/* 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([-39, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([12, 6, 2*w + 12]) primes_array = [ [3, 3, w + 6],\ [3, 3, -w + 7],\ [4, 2, 2],\ [11, 11, -3*w - 17],\ [11, 11, 3*w - 20],\ [13, 13, 2*w - 13],\ [13, 13, 2*w + 11],\ [17, 17, w + 7],\ [17, 17, -w + 8],\ [19, 19, -w - 4],\ [19, 19, -w + 5],\ [25, 5, 5],\ [31, 31, -6*w - 35],\ [31, 31, -6*w + 41],\ [37, 37, -w - 1],\ [37, 37, w - 2],\ [47, 47, 3*w + 16],\ [47, 47, -3*w + 19],\ [49, 7, -7],\ [67, 67, 3*w - 22],\ [67, 67, -3*w - 19],\ [71, 71, -w - 10],\ [71, 71, w - 11],\ [89, 89, -8*w - 47],\ [89, 89, 8*w - 55],\ [101, 101, 4*w - 29],\ [101, 101, -4*w - 25],\ [109, 109, -3*w - 20],\ [109, 109, 3*w - 23],\ [113, 113, 3*w - 17],\ [113, 113, -3*w - 14],\ [127, 127, -9*w - 53],\ [127, 127, 9*w - 62],\ [157, 157, 2*w - 1],\ [167, 167, 2*w - 19],\ [167, 167, -2*w - 17],\ [173, 173, 13*w - 89],\ [173, 173, -13*w - 76],\ [193, 193, 11*w - 73],\ [193, 193, -11*w - 62],\ [197, 197, 3*w - 14],\ [197, 197, -3*w - 11],\ [199, 199, 3*w - 25],\ [199, 199, -3*w - 22],\ [233, 233, -w - 16],\ [233, 233, w - 17],\ [239, 239, 7*w - 50],\ [239, 239, -7*w - 43],\ [257, 257, 6*w + 31],\ [257, 257, -6*w + 37],\ [263, 263, 3*w - 11],\ [263, 263, -3*w - 8],\ [277, 277, -12*w - 71],\ [277, 277, 12*w - 83],\ [281, 281, -3*w - 7],\ [281, 281, 3*w - 10],\ [283, 283, -7*w + 44],\ [283, 283, 7*w + 37],\ [311, 311, 3*w - 8],\ [311, 311, -3*w - 5],\ [313, 313, -13*w - 73],\ [313, 313, 13*w - 86],\ [317, 317, -9*w + 58],\ [317, 317, 9*w + 49],\ [331, 331, -5*w - 23],\ [331, 331, 5*w - 28],\ [347, 347, -3*w - 1],\ [347, 347, 3*w - 4],\ [349, 349, 3*w - 28],\ [349, 349, -3*w - 25],\ [353, 353, 3*w - 2],\ [353, 353, 3*w - 1],\ [389, 389, 6*w - 35],\ [389, 389, 6*w + 29],\ [419, 419, 2*w - 25],\ [419, 419, -2*w - 23],\ [431, 431, 10*w - 71],\ [431, 431, -10*w - 61],\ [457, 457, 27*w + 157],\ [457, 457, 27*w - 184],\ [461, 461, -4*w - 31],\ [461, 461, 4*w - 35],\ [467, 467, -w - 22],\ [467, 467, w - 23],\ [487, 487, 8*w + 41],\ [487, 487, -8*w + 49],\ [523, 523, -6*w - 41],\ [523, 523, 6*w - 47],\ [529, 23, -23],\ [547, 547, 4*w - 11],\ [547, 547, -4*w - 7],\ [557, 557, 28*w - 191],\ [557, 557, -28*w - 163],\ [571, 571, 20*w - 133],\ [571, 571, -20*w - 113],\ [577, 577, -3*w - 29],\ [577, 577, 3*w - 32],\ [593, 593, 19*w - 131],\ [593, 593, -19*w - 112],\ [601, 601, 5*w - 22],\ [601, 601, -5*w - 17],\ [617, 617, -18*w - 101],\ [617, 617, 18*w - 119],\ [619, 619, -4*w - 1],\ [619, 619, 4*w - 5],\ [631, 631, 17*w - 112],\ [631, 631, -17*w - 95],\ [641, 641, 15*w + 83],\ [641, 641, 15*w - 98],\ [647, 647, -17*w - 101],\ [647, 647, 17*w - 118],\ [653, 653, -11*w - 68],\ [653, 653, 11*w - 79],\ [659, 659, 5*w - 43],\ [659, 659, -5*w - 38],\ [661, 661, -25*w - 142],\ [661, 661, 25*w - 167],\ [677, 677, 13*w - 92],\ [677, 677, -13*w - 79],\ [709, 709, 5*w - 19],\ [709, 709, -5*w - 14],\ [727, 727, -9*w - 58],\ [727, 727, 9*w - 67],\ [733, 733, 7*w - 38],\ [733, 733, -7*w - 31],\ [739, 739, -18*w - 107],\ [739, 739, 18*w - 125],\ [743, 743, 2*w - 31],\ [743, 743, -2*w - 29],\ [769, 769, -3*w - 32],\ [769, 769, 3*w - 35],\ [773, 773, -w - 28],\ [773, 773, w - 29],\ [797, 797, 21*w - 139],\ [797, 797, -21*w - 118],\ [821, 821, -15*w + 97],\ [821, 821, 15*w + 82],\ [827, 827, 9*w - 53],\ [827, 827, -9*w - 44],\ [829, 829, -19*w - 106],\ [829, 829, 19*w - 125],\ [841, 29, -29],\ [853, 853, 9*w - 68],\ [853, 853, -9*w - 59],\ [907, 907, 3*w - 37],\ [907, 907, -3*w - 34],\ [911, 911, 5*w - 46],\ [911, 911, -5*w - 41],\ [929, 929, -6*w - 19],\ [929, 929, 6*w - 25],\ [941, 941, -20*w - 119],\ [941, 941, 20*w - 139],\ [953, 953, -w - 31],\ [953, 953, w - 32],\ [967, 967, 11*w + 56],\ [967, 967, -11*w + 67],\ [977, 977, 16*w - 113],\ [977, 977, -16*w - 97],\ [991, 991, -8*w - 35],\ [991, 991, 8*w - 43]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^6 - 3*x^5 - 10*x^4 + 33*x^3 - 8*x^2 - 8*x - 1 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1, e, -1, 12*e^5 - 39*e^4 - 111*e^3 + 425*e^2 - 194*e - 58, -3*e^5 + 10*e^4 + 27*e^3 - 109*e^2 + 57*e + 14, 4*e^5 - 13*e^4 - 37*e^3 + 141*e^2 - 65*e - 15, 4*e^5 - 13*e^4 - 37*e^3 + 142*e^2 - 63*e - 23, e^3 - 11*e + 4, 19*e^5 - 61*e^4 - 178*e^3 + 664*e^2 - 284*e - 91, 2*e^5 - 6*e^4 - 19*e^3 + 66*e^2 - 27*e - 11, 17*e^5 - 55*e^4 - 158*e^3 + 599*e^2 - 266*e - 83, 6*e^5 - 19*e^4 - 56*e^3 + 207*e^2 - 89*e - 30, 26*e^5 - 83*e^4 - 243*e^3 + 904*e^2 - 393*e - 123, 7*e^5 - 22*e^4 - 65*e^3 + 239*e^2 - 108*e - 28, 17*e^5 - 55*e^4 - 158*e^3 + 600*e^2 - 268*e - 86, 17*e^5 - 55*e^4 - 159*e^3 + 599*e^2 - 256*e - 86, -21*e^5 + 68*e^4 + 195*e^3 - 741*e^2 + 330*e + 104, 13*e^5 - 42*e^4 - 122*e^3 + 457*e^2 - 193*e - 59, -5*e^5 + 16*e^4 + 46*e^3 - 175*e^2 + 83*e + 21, -31*e^5 + 99*e^4 + 289*e^3 - 1078*e^2 + 475*e + 150, 7*e^5 - 22*e^4 - 65*e^3 + 240*e^2 - 109*e - 32, 8*e^5 - 27*e^4 - 73*e^3 + 293*e^2 - 142*e - 34, -3*e^5 + 10*e^4 + 26*e^3 - 108*e^2 + 66*e + 11, -30*e^5 + 95*e^4 + 282*e^3 - 1036*e^2 + 435*e + 144, -26*e^5 + 83*e^4 + 243*e^3 - 903*e^2 + 392*e + 117, -12*e^5 + 37*e^4 + 114*e^3 - 404*e^2 + 161*e + 60, -8*e^5 + 25*e^4 + 75*e^3 - 271*e^2 + 118*e + 33, 13*e^5 - 40*e^4 - 123*e^3 + 435*e^2 - 177*e - 48, -25*e^5 + 81*e^4 + 231*e^3 - 882*e^2 + 407*e + 118, -31*e^5 + 99*e^4 + 290*e^3 - 1078*e^2 + 464*e + 142, 26*e^5 - 83*e^4 - 243*e^3 + 903*e^2 - 392*e - 115, -45*e^5 + 144*e^4 + 420*e^3 - 1569*e^2 + 683*e + 219, 8*e^5 - 26*e^4 - 73*e^3 + 283*e^2 - 139*e - 47, -19*e^5 + 61*e^4 + 175*e^3 - 666*e^2 + 316*e + 94, 11*e^5 - 35*e^4 - 103*e^3 + 382*e^2 - 169*e - 56, 2*e^4 - 5*e^3 - 20*e^2 + 51*e - 11, -34*e^5 + 109*e^4 + 318*e^3 - 1187*e^2 + 517*e + 160, -11*e^5 + 36*e^4 + 101*e^3 - 393*e^2 + 180*e + 66, 42*e^5 - 134*e^4 - 392*e^3 + 1458*e^2 - 636*e - 194, 38*e^5 - 123*e^4 - 353*e^3 + 1341*e^2 - 598*e - 187, -41*e^5 + 132*e^4 + 382*e^3 - 1439*e^2 + 639*e + 209, -37*e^5 + 120*e^4 + 343*e^3 - 1307*e^2 + 592*e + 182, 62*e^5 - 197*e^4 - 581*e^3 + 2146*e^2 - 916*e - 291, -37*e^5 + 118*e^4 + 346*e^3 - 1285*e^2 + 557*e + 183, -19*e^5 + 64*e^4 + 172*e^3 - 695*e^2 + 347*e + 91, 7*e^5 - 22*e^4 - 67*e^3 + 238*e^2 - 90*e - 34, -44*e^5 + 142*e^4 + 411*e^3 - 1546*e^2 + 667*e + 219, 70*e^5 - 223*e^4 - 657*e^3 + 2429*e^2 - 1034*e - 331, 68*e^5 - 218*e^4 - 634*e^3 + 2375*e^2 - 1045*e - 334, 53*e^5 - 169*e^4 - 497*e^3 + 1838*e^2 - 785*e - 242, 9*e^5 - 30*e^4 - 85*e^3 + 325*e^2 - 130*e - 39, -2*e^5 + 8*e^4 + 16*e^3 - 88*e^2 + 63*e + 10, -14*e^5 + 45*e^4 + 134*e^3 - 491*e^2 + 185*e + 82, -6*e^5 + 20*e^4 + 52*e^3 - 216*e^2 + 132*e + 16, -65*e^5 + 209*e^4 + 606*e^3 - 2275*e^2 + 1004*e + 313, 38*e^5 - 118*e^4 - 360*e^3 + 1286*e^2 - 520*e - 172, -67*e^5 + 216*e^4 + 625*e^3 - 2352*e^2 + 1034*e + 322, e^5 - 4*e^4 - 9*e^3 + 43*e^2 - 18*e - 12, -12*e^5 + 36*e^4 + 115*e^3 - 392*e^2 + 147*e + 51, -4*e^5 + 13*e^4 + 39*e^3 - 139*e^2 + 46*e + 17, -22*e^5 + 71*e^4 + 206*e^3 - 773*e^2 + 329*e + 104, 43*e^5 - 135*e^4 - 406*e^3 + 1472*e^2 - 604*e - 203, 89*e^5 - 287*e^4 - 829*e^3 + 3126*e^2 - 1377*e - 432, 2*e^5 - 7*e^4 - 21*e^3 + 76*e^2 - 5*e - 23, -7*e^5 + 22*e^4 + 64*e^3 - 239*e^2 + 115*e + 25, 50*e^5 - 160*e^4 - 470*e^3 + 1742*e^2 - 725*e - 232, -16*e^5 + 52*e^4 + 147*e^3 - 564*e^2 + 267*e + 73, -47*e^5 + 148*e^4 + 442*e^3 - 1611*e^2 + 675*e + 209, 50*e^5 - 160*e^4 - 468*e^3 + 1745*e^2 - 752*e - 242, -26*e^5 + 84*e^4 + 244*e^3 - 915*e^2 + 388*e + 138, -18*e^5 + 60*e^4 + 166*e^3 - 655*e^2 + 297*e + 104, 27*e^5 - 87*e^4 - 251*e^3 + 944*e^2 - 423*e - 124, 11*e^5 - 35*e^4 - 104*e^3 + 383*e^2 - 156*e - 67, -80*e^5 + 257*e^4 + 745*e^3 - 2799*e^2 + 1238*e + 397, -41*e^5 + 132*e^4 + 384*e^3 - 1435*e^2 + 609*e + 189, -11*e^5 + 34*e^4 + 106*e^3 - 371*e^2 + 125*e + 53, -50*e^5 + 160*e^4 + 467*e^3 - 1742*e^2 + 763*e + 242, -5*e^5 + 13*e^4 + 51*e^3 - 141*e^2 + 32*e + 2, -77*e^5 + 247*e^4 + 719*e^3 - 2690*e^2 + 1163*e + 366, 6*e^5 - 22*e^4 - 54*e^3 + 238*e^2 - 122*e - 16, 49*e^5 - 158*e^4 - 458*e^3 + 1720*e^2 - 751*e - 233, 27*e^5 - 83*e^4 - 258*e^3 + 906*e^2 - 348*e - 131, -10*e^5 + 31*e^4 + 93*e^3 - 335*e^2 + 154*e + 21, 20*e^5 - 66*e^4 - 181*e^3 + 718*e^2 - 367*e - 87, -94*e^5 + 301*e^4 + 878*e^3 - 3279*e^2 + 1430*e + 448, -53*e^5 + 166*e^4 + 502*e^3 - 1809*e^2 + 737*e + 247, 93*e^5 - 300*e^4 - 864*e^3 + 3270*e^2 - 1466*e - 445, -2*e^5 + 4*e^4 + 22*e^3 - 46*e^2 - 8*e + 18, -55*e^5 + 177*e^4 + 510*e^3 - 1928*e^2 + 874*e + 258, -10*e^5 + 32*e^4 + 97*e^3 - 349*e^2 + 117*e + 47, -116*e^5 + 372*e^4 + 1083*e^3 - 4050*e^2 + 1767*e + 555, 28*e^5 - 90*e^4 - 262*e^3 + 978*e^2 - 424*e - 108, -6*e^5 + 20*e^4 + 58*e^3 - 214*e^2 + 66*e + 19, 110*e^5 - 352*e^4 - 1031*e^3 + 3833*e^2 - 1637*e - 523, 23*e^5 - 71*e^4 - 221*e^3 + 770*e^2 - 283*e - 106, 24*e^5 - 76*e^4 - 225*e^3 + 834*e^2 - 358*e - 147, 43*e^5 - 137*e^4 - 408*e^3 + 1492*e^2 - 584*e - 214, 10*e^5 - 27*e^4 - 104*e^3 + 299*e^2 - 39*e - 64, e^5 - 4*e^4 - 7*e^3 + 39*e^2 - 32*e + 26, -73*e^5 + 233*e^4 + 686*e^3 - 2535*e^2 + 1063*e + 355, 14*e^5 - 48*e^4 - 130*e^3 + 523*e^2 - 224*e - 98, 7*e^5 - 21*e^4 - 64*e^3 + 231*e^2 - 117*e - 39, -50*e^5 + 161*e^4 + 466*e^3 - 1755*e^2 + 769*e + 226, -6*e^5 + 18*e^4 + 59*e^3 - 196*e^2 + 67*e + 32, -9*e^5 + 30*e^4 + 76*e^3 - 325*e^2 + 213*e + 23, -16*e^5 + 54*e^4 + 140*e^3 - 588*e^2 + 338*e + 56, -5*e^5 + 18*e^4 + 45*e^3 - 197*e^2 + 96*e + 35, -100*e^5 + 324*e^4 + 928*e^3 - 3528*e^2 + 1578*e + 484, -52*e^5 + 163*e^4 + 492*e^3 - 1773*e^2 + 719*e + 245, 21*e^5 - 70*e^4 - 193*e^3 + 763*e^2 - 358*e - 98, -9*e^5 + 29*e^4 + 83*e^3 - 318*e^2 + 155*e + 42, 48*e^5 - 154*e^4 - 444*e^3 + 1675*e^2 - 771*e - 206, 14*e^5 - 45*e^4 - 129*e^3 + 491*e^2 - 232*e - 75, -90*e^5 + 288*e^4 + 843*e^3 - 3137*e^2 + 1341*e + 439, 62*e^5 - 199*e^4 - 582*e^3 + 2167*e^2 - 913*e - 281, 52*e^5 - 168*e^4 - 479*e^3 + 1824*e^2 - 860*e - 233, -58*e^5 + 184*e^4 + 547*e^3 - 2006*e^2 + 829*e + 286, 18*e^5 - 56*e^4 - 169*e^3 + 610*e^2 - 266*e - 41, -35*e^5 + 114*e^4 + 320*e^3 - 1243*e^2 + 601*e + 165, 73*e^5 - 231*e^4 - 687*e^3 + 2520*e^2 - 1059*e - 360, -26*e^5 + 84*e^4 + 238*e^3 - 910*e^2 + 446*e + 98, 37*e^5 - 119*e^4 - 345*e^3 + 1293*e^2 - 570*e - 148, -115*e^5 + 367*e^4 + 1077*e^3 - 3996*e^2 + 1721*e + 544, 6*e^5 - 19*e^4 - 56*e^3 + 205*e^2 - 89*e + 6, 25*e^5 - 80*e^4 - 234*e^3 + 871*e^2 - 375*e - 109, 80*e^5 - 256*e^4 - 751*e^3 + 2786*e^2 - 1167*e - 385, 61*e^5 - 195*e^4 - 571*e^3 + 2123*e^2 - 910*e - 306, 19*e^5 - 62*e^4 - 176*e^3 + 673*e^2 - 311*e - 85, -34*e^5 + 108*e^4 + 319*e^3 - 1174*e^2 + 491*e + 149, -15*e^5 + 48*e^4 + 146*e^3 - 527*e^2 + 173*e + 93, 31*e^5 - 99*e^4 - 290*e^3 + 1078*e^2 - 476*e - 139, 36*e^5 - 115*e^4 - 336*e^3 + 1251*e^2 - 546*e - 180, -78*e^5 + 249*e^4 + 730*e^3 - 2713*e^2 + 1169*e + 378, -25*e^5 + 80*e^4 + 234*e^3 - 871*e^2 + 367*e + 119, -51*e^5 + 167*e^4 + 471*e^3 - 1822*e^2 + 835*e + 272, -44*e^5 + 145*e^4 + 405*e^3 - 1578*e^2 + 735*e + 207, -64*e^5 + 203*e^4 + 600*e^3 - 2209*e^2 + 947*e + 298, -18*e^5 + 61*e^4 + 161*e^3 - 662*e^2 + 345*e + 83, -7*e^5 + 23*e^4 + 59*e^3 - 249*e^2 + 168*e + 34, 39*e^5 - 123*e^4 - 368*e^3 + 1340*e^2 - 550*e - 175, 12*e^5 - 39*e^4 - 110*e^3 + 427*e^2 - 213*e - 66, -72*e^5 + 234*e^4 + 666*e^3 - 2550*e^2 + 1166*e + 358, 54*e^5 - 175*e^4 - 504*e^3 + 1907*e^2 - 830*e - 270, -128*e^5 + 409*e^4 + 1196*e^3 - 4452*e^2 + 1931*e + 598, 17*e^5 - 59*e^4 - 153*e^3 + 642*e^2 - 333*e - 82, 147*e^5 - 469*e^4 - 1373*e^3 + 5109*e^2 - 2227*e - 696, 119*e^5 - 381*e^4 - 1111*e^3 + 4150*e^2 - 1819*e - 564, 9*e^5 - 28*e^4 - 86*e^3 + 301*e^2 - 123*e - 33, -100*e^5 + 321*e^4 + 933*e^3 - 3495*e^2 + 1528*e + 469, 7*e^5 - 18*e^4 - 72*e^3 + 199*e^2 - 37*e - 35, 36*e^5 - 111*e^4 - 346*e^3 + 1211*e^2 - 447*e - 176, 50*e^5 - 161*e^4 - 466*e^3 + 1751*e^2 - 772*e - 256, 43*e^5 - 135*e^4 - 408*e^3 + 1469*e^2 - 579*e - 203, 65*e^5 - 209*e^4 - 602*e^3 + 2278*e^2 - 1036*e - 313, 12*e^5 - 39*e^4 - 105*e^3 + 425*e^2 - 258*e - 44, -78*e^5 + 254*e^4 + 724*e^3 - 2764*e^2 + 1232*e + 404, 8*e^5 - 23*e^4 - 80*e^3 + 255*e^2 - 69*e - 46, -42*e^5 + 135*e^4 + 393*e^3 - 1471*e^2 + 637*e + 205, 6*e^5 - 16*e^4 - 58*e^3 + 178*e^2 - 60*e - 50, 81*e^5 - 261*e^4 - 751*e^3 + 2841*e^2 - 1285*e - 370] 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, w + 6])] = 1 AL_eigenvalues[ZF.ideal([4, 2, 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]