/* 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![-16, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, w], [2, 2, w + 1], [5, 5, w + 2], [7, 7, w + 1], [7, 7, w + 5], [9, 3, 3], [13, 13, w + 6], [29, 29, -2*w + 7], [29, 29, 2*w + 5], [37, 37, w + 9], [37, 37, w + 27], [47, 47, w + 10], [47, 47, w + 36], [61, 61, 2*w - 3], [61, 61, -2*w - 1], [67, 67, w + 23], [67, 67, w + 43], [73, 73, w + 24], [73, 73, w + 48], [79, 79, 2*w - 13], [79, 79, -2*w - 11], [83, 83, w + 13], [83, 83, w + 69], [97, 97, w + 14], [97, 97, w + 82], [101, 101, 4*w - 21], [101, 101, 6*w - 25], [121, 11, -11], [131, 131, 2*w - 15], [131, 131, -2*w - 13], [137, 137, w + 52], [137, 137, w + 84], [139, 139, -4*w - 9], [139, 139, 4*w - 13], [163, 163, w + 18], [163, 163, w + 144], [167, 167, w + 68], [167, 167, w + 98], [179, 179, 4*w - 11], [179, 179, -4*w - 7], [181, 181, 4*w + 19], [181, 181, 10*w + 33], [191, 191, 2*w - 17], [191, 191, -2*w - 15], [193, 193, w + 39], [193, 193, w + 153], [197, 197, w + 79], [197, 197, w + 117], [199, 199, 6*w - 31], [199, 199, 8*w - 33], [211, 211, -4*w - 5], [211, 211, 4*w - 9], [223, 223, w + 21], [223, 223, w + 201], [227, 227, w + 67], [227, 227, w + 159], [251, 251, -4*w - 1], [251, 251, 4*w - 5], [269, 269, 4*w - 25], [269, 269, -4*w - 21], [289, 17, -17], [293, 293, w + 123], [293, 293, w + 169], [307, 307, w + 133], [307, 307, w + 173], [311, 311, 8*w - 31], [311, 311, 8*w + 23], [317, 317, w + 25], [317, 317, w + 291], [353, 353, w + 99], [353, 353, w + 253], [361, 19, -19], [383, 383, w + 55], [383, 383, w + 327], [389, 389, 6*w - 17], [389, 389, -6*w - 11], [397, 397, w + 56], [397, 397, w + 340], [419, 419, 2*w - 23], [419, 419, -2*w - 21], [439, 439, 6*w + 29], [439, 439, 16*w + 53], [457, 457, w + 30], [457, 457, w + 426], [463, 463, w + 192], [463, 463, w + 270], [487, 487, w + 218], [487, 487, w + 268], [491, 491, 10*w - 51], [491, 491, 12*w - 49], [521, 521, 6*w - 11], [521, 521, -6*w - 5], [529, 23, -23], [557, 557, w + 197], [557, 557, w + 359], [569, 569, -6*w - 1], [569, 569, 6*w - 7], [571, 571, 6*w - 37], [571, 571, -6*w - 31], [577, 577, w + 107], [577, 577, w + 469], [587, 587, w + 34], [587, 587, w + 552], [593, 593, w + 185], [593, 593, w + 407], [599, 599, -8*w - 17], [599, 599, 8*w - 25], [601, 601, -16*w + 77], [601, 601, 10*w - 37], [613, 613, w + 273], [613, 613, w + 339], [617, 617, w + 131], [617, 617, w + 485], [641, 641, 18*w - 77], [641, 641, 18*w + 59], [643, 643, w + 143], [643, 643, w + 499], [659, 659, 12*w - 47], [659, 659, 12*w + 35], [683, 683, w + 188], [683, 683, w + 494], [701, 701, 4*w - 33], [701, 701, -4*w - 29], [719, 719, 2*w - 29], [719, 719, -2*w - 27], [733, 733, w + 38], [733, 733, w + 694], [743, 743, w + 288], [743, 743, w + 454], [751, 751, 8*w - 21], [751, 751, -8*w - 13], [773, 773, w + 349], [773, 773, w + 423], [787, 787, w + 148], [787, 787, w + 638], [809, 809, -8*w - 39], [809, 809, 8*w - 47], [827, 827, w + 361], [827, 827, w + 465], [829, 829, -4*w - 31], [829, 829, 4*w - 35], [853, 853, w + 41], [853, 853, w + 811], [859, 859, -6*w - 35], [859, 859, 6*w - 41], [863, 863, w + 155], [863, 863, w + 707], [877, 877, w + 132], [877, 877, w + 744], [881, 881, 14*w - 55], [881, 881, 14*w + 41], [911, 911, 16*w + 49], [911, 911, 16*w - 65], [919, 919, 8*w - 15], [919, 919, -8*w - 7], [947, 947, w + 257], [947, 947, w + 689], [961, 31, -31], [967, 967, w + 448], [967, 967, w + 518], [971, 971, 12*w - 43], [971, 971, -12*w - 31], [977, 977, w + 88], [977, 977, w + 888], [983, 983, w + 357], [983, 983, w + 625], [991, 991, -8*w - 3], [991, 991, 8*w - 11], [1031, 1031, 8*w - 7], [1031, 1031, 8*w - 1], [1033, 1033, w + 287], [1033, 1033, w + 745], [1039, 1039, 8*w - 5], [1039, 1039, 8*w - 3], [1049, 1049, 10*w - 29], [1049, 1049, -10*w - 19], [1069, 1069, 14*w - 53], [1069, 1069, 14*w + 39], [1087, 1087, w + 147], [1087, 1087, w + 939], [1091, 1091, 2*w - 35], [1091, 1091, -2*w - 33], [1097, 1097, w + 504], [1097, 1097, w + 592], [1103, 1103, w + 351], [1103, 1103, w + 751], [1109, 1109, -4*w - 35], [1109, 1109, 4*w - 39], [1123, 1123, w + 459], [1123, 1123, w + 663], [1163, 1163, w + 180], [1163, 1163, w + 982], [1171, 1171, 14*w - 73], [1171, 1171, 20*w - 83], [1217, 1217, w + 49], [1217, 1217, w + 1167], [1231, 1231, 2*w - 37], [1231, 1231, -2*w - 35], [1237, 1237, w + 474], [1237, 1237, w + 762], [1249, 1249, 14*w - 51], [1249, 1249, 14*w + 37], [1291, 1291, -10*w - 49], [1291, 1291, 10*w - 59], [1301, 1301, 10*w - 23], [1301, 1301, -10*w - 13], [1307, 1307, w + 302], [1307, 1307, w + 1004], [1361, 1361, -8*w - 45], [1361, 1361, 8*w - 53], [1367, 1367, w + 525], [1367, 1367, w + 841], [1373, 1373, w + 422], [1373, 1373, w + 950], [1381, 1381, 26*w - 111], [1381, 1381, 26*w + 85], [1423, 1423, w + 53], [1423, 1423, w + 1369], [1429, 1429, 10*w - 19], [1429, 1429, -10*w - 9], [1439, 1439, 24*w + 77], [1439, 1439, 24*w - 101], [1459, 1459, 20*w + 61], [1459, 1459, 20*w - 81], [1481, 1481, -10*w - 7], [1481, 1481, 10*w - 17], [1487, 1487, w + 172], [1487, 1487, w + 1314], [1493, 1493, w + 204], [1493, 1493, w + 1288], [1499, 1499, 12*w - 35], [1499, 1499, -12*w - 23], [1511, 1511, 10*w - 61], [1511, 1511, -10*w - 51], [1523, 1523, w + 526], [1523, 1523, w + 996], [1531, 1531, 6*w - 49], [1531, 1531, -6*w - 43], [1553, 1553, w + 352], [1553, 1553, w + 1200], [1559, 1559, -26*w + 125], [1559, 1559, 16*w - 59], [1567, 1567, w + 301], [1567, 1567, w + 1265], [1597, 1597, w + 211], [1597, 1597, w + 1385], [1607, 1607, w + 758], [1607, 1607, w + 848], [1609, 1609, 10*w - 9], [1609, 1609, 10*w - 1], [1621, 1621, 10*w - 3], [1621, 1621, 10*w - 7], [1627, 1627, w + 337], [1627, 1627, w + 1289], [1657, 1657, w + 774], [1657, 1657, w + 882], [1681, 41, -41], [1697, 1697, w + 585], [1697, 1697, w + 1111], [1699, 1699, 2*w - 43], [1699, 1699, -2*w - 41], [1723, 1723, w + 458], [1723, 1723, w + 1264], [1741, 1741, -14*w - 31], [1741, 1741, 14*w - 45], [1747, 1747, w + 508], [1747, 1747, w + 1238], [1753, 1753, w + 414], [1753, 1753, w + 1338], [1759, 1759, 16*w - 57], [1759, 1759, -16*w - 41], [1783, 1783, w + 119], [1783, 1783, w + 1663], [1787, 1787, w + 418], [1787, 1787, w + 1368], [1811, 1811, -12*w - 17], [1811, 1811, 12*w - 29], [1849, 43, -43], [1867, 1867, w + 855], [1867, 1867, w + 1011], [1871, 1871, 2*w - 45], [1871, 1871, -2*w - 43], [1877, 1877, w + 362], [1877, 1877, w + 1514], [1889, 1889, -14*w - 29], [1889, 1889, 14*w - 43], [1901, 1901, -28*w + 135], [1901, 1901, 18*w - 67], [1913, 1913, w + 898], [1913, 1913, w + 1014], [1949, 1949, 4*w - 49], [1949, 1949, -4*w - 45], [1951, 1951, -16*w - 39], [1951, 1951, 16*w - 55], [1979, 1979, -12*w - 13], [1979, 1979, 12*w - 25], [1987, 1987, w + 943], [1987, 1987, w + 1043], [1997, 1997, w + 236], [1997, 1997, w + 1760], [1999, 1999, 32*w - 137], [1999, 1999, 32*w + 105], [2011, 2011, 20*w - 77], [2011, 2011, 20*w + 57], [2017, 2017, w + 854], [2017, 2017, w + 1162], [2029, 2029, 14*w - 41], [2029, 2029, -14*w - 27], [2081, 2081, 30*w + 97], [2081, 2081, 30*w - 127], [2087, 2087, w + 904], [2087, 2087, w + 1182], [2089, 2089, 22*w - 87], [2089, 2089, 22*w + 65], [2113, 2113, w + 331], [2113, 2113, w + 1781], [2129, 2129, -18*w - 47], [2129, 2129, 18*w - 65], [2131, 2131, 28*w - 117], [2131, 2131, 18*w - 95], [2137, 2137, w + 65], [2137, 2137, w + 2071], [2141, 2141, -4*w - 47], [2141, 2141, 4*w - 51], [2143, 2143, w + 843], [2143, 2143, w + 1299], [2153, 2153, w + 564], [2153, 2153, w + 1588], [2161, 2161, 14*w - 39], [2161, 2161, -14*w - 25], [2203, 2203, w + 66], [2203, 2203, w + 2136], [2239, 2239, 2*w - 49], [2239, 2239, -2*w - 47], [2243, 2243, w + 792], [2243, 2243, w + 1450], [2267, 2267, w + 598], [2267, 2267, w + 1668], [2273, 2273, w + 780], [2273, 2273, w + 1492], [2293, 2293, w + 135], [2293, 2293, w + 2157], [2311, 2311, 16*w - 51], [2311, 2311, -16*w - 35], [2333, 2333, w + 686], [2333, 2333, w + 1646], [2339, 2339, 12*w - 7], [2339, 2339, 12*w - 5], [2341, 2341, 4*w - 53], [2341, 2341, -4*w - 49], [2347, 2347, w + 966], [2347, 2347, w + 1380], [2377, 2377, w + 668], [2377, 2377, w + 1708], [2389, 2389, 22*w + 63], [2389, 2389, 22*w - 85], [2423, 2423, w + 260], [2423, 2423, w + 2162], [2437, 2437, w + 899], [2437, 2437, w + 1537], [2441, 2441, -8*w - 55], [2441, 2441, 8*w - 63], [2477, 2477, w + 70], [2477, 2477, w + 2406], [2503, 2503, w + 447], [2503, 2503, w + 2055], [2521, 2521, 26*w + 79], [2521, 2521, 26*w - 105], [2531, 2531, 20*w - 73], [2531, 2531, 20*w + 53], [2539, 2539, 28*w - 115], [2539, 2539, 22*w - 113], [2543, 2543, w + 1180], [2543, 2543, w + 1362], [2549, 2549, -4*w - 51], [2549, 2549, 4*w - 55], [2551, 2551, -6*w - 53], [2551, 2551, 6*w - 59], [2591, 2591, -14*w - 69], [2591, 2591, 14*w - 83], [2593, 2593, w + 619], [2593, 2593, w + 1973], [2609, 2609, -14*w - 17], [2609, 2609, 14*w - 31], [2633, 2633, w + 1248], [2633, 2633, w + 1384], [2647, 2647, w + 544], [2647, 2647, w + 2102], [2657, 2657, w + 1117], [2657, 2657, w + 1539], [2663, 2663, w + 1174], [2663, 2663, w + 1488], [2683, 2683, w + 394], [2683, 2683, w + 2288], [2693, 2693, w + 73], [2693, 2693, w + 2619], [2729, 2729, 38*w + 125], [2729, 2729, 38*w - 163], [2731, 2731, 10*w - 71], [2731, 2731, -10*w - 61], [2767, 2767, w + 74], [2767, 2767, w + 2692], [2777, 2777, w + 900], [2777, 2777, w + 1876], [2791, 2791, -16*w - 29], [2791, 2791, 16*w - 45], [2797, 2797, w + 442], [2797, 2797, w + 2354], [2803, 2803, w + 665], [2803, 2803, w + 2137], [2809, 53, -53], [2851, 2851, 2*w - 55], [2851, 2851, -2*w - 53], [2861, 2861, 14*w - 25], [2861, 2861, -14*w - 11], [2897, 2897, w + 304], [2897, 2897, w + 2592], [2909, 2909, 34*w + 109], [2909, 2909, 34*w - 143], [2917, 2917, w + 1109], [2917, 2917, w + 1807], [2927, 2927, w + 924], [2927, 2927, w + 2002], [2939, 2939, 28*w + 85], [2939, 2939, 28*w - 113], [2953, 2953, w + 1404], [2953, 2953, w + 1548], [2957, 2957, w + 575], [2957, 2957, w + 2381], [2999, 2999, -10*w - 63], [2999, 2999, 10*w - 73], [3019, 3019, -20*w - 49], [3019, 3019, 20*w - 69], [3023, 3023, w + 396], [3023, 3023, w + 2626], [3037, 3037, w + 623], [3037, 3037, w + 2413], [3041, 3041, -14*w - 5], [3041, 3041, 14*w - 19], [3083, 3083, w + 675], [3083, 3083, w + 2407], [3119, 3119, 24*w - 91], [3119, 3119, 24*w + 67], [3121, 3121, -14*w - 1], [3121, 3121, 14*w - 15], [3167, 3167, w + 1082], [3167, 3167, w + 2084], [3169, 3169, 14*w - 11], [3169, 3169, 14*w - 3], [3181, 3181, 14*w - 9], [3181, 3181, 14*w - 5], [3187, 3187, w + 252], [3187, 3187, w + 2934], [3203, 3203, w + 299], [3203, 3203, w + 2903], [3217, 3217, w + 561], [3217, 3217, w + 2655], [3221, 3221, 4*w - 61], [3221, 3221, -4*w - 57], [3251, 3251, -20*w - 47], [3251, 3251, 20*w - 67], [3257, 3257, w + 477], [3257, 3257, w + 2779], [3259, 3259, -6*w - 59], [3259, 3259, 6*w - 65], [3299, 3299, 2*w - 59], [3299, 3299, -2*w - 57], [3301, 3301, 38*w + 123], [3301, 3301, 38*w - 161], [3307, 3307, w + 788], [3307, 3307, w + 2518], [3313, 3313, w + 81], [3313, 3313, w + 3231], [3319, 3319, 16*w - 37], [3319, 3319, -16*w - 21], [3323, 3323, w + 1557], [3323, 3323, w + 1765], [3329, 3329, 18*w - 53], [3329, 3329, -18*w - 35], [3331, 3331, 28*w - 111], [3331, 3331, 28*w + 83], [3343, 3343, w + 967], [3343, 3343, w + 2375], [3347, 3347, w + 1497], [3347, 3347, w + 1849], [3371, 3371, 36*w - 151], [3371, 3371, 22*w - 117], [3373, 3373, w + 641], [3373, 3373, w + 2731], [3389, 3389, 30*w + 91], [3389, 3389, 30*w - 121], [3413, 3413, w + 1165], [3413, 3413, w + 2247], [3449, 3449, -8*w - 63], [3449, 3449, 8*w - 71], [3461, 3461, -4*w - 59], [3461, 3461, 4*w - 63], [3463, 3463, w + 1318], [3463, 3463, w + 2144], [3481, 59, -59], [3511, 3511, 6*w - 67], [3511, 3511, -6*w - 61], [3517, 3517, w + 335], [3517, 3517, w + 3181], [3539, 3539, 14*w - 89], [3539, 3539, -14*w - 75], [3547, 3547, w + 168], [3547, 3547, w + 3378], [3557, 3557, w + 1699], [3557, 3557, w + 1857], [3559, 3559, 10*w - 77], [3559, 3559, -10*w - 67], [3571, 3571, 44*w + 145], [3571, 3571, 18*w + 85], [3583, 3583, w + 682], [3583, 3583, w + 2900], [3593, 3593, w + 864], [3593, 3593, w + 2728], [3607, 3607, w + 502], [3607, 3607, w + 3104], [3631, 3631, 16*w - 31], [3631, 3631, -16*w - 15], [3673, 3673, w + 1010], [3673, 3673, w + 2662], [3677, 3677, w + 1406], [3677, 3677, w + 2270], [3691, 3691, -20*w - 43], [3691, 3691, 20*w - 63], [3697, 3697, w + 1571], [3697, 3697, w + 2125], [3701, 3701, 42*w - 179], [3701, 3701, 42*w + 137], [3709, 3709, 4*w - 65], [3709, 3709, -4*w - 61], [3719, 3719, -16*w - 13], [3719, 3719, 16*w - 29], [3733, 3733, w + 86], [3733, 3733, w + 3646], [3761, 3761, 16*w - 97], [3761, 3761, -16*w - 81], [3769, 3769, -22*w - 53], [3769, 3769, 22*w - 75], [3779, 3779, 2*w - 63], [3779, 3779, -2*w - 61], [3803, 3803, w + 610], [3803, 3803, w + 3192], [3821, 3821, -18*w - 29], [3821, 3821, 18*w - 47], [3833, 3833, w + 1834], [3833, 3833, w + 1998], [3851, 3851, -10*w - 69], [3851, 3851, 10*w - 79], [3853, 3853, w + 614], [3853, 3853, w + 3238], [3863, 3863, w + 838], [3863, 3863, w + 3024], [3907, 3907, w + 1883], [3907, 3907, w + 2023], [3929, 3929, -40*w + 193], [3929, 3929, 26*w - 97], [3947, 3947, w + 1028], [3947, 3947, w + 2918], [3967, 3967, w + 1433], [3967, 3967, w + 2533], [4001, 4001, 8*w - 75], [4001, 4001, -8*w - 67], [4021, 4021, 22*w - 73], [4021, 4021, -22*w - 51], [4079, 4079, -16*w - 1], [4079, 4079, 16*w - 17], [4091, 4091, 28*w + 79], [4091, 4091, 28*w - 107], [4093, 4093, w + 1210], [4093, 4093, w + 2882], [4099, 4099, 20*w - 59], [4099, 4099, -20*w - 39], [4111, 4111, 16*w - 15], [4111, 4111, 16*w - 1], [4127, 4127, w + 537], [4127, 4127, w + 3589], [4153, 4153, w + 1074], [4153, 4153, w + 3078], [4159, 4159, 16*w - 9], [4159, 4159, 16*w - 7], [4211, 4211, -14*w - 79], [4211, 4211, 14*w - 93], [4217, 4217, w + 2022], [4217, 4217, w + 2194], [4229, 4229, 4*w - 69], [4229, 4229, -4*w - 65], [4241, 4241, 18*w - 41], [4241, 4241, -18*w - 23], [4243, 4243, w + 368], [4243, 4243, w + 3874], [4253, 4253, w + 184], [4253, 4253, w + 4068], [4261, 4261, 26*w - 95], [4261, 4261, 26*w + 69], [4283, 4283, w + 1489], [4283, 4283, w + 2793], [4289, 4289, -8*w - 69], [4289, 4289, 8*w - 77], [4297, 4297, w + 1681], [4297, 4297, w + 2615], [4327, 4327, w + 1600], [4327, 4327, w + 2726], [4337, 4337, w + 348], [4337, 4337, w + 3988], [4339, 4339, -18*w - 89], [4339, 4339, 18*w - 107], [4357, 4357, w + 890], [4357, 4357, w + 3466], [4363, 4363, w + 93], [4363, 4363, w + 4269], [4373, 4373, w + 804], [4373, 4373, w + 3568], [4391, 4391, 40*w - 167], [4391, 4391, 26*w - 137], [4421, 4421, 38*w - 157], [4421, 4421, 28*w - 145], [4457, 4457, w + 94], [4457, 4457, w + 4362], [4481, 4481, -18*w - 19], [4481, 4481, 18*w - 37], [4483, 4483, w + 1612], [4483, 4483, w + 2870], [4493, 4493, w + 510], [4493, 4493, w + 3982], [4513, 4513, w + 355], [4513, 4513, w + 4157], [4517, 4517, w + 2169], [4517, 4517, w + 2347], [4549, 4549, 12*w - 89], [4549, 4549, -12*w - 77], [4583, 4583, w + 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[9049, 9049, 26*w - 57], [9049, 9049, -26*w - 31], [9067, 9067, w + 2103], [9067, 9067, w + 6963], [9091, 9091, 68*w - 291], [9091, 9091, 68*w + 223], [9109, 9109, 12*w - 113], [9109, 9109, -12*w - 101], [9133, 9133, w + 1564], [9133, 9133, w + 7568], [9137, 9137, w + 427], [9137, 9137, w + 8709], [9151, 9151, 2*w - 97], [9151, 9151, -2*w - 95], [9157, 9157, w + 2883], [9157, 9157, w + 6273], [9161, 9161, 8*w - 105], [9161, 9161, -8*w - 97], [9173, 9173, w + 4194], [9173, 9173, w + 4978], [9181, 9181, 34*w - 115], [9181, 9181, -34*w - 81], [9221, 9221, -26*w - 29], [9221, 9221, 26*w - 55], [9239, 9239, 24*w - 23], [9239, 9239, 24*w - 1], [9277, 9277, w + 953], [9277, 9277, w + 8323], [9281, 9281, 66*w - 281], [9281, 9281, 66*w + 215], [9293, 9293, w + 1756], [9293, 9293, w + 7536], [9311, 9311, 24*w - 19], [9311, 9311, 24*w - 5], [9323, 9323, w + 1066], [9323, 9323, w + 8256], [9397, 9397, w + 1218], [9397, 9397, w + 8178], [9421, 9421, 46*w + 135], [9421, 9421, 46*w - 181], [9433, 9433, w + 2210], [9433, 9433, w + 7222], [9439, 9439, 64*w - 271], [9439, 9439, 64*w + 207], [9461, 9461, 28*w - 163], [9461, 9461, -28*w - 135], [9491, 9491, 28*w - 71], [9491, 9491, -28*w - 43], [9497, 9497, w + 2522], [9497, 9497, w + 6974], [9539, 9539, 2*w - 99], [9539, 9539, -2*w - 97], [9547, 9547, w + 817], [9547, 9547, w + 8729], [9551, 9551, 48*w - 191], [9551, 9551, 48*w + 143], [9587, 9587, w + 3938], [9587, 9587, w + 5648], [9613, 9613, w + 1393], [9613, 9613, w + 8219], [9619, 9619, -18*w - 113], [9619, 9619, 18*w - 131], [9629, 9629, -20*w - 117], [9629, 9629, 20*w - 137], [9649, 9649, 50*w + 151], [9649, 9649, 50*w - 201], [9677, 9677, w + 3318], [9677, 9677, w + 6358], [9689, 9689, 26*w - 49], [9689, 9689, -26*w - 23], [9721, 9721, 40*w - 209], [9721, 9721, 58*w - 241], [9743, 9743, w + 3061], [9743, 9743, w + 6681], [9749, 9749, 44*w - 225], [9749, 9749, 54*w - 221], [9787, 9787, w + 523], [9787, 9787, w + 9263], [9811, 9811, -14*w - 107], [9811, 9811, 14*w - 121], [9817, 9817, w + 560], [9817, 9817, w + 9256], [9829, 9829, -26*w - 21], [9829, 9829, 26*w - 47], [9833, 9833, w + 280], [9833, 9833, w + 9552], [9851, 9851, 44*w - 169], [9851, 9851, 44*w + 125], [9871, 9871, -66*w + 317], [9871, 9871, 40*w - 147], [9887, 9887, w + 3389], [9887, 9887, w + 6497], [9929, 9929, 70*w - 299], [9929, 9929, 70*w + 229], [9931, 9931, 28*w - 67], [9931, 9931, -28*w - 39], [9941, 9941, -4*w - 99], [9941, 9941, 4*w - 103], [9949, 9949, 34*w - 111], [9949, 9949, -34*w - 77], [9973, 9973, w + 2785], [9973, 9973, w + 7187]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [-2, -2, 1, -2, -2, -5, 4, 0, 0, 3, 3, 8, 8, 12, 12, -7, -7, 4, 4, -10, -10, -6, -6, -7, -7, 2, 2, 1, -18, -18, -7, -7, 10, 10, 4, 4, -12, -12, -15, -15, 7, 7, 17, 17, 4, 4, -2, -2, 0, 0, 12, 12, 19, 19, 18, 18, -23, -23, 10, 10, -30, 24, 24, 8, 8, 12, 12, 13, 13, -21, -21, -38, -1, -1, -15, -15, -2, -2, 20, 20, 40, 40, -12, -12, -11, -11, 23, 23, -8, -8, -3, -3, -45, -2, -2, 0, 0, -28, -28, 33, 33, 28, 28, 44, 44, 40, 40, 2, 2, -16, -16, 18, 18, -33, -33, 29, 29, 10, 10, -16, -16, 2, 2, 15, 15, -36, -36, 4, 4, -23, -23, -6, -6, -32, -32, 0, 0, -52, -52, 25, 25, 14, 14, -15, -15, 24, 24, -12, -12, -43, -43, 12, 12, 10, 10, -27, -27, -13, -32, -32, 47, 47, -27, -27, 39, 39, -8, -8, 32, 32, -16, -16, 5, 5, -55, -55, -20, -20, 8, 8, -58, -58, -42, -42, -51, -51, -30, -30, 24, 24, 34, 34, -3, -3, -42, -42, -18, -18, 18, 18, 40, 40, -8, -8, 27, 27, 28, 28, 12, 12, -72, -72, 39, 39, -68, -68, 29, 29, -70, -70, 0, 0, -20, -20, 32, 32, 58, 58, -36, -36, 55, 55, 37, 37, -41, -41, 32, 32, -56, -56, -60, -60, -52, -52, -32, -32, 33, 33, -10, -10, 22, 22, 78, 78, -2, -2, -18, -42, -42, 40, 40, -46, -46, 17, 17, -57, -57, 34, 34, -40, -40, 59, 59, -57, -57, 12, 12, -50, 28, 28, -3, -3, 18, 18, 70, 70, 77, 77, -36, -36, -40, -40, -23, -23, 30, 30, -22, -22, -72, -72, -20, -20, -13, -13, -17, -17, 45, 45, -18, -18, 48, 48, -10, -10, -86, -86, 20, 20, -68, -68, 73, 73, -58, -58, -91, -91, -26, -26, -13, -13, -1, -1, -70, -70, -56, -56, 93, 93, 4, 4, 29, 29, -13, -13, 59, 59, 10, 10, 67, 67, -37, -37, 3, 3, -50, -50, -31, -31, -82, -82, 42, 42, 48, 48, 14, 14, 72, 72, 57, 57, 0, 0, 34, 34, -20, -20, -98, -98, -58, -58, 14, 14, -30, -30, 39, 39, 38, 38, 38, 38, 39, 39, -16, -16, -41, -41, 30, 30, -68, -68, 48, 48, -42, -42, 42, 42, -42, -42, 44, 44, -70, 2, 2, -63, -63, 38, 38, -25, -25, 88, 88, -72, -72, -50, -50, -86, -86, 3, 3, -80, -80, 85, 85, 39, 39, 13, 13, 42, 42, 29, 29, -90, -90, 22, 22, 18, 18, 45, 45, 32, 32, -2, -2, -6, -6, 23, 23, -103, -103, -48, -48, 58, 58, -60, -60, 100, 100, -73, -73, 98, 98, 4, 4, 0, 0, -76, -76, -100, -100, -43, -43, 44, 44, -17, -17, -103, -103, 4, 4, 15, 15, 9, 9, 40, 40, -38, -38, -96, -96, -93, 12, 12, -22, -22, 20, 20, 53, 53, -27, -27, 0, 0, -28, -28, -96, -96, -26, -26, 58, 58, 32, 32, -76, -76, -62, -62, 92, 92, 23, 23, 102, 102, 20, 20, -55, -55, 114, 114, -88, -88, 40, 40, -30, -30, 74, 74, -3, -3, 19, 19, -73, -73, 74, 74, 54, 54, -22, -22, -60, -60, -107, -107, -92, -92, 102, 102, 22, 22, -60, -60, 92, 92, 94, 94, 20, 20, 62, 62, 48, 48, -36, -36, 85, 85, -63, -63, 33, 33, -55, -55, 92, 92, 64, 64, -16, -16, -113, -113, 39, 39, -60, -60, -62, -62, -107, -107, -2, -2, 55, 55, -117, -117, -6, -6, 84, 84, 42, 42, 72, 72, 78, 78, 82, 82, -106, -106, 29, 29, 59, 59, 8, 8, -10, -10, -36, -36, -52, -52, -81, -81, -73, -73, 114, 114, 25, 25, 30, 30, 30, 30, 77, 77, -132, -132, -88, -88, 72, 72, 93, 93, 5, 5, -126, -126, 32, 32, 94, 94, 100, 100, -1, -1, 28, 28, 119, 119, -45, -45, -3, -3, 54, 54, -133, 8, 8, -10, -10, -32, -32, 15, 15, -111, -111, 28, 28, 97, 97, -50, -50, -96, -96, 63, 63, 87, 87, -21, -21, -70, -70, 57, 57, -22, -22, -76, -76, 132, 132, -2, -2, -11, -11, -15, -15, 14, 14, 82, 82, 48, 48, 22, 22, 8, 8, -122, -122, 7, 7, -33, -33, 64, 64, 60, 60, 17, 17, 113, 113, -48, -48, 4, 4, -57, -57, -125, -125, -110, -110, 114, 114, -138, -138, -30, -30, 104, 104, 102, 102, -1, -1, -78, -78, -66, -66, 80, 80, 2, 2, -137, -137, -55, -55, -27, -27, 59, 59, -18, -18, 73, 73, 105, 105, -117, -117, -101, -101, -42, -42, -26, -26, -26, -26, -48, -48, 4, 4, -86, -86, 4, 4, -38, -38, -102, -102, -30, -30, -132, -132, -132, -132, -23, -23, 89, 89, 38, 38, -138, -138, 130, 130, -28, -28, -37, -37, 105, 105, 52, 52, 117, 117, -80, -80, 68, 68, -98, -98, -56, -56, 130, 130, -93, -93, -122, -122, -95, -95, 40, 40, -83, -83, -91, -91, 60, 60, -88, -88, 104, 104, -56, -56, -132, -132, 65, 65, 24, 24, -108, -108, -127, -127, -96, -96, -90, -90, -72, -72, 30, 30, -128, -128, -101, -101, 142, 142, -126, -126, 120, 120, 23, 23, -62, -62, -60, -60, 24, 24, -3, -3, -73, -73, -105, -105, 153, 153, 78, 78, 162, 162, -66, -66, 25, 25, -110, -110, -21, -21, -57, -57, -25, -25, -68, -68, -150, -150, -88, -88, 103, 103, -142, -142, 135, 135, 138, 138, 25, 25, -87, -87, 142, 142, 123, 123, -45, -45, 109, 109, -2, -2, 40, 40, -33, -33, -56, -56, 38, 38, 22, 22, -70, -70, 50, 50, -51, -51, 48, 48, 33, 33, 19, 19, -21, -21, 60, 60, 88, 88, -6, -6, -165, -165, -36, -36, 127, 127, 47, 18, 18, -136, -136, -26, -26, 85, 85, 137, 137, -46, -46, 130, 130, -40, -40, 135, 135, -86, -86, 112, 112, 168, 168, 84, 84, -38, -38, 92, 92, -120, -120, -17, -17, -46, -46, -100, -100, -26, -26, -72, -72, -108, -108, -38, -38, -110, -110, 59, 59, 130, 130, -92, -92, 168, 168, 110, 110, 54, 54, -23, -23, -36, -36, -126, -126, -146, -146, -93, -93, 130, 130, -137, -137, 145, 145, -108, -108, -127, -127, 124, 124, 98, 98, 132, 132, 40, 40, -32, -32, -13, -13, 55, 55, 69, 69, 124, 124, 48, 48, 157, 157, -70, -70, 90, 90, -62, -62, -36, -36, 109, 109, -133, -133, 129, 129, 15, 15, -128, -128, 25, 25, 153, 153, 144, 144, -130, -130, -147, -147, 67, 67, 85, 85, 79, 79, 78, 78, 102, 102, 63, 63, -138, -138, 24, 24, 82, 82, -103, -103, -60, -60, 43, 43, 132, 132, -66, -66, -13, -13, 134, 134, 83, 83, -78, -78, 94, 94, 65, 65, 62, 62, -108, -108, -87, -87, 0, 0, -2, -2, 27, 27, 18, 18, 64, 64, -140, -140, 25, 25, -90, -90, 138, 138, -175, -175, -178, -178, 164, 164, -170, -170, -2, -2, 62, 62, -102, -102, -80, -80, 134, 134, -48, -48, -128, -128, -117, -117, 0, 0, 57, 57, 42, 42, -10, -10, 4, 4]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); 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;