jsMath

# 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)`
 `423` `423`
`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 - 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`EllipticCurve('5077a').aplist(5)`
 `[-2, -3, -4]` `[-2, -3, -4]`