/* 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![-8, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, -w - 2], [2, 2, -w + 3], [3, 3, 2*w - 7], [11, 11, 4*w - 13], [17, 17, -2*w + 5], [17, 17, 2*w + 3], [25, 5, 5], [29, 29, -2*w + 3], [29, 29, 2*w + 1], [31, 31, -2*w + 9], [31, 31, 2*w + 7], [37, 37, -4*w - 11], [37, 37, 4*w - 15], [41, 41, -10*w + 33], [41, 41, 6*w - 19], [49, 7, -7], [67, 67, 2*w - 11], [67, 67, -2*w - 9], [83, 83, 4*w + 5], [83, 83, 4*w - 9], [97, 97, 8*w + 21], [97, 97, 8*w - 29], [101, 101, 6*w - 17], [101, 101, 6*w + 11], [103, 103, -6*w - 17], [103, 103, 6*w - 23], [107, 107, 4*w - 7], [107, 107, -4*w - 3], [131, 131, 4*w - 3], [131, 131, 4*w - 1], [149, 149, 10*w + 21], [149, 149, 10*w - 31], [157, 157, 4*w + 15], [157, 157, 4*w - 19], [163, 163, 2*w - 15], [163, 163, -2*w - 13], [167, 167, 8*w - 23], [167, 167, 8*w + 15], [169, 13, -13], [173, 173, 14*w + 31], [173, 173, 14*w - 45], [181, 181, -20*w + 69], [181, 181, -12*w + 43], [197, 197, -6*w - 7], [197, 197, 6*w - 13], [199, 199, 10*w - 37], [199, 199, -10*w - 27], [223, 223, 2*w - 17], [223, 223, -2*w - 15], [227, 227, 12*w + 25], [227, 227, 12*w - 37], [229, 229, -4*w + 21], [229, 229, -4*w - 17], [233, 233, 6*w - 11], [233, 233, -6*w - 5], [239, 239, -8*w + 21], [239, 239, 8*w + 13], [263, 263, 16*w - 51], [263, 263, 16*w + 35], [281, 281, -6*w - 1], [281, 281, 6*w - 7], [293, 293, 6*w - 5], [293, 293, 6*w - 1], [313, 313, -8*w + 33], [313, 313, -8*w - 25], [331, 331, -10*w - 29], [331, 331, 10*w - 39], [347, 347, 12*w - 35], [347, 347, 12*w + 23], [359, 359, -8*w - 9], [359, 359, 8*w - 17], [361, 19, -19], [367, 367, 2*w - 21], [367, 367, -2*w - 19], [379, 379, 6*w + 23], [379, 379, 6*w - 29], [397, 397, 4*w - 25], [397, 397, -4*w - 21], [421, 421, -28*w + 97], [421, 421, -20*w + 71], [431, 431, 16*w + 33], [431, 431, 16*w - 49], [433, 433, 8*w - 35], [433, 433, -8*w - 27], [461, 461, 14*w - 41], [461, 461, 14*w + 27], [463, 463, -18*w + 65], [463, 463, -34*w + 117], [479, 479, -8*w - 3], [479, 479, 8*w - 11], [487, 487, -6*w + 31], [487, 487, -6*w - 25], [491, 491, 36*w - 119], [491, 491, 20*w - 63], [499, 499, -14*w - 39], [499, 499, 14*w - 53], [503, 503, -8*w - 1], [503, 503, 8*w - 9], [529, 23, -23], [557, 557, 18*w + 37], [557, 557, 18*w - 55], [563, 563, -12*w + 31], [563, 563, 12*w + 19], [569, 569, -10*w - 11], [569, 569, 10*w - 21], [577, 577, -32*w + 111], [577, 577, -24*w + 85], [593, 593, 14*w - 39], [593, 593, 14*w + 25], [619, 619, -10*w - 33], [619, 619, 10*w - 43], [631, 631, -38*w + 131], [631, 631, -22*w + 79], [643, 643, 2*w - 27], [643, 643, -2*w - 25], [659, 659, 12*w + 17], [659, 659, 12*w - 29], [661, 661, -12*w + 49], [661, 661, -12*w - 37], [677, 677, 26*w - 83], [677, 677, 26*w + 57], [691, 691, 18*w - 67], [691, 691, -18*w - 49], [701, 701, 30*w + 67], [701, 701, 30*w - 97], [709, 709, -4*w - 27], [709, 709, 4*w - 31], [727, 727, -6*w - 29], [727, 727, 6*w - 35], [743, 743, 16*w + 29], [743, 743, 16*w - 45], [751, 751, 2*w - 29], [751, 751, -2*w - 27], [757, 757, -36*w + 125], [757, 757, -28*w + 99], [761, 761, 10*w - 13], [761, 761, -10*w - 3], [809, 809, 10*w - 9], [809, 809, 10*w - 1], [821, 821, 10*w - 3], [821, 821, 10*w - 7], [823, 823, -26*w + 93], [823, 823, -42*w + 145], [827, 827, -12*w - 13], [827, 827, 12*w - 25], [829, 829, 4*w - 33], [829, 829, -4*w - 29], [857, 857, 22*w + 45], [857, 857, 22*w - 67], [859, 859, 6*w - 37], [859, 859, -6*w - 31], [883, 883, -14*w + 57], [883, 883, -14*w - 43], [887, 887, 16*w + 27], [887, 887, 16*w - 43], [907, 907, 22*w - 81], [907, 907, -22*w - 59], [941, 941, 14*w + 19], [941, 941, 14*w - 33], [953, 953, 26*w + 55], [953, 953, 26*w - 81], [991, 991, 2*w - 33], [991, 991, -2*w - 31], [1019, 1019, 12*w - 19], [1019, 1019, -12*w - 7], [1021, 1021, 12*w - 53], [1021, 1021, -12*w - 41], [1031, 1031, 24*w + 49], [1031, 1031, 24*w - 73], [1039, 1039, -46*w + 159], [1039, 1039, -30*w + 107], [1087, 1087, -14*w + 59], [1087, 1087, -14*w - 45], [1091, 1091, 20*w - 57], [1091, 1091, 20*w + 37], [1093, 1093, 4*w - 37], [1093, 1093, -4*w - 33], [1097, 1097, 42*w - 137], [1097, 1097, 38*w - 123], [1123, 1123, 2*w - 35], [1123, 1123, -2*w - 33], [1151, 1151, 16*w + 23], [1151, 1151, 16*w - 39], [1153, 1153, -8*w - 37], [1153, 1153, 8*w - 45], [1163, 1163, 12*w - 11], [1163, 1163, 12*w - 1], [1171, 1171, -18*w - 53], [1171, 1171, 18*w - 71], [1187, 1187, 12*w - 7], [1187, 1187, 12*w - 5], [1213, 1213, -12*w - 43], [1213, 1213, 12*w - 55], [1217, 1217, 14*w - 27], [1217, 1217, -14*w - 13], [1223, 1223, 56*w - 185], [1223, 1223, 32*w - 101], [1229, 1229, 18*w + 29], [1229, 1229, -18*w + 47], [1237, 1237, -4*w - 35], [1237, 1237, 4*w - 39], [1279, 1279, -34*w + 121], [1279, 1279, -50*w + 173], [1283, 1283, 36*w - 115], [1283, 1283, 36*w + 79], [1289, 1289, 22*w + 41], [1289, 1289, 22*w - 63], [1291, 1291, -10*w + 51], [1291, 1291, -10*w - 41], [1303, 1303, 6*w - 43], [1303, 1303, -6*w - 37], [1319, 1319, 40*w + 89], [1319, 1319, 40*w - 129], [1321, 1321, -8*w - 39], [1321, 1321, 8*w - 47], [1361, 1361, 14*w - 23], [1361, 1361, -14*w - 9], [1423, 1423, -18*w + 73], [1423, 1423, -18*w - 55], [1427, 1427, 28*w + 57], [1427, 1427, 28*w - 85], [1451, 1451, -20*w + 53], [1451, 1451, -20*w - 33], [1453, 1453, 28*w - 103], [1453, 1453, -28*w - 75], [1481, 1481, 26*w - 77], [1481, 1481, 26*w + 51], [1483, 1483, 22*w - 85], [1483, 1483, -22*w - 63], [1487, 1487, -16*w - 17], [1487, 1487, 16*w - 33], [1489, 1489, -24*w - 67], [1489, 1489, 24*w - 91], [1493, 1493, 22*w - 61], [1493, 1493, 22*w + 39], [1543, 1543, -54*w + 187], [1543, 1543, -38*w + 135], [1549, 1549, -4*w - 39], [1549, 1549, 4*w - 43], [1553, 1553, -14*w - 1], [1553, 1553, 14*w - 15], [1559, 1559, 32*w + 67], [1559, 1559, 32*w - 99], [1567, 1567, 2*w - 41], [1567, 1567, -2*w - 39], [1583, 1583, 16*w - 31], [1583, 1583, -16*w - 15], [1601, 1601, 14*w - 11], [1601, 1601, 14*w - 3], [1609, 1609, -16*w - 53], [1609, 1609, 16*w - 69], [1613, 1613, 14*w - 9], [1613, 1613, 14*w - 5], [1619, 1619, -20*w + 51], [1619, 1619, 20*w + 31], [1621, 1621, 12*w + 47], [1621, 1621, -12*w + 59], [1667, 1667, 36*w - 113], [1667, 1667, 68*w - 225], [1699, 1699, -34*w + 123], [1699, 1699, -66*w + 227], [1733, 1733, 26*w - 75], [1733, 1733, 26*w + 49], [1741, 1741, -20*w + 81], [1741, 1741, -20*w - 61], [1747, 1747, 14*w + 51], [1747, 1747, 14*w - 65], [1753, 1753, 32*w + 85], [1753, 1753, 32*w - 117], [1783, 1783, -22*w - 65], [1783, 1783, 22*w - 87], [1811, 1811, 44*w + 97], [1811, 1811, 44*w - 141], [1823, 1823, 16*w - 25], [1823, 1823, -16*w - 9], [1831, 1831, -42*w + 149], [1831, 1831, -58*w + 201], [1847, 1847, 48*w - 155], [1847, 1847, 48*w + 107], [1849, 43, -43], [1873, 1873, -8*w - 45], [1873, 1873, 8*w - 53], [1877, 1877, -22*w + 57], [1877, 1877, 22*w + 35], [1879, 1879, 10*w - 57], [1879, 1879, -10*w - 47], [1889, 1889, -18*w - 19], [1889, 1889, 18*w - 37], [1913, 1913, 38*w - 119], [1913, 1913, 74*w - 245], [1931, 1931, 20*w + 27], [1931, 1931, 20*w - 47], [1949, 1949, 30*w - 89], [1949, 1949, 30*w + 59], [1951, 1951, -18*w - 59], [1951, 1951, 18*w - 77], [1979, 1979, 28*w + 53], [1979, 1979, 28*w - 81], [1997, 1997, 18*w - 35], [1997, 1997, -18*w - 17], [2011, 2011, -38*w + 137], [2011, 2011, -70*w + 241], [2017, 2017, -56*w + 195], [2017, 2017, -48*w + 169], [2029, 2029, -20*w + 83], [2029, 2029, -20*w - 63], [2063, 2063, 16*w - 15], [2063, 2063, 16*w - 1], [2081, 2081, 46*w - 147], [2081, 2081, 46*w + 101], [2083, 2083, 2*w - 47], [2083, 2083, -2*w - 45], [2087, 2087, 16*w - 3], [2087, 2087, 16*w - 13], [2111, 2111, 16*w - 9], [2111, 2111, 16*w - 7], [2113, 2113, -16*w - 57], [2113, 2113, 16*w - 73], [2129, 2129, 50*w - 161], [2129, 2129, 50*w + 111], [2137, 2137, -24*w - 71], [2137, 2137, 24*w - 95], [2141, 2141, 34*w + 69], [2141, 2141, 34*w - 103], [2143, 2143, -62*w + 215], [2143, 2143, -46*w + 163], [2153, 2153, 58*w - 189], [2153, 2153, 54*w - 175], [2161, 2161, 32*w - 119], [2161, 2161, -32*w - 87], [2179, 2179, -30*w - 83], [2179, 2179, 30*w - 113], [2203, 2203, -6*w - 47], [2203, 2203, 6*w - 53], [2207, 2207, 32*w - 95], [2207, 2207, 32*w + 63], [2209, 47, -47], [2213, 2213, 26*w + 45], [2213, 2213, 26*w - 71], [2243, 2243, 28*w + 51], [2243, 2243, 28*w - 79], [2269, 2269, -4*w - 47], [2269, 2269, 4*w - 51], [2273, 2273, -18*w - 11], [2273, 2273, 18*w - 29], [2281, 2281, 8*w - 57], [2281, 2281, -8*w - 49], [2293, 2293, 12*w - 65], [2293, 2293, -12*w - 53], [2309, 2309, 38*w + 79], [2309, 2309, 38*w - 117], [2311, 2311, 10*w - 61], [2311, 2311, -10*w - 51], [2339, 2339, -20*w - 21], [2339, 2339, 20*w - 41], [2341, 2341, -60*w + 209], [2341, 2341, -52*w + 183], [2347, 2347, -74*w + 255], [2347, 2347, -42*w + 151], [2351, 2351, -24*w + 61], [2351, 2351, 24*w + 37], [2377, 2377, 16*w - 75], [2377, 2377, 16*w + 59], [2393, 2393, 22*w - 51], [2393, 2393, 22*w + 29], [2411, 2411, 36*w + 73], [2411, 2411, 36*w - 109], [2417, 2417, 18*w - 25], [2417, 2417, -18*w - 7], [2441, 2441, -26*w + 69], [2441, 2441, 26*w + 43], [2459, 2459, 20*w - 39], [2459, 2459, -20*w - 19], [2467, 2467, 2*w - 51], [2467, 2467, -2*w - 49], [2473, 2473, -24*w + 97], [2473, 2473, -24*w - 73], [2477, 2477, 18*w - 23], [2477, 2477, -18*w - 5], [2539, 2539, -10*w - 53], [2539, 2539, 10*w - 63], [2543, 2543, 24*w + 35], [2543, 2543, 24*w - 59], [2549, 2549, 22*w + 27], [2549, 2549, 22*w - 49], [2557, 2557, 28*w - 109], [2557, 2557, -28*w - 81], [2591, 2591, 40*w - 123], [2591, 2591, 40*w + 83], [2609, 2609, 18*w - 17], [2609, 2609, 18*w - 1], [2657, 2657, 18*w - 5], [2657, 2657, 18*w - 13], [2671, 2671, 2*w - 53], [2671, 2671, -2*w - 51], [2677, 2677, -4*w - 51], [2677, 2677, 4*w - 55], [2689, 2689, -64*w + 223], [2689, 2689, -56*w + 197], [2707, 2707, -78*w + 269], [2707, 2707, -46*w + 165], [2731, 2731, 22*w - 93], [2731, 2731, -22*w - 71], [2741, 2741, 54*w + 119], [2741, 2741, 54*w - 173], [2789, 2789, 58*w + 129], [2789, 2789, 58*w - 187], [2797, 2797, -84*w + 289], [2797, 2797, -44*w + 159], [2801, 2801, 30*w - 83], [2801, 2801, 30*w + 53], [2803, 2803, 18*w - 83], [2803, 2803, 18*w + 65], [2809, 53, -53], [2837, 2837, -22*w - 23], [2837, 2837, 22*w - 45], [2879, 2879, 88*w - 291], [2879, 2879, 48*w - 151], [2887, 2887, -26*w + 105], [2887, 2887, -26*w - 79], [2903, 2903, 24*w + 31], [2903, 2903, 24*w - 55], [2939, 2939, -20*w - 9], [2939, 2939, 20*w - 29], [2953, 2953, -8*w - 55], [2953, 2953, 8*w - 63], [2969, 2969, 22*w - 43], [2969, 2969, -22*w - 21], [2971, 2971, -38*w - 103], [2971, 2971, 38*w - 141], [2999, 2999, 40*w + 81], [2999, 2999, 40*w - 121], [3001, 3001, -32*w - 91], [3001, 3001, 32*w - 123], [3011, 3011, -20*w - 7], [3011, 3011, 20*w - 27], [3019, 3019, 10*w - 67], [3019, 3019, -10*w - 57], [3037, 3037, -12*w - 59], [3037, 3037, 12*w - 71], [3061, 3061, -68*w + 237], [3061, 3061, -60*w + 211], [3067, 3067, 6*w - 61], [3067, 3067, -6*w - 55], [3119, 3119, 32*w + 57], [3119, 3119, 32*w - 89], [3137, 3137, 34*w - 97], [3137, 3137, 34*w + 63], [3167, 3167, 64*w - 207], [3167, 3167, 72*w - 235], [3169, 3169, -24*w + 101], [3169, 3169, -24*w - 77], [3203, 3203, 44*w + 91], [3203, 3203, 44*w - 135], [3209, 3209, 22*w - 39], [3209, 3209, -22*w - 17], [3217, 3217, -16*w + 81], [3217, 3217, -16*w - 65], [3251, 3251, 20*w - 17], [3251, 3251, 20*w - 3], [3259, 3259, -26*w + 107], [3259, 3259, -26*w - 81], [3271, 3271, -10*w - 59], [3271, 3271, 10*w - 69], [3299, 3299, 20*w - 11], [3299, 3299, 20*w - 9], [3301, 3301, 12*w - 73], [3301, 3301, -12*w - 61], [3329, 3329, -30*w + 79], [3329, 3329, 30*w + 49], [3331, 3331, 2*w - 59], [3331, 3331, -2*w - 57], [3391, 3391, -30*w - 89], [3391, 3391, 30*w - 119], [3407, 3407, 32*w + 55], [3407, 3407, 32*w - 87], [3433, 3433, -32*w - 93], [3433, 3433, 32*w - 125], [3449, 3449, 38*w - 111], [3449, 3449, 38*w + 73], [3457, 3457, -72*w + 251], [3457, 3457, -64*w + 225], [3461, 3461, 26*w - 59], [3461, 3461, 26*w + 33], [3463, 3463, -38*w - 105], [3463, 3463, 38*w - 143], [3467, 3467, 36*w + 67], [3467, 3467, 36*w - 103], [3469, 3469, 36*w - 137], [3469, 3469, -36*w - 101], [3481, 59, -59], [3499, 3499, -54*w + 193], [3499, 3499, -86*w + 297], [3527, 3527, 24*w - 47], [3527, 3527, -24*w - 23], [3529, 3529, 24*w - 103], [3529, 3529, -24*w - 79], [3533, 3533, 46*w - 141], [3533, 3533, 46*w + 95], [3539, 3539, 100*w - 331], [3539, 3539, 52*w - 163], [3547, 3547, -6*w - 59], [3547, 3547, 6*w - 65], [3581, 3581, -30*w + 77], [3581, 3581, 30*w + 47], [3593, 3593, -22*w - 9], [3593, 3593, 22*w - 31], [3613, 3613, -92*w + 317], [3613, 3613, -52*w + 187], [3631, 3631, -78*w + 271], [3631, 3631, -62*w + 219], [3659, 3659, 28*w - 67], [3659, 3659, 28*w + 39], [3671, 3671, 96*w - 317], [3671, 3671, 56*w - 177], [3697, 3697, -8*w - 61], [3697, 3697, 8*w - 69], [3721, 61, -61], [3727, 3727, 18*w + 71], [3727, 3727, -18*w + 89], [3733, 3733, -28*w - 87], [3733, 3733, -28*w + 115], [3761, 3761, 34*w + 59], [3761, 3761, 34*w - 93], [3779, 3779, 60*w + 131], [3779, 3779, 60*w - 191], [3793, 3793, -104*w + 357], [3793, 3793, -48*w + 175], [3797, 3797, 22*w - 25], [3797, 3797, -22*w - 3], [3803, 3803, 36*w + 65], [3803, 3803, 36*w - 101], [3853, 3853, 12*w - 77], [3853, 3853, -12*w - 65], [3863, 3863, 64*w + 141], [3863, 3863, 64*w - 205], [3877, 3877, -76*w + 265], [3877, 3877, -68*w + 239], [3911, 3911, 24*w - 41], [3911, 3911, -24*w - 17], [3919, 3919, 34*w - 133], [3919, 3919, -34*w - 99], [3923, 3923, 68*w - 219], [3923, 3923, 68*w + 151], [3929, 3929, 22*w - 3], [3929, 3929, 22*w - 19], [3931, 3931, -90*w + 311], [3931, 3931, -58*w + 207], [3943, 3943, 42*w - 157], [3943, 3943, -42*w - 115], [3989, 3989, 22*w - 13], [3989, 3989, 22*w - 9], [4001, 4001, 46*w + 93], [4001, 4001, 46*w - 139], [4027, 4027, -26*w - 85], [4027, 4027, 26*w - 111], [4051, 4051, -18*w + 91], [4051, 4051, -18*w - 73], [4057, 4057, -56*w + 201], [4057, 4057, -96*w + 331], [4091, 4091, 44*w - 131], [4091, 4091, 44*w + 87], [4093, 4093, -4*w - 63], [4093, 4093, 4*w - 67], [4127, 4127, 24*w - 37], [4127, 4127, -24*w - 13], [4129, 4129, 16*w - 87], [4129, 4129, -16*w - 71], [4133, 4133, 26*w - 51], [4133, 4133, -26*w - 25], [4157, 4157, -98*w + 323], [4157, 4157, 62*w - 197], [4159, 4159, -14*w - 69], [4159, 4159, 14*w - 83], [4241, 4241, 50*w + 103], [4241, 4241, 50*w - 153], [4253, 4253, 66*w - 211], [4253, 4253, 66*w + 145], [4259, 4259, 28*w - 61], [4259, 4259, -28*w - 33], [4261, 4261, -108*w + 371], [4261, 4261, -52*w + 189], [4273, 4273, 24*w - 107], [4273, 4273, -24*w - 83], [4289, 4289, 30*w + 41], [4289, 4289, 30*w - 71], [4327, 4327, -6*w - 65], [4327, 4327, 6*w - 71], [4339, 4339, 50*w + 133], [4339, 4339, 50*w - 183], [4357, 4357, 4*w - 69], [4357, 4357, -4*w - 65], [4373, 4373, 74*w - 239], [4373, 4373, 74*w + 165], [4391, 4391, -24*w - 7], [4391, 4391, 24*w - 31], [4397, 4397, 82*w - 267], [4397, 4397, 78*w - 253], [4421, 4421, -26*w - 21], [4421, 4421, 26*w - 47], [4423, 4423, 26*w - 113], [4423, 4423, -26*w - 87], [4447, 4447, 46*w 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8233, -64*w - 173], [8233, 8233, 64*w - 237], [8287, 8287, -62*w - 169], [8287, 8287, 62*w - 231], [8291, 8291, -36*w - 31], [8291, 8291, 36*w - 67], [8317, 8317, -52*w - 149], [8317, 8317, 52*w - 201], [8353, 8353, 56*w - 213], [8353, 8353, -56*w - 157], [8419, 8419, -34*w - 117], [8419, 8419, 34*w - 151], [8423, 8423, 32*w - 21], [8423, 8423, 32*w - 11], [8431, 8431, 2*w - 93], [8431, 8431, -2*w - 91], [8447, 8447, 32*w - 17], [8447, 8447, 32*w - 15], [8513, 8513, 34*w - 49], [8513, 8513, -34*w - 15], [8539, 8539, 6*w - 97], [8539, 8539, -6*w - 91], [8543, 8543, 72*w + 149], [8543, 8543, 72*w - 221], [8563, 8563, -18*w - 97], [8563, 8563, 18*w - 115], [8581, 8581, -20*w - 99], [8581, 8581, 20*w - 119], [8597, 8597, 118*w - 385], [8597, 8597, 106*w - 343], [8609, 8609, 62*w - 183], [8609, 8609, 62*w + 121], [8629, 8629, -36*w - 121], [8629, 8629, 36*w - 157], [8647, 8647, -154*w + 529], [8647, 8647, -74*w + 269], [8663, 8663, 88*w - 279], [8663, 8663, -144*w + 475], [8677, 8677, -44*w + 179], [8677, 8677, -44*w - 135], [8681, 8681, 54*w - 151], [8681, 8681, 54*w + 97], [8713, 8713, -32*w - 115], [8713, 8713, 32*w - 147], [8737, 8737, -112*w + 391], [8737, 8737, -104*w + 365], [8741, 8741, 70*w - 213], [8741, 8741, 70*w + 143], [8747, 8747, 44*w + 63], [8747, 8747, 44*w - 107], [8753, 8753, 34*w - 45], [8753, 8753, -34*w - 11], [8761, 8761, -72*w + 263], [8761, 8761, -160*w + 549], [8779, 8779, -10*w - 93], [8779, 8779, 10*w - 103], [8803, 8803, 2*w - 95], [8803, 8803, -2*w - 93], [8807, 8807, 48*w + 77], [8807, 8807, -48*w + 125], [8819, 8819, 92*w - 293], [8819, 8819, 92*w + 201], [8861, 8861, 34*w - 43], [8861, 8861, -34*w - 9], [8893, 8893, 4*w - 97], [8893, 8893, -4*w - 93], [8941, 8941, -68*w - 183], [8941, 8941, 68*w - 251], [8951, 8951, 96*w - 307], [8951, 8951, 96*w + 211], [9001, 9001, -40*w + 169], [9001, 9001, -40*w - 129], [9007, 9007, -66*w - 179], [9007, 9007, 66*w - 245], [9011, 9011, -36*w - 23], [9011, 9011, 36*w - 59], [9013, 9013, -12*w - 95], [9013, 9013, 12*w - 107], [9041, 9041, 66*w - 197], [9041, 9041, 66*w + 131], [9043, 9043, -146*w + 503], [9043, 9043, -82*w + 295], [9059, 9059, 100*w - 321], [9059, 9059, 100*w + 221], [9067, 9067, 54*w - 209], [9067, 9067, -54*w - 155], [9091, 9091, -62*w - 171], [9091, 9091, 62*w - 233], [9109, 9109, 60*w - 227], [9109, 9109, -60*w - 167], [9133, 9133, -132*w + 457], [9133, 9133, -92*w + 327], [9137, 9137, 34*w - 37], [9137, 9137, -34*w - 3], [9157, 9157, 28*w + 111], [9157, 9157, -28*w + 139], [9173, 9173, 54*w - 149], [9173, 9173, 54*w + 95], [9199, 9199, -14*w - 97], [9199, 9199, 14*w - 111], [9203, 9203, 108*w + 241], [9203, 9203, 108*w - 349], [9209, 9209, -38*w - 33], [9209, 9209, 38*w - 71], [9239, 9239, 112*w - 363], [9239, 9239, 120*w - 391], [9241, 9241, 32*w + 117], [9241, 9241, 32*w - 149], [9257, 9257, 58*w - 165], [9257, 9257, 58*w + 107], [9277, 9277, -4*w - 95], [9277, 9277, 4*w - 99], [9281, 9281, 34*w - 33], [9281, 9281, 34*w - 1], [9323, 9323, 36*w - 55], [9323, 9323, -36*w - 19], [9337, 9337, -16*w - 99], [9337, 9337, 16*w - 115], [9341, 9341, 34*w - 3], [9341, 9341, 34*w - 31], [9343, 9343, -158*w + 543], [9343, 9343, -78*w + 283], [9371, 9371, 164*w - 543], [9371, 9371, 84*w - 263], [9397, 9397, -116*w + 405], [9397, 9397, -108*w + 379], [9403, 9403, 38*w - 165], [9403, 9403, -38*w - 127], [9413, 9413, 38*w - 69], [9413, 9413, -38*w - 31], [9421, 9421, 12*w - 109], [9421, 9421, -12*w - 97], [9437, 9437, 34*w - 27], [9437, 9437, 34*w - 7], [9439, 9439, -46*w - 141], [9439, 9439, -46*w + 187], [9463, 9463, -22*w - 105], [9463, 9463, 22*w - 127], [9467, 9467, -36*w - 17], [9467, 9467, 36*w - 53], [9473, 9473, 34*w - 9], [9473, 9473, 34*w - 25], [9479, 9479, -40*w - 41], [9479, 9479, 40*w - 81], [9521, 9521, 34*w - 13], [9521, 9521, 34*w - 21], [9533, 9533, 34*w - 15], [9533, 9533, 34*w - 19], [9539, 9539, 52*w + 87], [9539, 9539, 52*w - 139], [9587, 9587, 68*w - 203], [9587, 9587, 68*w + 135], [9601, 9601, 40*w - 171], [9601, 9601, -40*w - 131], [9619, 9619, 14*w - 113], [9619, 9619, -14*w - 99], [9661, 9661, -28*w + 141], [9661, 9661, -28*w - 113], [9677, 9677, 82*w - 255], [9677, 9677, 82*w + 173], [9719, 9719, 40*w - 79], [9719, 9719, -40*w - 39], [9733, 9733, 52*w - 205], [9733, 9733, -52*w - 153], [9739, 9739, -86*w + 309], [9739, 9739, -150*w + 517], [9743, 9743, 56*w - 155], [9743, 9743, 56*w + 99], [9767, 9767, 64*w - 187], [9767, 9767, 64*w + 123], [9769, 9769, -16*w - 101], [9769, 9769, 16*w - 117], [9803, 9803, 156*w - 515], [9803, 9803, 92*w - 291], [9817, 9817, -96*w + 341], [9817, 9817, -136*w + 471], [9833, 9833, 74*w + 151], [9833, 9833, 74*w - 225], [9851, 9851, 36*w - 47], [9851, 9851, -36*w - 11], [9859, 9859, 66*w - 247], [9859, 9859, -66*w - 181], [9871, 9871, 18*w - 121], [9871, 9871, -18*w - 103], [9883, 9883, -58*w - 165], [9883, 9883, 58*w - 223], [9901, 9901, -60*w - 169], [9901, 9901, 60*w - 229], [9929, 9929, -42*w - 47], [9929, 9929, 42*w - 89], [9931, 9931, 22*w - 129], [9931, 9931, -22*w - 107], [9941, 9941, 86*w - 269], [9941, 9941, 170*w - 563], [9949, 9949, -44*w + 183], [9949, 9949, -44*w - 139], [9967, 9967, 2*w - 101], [9967, 9967, -2*w - 99]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [-1, -1, 2, 0, 5, 5, -9, -9, -9, -2, -2, -3, -3, 5, 5, -10, 2, 2, -6, -6, -13, -13, 10, 10, 8, 8, -6, -6, 0, 0, -17, -17, 2, 2, -2, -2, -12, -12, -25, -6, -6, 1, 1, 11, 11, 24, 24, -20, -20, 24, 24, 9, 9, 21, 21, -6, -6, 22, 22, -6, -6, -9, -9, 23, 23, -20, -20, -28, -28, 2, 2, -2, -14, -14, -32, -32, 13, 13, 13, 13, -12, -12, 19, 19, -33, -33, -20, -20, 16, 16, 2, 2, 2, 2, 8, 8, 38, 38, -42, 2, 2, -34, -34, -6, -6, -21, -21, -11, -11, 2, 2, -14, -14, -10, -10, -22, -22, 13, 13, 27, 27, 20, 20, -17, -17, -10, -10, -42, -42, 38, 38, -20, -20, 53, 53, 21, 21, -6, -6, 2, 2, -24, -24, 10, 10, -47, -47, 22, 22, 24, 24, -20, -20, 46, 46, 12, 12, 27, 27, -31, -31, -20, -20, 46, 46, -2, -2, -50, -50, -10, -10, -46, -46, 10, 10, 30, 30, -39, -39, 24, 24, 46, 46, -46, -46, 38, 38, 12, 12, -66, -66, 46, 46, 57, 57, -56, -56, -6, -6, 45, 45, 24, 24, -42, -42, 21, 21, -14, -14, 12, 12, 66, 66, 35, 35, -6, -6, -58, -58, 60, 60, -22, -22, 13, 13, 13, 13, 64, 64, -34, -34, 30, 30, 27, 27, 24, 24, 9, 9, -34, -34, -6, -6, -10, -10, 44, 44, 49, 49, 35, 35, -9, -9, 32, 32, -47, -47, -6, -6, 34, 34, -6, -6, -31, -31, -24, -24, 19, 19, -64, -64, -42, -42, -28, -28, -76, -76, 22, 22, -86, -42, -42, -42, -42, -2, -2, 5, 5, 33, 33, -72, -72, 10, 10, -14, -14, 0, 0, 27, 27, -46, -46, -14, -14, 45, 45, -6, -6, 21, 21, 74, 74, -72, -72, 44, 44, 46, 46, -83, -83, 79, 79, 79, 79, 20, 20, 5, 5, -10, -10, 24, 24, 68, 68, 24, 24, -90, -1, -1, -22, -22, 13, 13, 2, 2, 19, 19, -43, -43, -33, -33, 68, 68, -64, -64, -35, -35, 74, 74, -72, -72, -42, -42, -39, -39, 76, 76, -83, -83, 66, 66, 38, 38, -42, -42, 31, 31, -45, -45, -24, -24, -34, -34, 71, 71, 1, 1, 82, 82, 21, 21, -50, -50, 42, 42, 41, 41, -10, -10, 46, 46, 46, 46, -89, -89, 38, 38, -75, -75, -31, -31, -46, -46, -25, 11, 11, -50, -50, 78, 78, -66, -66, 32, 32, -65, -65, -22, -22, -20, -20, -64, -64, 75, 75, 16, 16, -98, -98, 46, 46, 46, 46, -2, -2, 60, 60, 10, 10, 0, 0, -42, -42, -12, -12, 82, 82, 23, 23, -6, -6, 24, 24, 30, 30, -22, -22, -31, -31, 46, 46, 20, 20, 2, 2, -72, -72, -9, -9, 5, 5, -42, -42, 35, 35, -24, -24, 10, 10, 41, 41, -54, 46, 46, -42, -42, -101, -101, 87, 87, 16, 16, -76, -76, -17, -17, -31, -31, 1, 1, 2, 2, -42, -42, -6, -6, -97, -97, -86, 86, 86, 30, 30, -11, -11, -72, -72, -2, -2, -89, -89, -50, -50, -31, -31, -78, -78, 1, 1, 60, 60, 24, 24, 24, 24, 21, 21, -58, -58, 78, 78, 90, 90, 93, 93, -20, -20, -42, -42, -101, -101, 88, 88, -119, -119, 54, 54, -25, -25, -61, -61, -66, -66, -20, -20, 49, 49, 79, 79, 32, 32, 85, 85, -65, -65, 33, 33, -80, -80, -32, -32, -42, -42, -105, -105, -12, -12, -105, -105, -33, -33, -86, -86, 2, 2, 24, 24, -111, -111, -6, -6, 35, 35, -53, -53, 86, 86, 120, 120, 24, 24, 118, 118, -76, -76, 101, 101, 27, 27, -31, -31, -46, -46, 19, 19, -6, -6, -67, -67, -36, -36, 66, 66, -2, -2, -56, -56, 74, 74, -6, -6, 23, 23, 33, 33, -119, -119, -12, -12, -98, -98, 90, 90, -50, -50, 52, 52, -32, -32, 2, 54, 54, 66, 66, 82, 82, 112, 112, -13, -13, -102, -102, 0, 0, -90, -90, -61, -61, -10, -10, -6, -6, -110, -110, -141, -141, -50, -50, -97, -97, -142, 68, 68, -78, -78, 104, 104, -31, -31, 145, 145, -75, -75, 64, 64, -113, -113, 55, 55, -20, -20, -86, -86, 24, 24, -6, -6, 122, 122, 35, 35, -86, -86, -14, -14, -42, -42, -13, -13, -58, -58, 126, 126, 100, 100, -61, -61, -31, -31, 120, 120, 27, 27, -22, -22, -86, -86, 118, 118, 16, 16, -10, -10, 110, 110, -68, -68, 120, 120, 82, 82, -65, -65, 24, 24, 122, 122, 44, 44, 27, 27, 134, 134, 42, 42, 52, 52, -72, -72, 46, 46, 49, 49, 90, 90, -83, -83, 67, 67, -64, -64, 131, 131, -154, -154, -94, -94, 101, 101, -58, -66, -66, -20, -20, -138, -138, 90, 90, -35, -35, -6, -6, 35, 35, 5, 5, 123, 123, -90, -90, 85, 85, 2, 2, 100, 100, 101, 101, -28, -28, 68, 68, -67, -67, 8, 8, -22, -22, 85, 85, -20, -20, 87, 87, -58, -58, -31, -31, -86, -86, 20, 20, 133, 133, -20, -20, -42, -42, 86, 86, 153, 153, 110, 110, -12, -12, 60, 60, 90, 90, -13, -13, 74, 74, 104, 104, 101, 101, -6, -6, 152, 152, -138, -138, 60, 60, 66, 66, 35, 35, 42, 42, 122, 122, -99, -99, -138, -138, -86, -86, -1, -1, 144, 144, 63, 63, -43, -43, -108, -108, -141, -141, -53, -53, -6, -6, -143, -143, 90, 90, -108, -108, -2, -2, 126, 126, 12, 12, -132, -132, 38, 38, 57, 57, 120, 120, 30, 30, 133, 133, 143, 143, -108, -108, 148, 148, -35, -35, -32, -32, -6, -6, 35, 35, -68, -68, -144, -144, -130, -130, -2, -2, 131, 131, 52, 52, -75, -75, -31, -31, -56, -56, 27, 27, -33, -33, -108, -108, 24, 24, -44, -44, -97, 93, 93, -9, -9, -24, -24, 2, 2, -101, -101, 134, 134, 115, 115, 90, 90, -78, -78, -69, -69, 71, 71, -43, -43, -33, -33, 46, 46, 12, 12, 46, 46, -144, -144, 30, 30, -98, -98, 30, 30, -138, -138, -75, -75, 107, 107, 30, 30, 126, 126, -10, -10, -22, -22, -66, -66, 112, 112, 90, 90, -142, -142, -42, -42, 27, 27, -130, -130, 1, 1, 2, 2, 170, 170, -79, -79, -111, -111, -97, -97, -53, -53, -141, -141, -100, -100, -171, -171, -109, -109, -20, -20, -20, -20, 90, 90, -116, -116, 126, 126, 89, 89, 53, 53, -160, -160, 79, 79, 108, 108, -34, -34, -91, -91, -66, -66, 24, 24, -160, -160, 24, 24, -98, -98, -86, -86, 46, 46, 57, 57, 177, 177, 66, 66, 134, 134, 112, 112, 21, 21, 0, 0, 35, 35, 137, 137, -179, -179, -171, -171, 16, 16, 86, 86, -45, -45, -58, -58, 0, 0, -130, -130, 152, 152, 82, 82, 45, 45, -77, -77, 134, 134, -86, -86, 112, 112, 21, 21, 192, 192, -39, -39, -9, -9, 120, 120, -72, -72, 31, 31, 100, 100, -163, -163, -105, -105, -50, -50, -35, -35, -124, -124, 82, 82, -66, -66, -86, -86, -12, -12, 78, 78, 154, 154, -6, -6, -64, -64, 118, 118, 78, 78, -42, -42, -42, -42, 42, 42, 27, 27, -175, -175, 112, 112]; 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;