/* 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![-1, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [4, 2, 2], [5, 5, -2*w + 1], [9, 3, 3], [11, 11, -3*w + 2], [11, 11, -3*w + 1], [19, 19, -4*w + 3], [19, 19, -4*w + 1], [29, 29, w + 5], [29, 29, -w + 6], [31, 31, -5*w + 2], [31, 31, -5*w + 3], [41, 41, -6*w + 5], [41, 41, w - 7], [49, 7, -7], [59, 59, 2*w - 9], [59, 59, 7*w - 5], [61, 61, 3*w - 10], [61, 61, -3*w - 7], [71, 71, -8*w + 7], [71, 71, w - 9], [79, 79, 8*w - 5], [79, 79, 3*w - 11], [89, 89, -10*w - 1], [89, 89, 10*w - 11], [101, 101, 4*w - 13], [101, 101, 5*w - 14], [109, 109, -11*w - 1], [109, 109, 11*w - 12], [131, 131, -12*w - 1], [131, 131, 12*w - 13], [139, 139, 2*w - 13], [139, 139, -11*w + 9], [149, 149, 7*w - 18], [149, 149, 4*w - 15], [151, 151, 5*w - 16], [151, 151, 6*w - 17], [169, 13, -13], [179, 179, 7*w - 19], [179, 179, 5*w - 17], [181, 181, -14*w - 1], [181, 181, 14*w - 15], [191, 191, 2*w - 15], [191, 191, -13*w + 11], [199, 199, 3*w - 16], [199, 199, -13*w + 10], [211, 211, 6*w - 19], [211, 211, 7*w - 20], [229, 229, -14*w + 11], [229, 229, 3*w - 17], [239, 239, -16*w - 1], [239, 239, 16*w - 17], [241, 241, 9*w - 23], [241, 241, 5*w - 19], [251, 251, 2*w - 17], [251, 251, -15*w + 13], [269, 269, 15*w - 11], [269, 269, 4*w - 19], [271, 271, -17*w - 1], [271, 271, 17*w - 18], [281, 281, 7*w - 22], [281, 281, 8*w - 23], [289, 17, -17], [311, 311, 5*w - 21], [311, 311, 16*w - 11], [331, 331, -17*w + 14], [331, 331, 3*w - 20], [349, 349, 17*w - 12], [349, 349, 5*w - 22], [359, 359, 10*w - 27], [359, 359, 7*w - 24], [379, 379, -20*w - 1], [379, 379, 20*w - 21], [389, 389, 5*w - 23], [389, 389, 18*w - 13], [401, 401, 7*w - 25], [401, 401, 11*w - 29], [409, 409, 3*w - 22], [409, 409, -19*w + 16], [419, 419, -21*w - 1], [419, 419, 21*w - 22], [421, 421, -19*w + 15], [421, 421, 4*w - 23], [431, 431, 19*w - 14], [431, 431, 5*w - 24], [439, 439, 6*w - 25], [439, 439, 19*w - 13], [449, 449, 8*w - 27], [449, 449, 11*w - 30], [461, 461, -22*w - 1], [461, 461, 22*w - 23], [479, 479, 23*w - 25], [479, 479, -23*w - 2], [491, 491, 13*w - 33], [491, 491, 7*w - 27], [499, 499, 11*w - 31], [499, 499, 9*w - 29], [509, 509, 4*w - 25], [509, 509, -21*w + 17], [521, 521, 5*w - 26], [521, 521, 21*w - 16], [529, 23, -23], [541, 541, 25*w - 28], [541, 541, -25*w - 3], [569, 569, -22*w + 17], [569, 569, 5*w - 27], [571, 571, 25*w - 27], [571, 571, -25*w - 2], [599, 599, -25*w - 1], [599, 599, 25*w - 26], [601, 601, 13*w - 35], [601, 601, 9*w - 31], [619, 619, 5*w - 28], [619, 619, -23*w + 18], [631, 631, 23*w - 17], [631, 631, 6*w - 29], [641, 641, 23*w - 16], [641, 641, 7*w - 30], [659, 659, 13*w - 36], [659, 659, 10*w - 33], [661, 661, 11*w - 34], [661, 661, 12*w - 35], [691, 691, 28*w - 31], [691, 691, -28*w - 3], [701, 701, -27*w - 1], [701, 701, 27*w - 28], [709, 709, 4*w - 29], [709, 709, -25*w + 21], [719, 719, 13*w - 37], [719, 719, 11*w - 35], [739, 739, 6*w - 31], [739, 739, 25*w - 19], [751, 751, 7*w - 32], [751, 751, 25*w - 18], [761, 761, 8*w - 33], [761, 761, 25*w - 17], [769, 769, 9*w - 34], [769, 769, 16*w - 41], [809, 809, 7*w - 33], [809, 809, 26*w - 19], [811, 811, -29*w - 1], [811, 811, 29*w - 30], [821, 821, -27*w + 23], [821, 821, 4*w - 31], [829, 829, 17*w - 43], [829, 829, 9*w - 35], [839, 839, -27*w + 22], [839, 839, 5*w - 32], [859, 859, 31*w - 34], [859, 859, -31*w - 3], [881, 881, 8*w - 35], [881, 881, 27*w - 19], [911, 911, 13*w - 40], [911, 911, 14*w - 41], [919, 919, -32*w - 3], [919, 919, 32*w - 35], [929, 929, -31*w - 1], [929, 929, 31*w - 32], [941, 941, 33*w - 37], [941, 941, -33*w - 4], [971, 971, 11*w - 39], [971, 971, 17*w - 45], [991, 991, -32*w - 1], [991, 991, 32*w - 33], [1009, 1009, 29*w - 21], [1009, 1009, 8*w - 37], [1019, 1019, 33*w - 35], [1019, 1019, -33*w - 2], [1021, 1021, 29*w - 20], [1021, 1021, 9*w - 38], [1031, 1031, 19*w - 48], [1031, 1031, 10*w - 39], [1039, 1039, 18*w - 47], [1039, 1039, 11*w - 40], [1049, 1049, 16*w - 45], [1049, 1049, 13*w - 42], [1051, 1051, 14*w - 43], [1051, 1051, 15*w - 44], [1061, 1061, -30*w + 23], [1061, 1061, 7*w - 37], [1069, 1069, -35*w - 4], [1069, 1069, 35*w - 39], [1091, 1091, 36*w - 41], [1091, 1091, -36*w - 5], [1109, 1109, 19*w - 49], [1109, 1109, 11*w - 41], [1129, 1129, -31*w + 24], [1129, 1129, 7*w - 38], [1151, 1151, 35*w - 37], [1151, 1151, -35*w - 2], [1171, 1171, 10*w - 41], [1171, 1171, 21*w - 52], [1181, 1181, 11*w - 42], [1181, 1181, 20*w - 51], [1201, 1201, 15*w - 46], [1201, 1201, 16*w - 47], [1229, 1229, 38*w - 43], [1229, 1229, -38*w - 5], [1231, 1231, 9*w - 41], [1231, 1231, 32*w - 23], [1249, 1249, 37*w - 40], [1249, 1249, -37*w - 3], [1259, 1259, -36*w - 1], [1259, 1259, 36*w - 37], [1279, 1279, 17*w - 49], [1279, 1279, 15*w - 47], [1289, 1289, 8*w - 41], [1289, 1289, 33*w - 25], [1291, 1291, 37*w - 39], [1291, 1291, -37*w - 2], [1301, 1301, -39*w - 5], [1301, 1301, 39*w - 44], [1319, 1319, 33*w - 23], [1319, 1319, 10*w - 43], [1321, 1321, -38*w - 3], [1321, 1321, 38*w - 41], [1361, 1361, 16*w - 49], [1361, 1361, 17*w - 50], [1369, 37, -37], [1381, 1381, 9*w - 43], [1381, 1381, 34*w - 25], [1399, 1399, -35*w + 29], [1399, 1399, 6*w - 41], [1409, 1409, 11*w - 45], [1409, 1409, 23*w - 57], [1429, 1429, 13*w - 47], [1429, 1429, 21*w - 55], [1439, 1439, 39*w - 41], [1439, 1439, -39*w - 2], [1451, 1451, 41*w - 46], [1451, 1451, -41*w - 5], [1459, 1459, 35*w - 26], [1459, 1459, 9*w - 44], [1471, 1471, 40*w - 43], [1471, 1471, -40*w - 3], [1481, 1481, -39*w - 1], [1481, 1481, 39*w - 40], [1489, 1489, 35*w - 24], [1489, 1489, 11*w - 46], [1499, 1499, 7*w - 43], [1499, 1499, -36*w + 29], [1511, 1511, 13*w - 48], [1511, 1511, 22*w - 57], [1531, 1531, 17*w - 52], [1531, 1531, 18*w - 53], [1549, 1549, -41*w - 3], [1549, 1549, 41*w - 44], [1559, 1559, -40*w - 1], [1559, 1559, 40*w - 41], [1571, 1571, 11*w - 47], [1571, 1571, 36*w - 25], [1579, 1579, -37*w + 30], [1579, 1579, 7*w - 44], [1601, 1601, -37*w + 29], [1601, 1601, 8*w - 45], [1609, 1609, 43*w - 48], [1609, 1609, -43*w - 5], [1619, 1619, 19*w - 55], [1619, 1619, 17*w - 53], [1621, 1621, 9*w - 46], [1621, 1621, 37*w - 28], [1669, 1669, 12*w - 49], [1669, 1669, 37*w - 25], [1699, 1699, 22*w - 59], [1699, 1699, 15*w - 52], [1709, 1709, 17*w - 54], [1709, 1709, 20*w - 57], [1721, 1721, -42*w - 1], [1721, 1721, 42*w - 43], [1741, 1741, 38*w - 27], [1741, 1741, 11*w - 49], [1759, 1759, 43*w - 45], [1759, 1759, -43*w - 2], [1789, 1789, 15*w - 53], [1789, 1789, 23*w - 61], [1801, 1801, 21*w - 59], [1801, 1801, 17*w - 55], [1811, 1811, 39*w - 29], [1811, 1811, 10*w - 49], [1831, 1831, 7*w - 47], [1831, 1831, -40*w + 33], [1849, 43, -43], [1861, 1861, 46*w - 51], [1861, 1861, -46*w - 5], [1871, 1871, 25*w - 64], [1871, 1871, 14*w - 53], [1879, 1879, -40*w + 31], [1879, 1879, 9*w - 49], [1889, 1889, 16*w - 55], [1889, 1889, 23*w - 62], [1901, 1901, 19*w - 58], [1901, 1901, 20*w - 59], [1931, 1931, 45*w - 47], [1931, 1931, -45*w - 2], [1949, 1949, -47*w - 5], [1949, 1949, 47*w - 52], [1951, 1951, 13*w - 53], [1951, 1951, 27*w - 67], [1979, 1979, -45*w - 1], [1979, 1979, 45*w - 46], [1999, 1999, 21*w - 61], [1999, 1999, 19*w - 59], [2011, 2011, 41*w - 30], [2011, 2011, 11*w - 52], [2029, 2029, 12*w - 53], [2029, 2029, 41*w - 29], [2039, 2039, 48*w - 53], [2039, 2039, -48*w - 5], [2069, 2069, -46*w - 1], [2069, 2069, 46*w - 47], [2081, 2081, 25*w - 66], [2081, 2081, 16*w - 57], [2089, 2089, 17*w - 58], [2089, 2089, 24*w - 65], [2099, 2099, 22*w - 63], [2099, 2099, 19*w - 60], [2111, 2111, 47*w - 49], [2111, 2111, -47*w - 2], [2129, 2129, 8*w - 51], [2129, 2129, -43*w + 35], [2131, 2131, -49*w - 5], [2131, 2131, 49*w - 54], [2141, 2141, 42*w - 29], [2141, 2141, 13*w - 55], [2161, 2161, -47*w - 1], [2161, 2161, 47*w - 48], [2179, 2179, -43*w + 33], [2179, 2179, 10*w - 53], [2209, 47, -47], [2221, 2221, 43*w - 31], [2221, 2221, 12*w - 55], [2239, 2239, 43*w - 30], [2239, 2239, 13*w - 56], [2251, 2251, -44*w + 35], [2251, 2251, 9*w - 53], [2269, 2269, 15*w - 58], [2269, 2269, 28*w - 71], [2281, 2281, 16*w - 59], [2281, 2281, 27*w - 70], [2309, 2309, 20*w - 63], [2309, 2309, 23*w - 66], [2311, 2311, 21*w - 64], [2311, 2311, 22*w - 65], [2339, 2339, 44*w - 31], [2339, 2339, 13*w - 57], [2341, 2341, -50*w - 3], [2341, 2341, 50*w - 53], [2351, 2351, -49*w - 1], [2351, 2351, 49*w - 50], [2371, 2371, 29*w - 73], [2371, 2371, 15*w - 59], [2381, 2381, -51*w - 4], [2381, 2381, 51*w - 55], [2389, 2389, 7*w - 53], [2389, 2389, -46*w + 39], [2399, 2399, 11*w - 56], [2399, 2399, 45*w - 34], [2411, 2411, 25*w - 69], [2411, 2411, 19*w - 63], [2441, 2441, 45*w - 32], [2441, 2441, 13*w - 58], [2459, 2459, 14*w - 59], [2459, 2459, 45*w - 31], [2521, 2521, -47*w + 39], [2521, 2521, 8*w - 55], [2531, 2531, 22*w - 67], [2531, 2531, 23*w - 68], [2539, 2539, 52*w - 55], [2539, 2539, -52*w - 3], [2549, 2549, -51*w - 1], [2549, 2549, 51*w - 52], [2551, 2551, -47*w + 38], [2551, 2551, 9*w - 56], [2579, 2579, 10*w - 57], [2579, 2579, -47*w + 37], [2591, 2591, -55*w - 7], [2591, 2591, 55*w - 62], [2609, 2609, 29*w - 75], [2609, 2609, 17*w - 63], [2621, 2621, -54*w - 5], [2621, 2621, 54*w - 59], [2659, 2659, 55*w - 61], [2659, 2659, -55*w - 6], [2671, 2671, 14*w - 61], [2671, 2671, 47*w - 33], [2689, 2689, 47*w - 32], [2689, 2689, 15*w - 62], [2699, 2699, 53*w - 55], [2699, 2699, -53*w - 2], [2711, 2711, 11*w - 59], [2711, 2711, -48*w + 37], [2719, 2719, 17*w - 64], [2719, 2719, 30*w - 77], [2729, 2729, 57*w - 65], [2729, 2729, -57*w - 8], [2731, 2731, 18*w - 65], [2731, 2731, 29*w - 76], [2741, 2741, 28*w - 75], [2741, 2741, 19*w - 66], [2749, 2749, 20*w - 67], [2749, 2749, 27*w - 74], [2789, 2789, -55*w - 4], [2789, 2789, 55*w - 59], [2791, 2791, -49*w + 39], [2791, 2791, 10*w - 59], [2801, 2801, 57*w - 64], [2801, 2801, -57*w - 7], [2809, 53, -53], [2819, 2819, -49*w + 38], [2819, 2819, 11*w - 60], [2851, 2851, 55*w - 58], [2851, 2851, -55*w - 3], [2861, 2861, -54*w - 1], [2861, 2861, 54*w - 55], [2879, 2879, 25*w - 73], [2879, 2879, 23*w - 71], [2909, 2909, -58*w - 7], [2909, 2909, 58*w - 65], [2939, 2939, -57*w - 5], [2939, 2939, 57*w - 62], [2969, 2969, -55*w - 1], [2969, 2969, 55*w - 56], [2971, 2971, 19*w - 68], [2971, 2971, 30*w - 79], [2999, 2999, 23*w - 72], [2999, 2999, 26*w - 75], [3001, 3001, 24*w - 73], [3001, 3001, 25*w - 74], [3011, 3011, 10*w - 61], [3011, 3011, -51*w + 41], [3019, 3019, -59*w - 7], [3019, 3019, 59*w - 66], [3041, 3041, -51*w + 40], [3041, 3041, 11*w - 62], [3049, 3049, 58*w - 63], [3049, 3049, -58*w - 5], [3061, 3061, 33*w - 83], [3061, 3061, 17*w - 67], [3079, 3079, -56*w - 1], [3079, 3079, 56*w - 57], [3089, 3089, 19*w - 69], [3089, 3089, 31*w - 81], [3109, 3109, 21*w - 71], [3109, 3109, 29*w - 79], [3119, 3119, 51*w - 37], [3119, 3119, 14*w - 65], [3121, 3121, 23*w - 73], [3121, 3121, 27*w - 77], [3169, 3169, -61*w - 8], [3169, 3169, 61*w - 69], [3181, 3181, 58*w - 61], [3181, 3181, -58*w - 3], [3191, 3191, -57*w - 1], [3191, 3191, 57*w - 58], [3209, 3209, 19*w - 70], [3209, 3209, 32*w - 83], [3221, 3221, 31*w - 82], [3221, 3221, 20*w - 71], [3229, 3229, -59*w - 4], [3229, 3229, 59*w - 63], [3251, 3251, 25*w - 76], [3251, 3251, 26*w - 77], [3259, 3259, 52*w - 37], [3259, 3259, 15*w - 67], [3271, 3271, 11*w - 64], [3271, 3271, -53*w + 42], [3299, 3299, 17*w - 69], [3299, 3299, 35*w - 87], [3301, 3301, 12*w - 65], [3301, 3301, -53*w + 41], [3319, 3319, 61*w - 67], [3319, 3319, -61*w - 6], [3329, 3329, 13*w - 66], [3329, 3329, 53*w - 40], [3331, 3331, 33*w - 85], [3331, 3331, 19*w - 71], [3359, 3359, 59*w - 61], [3359, 3359, -59*w - 2], [3361, 3361, -62*w - 7], [3361, 3361, 62*w - 69], [3371, 3371, 23*w - 75], [3371, 3371, 29*w - 81], [3389, 3389, -54*w + 43], [3389, 3389, 11*w - 65], [3391, 3391, 61*w - 66], [3391, 3391, -61*w - 5], [3449, 3449, 54*w - 41], [3449, 3449, 13*w - 67], [3461, 3461, 61*w - 65], [3461, 3461, -61*w - 4], [3469, 3469, 33*w - 86], [3469, 3469, 20*w - 73], [3491, 3491, 22*w - 75], [3491, 3491, 31*w - 84], [3499, 3499, 23*w - 76], [3499, 3499, 30*w - 83], [3511, 3511, 26*w - 79], [3511, 3511, 27*w - 80], [3529, 3529, 61*w - 64], [3529, 3529, -61*w - 3], [3539, 3539, -60*w - 1], [3539, 3539, 60*w - 61], [3541, 3541, -55*w + 43], [3541, 3541, 12*w - 67], [3559, 3559, -56*w + 47], [3559, 3559, 9*w - 65], [3571, 3571, 55*w - 42], [3571, 3571, 13*w - 68], [3581, 3581, 19*w - 73], [3581, 3581, 35*w - 89], [3631, 3631, 11*w - 67], [3631, 3631, -56*w + 45], [3659, 3659, -61*w - 1], [3659, 3659, 61*w - 62], [3671, 3671, 55*w - 38], [3671, 3671, 17*w - 72], [3691, 3691, 18*w - 73], [3691, 3691, 37*w - 92], [3701, 3701, -63*w - 4], [3701, 3701, 63*w - 67], [3709, 3709, 19*w - 74], [3709, 3709, 36*w - 91], [3719, 3719, 10*w - 67], [3719, 3719, -57*w + 47], [3739, 3739, 34*w - 89], [3739, 3739, 21*w - 76], [3761, 3761, 32*w - 87], [3761, 3761, 23*w - 78], [3769, 3769, 24*w - 79], [3769, 3769, 31*w - 86], [3779, 3779, 26*w - 81], [3779, 3779, 29*w - 84], [3821, 3821, -57*w + 44], [3821, 3821, 13*w - 70], [3851, 3851, 14*w - 71], [3851, 3851, 57*w - 43], [3881, 3881, 11*w - 69], [3881, 3881, -58*w + 47], [3889, 3889, 67*w - 75], [3889, 3889, -67*w - 8], [3911, 3911, 31*w - 87], [3911, 3911, 25*w - 81], [3919, 3919, 29*w - 85], [3919, 3919, 27*w - 83], [3929, 3929, 17*w - 74], [3929, 3929, 57*w - 40], [3931, 3931, 68*w - 77], [3931, 3931, -68*w - 9], [3989, 3989, 37*w - 94], [3989, 3989, 20*w - 77], [4001, 4001, 66*w - 71], [4001, 4001, -66*w - 5], [4019, 4019, 35*w - 92], [4019, 4019, 22*w - 79], [4021, 4021, -65*w - 3], [4021, 4021, 65*w - 68], [4049, 4049, 32*w - 89], [4049, 4049, 25*w - 82], [4051, 4051, 67*w - 73], [4051, 4051, -67*w - 6], [4079, 4079, 13*w - 72], [4079, 4079, -59*w + 46], [4091, 4091, 65*w - 67], [4091, 4091, -65*w - 2], [4099, 4099, 68*w - 75], [4099, 4099, -68*w - 7], [4111, 4111, 14*w - 73], [4111, 4111, 59*w - 45], [4129, 4129, -67*w - 5], [4129, 4129, 67*w - 72], [4139, 4139, -60*w + 49], [4139, 4139, 11*w - 71], [4159, 4159, -65*w - 1], [4159, 4159, 65*w - 66], [4201, 4201, 27*w - 85], [4201, 4201, 31*w - 89], [4211, 4211, 13*w - 73], [4211, 4211, -60*w + 47], [4219, 4219, 59*w - 41], [4219, 4219, 18*w - 77], [4229, 4229, -69*w - 7], [4229, 4229, 69*w - 76], [4231, 4231, 71*w - 81], [4231, 4231, -71*w - 10], [4241, 4241, 40*w - 99], [4241, 4241, 19*w - 78], [4259, 4259, 68*w - 73], [4259, 4259, -68*w - 5], [4261, 4261, 39*w - 98], [4261, 4261, 20*w - 79], [4271, 4271, 11*w - 72], [4271, 4271, -61*w + 50], [4289, 4289, -66*w - 1], [4289, 4289, 66*w - 67], [4339, 4339, 33*w - 92], [4339, 4339, 26*w - 85], [4349, 4349, 31*w - 90], [4349, 4349, 28*w - 87], [4391, 4391, -69*w - 5], [4391, 4391, 69*w - 74], [4409, 4409, -71*w - 8], [4409, 4409, 71*w - 79], [4421, 4421, -67*w - 1], [4421, 4421, 67*w - 68], [4441, 4441, 16*w - 77], [4441, 4441, 61*w - 45], [4451, 4451, 23*w - 83], [4451, 4451, 37*w - 97], [4481, 4481, 13*w - 75], [4481, 4481, -62*w + 49], [4489, 67, -67], [4519, 4519, 19*w - 80], [4519, 4519, 42*w - 103], [4549, 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8389, 45*w - 127], [8419, 8419, -95*w - 6], [8419, 8419, 95*w - 101], [8429, 8429, 55*w - 138], [8429, 8429, 28*w - 111], [8431, 8431, 18*w - 103], [8431, 8431, -85*w + 67], [8461, 8461, -86*w + 71], [8461, 8461, 15*w - 101], [8501, 8501, 31*w - 114], [8501, 8501, 52*w - 135], [8521, 8521, 51*w - 134], [8521, 8521, 32*w - 115], [8539, 8539, 33*w - 116], [8539, 8539, 50*w - 133], [8581, 8581, 47*w - 130], [8581, 8581, 36*w - 119], [8599, 8599, 38*w - 121], [8599, 8599, 45*w - 128], [8609, 8609, 43*w - 126], [8609, 8609, 40*w - 123], [8629, 8629, -95*w - 4], [8629, 8629, 95*w - 99], [8641, 8641, -98*w - 9], [8641, 8641, 98*w - 107], [8669, 8669, 19*w - 105], [8669, 8669, -86*w + 67], [8681, 8681, -97*w - 7], [8681, 8681, 97*w - 104], [8689, 8689, 85*w - 61], [8689, 8689, 24*w - 109], [8699, 8699, 53*w - 137], [8699, 8699, 31*w - 115], [8719, 8719, 101*w - 114], [8719, 8719, -101*w - 13], [8731, 8731, -95*w - 3], [8731, 8731, 95*w - 98], [8741, 8741, -94*w - 1], [8741, 8741, 94*w - 95], [8761, 8761, 86*w - 65], [8761, 8761, 21*w - 107], [8779, 8779, 100*w - 111], [8779, 8779, -100*w - 11], [8819, 8819, 43*w - 127], [8819, 8819, 41*w - 125], [8821, 8821, 28*w - 113], [8821, 8821, 57*w - 142], [8831, 8831, 95*w - 97], [8831, 8831, -95*w - 2], [8839, 8839, 15*w - 103], [8839, 8839, -88*w + 73], [8849, 8849, 29*w - 114], [8849, 8849, 56*w - 141], [8861, 8861, 19*w - 106], [8861, 8861, -87*w + 68], [8929, 8929, -95*w - 1], [8929, 8929, 95*w - 96], [8941, 8941, 33*w - 118], [8941, 8941, 52*w - 137], [8951, 8951, 17*w - 105], [8951, 8951, -88*w + 71], [8969, 8969, -101*w - 11], [8969, 8969, 101*w - 112], [8971, 8971, 103*w - 117], [8971, 8971, -103*w - 14], [8999, 8999, 87*w - 65], [8999, 8999, 22*w - 109], [9001, 9001, 48*w - 133], [9001, 9001, 37*w - 122], [9011, 9011, 38*w - 123], [9011, 9011, 47*w - 132], [9029, 9029, 41*w - 126], [9029, 9029, 44*w - 129], [9041, 9041, 23*w - 110], [9041, 9041, 87*w - 64], [9049, 9049, 57*w - 143], [9049, 9049, 29*w - 115], [9059, 9059, 99*w - 106], [9059, 9059, -99*w - 7], [9091, 9091, 101*w - 111], [9091, 9091, -101*w - 10], [9109, 9109, 97*w - 100], [9109, 9109, -97*w - 3], [9151, 9151, 88*w - 67], [9151, 9151, 21*w - 109], [9161, 9161, 102*w - 113], [9161, 9161, -102*w - 11], [9181, 9181, 35*w - 121], [9181, 9181, 51*w - 137], [9199, 9199, -89*w + 71], [9199, 9199, 18*w - 107], [9209, 9209, 49*w - 135], [9209, 9209, 37*w - 123], [9221, 9221, 59*w - 146], [9221, 9221, 28*w - 115], [9239, 9239, 88*w - 65], [9239, 9239, 23*w - 111], [9241, 9241, 45*w - 131], [9241, 9241, 41*w - 127], [9281, 9281, -99*w - 5], [9281, 9281, 99*w - 104], [9311, 9311, -97*w - 1], [9311, 9311, 97*w - 98], [9319, 9319, 25*w - 113], [9319, 9319, 88*w - 63], [9341, 9341, 17*w - 107], [9341, 9341, -90*w + 73], [9349, 9349, -89*w + 68], [9349, 9349, 21*w - 110], [9371, 9371, 34*w - 121], [9371, 9371, 53*w - 140], [9391, 9391, 27*w - 115], [9391, 9391, 88*w - 61], [9409, 97, -97], [9419, 9419, 50*w - 137], [9419, 9419, 37*w - 124], [9421, 9421, 106*w - 121], [9421, 9421, -106*w - 15], [9431, 9431, 38*w - 125], [9431, 9431, 49*w - 136], [9439, 9439, 89*w - 66], [9439, 9439, 23*w - 112], [9461, 9461, 43*w - 130], [9461, 9461, 44*w - 131], [9479, 9479, -103*w - 10], [9479, 9479, 103*w - 113], [9491, 9491, 105*w - 118], [9491, 9491, -105*w - 13], [9511, 9511, 31*w - 119], [9511, 9511, 57*w - 145], [9521, 9521, 25*w - 114], [9521, 9521, 89*w - 64], [9539, 9539, 17*w - 108], [9539, 9539, -91*w + 74], [9551, 9551, -104*w - 11], [9551, 9551, 104*w - 115], [9601, 9601, -103*w - 9], [9601, 9601, 103*w - 112], [9619, 9619, -107*w - 15], [9619, 9619, 107*w - 122], [9629, 9629, 61*w - 150], [9629, 9629, 28*w - 117], [9631, 9631, 37*w - 125], [9631, 9631, 51*w - 139], [9649, 9649, -91*w + 72], [9649, 9649, 19*w - 110], [9661, 9661, 60*w - 149], [9661, 9661, 29*w - 118], [9679, 9679, 45*w - 133], [9679, 9679, 43*w - 131], [9689, 9689, -106*w - 13], [9689, 9689, 106*w - 119], [9719, 9719, 58*w - 147], [9719, 9719, 31*w - 120], [9721, 9721, -103*w - 8], [9721, 9721, 103*w - 111], [9739, 9739, 17*w - 109], [9739, 9739, -92*w + 75], [9749, 9749, 105*w - 116], [9749, 9749, -105*w - 11], [9769, 9769, 56*w - 145], [9769, 9769, 33*w - 122], [9781, 9781, 101*w - 105], [9781, 9781, -101*w - 4], [9791, 9791, 34*w - 123], [9791, 9791, 55*w - 144], [9811, 9811, 54*w - 143], [9811, 9811, 35*w - 124], [9829, 9829, 36*w - 125], [9829, 9829, 53*w - 142], [9839, 9839, 103*w - 110], [9839, 9839, -103*w - 7], [9851, 9851, 19*w - 111], [9851, 9851, -92*w + 73], [9859, 9859, 38*w - 127], [9859, 9859, 51*w - 140], [9871, 9871, 50*w - 139], [9871, 9871, 39*w - 128], [9901, 9901, 44*w - 133], [9901, 9901, 45*w - 134], [9929, 9929, 31*w - 121], [9929, 9929, 59*w - 149], [9931, 9931, 25*w - 116], [9931, 9931, 91*w - 66], [9941, 9941, -93*w + 76], [9941, 9941, 17*w - 110], [9949, 9949, -106*w - 11], [9949, 9949, 106*w - 117]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [1, -2, 1, -4, 1, -4, 4, -2, 6, -8, 8, 2, -6, 2, 12, 12, -2, -2, -8, 8, 16, 8, -14, 2, -2, -10, -18, -2, -12, 20, -4, 4, 6, 14, 16, -24, 10, -12, -12, 22, -10, -16, -16, -24, -8, 12, 4, -26, 6, 0, 0, -14, -14, 28, -20, -18, 14, 16, -24, 10, -14, -14, 8, -24, 20, 20, -2, -18, 8, 24, -4, 28, 38, -10, 10, 10, -6, 26, -12, 20, 22, 22, 32, 0, -24, 32, -6, -22, 30, -42, 0, 0, -20, -20, 4, 20, -18, 14, -14, -14, -14, 30, -2, 10, -30, 20, 12, 24, 40, 26, 10, 20, 20, -8, 40, 42, -38, -12, 20, -10, -10, 4, 4, 6, 30, -26, -26, 16, -48, -4, 36, 0, 16, -6, -30, 2, 2, -6, -30, -28, 12, -18, -10, 14, -18, 24, -40, -28, -28, -30, 18, -32, 0, -32, 40, -14, 2, -10, 14, -36, -20, -24, -8, 2, 50, -36, 60, 14, 14, -24, 8, -40, 24, 26, 42, 12, 52, -42, -42, 46, 14, 20, 4, -66, -50, -22, 10, 0, -32, 4, 20, 14, -34, -30, -46, 38, -18, -32, -40, 34, -14, -52, 44, -16, 16, -6, 66, 20, -60, -2, 14, 24, -24, -22, 42, 26, -30, -38, 22, -26, -40, 48, 66, 50, 22, 22, 16, -16, -36, 12, 52, 60, 40, -16, 42, 66, -14, 18, -36, 76, -24, -40, -44, 28, -2, -34, -8, -24, 4, 20, 52, -4, 50, 42, -54, 10, -44, 36, -10, -10, -26, 70, -44, -28, -58, 54, -22, 58, 14, -50, -40, 0, 30, -50, -70, -22, -12, -12, -16, 32, -22, 38, -42, -48, 0, 16, -64, 58, 50, -66, -18, 44, -20, 22, 78, 48, 0, 44, 60, -8, -48, -12, -60, 46, 46, -72, 8, -74, 70, -22, 82, 10, -38, 68, -12, 48, 48, 74, 18, 20, -68, -42, 78, 18, 50, 28, 44, 2, -50, -66, -8, 64, 44, 68, -2, -34, -54, 42, -42, -66, 16, 16, 68, 36, -26, -90, 16, -32, -92, -4, 38, -42, -10, -58, 48, -32, -84, 92, 26, 50, 12, 44, 10, -6, -28, -60, 84, -44, 30, 6, -80, -56, -60, 36, -48, -32, -62, -6, 46, -66, 4, 12, -48, -64, -30, -62, -52, -4, -8, 24, 8, 0, 58, -6, 76, -20, 78, 22, 30, -2, -26, -50, 8, -16, -22, -46, 58, -60, -12, 76, -60, 38, 6, 64, 80, -10, 6, -100, -68, -70, 66, 76, -20, -8, 88, -22, 10, -12, 52, 92, 76, -6, 26, -86, -6, -10, 86, 8, -48, -30, -46, -26, -90, -64, -64, -78, -30, -30, -94, -98, 78, 56, 24, 74, -62, 30, -34, -34, 110, 20, -12, 68, 52, 24, -40, -44, 36, -10, -10, -104, -112, 2, -70, -100, 60, -16, 0, 66, -30, -52, 76, 54, -58, 72, 40, 66, -22, 54, -18, -18, 110, -44, -28, 100, 4, -32, -40, -54, 74, -12, 20, 22, 86, -40, 0, 44, 84, -26, -18, -96, 32, -100, -84, 24, -8, 44, -76, -42, 70, -98, -66, -72, -56, 20, 92, 114, 58, -38, -86, -28, -60, 6, -58, -36, 12, -6, -102, 2, 50, -104, -104, 40, -40, 66, -54, -44, 100, 86, -50, 106, -78, -60, -76, -10, 22, 66, -110, -84, -116, -112, 16, 60, -20, 76, 116, -24, 48, 2, -62, -116, -36, 40, 120, 26, 74, -60, 52, -60, 76, 94, -2, -104, 0, 50, 42, 20, 36, 38, 38, 112, 0, 18, -6, 116, -12, 54, -90, -72, -8, 114, 50, 6, 126, 58, 90, 100, 100, 66, 2, -22, 56, 24, 38, 54, -46, -14, 0, 80, -98, -98, 40, 16, -14, -6, -52, -68, -120, -8, -60, -60, 74, 18, 74, -70, -48, -128, -64, -104, -106, 22, 0, -48, 18, 50, 104, -112, -130, 30, -24, -72, -118, -6, 94, 126, 72, 120, 36, -108, 96, -48, -70, -22, 24, -40, -54, -70, -12, -20, 22, 54, 0, 128, 44, 12, 100, -4, -70, 98, 108, 60, -18, 110, -16, 96, 4, 20, -76, -60, 70, -114, -102, -38, 0, 64, -50, -18, 32, -80, 34, 98, 70, 78, -14, 8, 72, 142, -26, -120, 24, -20, 132, 16, 24, -6, 82, 90, -38, 80, 0, -8, -24, 30, 30, -128, -96, -14, -62, 12, 108, -30, 2, -114, 46, 136, -56, 56, -8, 10, 74, 20, 36, -12, -60, -58, -58, 90, 74, -58, -90, 128, 0, 78, 38, -10, 102, -52, 140, 72, -56, -30, -30, -50, -66, 8, 8, 26, -30, -84, 68, -122, 54, -50, 46, 88, 8, 58, -70, -92, 100, 14, -42, 12, -36, -114, 30, -32, -104, -110, 10, -28, 60, -66, 86, -86, 106, -12, -28, -152, -80, 32, -104, -12, -20, -58, -66, 86, -42, -90, -130, 64, -80, 124, 140, 78, -82, 40, -56, -54, -38, 40, -104, 138, 106, -140, 44, -130, -66, -74, 86, 26, -70, 44, -4, 6, -90, 34, 50, -52, 12, 114, -46, 2, 98, 72, -8, -70, -14, -20, -20, 78, -98, 72, 136, -68, -28, 4, 36, -58, -154, -16, 72, 26, -54, -124, 4, 30, -90, 38, -10, 16, 64, 146, 42, -84, 60, 62, -130, -24, -152, 110, 46, 74, 154, 118, 102, -128, 88, -6, -92, -60, 48, -96, 70, 6, -144, 64, -158, 130, -20, 60, -152, -64, -102, 90, 108, -36, -24, 80, -162, 30, 152, -72, 22, 86, 114, 130, 26, -70, -128, 96, -24, -8, 60, 60, 148, -60, -66, 102, 110, -146, 58, -38, -108, 68, 38, 54, 48, 48, -54, -54, -84, -60, -52, 76, -28, 100, -126, -126, 2, -126, -20, 60, -62, 2, -66, 150, -146, 14, -40, 120, -22, 74, -106, 14, 24, -40, -10, -74, -80, -144, 90, -102, 70, -10, -46, 18, -116, 12, 4, -148, 110, 30, 24, 88, 14, 14, -34, 134, 138, -150, 152, -152, 78, -50, 48, -112, -34, 22, 112, -16, 90, -22, 52, 4, 72, 24, 68, -148, 174, 54, 82, 42, 154, 26, 70, -122, 48, -48, 50, -30, 108, -20, 140, 4, 120, 80, -46, 18, -36, -84, 126, 126, -152, 168, 14, -178, 84, 52, -8, 16, 154, -118, -22, 26, 102, 38, -140, -124, -74, 150, 112, -144, 142, -162, 70, -42, 122, 10, 100, 84, 166, -58, -160, -152, 18, 10, 182, -74, -126, -158, -122, -154, -78, 90, 82, -62, 140, 12, 16, 8, -52, 100, -154, -114, 58, 58, -68, -20, 36, -28, -10, -186, 160, 48, -160, -72, 114, -30, -26, -66, -14, 66, 94, 126, -72, 24, -94, 18, 156, -12, 72, -184, -6, 58, -156, 36, 30, -114, -118, -14, 90, 58, -172, 36, -84, -20, -122, -26, 168, -80, -38, 154, -162, -178, 152, -104, -6, 50, 22, -90, 40, 72, 10, 138, 114, 138, -144, 160, -32, 104, 166, -66, 54, 166, -36, -36, 8, 112, 66, 140, 172, 94, -2, -8, 24, 64, 80, -98, 190, -72, -40, -60, -12, 80, -8, -94, -22, 164, 84, 64, -160, 66, 130, -12, 164, 142, 62, -184, 80, 178, 50, 158, 30, 56, 80, -30, 34, 24, 168, 26, -6, 148, 36, -98, 30, -86, 42, -26, 150, -96, 48, -44, -164, 102, -90, -96, -32, -52, -84, 44, 76, -32, 112, 46, 14, 18, 106, -20, 188, 78, 70, -162, -34]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); 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;