/* 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([5, -1, -6, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([25, 5, -w^3 + 4*w]) primes_array = [ [5, 5, w],\ [5, 5, -w^2 + w + 2],\ [7, 7, -w^2 + 2],\ [13, 13, -w^2 + 3],\ [13, 13, w^2 - w - 4],\ [16, 2, 2],\ [19, 19, -w^2 + w + 1],\ [23, 23, -w^3 + 4*w + 2],\ [25, 5, -w^3 + w^2 + 3*w - 1],\ [29, 29, w^3 - w^2 - 4*w + 1],\ [29, 29, -w + 3],\ [31, 31, -w^3 + w^2 + 5*w - 2],\ [37, 37, w^3 - 4*w - 1],\ [37, 37, w^3 - 3*w + 1],\ [53, 53, -w^3 + 2*w^2 + 4*w - 6],\ [59, 59, w^2 + w - 4],\ [61, 61, -2*w^2 + w + 8],\ [73, 73, -w^3 + 5*w - 1],\ [79, 79, 2*w^2 + w - 6],\ [79, 79, 2*w^3 - w^2 - 9*w + 3],\ [81, 3, -3],\ [83, 83, -w^3 + w^2 + 3*w - 4],\ [97, 97, -w^3 + 6*w + 2],\ [97, 97, -2*w^3 + w^2 + 10*w - 1],\ [97, 97, 3*w^3 - 4*w^2 - 15*w + 13],\ [97, 97, w^3 - w^2 - 6*w + 2],\ [103, 103, -2*w^3 + 3*w^2 + 11*w - 9],\ [109, 109, w^2 + 3*w - 3],\ [137, 137, -w^2 + 7],\ [149, 149, w^3 - 2*w^2 - 5*w + 4],\ [149, 149, w^3 + 2*w^2 - 6*w - 7],\ [151, 151, w^3 + w^2 - 5*w - 3],\ [163, 163, w^3 - 3*w - 4],\ [163, 163, -w^3 + 6*w - 3],\ [167, 167, w^3 - 2*w^2 - 6*w + 4],\ [169, 13, w^2 + 2*w - 4],\ [191, 191, w^3 - w^2 - 5*w - 2],\ [191, 191, 2*w^3 - 9*w - 2],\ [197, 197, w^3 - 4*w + 4],\ [211, 211, -w^3 + 2*w^2 + 6*w - 8],\ [223, 223, -w^3 + 3*w^2 + 4*w - 6],\ [229, 229, w^3 - w^2 - 5*w - 1],\ [251, 251, 2*w^3 - 2*w^2 - 7*w + 3],\ [257, 257, 3*w^2 - 14],\ [263, 263, w^2 - 3*w - 1],\ [263, 263, w^2 - 2*w - 7],\ [269, 269, -w^3 + w^2 + 2*w - 4],\ [269, 269, 2*w^3 - 11*w - 1],\ [271, 271, -w^3 + w^2 + 4*w - 7],\ [277, 277, w^3 - w^2 - 3*w + 8],\ [311, 311, w^3 + 2*w^2 - 7*w - 11],\ [311, 311, -3*w^2 - 2*w + 8],\ [317, 317, -2*w^3 + 2*w^2 + 7*w - 4],\ [331, 331, w^2 - w - 8],\ [331, 331, 3*w^3 - 2*w^2 - 12*w + 9],\ [337, 337, -2*w^3 + 2*w^2 + 11*w - 8],\ [343, 7, w^3 - 3*w^2 - 4*w + 11],\ [347, 347, -2*w^3 + 2*w^2 + 9*w - 4],\ [349, 349, -w^3 + 2*w^2 + 2*w - 6],\ [353, 353, 2*w^2 - w - 12],\ [353, 353, 2*w^3 - w^2 - 9*w - 1],\ [367, 367, -2*w^3 + w^2 + 10*w - 6],\ [367, 367, 2*w^3 - 11*w - 3],\ [379, 379, -3*w - 1],\ [379, 379, w^3 - 2*w - 3],\ [383, 383, -w^3 + 2*w^2 + 4*w - 11],\ [383, 383, -w^3 - w^2 + 7*w + 4],\ [389, 389, 2*w^3 - w^2 - 11*w + 4],\ [401, 401, w^3 + 2*w^2 - 7*w - 6],\ [401, 401, 2*w^3 - 11*w - 6],\ [419, 419, -2*w^3 + w^2 + 8*w - 1],\ [421, 421, -w^3 + 3*w^2 + 3*w - 12],\ [421, 421, -w^3 - 2*w^2 + 2*w + 6],\ [433, 433, w^3 - 2*w^2 - 6*w + 11],\ [433, 433, w^2 - 8],\ [439, 439, -w^3 + 3*w^2 + 3*w - 7],\ [443, 443, w^3 - w^2 - 2*w - 4],\ [443, 443, 2*w^3 + w^2 - 7*w - 2],\ [443, 443, 2*w^3 - w^2 - 8*w + 2],\ [443, 443, -w^2 - 2],\ [467, 467, 3*w^3 - 5*w^2 - 15*w + 19],\ [467, 467, 2*w^3 - 3*w^2 - 8*w + 13],\ [467, 467, -w^3 + 2*w^2 + 3*w - 9],\ [467, 467, -w^3 + 3*w^2 + 3*w - 8],\ [479, 479, -w^3 - 3*w^2 + 5*w + 13],\ [479, 479, -w^3 + w^2 + w - 4],\ [487, 487, 3*w^3 - w^2 - 13*w - 2],\ [491, 491, w^2 + w - 9],\ [499, 499, -3*w^3 + 3*w^2 + 13*w - 7],\ [503, 503, 2*w^2 + w - 9],\ [509, 509, -2*w^2 - 3*w + 6],\ [509, 509, -w^3 + w^2 - 4],\ [521, 521, -3*w^2 + w + 12],\ [523, 523, 2*w^3 - 8*w - 3],\ [529, 23, 2*w^3 - w^2 - 6*w - 1],\ [541, 541, -w^3 + w^2 + 7*w - 4],\ [547, 547, 2*w^3 - 3*w^2 - 9*w + 7],\ [557, 557, 3*w^2 - 2*w - 9],\ [557, 557, w^3 - 6*w - 7],\ [563, 563, -2*w^3 + w^2 + 6*w - 6],\ [577, 577, -w^3 + 4*w^2 + 2*w - 13],\ [587, 587, w^2 - 2*w + 3],\ [587, 587, w^2 + 2*w - 6],\ [593, 593, -2*w^3 + 2*w^2 + 11*w - 3],\ [599, 599, w^3 + 3*w^2 - 6*w - 12],\ [601, 601, -2*w^3 + w^2 + 9*w + 2],\ [607, 607, -w^3 + 3*w^2 + 4*w - 9],\ [617, 617, -2*w^3 + w^2 + 7*w - 3],\ [619, 619, -2*w^3 + 4*w^2 + 9*w - 11],\ [631, 631, 3*w^2 - w - 7],\ [631, 631, -w^3 - w^2 + 6*w + 1],\ [641, 641, -w^3 + 2*w^2 + 5*w - 1],\ [641, 641, -w^3 + 2*w^2 + 2*w - 7],\ [643, 643, -3*w^3 + w^2 + 14*w - 4],\ [647, 647, -3*w^3 + 5*w^2 + 14*w - 18],\ [647, 647, -w^3 + 7*w + 2],\ [647, 647, -2*w^3 + 7*w - 1],\ [647, 647, 2*w^3 - w^2 - 10*w - 1],\ [653, 653, -2*w^3 - w^2 + 11*w + 4],\ [659, 659, -w^3 - w^2 + 5*w - 1],\ [673, 673, -w^3 + w^2 + 7*w - 2],\ [673, 673, -w^3 + w^2 + 2*w - 8],\ [683, 683, 2*w^3 - 11*w - 8],\ [683, 683, 2*w^3 - 7*w - 3],\ [701, 701, 2*w^3 - 4*w^2 - 10*w + 17],\ [727, 727, -w^3 + w^2 + 4*w - 8],\ [727, 727, 3*w^2 - 11],\ [739, 739, 2*w^3 + 3*w^2 - 8*w - 13],\ [743, 743, 2*w^2 + w - 12],\ [743, 743, 3*w^2 - 2*w - 14],\ [751, 751, w^3 + w^2 - 3*w - 7],\ [757, 757, -2*w^3 + 10*w - 1],\ [761, 761, -2*w^3 + w^2 + 7*w - 1],\ [761, 761, 3*w^2 + w - 7],\ [773, 773, -2*w^3 + 2*w^2 + 8*w - 1],\ [797, 797, -2*w^3 + w^2 + 9*w + 3],\ [797, 797, 3*w^3 - 14*w - 8],\ [809, 809, 5*w^3 - 5*w^2 - 24*w + 16],\ [821, 821, -4*w^3 + 3*w^2 + 18*w - 9],\ [821, 821, w^3 - w - 3],\ [823, 823, 3*w^2 + w - 8],\ [829, 829, -2*w^3 + 3*w^2 + 10*w - 7],\ [829, 829, 3*w^2 - 2*w - 16],\ [841, 29, 2*w^2 + w - 11],\ [853, 853, 3*w^3 - 2*w^2 - 14*w + 3],\ [853, 853, 3*w^3 - 5*w^2 - 14*w + 17],\ [857, 857, 3*w^2 - w - 8],\ [857, 857, 3*w^3 - w^2 - 13*w + 3],\ [859, 859, w^3 - 3*w^2 - 6*w + 12],\ [877, 877, w^3 + 2*w^2 - 5*w - 4],\ [877, 877, w^2 - w - 9],\ [881, 881, w^3 + 2*w^2 - 4*w - 12],\ [881, 881, w^2 - 4*w - 4],\ [883, 883, -3*w^3 + 2*w^2 + 11*w - 8],\ [887, 887, -2*w^3 + w^2 + 6*w - 1],\ [911, 911, 3*w^2 - w - 17],\ [911, 911, 2*w^3 - 2*w^2 - 12*w + 3],\ [919, 919, 5*w^3 - 6*w^2 - 22*w + 24],\ [919, 919, -2*w^3 + w^2 + 6*w - 7],\ [929, 929, 2*w^3 - 7*w - 1],\ [937, 937, -w^3 - w^2 + 8*w - 2],\ [947, 947, 3*w^3 + w^2 - 15*w - 6],\ [947, 947, -2*w^3 + 9*w + 9],\ [953, 953, w^3 - 2*w^2 - 6*w + 1],\ [977, 977, w^3 - w^2 - 3*w + 9],\ [977, 977, w^3 + w^2 - 3*w - 8],\ [983, 983, w^3 - 5*w - 7],\ [991, 991, -w^3 + 2*w^2 + 2*w - 8],\ [991, 991, w^2 + 4*w - 1],\ [997, 997, 2*w^3 + w^2 - 12*w - 8]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^10 - 5*x^9 - 32*x^8 + 166*x^7 + 255*x^6 - 1555*x^5 - 220*x^4 + 5034*x^3 - 2076*x^2 - 4808*x + 3056 K. = NumberField(heckePol) hecke_eigenvalues_array = [1, 1, e, -994121/44377688*e^9 + 3209017/44377688*e^8 + 18419171/22188844*e^7 - 48522135/22188844*e^6 - 405263051/44377688*e^5 + 749568883/44377688*e^4 + 700651879/22188844*e^3 - 998116831/22188844*e^2 - 169536415/5547211*e + 201802122/5547211, 1628829/221888440*e^9 - 6360183/221888440*e^8 - 30534561/110944220*e^7 + 105675519/110944220*e^6 + 708885079/221888440*e^5 - 1964248693/221888440*e^4 - 1500667597/110944220*e^3 + 3020651727/110944220*e^2 + 494901836/27736055*e - 629558891/27736055, 1593779/221888440*e^9 - 5827313/221888440*e^8 - 7324359/27736055*e^7 + 92596929/110944220*e^6 + 645458429/221888440*e^5 - 1572637463/221888440*e^4 - 594787961/55472110*e^3 + 2088622887/110944220*e^2 + 651037427/55472110*e - 228115891/27736055, 1008327/221888440*e^9 - 5257709/221888440*e^8 - 16285463/110944220*e^7 + 86102837/110944220*e^6 + 287961457/221888440*e^5 - 1531418219/221888440*e^4 - 410904341/110944220*e^3 + 2018806411/110944220*e^2 + 33078623/27736055*e - 264731083/27736055, -1285389/110944220*e^9 + 899962/27736055*e^8 + 12430913/27736055*e^7 - 27129547/27736055*e^6 - 576830909/110944220*e^5 + 426013659/55472110*e^4 + 1008957127/55472110*e^3 - 1329923577/55472110*e^2 - 399201967/27736055*e + 647233777/27736055, -1305021/27736055*e^9 + 17934013/110944220*e^8 + 194142877/110944220*e^7 - 279547269/55472110*e^6 - 1086682757/55472110*e^5 + 4610101623/110944220*e^4 + 8063057819/110944220*e^3 - 6491764237/55472110*e^2 - 4418668659/55472110*e + 2624481182/27736055, -2278491/221888440*e^9 + 5342077/221888440*e^8 + 46050289/110944220*e^7 - 75449941/110944220*e^6 - 1162477041/221888440*e^5 + 981991347/221888440*e^4 + 2491406293/110944220*e^3 - 1190965043/110944220*e^2 - 811325689/27736055*e + 355351739/27736055, 1497285/44377688*e^9 - 4831933/44377688*e^8 - 28518793/22188844*e^7 + 75149297/22188844*e^6 + 662597079/44377688*e^5 - 1236342579/44377688*e^4 - 1284178497/22188844*e^3 + 1755865223/22188844*e^2 + 353248834/5547211*e - 346091832/5547211, -805381/27736055*e^9 + 9184543/110944220*e^8 + 125071377/110944220*e^7 - 138282529/55472110*e^6 - 370199661/27736055*e^5 + 2129341723/110944220*e^4 + 5698896469/110944220*e^3 - 1484858771/27736055*e^2 - 3127564849/55472110*e + 1313909572/27736055, -435247/27736055*e^9 + 5908191/110944220*e^8 + 33254897/55472110*e^7 - 47312379/27736055*e^6 - 196392097/27736055*e^5 + 1675020321/110944220*e^4 + 1623479249/55472110*e^3 - 1369039837/27736055*e^2 - 974840619/27736055*e + 1402766564/27736055, 259433/110944220*e^9 - 1165051/110944220*e^8 - 1984896/27736055*e^7 + 9044244/27736055*e^6 + 48832773/110944220*e^5 - 301544671/110944220*e^4 + 77135741/55472110*e^3 + 470718259/55472110*e^2 - 174773071/27736055*e - 103066134/27736055, 2473649/221888440*e^9 - 6558673/221888440*e^8 - 50224331/110944220*e^7 + 103146679/110944220*e^6 + 1283175319/221888440*e^5 - 1818622423/221888440*e^4 - 2836054157/110944220*e^3 + 3457352937/110944220*e^2 + 969498051/27736055*e - 1027091261/27736055, 907907/221888440*e^9 - 233339/221888440*e^8 - 18201203/110944220*e^7 - 15655483/110944220*e^6 + 420556517/221888440*e^5 + 966988031/221888440*e^4 - 553164981/110944220*e^3 - 2566311359/110944220*e^2 - 20313437/27736055*e + 798614447/27736055, -237348/5547211*e^9 + 3071847/22188844*e^8 + 17611961/11094422*e^7 - 23264231/5547211*e^6 - 193291471/11094422*e^5 + 718260601/22188844*e^4 + 326146095/5547211*e^3 - 467306583/5547211*e^2 - 287315300/5547211*e + 343425978/5547211, 609789/221888440*e^9 - 6504223/221888440*e^8 - 1104639/27736055*e^7 + 113987969/110944220*e^6 - 138734121/221888440*e^5 - 2305310053/221888440*e^4 + 196973832/27736055*e^3 + 3601564017/110944220*e^2 - 910471103/55472110*e - 610672431/27736055, -280427/44377688*e^9 + 1487391/44377688*e^8 + 4505051/22188844*e^7 - 24148051/22188844*e^6 - 78356405/44377688*e^5 + 416219789/44377688*e^4 + 106833669/22188844*e^3 - 489393699/22188844*e^2 - 13327098/5547211*e + 55411051/5547211, -1022161/44377688*e^9 + 3635313/44377688*e^8 + 19408871/22188844*e^7 - 58985007/22188844*e^6 - 456004371/44377688*e^5 + 1062857867/44377688*e^4 + 949525219/22188844*e^3 - 1677173371/22188844*e^2 - 293948491/5547211*e + 362087403/5547211, 1490275/44377688*e^9 - 4725359/44377688*e^8 - 7067842/5547211*e^7 + 72533579/22188844*e^6 + 649911749/44377688*e^5 - 1158020333/44377688*e^4 - 610980081/11094422*e^3 + 1591648299/22188844*e^2 + 627649997/11094422*e - 304633709/5547211, 2767037/221888440*e^9 - 3713549/221888440*e^8 - 58379863/110944220*e^7 + 36025777/110944220*e^6 + 1519590947/221888440*e^5 + 130103101/221888440*e^4 - 3101578801/110944220*e^3 - 699602749/110944220*e^2 + 921926803/27736055*e - 31542013/27736055, 1208879/44377688*e^9 - 4232971/44377688*e^8 - 21892277/22188844*e^7 + 64524731/22188844*e^6 + 464855885/44377688*e^5 - 1006090317/44377688*e^4 - 770563621/22188844*e^3 + 1284212773/22188844*e^2 + 182457722/5547211*e - 195970221/5547211, 332259/27736055*e^9 - 5048417/110944220*e^8 - 22528839/55472110*e^7 + 74331291/55472110*e^6 + 104952249/27736055*e^5 - 1029529547/110944220*e^4 - 523885713/55472110*e^3 + 797056163/55472110*e^2 + 56270708/27736055*e + 162630462/27736055, -2988793/221888440*e^9 + 12241231/221888440*e^8 + 52786037/110944220*e^7 - 196426643/110944220*e^6 - 1097172283/221888440*e^5 + 3430101581/221888440*e^4 + 1953257009/110944220*e^3 - 5025286979/110944220*e^2 - 667945392/27736055*e + 1045840527/27736055, -1639287/110944220*e^9 + 2799827/55472110*e^8 + 14826694/27736055*e^7 - 42058816/27736055*e^6 - 630270807/110944220*e^5 + 311486196/27736055*e^4 + 1068767341/55472110*e^3 - 1297978821/55472110*e^2 - 545829976/27736055*e + 146725236/27736055, -5533773/221888440*e^9 + 18235231/221888440*e^8 + 104550777/110944220*e^7 - 279764193/110944220*e^6 - 2416510763/221888440*e^5 + 4448882241/221888440*e^4 + 4771865539/110944220*e^3 - 5982496939/110944220*e^2 - 1413372072/27736055*e + 1239198337/27736055, 552114/27736055*e^9 - 2906251/55472110*e^8 - 20434017/27736055*e^7 + 76404181/55472110*e^6 + 433831623/55472110*e^5 - 371315591/55472110*e^4 - 1179900113/55472110*e^3 + 179614373/55472110*e^2 + 156409073/27736055*e + 273494762/27736055, 6167847/110944220*e^9 - 18350719/110944220*e^8 - 119183953/55472110*e^7 + 279857307/55472110*e^6 + 2814562127/110944220*e^5 - 4433879439/110944220*e^4 - 2732553258/27736055*e^3 + 3129798523/27736055*e^2 + 3032243936/27736055*e - 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390614605/44377688*e^4 - 2037936385/22188844*e^3 - 99052269/22188844*e^2 + 592971847/5547211*e - 55720027/5547211, 20448749/221888440*e^9 - 67681503/221888440*e^8 - 384839291/110944220*e^7 + 1049760189/110944220*e^6 + 8828295959/221888440*e^5 - 17106720713/221888440*e^4 - 17336168647/110944220*e^3 + 23244656187/110944220*e^2 + 5396552311/27736055*e - 4033889331/27736055, 40240283/221888440*e^9 - 135389491/221888440*e^8 - 747094797/110944220*e^7 + 2091384323/110944220*e^6 + 16628011233/221888440*e^5 - 33839208321/221888440*e^4 - 30323121949/110944220*e^3 + 46604639179/110944220*e^2 + 8408874852/27736055*e - 8806859692/27736055, 12036129/221888440*e^9 - 44618443/221888440*e^8 - 207635371/110944220*e^7 + 669126719/110944220*e^6 + 3955509719/221888440*e^5 - 10050255213/221888440*e^4 - 4759759717/110944220*e^3 + 12241623137/110944220*e^2 + 206319351/27736055*e - 2371122571/27736055, -2365029/44377688*e^9 + 8294193/44377688*e^8 + 21292095/11094422*e^7 - 127979823/22188844*e^6 - 880047715/44377688*e^5 + 2061537431/44377688*e^4 + 319065199/5547211*e^3 - 2786106057/22188844*e^2 - 304755203/11094422*e + 576372877/5547211, -1878171/22188844*e^9 + 3359091/11094422*e^8 + 17515256/5547211*e^7 - 53189150/5547211*e^6 - 796037099/22188844*e^5 + 898991541/11094422*e^4 + 1561842469/11094422*e^3 - 1230227609/5547211*e^2 - 893695140/5547211*e + 806396132/5547211] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([5, 5, w])] = -1 AL_eigenvalues[ZF.ideal([5, 5, -w^2 + w + 2])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]