/* 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![-41, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [3, 3, w + 1], [4, 2, 2], [5, 5, w + 2], [7, 7, w + 2], [7, 7, w + 4], [11, 11, w + 5], [13, 13, w + 1], [13, 13, w + 11], [23, 23, w + 10], [23, 23, w + 12], [29, 29, -w - 3], [29, 29, w - 4], [31, 31, -w - 8], [31, 31, w - 9], [41, 41, -w], [41, 41, w - 1], [43, 43, w + 18], [43, 43, w + 24], [47, 47, w + 13], [47, 47, w + 33], [53, 53, w + 17], [53, 53, w + 35], [73, 73, w + 23], [73, 73, w + 49], [101, 101, 2*w - 9], [101, 101, -2*w - 7], [113, 113, w + 19], [113, 113, w + 93], [127, 127, w + 30], [127, 127, w + 96], [131, 131, 3*w - 17], [131, 131, -3*w - 14], [137, 137, w + 56], [137, 137, w + 80], [149, 149, 2*w - 5], [149, 149, -2*w - 3], [181, 181, 3*w - 25], [181, 181, -3*w - 22], [193, 193, w + 50], [193, 193, w + 142], [199, 199, -w - 15], [199, 199, w - 16], [229, 229, -3*w - 23], [229, 229, 3*w - 26], [239, 239, -3*w - 10], [239, 239, 3*w - 13], [257, 257, w + 28], [257, 257, w + 228], [277, 277, w + 44], [277, 277, w + 232], [281, 281, 3*w - 11], [281, 281, -3*w - 8], [283, 283, w + 110], [283, 283, w + 172], [289, 17, -17], [307, 307, w + 63], [307, 307, w + 243], [317, 317, w + 31], [317, 317, w + 285], [331, 331, 3*w - 28], [331, 331, -3*w - 25], [337, 337, w + 66], [337, 337, w + 270], [353, 353, w + 42], [353, 353, w + 310], [359, 359, 3*w - 5], [359, 359, -3*w - 2], [361, 19, -19], [373, 373, w + 51], [373, 373, w + 321], [379, 379, -w - 20], [379, 379, w - 21], [383, 383, w + 34], [383, 383, w + 348], [421, 421, -w - 21], [421, 421, w - 22], [431, 431, 5*w - 27], [431, 431, -5*w - 22], [443, 443, w + 47], [443, 443, w + 395], [457, 457, w + 199], [457, 457, w + 257], [461, 461, 6*w - 35], [461, 461, -6*w - 29], [467, 467, w + 208], [467, 467, w + 258], [479, 479, -5*w - 21], [479, 479, 5*w - 26], [491, 491, 4*w - 15], [491, 491, -4*w - 11], [499, 499, 3*w - 31], [499, 499, -3*w - 28], [523, 523, w + 195], [523, 523, w + 327], [547, 547, w + 153], [547, 547, w + 393], [569, 569, 5*w - 24], [569, 569, -5*w - 19], [587, 587, w + 176], [587, 587, w + 410], [607, 607, w + 65], [607, 607, w + 541], [613, 613, w + 89], [613, 613, w + 523], [617, 617, w + 43], [617, 617, w + 573], [619, 619, 2*w - 29], [619, 619, -2*w - 27], [631, 631, 7*w + 48], [631, 631, 7*w - 55], [647, 647, w + 174], [647, 647, w + 472], [653, 653, w + 57], [653, 653, w + 595], [659, 659, 4*w - 3], [659, 659, 4*w - 1], [661, 661, -w - 26], [661, 661, w - 27], [673, 673, w + 192], [673, 673, w + 480], [683, 683, w + 125], [683, 683, w + 557], [691, 691, -5*w - 39], [691, 691, 5*w - 44], [701, 701, -6*w - 25], [701, 701, 6*w - 31], [709, 709, -4*w - 35], [709, 709, 4*w - 39], [733, 733, w + 277], [733, 733, w + 455], [751, 751, -3*w - 32], [751, 751, 3*w - 35], [761, 761, -7*w - 32], [761, 761, 7*w - 39], [773, 773, w + 62], [773, 773, w + 710], [787, 787, w + 74], [787, 787, w + 712], [797, 797, w + 330], [797, 797, w + 466], [809, 809, 6*w - 29], [809, 809, -6*w - 23], [821, 821, 5*w - 17], [821, 821, -5*w - 12], [829, 829, -w - 29], [829, 829, w - 30], [853, 853, w + 105], [853, 853, w + 747], [859, 859, 2*w - 33], [859, 859, -2*w - 31], [863, 863, w + 201], [863, 863, w + 661], [877, 877, w + 328], [877, 877, w + 548], [937, 937, w + 427], [937, 937, w + 509], [941, 941, 5*w - 12], [941, 941, -5*w - 7], [947, 947, w + 438], [947, 947, w + 508], [967, 967, w + 350], [967, 967, w + 616], [977, 977, w + 301], [977, 977, w + 675], [983, 983, w + 150], [983, 983, w + 832], [991, 991, 2*w - 35], [991, 991, -2*w - 33], [997, 997, w + 181], [997, 997, w + 815], [1013, 1013, w + 55], [1013, 1013, w + 957], [1019, 1019, -5*w - 1], [1019, 1019, 5*w - 6], [1021, 1021, -4*w - 39], [1021, 1021, 4*w - 43], [1031, 1031, 5*w - 2], [1031, 1031, 5*w - 3], [1033, 1033, w + 473], [1033, 1033, w + 559], [1039, 1039, 5*w - 48], [1039, 1039, -5*w - 43], [1063, 1063, w + 86], [1063, 1063, w + 976], [1091, 1091, 7*w - 34], [1091, 1091, -7*w - 27], [1103, 1103, w + 291], [1103, 1103, w + 811], [1117, 1117, w + 402], [1117, 1117, w + 714], [1151, 1151, -7*w - 26], [1151, 1151, 7*w - 33], [1171, 1171, 7*w - 60], [1171, 1171, -7*w - 53], [1193, 1193, w + 77], [1193, 1193, w + 1115], [1229, 1229, -6*w - 13], [1229, 1229, 6*w - 19], [1277, 1277, w + 171], [1277, 1277, w + 1105], [1279, 1279, 2*w - 39], [1279, 1279, -2*w - 37], [1289, 1289, 6*w - 17], [1289, 1289, -6*w - 11], [1291, 1291, -w - 36], [1291, 1291, w - 37], [1297, 1297, w + 95], [1297, 1297, w + 1201], [1307, 1307, w + 173], [1307, 1307, w + 1133], [1319, 1319, 7*w - 30], [1319, 1319, -7*w - 23], [1321, 1321, 5*w - 51], [1321, 1321, -5*w - 46], [1327, 1327, w + 131], [1327, 1327, w + 1195], [1361, 1361, 9*w - 49], [1361, 1361, -9*w - 40], [1367, 1367, w + 324], [1367, 1367, w + 1042], [1369, 37, -37], [1373, 1373, w + 64], [1373, 1373, w + 1308], [1433, 1433, w + 625], [1433, 1433, w + 807], [1447, 1447, w + 528], [1447, 1447, w + 918], [1451, 1451, 12*w + 61], [1451, 1451, 12*w - 73], [1481, 1481, 6*w - 5], [1481, 1481, 6*w - 1], [1489, 1489, 11*w - 86], [1489, 1489, -11*w - 75], [1523, 1523, w + 87], [1523, 1523, w + 1435], [1549, 1549, 4*w - 49], [1549, 1549, -4*w - 45], [1559, 1559, -7*w - 18], [1559, 1559, 7*w - 25], [1597, 1597, w + 341], [1597, 1597, w + 1255], [1601, 1601, 7*w - 24], [1601, 1601, -7*w - 17], [1607, 1607, w + 627], [1607, 1607, w + 979], [1609, 1609, 3*w - 46], [1609, 1609, -3*w - 43], [1619, 1619, -9*w - 37], [1619, 1619, 9*w - 46], [1621, 1621, -5*w - 49], [1621, 1621, 5*w - 54], [1627, 1627, w + 740], [1627, 1627, w + 886], [1637, 1637, w + 576], [1637, 1637, w + 1060], [1657, 1657, w + 451], [1657, 1657, w + 1205], [1663, 1663, w + 302], [1663, 1663, w + 1360], [1693, 1693, w + 654], [1693, 1693, w + 1038], [1697, 1697, w + 499], [1697, 1697, w + 1197], [1699, 1699, -3*w - 44], [1699, 1699, 3*w - 47], [1723, 1723, w + 611], [1723, 1723, w + 1111], [1741, 1741, -4*w - 47], [1741, 1741, 4*w - 51], [1777, 1777, w + 276], [1777, 1777, w + 1500], [1787, 1787, w + 73], [1787, 1787, w + 1713], [1811, 1811, -7*w - 11], [1811, 1811, 7*w - 18], [1831, 1831, 5*w - 56], [1831, 1831, -5*w - 51], [1867, 1867, w + 114], [1867, 1867, w + 1752], [1879, 1879, 6*w - 61], [1879, 1879, -6*w - 55], [1889, 1889, -7*w - 8], [1889, 1889, 7*w - 15], [1907, 1907, w + 529], [1907, 1907, w + 1377], [1931, 1931, 7*w - 13], [1931, 1931, -7*w - 6], [1933, 1933, w + 116], [1933, 1933, w + 1816], [1949, 1949, -7*w - 5], [1949, 1949, 7*w - 12], [1951, 1951, 2*w - 47], [1951, 1951, -2*w - 45], [1973, 1973, w + 947], [1973, 1973, w + 1025], [1979, 1979, 7*w - 10], [1979, 1979, -7*w - 3], [1987, 1987, w + 622], [1987, 1987, w + 1364], [1993, 1993, w + 457], [1993, 1993, w + 1535], [2003, 2003, w + 739], [2003, 2003, w + 1263], [2011, 2011, 7*w - 67], [2011, 2011, -7*w - 60], [2027, 2027, w + 641], [2027, 2027, w + 1385], [2029, 2029, -w - 45], [2029, 2029, w - 46], [2053, 2053, w + 629], [2053, 2053, w + 1423], [2081, 2081, 9*w - 40], [2081, 2081, -9*w - 31], [2111, 2111, -8*w - 19], [2111, 2111, 8*w - 27], [2129, 2129, -11*w - 48], [2129, 2129, 11*w - 59], [2141, 2141, 15*w - 92], [2141, 2141, -15*w - 77], [2161, 2161, -5*w - 54], [2161, 2161, 5*w - 59], [2179, 2179, 3*w - 52], [2179, 2179, -3*w - 49], [2237, 2237, w + 781], [2237, 2237, w + 1455], [2267, 2267, w + 924], [2267, 2267, w + 1342], [2269, 2269, -7*w - 62], [2269, 2269, 7*w - 69], [2281, 2281, -3*w - 50], [2281, 2281, 3*w - 53], [2287, 2287, w + 704], [2287, 2287, w + 1582], [2297, 2297, w + 1106], [2297, 2297, w + 1190], [2309, 2309, 15*w + 76], [2309, 2309, 15*w - 91], [2311, 2311, -w - 48], [2311, 2311, w - 49], [2333, 2333, w + 1050], [2333, 2333, w + 1282], [2339, 2339, 11*w - 57], [2339, 2339, -11*w - 46], [2341, 2341, 13*w + 90], [2341, 2341, 13*w - 103], [2351, 2351, 8*w - 21], [2351, 2351, -8*w - 13], [2357, 2357, w + 353], [2357, 2357, w + 2003], [2383, 2383, w + 280], [2383, 2383, w + 2102], [2411, 2411, 9*w - 35], [2411, 2411, -9*w - 26], [2423, 2423, w + 85], [2423, 2423, w + 2337], [2437, 2437, w + 460], [2437, 2437, w + 1976], [2441, 2441, -11*w - 45], [2441, 2441, 11*w - 56], [2447, 2447, w + 960], [2447, 2447, w + 1486], [2459, 2459, -12*w - 53], [2459, 2459, 12*w - 65], [2503, 2503, w + 180], [2503, 2503, w + 2322], [2539, 2539, 2*w - 53], [2539, 2539, -2*w - 51], [2549, 2549, -13*w - 60], [2549, 2549, 13*w - 73], [2591, 2591, -8*w - 3], [2591, 2591, 8*w - 11], [2593, 2593, w + 1009], [2593, 2593, w + 1583], [2609, 2609, 14*w - 81], [2609, 2609, -14*w - 67], [2617, 2617, w + 135], [2617, 2617, w + 2481], [2633, 2633, w + 545], [2633, 2633, w + 2087], [2647, 2647, w + 944], [2647, 2647, w + 1702], [2663, 2663, w + 642], [2663, 2663, w + 2020], [2671, 2671, -7*w - 65], [2671, 2671, 7*w - 72], [2683, 2683, w + 574], [2683, 2683, w + 2108], [2687, 2687, w + 856], [2687, 2687, w + 1830], [2689, 2689, -8*w - 69], [2689, 2689, 8*w - 77], [2693, 2693, w + 1168], [2693, 2693, w + 1524], [2713, 2713, w + 1113], [2713, 2713, w + 1599], [2731, 2731, 14*w + 97], [2731, 2731, 14*w - 111], [2741, 2741, 9*w - 29], [2741, 2741, -9*w - 20], [2753, 2753, w + 117], [2753, 2753, w + 2635], [2767, 2767, w + 875], [2767, 2767, w + 1891], [2777, 2777, w + 91], [2777, 2777, w + 2685], [2789, 2789, -9*w - 19], [2789, 2789, 9*w - 28], [2801, 2801, 17*w - 104], [2801, 2801, 17*w + 87], [2833, 2833, w + 496], [2833, 2833, w + 2336], [2843, 2843, w + 646], [2843, 2843, w + 2196], [2857, 2857, w + 1357], [2857, 2857, w + 1499], [2879, 2879, 9*w - 26], [2879, 2879, -9*w - 17], [2897, 2897, w + 842], [2897, 2897, w + 2054], [2917, 2917, w + 461], [2917, 2917, w + 2455], [2927, 2927, w + 1016], [2927, 2927, w + 1910], [2939, 2939, 13*w - 70], [2939, 2939, -13*w - 57], [2957, 2957, w + 898], [2957, 2957, w + 2058], [2963, 2963, w + 94], [2963, 2963, w + 2868], [2969, 2969, 10*w - 39], [2969, 2969, -10*w - 29], [2971, 2971, 2*w - 57], [2971, 2971, -2*w - 55], [2999, 2999, -9*w - 14], [2999, 2999, 9*w - 23], [3001, 3001, 5*w - 66], [3001, 3001, -5*w - 61], [3011, 3011, -11*w - 39], [3011, 3011, 11*w - 50], [3019, 3019, -11*w - 84], [3019, 3019, 11*w - 95], [3023, 3023, w + 584], [3023, 3023, w + 2438], [3061, 3061, -4*w - 59], [3061, 3061, 4*w - 63], [3083, 3083, w + 535], [3083, 3083, w + 2547], [3119, 3119, 15*w - 86], [3119, 3119, -15*w - 71], [3163, 3163, w + 828], [3163, 3163, w + 2334], [3169, 3169, 3*w - 61], [3169, 3169, -3*w - 58], [3187, 3187, w + 149], [3187, 3187, w + 3037], [3209, 3209, -9*w - 7], [3209, 3209, 9*w - 16], [3251, 3251, -9*w - 5], [3251, 3251, 9*w - 14], [3253, 3253, w + 959], [3253, 3253, w + 2293], [3257, 3257, w + 1491], [3257, 3257, w + 1765], [3259, 3259, 5*w - 68], [3259, 3259, -5*w - 63], [3271, 3271, 10*w - 91], [3271, 3271, -10*w - 81], [3299, 3299, 9*w - 11], [3299, 3299, -9*w - 2], [3301, 3301, 9*w - 86], [3301, 3301, -9*w - 77], [3307, 3307, w + 692], [3307, 3307, w + 2614], [3313, 3313, w + 1155], [3313, 3313, w + 2157], [3323, 3323, w + 1408], [3323, 3323, w + 1914], [3329, 3329, 9*w - 8], [3329, 3329, 9*w - 1], [3331, 3331, -13*w - 95], [3331, 3331, 13*w - 108], [3343, 3343, w + 592], [3343, 3343, w + 2750], [3347, 3347, w + 1608], [3347, 3347, w + 1738], [3373, 3373, w + 209], [3373, 3373, w + 3163], [3391, 3391, -5*w - 64], [3391, 3391, 5*w - 69], [3413, 3413, w + 1655], [3413, 3413, w + 1757], [3449, 3449, -10*w - 21], [3449, 3449, 10*w - 31], [3461, 3461, -14*w - 61], [3461, 3461, 14*w - 75], [3469, 3469, -12*w - 91], [3469, 3469, 12*w - 103], [3481, 59, -59], [3499, 3499, -w - 59], [3499, 3499, w - 60], [3517, 3517, w + 1538], [3517, 3517, w + 1978], [3529, 3529, 7*w - 78], [3529, 3529, -7*w - 71], [3539, 3539, 12*w - 55], [3539, 3539, -12*w - 43], [3557, 3557, w + 103], [3557, 3557, w + 3453], [3581, 3581, -15*w - 68], [3581, 3581, 15*w - 83], [3583, 3583, w + 158], [3583, 3583, w + 3424], [3607, 3607, w + 1373], [3607, 3607, w + 2233], [3617, 3617, w + 288], [3617, 3617, w + 3328], [3623, 3623, w + 580], [3623, 3623, w + 3042], [3631, 3631, 9*w - 88], [3631, 3631, -9*w - 79], [3637, 3637, w + 346], [3637, 3637, w + 3290], [3643, 3643, w + 1712], [3643, 3643, w + 1930], [3659, 3659, -11*w - 31], [3659, 3659, 11*w - 42], [3671, 3671, -16*w - 75], [3671, 3671, 16*w - 91], [3673, 3673, w + 1170], [3673, 3673, w + 2502], [3677, 3677, w + 441], [3677, 3677, w + 3235], [3721, 61, -61], [3761, 3761, 19*w + 96], [3761, 3761, 19*w - 115], [3767, 3767, w + 106], [3767, 3767, w + 3660], [3779, 3779, -13*w - 50], [3779, 3779, 13*w - 63], [3823, 3823, w + 405], [3823, 3823, w + 3417], [3833, 3833, w + 1862], [3833, 3833, w + 1970], [3847, 3847, w + 1811], [3847, 3847, w + 2035], [3907, 3907, w + 165], [3907, 3907, w + 3741], [3911, 3911, -17*w - 81], [3911, 3911, 17*w - 98], [3917, 3917, w + 1162], [3917, 3917, w + 2754], [3919, 3919, 3*w - 67], [3919, 3919, -3*w - 64], [3929, 3929, 10*w - 19], [3929, 3929, -10*w - 9], [3931, 3931, 2*w - 65], [3931, 3931, -2*w - 63], [3947, 3947, w + 1670], [3947, 3947, w + 2276], [3967, 3967, w + 1166], [3967, 3967, w + 2800], [3989, 3989, -14*w - 57], [3989, 3989, 14*w - 71], [4001, 4001, -13*w - 48], [4001, 4001, 13*w - 61], [4003, 4003, w + 1390], [4003, 4003, w + 2612], [4007, 4007, w + 555], [4007, 4007, w + 3451], [4013, 4013, w + 741], [4013, 4013, w + 3271], [4051, 4051, -3*w - 65], [4051, 4051, 3*w - 68], [4073, 4073, w + 1054], [4073, 4073, w + 3018], [4091, 4091, -12*w - 37], [4091, 4091, 12*w - 49], [4129, 4129, -11*w - 90], [4129, 4129, 11*w - 101], [4153, 4153, w + 1626], [4153, 4153, w + 2526], [4159, 4159, 13*w - 112], [4159, 4159, -13*w - 99], [4177, 4177, w + 1876], [4177, 4177, w + 2300], [4217, 4217, w + 311], [4217, 4217, w + 3905], [4241, 4241, 14*w - 69], [4241, 4241, -14*w - 55], [4243, 4243, w + 667], [4243, 4243, w + 3575], [4259, 4259, 12*w - 47], [4259, 4259, -12*w - 35], [4261, 4261, -12*w - 95], [4261, 4261, 12*w - 107], [4283, 4283, w + 793], [4283, 4283, w + 3489], [4289, 4289, 11*w - 32], [4289, 4289, -11*w - 21], [4297, 4297, w + 1490], [4297, 4297, w + 2806], [4337, 4337, w + 1758], [4337, 4337, w + 2578], [4339, 4339, 10*w - 97], [4339, 4339, -10*w - 87], [4363, 4363, w + 379], [4363, 4363, w + 3983], [4391, 4391, 11*w - 30], [4391, 4391, -11*w - 19], [4421, 4421, 13*w - 57], [4421, 4421, -13*w - 44], [4451, 4451, 15*w - 77], [4451, 4451, -15*w - 62], [4483, 4483, w + 1165], [4483, 4483, w + 3317], [4489, 67, -67], [4493, 4493, w + 1301], [4493, 4493, w + 3191], [4507, 4507, w + 937], [4507, 4507, w + 3569], [4519, 4519, -5*w - 72], [4519, 4519, 5*w - 77], [4547, 4547, w + 1252], [4547, 4547, w + 3294], [4567, 4567, w + 2161], [4567, 4567, w + 2405], [4591, 4591, 18*w + 125], [4591, 4591, 18*w - 143], [4597, 4597, w + 179], [4597, 4597, w + 4417], [4621, 4621, 7*w - 85], [4621, 4621, -7*w - 78], [4643, 4643, w + 152], [4643, 4643, w + 4490], [4649, 4649, 11*w - 24], [4649, 4649, -11*w - 13], [4651, 4651, -w - 68], [4651, 4651, w - 69], [4663, 4663, w + 1828], [4663, 4663, w + 2834], [4673, 4673, w + 1129], [4673, 4673, w + 3543], [4721, 4721, 14*w - 65], [4721, 4721, -14*w - 51], [4733, 4733, w + 1467], [4733, 4733, w + 3265], [4751, 4751, 11*w - 21], [4751, 4751, -11*w - 10], [4789, 4789, -w - 69], [4789, 4789, w - 70], [4801, 4801, 13*w - 115], [4801, 4801, -13*w - 102], [4813, 4813, w + 2314], [4813, 4813, w + 2498], [4877, 4877, w + 673], [4877, 4877, w + 4203], [4903, 4903, w + 1031], [4903, 4903, w + 3871], [4909, 4909, 11*w - 105], [4909, 4909, -11*w - 94], [4919, 4919, -11*w - 3], [4919, 4919, 11*w - 14], [4937, 4937, w + 2042], [4937, 4937, w + 2894], [4943, 4943, w + 2215], [4943, 4943, w + 2727], [4951, 4951, -7*w - 80], [4951, 4951, 7*w - 87], [4957, 4957, w + 1826], [4957, 4957, w + 3130], [4973, 4973, w + 483], [4973, 4973, w + 4489], [4993, 4993, w + 1749], [4993, 4993, w + 3243], [4999, 4999, -15*w - 112], [4999, 4999, 15*w - 127]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [1, -3, -1, 0, 0, -4, 2, 2, 0, 0, -2, -2, 0, 0, 10, 10, -4, -4, -8, -8, 10, 10, -10, -10, 6, 6, -2, -2, 8, 8, -12, -12, 6, 6, 22, 22, -10, -10, -2, -2, -8, -8, 6, 6, -16, -16, -18, -18, -6, -6, -6, -6, 12, 12, -30, -28, -28, 2, 2, 12, 12, 14, 14, -18, -18, -24, -24, -22, 26, 26, -20, -20, 24, 24, -26, -26, 0, 0, 12, 12, -10, -10, -18, -18, -28, -28, 0, 0, 28, 28, 4, 4, -4, -4, 20, 20, -6, -6, 12, 12, 8, 8, -22, -22, 6, 6, -4, -4, -8, -8, -32, -32, -46, -46, 20, 20, 22, 22, 30, 30, -36, -36, -44, -44, -2, -2, -26, -26, -14, -14, 16, 16, -6, -6, -6, -6, -28, -28, 2, 2, 10, 10, 54, 54, 30, 30, -6, -6, -20, -20, 56, 56, -30, -30, 54, 54, -50, -50, 36, 36, -32, -32, -2, -2, 0, 0, 32, 32, -54, -54, -22, -22, -36, -36, -2, -2, 24, 24, 54, 54, -32, -32, 16, 16, -44, -44, -24, -24, 18, 18, -16, -16, -12, -12, 38, 38, -18, -18, -30, -30, -16, -16, 42, 42, 44, 44, 46, 46, 28, 28, -24, -24, -22, -22, 56, 56, 18, 18, -16, -16, 26, 66, 66, 54, 54, 32, 32, -36, -36, -22, -22, 18, 18, -12, -12, -18, -18, 56, 56, 50, 50, 2, 2, 0, 0, -54, -54, -12, -12, 22, 22, 28, 28, -6, -6, -26, -26, 56, 56, 18, 18, 78, 78, 52, 52, -20, -20, 14, 14, -18, -18, -36, -36, -60, -60, -56, -56, 12, 12, 8, 8, -30, -30, -28, -28, -36, -36, 34, 34, 30, 30, 32, 32, 42, 42, 60, 60, -44, -44, -42, -42, -12, -12, 28, 28, 12, 12, 14, 14, -54, -54, -30, -30, -48, -48, 18, 18, -34, -34, -14, -14, 20, 20, -62, -62, -4, -4, 62, 62, -22, -22, -72, -72, -10, -10, 6, 6, 72, 72, -62, -62, -60, -60, 38, 38, 0, 0, 74, 74, -88, -88, -36, -36, -32, -32, -22, -22, 74, 74, 24, 24, -4, -4, 48, 48, -36, -36, 54, 54, 48, 48, -98, -98, -78, -78, -58, -58, 6, 6, -16, -16, 16, 16, -16, -16, -84, -84, 72, 72, 66, 66, 58, 58, -26, -26, -20, -20, 54, 54, -18, -18, -40, -40, -10, -10, -26, -26, -14, -14, -18, -18, -20, -20, 22, 22, 64, 64, 30, 30, 74, 74, 24, 24, -4, -4, 82, 82, 20, 20, 90, 90, -4, -4, 56, 56, 58, 58, -44, -44, 28, 28, -56, -56, -10, -10, 28, 28, -48, -48, 76, 76, -30, -30, 100, 100, 10, 10, 36, 36, 58, 58, -42, -42, 12, 12, 40, 40, -60, -60, 38, 38, -52, -52, 46, 46, 44, 44, -94, -94, -60, -60, -88, -88, 36, 36, 34, 34, -96, -96, -86, -86, 58, 58, 102, 102, -82, -82, -102, 28, 28, 82, 82, 10, 10, -44, -44, -38, -38, 30, 30, 88, 88, -112, -112, 46, 46, 16, 16, -80, -80, 26, 26, -116, -116, 12, 12, 8, 8, -58, -58, 2, 2, -118, 18, 18, 48, 48, 4, 4, 40, 40, 54, 54, 64, 64, -44, -44, 56, 56, 82, 82, 64, 64, -6, -6, 92, 92, -52, -52, 8, 8, -74, -74, -30, -30, -28, -28, 32, 32, -110, -110, -76, -76, -26, -26, -84, -84, 98, 98, -26, -26, -48, -48, -50, -50, -42, -42, -78, -78, 20, 20, 36, 36, -26, -26, 76, 76, 34, 34, 86, 86, -34, -34, -28, -28, 28, 28, 120, 120, 70, 70, 84, 84, -92, -92, 10, 18, 18, 28, 28, -40, -40, -12, -12, 112, 112, 112, 112, 58, 58, -50, -50, 36, 36, -54, -54, -20, -20, 64, 64, -82, -82, 82, 82, -30, -30, -128, -128, -106, -106, 2, 2, -30, -30, 50, 50, 16, 16, -50, -50, -72, -72, 102, 102, -120, -120, 56, 56, 82, 82, -46, -46, -34, -34, -8, -8]; 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;