Tutorial: Computing With Modular Forms Using SAGE

Contents, General, Modular Forms, Modular Symbols, Future

Defining Ambient and Cuspidal Spaces of Modular Forms

ModularForms(11)

Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight
2 over Rational Field

Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field

ModularForms(Gamma1(2007))

Modular Forms space of dimension 150960 for Congruence Subgroup Gamma1(2007) of
weight 2 over Rational Field

Modular Forms space of dimension 150960 for Congruence Subgroup Gamma1(2007) of weight 2 over Rational Field

CuspForms(Gamma1(13),3)

Cuspidal subspace of dimension 8 of Modular Forms space of dimension 20 for
Congruence Subgroup Gamma1(13) of weight 3 over Rational Field

Cuspidal subspace of dimension 8 of Modular Forms space of dimension 20 for Congruence Subgroup Gamma1(13) of weight 3 over Rational Field

G.<a> = DirichletGroup(13)
CuspForms(a^2,2)

Cuspidal subspace of dimension 1 of Modular Forms space of dimension 3,
character [zeta6] and weight 2 over Cyclotomic Field of order 6 and degree 2

Cuspidal subspace of dimension 1 of Modular Forms space of dimension 3, character [zeta6] and weight 2 over Cyclotomic Field of order 6 and degree 2

M = ModularForms(Gamma0(11),2); M

Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight
2 over Rational Field

Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field

M = ModularForms(Gamma1(13),2); M

Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight
2 over Rational Field

Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field

M = CuspForms(Gamma1(13),2); M

Cuspidal subspace of dimension 2 of Modular Forms space of dimension 13 for
Congruence Subgroup Gamma1(13) of weight 2 over Rational Field

Cuspidal subspace of dimension 2 of Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field

M = ModularForms(Gamma1(1000),2); M

Modular Forms space of dimension 31080 for Congruence Subgroup Gamma1(1000) of
weight 2 over Rational Field

Modular Forms space of dimension 31080 for Congruence Subgroup Gamma1(1000) of weight 2 over Rational Field

Explicit Basis

M = ModularForms(Gamma1(13),2, prec=20); M

Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight
2 over Rational Field

Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field

M.basis()

[
q - 4*q^3 - q^4 + 3*q^5 + 6*q^6 - 3*q^8 + q^9 - 6*q^10 - 2*q^12 + 2*q^13 +
10*q^16 - 3*q^17 - 3*q^18 - 6*q^19 + O(q^20),
q^2 - 2*q^3 - q^4 + 2*q^5 + 2*q^6 - 2*q^8 + q^9 - 3*q^10 + 3*q^13 - 2*q^15 +
5*q^16 - 3*q^17 - 2*q^18 - 2*q^19 + O(q^20),
1 + 21060/19*q^11 - 36504/19*q^12 + 75504/19*q^13 - 388440/19*q^14 +
595140/19*q^15 + 89856/19*q^16 - 323856/19*q^17 - 939120/19*q^18 +
617760/19*q^19 + O(q^20),
q + 11709/19*q^11 - 20687/19*q^12 + 43295/19*q^13 - 219842/19*q^14 +
336765/19*q^15 + 50783/19*q^16 - 183429/19*q^17 - 532306/19*q^18 +
350304/19*q^19 + O(q^20),
q^2 + 262*q^11 - 467*q^12 + 964*q^13 - 4779*q^14 + 7306*q^15 + 1097*q^16 -
3974*q^17 - 11550*q^18 + 7608*q^19 + O(q^20),
q^3 + 918/19*q^11 - 1215/19*q^12 + 2554/19*q^13 - 15203/19*q^14 + 23529/19*q^15
+ 3666/19*q^16 - 12783/19*q^17 - 36661/19*q^18 + 23964/19*q^19 + O(q^20),
q^4 - 882/19*q^11 + 2095/19*q^12 - 4330/19*q^13 + 18130/19*q^14 - 27230/19*q^15
- 4014/19*q^16 + 14916/19*q^17 + 43745/19*q^18 - 29064/19*q^19 + O(q^20),
q^5 - 1287/19*q^11 + 2607/19*q^12 - 5440/19*q^13 + 25201/19*q^14 - 38200/19*q^15
- 5761/19*q^16 + 20878/19*q^17 + 60950/19*q^18 - 40260/19*q^19 + O(q^20),
q^6 - 1080/19*q^11 + 2024/19*q^12 - 4138/19*q^13 + 20053/19*q^14 - 30520/19*q^15
- 4608/19*q^16 + 16608/19*q^17 + 48445/19*q^18 - 31908/19*q^19 + O(q^20),
q^7 - 675/19*q^11 + 1056/19*q^12 - 2192/19*q^13 + 12355/19*q^14 - 19075/19*q^15
- 2880/19*q^16 + 10380/19*q^17 + 29872/19*q^18 - 19553/19*q^19 + O(q^20),
q^8 - 360/19*q^11 + 453/19*q^12 - 898/19*q^13 + 5975/19*q^14 - 9350/19*q^15 -
1403/19*q^16 + 5042/19*q^17 + 14489/19*q^18 - 9420/19*q^19 + O(q^20),
q^9 - 153/19*q^11 + 98/19*q^12 - 204/19*q^13 + 2290/19*q^14 - 3665/19*q^15 -
592/19*q^16 + 2007/19*q^17 + 5575/19*q^18 - 3576/19*q^19 + O(q^20),
q^10 - 54/19*q^11 - 9/19*q^12 + 4/19*q^13 + 597/19*q^14 - 994/19*q^15 -
143/19*q^16 + 534/19*q^17 + 1420/19*q^18 - 900/19*q^19 + O(q^20)
]

[
q - 4*q^3 - q^4 + 3*q^5 + 6*q^6 - 3*q^8 + q^9 - 6*q^10 - 2*q^12 + 2*q^13 + 10*q^16 - 3*q^17 - 3*q^18 - 6*q^19 + O(q^20),
q^2 - 2*q^3 - q^4 + 2*q^5 + 2*q^6 - 2*q^8 + q^9 - 3*q^10 + 3*q^13 - 2*q^15 + 5*q^16 - 3*q^17 - 2*q^18 - 2*q^19 + O(q^20),
1 + 21060/19*q^11 - 36504/19*q^12 + 75504/19*q^13 - 388440/19*q^14 + 595140/19*q^15 + 89856/19*q^16 - 323856/19*q^17 - 939120/19*q^18 + 617760/19*q^19 + O(q^20),
q + 11709/19*q^11 - 20687/19*q^12 + 43295/19*q^13 - 219842/19*q^14 + 336765/19*q^15 + 50783/19*q^16 - 183429/19*q^17 - 532306/19*q^18 + 350304/19*q^19 + O(q^20),
q^2 + 262*q^11 - 467*q^12 + 964*q^13 - 4779*q^14 + 7306*q^15 + 1097*q^16 - 3974*q^17 - 11550*q^18 + 7608*q^19 + O(q^20),
q^3 + 918/19*q^11 - 1215/19*q^12 + 2554/19*q^13 - 15203/19*q^14 + 23529/19*q^15 + 3666/19*q^16 - 12783/19*q^17 - 36661/19*q^18 + 23964/19*q^19 + O(q^20),
q^4 - 882/19*q^11 + 2095/19*q^12 - 4330/19*q^13 + 18130/19*q^14 - 27230/19*q^15 - 4014/19*q^16 + 14916/19*q^17 + 43745/19*q^18 - 29064/19*q^19 + O(q^20),
q^5 - 1287/19*q^11 + 2607/19*q^12 - 5440/19*q^13 + 25201/19*q^14 - 38200/19*q^15 - 5761/19*q^16 + 20878/19*q^17 + 60950/19*q^18 - 40260/19*q^19 + O(q^20),
q^6 - 1080/19*q^11 + 2024/19*q^12 - 4138/19*q^13 + 20053/19*q^14 - 30520/19*q^15 - 4608/19*q^16 + 16608/19*q^17 + 48445/19*q^18 - 31908/19*q^19 + O(q^20),
q^7 - 675/19*q^11 + 1056/19*q^12 - 2192/19*q^13 + 12355/19*q^14 - 19075/19*q^15 - 2880/19*q^16 + 10380/19*q^17 + 29872/19*q^18 - 19553/19*q^19 + O(q^20),
q^8 - 360/19*q^11 + 453/19*q^12 - 898/19*q^13 + 5975/19*q^14 - 9350/19*q^15 - 1403/19*q^16 + 5042/19*q^17 + 14489/19*q^18 - 9420/19*q^19 + O(q^20),
q^9 - 153/19*q^11 + 98/19*q^12 - 204/19*q^13 + 2290/19*q^14 - 3665/19*q^15 - 592/19*q^16 + 2007/19*q^17 + 5575/19*q^18 - 3576/19*q^19 + O(q^20),
q^10 - 54/19*q^11 - 9/19*q^12 + 4/19*q^13 + 597/19*q^14 - 994/19*q^15 - 143/19*q^16 + 534/19*q^17 + 1420/19*q^18 - 900/19*q^19 + O(q^20)
]

S = M.cuspidal_subspace()

S.basis()

[
q - 4*q^3 - q^4 + 3*q^5 + 6*q^6 - 3*q^8 + q^9 - 6*q^10 - 2*q^12 + 2*q^13 +
10*q^16 - 3*q^17 - 3*q^18 - 6*q^19 + O(q^20),
q^2 - 2*q^3 - q^4 + 2*q^5 + 2*q^6 - 2*q^8 + q^9 - 3*q^10 + 3*q^13 - 2*q^15 +
5*q^16 - 3*q^17 - 2*q^18 - 2*q^19 + O(q^20)
]

[
q - 4*q^3 - q^4 + 3*q^5 + 6*q^6 - 3*q^8 + q^9 - 6*q^10 - 2*q^12 + 2*q^13 + 10*q^16 - 3*q^17 - 3*q^18 - 6*q^19 + O(q^20),
q^2 - 2*q^3 - q^4 + 2*q^5 + 2*q^6 - 2*q^8 + q^9 - 3*q^10 + 3*q^13 - 2*q^15 + 5*q^16 - 3*q^17 - 2*q^18 - 2*q^19 + O(q^20)
]

Cutting Out Submodules: Eisenstein, Cuspidal, New, Old

M = ModularForms(Gamma0(54)); M

Modular Forms space of dimension 15 for Congruence Subgroup Gamma0(54) of weight
2 over Rational Field

Modular Forms space of dimension 15 for Congruence Subgroup Gamma0(54) of weight 2 over Rational Field

M.eisenstein_subspace()

Eisenstein subspace of dimension 11 of Modular Forms space of dimension 15 for
Congruence Subgroup Gamma0(54) of weight 2 over Rational Field

Eisenstein subspace of dimension 11 of Modular Forms space of dimension 15 for Congruence Subgroup Gamma0(54) of weight 2 over Rational Field

M.cuspidal_subspace()

Cuspidal subspace of dimension 4 of Modular Forms space of dimension 15 for
Congruence Subgroup Gamma0(54) of weight 2 over Rational Field

Cuspidal subspace of dimension 4 of Modular Forms space of dimension 15 for Congruence Subgroup Gamma0(54) of weight 2 over Rational Field

n = M.new_subspace(); n

Modular Forms subspace of dimension 4 of Modular Forms space of dimension 15 for
Congruence Subgroup Gamma0(54) of weight 2 over Rational Field

Modular Forms subspace of dimension 4 of Modular Forms space of dimension 15 for Congruence Subgroup Gamma0(54) of weight 2 over Rational Field

Newforms

Not implemented in SAGE yet. But can do with modular symbols (next slide).

Numerical Newforms

Jon Hanke and I wrote this at the last Arizona Winter School

n = numerical_eigenforms(23); n

Numerical Hecke eigenvalues for Congruence Subgroup Gamma0(23) of weight 2

Numerical Hecke eigenvalues for Congruence Subgroup Gamma0(23) of weight 2

time n.systems_of_eigenvalues(10)

[
[-1.61803398875, 2.2360679775, -3.2360679775, -1.2360679775],
[0.61803398875, -2.2360679775, 1.2360679775, 3.2360679775],
[3.0, 4.0, 6.0, 8.0]
]
CPU time: 0.03 s,  Wall time: 0.35 s

[
[-1.61803398875, 2.2360679775, -3.2360679775, -1.2360679775],
[0.61803398875, -2.2360679775, 1.2360679775, 3.2360679775],
[3.0, 4.0, 6.0, 8.0]
]
CPU time: 0.03 s,  Wall time: 0.35 s

time n = numerical_eigenforms(5077); n

Numerical Hecke eigenvalues for Congruence Subgroup Gamma0(5077) of weight 2
CPU time: 0.00 s,  Wall time: 0.00 s

Numerical Hecke eigenvalues for Congruence Subgroup Gamma0(5077) of weight 2
CPU time: 0.00 s,  Wall time: 0.00 s

time v = n.systems_of_eigenvalues(5)

CPU time: 12.62 s,  Wall time: 17.53 s

CPU time: 12.62 s,  Wall time: 17.53 s

len(v)

[
[-2.81633052252 + 1.89736766871e-16*I, -3.09812140794 + 3.79724406674e-16*I],
[-2.81173619731 + 1.13637238565e-16*I, -2.29295611838 + 4.35265891993e-16*I],
[-2.78059895058 + 8.30750845558e-17*I, 0.0708989774359 - 3.23606417627e-16*I],
[-2.77683512072 + 3.41777278382e-16*I, 2.42900843479 - 8.3645069403e-17*I],
[-2.77000668171 - 3.18586165736e-16*I, -0.453814884692 + 8.67093310421e-17*I],
[-2.75098907972 + 3.65596841088e-16*I, 0.880999648227 + 9.80569116184e-17*I],
[-2.74765329171 + 5.81903177775e-18*I, 1.7968889597 + 8.86987623814e-17*I],
[-2.72893048103 + 3.21675817418e-16*I, 1.44573121615 - 5.10604823232e-17*I],
[-2.71480605647 + 1.66916315359e-16*I, 1.77106552601 - 7.13891853498e-16*I],
[-2.71273384616 - 1.90166897426e-16*I, -1.56060940103 + 5.91479787978e-16*I],
[-2.70901336297 + 8.15777725248e-16*I, 2.53540347892 + 3.45814569298e-16*I],
[-2.68287722463 - 4.00499161783e-16*I, 3.14005787872 - 5.57887777961e-15*I],
[-2.66174229747 + 3.00423351958e-16*I, -1.61696460545 + 3.00412610409e-16*I],
[-2.65977747495 + 2.11573347183e-16*I, 0.381673602422 - 4.2056230512e-18*I],
[-2.65566171379 + 5.14733208106e-17*I, -0.0062129859614 + 5.48801255179e-16*I],
[-2.65348266346 + 2.30644971061e-16*I, -0.258722740207 + 3.36723951946e-16*I],
[-2.63505518173 + 8.98306402081e-17*I, -2.68532007899 + 7.3208517693e-16*I],
[-2.63322403587 + 5.37649782699e-29*I, -3.42055050855 - 1.95800770707e-28*I],
[-2.62320319425 + 2.58498055779e-16*I, -1.94038121344 + 3.77344051025e-16*I],
[-2.61106616068 + 4.03119861071e-16*I, -0.462954048383 + 1.57334467245e-16*I],
[-2.60586482991 - 3.52367631513e-17*I, -2.49080849205 + 3.87264039519e-16*I],
[-2.60140280963 + 3.92818454892e-16*I, 3.18806665251 + 5.85395912464e-17*I],
[-2.58891316215 - 2.88983064295e-15*I, 3.03601424595 - 1.7623405329e-15*I],
[-2.58761752277 + 2.15772975285e-17*I, 0.766218993791 + 1.28295199699e-16*I],
[-2.54247577065 + 3.73975014015e-16*I, 2.75173699257 - 2.21282569308e-16*I],
[-2.52071434403 + 9.47818215278e-17*I, -0.980646564103 + 2.6965671829e-16*I],
[-2.50571925273 + 2.90446714962e-16*I, 0.31357006254 + 2.01529439686e-17*I],
[-2.49199341486 + 9.65330949652e-19*I, 1.36066717585 + 2.0246449024e-17*I],
[-2.48492243208 + 1.00422599011e-16*I, -1.01672191379 + 1.1221921138e-15*I],
[-2.48214548341 - 2.84763360179e-16*I, -2.1841996205 - 2.3502871757e-16*I],
[-2.47319189548 + 2.99582433118e-16*I, -1.27283191477 + 2.89073318964e-16*I],
[-2.46209700631 - 2.54695742524e-16*I, -1.32035761826 - 5.334925691e-17*I],
[-2.46114369382 - 2.81554907691e-16*I, 2.54758298437 + 4.17874365616e-16*I],
[-2.46112281537 - 1.13938507308e-17*I, -3.25744921526 - 2.10110742169e-16*I],
[-2.46104836129 + 4.06266960343e-14*I, -2.68177080965 + 1.55832240661e-16*I],
[-2.45990149172 + 1.96098407487e-17*I, 1.58814672819 + 9.62954558376e-17*I],
[-2.45125543618 + 1.97616812962e-16*I, 0.78055724384 - 1.60389408929e-17*I],
[-2.4355920403 - 6.53526417063e-16*I, 3.36409962223 - 3.86175652777e-16*I],
[-2.43502272649 + 1.92192301498e-16*I, 2.39260931227 - 2.19062704832e-16*I],
[-2.42778047743 + 8.32653233978e-16*I, -2.9047201583 + 8.92877014338e-16*I],
[-2.42574471381 + 3.59861305115e-17*I, -1.56213086811 + 2.34669021656e-16*I],
[-2.42052846938 - 4.49370366763e-18*I, 2.46085583516 - 3.31056195653e-16*I],
[-2.39213731273 + 4.69803793752e-17*I, -1.70029779954 + 3.2241482347e-16*I],
[-2.38830367541 + 3.50494722116e-16*I, -1.75089303916 - 2.19197434945e-15*I],
[-2.37963368604 + 2.44289752357e-16*I, 1.69226962807 + 1.95267626208e-16*I],
[-2.3761757314 + 2.59026968674e-16*I, -2.47113014429 + 2.3979380101e-16*I],
[-2.36142007241 + 2.39538932285e-16*I, -0.634334854137 - 2.21562689974e-17*I],
[-2.32461527546 + 2.56240232106e-16*I, 0.383241805373 + 6.31323042638e-17*I],
[-2.32182772033 - 2.30181101959e-16*I, 1.31865283886 + 3.62519500132e-16*I],
[-2.31115080111 - 2.64162413798e-16*I, -0.0637202318872 + 1.27065310606e-16*I],
[-2.26949552547 + 2.5968588986e-16*I, -0.426411160791 + 1.17949573149e-16*I],
[-2.26893795418 + 2.77855583511e-16*I, -3.1907564852 + 3.90742066527e-16*I],
[-2.26451044952 + 8.14989897032e-18*I, 2.7761273308 - 1.81302503337e-16*I],
[-2.23097207875 - 8.54299320783e-17*I, 2.71709855484 - 6.36562320239e-17*I],
[-2.22561757658 + 2.98728859283e-17*I, 1.33552046572 + 2.16186218344e-17*I],
[-2.21532639395 - 7.03027868794e-16*I, 1.18268933734 + 4.1557364891e-15*I],
[-2.20903059724 + 3.29169249349e-17*I, 0.731629713922 + 1.7765997453e-16*I],
[-2.17895915938 + 3.66870344136e-16*I, -1.49311796926 + 6.75433334913e-15*I],
[-2.17858178396 - 1.01983965266e-16*I, 0.857575960376 + 4.44177326199e-16*I],
[-2.17778306399 + 2.97903133611e-16*I, -2.87639846723 + 4.72295364332e-16*I],
[-2.14131869717 - 2.21830272298e-16*I, 0.269273046076 + 7.18935982612e-16*I],
[-2.12184206019 + 2.78012908634e-16*I, 1.18025165812 + 2.70325279537e-17*I],
[-2.0895497735 + 1.00977702995e-17*I, -0.710981893439 + 1.10396649515e-16*I],
[-2.08306054943 + 7.1417808878e-16*I, -1.91349767824 - 2.13698137174e-15*I],
[-2.0806373044 - 7.95899029239e-17*I, -2.69926173561 - 4.17528414608e-16*I],
[-2.07910895974 - 1.65835928489e-21*I, -1.49426546337 - 1.01303259544e-15*I],
[-2.07709647034 - 2.11370288423e-16*I, -3.15035495569 - 1.08349532007e-15*I],
[-2.06366402736 + 1.45356111935e-17*I, 1.97433010288 - 8.12037967909e-17*I],
[-2.05628359423 - 6.55310332547e-16*I, -0.386448204619 + 4.34781370075e-15*I],
[-2.05092069612 + 2.23392849494e-16*I, -0.896788321741 + 2.85289067088e-16*I],
[-2.05026134909 + 2.9756988939e-16*I, 2.79926610226 - 2.90524004368e-16*I],
[-2.04898509589 - 1.96704497991e-16*I, -3.12953078829 - 2.60573862959e-16*I],
[-2.03701176355 - 1.24111333052e-16*I, 2.66802565266 - 2.45060086573e-16*I],
[-2.0187224494 - 1.74125938161e-16*I, -0.450815314676 + 1.98294834428e-16*I],
[-2.00823353583 - 2.15364456545e-16*I, 2.56690564518 + 4.44721890948e-16*I],
[-2.00614264144 + 2.08283964651e-16*I, 0.998741941826 + 2.07754245106e-17*I],
[-2.0 + 2.43846210326e-16*I, -3.0 + 3.25056741177e-16*I],
[-1.99870575955 - 1.9968326588e-18*I, 2.01951718689 - 1.60948283638e-17*I],
[-1.99459817941 - 1.30624476471e-16*I, -0.160953435418 + 2.38642605013e-16*I],
[-1.97222550908 + 4.32699549135e-16*I, -1.30684721618 + 3.79100036455e-15*I],
[-1.96942631012 - 4.25634944826e-18*I, -2.05248578624 - 3.61411279791e-16*I],
[-1.95695190588 - 3.53723882489e-17*I, 2.46660960993 - 1.74560241254e-16*I],
[-1.87444460541 - 1.21324596997e-18*I, -1.43394678767 + 8.68991491306e-18*I],
[-1.85343551 + 1.27422818391e-16*I, -0.654538736028 + 3.63938818219e-16*I],
[-1.84746660312 - 4.86295554714e-16*I, -2.66538574934 - 1.29438343695e-15*I],
[-1.83947355913 + 1.93482290961e-16*I, -0.447528045194 + 1.40574472348e-16*I],
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[3.0, 4.0]
]

EllipticCurve('5077a').aplist(5)

[-2, -3, -4]

[-2, -3, -4]