/* This code can be loaded, or copied and paste using cpaste, into Sage. It will load the data associated to the HMF, including the field, level, and Hecke and Atkin-Lehner eigenvalue data. */ P. = PolynomialRing(QQ) g = P([-98, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([9, 3, 3]) primes_array = [ [2, 2, -17*w - 160],\ [2, 2, -17*w + 177],\ [3, 3, -842*w + 8767],\ [7, 7, -2*w + 21],\ [7, 7, 2*w + 19],\ [13, 13, -12*w - 113],\ [13, 13, 12*w - 125],\ [17, 17, 182*w - 1895],\ [17, 17, 182*w + 1713],\ [23, 23, -512*w - 4819],\ [23, 23, 512*w - 5331],\ [25, 5, -5],\ [29, 29, 22*w - 229],\ [29, 29, 22*w + 207],\ [43, 43, 114*w + 1073],\ [43, 43, 114*w - 1187],\ [47, 47, 8*w - 83],\ [47, 47, -8*w - 75],\ [61, 61, -1172*w - 11031],\ [61, 61, 1172*w - 12203],\ [71, 71, 216*w + 2033],\ [71, 71, 216*w - 2249],\ [83, 83, 8932*w - 93001],\ [83, 83, -2196*w + 22865],\ [109, 109, -4*w - 39],\ [109, 109, 4*w - 43],\ [121, 11, -11],\ [131, 131, 5564*w - 57933],\ [137, 137, 2*w - 17],\ [137, 137, -2*w - 15],\ [149, 149, 18*w - 187],\ [149, 149, 18*w + 169],\ [151, 151, -1502*w - 14137],\ [151, 151, -1502*w + 15639],\ [173, 173, -6*w - 55],\ [173, 173, 6*w - 61],\ [193, 193, 808*w - 8413],\ [193, 193, -808*w - 7605],\ [197, 197, 2*w - 15],\ [197, 197, -2*w - 13],\ [211, 211, -70*w + 729],\ [211, 211, -70*w - 659],\ [227, 227, 284*w + 2673],\ [227, 227, 284*w - 2957],\ [251, 251, 124*w - 1291],\ [251, 251, 124*w + 1167],\ [257, 257, 42*w + 395],\ [257, 257, 42*w - 437],\ [271, 271, 410*w + 3859],\ [271, 271, 410*w - 4269],\ [277, 277, 4*w - 45],\ [277, 277, -4*w - 41],\ [281, 281, -3550*w - 33413],\ [281, 281, 3550*w - 36963],\ [283, 283, 2*w - 27],\ [283, 283, -2*w - 25],\ [293, 293, 2*w - 11],\ [293, 293, -2*w - 9],\ [307, 307, -6*w - 59],\ [307, 307, 6*w - 65],\ [337, 337, 376*w - 3915],\ [337, 337, 376*w + 3539],\ [347, 347, -4*w - 33],\ [347, 347, -4*w + 37],\ [359, 359, -8*w - 73],\ [359, 359, -8*w + 81],\ [361, 19, -19],\ [367, 367, -774*w + 8059],\ [367, 367, -774*w - 7285],\ [379, 379, -30*w + 313],\ [379, 379, -30*w - 283],\ [389, 389, 2*w - 3],\ [389, 389, -2*w - 1],\ [397, 397, -172*w + 1791],\ [397, 397, -172*w - 1619],\ [401, 401, 6*w - 59],\ [401, 401, 6*w + 53],\ [409, 409, -40*w - 377],\ [409, 409, 40*w - 417],\ [419, 419, 100*w - 1041],\ [419, 419, -100*w - 941],\ [421, 421, 308*w + 2899],\ [421, 421, 308*w - 3207],\ [439, 439, 274*w - 2853],\ [439, 439, 274*w + 2579],\ [443, 443, -52*w - 489],\ [443, 443, 52*w - 541],\ [449, 449, 15338*w - 159701],\ [449, 449, -6918*w + 72031],\ [457, 457, 17352*w - 180671],\ [457, 457, 6224*w - 64805],\ [461, 461, 614*w - 6393],\ [461, 461, 614*w + 5779],\ [479, 479, 192*w - 1999],\ [479, 479, 192*w + 1807],\ [487, 487, -94*w + 979],\ [487, 487, -94*w - 885],\ [491, 491, -20*w - 187],\ [491, 491, 20*w - 207],\ [503, 503, -16*w + 165],\ [503, 503, 16*w + 149],\ [509, 509, 2890*w - 30091],\ [509, 509, -2890*w - 27201],\ [521, 521, 11970*w - 124633],\ [521, 521, -10286*w + 107099],\ [577, 577, -1104*w - 10391],\ [577, 577, 1104*w - 11495],\ [593, 593, 134*w + 1261],\ [593, 593, 134*w - 1395],\ [601, 601, -8*w - 79],\ [601, 601, 8*w - 87],\ [613, 613, -2492*w - 23455],\ [613, 613, 2492*w - 25947],\ [617, 617, 226*w - 2353],\ [617, 617, 226*w + 2127],\ [631, 631, 2*w - 33],\ [631, 631, -2*w - 31],\ [677, 677, -62*w - 583],\ [677, 677, 62*w - 645],\ [691, 691, 1798*w - 18721],\ [691, 691, -1798*w - 16923],\ [739, 739, 74*w + 697],\ [739, 739, 74*w - 771],\ [743, 743, 48*w - 499],\ [743, 743, 48*w + 451],\ [757, 757, -44*w + 459],\ [757, 757, -44*w - 415],\ [761, 761, 14*w + 129],\ [761, 761, 14*w - 143],\ [769, 769, 16*w + 153],\ [769, 769, 16*w - 169],\ [773, 773, 4574*w - 47625],\ [773, 773, -39938*w + 415839],\ [787, 787, 3846*w + 36199],\ [787, 787, 3846*w - 40045],\ [809, 809, 30*w + 281],\ [809, 809, -30*w + 311],\ [811, 811, 1070*w - 11141],\ [811, 811, -1070*w - 10071],\ [829, 829, -4*w - 47],\ [829, 829, 4*w - 51],\ [857, 857, -18*w + 185],\ [857, 857, 18*w + 167],\ [877, 877, -196*w - 1845],\ [877, 877, -196*w + 2041],\ [881, 881, -26*w - 243],\ [881, 881, 26*w - 269],\ [907, 907, -14*w - 135],\ [907, 907, -14*w + 149],\ [937, 937, 8*w - 89],\ [937, 937, -8*w - 81],\ [941, 941, 294*w - 3061],\ [941, 941, 294*w + 2767],\ [947, 947, 4*w - 27],\ [947, 947, -4*w - 23],\ [961, 31, -31],\ [971, 971, -1604*w + 16701],\ [971, 971, 1604*w + 15097],\ [983, 983, 144*w - 1499],\ [983, 983, 144*w + 1355],\ [991, 991, -10*w - 99],\ [991, 991, 10*w - 109],\ [997, 997, 604*w - 6289],\ [997, 997, 604*w + 5685]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^20 - 34*x^18 + 483*x^16 - 3726*x^14 + 16964*x^12 - 46264*x^10 + 72948*x^8 - 60398*x^6 + 21935*x^4 - 3328*x^2 + 169 K. = NumberField(heckePol) hecke_eigenvalues_array = [3559466/120036475*e^19 - 23829038/24007295*e^17 + 1659081283/120036475*e^15 - 12468478199/120036475*e^13 + 54785135053/120036475*e^11 - 141859152367/120036475*e^9 + 41065884588/24007295*e^7 - 10976057561/9233575*e^5 + 30691469683/120036475*e^3 - 87591177/9233575*e, e, 0, 204729/9233575*e^18 - 1273787/1846715*e^16 + 81131502/9233575*e^14 - 548135381/9233575*e^12 + 2130047707/9233575*e^10 - 4840042348/9233575*e^8 + 1257614987/1846715*e^6 - 339362084/710275*e^4 + 1452652052/9233575*e^2 - 10857413/710275, -111599/9233575*e^18 + 585032/1846715*e^16 - 27823537/9233575*e^14 + 100566886/9233575*e^12 + 74696083/9233575*e^10 - 1579646362/9233575*e^8 + 844600893/1846715*e^6 - 310800371/710275*e^4 + 949496663/9233575*e^2 - 2403072/710275, 332064/9233575*e^18 - 2176624/1846715*e^16 + 148042047/9233575*e^14 - 1085190211/9233575*e^12 + 4650808937/9233575*e^10 - 11776114998/9233575*e^8 + 671594821/369343*e^6 - 902575339/710275*e^4 + 2792902002/9233575*e^2 - 10979828/710275, 199604/9233575*e^18 - 1255694/1846715*e^16 + 81068342/9233575*e^14 - 556434271/9233575*e^12 + 2197183207/9233575*e^10 - 5033567728/9233575*e^8 + 254656327/369343*e^6 - 295335879/710275*e^4 + 695008647/9233575*e^2 - 2550008/710275, -972184/120036475*e^19 + 7157353/24007295*e^17 - 550270237/120036475*e^15 + 4580222271/120036475*e^13 - 22347678647/120036475*e^11 + 64582021398/120036475*e^9 - 21185968536/24007295*e^7 + 6766376674/9233575*e^5 - 29090338102/120036475*e^3 + 297929583/9233575*e, -7936442/120036475*e^19 + 10951872/4801459*e^17 - 3944998851/120036475*e^15 + 30793359718/120036475*e^13 - 140999100586/120036475*e^11 + 381626937814/120036475*e^9 - 115959563987/24007295*e^7 + 32967572697/9233575*e^5 - 107158710011/120036475*e^3 + 554893689/9233575*e, -13331251/120036475*e^19 + 18123849/4801459*e^17 - 6426305778/120036475*e^15 + 49376164354/120036475*e^13 - 222862516008/120036475*e^11 + 596903089217/120036475*e^9 - 181089822821/24007295*e^7 + 52602856941/9233575*e^5 - 189934572833/120036475*e^3 + 1208761142/9233575*e, -225649/4801459*e^19 + 39069038/24007295*e^17 - 565340687/24007295*e^15 + 4436327592/24007295*e^13 - 4093510790/4801459*e^11 + 56117809134/24007295*e^9 - 87549996747/24007295*e^7 + 5315528206/1846715*e^5 - 21268879636/24007295*e^3 + 149826441/1846715*e, -195586/9233575*e^18 + 1161731/1846715*e^16 - 68860053/9233575*e^14 + 413607839/9233575*e^12 - 1305407388/9233575*e^10 + 1938234802/9233575*e^8 - 24454542/369343*e^6 - 89388289/710275*e^4 + 754884477/9233575*e^2 - 10403203/710275, -1609961/24007295*e^19 + 54818117/24007295*e^17 - 779313266/24007295*e^15 + 6006771787/24007295*e^13 - 27236049293/24007295*e^11 + 73478164468/24007295*e^9 - 112922936123/24007295*e^7 + 1349307456/369343*e^5 - 26370090612/24007295*e^3 + 200469746/1846715*e, -7324037/120036475*e^19 + 50869917/24007295*e^17 - 3691692601/120036475*e^15 + 29052469988/120036475*e^13 - 134306922146/120036475*e^11 + 368121695209/120036475*e^9 - 22831341100/4801459*e^7 + 33908666512/9233575*e^5 - 126328650316/120036475*e^3 + 799683899/9233575*e, -273168/9233575*e^18 + 1763022/1846715*e^16 - 117554019/9233575*e^14 + 840885037/9233575*e^12 - 3506118019/9233575*e^10 + 8666471661/9233575*e^8 - 2465452001/1846715*e^6 + 709948408/710275*e^4 - 2934256464/9233575*e^2 + 21339201/710275, 664496/9233575*e^18 - 872812/369343*e^16 + 297788688/9233575*e^14 - 2194747784/9233575*e^12 + 9490810393/9233575*e^10 - 24392525557/9233575*e^8 + 7135472371/1846715*e^6 - 2012196711/710275*e^4 + 7039194543/9233575*e^2 - 40711407/710275, 269827/24007295*e^19 - 7653872/24007295*e^17 + 84131084/24007295*e^15 - 436149251/24007295*e^13 + 915306146/24007295*e^11 + 141701094/4801459*e^9 - 6750930607/24007295*e^7 + 848490124/1846715*e^5 - 1229894672/4801459*e^3 + 36475697/1846715*e, -14754581/120036475*e^19 + 98160663/24007295*e^17 - 6792859678/120036475*e^15 + 50794414634/120036475*e^13 - 222692210523/120036475*e^11 + 579003981422/120036475*e^9 - 170807970993/24007295*e^7 + 48553198951/9233575*e^5 - 175150149903/120036475*e^3 + 1156261632/9233575*e, -1178783/9233575*e^18 + 1480428/369343*e^16 - 476569174/9233575*e^14 + 3257226907/9233575*e^12 - 12781135489/9233575*e^10 + 29044045461/9233575*e^8 - 7309934198/1846715*e^6 + 1740864328/710275*e^4 - 5189631414/9233575*e^2 + 32187561/710275, 709539/9233575*e^18 - 4808246/1846715*e^16 + 339653637/9233575*e^14 - 2597749926/9233575*e^12 + 11670720062/9233575*e^10 - 31170337903/9233575*e^8 + 9492185663/1846715*e^6 - 2818369459/710275*e^4 + 10900254647/9233575*e^2 - 69748023/710275, 3309998/120036475*e^19 - 22216347/24007295*e^17 + 1553872534/120036475*e^15 - 11771842457/120036475*e^13 + 52460116459/120036475*e^11 - 139302117621/120036475*e^9 + 42243066401/24007295*e^7 - 12415699013/9233575*e^5 + 45533067254/120036475*e^3 - 270285386/9233575*e, -2656282/120036475*e^19 + 18329592/24007295*e^17 - 1328699811/120036475*e^15 + 10547632993/120036475*e^13 - 50030235731/120036475*e^11 + 144887371224/120036475*e^9 - 10009081265/4801459*e^7 + 18370987432/9233575*e^5 - 106080183151/120036475*e^3 + 1062779539/9233575*e, -29863514/120036475*e^19 + 39952600/4801459*e^17 - 13903803417/120036475*e^15 + 104563096756/120036475*e^13 - 460723716162/120036475*e^11 + 1201714530963/120036475*e^9 - 354226815064/24007295*e^7 + 99616134324/9233575*e^5 - 343301315637/120036475*e^3 + 2069649663/9233575*e, 29896504/120036475*e^19 - 200258538/24007295*e^17 + 13969156697/120036475*e^15 - 105424400076/120036475*e^13 + 467100173032/120036475*e^11 - 1229075735313/120036475*e^9 + 367494942231/24007295*e^7 - 105986963369/9233575*e^5 + 385622691562/120036475*e^3 - 2420375748/9233575*e, 114171/9233575*e^18 - 805124/1846715*e^16 + 59689718/9233575*e^14 - 483928264/9233575*e^12 + 2330833968/9233575*e^10 - 6757910742/9233575*e^8 + 2266956697/1846715*e^6 - 759873476/710275*e^4 + 3584730533/9233575*e^2 - 31781572/710275, 697093/9233575*e^18 - 1002530/369343*e^16 + 376607154/9233575*e^14 - 3063379622/9233575*e^12 + 14582309319/9233575*e^10 - 40877599456/9233575*e^8 + 12811508938/1846715*e^6 - 3754317638/710275*e^4 + 12890641519/9233575*e^2 - 71617506/710275, -125732/1846715*e^18 + 787492/369343*e^16 - 50422416/1846715*e^14 + 341074468/1846715*e^12 - 1314126831/1846715*e^10 + 2895024014/1846715*e^8 - 691801119/369343*e^6 + 150354592/142055*e^4 - 343202376/1846715*e^2 - 513181/142055, -13054326/120036475*e^19 + 88942556/24007295*e^17 - 6325944948/120036475*e^15 + 48790409374/120036475*e^13 - 221338241008/120036475*e^11 + 596982349382/120036475*e^9 - 36592992883/4801459*e^7 + 54024011651/9233575*e^5 - 201134839343/120036475*e^3 + 1203991427/9233575*e, 765276/120036475*e^19 - 1175953/4801459*e^17 + 468128528/120036475*e^15 - 3999659704/120036475*e^13 + 19835744833/120036475*e^11 - 57669305042/120036475*e^9 + 18840671491/24007295*e^7 - 5894006916/9233575*e^5 + 21525518658/120036475*e^3 + 139556258/9233575*e, 31832624/120036475*e^19 - 210958652/24007295*e^17 + 14522842487/120036475*e^15 - 107830122336/120036475*e^13 + 468118350617/120036475*e^11 - 1199836495713/120036475*e^9 + 345927480852/24007295*e^7 - 93933005379/9233575*e^5 + 294215983237/120036475*e^3 - 1424290728/9233575*e, 23291988/120036475*e^19 - 156890872/24007295*e^17 + 11009152229/120036475*e^15 - 83586769917/120036475*e^13 + 372466229104/120036475*e^11 - 985069712001/120036475*e^9 + 295934447346/24007295*e^7 - 85931069528/9233575*e^5 + 318620019049/120036475*e^3 - 1965524941/9233575*e, -19638546/120036475*e^19 + 133367931/24007295*e^17 - 9445419058/120036475*e^15 + 72444177079/120036475*e^13 - 326215594518/120036475*e^11 + 870989973072/120036475*e^9 - 52592807788/4801459*e^7 + 75537259946/9233575*e^5 - 260379402303/120036475*e^3 + 1347952492/9233575*e, -446749/9233575*e^18 + 2153784/1846715*e^16 - 82950252/9233575*e^14 + 55626226/9233575*e^12 + 2495576308/9233575*e^10 - 14030529057/9233575*e^8 + 1267383917/369343*e^6 - 2332072876/710275*e^4 + 9241184443/9233575*e^2 - 58823452/710275, 98811/9233575*e^18 - 948001/1846715*e^16 + 89548728/9233575*e^14 - 879833614/9233575*e^12 + 4918224963/9233575*e^10 - 15899959627/9233575*e^8 + 1143939508/369343*e^6 - 1973431536/710275*e^4 + 9122512573/9233575*e^2 - 67009247/710275, -82470/369343*e^19 + 13692638/1846715*e^17 - 189059617/1846715*e^15 + 1408918977/1846715*e^13 - 1228969770/369343*e^11 + 15835646519/1846715*e^9 - 22953558617/1846715*e^7 + 1250733841/142055*e^5 - 3898935416/1846715*e^3 + 17493166/142055*e, 34683118/120036475*e^19 - 234255591/24007295*e^17 + 16491678324/120036475*e^15 - 125678095117/120036475*e^13 + 562097183594/120036475*e^11 - 1490033557396/120036475*e^9 + 446365691007/24007295*e^7 - 127107630873/9233575*e^5 + 435769498354/120036475*e^3 - 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607432232/710275*e^4 - 127338198/710275*e^2 + 43474711/710275, 665726/9233575*e^18 - 855952/369343*e^16 + 285320053/9233575*e^14 - 2052479579/9233575*e^12 + 8660296558/9233575*e^10 - 21658518192/9233575*e^8 + 6061592741/1846715*e^6 - 1513291266/710275*e^4 + 2916188558/9233575*e^2 - 16845492/710275, 2727472/9233575*e^18 - 17747463/1846715*e^16 + 1196997426/9233575*e^14 - 8693061498/9233575*e^12 + 36892329701/9233575*e^10 - 92529583944/9233575*e^8 + 26183957144/1846715*e^6 - 7022378407/710275*e^4 + 22123139131/9233575*e^2 - 115763729/710275, 3917139/120036475*e^19 - 6221175/4801459*e^17 + 2548299192/120036475*e^15 - 22309747981/120036475*e^13 + 112843089562/120036475*e^11 - 332379986313/120036475*e^9 + 108822489859/24007295*e^7 - 33774933249/9233575*e^5 + 138091744837/120036475*e^3 - 1296219538/9233575*e, -41275999/120036475*e^19 + 55028327/4801459*e^17 - 19079039872/120036475*e^15 + 142950718271/120036475*e^13 - 627796068642/120036475*e^11 + 1633673985183/120036475*e^9 - 480967593494/24007295*e^7 + 134772333109/9233575*e^5 - 443870108167/120036475*e^3 + 1645438383/9233575*e, 2631038/24007295*e^19 - 90011094/24007295*e^17 + 257732795/4801459*e^15 - 10035316113/24007295*e^13 + 46183463774/24007295*e^11 - 127312507678/24007295*e^9 + 202136152006/24007295*e^7 - 12763420391/1846715*e^5 + 56231807452/24007295*e^3 - 493485459/1846715*e, -2700731/4801459*e^19 + 452552679/24007295*e^17 - 6314836416/24007295*e^15 + 47622658376/24007295*e^13 - 42091595487/4801459*e^11 + 550335695077/24007295*e^9 - 811193961701/24007295*e^7 + 45181792288/1846715*e^5 - 146623472538/24007295*e^3 + 703519643/1846715*e, 3798196/9233575*e^18 - 24186826/1846715*e^16 + 1584859508/9233575*e^14 - 11073851404/9233575*e^12 + 44652464468/9233575*e^10 - 104774833972/9233575*e^8 + 5448061520/369343*e^6 - 6568529396/710275*e^4 + 17591915428/9233575*e^2 - 69627067/710275, -2894715/4801459*e^19 + 481133866/24007295*e^17 - 6650485424/24007295*e^15 + 49614148394/24007295*e^13 - 43324499629/4801459*e^11 + 559060859893/24007295*e^9 - 812773652504/24007295*e^7 + 44633144217/1846715*e^5 - 141921240197/24007295*e^3 + 600350207/1846715*e, 76548133/120036475*e^19 - 522591824/24007295*e^17 + 37230864929/120036475*e^15 - 287400212637/120036475*e^13 + 1302879061264/120036475*e^11 - 3501616818196/120036475*e^9 + 1063678452719/24007295*e^7 - 307663878768/9233575*e^5 + 1086255298904/120036475*e^3 - 6579711426/9233575*e, -45349632/120036475*e^19 + 303016906/24007295*e^17 - 21051396866/120036475*e^15 + 157840459398/120036475*e^13 - 692157516931/120036475*e^11 + 1791952956109/120036475*e^9 - 522095057526/24007295*e^7 + 144031767272/9233575*e^5 - 477989676891/120036475*e^3 + 2991577479/9233575*e, 12642759/24007295*e^19 - 424491007/24007295*e^17 + 1188139131/4801459*e^15 - 45009353199/24007295*e^13 + 200379435322/24007295*e^11 - 530271335844/24007295*e^9 + 797749208563/24007295*e^7 - 46256604393/1846715*e^5 + 167531457791/24007295*e^3 - 975472357/1846715*e, 1894618/9233575*e^18 - 13904854/1846715*e^16 + 1065064984/9233575*e^14 - 8817205927/9233575*e^12 + 42617978444/9233575*e^10 - 120967338316/9233575*e^8 + 38265232159/1846715*e^6 - 11276398103/710275*e^4 + 38801309409/9233575*e^2 - 224225546/710275, 1198582/9233575*e^18 - 8405376/1846715*e^16 + 618009941/9233575*e^14 - 4949567323/9233575*e^12 + 23417088531/9233575*e^10 - 66136376984/9233575*e^8 + 21311030996/1846715*e^6 - 6658645847/710275*e^4 + 26858197216/9233575*e^2 - 186209704/710275, -2007242/9233575*e^18 + 14419636/1846715*e^16 - 1084036921/9233575*e^14 + 8840767063/9233575*e^12 - 42289782311/9233575*e^10 + 119388506579/9233575*e^8 - 37723018466/1846715*e^6 + 11097820357/710275*e^4 - 36944435046/9233575*e^2 + 200158524/710275, -1137788/9233575*e^18 + 7446229/1846715*e^16 - 507009494/9233575*e^14 + 3739068557/9233575*e^12 - 16255737054/9233575*e^10 + 42299840031/9233575*e^8 - 12657171529/1846715*e^6 + 3725868823/710275*e^4 - 14734159669/9233575*e^2 + 120067611/710275] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([3, 3, -842*w + 8767])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]