/* 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![-15, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, w + 1], [3, 3, w], [5, 5, w], [7, 7, w + 1], [7, 7, w + 6], [11, 11, -w - 2], [11, 11, w - 2], [17, 17, w + 7], [17, 17, w + 10], [43, 43, w + 12], [43, 43, w + 31], [53, 53, w + 11], [53, 53, w + 42], [59, 59, 2*w - 1], [59, 59, -2*w - 1], [61, 61, 2*w - 11], [61, 61, -2*w - 11], [67, 67, w + 22], [67, 67, w + 45], [71, 71, 3*w - 8], [71, 71, -3*w - 8], [103, 103, w + 18], [103, 103, w + 85], [109, 109, 2*w - 13], [109, 109, -2*w - 13], [113, 113, w + 44], [113, 113, w + 69], [127, 127, w + 53], [127, 127, w + 74], [131, 131, 3*w - 2], [131, 131, -3*w - 2], [137, 137, w + 17], [137, 137, w + 120], [163, 163, w + 34], [163, 163, w + 129], [169, 13, -13], [173, 173, w + 19], [173, 173, w + 154], [179, 179, 5*w - 14], [179, 179, -5*w - 14], [181, 181, -w - 14], [181, 181, w - 14], [191, 191, 4*w - 7], [191, 191, -4*w - 7], [197, 197, w + 58], [197, 197, w + 139], [223, 223, w + 98], [223, 223, w + 125], [229, 229, 2*w - 17], [229, 229, -2*w - 17], [233, 233, w + 99], [233, 233, w + 134], [239, 239, -4*w - 1], [239, 239, 4*w - 1], [241, 241, -w - 16], [241, 241, w - 16], [251, 251, 6*w - 17], [251, 251, -6*w - 17], [257, 257, w + 23], [257, 257, w + 234], [283, 283, w + 79], [283, 283, w + 204], [293, 293, w + 111], [293, 293, w + 182], [307, 307, w + 130], [307, 307, w + 177], [311, 311, -5*w - 8], [311, 311, 5*w - 8], [317, 317, w + 40], [317, 317, w + 277], [349, 349, 3*w - 22], [349, 349, -3*w - 22], [353, 353, w + 108], [353, 353, w + 245], [359, 359, 5*w - 4], [359, 359, -5*w - 4], [361, 19, -19], [367, 367, w + 105], [367, 367, w + 262], [409, 409, 5*w - 28], [409, 409, -5*w - 28], [419, 419, 6*w - 11], [419, 419, -6*w - 11], [421, 421, 6*w - 31], [421, 421, -6*w - 31], [431, 431, 8*w - 23], [431, 431, -8*w - 23], [463, 463, w + 101], [463, 463, w + 362], [479, 479, 7*w - 16], [479, 479, -7*w - 16], [487, 487, w + 224], [487, 487, w + 263], [491, 491, -6*w - 7], [491, 491, 6*w - 7], [523, 523, w + 231], [523, 523, w + 292], [529, 23, -23], [541, 541, -3*w - 26], [541, 541, 3*w - 26], [547, 547, w + 62], [547, 547, w + 485], [557, 557, w + 153], [557, 557, w + 404], [593, 593, w + 269], [593, 593, w + 324], [599, 599, -8*w - 19], [599, 599, 8*w - 19], [601, 601, 4*w - 29], [601, 601, -4*w - 29], [607, 607, w + 135], [607, 607, w + 472], [617, 617, w + 218], [617, 617, w + 399], [643, 643, w + 119], [643, 643, w + 524], [653, 653, w + 253], [653, 653, w + 400], [659, 659, 10*w - 29], [659, 659, -10*w - 29], [661, 661, -w - 26], [661, 661, w - 26], [677, 677, w + 37], [677, 677, w + 640], [709, 709, -7*w - 38], [709, 709, 7*w - 38], [719, 719, -7*w - 4], [719, 719, 7*w - 4], [727, 727, w + 200], [727, 727, w + 527], [769, 769, -w - 28], [769, 769, w - 28], [773, 773, w + 244], [773, 773, w + 529], [787, 787, w + 184], [787, 787, w + 603], [797, 797, w + 183], [797, 797, w + 614], [823, 823, w + 76], [823, 823, w + 747], [829, 829, 6*w - 37], [829, 829, -6*w - 37], [839, 839, -8*w - 11], [839, 839, 8*w - 11], [841, 29, -29], [857, 857, w + 300], [857, 857, w + 557], [883, 883, w + 402], [883, 883, w + 481], [907, 907, w + 165], [907, 907, w + 742], [911, 911, 8*w - 7], [911, 911, -8*w - 7], [953, 953, w + 341], [953, 953, w + 612], [961, 31, -31], [967, 967, w + 54], [967, 967, w + 913], [971, 971, 10*w - 23], [971, 971, -10*w - 23], [977, 977, w + 70], [977, 977, w + 907], [1009, 1009, -w - 32], [1009, 1009, w - 32], [1013, 1013, w + 415], [1013, 1013, w + 598], [1019, 1019, -9*w - 14], [1019, 1019, 9*w - 14], [1021, 1021, 3*w - 34], [1021, 1021, -3*w - 34], [1031, 1031, -11*w - 28], [1031, 1031, 11*w - 28], [1063, 1063, w + 503], [1063, 1063, w + 560], [1069, 1069, 5*w - 38], [1069, 1069, -5*w - 38], [1087, 1087, w + 333], [1087, 1087, w + 754], [1091, 1091, 13*w - 38], [1091, 1091, -13*w - 38], [1097, 1097, w + 47], [1097, 1097, w + 1050], [1123, 1123, w + 517], [1123, 1123, w + 606], [1129, 1129, 4*w - 37], [1129, 1129, -4*w - 37], [1151, 1151, 9*w - 8], [1151, 1151, -9*w - 8], [1193, 1193, w + 49], [1193, 1193, w + 1144], [1201, 1201, -7*w - 44], [1201, 1201, 7*w - 44], [1217, 1217, w + 254], [1217, 1217, w + 963], [1249, 1249, -8*w - 47], [1249, 1249, 8*w - 47], [1259, 1259, 14*w - 41], [1259, 1259, -14*w - 41], [1277, 1277, w + 80], [1277, 1277, w + 1197], [1303, 1303, w + 620], [1303, 1303, w + 683], [1319, 1319, -12*w - 29], [1319, 1319, 12*w - 29], [1321, 1321, 11*w - 56], [1321, 1321, -11*w - 56], [1327, 1327, w + 544], [1327, 1327, w + 783], [1369, 37, -37], [1373, 1373, w + 580], [1373, 1373, w + 793], [1381, 1381, 7*w - 46], [1381, 1381, -7*w - 46], [1423, 1423, w + 381], [1423, 1423, w + 1042], [1429, 1429, -w - 38], [1429, 1429, w - 38], [1433, 1433, w + 674], [1433, 1433, w + 759], [1439, 1439, 15*w - 44], [1439, 1439, -15*w - 44], [1447, 1447, w + 66], [1447, 1447, w + 1381], [1451, 1451, -10*w - 7], [1451, 1451, 10*w - 7], [1483, 1483, w + 434], [1483, 1483, w + 1049], [1489, 1489, 9*w - 52], [1489, 1489, -9*w - 52], [1493, 1493, w + 222], [1493, 1493, w + 1271], [1499, 1499, -10*w - 1], [1499, 1499, 10*w - 1], [1511, 1511, 13*w - 32], [1511, 1511, -13*w - 32], [1543, 1543, w + 104], [1543, 1543, w + 1439], [1549, 1549, 11*w - 58], [1549, 1549, -11*w - 58], [1553, 1553, w + 633], [1553, 1553, w + 920], [1559, 1559, 11*w - 16], [1559, 1559, -11*w - 16], [1567, 1567, w + 749], [1567, 1567, w + 818], [1571, 1571, 14*w - 37], [1571, 1571, -14*w - 37], [1609, 1609, -4*w - 43], [1609, 1609, 4*w - 43], [1613, 1613, w + 593], [1613, 1613, w + 1020], [1619, 1619, -11*w - 14], [1619, 1619, 11*w - 14], [1621, 1621, 2*w - 41], [1621, 1621, -2*w - 41], [1627, 1627, w + 760], [1627, 1627, w + 867], [1637, 1637, w + 735], [1637, 1637, w + 902], [1663, 1663, w + 443], [1663, 1663, w + 1220], [1669, 1669, -6*w - 47], [1669, 1669, 6*w - 47], [1681, 41, -41], [1697, 1697, w + 267], [1697, 1697, w + 1430], [1723, 1723, w + 72], [1723, 1723, w + 1651], [1733, 1733, w + 59], [1733, 1733, w + 1674], [1741, 1741, 5*w - 46], [1741, 1741, -5*w - 46], [1747, 1747, w + 310], [1747, 1747, w + 1437], [1783, 1783, w + 753], [1783, 1783, w + 1030], [1789, 1789, 2*w - 43], [1789, 1789, -2*w - 43], [1801, 1801, -3*w - 44], [1801, 1801, 3*w - 44], [1811, 1811, 11*w - 2], [1811, 1811, -11*w - 2], [1861, 1861, 6*w - 49], [1861, 1861, -6*w - 49], [1867, 1867, w + 690], [1867, 1867, w + 1177], [1871, 1871, -12*w - 17], [1871, 1871, 12*w - 17], [1877, 1877, w + 814], [1877, 1877, w + 1063], [1913, 1913, w + 433], [1913, 1913, w + 1480], [1931, 1931, -15*w - 38], [1931, 1931, 15*w - 38], [1973, 1973, w + 691], [1973, 1973, w + 1282], [1979, 1979, -14*w - 31], [1979, 1979, 14*w - 31], [1987, 1987, w + 118], [1987, 1987, w + 1869], [1997, 1997, w + 100], [1997, 1997, w + 1897], [2029, 2029, -11*w - 62], [2029, 2029, 11*w - 62], [2039, 2039, 12*w - 11], [2039, 2039, -12*w - 11], [2083, 2083, w + 981], [2083, 2083, w + 1102], [2089, 2089, -13*w - 68], [2089, 2089, 13*w - 68], [2099, 2099, -14*w - 29], [2099, 2099, 14*w - 29], [2111, 2111, 12*w - 7], [2111, 2111, -12*w - 7], [2143, 2143, w + 882], [2143, 2143, w + 1261], [2153, 2153, w + 605], [2153, 2153, w + 1548], [2161, 2161, 4*w - 49], [2161, 2161, -4*w - 49], [2203, 2203, w + 837], [2203, 2203, w + 1366], [2209, 47, -47], [2213, 2213, w + 194], [2213, 2213, w + 2019], [2221, 2221, 10*w - 61], [2221, 2221, -10*w - 61], [2237, 2237, w + 67], [2237, 2237, w + 2170], [2269, 2269, -6*w - 53], [2269, 2269, 6*w - 53], [2273, 2273, w + 309], [2273, 2273, w + 1964], [2281, 2281, 11*w - 64], [2281, 2281, -11*w - 64], [2287, 2287, w + 483], [2287, 2287, w + 1804], [2297, 2297, w + 974], [2297, 2297, w + 1323], [2333, 2333, w + 919], [2333, 2333, w + 1414], [2339, 2339, -13*w - 14], [2339, 2339, 13*w - 14], [2341, 2341, 2*w - 49], [2341, 2341, -2*w - 49], [2347, 2347, w + 84], [2347, 2347, w + 2263], [2351, 2351, 15*w - 32], [2351, 2351, -15*w - 32], [2357, 2357, w + 1039], [2357, 2357, w + 1318], [2383, 2383, w + 229], [2383, 2383, w + 2154], [2389, 2389, 14*w - 73], [2389, 2389, -14*w - 73], [2393, 2393, w + 761], [2393, 2393, w + 1632], [2399, 2399, 17*w - 44], [2399, 2399, -17*w - 44], [2411, 2411, 14*w - 23], [2411, 2411, -14*w - 23], [2417, 2417, w + 110], [2417, 2417, w + 2307], [2459, 2459, -18*w - 49], [2459, 2459, 18*w - 49], [2467, 2467, w + 233], [2467, 2467, w + 2234], [2477, 2477, w + 510], [2477, 2477, w + 1967], [2503, 2503, w + 1208], [2503, 2503, w + 1295], [2521, 2521, 8*w - 59], [2521, 2521, -8*w - 59], [2531, 2531, 13*w - 2], [2531, 2531, -13*w - 2], [2579, 2579, -14*w - 19], [2579, 2579, 14*w - 19], [2591, 2591, -15*w - 28], [2591, 2591, 15*w - 28], [2633, 2633, w + 1259], [2633, 2633, w + 1374], [2647, 2647, w + 696], [2647, 2647, w + 1951], [2657, 2657, w + 73], [2657, 2657, w + 2584], [2683, 2683, w + 424], [2683, 2683, w + 2259], [2689, 2689, -w - 52], [2689, 2689, w - 52], [2693, 2693, w + 214], [2693, 2693, w + 2479], [2699, 2699, 15*w - 26], [2699, 2699, -15*w - 26], [2707, 2707, w + 285], [2707, 2707, w + 2422], [2711, 2711, 19*w - 52], [2711, 2711, -19*w - 52], [2749, 2749, 2*w - 53], [2749, 2749, -2*w - 53], [2753, 2753, w + 382], [2753, 2753, w + 2371], [2767, 2767, w + 840], [2767, 2767, w + 1927], [2777, 2777, w + 540], [2777, 2777, w + 2237], [2803, 2803, w + 724], [2803, 2803, w + 2079], [2819, 2819, 14*w - 11], [2819, 2819, -14*w - 11], [2837, 2837, w + 306], [2837, 2837, w + 2531], [2879, 2879, 16*w - 31], [2879, 2879, -16*w - 31], [2887, 2887, w + 686], [2887, 2887, w + 2201], [2897, 2897, w + 940], [2897, 2897, w + 1957], [2939, 2939, -14*w - 1], [2939, 2939, 14*w - 1], [2957, 2957, w + 77], [2957, 2957, w + 2880], [2999, 2999, -16*w - 29], [2999, 2999, 16*w - 29], [3001, 3001, -3*w - 56], [3001, 3001, 3*w - 56], [3011, 3011, 18*w - 43], [3011, 3011, -18*w - 43], [3049, 3049, -16*w - 83], [3049, 3049, 16*w - 83], [3061, 3061, 17*w - 86], [3061, 3061, -17*w - 86], [3067, 3067, w + 96], [3067, 3067, w + 2971], [3109, 3109, -7*w - 62], [3109, 3109, 7*w - 62], [3119, 3119, -15*w - 16], [3119, 3119, 15*w - 16], [3121, 3121, -w - 56], [3121, 3121, w - 56], [3137, 3137, w + 363], [3137, 3137, w + 2774], [3163, 3163, w + 1507], [3163, 3163, w + 1656], [3169, 3169, 12*w - 73], [3169, 3169, -12*w - 73], [3181, 3181, 6*w - 61], [3181, 3181, -6*w - 61], [3187, 3187, w + 772], [3187, 3187, w + 2415], [3191, 3191, 20*w - 53], [3191, 3191, -20*w - 53], [3229, 3229, 3*w - 58], [3229, 3229, -3*w - 58], [3251, 3251, -21*w - 58], [3251, 3251, 21*w - 58], [3257, 3257, w + 668], [3257, 3257, w + 2589], [3299, 3299, 19*w - 46], [3299, 3299, -19*w - 46], [3301, 3301, -14*w - 79], [3301, 3301, 14*w - 79], [3307, 3307, w + 315], [3307, 3307, w + 2992], [3343, 3343, w + 689], [3343, 3343, w + 2654], [3359, 3359, -15*w - 4], [3359, 3359, 15*w - 4], [3361, 3361, 7*w - 64], [3361, 3361, -7*w - 64], [3371, 3371, 15*w - 2], [3371, 3371, -15*w - 2], [3413, 3413, w + 1586], [3413, 3413, w + 1827], [3463, 3463, w + 102], [3463, 3463, w + 3361], [3469, 3469, -5*w - 62], [3469, 3469, 5*w - 62], [3491, 3491, -18*w - 37], [3491, 3491, 18*w - 37], [3529, 3529, 8*w - 67], [3529, 3529, -8*w - 67], [3533, 3533, w + 775], [3533, 3533, w + 2758], [3539, 3539, 22*w - 61], [3539, 3539, -22*w - 61], [3541, 3541, 10*w - 71], [3541, 3541, -10*w - 71], [3547, 3547, w + 1235], [3547, 3547, w + 2312], [3557, 3557, w + 634], [3557, 3557, w + 2923], [3583, 3583, w + 1288], [3583, 3583, w + 2295], [3593, 3593, w + 526], [3593, 3593, w + 3067], [3607, 3607, w + 1071], [3607, 3607, w + 2536], [3617, 3617, w + 248], [3617, 3617, w + 3369], [3643, 3643, w + 505], [3643, 3643, w + 3138], [3659, 3659, -17*w - 26], [3659, 3659, 17*w - 26], [3671, 3671, -16*w - 13], [3671, 3671, 16*w - 13], [3677, 3677, w + 393], [3677, 3677, w + 3284], [3709, 3709, -3*w - 62], [3709, 3709, 3*w - 62], [3719, 3719, -16*w - 11], [3719, 3719, 16*w - 11], [3727, 3727, w + 688], [3727, 3727, w + 3039], [3769, 3769, 12*w - 77], [3769, 3769, -12*w - 77], [3779, 3779, -22*w - 59], [3779, 3779, 22*w - 59], [3797, 3797, w + 354], [3797, 3797, w + 3443], [3823, 3823, w + 1329], [3823, 3823, w + 2494], [3833, 3833, w + 1482], [3833, 3833, w + 2351], [3847, 3847, w + 460], [3847, 3847, w + 3387], [3851, 3851, -17*w - 22], [3851, 3851, 17*w - 22], [3889, 3889, 7*w - 68], [3889, 3889, -7*w - 68], [3907, 3907, w + 679], [3907, 3907, w + 3228], [3911, 3911, 21*w - 52], [3911, 3911, -21*w - 52], [3917, 3917, w + 140], [3917, 3917, w + 3777], [3943, 3943, w + 514], [3943, 3943, w + 3429], [3967, 3967, w + 345], [3967, 3967, w + 3622], [4003, 4003, w + 639], [4003, 4003, w + 3364], [4013, 4013, w + 1523], [4013, 4013, w + 2490], [4019, 4019, -18*w - 29], [4019, 4019, 18*w - 29], [4021, 4021, -15*w - 86], [4021, 4021, 15*w - 86], [4027, 4027, w + 979], [4027, 4027, w + 3048], [4073, 4073, w + 1388], [4073, 4073, w + 2685], [4079, 4079, 17*w - 16], [4079, 4079, -17*w - 16], [4091, 4091, 23*w - 62], [4091, 4091, -23*w - 62], [4129, 4129, 17*w - 92], [4129, 4129, -17*w - 92], [4133, 4133, w + 91], [4133, 4133, w + 4042], [4139, 4139, 17*w - 14], [4139, 4139, -17*w - 14], [4157, 4157, w + 1003], [4157, 4157, w + 3154], [4201, 4201, 20*w - 101], [4201, 4201, -20*w - 101], [4211, 4211, 26*w - 77], [4211, 4211, -26*w - 77], [4217, 4217, w + 1495], [4217, 4217, w + 2722], [4243, 4243, w + 545], [4243, 4243, w + 3698], [4253, 4253, w + 1992], [4253, 4253, w + 2261], [4259, 4259, 19*w - 34], [4259, 4259, -19*w - 34], [4261, 4261, -9*w - 74], [4261, 4261, 9*w - 74], [4271, 4271, 17*w - 8], [4271, 4271, -17*w - 8], [4327, 4327, w + 114], [4327, 4327, w + 4213], [4337, 4337, w + 1253], [4337, 4337, w + 3084], [4363, 4363, w + 1291], [4363, 4363, w + 3072], [4373, 4373, w + 1835], [4373, 4373, w + 2538], [4391, 4391, -19*w - 32], [4391, 4391, 19*w - 32], [4397, 4397, w + 2008], [4397, 4397, w + 2389], [4423, 4423, w + 176], [4423, 4423, w + 4247], [4441, 4441, 16*w - 91], [4441, 4441, 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7460], [9293, 9293, w + 2675], [9293, 9293, w + 6618], [9311, 9311, 25*w - 8], [9311, 9311, -25*w - 8], [9343, 9343, w + 3029], [9343, 9343, w + 6314], [9349, 9349, 2*w - 97], [9349, 9349, -2*w - 97], [9371, 9371, 25*w - 2], [9371, 9371, -25*w - 2], [9377, 9377, w + 137], [9377, 9377, w + 9240], [9403, 9403, w + 168], [9403, 9403, w + 9235], [9409, 97, -97], [9413, 9413, w + 1265], [9413, 9413, w + 8148], [9419, 9419, -34*w - 89], [9419, 9419, 34*w - 89], [9421, 9421, -11*w - 106], [9421, 9421, 11*w - 106], [9431, 9431, -32*w - 77], [9431, 9431, 32*w - 77], [9437, 9437, w + 1073], [9437, 9437, w + 8364], [9463, 9463, w + 3367], [9463, 9463, w + 6096], [9473, 9473, w + 3817], [9473, 9473, w + 5656], [9479, 9479, -29*w - 56], [9479, 9479, 29*w - 56], [9491, 9491, -27*w - 38], [9491, 9491, 27*w - 38], [9497, 9497, w + 3511], [9497, 9497, w + 5986], [9533, 9533, w + 4117], [9533, 9533, w + 5416], [9539, 9539, 35*w - 94], [9539, 9539, -35*w - 94], [9547, 9547, w + 1712], [9547, 9547, w + 7835], [9551, 9551, -28*w - 47], [9551, 9551, 28*w - 47], [9601, 9601, 9*w - 104], [9601, 9601, -9*w - 104], [9643, 9643, w + 3765], [9643, 9643, w + 5878], [9649, 9649, -8*w - 103], [9649, 9649, 8*w - 103], [9661, 9661, -6*w - 101], [9661, 9661, 6*w - 101], [9677, 9677, w + 220], [9677, 9677, w + 9457], [9719, 9719, 37*w - 104], [9719, 9719, -37*w - 104], [9721, 9721, 12*w - 109], [9721, 9721, -12*w - 109], [9769, 9769, -21*w - 128], [9769, 9769, 21*w - 128], [9781, 9781, 18*w - 121], [9781, 9781, -18*w - 121], [9787, 9787, w + 3175], [9787, 9787, w + 6612], [9791, 9791, 31*w - 68], [9791, 9791, -31*w - 68], [9829, 9829, 14*w - 113], [9829, 9829, -14*w - 113], [9833, 9833, w + 3037], [9833, 9833, w + 6796], [9839, 9839, 40*w - 119], [9839, 9839, -40*w - 119], [9851, 9851, 26*w - 17], [9851, 9851, -26*w - 17], [9857, 9857, w + 4204], [9857, 9857, w + 5653], [9883, 9883, w + 3077], [9883, 9883, w + 6806], [9901, 9901, -22*w - 131], [9901, 9901, 22*w - 131], [9907, 9907, w + 3458], [9907, 9907, w + 6449], [9949, 9949, 10*w - 107], [9949, 9949, -10*w - 107], [9967, 9967, w + 3952], [9967, 9967, w + 6015]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [-1, 1, -1, -4, -4, 0, 0, 6, 6, -4, -4, -6, -6, 0, 0, -10, -10, -4, -4, 0, 0, -4, -4, -10, -10, -18, -18, 20, 20, 0, 0, 6, 6, -4, -4, -22, 18, 18, 24, 24, 14, 14, -24, -24, -6, -6, 20, 20, -10, -10, -18, -18, 24, 24, 2, 2, -24, -24, -18, -18, -28, -28, -6, -6, 20, 20, 0, 0, 18, 18, -10, -10, 6, 6, -24, -24, -22, -28, -28, 26, 26, 0, 0, -10, -10, 0, 0, -4, -4, -24, -24, -28, -28, 24, 24, 20, 20, -46, -10, -10, -28, -28, 18, 18, 30, 30, -24, -24, -22, -22, -4, -4, 30, 30, -4, -4, 18, 18, 48, 48, 14, 14, -6, -6, 38, 38, 0, 0, -28, -28, 2, 2, 42, 42, -4, -4, -30, -30, 20, 20, 38, 38, 24, 24, -22, -18, -18, -4, -4, 44, 44, -48, -48, 6, 6, 2, -4, -4, 24, 24, -42, -42, 2, 2, -54, -54, 48, 48, 14, 14, 24, 24, -28, -28, 14, 14, -52, -52, 48, 48, 6, 6, -52, -52, 26, 26, -24, -24, 6, 6, 50, 50, 30, 30, -46, -46, 0, 0, -30, -30, 44, 44, -48, -48, 26, 26, 20, 20, -70, 66, 66, -10, -10, -52, -52, -58, -58, 6, 6, 0, 0, 20, 20, 48, 48, -28, -28, 50, 50, -54, -54, 48, 48, -48, -48, 68, 68, 38, 38, 6, 6, -48, -48, 68, 68, 24, 24, 26, 26, 66, 66, -48, -48, -34, -34, -28, -28, 42, 42, 20, 20, -58, -58, -46, -18, -18, -76, -76, -6, -6, -34, -34, 20, 20, -52, -52, 38, 38, -22, -22, 0, 0, -10, -10, -4, -4, 24, 24, 42, 42, -66, -66, 0, 0, 42, 42, 0, 0, -28, -28, -78, -78, 14, 14, 48, 48, -28, -28, 26, 26, -72, -72, 24, 24, -4, -4, -66, -66, 2, 2, 44, 44, -94, -6, -6, 62, 62, 66, 66, -34, -34, -42, -42, 74, 74, 20, 20, 78, 78, 18, 18, 24, 24, 38, 38, -4, -4, -72, -72, 42, 42, -28, -28, 14, 14, -66, -66, -48, -48, -48, -48, -66, -66, 0, 0, 92, 92, -30, -30, -52, -52, 26, 26, -24, -24, 24, 24, 48, 48, 6, 6, 92, 92, -42, -42, 92, 92, 2, 2, -6, -6, -48, -48, -52, -52, 24, 24, 14, 14, 30, 30, -28, -28, 54, 54, -4, -4, -48, -48, -54, -54, -96, -96, 92, 92, 78, 78, -72, -72, -78, -78, -24, -24, -22, -22, 24, 24, 26, 26, -34, -34, -28, -28, 86, 86, 72, 72, 2, 2, 102, 102, -52, -52, -46, -46, 14, 14, -52, -52, 96, 96, -10, -10, -96, -96, -90, -90, -24, -24, 38, 38, -100, -100, -76, -76, 96, 96, 98, 98, 72, 72, 42, 42, 68, 68, 62, 62, -48, -48, 74, 74, 18, 18, 0, 0, 14, 14, -28, -28, 90, 90, 20, 20, -18, -18, 44, 44, -42, -42, -52, -52, 24, 24, -24, -24, 66, 66, 62, 62, 72, 72, -4, -4, -70, -70, 48, 48, -6, -6, -28, -28, 30, 30, 68, 68, 96, 96, 2, 2, -124, -124, -24, -24, 18, 18, -76, -76, -28, -28, 68, 68, 18, 18, 48, 48, 62, 62, -28, -28, 6, 6, 72, 72, 96, 96, 50, 50, -102, -102, -48, -48, 66, 66, -70, -70, -72, -72, -42, -42, -124, -124, 66, 66, 0, 0, 62, 62, -120, -120, 68, 68, 54, 54, -4, -4, -6, -6, 0, 0, 18, 18, -28, -28, 74, 74, -28, -28, -48, -48, 6, 6, 92, 92, -30, -30, 116, 116, 90, 90, 14, 14, 2, 2, -28, -28, -124, -124, 62, 62, -78, -78, -52, -52, -114, -114, -24, -24, -48, -48, 68, 68, -22, -22, -78, -78, -48, -48, 68, 68, -34, -34, -66, -66, 72, 72, 50, 50, -18, -18, -58, -58, 0, 0, -126, -126, -28, -28, -58, -58, -24, -24, 48, 48, 78, 78, -70, -70, 114, 114, -100, -100, 20, 20, 72, 72, 96, 96, -48, -48, -10, -10, 92, 92, 78, 78, 44, 44, 0, 0, 74, 74, 44, 44, 48, 48, -102, -102, -66, -66, -24, -24, 50, 50, 6, 6, -4, -4, -142, -54, -54, 68, 68, -96, -96, 102, 102, -24, -24, 44, 44, 54, 54, -100, -100, -118, -118, -24, -24, -6, -6, -28, -28, -72, -72, 50, 50, -52, -52, 72, 72, 44, 44, 98, 98, -6, -6, 86, 86, 48, 48, 44, 44, 0, 0, -118, -118, -148, -148, 24, 24, -42, -42, 116, 116, -70, -70, 66, 66, 38, 38, 120, 120, 42, 42, -76, -76, 14, 14, -54, -54, -130, -130, -100, -100, -130, -130, 0, 0, -70, -70, 78, 78, 116, 116, 0, 0, 20, 20, 120, 120, -4, -4, 90, 90, 92, 92, -18, -18, -22, -22, 120, 120, 44, 44, -126, -126, -54, -54, -106, -106, -94, -124, -124, 126, 126, -96, -96, -34, -34, 120, 120, -78, -78, 92, 92, -66, -66, -144, -144, 122, 122, -100, -100, -130, -130, 20, 20, 14, 14, -66, -66, 2, 2, -96, -96, 50, 50, -4, -4, -144, -144, 120, 120, 20, 20, 66, 66, -24, -24, 62, 62, 44, 44, 14, 14, -120, -120, -66, -66, 68, 68, -24, -24, -34, -34, -72, -72, -4, -4, -34, -34, -42, -42, 122, 122, -18, -18, -28, -28, -22, -72, -72, 68, 68, 24, 24, 90, 90, -10, -10, 0, 0, -142, -142, 92, 92, 24, 24, -90, -90, -102, -102, 120, 120, 116, 116, -130, -130, 24, 24, 74, 74, 48, 48, 54, 54, 68, 68, -72, -72, 20, 20, 138, 138, -10, -10, -118, -118, -168, -168, 26, 26, -42, -42, -24, -24, 6, 6, -94, -94, 24, 24, -124, -124, 162, 162, 134, 134, 96, 96, -70, -70, -114, -114, -100, -100, 134, 134, 110, 110, 30, 30, 50, 50, 20, 20, 0, 0, 164, 164, 110, 110, 18, 18, -130, -130, 6, 6, -162, -162, 114, 114, -100, -100, -6, -6, -144, -144, 146, -52, -52, 54, 54, -28, -28, -48, -48, -118, -118, 66, 66, -154, -154, -96, -96, 138, 138, -46, -46, -4, -4, 96, 96, -94, -94, 144, 144, 14, 14, 168, 168, -78, -78, -4, -4, -34, -34, 6, 6, 140, 140, 24, 24, -114, -114, 74, 74, -154, -154, 116, 116, -34, -34, 68, 68, -90, -90, -118, -118, 68, 68, -90, -90, -52, -52, -30, -30, 14, 14, 90, 90, -4, -4, 134, 134, -94, -94, 20, 20, 2, 2, 138, 138, -72, -72, -148, -148, -114, -114, 170, 170, 92, 92, 168, 168, 86, 86, -72, -72, -6, -6, 68, 68, -28, -28, -4, -4, 146, 146, 42, 42, -130, -130, -48, -48, 72, 72, -70, -70, -28, -28, -96, -96, 140, 140, -70, -70, -48, -48, 92, 92, -76, -76, -34, -34, -52, -52, 174, 174, -54, -54, -178, -178, -4, -4, 96, 96, 122, 122, -42, -42, -148, -148, -174, -174, -72, -72, -124, -124, 62, 62, 96, 96, 174, 174, -28, -28, -190, 90, 90, 120, 120, 62, 62, 72, 72, 162, 162, -4, -4, 126, 126, -96, -96, 120, 120, -90, -90, -30, -30, -168, -168, 20, 20, -24, -24, 146, 146, -172, -172, 50, 50, 134, 134, -126, -126, -144, -144, 26, 26, 74, 74, -58, -58, -52, -52, -48, -48, -10, -10, 126, 126, 192, 192, 0, 0, 30, 30, -4, -4, -106, -106, -52, -52, -130, -130, 44, 44]; 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;