/* 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([-55, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([6, 6, -w - 7]) primes_array = [ [2, 2, w + 1],\ [3, 3, w + 1],\ [3, 3, w + 2],\ [5, 5, -2*w + 15],\ [11, 11, 3*w - 22],\ [13, 13, w + 4],\ [13, 13, w + 9],\ [17, 17, w + 2],\ [17, 17, w + 15],\ [19, 19, -w - 6],\ [19, 19, w - 6],\ [23, 23, w + 3],\ [23, 23, w + 20],\ [47, 47, w + 14],\ [47, 47, w + 33],\ [49, 7, -7],\ [67, 67, w + 16],\ [67, 67, w + 51],\ [73, 73, w + 36],\ [73, 73, w + 37],\ [79, 79, 5*w + 36],\ [79, 79, 5*w - 36],\ [89, 89, -w - 12],\ [89, 89, w - 12],\ [103, 103, w + 40],\ [103, 103, w + 63],\ [131, 131, -6*w - 43],\ [131, 131, -6*w + 43],\ [139, 139, 2*w - 9],\ [139, 139, -2*w - 9],\ [151, 151, 4*w - 27],\ [151, 151, 4*w + 27],\ [163, 163, w + 50],\ [163, 163, w + 113],\ [173, 173, w + 48],\ [173, 173, w + 125],\ [181, 181, -3*w - 26],\ [181, 181, 3*w - 26],\ [193, 193, w + 21],\ [193, 193, w + 172],\ [197, 197, w + 45],\ [197, 197, w + 152],\ [211, 211, 2*w - 3],\ [211, 211, -2*w - 3],\ [223, 223, w + 72],\ [223, 223, w + 151],\ [229, 229, 6*w + 47],\ [229, 229, 6*w - 47],\ [233, 233, w + 88],\ [233, 233, w + 145],\ [239, 239, -3*w - 16],\ [239, 239, 3*w - 16],\ [269, 269, -w - 18],\ [269, 269, w - 18],\ [271, 271, -8*w - 57],\ [271, 271, -8*w + 57],\ [277, 277, w + 71],\ [277, 277, w + 206],\ [293, 293, w + 73],\ [293, 293, w + 220],\ [337, 337, w + 27],\ [337, 337, w + 310],\ [359, 359, 9*w + 64],\ [359, 359, 9*w - 64],\ [367, 367, w + 34],\ [367, 367, w + 333],\ [373, 373, w + 146],\ [373, 373, w + 227],\ [383, 383, w + 173],\ [383, 383, w + 210],\ [389, 389, -5*w - 42],\ [389, 389, 5*w - 42],\ [401, 401, 29*w - 216],\ [401, 401, 11*w - 84],\ [421, 421, -6*w + 49],\ [421, 421, -6*w - 49],\ [431, 431, 3*w - 8],\ [431, 431, -3*w - 8],\ [439, 439, -4*w - 21],\ [439, 439, 4*w - 21],\ [443, 443, w + 99],\ [443, 443, w + 344],\ [449, 449, -8*w - 63],\ [449, 449, -8*w + 63],\ [457, 457, w + 91],\ [457, 457, w + 366],\ [463, 463, w + 38],\ [463, 463, w + 425],\ [467, 467, w + 191],\ [467, 467, w + 276],\ [479, 479, -3*w - 4],\ [479, 479, 3*w - 4],\ [487, 487, w + 223],\ [487, 487, w + 264],\ [491, 491, 3*w - 2],\ [491, 491, -3*w - 2],\ [509, 509, 2*w - 27],\ [509, 509, -2*w - 27],\ [521, 521, -w - 24],\ [521, 521, w - 24],\ [557, 557, w + 75],\ [557, 557, w + 482],\ [571, 571, 11*w + 78],\ [571, 571, 11*w - 78],\ [587, 587, w + 246],\ [587, 587, w + 341],\ [593, 593, w + 184],\ [593, 593, w + 409],\ [613, 613, w + 235],\ [613, 613, w + 378],\ [641, 641, -4*w - 39],\ [641, 641, 4*w - 39],\ [643, 643, w + 174],\ [643, 643, w + 469],\ [647, 647, w + 257],\ [647, 647, w + 390],\ [659, 659, -18*w + 131],\ [659, 659, 30*w - 221],\ [661, 661, 3*w - 34],\ [661, 661, -3*w - 34],\ [673, 673, w + 196],\ [673, 673, w + 477],\ [677, 677, w + 240],\ [677, 677, w + 437],\ [683, 683, w + 136],\ [683, 683, w + 547],\ [709, 709, 21*w - 158],\ [709, 709, 27*w - 202],\ [727, 727, w + 339],\ [727, 727, w + 388],\ [733, 733, w + 39],\ [733, 733, w + 694],\ [739, 739, 10*w - 69],\ [739, 739, 10*w + 69],\ [811, 811, 35*w - 258],\ [811, 811, -19*w + 138],\ [823, 823, w + 177],\ [823, 823, w + 646],\ [829, 829, -6*w - 53],\ [829, 829, 6*w - 53],\ [841, 29, -29],\ [853, 853, w + 316],\ [853, 853, w + 537],\ [857, 857, w + 270],\ [857, 857, w + 587],\ [863, 863, w + 405],\ [863, 863, w + 458],\ [877, 877, w + 377],\ [877, 877, w + 500],\ [881, 881, -19*w + 144],\ [881, 881, -37*w + 276],\ [883, 883, w + 52],\ [883, 883, w + 831],\ [907, 907, w + 350],\ [907, 907, w + 557],\ [919, 919, -8*w - 51],\ [919, 919, 8*w - 51],\ [929, 929, 5*w - 48],\ [929, 929, -5*w - 48],\ [937, 937, w + 176],\ [937, 937, w + 761],\ [947, 947, w + 252],\ [947, 947, w + 695],\ [953, 953, w + 334],\ [953, 953, w + 619],\ [961, 31, -31],\ [983, 983, w + 277],\ [983, 983, w + 706],\ [997, 997, w + 371],\ [997, 997, w + 626]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^8 + 18*x^6 + 81*x^4 + 53*x^2 + 4 K. = NumberField(heckePol) hecke_eigenvalues_array = [-5/106*e^7 - 47/53*e^5 - 459/106*e^3 - 399/106*e, 5/106*e^7 + 47/53*e^5 + 459/106*e^3 + 399/106*e, e, -7/53*e^6 - 121/53*e^4 - 473/53*e^2 - 71/53, -10/53*e^6 - 188/53*e^4 - 865/53*e^2 - 321/53, -67/106*e^7 - 598/53*e^5 - 5239/106*e^3 - 2739/106*e, 12/53*e^7 + 215/53*e^5 + 932/53*e^3 + 417/53*e, -29/106*e^7 - 262/53*e^5 - 2429/106*e^3 - 2081/106*e, 12/53*e^7 + 215/53*e^5 + 932/53*e^3 + 258/53*e, -2/53*e^6 - 27/53*e^4 - 14/53*e^2 + 116/53, 2/53*e^6 + 27/53*e^4 + 67/53*e^2 - 63/53, 19/53*e^7 + 336/53*e^5 + 1458/53*e^3 + 859/53*e, 5/53*e^7 + 94/53*e^5 + 512/53*e^3 + 876/53*e, 77/106*e^7 + 692/53*e^5 + 6157/106*e^3 + 3113/106*e, 60/53*e^7 + 1075/53*e^5 + 4713/53*e^3 + 2297/53*e, 14/53*e^6 + 242/53*e^4 + 999/53*e^2 + 301/53, -44/53*e^7 - 806/53*e^5 - 3753/53*e^3 - 2642/53*e, -99/106*e^7 - 867/53*e^5 - 7371/106*e^3 - 3533/106*e, 70/53*e^7 + 1263/53*e^5 + 5684/53*e^3 + 3519/53*e, 1/106*e^7 + 20/53*e^5 + 431/106*e^3 + 1373/106*e, 8/53*e^6 + 108/53*e^4 + 321/53*e^2 + 596/53, -1/53*e^6 + 13/53*e^4 + 205/53*e^2 - 154/53, 9/53*e^6 + 201/53*e^4 + 1176/53*e^2 + 909/53, 28/53*e^6 + 484/53*e^4 + 1945/53*e^2 + 814/53, 201/106*e^7 + 1794/53*e^5 + 15717/106*e^3 + 8111/106*e, -209/106*e^7 - 1848/53*e^5 - 15985/106*e^3 - 8177/106*e, 30/53*e^6 + 564/53*e^4 + 2701/53*e^2 + 1228/53, -20/53*e^6 - 323/53*e^4 - 1094/53*e^2 + 100/53, -29/53*e^6 - 524/53*e^4 - 2270/53*e^2 - 650/53, 49/53*e^6 + 847/53*e^4 + 3470/53*e^2 + 1292/53, 49/53*e^6 + 847/53*e^4 + 3470/53*e^2 + 1133/53, -8/53*e^6 - 161/53*e^4 - 798/53*e^2 + 411/53, -68/53*e^7 - 1183/53*e^5 - 4875/53*e^3 - 1727/53*e, -101/106*e^7 - 907/53*e^5 - 8127/106*e^3 - 5113/106*e, -127/106*e^7 - 1109/53*e^5 - 9157/106*e^3 - 2545/106*e, 233/106*e^7 + 2063/53*e^5 + 17743/106*e^3 + 7421/106*e, 9/53*e^6 + 148/53*e^4 + 487/53*e^2 - 416/53, -4/53*e^6 - 54/53*e^4 - 28/53*e^2 + 338/53, 89/53*e^7 + 1599/53*e^5 + 7142/53*e^3 + 4060/53*e, -98/53*e^7 - 1747/53*e^5 - 7629/53*e^3 - 3697/53*e, -70/53*e^7 - 1210/53*e^5 - 4942/53*e^3 - 1558/53*e, -13/53*e^7 - 255/53*e^5 - 1469/53*e^3 - 2373/53*e, 9/53*e^6 + 148/53*e^4 + 540/53*e^2 - 628/53, e^6 + 17*e^4 + 65*e^2 + 16, -23/106*e^7 - 195/53*e^5 - 1539/106*e^3 - 1051/106*e, 181/106*e^7 + 1606/53*e^5 + 13775/106*e^3 + 5985/106*e, 19/53*e^6 + 336/53*e^4 + 1511/53*e^2 + 329/53, 29/53*e^6 + 524/53*e^4 + 2376/53*e^2 + 1286/53, 19/53*e^7 + 336/53*e^5 + 1458/53*e^3 + 859/53*e, -291/106*e^7 - 2587/53*e^5 - 22601/106*e^3 - 12537/106*e, -26/53*e^6 - 510/53*e^4 - 2408/53*e^2 - 559/53, 55/53*e^6 + 928/53*e^4 + 3724/53*e^2 + 1156/53, -54/53*e^6 - 888/53*e^4 - 3399/53*e^2 - 1320/53, -38/53*e^6 - 619/53*e^4 - 2280/53*e^2 + 296/53, 2/53*e^6 + 27/53*e^4 + 120/53*e^2 + 626/53, 24/53*e^6 + 377/53*e^4 + 1228/53*e^2 - 544/53, -3/53*e^7 - 67/53*e^5 - 445/53*e^3 - 1257/53*e, -63/53*e^7 - 1089/53*e^5 - 4363/53*e^3 - 1063/53*e, 86/53*e^7 + 1532/53*e^5 + 6591/53*e^3 + 2538/53*e, 83/53*e^7 + 1465/53*e^5 + 6199/53*e^3 + 1970/53*e, -3/2*e^7 - 27*e^5 - 243/2*e^3 - 171/2*e, 347/106*e^7 + 3071/53*e^5 + 26491/106*e^3 + 13529/106*e, 37/53*e^6 + 632/53*e^4 + 2697/53*e^2 + 1352/53, 28/53*e^6 + 484/53*e^4 + 1892/53*e^2 - 829/53, 58/53*e^7 + 995/53*e^5 + 3851/53*e^3 - 237/53*e, 118/53*e^7 + 2123/53*e^5 + 9518/53*e^3 + 5717/53*e, -28/53*e^7 - 484/53*e^5 - 1998/53*e^3 - 549/53*e, 111/106*e^7 + 948/53*e^5 + 7455/106*e^3 + 1671/106*e, 60/53*e^7 + 1075/53*e^5 + 4766/53*e^3 + 2509/53*e, 1/106*e^7 + 20/53*e^5 + 431/106*e^3 + 1373/106*e, 49/53*e^6 + 900/53*e^4 + 3894/53*e^2 + 868/53, 45/53*e^6 + 793/53*e^4 + 3336/53*e^2 + 1418/53, 43/53*e^6 + 713/53*e^4 + 2739/53*e^2 + 50/53, -11/53*e^6 - 175/53*e^4 - 660/53*e^2 + 320/53, -65/53*e^6 - 1063/53*e^4 - 4006/53*e^2 - 1212/53, 55/53*e^6 + 928/53*e^4 + 3618/53*e^2 + 626/53, -5/53*e^6 - 147/53*e^4 - 1148/53*e^2 - 2095/53, 22/53*e^6 + 350/53*e^4 + 1002/53*e^2 - 640/53, -59/53*e^6 - 982/53*e^4 - 3434/53*e^2 + 189/53, -56/53*e^6 - 968/53*e^4 - 3996/53*e^2 - 144/53, 233/106*e^7 + 2116/53*e^5 + 19015/106*e^3 + 10177/106*e, 191/106*e^7 + 1700/53*e^5 + 15011/106*e^3 + 9963/106*e, -68/53*e^6 - 1183/53*e^4 - 5034/53*e^2 - 1939/53, 41/53*e^6 + 739/53*e^4 + 3149/53*e^2 + 2286/53, -181/53*e^7 - 3265/53*e^5 - 14623/53*e^3 - 8635/53*e, -38/53*e^7 - 672/53*e^5 - 2810/53*e^3 - 817/53*e, 138/53*e^7 + 2499/53*e^5 + 11460/53*e^3 + 8002/53*e, -165/106*e^7 - 1498/53*e^5 - 13557/106*e^3 - 8715/106*e, -47/106*e^7 - 410/53*e^5 - 3615/106*e^3 - 2839/106*e, -415/106*e^7 - 3689/53*e^5 - 31843/106*e^3 - 13507/106*e, -26/53*e^6 - 510/53*e^4 - 2461/53*e^2 - 2202/53, -9/53*e^6 - 148/53*e^4 - 699/53*e^2 - 485/53, -218/53*e^7 - 3897/53*e^5 - 17161/53*e^3 - 8768/53*e, 52/53*e^7 + 967/53*e^5 + 4763/53*e^3 + 4987/53*e, 29/53*e^6 + 471/53*e^4 + 1634/53*e^2 + 703/53, 88/53*e^6 + 1612/53*e^4 + 7294/53*e^2 + 2581/53, 52/53*e^6 + 914/53*e^4 + 3809/53*e^2 + 1012/53, -87/53*e^6 - 1572/53*e^4 - 6757/53*e^2 - 2692/53, -40/53*e^6 - 752/53*e^4 - 3619/53*e^2 - 966/53, 67/53*e^6 + 1143/53*e^4 + 4603/53*e^2 + 1838/53, 219/106*e^7 + 1942/53*e^5 + 16691/106*e^3 + 7809/106*e, -141/53*e^7 - 2513/53*e^5 - 11057/53*e^3 - 6291/53*e, -32/53*e^6 - 538/53*e^4 - 2238/53*e^2 - 211/53, 15/53*e^6 + 229/53*e^4 + 741/53*e^2 - 1612/53, 43/106*e^7 + 330/53*e^5 + 1573/106*e^3 - 4561/106*e, 273/106*e^7 + 2439/53*e^5 + 21309/106*e^3 + 9447/106*e, 71/53*e^7 + 1250/53*e^5 + 5479/53*e^3 + 3673/53*e, 43/53*e^7 + 713/53*e^5 + 2527/53*e^3 - 480/53*e, 261/106*e^7 + 2358/53*e^5 + 21119/106*e^3 + 11945/106*e, -2/53*e^7 - 27/53*e^5 + 198/53*e^3 + 2236/53*e, 24/53*e^6 + 483/53*e^4 + 2553/53*e^2 + 675/53, -31/53*e^6 - 604/53*e^4 - 2867/53*e^2 - 428/53, -128/53*e^7 - 2311/53*e^5 - 10489/53*e^3 - 7363/53*e, -329/106*e^7 - 2923/53*e^5 - 25411/106*e^3 - 12453/106*e, 197/53*e^7 + 3481/53*e^5 + 14894/53*e^3 + 6594/53*e, 35/106*e^7 + 276/53*e^5 + 1941/106*e^3 + 1097/106*e, -68/53*e^6 - 1130/53*e^4 - 4186/53*e^2 - 1144/53, 54/53*e^6 + 994/53*e^4 + 4459/53*e^2 + 2274/53, -61/53*e^6 - 1115/53*e^4 - 4826/53*e^2 - 596/53, 56/53*e^6 + 1021/53*e^4 + 4314/53*e^2 - 15/53, 152/53*e^7 + 2741/53*e^5 + 12459/53*e^3 + 8568/53*e, 161/53*e^7 + 2836/53*e^5 + 12204/53*e^3 + 6562/53*e, e^7 + 19*e^5 + 98*e^3 + 113*e, 66/53*e^7 + 1209/53*e^5 + 5444/53*e^3 + 2585/53*e, 165/106*e^7 + 1498/53*e^5 + 13981/106*e^3 + 11471/106*e, 188/53*e^7 + 3333/53*e^5 + 14248/53*e^3 + 5526/53*e, 132/53*e^6 + 2365/53*e^4 + 10464/53*e^2 + 3421/53, -42/53*e^6 - 779/53*e^4 - 3580/53*e^2 - 108/53, -141/53*e^7 - 2513/53*e^5 - 11004/53*e^3 - 5973/53*e, -e^7 - 17*e^5 - 62*e^3 + 27*e, -91/53*e^7 - 1573/53*e^5 - 6414/53*e^3 - 1824/53*e, 413/106*e^7 + 3702/53*e^5 + 33207/106*e^3 + 21467/106*e, -1/53*e^6 + 13/53*e^4 - 7/53*e^2 - 366/53, -97/53*e^6 - 1601/53*e^4 - 6138/53*e^2 - 2324/53, 86/53*e^6 + 1532/53*e^4 + 6750/53*e^2 + 3015/53, -8/53*e^6 - 108/53*e^4 - 427/53*e^2 - 66/53, 37/106*e^7 + 369/53*e^5 + 3757/106*e^3 + 2677/106*e, 111/53*e^7 + 2002/53*e^5 + 9257/53*e^3 + 7554/53*e, -23/53*e^6 - 496/53*e^4 - 2758/53*e^2 - 1210/53, -102/53*e^6 - 1695/53*e^4 - 6279/53*e^2 - 656/53, 66/53*e^6 + 1103/53*e^4 + 4225/53*e^2 + 889/53, 50/53*e^7 + 887/53*e^5 + 3954/53*e^3 + 3195/53*e, 1/53*e^7 + 40/53*e^5 + 325/53*e^3 + 154/53*e, 142/53*e^7 + 2500/53*e^5 + 10693/53*e^3 + 5014/53*e, -223/106*e^7 - 1969/53*e^5 - 16825/106*e^3 - 7047/106*e, -275/106*e^7 - 2373/53*e^5 - 19309/106*e^3 - 5727/106*e, -100/53*e^7 - 1774/53*e^5 - 7537/53*e^3 - 2680/53*e, -21/53*e^7 - 310/53*e^5 - 730/53*e^3 + 2013/53*e, -157/53*e^7 - 2782/53*e^5 - 11805/53*e^3 - 3561/53*e, 6/53*e^6 + 187/53*e^4 + 1261/53*e^2 + 606/53, 38/53*e^6 + 619/53*e^4 + 2280/53*e^2 - 455/53, 145/53*e^7 + 2567/53*e^5 + 10979/53*e^3 + 3780/53*e, 167/53*e^7 + 3023/53*e^5 + 13783/53*e^3 + 9500/53*e, -37/53*e^7 - 685/53*e^5 - 3174/53*e^3 - 1882/53*e, -42/53*e^7 - 726/53*e^5 - 2732/53*e^3 + 1376/53*e, 11/53*e^6 + 175/53*e^4 + 607/53*e^2 - 532/53, 61/53*e^6 + 1062/53*e^4 + 4084/53*e^2 - 517/53, -87/53*e^6 - 1466/53*e^4 - 6015/53*e^2 - 2851/53, -72/53*e^6 - 1343/53*e^4 - 5963/53*e^2 - 912/53, -164/53*e^7 - 2956/53*e^5 - 13444/53*e^3 - 9038/53*e, 18/53*e^7 + 296/53*e^5 + 762/53*e^3 - 2687/53*e, 268/53*e^7 + 4784/53*e^5 + 20956/53*e^3 + 11274/53*e, -168/53*e^7 - 3010/53*e^5 - 13313/53*e^3 - 7852/53*e, 182/53*e^7 + 3252/53*e^5 + 14418/53*e^3 + 8471/53*e, 161/53*e^7 + 2889/53*e^5 + 12840/53*e^3 + 7940/53*e, -87/53*e^6 - 1519/53*e^4 - 6386/53*e^2 - 3063/53, 229/53*e^7 + 4072/53*e^5 + 17609/53*e^3 + 7229/53*e, -104/53*e^7 - 1881/53*e^5 - 8519/53*e^3 - 5734/53*e, -250/53*e^7 - 4488/53*e^5 - 19982/53*e^3 - 11205/53*e, -551/106*e^7 - 4872/53*e^5 - 41487/106*e^3 - 16961/106*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([2, 2, w + 1])] = 5/106*e^7 + 47/53*e^5 + 459/106*e^3 + 399/106*e AL_eigenvalues[ZF.ideal([3, 3, w + 1])] = -5/106*e^7 - 47/53*e^5 - 459/106*e^3 - 399/106*e # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]