/* 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![-7, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, w - 3], [3, 3, w - 2], [3, 3, w + 2], [7, 7, w], [19, 19, 2*w - 3], [19, 19, 2*w + 3], [25, 5, 5], [29, 29, -w - 6], [29, 29, w - 6], [31, 31, 4*w + 9], [31, 31, -4*w + 9], [37, 37, -3*w + 10], [37, 37, -6*w + 17], [47, 47, -3*w - 4], [47, 47, 3*w - 4], [53, 53, 2*w - 9], [53, 53, 2*w + 9], [59, 59, 3*w - 2], [59, 59, -3*w - 2], [83, 83, -6*w - 13], [83, 83, -6*w + 13], [103, 103, 4*w - 3], [103, 103, -4*w - 3], [109, 109, -6*w + 19], [109, 109, -6*w - 19], [113, 113, -13*w + 36], [113, 113, -4*w + 15], [121, 11, -11], [131, 131, -6*w + 11], [131, 131, -6*w - 11], [137, 137, -w - 12], [137, 137, w - 12], [139, 139, -5*w - 6], [139, 139, 5*w - 6], [149, 149, -5*w + 18], [149, 149, -14*w + 39], [167, 167, -9*w + 20], [167, 167, -9*w - 20], [169, 13, -13], [193, 193, 3*w - 16], [193, 193, -3*w - 16], [197, 197, 2*w - 15], [197, 197, -2*w - 15], [199, 199, -7*w - 12], [199, 199, -7*w + 12], [223, 223, -8*w + 15], [223, 223, -8*w - 15], [227, 227, 6*w - 5], [227, 227, -6*w - 5], [233, 233, -16*w + 45], [233, 233, -7*w + 24], [251, 251, -6*w - 1], [251, 251, 6*w - 1], [271, 271, -11*w + 24], [271, 271, -11*w - 24], [277, 277, -21*w + 58], [277, 277, -6*w + 23], [281, 281, -8*w + 27], [281, 281, -17*w + 48], [283, 283, 13*w - 30], [283, 283, -14*w + 33], [289, 17, -17], [307, 307, 7*w - 6], [307, 307, -7*w - 6], [311, 311, -9*w - 16], [311, 311, -9*w + 16], [317, 317, -w - 18], [317, 317, w - 18], [337, 337, -3*w - 20], [337, 337, 3*w - 20], [367, 367, -8*w - 9], [367, 367, 8*w - 9], [373, 373, -6*w + 25], [373, 373, -27*w + 74], [383, 383, -12*w - 25], [383, 383, -12*w + 25], [389, 389, -19*w + 54], [389, 389, -10*w + 33], [401, 401, -5*w - 24], [401, 401, 5*w - 24], [419, 419, -15*w - 34], [419, 419, -15*w + 34], [421, 421, 3*w - 22], [421, 421, -3*w - 22], [439, 439, -8*w - 3], [439, 439, 8*w - 3], [449, 449, -11*w + 36], [449, 449, -11*w - 36], [457, 457, -9*w + 32], [457, 457, -24*w + 67], [467, 467, -9*w - 10], [467, 467, 9*w - 10], [479, 479, -12*w + 23], [479, 479, -12*w - 23], [503, 503, 9*w - 8], [503, 503, -9*w - 8], [523, 523, -11*w + 18], [523, 523, -11*w - 18], [529, 23, -23], [541, 541, -15*w + 46], [541, 541, -15*w - 46], [557, 557, 7*w + 30], [557, 557, 7*w - 30], [563, 563, 9*w - 2], [563, 563, -9*w - 2], [569, 569, -w - 24], [569, 569, w - 24], [587, 587, -18*w + 41], [587, 587, -18*w - 41], [607, 607, -13*w - 24], [607, 607, -13*w + 24], [613, 613, -3*w - 26], [613, 613, 3*w - 26], [617, 617, -4*w - 27], [617, 617, 4*w - 27], [619, 619, 10*w - 9], [619, 619, -10*w - 9], [641, 641, -8*w + 33], [641, 641, -35*w + 96], [643, 643, -14*w + 27], [643, 643, -14*w - 27], [647, 647, -12*w + 19], [647, 647, -12*w - 19], [653, 653, -14*w + 45], [653, 653, -14*w - 45], [673, 673, -12*w + 41], [673, 673, -27*w + 76], [691, 691, 10*w - 3], [691, 691, -10*w - 3], [701, 701, 2*w - 27], [701, 701, -2*w - 27], [709, 709, 6*w - 31], [709, 709, -6*w - 31], [719, 719, -12*w - 17], [719, 719, 12*w - 17], [727, 727, -17*w + 36], [727, 727, -17*w - 36], [757, 757, -18*w + 55], [757, 757, -18*w - 55], [787, 787, -22*w + 51], [787, 787, -22*w - 51], [809, 809, -16*w - 51], [809, 809, -16*w + 51], [811, 811, -11*w - 6], [811, 811, 11*w - 6], [821, 821, -37*w + 102], [821, 821, -10*w + 39], [839, 839, -12*w - 13], [839, 839, 12*w - 13], [859, 859, 13*w - 18], [859, 859, -13*w - 18], [877, 877, 9*w + 38], [877, 877, 9*w - 38], [887, 887, 12*w - 11], [887, 887, -12*w - 11], [953, 953, 7*w - 36], [953, 953, -7*w - 36], [971, 971, -21*w + 46], [971, 971, -21*w - 46], [977, 977, 4*w - 33], [977, 977, -4*w - 33], [983, 983, -12*w - 5], [983, 983, 12*w - 5], [1009, 1009, -21*w + 64], [1009, 1009, -21*w - 64], [1033, 1033, 9*w + 40], [1033, 1033, 9*w - 40], [1039, 1039, 13*w - 12], [1039, 1039, -13*w - 12], [1061, 1061, 2*w - 33], [1061, 1061, -2*w - 33], [1063, 1063, -16*w - 27], [1063, 1063, -16*w + 27], [1091, 1091, 15*w - 22], [1091, 1091, -15*w - 22], [1093, 1093, 3*w - 34], [1093, 1093, -3*w - 34], [1117, 1117, 6*w - 37], [1117, 1117, -6*w - 37], [1123, 1123, -17*w - 30], [1123, 1123, -17*w + 30], [1129, 1129, -15*w - 52], [1129, 1129, 15*w - 52], [1151, 1151, -21*w + 44], [1151, 1151, -21*w - 44], [1201, 1201, -45*w + 124], [1201, 1201, -12*w + 47], [1213, 1213, -18*w - 59], [1213, 1213, -18*w + 59], [1223, 1223, -24*w - 53], [1223, 1223, -24*w + 53], [1229, 1229, -41*w + 114], [1229, 1229, -14*w + 51], [1231, 1231, -19*w - 36], [1231, 1231, -19*w + 36], [1259, 1259, -27*w + 62], [1259, 1259, -27*w - 62], [1279, 1279, -20*w + 39], [1279, 1279, -20*w - 39], [1289, 1289, -w - 36], [1289, 1289, w - 36], [1291, 1291, 14*w - 9], [1291, 1291, -14*w - 9], [1297, 1297, -24*w + 73], [1297, 1297, -24*w - 73], [1307, 1307, -18*w - 31], [1307, 1307, -18*w + 31], [1319, 1319, -15*w - 16], [1319, 1319, 15*w - 16], [1373, 1373, -22*w - 69], [1373, 1373, -22*w + 69], [1381, 1381, -3*w - 38], [1381, 1381, 3*w - 38], [1399, 1399, -23*w + 48], [1399, 1399, -23*w - 48], [1409, 1409, -4*w - 39], [1409, 1409, 4*w - 39], [1427, 1427, -18*w - 29], [1427, 1427, -18*w + 29], [1429, 1429, -6*w - 41], [1429, 1429, 6*w - 41], [1447, 1447, -17*w - 24], [1447, 1447, 17*w - 24], [1453, 1453, -39*w + 110], [1453, 1453, -18*w + 61], [1459, 1459, -25*w + 54], [1459, 1459, -25*w - 54], [1481, 1481, -23*w + 72], [1481, 1481, -23*w - 72], [1483, 1483, -26*w + 57], [1483, 1483, -26*w - 57], [1487, 1487, -21*w - 40], [1487, 1487, -21*w + 40], [1493, 1493, 2*w - 39], [1493, 1493, -2*w - 39], [1511, 1511, -15*w - 8], [1511, 1511, 15*w - 8], [1531, 1531, -29*w - 66], [1531, 1531, -29*w + 66], [1543, 1543, -31*w + 72], [1543, 1543, -31*w - 72], [1549, 1549, 9*w - 46], [1549, 1549, -9*w - 46], [1559, 1559, -15*w - 4], [1559, 1559, 15*w - 4], [1567, 1567, 16*w - 15], [1567, 1567, -16*w - 15], [1571, 1571, 15*w - 2], [1571, 1571, -15*w - 2], [1597, 1597, 6*w - 43], [1597, 1597, -6*w - 43], [1621, 1621, -27*w + 82], [1621, 1621, -27*w - 82], [1627, 1627, -19*w - 30], [1627, 1627, -19*w + 30], [1681, 41, -41], [1699, 1699, -17*w - 18], [1699, 1699, 17*w - 18], [1709, 1709, -25*w - 78], [1709, 1709, -25*w + 78], [1733, 1733, -58*w + 159], [1733, 1733, -13*w + 54], [1783, 1783, 16*w - 3], [1783, 1783, -16*w - 3], [1789, 1789, -15*w + 58], [1789, 1789, -54*w + 149], [1801, 1801, 12*w - 53], [1801, 1801, 12*w + 53], [1811, 1811, -30*w - 67], [1811, 1811, -30*w + 67], [1823, 1823, -24*w + 47], [1823, 1823, -24*w - 47], [1847, 1847, -33*w - 76], [1847, 1847, -33*w + 76], [1849, 43, -43], [1867, 1867, -22*w + 39], [1867, 1867, -22*w - 39], [1873, 1873, -3*w - 44], [1873, 1873, 3*w - 44], [1877, 1877, -14*w + 57], [1877, 1877, -59*w + 162], [1879, 1879, 17*w - 12], [1879, 1879, -17*w - 12], [1901, 1901, 10*w - 51], [1901, 1901, -10*w - 51], [1907, 1907, -18*w - 19], [1907, 1907, 18*w - 19], [1913, 1913, 4*w - 45], [1913, 1913, -4*w - 45], [1931, 1931, -21*w + 34], [1931, 1931, -21*w - 34], [1933, 1933, 9*w - 50], [1933, 1933, -9*w - 50], [1951, 1951, -19*w - 24], [1951, 1951, 19*w - 24], [1979, 1979, 18*w - 17], [1979, 1979, -18*w - 17], [1987, 1987, -17*w - 6], [1987, 1987, 17*w - 6], [1997, 1997, 2*w - 45], [1997, 1997, -2*w - 45], [2017, 2017, 12*w + 55], [2017, 2017, 12*w - 55], [2053, 2053, 3*w - 46], [2053, 2053, -3*w - 46], [2063, 2063, -21*w - 32], [2063, 2063, -21*w + 32], [2069, 2069, -11*w - 54], [2069, 2069, 11*w - 54], [2081, 2081, -28*w - 87], [2081, 2081, -28*w + 87], [2099, 2099, -18*w - 13], [2099, 2099, 18*w - 13], [2129, 2129, 5*w - 48], [2129, 2129, -5*w - 48], [2131, 2131, -26*w + 51], [2131, 2131, -26*w - 51], [2137, 2137, 9*w - 52], [2137, 2137, -9*w - 52], [2153, 2153, 8*w - 51], [2153, 2153, -8*w - 51], [2203, 2203, 19*w - 18], [2203, 2203, -19*w - 18], [2213, 2213, -29*w + 90], [2213, 2213, -29*w - 90], [2221, 2221, -18*w + 67], [2221, 2221, -57*w + 158], [2237, 2237, -22*w + 75], [2237, 2237, -49*w + 138], [2239, 2239, -28*w - 57], [2239, 2239, -28*w + 57], [2243, 2243, 18*w - 5], [2243, 2243, -18*w - 5], [2267, 2267, -18*w - 1], [2267, 2267, 18*w - 1], [2269, 2269, -66*w + 181], [2269, 2269, -15*w + 62], [2287, 2287, -29*w + 60], [2287, 2287, -29*w - 60], [2293, 2293, -27*w + 86], [2293, 2293, -27*w - 86], [2297, 2297, -w - 48], [2297, 2297, w - 48], [2333, 2333, -17*w + 66], [2333, 2333, -62*w + 171], [2351, 2351, -24*w + 41], [2351, 2351, -24*w - 41], [2371, 2371, -31*w - 66], [2371, 2371, -31*w + 66], [2377, 2377, -33*w + 100], [2377, 2377, -33*w - 100], [2381, 2381, -50*w + 141], [2381, 2381, -23*w + 78], [2383, 2383, -19*w - 12], [2383, 2383, 19*w - 12], [2389, 2389, -21*w + 74], [2389, 2389, -54*w + 151], [2399, 2399, -27*w - 52], [2399, 2399, -27*w + 52], [2411, 2411, 21*w - 26], [2411, 2411, -21*w - 26], [2417, 2417, 13*w - 60], [2417, 2417, -13*w - 60], [2437, 2437, -3*w - 50], [2437, 2437, 3*w - 50], [2467, 2467, -34*w - 75], [2467, 2467, -34*w + 75], [2473, 2473, -12*w - 59], [2473, 2473, 12*w - 59], [2521, 2521, 15*w - 64], [2521, 2521, 15*w + 64], [2539, 2539, -38*w - 87], [2539, 2539, -38*w + 87], [2549, 2549, 10*w - 57], [2549, 2549, -10*w - 57], [2551, 2551, 40*w - 93], [2551, 2551, -41*w + 96], [2557, 2557, -6*w - 53], [2557, 2557, 6*w - 53], [2579, 2579, -30*w - 61], [2579, 2579, -30*w + 61], [2633, 2633, -32*w + 99], [2633, 2633, -32*w - 99], [2657, 2657, -64*w + 177], [2657, 2657, -19*w + 72], [2659, 2659, -22*w - 27], [2659, 2659, 22*w - 27], [2663, 2663, -24*w + 37], [2663, 2663, -24*w - 37], [2687, 2687, 21*w - 20], [2687, 2687, -21*w - 20], [2689, 2689, -60*w + 167], [2689, 2689, -21*w + 76], [2707, 2707, -26*w - 45], [2707, 2707, -26*w + 45], [2713, 2713, 12*w - 61], [2713, 2713, -12*w - 61], [2719, 2719, -20*w - 9], [2719, 2719, 20*w - 9], [2741, 2741, -5*w - 54], [2741, 2741, 5*w - 54], [2753, 2753, -11*w - 60], [2753, 2753, 11*w - 60], [2791, 2791, -20*w - 3], [2791, 2791, 20*w - 3], [2797, 2797, -9*w - 58], [2797, 2797, 9*w - 58], [2801, 2801, 8*w - 57], [2801, 2801, -8*w - 57], [2803, 2803, -23*w - 30], [2803, 2803, 23*w - 30], [2819, 2819, -30*w + 59], [2819, 2819, -30*w - 59], [2837, 2837, -26*w + 87], [2837, 2837, -53*w + 150], [2857, 2857, 24*w - 83], [2857, 2857, -24*w - 83], [2887, 2887, -28*w - 51], [2887, 2887, -28*w + 51], [2903, 2903, -39*w - 88], [2903, 2903, -39*w + 88], [2909, 2909, -w - 54], [2909, 2909, w - 54], [2939, 2939, -42*w - 97], [2939, 2939, -42*w + 97], [2969, 2969, 16*w + 69], [2969, 2969, 16*w - 69], [2971, 2971, -29*w - 54], [2971, 2971, -29*w + 54], [2999, 2999, -33*w + 68], [2999, 2999, -33*w - 68], [3023, 3023, -21*w - 8], [3023, 3023, 21*w - 8], [3049, 3049, 15*w - 68], [3049, 3049, 15*w + 68], [3061, 3061, -18*w + 73], [3061, 3061, -75*w + 206], [3079, 3079, 25*w - 36], [3079, 3079, -25*w - 36], [3083, 3083, 21*w - 2], [3083, 3083, -21*w - 2], [3089, 3089, -35*w + 108], [3089, 3089, -35*w - 108], [3109, 3109, -30*w + 97], [3109, 3109, -30*w - 97], [3137, 3137, 4*w - 57], [3137, 3137, -4*w - 57], [3163, 3163, -22*w - 15], [3163, 3163, 22*w - 15], [3167, 3167, -27*w - 44], [3167, 3167, -27*w + 44], [3191, 3191, -24*w - 29], [3191, 3191, 24*w - 29], [3217, 3217, 12*w - 65], [3217, 3217, -12*w - 65], [3221, 3221, 2*w - 57], [3221, 3221, -2*w - 57], [3229, 3229, -6*w - 59], [3229, 3229, 6*w - 59], [3251, 3251, -39*w - 86], [3251, 3251, -39*w + 86], [3257, 3257, 7*w - 60], [3257, 3257, -7*w - 60], [3301, 3301, 3*w - 58], [3301, 3301, -3*w - 58], [3307, 3307, -22*w - 9], [3307, 3307, 22*w - 9], [3313, 3313, -21*w + 80], [3313, 3313, -72*w + 199], [3323, 3323, -42*w + 95], [3323, 3323, -42*w - 95], [3329, 3329, -29*w + 96], [3329, 3329, -56*w + 159], [3331, 3331, -34*w - 69], [3331, 3331, -34*w + 69], [3359, 3359, -45*w + 104], [3359, 3359, -45*w - 104], [3361, 3361, -27*w + 92], [3361, 3361, -60*w + 169], [3389, 3389, -14*w - 69], [3389, 3389, 14*w - 69], [3391, 3391, -35*w + 72], [3391, 3391, -35*w - 72], [3407, 3407, -24*w - 25], [3407, 3407, 24*w - 25], [3413, 3413, -37*w - 114], [3413, 3413, -37*w + 114], [3463, 3463, -28*w - 45], [3463, 3463, -28*w + 45], [3469, 3469, 6*w - 61], [3469, 3469, -6*w - 61], [3491, 3491, -30*w + 53], [3491, 3491, -30*w - 53], [3499, 3499, -37*w + 78], [3499, 3499, -37*w - 78], [3527, 3527, -33*w - 64], [3527, 3527, -33*w + 64], [3529, 3529, 9*w - 64], [3529, 3529, -9*w - 64], [3547, 3547, -38*w + 81], [3547, 3547, -38*w - 81], [3557, 3557, -82*w + 225], [3557, 3557, -19*w + 78], [3559, 3559, -23*w - 12], [3559, 3559, 23*w - 12], [3581, 3581, -38*w + 117], [3581, 3581, -38*w - 117], [3583, 3583, -29*w + 48], [3583, 3583, -29*w - 48], [3593, 3593, -w - 60], [3593, 3593, w - 60], [3613, 3613, -33*w - 106], [3613, 3613, -33*w + 106], [3631, 3631, -40*w + 87], [3631, 3631, -40*w - 87], [3637, 3637, -21*w + 82], [3637, 3637, -78*w + 215], [3643, 3643, 26*w - 33], [3643, 3643, -26*w - 33], [3659, 3659, -27*w - 38], [3659, 3659, 27*w - 38], [3671, 3671, 24*w - 19], [3671, 3671, -24*w - 19], [3677, 3677, -31*w + 102], [3677, 3677, -31*w - 102], [3697, 3697, -36*w - 113], [3697, 3697, -36*w + 113], [3721, 61, -61], [3727, 3727, -43*w - 96], [3727, 3727, -43*w + 96], [3733, 3733, -27*w - 94], [3733, 3733, 27*w - 94], [3761, 3761, -20*w + 81], [3761, 3761, -83*w + 228], [3779, 3779, -33*w - 62], [3779, 3779, -33*w + 62], [3833, 3833, 16*w - 75], [3833, 3833, -16*w - 75], [3863, 3863, 24*w - 13], [3863, 3863, -24*w - 13], [3889, 3889, -75*w + 208], [3889, 3889, -24*w + 89], [3911, 3911, -24*w - 11], [3911, 3911, 24*w - 11], [3917, 3917, -26*w + 93], [3917, 3917, -71*w + 198], [3919, 3919, -32*w - 57], [3919, 3919, -32*w + 57], [3923, 3923, -39*w - 82], [3923, 3923, -39*w + 82], [3929, 3929, -40*w - 123], [3929, 3929, -40*w + 123], [3947, 3947, -27*w - 34], [3947, 3947, 27*w - 34], [3967, 3967, 28*w - 39], [3967, 3967, -28*w - 39], [4001, 4001, 13*w - 72], [4001, 4001, -13*w - 72], [4003, 4003, -26*w - 27], [4003, 4003, 26*w - 27], [4007, 4007, -24*w - 5], [4007, 4007, 24*w - 5], [4013, 4013, -7*w - 66], [4013, 4013, 7*w - 66], [4051, 4051, -25*w - 18], [4051, 4051, 25*w - 18], [4057, 4057, 9*w - 68], [4057, 4057, -9*w - 68], [4079, 4079, -27*w - 32], [4079, 4079, 27*w - 32], [4091, 4091, -30*w + 47], [4091, 4091, -30*w - 47], [4153, 4153, -36*w - 115], [4153, 4153, -36*w + 115], [4201, 4201, 15*w - 76], [4201, 4201, -15*w - 76], [4219, 4219, -35*w + 66], [4219, 4219, -35*w - 66], [4229, 4229, -34*w - 111], [4229, 4229, -34*w + 111], [4231, 4231, 25*w - 12], [4231, 4231, -25*w - 12], [4253, 4253, 14*w - 75], [4253, 4253, -14*w - 75], [4259, 4259, -33*w - 58], [4259, 4259, -33*w + 58], [4283, 4283, -51*w - 118], [4283, 4283, -51*w + 118], [4337, 4337, 11*w - 72], [4337, 4337, -11*w - 72], [4339, 4339, 25*w - 6], [4339, 4339, -25*w - 6], [4349, 4349, -w - 66], [4349, 4349, w - 66], [4397, 4397, -23*w + 90], [4397, 4397, -86*w + 237], [4421, 4421, -35*w + 114], [4421, 4421, -35*w - 114], [4423, 4423, -31*w - 48], [4423, 4423, -31*w + 48], [4451, 4451, -30*w - 43], [4451, 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+ 114], [8231, 8231, -36*w - 29], [8231, 8231, 36*w - 29], [8233, 8233, -57*w + 176], [8233, 8233, -57*w - 176], [8263, 8263, -56*w + 117], [8263, 8263, -56*w - 117], [8269, 8269, -9*w - 94], [8269, 8269, 9*w - 94], [8287, 8287, 37*w - 36], [8287, 8287, -37*w - 36], [8291, 8291, -66*w + 149], [8291, 8291, -66*w - 149], [8297, 8297, -52*w - 165], [8297, 8297, -52*w + 165], [8317, 8317, -63*w + 190], [8317, 8317, -63*w - 190], [8353, 8353, 27*w + 116], [8353, 8353, 27*w - 116], [8363, 8363, -69*w + 158], [8363, 8363, -69*w - 158], [8369, 8369, -11*w - 96], [8369, 8369, 11*w - 96], [8419, 8419, -58*w - 123], [8419, 8419, -58*w + 123], [8429, 8429, 14*w - 99], [8429, 8429, -14*w - 99], [8431, 8431, -35*w - 12], [8431, 8431, 35*w - 12], [8447, 8447, -36*w - 25], [8447, 8447, 36*w - 25], [8513, 8513, -128*w + 351], [8513, 8513, -29*w + 120], [8521, 8521, -48*w - 157], [8521, 8521, -48*w + 157], [8537, 8537, 4*w - 93], [8537, 8537, -4*w - 93], [8539, 8539, -35*w - 6], [8539, 8539, 35*w - 6], [8543, 8543, -36*w - 23], [8543, 8543, 36*w - 23], [8597, 8597, -91*w + 258], [8597, 8597, -46*w + 153], [8599, 8599, -40*w - 51], [8599, 8599, 40*w - 51], [8623, 8623, -61*w + 132], [8623, 8623, -61*w - 132], [8627, 8627, 42*w - 61], [8627, 8627, -42*w - 61], [8677, 8677, -42*w - 145], [8677, 8677, 42*w - 145], [8681, 8681, -103*w + 288], [8681, 8681, -40*w + 141], [8689, 8689, -51*w + 164], [8689, 8689, -51*w - 164], [8699, 8699, -45*w - 74], [8699, 8699, -45*w + 74], [8707, 8707, -49*w - 90], [8707, 8707, -49*w + 90], [8737, 8737, -24*w - 113], [8737, 8737, 24*w - 113], [8761, 8761, -33*w + 128], [8761, 8761, -120*w + 331], [8783, 8783, -36*w - 17], [8783, 8783, 36*w - 17], [8819, 8819, -57*w + 118], [8819, 8819, -57*w - 118], [8821, 8821, 27*w + 118], [8821, 8821, 27*w - 118], [8839, 8839, -65*w - 144], [8839, 8839, -65*w + 144], [8849, 8849, -35*w + 132], [8849, 8849, -116*w + 321], [8867, 8867, -42*w - 59], [8867, 8867, 42*w - 59], [8923, 8923, -67*w - 150], [8923, 8923, -67*w + 150], [8929, 8929, -45*w + 152], [8929, 8929, -96*w + 271], [8933, 8933, 22*w - 111], [8933, 8933, -22*w - 111], [8941, 8941, -57*w - 178], [8941, 8941, -57*w + 178], [8951, 8951, 36*w - 11], [8951, 8951, -36*w - 11], [8963, 8963, -54*w + 107], [8963, 8963, -54*w - 107], [8969, 8969, -41*w + 144], [8969, 8969, -104*w + 291], [9007, 9007, -37*w - 24], [9007, 9007, 37*w - 24], [9013, 9013, -21*w - 110], [9013, 9013, 21*w - 110], [9041, 9041, 5*w - 96], [9041, 9041, -5*w - 96], [9043, 9043, -71*w - 162], [9043, 9043, -71*w + 162], [9091, 9091, -74*w - 171], [9091, 9091, -74*w + 171], [9103, 9103, -76*w + 177], [9103, 9103, -76*w - 177], [9109, 9109, -66*w + 199], [9109, 9109, -66*w - 199], [9127, 9127, -52*w + 99], [9127, 9127, -52*w - 99], [9137, 9137, 19*w - 108], [9137, 9137, -19*w - 108], [9157, 9157, 6*w - 97], [9157, 9157, -6*w - 97], [9181, 9181, -18*w - 107], [9181, 9181, 18*w - 107], [9187, 9187, -46*w + 75], [9187, 9187, -46*w - 75], [9203, 9203, 39*w - 38], [9203, 9203, -39*w - 38], [9209, 9209, -w - 96], [9209, 9209, w - 96], [9221, 9221, 13*w - 102], [9221, 9221, -13*w - 102], [9239, 9239, -48*w - 83], [9239, 9239, -48*w + 83], [9241, 9241, -15*w - 104], [9241, 9241, 15*w - 104], [9277, 9277, -33*w + 130], [9277, 9277, -126*w + 347], [9293, 9293, -23*w - 114], [9293, 9293, 23*w - 114], [9323, 9323, -42*w - 55], [9323, 9323, 42*w - 55], [9343, 9343, 43*w - 60], [9343, 9343, -43*w - 60], [9349, 9349, -51*w + 166], [9349, 9349, -51*w - 166], [9371, 9371, -51*w - 94], [9371, 9371, -51*w + 94], [9377, 9377, -56*w + 177], [9377, 9377, -56*w - 177], [9409, 97, -97], [9433, 9433, 9*w - 100], [9433, 9433, -9*w - 100], [9437, 9437, -94*w + 267], [9437, 9437, -49*w + 162], [9439, 9439, 37*w - 12], [9439, 9439, -37*w - 12], [9461, 9461, -37*w + 138], [9461, 9461, -118*w + 327], [9463, 9463, -41*w - 48], [9463, 9463, 41*w - 48], [9467, 9467, -66*w + 145], [9467, 9467, -66*w - 145], [9473, 9473, -32*w + 129], [9473, 9473, -131*w + 360], [9491, 9491, 39*w - 34], [9491, 9491, -39*w - 34], [9511, 9511, -55*w - 108], [9511, 9511, -55*w + 108], [9521, 9521, -20*w - 111], [9521, 9521, 20*w - 111], [9539, 9539, 42*w - 53], [9539, 9539, -42*w - 53], [9547, 9547, 37*w - 6], [9547, 9547, -37*w - 6], [9551, 9551, -45*w - 68], [9551, 9551, 45*w - 68], [9601, 9601, -12*w - 103], [9601, 9601, 12*w - 103], [9613, 9613, 18*w - 109], [9613, 9613, -18*w - 109], [9623, 9623, -39*w - 32], [9623, 9623, 39*w - 32], [9629, 9629, -65*w + 198], [9629, 9629, -65*w - 198], [9631, 9631, -56*w + 111], [9631, 9631, -56*w - 111], [9661, 9661, 15*w - 106], [9661, 9661, -15*w - 106], [9679, 9679, 40*w - 39], [9679, 9679, -40*w - 39], [9689, 9689, -4*w - 99], [9689, 9689, 4*w - 99], [9697, 9697, -36*w + 137], [9697, 9697, -123*w + 340], [9719, 9719, -72*w - 163], [9719, 9719, -72*w + 163], [9743, 9743, -51*w - 92], [9743, 9743, -51*w + 92], [9769, 9769, -60*w - 187], [9769, 9769, -60*w + 187], [9781, 9781, 27*w - 122], [9781, 9781, 27*w + 122], [9791, 9791, -75*w + 172], [9791, 9791, -75*w - 172], [9803, 9803, -54*w + 103], [9803, 9803, -54*w - 103], [9829, 9829, -30*w + 127], [9829, 9829, -141*w + 386], [9857, 9857, 44*w - 153], [9857, 9857, -44*w - 153], [9859, 9859, -58*w - 117], [9859, 9859, -58*w + 117], [9883, 9883, 38*w - 15], [9883, 9883, -38*w - 15], [9887, 9887, -48*w + 79], [9887, 9887, -48*w - 79], [9931, 9931, -50*w - 87], [9931, 9931, -50*w + 87], [9941, 9941, -58*w + 183], [9941, 9941, -58*w - 183], [9949, 9949, -6*w - 101], [9949, 9949, 6*w - 101], [9967, 9967, -59*w + 120], [9967, 9967, -59*w - 120]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [0, 1, 1, -1, -4, -4, -6, 6, 6, -8, -8, -10, -10, -8, -8, 6, 6, 4, 4, 4, 4, 0, 0, 14, 14, -14, -14, -22, -12, -12, -14, -14, -4, -4, 6, 6, 16, 16, -22, 18, 18, 22, 22, -24, -24, 16, 16, 12, 12, 18, 18, 20, 20, -8, -8, -10, -10, -30, -30, -12, -12, 2, -4, -4, 0, 0, -2, -2, -30, -30, 32, 32, 22, 22, -16, -16, 14, 14, 10, 10, -36, -36, 6, 6, 24, 24, 34, 34, 10, 10, -4, -4, -24, -24, -24, -24, 12, 12, -30, -2, -2, -18, -18, 12, 12, 18, 18, 36, 36, -32, -32, -10, -10, 18, 18, 12, 12, 26, 26, -20, -20, -32, -32, 6, 6, -46, -46, 36, 36, 38, 38, 6, 6, -16, -16, -16, -16, -10, -10, 20, 20, 26, 26, -4, -4, -42, -42, 48, 48, -44, -44, -2, -2, 0, 0, -38, -38, 28, 28, -22, -22, -40, -40, -30, -30, 42, 42, 16, 16, -18, -18, 0, 0, -12, -12, -26, -26, -2, -2, -60, -60, -22, -22, -32, -32, -62, -62, 30, 30, 32, 32, 14, 14, 64, 64, -68, -68, 48, 48, 26, 26, -28, -28, -14, -14, -68, -68, -32, -32, -26, -26, -58, -58, -16, -16, -14, -14, -12, -12, -26, -26, -48, -48, -18, -18, -68, -68, 42, 42, -44, -44, -48, -48, 54, 54, -40, -40, -20, -20, 56, 56, 14, 14, 48, 48, -8, -8, 12, 12, 62, 62, -42, -42, 20, 20, 18, 68, 68, 46, 46, 46, 46, 16, 16, 62, 62, -6, -6, 28, 28, 32, 32, -72, -72, 58, -52, -52, -14, -14, 6, 6, -64, -64, -50, -50, 68, 68, -30, -30, -28, -28, -50, -50, 32, 32, 36, 36, -44, -44, 38, 38, 82, 82, -26, -26, 48, 48, 30, 30, -22, -22, 84, 84, 10, 10, -20, -20, 10, 10, -6, -6, 4, 4, -66, -66, 78, 78, -42, -42, 72, 72, -4, -4, 52, 52, -50, -50, -8, -8, -10, -10, -30, -30, 22, 22, 0, 0, 28, 28, -6, -6, -18, -18, 0, 0, 22, 22, 0, 0, 28, 28, 18, 18, 6, 6, 52, 52, 58, 58, -54, -54, 20, 20, -10, -10, 32, 32, -34, -34, -28, -28, -6, -6, -78, -78, -60, -60, -56, -56, 72, 72, -62, -62, 28, 28, -86, -86, 32, 32, -18, -18, 66, 66, 88, 88, 78, 78, 34, 34, 4, 4, -36, -36, 30, 30, -86, -86, 80, 80, 48, 48, -10, -10, -20, -20, 18, 18, 60, 60, 64, 64, 0, 0, -70, -70, -90, -90, 40, 40, 4, 4, -6, -6, 86, 86, 18, 18, -68, -68, 0, 0, 48, 48, 34, 34, -10, -10, 30, 30, -4, -4, 26, 26, -58, -58, 108, 108, 66, 66, -100, -100, -54, -54, 84, 84, -24, -24, 2, 2, 102, 102, 24, 24, -16, -16, 54, 54, -112, -112, 46, 46, -12, -12, -44, -44, 48, 48, 58, 58, -12, -12, -26, -26, -112, -112, -10, -10, -56, -56, 42, 42, -34, -34, 8, 8, 54, 54, -12, -12, 92, 92, 64, 64, -34, -34, 2, 2, -22, -64, -64, -106, -106, -30, -30, 100, 100, 66, 66, -16, -16, -62, -62, -48, -48, -98, -98, -48, -48, 60, 60, 90, 90, 68, 68, 24, 24, -14, -14, -52, -52, -120, -120, -74, -74, 28, 28, 26, 26, 96, 96, -12, -12, -38, -38, -86, -86, 92, 92, -82, -82, -88, -88, -90, -90, 4, 4, -12, -12, 2, 2, -68, -68, 118, 118, -90, -90, 86, 86, 72, 72, 12, 12, -102, -102, 28, 28, 10, -28, -28, 38, 38, -14, -14, 72, 72, 56, 56, -40, -40, 62, 62, -40, -40, 98, 98, -60, -60, 34, 34, -126, -126, 0, 0, 56, 56, 12, 12, -38, -38, 118, 118, -96, -96, 80, 80, 92, 92, 86, 86, 30, 30, -78, -78, -40, -40, -88, -88, 46, 46, 40, 40, -52, -52, 50, 50, -98, -98, -4, -4, 18, 18, 4, 4, -14, -14, -92, -92, -50, -50, 48, 48, -126, -76, -76, 38, 38, 120, 120, 52, 52, -118, -118, 12, 12, -12, -12, -138, -138, -86, -86, -76, -76, -14, -14, 22, 22, 126, 126, -54, -54, 108, 108, -142, 76, 76, 48, 48, -72, -72, -58, -58, 88, 88, -22, -22, -112, -112, 36, 36, 72, 72, -84, -84, 2, 2, 14, 14, -50, -50, -40, -40, 32, 32, 134, 134, -78, -78, -100, -100, 68, 68, 54, 54, -112, -112, 10, 10, 78, 78, 8, 8, -74, -74, -34, -34, -20, -20, 36, 36, 10, 10, -20, -20, -74, -74, -48, -48, 90, 90, -136, -136, 36, 36, 4, 4, -82, -82, -56, -56, 20, 20, 106, 106, -32, -32, -18, -18, -94, -94, -140, -140, 22, 22, -152, -152, 100, 100, 126, 126, -22, -22, -94, 0, 0, -16, -16, -114, -114, 120, 120, 20, 20, -34, -34, 106, 106, 2, 2, 26, 26, 120, 120, -138, -138, -78, -78, 70, 70, 18, 18, 144, 144, 74, 74, -52, -52, 2, 2, 22, 22, 24, 24, -104, -104, -34, -34, -58, -58, -78, -78, -102, -102, 44, 44, -114, -114, -88, -88, 84, 84, 100, 100, -104, -104, -82, -82, 26, 26, -150, -150, -38, -38, -144, -144, 110, 110, -52, -52, -98, -98, -60, -60, -36, -36, 64, 64, 118, 118, 74, 74, -84, -84, -52, -52, 18, 18, -104, -104, 102, 102, -38, -38, -24, -24, -6, -6, 52, 52, 114, 114, 92, 92, 78, 78, -108, -108, 76, 76, -50, -50, -122, -122, -14, -14, 4, 4, -86, -86, -124, -124, -6, -6, -42, -42, 60, 60, -36, -36, 122, 122, 78, 78, 24, 24, -70, -70, -114, -114, 24, 24, -48, -48, -20, -20, 22, 22, -22, -22, 146, 146, -108, -108, 92, 92, 120, 120, -128, -128, -22, -22, -58, -58, -16, -16, -94, -94, 122, 122, -148, -148, -26, -26, -142, 8, 8, 46, 46, -66, -66, 104, 104, -6, -6, 76, 76, 98, 98, 88, 88, -86, -86, 22, 22, 22, 22, -24, -24, 118, 118, 156, 156, -84, -84, 64, 64, 156, 156, 80, 80, 170, 170, 8, 8, 30, 30, 96, 96, -12, -12, -38, -38, -98, -98, 2, 2, 36, 36, -150, -150, -60, -60, -18, -18, 8, 8, 0, 0, 114, 114, -118, -118, -118, -118, -100, -100, -8, -8, -2, -2, -64, -64, 16, 16, -60, -60, 166, 166, 106, 106, -62, -62, -92, -92, -100, -100, 114, 114, 90, 90, 144, 144, -68, -68, 6, 6, 16, 16, 42, 42, -68, -68, 116, 116, -94, -94, -18, -18, 94, 94, 160, 160, 148, 148, 74, 74, -128, -128, 86, 86, 122, 122, -156, -156, 76, 76, -168, -168, -26, -26, -16, -16, 10, 10, 22, 22, -2, -2, -124, -124, -108, -108, -46, -46, -58, -58, 40, 40, 58, 58, -2, -2, 62, 62, -36, -36, 32, 32, -10, -10, 180, 180, -126, -126, -94, -86, -86, 78, 78, 64, 64, -66, -66, -112, -112, 60, 60, 90, 90, 28, 28, -136, -136, 10, 10, 116, 116, 36, 36, -72, -72, 66, 66, -82, -82, 152, 152, -66, -66, -80, -80, 94, 94, 56, 56, -86, -86, -78, -78, 88, 88, -96, -96, -6, -6, 166, 166, -120, -120, -84, -84, 38, 38, -38, -38, -188, -188, 148, 148, -48, -48, 164, 164, 118, 118, 14, 14, -72, -72]; 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; 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;