/* 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([-21, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([15, 15, w + 2]) primes_array = [ [3, 3, w],\ [3, 3, w + 2],\ [4, 2, 2],\ [5, 5, w + 2],\ [7, 7, w],\ [7, 7, w + 6],\ [17, 17, w + 8],\ [19, 19, w + 1],\ [19, 19, w - 2],\ [23, 23, w + 9],\ [23, 23, w + 13],\ [37, 37, w + 11],\ [37, 37, w + 25],\ [59, 59, 3*w + 10],\ [59, 59, 3*w - 13],\ [73, 73, w + 15],\ [73, 73, w + 57],\ [89, 89, -w - 10],\ [89, 89, w - 11],\ [97, 97, w + 22],\ [97, 97, w + 74],\ [101, 101, 3*w - 11],\ [101, 101, -3*w - 8],\ [107, 107, w + 18],\ [107, 107, w + 88],\ [113, 113, w + 28],\ [113, 113, w + 84],\ [121, 11, -11],\ [149, 149, 3*w - 8],\ [149, 149, -3*w - 5],\ [151, 151, -3*w - 17],\ [151, 151, 3*w - 20],\ [163, 163, w + 66],\ [163, 163, w + 96],\ [167, 167, w + 34],\ [167, 167, w + 132],\ [169, 13, -13],\ [173, 173, w + 68],\ [173, 173, w + 104],\ [179, 179, 3*w - 5],\ [179, 179, -3*w - 2],\ [191, 191, 3*w - 2],\ [191, 191, 3*w - 1],\ [193, 193, w + 24],\ [193, 193, w + 168],\ [197, 197, w + 85],\ [197, 197, w + 111],\ [227, 227, w + 26],\ [227, 227, w + 200],\ [229, 229, 3*w - 22],\ [229, 229, -3*w - 19],\ [233, 233, w + 102],\ [233, 233, w + 130],\ [239, 239, 2*w - 19],\ [239, 239, -2*w - 17],\ [251, 251, -w - 16],\ [251, 251, w - 17],\ [271, 271, -3*w - 20],\ [271, 271, 3*w - 23],\ [277, 277, w + 37],\ [277, 277, w + 239],\ [281, 281, 5*w - 31],\ [281, 281, -5*w - 26],\ [283, 283, w + 29],\ [283, 283, w + 253],\ [313, 313, w + 140],\ [313, 313, w + 172],\ [317, 317, w + 134],\ [317, 317, w + 182],\ [331, 331, -4*w - 1],\ [331, 331, 4*w - 5],\ [337, 337, w + 95],\ [337, 337, w + 241],\ [347, 347, w + 49],\ [347, 347, w + 297],\ [349, 349, -5*w - 11],\ [349, 349, 5*w - 16],\ [359, 359, -w - 19],\ [359, 359, w - 20],\ [367, 367, w + 33],\ [367, 367, w + 333],\ [389, 389, 4*w - 29],\ [389, 389, -4*w - 25],\ [397, 397, w + 150],\ [397, 397, w + 246],\ [409, 409, -3*w - 23],\ [409, 409, 3*w - 26],\ [421, 421, -5*w - 8],\ [421, 421, 5*w - 13],\ [461, 461, -5*w - 29],\ [461, 461, 5*w - 34],\ [487, 487, w + 38],\ [487, 487, w + 448],\ [491, 491, 2*w - 25],\ [491, 491, -2*w - 23],\ [503, 503, w + 136],\ [503, 503, w + 366],\ [509, 509, -6*w - 13],\ [509, 509, 6*w - 19],\ [547, 547, w + 52],\ [547, 547, w + 494],\ [569, 569, 6*w - 17],\ [569, 569, -6*w - 11],\ [599, 599, 9*w - 38],\ [599, 599, 9*w + 29],\ [607, 607, w + 252],\ [607, 607, w + 354],\ [617, 617, w + 286],\ [617, 617, w + 330],\ [631, 631, 8*w - 31],\ [631, 631, -8*w - 23],\ [643, 643, w + 191],\ [643, 643, w + 451],\ [653, 653, w + 44],\ [653, 653, w + 608],\ [659, 659, -5*w - 32],\ [659, 659, 5*w - 37],\ [661, 661, 7*w - 23],\ [661, 661, -7*w - 16],\ [673, 673, w + 124],\ [673, 673, w + 548],\ [677, 677, w + 315],\ [677, 677, w + 361],\ [683, 683, w + 45],\ [683, 683, w + 637],\ [701, 701, 6*w - 11],\ [701, 701, -6*w - 5],\ [739, 739, -3*w - 29],\ [739, 739, 3*w - 32],\ [743, 743, w + 268],\ [743, 743, w + 474],\ [761, 761, 6*w - 5],\ [761, 761, 6*w - 1],\ [769, 769, 7*w - 20],\ [769, 769, -7*w - 13],\ [787, 787, w + 287],\ [787, 787, w + 499],\ [823, 823, w + 192],\ [823, 823, w + 630],\ [827, 827, w + 149],\ [827, 827, w + 677],\ [829, 829, -9*w + 55],\ [829, 829, -9*w - 46],\ [841, 29, -29],\ [853, 853, w + 310],\ [853, 853, w + 542],\ [857, 857, w + 389],\ [857, 857, w + 467],\ [859, 859, 7*w - 17],\ [859, 859, -7*w - 10],\ [877, 877, w + 51],\ [877, 877, w + 825],\ [887, 887, w + 343],\ [887, 887, w + 543],\ [907, 907, w + 67],\ [907, 907, w + 839],\ [919, 919, -8*w - 17],\ [919, 919, 8*w - 25],\ [947, 947, w + 53],\ [947, 947, w + 893],\ [961, 31, -31],\ [971, 971, -w - 31],\ [971, 971, w - 32],\ [983, 983, w + 54],\ [983, 983, w + 928],\ [997, 997, w + 151],\ [997, 997, w + 845],\ [1013, 1013, w + 313],\ [1013, 1013, w + 699],\ [1019, 1019, 9*w - 31],\ [1019, 1019, -9*w - 22],\ [1021, 1021, -7*w - 1],\ [1021, 1021, 7*w - 8],\ [1039, 1039, 7*w - 2],\ [1039, 1039, 7*w - 5],\ [1069, 1069, 3*w - 37],\ [1069, 1069, -3*w - 34],\ [1093, 1093, w + 517],\ [1093, 1093, w + 575],\ [1109, 1109, 5*w - 43],\ [1109, 1109, -5*w - 38],\ [1117, 1117, w + 342],\ [1117, 1117, w + 774],\ [1153, 1153, w + 176],\ [1153, 1153, w + 976],\ [1163, 1163, w + 413],\ [1163, 1163, w + 749],\ [1171, 1171, -6*w - 41],\ [1171, 1171, 6*w - 47],\ [1181, 1181, 4*w - 41],\ [1181, 1181, -4*w - 37],\ [1187, 1187, w + 273],\ [1187, 1187, w + 913],\ [1193, 1193, w + 91],\ [1193, 1193, w + 1101],\ [1213, 1213, w + 60],\ [1213, 1213, w + 1152],\ [1217, 1217, w + 357],\ [1217, 1217, w + 859],\ [1249, 1249, 9*w - 59],\ [1249, 1249, -9*w - 50],\ [1259, 1259, 9*w - 26],\ [1259, 1259, -9*w - 17],\ [1279, 1279, 8*w - 13],\ [1279, 1279, -8*w - 5],\ [1291, 1291, 3*w - 40],\ [1291, 1291, -3*w - 37],\ [1297, 1297, w + 505],\ [1297, 1297, w + 791],\ [1301, 1301, -9*w - 16],\ [1301, 1301, 9*w - 25],\ [1303, 1303, w + 466],\ [1303, 1303, w + 836],\ [1361, 1361, 5*w - 46],\ [1361, 1361, -5*w - 41],\ [1367, 1367, w + 293],\ [1367, 1367, w + 1073],\ [1381, 1381, 11*w - 40],\ [1381, 1381, -11*w - 29],\ [1409, 1409, 7*w - 53],\ [1409, 1409, -7*w - 46],\ [1423, 1423, w + 65],\ [1423, 1423, w + 1357],\ [1429, 1429, -12*w + 73],\ [1429, 1429, -12*w - 61],\ [1433, 1433, w + 300],\ [1433, 1433, w + 1132],\ [1471, 1471, 9*w - 61],\ [1471, 1471, -9*w - 52],\ [1481, 1481, 9*w - 20],\ [1481, 1481, -9*w - 11],\ [1493, 1493, w + 468],\ [1493, 1493, w + 1024],\ [1511, 1511, -9*w - 10],\ [1511, 1511, 9*w - 19],\ [1523, 1523, w + 561],\ [1523, 1523, w + 961],\ [1531, 1531, 3*w - 43],\ [1531, 1531, -3*w - 40],\ [1549, 1549, 10*w - 29],\ [1549, 1549, -10*w - 19],\ [1553, 1553, w + 454],\ [1553, 1553, w + 1098],\ [1567, 1567, w + 409],\ [1567, 1567, w + 1157],\ [1579, 1579, -11*w - 26],\ [1579, 1579, 11*w - 37],\ [1619, 1619, -w - 40],\ [1619, 1619, w - 41],\ [1627, 1627, w + 193],\ [1627, 1627, w + 1433],\ [1637, 1637, w + 305],\ [1637, 1637, w + 1331],\ [1663, 1663, w + 397],\ [1663, 1663, w + 1265],\ [1681, 41, -41],\ [1693, 1693, w + 739],\ [1693, 1693, w + 953],\ [1697, 1697, w + 71],\ [1697, 1697, w + 1625],\ [1699, 1699, -13*w - 37],\ [1699, 1699, 13*w - 50],\ [1709, 1709, 9*w - 8],\ [1709, 1709, 9*w - 1],\ [1721, 1721, 9*w - 4],\ [1721, 1721, 9*w - 5],\ [1723, 1723, w + 329],\ [1723, 1723, w + 1393],\ [1759, 1759, -11*w - 23],\ [1759, 1759, 11*w - 34],\ [1789, 1789, 3*w - 46],\ [1789, 1789, -3*w - 43],\ [1801, 1801, 10*w - 23],\ [1801, 1801, -10*w - 13],\ [1811, 1811, -13*w - 67],\ [1811, 1811, 13*w - 80],\ [1847, 1847, w + 324],\ [1847, 1847, w + 1522],\ [1849, 43, -43],\ [1861, 1861, 14*w - 55],\ [1861, 1861, -14*w - 41],\ [1867, 1867, w + 788],\ [1867, 1867, w + 1078],\ [1871, 1871, -w - 43],\ [1871, 1871, w - 44],\ [1873, 1873, w + 178],\ [1873, 1873, w + 1694],\ [1877, 1877, w + 766],\ [1877, 1877, w + 1110],\ [1879, 1879, -3*w - 44],\ [1879, 1879, 3*w - 47],\ [1889, 1889, -8*w - 53],\ [1889, 1889, 8*w - 61],\ [1907, 1907, w + 447],\ [1907, 1907, w + 1459],\ [1933, 1933, w + 228],\ [1933, 1933, w + 1704],\ [1951, 1951, -13*w - 34],\ [1951, 1951, 13*w - 47],\ [2003, 2003, w + 118],\ [2003, 2003, w + 1884],\ [2017, 2017, w + 356],\ [2017, 2017, w + 1660],\ [2039, 2039, -7*w - 52],\ [2039, 2039, 7*w - 59],\ [2063, 2063, w + 653],\ [2063, 2063, w + 1409],\ [2089, 2089, -10*w - 1],\ [2089, 2089, 10*w - 11],\ [2099, 2099, -12*w - 25],\ [2099, 2099, 12*w - 37],\ [2113, 2113, w + 220],\ [2113, 2113, w + 1892],\ [2129, 2129, 15*w - 59],\ [2129, 2129, -15*w - 44],\ [2137, 2137, w + 560],\ [2137, 2137, w + 1576],\ [2141, 2141, -w - 46],\ [2141, 2141, w - 47],\ [2153, 2153, w + 80],\ [2153, 2153, w + 2072],\ [2161, 2161, -3*w - 47],\ [2161, 2161, 3*w - 50],\ [2203, 2203, w + 457],\ [2203, 2203, w + 1745],\ [2207, 2207, w + 81],\ [2207, 2207, w + 2125],\ [2209, 47, -47],\ [2213, 2213, w + 1065],\ [2213, 2213, w + 1147],\ [2237, 2237, w + 680],\ [2237, 2237, w + 1556],\ [2267, 2267, w + 359],\ [2267, 2267, w + 1907],\ [2269, 2269, -12*w - 67],\ [2269, 2269, 12*w - 79],\ [2273, 2273, w + 793],\ [2273, 2273, w + 1479],\ [2311, 2311, 9*w - 68],\ [2311, 2311, -9*w - 59],\ [2357, 2357, w + 497],\ [2357, 2357, w + 1859],\ [2371, 2371, -6*w - 53],\ [2371, 2371, 6*w - 59],\ [2377, 2377, w + 1071],\ [2377, 2377, w + 1305],\ [2381, 2381, -7*w - 55],\ [2381, 2381, 7*w - 62],\ [2383, 2383, w + 327],\ [2383, 2383, w + 2055],\ [2389, 2389, 11*w - 19],\ [2389, 2389, -11*w - 8],\ [2399, 2399, -11*w - 65],\ [2399, 2399, 11*w - 76],\ [2417, 2417, w + 255],\ [2417, 2417, w + 2161],\ [2437, 2437, w + 1175],\ [2437, 2437, w + 1261],\ [2477, 2477, w + 1172],\ [2477, 2477, w + 1304],\ [2531, 2531, -12*w - 17],\ [2531, 2531, 12*w - 29],\ [2543, 2543, w + 133],\ [2543, 2543, w + 2409],\ [2549, 2549, 5*w - 58],\ [2549, 2549, -5*w - 53],\ [2551, 2551, 11*w - 10],\ [2551, 2551, 11*w - 1],\ [2557, 2557, w + 807],\ [2557, 2557, w + 1749],\ [2609, 2609, 8*w - 67],\ [2609, 2609, -8*w - 59],\ [2647, 2647, w + 684],\ [2647, 2647, w + 1962],\ [2657, 2657, w + 1194],\ [2657, 2657, w + 1462],\ [2663, 2663, w + 89],\ [2663, 2663, w + 2573],\ [2671, 2671, 3*w - 55],\ [2671, 2671, -3*w - 52],\ [2683, 2683, w + 347],\ [2683, 2683, w + 2335],\ [2693, 2693, w + 1140],\ [2693, 2693, w + 1552],\ [2699, 2699, -12*w - 13],\ [2699, 2699, 12*w - 25],\ [2711, 2711, -15*w - 38],\ [2711, 2711, 15*w - 53],\ [2713, 2713, w + 950],\ [2713, 2713, w + 1762],\ [2719, 2719, 17*w - 67],\ [2719, 2719, -17*w - 50],\ [2729, 2729, 13*w - 86],\ [2729, 2729, -13*w - 73],\ [2741, 2741, -7*w - 58],\ [2741, 2741, 7*w - 65],\ [2777, 2777, w + 139],\ [2777, 2777, w + 2637],\ [2789, 2789, -17*w + 103],\ [2789, 2789, -17*w - 86],\ [2801, 2801, 15*w - 52],\ [2801, 2801, -15*w - 37],\ [2809, 53, -53],\ [2833, 2833, w + 991],\ [2833, 2833, w + 1841],\ [2887, 2887, w + 426],\ [2887, 2887, w + 2460],\ [2897, 2897, w + 879],\ [2897, 2897, w + 2017],\ [2909, 2909, -4*w - 55],\ [2909, 2909, 4*w - 59],\ [2917, 2917, w + 1411],\ [2917, 2917, w + 1505],\ [2927, 2927, w + 1151],\ [2927, 2927, w + 1775],\ [2939, 2939, -12*w - 5],\ [2939, 2939, 12*w - 17],\ [2953, 2953, w + 829],\ [2953, 2953, w + 2123],\ [2963, 2963, w + 1028],\ [2963, 2963, w + 1934],\ [2971, 2971, 9*w - 73],\ [2971, 2971, -9*w - 64],\ [3001, 3001, 3*w - 58],\ [3001, 3001, -3*w - 55],\ [3011, 3011, -12*w - 1],\ [3011, 3011, 12*w - 13],\ [3023, 3023, w + 731],\ [3023, 3023, w + 2291],\ [3037, 3037, w + 900],\ [3037, 3037, w + 2136],\ [3041, 3041, 18*w - 71],\ [3041, 3041, -18*w - 53],\ [3061, 3061, -17*w - 47],\ [3061, 3061, 17*w - 64],\ [3067, 3067, w + 567],\ [3067, 3067, w + 2499],\ [3079, 3079, -6*w - 59],\ [3079, 3079, 6*w - 65],\ [3083, 3083, w + 372],\ [3083, 3083, w + 2710],\ [3109, 3109, -9*w - 65],\ [3109, 3109, 9*w - 74],\ [3119, 3119, 7*w - 68],\ [3119, 3119, -7*w - 61],\ [3167, 3167, w + 718],\ [3167, 3167, w + 2448],\ [3181, 3181, 12*w - 85],\ [3181, 3181, -12*w - 73],\ [3203, 3203, w + 1311],\ [3203, 3203, w + 1891],\ [3209, 3209, 21*w - 89],\ [3209, 3209, 21*w + 68],\ [3221, 3221, 15*w - 47],\ [3221, 3221, -15*w - 32],\ [3229, 3229, 22*w - 95],\ [3229, 3229, 22*w + 73],\ [3251, 3251, -5*w - 59],\ [3251, 3251, 5*w - 64],\ [3253, 3253, w + 705],\ [3253, 3253, w + 2547],\ [3257, 3257, w + 1282],\ [3257, 3257, w + 1974],\ [3299, 3299, -15*w - 31],\ [3299, 3299, 15*w - 46],\ [3319, 3319, -13*w - 10],\ [3319, 3319, 13*w - 23],\ [3331, 3331, 6*w - 67],\ [3331, 3331, -6*w - 61],\ [3343, 3343, w + 760],\ [3343, 3343, w + 2582],\ [3373, 3373, w + 438],\ [3373, 3373, w + 2934],\ [3391, 3391, -9*w - 67],\ [3391, 3391, 9*w - 76],\ [3407, 3407, w + 154],\ [3407, 3407, w + 3252],\ [3449, 3449, 15*w - 44],\ [3449, 3449, -15*w - 29],\ [3457, 3457, w + 602],\ [3457, 3457, w + 2854],\ [3463, 3463, w + 713],\ [3463, 3463, w + 2749],\ [3469, 3469, -3*w - 59],\ [3469, 3469, 3*w - 62],\ [3511, 3511, -19*w - 55],\ [3511, 3511, 19*w - 74],\ [3533, 3533, w + 685],\ [3533, 3533, w + 2847],\ [3547, 3547, w + 309],\ [3547, 3547, w + 3237],\ [3571, 3571, 13*w - 2],\ [3571, 3571, 13*w - 11],\ [3593, 3593, w + 311],\ [3593, 3593, w + 3281],\ [3607, 3607, w + 1255],\ [3607, 3607, w + 2351],\ [3643, 3643, w + 289],\ [3643, 3643, w + 3353],\ [3659, 3659, 15*w - 41],\ [3659, 3659, -15*w - 26],\ [3671, 3671, -19*w - 97],\ [3671, 3671, -19*w + 116],\ [3677, 3677, w + 160],\ [3677, 3677, w + 3516],\ [3691, 3691, -17*w - 41],\ [3691, 3691, 17*w - 58],\ [3719, 3719, 13*w - 92],\ [3719, 3719, -13*w - 79],\ [3721, 61, -61],\ [3733, 3733, w + 649],\ [3733, 3733, w + 3083],\ [3739, 3739, 20*w - 79],\ [3739, 3739, -20*w - 59],\ [3761, 3761, -w - 61],\ [3761, 3761, w - 62],\ [3767, 3767, w + 604],\ [3767, 3767, w + 3162],\ [3797, 3797, w + 526],\ [3797, 3797, w + 3270],\ [3803, 3803, w + 489],\ [3803, 3803, w + 3313],\ [3821, 3821, -20*w + 121],\ [3821, 3821, -20*w - 101],\ [3847, 3847, w + 934],\ [3847, 3847, w + 2912],\ [3851, 3851, -15*w - 23],\ [3851, 3851, 15*w - 38],\ [3853, 3853, w + 1080],\ [3853, 3853, w + 2772],\ [3889, 3889, 19*w - 71],\ [3889, 3889, -19*w - 52],\ [3907, 3907, w + 1717],\ [3907, 3907, w + 2189],\ [3911, 3911, 15*w - 37],\ [3911, 3911, -15*w - 22],\ [3917, 3917, w + 108],\ [3917, 3917, w + 3808],\ [3919, 3919, 16*w - 47],\ [3919, 3919, -16*w - 31],\ [3929, 3929, 7*w - 74],\ [3929, 3929, -7*w - 67],\ [3931, 3931, 23*w - 97],\ [3931, 3931, 23*w + 74],\ [3947, 3947, w + 1111],\ [3947, 3947, w + 2835],\ [3967, 3967, w + 1564],\ [3967, 3967, w + 2402],\ [4007, 4007, w + 1764],\ [4007, 4007, w + 2242],\ [4021, 4021, -14*w - 5],\ [4021, 4021, 14*w - 19],\ [4057, 4057, w + 142],\ [4057, 4057, w + 3914],\ [4073, 4073, w + 1445],\ [4073, 4073, w + 2627],\ [4079, 4079, -15*w - 19],\ [4079, 4079, 15*w - 34],\ [4099, 4099, 3*w - 67],\ [4099, 4099, -3*w - 64],\ [4129, 4129, 14*w - 13],\ [4129, 4129, 14*w - 1],\ [4139, 4139, -w - 64],\ [4139, 4139, w - 65],\ [4153, 4153, w + 2020],\ [4153, 4153, w + 2132],\ [4177, 4177, w + 1773],\ [4177, 4177, w + 2403],\ [4201, 4201, -22*w - 67],\ [4201, 4201, 22*w - 89],\ [4229, 4229, -15*w - 16],\ [4229, 4229, 15*w - 31],\ [4231, 4231, -3*w - 65],\ [4231, 4231, 3*w - 68],\ [4241, 4241, -13*w - 82],\ [4241, 4241, 13*w - 95],\ [4243, 4243, w + 338],\ [4243, 4243, w + 3904],\ [4253, 4253, w + 437],\ [4253, 4253, w + 3815],\ [4259, 4259, -21*w - 61],\ [4259, 4259, 21*w - 82],\ [4271, 4271, 2*w - 67],\ [4271, 4271, -2*w - 65],\ [4273, 4273, w + 1366],\ [4273, 4273, w + 2906],\ [4339, 4339, 15*w - 103],\ [4339, 4339, -15*w - 88],\ [4357, 4357, w + 1916],\ [4357, 4357, w + 2440],\ [4363, 4363, w + 114],\ [4363, 4363, w + 4248],\ [4397, 4397, w + 175],\ [4397, 4397, w + 4221],\ [4421, 4421, -4*w - 67],\ [4421, 4421, 4*w - 71],\ [4423, 4423, w + 1415],\ [4423, 4423, w + 3007],\ [4441, 4441, -9*w - 74],\ [4441, 4441, 9*w - 83],\ [4447, 4447, w + 1198],\ [4447, 4447, w + 3248],\ [4457, 4457, w + 2170],\ [4457, 4457, w + 2286],\ [4483, 4483, w + 2166],\ [4483, 4483, w + 2316],\ [4489, 67, -67],\ [4493, 4493, w + 1266],\ [4493, 4493, w + 3226],\ [4517, 4517, w + 116],\ [4517, 4517, w + 4400],\ [4567, 4567, w + 692],\ [4567, 4567, w + 3874],\ [4583, 4583, w + 940],\ [4583, 4583, w + 3642],\ [4591, 4591, 19*w - 65],\ [4591, 4591, -19*w - 46],\ [4597, 4597, w + 975],\ [4597, 4597, w + 3621],\ [4639, 4639, -3*w - 68],\ [4639, 4639, 3*w - 71],\ [4649, 4649, -15*w - 4],\ [4649, 4649, 15*w - 19],\ [4663, 4663, w + 908],\ [4663, 4663, w + 3754],\ [4679, 4679, -21*w - 58],\ [4679, 4679, 21*w - 79],\ [4691, 4691, 15*w - 17],\ [4691, 4691, -15*w - 2],\ [4703, 4703, w + 181],\ [4703, 4703, w + 4521],\ [4723, 4723, w + 545],\ [4723, 4723, w + 4177],\ [4733, 4733, w + 418],\ [4733, 4733, w + 4314],\ [4751, 4751, 15*w - 2],\ [4751, 4751, 15*w - 13],\ [4759, 4759, -9*w - 76],\ [4759, 4759, 9*w - 85],\ [4783, 4783, w + 1655],\ [4783, 4783, w + 3127],\ [4787, 4787, w + 920],\ [4787, 4787, w + 3866],\ [4817, 4817, w + 858],\ [4817, 4817, w + 3958],\ [4861, 4861, -12*w - 83],\ [4861, 4861, 12*w - 95],\ [4871, 4871, 5*w - 76],\ [4871, 4871, -5*w - 71],\ [4903, 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8034],\ [9011, 9011, 27*w - 94],\ [9011, 9011, -27*w - 67],\ [9013, 9013, w + 164],\ [9013, 9013, w + 8848],\ [9029, 9029, -21*w - 8],\ [9029, 9029, 21*w - 29],\ [9059, 9059, -7*w - 97],\ [9059, 9059, 7*w - 104],\ [9067, 9067, w + 1154],\ [9067, 9067, w + 7912],\ [9091, 9091, -28*w - 73],\ [9091, 9091, 28*w - 101],\ [9157, 9157, w + 4495],\ [9157, 9157, w + 4661],\ [9161, 9161, 21*w - 25],\ [9161, 9161, -21*w - 4],\ [9173, 9173, w + 642],\ [9173, 9173, w + 8530],\ [9181, 9181, 26*w - 85],\ [9181, 9181, -26*w - 59],\ [9187, 9187, w + 2106],\ [9187, 9187, w + 7080],\ [9199, 9199, 9*w - 109],\ [9199, 9199, -9*w - 100],\ [9203, 9203, w + 498],\ [9203, 9203, w + 8704],\ [9239, 9239, -21*w - 1],\ [9239, 9239, 21*w - 22],\ [9277, 9277, w + 500],\ [9277, 9277, w + 8776],\ [9281, 9281, 21*w - 20],\ [9281, 9281, 21*w - 1],\ [9293, 9293, w + 1705],\ [9293, 9293, w + 7587],\ [9323, 9323, w + 3780],\ [9323, 9323, w + 5542],\ [9341, 9341, 21*w - 5],\ [9341, 9341, 21*w - 16],\ [9343, 9343, w + 4028],\ [9343, 9343, w + 5314],\ [9349, 9349, 23*w - 55],\ [9349, 9349, -23*w - 32],\ [9371, 9371, 21*w - 10],\ [9371, 9371, 21*w - 11],\ [9377, 9377, w + 2992],\ [9377, 9377, w + 6384],\ [9413, 9413, w + 2982],\ [9413, 9413, w + 6430],\ [9419, 9419, -11*w - 104],\ [9419, 9419, 11*w - 115],\ [9431, 9431, 24*w - 65],\ [9431, 9431, -24*w - 41],\ [9439, 9439, 32*w - 127],\ [9439, 9439, -32*w - 95],\ [9461, 9461, 4*w - 101],\ [9461, 9461, -4*w - 97],\ [9463, 9463, w + 1279],\ [9463, 9463, w + 8183],\ [9497, 9497, w + 4619],\ [9497, 9497, w + 4877],\ [9511, 9511, 3*w - 100],\ [9511, 9511, -3*w - 97],\ [9521, 9521, 30*w - 113],\ [9521, 9521, -30*w - 83],\ [9539, 9539, -10*w - 103],\ [9539, 9539, 10*w - 113],\ [9547, 9547, w + 3380],\ [9547, 9547, w + 6166],\ [9601, 9601, -23*w - 29],\ [9601, 9601, 23*w - 52],\ [9631, 9631, 37*w - 158],\ [9631, 9631, 37*w + 121],\ [9689, 9689, -16*w - 115],\ [9689, 9689, 16*w - 131],\ [9697, 9697, w + 1572],\ [9697, 9697, w + 8124],\ [9739, 9739, 15*w - 128],\ [9739, 9739, -15*w - 113],\ [9749, 9749, -13*w - 109],\ [9749, 9749, 13*w - 122],\ [9787, 9787, w + 4689],\ [9787, 9787, w + 5097],\ [9791, 9791, 27*w - 89],\ [9791, 9791, -27*w - 62],\ [9803, 9803, w + 1375],\ [9803, 9803, w + 8427],\ [9811, 9811, -21*w - 128],\ [9811, 9811, 21*w - 149],\ [9833, 9833, w + 4830],\ [9833, 9833, w + 5002],\ [9839, 9839, 24*w - 61],\ [9839, 9839, -24*w - 37],\ [9851, 9851, -29*w - 152],\ [9851, 9851, 29*w - 181],\ [9857, 9857, w + 3977],\ [9857, 9857, w + 5879],\ [9859, 9859, -25*w - 46],\ [9859, 9859, 25*w - 71],\ [9883, 9883, w + 3577],\ [9883, 9883, w + 6305],\ [9887, 9887, w + 2342],\ [9887, 9887, w + 7544],\ [9923, 9923, w + 4658],\ [9923, 9923, w + 5264],\ [9929, 9929, -23*w - 134],\ [9929, 9929, 23*w - 157],\ [9941, 9941, -27*w - 61],\ [9941, 9941, 27*w - 88],\ [9949, 9949, 38*w + 125],\ [9949, 9949, 38*w - 163],\ [9967, 9967, w + 3410],\ [9967, 9967, w + 6556],\ [9973, 9973, w + 2029],\ [9973, 9973, w + 7943]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 + 4 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, 1/2*e, 1, 1/2*e, 2*e, -2*e, 3*e, 2, -4, 0, 0, 4*e, -e, 0, 12, -5*e, -7*e, -6, 12, 5*e, 4*e, 12, -18, 6*e, -3*e, -9*e, 9*e, 4, 6, 0, -20, 10, 8*e, -2*e, -3*e, 6*e, -10, -3*e, 3*e, -12, -6, 18, 0, 7*e, 11*e, -6*e, -12*e, 6*e, -6*e, 20, -16, -6*e, 0, 0, 30, 6, 12, 16, -2, 5*e, 13*e, -12, -6, -2*e, -7*e, 13*e, -e, -15*e, 9*e, 20, 20, -13*e, 10*e, 0, -6*e, 22, 10, -36, -12, 13*e, -4*e, 18, 6, -11*e, -7*e, 34, 22, 34, 22, -6, -30, e, 8*e, -18, 36, -12*e, -3*e, 18, 42, 2*e, 4*e, 6, 24, 0, -6, 14*e, 16*e, 3*e, 21*e, 46, 4, -14*e, -13*e, -24*e, 0, -6, 12, -10, 2, 13*e, 5*e, 3*e, 21*e, -6*e, -18*e, 18, 30, -20, -20, 24*e, 0, -12, 6, 20, -40, -20*e, 8*e, 23*e, -20*e, -24*e, 21*e, -16, -16, -50, -22*e, 13*e, 3*e, 9*e, -40, -46, -5*e, 5*e, 6*e, -12*e, -10*e, -8*e, -20, -8, 6*e, 12*e, -10, -18, 12, 12*e, -24*e, -20*e, 11*e, -12*e, -6*e, 54, -36, 14, -58, 14, -4, 26, -10, -5*e, -19*e, 30, 0, 7*e, 29*e, 14*e, 25*e, 18*e, 6*e, 22, 4, -30, 30, 6*e, -6*e, -3*e, 3*e, -23*e, 23*e, 9*e, -33*e, -62, -38, -60, 6, -56, -20, -40, 20, -20*e, 5*e, 6, -6, 28*e, 26*e, 18, 66, -30*e, -21*e, 40, 4, -6, -18, -8*e, -37*e, -58, -46, -21*e, -3*e, 32, 20, -30, -18, 9*e, -27*e, 48, 12, -30*e, -6*e, 56, 20, 20, 20, -27*e, -3*e, 19*e, -4*e, 4, 4, -60, 12, -11*e, 26*e, 9*e, -3*e, -22*e, -32*e, 34, 17*e, -17*e, -9*e, -15*e, -10, 44, 30, -66, 42, 30, -20*e, -16*e, 16, -20, 32, 32, 38, -70, 78, -12, -15*e, -18*e, 32, 10, -26, 38*e, -11*e, 30, -48, -31*e, 37*e, 6*e, -18*e, -8, -68, -54, -30, -12*e, 9*e, 41*e, 43*e, -16, -40, 24*e, -6*e, 35*e, -14*e, -72, 78, -45*e, -18*e, 62, -34, 36, 0, -41*e, -40*e, 6, 0, 11*e, 16*e, 0, -30, -36*e, 42*e, 64, -56, 11*e, 10*e, 9*e, 24*e, -2, -3*e, 21*e, -3*e, 45*e, 24*e, -6*e, 92, -40, 15*e, -39*e, -8, -62, 27*e, -3*e, -50, 4, 34*e, -43*e, 60, -30, 23*e, 28*e, -50, -14, 36, -24, -3*e, -45*e, 4*e, -31*e, -9*e, 39*e, -24, -84, -45*e, -18*e, -78, -78, 22, -32, -e, -26*e, -42, -36, -26*e, -4*e, 6*e, 24*e, 0, 33*e, 44, -16, -37*e, 10*e, -39*e, 21*e, 84, -84, -12, 60, 7*e, -25*e, 4, -56, 42, -6, 42, 54, -42*e, -42*e, 90, 84, -66, -72, 34, -28*e, -17*e, -17*e, -16*e, 30*e, 30*e, 90, -24, -29*e, -13*e, -6*e, -21*e, -78, -36, 44*e, 25*e, -42*e, -6*e, 20, -16, 14, 50, 30, 36, -42*e, -21*e, -e, 10*e, -18, -54, 10, 70, -34*e, -8*e, 16, 16, -15*e, 12*e, -86, -26, 42, -24, 6*e, -12*e, -94, 38, -18*e, -21*e, 6, 54, -48, -6, -82, 26, -48, 0, 56*e, -23*e, 39*e, -45*e, 12, 102, -28, -82, 20, 56, -20*e, -31*e, 7*e, 32*e, -28, 8, -36*e, -54*e, 42, 60, 13*e, 17*e, -4*e, 10*e, 10, -38, -40, 32, -15*e, -21*e, 8*e, -44*e, 20, 20, 3*e, 3*e, 4*e, -58*e, -26*e, 5*e, -84, 6, -36, -60, -6*e, -36*e, 82, -8, 96, 96, -94, -7*e, -11*e, 112, -68, 114, -66, -36*e, 6*e, 54*e, -6*e, -30*e, -48*e, 30, -66, 56*e, -35*e, -60, -30, -53*e, 44*e, -64, 56, -50*e, -28*e, -72, -90, 51*e, 27*e, -10, 20, -12, 66, 16, -26, 24*e, -6*e, 7*e, 20*e, -12*e, -6*e, 10, -50, 23*e, 46*e, -21*e, 57*e, 0, 90, 50, -112, 118, -26, 48, 84, 7*e, 2*e, -e, 37*e, -82, 98, 30, 36, -20, -38, 12, -42, 11*e, -38*e, -57*e, -33*e, -120, -108, -48, 120, -23*e, -13*e, 14, 44, -e, 37*e, -38*e, -7*e, -39*e, 57*e, -60, 30, 47*e, -56*e, 112, -32, -28*e, 28*e, -39*e, -33*e, -28*e, -32*e, -100, -15*e, -27*e, 27*e, 63*e, -28*e, -44*e, 21*e, -6*e, -112, -16, 49*e, -13*e, 16, 28, 30, 48, -32*e, 17*e, 96, 78, 72, 0, -24*e, 48*e, 56*e, 4*e, -15*e, 27*e, 42, 72, 2, 44, -7*e, 52*e, 27*e, 54*e, 9*e, -63*e, -44, -20, -132, 36, 23*e, -32*e, 74, -10, 60, -96, -17*e, 26*e, 33*e, 33*e, 40, -2, 47*e, 19*e, -9*e, 6*e, 46*e, 44*e, -11*e, -64*e, -58, 68, 24*e, 66*e, 92, -64, 52, -36, -12, 31*e, 59*e, 138, -66, 108, -72, -118, 50, 38*e, -47*e, 88, -8, 6, -108, 7*e, 47*e, 20*e, 61*e, 12, -6, 54*e, 0, -72, -6, -42*e, -12*e, 42*e, 0, 138, 48, -102, -42, 2*e, -35*e, -69*e, 27*e, -56, -44, -40, 140, -62*e, 53*e, -6, -42, 22*e, 41*e, -22, -22, -9*e, 27*e, 5*e, 4*e, -34, -10, -51*e, -27*e, -24, -84, -52*e, -65*e, -46, -100, 30, 30, 2*e, 52*e, -48, 54, 21*e, 9*e, 14*e, -44*e, -4, -82, 24*e, -21*e, -78, -18, -69*e, -66*e, 40, 64, 18*e, 42*e, 90, -66, 48, -54, -26, -50, -26, -98, 62*e, -8*e, 36*e, -3*e, -52*e, 37*e, -66*e, -12*e, -4*e, -26*e, 48*e, -33*e, 96, 6, 63*e, -39*e, 106, -134, 60*e, 12*e, -43*e, 16*e, -108, 150, -22, 57*e, 0, 114, -6, -110, -56, -36*e, 24*e, 120, 12, 24, 132, -66*e, 48*e, 21*e, 57*e, -84, -36, -82, 116, 47*e, 31*e, -68, 70, -146, 106, -106, 74, -122, 94, 88, 70, 18, -42, 61*e, 56*e, 41*e, 16*e, 9*e, 9*e, 56, -94, 18, 42, -32*e, -52*e, 120, 42, 39*e, 27*e, -2*e, -52*e, 84, 18, -94, -130, 132, -36, -46*e, -23*e, -69*e, 30*e, -58*e, 46*e, 9*e, 24*e, 33*e, -15*e, 72*e, -21*e, -42, 108, 94, 52*e, 38*e, -24, 138, -3*e, -72*e, 106, 106, -22, 98, 8*e, 28*e, -66, -108, -39*e, -33*e, 16, -158, 43*e, 65*e, -60, 24, -22*e, -50*e, -160, -70, -15*e, 36*e, -78*e, -45*e, 126, -18, -104, 76, -8*e, -19*e, 25*e, -31*e, -90, -30, 35*e, -44*e, -6*e, -15*e, -33*e, -3*e, 24*e, 42*e, 69*e, 6*e, -146, -14, -24, -108, -40*e, 55*e, -4, -136, 64, -38, 53*e, -44*e, 39*e, 9*e, -32, -164, -e, -59*e, -114, -72, -72*e, 18*e, 56, 140, 132, -66, 14*e, -59*e, -9*e, 21*e, -24, 42, 79*e, 35*e, 74, -106, 110, -34, 12*e, 6*e, -98, 16, -75*e, 72*e, 174, 42, -16, 32, -3*e, 63*e, 4*e, 74*e, 100, -44, 10*e, 23*e, -24*e, 42*e, -9*e, -45*e, 30*e, -48*e, 39*e, 6*e, 24, 150, 102, 18, -45*e, 3*e, 80, 14, -36*e, 81*e, -102, -84, -40*e, -23*e, -25*e, 7*e, 58*e, -13*e, 32*e, -83*e, 102, -36, -52, 128, -44*e, -55*e, 108, -72, 67*e, 68*e, -140, -92, 24*e, 24*e, -10, 122, -48, -162, -60*e, 3*e, 88, 52, 44*e, -83*e, 28, 4, 178, -98, 60, 24, 35*e, 19*e, 57*e, -9*e, 61*e, -10*e, 140, 104, -122, 166, -55*e, -23*e, 90*e, -21*e, 70, -98, 40, 28, 8, 8, -38*e, -19*e, -102, 90, -94, 158, 74*e, -2*e, -24*e, 24*e, -43*e, -2*e, -3*e, -87*e, 74, 122, -42*e, 54*e, -22*e, -50*e, 24*e, -54*e, -90, 78, -e, 85*e, 62, -130, 27*e, 87*e, -56*e, -88*e, -136, 26, -12*e, 18*e, 8*e, 76*e, -72, 12, -46, -94, 48, 30, -3*e, -39*e, -104, -128, 108, -66, 54, 78, 50*e, -50*e, -54*e, -3*e, -178, 14, 22, -14, 120, -180, 124, -56, -68*e, 32*e, -108, -48, 19*e, -46*e, 18, -6, 84, -42, -82*e, 34*e, 92, -52, 61*e, 11*e, 186, 42, -12*e, -66*e, 16, 52, -58*e, -17*e, 152, 2, -42*e, 21*e, -60, -120, -31*e, -74*e, -6, -18, 15*e, 21*e, 9*e, 18*e, 42, 66, -92*e, -28*e, 10, 190, -156, -162, 27*e, 81*e, -63*e, 33*e, -24, -120, 0, 36, -68, 112, 150, -102, -56*e, -19*e, -33*e, 57*e, 8, 8, 108, -102, 60, 66, 86*e, -68*e, -56, 4, 80, -148, 66, 120, -31*e, 70*e, -20, -164, 90, -90, 74*e, -11*e, -180, 72, -6*e, 6*e, -62, -8, 30*e, 0, 180, 60, -132, 84, -15*e, 57*e, -4, -22, -50*e, -e, -84*e, -63*e, 72*e, -42*e, 102, 42, 72, -186, 190, -86, 55*e, -76*e, -46*e, -17*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 + 2])] = -1/2*e AL_eigenvalues[ZF.ideal([5, 5, w + 2])] = -1/2*e # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]