/* 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![-6, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, -w + 2], [3, 3, w - 3], [5, 5, w + 1], [5, 5, w - 1], [19, 19, w + 5], [19, 19, -w + 5], [23, 23, -2*w + 1], [23, 23, -2*w - 1], [29, 29, -3*w + 5], [29, 29, -3*w - 5], [43, 43, -w - 7], [43, 43, w - 7], [47, 47, 4*w - 7], [47, 47, 6*w - 13], [49, 7, -7], [53, 53, -3*w - 1], [53, 53, 3*w - 1], [67, 67, -7*w + 19], [67, 67, -3*w + 11], [71, 71, -4*w - 5], [71, 71, 4*w - 5], [73, 73, -4*w + 13], [73, 73, -6*w + 17], [97, 97, 2*w - 11], [97, 97, -2*w - 11], [101, 101, 5*w - 7], [101, 101, -5*w - 7], [121, 11, -11], [139, 139, -9*w + 25], [139, 139, -5*w + 17], [149, 149, -5*w - 1], [149, 149, 5*w - 1], [163, 163, -w - 13], [163, 163, w - 13], [167, 167, -6*w - 7], [167, 167, 6*w - 7], [169, 13, -13], [173, 173, 13*w - 29], [173, 173, 7*w - 11], [191, 191, 6*w - 5], [191, 191, -6*w - 5], [193, 193, -4*w + 17], [193, 193, 14*w - 37], [197, 197, 9*w - 17], [197, 197, -11*w + 23], [211, 211, -5*w - 19], [211, 211, 5*w - 19], [239, 239, 10*w - 19], [239, 239, -12*w + 25], [241, 241, -8*w + 25], [241, 241, -10*w + 29], [263, 263, -8*w - 11], [263, 263, 8*w - 11], [269, 269, -7*w - 5], [269, 269, 7*w - 5], [283, 283, -w - 17], [283, 283, w - 17], [289, 17, -17], [293, 293, -7*w - 1], [293, 293, 7*w - 1], [307, 307, 3*w - 19], [307, 307, -3*w - 19], [311, 311, -16*w + 35], [311, 311, 10*w - 17], [313, 313, -6*w - 23], [313, 313, 6*w - 23], [317, 317, 9*w - 13], [317, 317, 19*w - 43], [331, 331, -7*w + 25], [331, 331, -15*w + 41], [337, 337, 2*w - 19], [337, 337, -2*w - 19], [359, 359, 8*w - 5], [359, 359, -8*w - 5], [379, 379, 5*w - 23], [379, 379, -5*w - 23], [383, 383, -8*w - 1], [383, 383, 8*w - 1], [389, 389, 13*w - 25], [389, 389, -15*w + 31], [409, 409, -6*w + 25], [409, 409, 20*w - 53], [431, 431, 10*w - 13], [431, 431, -10*w - 13], [433, 433, -4*w - 23], [433, 433, 4*w - 23], [457, 457, -18*w + 49], [457, 457, -8*w + 29], [461, 461, -9*w - 5], [461, 461, 9*w - 5], [479, 479, -10*w - 11], [479, 479, 10*w - 11], [499, 499, -15*w + 43], [499, 499, -11*w + 35], [503, 503, 12*w - 19], [503, 503, 22*w - 49], [509, 509, 15*w - 29], [509, 509, -17*w + 35], [523, 523, -w - 23], [523, 523, w - 23], [547, 547, -7*w + 29], [547, 547, 23*w - 61], [557, 557, 11*w - 13], [557, 557, -11*w - 13], [571, 571, 3*w - 25], [571, 571, -3*w - 25], [577, 577, -8*w - 31], [577, 577, 8*w - 31], [599, 599, -10*w - 1], [599, 599, 10*w - 1], [601, 601, 2*w - 25], [601, 601, -2*w - 25], [619, 619, -w - 25], [619, 619, w - 25], [643, 643, -11*w + 37], [643, 643, -19*w + 53], [647, 647, 14*w - 23], [647, 647, -24*w + 53], [653, 653, 13*w - 19], [653, 653, 27*w - 61], [673, 673, -14*w + 43], [673, 673, -16*w + 47], [677, 677, 11*w - 7], [677, 677, -11*w - 7], [691, 691, -5*w - 29], [691, 691, 5*w - 29], [701, 701, 11*w - 5], [701, 701, -11*w - 5], [719, 719, 18*w - 35], [719, 719, -20*w + 41], [739, 739, -9*w - 35], [739, 739, 9*w - 35], [743, 743, 12*w - 11], [743, 743, -12*w - 11], [769, 769, -10*w + 37], [769, 769, -24*w + 65], [773, 773, 17*w - 31], [773, 773, -23*w + 49], [787, 787, -3*w - 29], [787, 787, 3*w - 29], [797, 797, 19*w - 37], [797, 797, -21*w + 43], [811, 811, 5*w - 31], [811, 811, -5*w - 31], [821, 821, 29*w - 65], [821, 821, 15*w - 23], [839, 839, -12*w - 5], [839, 839, 12*w - 5], [859, 859, -19*w + 55], [859, 859, -15*w + 47], [863, 863, -12*w - 1], [863, 863, 12*w - 1], [883, 883, -9*w + 37], [883, 883, 29*w - 77], [887, 887, -14*w - 17], [887, 887, 14*w - 17], [907, 907, 3*w - 31], [907, 907, -3*w - 31], [911, 911, 16*w - 25], [911, 911, 30*w - 67], [937, 937, 2*w - 31], [937, 937, -2*w - 31], [941, 941, -25*w + 53], [941, 941, 19*w - 35], [961, 31, -31], [983, 983, -28*w + 61], [983, 983, 18*w - 31], [1009, 1009, -6*w - 35], [1009, 1009, 6*w - 35], [1013, 1013, -13*w - 1], [1013, 1013, 13*w - 1], [1031, 1031, 20*w - 37], [1031, 1031, -26*w + 55], [1033, 1033, -14*w + 47], [1033, 1033, -24*w + 67], [1051, 1051, -15*w + 49], [1051, 1051, -23*w + 65], [1061, 1061, 15*w - 17], [1061, 1061, -15*w - 17], [1103, 1103, 32*w - 71], [1103, 1103, 18*w - 29], [1109, 1109, 17*w - 25], [1109, 1109, 35*w - 79], [1123, 1123, -11*w - 43], [1123, 1123, 11*w - 43], [1129, 1129, -4*w - 35], [1129, 1129, 4*w - 35], [1151, 1151, -14*w - 5], [1151, 1151, 14*w - 5], [1153, 1153, 6*w - 37], [1153, 1153, -6*w - 37], [1171, 1171, -3*w - 35], [1171, 1171, 3*w - 35], [1181, 1181, -15*w - 13], [1181, 1181, 15*w - 13], [1201, 1201, 2*w - 35], [1201, 1201, -2*w - 35], [1223, 1223, -28*w + 59], [1223, 1223, 22*w - 41], [1229, 1229, 15*w - 11], [1229, 1229, -15*w - 11], [1249, 1249, -10*w - 43], [1249, 1249, 10*w - 43], [1277, 1277, -31*w + 67], [1277, 1277, 21*w - 37], [1291, 1291, -25*w + 71], [1291, 1291, -17*w + 55], [1297, 1297, 8*w - 41], [1297, 1297, -8*w - 41], [1301, 1301, 15*w - 7], [1301, 1301, -15*w - 7], [1319, 1319, 18*w - 25], [1319, 1319, -18*w - 25], [1321, 1321, -20*w + 61], [1321, 1321, -22*w + 65], [1367, 1367, -16*w - 13], [1367, 1367, 16*w - 13], [1369, 37, -37], [1373, 1373, 17*w - 19], [1373, 1373, -17*w - 19], [1439, 1439, 20*w - 31], [1439, 1439, 38*w - 85], [1459, 1459, -15*w + 53], [1459, 1459, -31*w + 85], [1483, 1483, -11*w + 47], [1483, 1483, 39*w - 103], [1487, 1487, -16*w - 7], [1487, 1487, 16*w - 7], [1489, 1489, -16*w + 55], [1489, 1489, -30*w + 83], [1493, 1493, -33*w + 71], [1493, 1493, 23*w - 41], [1511, 1511, 16*w - 5], [1511, 1511, -16*w - 5], [1531, 1531, 5*w - 41], [1531, 1531, -5*w - 41], [1559, 1559, 42*w - 95], [1559, 1559, 20*w - 29], [1579, 1579, -25*w + 73], [1579, 1579, -21*w + 65], [1583, 1583, -18*w - 19], [1583, 1583, 18*w - 19], [1607, 1607, 24*w - 43], [1607, 1607, -34*w + 73], [1609, 1609, 10*w - 47], [1609, 1609, -10*w - 47], [1613, 1613, 17*w - 11], [1613, 1613, -17*w - 11], [1627, 1627, -3*w - 41], [1627, 1627, 3*w - 41], [1637, 1637, -19*w - 23], [1637, 1637, 19*w - 23], [1657, 1657, 2*w - 41], [1657, 1657, -2*w - 41], [1681, 41, -41], [1699, 1699, 5*w - 43], [1699, 1699, -5*w - 43], [1709, 1709, -17*w - 5], [1709, 1709, 17*w - 5], [1723, 1723, -9*w - 47], [1723, 1723, 9*w - 47], [1733, 1733, -17*w - 1], [1733, 1733, 17*w - 1], [1747, 1747, -33*w + 91], [1747, 1747, -17*w + 59], [1753, 1753, -4*w - 43], [1753, 1753, 4*w - 43], [1777, 1777, -32*w + 89], [1777, 1777, -18*w + 61], [1801, 1801, -10*w - 49], [1801, 1801, 10*w - 49], [1823, 1823, -18*w - 11], [1823, 1823, 18*w - 11], [1847, 1847, -36*w + 77], [1847, 1847, 26*w - 47], [1867, 1867, -27*w + 79], [1867, 1867, -23*w + 71], [1871, 1871, -20*w - 23], [1871, 1871, 20*w - 23], [1873, 1873, -24*w + 73], [1873, 1873, -26*w + 77], [1877, 1877, -19*w - 17], [1877, 1877, 19*w - 17], [1901, 1901, -39*w + 85], [1901, 1901, 25*w - 43], [1949, 1949, 45*w - 101], [1949, 1949, 23*w - 35], [1973, 1973, 27*w - 49], [1973, 1973, -37*w + 79], [1987, 1987, -37*w + 101], [1987, 1987, -17*w + 61], [1993, 1993, -6*w - 47], [1993, 1993, 6*w - 47], [1997, 1997, -19*w - 13], [1997, 1997, 19*w - 13], [2011, 2011, 45*w - 119], [2011, 2011, -13*w + 55], [2017, 2017, 8*w - 49], [2017, 2017, -8*w - 49], [2039, 2039, 20*w - 19], [2039, 2039, -20*w - 19], [2063, 2063, 22*w - 29], [2063, 2063, -22*w - 29], [2069, 2069, -43*w + 95], [2069, 2069, 25*w - 41], [2083, 2083, 11*w - 53], [2083, 2083, -11*w - 53], [2087, 2087, 24*w - 37], [2087, 2087, 46*w - 103], [2089, 2089, -20*w + 67], [2089, 2089, -34*w + 95], [2111, 2111, 20*w - 17], [2111, 2111, -20*w - 17], [2113, 2113, -4*w - 47], [2113, 2113, 4*w - 47], [2131, 2131, -15*w - 59], [2131, 2131, 15*w - 59], [2137, 2137, -22*w + 71], [2137, 2137, -32*w + 91], [2141, 2141, -19*w - 5], [2141, 2141, 19*w - 5], [2161, 2161, -12*w - 55], [2161, 2161, 12*w - 55], [2179, 2179, -29*w + 85], [2179, 2179, -25*w + 77], [2203, 2203, -w - 47], [2203, 2203, w - 47], [2207, 2207, 26*w - 43], [2207, 2207, -44*w + 97], [2213, 2213, -23*w - 31], [2213, 2213, 23*w - 31], [2237, 2237, 29*w - 53], [2237, 2237, -39*w + 83], [2251, 2251, -5*w - 49], [2251, 2251, 5*w - 49], [2281, 2281, -18*w + 65], [2281, 2281, -40*w + 109], [2309, 2309, 33*w - 65], [2309, 2309, -35*w + 71], [2333, 2333, -23*w - 29], [2333, 2333, 23*w - 29], [2347, 2347, 3*w - 49], [2347, 2347, -3*w - 49], [2351, 2351, -20*w - 7], [2351, 2351, 20*w - 7], [2357, 2357, -21*w - 17], [2357, 2357, 21*w - 17], [2371, 2371, -15*w + 61], [2371, 2371, 47*w - 125], [2377, 2377, 2*w - 49], [2377, 2377, -2*w - 49], [2381, 2381, 25*w - 37], [2381, 2381, 51*w - 115], [2399, 2399, -20*w - 1], [2399, 2399, 20*w - 1], [2423, 2423, 32*w - 61], [2423, 2423, -38*w + 79], [2447, 2447, 34*w - 67], [2447, 2447, -36*w + 73], [2467, 2467, 13*w - 59], [2467, 2467, -13*w - 59], [2473, 2473, -24*w + 77], [2473, 2473, -34*w + 97], [2477, 2477, 21*w - 13], [2477, 2477, -21*w - 13], [2521, 2521, -28*w + 85], [2521, 2521, -30*w + 89], [2539, 2539, 9*w - 55], [2539, 2539, -9*w - 55], [2543, 2543, -22*w - 19], [2543, 2543, 22*w - 19], [2549, 2549, 23*w - 25], [2549, 2549, -23*w - 25], [2591, 2591, 30*w - 53], [2591, 2591, -44*w + 95], [2593, 2593, -6*w - 53], [2593, 2593, 6*w - 53], [2617, 2617, -12*w - 59], [2617, 2617, 12*w - 59], [2621, 2621, 21*w - 5], [2621, 2621, -21*w - 5], [2659, 2659, 5*w - 53], [2659, 2659, -5*w - 53], [2663, 2663, 32*w - 59], [2663, 2663, -42*w + 89], [2683, 2683, -41*w + 113], [2683, 2683, -21*w + 73], [2687, 2687, 26*w - 37], [2687, 2687, 56*w - 127], [2689, 2689, -16*w + 65], [2689, 2689, 50*w - 133], [2693, 2693, 53*w - 119], [2693, 2693, 27*w - 41], [2707, 2707, 13*w - 61], [2707, 2707, -13*w - 61], [2711, 2711, -40*w + 83], [2711, 2711, 34*w - 65], [2713, 2713, 4*w - 53], [2713, 2713, -4*w - 53], [2731, 2731, -7*w - 55], [2731, 2731, 7*w - 55], [2741, 2741, 31*w - 55], [2741, 2741, -45*w + 97], [2789, 2789, 25*w - 31], [2789, 2789, -25*w - 31], [2803, 2803, -w - 53], [2803, 2803, w - 53], [2833, 2833, -36*w + 103], [2833, 2833, -26*w + 83], [2837, 2837, 29*w - 47], [2837, 2837, -51*w + 113], [2851, 2851, -27*w + 85], [2851, 2851, -35*w + 101], [2857, 2857, 12*w - 61], [2857, 2857, -12*w - 61], [2861, 2861, 35*w - 67], [2861, 2861, -41*w + 85], [2879, 2879, -22*w - 5], [2879, 2879, 22*w - 5], [2903, 2903, -22*w - 1], [2903, 2903, 22*w - 1], [2909, 2909, 25*w - 29], [2909, 2909, -25*w - 29], [2927, 2927, 24*w - 23], [2927, 2927, -24*w - 23], [2953, 2953, 54*w - 143], [2953, 2953, -16*w + 67], [2957, 2957, 31*w - 53], [2957, 2957, -49*w + 107], [2971, 2971, 3*w - 55], [2971, 2971, -3*w - 55], [2999, 2999, 30*w - 49], [2999, 2999, -52*w + 115], [3001, 3001, 2*w - 55], [3001, 3001, -2*w - 55], [3019, 3019, -w - 55], [3019, 3019, w - 55], [3023, 3023, 58*w - 131], [3023, 3023, 28*w - 41], [3049, 3049, -14*w - 65], [3049, 3049, 14*w - 65], [3067, 3067, -23*w + 79], [3067, 3067, -43*w + 119], [3119, 3119, -50*w + 109], [3119, 3119, 32*w - 55], [3121, 3121, 10*w - 61], [3121, 3121, -10*w - 61], [3163, 3163, 19*w - 73], [3163, 3163, -19*w - 73], [3167, 3167, 24*w - 17], [3167, 3167, -24*w - 17], [3169, 3169, -40*w + 113], [3169, 3169, -26*w + 85], [3187, 3187, -7*w - 59], [3187, 3187, 7*w - 59], [3191, 3191, 30*w - 47], [3191, 3191, 56*w - 125], [3217, 3217, -38*w + 109], [3217, 3217, -28*w + 89], [3221, 3221, -25*w - 23], [3221, 3221, 25*w - 23], [3259, 3259, -35*w + 103], [3259, 3259, -31*w + 95], [3307, 3307, -17*w + 71], [3307, 3307, 57*w - 151], [3313, 3313, -14*w - 67], [3313, 3313, 14*w - 67], [3331, 3331, -5*w - 59], [3331, 3331, 5*w - 59], [3359, 3359, 40*w - 79], [3359, 3359, -42*w + 85], [3361, 3361, 12*w - 65], [3361, 3361, -12*w - 65], [3389, 3389, -25*w - 19], [3389, 3389, 25*w - 19], [3407, 3407, -24*w - 7], [3407, 3407, 24*w - 7], [3413, 3413, -27*w - 31], [3413, 3413, 27*w - 31], [3433, 3433, -24*w + 83], [3433, 3433, -46*w + 127], [3457, 3457, 2*w - 59], [3457, 3457, -2*w - 59], [3461, 3461, -25*w - 17], [3461, 3461, 25*w - 17], [3481, 59, -59], [3499, 3499, -11*w - 65], [3499, 3499, 11*w - 65], [3527, 3527, 26*w - 23], [3527, 3527, -26*w - 23], [3529, 3529, -20*w - 77], [3529, 3529, 20*w - 77], [3533, 3533, 27*w - 29], [3533, 3533, -27*w - 29], [3547, 3547, -43*w + 121], [3547, 3547, -27*w + 89], [3557, 3557, 61*w - 137], [3557, 3557, 31*w - 47], [3571, 3571, 5*w - 61], [3571, 3571, -5*w - 61], [3581, 3581, -25*w - 13], [3581, 3581, 25*w - 13], [3623, 3623, -48*w + 101], [3623, 3623, 38*w - 71], [3643, 3643, -31*w + 97], [3643, 3643, -39*w + 113], [3671, 3671, -46*w + 95], [3671, 3671, 40*w - 77], [3673, 3673, -34*w + 103], [3673, 3673, -36*w + 107], [3677, 3677, -29*w - 37], [3677, 3677, 29*w - 37], [3691, 3691, -15*w - 71], [3691, 3691, 15*w - 71], [3697, 3697, 2*w - 61], [3697, 3697, -2*w - 61], [3701, 3701, 25*w - 7], [3701, 3701, -25*w - 7], [3719, 3719, -30*w - 41], [3719, 3719, 30*w - 41], [3721, 61, -61], [3739, 3739, -9*w - 65], [3739, 3739, 9*w - 65], [3767, 3767, 26*w - 17], [3767, 3767, -26*w - 17], [3769, 3769, -50*w + 137], [3769, 3769, -24*w + 85], [3793, 3793, -16*w - 73], [3793, 3793, 16*w - 73], [3797, 3797, 39*w - 73], [3797, 3797, -49*w + 103], [3821, 3821, -29*w - 35], [3821, 3821, 29*w - 35], [3863, 3863, -28*w - 29], [3863, 3863, 28*w - 29], [3889, 3889, 10*w - 67], [3889, 3889, -10*w - 67], [3907, 3907, -47*w + 131], [3907, 3907, -27*w + 91], [3911, 3911, -60*w + 133], [3911, 3911, 34*w - 55], [3917, 3917, -31*w - 43], [3917, 3917, 31*w - 43], [3931, 3931, -7*w - 65], [3931, 3931, 7*w - 65], [3989, 3989, -55*w + 119], [3989, 3989, 37*w - 65], [4003, 4003, -9*w - 67], [4003, 4003, 9*w - 67], [4007, 4007, 26*w - 7], [4007, 4007, -26*w - 7], [4013, 4013, -27*w - 19], [4013, 4013, 27*w - 19], [4027, 4027, -13*w - 71], [4027, 4027, 13*w - 71], [4051, 4051, -55*w + 149], [4051, 4051, -23*w + 85], [4057, 4057, -32*w + 101], [4057, 4057, -42*w + 121], [4079, 4079, -28*w - 25], [4079, 4079, 28*w - 25], [4099, 4099, -39*w + 115], [4099, 4099, -35*w + 107], [4127, 4127, 64*w - 143], [4127, 4127, 34*w - 53], [4129, 4129, 4*w - 65], [4129, 4129, -4*w - 65], [4133, 4133, 33*w - 49], [4133, 4133, 67*w - 151], [4153, 4153, 14*w - 73], [4153, 4153, -14*w - 73], [4157, 4157, -51*w + 107], [4157, 4157, 41*w - 77], [4177, 4177, -12*w - 71], [4177, 4177, 12*w - 71], [4201, 4201, 2*w - 65], [4201, 4201, -2*w - 65], [4219, 4219, -w - 65], [4219, 4219, w - 65], [4229, 4229, 45*w - 89], [4229, 4229, -47*w + 95], [4243, 4243, -21*w - 83], [4243, 4243, 21*w - 83], [4253, 4253, -27*w - 11], [4253, 4253, 27*w - 11], [4271, 4271, -54*w + 115], [4271, 4271, 40*w - 73], [4273, 4273, 6*w - 67], [4273, 4273, -6*w - 67], [4297, 4297, -18*w - 79], [4297, 4297, 18*w - 79], [4339, 4339, -5*w - 67], [4339, 4339, 5*w - 67], [4349, 4349, -27*w - 5], [4349, 4349, 27*w - 5], [4363, 4363, -29*w + 97], [4363, 4363, -49*w + 137], [4373, 4373, -27*w - 1], [4373, 4373, 27*w - 1], [4391, 4391, 44*w - 85], [4391, 4391, -50*w + 103], [4397, 4397, 31*w - 37], [4397, 4397, -31*w - 37], [4421, 4421, -29*w - 25], [4421, 4421, 29*w - 25], [4441, 4441, 10*w - 71], [4441, 4441, -10*w - 71], [4463, 4463, 32*w - 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8209, 20*w - 103], [8209, 8209, -20*w - 103], [8231, 8231, -40*w - 37], [8231, 8231, 40*w - 37], [8233, 8233, -14*w - 97], [8233, 8233, 14*w - 97], [8237, 8237, 41*w - 43], [8237, 8237, -41*w - 43], [8329, 8329, -60*w + 173], [8329, 8329, -46*w + 145], [8353, 8353, -74*w + 203], [8353, 8353, -36*w + 127], [8377, 8377, -58*w + 169], [8377, 8377, -48*w + 149], [8419, 8419, -55*w + 163], [8419, 8419, -51*w + 155], [8423, 8423, 62*w - 121], [8423, 8423, -68*w + 139], [8429, 8429, 45*w - 61], [8429, 8429, -45*w - 61], [8443, 8443, -19*w - 103], [8443, 8443, 19*w - 103], [8447, 8447, 64*w - 127], [8447, 8447, -66*w + 133], [8467, 8467, -17*w - 101], [8467, 8467, 17*w - 101], [8501, 8501, 39*w - 25], [8501, 8501, -39*w - 25], [8521, 8521, 90*w - 239], [8521, 8521, -28*w + 115], [8539, 8539, 9*w - 95], [8539, 8539, -9*w - 95], [8543, 8543, -38*w - 11], [8543, 8543, 38*w - 11], [8563, 8563, -71*w + 197], [8563, 8563, -39*w + 133], [8573, 8573, -83*w + 181], [8573, 8573, 53*w - 91], [8597, 8597, -39*w - 23], [8597, 8597, 39*w - 23], [8641, 8641, -8*w - 95], [8641, 8641, 8*w - 95], [8663, 8663, -38*w - 1], [8663, 8663, 38*w - 1], [8669, 8669, 45*w - 59], [8669, 8669, -45*w - 59], [8689, 8689, -34*w + 125], [8689, 8689, -80*w + 217], [8693, 8693, 43*w - 49], [8693, 8693, -43*w - 49], [8707, 8707, -23*w - 109], [8707, 8707, 23*w - 109], [8713, 8713, 26*w - 113], [8713, 8713, -26*w - 113], [8731, 8731, 7*w - 95], [8731, 8731, -7*w - 95], [8737, 8737, -42*w + 139], [8737, 8737, -68*w + 191], [8741, 8741, 55*w - 97], [8741, 8741, -81*w + 175], [8761, 8761, -30*w - 119], [8761, 8761, 30*w - 119], [8779, 8779, -35*w + 127], [8779, 8779, -79*w + 215], [8783, 8783, 98*w - 221], [8783, 8783, 48*w - 71], [8803, 8803, -21*w - 107], [8803, 8803, 21*w - 107], [8807, 8807, -44*w - 53], [8807, 8807, 44*w - 53], [8831, 8831, -74*w + 155], [8831, 8831, 60*w - 113], [8837, 8837, 39*w - 17], [8837, 8837, -39*w - 17], [8861, 8861, 41*w - 35], [8861, 8861, -41*w - 35], [8923, 8923, 9*w - 97], [8923, 8923, -9*w - 97], [8929, 8929, -4*w - 95], [8929, 8929, 4*w - 95], [8933, 8933, 53*w - 89], [8933, 8933, -87*w + 191], [8951, 8951, -70*w + 143], [8951, 8951, 64*w - 125], [8971, 8971, -3*w - 95], [8971, 8971, 3*w - 95], [8999, 8999, 52*w - 85], [8999, 8999, -90*w + 199], [9001, 9001, 2*w - 95], [9001, 9001, -2*w - 95], [9029, 9029, 47*w - 65], [9029, 9029, -47*w - 65], [9043, 9043, -51*w + 157], [9043, 9043, -59*w + 173], [9049, 9049, 20*w - 107], [9049, 9049, -20*w - 107], [9067, 9067, -57*w + 169], [9067, 9067, -53*w + 161], [9091, 9091, -33*w + 125], [9091, 9091, -85*w + 229], [9173, 9173, -73*w + 151], [9173, 9173, 63*w - 121], [9187, 9187, 13*w - 101], [9187, 9187, -13*w - 101], [9221, 9221, 65*w - 127], [9221, 9221, -71*w + 145], [9239, 9239, -40*w - 19], [9239, 9239, 40*w - 19], [9241, 9241, 30*w - 121], [9241, 9241, -30*w - 121], [9283, 9283, 19*w - 107], [9283, 9283, -19*w - 107], [9293, 9293, 57*w - 101], [9293, 9293, -83*w + 179], [9311, 9311, 40*w - 17], [9311, 9311, -40*w - 17], [9337, 9337, 12*w - 101], [9337, 9337, -12*w - 101], [9341, 9341, -45*w - 53], [9341, 9341, 45*w - 53], [9403, 9403, -w - 97], [9403, 9403, w - 97], [9413, 9413, -43*w - 41], [9413, 9413, 43*w - 41], [9431, 9431, 40*w - 13], [9431, 9431, -40*w - 13], [9433, 9433, 14*w - 103], [9433, 9433, -14*w - 103], [9437, 9437, -81*w + 173], [9437, 9437, 59*w - 107], [9461, 9461, 41*w - 25], [9461, 9461, -41*w - 25], [9479, 9479, -40*w - 11], [9479, 9479, 40*w - 11], [9533, 9533, 47*w - 61], [9533, 9533, -47*w - 61], [9547, 9547, -67*w + 191], [9547, 9547, -47*w + 151], [9551, 9551, 40*w - 7], [9551, 9551, -40*w - 7], [9601, 9601, 10*w - 101], [9601, 9601, -10*w - 101], [9619, 9619, -65*w + 187], [9619, 9619, -49*w + 155], [9623, 9623, -42*w - 31], [9623, 9623, 42*w - 31], [9629, 9629, 95*w - 211], [9629, 9629, 53*w - 85], [9643, 9643, -39*w + 137], [9643, 9643, -79*w + 217], [9649, 9649, -64*w + 185], [9649, 9649, -50*w + 157], [9677, 9677, 101*w - 227], [9677, 9677, 51*w - 77], [9697, 9697, -52*w + 161], [9697, 9697, -62*w + 181], [9719, 9719, 60*w - 109], [9719, 9719, -82*w + 175], [9721, 9721, -40*w + 139], [9721, 9721, -78*w + 215], [9739, 9739, -59*w + 175], [9739, 9739, -55*w + 167], [9743, 9743, 42*w - 29], [9743, 9743, -42*w - 29], [9749, 9749, 45*w - 49], [9749, 9749, -45*w - 49], [9767, 9767, 44*w - 43], [9767, 9767, -44*w - 43], [9769, 9769, -24*w - 115], [9769, 9769, 24*w - 115], [9787, 9787, 27*w - 119], [9787, 9787, -27*w - 119], [9791, 9791, -90*w + 197], [9791, 9791, 56*w - 95], [9811, 9811, -87*w + 235], [9811, 9811, -35*w + 131], [9817, 9817, -8*w - 101], [9817, 9817, 8*w - 101], [9839, 9839, 62*w - 115], [9839, 9839, -80*w + 169], [9859, 9859, 31*w - 125], [9859, 9859, -31*w - 125], [9883, 9883, 11*w - 103], [9883, 9883, -11*w - 103], [9887, 9887, 46*w - 53], [9887, 9887, -46*w - 53], [9907, 9907, -7*w - 101], [9907, 9907, 7*w - 101], [9931, 9931, -75*w + 209], [9931, 9931, -43*w + 145], [9941, 9941, 45*w - 47], [9941, 9941, -45*w - 47]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [0, 2, -1, -1, 4, 4, -6, -6, 6, 6, 10, 10, 6, 6, -10, -6, -6, -2, -2, 12, 12, 2, 2, 2, 2, 6, 6, -22, 4, 4, -6, -6, 10, 10, -18, -18, -22, -6, -6, 12, 12, 26, 26, 18, 18, 16, 16, 24, 24, 14, 14, 18, 18, 18, 18, -14, -14, 2, -30, -30, -2, -2, -12, -12, -22, -22, -6, -6, -8, -8, 2, 2, -24, -24, 28, 28, -6, -6, -6, -6, -34, -34, -36, -36, 2, 2, 26, 26, 30, 30, 24, 24, 4, 4, 18, 18, 6, 6, -14, -14, -26, -26, -30, -30, -8, -8, -22, -22, 0, 0, -10, -10, -20, -20, -14, -14, -42, -42, 42, 42, -46, -46, 18, 18, -8, -8, -30, -30, 24, 24, -20, -20, -6, -6, 2, 2, -30, -30, -26, -26, 42, 42, 16, 16, -54, -54, 48, 48, 4, 4, -6, -6, -14, -14, -18, -18, 46, 46, -12, -12, -22, -22, -18, -18, -46, 18, 18, 50, 50, 18, 18, -12, -12, 2, 2, -8, -8, 54, 54, 42, 42, 30, 30, -14, -14, 62, 62, -12, -12, -22, -22, 40, 40, -30, -30, 14, 14, -30, -30, 6, 6, -46, -46, -6, -6, -56, -56, 26, 26, 42, 42, 48, 48, -10, -10, -18, -18, -70, 18, 18, 0, 0, 52, 52, -62, -62, 30, 30, -58, -58, 42, 42, -60, -60, 64, 64, -24, -24, 4, 4, -78, -78, 6, 6, 14, 14, -6, -6, 46, 46, 42, 42, -70, -70, -46, -44, -44, 6, 6, 58, 58, -78, -78, -2, -2, 26, 26, -22, -22, 38, 38, 42, 42, -42, -42, -74, -74, 36, 36, 26, 26, -30, -30, 18, 18, 18, 18, -78, -78, 46, 46, 50, 50, -54, -54, -32, -32, 2, 2, -24, -24, -54, -54, 30, 30, 82, 82, 30, 30, 74, 74, -12, -12, 26, 26, -8, -8, 2, 2, -30, -30, -34, -34, 52, 52, 10, 10, 54, 54, -54, -54, 42, 42, 16, 16, 86, 86, -54, -54, 18, 18, 46, 46, 36, 36, 18, 18, -80, -80, 74, 74, -78, -78, -72, -72, -54, -54, 54, 54, 70, 70, 74, 74, -30, -30, -10, -10, -68, -68, -78, -78, 90, 90, -12, -12, -46, -46, 2, 2, -18, -18, 4, 4, -6, -6, -38, -38, -42, -42, -10, -10, 42, 42, -50, -50, -36, -36, -46, -46, -32, -32, -54, -54, 90, 90, -38, -38, -46, -46, 18, 18, -32, -32, -46, -46, -66, -66, -96, -96, -6, -6, -30, -30, 6, 6, 74, 74, -102, -102, 88, 88, 24, 24, -70, -70, -44, -44, 66, 66, 26, 26, -2, -2, 72, 72, 2, 2, -14, -14, -66, -66, 50, 50, -2, -2, -108, -108, 50, 50, 42, 42, -20, -20, 46, 46, -46, -46, 88, 88, -72, -72, -34, -34, -90, -90, -18, -18, -102, -102, 26, 26, -70, -70, -42, -42, 26, -68, -68, -18, -18, -22, -22, 90, 90, 94, 94, 42, 42, 88, 88, -66, -66, 18, 18, 34, 34, -12, -12, 50, 50, 18, 18, 88, 88, -46, -46, 42, 42, 72, 72, -118, 76, 76, 102, 102, -34, -34, -22, -22, 18, 18, 30, 30, 66, 66, 38, 38, 22, 22, 60, 60, -78, -78, -80, -80, 78, 78, 34, 34, -114, -114, 42, 42, -50, -50, -56, -56, 26, 26, 48, 48, -44, -44, 54, 54, -10, -10, 66, 66, 50, 50, -30, -30, 2, 2, -22, -22, 76, 76, 30, 30, 106, 106, 42, 42, -60, -60, -46, -46, -22, -22, -20, -20, 18, 18, 106, 106, 18, 18, 60, 60, -30, -30, 90, 90, -118, -118, -6, -6, 106, 106, 42, 42, -26, -26, -22, -22, -54, -54, 50, 50, -6, -6, -62, -62, 114, 114, -8, -8, 122, 122, 48, 48, -126, -126, 58, 58, 122, 122, -78, -78, 84, 84, -24, -24, -130, -130, -12, -12, -126, -126, 48, 48, -54, -54, 54, 54, -34, -34, 42, 42, 22, 22, 98, 98, -80, -80, 78, 78, 24, 24, -116, -116, -90, -90, -26, -26, -70, -70, -20, -20, 90, 90, 26, 26, -122, -122, -60, -60, 74, 74, 42, 42, 78, 78, 96, 96, 50, 50, -102, -102, -126, -126, 130, 130, -54, -54, 70, 70, 36, 36, -6, -6, 72, 72, 52, 52, 10, 10, 14, 14, 84, 84, -30, -30, -126, -126, 48, 48, 62, 62, -86, -86, 134, 134, -6, -6, -108, -108, 0, 0, -10, -10, -20, -20, 30, 30, 34, 34, 74, 74, -126, -126, 108, 108, -102, -102, -94, -94, 30, 30, 100, 100, 66, 66, -114, -114, 42, 42, -2, -2, -56, -56, 50, 50, -42, -42, 0, 0, 134, 134, 138, 138, -86, -86, -42, -42, -94, -94, 30, 30, 66, 66, 10, 10, 6, 6, -6, -6, 118, 118, 50, 50, 64, 64, 138, 138, -58, -58, 114, 114, -134, -134, -6, -6, -102, -102, -80, -80, 26, 26, 30, 30, -94, -126, -126, -78, -78, 150, 150, -60, -60, -102, -102, 98, 98, 48, 48, -118, -118, -44, -44, 78, 78, 70, 70, 64, 64, -142, -142, 50, 50, -122, -122, 84, 84, -46, -46, 64, 64, -118, -118, 6, 6, -48, -48, 100, 100, -54, -54, -94, -94, -152, -152, 114, 114, 96, 96, -14, -14, -60, -60, 122, 122, -22, -22, 114, 114, 126, 126, -86, -86, -130, -50, -50, -36, -36, 66, 66, -24, -24, 50, 50, 66, 66, 114, 114, -98, -98, 74, 74, 0, 0, -102, -102, -54, -54, -66, -66, -34, -34, -12, -12, -142, -142, 28, 28, 6, 6, -14, -14, -162, -162, -30, -30, -118, -118, 122, 122, 78, 78, -118, -118, 26, 26, 16, 16, 74, 74, 124, 124, 6, 6, 86, 86, 46, 46, 90, 90, -22, -22, -42, -42, 96, 96, -58, -58, 114, 114, -114, -114, 154, 154, -90, -90, 110, 110, -140, -140, 114, 114, -38, -38, -42, -42, -94, -94, -78, -78, -6, -6, -6, -6, -54, -54, -2, -2, 98, 98, 42, 42, -30, -30, 120, 120, -142, -90, -90, 82, 82, 74, 74, -56, -56, -142, -142, 0, 0, 4, 4, -114, -114, -90, -90, 62, 62, -174, -174, -36, -36, -6, -6, 50, 50, -140, -140, 50, 50, 12, 12, -94, -94, -102, -102, 26, 26, -70, -70, 98, 98, 148, 148, 138, 138, -30, -30, 10, 10, -42, -42, -2, -2, -54, -54, -154, -154, 28, 28, 66, 66, 106, 106, -30, -30, 42, 42, -46, -46, -126, -126, 54, 54, -142, -142, 90, 90, -50, -50, -70, -70, 136, 136, -118, -118, -42, -42, -58, -58, 4, 4, -126, -126, 34, 34, 30, 30, -132, -132, -78, -78, -30, -30, -14, -14, -106, -106, -6, -6, 132, 132, -32, -32, -120, -120, 170, 170, -66, -66, 10, 10, 110, 110, -122, -122, 160, 160, 186, 186, 46, 46, -102, -102, -120, -120, 182, 182, -14, -14, 18, 18, -132, -132, 50, 50, 66, 66, -134, -134, -126, -126, -180, -180, -70, -70, 90, 90, -150, -150, 120, 120, -6, -6, -194, -194, 108, 108, -130, -130, 100, 100, -6, -6, -42, -42, 10, 10, -106, -106, 42, 42, 74, 74, -144, -144, 38, 38, -92, -92, -78, -78, 78, 78, -66, -66, -70, -70, 166, 166, -36, -36, -56, -56, 170, 170, 0, 0, 76, 76, 58, 58, -90, -90, 142, 142, 112, 112, -42, -42]; 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;