/* 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![-11, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, -w + 3], [5, 5, w - 4], [5, 5, -w - 4], [7, 7, w + 2], [7, 7, w - 2], [9, 3, 3], [11, 11, -w], [19, 19, 2*w - 5], [19, 19, -2*w - 5], [37, 37, 2*w - 9], [37, 37, -2*w - 9], [43, 43, 2*w - 1], [43, 43, -2*w - 1], [53, 53, -w - 8], [53, 53, w - 8], [79, 79, 5*w - 14], [79, 79, 8*w - 25], [83, 83, -3*w - 4], [83, 83, 3*w - 4], [89, 89, -w - 10], [89, 89, w - 10], [97, 97, -3*w - 14], [97, 97, 3*w - 14], [107, 107, 6*w - 17], [107, 107, 9*w - 28], [113, 113, -4*w + 17], [113, 113, 11*w - 38], [127, 127, 4*w - 7], [127, 127, -4*w - 7], [131, 131, 5*w - 12], [131, 131, 14*w - 45], [137, 137, -7*w + 26], [137, 137, 8*w - 29], [139, 139, 7*w - 20], [139, 139, 10*w - 31], [151, 151, -4*w - 5], [151, 151, 4*w - 5], [157, 157, 3*w - 16], [157, 157, -3*w - 16], [167, 167, 4*w - 3], [167, 167, -4*w - 3], [169, 13, -13], [181, 181, 2*w - 15], [181, 181, -2*w - 15], [211, 211, -5*w - 8], [211, 211, 5*w - 8], [227, 227, -6*w - 13], [227, 227, 6*w - 13], [229, 229, -6*w + 25], [229, 229, 15*w - 52], [239, 239, -5*w - 6], [239, 239, 5*w - 6], [257, 257, 13*w - 46], [257, 257, -8*w + 31], [263, 263, 8*w - 21], [263, 263, 17*w - 54], [269, 269, -10*w + 37], [269, 269, 11*w - 40], [271, 271, 5*w - 2], [271, 271, -5*w - 2], [283, 283, 7*w - 16], [283, 283, -7*w - 16], [289, 17, -17], [307, 307, 11*w - 32], [307, 307, 14*w - 43], [313, 313, -w - 18], [313, 313, w - 18], [317, 317, 2*w - 19], [317, 317, -2*w - 19], [347, 347, -6*w - 7], [347, 347, 6*w - 7], [353, 353, -4*w - 23], [353, 353, 4*w - 23], [359, 359, 12*w - 35], [359, 359, 15*w - 46], [389, 389, -w - 20], [389, 389, w - 20], [397, 397, 2*w - 21], [397, 397, -2*w - 21], [401, 401, 5*w - 26], [401, 401, -5*w - 26], [421, 421, 17*w - 60], [421, 421, -10*w + 39], [431, 431, 11*w - 30], [431, 431, 20*w - 63], [433, 433, -11*w + 42], [433, 433, 16*w - 57], [439, 439, 7*w - 10], [439, 439, -7*w - 10], [449, 449, 4*w - 25], [449, 449, -4*w - 25], [479, 479, 8*w - 15], [479, 479, -8*w - 15], [491, 491, 9*w - 20], [491, 491, -9*w - 20], [503, 503, 7*w - 6], [503, 503, -7*w - 6], [509, 509, 5*w - 28], [509, 509, -5*w - 28], [521, 521, 8*w - 35], [521, 521, -8*w - 35], [523, 523, -7*w - 4], [523, 523, 7*w - 4], [529, 23, -23], [547, 547, 11*w + 28], [547, 547, -11*w + 28], [563, 563, 22*w - 69], [563, 563, 13*w - 36], [571, 571, 10*w - 23], [571, 571, -10*w - 23], [577, 577, -3*w - 26], [577, 577, 3*w - 26], [607, 607, 16*w - 47], [607, 607, 19*w - 58], [617, 617, -7*w - 34], [617, 617, 7*w - 34], [641, 641, -13*w + 50], [641, 641, 20*w - 71], [653, 653, 19*w - 68], [653, 653, -14*w + 53], [659, 659, -10*w - 21], [659, 659, 10*w - 21], [661, 661, 18*w - 65], [661, 661, -15*w + 56], [709, 709, -9*w - 40], [709, 709, 9*w - 40], [739, 739, 10*w - 19], [739, 739, -10*w - 19], [743, 743, 33*w - 106], [743, 743, 12*w - 29], [757, 757, 7*w - 36], [757, 757, -7*w - 36], [773, 773, -w - 28], [773, 773, w - 28], [787, 787, 14*w - 37], [787, 787, 29*w - 92], [797, 797, 2*w - 29], [797, 797, -2*w - 29], [811, 811, -10*w - 17], [811, 811, 10*w - 17], [827, 827, 9*w - 8], [827, 827, -9*w - 8], [829, 829, -6*w - 35], [829, 829, 6*w - 35], [841, 29, -29], [881, 881, -5*w - 34], [881, 881, 5*w - 34], [887, 887, 9*w - 2], [887, 887, -9*w - 2], [919, 919, 20*w - 59], [919, 919, 23*w - 70], [929, 929, -19*w + 70], [929, 929, 20*w - 73], [961, 31, -31], [967, 967, 31*w - 98], [967, 967, 16*w - 43], [977, 977, 8*w - 41], [977, 977, -8*w - 41], [1013, 1013, -w - 32], [1013, 1013, w - 32], [1019, 1019, 10*w - 9], [1019, 1019, -10*w - 9], [1021, 1021, 5*w - 36], [1021, 1021, -5*w - 36], [1031, 1031, 15*w + 38], [1031, 1031, -15*w + 38], [1049, 1049, -4*w - 35], [1049, 1049, 4*w - 35], [1051, 1051, -10*w - 7], [1051, 1051, 10*w - 7], [1061, 1061, 7*w - 40], [1061, 1061, -7*w - 40], [1063, 1063, 17*w - 46], [1063, 1063, 32*w - 101], [1091, 1091, 10*w - 3], [1091, 1091, -10*w - 3], [1093, 1093, -14*w + 57], [1093, 1093, 31*w - 108], [1109, 1109, 10*w - 47], [1109, 1109, -41*w + 140], [1151, 1151, 20*w - 57], [1151, 1151, 29*w - 90], [1153, 1153, 29*w - 102], [1153, 1153, -16*w + 63], [1163, 1163, 33*w - 104], [1163, 1163, 18*w - 49], [1181, 1181, 2*w - 35], [1181, 1181, -2*w - 35], [1187, 1187, -11*w - 12], [1187, 1187, 11*w - 12], [1193, 1193, 4*w - 37], [1193, 1193, -4*w - 37], [1213, 1213, 26*w - 93], [1213, 1213, -19*w + 72], [1223, 1223, 12*w - 19], [1223, 1223, -12*w - 19], [1231, 1231, 11*w - 10], [1231, 1231, -11*w - 10], [1237, 1237, -22*w + 81], [1237, 1237, 23*w - 84], [1277, 1277, 38*w - 131], [1277, 1277, -13*w + 56], [1283, 1283, -13*w - 24], [1283, 1283, 13*w - 24], [1301, 1301, -10*w - 49], [1301, 1301, 10*w - 49], [1319, 1319, -15*w - 34], [1319, 1319, 15*w - 34], [1321, 1321, -8*w - 45], [1321, 1321, 8*w - 45], [1327, 1327, 11*w - 2], [1327, 1327, -11*w - 2], [1373, 1373, -46*w + 157], [1373, 1373, 11*w - 52], [1399, 1399, 25*w - 74], [1399, 1399, 28*w - 85], [1409, 1409, -16*w + 65], [1409, 1409, 35*w - 122], [1427, 1427, 14*w - 27], [1427, 1427, -14*w - 27], [1433, 1433, -w - 38], [1433, 1433, w - 38], [1447, 1447, 16*w - 37], [1447, 1447, 49*w - 158], [1451, 1451, 15*w - 32], [1451, 1451, -15*w - 32], [1453, 1453, 6*w - 43], [1453, 1453, -6*w - 43], [1459, 1459, -13*w - 20], [1459, 1459, 13*w - 20], [1471, 1471, 40*w - 127], [1471, 1471, 19*w - 50], [1487, 1487, 36*w - 113], [1487, 1487, 21*w - 58], [1489, 1489, -5*w - 42], [1489, 1489, 5*w - 42], [1531, 1531, 14*w - 25], [1531, 1531, -14*w - 25], [1549, 1549, -21*w + 80], [1549, 1549, 30*w - 107], [1559, 1559, -12*w - 5], [1559, 1559, 12*w - 5], [1579, 1579, 50*w - 161], [1579, 1579, 17*w - 40], [1583, 1583, -12*w - 1], [1583, 1583, 12*w - 1], [1609, 1609, 9*w - 50], [1609, 1609, -9*w - 50], [1619, 1619, 27*w - 80], [1619, 1619, 30*w - 91], [1621, 1621, -15*w + 64], [1621, 1621, 42*w - 145], [1627, 1627, 14*w - 23], [1627, 1627, -14*w - 23], [1637, 1637, 2*w - 41], [1637, 1637, -2*w - 41], [1663, 1663, -13*w - 14], [1663, 1663, 13*w - 14], [1667, 1667, 19*w + 48], [1667, 1667, -19*w + 48], [1681, 41, -41], [1697, 1697, 8*w - 49], [1697, 1697, -8*w - 49], [1709, 1709, -10*w - 53], [1709, 1709, 10*w - 53], [1721, 1721, 40*w - 139], [1721, 1721, -17*w + 70], [1723, 1723, 38*w - 119], [1723, 1723, 23*w - 64], [1741, 1741, 13*w - 60], [1741, 1741, -50*w + 171], [1753, 1753, -w - 42], [1753, 1753, w - 42], [1759, 1759, -13*w - 10], [1759, 1759, 13*w - 10], [1811, 1811, 26*w - 75], [1811, 1811, 35*w - 108], [1823, 1823, 13*w - 6], [1823, 1823, -13*w - 6], [1847, 1847, 24*w - 67], [1847, 1847, 39*w - 122], [1867, 1867, -14*w - 17], [1867, 1867, 14*w - 17], [1873, 1873, -21*w + 82], [1873, 1873, 36*w - 127], [1901, 1901, -22*w + 85], [1901, 1901, 35*w - 124], [1931, 1931, -14*w - 15], [1931, 1931, 14*w - 15], [1973, 1973, -26*w + 97], [1973, 1973, 31*w - 112], [1979, 1979, 30*w - 89], [1979, 1979, 33*w - 100], [1987, 1987, 14*w - 13], [1987, 1987, -14*w - 13], [1999, 1999, 20*w + 49], [1999, 1999, -20*w + 49], [2017, 2017, 3*w - 46], [2017, 2017, -3*w - 46], [2029, 2029, 5*w - 48], [2029, 2029, -5*w - 48], [2063, 2063, 37*w - 114], [2063, 2063, 28*w - 81], [2069, 2069, -55*w + 188], [2069, 2069, 14*w - 65], [2087, 2087, -16*w - 27], [2087, 2087, 16*w - 27], [2111, 2111, 24*w - 65], [2111, 2111, 45*w - 142], [2113, 2113, 44*w - 153], [2113, 2113, -19*w + 78], [2131, 2131, -14*w - 5], [2131, 2131, 14*w - 5], [2137, 2137, 12*w - 61], [2137, 2137, -12*w - 61], [2161, 2161, -20*w + 81], [2161, 2161, 43*w - 150], [2207, 2207, -19*w - 42], [2207, 2207, 19*w - 42], [2209, 47, -47], [2237, 2237, -13*w - 64], [2237, 2237, 13*w - 64], [2239, 2239, 32*w - 95], [2239, 2239, 35*w - 106], [2243, 2243, 27*w - 76], [2243, 2243, 42*w - 131], [2251, 2251, 46*w - 145], [2251, 2251, 25*w - 68], [2269, 2269, -11*w - 60], [2269, 2269, 11*w - 60], [2281, 2281, -23*w + 90], [2281, 2281, 40*w - 141], [2287, 2287, -16*w - 23], [2287, 2287, 16*w - 23], [2293, 2293, -w - 48], [2293, 2293, w - 48], [2297, 2297, 17*w - 74], [2297, 2297, -17*w - 74], [2333, 2333, -61*w + 208], [2333, 2333, 14*w - 67], [2339, 2339, 18*w - 35], [2339, 2339, -18*w - 35], [2341, 2341, -25*w + 96], [2341, 2341, 38*w - 135], [2351, 2351, 60*w - 193], [2351, 2351, 21*w - 50], [2357, 2357, 2*w - 49], [2357, 2357, -2*w - 49], [2371, 2371, 19*w - 40], [2371, 2371, -19*w - 40], [2377, 2377, 7*w - 54], [2377, 2377, -7*w - 54], [2381, 2381, -10*w - 59], [2381, 2381, 10*w - 59], [2383, 2383, 43*w - 134], [2383, 2383, 28*w - 79], [2411, 2411, -15*w - 8], [2411, 2411, 15*w - 8], [2459, 2459, -15*w - 4], [2459, 2459, 15*w - 4], [2473, 2473, -9*w - 58], [2473, 2473, 9*w - 58], [2503, 2503, -17*w - 26], [2503, 2503, 17*w - 26], [2543, 2543, 27*w - 74], [2543, 2543, 48*w - 151], [2551, 2551, -20*w - 43], [2551, 2551, 20*w - 43], [2557, 2557, 2*w - 51], [2557, 2557, -2*w - 51], [2591, 2591, 16*w - 15], [2591, 2591, -16*w - 15], [2621, 2621, 10*w - 61], [2621, 2621, -10*w - 61], [2633, 2633, 4*w - 53], [2633, 2633, -4*w - 53], [2647, 2647, -16*w - 13], [2647, 2647, 16*w - 13], [2659, 2659, 35*w - 104], [2659, 2659, 38*w - 115], [2677, 2677, -18*w - 79], [2677, 2677, 18*w - 79], [2683, 2683, 62*w - 199], [2683, 2683, 23*w - 56], [2689, 2689, 45*w - 158], [2689, 2689, -24*w + 95], [2693, 2693, -w - 52], [2693, 2693, w - 52], [2719, 2719, -20*w - 41], [2719, 2719, 20*w - 41], [2729, 2729, -25*w + 98], [2729, 2729, 44*w - 155], [2753, 2753, -19*w - 82], [2753, 2753, 19*w - 82], [2767, 2767, -16*w - 7], [2767, 2767, 16*w - 7], [2777, 2777, 8*w - 59], [2777, 2777, -8*w - 59], [2791, 2791, 16*w - 5], [2791, 2791, -16*w - 5], [2797, 2797, 42*w - 149], [2797, 2797, -27*w + 104], [2851, 2851, 50*w - 157], [2851, 2851, 29*w - 80], [2861, 2861, 5*w - 56], [2861, 2861, -5*w - 56], [2879, 2879, 20*w - 39], [2879, 2879, -20*w - 39], [2897, 2897, -32*w + 119], [2897, 2897, 37*w - 134], [2903, 2903, 68*w - 219], [2903, 2903, 23*w - 54], [2909, 2909, -34*w + 125], [2909, 2909, 35*w - 128], [2939, 2939, 18*w - 25], [2939, 2939, -18*w - 25], [2953, 2953, -9*w - 62], [2953, 2953, 9*w - 62], [2957, 2957, 53*w - 184], [2957, 2957, -22*w + 91], [2999, 2999, 55*w - 174], [2999, 2999, 28*w - 75], [3001, 3001, -15*w - 74], [3001, 3001, 15*w - 74], [3011, 3011, 30*w - 83], [3011, 3011, 51*w - 160], [3037, 3037, -3*w - 56], [3037, 3037, 3*w - 56], [3041, 3041, -13*w - 70], [3041, 3041, 13*w - 70], [3061, 3061, 7*w - 60], [3061, 3061, -7*w - 60], [3079, 3079, 17*w - 10], [3079, 3079, -17*w - 10], [3089, 3089, 5*w - 58], [3089, 3089, -5*w - 58], [3119, 3119, 27*w - 70], [3119, 3119, 60*w - 191], [3163, 3163, 17*w - 4], [3163, 3163, -17*w - 4], [3167, 3167, 29*w - 78], [3167, 3167, 56*w - 177], [3169, 3169, 20*w - 87], [3169, 3169, -20*w - 87], [3187, 3187, 19*w - 28], [3187, 3187, -19*w - 28], [3203, 3203, -18*w - 19], [3203, 3203, 18*w - 19], [3217, 3217, -27*w + 106], [3217, 3217, 48*w - 169], [3221, 3221, -70*w + 239], [3221, 3221, 17*w - 80], [3251, 3251, -21*w - 40], [3251, 3251, 21*w - 40], [3257, 3257, -28*w + 109], [3257, 3257, 47*w - 166], [3299, 3299, -22*w - 45], [3299, 3299, 22*w - 45], [3301, 3301, 15*w - 76], [3301, 3301, -15*w - 76], [3307, 3307, 34*w - 97], [3307, 3307, 49*w - 152], [3319, 3319, -23*w - 50], [3319, 3319, 23*w - 50], [3343, 3343, 53*w - 166], [3343, 3343, 32*w - 89], [3389, 3389, 10*w - 67], [3389, 3389, -10*w - 67], [3407, 3407, -21*w - 38], [3407, 3407, 21*w - 38], [3413, 3413, -34*w + 127], [3413, 3413, 41*w - 148], [3433, 3433, -36*w + 133], [3433, 3433, 39*w - 142], [3457, 3457, -12*w - 71], [3457, 3457, 12*w - 71], [3467, 3467, 26*w - 63], [3467, 3467, 71*w - 228], [3469, 3469, 14*w - 75], [3469, 3469, -14*w - 75], [3481, 59, -59], [3511, 3511, 25*w - 58], [3511, 3511, 76*w - 245], [3527, 3527, -24*w - 53], [3527, 3527, 24*w - 53], [3529, 3529, 56*w - 195], [3529, 3529, -25*w + 102], [3539, 3539, 18*w - 5], [3539, 3539, -18*w - 5], [3557, 3557, 7*w - 64], [3557, 3557, -7*w - 64], [3559, 3559, -20*w - 29], [3559, 3559, 20*w - 29], [3571, 3571, -19*w - 20], [3571, 3571, 19*w - 20], [3583, 3583, -28*w + 71], [3583, 3583, 28*w + 71], [3607, 3607, 41*w - 122], [3607, 3607, 44*w - 133], [3613, 3613, 21*w - 92], [3613, 3613, -21*w - 92], [3617, 3617, 13*w - 74], [3617, 3617, -13*w - 74], [3643, 3643, -22*w - 41], [3643, 3643, 22*w - 41], [3659, 3659, 30*w - 79], [3659, 3659, 63*w - 200], [3671, 3671, -20*w - 27], [3671, 3671, 20*w - 27], [3677, 3677, 2*w - 61], [3677, 3677, -2*w - 61], [3691, 3691, 34*w - 95], [3691, 3691, 55*w - 172], [3697, 3697, 53*w - 186], [3697, 3697, -28*w + 111], [3701, 3701, -22*w - 95], [3701, 3701, 22*w - 95], [3721, 61, -61], [3733, 3733, 9*w - 68], [3733, 3733, -9*w - 68], [3739, 3739, 25*w - 56], [3739, 3739, -25*w - 56], [3779, 3779, 42*w - 125], [3779, 3779, 45*w - 136], [3793, 3793, -4*w - 63], [3793, 3793, 4*w - 63], [3803, 3803, 22*w - 39], [3803, 3803, -22*w - 39], [3821, 3821, -5*w - 64], [3821, 3821, 5*w - 64], [3823, 3823, 37*w - 106], [3823, 3823, 52*w - 161], [3833, 3833, -w - 62], [3833, 3833, w - 62], [3847, 3847, 64*w - 203], [3847, 3847, 31*w - 82], [3853, 3853, 11*w - 72], [3853, 3853, -11*w - 72], [3863, 3863, 28*w + 69], [3863, 3863, -28*w + 69], [3877, 3877, -17*w - 84], [3877, 3877, 17*w - 84], [3881, 3881, -20*w - 91], [3881, 3881, 20*w - 91], [3907, 3907, 19*w - 8], [3907, 3907, -19*w - 8], [3911, 3911, 49*w - 150], [3911, 3911, 40*w - 117], [3917, 3917, 13*w - 76], [3917, 3917, -13*w - 76], [3923, 3923, 27*w - 64], [3923, 3923, 78*w - 251], [3967, 3967, 19*w - 2], [3967, 3967, -19*w - 2], [4003, 4003, 53*w - 164], [4003, 4003, 38*w - 109], [4013, 4013, -26*w + 107], [4013, 4013, 61*w - 212], [4049, 4049, 4*w - 65], [4049, 4049, -4*w - 65], [4057, 4057, -8*w - 69], [4057, 4057, 8*w - 69], [4073, 4073, -16*w - 83], [4073, 4073, 16*w - 83], [4091, 4091, 41*w - 120], [4091, 4091, 50*w - 153], [4093, 4093, 6*w - 67], [4093, 4093, -6*w - 67], [4099, 4099, 22*w - 35], [4099, 4099, -22*w - 35], [4111, 4111, 20*w - 17], [4111, 4111, -20*w - 17], [4127, 4127, 24*w - 47], [4127, 4127, -24*w - 47], [4129, 4129, -80*w + 273], [4129, 4129, 19*w - 90], [4217, 4217, 17*w - 86], [4217, 4217, -17*w - 86], [4219, 4219, 23*w - 40], [4219, 4219, -23*w - 40], [4229, 4229, -10*w - 73], [4229, 4229, 10*w - 73], [4231, 4231, 20*w - 13], [4231, 4231, -20*w - 13], [4243, 4243, 37*w - 104], [4243, 4243, 58*w - 181], [4259, 4259, 35*w - 96], [4259, 4259, 62*w - 195], [4261, 4261, 57*w - 200], [4261, 4261, -30*w + 119], [4273, 4273, -24*w + 103], [4273, 4273, 69*w - 238], [4337, 4337, -8*w - 71], [4337, 4337, 8*w - 71], [4349, 4349, 5*w - 68], [4349, 4349, -5*w - 68], [4357, 4357, -18*w - 89], [4357, 4357, 18*w - 89], [4363, 4363, -22*w - 31], [4363, 4363, 22*w - 31], [4391, 4391, -20*w - 3], [4391, 4391, 20*w - 3], [4409, 4409, -16*w - 85], [4409, 4409, 16*w - 85], [4451, 4451, 21*w - 20], [4451, 4451, -21*w - 20], [4463, 4463, 52*w - 159], [4463, 4463, 43*w - 126], [4481, 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- 240], [8233, 8233, 84*w - 293], [8233, 8233, -39*w + 158], [8237, 8237, 2*w - 91], [8237, 8237, -2*w - 91], [8263, 8263, -28*w - 19], [8263, 8263, 28*w - 19], [8273, 8273, -11*w - 98], [8273, 8273, 11*w - 98], [8291, 8291, 35*w - 72], [8291, 8291, -35*w - 72], [8297, 8297, -7*w - 94], [8297, 8297, 7*w - 94], [8311, 8311, 55*w - 158], [8311, 8311, 76*w - 235], [8317, 8317, 18*w - 109], [8317, 8317, -18*w - 109], [8353, 8353, -56*w + 207], [8353, 8353, 61*w - 222], [8369, 8369, -20*w - 113], [8369, 8369, 20*w - 113], [8423, 8423, 47*w - 126], [8423, 8423, 92*w - 291], [8429, 8429, 94*w - 325], [8429, 8429, -35*w + 148], [8443, 8443, 73*w - 224], [8443, 8443, 58*w - 169], [8447, 8447, 51*w - 142], [8447, 8447, 84*w - 263], [8467, 8467, 29*w - 28], [8467, 8467, -29*w - 28], [8501, 8501, -25*w - 124], [8501, 8501, 25*w - 124], [8527, 8527, 44*w - 113], [8527, 8527, 101*w - 322], [8537, 8537, 28*w - 131], [8537, 8537, -113*w + 386], [8543, 8543, 28*w - 9], [8543, 8543, -28*w - 9], [8573, 8573, 19*w - 112], [8573, 8573, -19*w - 112], [8581, 8581, -23*w - 120], [8581, 8581, 23*w - 120], [8599, 8599, 28*w - 5], [8599, 8599, -28*w - 5], [8623, 8623, -28*w - 1], [8623, 8623, 28*w - 1], [8629, 8629, -6*w - 95], [8629, 8629, 6*w - 95], [8663, 8663, 32*w - 51], [8663, 8663, -32*w - 51], [8669, 8669, 11*w - 100], [8669, 8669, -11*w - 100], [8677, 8677, 7*w - 96], [8677, 8677, -7*w - 96], [8693, 8693, -26*w - 127], [8693, 8693, 26*w - 127], [8707, 8707, 59*w - 172], [8707, 8707, 74*w - 227], [8713, 8713, -9*w - 98], [8713, 8713, 9*w - 98], [8719, 8719, 52*w - 145], [8719, 8719, 85*w - 266], [8731, 8731, 89*w - 280], [8731, 8731, 50*w - 137], [8737, 8737, 3*w - 94], [8737, 8737, -3*w - 94], [8747, 8747, 34*w - 63], [8747, 8747, -34*w - 63], [8761, 8761, 15*w - 106], [8761, 8761, -15*w - 106], [8807, 8807, 31*w - 42], [8807, 8807, -31*w - 42], [8819, 8819, 45*w - 116], [8819, 8819, 102*w - 325], [8837, 8837, 22*w - 119], [8837, 8837, -22*w - 119], [8839, 8839, 65*w - 194], [8839, 8839, 68*w - 205], [8849, 8849, -4*w - 95], [8849, 8849, 4*w - 95], [8863, 8863, 32*w - 49], [8863, 8863, -32*w - 49], [8887, 8887, 116*w - 373], [8887, 8887, 41*w - 98], [8893, 8893, -34*w - 147], [8893, 8893, 34*w - 147], [8923, 8923, 98*w - 311], [8923, 8923, 47*w - 124], [8933, 8933, -49*w + 188], [8933, 8933, 74*w - 263], [8941, 8941, 5*w - 96], [8941, 8941, -5*w - 96], [8951, 8951, 40*w - 93], [8951, 8951, 121*w - 390], [8969, 8969, 89*w - 310], [8969, 8969, -40*w + 163], [8971, 8971, -31*w - 40], [8971, 8971, 31*w - 40], [9001, 9001, 25*w - 126], [9001, 9001, -25*w - 126], [9011, 9011, 51*w - 140], [9011, 9011, 90*w - 283], [9013, 9013, 6*w - 97], [9013, 9013, -6*w - 97], [9029, 9029, 31*w - 140], [9029, 9029, -31*w - 140], [9059, 9059, 30*w - 29], [9059, 9059, -30*w - 29], [9103, 9103, -43*w + 106], [9103, 9103, 43*w + 106], [9109, 9109, 9*w - 100], [9109, 9109, -9*w - 100], [9127, 9127, -31*w - 38], [9127, 9127, 31*w - 38], [9133, 9133, 69*w - 248], [9133, 9133, -54*w + 203], [9151, 9151, -29*w - 10], [9151, 9151, 29*w - 10], [9157, 9157, -42*w + 169], [9157, 9157, 87*w - 304], [9161, 9161, 68*w - 245], [9161, 9161, -55*w + 206], [9187, 9187, -29*w - 8], [9187, 9187, 29*w - 8], [9203, 9203, 117*w - 376], [9203, 9203, 42*w - 101], [9221, 9221, -58*w + 215], [9221, 9221, 65*w - 236], [9239, 9239, -32*w - 45], [9239, 9239, 32*w - 45], [9241, 9241, 63*w - 230], [9241, 9241, -60*w + 221], [9277, 9277, -37*w + 156], [9277, 9277, 98*w - 339], [9283, 9283, 37*w - 76], [9283, 9283, -37*w - 76], [9293, 9293, -14*w - 107], [9293, 9293, 14*w - 107], [9319, 9319, -40*w - 91], [9319, 9319, 40*w - 91], [9323, 9323, 38*w - 81], [9323, 9323, -38*w - 81], [9337, 9337, -33*w - 146], [9337, 9337, 33*w - 146], [9371, 9371, 30*w - 23], [9371, 9371, -30*w - 23], [9377, 9377, -13*w - 106], [9377, 9377, 13*w - 106], [9391, 9391, 59*w - 170], [9391, 9391, 80*w - 247], [9397, 9397, -38*w + 159], [9397, 9397, 97*w - 336], [9421, 9421, -117*w + 400], [9421, 9421, 30*w - 139], [9461, 9461, -7*w - 100], [9461, 9461, 7*w - 100], [9467, 9467, -34*w - 57], [9467, 9467, 34*w - 57], [9479, 9479, -33*w - 50], [9479, 9479, 33*w - 50], [9497, 9497, -8*w - 101], [9497, 9497, 8*w - 101], [9511, 9511, 100*w - 317], [9511, 9511, 49*w - 130], [9539, 9539, 30*w - 19], [9539, 9539, -30*w - 19], [9547, 9547, -31*w - 32], [9547, 9547, 31*w - 32], [9587, 9587, -46*w + 117], [9587, 9587, 46*w + 117], [9601, 9601, 28*w - 135], [9601, 9601, -28*w - 135], [9629, 9629, 35*w - 152], [9629, 9629, -35*w - 152], [9631, 9631, -35*w - 62], [9631, 9631, 35*w - 62], [9643, 9643, -38*w - 79], [9643, 9643, 38*w - 79], [9661, 9661, -18*w - 115], [9661, 9661, 18*w - 115], [9679, 9679, -40*w - 89], [9679, 9679, 40*w - 89], [9689, 9689, -49*w + 190], [9689, 9689, 80*w - 283], [9719, 9719, 105*w - 334], [9719, 9719, 48*w - 125], [9733, 9733, 94*w - 327], [9733, 9733, -41*w + 168], [9743, 9743, 32*w - 39], [9743, 9743, -32*w - 39], [9749, 9749, -50*w + 193], [9749, 9749, 79*w - 280], [9767, 9767, -36*w - 67], [9767, 9767, 36*w - 67], [9769, 9769, -36*w - 155], [9769, 9769, 36*w - 155], [9787, 9787, -31*w - 28], [9787, 9787, 31*w - 28], [9803, 9803, 78*w - 239], [9803, 9803, 63*w - 184], [9811, 9811, 50*w - 133], [9811, 9811, 101*w - 320], [9817, 9817, -17*w - 114], [9817, 9817, 17*w - 114], [9851, 9851, -30*w - 7], [9851, 9851, 30*w - 7], [9857, 9857, -52*w + 199], [9857, 9857, 77*w - 274], [9901, 9901, 3*w - 100], [9901, 9901, -3*w - 100], [9907, 9907, -34*w - 53], [9907, 9907, 34*w - 53], [9949, 9949, 75*w - 268], [9949, 9949, -54*w + 205]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [-1, -2, -2, -1, -1, 1, 4, 4, 4, 6, 6, -4, -4, 6, 6, -16, -16, -12, -12, -14, -14, 18, 18, 4, 4, -14, -14, 0, 0, 4, 4, -6, -6, 12, 12, 8, 8, -2, -2, -8, -8, -22, -26, -26, 4, 4, -12, -12, -10, -10, 24, 24, 26, 26, 16, 16, 6, 6, 16, 16, -20, -20, 2, 4, 4, 26, 26, -18, -18, -28, -28, 10, 10, 32, 32, 6, 6, -18, -18, -30, -30, 38, 38, -24, -24, -14, -14, -24, -24, -30, -30, -16, -16, 20, 20, 24, 24, -10, -10, 18, 18, -20, -20, -46, 4, 4, 4, 4, -4, -4, 34, 34, -16, -16, -6, -6, 18, 18, -18, -18, 12, 12, 22, 22, 6, 6, 36, 36, 0, 0, -10, -10, 14, 14, -44, -44, -26, -26, 44, 44, -12, -12, 14, 14, -54, -6, -6, 8, 8, 8, 8, 26, 26, -62, 40, 40, -30, -30, -34, -34, -12, -12, 30, 30, 0, 0, 2, 2, 28, 28, -26, -26, -24, -24, -60, -60, 38, 38, -34, -34, -32, -32, -14, -14, -60, -60, 6, 6, 28, 28, 18, 18, 30, 30, 24, 24, 32, 32, 6, 6, -58, -58, -36, -36, -18, -18, -40, -40, 10, 10, -64, -64, 30, 30, -8, -8, 2, 2, 20, 20, -14, -14, 8, 8, -12, -12, 46, 46, 4, 4, -16, -16, 48, 48, -62, -62, -36, -36, 46, 46, 40, 40, -20, -20, 24, 24, 58, 58, 44, 44, -10, -10, 12, 12, -18, -18, 32, 32, -52, -52, -78, -22, -22, -66, -66, -62, -62, 28, 28, -50, -50, 10, 10, -16, -16, 68, 68, -64, -64, -40, -40, 44, 44, 82, 82, -2, -2, 60, 60, 14, 14, -52, -52, 68, 68, -80, -80, 66, 66, 30, 30, -48, -48, -10, -10, -16, -16, -24, -24, -14, -14, -28, -28, -38, -38, 34, 34, 56, 56, -94, 30, 30, 32, 32, -12, -12, -20, -20, -34, -34, -70, -70, 16, 16, -74, -74, -22, -22, -34, -34, 28, 28, -74, -74, 48, 48, -34, -34, 52, 52, -54, -54, -2, -2, 96, 96, 12, 12, -76, -76, -86, -86, 40, 40, 72, 72, 56, 56, -2, -2, 40, 40, 38, 38, -54, -54, 40, 40, -60, -60, 70, 70, -4, -4, 34, 34, -66, -66, 16, 16, 18, 18, -94, -94, -80, -80, -78, -78, -8, -8, -18, -18, 52, 52, 6, 6, -24, -24, 90, 90, 72, 72, -66, -66, -100, -100, 74, 74, -42, -42, -24, -24, 42, 42, -36, -36, -34, -34, -102, -102, -10, -10, -40, -40, 34, 34, 104, 104, 92, 92, 96, 96, -62, -62, -108, -108, -36, -36, 18, 18, 38, 38, -28, -28, -6, -6, -20, -20, -58, -58, 108, 108, -24, -24, -16, -16, 62, 62, 112, 112, -26, -26, 26, 26, 18, 18, -60, -60, 14, 14, 26, 56, 56, -104, -104, -54, -54, 44, 44, -42, -42, 72, 72, -12, -12, 32, 32, -88, -88, 62, 62, 106, 106, 44, 44, 12, 12, 40, 40, -114, -114, -20, -20, 50, 50, -114, -114, -118, 54, 54, 44, 44, -60, -60, 34, 34, -44, -44, 118, 118, -64, -64, -54, -54, -40, -40, 62, 62, -56, -56, -58, -58, 34, 34, 36, 36, -72, -72, 94, 94, 36, 36, -64, -64, -76, -76, 78, 78, 90, 90, -38, -38, 82, 82, -52, -52, -50, -50, -12, -12, -48, -48, -88, -88, 34, 34, -30, -30, 28, 28, -90, -90, -40, -40, -76, -76, -60, -60, -58, -58, 34, 34, -14, -14, -66, -66, 38, 38, 92, 92, 112, 112, 66, 66, 84, 84, 56, 56, -78, -78, 84, 84, -118, -10, -10, -116, -116, -62, -62, 84, 84, 8, 8, 8, 8, 46, 46, 32, 32, 114, 114, 88, 88, -30, -30, -16, -16, -2, -2, 48, 48, 56, 56, 100, 100, 22, 22, -30, -30, 112, 112, 6, 6, 114, 114, 88, 88, -50, -50, -40, -40, 54, 54, 26, 26, -128, -128, 118, 118, -78, -78, 20, 20, 78, 78, -64, -64, -142, 68, 68, -60, -60, -36, -36, -102, -102, 36, 36, -110, -110, 112, 112, -82, -82, -132, -132, 24, 24, 38, 38, -98, -98, 90, 90, -88, -88, -62, -62, -84, -84, -110, -114, -114, -92, -92, -54, -54, 0, 0, -122, -122, -30, -30, 76, 76, 72, 72, 14, 14, 10, 10, 54, 54, -60, -60, 64, 64, 28, 28, -62, -62, -50, -50, -88, -88, -96, -96, 58, 58, -132, -132, 42, 42, -66, -66, -76, -76, 118, 118, -48, -48, 48, 48, 34, 34, 88, 88, 14, 14, 20, 20, -76, -76, -4, -4, -46, -46, 54, 54, -94, -94, -24, -24, -140, -140, 30, 30, -74, -74, 112, 112, -66, -66, 52, 52, -70, -70, -80, -80, -68, -68, -6, -6, 24, 24, 150, 150, -8, -8, 52, 52, 20, 20, 54, 54, -88, -88, 138, 138, -56, -56, 108, 108, 62, 62, 0, 0, -90, -90, 58, 58, 130, 130, -24, -24, -86, -86, -58, -58, -100, -100, 14, 14, 50, 50, 6, 6, 34, 34, 42, 42, -12, -12, 40, 40, -68, -68, 38, 38, -120, -120, -112, -112, 76, 76, -50, -50, 86, 86, -56, -56, -110, -110, 94, 94, 42, 42, -50, -50, -108, -108, -18, -18, -86, -86, 16, 16, 118, 118, -120, -120, 84, 84, 60, 60, 60, 60, 6, 6, 132, 132, -88, -88, 2, 2, 92, 92, -102, -102, 96, 96, 22, 22, -102, -102, 96, 96, 32, 32, 16, 16, -42, -42, 34, 34, -112, -112, 10, 10, -118, -118, -72, -72, 132, 132, -42, -42, 82, 82, 14, 14, -2, -2, 98, 98, -108, -108, -134, -134, 18, 18, -24, -24, 82, 82, 20, 20, -138, -138, -108, -108, -54, -54, -98, -98, 136, 136, -22, -22, 38, 38, 18, 18, 84, 84, 40, 40, 86, 86, 74, 74, -62, -62, 28, 28, 68, 68, 42, 42, -18, -18, -46, -46, 56, 56, 82, 82, 124, 124, 86, 86, 84, 84, -90, -90, -24, -24, 168, 168, -4, -4, 74, 74, 18, 18, -58, -58, 44, 44, 144, 144, 90, 90, 134, 134, 36, 36, -60, -60, 80, 80, -174, -174, -124, -124, -130, -130, 42, 42, -42, -42, 56, 56, -6, -6, -92, -92, 42, 42, -104, -104, -98, -98, -94, -94, 34, 34, 160, 160, 30, 30, -148, -148, -128, -128, -44, -44, 174, 174, 112, 112, -6, -6, -112, -112, 118, 118, -58, -58, 56, 56, -96, -96, 22, 22, -128, -128, 150, 150, 102, 102, -66, -66, 148, 148, -38, -38, 64, 64, -52, -52, 98, 98, 20, 20, 122, 122, -48, -48, 100, 100, -50, -50, -56, -56, -94, -94, 96, 96, 40, 40, 158, 158, -20, -20, -122, -122, -114, -114, -24, -24, -54, -54, -116, -116, -70, -70, -164, -164, 118, 118, 126, 126, 108, 108, 16, 16, -106, -106, 72, 72, 62, 62, 16, 16, -58, -58, -142, -142, -60, -60, -44, -44, -138, -138, 120, 120, -38, -38, -2, -2, -76, -76, -66, -66, 104, 104, -116, -116, -6, -6, 12, 12, 98, 98, -32, -32, 86, 86, 30, 30, -138, -138, -4, -4, 160, 160, -126, -126, -8, -8, 132, 132, -164, -164, -84, -84, 130, 130, 14, 14, -16, -16, 188, 188, -2, -2, 112, 112, 154, 154, -168, -168, -154, -154, -48, -48, -162, -162, 96, 96, -54, -54, -52, -52, 12, 12, 148, 148, 42, 42, 116, 116, 18, 18, 158, 158, 132, 132, -66, -66]; 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;