/* 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![-3, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [3, 3, -w], [3, 3, -w + 1], [4, 2, 2], [13, 13, -2*w + 1], [17, 17, w + 4], [17, 17, -w + 5], [23, 23, 3*w + 1], [23, 23, -3*w + 4], [25, 5, 5], [29, 29, 3*w - 2], [29, 29, -3*w + 1], [43, 43, -4*w - 1], [43, 43, 4*w - 5], [49, 7, -7], [53, 53, -w - 7], [53, 53, w - 8], [61, 61, -3*w - 8], [61, 61, 3*w - 11], [79, 79, 5*w - 4], [79, 79, 5*w - 1], [101, 101, -6*w - 1], [101, 101, 6*w - 7], [103, 103, 3*w - 13], [103, 103, -3*w - 10], [107, 107, -w - 10], [107, 107, w - 11], [113, 113, 6*w - 5], [113, 113, 6*w - 1], [121, 11, -11], [127, 127, -3*w - 11], [127, 127, 3*w - 14], [131, 131, 2*w - 13], [131, 131, -2*w - 11], [139, 139, -7*w - 1], [139, 139, 7*w - 8], [157, 157, 7*w - 2], [157, 157, 7*w - 5], [173, 173, 4*w - 17], [173, 173, -4*w - 13], [179, 179, -w - 13], [179, 179, w - 14], [181, 181, 3*w - 16], [181, 181, -3*w - 13], [191, 191, 9*w - 13], [191, 191, -9*w - 4], [199, 199, 8*w - 7], [199, 199, 8*w - 1], [211, 211, -3*w - 14], [211, 211, 3*w - 17], [233, 233, -9*w - 1], [233, 233, 9*w - 10], [251, 251, 9*w - 8], [251, 251, 9*w - 1], [257, 257, 9*w - 2], [257, 257, 9*w - 7], [263, 263, 9*w - 4], [263, 263, 9*w - 5], [269, 269, -w - 16], [269, 269, w - 17], [277, 277, 3*w - 19], [277, 277, -3*w - 16], [283, 283, 11*w - 16], [283, 283, -11*w - 5], [311, 311, 2*w - 19], [311, 311, -2*w - 17], [313, 313, -3*w - 17], [313, 313, 3*w - 20], [337, 337, 11*w - 13], [337, 337, -11*w - 2], [347, 347, -12*w - 5], [347, 347, 12*w - 17], [361, 19, -19], [367, 367, 6*w - 25], [367, 367, -6*w - 19], [373, 373, 11*w - 10], [373, 373, 11*w - 1], [389, 389, -4*w - 19], [389, 389, 4*w - 23], [419, 419, -12*w - 1], [419, 419, 12*w - 13], [433, 433, -3*w - 20], [433, 433, 3*w - 23], [439, 439, 13*w - 17], [439, 439, -13*w - 4], [443, 443, 12*w - 11], [443, 443, 12*w - 1], [467, 467, 12*w - 7], [467, 467, 12*w - 5], [491, 491, 7*w - 29], [491, 491, -7*w - 22], [503, 503, -w - 22], [503, 503, w - 23], [521, 521, -15*w + 22], [521, 521, 8*w - 31], [523, 523, 3*w - 25], [523, 523, -3*w - 22], [547, 547, 13*w - 5], [547, 547, 13*w - 8], [563, 563, 2*w - 25], [563, 563, -2*w - 23], [569, 569, 5*w - 28], [569, 569, -5*w - 23], [571, 571, -3*w - 23], [571, 571, 3*w - 26], [599, 599, -15*w - 4], [599, 599, 15*w - 19], [601, 601, 14*w - 13], [601, 601, 14*w - 1], [607, 607, 9*w - 34], [607, 607, -9*w - 25], [641, 641, 15*w - 17], [641, 641, -15*w - 2], [647, 647, -w - 25], [647, 647, w - 26], [653, 653, 7*w - 32], [653, 653, -7*w - 25], [659, 659, -15*w - 1], [659, 659, 15*w - 16], [673, 673, 3*w - 28], [673, 673, -3*w - 25], [677, 677, 4*w - 29], [677, 677, -4*w - 25], [701, 701, 15*w - 2], [701, 701, 15*w - 13], [719, 719, 15*w - 11], [719, 719, 15*w - 4], [727, 727, -3*w - 26], [727, 727, 3*w - 29], [751, 751, -16*w - 1], [751, 751, 16*w - 17], [757, 757, -17*w - 5], [757, 757, 17*w - 22], [797, 797, 18*w - 25], [797, 797, -18*w - 7], [809, 809, -w - 28], [809, 809, w - 29], [823, 823, 16*w - 5], [823, 823, 16*w - 11], [829, 829, 17*w - 19], [829, 829, -17*w - 2], [857, 857, 18*w - 23], [857, 857, -18*w - 5], [859, 859, -9*w - 29], [859, 859, 9*w - 38], [881, 881, -8*w - 29], [881, 881, 8*w - 37], [883, 883, 17*w - 16], [883, 883, 17*w - 1], [887, 887, 2*w - 31], [887, 887, -2*w - 29], [907, 907, -6*w - 29], [907, 907, 6*w - 35], [911, 911, -5*w - 29], [911, 911, 5*w - 34], [919, 919, 17*w - 4], [919, 919, 17*w - 13], [937, 937, 17*w - 10], [937, 937, 17*w - 7], [953, 953, -18*w - 1], [953, 953, 18*w - 19], [961, 31, -31], [971, 971, 10*w - 41], [971, 971, 21*w - 32], [991, 991, -19*w - 4], [991, 991, 19*w - 23], [997, 997, -9*w - 31], [997, 997, 9*w - 40], [1013, 1013, -21*w + 31], [1013, 1013, 11*w - 43], [1031, 1031, -7*w - 31], [1031, 1031, 7*w - 38], [1039, 1039, 6*w - 37], [1039, 1039, -6*w - 31], [1049, 1049, 18*w - 7], [1049, 1049, 18*w - 11], [1063, 1063, -19*w - 1], [1063, 1063, 19*w - 20], [1069, 1069, 9*w - 41], [1069, 1069, -9*w - 32], [1091, 1091, -21*w - 8], [1091, 1091, 21*w - 29], [1093, 1093, -3*w - 32], [1093, 1093, 3*w - 35], [1109, 1109, -5*w - 32], [1109, 1109, 5*w - 37], [1117, 1117, 19*w - 2], [1117, 1117, 19*w - 17], [1153, 1153, 19*w - 14], [1153, 1153, 19*w - 5], [1171, 1171, 19*w - 8], [1171, 1171, 19*w - 11], [1187, 1187, -w - 34], [1187, 1187, w - 35], [1193, 1193, 21*w - 26], [1193, 1193, -21*w - 5], [1213, 1213, -23*w + 34], [1213, 1213, 12*w - 47], [1223, 1223, 21*w - 25], [1223, 1223, -21*w - 4], [1231, 1231, 3*w - 37], [1231, 1231, -3*w - 34], [1249, 1249, 22*w - 29], [1249, 1249, -22*w - 7], [1277, 1277, 21*w - 23], [1277, 1277, -21*w - 2], [1283, 1283, 2*w - 37], [1283, 1283, -2*w - 35], [1291, 1291, 20*w - 13], [1291, 1291, 20*w - 7], [1297, 1297, -9*w - 35], [1297, 1297, 9*w - 44], [1301, 1301, -21*w - 1], [1301, 1301, 21*w - 22], [1303, 1303, -3*w - 35], [1303, 1303, 3*w - 38], [1327, 1327, -6*w - 35], [1327, 1327, 6*w - 41], [1361, 1361, 21*w - 2], [1361, 1361, 21*w - 19], [1369, 37, -37], [1381, 1381, 12*w - 49], [1381, 1381, 25*w - 38], [1427, 1427, 21*w - 13], [1427, 1427, 21*w - 8], [1429, 1429, -22*w - 1], [1429, 1429, 22*w - 23], [1433, 1433, 21*w - 10], [1433, 1433, 21*w - 11], [1439, 1439, 10*w - 47], [1439, 1439, -10*w - 37], [1447, 1447, 23*w - 28], [1447, 1447, -23*w - 5], [1453, 1453, 3*w - 40], [1453, 1453, -3*w - 37], [1459, 1459, 9*w - 46], [1459, 1459, -9*w - 37], [1481, 1481, -7*w - 37], [1481, 1481, 7*w - 44], [1483, 1483, 6*w - 43], [1483, 1483, -6*w - 37], [1499, 1499, -11*w - 38], [1499, 1499, 11*w - 49], [1511, 1511, -24*w - 7], [1511, 1511, 24*w - 31], [1531, 1531, -3*w - 38], [1531, 1531, 3*w - 41], [1543, 1543, -9*w - 38], [1543, 1543, 9*w - 47], [1559, 1559, 5*w - 43], [1559, 1559, -5*w - 38], [1583, 1583, -24*w - 5], [1583, 1583, 24*w - 29], [1609, 1609, 23*w - 22], [1609, 1609, 23*w - 1], [1613, 1613, 13*w - 53], [1613, 1613, 27*w - 41], [1621, 1621, -15*w - 41], [1621, 1621, 15*w - 56], [1637, 1637, -w - 40], [1637, 1637, w - 41], [1663, 1663, 23*w - 19], [1663, 1663, 23*w - 4], [1667, 1667, -27*w + 40], [1667, 1667, 14*w - 55], [1681, 41, -41], [1693, 1693, 3*w - 43], [1693, 1693, -3*w - 40], [1699, 1699, 23*w - 16], [1699, 1699, 23*w - 7], [1733, 1733, 7*w - 47], [1733, 1733, -7*w - 40], [1741, 1741, 12*w - 53], [1741, 1741, -12*w - 41], [1759, 1759, 25*w - 29], [1759, 1759, -25*w - 4], [1777, 1777, -3*w - 41], [1777, 1777, 3*w - 44], [1811, 1811, 5*w - 46], [1811, 1811, -5*w - 41], [1823, 1823, 24*w - 19], [1823, 1823, 24*w - 5], [1847, 1847, 24*w - 17], [1847, 1847, 24*w - 7], [1871, 1871, 24*w - 13], [1871, 1871, 24*w - 11], [1873, 1873, 26*w - 31], [1873, 1873, -26*w - 5], [1889, 1889, -w - 43], [1889, 1889, w - 44], [1901, 1901, -13*w - 43], [1901, 1901, 13*w - 56], [1907, 1907, 27*w - 35], [1907, 1907, -27*w - 8], [1933, 1933, -12*w - 43], [1933, 1933, 12*w - 55], [1949, 1949, -27*w - 7], [1949, 1949, 27*w - 34], [1951, 1951, 3*w - 46], [1951, 1951, -3*w - 43], [1973, 1973, -4*w - 43], [1973, 1973, 4*w - 47], [1979, 1979, -10*w - 43], [1979, 1979, 10*w - 53], [1993, 1993, 9*w - 52], [1993, 1993, -9*w - 43], [1999, 1999, 6*w - 49], [1999, 1999, -6*w - 43], [2003, 2003, 7*w - 50], [2003, 2003, -7*w - 43], [2011, 2011, 25*w - 8], [2011, 2011, 25*w - 17], [2027, 2027, -27*w - 5], [2027, 2027, 27*w - 32], [2029, 2029, 25*w - 11], [2029, 2029, 25*w - 14], [2053, 2053, 26*w - 25], [2053, 2053, 26*w - 1], [2063, 2063, -27*w - 4], [2063, 2063, 27*w - 31], [2081, 2081, -5*w - 44], [2081, 2081, 5*w - 49], [2083, 2083, -29*w - 11], [2083, 2083, 29*w - 40], [2089, 2089, -9*w - 44], [2089, 2089, 9*w - 53], [2129, 2129, 27*w - 29], [2129, 2129, -27*w - 2], [2131, 2131, 15*w - 61], [2131, 2131, 31*w - 47], [2141, 2141, 17*w - 64], [2141, 2141, -17*w - 47], [2161, 2161, 26*w - 7], [2161, 2161, 26*w - 19], [2207, 2207, 13*w - 59], [2207, 2207, -13*w - 46], [2209, 47, -47], [2213, 2213, 27*w - 26], [2213, 2213, 27*w - 1], [2237, 2237, 27*w - 2], [2237, 2237, 27*w - 25], [2239, 2239, 32*w - 49], [2239, 2239, 15*w - 62], [2287, 2287, 9*w - 55], [2287, 2287, -9*w - 46], [2297, 2297, 27*w - 22], [2297, 2297, 27*w - 5], [2311, 2311, 18*w - 67], [2311, 2311, -18*w - 49], [2339, 2339, 27*w - 8], [2339, 2339, 27*w - 19], [2341, 2341, -12*w - 47], [2341, 2341, 12*w - 59], [2357, 2357, 27*w - 10], [2357, 2357, 27*w - 17], [2383, 2383, -6*w - 47], [2383, 2383, 6*w - 53], [2389, 2389, -9*w - 47], [2389, 2389, 9*w - 56], [2393, 2393, -8*w - 47], [2393, 2393, 8*w - 55], [2417, 2417, 16*w - 65], [2417, 2417, 33*w - 50], [2441, 2441, -30*w - 7], [2441, 2441, 30*w - 37], [2447, 2447, -w - 49], [2447, 2447, w - 50], [2467, 2467, 28*w - 5], [2467, 2467, 28*w - 23], [2473, 2473, 31*w - 41], [2473, 2473, -31*w - 10], [2521, 2521, 3*w - 52], [2521, 2521, -3*w - 49], [2531, 2531, 13*w - 62], [2531, 2531, -13*w - 49], [2539, 2539, 28*w - 11], [2539, 2539, 28*w - 17], [2549, 2549, 4*w - 53], [2549, 2549, -4*w - 49], [2551, 2551, 29*w - 28], [2551, 2551, 29*w - 1], [2557, 2557, 12*w - 61], [2557, 2557, -12*w - 49], [2591, 2591, -10*w - 49], [2591, 2591, 10*w - 59], [2609, 2609, -19*w - 52], [2609, 2609, 19*w - 71], [2617, 2617, -31*w - 7], [2617, 2617, 31*w - 38], [2677, 2677, 29*w - 7], [2677, 2677, 29*w - 22], [2687, 2687, 11*w - 61], [2687, 2687, -11*w - 50], [2707, 2707, 9*w - 59], [2707, 2707, -9*w - 50], [2713, 2713, 29*w - 19], [2713, 2713, 29*w - 10], [2729, 2729, 30*w - 29], [2729, 2729, 30*w - 1], [2731, 2731, 29*w - 16], [2731, 2731, 29*w - 13], [2753, 2753, -w - 52], [2753, 2753, w - 53], [2791, 2791, -35*w + 52], [2791, 2791, 18*w - 71], [2833, 2833, 3*w - 55], [2833, 2833, -3*w - 52], [2837, 2837, -33*w - 10], [2837, 2837, 33*w - 43], [2843, 2843, 36*w - 55], [2843, 2843, 17*w - 70], [2851, 2851, -31*w - 1], [2851, 2851, 31*w - 32], [2857, 2857, 34*w - 47], [2857, 2857, -34*w - 13], [2861, 2861, 30*w - 23], [2861, 2861, 30*w - 7], [2887, 2887, -32*w - 5], [2887, 2887, 32*w - 37], [2903, 2903, 2*w - 55], [2903, 2903, -2*w - 53], [2909, 2909, 30*w - 19], [2909, 2909, 30*w - 11], [2939, 2939, 33*w - 41], [2939, 2939, -33*w - 8], [2963, 2963, -14*w - 53], [2963, 2963, 14*w - 67], [2999, 2999, 5*w - 58], [2999, 2999, -5*w - 53], [3019, 3019, -6*w - 53], [3019, 3019, 6*w - 59], [3041, 3041, -8*w - 53], [3041, 3041, 8*w - 61], [3067, 3067, 31*w - 23], [3067, 3067, 31*w - 8], [3119, 3119, 33*w - 37], [3119, 3119, -33*w - 4], [3121, 3121, 31*w - 14], [3121, 3121, 31*w - 17], [3137, 3137, 16*w - 71], [3137, 3137, -16*w - 55], [3163, 3163, 3*w - 58], [3163, 3163, -3*w - 55], [3169, 3169, -35*w - 11], [3169, 3169, 35*w - 46], [3181, 3181, -34*w - 7], [3181, 3181, 34*w - 41], [3221, 3221, 17*w - 73], [3221, 3221, -17*w - 56], [3251, 3251, 36*w - 49], [3251, 3251, -36*w - 13], [3253, 3253, -12*w - 55], [3253, 3253, 12*w - 67], [3259, 3259, 21*w - 79], [3259, 3259, -21*w - 58], [3299, 3299, 33*w - 32], [3299, 3299, 33*w - 1], [3301, 3301, -15*w - 56], [3301, 3301, 15*w - 71], [3319, 3319, 32*w - 19], [3319, 3319, 32*w - 13], [3329, 3329, 33*w - 2], [3329, 3329, 33*w - 31], [3331, 3331, 35*w - 43], [3331, 3331, -35*w - 8], [3371, 3371, 36*w - 47], [3371, 3371, -36*w - 11], [3389, 3389, 11*w - 67], [3389, 3389, -11*w - 56], [3407, 3407, 33*w - 5], [3407, 3407, 33*w - 28], [3433, 3433, -34*w - 1], [3433, 3433, 34*w - 35], [3449, 3449, 33*w - 7], [3449, 3449, 33*w - 26], [3457, 3457, 37*w - 50], [3457, 3457, -37*w - 13], [3461, 3461, -39*w + 58], [3461, 3461, 20*w - 79], [3467, 3467, 33*w - 8], [3467, 3467, 33*w - 25], [3481, 59, -59], [3511, 3511, 3*w - 61], [3511, 3511, -3*w - 58], [3527, 3527, 33*w - 13], [3527, 3527, 33*w - 20], [3533, 3533, 33*w - 14], [3533, 3533, 33*w - 19], [3539, 3539, 33*w - 16], [3539, 3539, 33*w - 17], [3559, 3559, 15*w - 73], [3559, 3559, -15*w - 58], [3571, 3571, -18*w - 59], [3571, 3571, 18*w - 77], [3613, 3613, 34*w - 29], [3613, 3613, 34*w - 5], [3617, 3617, -17*w - 59], [3617, 3617, 17*w - 76], [3623, 3623, -7*w - 58], [3623, 3623, 7*w - 65], [3631, 3631, -3*w - 59], [3631, 3631, 3*w - 62], [3637, 3637, 37*w - 47], [3637, 3637, -37*w - 10], [3643, 3643, -9*w - 58], [3643, 3643, 9*w - 67], [3691, 3691, -15*w - 59], [3691, 3691, 15*w - 74], [3701, 3701, -5*w - 59], [3701, 3701, 5*w - 64], [3709, 3709, 35*w - 34], [3709, 3709, 35*w - 1], [3719, 3719, 14*w - 73], [3719, 3719, -14*w - 59], [3727, 3727, -6*w - 59], [3727, 3727, 6*w - 65], [3761, 3761, 8*w - 67], [3761, 3761, -8*w - 59], [3767, 3767, 11*w - 70], [3767, 3767, -11*w - 59], [3769, 3769, 9*w - 68], [3769, 3769, -9*w - 59], [3779, 3779, -w - 61], [3779, 3779, w - 62], [3793, 3793, -38*w - 11], [3793, 3793, 38*w - 49], [3797, 3797, 19*w - 80], [3797, 3797, -19*w - 61], [3821, 3821, -39*w - 14], [3821, 3821, 39*w - 53], [3823, 3823, -41*w + 61], [3823, 3823, 21*w - 83], [3847, 3847, -18*w - 61], [3847, 3847, 18*w - 79], [3851, 3851, -36*w - 1], [3851, 3851, 36*w - 37], [3877, 3877, 3*w - 64], [3877, 3877, -3*w - 61], [3917, 3917, 4*w - 65], [3917, 3917, -4*w - 61], [3923, 3923, 36*w - 35], [3923, 3923, 36*w - 1], [3929, 3929, 16*w - 77], [3929, 3929, -16*w - 61], [3943, 3943, 37*w - 41], [3943, 3943, -37*w - 4], [4001, 4001, 7*w - 68], [4001, 4001, -7*w - 61], [4003, 4003, -3*w - 62], [4003, 4003, 3*w - 65], [4007, 4007, 13*w - 74], [4007, 4007, -13*w - 61], [4013, 4013, 39*w - 50], [4013, 4013, -39*w - 11], [4021, 4021, 12*w - 73], [4021, 4021, -12*w - 61], [4027, 4027, 9*w - 70], [4027, 4027, -9*w - 61], [4057, 4057, -24*w - 65], [4057, 4057, 24*w - 89], [4073, 4073, -39*w - 10], [4073, 4073, 39*w - 49], [4079, 4079, -5*w - 62], [4079, 4079, 5*w - 67], [4091, 4091, 36*w - 7], [4091, 4091, 36*w - 29], [4099, 4099, -15*w - 62], [4099, 4099, 15*w - 77], [4111, 4111, 40*w - 53], [4111, 4111, -40*w - 13], [4133, 4133, -23*w - 65], [4133, 4133, 23*w - 88], [4157, 4157, -w - 64], [4157, 4157, w - 65], [4159, 4159, -9*w - 62], [4159, 4159, 9*w - 71], [4177, 4177, 37*w - 2], [4177, 4177, 37*w - 35], [4211, 4211, 36*w - 19], [4211, 4211, 36*w - 17], [4229, 4229, -19*w - 64], [4229, 4229, 19*w - 83], [4241, 4241, 39*w - 46], [4241, 4241, -39*w - 7], [4261, 4261, 3*w - 67], [4261, 4261, -3*w - 64], [4273, 4273, -41*w - 14], [4273, 4273, 41*w - 55], [4289, 4289, 25*w - 92], [4289, 4289, -25*w - 67], [4339, 4339, 37*w - 29], [4339, 4339, 37*w - 8], [4391, 4391, -39*w - 4], [4391, 4391, 39*w - 43], [4397, 4397, 7*w - 71], [4397, 4397, -7*w - 64], [4421, 4421, -13*w - 64], [4421, 4421, 13*w - 77], [4423, 4423, 18*w - 83], [4423, 4423, -18*w - 65], [4447, 4447, 37*w - 17], [4447, 4447, 37*w - 20], [4463, 4463, -17*w - 65], [4463, 4463, 17*w - 82], [4481, 4481, 39*w - 41], [4481, 4481, -39*w - 2], [4489, 67, -67], [4507, 4507, -6*w - 65], [4507, 4507, 6*w - 71], [4523, 4523, -39*w - 1], [4523, 4523, 39*w - 40], [4547, 4547, -14*w - 65], [4547, 4547, 14*w - 79], [4549, 4549, 38*w - 31], [4549, 4549, 38*w - 7], [4567, 4567, -9*w - 65], [4567, 4567, 9*w - 74], [4603, 4603, 43*w - 59], [4603, 4603, -43*w - 16], [4637, 4637, 39*w - 2], [4637, 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[8501, 8501, 20*w - 109], [8501, 8501, -20*w - 89], [8527, 8527, 3*w - 94], [8527, 8527, -3*w - 91], [8537, 8537, 11*w - 100], [8537, 8537, -11*w - 89], [8563, 8563, 52*w - 41], [8563, 8563, 52*w - 11], [8581, 8581, -15*w - 89], [8581, 8581, 15*w - 104], [8597, 8597, -4*w - 91], [8597, 8597, 4*w - 95], [8609, 8609, 29*w - 121], [8609, 8609, -29*w - 92], [8623, 8623, 53*w - 4], [8623, 8623, 53*w - 49], [8629, 8629, 58*w - 77], [8629, 8629, -58*w - 19], [8641, 8641, -55*w - 7], [8641, 8641, 55*w - 62], [8681, 8681, 25*w - 116], [8681, 8681, -25*w - 91], [8693, 8693, -54*w - 1], [8693, 8693, 54*w - 55], [8707, 8707, 52*w - 17], [8707, 8707, 52*w - 35], [8713, 8713, -3*w - 92], [8713, 8713, 3*w - 95], [8719, 8719, 6*w - 97], [8719, 8719, -6*w - 91], [8737, 8737, -24*w - 91], [8737, 8737, 24*w - 115], [8753, 8753, 57*w - 71], [8753, 8753, -57*w - 14], [8761, 8761, 27*w - 119], [8761, 8761, -27*w - 92], [8779, 8779, 52*w - 23], [8779, 8779, 52*w - 29], [8831, 8831, -22*w - 91], [8831, 8831, 22*w - 113], [8837, 8837, -57*w - 13], [8837, 8837, 57*w - 70], [8839, 8839, -55*w - 4], [8839, 8839, 55*w - 59], [8849, 8849, -5*w - 92], [8849, 8849, 5*w - 97], [8863, 8863, -59*w - 20], [8863, 8863, 59*w - 79], [8867, 8867, 31*w - 125], [8867, 8867, 63*w - 95], [8893, 8893, 33*w - 128], [8893, 8893, -62*w + 91], [8941, 8941, -12*w - 91], [8941, 8941, 12*w - 103], [8969, 8969, 16*w - 107], [8969, 8969, -16*w - 91], [8971, 8971, 15*w - 106], [8971, 8971, -15*w - 91], [8999, 8999, -57*w - 11], [8999, 8999, 57*w - 68], [9013, 9013, 36*w - 133], [9013, 9013, -36*w - 97], [9049, 9049, -9*w - 92], [9049, 9049, 9*w - 101], [9091, 9091, 3*w - 97], [9091, 9091, -3*w - 94], [9103, 9103, 56*w - 61], [9103, 9103, -56*w - 5], [9109, 9109, 53*w - 22], [9109, 9109, 53*w - 31], [9127, 9127, 53*w - 28], [9127, 9127, 53*w - 25], [9151, 9151, -59*w - 17], [9151, 9151, 59*w - 76], [9161, 9161, 17*w - 109], [9161, 9161, -17*w - 92], [9181, 9181, 55*w - 2], [9181, 9181, 55*w - 53], [9187, 9187, -27*w - 94], [9187, 9187, 27*w - 121], [9203, 9203, 2*w - 97], [9203, 9203, -2*w - 95], [9221, 9221, 54*w - 43], [9221, 9221, 54*w - 11], [9227, 9227, 57*w - 65], [9227, 9227, -57*w - 8], [9239, 9239, -34*w - 97], [9239, 9239, 34*w - 131], [9257, 9257, 29*w - 124], [9257, 9257, -29*w - 95], [9281, 9281, 54*w - 41], [9281, 9281, 54*w - 13], [9283, 9283, -3*w - 95], [9283, 9283, 3*w - 98], [9311, 9311, 25*w - 119], [9311, 9311, -25*w - 94], [9337, 9337, -61*w - 22], [9337, 9337, 61*w - 83], [9343, 9343, 33*w - 130], [9343, 9343, -64*w + 95], [9377, 9377, 54*w - 17], [9377, 9377, 54*w - 37], [9403, 9403, 27*w - 122], [9403, 9403, -27*w - 95], [9409, 97, -97], [9413, 9413, 54*w - 19], [9413, 9413, 54*w - 35], [9421, 9421, 59*w - 73], [9421, 9421, -59*w - 14], [9437, 9437, -57*w - 5], [9437, 9437, 57*w - 62], [9439, 9439, -9*w - 94], [9439, 9439, 9*w - 103], [9461, 9461, 54*w - 23], [9461, 9461, 54*w - 31], [9463, 9463, 56*w - 55], [9463, 9463, 56*w - 1], [9467, 9467, -26*w - 95], [9467, 9467, 26*w - 121], [9473, 9473, 54*w - 29], [9473, 9473, 54*w - 25], [9491, 9491, 60*w - 77], [9491, 9491, -60*w - 17], [9533, 9533, 31*w - 128], [9533, 9533, 66*w - 101], [9539, 9539, -19*w - 94], [9539, 9539, 19*w - 113], [9551, 9551, -13*w - 94], [9551, 9551, 13*w - 107], [9619, 9619, 30*w - 127], [9619, 9619, -30*w - 97], [9623, 9623, 23*w - 118], [9623, 9623, -23*w - 95], [9629, 9629, 57*w - 59], [9629, 9629, -57*w - 2], [9643, 9643, 61*w - 80], [9643, 9643, -61*w - 19], [9649, 9649, 55*w - 41], [9649, 9649, 55*w - 14], [9689, 9689, -57*w - 1], [9689, 9689, 57*w - 58], [9697, 9697, 21*w - 116], [9697, 9697, -21*w - 95], [9721, 9721, 55*w - 38], [9721, 9721, 55*w - 17], [9733, 9733, -12*w - 95], [9733, 9733, 12*w - 107], [9749, 9749, 4*w - 101], [9749, 9749, -4*w - 97], [9767, 9767, -14*w - 95], [9767, 9767, 14*w - 109], [9803, 9803, 57*w - 56], [9803, 9803, 57*w - 1], [9811, 9811, 55*w - 32], [9811, 9811, 55*w - 23], [9829, 9829, 55*w - 26], [9829, 9829, 55*w - 29], [9851, 9851, 60*w - 73], [9851, 9851, -60*w - 13], [9857, 9857, 57*w - 2], [9857, 9857, 57*w - 55], [9871, 9871, -3*w - 98], [9871, 9871, 3*w - 101], [9883, 9883, 6*w - 103], [9883, 9883, -6*w - 97], [9907, 9907, 59*w - 67], [9907, 9907, -59*w - 8], [9923, 9923, 29*w - 127], [9923, 9923, -29*w - 98], [9929, 9929, -63*w - 23], [9929, 9929, 63*w - 86], [9931, 9931, 61*w - 77], [9931, 9931, -61*w - 16], [9941, 9941, -7*w - 97], [9941, 9941, 7*w - 104], [9949, 9949, 36*w - 137], [9949, 9949, -36*w - 101], [9967, 9967, 56*w - 43], [9967, 9967, 56*w - 13]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [0, 2, 1, -5, 3, -3, 6, 6, 1, 9, -6, -4, 2, -13, -3, -9, -14, -5, 10, 10, -3, 3, 2, 2, 6, 0, 6, 9, 13, 22, -8, 0, 12, 14, -10, 2, 5, -18, -3, 18, 6, 5, -19, 6, -12, -2, -8, 10, 16, -18, -3, 6, -30, -21, 18, 6, 24, 21, 3, -7, 17, -14, -2, 0, 18, 10, 13, 10, -14, 12, 30, 11, 26, -28, -29, 31, -18, -30, 18, -12, -11, -2, 8, -10, 24, 18, -36, 6, 42, -36, -6, -12, -3, 3, -4, -22, 8, -28, -30, 12, -45, 33, -14, -8, -24, 6, -26, -11, -4, -40, 27, 18, 42, 12, -6, 30, -24, -18, -46, -13, 15, -39, -30, 15, 6, -18, 28, -44, -22, 14, -29, 7, 39, 6, 3, -30, 26, 14, 7, 31, -45, -30, 40, -32, -30, -18, 16, -2, -18, 42, 34, -26, 48, -6, -26, -44, -23, 25, 15, 21, -10, 24, -36, 8, -28, -22, 5, -30, -42, 48, 0, 26, 2, 21, 15, -34, 26, -17, -17, 18, -36, -29, 34, -9, 33, -43, -43, -13, 47, 56, -4, -48, -6, -21, -3, -41, -29, -12, 42, 8, 68, -25, 11, 45, 39, -6, 54, -20, 64, 61, 25, 21, -42, 52, -56, 22, 40, 54, 45, -25, 5, 29, 36, -54, -55, -10, -54, 6, -36, 18, 52, -14, 41, 41, 38, 32, 30, 69, -16, -22, 24, 48, 0, -48, 34, -20, 4, 4, 36, -72, -42, 30, 10, 37, -30, -39, -29, -65, -66, -57, 4, 34, 66, -12, 10, -7, 65, 4, -8, -21, -15, -77, -14, 50, -40, -23, 34, 42, -24, -84, 36, 18, -12, 6, -60, -23, -38, -69, -66, 21, -33, -30, -18, -55, -19, 66, -33, -28, 62, 78, -21, 36, 60, -25, -10, -58, 56, -48, 30, -22, 20, -6, -24, 86, 14, 34, -29, 18, 0, -15, 78, 64, -14, 1, -23, -45, 45, 20, -88, 57, 75, 37, 58, -48, 42, -2, -15, 27, -15, -9, 28, -8, -52, -16, 42, 78, -70, 80, -36, -24, -14, 67, 75, -54, 34, -38, -41, -14, -51, -42, 57, 27, 75, 45, -18, -12, -52, 20, 71, -22, 62, -34, -60, -12, -16, 98, -81, 9, 16, 10, 17, 14, 18, 48, 3, 57, -37, 62, -26, 7, -30, 60, -56, 22, 22, -23, 78, 81, 40, -80, -69, -51, -20, -74, -82, 23, 93, 3, -66, 54, 62, -10, -73, -13, 102, 63, -74, -98, -6, 66, 66, 6, 66, -66, -90, 24, -78, -60, 58, -50, 21, 15, -58, 50, -54, -12, 38, 23, -27, 39, 14, -58, 61, -11, -91, -55, 54, -93, 18, 78, -58, -31, 56, -70, 42, -42, 55, -62, 82, -26, -81, -39, 94, -38, 78, 78, 81, -69, -48, -102, -73, -61, -51, 6, -97, 62, -105, 45, -90, -72, -35, -58, -100, 108, 72, -42, -102, 36, -66, 104, -40, 100, -62, -58, 77, 30, 3, 60, 84, 76, -50, -19, -22, -82, -10, -92, -44, -105, -87, -113, 82, 102, 24, 52, 58, -78, 90, -54, 6, 34, 85, 72, 6, -71, 58, -6, 57, -33, -30, 16, 4, 8, -46, -48, 36, 89, -7, -9, -87, 24, 42, -30, -9, 56, 38, 30, 6, 46, 118, 78, -42, -39, 27, -7, -67, 2, -58, -11, -83, 39, 81, 96, -18, -12, 30, -104, 22, -40, 8, 18, -6, 117, -117, -92, 94, -58, 86, 96, -60, -39, 27, 81, 30, 77, 53, 73, -62, -87, 15, -28, -46, 48, 90, -129, 18, 21, 87, -26, 130, 20, -124, -18, -102, 39, 129, 62, 64, 94, -18, -96, 72, -132, -101, -89, -32, -14, 86, 104, -87, 66, 28, -116, 58, 109, 56, 44, -60, -84, 24, -66, 74, -82, -59, -23, -30, 21, -64, -118, 80, 116, 60, 48, -21, 9, 85, -131, -17, 115, -19, -34, 108, -30, 51, 66, 96, 12, -45, -51, -120, -6, -62, -53, 78, -30, 121, 22, 25, 97, 93, 63, -30, 54, 13, 48, -108, -18, 108, -37, 35, 74, 26, 0, 42, 108, 66, 53, -19, -11, -71, -26, -98, 15, 105, 48, -66, 25, -71, 90, 48, 47, -66, -21, -80, -2, 69, -117, 24, -66, -116, -32, 69, 42, 76, -20, 67, 91, 82, 10, -90, -87, -120, -66, 110, -58, 83, -133, 118, 34, -93, 33, -91, 29, 30, 24, 114, -138, 35, 26, 90, 126, 44, -100, 9, 3, -30, 15, -36, -60, 63, 57, -22, 23, -22, 116, -130, -79, 81, 123, -114, -18, -41, 19, -94, 50, 99, 129, 58, 76, -126, 60, -126, -60, 0, 0, 90, -90, -145, 119, 6, 90, 92, 50, 78, 63, -80, 64, 3, 9, -81, -39, 7, 154, 56, -46, 57, 15, -20, 70, -58, -25, 81, -18, -30, -144, -114, -87, -94, -67, 105, 3, -110, -44, -3, 54, 130, 145, -158, -110, -53, -77, 76, 34, -151, -91, -77, 31, 18, -39, 8, 86, -102, 75, 108, 18, -35, 49, 126, 120, 71, 35, 39, 66, -73, 71, -57, 57, -112, 110, 78, 102, -6, 42, 34, -62, -110, -8, 62, 110, 137, -70, 29, -34, -21, 21, 45, 63, 122, 26, -54, 36, -65, -89, -106, 47, -96, -18, -119, 66, -126, -104, 76, 30, -153, -90, -42, 16, 142, 30, -36, 162, 126, -88, -16, 34, -5, -54, -30, 120, -30, -158, 67, -63, 147, -78, 120, 84, 96, 16, -44, 121, -122, 66, 51, 60, -18, 20, 74, 66, 111, 70, 106, -69, 150, -114, 36, -73, -73, -18, -90, -29, -110, 30, -90, -163, -127, -3, -9, -86, 130, -128, 130, -153, 66, -38, 22, -24, -102, 61, 34, -93, -123, 36, 66, -109, 59, 117, 99, 91, 91, -12, -132, -33, -39, 136, -8, 14, -115, -156, 42, -91, -82, -54, -129, -34, 20, 110, 134, 16, -26, 9, -153, -153, -123, -12, 66, 21, -9, -3, 39, 57, -57, 40, -164, -15, -54, -42, 108, 37, 112, -2, 65, -19, 63, 102, -56, 52, -46, 107, 32, -112, 6, 165, -6, 66, 10, -179, -66, 84, -6, 108, 37, -143, -44, 22, 152, 152, -114, 120, -26, 1, 36, 66, -55, 77, 78, 48, -139, -130, 87, -78, -4, 38, 7, 139, -47, -86, -114, -120, 51, 117, 53, -151, -12, 66, 48, -162, -62, 118, 45, -126, 68, 104, -153, 66, 128, -40, 58, -149, -66, 42, -69, -135, -110, -164, 17, -55, 146, 47, -15, -69, 57, 18, 104, -58, -26, 157, -46, -112, 143, -109, -78, -111, 22, 13, -94, 86, 144, -30, 21, 75, 20, -16, -75, -126, 124, 154, -6, -126, 31, -158, 158, -82, -39, -141, -46, 152, 36, -30, 125, 182, 49, -86, -106, 122, -86, 22, -29, -65, -14, 16, -164, 4, 27, -78, 29, 110, 62, 44, 48, -186, -165, -135, 18, -78, -108, -174, -114, 18, 30, 153, -80, 178, -186, 42, -58, 23, 74, -142, 27, -99, 64, 100, 14, -102, 6, 10, 46, -102, -99, 38, 56, -114, 135, -104, 76, 162, 126, -81, -39, 24, -48, -117, 93, -12, 168, 168, 18, 50, -22, 174, -42, -159, -189, -10, 68, 155, 86, -165, 153, 157, 133, -82, 83, 166, 10, -81, 21, -18, -60, -168, -78, 68, -16, 134, -7, -144, -108, 135, 105, 22, 136, -16, 146, -122, -170, -102, 6, 141, -21, 104, -76, -18, -81, 55, 55, -62, -38]; 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;