/* 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![-2, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, -w], [7, 7, -2*w + 1], [7, 7, -2*w - 1], [9, 3, 3], [17, 17, 3*w + 1], [17, 17, 3*w - 1], [23, 23, w + 5], [23, 23, -w + 5], [25, 5, 5], [31, 31, 4*w + 1], [31, 31, -4*w + 1], [41, 41, 2*w - 7], [41, 41, -2*w - 7], [47, 47, -w - 7], [47, 47, w - 7], [71, 71, -6*w - 1], [71, 71, 6*w - 1], [73, 73, -7*w - 5], [73, 73, 7*w - 5], [79, 79, -w - 9], [79, 79, w - 9], [89, 89, -4*w - 11], [89, 89, 4*w - 11], [97, 97, -7*w - 1], [97, 97, 7*w - 1], [103, 103, -3*w - 11], [103, 103, 3*w - 11], [113, 113, 2*w - 11], [113, 113, -2*w - 11], [121, 11, -11], [127, 127, -8*w - 1], [127, 127, 8*w - 1], [137, 137, 4*w - 13], [137, 137, -4*w - 13], [151, 151, 3*w - 13], [151, 151, -3*w - 13], [167, 167, -w - 13], [167, 167, w - 13], [169, 13, -13], [191, 191, 10*w - 3], [191, 191, -10*w - 3], [193, 193, -4*w - 15], [193, 193, 4*w - 15], [199, 199, -10*w - 1], [199, 199, 10*w - 1], [223, 223, -w - 15], [223, 223, w - 15], [233, 233, -11*w - 3], [233, 233, 11*w - 3], [239, 239, -5*w - 17], [239, 239, 5*w - 17], [241, 241, -11*w - 1], [241, 241, 11*w - 1], [257, 257, 4*w - 17], [257, 257, -4*w - 17], [263, 263, 7*w - 19], [263, 263, -7*w - 19], [271, 271, -3*w - 17], [271, 271, 3*w - 17], [281, 281, 2*w - 17], [281, 281, -2*w - 17], [311, 311, -5*w - 19], [311, 311, 5*w - 19], [313, 313, -8*w - 21], [313, 313, 8*w - 21], [337, 337, -13*w - 1], [337, 337, 13*w - 1], [353, 353, 2*w - 19], [353, 353, -2*w - 19], [359, 359, -w - 19], [359, 359, w - 19], [361, 19, -19], [367, 367, 9*w - 23], [367, 367, -9*w - 23], [383, 383, -14*w - 3], [383, 383, 14*w - 3], [401, 401, -8*w - 23], [401, 401, 8*w - 23], [409, 409, -17*w - 13], [409, 409, 17*w - 13], [431, 431, -7*w - 23], [431, 431, 7*w - 23], [433, 433, 2*w - 21], [433, 433, -2*w - 21], [439, 439, -w - 21], [439, 439, w - 21], [449, 449, -15*w - 1], [449, 449, 15*w - 1], [457, 457, -6*w - 23], [457, 457, 6*w - 23], [463, 463, 9*w - 25], [463, 463, -9*w - 25], [479, 479, -18*w - 13], [479, 479, 18*w - 13], [487, 487, 16*w - 5], [487, 487, -16*w - 5], [503, 503, 16*w - 3], [503, 503, -16*w - 3], [521, 521, 2*w - 23], [521, 521, -2*w - 23], [569, 569, -17*w - 3], [569, 569, 17*w - 3], [577, 577, -17*w - 1], [577, 577, 17*w - 1], [593, 593, 4*w - 25], [593, 593, -4*w - 25], [599, 599, -11*w - 29], [599, 599, 11*w - 29], [601, 601, 8*w - 27], [601, 601, -8*w - 27], [607, 607, 3*w - 25], [607, 607, -3*w - 25], [617, 617, 2*w - 25], [617, 617, -2*w - 25], [631, 631, -7*w - 27], [631, 631, 7*w - 27], [641, 641, -10*w - 29], [641, 641, 10*w - 29], [647, 647, -18*w - 1], [647, 647, 18*w - 1], [673, 673, -12*w - 31], [673, 673, 12*w - 31], [719, 719, 11*w - 31], [719, 719, -11*w - 31], [727, 727, -w - 27], [727, 727, w - 27], [743, 743, 7*w - 29], [743, 743, -7*w - 29], [751, 751, 13*w - 33], [751, 751, -13*w - 33], [761, 761, 10*w - 31], [761, 761, -10*w - 31], [769, 769, 23*w - 17], [769, 769, -23*w - 17], [809, 809, 4*w - 29], [809, 809, -4*w - 29], [823, 823, -3*w - 29], [823, 823, 3*w - 29], [839, 839, -w - 29], [839, 839, w - 29], [841, 29, -29], [857, 857, 21*w - 5], [857, 857, -21*w - 5], [863, 863, 24*w - 17], [863, 863, -24*w - 17], [881, 881, -21*w - 1], [881, 881, 21*w - 1], [887, 887, 13*w - 35], [887, 887, -13*w - 35], [911, 911, 26*w + 21], [911, 911, -26*w + 21], [919, 919, 22*w - 7], [919, 919, -22*w - 7], [929, 929, -4*w - 31], [929, 929, 4*w - 31], [937, 937, -12*w - 35], [937, 937, 12*w - 35], [953, 953, 2*w - 31], [953, 953, -2*w - 31], [967, 967, -22*w - 1], [967, 967, 22*w - 1], [977, 977, -14*w - 37], [977, 977, 14*w - 37], [983, 983, -11*w - 35], [983, 983, 11*w - 35], [991, 991, -26*w - 19], [991, 991, 26*w - 19], [1009, 1009, -23*w - 7], [1009, 1009, 23*w - 7], [1031, 1031, 13*w - 37], [1031, 1031, -13*w - 37], [1033, 1033, 23*w - 5], [1033, 1033, -23*w - 5], [1039, 1039, 5*w - 33], [1039, 1039, -5*w - 33], [1049, 1049, -23*w - 3], [1049, 1049, 23*w - 3], [1063, 1063, -9*w - 35], [1063, 1063, 9*w - 35], [1087, 1087, -w - 33], [1087, 1087, w - 33], [1097, 1097, -27*w - 19], [1097, 1097, 27*w - 19], [1103, 1103, -24*w - 7], [1103, 1103, 24*w - 7], [1129, 1129, -14*w - 39], [1129, 1129, 14*w - 39], [1151, 1151, -24*w - 1], [1151, 1151, 24*w - 1], [1153, 1153, -6*w - 35], [1153, 1153, 6*w - 35], [1193, 1193, -4*w - 35], [1193, 1193, 4*w - 35], [1201, 1201, 25*w - 7], [1201, 1201, -25*w - 7], [1217, 1217, 2*w - 35], [1217, 1217, -2*w - 35], [1223, 1223, -w - 35], [1223, 1223, w - 35], [1231, 1231, -15*w - 41], [1231, 1231, 15*w - 41], [1249, 1249, -25*w - 1], [1249, 1249, 25*w - 1], [1279, 1279, 11*w - 39], [1279, 1279, -11*w - 39], [1289, 1289, -14*w - 41], [1289, 1289, 14*w - 41], [1297, 1297, 31*w + 25], [1297, 1297, -31*w + 25], [1303, 1303, 26*w - 7], [1303, 1303, -26*w - 7], [1319, 1319, -5*w - 37], [1319, 1319, 5*w - 37], [1321, 1321, -10*w - 39], [1321, 1321, 10*w - 39], [1327, 1327, 26*w - 5], [1327, 1327, -26*w - 5], [1361, 1361, 2*w - 37], [1361, 1361, -2*w - 37], [1367, 1367, -w - 37], [1367, 1367, w - 37], [1369, 37, -37], [1399, 1399, 15*w - 43], [1399, 1399, -15*w - 43], [1409, 1409, -27*w - 7], [1409, 1409, 27*w - 7], [1423, 1423, -32*w - 25], [1423, 1423, 32*w - 25], [1433, 1433, -27*w - 5], [1433, 1433, 27*w - 5], [1439, 1439, -11*w - 41], [1439, 1439, 11*w - 41], [1447, 1447, -17*w - 45], [1447, 1447, 17*w - 45], [1471, 1471, -5*w - 39], [1471, 1471, 5*w - 39], [1481, 1481, 10*w - 41], [1481, 1481, -10*w - 41], [1487, 1487, 28*w - 9], [1487, 1487, -28*w - 9], [1489, 1489, -4*w - 39], [1489, 1489, 4*w - 39], [1511, 1511, -13*w - 43], [1511, 1511, 13*w - 43], [1543, 1543, 28*w - 5], [1543, 1543, -28*w - 5], [1553, 1553, -33*w - 25], [1553, 1553, 33*w - 25], [1559, 1559, 28*w - 3], [1559, 1559, -28*w - 3], [1567, 1567, -28*w - 1], [1567, 1567, 28*w - 1], [1583, 1583, -34*w + 27], [1583, 1583, 34*w + 27], [1601, 1601, -29*w - 9], [1601, 1601, 29*w - 9], [1607, 1607, -11*w - 43], [1607, 1607, 11*w - 43], [1609, 1609, -6*w - 41], [1609, 1609, 6*w - 41], [1657, 1657, -29*w - 5], [1657, 1657, 29*w - 5], [1663, 1663, -3*w - 41], [1663, 1663, 3*w - 41], [1697, 1697, -16*w - 47], [1697, 1697, 16*w - 47], [1721, 1721, -35*w - 27], [1721, 1721, 35*w - 27], [1753, 1753, 18*w - 49], [1753, 1753, -18*w - 49], [1759, 1759, -15*w - 47], [1759, 1759, 15*w - 47], [1777, 1777, 6*w - 43], [1777, 1777, -6*w - 43], [1783, 1783, 11*w - 45], [1783, 1783, -11*w - 45], [1801, 1801, -20*w - 51], [1801, 1801, 20*w - 51], [1823, 1823, 17*w - 49], [1823, 1823, -17*w - 49], [1831, 1831, 3*w - 43], [1831, 1831, -3*w - 43], [1847, 1847, -w - 43], [1847, 1847, w - 43], [1849, 43, -43], [1871, 1871, 13*w - 47], [1871, 1871, -13*w - 47], [1873, 1873, -31*w - 7], [1873, 1873, 31*w - 7], [1879, 1879, 19*w - 51], [1879, 1879, -19*w - 51], [1889, 1889, 16*w - 49], [1889, 1889, -16*w - 49], [1913, 1913, 31*w - 3], [1913, 1913, -31*w - 3], [1951, 1951, 15*w - 49], [1951, 1951, -15*w - 49], [1993, 1993, 4*w - 45], [1993, 1993, -4*w - 45], [1999, 1999, 32*w - 7], [1999, 1999, -32*w - 7], [2017, 2017, 2*w - 45], [2017, 2017, -2*w - 45], [2039, 2039, -32*w - 3], [2039, 2039, 32*w - 3], [2063, 2063, 13*w - 49], [2063, 2063, -13*w - 49], [2081, 2081, -39*w + 31], [2081, 2081, 39*w + 31], [2087, 2087, -19*w - 53], [2087, 2087, 19*w - 53], [2089, 2089, -16*w - 51], [2089, 2089, 16*w - 51], [2111, 2111, 7*w - 47], [2111, 2111, -7*w - 47], [2113, 2113, 12*w - 49], [2113, 2113, -12*w - 49], [2129, 2129, 33*w - 7], [2129, 2129, -33*w - 7], [2137, 2137, -6*w - 47], [2137, 2137, 6*w - 47], [2143, 2143, -21*w - 55], [2143, 2143, 21*w - 55], [2153, 2153, 33*w - 5], [2153, 2153, -33*w - 5], [2161, 2161, -18*w - 53], [2161, 2161, 18*w - 53], [2207, 2207, -w - 47], [2207, 2207, w - 47], [2239, 2239, 40*w - 31], [2239, 2239, -40*w - 31], [2273, 2273, 8*w - 49], [2273, 2273, -8*w - 49], [2281, 2281, -22*w - 57], [2281, 2281, 22*w - 57], [2287, 2287, -34*w - 5], [2287, 2287, 34*w - 5], [2297, 2297, -16*w - 53], [2297, 2297, 16*w - 53], [2311, 2311, -34*w - 1], [2311, 2311, 34*w - 1], [2351, 2351, -5*w - 49], [2351, 2351, 5*w - 49], [2377, 2377, 18*w - 55], [2377, 2377, -18*w - 55], [2383, 2383, 3*w - 49], [2383, 2383, -3*w - 49], [2393, 2393, 2*w - 49], [2393, 2393, -2*w - 49], [2399, 2399, -w - 49], [2399, 2399, w - 49], [2417, 2417, -14*w - 53], [2417, 2417, 14*w - 53], [2423, 2423, -23*w - 59], [2423, 2423, 23*w - 59], [2441, 2441, -35*w - 3], [2441, 2441, 35*w - 3], [2447, 2447, -17*w - 55], [2447, 2447, 17*w - 55], [2473, 2473, 8*w - 51], [2473, 2473, -8*w - 51], [2503, 2503, -7*w - 51], [2503, 2503, 7*w - 51], [2521, 2521, -41*w - 29], [2521, 2521, 41*w - 29], [2543, 2543, 36*w - 7], [2543, 2543, -36*w - 7], [2551, 2551, 5*w - 51], [2551, 2551, -5*w - 51], [2591, 2591, -36*w - 1], [2591, 2591, 36*w - 1], [2593, 2593, 2*w - 51], [2593, 2593, -2*w - 51], [2609, 2609, 43*w - 33], [2609, 2609, -43*w - 33], [2617, 2617, 37*w - 11], [2617, 2617, -37*w - 11], [2633, 2633, -14*w - 55], [2633, 2633, 14*w - 55], [2647, 2647, -44*w + 35], [2647, 2647, 44*w + 35], [2657, 2657, 37*w - 9], [2657, 2657, -37*w - 9], [2663, 2663, -23*w - 61], [2663, 2663, 23*w - 61], [2671, 2671, 17*w - 57], [2671, 2671, -17*w - 57], [2687, 2687, -13*w - 55], [2687, 2687, 13*w - 55], [2689, 2689, -37*w - 7], [2689, 2689, 37*w - 7], [2711, 2711, 7*w - 53], [2711, 2711, -7*w - 53], [2713, 2713, -37*w - 5], [2713, 2713, 37*w - 5], [2719, 2719, 25*w - 63], [2719, 2719, -25*w - 63], [2729, 2729, 37*w - 3], [2729, 2729, -37*w - 3], [2753, 2753, -22*w - 61], [2753, 2753, 22*w - 61], [2767, 2767, -38*w - 11], [2767, 2767, 38*w - 11], [2777, 2777, 4*w - 53], [2777, 2777, -4*w - 53], [2791, 2791, -3*w - 53], [2791, 2791, 3*w - 53], [2801, 2801, 2*w - 53], [2801, 2801, -2*w - 53], [2809, 53, -53], [2833, 2833, -18*w - 59], [2833, 2833, 18*w - 59], [2857, 2857, 14*w - 57], [2857, 2857, -14*w - 57], [2879, 2879, -38*w - 3], [2879, 2879, 38*w - 3], [2887, 2887, -38*w - 1], [2887, 2887, 38*w - 1], [2897, 2897, -8*w - 55], [2897, 2897, 8*w - 55], [2903, 2903, -17*w - 59], [2903, 2903, 17*w - 59], [2927, 2927, -7*w - 55], [2927, 2927, 7*w - 55], [2953, 2953, 6*w - 55], [2953, 2953, -6*w - 55], [2969, 2969, 16*w - 59], [2969, 2969, -16*w - 59], [2999, 2999, 19*w - 61], [2999, 2999, -19*w - 61], [3001, 3001, 22*w - 63], [3001, 3001, -22*w - 63], [3023, 3023, -w - 55], [3023, 3023, w - 55], [3041, 3041, -39*w - 1], [3041, 3041, 39*w - 1], [3049, 3049, -47*w - 37], [3049, 3049, 47*w - 37], [3079, 3079, -40*w - 11], [3079, 3079, 40*w - 11], [3089, 3089, 14*w - 59], [3089, 3089, -14*w - 59], [3119, 3119, -40*w - 9], [3119, 3119, 40*w - 9], [3121, 3121, 8*w - 57], [3121, 3121, -8*w - 57], [3137, 3137, 26*w - 67], [3137, 3137, -26*w - 67], [3167, 3167, -23*w - 65], [3167, 3167, 23*w - 65], [3169, 3169, 20*w - 63], [3169, 3169, -20*w - 63], [3191, 3191, 40*w - 3], [3191, 3191, -40*w - 3], [3209, 3209, 16*w - 61], [3209, 3209, -16*w - 61], [3217, 3217, 4*w - 57], [3217, 3217, -4*w - 57], [3257, 3257, -22*w - 65], [3257, 3257, 22*w - 65], [3271, 3271, 15*w - 61], [3271, 3271, -15*w - 61], [3313, 3313, -41*w - 7], [3313, 3313, 41*w - 7], [3319, 3319, 9*w - 59], [3319, 3319, -9*w - 59], [3329, 3329, -47*w - 33], [3329, 3329, 47*w - 33], [3343, 3343, -21*w - 65], [3343, 3343, 21*w - 65], [3359, 3359, -42*w - 13], [3359, 3359, 42*w - 13], [3361, 3361, -41*w - 1], [3361, 3361, 41*w - 1], [3391, 3391, -17*w - 63], [3391, 3391, 17*w - 63], [3407, 3407, 42*w - 11], [3407, 3407, -42*w - 11], [3433, 3433, -49*w - 37], [3433, 3433, 49*w - 37], [3449, 3449, -4*w - 59], [3449, 3449, 4*w - 59], [3457, 3457, -16*w - 63], [3457, 3457, 16*w - 63], [3463, 3463, -3*w - 59], [3463, 3463, 3*w - 59], [3481, 59, -59], [3511, 3511, 25*w - 69], [3511, 3511, -25*w - 69], [3527, 3527, -42*w - 1], [3527, 3527, 42*w - 1], [3529, 3529, -43*w - 13], [3529, 3529, 43*w - 13], [3559, 3559, 9*w - 61], [3559, 3559, -9*w - 61], [3583, 3583, 27*w - 71], [3583, 3583, -27*w - 71], [3593, 3593, -8*w - 61], [3593, 3593, 8*w - 61], [3607, 3607, -21*w - 67], [3607, 3607, 21*w - 67], [3617, 3617, 43*w - 9], [3617, 3617, -43*w - 9], [3623, 3623, 7*w - 61], [3623, 3623, -7*w - 61], [3631, 3631, -50*w - 37], [3631, 3631, 50*w - 37], [3671, 3671, 5*w - 61], [3671, 3671, -5*w - 61], [3673, 3673, 43*w - 5], [3673, 3673, -43*w - 5], [3697, 3697, -43*w - 1], [3697, 3697, 43*w - 1], [3719, 3719, -w - 61], [3719, 3719, w - 61], [3721, 61, -61], [3727, 3727, 52*w - 41], [3727, 3727, -52*w - 41], [3761, 3761, 28*w - 73], [3761, 3761, -28*w - 73], [3767, 3767, 19*w - 67], [3767, 3767, -19*w - 67], [3769, 3769, -10*w - 63], [3769, 3769, 10*w - 63], [3793, 3793, -22*w - 69], [3793, 3793, 22*w - 69], [3823, 3823, -44*w - 7], [3823, 3823, 44*w - 7], [3833, 3833, 51*w - 37], [3833, 3833, -51*w - 37], [3847, 3847, -44*w - 5], [3847, 3847, 44*w - 5], [3863, 3863, -44*w - 3], [3863, 3863, 44*w - 3], [3881, 3881, -45*w - 13], [3881, 3881, 45*w - 13], [3889, 3889, -24*w - 71], [3889, 3889, 24*w - 71], [3911, 3911, -17*w - 67], [3911, 3911, 17*w - 67], [3919, 3919, 5*w - 63], [3919, 3919, -5*w - 63], [3929, 3929, 45*w - 11], [3929, 3929, -45*w - 11], [3943, 3943, 29*w - 75], [3943, 3943, -29*w - 75], [3967, 3967, -w - 63], [3967, 3967, w - 63], [4001, 4001, -45*w - 7], [4001, 4001, 45*w - 7], [4007, 4007, -31*w - 77], [4007, 4007, 31*w - 77], [4049, 4049, -45*w - 1], [4049, 4049, 45*w - 1], [4057, 4057, 28*w - 75], [4057, 4057, -28*w - 75], [4073, 4073, 22*w - 71], [4073, 4073, -22*w - 71], [4079, 4079, 25*w - 73], [4079, 4079, -25*w - 73], [4111, 4111, -46*w - 11], [4111, 4111, 46*w - 11], [4127, 4127, -7*w - 65], [4127, 4127, 7*w - 65], [4129, 4129, -30*w - 77], [4129, 4129, 30*w - 77], [4153, 4153, -6*w - 65], [4153, 4153, 6*w - 65], [4159, 4159, 21*w - 71], [4159, 4159, -21*w - 71], [4177, 4177, 24*w - 73], [4177, 4177, -24*w - 73], [4201, 4201, 55*w - 43], [4201, 4201, -55*w - 43], [4217, 4217, 2*w - 65], [4217, 4217, -2*w - 65], [4231, 4231, -46*w - 1], [4231, 4231, 46*w - 1], [4241, 4241, -20*w - 71], [4241, 4241, 20*w - 71], [4271, 4271, 23*w - 73], [4271, 4271, -23*w - 73], [4273, 4273, -26*w - 75], [4273, 4273, 26*w - 75], [4289, 4289, 10*w - 67], [4289, 4289, -10*w - 67], [4297, 4297, 47*w - 11], [4297, 4297, -47*w - 11], [4327, 4327, -9*w - 67], [4327, 4327, 9*w - 67], [4337, 4337, -47*w - 9], [4337, 4337, 47*w - 9], [4391, 4391, 7*w - 67], [4391, 4391, -7*w - 67], [4409, 4409, -47*w - 3], [4409, 4409, 47*w - 3], [4423, 4423, 56*w - 43], [4423, 4423, -56*w - 43], [4441, 4441, -30*w - 79], [4441, 4441, 30*w - 79], [4447, 4447, -21*w - 73], [4447, 4447, 21*w - 73], [4457, 4457, -4*w - 67], [4457, 4457, 4*w - 67], [4463, 4463, 17*w - 71], [4463, 4463, -17*w - 71], [4481, 4481, 2*w - 67], [4481, 4481, -2*w - 67], [4489, 67, -67], [4513, 4513, -32*w - 81], [4513, 4513, 32*w - 81], [4519, 4519, 58*w + 47], [4519, 4519, -58*w + 47], [4561, 4561, -10*w - 69], [4561, 4561, 10*w - 69], [4567, 4567, 23*w - 75], [4567, 4567, -23*w - 75], 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8431, -33*w - 103], [8431, 8431, 33*w - 103], [8447, 8447, 78*w - 61], [8447, 8447, -78*w - 61], [8513, 8513, -79*w + 63], [8513, 8513, 79*w + 63], [8521, 8521, -8*w - 93], [8521, 8521, 8*w - 93], [8527, 8527, 76*w - 55], [8527, 8527, -76*w - 55], [8537, 8537, -67*w - 21], [8537, 8537, 67*w - 21], [8543, 8543, 66*w - 13], [8543, 8543, -66*w - 13], [8599, 8599, 5*w - 93], [8599, 8599, -5*w - 93], [8609, 8609, 77*w - 57], [8609, 8609, -77*w - 57], [8623, 8623, -43*w - 111], [8623, 8623, 43*w - 111], [8641, 8641, 2*w - 93], [8641, 8641, -2*w - 93], [8647, 8647, -w - 93], [8647, 8647, w - 93], [8663, 8663, -66*w - 7], [8663, 8663, 66*w - 7], [8681, 8681, -40*w - 109], [8681, 8681, 40*w - 109], [8689, 8689, -67*w - 17], [8689, 8689, 67*w - 17], [8713, 8713, -34*w - 105], [8713, 8713, 34*w - 105], [8719, 8719, 45*w - 113], [8719, 8719, -45*w - 113], [8737, 8737, -12*w - 95], [8737, 8737, 12*w - 95], [8753, 8753, -67*w - 15], [8753, 8753, 67*w - 15], [8761, 8761, 79*w - 61], [8761, 8761, -79*w - 61], [8783, 8783, -11*w - 95], [8783, 8783, 11*w - 95], [8807, 8807, -68*w - 21], [8807, 8807, 68*w - 21], [8831, 8831, 80*w - 63], [8831, 8831, -80*w - 63], [8839, 8839, -39*w - 109], [8839, 8839, 39*w - 109], [8849, 8849, -26*w - 101], [8849, 8849, 26*w - 101], [8863, 8863, 9*w - 95], [8863, 8863, -9*w - 95], [8887, 8887, 68*w - 19], [8887, 8887, -68*w - 19], [8929, 8929, 67*w - 7], [8929, 8929, -67*w - 7], [8951, 8951, 25*w - 101], [8951, 8951, -25*w - 101], [8969, 8969, 67*w - 3], [8969, 8969, -67*w - 3], [8999, 8999, -35*w - 107], [8999, 8999, 35*w - 107], [9001, 9001, 79*w - 59], [9001, 9001, -79*w - 59], [9007, 9007, -3*w - 95], [9007, 9007, 3*w - 95], [9041, 9041, 28*w - 103], [9041, 9041, -28*w - 103], [9049, 9049, 24*w - 101], [9049, 9049, -24*w - 101], [9103, 9103, 31*w - 105], [9103, 9103, -31*w - 105], [9127, 9127, -68*w - 11], [9127, 9127, 68*w - 11], [9137, 9137, 34*w - 107], [9137, 9137, -34*w - 107], [9151, 9151, -27*w - 103], [9151, 9151, 27*w - 103], [9161, 9161, 69*w - 19], [9161, 9161, -69*w - 19], [9199, 9199, 68*w - 7], [9199, 9199, -68*w - 7], [9209, 9209, 10*w - 97], [9209, 9209, -10*w - 97], [9239, 9239, -68*w - 3], [9239, 9239, 68*w - 3], [9241, 9241, -42*w - 113], [9241, 9241, 42*w - 113], [9257, 9257, -26*w - 103], [9257, 9257, 26*w - 103], [9281, 9281, 8*w - 97], [9281, 9281, -8*w - 97], [9311, 9311, -7*w - 97], [9311, 9311, 7*w - 97], [9319, 9319, -80*w - 59], [9319, 9319, 80*w - 59], [9337, 9337, 6*w - 97], [9337, 9337, -6*w - 97], [9343, 9343, -29*w - 105], [9343, 9343, 29*w - 105], [9377, 9377, 4*w - 97], [9377, 9377, -4*w - 97], [9391, 9391, 3*w - 97], [9391, 9391, -3*w - 97], [9431, 9431, 35*w - 109], [9431, 9431, -35*w - 109], [9433, 9433, -38*w - 111], [9433, 9433, 38*w - 111], [9439, 9439, 70*w - 19], [9439, 9439, -70*w - 19], [9463, 9463, 13*w - 99], [9463, 9463, -13*w - 99], [9473, 9473, -69*w - 7], [9473, 9473, 69*w - 7], [9479, 9479, 82*w - 63], [9479, 9479, -82*w - 63], [9497, 9497, -69*w - 5], [9497, 9497, 69*w - 5], [9511, 9511, -70*w - 17], [9511, 9511, 70*w - 17], [9521, 9521, -69*w - 1], [9521, 9521, 69*w - 1], [9551, 9551, 80*w - 57], [9551, 9551, -80*w - 57], [9601, 9601, -10*w - 99], [9601, 9601, 10*w - 99], [9623, 9623, -84*w + 67], [9623, 9623, 84*w + 67], [9631, 9631, -70*w - 13], [9631, 9631, 70*w - 13], [9649, 9649, -30*w - 107], [9649, 9649, 30*w - 107], [9679, 9679, 70*w - 11], [9679, 9679, -70*w - 11], [9689, 9689, -16*w - 101], [9689, 9689, 16*w - 101], [9697, 9697, 42*w - 115], [9697, 9697, -42*w - 115], [9719, 9719, 70*w - 9], [9719, 9719, -70*w - 9], [9721, 9721, -71*w - 19], [9721, 9721, 71*w - 19], [9743, 9743, -47*w - 119], [9743, 9743, 47*w - 119], [9767, 9767, -29*w - 107], [9767, 9767, 29*w - 107], [9769, 9769, -4*w - 99], [9769, 9769, 4*w - 99], [9791, 9791, 70*w - 3], [9791, 9791, -70*w - 3], [9817, 9817, -44*w - 117], [9817, 9817, 44*w - 117], [9833, 9833, 32*w - 109], [9833, 9833, -32*w - 109], [9839, 9839, -49*w - 121], [9839, 9839, 49*w - 121], [9857, 9857, -71*w - 15], [9857, 9857, 71*w - 15], [9871, 9871, 35*w - 111], [9871, 9871, -35*w - 111], [9887, 9887, 84*w - 65], [9887, 9887, -84*w - 65], [9929, 9929, -46*w - 119], [9929, 9929, 46*w - 119], [9967, 9967, 82*w - 59], [9967, 9967, -82*w - 59]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [0, -4, -4, -2, -2, -2, 4, 4, -6, 0, 0, -6, -6, -8, -8, 12, 12, 14, 14, -8, -8, -2, -2, -2, -2, -4, -4, 2, 2, -18, 16, 16, 10, 10, 4, 4, -20, -20, -22, -16, -16, -2, -2, -4, -4, 0, 0, -18, -18, -24, -24, -2, -2, 18, 18, 12, 12, 8, 8, -18, -18, -28, -28, 10, 10, -14, -14, 18, 18, -4, -4, -34, -8, -8, 0, 0, 30, 30, 10, 10, 40, 40, 30, 30, 36, 36, -34, -34, -6, -6, -8, -8, 0, 0, -20, -20, 20, 20, -22, -22, 26, 26, 2, 2, 18, 18, -12, -12, 30, 30, -16, -16, -2, -2, -44, -44, 30, 30, 12, 12, -2, -2, 24, 24, -12, -12, 44, 44, 8, 8, 10, 10, -34, -34, -6, -6, -28, -28, -36, -36, -22, -6, -6, -32, -32, -46, -46, 36, 36, -24, -24, 36, 36, 30, 30, 46, 46, 58, 58, 28, 28, 30, 30, 20, 20, 16, 16, -34, -34, -20, -20, -34, -34, 40, 40, -18, -18, -52, -52, -32, -32, -2, -2, -24, -24, -38, -38, 64, 64, -14, -14, -34, -34, 50, 50, 2, 2, 12, 12, -40, -40, -14, -14, 64, 64, 14, 14, -34, -34, -44, -44, 12, 12, 10, 10, -24, -24, -2, -2, -12, -12, 26, -12, -12, 30, 30, 24, 24, -34, -34, -16, -16, 28, 28, 32, 32, -2, -2, -8, -8, -34, -34, 12, 12, -20, -20, -14, -14, 52, 52, -64, -64, -24, -24, -14, -14, -20, -20, 46, 46, 42, 42, -64, -64, -66, -66, 30, 30, 46, 46, -16, -16, 66, 66, -76, -76, -18, -18, 32, 32, -4, -4, -12, -12, -50, 40, 40, -2, -2, 4, 4, 34, 34, -66, -66, 64, 64, -22, -22, -24, -24, -66, -66, -60, -60, -56, -56, 30, 30, -52, -52, -18, -18, 64, 64, 66, 66, -46, -46, -22, -22, -16, -16, -54, -54, 62, 62, 32, 32, -64, -64, -46, -46, 30, 30, 56, 56, 74, 74, -20, -20, -24, -24, -22, -22, 24, 24, 14, 14, 48, 48, 30, 30, 36, 36, 78, 78, 72, 72, -34, -34, -20, -20, -70, -70, 24, 24, 52, 52, 0, 0, -14, -14, -66, -66, -86, -86, -6, -6, 68, 68, -78, -78, -36, -36, -8, -8, -32, -32, -62, -62, -76, -76, 10, 10, -80, -80, -86, -86, 30, 30, 24, 24, -102, -102, 60, 60, 30, 30, -70, -30, -30, 62, 62, -32, -32, -52, -52, -66, -66, -12, -12, 56, 56, 90, 90, 30, 30, 36, 36, 26, 26, -72, -72, -34, -34, -2, -2, 76, 76, 34, 34, -72, -72, -62, -62, -14, -14, 48, 48, 62, 62, 36, 36, 42, 42, 34, 34, -2, -2, 60, 60, 2, 2, 52, 52, -2, -2, 24, 24, -16, -16, 82, 82, -48, -48, -104, -104, 26, 26, -34, -34, 30, 30, -100, -100, 78, 20, 20, 44, 44, -34, -34, -116, -116, 64, 64, 58, 58, -44, -44, 94, 94, 12, 12, 24, 24, -44, -44, 14, 14, -2, -2, -36, -36, -118, -104, -104, -78, -78, 52, 52, -86, -86, 62, 62, 24, 24, -114, -114, 28, 28, -12, -12, -38, -38, -66, -66, 76, 76, -56, -56, 90, 90, 108, 108, 64, 64, 34, 34, 12, 12, -14, -14, 30, 30, 42, 42, -104, -104, 104, 104, 64, 64, 30, 30, -18, -18, -112, -112, -30, -30, -82, -82, -18, -18, 44, 44, 126, 126, 8, 8, -46, -46, -78, -78, -50, -50, -52, -52, -2, -2, 92, 92, 30, 30, 12, 12, -86, -86, 32, 32, 126, 126, 88, 88, 2, 2, -34, 114, 114, -84, -84, -34, -34, 132, 132, 12, 12, 8, 8, 112, 112, 10, 10, -14, -14, 36, 36, -2, -2, -132, -132, -16, -16, 2, 2, 90, 90, 8, 8, -60, -60, 56, 56, 106, 106, 112, 112, -62, -62, -30, -30, 16, 16, -52, -52, 90, 90, -4, -4, 68, 68, -34, -34, -120, -120, 52, 52, -84, -84, 10, 10, -46, -46, -4, -4, 62, 62, -16, -16, 88, 88, -2, -2, -32, -32, -18, -18, -64, -64, 94, 94, -72, -72, 30, 30, 104, 104, -14, -14, -6, -6, 0, 0, -98, -98, 66, 66, 36, 36, -20, -20, -94, -94, 36, 36, 80, 80, -102, -102, -92, -92, -34, -34, 138, 138, 96, 96, -68, -68, -48, -48, 104, 104, -130, -130, -92, -92, 114, 114, -12, -12, -28, -28, -132, -132, 74, 74, -8, -8, 74, 74, -2, -2, 8, 8, -50, -50, -8, -8, -60, -60, 16, 16, 42, 42, 8, 8, 56, 56, 42, 42, -34, -34, -12, -12, 110, 110, -102, -102, 72, 72, 92, 92, -66, -66, 132, 132, -112, -112, 10, 10, -48, -48, -82, -82, -66, -66, -98, -98, -16, -16, -132, -132, -76, -76, 90, 90, 140, 140, 30, 30, 116, 116, 48, 48, 104, 104, 12, 12, -54, -54, 94, 94, 92, 92, -30, -30, 52, 52, 30, 30, -16, -16, 130, 130, 58, 58, 34, 34, -114, -114, -110, -110, -44, -44, -34, -34, -66, -66, -98, -98, 60, 60, 152, 152, -130, -130, -156, -156, 34, 34, 56, 56, -80, -80, -130, -130, 158, 158, 92, 92, -34, -34, -4, -4, 30, 30, -50, -50, 78, 78, 24, 24, -28, -28, -130, -48, -48, -72, -72, 30, 30, 84, 84, -130, -130, 108, 108, -56, -56, 26, 26, 80, 80, -66, -66, -4, -4, -144, -144, -78, -78, -156, -156, 14, 14, 40, 40, 52, 52, -38, -38, 74, 74, 76, 76, 72, 72, 62, 62, -150, -150, -92, -92, 90, 90, 146, 146, 26, 26, -6, -6, -34, -34, 142, 142, -80, -80, -46, -46, 78, 78, 30, 30, -4, -4, -38, -38, 78, 78, -32, -32, 92, 92, 68, 68, 36, 36, 126, 126, -134, -134, 62, 62, -68, -68, -44, -44, -24, -24, -114, -114, -56, -56, -14, -14, -162, -162, 56, 56, 98, 98, 126, 126, -4, -4, -72, -72, -108, -108, 62, 62, 56, 56, -34, -34, -70, -70, 62, 62, 76, 76, 162, 162, -60, -60, 122, 122, -120, -120, 34, 34, -132, -132, 0, 0, 34, 34, -4, -4, 94, 94, -148, -148, -110, -110, -80, -80, 138, 138, -92, -92, -86, -86, -66, -66, 66, 66, 90, 90, -116, -116, -136, -136, 96, 96, -162, -162, 122, 122, -56, -56, 110, 110, 96, 96, -140, -140, -30, -30, -104, -104, -34, -34, -52, -52, -108, -108, -102, -102, -98, -98, 110, 110, -168, -168, -2, -2, 126, 126, -66, -66, -24, -24, 156, 156, 48, 48, -20, -20, -98, -98, -144, -144, 68, 68, -34, -34, 180, 180, -18, -18, -164, -164, -2, -2, 40, 40, -66, -66, -182, -182, -56, -56, 44, 44, -174, -174, 32, 32, -150, -150, 136, 136, -114, -114, -44, -44, 14, 14, 90, 90, -62, -62, 160, 160, -20, -20, 142, 142, -64, -64, -30, -30, 152, 152, 4, 4, 122, 122, -144, -144, 132, 132, 66, 66, 124, 124, 74, 74, -116, -116, 114, 114, -8, -8, 2, 2, -60, -60, 96, 96, 66, 66, -8, -8, -86, -86, 98, 98, -108, -108, 78, 78, -120, -120, -148, -148, 14, 14, -48, -48, 170, 170, -6, -6, -152, -152, 30, 30, 8, 8, 32, 32, -86, -86, -8, -8]; 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;