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David-Berghaus / q-expansion

Code to compute modular forms on noncongruence subgroups
GNU General Public License v3.0
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Code to compute Modular Forms and Eisenstein Series on Noncongruence Subgroups of PSL(2,ℤ)

This repository hosts code that has been developed for my PhD-thesis. For further information, see arXiv:2207.13365, Noncongruence Database.

Dependencies

The code has been developed using SageMath version 9.2 and has also been tested to work on version 9.5 (other versions of Sage might or might not work) as well as Psage. For the case that you cannot/do not want to install the later, we have also copied the required files into the folder psage to make this repository "self sufficient".

Installation

To install the code run

sage setup.py develop

Once the compilation is finished, go to the folder unit_tests and run

sage: load("unit_tests.sage")
sage: run_unit_tests()

to make sure that everything works correctly.

Example usage

Belyi Maps

Let G be a genus zero subgroup

sage: from psage.modform.arithgroup.mysubgroup import MySubgroup
sage: G = MySubgroup(o2='(1 2)(3 4)(5)(6 7)',o3='(1)(2 3 5)(4 6 7)')

To compute the associated Belyi map we run

sage: from classes.belyi_map import BelyiMap
sage: B = BelyiMap(G)
sage: B
(x + 14*u)(x^2 + ((2*v - 10)*u)*x + (184/3*v - 164/3)*u^2)^3 / (x + (6*v - 16)*u)

The Belyi map object contains a variety of functions and parameters. For example, to determine the expressions of v and u from the above example we run

sage: B._Kv
Number Field in v with defining polynomial x^2 - x + 1 with v = 0.50000000000000000? - 0.866025403784439?*I
sage: B.get_u_str()
'(125248356/96889010407*v - 199546416/96889010407)^(1/6)'

Because G is a genus zero subgroup, we can compute the Fourier expansion of its Hauptmodul

sage: j_G = B.get_hauptmodul_q_expansion(15)
sage: j_G
Hauptmodul of weight 0 with leading order expansion at infinity given by:
q_6^-1 + ((-80/3*v + 148/3)*u^2)*q_6 + ((-220*v + 64)*u^3)*q_6^2 + ((464*v + 992/3)*u^4)*q_6^3 + ((10928/3*v - 4576/3)*u^5)*q_6^4 + ((-875839/243*v - 486421/243)*u^6)*q_6^5 + ((5312*v + 48448/3)*u^7)*q_6^6 + ((-3552116/81*v - 40255396/243)*u^8)*q_6^7 + ((196309792/243*v - 143118584/243)*u^9)*q_6^8 + ((-786407200/243*v + 338569424/243)*u^10)*q_6^9 + O(q_6^10)

Fourier expansions are returned as instances of the FourierExpansion class. By default, the expansions up to 10-th order are printed. We can access the full expansion via

sage: j_G.get_cusp_expansion(Cusp(1,0))
q_6^-1 + (-1937780208140/93936267*w^8 + 2647164/386569*w^2)*q_6 + (-5328895572385/31312089*w^9 - 110754064/386569*w^3)*q_6^2 + (11239125207212/31312089*w^10 + 1240786976/1159707*w^4)*q_6^3 + (264700776431924/93936267*w^11 + 1653813536/386569*w^5)*q_6^4 + (52706752/386569*w^6 + 3503356/386569)*q_6^5 + (15074131072/1159707*w^7 - 5163264/386569*w)*q_6^6 + (-13117827734704/93936267*w^8 + 42625392/386569*w^2)*q_6^7 + (-33447329283592/31312089*w^9 - 785239168/386569*w^3)*q_6^8 + (34579490081584/10437363*w^10 + 3145628800/386569*w^4)*q_6^9 + (1499731150802752/93936267*w^11 + 9987105280/386569*w^5)*q_6^10 + (-151696640/386569*w^6 + 18143414/386569)*q_6^11 + (22798246912/386569*w^7 - 
28366848/386569*w)*q_6^12 + (-67436944554616/93936267*w^8 + 171935448/386569*w^2)*q_6^13 + (-155884848364196/31312089*w^9 - 3547563584/386569*w^3)*q_6^14 + O(q_6^15)

Moreover, the Belyi map can be used to compute Fourier expansions of spaces of cusp forms and modular forms rigorously. For this we run (the first parameter corresponds to the weight while the second parameter denotes the truncation order)

sage: B.get_cuspforms(4,15)
[CuspForm of weight 4 with leading order expansion at infinity given by:
 q_6 + ((4*v - 6)*u)*q_6^2 + ((28/3*v - 140/3)*u^2)*q_6^3 + ((-84*v + 224)*u^3)*q_6^4 + ((-672*v - 112/3)*u^4)*q_6^5 + ((3776/3*v - 400/3)*u^5)*q_6^6 + ((-102701/243*v - 27791/243)*u^6)*q_6^7 + ((-1054412/243*v + 7219348/243)*u^7)*q_6^8 + ((538636/27*v - 1477280/81)*u^8)*q_6^9 + O(q_6^10)])

sage: B.get_modforms(4,15)
[ModForm of weight 4 with leading order expansion at infinity given by:
 1 + 240*q_6^6 + O(q_6^10),
 ModForm of weight 4 with leading order expansion at infinity given by:
 q_6 + ((-104/3*v + 16/3)*u^2)*q_6^3 + ((-380*v + 360)*u^3)*q_6^4 + ((-2432*v + 1424/3)*u^4)*q_6^5 + ((-15904/3*v - 5152/3)*u^5)*q_6^6 + ((-4550573/243*v - 6201935/243)*u^6)*q_6^7 + ((-47168/3*v - 355072/3)*u^7)*q_6^8 + ((2876464/9*v - 58380968/81)*u^8)*q_6^9 + O(q_6^10),
 ModForm of weight 4 with leading order expansion at infinity given by:
 q_6^2 + ((-2*v + 10)*u)*q_6^3 + ((-44*v + 52)*u^2)*q_6^4 + ((-304*v + 288)*u^3)*q_6^5 + ((-1632*v + 824)*u^4)*q_6^6 + ((-7552*v + 800)*u^5)*q_6^7 + ((-5732782/243*v - 2174842/243)*u^6)*q_6^8 + ((-325322/9*v - 2510672/27)*u^7)*q_6^9 + O(q_6^10)]

If we are interested in computing Fourier expansions using floating-point numbers with rigorous error bounds, we specify the additional parameter digit_prec

sage: B.get_cuspforms(4,10,digit_prec=100)
[CuspForm of weight 4 with leading order expansion at infinity given by:
 1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000*q_6 + ([0.40612586902423187183335550026487686820483221051224283744087953569362805591694546003920748787847540 +/- 4.83e-99] + [1.8008094153146945892280356287437496429473684304866141043864220502882601089687405949482072680314734 +/- 3.36e-98]*I)*q_6^2 + ([-0.57225269974370032162829055456220172105456532610557227329887099527821812923086466881592822071445 +/- 3.71e-96] + [-5.17403144371494882322435101719851179904582529787733639632235041741687734522376882410268813211592 +/- 5.20e-96]*I)*q_6^3 + ([5.4576579687657888278877119045753255017920315739170005707736919028138204813423995872513929184125 +/- 8.37e-95] + [-6.2828063634869774428704267575373877394923210587696171244007797432784081535418607599470865524615 +/- 3.11e-95]*I)*q_6^4 + ([-0.309068825274934491197609857874297334630224430653008054218849916216846626768493851407892260752 +/- 7.57e-94] + [-10.237445123663256040609619379942973631040702980202373806268081493946088592298259509418266439239 +/- 7.96e-94]*I)*q_6^5 + ([2.7571289971805312209164723044834801503807421684934001520786426794791312225744810264040245499 +/- 3.00e-92] + [-5.5407910114108948034342876705765995745842409826484591193637050689354290971474137146681017555 +/- 3.61e-92]*I)*q_6^6 + ([0.870011644807860670298940067488886433373849331486030480497071798315327797091347992758119491 +/- 1.23e-91] + [-0.152639655015122268606554783030309519846858068249675104350080381158850831875412785356928862 +/- 2.85e-91]*I)*q_6^7 + ([2.26566749722255701948458971144637187642563486125791422815896227471733802372053099487069485 +/- 1.57e-90] + [17.33470003266323465504954059357681226192948590462371810751914208942639532198253465082752925 +/- 3.45e-90]*I)*q_6^8 + ([-4.1900260618523146392424799564699111199182023563424398414215427932180154369708621521746875 +/- 3.67e-89] + [0.3209425108888763297411129942038273585424564998133946104228282854323256367001945505625534 +/- 2.70e-89]*I)*q_6^9 + O(q_6^10)]

Hejhal's method

To compute Fourier expansions of modular forms numerically, using an optimized version of Hejhal's method, we can run

sage: from psage.modform.maass.automorphic_forms import AutomorphicFormSpace
sage: from classes.fourier_expansion import get_hauptmodul_q_expansion_approx, get_cuspform_basis_approx, get_modform_basis_approx

sage: digit_prec = 50 #Compute coefficients to about 50 digits precision

sage: j_G = get_hauptmodul_q_expansion_approx(AutomorphicFormSpace(G,0),digit_prec)
sage: j_G
Hauptmodul of weight 0 with leading order expansion at infinity given by:
1.000000000000000000000000000000000000000000000000*q_6^-1 + 0.0000000000000000000000000000000000000000000000000 + (-1.388983637482258528107138076635774326727017422309 + 5.016851501461998136341128266268848850335043646534*I)*q_6 + (6.998031044495970114687963449848038883098420993437 + 4.504136743636433523935354884887510289659450352209*I)*q_6^2 + (-3.512673610933919578738824203927641677521862598476 + 9.620911095204126577534732242676748912025407787146*I)*q_6^3 + (2.094356720009019678564632648311143927361526949870 - 16.23972128159629096243530616166816650495984318081*I)*q_6^4 + (8.870011644807860670298940067488886433373885562314 - 0.1526396550151222686065547830303095198469284518880*I)*q_6^5 + (5.896478618157735372967788703995928301000169284024 + 10.65387003159377190681326352327856535598346781938*I)*q_6^6 + (-7.102011599173434322441160409328590357785970189447 + 41.39193782819187367845381988814291456296782534338*I)*q_6^7 + (40.67104315040637664617150784384462137410245416033 + 37.61506266854387755161499630632108573866992332052*I)*q_6^8 + (-11.18048794365418621081119907300687097511058015156 + 74.25239079134411962653170377372871107862602489451*I)*q_6^9 + O(q_6^10)

sage: cuspforms = get_cuspform_basis_approx(AutomorphicFormSpace(G,4),digit_prec)
sage: cuspforms
[CuspForm of weight 4 with leading order expansion at infinity given by:
 0.0000000000000000000000000000000000000000000000000 + 1.000000000000000000000000000000000000000000000000*q_6 + (0.4061258690242318718333555002648768682048322105118 + 1.800809415314694589228035628743749642947368430484*I)*q_6^2 + (-0.5722526997437003216282905545622017210545653261049 - 5.174031443714948823224351017198511799045825297871*I)*q_6^3 + (5.457657968765788827887711904575325501792031573910 - 6.282806363486977442870426757537387739492321058763*I)*q_6^4 + (-0.3090688252749344911976098578742973346302244306591 - 10.23744512366325604060961937994297363104070298019*I)*q_6^5 + (2.757128997180531220916472304483480150380742168491 - 5.540791011410894803434287670576599574584240982661*I)*q_6^6 + (0.8700116448078606702989400674888864333738493314920 - 0.1526396550151222686065547830303095198468580682568*I)*q_6^7 + (2.265667497222557019484589711446371876425634861251 + 17.33470003266323465504954059357681226192948590460*I)*q_6^8 + (-4.190026061852314639242479956469911119918202356384 + 0.3209425108888763297411129942038273585424564997926*I)*q_6^9 + O(q_6^10)]

 sage: modforms = get_modform_basis_approx(AutomorphicFormSpace(G,4),digit_prec)
 sage: modforms
[ModForm of weight 4 with leading order expansion at infinity given by:
 1.000000000000000000000000000000000000000000000000 + 0.0000000000000000000000000000000000000000000000000*q_6 + 0.0000000000000000000000000000000000000000000000000*q_6^2 + (5.173867968909958021375225882611907408025101824397e-48 - 7.970323278073582313339687657733545079142527762251e-49*I)*q_6^3 + (1.521600339715448373107679223977448577684990119567e-48 - 1.690006571972357747090536448860321771465182930549e-47*I)*q_6^4 + (-2.481823997698783086166639991159310772652454396656e-47 + 8.824315575114062602244189376941089893248108439657e-48*I)*q_6^5 + (239.9999999999999999999999999999999999999999999997 + 2.710574002751411907670920075732540763231046905499e-49*I)*q_6^6 + (-2.496990767074187374580873494609326250935329826270e-48 - 9.332762177652844612402267175999807446596701979134e-47*I)*q_6^7 + (-6.223461591778605168274369519418530624655598066330e-47 + 1.816222685565602054010243890294696748061962944949e-47*I)*q_6^8 + (5.721009063778472937971074082829849712664081280362e-47 + 1.110029227581787666878157992006010749856859285721e-45*I)*q_6^9 + O(q_6^10),
 ModForm of weight 4 with leading order expansion at infinity given by:
 0.0000000000000000000000000000000000000000000000000 + 1.000000000000000000000000000000000000000000000000*q_6 + 0.0000000000000000000000000000000000000000000000000*q_6^2 + (-3.922472674451917699470857262395952095563173685918 - 0.3143598845059013737664455018593258974215676047610*I)*q_6^3 + (15.56961126657719867821959419302920548111307097088 
- 2.223337024813179898668839840812346812291088352924*I)*q_6^4 + (-8.570691348242511613465697697226770028194863625042 - 31.94540342790802704797863241436137033115271776987*I)*q_6^5 + (-23.38257653178727285203545774821251364908427233504 + 22.92846755091636610947226438242966273592271078635*I)*q_6^6 + (67.45591183459758628268347857974629132905021352851 + 16.13619210159863982412150563463272066952499578644*I)*q_6^7 + (-29.10058892204735348602258852954910002606495774264 - 74.35665548155587899612687971664107082944201905954*I)*q_6^8 + (-101.1055988790414513948803130837117586125454121268 + 92.82761528176435592570644430612204498481043277230*I)*q_6^9 + O(q_6^10),
 ModForm of weight 4 with leading order expansion at infinity given by:
 0.0000000000000000000000000000000000000000000000000 + 0.0000000000000000000000000000000000000000000000000*q_6 + 1.000000000000000000000000000000000000000000000000*q_6^2 + (-2.168735504573075062036992828741925592016852219463 - 2.349498803263025433159017870009486907820436436223*I)*q_6^3 + (-3.350219974708217377842566707833750374508608359810 
+ 4.859671559209047449457905515339185901624257693116*I)*q_6^4 + (12.45568901326175894257567535442336438489045677671 - 1.778669619850543918935071872649877449832870682352*I)*q_6^5 + (-11.92883054653896394660571697221726303840162214564 - 17.20576977087227245013909048171313460360696771426*I)*q_6^6 + (-16.54277398308318732549883382690088090228445301094 + 33.24474606846536882060572602345959744750544589586*I)*q_6^7 + (52.19043085789084480106078249708879803483242526495 - 5.647667235559523938442526972121452234333748525241*I)*q_6^8 + (-37.33346907663913873807894495454271160265929787356 - 62.23738027826932901253870681918951924693253664783*I)*q_6^9 + O(q_6^10)]

Eisenstein series

To compute Eisenstein series numerically we run

sage: from eisenstein.eisenstein_computation import compute_eisenstein_series
sage: eisenstein_series = compute_eisenstein_series(cuspforms,modforms)
sage: eisenstein_series
[ModForm of weight 4 with leading order expansion at infinity given by:
 1.000000000000000000000000000000000000000000000000 + 0.0000000000000000000000000000000000000000000000000*q_6 + 0.0000000000000000000000000000000000000000000000000*q_6^2 + (5.173867968909958021375225882611907408025101824397e-48 - 7.970323278073582313339687657733545079142527762251e-49*I)*q_6^3 + (1.521600339715448373107679223977448577684990119567e-48 - 1.690006571972357747090536448860321771465182930549e-47*I)*q_6^4 + (-2.481823997698783086166639991159310772652454396656e-47 + 8.824315575114062602244189376941089893248108439657e-48*I)*q_6^5 + (239.9999999999999999999999999999999999999999999997 + 2.710574002751411907670920075732540763231046905499e-49*I)*q_6^6 + (-2.496990767074187374580873494609326250935329826270e-48 - 9.332762177652844612402267175999807446596701979134e-47*I)*q_6^7 + (-6.223461591778605168274369519418530624655598066330e-47 + 1.816222685565602054010243890294696748061962944949e-47*I)*q_6^8 + (5.721009063778472937971074082829849712664081280362e-47 + 1.110029227581787666878157992006010749856859285721e-45*I)*q_6^9 + O(q_6^10), 
 ModForm of weight 4 with leading order expansion at infinity given by:
 0.0000000000000000000000000000000000000000000000000 + 1.000000000000000000000000000000000000000000000000*q_6 + (-3.959109788096710882221472451439743149088870430971 - 7.795413438144329803802188079360076192337431485404*I)*q_6^2 + (-13.65152525436440654764120628976826327472286261639 + 25.89375372074392402064795167715072324903597684807*I)*q_6^3 + (66.71664893834763385362532291318665737264181200048 + 4.653039549767070150131102051030926738554270863798*I)*q_6^4 + (-71.74959669473780057680001369295817005615242718986 - 122.0007006414948858372588629873477037325319571613*I)*q_6^5 + (-110.2811156399130224563304862496987597331715929018 + 184.0381650063516921508005160942826614490270642449*I)*q_6^6 + (392.1071104831046577268891085343816117261130389296 + 13.47435515105882973830847314756741893350040983067*I)*q_6^7 + (-279.7541356387557678479425397874621868205459842135 - 458.8429069014838697794072043238070499869638942437*I)*q_6^8 + (-438.4644066102331779945844497978370662214171744273 + 630.2620632595361783359588630807033599555620303355*I)*q_6^9 + O(q_6^10)]

These results are normalized to a reduced row echelon basis. To obtain a more canonical normalization for which the leading order coefficients are 1 for one cusp and zero at the others we can run

sage: from eisenstein.eisenstein_computation import echelon_basis_to_eisenstein_basis
sage: echelon_basis_to_eisenstein_basis(eisenstein_series)
[ModForm of weight 4 with leading order expansion at infinity given by:
 1.000000000000000000000000000000000000000000000000 + (0.09600261606478410651446132739708791352637943378200 + 0.1600435981787040832542649659356889835823386710947*I)*q_6 + (0.8675211189862639012472634923855411991140381793971 - 1.382010259439953007974070885363760895325463350190*I)*q_6^2 + (-5.454711653514514669605066818403045202708050477885 + 0.3010288745927257385684503677435384241093328711083*I)*q_6^3 + (5.660283641144634922635684610893993488920887039381 + 11.12427652394911701792105552660416447074543118740*I)*q_6^4 + (12.63728212669975718840986391144647481930810247703 - 23.19545004621678474830962404877918535941417230443*I)*q_6^5 + (199.9585942662027830631160760894444598037093026086 + 0.01835873819872895534684551182952741287722945056718*I)*q_6^6 + (35.48682410246816884848043828010572232945921714869 + 64.04780617745837623362792493106383419497741385541*I)*q_6^7 + (46.57774094302660467443812829326969555762071980609 - 88.82297789831234121028781356031453273507262880430*I)*q_6^8 + (-142.9631384854657842949046435700282511408229961404 - 9.666614427888156543010183076143852706846395848912*I)*q_6^9 + O(q_6^10),
 ModForm of weight 4 with leading order expansion at infinity given by:
 0.0000000000000000000000000000000000000000000000000 + (-3.456094178332227834520607786295164886949659616155 - 5.761569534433346997153538773684803408964192159400*I)*q_6 + (-31.23076028350550044490148572587948316810537445821 + 49.75236933983830828706655187309539223171668060684*I)*q_6^2 + (196.3696195265225281057824054625096272974898172039 - 10.83703948533812658846421323876738326793598336013*I)*q_6^3 + (-203.7702110812068572148846459921837656011519334179 - 400.4739548621682126451579989577499209468355227462*I)*q_6^4 + (-454.9421565611912587827551008120730934950916891726 + 835.0362016638042509391464657560506729389102029594*I)*q_6^5 + (1441.490606416699809727821260779999447066465106076 - 0.6609145751542423924864384258629868635802602219901*I)*q_6^6 + (-1277.525667688854078545295778083806003860531817354 - 2305.721022388501544410605297518298031019186898795*I)*q_6^7 + (-1676.798673948957768279772618557709040074345913016 + 3197.627204339244283570361288171323178462614636954*I)*q_6^8 + (5146.672985476768234616567168521017041069627861052 + 347.9981194039736355483665907411786974464702505948*I)*q_6^9 + O(q_6^10)]

Computing Passports (experimental)

Functions for computing full passports are implemented in passports.sage. For example to compute the passport 7_0_2_1_1_0_a with 50 Fourier expansion terms, Eisenstein series with 100 digits precision, up to weight 6, run

sage: from psage.modform.arithgroup.mysubgroup import MySubgroup
sage: passport_els = [MySubgroup(o2="(1 6)(2)(3 4)(5 7)",o3="(1 7 6)(2 3 5)(4)"),MySubgroup(o2="(1 4)(2)(3 5)(6 7)",o3="(1 5 4)(2 3 6)(7)")]    
sage: load("passports.sage")
sage: res = compute_passport_data_genus_zero(passport_els,50,100,6)