/* 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 - 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 := [ideal : I in primesArray]; heckePol := x^8 + 12*x^6 + 38*x^4 + 13*x^2 + 1; K := NumberField(heckePol); heckeEigenvaluesArray := [-e^3 - 5*e, e, e^6 + 12*e^4 + 37*e^2 + 6, -4*e^7 - 47*e^5 - 142*e^3 - 27*e, e^7 + 12*e^5 + 37*e^3 + 7*e, -e^7 - 12*e^5 - 36*e^3 - 2*e, 4*e^7 + 47*e^5 + 142*e^3 + 27*e, -2*e^4 - 12*e^2 - 5, -e^4 - 7*e^2 - 1, e^5 + 8*e^3 + 13*e, -4*e^7 - 48*e^5 - 150*e^3 - 40*e, -5*e^7 - 59*e^5 - 178*e^3 - 26*e, -3*e^7 - 34*e^5 - 97*e^3 - 8*e, e^6 + 12*e^4 + 38*e^2 + 12, 2*e^6 + 28*e^4 + 97*e^2 + 11, -e^5 - 2*e^3 + 25*e, 9*e^7 + 107*e^5 + 327*e^3 + 65*e, 5*e^6 + 59*e^4 + 177*e^2 + 33, 3*e^6 + 31*e^4 + 79*e^2 + 6, e^5 + 13*e^3 + 40*e, -3*e^7 - 35*e^5 - 103*e^3 - 3*e, e^6 + 14*e^4 + 51*e^2 + 5, -3*e^6 - 38*e^4 - 125*e^2 - 33, -9*e^7 - 106*e^5 - 323*e^3 - 73*e, 2*e^7 + 23*e^5 + 72*e^3 + 39*e, -5*e^7 - 58*e^5 - 170*e^3 - 21*e, 9*e^7 + 104*e^5 + 305*e^3 + 36*e, -5*e^6 - 57*e^4 - 166*e^2 - 32, 3*e^6 + 29*e^4 + 70*e^2 + 14, -6*e^6 - 68*e^4 - 196*e^2 - 29, 3*e^6 + 37*e^4 + 119*e^2 + 32, -10*e^6 - 115*e^4 - 338*e^2 - 61, e^7 + 13*e^5 + 38*e^3 - 18*e, -14*e^7 - 165*e^5 - 501*e^3 - 99*e, e^7 + 12*e^5 + 41*e^3 + 33*e, 17*e^7 + 205*e^5 + 638*e^3 + 138*e, -3*e^6 - 28*e^4 - 64*e^2 - 19, -8*e^7 - 94*e^5 - 288*e^3 - 76*e, 9*e^7 + 110*e^5 + 346*e^3 + 71*e, -e^6 - 7*e^4 - 5*e^2 + 16, -e^6 - 21*e^4 - 90*e^2 - 10, -e^6 - 13*e^4 - 43*e^2 - 4, 6*e^6 + 69*e^4 + 205*e^2 + 34, 5*e^7 + 55*e^5 + 148*e^3 - 11*e, -12*e^7 - 142*e^5 - 432*e^3 - 91*e, 13*e^7 + 154*e^5 + 475*e^3 + 127*e, 7*e^7 + 81*e^5 + 237*e^3 + 16*e, 9*e^7 + 111*e^5 + 356*e^3 + 92*e, -9*e^7 - 107*e^5 - 334*e^3 - 113*e, 5*e^6 + 57*e^4 + 165*e^2 + 28, 2*e^6 + 26*e^4 + 88*e^2 + 24, 17*e^7 + 203*e^5 + 626*e^3 + 131*e, -6*e^7 - 71*e^5 - 213*e^3 - 22*e, 7*e^6 + 84*e^4 + 258*e^2 + 46, -2*e^6 - 19*e^4 - 43*e^2 + 4, 3*e^6 + 31*e^4 + 80*e^2 + 4, 3*e^4 + 18*e^2 + 12, -3*e^6 - 34*e^4 - 95*e^2 - 12, -6*e^6 - 74*e^4 - 237*e^2 - 42, -4*e^7 - 49*e^5 - 163*e^3 - 75*e, 27*e^7 + 319*e^5 + 967*e^3 + 173*e, e^6 + 12*e^4 + 30*e^2 - 14, -4*e^6 - 46*e^4 - 135*e^2 - 21, -2*e^7 - 23*e^5 - 71*e^3 - 24*e, -9*e^7 - 107*e^5 - 324*e^3 - 45*e, 5*e^7 + 59*e^5 + 187*e^3 + 84*e, 10*e^7 + 121*e^5 + 372*e^3 + 52*e, -7*e^7 - 83*e^5 - 252*e^3 - 38*e, -12*e^7 - 141*e^5 - 427*e^3 - 85*e, -13*e^6 - 150*e^4 - 443*e^2 - 79, 4*e^6 + 55*e^4 + 185*e^2 + 16, 28*e^7 + 331*e^5 + 1010*e^3 + 210*e, -8*e^7 - 96*e^5 - 298*e^3 - 67*e, -29*e^7 - 340*e^5 - 1026*e^3 - 202*e, 5*e^7 + 58*e^5 + 172*e^3 + 32*e, 6*e^6 + 62*e^4 + 163*e^2 + 27, 5*e^6 + 55*e^4 + 149*e^2 + 19, -e^6 - 7*e^4 - e^2 + 20, -10*e^6 - 116*e^4 - 347*e^2 - 61, -6*e^7 - 71*e^5 - 223*e^3 - 90*e, 18*e^7 + 217*e^5 + 678*e^3 + 162*e, -6*e^6 - 70*e^4 - 213*e^2 - 44, -10*e^6 - 118*e^4 - 354*e^2 - 51, -26*e^7 - 308*e^5 - 936*e^3 - 166*e, 5*e^7 + 60*e^5 + 194*e^3 + 92*e, 3*e^6 + 43*e^4 + 157*e^2 + 37, -5*e^6 - 63*e^4 - 200*e^2 - 27, 2*e^6 + 19*e^4 + 46*e^2 + 36, -5*e^6 - 50*e^4 - 127*e^2 - 45, 4*e^6 + 57*e^4 + 206*e^2 + 60, -2*e^6 - 23*e^4 - 68*e^2 - 5, -26*e^7 - 308*e^5 - 940*e^3 - 200*e, 9*e^7 + 100*e^5 + 279*e^3 + 21*e, -9*e^6 - 105*e^4 - 311*e^2 - 38, 7*e^6 + 75*e^4 + 208*e^2 + 47, 24*e^7 + 281*e^5 + 848*e^3 + 177*e, -6*e^7 - 65*e^5 - 177*e^3 - 22*e, -7*e^6 - 81*e^4 - 248*e^2 - 69, -9*e^6 - 116*e^4 - 376*e^2 - 59, -11*e^7 - 125*e^5 - 361*e^3 - 54*e, -22*e^7 - 262*e^5 - 799*e^3 - 146*e, -10*e^6 - 118*e^4 - 353*e^2 - 80, 11*e^6 + 131*e^4 + 406*e^2 + 91, 7*e^4 + 50*e^2 + 43, 8*e^6 + 93*e^4 + 282*e^2 + 62, 16*e^7 + 194*e^5 + 611*e^3 + 146*e, -21*e^7 - 250*e^5 - 769*e^3 - 157*e, -7*e^7 - 82*e^5 - 258*e^3 - 105*e, -17*e^7 - 202*e^5 - 616*e^3 - 113*e, -4*e^6 - 47*e^4 - 148*e^2 - 53, -5*e^6 - 58*e^4 - 167*e^2 - 17, 5*e^7 + 53*e^5 + 137*e^3 - 5*e, -25*e^7 - 290*e^5 - 868*e^3 - 182*e, 3*e^5 + 10*e^3 - 47*e, -11*e^7 - 129*e^5 - 385*e^3 - 51*e, 6*e^6 + 81*e^4 + 278*e^2 + 53, -2*e^6 - 19*e^4 - 39*e^2 + 8, -5*e^6 - 60*e^4 - 186*e^2 - 9, -3*e^6 - 32*e^4 - 83*e^2 - 7, 6*e^7 + 66*e^5 + 188*e^3 + 49*e, -5*e^7 - 65*e^5 - 235*e^3 - 144*e, 12*e^7 + 145*e^5 + 456*e^3 + 116*e, 20*e^7 + 233*e^5 + 694*e^3 + 124*e, 16*e^7 + 186*e^5 + 549*e^3 + 55*e, 21*e^7 + 249*e^5 + 753*e^3 + 111*e, -4*e^6 - 46*e^4 - 138*e^2 - 24, 4*e^6 + 40*e^4 + 89*e^2 - 25, -8*e^6 - 96*e^4 - 297*e^2 - 49, 4*e^6 + 48*e^4 + 151*e^2 + 27, -3*e^7 - 34*e^5 - 86*e^3 + 56*e, 30*e^7 + 357*e^5 + 1093*e^3 + 217*e, 3*e^6 + 49*e^4 + 188*e^2 + 19, -e^6 - 8*e^4 - 18*e^2 - 14, -2*e^6 - 17*e^4 - 36*e^2 - 31, -3*e^6 - 20*e^4 - 5*e^2 + 32, e^5 + 6*e^3 - 17*e, 2*e^7 + 22*e^5 + 59*e^3 - 8*e, -18*e^7 - 208*e^5 - 611*e^3 - 90*e, -7*e^7 - 78*e^5 - 215*e^3 - 13*e, 10*e^7 + 115*e^5 + 337*e^3 + 55*e, 42*e^7 + 501*e^5 + 1533*e^3 + 267*e, 5*e^6 + 58*e^4 + 182*e^2 + 59, -19*e^6 - 218*e^4 - 639*e^2 - 120, -9*e^6 - 93*e^4 - 247*e^2 - 81, -4*e^7 - 52*e^5 - 178*e^3 - 69*e, -9*e^7 - 103*e^5 - 315*e^3 - 120*e, -25*e^7 - 292*e^5 - 865*e^3 - 100*e, -9*e^7 - 112*e^5 - 379*e^3 - 183*e, 14*e^6 + 173*e^4 + 543*e^2 + 89, -11*e^6 - 141*e^4 - 457*e^2 - 82, 4*e^3 + 27*e, -22*e^7 - 259*e^5 - 800*e^3 - 250*e, 5*e^7 + 53*e^5 + 141*e^3 + 21*e, -3*e^7 - 35*e^5 - 106*e^3 - 41*e, -2*e^7 - 23*e^5 - 80*e^3 - 76*e, -43*e^7 - 501*e^5 - 1503*e^3 - 302*e, -7*e^6 - 62*e^4 - 119*e^2 + 21, 8*e^6 + 83*e^4 + 221*e^2 + 66, 15*e^7 + 175*e^5 + 514*e^3 + 57*e, 8*e^7 + 95*e^5 + 280*e^3 + e, 12*e^6 + 152*e^4 + 494*e^2 + 86, -14*e^6 - 155*e^4 - 440*e^2 - 77, 2*e^6 + 20*e^4 + 54*e^2 + 33, -30*e^7 - 353*e^5 - 1074*e^3 - 250*e, -17*e^7 - 196*e^5 - 570*e^3 - 55*e, -e^7 - 7*e^5 + 10*e^3 + 84*e, -24*e^7 - 290*e^5 - 908*e^3 - 209*e, 10*e^7 + 118*e^5 + 355*e^3 + 43*e, -5*e^7 - 52*e^5 - 146*e^3 - 78*e, -10*e^6 - 109*e^4 - 292*e^2 - 28, -10*e^6 - 106*e^4 - 282*e^2 - 36, 10*e^6 + 127*e^4 + 411*e^2 + 86, 15*e^6 + 188*e^4 + 601*e^2 + 100, -18*e^6 - 197*e^4 - 547*e^2 - 93, -18*e^6 - 221*e^4 - 692*e^2 - 109, 12*e^6 + 151*e^4 + 483*e^2 + 94, -9*e^6 - 96*e^4 - 271*e^2 - 89, 3*e^7 + 38*e^5 + 117*e^3 - 27*e, -11*e^7 - 129*e^5 - 390*e^3 - 76*e, -e^6 - 26*e^4 - 122*e^2 - 37, -5*e^6 - 62*e^4 - 198*e^2 - 21, -6*e^7 - 79*e^5 - 268*e^3 - 67*e, -37*e^7 - 443*e^5 - 1377*e^3 - 328*e, 10*e^7 + 119*e^5 + 387*e^3 + 204*e, 19*e^7 + 222*e^5 + 673*e^3 + 166*e, e^7 + 5*e^5 - 21*e^3 - 79*e, 10*e^7 + 115*e^5 + 329*e^3 + 5*e, -e^6 - 8*e^4 - 18*e^2 - 19, -16*e^6 - 171*e^4 - 463*e^2 - 61, -5*e^6 - 69*e^4 - 248*e^2 - 67, -16*e^6 - 181*e^4 - 527*e^2 - 95, -15*e^7 - 184*e^5 - 586*e^3 - 144*e, -46*e^7 - 547*e^5 - 1681*e^3 - 367*e, -17*e^7 - 196*e^5 - 593*e^3 - 183*e, 3*e^7 + 39*e^5 + 146*e^3 + 101*e, -16*e^7 - 187*e^5 - 566*e^3 - 129*e, 47*e^7 + 553*e^5 + 1684*e^3 + 369*e, -5*e^7 - 54*e^5 - 131*e^3 + 71*e, 13*e^7 + 151*e^5 + 470*e^3 + 200*e, 7*e^6 + 79*e^4 + 230*e^2 + 78, -12*e^6 - 141*e^4 - 426*e^2 - 107, -3*e^6 - 23*e^4 - 34*e^2 - 14, e^6 + 19*e^4 + 92*e^2 + 74, 12*e^6 + 129*e^4 + 346*e^2 + 59, -9*e^6 - 108*e^4 - 338*e^2 - 94, 2*e^6 + 12*e^4 - 2*e^2 - 40, -6*e^6 - 63*e^4 - 164*e^2, 8*e^7 + 105*e^5 + 370*e^3 + 176*e, 34*e^7 + 407*e^5 + 1255*e^3 + 235*e, 5*e^6 + 56*e^4 + 165*e^2 + 21, -20*e^6 - 232*e^4 - 690*e^2 - 121, -10*e^7 - 114*e^5 - 330*e^3 - 50*e, 35*e^7 + 416*e^5 + 1274*e^3 + 279*e, -3*e^6 - 38*e^4 - 138*e^2 - 76, 6*e^6 + 44*e^4 + 48*e^2 - 23, e^5 - 11*e^3 - 114*e, -19*e^7 - 218*e^5 - 638*e^3 - 91*e, 13*e^6 + 149*e^4 + 437*e^2 + 38, e^6 + 7*e^4 - e^2 - 30, 22*e^6 + 257*e^4 + 777*e^2 + 152, 15*e^6 + 151*e^4 + 384*e^2 + 98, 7*e^7 + 72*e^5 + 167*e^3 - 82*e, -53*e^7 - 629*e^5 - 1927*e^3 - 405*e, -5*e^6 - 61*e^4 - 192*e^2 - 41, -7*e^6 - 107*e^4 - 405*e^2 - 105, 36*e^7 + 429*e^5 + 1316*e^3 + 254*e, -32*e^7 - 375*e^5 - 1121*e^3 - 172*e, 9*e^6 + 101*e^4 + 283*e^2 + 43, -13*e^6 - 174*e^4 - 590*e^2 - 97, 16*e^6 + 204*e^4 + 667*e^2 + 131, 2*e^6 - 3*e^4 - 94*e^2 - 47, 4*e^7 + 55*e^5 + 197*e^3 + 62*e, 20*e^7 + 233*e^5 + 703*e^3 + 151*e, 17*e^6 + 223*e^4 + 747*e^2 + 148, -10*e^6 - 121*e^4 - 381*e^2 - 93, 20*e^7 + 236*e^5 + 731*e^3 + 206*e, 10*e^7 + 123*e^5 + 393*e^3 + 106*e, -2*e^6 - 31*e^4 - 125*e^2 - 49, 10*e^6 + 141*e^4 + 503*e^2 + 119, 2*e^6 + 37*e^4 + 156*e^2 + 81, -e^6 - 3*e^4 + 19*e^2 - 19, -20*e^7 - 232*e^5 - 683*e^3 - 75*e, -6*e^7 - 71*e^5 - 212*e^3 - 6*e, -7*e^5 - 67*e^3 - 157*e, 28*e^7 + 331*e^5 + 1001*e^3 + 186*e, 2*e^6 + 23*e^4 + 67*e^2 - 4, 4*e^6 + 40*e^4 + 100*e^2 + 29, 22*e^6 + 252*e^4 + 731*e^2 + 111, -25*e^6 - 279*e^4 - 797*e^2 - 153, -34*e^7 - 404*e^5 - 1228*e^3 - 205*e, 20*e^7 + 240*e^5 + 749*e^3 + 197*e, 12*e^7 + 138*e^5 + 417*e^3 + 144*e, -25*e^7 - 304*e^5 - 947*e^3 - 170*e, 5*e^7 + 62*e^5 + 202*e^3 + 86*e, -31*e^7 - 379*e^5 - 1207*e^3 - 338*e, 21*e^6 + 242*e^4 + 720*e^2 + 101, 23*e^7 + 271*e^5 + 837*e^3 + 260*e, -15*e^7 - 173*e^5 - 516*e^3 - 122*e, -27*e^7 - 307*e^5 - 885*e^3 - 107*e, 29*e^7 + 343*e^5 + 1055*e^3 + 294*e, 6*e^6 + 67*e^4 + 185*e^2 + 24, 22*e^6 + 260*e^4 + 799*e^2 + 146, 14*e^6 + 150*e^4 + 402*e^2 + 41, -5*e^6 - 54*e^4 - 139*e^2 - 30, 8*e^6 + 99*e^4 + 319*e^2 + 105, 19*e^6 + 219*e^4 + 648*e^2 + 115, 25*e^7 + 300*e^5 + 937*e^3 + 253*e, 20*e^7 + 241*e^5 + 757*e^3 + 207*e, -16*e^6 - 163*e^4 - 419*e^2 - 97, -28*e^6 - 325*e^4 - 972*e^2 - 170, -7*e^6 - 71*e^4 - 175*e^2 - 31, 12*e^6 + 119*e^4 + 286*e^2 - 11, -17*e^6 - 198*e^4 - 594*e^2 - 141, -16*e^6 - 188*e^4 - 550*e^2 - 61, -14*e^6 - 148*e^4 - 386*e^2 - 15, -4*e^6 - 55*e^4 - 206*e^2 - 98, 27*e^7 + 318*e^5 + 958*e^3 + 145*e, 15*e^7 + 179*e^5 + 572*e^3 + 216*e, -2*e^6 - 29*e^4 - 99*e^2 + 8, -10*e^6 - 145*e^4 - 519*e^2 - 107, -23*e^6 - 276*e^4 - 851*e^2 - 126, 8*e^7 + 87*e^5 + 225*e^3 - 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334*e, -75*e^7 - 886*e^5 - 2712*e^3 - 638*e, -163*e^7 - 1922*e^5 - 5823*e^3 - 1061*e, 22*e^6 + 244*e^4 + 715*e^2 + 168, 13*e^6 + 155*e^4 + 434*e^2 - 63, 28*e^6 + 308*e^4 + 886*e^2 + 223, -16*e^6 - 134*e^4 - 250*e^2 - 65, 35*e^6 + 416*e^4 + 1275*e^2 + 321, -5*e^6 - 41*e^4 - 35*e^2 + 58, 24*e^6 + 277*e^4 + 822*e^2 + 66, -6*e^6 - 92*e^4 - 338*e^2 - 95, -113*e^7 - 1329*e^5 - 3983*e^3 - 599*e, 58*e^7 + 702*e^5 + 2210*e^3 + 559*e, -53*e^6 - 607*e^4 - 1751*e^2 - 270, 15*e^6 + 139*e^4 + 286*e^2 + 24, 30*e^7 + 349*e^5 + 1056*e^3 + 283*e, 29*e^7 + 321*e^5 + 874*e^3 - 2*e, -52*e^6 - 652*e^4 - 2100*e^2 - 345, -18*e^6 - 229*e^4 - 764*e^2 - 178, 19*e^6 + 205*e^4 + 589*e^2 + 133, 39*e^6 + 482*e^4 + 1519*e^2 + 251, -50*e^7 - 593*e^5 - 1874*e^3 - 688*e, 82*e^7 + 974*e^5 + 3019*e^3 + 860*e, 3*e^6 + 56*e^4 + 240*e^2 + 114, -8*e^6 - 54*e^4 - 74*e^2 - 176, -82*e^7 - 983*e^5 - 3049*e^3 - 715*e, 94*e^7 + 1136*e^5 + 3544*e^3 + 745*e, 9*e^6 + 110*e^4 + 338*e^2 + 39, 39*e^6 + 503*e^4 + 1643*e^2 + 306, -5*e^7 - 58*e^5 - 139*e^3 + 205*e, -98*e^7 - 1145*e^5 - 3479*e^3 - 896*e, -39*e^6 - 490*e^4 - 1583*e^2 - 315, 36*e^6 + 432*e^4 + 1317*e^2 + 253, -8*e^7 - 104*e^5 - 397*e^3 - 423*e, -26*e^7 - 313*e^5 - 987*e^3 - 302*e, -10*e^6 - 156*e^4 - 586*e^2 - 45, -23*e^6 - 259*e^4 - 733*e^2 - 42, -76*e^7 - 901*e^5 - 2727*e^3 - 456*e, 47*e^7 + 554*e^5 + 1705*e^3 + 469*e, -21*e^6 - 202*e^4 - 496*e^2 - 180, -16*e^6 - 194*e^4 - 626*e^2 - 148, -80*e^7 - 937*e^5 - 2827*e^3 - 532*e, 101*e^7 + 1194*e^5 + 3673*e^3 + 923*e, -23*e^6 - 274*e^4 - 832*e^2 - 149, -25*e^6 - 275*e^4 - 760*e^2 - 63, 146*e^7 + 1722*e^5 + 5203*e^3 + 903*e, 41*e^7 + 487*e^5 + 1549*e^3 + 659*e, -79*e^7 - 927*e^5 - 2827*e^3 - 698*e, 91*e^7 + 1071*e^5 + 3254*e^3 + 690*e, 24*e^6 + 252*e^4 + 691*e^2 + 220, -32*e^6 - 391*e^4 - 1223*e^2 - 192, 41*e^7 + 499*e^5 + 1555*e^3 + 283*e, -26*e^7 - 296*e^5 - 866*e^3 - 130*e, -33*e^6 - 382*e^4 - 1132*e^2 - 203, -17*e^6 - 196*e^4 - 588*e^2 - 159, -13*e^6 - 116*e^4 - 262*e^2 - 162, -38*e^6 - 386*e^4 - 962*e^2 - 107, -11*e^7 - 106*e^5 - 235*e^3 + 58*e, 98*e^7 + 1159*e^5 + 3545*e^3 + 822*e, -29*e^7 - 340*e^5 - 1044*e^3 - 277*e, -2*e^7 - 35*e^5 - 151*e^3 - 127*e, -41*e^6 - 416*e^4 - 1052*e^2 - 182, 20*e^6 + 259*e^4 + 862*e^2 + 83, 49*e^6 + 567*e^4 + 1672*e^2 + 251, 26*e^6 + 304*e^4 + 920*e^2 + 204, 7*e^6 + 114*e^4 + 442*e^2 + 60, -17*e^6 - 188*e^4 - 519*e^2 - 83, 33*e^6 + 322*e^4 + 787*e^2 + 225, -59*e^6 - 709*e^4 - 2176*e^2 - 389, -64*e^7 - 757*e^5 - 2280*e^3 - 346*e, -38*e^7 - 427*e^5 - 1204*e^3 - 94*e, -132*e^7 - 1583*e^5 - 4871*e^3 - 897*e, -50*e^7 - 583*e^5 - 1760*e^3 - 459*e, 6*e^6 + 30*e^4 - 38*e^2 - 15, -53*e^6 - 652*e^4 - 2054*e^2 - 358, 20*e^6 + 246*e^4 + 764*e^2 + 235, 8*e^6 + 91*e^4 + 261*e^2 - 20, 47*e^6 + 550*e^4 + 1648*e^2 + 260, -15*e^6 - 145*e^4 - 350*e^2 - 40, 4*e^7 + 48*e^5 + 127*e^3 - 99*e, -45*e^7 - 541*e^5 - 1694*e^3 - 410*e, 21*e^6 + 276*e^4 + 928*e^2 + 155, 28*e^6 + 293*e^4 + 791*e^2 + 188, 23*e^6 + 259*e^4 + 702*e^2 + 3, -28*e^6 - 340*e^4 - 1042*e^2 - 113, -3*e^6 - 79*e^4 - 362*e^2 - 31, -41*e^6 - 455*e^4 - 1289*e^2 - 245, 65*e^7 + 765*e^5 + 2370*e^3 + 723*e, 30*e^7 + 373*e^5 + 1197*e^3 + 292*e, -11*e^6 - 164*e^4 - 596*e^2 - 178, 14*e^6 + 189*e^4 + 679*e^2 + 244, 27*e^6 + 307*e^4 + 883*e^2 + 48, 9*e^6 + 110*e^4 + 391*e^2 + 215, 111*e^7 + 1319*e^5 + 4054*e^3 + 909*e, 145*e^7 + 1699*e^5 + 5107*e^3 + 906*e, 10*e^6 + 139*e^4 + 481*e^2 + 124, -37*e^6 - 460*e^4 - 1452*e^2 - 312, -98*e^7 - 1158*e^5 - 3519*e^3 - 683*e, -9*e^7 - 110*e^5 - 365*e^3 - 198*e, -17*e^6 - 191*e^4 - 556*e^2 - 195, 31*e^4 + 177*e^2 - 42, -112*e^7 - 1326*e^5 - 4051*e^3 - 806*e, -24*e^7 - 278*e^5 - 839*e^3 - 264*e, -32*e^6 - 403*e^4 - 1293*e^2 - 202, 59*e^6 + 704*e^4 + 2151*e^2 + 381, -13*e^6 - 129*e^4 - 305*e^2 - 87, -11*e^6 - 114*e^4 - 287*e^2 - 37, 84*e^7 + 992*e^5 + 2979*e^3 + 394*e, 52*e^7 + 612*e^5 + 1832*e^3 + 225*e, -44*e^6 - 468*e^4 - 1243*e^2 - 124, -12*e^6 - 157*e^4 - 505*e^2 - 55, 16*e^7 + 202*e^5 + 700*e^3 + 420*e, 45*e^7 + 532*e^5 + 1643*e^3 + 482*e, 58*e^7 + 682*e^5 + 2055*e^3 + 369*e, 70*e^7 + 836*e^5 + 2564*e^3 + 526*e, 131*e^7 + 1540*e^5 + 4641*e^3 + 838*e, 13*e^7 + 137*e^5 + 374*e^3 + 121*e, 7*e^6 + 74*e^4 + 230*e^2 + 133, -39*e^6 - 432*e^4 - 1239*e^2 - 231, 2*e^6 + 14*e^4 + 27*e^2 + 12, -37*e^6 - 409*e^4 - 1124*e^2 - 95, 37*e^6 + 409*e^4 + 1177*e^2 + 275, -20*e^6 - 267*e^4 - 871*e^2 - 74, 15*e^7 + 183*e^5 + 593*e^3 + 253*e, -121*e^7 - 1449*e^5 - 4470*e^3 - 894*e, 83*e^7 + 991*e^5 + 3047*e^3 + 565*e, 73*e^7 + 861*e^5 + 2611*e^3 + 484*e]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := -e^7 - 12*e^5 - 37*e^3 - 7*e; // 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;