/* 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([-114, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([3, 3, w]) primes_array = [ [2, 2, -3*w - 32],\ [3, 3, w],\ [5, 5, w + 2],\ [5, 5, w + 3],\ [7, 7, -w + 11],\ [7, 7, w + 11],\ [11, 11, w + 2],\ [11, 11, w + 9],\ [13, 13, w + 6],\ [13, 13, w + 7],\ [19, 19, w],\ [37, 37, w + 15],\ [37, 37, w + 22],\ [41, 41, 37*w + 395],\ [41, 41, 5*w + 53],\ [67, 67, w + 28],\ [67, 67, w + 39],\ [71, 71, 40*w + 427],\ [71, 71, 8*w + 85],\ [73, 73, -2*w + 23],\ [73, 73, 2*w + 23],\ [83, 83, w + 23],\ [83, 83, w + 60],\ [89, 89, -w - 5],\ [89, 89, w - 5],\ [101, 101, w + 35],\ [101, 101, w + 66],\ [109, 109, w + 21],\ [109, 109, w + 88],\ [113, 113, -w - 1],\ [113, 113, w - 1],\ [131, 131, w + 30],\ [131, 131, w + 101],\ [149, 149, w + 73],\ [149, 149, w + 76],\ [167, 167, 2*w - 17],\ [167, 167, -2*w - 17],\ [181, 181, w + 64],\ [181, 181, w + 117],\ [197, 197, w + 36],\ [197, 197, w + 161],\ [199, 199, -3*w - 35],\ [199, 199, 3*w - 35],\ [211, 211, w + 89],\ [211, 211, w + 122],\ [251, 251, w + 37],\ [251, 251, w + 214],\ [257, 257, 7*w - 73],\ [257, 257, -7*w - 73],\ [271, 271, -15*w - 161],\ [271, 271, 81*w + 865],\ [281, 281, 55*w + 587],\ [281, 281, 23*w + 245],\ [289, 17, -17],\ [307, 307, w + 81],\ [307, 307, w + 226],\ [313, 313, 78*w + 833],\ [313, 313, -18*w - 193],\ [331, 331, w + 86],\ [331, 331, w + 245],\ [347, 347, w + 43],\ [347, 347, w + 304],\ [367, 367, 11*w - 119],\ [367, 367, 11*w + 119],\ [373, 373, w + 62],\ [373, 373, w + 311],\ [379, 379, w + 71],\ [379, 379, w + 308],\ [383, 383, 6*w + 61],\ [383, 383, 6*w - 61],\ [389, 389, w + 170],\ [389, 389, w + 219],\ [401, 401, -3*w - 25],\ [401, 401, 3*w - 25],\ [419, 419, w + 47],\ [419, 419, w + 372],\ [421, 421, w + 172],\ [421, 421, w + 249],\ [431, 431, 2*w - 5],\ [431, 431, -2*w - 5],\ [443, 443, w + 185],\ [443, 443, w + 258],\ [449, 449, 5*w - 49],\ [449, 449, 5*w + 49],\ [457, 457, -14*w - 151],\ [457, 457, 14*w - 151],\ [461, 461, w + 110],\ [461, 461, w + 351],\ [463, 463, -33*w - 353],\ [463, 463, 63*w + 673],\ [467, 467, w + 54],\ [467, 467, w + 413],\ [491, 491, w + 144],\ [491, 491, w + 347],\ [521, 521, 67*w + 715],\ [521, 521, 35*w + 373],\ [523, 523, w + 239],\ [523, 523, w + 284],\ [529, 23, -23],\ [547, 547, w + 85],\ [547, 547, w + 462],\ [557, 557, w + 79],\ [557, 557, w + 478],\ [569, 569, 11*w - 115],\ [569, 569, -11*w - 115],\ [577, 577, 4*w - 49],\ [577, 577, -4*w - 49],\ [587, 587, w + 124],\ [587, 587, w + 463],\ [599, 599, 4*w - 35],\ [599, 599, -4*w - 35],\ [631, 631, -5*w - 59],\ [631, 631, 5*w - 59],\ [641, 641, 5*w - 47],\ [641, 641, -5*w - 47],\ [653, 653, w + 296],\ [653, 653, w + 357],\ [661, 661, w + 82],\ [661, 661, w + 579],\ [701, 701, w + 157],\ [701, 701, w + 544],\ [727, 727, -w - 29],\ [727, 727, w - 29],\ [743, 743, 76*w + 811],\ [743, 743, 44*w + 469],\ [769, 769, 2*w - 35],\ [769, 769, -2*w - 35],\ [787, 787, w + 129],\ [787, 787, w + 658],\ [811, 811, w + 134],\ [811, 811, w + 677],\ [821, 821, w + 186],\ [821, 821, w + 635],\ [823, 823, 3*w - 43],\ [823, 823, -3*w - 43],\ [829, 829, w + 51],\ [829, 829, w + 778],\ [839, 839, 126*w + 1345],\ [839, 839, 30*w + 319],\ [841, 29, -29],\ [857, 857, -3*w - 13],\ [857, 857, 3*w - 13],\ [863, 863, 4*w - 31],\ [863, 863, -4*w - 31],\ [877, 877, w + 248],\ [877, 877, w + 629],\ [887, 887, -22*w + 233],\ [887, 887, 22*w + 233],\ [907, 907, w + 387],\ [907, 907, w + 520],\ [911, 911, 82*w + 875],\ [911, 911, 50*w + 533],\ [919, 919, -15*w + 163],\ [919, 919, -15*w - 163],\ [937, 937, -6*w - 71],\ [937, 937, 6*w - 71],\ [947, 947, w + 190],\ [947, 947, w + 757],\ [953, 953, 9*w + 91],\ [953, 953, 9*w - 91],\ [961, 31, -31],\ [967, 967, -9*w - 101],\ [967, 967, 9*w - 101],\ [977, 977, -3*w - 7],\ [977, 977, 3*w - 7],\ [983, 983, -4*w - 29],\ [983, 983, 4*w - 29]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^8 - 19*x^6 + 87*x^4 - 116*x^2 + 16 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1/76*e^7 + 9/76*e^5 + 79/76*e^3 - 155/38*e, -1, e, -1/76*e^7 + 9/76*e^5 + 79/76*e^3 - 193/38*e, -3/76*e^6 + 65/76*e^4 - 333/76*e^2 + 62/19, 2/19*e^6 - 37/19*e^4 + 146/19*e^2 - 83/19, 5/76*e^7 - 83/76*e^5 + 251/76*e^3 - 40/19*e, -21/152*e^7 + 379/152*e^5 - 1495/152*e^3 + 179/19*e, -1/38*e^6 + 9/38*e^4 + 79/38*e^2 - 117/19, 3/19*e^6 - 46/19*e^4 + 86/19*e^2 + 56/19, 5/38*e^6 - 83/38*e^4 + 251/38*e^2 - 23/19, 17/76*e^6 - 267/76*e^4 + 595/76*e^2 + 111/19, 3/76*e^6 - 65/76*e^4 + 409/76*e^2 - 62/19, 15/152*e^7 - 249/152*e^5 + 829/152*e^3 - 174/19*e, -37/76*e^7 + 675/76*e^5 - 2739/76*e^3 + 714/19*e, -4/19*e^6 + 74/19*e^4 - 311/19*e^2 + 318/19, -1/4*e^6 + 15/4*e^4 - 27/4*e^2, -5/38*e^7 + 32/19*e^5 + 36/19*e^3 - 809/38*e, 17/152*e^7 - 343/152*e^5 + 1811/152*e^3 - 277/19*e, -8/19*e^6 + 129/19*e^4 - 337/19*e^2 + 47/19, -9/38*e^6 + 157/38*e^4 - 581/38*e^2 + 220/19, 11/152*e^7 - 213/152*e^5 + 993/152*e^3 - 158/19*e, 13/152*e^7 - 231/152*e^5 + 835/152*e^3 - 85/38*e, 39/152*e^7 - 693/152*e^5 + 2505/152*e^3 - 331/38*e, 37/152*e^7 - 675/152*e^5 + 2663/152*e^3 - 281/19*e, 7/76*e^7 - 139/76*e^5 + 663/76*e^3 - 283/38*e, 43/152*e^7 - 805/152*e^5 + 3481/152*e^3 - 476/19*e, 3/19*e^6 - 46/19*e^4 + 67/19*e^2 + 360/19, -27/76*e^6 + 433/76*e^4 - 1021/76*e^2 + 159/19, 43/152*e^7 - 805/152*e^5 + 3329/152*e^3 - 267/19*e, 101/152*e^7 - 1859/152*e^5 + 7639/152*e^3 - 936/19*e, 9/152*e^7 - 271/152*e^5 + 2291/152*e^3 - 549/19*e, 77/152*e^7 - 1415/152*e^5 + 5811/152*e^3 - 1281/38*e, -93/152*e^7 + 1635/152*e^5 - 5991/152*e^3 + 771/19*e, 81/152*e^7 - 1527/152*e^5 + 6939/152*e^3 - 1236/19*e, 81/152*e^7 - 1375/152*e^5 + 4507/152*e^3 - 400/19*e, -69/76*e^7 + 1229/76*e^5 - 4657/76*e^3 + 2035/38*e, -9/19*e^6 + 157/19*e^4 - 562/19*e^2 + 497/19, -39/76*e^6 + 617/76*e^4 - 1517/76*e^2 + 179/19, 16/19*e^7 - 277/19*e^5 + 978/19*e^3 - 873/19*e, -213/152*e^7 + 3779/152*e^5 - 14143/152*e^3 + 1536/19*e, 53/76*e^6 - 819/76*e^4 + 1779/76*e^2 + 203/19, 47/76*e^6 - 841/76*e^4 + 3241/76*e^2 - 433/19, 7/38*e^6 - 101/38*e^4 + 131/38*e^2 + 325/19, 11/76*e^6 - 213/76*e^4 + 993/76*e^2 + 7/19, 15/19*e^7 - 517/38*e^5 + 1715/38*e^3 - 941/38*e, -1/76*e^7 + 47/76*e^5 - 643/76*e^3 + 521/19*e, -79/76*e^7 + 1395/76*e^5 - 5159/76*e^3 + 2423/38*e, 111/152*e^7 - 2025/152*e^5 + 8445/152*e^3 - 1337/19*e, -49/38*e^6 + 821/38*e^4 - 2437/38*e^2 + 442/19, -3/19*e^6 + 46/19*e^4 - 162/19*e^2 + 381/19, -35/76*e^7 + 733/76*e^5 - 4113/76*e^3 + 1648/19*e, -5/38*e^7 + 121/38*e^5 - 821/38*e^3 + 517/19*e, 5/38*e^6 - 83/38*e^4 + 251/38*e^2 - 61/19, -3/38*e^6 + 27/38*e^4 + 313/38*e^2 - 408/19, 43/38*e^6 - 691/38*e^4 + 1695/38*e^2 + 167/19, -29/38*e^6 + 489/38*e^4 - 1433/38*e^2 + 122/19, 7/19*e^6 - 120/19*e^4 + 340/19*e^2 + 61/19, -5/19*e^6 + 64/19*e^4 + 15/19*e^2 - 448/19, -25/76*e^6 + 491/76*e^4 - 2319/76*e^2 + 333/19, -16/19*e^7 + 277/19*e^5 - 940/19*e^3 + 569/19*e, 27/152*e^7 - 509/152*e^5 + 2465/152*e^3 - 526/19*e, 75/76*e^6 - 1245/76*e^4 + 3917/76*e^2 - 695/19, 125/76*e^6 - 2075/76*e^4 + 6123/76*e^2 - 639/19, 1/2*e^6 - 17/2*e^4 + 53/2*e^2 - 24, 1/38*e^6 - 9/38*e^4 - 3/38*e^2 - 339/19, 3/76*e^6 + 11/76*e^4 - 807/76*e^2 + 394/19, -7/19*e^6 + 101/19*e^4 - 112/19*e^2 - 270/19, 167/152*e^7 - 3061/152*e^5 + 12457/152*e^3 - 3027/38*e, 49/152*e^7 - 935/152*e^5 + 3995/152*e^3 - 291/19*e, 139/152*e^7 - 2581/152*e^5 + 10945/152*e^3 - 1392/19*e, 13/152*e^7 - 383/152*e^5 + 3115/152*e^3 - 1263/38*e, 73/152*e^7 - 1227/152*e^5 + 3847/152*e^3 - 451/38*e, -121/152*e^7 + 2115/152*e^5 - 7503/152*e^3 + 1253/38*e, -5/38*e^7 + 51/19*e^5 - 249/19*e^3 + 65/38*e, -5/8*e^7 + 99/8*e^5 - 495/8*e^3 + 91*e, 29/38*e^6 - 489/38*e^4 + 1509/38*e^2 - 8/19, 11/38*e^6 - 175/38*e^4 + 499/38*e^2 + 109/19, 31/76*e^7 - 659/76*e^5 + 3859/76*e^3 - 3327/38*e, -13/38*e^7 + 193/38*e^5 - 265/38*e^3 - 210/19*e, 7/19*e^7 - 183/38*e^5 - 99/38*e^3 + 1471/38*e, -65/76*e^7 + 1269/76*e^5 - 6189/76*e^3 + 4099/38*e, -20/19*e^7 + 759/38*e^5 - 3471/38*e^3 + 4491/38*e, 111/152*e^7 - 1797/152*e^5 + 4721/152*e^3 - 375/38*e, -25/38*e^6 + 491/38*e^4 - 2167/38*e^2 + 761/19, 15/19*e^6 - 287/19*e^4 + 1209/19*e^2 - 689/19, 35/76*e^7 - 467/76*e^5 - 181/76*e^3 + 2005/38*e, -27/76*e^7 + 623/76*e^5 - 4175/76*e^3 + 3757/38*e, 23/76*e^6 - 473/76*e^4 + 2477/76*e^2 - 735/19, 8/19*e^6 - 110/19*e^4 + 71/19*e^2 + 219/19, -77/76*e^7 + 1415/76*e^5 - 5811/76*e^3 + 1395/19*e, -15/76*e^7 + 325/76*e^5 - 1817/76*e^3 + 519/19*e, 143/152*e^7 - 2617/152*e^5 + 10781/152*e^3 - 1294/19*e, -43/76*e^7 + 653/76*e^5 - 1125/76*e^3 - 454/19*e, 27/76*e^7 - 547/76*e^5 + 3035/76*e^3 - 3035/38*e, -87/76*e^7 + 1543/76*e^5 - 5743/76*e^3 + 2779/38*e, -79/38*e^6 + 1319/38*e^4 - 3905/38*e^2 + 941/19, -47/76*e^6 + 765/76*e^4 - 2481/76*e^2 + 908/19, 30/19*e^6 - 498/19*e^4 + 1506/19*e^2 - 561/19, -151/76*e^6 + 2537/76*e^4 - 7565/76*e^2 + 790/19, -1/19*e^6 + 9/19*e^4 - 54/19*e^2 + 640/19, 73/152*e^7 - 1607/152*e^5 + 9851/152*e^3 - 1983/19*e, 11/152*e^7 - 517/152*e^5 + 5857/152*e^3 - 1545/19*e, 65/76*e^7 - 1117/76*e^5 + 3757/76*e^3 - 1781/38*e, -23/38*e^7 + 473/38*e^5 - 2629/38*e^3 + 2363/19*e, -79/76*e^6 + 1357/76*e^4 - 4437/76*e^2 + 347/19, -21/76*e^6 + 303/76*e^4 - 583/76*e^2 - 60/19, -3/38*e^7 + 27/38*e^5 + 199/38*e^3 - 332/19*e, 83/152*e^7 - 1545/152*e^5 + 6629/152*e^3 - 1747/38*e, -11/19*e^7 + 369/38*e^5 - 1131/38*e^3 + 913/38*e, 10/19*e^7 - 204/19*e^5 + 1110/19*e^3 - 1897/19*e, -43/76*e^6 + 729/76*e^4 - 2189/76*e^2 - 264/19, 9/38*e^6 - 157/38*e^4 + 467/38*e^2 - 353/19, 25/76*e^7 - 377/76*e^5 + 685/76*e^3 + 227/38*e, -9/8*e^7 + 163/8*e^5 - 647/8*e^3 + 149/2*e, 85/152*e^7 - 1411/152*e^5 + 4039/152*e^3 - 74/19*e, -43/152*e^7 + 805/152*e^5 - 3633/152*e^3 + 571/19*e, 63/76*e^6 - 985/76*e^4 + 2053/76*e^2 - 181/19, -9/38*e^6 + 119/38*e^4 + 103/38*e^2 - 901/19, 245/152*e^7 - 4523/152*e^5 + 19063/152*e^3 - 2823/19*e, -17/38*e^7 + 305/38*e^5 - 1279/38*e^3 + 1203/19*e, -1/76*e^6 + 47/76*e^4 - 643/76*e^2 + 692/19, -27/38*e^6 + 433/38*e^4 - 1059/38*e^2 + 318/19, -49/76*e^7 + 707/76*e^5 - 803/76*e^3 - 558/19*e, 311/152*e^7 - 5725/152*e^5 + 23881/152*e^3 - 6117/38*e, 7/19*e^6 - 82/19*e^4 - 78/19*e^2 + 232/19, 51/38*e^6 - 915/38*e^4 + 3419/38*e^2 - 1101/19, -3/76*e^6 + 141/76*e^4 - 1397/76*e^2 + 195/19, 13/76*e^6 - 307/76*e^4 + 1899/76*e^2 - 731/19, -31/76*e^6 + 469/76*e^4 - 1161/76*e^2 + 191/19, -3/2*e^6 + 51/2*e^4 - 161/2*e^2 + 30, -65/152*e^7 + 1003/152*e^5 - 2199/152*e^3 - 31/38*e, 253/152*e^7 - 4519/152*e^5 + 17139/152*e^3 - 3487/38*e, -21/19*e^6 + 341/19*e^4 - 1001/19*e^2 + 710/19, -39/19*e^6 + 655/19*e^4 - 2011/19*e^2 + 944/19, -17/19*e^6 + 286/19*e^4 - 823/19*e^2 + 278/19, 43/76*e^6 - 729/76*e^4 + 2037/76*e^2 + 245/19, 59/76*e^7 - 1139/76*e^5 + 5599/76*e^3 - 4307/38*e, -281/152*e^7 + 4771/152*e^5 - 15079/152*e^3 + 1849/38*e, -30/19*e^6 + 498/19*e^4 - 1506/19*e^2 + 979/19, -4/19*e^7 + 91/38*e^5 + 233/38*e^3 - 1359/38*e, 203/152*e^7 - 3613/152*e^5 + 13489/152*e^3 - 1173/19*e, -15/76*e^7 + 363/76*e^5 - 2615/76*e^3 + 2501/38*e, 63/152*e^7 - 909/152*e^5 + 761/152*e^3 + 1187/38*e, -21/38*e^6 + 379/38*e^4 - 1571/38*e^2 + 868/19, -49/38*e^6 + 783/38*e^4 - 1943/38*e^2 + 176/19, 31/152*e^7 - 849/152*e^5 + 6557/152*e^3 - 1435/19*e, 229/152*e^7 - 4227/152*e^5 + 17591/152*e^3 - 2056/19*e, -1/38*e^6 + 9/38*e^4 - 35/38*e^2 + 187/19, -63/76*e^6 + 1061/76*e^4 - 3193/76*e^2 + 276/19, 195/152*e^7 - 3389/152*e^5 + 11841/152*e^3 - 1331/19*e, -61/76*e^7 + 1195/76*e^5 - 6011/76*e^3 + 2464/19*e, 33/76*e^6 - 487/76*e^4 + 927/76*e^2 - 283/19, 77/76*e^6 - 1339/76*e^4 + 4595/76*e^2 - 863/19, -89/76*e^6 + 1447/76*e^4 - 3799/76*e^2 + 218/19, -11/76*e^6 + 213/76*e^4 - 1221/76*e^2 + 620/19, 63/152*e^7 - 1213/152*e^5 + 5321/152*e^3 - 1131/38*e, 28/19*e^7 - 499/19*e^5 + 1854/19*e^3 - 1314/19*e, 273/152*e^7 - 4699/152*e^5 + 15863/152*e^3 - 2697/38*e, -223/152*e^7 + 4021/152*e^5 - 16089/152*e^3 + 4311/38*e, -125/76*e^6 + 2075/76*e^4 - 6275/76*e^2 + 506/19, -16/19*e^6 + 239/19*e^4 - 465/19*e^2 + 189/19, -29/19*e^6 + 508/19*e^4 - 1794/19*e^2 + 1232/19, 1/38*e^7 + 29/38*e^5 - 687/38*e^3 + 1010/19*e, -43/152*e^7 + 805/152*e^5 - 3481/152*e^3 + 305/19*e, 85/76*e^7 - 1601/76*e^5 + 7193/76*e^3 - 5255/38*e, -12/19*e^7 + 463/38*e^5 - 2265/38*e^3 + 4207/38*e] 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])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]