/* This code can be loaded, or copied and pasted, into Magma. It will load the data associated to the HMF, including the field, level, and Hecke and Atkin-Lehner eigenvalue data. At the *bottom* of the file, there is code to recreate the Hilbert modular form in Magma, by creating the HMF space and cutting out the corresponding Hecke irreducible subspace. From there, you can ask for more eigenvalues or modify as desired. It is commented out, as this computation may be lengthy. */ P := PolynomialRing(Rationals()); g := P![-21, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [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 - 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97], [8929, 8929, -3*w - 94], [8941, 8941, -29*w - 80], [8941, 8941, 29*w - 109], [8951, 8951, 21*w - 31], [8951, 8951, -21*w - 10], [9001, 9001, 35*w - 148], [9001, 9001, 35*w + 113], [9007, 9007, w + 972], [9007, 9007, w + 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 - 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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 := [ideal : I in primesArray]; heckePol := x^3 - 8*x + 4; K := NumberField(heckePol); heckeEigenvaluesArray := [1, e, 1, 1/2*e^2 - 2, 1/2*e^2, -1/2*e^2 - e + 2, -e + 4, -e^2 - e + 4, e - 2, -1/2*e^2 - 2*e + 4, -3/2*e^2 + e + 8, -3/2*e^2 - e + 12, 1/2*e^2 - e - 2, e^2 + 4*e - 8, -e^2 + 4, e^2 + 2*e - 4, 4*e + 4, -2*e^2 - 5*e + 16, -2*e^2 - 3*e + 8, 2*e^2 + 5*e - 8, -e^2 - 3*e, 2*e + 4, 4*e - 4, 2*e^2 + 5*e - 16, -e^2 + 2*e + 8, -4*e + 4, -2*e^2 - 6*e + 8, e^2 + 2*e - 10, 2*e - 2, 2*e^2 - 6*e - 20, -4*e^2 + 18, 2*e^2 - 24, -3*e^2 - 3*e + 8, 2*e^2 + 2*e - 20, 5/2*e^2 + 6*e - 22, 1/2*e^2 - 3*e + 10, 2*e^2 + 4*e - 22, -5/2*e^2 - 3*e + 10, 3/2*e^2 + e + 2, -2*e^2 - e + 18, -5*e^2 - 3*e + 20, 3*e^2 - 2*e - 10, -2*e^2 + 8, -4*e^2 + 24, -3*e^2 - 2*e + 16, 1/2*e^2 + 5*e + 8, 1/2*e^2 + e - 12, -e^2 + 4*e + 12, e^2 - e, -4*e^2 - 8*e + 20, 2*e^2 - 18, e^2 - e - 12, 3*e^2 + 3*e - 24, -2*e^2 - 8*e + 16, 5*e^2 + 4*e - 18, e^2 + 5*e, -5*e + 2, -4*e^2 - 6*e + 18, 3*e^2 + 4*e - 26, 11/2*e^2 + 6*e - 32, 5/2*e^2 - 14, e + 8, -2*e^2 - 9*e + 8, -e^2 - 3*e + 24, 4*e^2 + 6*e - 20, 5*e^2 + 6*e - 24, 8, -3/2*e^2 + 18, 9/2*e^2 + 6*e - 30, 2*e^2 - 2*e - 16, -e^2 + 24, 6*e^2 + 9*e - 28, 5*e + 12, 5*e^2 + 10*e - 24, -3*e, e^2 - 6*e - 20, -e^2 + 4*e + 8, 2*e^2 - 4*e - 16, 6*e^2 + 4*e - 22, -3/2*e^2 - 12*e + 8, 7/2*e^2 + 7*e - 22, -2*e^2 - 4*e + 6, 6*e^2 + 10*e - 34, 17/2*e^2 + 6*e - 32, -1/2*e^2 + 10, 8*e^2 + 5*e - 44, -3*e - 28, 3*e^2 + 6*e - 28, 2*e^2 + 6*e + 6, 7*e^2 + 10*e - 32, -6*e^2 + 4*e + 38, -15/2*e^2 - 6*e + 38, 1/2*e^2 - 13*e - 8, 5*e + 4, 7*e + 14, 7/2*e^2 - 5*e - 36, -7/2*e^2 - 6*e + 8, 5*e^2 + 12*e - 32, 2*e^2 - 2*e + 18, e^2 + 2*e + 20, -e^2 - 5*e + 8, 2*e^2 - 2, 3*e^2 + 2*e - 26, -6*e^2 + 10*e + 44, 8*e^2 + 8*e - 28, 1/2*e^2 - 8*e + 2, 17/2*e^2 + 3*e - 44, -e^2 - 4*e + 20, 2*e^2 + 12*e - 20, -2*e^2 + 4*e, -2*e^2 + 6*e - 8, 3*e - 16, 2*e^2 + 10*e - 4, 1/2*e^2 - 3*e + 28, 1/2*e^2 + 3*e + 4, -2*e + 20, 8*e^2 + 4*e - 24, 4*e^2 + 4*e - 48, 2*e^2 + 6*e + 12, -4*e^2 - 2*e + 8, -2*e + 16, 3/2*e^2 - 10*e - 8, 3/2*e^2 - 6*e - 24, -3*e^2 + 4*e + 32, -4*e^2 - 7*e + 20, 2*e^2 - 6*e - 26, 4*e^2 + 2*e - 18, -11*e + 10, -8*e^2 - e + 44, 3/2*e^2 - 11*e - 28, -15/2*e^2 + 36, -11*e^2 - 14*e + 58, -2*e^2 - 8*e + 34, -4*e^2 - 12*e + 18, -2*e^2 - 4*e + 26, 2*e^2 - 9*e - 36, -6*e^2 - 4*e, 7/2*e^2 + 2*e - 32, -11/2*e^2 - 3*e + 42, -6*e^2 - 5*e + 8, e^2 + 4*e + 4, 2*e^2 + 8*e - 8, -2*e^2 + 10*e + 12, -11*e^2 - e + 50, 5/2*e^2 + 3*e - 44, -7/2*e^2 + e + 42, 2*e^2 - 4*e - 4, 2*e^2 - 2*e - 48, e^2 - 8*e - 24, -9*e^2 - 10*e + 36, -1/2*e^2 + 8*e + 8, -19/2*e^2 - 18*e + 58, -11/2*e^2 - 8*e + 54, 1/2*e^2 - 5*e + 18, -7*e^2 - 16*e + 40, -5*e^2 - 11*e + 24, -10*e^2 - 6*e + 60, -2*e^2 + 2*e + 8, 4*e^2 + 3*e - 28, -e^2 + 6*e + 4, -5*e - 8, -8*e - 4, -e^2 + 8*e + 20, -5/2*e^2 - 3*e + 46, 11/2*e^2 - 4*e - 6, -17/2*e^2 - e + 40, 3/2*e^2 + 5*e - 18, -5/2*e^2 + 13*e + 12, -1/2*e^2 + 11*e + 24, 6*e^2 + 8*e - 44, 5*e^2 - 16*e - 40, -4*e^2 + 6*e + 18, 4*e^2 + 16*e - 24, e^2 - 12*e + 10, -4*e^2 - 8*e + 20, -e^2 + 10*e - 16, 2*e^2 - 4*e - 26, 7/2*e^2 - 18*e - 24, 17/2*e^2 + 4*e - 66, -2*e^2 + 8*e + 12, -4*e^2 + 4, -13/2*e^2 - 12*e + 46, -11/2*e^2 - 2*e + 44, e^2 - 3*e + 16, 8*e^2 + 5*e - 20, -17*e^2 - 9*e + 68, 5*e^2 + 10*e - 36, 5*e^2 + 17*e - 20, -4*e^2 - 5*e + 50, 2*e^2 - 6*e + 22, -2*e^2 + 6*e + 38, 9*e^2 + 6*e - 60, -3*e^2 + 9*e + 12, 9*e^2 + 12*e - 60, 6*e^2, -7/2*e^2 - 8*e + 60, -1/2*e^2 + 16*e + 6, -2*e^2 - 8*e - 32, 6*e^2 + 6*e, 5*e^2 + 18*e - 18, -7*e^2 - 14*e + 14, -8*e^2 - 4*e + 36, e^2 - 2*e + 28, -6*e^2 + 2*e + 36, -15*e^2 - 10*e + 66, -5*e^2 + 8*e + 8, 2*e^2 + 4*e - 16, -7*e^2 + 3*e + 12, -5*e^2 + 11*e + 20, 11*e^2 - 4*e - 64, -5*e^2 - 6*e + 20, -13/2*e^2 + 4*e + 42, 3/2*e^2 - 17*e - 20, -4*e^2 + 58, -8*e^2 + 4*e + 34, 9/2*e^2 + 5*e - 2, -11/2*e^2 + 12*e + 22, 8*e + 18, 2*e^2 - 10*e - 14, -12*e^2 - 25*e + 82, 3*e^2 + 13*e - 22, -11/2*e^2 - 7*e + 28, -11/2*e^2 + 12*e + 42, 6*e^2 + 8*e - 66, e^2 + 6*e + 40, -13*e^2 - 15*e + 64, -8*e^2 + 9*e + 68, -8*e^2 + 16, 4*e^2 + 2*e - 40, 4*e^2 + 9*e + 14, 4*e^2 + 27*e - 22, 19/2*e^2 + 6*e - 14, 3/2*e^2 - 4*e + 10, -2*e^2 + 10*e - 26, -8*e^2 - 6*e + 80, -e^2 - 4*e + 8, -11*e^2 - 11*e + 52, 10*e^2 - 2*e - 60, 3*e^2 - 16*e - 36, -2*e^2 - 2*e + 24, 6*e^2 + 20*e - 66, 11*e^2 - 5*e - 60, 6*e^2 - e + 4, -5/2*e^2 + 4*e + 62, 23/2*e^2 + 15*e - 80, 9*e^2 + 20*e - 48, 2*e^2 - 16*e - 8, -2*e^2 - 21*e + 8, -6*e^2 + 5*e + 58, 6*e^2 - 2*e - 44, -5*e^2 - 3*e - 8, 3/2*e^2 - 11*e + 2, 27/2*e^2 + 27*e - 90, -9/2*e^2 - e - 24, 7/2*e^2 - 14*e - 46, 11*e^2 + 8*e - 38, -7/2*e^2 - 12*e + 16, 3/2*e^2 + 14*e - 42, 11*e^2 + 3*e - 20, -10*e^2 - 21*e + 52, 8*e^2 + 13*e - 28, 4*e^2 - 9*e - 38, 5*e^2 + 14*e - 64, -10*e^2 - 12*e + 46, -6*e^2 - 19*e + 22, 11*e^2 + 7*e - 66, e^2 - 4*e + 32, 9*e - 40, 20*e + 6, e^2 - 10*e - 34, 8*e^2 - 2*e - 76, 6*e^2 + 10*e - 38, e^2 + 9*e + 22, -2*e^2 - 19*e + 2, 9*e^2 - 24, 11*e^2 + 8*e - 52, 9/2*e^2 + 30, 17/2*e^2 + 11*e - 78, 8*e^2 + 20*e - 86, -4*e^2 + 6*e - 6, 2*e^2 - 4*e + 10, -17*e + 28, 11*e^2 + 8*e - 68, -15*e^2 - 10*e + 82, -10*e^2 - 4*e + 50, 10*e^2 - 18*e - 68, -13*e^2 + 4*e + 68, 15/2*e^2 + e - 82, 7/2*e^2 + 9*e - 14, 14*e^2 + 20*e - 98, -6*e^2 - 30*e + 44, -e^2 - 17*e + 30, -3*e^2 - 11*e + 2, 3*e^2 - 11*e - 40, 9*e^2 + 24*e - 60, 1/2*e^2 - 6*e - 12, 11/2*e^2 + 2*e + 38, -4*e^2 - 14*e - 16, -7*e^2 - 16*e + 58, -8*e + 56, -7*e^2 - e + 32, -12*e^2 - 23*e + 88, 7*e^2 + 13*e, 8*e^2 + 16*e - 24, 14*e^2 + 4*e - 36, 17/2*e^2 - 4*e - 36, -25/2*e^2 - 3*e + 44, 10*e^2 - 5*e - 66, -4*e^2 + 15*e + 42, 3*e^2 - 11*e - 28, -14*e^2 + 3*e + 62, -7*e^2 - 7*e + 40, -10*e^2 - 7*e + 88, 10*e^2 + 25*e - 56, -14*e^2 - 5*e + 52, -5*e^2 + 9*e + 40, 3*e^2 + 5*e - 72, 8*e^2 - 10*e - 52, -8*e^2 - 8*e + 70, -6*e^2 - 7*e + 28, -9*e^2 - 19*e + 52, 8*e - 30, 10*e^2 - 4*e - 34, -4*e^2 + 22*e + 56, 11*e^2 - 7*e - 32, -33/2*e^2 - 22*e + 76, -15/2*e^2 - 9*e + 24, 6*e^2 + 20*e - 18, -17/2*e^2 + 2*e + 92, -13/2*e^2 + 4*e + 16, 27/2*e^2 + 15*e - 36, -29/2*e^2 - 25*e + 68, -5*e^2 - 14*e + 28, 4*e^2 + 19*e - 32, 18*e^2 + 30*e - 100, -2*e^2 + 14*e - 4, 10*e^2 + 14*e - 84, 5*e^2 + 18*e - 56, 2*e^2 - 18*e + 12, 13*e^2 + 4*e - 18, 3/2*e^2 - 13*e + 28, -1/2*e^2 - 21*e + 20, 10*e^2 + 22*e - 36, -15*e^2 - 8*e + 88, 5*e^2 + 11*e - 32, 4*e^2 + 7*e + 36, 8*e^2 + 18*e - 50, 12*e + 12, 13/2*e^2 - 24, -17/2*e^2 - 19*e + 94, 6*e^2 + 20*e - 36, 10*e^2 + 8*e - 34, 16*e + 14, -2*e^2 + 10*e - 20, -5*e^2 + 12*e + 20, -4*e^2 - 20, 5/2*e^2 + e + 18, 1/2*e^2 - 7*e + 28, -7/2*e^2 - 4*e + 12, -11/2*e^2 + 12*e + 16, -3*e^2 - 18*e + 60, 14*e^2 + 4*e - 84, -11/2*e^2 - 14*e + 42, -31/2*e^2 - 3*e + 86, 6*e^2 + 18*e - 30, 12*e^2 - 6*e - 66, -14*e^2 - 10*e + 44, -8*e^2 - 14*e + 36, -19/2*e^2 - 11*e + 54, -27/2*e^2 - 9*e + 56, 8*e^2 - e - 88, -4*e^2 + e - 48, 13/2*e^2 + e - 70, 31/2*e^2 + 14*e - 104, -6*e^2 - 11*e + 44, 5*e^2 - 7*e - 76, -15/2*e^2 - e + 16, -13/2*e^2 + 12*e + 56, -2*e^2 + 10*e, 2*e^2 - 20*e - 34, 7*e^2 + 16*e - 84, -3*e^2 + 7*e + 40, 5/2*e^2 + 24*e - 34, -3/2*e^2 - 16*e + 10, 7*e^2 - 22*e - 60, -6*e^2 - 8*e - 16, 2*e^2 - 20, 14*e^2 + 18*e - 104, -e^2 - 6*e + 36, 32*e + 12, -17*e^2 - 20*e + 66, 2*e^2 - 12*e + 18, -3*e^2 - 17*e - 34, -14*e^2 + e + 106, -7*e^2 - 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8, 10*e^2 + 35*e - 48, -16*e^2 + 40*e + 152, 6*e^2 - 16*e - 84, 14*e^2 + 11*e - 8, 12*e^2 - 17*e - 80, -4*e^2 + 38*e + 68, -34*e^2 - 40*e + 158, 7/2*e^2 + 19*e + 2, -5/2*e^2 - 3*e - 120, 25*e^2 + 19*e - 120, -2*e^2 - 59*e - 6, -57/2*e^2 - 7*e + 94, -1/2*e^2 + 12*e + 2, -21*e^2 - 14*e + 88, 15*e^2 - 8*e - 116, 2*e^2 + 10*e + 120, -10*e^2 - 18*e - 8, -13*e^2 + 23*e + 140, -27*e^2 - 70*e + 196, 7*e^2 - 10*e - 82, 13*e^2 + 26*e + 2, 1/2*e^2 - 35*e + 20, -11/2*e^2 + 42*e + 6, 12*e^2 - 28*e - 108, -2*e^2 - 32*e - 12, -26*e^2 - 19*e + 116, -12*e^2 + 21*e + 92, 26*e^2 + 23*e - 118, 8*e^2 - 3*e - 68, -16*e^2 - 26*e + 80, 2*e^2 - 10*e + 88, 6*e^2 - 16*e - 92, 8*e^2 + 10*e + 96, -27/2*e^2 + 25*e + 58, 25/2*e^2 + 5*e - 116, 9*e^2 + 22*e - 38, -18*e^2 + 8*e + 54, -28*e^2 - 21*e + 96, -6*e^2 + 19*e + 52, 8*e^2 + 18*e, 19*e^2 + 28*e - 180, 5*e^2 - 23*e + 12, -6*e^2 + 2*e - 92, -6*e^2 + 8*e + 66, -17*e^2 - 4*e + 164, -9*e^2 - 44*e + 28, -20*e^2 - 24*e + 124, -13*e^2 - 2*e + 134, 18*e - 28, 16*e^2 + 35*e - 160, -7*e^2 + 8*e - 8, 22*e + 62, -28*e^2 + 2*e + 122, -2*e^2 - e + 116, 12*e^2 + 15*e - 136, 55/2*e^2 + 12*e - 186, -9/2*e^2 - 7*e + 168, -4*e^2 + 19*e - 24, -e^2 - 45*e + 4, 15*e^2 + 18*e - 64, 4*e^2 + 21*e - 44, 33/2*e^2 + 46*e - 118, -15/2*e^2 - 12*e - 42, 14*e^2 + 6*e - 54, 9*e^2 + 14*e - 132, -27/2*e^2 - 31*e + 118, 21/2*e^2 - 11*e - 58, -69/2*e^2 - 2*e + 172, 37/2*e^2 + 53*e - 122, 25/2*e^2 - 7*e - 58, -31/2*e^2 - 10*e + 114, 8*e^2 + 20*e - 34, -21*e^2 + 32*e + 148, 23/2*e^2 + 36*e - 110, -51/2*e^2 - 54*e + 176, -5*e^2 + 28*e - 18, 7*e^2 - 40*e - 82, -27/2*e^2 + 2*e - 8, 57/2*e^2 + 24*e - 120, 4*e^2 + e - 108, -6*e^2 - 50*e + 76, -14*e^2 + 32*e + 104, -4*e^2 + 2*e + 136, -7/2*e^2 - 19*e + 150, -15/2*e^2 - 28*e + 34, 13*e^2 + 33*e - 104, 11*e^2 - 28*e - 8, 12*e^2 + 23*e - 50, 9*e^2 + 21*e - 156, -10*e^2 + 28*e + 140, -6*e^2 + 14*e + 114, -6*e^2 - 20*e - 46, -11*e^2 - 44*e + 22, -19/2*e^2 + 43*e + 70, -35/2*e^2 - 13*e + 170, 6*e + 122, 12*e^2 + 6*e - 52, -20*e^2 + 11*e + 108, -38*e^2 - 31*e + 168, 15*e^2 - 22*e - 164, -9*e^2 - 6*e + 120, 13/2*e^2 - 29*e - 16, 49/2*e^2 + 30*e - 198, -16*e^2 + 17*e + 56, 18*e^2 + 36*e - 204, 13*e^2 - 134, 12*e - 46, -5*e^2 + 12*e - 80, 3*e^2 - 12*e + 32, 24*e^2 + 40*e - 160, 30*e^2 - 6*e - 162, 10*e^2 - 39*e - 92, -24*e^2 - 23*e + 136, 33/2*e^2 - 3*e - 46, 55/2*e^2 + 26*e - 176, -4*e^2 + 29*e + 32, 10*e^2 + 51*e - 46, -11/2*e^2 - e + 34, 53/2*e^2 + 11*e - 40, 6*e^2 - 24*e + 30, -33*e^2 - 4*e + 160, 18*e^2 + 32*e - 104, 7*e^2 + 14*e + 36, 31*e^2 + 10*e - 180, 9*e^2 - 15*e - 124, 13*e^2 + 15*e - 68, 16*e^2 - 27*e - 110, 27/2*e^2 - 12*e - 118, 49/2*e^2 + 20*e - 44, -16*e^2 - 36*e + 58, -27*e^2 - 4*e + 106, -13/2*e^2 - 5*e - 26, 35/2*e^2 - 33*e - 82, -24*e^2 - 22*e + 168, -16*e^2 - 34*e + 70, -8*e^2 - 65*e + 36, -11*e^2 - 38*e + 120, 10*e^2 + 8*e - 64, -22*e^2 - 50*e + 140, 4*e^2 + 19*e - 8, 8*e^2 + 10*e - 36, e^2 - 8*e - 122, 10*e^2 - 42*e - 76, 35/2*e^2 + 5*e - 58, 19/2*e^2 + 23*e + 16, 13*e^2 + 49*e - 50, e^2 - 13*e + 66, -7/2*e^2 - 12*e - 124, 25/2*e^2 + 12*e - 8, -24*e^2 + 14*e + 112, -34*e^2 - 11*e + 156, 4*e^2 - 8*e - 50, 30*e^2 + 18*e - 66, 35/2*e^2 + 52*e - 72, -11/2*e^2 + 13*e + 38, 8*e^2 - 6*e - 114, 8*e^2 - 12*e - 144, 10*e^2 + 11*e + 6, 21*e^2 + 5*e - 92, 14*e^2 - 15*e - 56, -9*e^2 + 17*e + 148, -5/2*e^2 + 5*e - 16, -57/2*e^2 - 19*e + 100, 26*e^2 + 29*e - 130, -5*e^2 + 7*e + 4, 18*e^2 + 16*e - 190, 8*e^2 - 14*e, 24*e^2 + 50*e - 108, 22*e^2 + 12*e - 140, 14*e^2 + 56*e - 94, -12*e^2 - 48*e + 54, 21/2*e^2 + 47*e - 126, 15/2*e^2 + 10*e - 92, 31*e^2 + 13*e - 104, -2*e^2 + e - 68, -2*e^2 + 4*e + 24, 30*e^2 + 66*e - 196, 2*e^2 + 39*e + 28, -4*e^2 + 9*e - 44, -20*e^2 - 33*e + 182, -9*e^2 - 19*e + 40, 3*e^2 - 26*e - 80, 9*e^2 - 23*e - 80, -8*e^2 - 21*e + 148, -8*e^2 + 7*e + 36, 30*e^2 + 28*e - 122, 8*e^2 - 12*e - 24, -38*e^2 - 40*e + 198, 11*e^2 - 26*e - 102, -27*e^2 + 15*e + 152, 2*e^2 - 49*e - 20, 50*e, 7*e^2 + 20*e - 28, 12*e^2 + 30*e - 138, -9*e^2 + 28*e + 116, 25*e^2 + 13*e - 60, 15*e^2 + 22*e - 116, 22*e^2 + 16*e - 2, 31*e^2 + 46*e - 194, -14*e^2 - 14*e + 148, -22*e^2 - 7*e + 140, -6*e^2 - 35*e + 40, 2*e^2 - 21*e - 126, -6*e^2 - 43*e + 4, -35*e^2 - 15*e + 140, -12*e^2 + 36*e + 78, -5*e^2 + 6*e - 22, -2*e^2 - 52, 9*e^2 + 12*e + 12, 2*e^2 + 18*e + 16, -11*e^2 - 22*e + 96, 23*e^2 - 8*e - 124, -16*e^2 - 42*e + 168, 32*e^2 + 3*e - 180, -20*e^2 - 34*e + 80, 9/2*e^2 - 43*e - 2, -19/2*e^2 - 2*e - 14, -8*e^2 - 18*e - 28, 26*e^2 - 33*e - 152, 9*e^2 - 8*e - 82, 11*e^2 - 44*e - 114, 28*e^2 - 6*e - 130, 2*e^2 + 62*e + 14, 7*e^2 + 60*e - 68, -32*e^2 - 32*e + 150, 5/2*e^2 + 6*e + 22, -3/2*e^2 - 15*e + 116, -15/2*e^2 + 3*e - 70, -47/2*e^2 - 23*e + 200]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := -1; ALEigenvalues[ideal] := -1; // EXAMPLE: // pp := Factorization(2*ZF)[1][1]; // heckeEigenvalues[pp]; print "To reconstruct the Hilbert newform f, type f, iso := Explode(make_newform());"; function make_newform(); M := HilbertCuspForms(F, NN); S := NewSubspace(M); // SetVerbose("ModFrmHil", 1); NFD := NewformDecomposition(S); newforms := [* Eigenform(U) : U in NFD *]; if #newforms eq 0 then; print "No Hilbert newforms at this level"; return 0; end if; print "Testing ", #newforms, " possible newforms"; newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *]; print #newforms, " newforms have the correct Hecke field"; if #newforms eq 0 then; print "No Hilbert newform found with the correct Hecke field"; return 0; end if; autos := Automorphisms(K); xnewforms := [* *]; for f in newforms do; if K eq RationalField() then; Append(~xnewforms, [* f, autos[1] *]); else; flag, iso := IsIsomorphic(K,BaseField(f)); for a in autos do; Append(~xnewforms, [* f, a*iso *]); end for; end if; end for; newforms := xnewforms; for P in primes do; xnewforms := [* *]; for f_iso in newforms do; f, iso := Explode(f_iso); if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then; Append(~xnewforms, f_iso); end if; end for; newforms := xnewforms; if #newforms eq 0 then; print "No Hilbert newform found which matches the Hecke eigenvalues"; return 0; else if #newforms eq 1 then; print "success: unique match"; return newforms[1]; end if; end if; end for; print #newforms, "Hilbert newforms found which match the Hecke eigenvalues"; return newforms[1]; end function;