_groups.py	96.47% <0.00%> (-0.59%)	⬇️
src/sage/modular/arithgroup/congroup_gamma0.py	94.41% <0.00%> (-0.56%)	⬇️
src/sage/combinat/tableau.py	95.72% <0.00%> (-0.50%)	⬇️
src/sage/graphs/generic_graph.py	89.12% <0.00%> (-0.42%)	⬇️
src/sage/modular/arithgroup/arithgroup_perm.py	92.57% <0.00%> (-0.38%)	⬇️
... and 15 more

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[DavidAyotte]

About commit 0f486e3 : Oops my bad, I wanted to implement the depth of a quasimodular form, but accidently pushed the changes to this branch.

[DavidAyotte]

This is a really cool feature.

It would be nice if this was used to refactor a bit _homogeneous_to_polynomial.

Hello Vincent, I was wondering what do you have in mind when you say to refactor _homogeneous_to_polynomial? This method is implemented in the modular forms ring element class. Do you mean to use the method monomials_of_weight at the polynomial ring level in order to first compute a weight $k$ basis?

[videlec]

Indeed, _homogeneous_to_polynomial is a method of GradedModularFormElement. Though it does need to generate the list of elements of given weight.

[DavidAyotte]

Indeed, _homogeneous_to_polynomial is a method of GradedModularFormElement. Though it does need to generate the list of elements of given weight.

Ok I see. I'll do the changes in a seperate PR. Also, there are some more refactoring and enhancement that could be done in the polynomial and polynomial_ring methods for (quasi)modular forms.

[grhkm21]

Hey, you'll be receiving a few notifications from me, since I'm going through old PRs. Nevermind, there's only three PRs still open, and one I don't understand :)

I gave some feedback.

[DavidAyotte]

Hey, you'll be receiving a few notifications from me, since I'm going through old PRs. Nevermind, there's only three PRs still open, and one I don't understand :)

I gave some feedback.

Hello, thank you for the reviews, I will address the comments in a few weeks.

[grhkm21]

Looks good to me. Thanks!

[github-actions]

Documentation preview for this PR (built with commit 4df388b; changes) is ready! 🎉