/* 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([-28, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([13, 13, -2*w + 11]) primes_array = [ [2, 2, -w + 6],\ [2, 2, w + 5],\ [7, 7, 6*w - 35],\ [7, 7, -6*w - 29],\ [9, 3, 3],\ [11, 11, 4*w + 19],\ [11, 11, 4*w - 23],\ [13, 13, -2*w + 11],\ [13, 13, 2*w + 9],\ [25, 5, -5],\ [31, 31, 2*w - 13],\ [31, 31, -2*w - 11],\ [41, 41, -8*w - 39],\ [41, 41, 8*w - 47],\ [53, 53, -26*w - 125],\ [53, 53, 26*w - 151],\ [61, 61, -14*w + 81],\ [61, 61, -14*w - 67],\ [83, 83, 2*w - 15],\ [83, 83, -2*w - 13],\ [97, 97, 2*w - 5],\ [97, 97, -2*w - 3],\ [109, 109, 2*w - 3],\ [109, 109, -2*w - 1],\ [113, 113, 2*w - 1],\ [127, 127, 8*w - 45],\ [127, 127, 8*w + 37],\ [131, 131, -50*w + 291],\ [131, 131, 50*w + 241],\ [139, 139, 6*w - 37],\ [139, 139, -6*w - 31],\ [149, 149, 20*w + 97],\ [149, 149, 20*w - 117],\ [157, 157, -12*w - 59],\ [157, 157, 12*w - 71],\ [163, 163, 4*w - 19],\ [163, 163, 4*w + 15],\ [173, 173, 4*w + 23],\ [173, 173, 4*w - 27],\ [211, 211, 2*w - 19],\ [211, 211, -2*w - 17],\ [227, 227, -4*w - 13],\ [227, 227, 4*w - 17],\ [233, 233, 6*w + 25],\ [233, 233, -6*w + 31],\ [239, 239, -14*w - 69],\ [239, 239, 14*w - 83],\ [241, 241, 46*w + 221],\ [241, 241, -46*w + 267],\ [251, 251, 22*w - 129],\ [251, 251, 22*w + 107],\ [257, 257, -34*w + 197],\ [257, 257, -34*w - 163],\ [277, 277, 4*w - 29],\ [277, 277, -4*w - 25],\ [283, 283, 4*w - 15],\ [283, 283, -4*w - 11],\ [289, 17, -17],\ [307, 307, 68*w - 395],\ [307, 307, 68*w + 327],\ [311, 311, 10*w - 61],\ [311, 311, -10*w - 51],\ [313, 313, -32*w - 155],\ [313, 313, 32*w - 187],\ [317, 317, -18*w - 85],\ [317, 317, 18*w - 103],\ [331, 331, -4*w - 9],\ [331, 331, 4*w - 13],\ [337, 337, -16*w - 79],\ [337, 337, 16*w - 95],\ [347, 347, -12*w + 67],\ [347, 347, 12*w + 55],\ [353, 353, 14*w - 79],\ [353, 353, 14*w + 65],\ [361, 19, -19],\ [367, 367, -32*w - 153],\ [367, 367, -32*w + 185],\ [383, 383, 56*w + 269],\ [383, 383, 56*w - 325],\ [389, 389, -4*w - 27],\ [389, 389, 4*w - 31],\ [401, 401, 8*w - 51],\ [401, 401, -8*w - 43],\ [421, 421, 12*w - 73],\ [421, 421, -12*w - 61],\ [439, 439, -8*w - 33],\ [439, 439, 8*w - 41],\ [443, 443, -4*w - 1],\ [443, 443, 4*w - 5],\ [461, 461, -30*w + 173],\ [461, 461, 30*w + 143],\ [463, 463, 2*w - 25],\ [463, 463, -2*w - 23],\ [467, 467, 34*w + 165],\ [467, 467, 34*w - 199],\ [503, 503, -26*w - 127],\ [503, 503, 26*w - 153],\ [509, 509, 4*w - 33],\ [509, 509, -4*w - 29],\ [521, 521, -10*w + 53],\ [521, 521, 10*w + 43],\ [529, 23, -23],\ [547, 547, 14*w - 85],\ [547, 547, -14*w - 71],\ [557, 557, 66*w + 317],\ [557, 557, 66*w - 383],\ [563, 563, 2*w - 27],\ [563, 563, -2*w - 25],\ [569, 569, -64*w + 373],\ [569, 569, 64*w + 309],\ [587, 587, 12*w + 53],\ [587, 587, -12*w + 65],\ [593, 593, 8*w + 45],\ [593, 593, 8*w - 53],\ [601, 601, -26*w - 123],\ [601, 601, 26*w - 149],\ [617, 617, 6*w - 23],\ [617, 617, -6*w - 17],\ [647, 647, 24*w - 137],\ [647, 647, -24*w - 113],\ [653, 653, 28*w - 165],\ [653, 653, -28*w - 137],\ [677, 677, -22*w - 103],\ [677, 677, 22*w - 125],\ [691, 691, 20*w - 113],\ [691, 691, 20*w + 93],\ [709, 709, -10*w - 41],\ [709, 709, 10*w - 51],\ [719, 719, -8*w + 37],\ [719, 719, -8*w - 29],\ [727, 727, -22*w - 109],\ [727, 727, 22*w - 131],\ [739, 739, 46*w - 269],\ [739, 739, -46*w - 223],\ [761, 761, -6*w - 13],\ [761, 761, 6*w - 19],\ [769, 769, 110*w - 639],\ [769, 769, 110*w + 529],\ [773, 773, 4*w - 37],\ [773, 773, -4*w - 33],\ [787, 787, 2*w - 31],\ [787, 787, -2*w - 29],\ [809, 809, 56*w + 271],\ [809, 809, -56*w + 327],\ [821, 821, 6*w - 17],\ [821, 821, -6*w - 11],\ [823, 823, 38*w - 223],\ [823, 823, -38*w - 185],\ [827, 827, 132*w - 767],\ [827, 827, 132*w + 635],\ [841, 29, -29],\ [853, 853, -76*w + 443],\ [853, 853, 76*w + 367],\ [863, 863, 14*w - 87],\ [863, 863, -14*w - 73],\ [911, 911, 2*w - 33],\ [911, 911, -2*w - 31],\ [919, 919, -6*w - 41],\ [919, 919, 6*w - 47],\ [929, 929, 50*w + 239],\ [929, 929, 50*w - 289],\ [953, 953, 6*w - 11],\ [953, 953, -6*w - 5],\ [967, 967, -8*w - 25],\ [967, 967, 8*w - 33],\ [991, 991, 16*w + 71],\ [991, 991, -16*w + 87]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^13 + x^12 - 19*x^11 - 15*x^10 + 134*x^9 + 73*x^8 - 433*x^7 - 125*x^6 + 631*x^5 + 54*x^4 - 337*x^3 - 35*x^2 + 58*x + 11 K. = NumberField(heckePol) hecke_eigenvalues_array = [-3912/611*e^12 - 8208/611*e^11 + 65453/611*e^10 + 130764/611*e^9 - 383101/611*e^8 - 709352/611*e^7 + 931059/611*e^6 + 1526224/611*e^5 - 837896/611*e^4 - 1155734/611*e^3 + 100409/611*e^2 + 253535/611*e + 39602/611, e, -424/47*e^12 - 900/47*e^11 + 7074/47*e^10 + 14353/47*e^9 - 41198/47*e^8 - 78014/47*e^7 + 99122/47*e^6 + 168567/47*e^5 - 86867/47*e^4 - 128994/47*e^3 + 7795/47*e^2 + 28757/47*e + 4976/47, -7767/611*e^12 - 16064/611*e^11 + 130337/611*e^10 + 255520/611*e^9 - 766365/611*e^8 - 1382339/611*e^7 + 1876322/611*e^6 + 2958292/611*e^5 - 1710368/611*e^4 - 2213683/611*e^3 + 216351/611*e^2 + 482935/611*e + 75974/611, 8291/611*e^12 + 17006/611*e^11 - 139341/611*e^10 - 270404/611*e^9 + 821443/611*e^8 + 1461962/611*e^7 - 2021263/611*e^6 - 3125129/611*e^5 + 1864793/611*e^4 + 2332822/611*e^3 - 258132/611*e^2 - 508548/611*e - 77154/611, 17417/611*e^12 + 36285/611*e^11 - 291766/611*e^10 - 577646/611*e^9 + 1710656/611*e^8 + 3129615/611*e^7 - 4166523/611*e^6 - 6717445/611*e^5 + 3752965/611*e^4 + 5061756/611*e^3 - 429485/611*e^2 - 1113100/611*e - 183702/611, -3087/611*e^12 - 6522/611*e^11 + 51622/611*e^10 + 104070/611*e^9 - 301914/611*e^8 - 566017/611*e^7 + 732933/611*e^6 + 1223364/611*e^5 - 658681/611*e^4 - 933547/611*e^3 + 79253/611*e^2 + 203474/611*e + 31345/611, 1, 18663/611*e^12 + 39038/611*e^11 - 312458/611*e^10 - 621788/611*e^9 + 1830472/611*e^8 + 3371897/611*e^7 - 4453340/611*e^6 - 7251260/611*e^5 + 4006700/611*e^4 + 5487730/611*e^3 - 461277/611*e^2 - 1209799/611*e - 197893/611, -9066/611*e^12 - 18812/611*e^11 + 152019/611*e^10 + 299535/611*e^9 - 892729/611*e^8 - 1623457/611*e^7 + 2180664/611*e^6 + 3488027/611*e^5 - 1976953/611*e^4 - 2637322/611*e^3 + 237902/611*e^2 + 587347/611*e + 93372/611, 1794/47*e^12 + 3722/47*e^11 - 30084/47*e^10 - 59223/47*e^9 + 176693/47*e^8 + 320586/47*e^7 - 431734/47*e^6 - 686964/47*e^5 + 391645/47*e^4 + 515751/47*e^3 - 47320/47*e^2 - 113214/47*e - 18235/47, -10731/611*e^12 - 22148/611*e^11 + 180088/611*e^10 + 352253/611*e^9 - 1059112/611*e^8 - 1905290/611*e^7 + 2594338/611*e^6 + 4075980/611*e^5 - 2366803/611*e^4 - 3047776/611*e^3 + 296418/611*e^2 + 666261/611*e + 108292/611, -5995/611*e^12 - 12496/611*e^11 + 100546/611*e^10 + 199231/611*e^9 - 590855/611*e^8 - 1082301/611*e^7 + 1446681/611*e^6 + 2335610/611*e^5 - 1325310/611*e^4 - 1781647/611*e^3 + 182695/611*e^2 + 395793/611*e + 58453/611, 33967/611*e^12 + 70496/611*e^11 - 569539/611*e^10 - 1121920/611*e^9 + 3344908/611*e^8 + 6075304/611*e^7 - 8175100/611*e^6 - 13028050/611*e^5 + 7429982/611*e^4 + 9798337/611*e^3 - 922097/611*e^2 - 2150348/611*e - 341492/611, -44344/611*e^12 - 91905/611*e^11 + 743810/611*e^10 + 1462490/611*e^9 - 4370752/611*e^8 - 7918046/611*e^7 + 10690174/611*e^6 + 16973309/611*e^5 - 9721263/611*e^4 - 12757620/611*e^3 + 1193780/611*e^2 + 2810721/611*e + 451720/611, -26149/611*e^12 - 54310/611*e^11 + 438307/611*e^10 + 864401/611*e^9 - 2572611/611*e^8 - 4681427/611*e^7 + 6279747/611*e^6 + 10040471/611*e^5 - 5689772/611*e^4 - 7549065/611*e^3 + 688901/611*e^2 + 1648359/611*e + 266905/611, 1604/47*e^12 + 3332/47*e^11 - 26887/47*e^10 - 53030/47*e^9 + 157794/47*e^8 + 287181/47*e^7 - 384924/47*e^6 - 615897/47*e^5 + 347678/47*e^4 + 463247/47*e^3 - 40331/47*e^2 - 101572/47*e - 16978/47, -20862/611*e^12 - 42681/611*e^11 + 350617/611*e^10 + 678149/611*e^9 - 2066622/611*e^8 - 3661900/611*e^7 + 5080800/611*e^6 + 7809887/611*e^5 - 4667034/611*e^4 - 5805096/611*e^3 + 607611/611*e^2 + 1266175/611*e + 202136/611, 4155/47*e^12 + 8734/47*e^11 - 69499/47*e^10 - 139180/47*e^9 + 406527/47*e^8 + 755364/47*e^7 - 986370/47*e^6 - 1626769/47*e^5 + 882493/47*e^4 + 1234573/47*e^3 - 97956/47*e^2 - 272420/47*e - 44489/47, -54927/611*e^12 - 114897/611*e^11 + 919312/611*e^10 + 1829704/611*e^9 - 5382447/611*e^8 - 9918882/611*e^7 + 13078236/611*e^6 + 21315181/611*e^5 - 11724491/611*e^4 - 16103006/611*e^3 + 1300461/611*e^2 + 3542079/611*e + 587878/611, -53571/611*e^12 - 109964/611*e^11 + 900116/611*e^10 + 1747909/611*e^9 - 5303904/611*e^8 - 9444657/611*e^7 + 13036164/611*e^6 + 20166441/611*e^5 - 11978227/611*e^4 - 15023128/611*e^3 + 1576725/611*e^2 + 3280945/611*e + 517759/611, 21241/611*e^12 + 43971/611*e^11 - 356094/611*e^10 - 699306/611*e^9 + 2090482/611*e^8 + 3781967/611*e^7 - 5103968/611*e^6 - 8088390/611*e^5 + 4624861/611*e^4 + 6044592/611*e^3 - 561389/611*e^2 - 1317572/611*e - 209356/611, -12310/611*e^12 - 24835/611*e^11 + 207127/611*e^10 + 393741/611*e^9 - 1222836/611*e^8 - 2118032/611*e^7 + 3012281/611*e^6 + 4483230/611*e^5 - 2766940/611*e^4 - 3276125/611*e^3 + 341624/611*e^2 + 706767/611*e + 121190/611, -12788/611*e^12 - 25643/611*e^11 + 215779/611*e^10 + 406810/611*e^9 - 1280243/611*e^8 - 2191022/611*e^7 + 3184528/611*e^6 + 4650703/611*e^5 - 2996905/611*e^4 - 3426780/611*e^3 + 448540/611*e^2 + 756766/611*e + 112635/611, 39612/611*e^12 + 82610/611*e^11 - 663236/611*e^10 - 1314992/611*e^9 + 3885397/611*e^8 + 7123421/611*e^7 - 9449077/611*e^6 - 15285961/611*e^5 + 8482038/611*e^4 + 11510959/611*e^3 - 945844/611*e^2 - 2524292/611*e - 415346/611, 4722/611*e^12 + 9930/611*e^11 - 78877/611*e^10 - 158128/611*e^9 + 460279/611*e^8 + 856913/611*e^7 - 1111152/611*e^6 - 1838908/611*e^5 + 979581/611*e^4 + 1383124/611*e^3 - 90697/611*e^2 - 304030/611*e - 54952/611, -44454/611*e^12 - 92252/611*e^11 + 745369/611*e^10 + 1467760/611*e^9 - 4377137/611*e^8 - 7944082/611*e^7 + 10693495/611*e^6 + 17018171/611*e^5 - 9699537/611*e^4 - 12771807/611*e^3 + 1168617/611*e^2 + 2804687/611*e + 453269/611, -68960/611*e^12 - 142440/611*e^11 + 1157263/611*e^10 + 2265222/611*e^9 - 6805416/611*e^8 - 12250390/611*e^7 + 16666424/611*e^6 + 26201412/611*e^5 - 15195033/611*e^4 - 19591014/611*e^3 + 1898206/611*e^2 + 4288485/611*e + 684301/611, -65946/611*e^12 - 136476/611*e^11 + 1106359/611*e^10 + 2171537/611*e^9 - 6503379/611*e^8 - 11755375/611*e^7 + 15918848/611*e^6 + 25195086/611*e^5 - 14515535/611*e^4 - 18932717/611*e^3 + 1844574/611*e^2 + 4162614/611*e + 655056/611, 11881/611*e^12 + 26720/611*e^11 - 195609/611*e^10 - 428178/611*e^9 + 1113922/611*e^8 + 2346676/611*e^7 - 2569124/611*e^6 - 5151326/611*e^5 + 2035742/611*e^4 + 4072102/611*e^3 - 3078/611*e^2 - 918121/611*e - 175088/611, 37030/611*e^12 + 78542/611*e^11 - 618188/611*e^10 - 1252724/611*e^9 + 3604603/611*e^8 + 6809631/611*e^7 - 8696369/611*e^6 - 14711696/611*e^5 + 7682789/611*e^4 + 11243795/611*e^3 - 761818/611*e^2 - 2491136/611*e - 425245/611, 16778/611*e^12 + 35008/611*e^11 - 281271/611*e^10 - 557919/611*e^9 + 1651930/611*e^8 + 3028827/611*e^7 - 4041578/611*e^6 - 6528369/611*e^5 + 3697573/611*e^4 + 4967028/611*e^3 - 503189/611*e^2 - 1102016/611*e - 166433/611, -8454/611*e^12 - 17348/611*e^11 + 141890/611*e^10 + 275547/611*e^9 - 834376/611*e^8 - 1486834/611*e^7 + 2042109/611*e^6 + 3165300/611*e^5 - 1855917/611*e^4 - 2339175/611*e^3 + 222117/611*e^2 + 501062/611*e + 83132/611, -29896/611*e^12 - 62914/611*e^11 + 499894/611*e^10 + 1002535/611*e^9 - 2922611/611*e^8 - 5440931/611*e^7 + 7085152/611*e^6 + 11719362/611*e^5 - 6326244/611*e^4 - 8905006/611*e^3 + 685608/611*e^2 + 1978766/611*e + 325874/611, -16416/611*e^12 - 33555/611*e^11 + 276296/611*e^10 + 533355/611*e^9 - 1633306/611*e^8 - 2882056/611*e^7 + 4042880/611*e^6 + 6154740/611*e^5 - 3791445/611*e^4 - 4584201/611*e^3 + 594741/611*e^2 + 994852/611*e + 140828/611, -21033/611*e^12 - 44959/611*e^11 + 350491/611*e^10 + 717558/611*e^9 - 2037416/611*e^8 - 3904804/611*e^7 + 4887271/611*e^6 + 8452234/611*e^5 - 4260861/611*e^4 - 6479715/611*e^3 + 374046/611*e^2 + 1426564/611*e + 243598/611, -56027/611*e^12 - 117145/611*e^11 + 937957/611*e^10 + 1865296/611*e^9 - 5494566/611*e^8 - 10109588/611*e^7 + 13368066/611*e^6 + 21713699/611*e^5 - 12032691/611*e^4 - 16378074/611*e^3 + 1395879/611*e^2 + 3586220/611*e + 581983/611, -707/13*e^12 - 1444/13*e^11 + 11892/13*e^10 + 22947/13*e^9 - 70193/13*e^8 - 123933/13*e^7 + 173035/13*e^6 + 264339/13*e^5 - 159995/13*e^4 - 196325/13*e^3 + 22011/13*e^2 + 42660/13*e + 6519/13, 53613/611*e^12 + 109741/611*e^11 - 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5851700/611*e^2 - 13015440/611*e - 2067641/611, 7820/611*e^12 + 14226/611*e^11 - 134382/611*e^10 - 223621/611*e^9 + 820571/611*e^8 + 1184433/611*e^7 - 2140691/611*e^6 - 2428141/611*e^5 + 2197354/611*e^4 + 1644340/611*e^3 - 463952/611*e^2 - 348896/611*e - 33356/611, 11291/611*e^12 + 17749/611*e^11 - 198745/611*e^10 - 273268/611*e^9 + 1256643/611*e^8 + 1393343/611*e^7 - 3454536/611*e^6 - 2629173/611*e^5 + 3865072/611*e^4 + 1400980/611*e^3 - 1068374/611*e^2 - 253112/611*e + 21075/611, 90768/611*e^12 + 189205/611*e^11 - 1520432/611*e^10 - 3011911/611*e^9 + 8914118/611*e^8 + 16315467/611*e^7 - 21713707/611*e^6 - 35001269/611*e^5 + 19575386/611*e^4 + 26320239/611*e^3 - 2270963/611*e^2 - 5745579/611*e - 955726/611, 2615/47*e^12 + 5474/47*e^11 - 43772/47*e^10 - 87161/47*e^9 + 256319/47*e^8 + 472452/47*e^7 - 623049/47*e^6 - 1015198/47*e^5 + 559865/47*e^4 + 766504/47*e^3 - 65590/47*e^2 - 167486/47*e - 27033/47, 163456/611*e^12 + 338706/611*e^11 - 2741129/611*e^10 - 5389028/611*e^9 + 16100953/611*e^8 + 29168636/611*e^7 - 39353204/611*e^6 - 62488882/611*e^5 + 35750393/611*e^4 + 46880830/611*e^3 - 4412195/611*e^2 - 10255000/611*e - 1629080/611, -253146/611*e^12 - 524877/611*e^11 + 4245183/611*e^10 + 8351737/611*e^9 - 24936001/611*e^8 - 45210498/611*e^7 + 60952369/611*e^6 + 96884067/611*e^5 - 55380567/611*e^4 - 72753848/611*e^3 + 6817087/611*e^2 + 15969773/611*e + 2539809/611, 177454/611*e^12 + 366333/611*e^11 - 2978334/611*e^10 - 5826924/611*e^9 + 17518030/611*e^8 + 31522632/611*e^7 - 42918867/611*e^6 - 67461987/611*e^5 + 39173691/611*e^4 + 50492384/611*e^3 - 4956652/611*e^2 - 11029710/611*e - 1762420/611, 1364/47*e^12 + 3062/47*e^11 - 22490/47*e^10 - 49086/47*e^9 + 128383/47*e^8 + 269163/47*e^7 - 297274/47*e^6 - 591426/47*e^5 + 236421/47*e^4 + 469509/47*e^3 + 2874/47*e^2 - 109597/47*e - 22084/47, 47074/611*e^12 + 98184/611*e^11 - 789167/611*e^10 - 1563565/611*e^9 + 4634197/611*e^8 + 8476006/611*e^7 - 11328994/611*e^6 - 18210402/611*e^5 + 10320745/611*e^4 + 13733491/611*e^3 - 1328642/611*e^2 - 2995685/611*e - 457947/611, 64210/611*e^12 + 138565/611*e^11 - 1069113/611*e^10 - 2214455/611*e^9 + 6206981/611*e^8 + 12077824/611*e^7 - 14860264/611*e^6 - 26256327/611*e^5 + 12922785/611*e^4 + 20321927/611*e^3 - 1144347/611*e^2 - 4537935/611*e - 763608/611, 4744/13*e^12 + 9820/13*e^11 - 79571/13*e^10 - 156218/13*e^9 + 467523/13*e^8 + 845306/13*e^7 - 1143180/13*e^6 - 1809923/13*e^5 + 1038837/13*e^4 + 1356402/13*e^3 - 127072/13*e^2 - 296877/13*e - 47974/13] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([13, 13, -2*w + 11])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]