/* 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![-23, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [3, 3, -w + 5], [4, 2, 2], [7, 7, w - 6], [7, 7, -w - 5], [11, 11, -w - 3], [11, 11, w - 4], [17, 17, w + 2], [17, 17, w - 3], [19, 19, w + 6], [19, 19, -w + 7], [23, 23, w], [23, 23, w - 1], [25, 5, -5], [29, 29, -2*w + 9], [29, 29, 2*w + 7], [31, 31, 3*w - 17], [53, 53, 3*w - 14], [53, 53, -3*w - 11], [67, 67, -w - 9], [67, 67, w - 10], [83, 83, 4*w - 19], [83, 83, 7*w - 36], [89, 89, 2*w - 3], [89, 89, -2*w - 1], [97, 97, 3*w - 19], [97, 97, -3*w - 16], [103, 103, 2*w - 15], [103, 103, -2*w - 13], [109, 109, -w - 11], [109, 109, w - 12], [137, 137, -3*w - 7], [137, 137, 3*w - 10], [157, 157, -4*w + 25], [157, 157, 4*w + 21], [163, 163, 2*w - 17], [163, 163, -2*w - 15], [167, 167, 3*w - 8], [167, 167, -3*w - 5], [169, 13, -13], [179, 179, -3*w - 4], [179, 179, 3*w - 7], [193, 193, 7*w - 40], [193, 193, 8*w - 45], [197, 197, 3*w - 5], [197, 197, -3*w - 2], [211, 211, 3*w - 22], [211, 211, -3*w - 19], [239, 239, 17*w - 89], [239, 239, 5*w - 21], [251, 251, -4*w - 9], [251, 251, 4*w - 13], [263, 263, 8*w - 39], [263, 263, 11*w - 56], [269, 269, 13*w - 67], [269, 269, 7*w - 33], [283, 283, -w - 17], [283, 283, w - 18], [307, 307, 2*w - 21], [307, 307, -2*w - 19], [347, 347, 4*w - 7], [347, 347, -4*w - 3], [349, 349, 5*w - 33], [349, 349, -19*w + 103], [353, 353, 21*w - 110], [353, 353, 6*w - 25], [373, 373, -9*w + 52], [373, 373, 12*w - 67], [379, 379, 10*w - 57], [379, 379, 11*w - 62], [383, 383, -7*w + 31], [383, 383, -19*w + 99], [389, 389, 10*w - 49], [389, 389, 13*w - 66], [397, 397, -w - 20], [397, 397, w - 21], [401, 401, 15*w - 77], [401, 401, 9*w - 43], [421, 421, -7*w + 43], [421, 421, 7*w + 36], [439, 439, -w - 21], [439, 439, w - 22], [449, 449, 5*w - 14], [449, 449, -5*w - 9], [461, 461, 11*w - 54], [461, 461, 14*w - 71], [491, 491, 5*w - 12], [491, 491, -5*w - 7], [509, 509, -5*w - 6], [509, 509, 5*w - 11], [541, 541, 5*w - 36], [541, 541, -5*w - 31], [547, 547, -3*w - 26], [547, 547, 3*w - 29], [557, 557, 17*w - 87], [557, 557, 11*w - 53], [569, 569, -5*w - 1], [569, 569, 5*w - 6], [577, 577, -w - 24], [577, 577, w - 25], [587, 587, 7*w - 27], [587, 587, -7*w - 20], [607, 607, -6*w - 35], [607, 607, 6*w - 41], [641, 641, 6*w - 17], [641, 641, -6*w - 11], [647, 647, -8*w - 25], [647, 647, 8*w - 33], [661, 661, 3*w - 31], [661, 661, -3*w - 28], [677, 677, -7*w - 18], [677, 677, 7*w - 25], [691, 691, 2*w - 29], [691, 691, -2*w - 27], [719, 719, 7*w - 24], [719, 719, -7*w - 17], [727, 727, 19*w - 105], [727, 727, -11*w + 65], [733, 733, -w - 27], [733, 733, w - 28], [743, 743, 11*w - 51], [743, 743, -11*w - 40], [751, 751, -5*w - 34], [751, 751, 5*w - 39], [761, 761, 9*w - 38], [761, 761, 30*w - 157], [769, 769, 9*w + 47], [769, 769, -9*w + 56], [773, 773, 6*w - 11], [773, 773, -6*w - 5], [797, 797, -7*w - 15], [797, 797, 7*w - 22], [809, 809, 15*w - 74], [809, 809, 18*w - 91], [811, 811, -10*w + 61], [811, 811, 10*w + 51], [821, 821, -6*w - 1], [821, 821, 6*w - 7], [827, 827, -9*w - 28], [827, 827, 9*w - 37], [853, 853, 4*w - 37], [853, 853, -4*w - 33], [863, 863, 8*w - 29], [863, 863, -8*w - 21], [877, 877, 21*w - 116], [877, 877, -12*w + 71], [881, 881, 10*w - 43], [881, 881, -31*w + 162], [907, 907, -w - 30], [907, 907, w - 31], [911, 911, 16*w - 79], [911, 911, 19*w - 96], [919, 919, 19*w - 106], [919, 919, -14*w + 81], [929, 929, -7*w - 11], [929, 929, 7*w - 18], [937, 937, 16*w - 91], [937, 937, 17*w - 96], [941, 941, 21*w - 107], [941, 941, 15*w - 73], [947, 947, 12*w - 55], [947, 947, -12*w - 43], [953, 953, 9*w - 35], [953, 953, -9*w - 26], [983, 983, 7*w - 16], [983, 983, -7*w - 9], [997, 997, -4*w - 35], [997, 997, 4*w - 39], [1013, 1013, -9*w - 25], [1013, 1013, 9*w - 34], [1019, 1019, 17*w - 84], [1019, 1019, 20*w - 101], [1033, 1033, -w - 32], [1033, 1033, w - 33], [1039, 1039, 10*w + 53], [1039, 1039, -10*w + 63], [1049, 1049, 7*w - 13], [1049, 1049, -7*w - 6], [1051, 1051, 3*w - 37], [1051, 1051, -3*w - 34], [1063, 1063, 2*w - 35], [1063, 1063, -2*w - 33], [1087, 1087, 9*w - 59], [1087, 1087, -33*w + 179], [1091, 1091, 35*w - 183], [1091, 1091, 11*w - 47], [1093, 1093, 11*w + 57], [1093, 1093, -11*w + 68], [1097, 1097, 7*w - 10], [1097, 1097, -7*w - 3], [1109, 1109, 7*w - 9], [1109, 1109, -7*w - 2], [1117, 1117, -7*w - 44], [1117, 1117, 7*w - 51], [1123, 1123, -3*w - 35], [1123, 1123, 3*w - 38], [1181, 1181, 9*w - 31], [1181, 1181, -9*w - 22], [1193, 1193, 14*w - 65], [1193, 1193, -14*w - 51], [1213, 1213, -7*w - 45], [1213, 1213, 7*w - 52], [1237, 1237, -w - 35], [1237, 1237, w - 36], [1249, 1249, -15*w + 88], [1249, 1249, 24*w - 133], [1277, 1277, 13*w - 58], [1277, 1277, -13*w - 45], [1279, 1279, 6*w - 49], [1279, 1279, -6*w - 43], [1283, 1283, 9*w - 29], [1283, 1283, -9*w - 20], [1291, 1291, -17*w + 98], [1291, 1291, 22*w - 123], [1301, 1301, -10*w - 27], [1301, 1301, 10*w - 37], [1303, 1303, 18*w - 103], [1303, 1303, 21*w - 118], [1319, 1319, -8*w - 9], [1319, 1319, 8*w - 17], [1321, 1321, 8*w - 57], [1321, 1321, -8*w - 49], [1327, 1327, -13*w + 79], [1327, 1327, 29*w - 159], [1367, 1367, 8*w - 15], [1367, 1367, -8*w - 7], [1369, 37, -37], [1399, 1399, -5*w - 42], [1399, 1399, 5*w - 47], [1423, 1423, -9*w - 53], [1423, 1423, 9*w - 62], [1439, 1439, -8*w - 3], [1439, 1439, 8*w - 11], [1459, 1459, -w - 38], [1459, 1459, w - 39], [1471, 1471, 25*w - 139], [1471, 1471, -17*w + 99], [1481, 1481, -11*w - 31], [1481, 1481, 11*w - 42], [1487, 1487, 8*w - 5], [1487, 1487, 8*w - 3], [1489, 1489, 5*w - 48], [1489, 1489, -5*w - 43], [1499, 1499, -12*w - 37], [1499, 1499, 12*w - 49], [1511, 1511, 21*w - 104], [1511, 1511, 24*w - 121], [1553, 1553, 11*w - 41], [1553, 1553, -11*w - 30], [1571, 1571, -15*w - 53], [1571, 1571, 15*w - 68], [1597, 1597, -3*w - 41], [1597, 1597, 3*w - 44], [1607, 1607, 17*w - 80], [1607, 1607, 32*w - 165], [1609, 1609, 11*w - 72], [1609, 1609, -40*w + 217], [1621, 1621, 13*w + 68], [1621, 1621, -13*w + 81], [1657, 1657, 9*w - 64], [1657, 1657, -9*w - 55], [1663, 1663, -6*w - 47], [1663, 1663, 6*w - 53], [1667, 1667, 12*w - 47], [1667, 1667, -12*w - 35], [1681, 41, -41], [1693, 1693, -12*w + 77], [1693, 1693, 12*w + 65], [1697, 1697, 27*w - 137], [1697, 1697, 21*w - 103], [1699, 1699, -w - 41], [1699, 1699, w - 42], [1723, 1723, 7*w - 57], [1723, 1723, -7*w - 50], [1741, 1741, -15*w + 91], [1741, 1741, 33*w - 181], [1777, 1777, -9*w - 56], [1777, 1777, 9*w - 65], [1783, 1783, -w - 42], [1783, 1783, w - 43], [1811, 1811, 9*w - 13], [1811, 1811, -9*w - 4], [1831, 1831, -7*w - 51], [1831, 1831, 7*w - 58], [1849, 43, -43], [1861, 1861, -3*w - 44], [1861, 1861, 3*w - 47], [1867, 1867, 6*w - 55], [1867, 1867, -6*w - 49], [1871, 1871, 9*w - 8], [1871, 1871, 9*w - 1], [1877, 1877, 9*w - 2], [1877, 1877, 9*w - 7], [1879, 1879, -46*w + 249], [1879, 1879, 11*w - 74], [1889, 1889, 17*w - 78], [1889, 1889, -17*w - 61], [1913, 1913, 19*w - 90], [1913, 1913, 34*w - 175], [1949, 1949, 13*w - 51], [1949, 1949, -13*w - 38], [1979, 1979, 12*w - 43], [1979, 1979, -12*w - 31], [1993, 1993, -8*w - 55], [1993, 1993, 8*w - 63], [1997, 1997, 29*w - 147], [1997, 1997, 23*w - 113], [2017, 2017, 11*w - 75], [2017, 2017, -11*w - 64], [2027, 2027, -17*w - 60], [2027, 2027, 17*w - 77], [2029, 2029, -4*w - 47], [2029, 2029, 4*w - 51], [2039, 2039, 48*w - 251], [2039, 2039, 15*w - 64], [2053, 2053, 7*w - 60], [2053, 2053, -7*w - 53], [2063, 2063, 33*w - 169], [2063, 2063, 21*w - 101], [2069, 2069, -10*w - 11], [2069, 2069, 10*w - 21], [2099, 2099, 25*w - 124], [2099, 2099, 28*w - 141], [2111, 2111, 11*w - 32], [2111, 2111, -11*w - 21], [2113, 2113, -19*w + 112], [2113, 2113, 32*w - 177], [2129, 2129, 10*w - 19], [2129, 2129, -10*w - 9], [2143, 2143, -3*w - 47], [2143, 2143, 3*w - 50], [2179, 2179, -21*w + 122], [2179, 2179, 30*w - 167], [2203, 2203, 29*w - 162], [2203, 2203, -22*w + 127], [2207, 2207, -13*w - 35], [2207, 2207, 13*w - 48], [2209, 47, -47], [2213, 2213, 11*w - 30], [2213, 2213, -11*w - 19], [2221, 2221, 28*w - 157], [2221, 2221, 23*w - 132], [2239, 2239, 25*w - 142], [2239, 2239, 26*w - 147], [2243, 2243, 36*w - 185], [2243, 2243, 21*w - 100], [2251, 2251, -15*w + 94], [2251, 2251, 15*w + 79], [2281, 2281, 5*w - 56], [2281, 2281, -5*w - 51], [2297, 2297, -47*w + 245], [2297, 2297, 17*w - 75], [2309, 2309, 10*w - 9], [2309, 2309, 10*w - 1], [2341, 2341, 3*w - 52], [2341, 2341, -3*w - 49], [2351, 2351, 11*w - 27], [2351, 2351, -11*w - 16], [2389, 2389, -5*w - 52], [2389, 2389, 5*w - 57], [2393, 2393, -11*w - 15], [2393, 2393, 11*w - 26], [2399, 2399, 23*w - 111], [2399, 2399, 35*w - 179], [2411, 2411, 20*w - 93], [2411, 2411, 41*w - 212], [2417, 2417, 22*w - 105], [2417, 2417, 37*w - 190], [2437, 2437, -4*w - 51], [2437, 2437, 4*w - 55], [2441, 2441, 14*w - 53], [2441, 2441, -14*w - 39], [2447, 2447, 13*w - 45], [2447, 2447, -13*w - 32], [2467, 2467, -18*w + 109], [2467, 2467, 39*w - 214], [2521, 2521, 7*w - 64], [2521, 2521, -7*w - 57], [2539, 2539, -9*w - 62], [2539, 2539, 9*w - 71], [2551, 2551, 13*w - 87], [2551, 2551, -53*w + 287], [2579, 2579, 15*w - 59], [2579, 2579, -15*w - 44], [2593, 2593, -17*w + 105], [2593, 2593, 17*w + 88], [2621, 2621, 14*w - 51], [2621, 2621, -14*w - 37], [2633, 2633, 18*w - 79], [2633, 2633, 18*w + 61], [2657, 2657, 11*w - 18], [2657, 2657, -11*w - 7], [2671, 2671, -22*w + 129], [2671, 2671, 35*w - 194], [2687, 2687, 17*w - 72], [2687, 2687, 56*w - 293], [2693, 2693, 19*w - 85], [2693, 2693, -19*w - 66], [2707, 2707, 34*w - 189], [2707, 2707, -23*w + 134], [2713, 2713, 13*w - 88], [2713, 2713, -13*w - 75], [2741, 2741, -11*w - 3], [2741, 2741, 11*w - 14], [2767, 2767, -7*w - 59], [2767, 2767, 7*w - 66], [2789, 2789, 43*w - 222], [2789, 2789, 22*w - 103], [2791, 2791, 27*w - 154], [2791, 2791, 30*w - 169], [2797, 2797, 28*w - 159], [2797, 2797, 29*w - 164], [2801, 2801, 11*w - 2], [2801, 2801, 11*w - 9], [2819, 2819, -12*w - 17], [2819, 2819, 12*w - 29], [2843, 2843, 19*w - 84], [2843, 2843, 19*w + 65], [2857, 2857, 39*w - 215], [2857, 2857, -21*w + 125], [2879, 2879, 15*w - 56], [2879, 2879, -15*w - 41], [2887, 2887, 11*w - 81], [2887, 2887, -11*w - 70], [2909, 2909, 18*w - 77], [2909, 2909, 57*w - 298], [2927, 2927, 16*w - 63], [2927, 2927, -16*w - 47], [2953, 2953, 37*w - 205], [2953, 2953, -23*w + 135], [2957, 2957, -14*w - 33], [2957, 2957, 14*w - 47], [2969, 2969, 30*w - 149], [2969, 2969, 33*w - 166], [2999, 2999, -13*w - 24], [2999, 2999, 13*w - 37], [3001, 3001, -8*w - 63], [3001, 3001, 8*w - 71], [3041, 3041, 35*w - 177], [3041, 3041, 29*w - 143], [3079, 3079, 5*w - 63], [3079, 3079, -5*w - 58], [3109, 3109, 4*w - 61], [3109, 3109, -4*w - 57], [3137, 3137, 19*w - 82], [3137, 3137, -58*w + 303], [3163, 3163, 19*w + 98], [3163, 3163, -19*w + 117], [3169, 3169, -w - 56], [3169, 3169, w - 57], [3181, 3181, -17*w + 108], [3181, 3181, 17*w + 91], [3187, 3187, -14*w - 81], [3187, 3187, 14*w - 95], [3191, 3191, 24*w - 113], [3191, 3191, 45*w - 232], [3229, 3229, -9*w - 67], [3229, 3229, 9*w - 76], [3251, 3251, 15*w - 52], [3251, 3251, -15*w - 37], [3259, 3259, 6*w - 67], [3259, 3259, -6*w - 61], [3271, 3271, 2*w - 59], [3271, 3271, -2*w - 57], [3299, 3299, -12*w - 1], [3299, 3299, 12*w - 13], [3319, 3319, -21*w + 127], [3319, 3319, 45*w - 247], [3323, 3323, 12*w - 11], [3323, 3323, 12*w - 1], [3329, 3329, 13*w - 31], [3329, 3329, -13*w - 18], [3331, 3331, 3*w - 61], [3331, 3331, -3*w - 58], [3347, 3347, 12*w - 7], [3347, 3347, 12*w - 5], [3359, 3359, 32*w - 159], [3359, 3359, 35*w - 176], [3371, 3371, -59*w + 308], [3371, 3371, 20*w - 87], [3373, 3373, 9*w - 77], [3373, 3373, -9*w - 68], [3413, 3413, -17*w - 49], [3413, 3413, 17*w - 66], [3457, 3457, -59*w + 320], [3457, 3457, 16*w - 105], [3467, 3467, 13*w - 28], [3467, 3467, -13*w - 15], [3469, 3469, -20*w + 123], [3469, 3469, 20*w + 103], [3481, 59, -59], [3511, 3511, -63*w + 341], [3511, 3511, 15*w - 101], [3517, 3517, -w - 59], [3517, 3517, w - 60], [3527, 3527, 17*w - 65], [3527, 3527, -17*w - 48], [3533, 3533, 14*w - 39], [3533, 3533, -14*w - 25], [3541, 3541, 12*w - 89], [3541, 3541, -12*w - 77], [3557, 3557, 23*w - 105], [3557, 3557, -23*w - 82], [3559, 3559, -25*w + 147], [3559, 3559, 41*w - 227], [3583, 3583, -58*w + 315], [3583, 3583, 17*w - 110], [3617, 3617, 47*w - 242], [3617, 3617, 26*w - 123], [3623, 3623, -13*w - 11], [3623, 3623, 13*w - 24], [3631, 3631, -22*w + 133], [3631, 3631, 47*w - 258], [3637, 3637, -w - 60], [3637, 3637, w - 61], [3643, 3643, 39*w - 217], [3643, 3643, -27*w + 157], [3671, 3671, 15*w - 47], [3671, 3671, -15*w - 32], [3691, 3691, -7*w - 66], [3691, 3691, 7*w - 73], [3697, 3697, 3*w - 64], [3697, 3697, -3*w - 61], [3701, 3701, 19*w - 78], [3701, 3701, -19*w - 59], [3709, 3709, 5*w - 68], [3709, 3709, -5*w - 63], [3719, 3719, -13*w - 8], [3719, 3719, 13*w - 21], [3721, 61, -61], [3727, 3727, -13*w - 81], [3727, 3727, 13*w - 94], [3739, 3739, 35*w - 197], [3739, 3739, 31*w - 177], [3769, 3769, 45*w - 248], [3769, 3769, -24*w + 143], [3797, 3797, 13*w - 18], [3797, 3797, -13*w - 5], [3803, 3803, 29*w - 140], [3803, 3803, 44*w - 225], [3823, 3823, -3*w - 62], [3823, 3823, 3*w - 65], [3853, 3853, -4*w - 63], [3853, 3853, 4*w - 67], [3877, 3877, -12*w - 79], [3877, 3877, 12*w - 91], [3881, 3881, -14*w - 19], [3881, 3881, 14*w - 33], [3889, 3889, 15*w - 103], [3889, 3889, -15*w - 88], [3907, 3907, 14*w - 99], [3907, 3907, -14*w - 85], [3917, 3917, 13*w - 3], [3917, 3917, 13*w - 10], [3923, 3923, 13*w - 4], [3923, 3923, 13*w - 9], [3929, 3929, 13*w - 6], [3929, 3929, 13*w - 7], [3931, 3931, 42*w - 233], [3931, 3931, -27*w + 158], [3989, 3989, 35*w - 174], [3989, 3989, 38*w - 191], [4003, 4003, 2*w - 65], [4003, 4003, -2*w - 63], [4027, 4027, 21*w + 109], [4027, 4027, -21*w + 130], [4073, 4073, -14*w - 15], [4073, 4073, 14*w - 29], [4091, 4091, 21*w - 89], [4091, 4091, 69*w - 361], [4093, 4093, 36*w - 203], [4093, 4093, 33*w - 188], [4099, 4099, 34*w - 193], [4099, 4099, 35*w - 198], [4111, 4111, 5*w - 71], [4111, 4111, -5*w - 66], [4157, 4157, 14*w - 27], [4157, 4157, -14*w - 13], [4159, 4159, 11*w - 89], [4159, 4159, -11*w - 78], [4201, 4201, -67*w + 363], [4201, 4201, -17*w + 113], [4211, 4211, 36*w - 179], [4211, 4211, 39*w - 196], [4229, 4229, -25*w - 89], [4229, 4229, 25*w - 114], [4253, 4253, 46*w - 235], [4253, 4253, 31*w - 150], [4259, 4259, -20*w - 61], [4259, 4259, 20*w - 81], [4261, 4261, -13*w - 84], [4261, 4261, 13*w - 97], [4271, 4271, -16*w - 33], [4271, 4271, 16*w - 49], [4289, 4289, 23*w - 101], [4289, 4289, 23*w + 78], [4297, 4297, -24*w + 145], [4297, 4297, 51*w - 280], [4327, 4327, -11*w - 79], [4327, 4327, 11*w - 90], [4397, 4397, -18*w - 47], [4397, 4397, 18*w - 65], [4441, 4441, 8*w - 81], [4441, 4441, -8*w - 73], [4447, 4447, 49*w - 270], [4447, 4447, -26*w + 155], [4457, 4457, -14*w - 3], [4457, 4457, 14*w - 17], [4463, 4463, -17*w - 39], [4463, 4463, 17*w - 56], [4481, 4481, -26*w - 93], [4481, 4481, 26*w - 119], [4483, 4483, 3*w - 70], 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[8237, 8237, 19*w - 22], [8237, 8237, -19*w - 3], [8273, 8273, 21*w - 55], [8273, 8273, -21*w - 34], [8287, 8287, -17*w - 114], [8287, 8287, 17*w - 131], [8293, 8293, -23*w - 132], [8293, 8293, 23*w - 155], [8317, 8317, 11*w - 111], [8317, 8317, -11*w - 100], [8353, 8353, 72*w - 395], [8353, 8353, -33*w + 200], [8363, 8363, 19*w - 15], [8363, 8363, 19*w - 4], [8369, 8369, -25*w - 66], [8369, 8369, 25*w - 91], [8377, 8377, -19*w - 120], [8377, 8377, 19*w - 139], [8387, 8387, 19*w - 7], [8387, 8387, 19*w - 12], [8389, 8389, 20*w - 143], [8389, 8389, -20*w - 123], [8419, 8419, 9*w - 106], [8419, 8419, -9*w - 97], [8423, 8423, 29*w - 120], [8423, 8423, -29*w - 91], [8447, 8447, -21*w - 32], [8447, 8447, 21*w - 53], [8467, 8467, -34*w + 205], [8467, 8467, 71*w - 390], [8527, 8527, 63*w - 349], [8527, 8527, -39*w + 229], [8537, 8537, -97*w + 507], [8537, 8537, 31*w - 133], [8539, 8539, 5*w - 98], [8539, 8539, -5*w - 93], [8563, 8563, 7*w - 102], [8563, 8563, -7*w - 95], [8573, 8573, 37*w - 171], [8573, 8573, -37*w - 134], [8581, 8581, 23*w - 156], [8581, 8581, -23*w - 133], [8609, 8609, 38*w - 177], [8609, 8609, 77*w - 398], [8623, 8623, -9*w - 98], [8623, 8623, 9*w - 107], [8677, 8677, 69*w - 380], [8677, 8677, -36*w + 215], [8689, 8689, 16*w - 129], [8689, 8689, -16*w - 113], [8693, 8693, -21*w - 29], [8693, 8693, 21*w - 50], [8713, 8713, 8*w - 105], [8713, 8713, -8*w - 97], [8719, 8719, -w - 93], [8719, 8719, w - 94], [8731, 8731, -5*w - 94], [8731, 8731, 5*w - 99], [8741, 8741, -35*w - 123], [8741, 8741, 35*w - 158], [8753, 8753, 22*w - 61], [8753, 8753, -22*w - 39], [8761, 8761, 7*w - 103], [8761, 8761, -7*w - 96], [8807, 8807, 33*w - 145], [8807, 8807, 33*w + 112], [8819, 8819, 28*w - 111], [8819, 8819, -28*w - 83], [8831, 8831, -25*w - 63], [8831, 8831, 25*w - 88], [8839, 8839, -29*w + 183], [8839, 8839, 29*w + 154], [8861, 8861, 45*w - 218], [8861, 8861, 66*w - 337], [8863, 8863, 67*w - 370], [8863, 8863, -38*w + 225], [8929, 8929, -21*w - 128], [8929, 8929, 21*w - 149], [8951, 8951, 61*w - 309], [8951, 8951, 49*w - 241], [9011, 9011, -20*w - 7], [9011, 9011, 20*w - 27], [9049, 9049, -15*w - 112], [9049, 9049, 15*w - 127], [9091, 9091, 33*w + 169], [9091, 9091, -33*w + 202], [9103, 9103, -3*w - 95], [9103, 9103, 3*w - 98], [9133, 9133, -93*w + 505], [9133, 9133, -27*w + 175], [9137, 9137, 22*w - 57], [9137, 9137, -22*w - 35], [9181, 9181, -13*w - 108], [9181, 9181, 13*w - 121], [9203, 9203, -28*w - 81], [9203, 9203, 28*w - 109], [9277, 9277, 61*w - 340], [9277, 9277, -44*w + 255], [9281, 9281, 39*w - 181], [9281, 9281, -39*w - 142], [9283, 9283, -17*w - 118], [9283, 9283, 17*w - 135], [9293, 9293, 27*w - 101], [9293, 9293, -27*w - 74], [9311, 9311, 79*w - 408], [9311, 9311, 40*w - 187], [9319, 9319, -5*w - 97], [9319, 9319, 5*w - 102], [9323, 9323, 21*w - 41], [9323, 9323, -21*w - 20], [9349, 9349, 76*w - 417], [9349, 9349, -35*w + 212], [9377, 9377, 59*w - 297], [9377, 9377, 53*w - 263], [9397, 9397, 11*w - 116], [9397, 9397, -11*w - 105], [9403, 9403, 58*w - 325], [9403, 9403, 47*w - 270], [9419, 9419, 25*w - 84], [9419, 9419, -25*w - 59], [9421, 9421, -12*w - 107], [9421, 9421, 12*w - 119], [9433, 9433, 48*w - 275], [9433, 9433, 57*w - 320], [9437, 9437, 23*w - 65], [9437, 9437, -23*w - 42], [9461, 9461, 31*w - 129], [9461, 9461, -31*w - 98], [9463, 9463, 33*w + 170], [9463, 9463, -33*w + 203], [9467, 9467, -27*w - 73], [9467, 9467, 27*w - 100], [9479, 9479, -37*w - 131], [9479, 9479, 37*w - 168], [9497, 9497, -21*w - 17], [9497, 9497, 21*w - 38], [9511, 9511, 2*w - 99], [9511, 9511, -2*w - 97], [9539, 9539, 68*w - 347], [9539, 9539, 47*w - 228], [9551, 9551, -21*w - 16], [9551, 9551, 21*w - 37], [9619, 9619, -11*w - 106], [9619, 9619, 11*w - 117], [9623, 9623, 51*w - 251], [9623, 9623, 63*w - 319], [9643, 9643, -13*w - 110], [9643, 9643, 13*w - 123], [9649, 9649, 25*w - 168], [9649, 9649, -104*w + 563], [9661, 9661, -28*w + 181], [9661, 9661, -95*w + 516], [9679, 9679, -w - 98], [9679, 9679, w - 99], [9689, 9689, -30*w - 91], [9689, 9689, 30*w - 121], [9697, 9697, 19*w - 144], [9697, 9697, -19*w - 125], [9721, 9721, -5*w - 99], [9721, 9721, 5*w - 104], [9739, 9739, -35*w + 213], [9739, 9739, 79*w - 433], [9749, 9749, 23*w - 62], [9749, 9749, -23*w - 39], [9769, 9769, -21*w - 131], [9769, 9769, 21*w - 152], [9781, 9781, -7*w - 101], [9781, 9781, 7*w - 108], [9791, 9791, -21*w - 11], [9791, 9791, 21*w - 32], [9829, 9829, -4*w - 99], [9829, 9829, 4*w - 103], [9833, 9833, 21*w - 31], [9833, 9833, -21*w - 10], [9839, 9839, 25*w - 81], [9839, 9839, -25*w - 56], [9851, 9851, 29*w - 113], [9851, 9851, -29*w - 84], [9857, 9857, 38*w - 173], [9857, 9857, -38*w - 135], [9859, 9859, 29*w + 157], [9859, 9859, -29*w + 186], [9883, 9883, -103*w + 558], [9883, 9883, -26*w + 173], [9887, 9887, 48*w - 233], [9887, 9887, 69*w - 352], [9907, 9907, 2*w - 101], [9907, 9907, -2*w - 99], [9923, 9923, -39*w - 140], [9923, 9923, 39*w - 179], [9941, 9941, 26*w - 89], [9941, 9941, -26*w - 63], [9967, 9967, 19*w - 145], [9967, 9967, -19*w - 126]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [0, 1, 0, 0, 0, 0, -6, -6, 4, 4, 8, 8, -6, 2, 2, -1, -6, -6, -12, -12, 8, 8, -6, -6, 2, 2, 8, 8, -2, -2, 2, 2, 14, 14, -4, -4, 8, 8, -22, 16, 16, -14, -14, -6, -6, -4, -4, 8, 8, 8, 8, -16, -16, -6, -6, 4, 4, -20, -20, -24, -24, -26, -26, -14, -14, -26, -26, 20, 20, -24, -24, 10, 10, 6, 6, -6, -6, -18, -18, 32, 32, -6, -6, 42, 42, 0, 0, 10, 10, -34, -34, -20, -20, -14, -14, -46, -46, 18, 18, -16, -16, 16, 16, -22, -22, 16, 16, 22, 22, 42, 42, 36, 36, -16, -16, -8, -8, -26, -26, -16, -16, 8, 8, 34, 34, -30, -30, 34, 34, 18, 18, -30, -30, -28, -28, 26, 26, 8, 8, -34, -34, -24, -24, 22, 22, -14, -14, 36, 36, 0, 0, 8, 8, -38, -38, -22, -22, 18, 18, -16, -16, 34, 34, -16, -16, -26, -26, 18, 18, -48, -48, -6, -6, 24, 24, 10, 10, -20, -20, -8, -8, 32, 32, -8, -8, 6, 6, -6, -6, 50, 50, 38, 38, 4, 4, -30, -30, -6, -6, -58, -58, -18, -18, -46, -46, 42, 42, -40, -40, 24, 24, -44, -44, 18, 18, -32, -32, -56, -56, 26, 26, -8, -8, 48, 48, 26, 56, 56, 16, 16, 24, 24, 52, 52, -24, -24, -38, -38, 48, 48, 34, 34, -64, -64, -72, -72, 50, 50, 24, 24, 14, 14, -24, -24, -38, -38, 6, 6, -22, -22, 56, 56, 0, 0, -46, -26, -26, -22, -22, 12, 12, 12, 12, 70, 70, 34, 34, 40, 40, 56, 56, -40, -40, -22, 70, 70, -76, -76, -72, -72, 2, 2, -32, -32, -14, -14, -6, -6, -62, -62, -88, -88, 58, 58, 26, 26, 2, 2, 32, 32, -34, -34, -16, -16, -34, -34, 72, 72, 34, 34, 0, 0, 88, 88, -14, -14, -30, -30, -40, -40, -84, -84, -76, -76, -8, -8, -30, -30, -30, -18, -18, 0, 0, 56, 56, 20, 20, 10, 10, -78, -78, -30, -30, -26, -26, 72, 72, -26, -26, 42, 42, -88, -88, 0, 0, -22, -22, 46, 46, 42, 42, 80, 80, -52, -52, -70, -70, 60, 60, 80, 80, 72, 72, 34, 34, 26, 26, -30, -30, -62, -62, -32, -32, -32, -32, -6, -6, -68, -68, -86, -86, 50, 50, -32, -32, 74, 74, 16, 16, 86, 86, -86, -86, -8, -8, -72, -72, -22, -22, 48, 48, 48, 48, 82, 82, 8, 8, -22, -22, 50, 50, 90, 90, -80, -80, 106, 106, -38, -38, -48, -48, 14, 14, -14, -14, -12, -12, 98, 98, -74, -74, -20, -20, 48, 48, -10, -10, -72, -72, 84, 84, 32, 32, -8, -8, -40, -40, -24, -24, 58, 58, 20, 20, 32, 32, 40, 40, -8, -8, -34, -34, 18, 18, -78, -78, 80, 80, -26, -26, 26, 32, 32, 6, 6, 0, 0, -86, -86, 6, 6, -6, -6, 24, 24, 112, 112, 90, 90, 96, 96, -40, -40, -98, -98, 44, 44, -48, -48, 4, 4, 34, 34, -102, -102, 78, 78, -48, -48, -86, 8, 8, 60, 60, 26, 26, -94, -94, 72, 72, -32, -32, -18, -18, -26, -26, -94, -94, -94, -94, 36, 36, -102, -102, -72, -72, 34, 34, -20, -20, -46, -46, 68, 68, 92, 92, 66, 66, 24, 24, -98, -98, -4, -4, -64, -64, -14, -14, 88, 88, 106, 106, -40, -40, 82, 82, 106, 106, 24, 24, 110, 110, 8, 8, 18, 18, -70, -70, 80, 80, 10, 10, -38, -38, -40, -40, -38, -38, 88, 88, 2, 2, 84, 84, -86, -86, -46, -46, -38, -38, 40, 40, 82, 82, 96, 96, 88, 88, -58, -58, -18, -18, -96, -96, 72, 72, 66, 66, -60, -60, 66, 66, -38, -38, -8, -8, 32, 32, 34, 34, 32, 32, 112, 112, -64, -64, 6, 6, 2, 2, -74, -74, -94, -94, -24, -24, 48, 48, -26, -26, 102, 102, -54, -54, 34, 34, 82, 82, 48, 48, 48, 48, 100, 100, -118, -118, 64, 64, -104, -104, -78, 64, 64, -96, -96, 80, 80, 40, 40, -4, -4, 34, 34, 118, 118, 26, 26, 92, 92, 24, 24, -110, -110, 66, 66, 18, 18, 34, 34, -78, -78, -46, 120, 120, -62, -62, 86, 86, 114, 114, -44, -44, -12, -12, -72, -72, -78, -78, -96, -96, -56, -56, -8, -8, -48, -48, 70, 70, 116, 116, 66, 66, -14, -14, -50, -50, -40, -40, -16, -16, -78, -78, -46, -46, -36, -36, -6, -6, -42, -42, 42, 42, 26, 26, 34, 34, 104, 104, 110, 110, -96, -96, 8, 8, 64, 64, -30, -30, 110, 110, -80, -80, -20, -20, -56, -56, -14, -14, 122, 122, -86, -86, 10, 10, -22, -22, 42, 42, 42, 42, -120, -120, -44, -44, -128, -128, -54, -54, -94, 24, 24, 138, 138, -8, -8, 64, 64, 6, 6, -46, -46, 24, 24, -136, -136, -46, -46, 14, 14, 154, 154, -106, -106, 76, 76, 66, 66, 24, 24, -22, -22, 130, 130, 136, 136, -14, -14, -8, -8, 96, 96, 4, 4, -110, -110, -128, -128, -30, -30, -64, -64, -96, -96, 90, 90, 148, 148, 0, 0, 74, 74, -10, -10, 90, 90, 42, 42, 56, 56, 40, 40, -20, -20, 48, 48, 116, 116, 48, 48, 72, 72, -74, -74, 24, 24, -40, -40, 120, 120, 82, 82, -120, -120, 112, 112, 56, 56, 2, 2, 14, 14, -112, -112, 50, 50, 0, 0, 74, 74, -24, -24, -62, -62, -50, -50, 116, 116, 16, 16, 18, 18, -24, -24, 0, 0, 154, 154, 24, 24, -24, -24, -116, -116, 26, 26, -126, -126, 120, 120, -70, -70, -140, -140, -46, -46, 52, 52, 58, 58, -112, -112, 42, 42, -158, -158, -10, -10, -80, -80, 10, 10, -114, -114, -22, -22, -100, -100, 104, 104, -40, -40, -54, -54, 80, 80, -112, -112, 36, 36, 104, 104, -154, -154, -30, -30, 16, 16, 154, 154, 114, 114, -126, -126, 32, 32, 34, 34, 174, 174, -30, -30, 130, 130, 146, 146, -174, -174, -32, -32, -162, -162, -166, -166, 98, 98, 16, 16, -88, -88, -94, -94, 90, 90, -102, -102, -30, -30, 72, 72, 70, 70, -74, -74, -94, -94, 136, 136, -126, -126, 58, 58, 56, 56, 110, 110, 44, 44, 168, 168, 32, 32, -124, -124, 168, 168, -118, -118, -44, -44, 4, 4, -62, -62, 22, 22, 66, 66, -104, -104, 22, 22, 18, 18, 26, 26, 10, 10, 80, 80, -44, -44, -62, -62, -22, -22, -54, -54, -64, -64, -24, -24, -96, -96, -104, -104, 138, 138, -104, -104, -142, -142, 120, 120, -24, -24, 26, 26, -140, -140, -56, -56, 142, 142, -142, -142, 70, 70, -168, -168, 118, 118, 162, 162, -76, -76, 90, 90, 0, 0, 80, 80, 96, 96, -106, -106, 82, 82, -34, -34, 36, 36, -104, -104, -138, -138, -182, -182, 50, 50, 42, 42, -16, -16, 104, 104, -104, -104, 66, 66, -8, -8, -144, -144, -120, -120, -28, -28, -176, -176, 172, 172, 34, 34, 86, 86, 56, 56, 178, 178, -78, -78, -22, -22, -172, -172, 74, 74, -86, -86, 6, 6, 96, 96, -50, -50, -126, -126, 88, 88, 48, 48, -102, -102, -44, -44, 132, 132, 8, 8, 4, 4, -64, -64, -70, -70, 112, 112]; 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;