/* 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![-14, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, -w + 4], [2, 2, -w - 3], [3, 3, -4*w - 13], [7, 7, -2*w - 7], [7, 7, -2*w + 9], [19, 19, 10*w + 33], [25, 5, 5], [29, 29, -6*w - 19], [29, 29, -6*w + 25], [41, 41, 2*w - 5], [41, 41, -2*w - 3], [43, 43, 2*w - 11], [43, 43, 2*w + 9], [53, 53, 2*w - 3], [53, 53, -2*w - 1], [59, 59, 4*w - 15], [59, 59, 4*w + 11], [61, 61, -4*w - 15], [61, 61, -4*w + 19], [71, 71, 8*w + 25], [71, 71, 8*w - 33], [73, 73, -16*w - 53], [73, 73, 24*w + 79], [89, 89, 14*w + 45], [89, 89, 14*w - 59], [107, 107, -4*w - 9], [107, 107, 4*w - 13], [113, 113, 6*w - 23], [113, 113, 6*w + 17], [121, 11, -11], [139, 139, 2*w - 15], [139, 139, -2*w - 13], [157, 157, 12*w - 53], [157, 157, 12*w + 41], [163, 163, -6*w - 23], [163, 163, -6*w + 29], [167, 167, 56*w + 183], [167, 167, 16*w + 51], [169, 13, -13], [173, 173, 22*w + 71], [173, 173, 42*w + 137], [179, 179, -4*w - 5], [179, 179, 4*w - 9], [199, 199, 2*w - 17], [199, 199, -2*w - 15], [227, 227, 4*w - 3], [227, 227, 4*w - 1], [229, 229, 20*w - 87], [229, 229, -20*w - 67], [257, 257, -6*w - 13], [257, 257, 6*w - 19], [269, 269, 10*w - 39], [269, 269, 10*w + 29], [271, 271, -6*w + 31], [271, 271, -6*w - 25], [277, 277, -28*w - 93], [277, 277, 52*w + 171], [281, 281, 30*w + 97], [281, 281, 50*w + 163], [283, 283, 18*w + 61], [283, 283, 18*w - 79], [289, 17, -17], [293, 293, 14*w + 43], [293, 293, 14*w - 57], [313, 313, -8*w - 31], [313, 313, -8*w + 39], [317, 317, 6*w - 17], [317, 317, -6*w - 11], [349, 349, -12*w - 43], [349, 349, 12*w - 55], [367, 367, 26*w - 113], [367, 367, -26*w - 87], [383, 383, -8*w - 19], [383, 383, 8*w - 27], [397, 397, -4*w - 23], [397, 397, 4*w - 27], [401, 401, 10*w + 27], [401, 401, 10*w - 37], [431, 431, 32*w - 135], [431, 431, 32*w + 103], [449, 449, 6*w - 11], [449, 449, -6*w - 5], [457, 457, -8*w + 41], [457, 457, -8*w - 33], [463, 463, 58*w + 191], [463, 463, -42*w - 139], [499, 499, 22*w - 97], [499, 499, 22*w + 75], [509, 509, 6*w - 5], [509, 509, 6*w - 1], [521, 521, 18*w + 55], [521, 521, 18*w - 73], [529, 23, -23], [541, 541, 20*w - 89], [541, 541, 20*w + 69], [563, 563, 28*w - 117], [563, 563, 28*w + 89], [569, 569, 46*w + 149], [569, 569, 66*w + 215], [571, 571, -14*w - 51], [571, 571, 14*w - 65], [577, 577, -16*w - 57], [577, 577, 16*w - 73], [599, 599, 40*w + 129], [599, 599, 80*w + 261], [613, 613, -4*w - 27], [613, 613, 4*w - 31], [619, 619, 2*w - 27], [619, 619, -2*w - 25], [631, 631, 30*w - 131], [631, 631, 30*w + 101], [641, 641, 10*w - 33], [641, 641, -10*w - 23], [643, 643, 6*w - 37], [643, 643, -6*w - 31], [659, 659, 20*w + 61], [659, 659, 20*w - 81], [677, 677, 14*w - 53], [677, 677, 14*w + 39], [683, 683, 12*w - 43], [683, 683, 12*w + 31], [691, 691, -10*w + 51], [691, 691, -10*w - 41], [709, 709, 28*w + 95], [709, 709, 28*w - 123], [727, 727, 2*w - 29], [727, 727, -2*w - 27], [733, 733, 4*w - 33], [733, 733, -4*w - 29], [739, 739, 38*w - 165], [739, 739, -38*w - 127], [743, 743, -8*w - 9], [743, 743, 8*w - 17], [757, 757, -12*w - 47], [757, 757, -12*w + 59], [769, 769, -8*w - 37], [769, 769, 8*w - 45], [773, 773, 18*w - 71], [773, 773, 18*w + 53], [797, 797, 26*w + 81], [797, 797, 26*w - 107], [823, 823, -46*w - 153], [823, 823, 94*w + 309], [827, 827, 12*w + 29], [827, 827, 12*w - 41], [839, 839, 16*w + 45], [839, 839, 16*w - 61], [853, 853, 36*w + 121], [853, 853, 36*w - 157], [857, 857, 14*w - 51], [857, 857, 14*w + 37], [859, 859, 18*w - 83], [859, 859, -18*w - 65], [863, 863, -8*w - 3], [863, 863, 8*w - 11], [883, 883, -54*w - 179], [883, 883, 86*w + 283], [887, 887, -8*w - 1], [887, 887, 8*w - 9], [911, 911, 8*w - 5], [911, 911, 8*w - 3], [919, 919, -62*w - 205], [919, 919, -78*w - 257], [937, 937, -8*w - 39], [937, 937, 8*w - 47], [941, 941, -10*w - 17], [941, 941, 10*w - 27], [953, 953, 62*w + 201], [953, 953, 82*w + 267], [961, 31, -31], [967, 967, 2*w - 33], [967, 967, -2*w - 31], [971, 971, 28*w - 115], [971, 971, 28*w + 87], [977, 977, 38*w + 121], [977, 977, 38*w - 159], [983, 983, 24*w + 73], [983, 983, 24*w - 97], [997, 997, 4*w - 37], [997, 997, -4*w - 33], [1019, 1019, 44*w + 141], [1019, 1019, 44*w - 185], [1033, 1033, 32*w - 141], [1033, 1033, 32*w + 109], [1051, 1051, -14*w - 55], [1051, 1051, -14*w + 69], [1069, 1069, -52*w - 173], [1069, 1069, 108*w + 355], [1087, 1087, 6*w - 43], [1087, 1087, -6*w - 37], [1091, 1091, -12*w - 25], [1091, 1091, 12*w - 37], [1097, 1097, 34*w + 107], [1097, 1097, 34*w - 141], [1181, 1181, 70*w + 227], [1181, 1181, 90*w + 293], [1193, 1193, -14*w - 33], [1193, 1193, 14*w - 47], [1201, 1201, -24*w - 85], [1201, 1201, 24*w - 109], [1213, 1213, -76*w - 251], [1213, 1213, -84*w - 277], [1229, 1229, 10*w - 19], [1229, 1229, -10*w - 9], [1279, 1279, 10*w - 57], [1279, 1279, -10*w - 47], [1283, 1283, 52*w + 167], [1283, 1283, 52*w - 219], [1297, 1297, 8*w - 51], [1297, 1297, -8*w - 43], [1303, 1303, -14*w + 71], [1303, 1303, -14*w - 57], [1307, 1307, 36*w - 149], [1307, 1307, 36*w + 113], [1319, 1319, 24*w - 95], [1319, 1319, 24*w + 71], [1327, 1327, 38*w + 129], [1327, 1327, 38*w - 167], [1361, 1361, 10*w - 13], [1361, 1361, -10*w - 3], [1367, 1367, 32*w + 99], [1367, 1367, 32*w - 131], [1369, 37, -37], [1409, 1409, 10*w - 9], [1409, 1409, 10*w - 1], [1423, 1423, -6*w - 41], [1423, 1423, 6*w - 47], [1427, 1427, 12*w - 31], [1427, 1427, -12*w - 19], [1429, 1429, -12*w + 65], [1429, 1429, -12*w - 53], [1433, 1433, 78*w + 253], [1433, 1433, 98*w + 319], [1439, 1439, 16*w + 39], [1439, 1439, 16*w - 55], [1453, 1453, -4*w - 39], [1453, 1453, 4*w - 43], [1481, 1481, 18*w + 47], [1481, 1481, 18*w - 65], [1483, 1483, 34*w + 117], [1483, 1483, 34*w - 151], [1489, 1489, -8*w - 45], [1489, 1489, 8*w - 53], [1511, 1511, 72*w + 233], [1511, 1511, 112*w + 365], [1523, 1523, -12*w - 17], [1523, 1523, 12*w - 29], [1531, 1531, 46*w - 201], [1531, 1531, 46*w + 155], [1543, 1543, 2*w - 41], [1543, 1543, -2*w - 39], [1553, 1553, 54*w + 173], [1553, 1553, 54*w - 227], [1567, 1567, -22*w + 103], [1567, 1567, -22*w - 81], [1571, 1571, 60*w - 253], [1571, 1571, 60*w + 193], [1597, 1597, -28*w - 99], [1597, 1597, 28*w - 127], [1621, 1621, 4*w - 45], [1621, 1621, -4*w - 41], [1637, 1637, 14*w - 41], [1637, 1637, -14*w - 27], [1657, 1657, -64*w - 213], [1657, 1657, 64*w - 277], [1667, 1667, 44*w + 139], [1667, 1667, 44*w - 183], [1669, 1669, -12*w - 55], [1669, 1669, 12*w - 67], [1709, 1709, 86*w + 279], [1709, 1709, 106*w + 345], [1753, 1753, -72*w - 239], [1753, 1753, 128*w + 421], [1759, 1759, 42*w - 185], [1759, 1759, 42*w + 143], [1783, 1783, -18*w - 71], [1783, 1783, -18*w + 89], [1823, 1823, 32*w + 97], [1823, 1823, 32*w - 129], [1831, 1831, -14*w + 75], [1831, 1831, -14*w - 61], [1867, 1867, 62*w - 269], [1867, 1867, -62*w - 207], [1873, 1873, -88*w - 291], [1873, 1873, -112*w - 369], [1877, 1877, 62*w - 261], [1877, 1877, 62*w + 199], [1879, 1879, 2*w - 45], [1879, 1879, -2*w - 43], [1889, 1889, 26*w - 101], [1889, 1889, 26*w + 75], [1913, 1913, -18*w + 61], [1913, 1913, 18*w + 43], [1931, 1931, -12*w - 5], [1931, 1931, 12*w - 17], [1979, 1979, 20*w - 71], [1979, 1979, 20*w + 51], [1987, 1987, -6*w - 47], [1987, 1987, 6*w - 53], [1993, 1993, -24*w - 89], [1993, 1993, -24*w + 113], [1997, 1997, 22*w + 59], [1997, 1997, 22*w - 81], [1999, 1999, -70*w - 233], [1999, 1999, 70*w - 303], [2003, 2003, -12*w - 1], [2003, 2003, 12*w - 13], [2011, 2011, 50*w + 169], [2011, 2011, 50*w - 219], [2027, 2027, 12*w - 11], [2027, 2027, 12*w - 1], [2053, 2053, 28*w - 129], [2053, 2053, -28*w - 101], [2081, 2081, 38*w + 117], [2081, 2081, 38*w - 155], [2111, 2111, 88*w + 285], [2111, 2111, 128*w + 417], [2113, 2113, 8*w - 59], [2113, 2113, -8*w - 51], [2137, 2137, 48*w + 163], [2137, 2137, 48*w - 211], [2141, 2141, 58*w - 243], [2141, 2141, 58*w + 185], [2207, 2207, 48*w + 151], [2207, 2207, 48*w - 199], [2209, 47, -47], [2221, 2221, -20*w - 79], [2221, 2221, -20*w + 99], [2237, 2237, 26*w + 73], [2237, 2237, 26*w - 99], [2239, 2239, 58*w + 195], [2239, 2239, 58*w - 253], [2251, 2251, 126*w + 415], [2251, 2251, -94*w - 311], [2273, 2273, 22*w - 79], [2273, 2273, 22*w + 57], [2281, 2281, -16*w - 69], [2281, 2281, -16*w + 85], [2287, 2287, -102*w - 337], [2287, 2287, -118*w - 389], [2309, 2309, -14*w - 15], [2309, 2309, 14*w - 29], [2333, 2333, 102*w + 331], [2333, 2333, 122*w + 397], [2339, 2339, 36*w + 109], [2339, 2339, 36*w - 145], [2341, 2341, 44*w - 195], [2341, 2341, 44*w + 151], [2351, 2351, 40*w - 163], [2351, 2351, 40*w + 123], [2393, 2393, 14*w - 27], [2393, 2393, -14*w - 13], [2423, 2423, -16*w - 27], [2423, 2423, 16*w - 43], [2437, 2437, 12*w - 73], [2437, 2437, -12*w - 61], [2447, 2447, 136*w + 443], [2447, 2447, 96*w + 311], [2459, 2459, 60*w + 191], [2459, 2459, 60*w - 251], [2467, 2467, -26*w + 123], [2467, 2467, -26*w - 97], [2549, 2549, 34*w - 135], [2549, 2549, 34*w + 101], [2551, 2551, -30*w - 109], [2551, 2551, 30*w - 139], [2557, 2557, -36*w - 127], [2557, 2557, 36*w - 163], [2579, 2579, 84*w + 271], [2579, 2579, 164*w + 535], [2593, 2593, 16*w - 87], [2593, 2593, 16*w + 71], [2621, 2621, 38*w + 115], [2621, 2621, 38*w - 153], [2647, 2647, 2*w - 53], [2647, 2647, -2*w - 51], [2663, 2663, 56*w + 177], [2663, 2663, 56*w - 233], [2671, 2671, -10*w - 59], [2671, 2671, 10*w - 69], [2677, 2677, 52*w - 229], [2677, 2677, 52*w + 177], [2683, 2683, -14*w - 67], [2683, 2683, 14*w - 81], [2687, 2687, -16*w - 23], [2687, 2687, 16*w - 39], [2693, 2693, -14*w - 3], [2693, 2693, 14*w - 17], [2707, 2707, -22*w - 87], [2707, 2707, -22*w + 109], [2711, 2711, 32*w - 125], [2711, 2711, 32*w + 93], [2729, 2729, -14*w - 1], [2729, 2729, 14*w - 15], [2777, 2777, 14*w - 11], [2777, 2777, 14*w - 3], [2789, 2789, 14*w - 9], [2789, 2789, 14*w - 5], [2791, 2791, 50*w + 171], [2791, 2791, 50*w - 221], [2797, 2797, 4*w - 57], [2797, 2797, -4*w - 53], [2801, 2801, 22*w + 53], [2801, 2801, 22*w - 75], [2843, 2843, 36*w - 143], [2843, 2843, 36*w + 107], [2851, 2851, 6*w - 61], [2851, 2851, -6*w - 55], [2857, 2857, -72*w - 241], [2857, 2857, 72*w - 313], [2879, 2879, 24*w - 85], [2879, 2879, 24*w + 61], [2903, 2903, 48*w - 197], [2903, 2903, 48*w + 149], [2909, 2909, 26*w - 95], [2909, 2909, 26*w + 69], [2939, 2939, 44*w + 135], [2939, 2939, 44*w - 179], [2957, 2957, 74*w + 237], [2957, 2957, 74*w - 311], [2963, 2963, 172*w + 561], [2963, 2963, 92*w + 297], [2971, 2971, 46*w - 205], [2971, 2971, 46*w + 159], [3019, 3019, 162*w + 533], [3019, 3019, -98*w - 325], [3023, 3023, -16*w - 17], [3023, 3023, 16*w - 33], [3037, 3037, 44*w - 197], [3037, 3037, 44*w + 153], [3049, 3049, 80*w - 347], [3049, 3049, -80*w - 267], [3079, 3079, 2*w - 57], [3079, 3079, -2*w - 55], [3119, 3119, 16*w - 31], [3119, 3119, -16*w - 15], [3121, 3121, -40*w - 141], [3121, 3121, 40*w - 181], [3137, 3137, 54*w - 223], [3137, 3137, 54*w + 169], [3163, 3163, -114*w - 377], [3163, 3163, -146*w - 481], [3167, 3167, 24*w + 59], [3167, 3167, 24*w - 83], [3191, 3191, 152*w + 495], [3191, 3191, 112*w + 363], [3217, 3217, -88*w - 293], [3217, 3217, 88*w - 381], [3221, 3221, 30*w - 113], [3221, 3221, 30*w + 83], [3251, 3251, 28*w + 75], [3251, 3251, 28*w - 103], [3253, 3253, 4*w - 61], [3253, 3253, -4*w - 57], [3257, 3257, 46*w - 187], [3257, 3257, 46*w + 141], [3299, 3299, 20*w - 59], [3299, 3299, -20*w - 39], [3307, 3307, 2*w - 59], [3307, 3307, -2*w - 57], [3313, 3313, -8*w - 61], [3313, 3313, 8*w - 69], [3331, 3331, -6*w - 59], [3331, 3331, 6*w - 65], [3347, 3347, 236*w + 771], [3347, 3347, 76*w + 243], [3359, 3359, 16*w - 25], [3359, 3359, -16*w - 9], [3361, 3361, 184*w + 605], [3361, 3361, -96*w - 319], [3371, 3371, 180*w + 587], [3371, 3371, 100*w + 323], [3391, 3391, 54*w + 185], [3391, 3391, 54*w - 239], [3413, 3413, 94*w + 303], [3413, 3413, 194*w + 633], [3449, 3449, 126*w + 409], [3449, 3449, 146*w + 475], [3461, 3461, 18*w - 43], [3461, 3461, -18*w - 25], [3463, 3463, 66*w + 223], [3463, 3463, 66*w - 289], [3469, 3469, -28*w + 135], [3469, 3469, -28*w - 107], [3491, 3491, 20*w - 57], [3491, 3491, -20*w - 37], [3527, 3527, 16*w - 19], [3527, 3527, -16*w - 3], [3533, 3533, 22*w - 69], [3533, 3533, -22*w - 47], [3541, 3541, 76*w - 331], [3541, 3541, 76*w + 255], [3559, 3559, -30*w - 113], [3559, 3559, -30*w + 143], [3583, 3583, 6*w - 67], [3583, 3583, -6*w - 61], [3593, 3593, 18*w - 41], [3593, 3593, -18*w - 23], [3607, 3607, 14*w - 87], [3607, 3607, -14*w - 73], [3623, 3623, 16*w - 3], [3623, 3623, 16*w - 13], [3673, 3673, -24*w + 121], [3673, 3673, -24*w - 97], [3677, 3677, 38*w - 149], [3677, 3677, 38*w + 111], [3691, 3691, -34*w + 159], [3691, 3691, -34*w - 125], [3697, 3697, -152*w - 501], [3697, 3697, -128*w - 423], [3701, 3701, 62*w + 195], [3701, 3701, 62*w - 257], [3709, 3709, -20*w + 107], [3709, 3709, -20*w - 87], [3719, 3719, -24*w - 55], [3719, 3719, 24*w - 79], [3733, 3733, 36*w - 167], [3733, 3733, -36*w - 131], [3761, 3761, -22*w - 45], [3761, 3761, 22*w - 67], [3769, 3769, 40*w - 183], [3769, 3769, -40*w - 143], [3803, 3803, 108*w + 349], [3803, 3803, 188*w + 613], [3821, 3821, 42*w - 167], [3821, 3821, 42*w + 125], [3823, 3823, -26*w - 103], [3823, 3823, -26*w + 129], [3833, 3833, -18*w - 19], [3833, 3833, 18*w - 37], [3847, 3847, 18*w - 101], [3847, 3847, -18*w - 83], [3851, 3851, 20*w - 53], [3851, 3851, -20*w - 33], [3877, 3877, -12*w - 71], [3877, 3877, 12*w - 83], [3917, 3917, 54*w + 167], [3917, 3917, 54*w - 221], [3919, 3919, -22*w + 115], [3919, 3919, -22*w - 93], [3929, 3929, 34*w - 129], [3929, 3929, 34*w + 95], [3931, 3931, 14*w - 89], [3931, 3931, -14*w - 75], [3947, 3947, 28*w - 99], [3947, 3947, 28*w + 71], [3989, 3989, 30*w - 109], [3989, 3989, 30*w + 79], [4019, 4019, 20*w - 51], [4019, 4019, -20*w - 31], [4049, 4049, 74*w - 309], [4049, 4049, 74*w + 235], [4051, 4051, -10*w - 69], [4051, 4051, 10*w - 79], [4079, 4079, 64*w - 265], [4079, 4079, 64*w + 201], [4111, 4111, -6*w - 65], [4111, 4111, 6*w - 71], [4129, 4129, 8*w - 75], [4129, 4129, -8*w - 67], [4133, 4133, -18*w - 13], [4133, 4133, 18*w - 31], [4153, 4153, 56*w - 249], [4153, 4153, 56*w + 193], [4157, 4157, -26*w - 61], [4157, 4157, 26*w - 87], [4159, 4159, 70*w - 307], [4159, 4159, 70*w + 237], [4177, 4177, -32*w + 153], [4177, 4177, -32*w - 121], [4211, 4211, 60*w - 247], [4211, 4211, 60*w + 187], [4217, 4217, -18*w - 11], [4217, 4217, 18*w - 29], [4219, 4219, 18*w - 103], [4219, 4219, -18*w - 85], [4243, 4243, 54*w - 241], [4243, 4243, 54*w + 187], [4259, 4259, 116*w + 375], [4259, 4259, 196*w + 639], [4261, 4261, 4*w - 69], [4261, 4261, -4*w - 65], [4271, 4271, 48*w + 145], [4271, 4271, 48*w - 193], [4273, 4273, 16*w - 97], [4273, 4273, -16*w - 81], [4283, 4283, 28*w + 69], [4283, 4283, 28*w - 97], [4289, 4289, 70*w - 291], [4289, 4289, 70*w + 221], [4339, 4339, -22*w - 95], [4339, 4339, -22*w + 117], [4357, 4357, -108*w - 359], [4357, 4357, 212*w + 697], [4373, 4373, 34*w + 93], [4373, 4373, 34*w - 127], [4391, 4391, 32*w - 117], [4391, 4391, 32*w + 85], [4397, 4397, 22*w - 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437], [8543, 8543, 136*w - 573], [8599, 8599, -142*w - 473], [8599, 8599, 142*w - 615], [8609, 8609, -26*w - 19], [8609, 8609, 26*w - 45], [8623, 8623, 54*w - 251], [8623, 8623, -54*w - 197], [8663, 8663, 32*w - 93], [8663, 8663, -32*w - 61], [8689, 8689, -168*w - 557], [8689, 8689, 272*w + 895], [8693, 8693, 98*w - 407], [8693, 8693, 98*w + 309], [8707, 8707, 58*w - 267], [8707, 8707, -58*w - 209], [8713, 8713, 64*w - 291], [8713, 8713, -64*w - 227], [8719, 8719, -42*w - 163], [8719, 8719, -42*w + 205], [8737, 8737, 112*w - 489], [8737, 8737, 112*w + 377], [8753, 8753, 42*w + 107], [8753, 8753, 42*w - 149], [8779, 8779, 2*w - 95], [8779, 8779, -2*w - 93], [8803, 8803, -122*w - 409], [8803, 8803, 122*w - 531], [8807, 8807, 48*w + 131], [8807, 8807, 48*w - 179], [8819, 8819, 44*w - 159], [8819, 8819, 44*w + 115], [8821, 8821, 100*w + 339], [8821, 8821, 100*w - 439], [8831, 8831, 80*w + 247], [8831, 8831, 80*w - 327], [8837, 8837, 46*w + 123], [8837, 8837, 46*w - 169], [8839, 8839, -18*w - 107], [8839, 8839, 18*w - 125], [8849, 8849, -26*w - 15], [8849, 8849, 26*w - 41], [8863, 8863, 26*w - 149], [8863, 8863, -26*w - 123], [8867, 8867, 76*w - 309], [8867, 8867, 76*w + 233], [8893, 8893, -44*w - 169], [8893, 8893, -44*w + 213], [8933, 8933, 174*w + 563], [8933, 8933, 274*w + 893], [8941, 8941, -20*w - 111], [8941, 8941, 20*w - 131], [8951, 8951, 120*w + 383], [8951, 8951, 120*w - 503], [8963, 8963, 28*w - 61], [8963, 8963, -28*w - 33], [8999, 8999, 200*w + 649], [8999, 8999, 240*w + 781], [9007, 9007, -38*w + 191], [9007, 9007, -38*w - 153], [9013, 9013, 84*w - 373], [9013, 9013, 84*w + 289], [9049, 9049, -192*w - 635], [9049, 9049, -248*w - 817], [9059, 9059, 36*w - 115], [9059, 9059, -36*w - 79], [9067, 9067, 98*w + 333], [9067, 9067, 98*w - 431], [9091, 9091, -6*w - 95], [9091, 9091, 6*w - 101], [9127, 9127, -34*w + 177], [9127, 9127, -34*w - 143], [9161, 9161, 302*w + 985], [9161, 9161, 162*w + 523], [9173, 9173, 66*w - 263], [9173, 9173, 66*w + 197], [9181, 9181, -4*w - 95], [9181, 9181, 4*w - 99], [9209, 9209, 110*w + 349], [9209, 9209, 110*w - 459], [9227, 9227, 316*w + 1031], [9227, 9227, 156*w + 503], [9241, 9241, -40*w - 159], [9241, 9241, -40*w + 199], [9277, 9277, 28*w - 157], [9277, 9277, -28*w - 129], [9283, 9283, 294*w + 967], [9283, 9283, -166*w - 551], [9293, 9293, 86*w - 353], [9293, 9293, 86*w + 267], [9319, 9319, -50*w + 237], [9319, 9319, -50*w - 187], [9323, 9323, -28*w - 29], [9323, 9323, 28*w - 57], [9341, 9341, -38*w - 87], [9341, 9341, 38*w - 125], [9349, 9349, 108*w - 473], [9349, 9349, 108*w + 365], [9377, 9377, -26*w - 3], [9377, 9377, 26*w - 29], [9391, 9391, -10*w - 99], [9391, 9391, 10*w - 109], [9397, 9397, 12*w - 113], [9397, 9397, -12*w - 101], [9403, 9403, 78*w - 349], [9403, 9403, 78*w + 271], [9409, 97, -97], [9413, 9413, 78*w - 317], [9413, 9413, 78*w + 239], [9419, 9419, 116*w - 485], [9419, 9419, 116*w + 369], [9421, 9421, -36*w - 149], [9421, 9421, -36*w + 185], [9433, 9433, -32*w + 171], [9433, 9433, -32*w - 139], [9437, 9437, -26*w - 1], [9437, 9437, 26*w - 27], [9461, 9461, -30*w - 43], [9461, 9461, 30*w - 73], [9463, 9463, 286*w + 941], [9463, 9463, -174*w - 577], [9491, 9491, 28*w - 55], [9491, 9491, -28*w - 27], [9511, 9511, 94*w - 415], [9511, 9511, 94*w + 321], [9521, 9521, 58*w - 225], [9521, 9521, 58*w + 167], [9533, 9533, 26*w - 23], [9533, 9533, 26*w - 3], [9547, 9547, 2*w - 99], [9547, 9547, -2*w - 97], [9551, 9551, -32*w - 55], [9551, 9551, 32*w - 87], [9601, 9601, -56*w - 205], [9601, 9601, 56*w - 261], [9619, 9619, -182*w - 603], [9619, 9619, 278*w + 915], [9629, 9629, 26*w - 11], [9629, 9629, 26*w - 15], [9631, 9631, 106*w + 359], [9631, 9631, 106*w - 465], [9649, 9649, -72*w - 253], [9649, 9649, 72*w - 325], [9661, 9661, -44*w - 171], [9661, 9661, -44*w + 215], [9689, 9689, -30*w - 41], [9689, 9689, 30*w - 71], [9697, 9697, 8*w - 107], [9697, 9697, -8*w - 99], [9719, 9719, 56*w - 215], [9719, 9719, 56*w + 159], [9733, 9733, -68*w - 241], [9733, 9733, 68*w - 309], [9739, 9739, -62*w - 223], [9739, 9739, 62*w - 285], [9743, 9743, 128*w + 409], [9743, 9743, 128*w - 537], [9749, 9749, -34*w - 65], [9749, 9749, 34*w - 99], [9803, 9803, -28*w - 23], [9803, 9803, 28*w - 51], [9811, 9811, 10*w - 111], [9811, 9811, -10*w - 101], [9829, 9829, -12*w - 103], [9829, 9829, 12*w - 115], [9833, 9833, 62*w - 243], [9833, 9833, 62*w + 181], [9857, 9857, 134*w + 429], [9857, 9857, 134*w - 563], [9859, 9859, 262*w + 863], [9859, 9859, -198*w - 655], [9967, 9967, -26*w - 127], [9967, 9967, 26*w - 153], [9973, 9973, -4*w - 99], [9973, 9973, 4*w - 103]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [1, 1, -1, -1, -1, 4, -6, 2, 2, -2, -2, -4, -4, -6, -6, -12, -12, -2, -2, 0, 0, -6, -6, 14, 14, -4, -4, 14, 14, -6, 12, 12, -2, -2, 4, 4, 8, 8, -22, 10, 10, 4, 4, 24, 24, 12, 12, -10, -10, -26, -26, -6, -6, 16, 16, 22, 22, 22, 22, -20, -20, 2, -14, -14, 26, 26, 18, 18, -2, -2, 0, 0, 0, 0, -18, -18, 30, 30, 24, 24, 30, 30, 10, 10, 16, 16, 4, 4, 10, 10, -18, -18, -46, -34, -34, -4, -4, -10, -10, -4, -4, 34, 34, -48, -48, -26, -26, -20, -20, -40, -40, -18, -18, 20, 20, -12, -12, 18, 18, 12, 12, 20, 20, 6, 6, -40, -40, -18, -18, 36, 36, 0, 0, -10, -10, 2, 2, -14, -14, 26, 26, 24, 24, 12, 12, 8, 8, -10, -10, 14, 14, 44, 44, 24, 24, -28, -28, -8, -8, 24, 24, 8, 8, 42, 42, -38, -38, -26, -26, -62, 40, 40, -12, -12, 30, 30, -24, -24, -26, -26, 12, 12, -22, -22, 28, 28, 46, 46, 32, 32, 60, 60, 46, 46, -6, -6, -18, -18, 50, 50, 30, 30, 34, 34, -48, -48, 36, 36, 18, 18, -56, -56, -28, -28, 40, 40, -64, -64, 38, 38, 32, 32, -38, -2, -2, 32, 32, -20, -20, -10, -10, 14, 14, -8, -8, 46, 46, -10, -10, -36, -36, -62, -62, -8, -8, -60, -60, -36, -36, -72, -72, 54, 54, -32, -32, -20, -20, -34, -34, -10, -10, 18, 18, 26, 26, 52, 52, -10, -10, 66, 66, 10, 10, -16, -16, 56, 56, 64, 64, -24, -24, 44, 44, 82, 82, -6, -6, 56, 56, 6, 6, 54, 54, -60, -60, 52, 52, 68, 68, -54, -54, 18, 18, -80, -80, 20, 20, 60, 60, 12, 12, -26, -26, -18, -18, 24, 24, -14, -14, -38, -38, -6, -6, -56, -56, -94, -50, -50, -30, -30, 32, 32, -20, -20, 6, 6, -70, -70, 16, 16, -30, -30, 34, 34, -28, -28, -74, -74, -48, -48, -2, -2, 48, 48, 6, 6, 24, 24, 76, 76, -28, -28, 90, 90, 56, 56, -2, -2, -52, -52, 2, 2, -38, -38, 40, 40, -8, -8, 32, 32, 70, 70, -4, -4, 48, 48, 66, 66, -28, -28, -80, -80, -18, -18, 78, 78, -30, -30, -8, -8, -18, -18, -50, -50, 28, 28, 52, 52, -54, -54, 24, 24, -72, -72, 66, 66, 100, 100, 42, 42, -28, -28, -36, -36, 12, 12, -16, -16, -34, -34, 10, 10, -40, -40, -104, -104, -30, -30, -98, -98, 92, 92, -96, -96, 88, 88, 18, 18, -38, -38, 28, 28, -26, -26, 6, 6, 20, 20, 108, 108, 18, 18, 100, 100, -76, -76, -80, -80, -30, -30, 12, 12, 32, 32, 26, 26, -66, -66, -14, -14, -56, -56, 14, 14, -36, -36, 104, 104, -6, -6, 86, 86, 72, 72, 32, 32, -42, -42, -88, -88, 112, 112, -70, -70, 114, 114, -20, -20, 50, 50, 114, 114, 62, 62, -80, -80, 54, 54, -18, -18, -70, -70, 44, 44, -118, -118, -64, -64, 54, 54, -40, -40, -20, -20, -58, -58, -94, -94, -32, -32, -122, -122, -4, -4, 68, 68, 114, 114, -108, -108, -90, -90, -28, -28, -16, -16, -48, -48, 34, 34, 18, 18, -6, -6, -54, -54, 96, 96, 98, 98, 4, 4, 30, 30, 28, 28, -76, -76, 60, 60, -58, -58, -72, -72, 34, 34, -44, -44, -10, -10, 100, 100, 38, 38, 66, 66, -112, -112, -78, -78, 10, 10, 112, 112, -118, -116, -116, 74, 74, -40, -40, 82, 82, 8, 8, 12, 12, 46, 46, 134, 134, 30, 30, 16, 16, 100, 100, -38, -38, 2, 2, 56, 56, -100, -100, 22, 22, 46, 46, -82, -82, 112, 112, 78, 78, -6, -6, 88, 88, -50, -50, 108, 108, 104, 104, -88, -88, -66, -66, 128, 128, -118, -118, -36, -36, 78, 78, -64, -64, -60, -60, 22, 22, 78, 78, -48, -48, 62, 62, 60, 60, 76, 76, -86, -86, -38, -38, -90, -90, -58, -58, -16, -16, -102, -102, 114, 114, -120, -120, 92, 92, 48, 48, 0, 0, -122, -122, 30, 30, 76, 76, 72, 72, -44, -44, -72, -72, -120, -120, -54, -54, 114, 114, -56, -56, -20, -20, -58, -58, 96, 96, 58, 58, -128, -128, 132, 132, -42, -42, 28, 28, -94, -94, 118, 118, -94, -94, -80, -80, -42, -42, -88, -88, -14, -14, 94, 94, -80, -80, 76, 76, -46, -46, 60, 60, 62, 62, 88, 88, 24, 24, -8, -8, -110, -110, 106, 106, -12, -12, 28, 28, 52, 52, -68, -68, -54, -54, -66, -66, -84, -84, 116, 116, -150, -150, -8, -8, 20, 20, -54, -54, -70, -70, 54, 54, 98, -8, -8, 18, 18, 96, 96, 38, 38, -108, -108, 0, 0, 20, 20, -58, -58, -24, -24, -120, -120, -52, -52, -50, -50, 6, 6, -34, -34, 92, 92, -12, -12, -40, -40, -38, -38, 68, 68, -2, -2, 12, 12, 76, 76, -50, -50, -86, -86, -154, -154, 146, 146, 66, 66, 48, 48, 94, 94, 84, 84, -50, -50, -48, -48, -34, -34, -6, -6, -118, -118, 84, 84, -22, -60, -60, -40, -40, -132, -132, 2, 2, 128, 128, 22, 22, -142, -142, -44, -44, 36, 36, 96, 96, -12, -12, 46, 46, 42, 42, -34, -34, 112, 112, 10, 10, -72, -72, -132, -132, 54, 54, 140, 140, 72, 72, 42, 42, 82, 82, 26, 26, 2, 2, 56, 56, 10, 10, -108, -108, -134, -134, -140, -140, -156, -156, -18, -18, -20, -20, 138, 138, 108, 108, -98, -98, 38, 38, -18, -18, 24, 24, 90, 90, -136, -136, -62, -62, 136, 136, -28, -28, 68, 68, -24, -24, 76, 76, -48, -48, 42, 42, 160, 160, 46, 46, 126, 126, -56, -56, -82, -82, 124, 124, -46, -46, -60, -60, 24, 24, 168, 168, 6, 6, -144, -144, -74, -74, 152, 152, -58, -58, 18, 18, -144, -144, 98, 98, 134, 134, 4, 4, -36, -36, 40, 40, -60, -60, -174, -174, 42, 42, 42, 42, 56, 56, 46, 46, 6, 6, 102, 102, -42, -42, -70, -70, 132, 132, -86, -86, 180, 180, -30, -30, -148, -148, 174, 174, -174, -174, 10, 10, 112, 112, 56, 56, -26, -26, -96, -96, 128, 128, -110, -110, 66, 66, 148, 148, -38, -38, 64, 64, 98, 98, -26, -26, -148, -148, 116, 116, 48, 48, -100, -100, -106, -106, -136, -136, 50, 50, -56, -56, 94, 94, 96, 96, 76, 76, 158, 158, 122, 122, -114, -114, 24, 24, -36, -36, 48, 48, -64, -64, 118, 118, -86, -86, -108, -108, 140, 140, -76, -76, 72, 72, 142, 142, -46, -46, -130, -130, -106, -106, 12, 12, -38, -38, -2, -2, -76, -76, 66, 66, 104, 104, 116, 116, 42, 42, -26, -26, -98, -98, -32, -32, 86, 86, 60, 60, 130, -46, -46, -148, -148, 30, 30, 26, 26, 178, 178, 138, 138, -40, -40, 92, 92, -8, -8, 30, 30, 42, 42, -164, -164, 160, 160, 130, 130, 100, 100, -14, -14, -16, -16, -30, -30, -2, -2, -154, -154, -30, -30, 168, 168, -154, -154, -52, -52, 48, 48, 162, 162, -12, -12, 148, 148, 134, 134, -2, -2, -18, -18, -28, -28, -48, -48, 6, 6]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := 1; ALEigenvalues[ideal] := 1; ALEigenvalues[ideal] := 1; // EXAMPLE: // pp := Factorization(2*ZF)[1][1]; // heckeEigenvalues[pp]; print "To reconstruct the Hilbert newform f, type f, iso := Explode(make_newform());"; function make_newform(); M := HilbertCuspForms(F, NN); S := NewSubspace(M); // SetVerbose("ModFrmHil", 1); NFD := NewformDecomposition(S); newforms := [* Eigenform(U) : U in NFD *]; if #newforms eq 0 then; print "No Hilbert newforms at this level"; return 0; end if; print "Testing ", #newforms, " possible newforms"; newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *]; print #newforms, " newforms have the correct Hecke field"; if #newforms eq 0 then; print "No Hilbert newform found with the correct Hecke field"; return 0; end if; autos := Automorphisms(K); xnewforms := [* *]; for f in newforms do; if K eq RationalField() then; Append(~xnewforms, [* f, autos[1] *]); else; flag, iso := IsIsomorphic(K,BaseField(f)); for a in autos do; Append(~xnewforms, [* f, a*iso *]); end for; end if; end for; newforms := xnewforms; for P in primes do; xnewforms := [* *]; for f_iso in newforms do; f, iso := Explode(f_iso); if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then; Append(~xnewforms, f_iso); end if; end for; newforms := xnewforms; if #newforms eq 0 then; print "No Hilbert newform found which matches the Hecke eigenvalues"; return 0; else if #newforms eq 1 then; print "success: unique match"; return newforms[1]; end if; end if; end for; print #newforms, "Hilbert newforms found which match the Hecke eigenvalues"; return newforms[1]; end function;