/* 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![-10, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, w], [3, 3, w + 1], [3, 3, w + 2], [5, 5, w], [13, 13, w + 6], [13, 13, w + 7], [31, 31, -2*w + 3], [31, 31, 2*w + 3], [37, 37, w + 11], [37, 37, w + 26], [41, 41, 3*w + 7], [41, 41, -3*w + 7], [43, 43, w + 15], [43, 43, w + 28], [49, 7, -7], [53, 53, w + 13], [53, 53, w + 40], [67, 67, w + 12], [67, 67, w + 55], [71, 71, -w - 9], [71, 71, w - 9], [79, 79, -4*w - 9], [79, 79, 4*w - 9], [83, 83, w + 33], [83, 83, w + 50], [89, 89, -3*w - 1], [89, 89, 3*w - 1], [107, 107, w + 44], [107, 107, w + 63], [121, 11, -11], [151, 151, 4*w - 3], [151, 151, -4*w - 3], [157, 157, w + 18], [157, 157, w + 139], [163, 163, w + 70], [163, 163, w + 93], [173, 173, w + 23], [173, 173, w + 150], [191, 191, -6*w - 13], [191, 191, 6*w - 13], [197, 197, w + 73], [197, 197, w + 124], [199, 199, -3*w - 17], [199, 199, 3*w - 17], [227, 227, w + 64], [227, 227, w + 163], [239, 239, 6*w - 11], [239, 239, -6*w - 11], [241, 241, -5*w - 3], [241, 241, 5*w - 3], [271, 271, 3*w - 19], [271, 271, -3*w - 19], [277, 277, w + 29], [277, 277, w + 248], [281, 281, 4*w - 21], [281, 281, -4*w - 21], [283, 283, w + 24], [283, 283, w + 259], [289, 17, -17], [293, 293, w + 89], [293, 293, w + 204], [307, 307, w + 138], [307, 307, w + 169], [311, 311, -6*w - 7], [311, 311, 6*w - 7], [317, 317, w + 31], [317, 317, w + 286], [347, 347, w + 125], [347, 347, w + 222], [359, 359, -6*w - 1], [359, 359, 6*w - 1], [361, 19, -19], [373, 373, w + 82], [373, 373, w + 291], [397, 397, w + 176], [397, 397, w + 221], [401, 401, 2*w - 21], [401, 401, -2*w - 21], [409, 409, 7*w - 9], [409, 409, -7*w - 9], [431, 431, -w - 21], [431, 431, w - 21], [439, 439, -3*w - 23], [439, 439, 3*w - 23], [443, 443, w + 203], [443, 443, w + 240], [449, 449, 8*w - 33], [449, 449, -8*w - 33], [467, 467, w + 145], [467, 467, w + 322], [479, 479, -5*w - 27], [479, 479, 5*w - 27], [521, 521, 9*w - 17], [521, 521, -9*w - 17], [523, 523, w + 202], [523, 523, w + 321], [529, 23, -23], [547, 547, w + 253], [547, 547, w + 294], [557, 557, w + 41], [557, 557, w + 516], [563, 563, w + 130], [563, 563, w + 433], [569, 569, -4*w - 27], [569, 569, 4*w - 27], [587, 587, w + 214], [587, 587, w + 373], [599, 599, 12*w + 29], [599, 599, -12*w + 29], [601, 601, 6*w - 31], [601, 601, -6*w - 31], [613, 613, w + 43], [613, 613, w + 570], [631, 631, -8*w - 3], [631, 631, 8*w - 3], [641, 641, 9*w - 13], [641, 641, -9*w - 13], [643, 643, w + 36], [643, 643, w + 607], [653, 653, w + 140], [653, 653, w + 513], [677, 677, w + 213], [677, 677, w + 464], [683, 683, w + 159], [683, 683, w + 524], [719, 719, -w - 27], [719, 719, w - 27], [733, 733, w + 47], [733, 733, w + 686], [751, 751, -3*w - 29], [751, 751, 3*w - 29], [757, 757, w + 143], [757, 757, w + 614], [761, 761, -9*w - 7], [761, 761, 9*w - 7], [769, 769, 11*w - 21], [769, 769, -11*w - 21], [773, 773, w + 118], [773, 773, w + 655], [787, 787, w + 369], [787, 787, w + 418], [797, 797, w + 49], [797, 797, w + 748], [809, 809, -9*w - 1], [809, 809, 9*w - 1], [827, 827, w + 254], [827, 827, w + 573], [839, 839, 5*w - 33], [839, 839, -5*w - 33], [841, 29, -29], [853, 853, w + 160], [853, 853, w + 693], [877, 877, w + 42], [877, 877, w + 835], [881, 881, -8*w - 39], [881, 881, 8*w - 39], [883, 883, w + 234], [883, 883, w + 649], [907, 907, w + 266], [907, 907, w + 641], [911, 911, 12*w - 23], [911, 911, -12*w - 23], [919, 919, 10*w - 9], [919, 919, -10*w - 9], [929, 929, 4*w - 33], [929, 929, -4*w - 33], [947, 947, w + 111], [947, 947, w + 836], [991, 991, 10*w - 3], [991, 991, -10*w - 3], [997, 997, w + 134], [997, 997, w + 863], [1009, 1009, 6*w - 37], [1009, 1009, -6*w - 37], [1013, 1013, w + 353], [1013, 1013, w + 660], [1031, 1031, 7*w - 39], [1031, 1031, -7*w - 39], [1039, 1039, 16*w + 39], [1039, 1039, -16*w + 39], [1049, 1049, 2*w - 33], [1049, 1049, -2*w - 33], [1093, 1093, w + 292], [1093, 1093, w + 801], [1117, 1117, w + 521], [1117, 1117, w + 596], [1123, 1123, w + 75], [1123, 1123, w + 1048], [1129, 1129, -11*w - 9], [1129, 1129, 11*w - 9], [1151, 1151, -12*w - 17], [1151, 1151, 12*w - 17], [1163, 1163, w + 123], [1163, 1163, w + 1040], [1187, 1187, w + 312], [1187, 1187, w + 875], [1201, 1201, -11*w - 3], [1201, 1201, 11*w - 3], [1213, 1213, w + 181], [1213, 1213, w + 1032], [1231, 1231, 14*w - 27], [1231, 1231, -14*w - 27], [1237, 1237, w + 61], [1237, 1237, w + 1176], [1249, 1249, -13*w - 21], [1249, 1249, 13*w - 21], [1277, 1277, w + 574], [1277, 1277, w + 703], [1279, 1279, 3*w - 37], [1279, 1279, -3*w - 37], [1283, 1283, w + 152], [1283, 1283, w + 1131], [1289, 1289, -15*w - 31], [1289, 1289, 15*w - 31], [1307, 1307, w + 453], [1307, 1307, w + 854], [1319, 1319, 12*w - 11], [1319, 1319, -12*w - 11], [1321, 1321, -6*w - 41], [1321, 1321, 6*w - 41], [1361, 1361, -4*w - 39], [1361, 1361, 4*w - 39], [1373, 1373, w + 243], [1373, 1373, w + 1130], [1399, 1399, -9*w - 47], [1399, 1399, 9*w - 47], [1409, 1409, 15*w - 29], [1409, 1409, -15*w - 29], [1427, 1427, w + 504], [1427, 1427, w + 923], [1439, 1439, -12*w - 1], [1439, 1439, 12*w - 1], [1453, 1453, w + 54], [1453, 1453, w + 1399], [1471, 1471, -16*w - 33], [1471, 1471, 16*w - 33], [1481, 1481, 2*w - 39], [1481, 1481, -2*w - 39], [1483, 1483, w + 708], [1483, 1483, w + 775], [1489, 1489, 6*w - 43], [1489, 1489, -6*w - 43], [1493, 1493, w + 67], [1493, 1493, w + 1426], [1511, 1511, -w - 39], [1511, 1511, w - 39], [1523, 1523, w + 489], [1523, 1523, w + 1034], [1559, 1559, -13*w - 57], [1559, 1559, 13*w - 57], [1597, 1597, w + 653], [1597, 1597, w + 944], [1601, 1601, 10*w - 51], [1601, 1601, -10*w - 51], [1609, 1609, -13*w - 9], [1609, 1609, 13*w - 9], [1613, 1613, w + 220], [1613, 1613, w + 1393], [1627, 1627, w + 389], [1627, 1627, w + 1238], [1637, 1637, w + 565], [1637, 1637, w + 1072], [1667, 1667, w + 798], [1667, 1667, w + 869], [1693, 1693, w + 324], [1693, 1693, w + 1369], [1721, 1721, -15*w - 23], [1721, 1721, 15*w - 23], [1723, 1723, w + 449], [1723, 1723, w + 1274], [1733, 1733, w + 273], [1733, 1733, w + 1460], [1747, 1747, w + 532], [1747, 1747, w + 1215], [1759, 1759, 3*w - 43], [1759, 1759, -3*w - 43], [1787, 1787, w + 564], [1787, 1787, w + 1223], [1801, 1801, 17*w - 33], [1801, 1801, -17*w - 33], [1831, 1831, -16*w - 27], [1831, 1831, 16*w - 27], [1867, 1867, w + 710], [1867, 1867, w + 1157], [1871, 1871, -18*w - 37], [1871, 1871, 18*w - 37], [1877, 1877, w + 605], [1877, 1877, w + 1272], [1879, 1879, 14*w - 9], [1879, 1879, -14*w - 9], [1889, 1889, -15*w - 19], [1889, 1889, 15*w - 19], [1907, 1907, w + 840], [1907, 1907, w + 1067], [1933, 1933, w + 819], [1933, 1933, w + 1114], [1951, 1951, -14*w - 3], [1951, 1951, 14*w - 3], [1973, 1973, w + 77], [1973, 1973, w + 1896], [1987, 1987, w + 716], [1987, 1987, w + 1271], [1997, 1997, w + 783], [1997, 1997, w + 1214], [1999, 1999, -9*w - 53], [1999, 1999, 9*w - 53], [2003, 2003, w + 689], [2003, 2003, w + 1314], [2027, 2027, w + 487], [2027, 2027, w + 1540], [2039, 2039, -11*w - 57], [2039, 2039, 11*w - 57], [2053, 2053, w + 808], [2053, 2053, w + 1245], [2081, 2081, -15*w - 13], [2081, 2081, 15*w - 13], [2083, 2083, w + 250], [2083, 2083, w + 1833], [2089, 2089, -19*w - 39], [2089, 2089, 19*w - 39], [2111, 2111, -7*w - 51], [2111, 2111, 7*w - 51], [2129, 2129, 15*w - 11], [2129, 2129, -15*w - 11], [2161, 2161, 17*w - 27], [2161, 2161, -17*w - 27], [2203, 2203, w + 105], [2203, 2203, w + 2098], [2209, 47, -47], [2213, 2213, w + 426], [2213, 2213, w + 1787], [2237, 2237, w + 602], [2237, 2237, w + 1635], [2239, 2239, -22*w + 51], [2239, 2239, 22*w + 51], [2243, 2243, w + 540], [2243, 2243, w + 1703], [2267, 2267, w + 733], [2267, 2267, w + 1534], [2281, 2281, 12*w - 61], [2281, 2281, -12*w - 61], [2293, 2293, w + 83], [2293, 2293, w + 2210], [2311, 2311, 3*w - 49], [2311, 2311, -3*w - 49], [2333, 2333, w + 251], [2333, 2333, w + 2082], [2347, 2347, w + 325], [2347, 2347, w + 2022], [2351, 2351, 5*w - 51], [2351, 2351, -5*w - 51], [2357, 2357, w + 206], [2357, 2357, w + 2151], [2399, 2399, -18*w - 29], [2399, 2399, 18*w - 29], [2437, 2437, w + 436], [2437, 2437, w + 2001], [2441, 2441, -4*w - 51], [2441, 2441, 4*w - 51], [2467, 2467, w + 479], [2467, 2467, w + 1988], [2477, 2477, w + 695], [2477, 2477, w + 1782], [2521, 2521, 19*w - 33], [2521, 2521, -19*w - 33], [2551, 2551, 16*w - 3], [2551, 2551, -16*w - 3], [2557, 2557, w + 1005], [2557, 2557, w + 1552], [2591, 2591, -w - 51], [2591, 2591, w - 51], [2609, 2609, 8*w - 57], [2609, 2609, -8*w - 57], [2671, 2671, 9*w - 59], [2671, 2671, -9*w - 59], [2677, 2677, w + 1089], [2677, 2677, w + 1588], [2683, 2683, w + 832], [2683, 2683, w + 1851], [2689, 2689, -18*w - 77], [2689, 2689, 18*w - 77], [2693, 2693, w + 873], [2693, 2693, w + 1820], [2707, 2707, w + 654], [2707, 2707, w + 2053], [2711, 2711, 18*w - 23], [2711, 2711, -18*w - 23], [2719, 2719, -3*w - 53], [2719, 2719, 3*w - 53], [2729, 2729, 21*w - 41], [2729, 2729, -21*w - 41], [2791, 2791, -26*w + 63], [2791, 2791, 26*w + 63], [2797, 2797, w + 788], [2797, 2797, w + 2009], [2801, 2801, -14*w - 69], [2801, 2801, 14*w - 69], [2803, 2803, w + 290], [2803, 2803, w + 2513], [2837, 2837, w + 226], [2837, 2837, w + 2611], [2843, 2843, w + 973], [2843, 2843, w + 1870], [2879, 2879, -18*w - 19], [2879, 2879, 18*w - 19], [2917, 2917, w + 599], [2917, 2917, w + 2318], [2957, 2957, w + 648], [2957, 2957, w + 2309], [2963, 2963, w + 1303], [2963, 2963, w + 1660], [2969, 2969, -10*w - 63], [2969, 2969, 10*w - 63], [2999, 2999, -5*w - 57], [2999, 2999, 5*w - 57], [3001, 3001, -25*w + 57], [3001, 3001, 25*w + 57], [3037, 3037, w + 78], [3037, 3037, w + 2959], [3041, 3041, -21*w - 37], [3041, 3041, 21*w - 37], [3049, 3049, -12*w - 67], [3049, 3049, 12*w - 67], [3067, 3067, w + 910], [3067, 3067, w + 2157], [3079, 3079, 15*w - 73], [3079, 3079, -15*w - 73], [3083, 3083, w + 795], [3083, 3083, w + 2288], [3089, 3089, 4*w - 57], [3089, 3089, -4*w - 57], [3119, 3119, -18*w - 11], [3119, 3119, 18*w - 11], [3121, 3121, -6*w - 59], [3121, 3121, 6*w - 59], [3163, 3163, w + 1122], [3163, 3163, w + 2041], [3169, 3169, 19*w - 21], [3169, 3169, -19*w - 21], [3187, 3187, w + 411], [3187, 3187, w + 2776], [3191, 3191, 18*w - 7], [3191, 3191, -18*w - 7], [3203, 3203, w + 310], [3203, 3203, w + 2893], [3209, 3209, 2*w - 57], [3209, 3209, -2*w - 57], [3253, 3253, w + 242], [3253, 3253, w + 3011], [3271, 3271, -20*w - 27], [3271, 3271, 20*w - 27], [3307, 3307, w + 244], [3307, 3307, w + 3063], [3319, 3319, 22*w - 39], [3319, 3319, -22*w - 39], [3323, 3323, w + 1259], [3323, 3323, w + 2064], [3329, 3329, -8*w - 63], [3329, 3329, 8*w - 63], [3347, 3347, w + 862], [3347, 3347, w + 2485], [3359, 3359, -24*w - 49], [3359, 3359, 24*w - 49], [3361, 3361, 6*w - 61], [3361, 3361, -6*w - 61], [3373, 3373, w + 949], [3373, 3373, w + 2424], [3391, 3391, -3*w - 59], [3391, 3391, 3*w - 59], [3413, 3413, w + 320], [3413, 3413, w + 3093], [3449, 3449, 21*w - 31], [3449, 3449, -21*w - 31], [3467, 3467, w + 395], [3467, 3467, w + 3072], [3481, 59, -59], [3511, 3511, -21*w - 89], [3511, 3511, 21*w - 89], [3517, 3517, w + 1302], [3517, 3517, w + 2215], [3529, 3529, 19*w - 9], [3529, 3529, -19*w - 9], [3533, 3533, w + 103], [3533, 3533, w + 3430], [3547, 3547, w + 526], [3547, 3547, w + 3021], [3557, 3557, w + 980], [3557, 3557, w + 2577], [3559, 3559, -20*w - 21], [3559, 3559, 20*w - 21], [3613, 3613, w + 937], [3613, 3613, w + 2676], [3631, 3631, 3*w - 61], [3631, 3631, -3*w - 61], [3637, 3637, w + 1751], [3637, 3637, w + 1886], [3643, 3643, w + 135], [3643, 3643, w + 3508], [3671, 3671, 30*w + 73], [3671, 3671, -30*w + 73], [3677, 3677, w + 974], [3677, 3677, w + 2703], [3719, 3719, 5*w - 63], [3719, 3719, -5*w - 63], [3721, 61, -61], [3733, 3733, w + 768], [3733, 3733, w + 2965], [3761, 3761, -10*w - 69], [3761, 3761, 10*w - 69], [3769, 3769, 23*w - 39], [3769, 3769, -23*w - 39], [3797, 3797, w + 558], [3797, 3797, w + 3239], [3803, 3803, w + 1848], [3803, 3803, w + 1955], [3853, 3853, w + 340], [3853, 3853, w + 3513], [3877, 3877, w + 1529], [3877, 3877, w + 2348], [3881, 3881, 21*w - 23], [3881, 3881, -21*w - 23], [3889, 3889, 12*w - 73], [3889, 3889, -12*w - 73], [3907, 3907, w + 1004], [3907, 3907, w + 2903], [3911, 3911, 24*w - 43], [3911, 3911, -24*w - 43], [3917, 3917, w + 1620], [3917, 3917, w + 2297], [3919, 3919, -20*w - 9], [3919, 3919, 20*w - 9], [3923, 3923, w + 381], [3923, 3923, w + 3542], [3929, 3929, 2*w - 63], [3929, 3929, -2*w - 63], [3947, 3947, w + 1919], [3947, 3947, w + 2028], [4001, 4001, 16*w - 81], [4001, 4001, -16*w - 81], [4003, 4003, w + 1083], [4003, 4003, w + 2920], [4013, 4013, w + 1794], [4013, 4013, w + 2219], [4027, 4027, w + 1266], [4027, 4027, w + 2761], [4049, 4049, -21*w - 19], [4049, 4049, 21*w - 19], [4079, 4079, 24*w - 41], [4079, 4079, -24*w - 41], [4093, 4093, w + 1738], [4093, 4093, w + 2355], [4111, 4111, -22*w - 27], [4111, 4111, 22*w - 27], [4129, 4129, 6*w - 67], [4129, 4129, -6*w - 67], [4133, 4133, w + 713], [4133, 4133, w + 3420], [4157, 4157, w + 1608], [4157, 4157, w + 2549], [4159, 4159, 26*w - 51], [4159, 4159, -26*w - 51], [4201, 4201, -23*w - 33], [4201, 4201, 23*w - 33], [4231, 4231, -9*w - 71], [4231, 4231, 9*w - 71], [4241, 4241, 21*w - 13], [4241, 4241, -21*w - 13], [4243, 4243, w + 2065], [4243, 4243, w + 2178], [4253, 4253, w + 113], [4253, 4253, w + 4140], [4271, 4271, -7*w - 69], [4271, 4271, 7*w - 69], [4283, 4283, w + 578], [4283, 4283, w + 3705], [4289, 4289, -21*w - 11], [4289, 4289, 21*w - 11], [4357, 4357, w + 343], [4357, 4357, w + 4014], [4363, 4363, w + 637], [4363, 4363, w + 3726], [4373, 4373, w + 1489], [4373, 4373, w + 2884], [4391, 4391, 24*w - 37], [4391, 4391, -24*w - 37], [4397, 4397, w + 988], [4397, 4397, w + 3409], [4409, 4409, -21*w - 1], [4409, 4409, 21*w - 1], [4441, 4441, 24*w - 101], [4441, 4441, -24*w - 101], [4481, 4481, 27*w - 53], [4481, 4481, -27*w - 53], [4483, 4483, w + 1774], [4483, 4483, w + 2709], [4493, 4493, w + 592], [4493, 4493, w + 3901], [4507, 4507, w + 1705], [4507, 4507, w + 2802], [4517, 4517, w + 1895], [4517, 4517, w + 2622], [4519, 4519, 9*w - 73], [4519, 4519, -9*w - 73], [4523, 4523, w + 856], [4523, 4523, w + 3667], [4547, 4547, w + 1352], [4547, 4547, w + 3195], [4561, 4561, -23*w - 27], [4561, 4561, 23*w - 27], [4591, 4591, -28*w - 57], [4591, 4591, 28*w - 57], [4597, 4597, w + 1263], [4597, 4597, w + 3334], [4603, 4603, w + 96], [4603, 4603, w + 4507], [4637, 4637, w + 1831], [4637, 4637, w + 2806], [4639, 4639, 15*w - 83], [4639, 4639, -15*w - 83], [4643, 4643, w + 1049], [4643, 4643, w + 3594], [4649, 4649, -33*w + 79], [4649, 4649, 33*w + 79], [4679, 4679, -17*w - 87], [4679, 4679, 17*w - 87], [4721, 4721, 2*w - 69], [4721, 4721, -2*w - 69], [4723, 4723, w + 1003], [4723, 4723, w + 3720], [4729, 4729, -25*w - 39], [4729, 4729, 25*w - 39], [4733, 4733, w + 763], [4733, 4733, w + 3970], [4751, 4751, -w - 69], [4751, 4751, w - 69], [4759, 4759, -22*w - 9], [4759, 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+ 8575], [8719, 8719, 32*w - 39], [8719, 8719, -32*w - 39], [8747, 8747, w + 826], [8747, 8747, w + 7921], [8761, 8761, 12*w - 101], [8761, 8761, -12*w - 101], [8803, 8803, w + 2160], [8803, 8803, w + 6643], [8831, 8831, -30*w - 13], [8831, 8831, 30*w - 13], [8837, 8837, w + 4249], [8837, 8837, w + 4588], [8839, 8839, 46*w + 111], [8839, 8839, -46*w + 111], [8849, 8849, -22*w - 117], [8849, 8849, 22*w - 117], [8867, 8867, w + 1888], [8867, 8867, w + 6979], [8893, 8893, w + 3249], [8893, 8893, w + 5644], [8923, 8923, w + 2108], [8923, 8923, w + 6815], [8929, 8929, 37*w - 69], [8929, 8929, -37*w - 69], [8933, 8933, w + 2933], [8933, 8933, w + 6000], [8951, 8951, -30*w - 7], [8951, 8951, 30*w - 7], [8963, 8963, w + 913], [8963, 8963, w + 8050], [8969, 8969, -39*w - 79], [8969, 8969, 39*w - 79], [8999, 8999, -30*w - 1], [8999, 8999, 30*w - 1], [9001, 9001, -35*w - 57], [9001, 9001, 35*w - 57], [9013, 9013, w + 520], [9013, 9013, w + 8493], [9041, 9041, -33*w - 43], [9041, 9041, 33*w - 43], [9043, 9043, w + 2900], [9043, 9043, w + 6143], [9049, 9049, 6*w - 97], [9049, 9049, -6*w - 97], [9067, 9067, w + 404], [9067, 9067, w + 8663], [9133, 9133, w + 3938], [9133, 9133, w + 5195], [9151, 9151, -32*w - 33], [9151, 9151, 32*w - 33], [9157, 9157, w + 406], [9157, 9157, w + 8751], [9161, 9161, 8*w - 99], [9161, 9161, -8*w - 99], [9173, 9173, w + 1493], [9173, 9173, w + 7680], [9187, 9187, w + 2773], [9187, 9187, w + 6414], [9199, 9199, -15*w - 107], [9199, 9199, 15*w - 107], [9203, 9203, w + 3113], [9203, 9203, w + 6090], [9209, 9209, 33*w - 41], [9209, 9209, -33*w - 41], [9227, 9227, w + 3004], [9227, 9227, w + 6223], [9239, 9239, -36*w - 61], [9239, 9239, 36*w - 61], [9241, 9241, -36*w - 149], [9241, 9241, 36*w - 149], [9277, 9277, w + 1345], [9277, 9277, w + 7932], [9281, 9281, 39*w - 77], [9281, 9281, -39*w - 77], [9283, 9283, w + 4558], [9283, 9283, w + 4725], [9293, 9293, w + 167], [9293, 9293, w + 9126], [9311, 9311, 7*w - 99], [9311, 9311, -7*w - 99], [9319, 9319, 3*w - 97], [9319, 9319, -3*w - 97], [9323, 9323, w + 3342], [9323, 9323, w + 5981], [9391, 9391, -9*w - 101], [9391, 9391, 9*w - 101], [9397, 9397, w + 1907], [9397, 9397, w + 7490], [9403, 9403, w + 3971], [9403, 9403, w + 5432], [9409, 97, -97], [9413, 9413, w + 2883], [9413, 9413, w + 6530], [9431, 9431, 17*w - 111], [9431, 9431, -17*w - 111], [9437, 9437, w + 3741], [9437, 9437, w + 5696], [9439, 9439, -40*w - 81], [9439, 9439, 40*w - 81], [9467, 9467, w + 2409], [9467, 9467, w + 7058], [9479, 9479, -36*w - 59], [9479, 9479, 36*w - 59], [9511, 9511, 32*w - 27], [9511, 9511, -32*w - 27], [9521, 9521, 33*w - 37], [9521, 9521, -33*w - 37], [9533, 9533, w + 4439], [9533, 9533, w + 5094], [9547, 9547, w + 1698], [9547, 9547, w + 7849], [9551, 9551, -5*w - 99], [9551, 9551, 5*w - 99], [9587, 9587, w + 1429], [9587, 9587, w + 8158], [9601, 9601, 31*w - 3], [9601, 9601, -31*w - 3], [9613, 9613, w + 2990], [9613, 9613, w + 6623], [9631, 9631, 15*w - 109], [9631, 9631, -15*w - 109], [9643, 9643, w + 947], [9643, 9643, w + 8696], [9649, 9649, -35*w - 51], [9649, 9649, 35*w - 51], [9677, 9677, w + 1091], [9677, 9677, w + 8586], [9679, 9679, 38*w - 69], [9679, 9679, -38*w - 69], [9689, 9689, -20*w - 117], [9689, 9689, 20*w - 117], [9719, 9719, 37*w - 153], [9719, 9719, -37*w - 153], [9721, 9721, -37*w - 63], [9721, 9721, 37*w - 63], [9733, 9733, w + 3460], [9733, 9733, w + 6273], [9769, 9769, -47*w + 111], [9769, 9769, 47*w + 111], [9787, 9787, w + 1523], [9787, 9787, w + 8264], [9791, 9791, -w - 99], [9791, 9791, w - 99], [9803, 9803, w + 357], [9803, 9803, w + 9446], [9839, 9839, 23*w - 123], [9839, 9839, -23*w - 123], [9871, 9871, -27*w - 131], [9871, 9871, 27*w - 131], [9883, 9883, w + 878], [9883, 9883, w + 9005], [9907, 9907, w + 3422], [9907, 9907, w + 6485], [9923, 9923, w + 2524], [9923, 9923, w + 7399], [9929, 9929, -33*w - 31], [9929, 9929, 33*w - 31], [9973, 9973, w + 173], [9973, 9973, w + 9800]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [1, -1, -1, 3, -1, -1, -4, -4, 7, 7, 0, 0, 1, 1, -13, 0, 0, -14, -14, -3, -3, 8, 8, -12, -12, -6, -6, -12, -12, 14, 17, 17, -14, -14, 16, 16, 0, 0, -18, -18, -3, -3, 2, 2, 0, 0, 15, 15, -10, -10, 11, 11, 28, 28, -6, -6, 4, 4, -25, -21, -21, -2, -2, -30, -30, 6, 6, -3, -3, 0, 0, -34, 4, 4, 34, 34, 36, 36, 32, 32, -33, -33, 26, 26, -21, -21, 6, 6, -36, -36, -21, -21, -9, -9, -20, -20, -46, -17, -17, -3, -3, -39, -39, 15, 15, -24, -24, 6, 6, -19, -19, -38, -38, 29, 29, -18, -18, -14, -14, 0, 0, -48, -48, -24, -24, 6, 6, -23, -23, -40, -40, 16, 16, -6, -6, 32, 32, 39, 39, 40, 40, 42, 42, -33, -33, -18, -18, 0, 0, -22, 37, 37, 13, 13, 21, 21, -29, -29, 37, 37, 30, 30, -16, -16, 36, 36, -6, -6, 2, 2, 46, 46, 38, 38, -6, -6, -24, -24, 14, 14, -33, -33, 22, 22, -38, -38, 10, 10, -4, -4, 24, 24, -36, -36, 3, 3, -40, -40, -44, -44, 8, 8, 13, 13, 17, 17, 6, 6, -7, -7, -12, -12, 42, 42, 18, 18, 57, 57, -28, -28, -27, -27, 21, 21, 41, 41, 60, 60, -9, -9, -12, -12, 58, 58, 32, 32, 51, 51, -53, -53, -22, -22, 33, 33, 30, 30, 66, 66, 12, 12, -41, -41, -6, -6, -1, -1, -48, -48, 46, 46, 36, 36, 21, 21, -2, -2, -24, -24, 4, 4, -12, -12, -2, -2, 32, 32, 6, 6, 20, 20, -67, -67, -32, -32, 18, 18, 57, 57, 71, 71, 42, 42, -15, -15, -26, -26, -46, -46, 12, 12, 82, 82, -9, -9, 38, 38, -60, -60, -39, -39, -36, -36, 52, 52, -33, -33, 76, 76, -43, -43, -48, -48, -18, -18, 74, 74, -8, -8, -85, -48, -48, 48, 48, 80, 80, -36, -36, -72, -72, 68, 68, -53, -53, 26, 26, 81, 81, -44, -44, 33, 33, -42, -42, -51, -51, -26, -26, 78, 78, 52, 52, 81, 81, 65, 65, 32, 32, -2, -2, 42, 42, -30, -30, -37, -37, -68, -68, -38, -38, -10, -10, -87, -87, -35, -35, 63, 63, 29, 29, 21, 21, -70, -70, 37, 37, 0, 0, 58, 58, 66, 66, -69, -69, -39, -39, -74, -74, 78, 78, -3, -3, -72, -72, 90, 90, 32, 32, 85, 85, 54, 54, 80, 80, 64, 64, -88, -88, -54, -54, 30, 30, -48, -48, 17, 17, 85, 85, 29, 29, 16, 16, 3, 3, -72, -72, 0, 0, 22, 22, 59, 59, -14, -14, -82, -82, 96, 96, -18, -18, -42, -42, 21, 21, -4, -4, 46, 46, 35, 35, -81, -81, 9, 9, 81, 81, -82, 92, 92, -107, -107, 56, 56, -18, -18, 52, 52, 33, 33, 14, 14, 4, 4, 32, 32, -110, -110, 109, 109, -63, -63, 63, 63, -114, -114, -58, -71, -71, -57, -57, -82, -82, -18, -18, 24, 24, -74, -74, -44, -44, 54, 54, -16, -16, -56, -56, 117, 117, -36, -36, -40, -40, 33, 33, 81, 81, 12, 12, -45, -45, -89, -89, 102, 102, 31, 31, -66, -66, 6, 6, 91, 91, -46, -46, -10, -10, -66, -66, -84, -84, -16, -16, -82, -82, 65, 65, -93, -93, 22, 22, -54, -54, -33, -33, 30, 30, 105, 105, -101, -101, 28, 28, 63, 63, -6, -6, -54, -54, -24, -24, 122, 122, -75, -75, -92, -92, 6, 6, -44, -44, 57, 57, -25, -25, 12, 12, -123, -123, 80, 80, -61, -61, 73, 73, -41, -41, -126, -126, -4, -4, -84, -84, -30, -30, -18, -18, -54, -54, 28, 28, 41, 41, 54, 54, 96, 96, 8, 8, 96, 96, -39, -39, -31, -31, -8, -8, 89, 89, 108, 108, -39, -39, -105, -105, 24, 24, 70, 70, -25, -25, -44, -44, 38, 38, 6, 6, -2, -2, 5, 5, 132, 132, -33, -33, 45, 45, 43, 43, -72, -72, 112, 112, -64, -64, -117, -117, -32, -32, 35, 35, 37, 37, 21, 21, 42, 42, -12, -12, -91, -91, -32, -32, -142, 72, 72, -11, -11, -3, -3, -96, -96, -12, -12, -113, -113, -58, -58, 16, 16, -132, -132, -89, -89, -22, -22, 120, 120, -12, -12, 71, 71, -132, -132, -18, -18, 135, 135, 131, 131, 58, 58, 52, 52, -28, -28, 54, 54, -60, -60, -54, -54, -109, -109, -11, -11, 22, 22, 20, 20, -6, -6, 120, 120, -78, -78, -31, -31, 63, 63, 114, 114, -104, -104, -64, -64, 36, 36, 18, 18, -123, -123, -138, -138, 50, 50, -38, -38, 36, 36, -89, -89, -14, -14, -114, -114, 97, 97, 113, 113, -66, -66, -52, -52, 40, 40, 32, 32, 100, 100, -111, -111, -30, -30, -91, -91, 48, 48, -109, -109, -59, -59, -108, -108, 36, 36, 48, 48, 6, 6, 9, 9, 53, 53, -74, -74, 82, 82, 34, 34, -57, -57, 38, 38, 6, 6, 41, 41, -14, -14, 0, 0, -114, -114, 114, 114, -15, -15, -113, -113, -132, -132, 74, 74, 126, 126, 117, 117, 124, 124, -105, -105, -41, -41, 132, 132, 21, 21, 72, 72, -115, -115, 68, 68, -152, -152, -23, -23, 45, 45, -6, -6, -18, -18, -36, -36, 68, 68, -28, -28, -68, -68, 150, 150, 63, 63, 82, 82, 56, 56, 63, 63, 48, 48, -30, -30, 56, 56, 42, 42, -148, -148, 72, 72, -14, -14, -128, -128, 10, 10, -90, -90, 111, 111, 99, 99, 122, 122, 46, 46, -163, -163, 98, 98, 127, 127, -144, -144, 155, 155, -128, -128, 66, 66, 120, 120, -90, -90, -66, -66, 96, 96, -106, -106, -41, -41, -148, -148, -152, -152, 71, 71, 60, 60, -18, -18, -76, -76, -77, -77, -47, -47, -6, -6, -88, -88, -12, -12, -132, -132, -38, -38, 132, 132, -112, -112, -126, -126, 36, 36, 21, 21, 166, 166, -112, -112, 154, 154, -27, -27, 147, 147, -161, -161, -90, -90, -91, -91, -99, -99, -60, -60, -153, -153, -54, -54, 27, 27, -154, -154, -52, -52, 38, 38, 135, 135, -111, -111, -108, -108, 76, 76, -4, -4, 52, 52, 41, 41, -144, -144, -87, -87, -66, -66, 35, 35, 160, 160, 91, 91, -46, -46, 67, 67, 33, 33, 90, 90, -67, -67, 45, 45, -168, -168, 11, 11, 34, 34, -18, -18, 86, 86, -78, -78, -77, -77, 116, 116, -102, -102, 5, 5, -146, -146, -108, -108, 186, 186, 20, 20, -9, -9, 69, 69, 106, 106, 58, 58, 74, 74, -18, -18, 3, 3, -114, -114, -21, -21, 156, 156, 128, 128, 94, 94, 84, 84, -26, -26, 86, 86, -14, -14, -125, -125, 80, 80, 103, 103, 6, 6, 123, 123, -53, -53, -115, -115, -15, -15, -132, -132, 39, 39, -78, -78, -154, -154, 46, 46, 111, 111, -77, -77, -9, -9, -90, -90, -49, -49, 6, 6, 23, 23, 34, 34, -71, -71, -94, 144, 144, 168, 168, -114, -114, -100, -100, -84, -84, 24, 24, 80, 80, 90, 90, -162, -162, 28, 28, 0, 0, -72, -72, 62, 62, 43, 43, -7, -7, 76, 76, -70, -70, -174, -174, -175, -175, -105, -105, 108, 108, 83, 83, -122, -122, 86, 86, 112, 112, -105, -105, -39, -39, -165, -165, 44, 44, -119, -119, -53, -53, 60, 60, -75, -75, -11, -11]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := -1; 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;