/* 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![-4, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, -w + 2], [2, 2, -w - 1], [9, 3, 3], [13, 13, -2*w + 3], [13, 13, 2*w + 1], [17, 17, -2*w + 1], [19, 19, -2*w + 7], [19, 19, 2*w + 5], [25, 5, -5], [43, 43, 4*w - 7], [43, 43, 4*w + 3], [47, 47, -2*w + 9], [47, 47, 2*w + 7], [49, 7, -7], [53, 53, 4*w - 13], [53, 53, 6*w - 13], [59, 59, -4*w - 1], [59, 59, 4*w - 5], [67, 67, 4*w - 3], [67, 67, 4*w - 1], [83, 83, 2*w - 11], [83, 83, -2*w - 9], [89, 89, 6*w - 11], [89, 89, 6*w + 5], [101, 101, 10*w - 23], [101, 101, 4*w - 15], [103, 103, 8*w + 9], [103, 103, 8*w - 17], [121, 11, -11], [127, 127, 2*w - 13], [127, 127, -2*w - 11], [137, 137, -6*w - 1], [137, 137, 6*w - 7], [149, 149, 6*w - 5], [149, 149, 6*w - 1], [151, 151, 8*w - 15], [151, 151, 8*w + 7], [157, 157, 14*w + 19], [157, 157, 4*w + 13], [179, 179, 2*w - 15], [179, 179, -2*w - 13], [191, 191, 8*w - 13], [191, 191, 8*w + 5], [223, 223, -8*w - 3], [223, 223, 8*w - 11], [229, 229, 10*w - 19], [229, 229, 10*w + 9], [239, 239, 2*w - 17], [239, 239, -2*w - 15], [251, 251, 12*w + 13], [251, 251, 12*w - 25], [257, 257, 8*w - 27], [257, 257, 14*w - 31], [263, 263, 8*w - 7], [263, 263, 8*w - 1], [271, 271, 8*w - 5], [271, 271, 8*w - 3], [281, 281, 10*w + 7], [281, 281, 10*w - 17], [293, 293, 4*w - 21], [293, 293, -4*w - 17], [307, 307, 2*w - 19], [307, 307, -2*w - 17], [331, 331, 6*w - 25], [331, 331, -6*w - 19], [349, 349, 14*w + 15], [349, 349, 14*w - 29], [353, 353, 18*w - 41], [353, 353, 8*w - 29], [359, 359, 16*w - 35], [359, 359, 10*w - 33], [373, 373, -4*w - 19], [373, 373, 4*w - 23], [383, 383, 2*w - 21], [383, 383, -2*w - 19], [389, 389, -10*w - 1], [389, 389, 10*w - 11], [409, 409, 10*w - 9], [409, 409, 10*w - 1], [421, 421, 10*w - 3], [421, 421, 10*w - 7], [433, 433, 14*w - 27], [433, 433, -14*w - 13], [443, 443, 12*w - 19], [443, 443, -12*w - 7], [457, 457, 22*w - 51], [457, 457, 8*w - 31], [461, 461, 4*w - 25], [461, 461, -4*w - 21], [463, 463, 16*w + 17], [463, 463, 16*w - 33], [467, 467, 2*w - 23], [467, 467, -2*w - 21], [491, 491, -12*w - 5], [491, 491, 12*w - 17], [509, 509, 14*w - 25], [509, 509, 14*w + 11], [523, 523, -6*w - 23], [523, 523, 6*w - 29], [529, 23, -23], [557, 557, -4*w - 23], [557, 557, 4*w - 27], [563, 563, -12*w - 1], [563, 563, 12*w - 13], [569, 569, 26*w + 35], [569, 569, 26*w - 61], [577, 577, 14*w - 23], [577, 577, -14*w - 9], [587, 587, 12*w - 11], [587, 587, 12*w - 1], [593, 593, 18*w + 19], [593, 593, 18*w - 37], [599, 599, 10*w - 37], [599, 599, 24*w - 55], [613, 613, 12*w - 41], [613, 613, 22*w - 49], [631, 631, 6*w - 31], [631, 631, -6*w - 25], [647, 647, 16*w + 13], [647, 647, 16*w - 29], [659, 659, 2*w - 27], [659, 659, -2*w - 25], [661, 661, 4*w - 29], [661, 661, -4*w - 25], [701, 701, 18*w - 35], [701, 701, -18*w - 17], [727, 727, 16*w + 11], [727, 727, 16*w - 27], [733, 733, -14*w - 3], [733, 733, 14*w - 17], [739, 739, 20*w + 21], [739, 739, 20*w - 41], [757, 757, 26*w - 59], [757, 757, 12*w - 43], [761, 761, 22*w - 47], [761, 761, 22*w + 25], [769, 769, -14*w - 1], [769, 769, 14*w - 15], [773, 773, -4*w - 27], [773, 773, 4*w - 31], [797, 797, 14*w - 13], [797, 797, 14*w - 1], [829, 829, 14*w - 9], [829, 829, 14*w - 5], [841, 29, -29], [859, 859, 20*w - 39], [859, 859, -20*w - 19], [863, 863, -16*w - 7], [863, 863, 16*w - 23], [883, 883, 2*w - 31], [883, 883, -2*w - 29], [919, 919, 16*w - 21], [919, 919, -16*w - 5], [937, 937, 26*w - 57], [937, 937, 16*w - 53], [953, 953, -8*w - 31], [953, 953, 8*w - 39], [961, 31, -31], [967, 967, 16*w - 19], [967, 967, -16*w - 3], [971, 971, 20*w - 37], [971, 971, 20*w + 17], [977, 977, -18*w - 11], [977, 977, 18*w - 29], [1019, 1019, -10*w - 33], [1019, 1019, 10*w - 43], [1021, 1021, -4*w - 31], [1021, 1021, 4*w - 35], [1033, 1033, 22*w - 43], [1033, 1033, -22*w - 21], [1039, 1039, 16*w - 15], [1039, 1039, 16*w - 1], [1063, 1063, 16*w - 3], [1063, 1063, 16*w - 13], [1069, 1069, 34*w - 79], [1069, 1069, 12*w - 47], [1087, 1087, 16*w - 9], [1087, 1087, 16*w - 7], [1097, 1097, 8*w - 41], [1097, 1097, -8*w - 33], [1103, 1103, 32*w - 73], [1103, 1103, 14*w - 51], [1109, 1109, 26*w + 29], [1109, 1109, 26*w - 55], [1123, 1123, 28*w - 61], [1123, 1123, 18*w - 59], [1171, 1171, 20*w - 33], [1171, 1171, -20*w - 13], [1181, 1181, 18*w - 23], [1181, 1181, -18*w - 5], [1223, 1223, 24*w - 47], [1223, 1223, -24*w - 23], [1237, 1237, 38*w + 51], [1237, 1237, 38*w - 89], [1249, 1249, 8*w - 43], [1249, 1249, -8*w - 35], [1259, 1259, 20*w - 31], [1259, 1259, -20*w - 11], [1277, 1277, -18*w - 1], [1277, 1277, 18*w - 19], [1279, 1279, 2*w - 37], [1279, 1279, -2*w - 35], [1283, 1283, 36*w - 83], [1283, 1283, 14*w - 53], [1291, 1291, -6*w - 35], [1291, 1291, 6*w - 41], [1301, 1301, -4*w - 35], [1301, 1301, 4*w - 39], [1307, 1307, 28*w - 59], [1307, 1307, 28*w + 31], [1327, 1327, 32*w - 71], [1327, 1327, 18*w - 61], [1361, 1361, 18*w - 5], [1361, 1361, 18*w - 13], [1369, 37, -37], [1373, 1373, 18*w - 7], [1373, 1373, 18*w - 11], [1381, 1381, -22*w - 15], [1381, 1381, 22*w - 37], [1409, 1409, -8*w - 37], [1409, 1409, 8*w - 45], [1427, 1427, 2*w - 39], [1427, 1427, -2*w - 37], [1429, 1429, 26*w - 51], [1429, 1429, -26*w - 25], [1447, 1447, 6*w - 43], [1447, 1447, -6*w - 37], [1453, 1453, 4*w - 41], [1453, 1453, -4*w - 37], [1471, 1471, 40*w - 93], [1471, 1471, 14*w - 55], [1481, 1481, 22*w - 35], [1481, 1481, -22*w - 13], [1483, 1483, 28*w + 29], [1483, 1483, 28*w - 57], [1487, 1487, 24*w - 43], [1487, 1487, 24*w + 19], [1511, 1511, -10*w - 39], [1511, 1511, 10*w - 49], [1531, 1531, -20*w - 3], [1531, 1531, 20*w - 23], [1543, 1543, 22*w - 71], [1543, 1543, 32*w - 69], [1549, 1549, 20*w - 67], [1549, 1549, 34*w - 75], [1579, 1579, -20*w - 1], [1579, 1579, 20*w - 21], [1583, 1583, 2*w - 41], [1583, 1583, -2*w - 39], [1597, 1597, 12*w - 53], [1597, 1597, -12*w - 41], [1607, 1607, 24*w - 41], [1607, 1607, 24*w + 17], [1613, 1613, -4*w - 39], [1613, 1613, 4*w - 43], [1619, 1619, 20*w - 19], [1619, 1619, 20*w - 1], [1657, 1657, -22*w - 9], [1657, 1657, 22*w - 31], [1667, 1667, 44*w + 59], [1667, 1667, 44*w - 103], [1681, 41, -41], [1699, 1699, 20*w - 11], [1699, 1699, 20*w - 9], [1709, 1709, 30*w + 31], [1709, 1709, 30*w - 61], [1721, 1721, 16*w - 61], [1721, 1721, 42*w - 97], [1733, 1733, 22*w - 29], [1733, 1733, -22*w - 7], [1747, 1747, 2*w - 43], [1747, 1747, -2*w - 41], [1753, 1753, 8*w - 49], [1753, 1753, -8*w - 41], [1759, 1759, 18*w - 65], [1759, 1759, 40*w - 91], [1777, 1777, 24*w - 77], [1777, 1777, 34*w - 73], [1783, 1783, -6*w - 41], [1783, 1783, 6*w - 47], [1787, 1787, 36*w - 79], [1787, 1787, 22*w - 73], [1789, 1789, -12*w - 43], [1789, 1789, 12*w - 55], [1801, 1801, -22*w - 5], [1801, 1801, 22*w - 27], [1811, 1811, 28*w + 25], [1811, 1811, 28*w - 53], [1823, 1823, 24*w - 37], [1823, 1823, -24*w - 13], [1861, 1861, 22*w - 25], [1861, 1861, -22*w - 3], [1871, 1871, 14*w - 59], [1871, 1871, -14*w - 45], [1879, 1879, -10*w - 43], [1879, 1879, 10*w - 53], [1889, 1889, 30*w - 59], [1889, 1889, -30*w - 29], [1913, 1913, -22*w - 1], [1913, 1913, 22*w - 23], [1951, 1951, 32*w + 33], [1951, 1951, 32*w - 65], [1973, 1973, 26*w - 43], [1973, 1973, -26*w - 17], [1987, 1987, 18*w - 67], [1987, 1987, 44*w - 101], [1993, 1993, 22*w - 3], [1993, 1993, 22*w - 19], [1997, 1997, 34*w - 71], [1997, 1997, 34*w + 37], [2027, 2027, 36*w - 77], [2027, 2027, 26*w - 83], [2039, 2039, 22*w - 75], [2039, 2039, 40*w - 89], [2053, 2053, 22*w - 13], [2053, 2053, 22*w - 9], [2083, 2083, 14*w - 61], [2083, 2083, -14*w - 47], [2087, 2087, -24*w - 7], [2087, 2087, 24*w - 31], [2089, 2089, -26*w - 15], [2089, 2089, 26*w - 41], [2099, 2099, 2*w - 47], [2099, 2099, -2*w - 45], [2129, 2129, -8*w - 45], [2129, 2129, 8*w - 53], [2141, 2141, 4*w - 49], [2141, 2141, -4*w - 45], [2143, 2143, 32*w - 63], [2143, 2143, -32*w - 31], [2161, 2161, 50*w + 67], [2161, 2161, 50*w - 117], [2243, 2243, 28*w + 19], [2243, 2243, 28*w - 47], [2269, 2269, 46*w - 105], [2269, 2269, 20*w - 73], [2287, 2287, 2*w - 49], [2287, 2287, -2*w - 47], [2293, 2293, 38*w - 81], [2293, 2293, 38*w + 43], [2297, 2297, 26*w - 37], [2297, 2297, -26*w - 11], [2311, 2311, 40*w - 87], [2311, 2311, 26*w - 85], [2333, 2333, -4*w - 47], [2333, 2333, 4*w - 51], [2347, 2347, -6*w - 47], [2347, 2347, 6*w - 53], [2371, 2371, -28*w - 17], [2371, 2371, 28*w - 45], [2381, 2381, 30*w - 53], [2381, 2381, 30*w + 23], [2389, 2389, -26*w - 9], [2389, 2389, 26*w - 35], [2393, 2393, -16*w - 51], [2393, 2393, 16*w - 67], [2399, 2399, 24*w - 19], [2399, 2399, 24*w - 5], [2423, 2423, 24*w - 17], [2423, 2423, 24*w - 7], [2447, 2447, 24*w - 13], [2447, 2447, 24*w - 11], [2467, 2467, 52*w - 121], [2467, 2467, 18*w - 71], [2473, 2473, 26*w - 33], [2473, 2473, -26*w - 7], [2503, 2503, 32*w - 59], [2503, 2503, 32*w + 27], [2531, 2531, -14*w - 51], [2531, 2531, 14*w - 65], [2549, 2549, 26*w - 31], [2549, 2549, -26*w - 5], [2551, 2551, 6*w - 55], [2551, 2551, -6*w - 49], [2593, 2593, 24*w - 83], [2593, 2593, 46*w - 103], [2609, 2609, -34*w - 31], [2609, 2609, 34*w - 65], [2617, 2617, -26*w - 3], [2617, 2617, 26*w - 29], [2633, 2633, -16*w - 53], [2633, 2633, 16*w - 69], [2671, 2671, 32*w + 25], [2671, 2671, 32*w - 57], [2677, 2677, -26*w - 1], [2677, 2677, 26*w - 27], [2687, 2687, 2*w - 53], [2687, 2687, -2*w - 51], [2699, 2699, 36*w - 71], [2699, 2699, -36*w - 35], [2707, 2707, 28*w - 39], [2707, 2707, -28*w - 11], [2711, 2711, 10*w - 61], [2711, 2711, -10*w - 51], [2719, 2719, 56*w + 75], [2719, 2719, 56*w - 131], [2729, 2729, 26*w - 25], [2729, 2729, 26*w - 1], [2741, 2741, -4*w - 51], [2741, 2741, 4*w - 55], [2753, 2753, 8*w - 59], [2753, 2753, -8*w - 51], [2767, 2767, -14*w - 53], [2767, 2767, 14*w - 67], [2789, 2789, 54*w - 125], [2789, 2789, 20*w - 77], [2797, 2797, 34*w - 63], [2797, 2797, 34*w + 29], [2801, 2801, 30*w - 47], [2801, 2801, -30*w - 17], [2803, 2803, 28*w - 37], [2803, 2803, -28*w - 9], [2837, 2837, 26*w - 7], [2837, 2837, 26*w - 19], [2843, 2843, 22*w - 81], [2843, 2843, 52*w - 119], [2857, 2857, 26*w - 17], [2857, 2857, 26*w - 9], [2903, 2903, 48*w - 107], [2903, 2903, 26*w - 89], [2909, 2909, 46*w - 101], [2909, 2909, 28*w - 93], [2939, 2939, -10*w - 53], [2939, 2939, 10*w - 63], [2957, 2957, 4*w - 57], [2957, 2957, -4*w - 53], [2971, 2971, 28*w - 33], [2971, 2971, -28*w - 5], [3001, 3001, 38*w - 75], [3001, 3001, -38*w - 37], [3011, 3011, -14*w - 55], [3011, 3011, 14*w - 69], [3041, 3041, -30*w - 13], [3041, 3041, 30*w - 43], [3061, 3061, 58*w - 135], [3061, 3061, 20*w - 79], [3079, 3079, 40*w + 41], [3079, 3079, 40*w - 81], [3109, 3109, -12*w - 55], [3109, 3109, 12*w - 67], [3119, 3119, 2*w - 57], [3119, 3119, -2*w - 55], [3137, 3137, -16*w - 57], [3137, 3137, 16*w - 73], [3163, 3163, 28*w - 27], [3163, 3163, 28*w - 1], [3181, 3181, -4*w - 55], [3181, 3181, 4*w - 59], [3187, 3187, 44*w + 49], [3187, 3187, 44*w - 93], [3209, 3209, -8*w - 55], [3209, 3209, 8*w - 63], [3217, 3217, 46*w - 99], [3217, 3217, 32*w - 103], [3221, 3221, 38*w - 73], [3221, 3221, -38*w - 35], [3229, 3229, 50*w - 111], [3229, 3229, 28*w - 95], [3251, 3251, 28*w - 5], [3251, 3251, 28*w - 23], [3299, 3299, 36*w + 29], [3299, 3299, 36*w - 65], [3307, 3307, 28*w - 9], [3307, 3307, 28*w - 19], [3313, 3313, 34*w + 23], [3313, 3313, 34*w - 57], [3319, 3319, 40*w - 79], [3319, 3319, -40*w - 39], [3323, 3323, 28*w - 11], [3323, 3323, 28*w - 17], [3331, 3331, 28*w - 15], [3331, 3331, 28*w - 13], [3347, 3347, 2*w - 59], [3347, 3347, -2*w - 57], [3391, 3391, -32*w - 15], [3391, 3391, 32*w - 47], [3413, 3413, 4*w - 61], [3413, 3413, -4*w - 57], [3433, 3433, 38*w + 33], [3433, 3433, 38*w - 71], [3449, 3449, 8*w - 65], [3449, 3449, -8*w - 57], [3467, 3467, 44*w + 47], [3467, 3467, 44*w - 91], [3469, 3469, -34*w - 21], [3469, 3469, 34*w - 55], [3511, 3511, -32*w - 13], [3511, 3511, 32*w - 45], [3517, 3517, 46*w - 97], [3517, 3517, 46*w + 51], [3527, 3527, 26*w - 93], [3527, 3527, 56*w - 127], [3557, 3557, 28*w - 97], [3557, 3557, 54*w - 121], [3571, 3571, 30*w - 101], [3571, 3571, 52*w - 115], [3583, 3583, 2*w - 61], [3583, 3583, -2*w - 59], [3613, 3613, -12*w - 59], [3613, 3613, 12*w - 71], [3617, 3617, -34*w - 19], [3617, 3617, 34*w - 53], [3623, 3623, -32*w - 11], [3623, 3623, 32*w - 43], [3637, 3637, 38*w - 69], [3637, 3637, 38*w + 31], [3659, 3659, 36*w + 25], [3659, 3659, 36*w - 61], [3671, 3671, -10*w - 59], [3671, 3671, 10*w - 69], [3673, 3673, 16*w - 77], [3673, 3673, -16*w - 61], [3691, 3691, -6*w - 59], [3691, 3691, 6*w - 65], [3697, 3697, 8*w - 67], [3697, 3697, -8*w - 59], [3719, 3719, 64*w - 149], [3719, 3719, 22*w - 87], [3721, 61, -61], [3727, 3727, 32*w - 41], [3727, 3727, -32*w - 9], [3739, 3739, 44*w + 45], [3739, 3739, 44*w - 89], [3761, 3761, 30*w - 23], [3761, 3761, 30*w - 7], [3793, 3793, 62*w - 143], [3793, 3793, 24*w - 91], [3821, 3821, 30*w - 17], [3821, 3821, 30*w - 13], [3823, 3823, 32*w - 39], [3823, 3823, -32*w - 7], [3833, 3833, 38*w - 67], [3833, 3833, 38*w + 29], [3851, 3851, 60*w - 137], [3851, 3851, 26*w - 95], [3863, 3863, 48*w - 101], [3863, 3863, 48*w + 53], [3877, 3877, 12*w - 73], [3877, 3877, -12*w - 61], [3889, 3889, 34*w - 49], [3889, 3889, -34*w - 15], [3911, 3911, -32*w - 5], [3911, 3911, 32*w - 37], [3919, 3919, 30*w - 103], [3919, 3919, 56*w - 125], [3923, 3923, 52*w - 113], [3923, 3923, 34*w - 111], [3929, 3929, 32*w - 107], [3929, 3929, 54*w - 119], [3931, 3931, 10*w - 71], [3931, 3931, -10*w - 61], [3943, 3943, 6*w - 67], [3943, 3943, -6*w - 61], [4003, 4003, 44*w - 87], [4003, 4003, -44*w - 43], [4013, 4013, 34*w - 47], [4013, 4013, -34*w - 13], [4021, 4021, 38*w - 65], [4021, 4021, 38*w + 27], [4027, 4027, 68*w + 91], [4027, 4027, 68*w - 159], [4079, 4079, 2*w - 65], [4079, 4079, -2*w - 63], [4093, 4093, 46*w + 47], [4093, 4093, 46*w - 93], [4099, 4099, 18*w - 83], [4099, 4099, -18*w - 65], [4127, 4127, 32*w - 31], [4127, 4127, 32*w - 1], [4129, 4129, 34*w - 45], [4129, 4129, -34*w - 11], [4133, 4133, 42*w - 79], [4133, 4133, 42*w + 37], [4139, 4139, 36*w - 55], [4139, 4139, -36*w - 19], [4157, 4157, -4*w - 63], [4157, 4157, 4*w - 67], [4201, 4201, 38*w - 63], [4201, 4201, -38*w - 25], [4217, 4217, -8*w - 63], [4217, 4217, 8*w - 71], [4229, 4229, 20*w - 87], [4229, 4229, -20*w - 67], [4231, 4231, 32*w - 27], [4231, 4231, 32*w - 5], [4241, 4241, 16*w - 81], [4241, 4241, -16*w - 65], [4259, 4259, 44*w - 85], [4259, 4259, -44*w - 41], [4271, 4271, 32*w - 7], [4271, 4271, 32*w - 25], [4283, 4283, -36*w - 17], [4283, 4283, 36*w - 53], [4297, 4297, 32*w - 109], [4297, 4297, 58*w - 129], [4327, 4327, 32*w - 21], [4327, 4327, 32*w - 11], [4337, 4337, -34*w - 7], [4337, 4337, 34*w - 41], [4339, 4339, 2*w - 67], [4339, 4339, -2*w - 65], [4373, 4373, -38*w - 23], [4373, 4373, 38*w - 61], [4421, 4421, 4*w - 69], [4421, 4421, -4*w - 65], [4441, 4441, 70*w - 163], [4441, 4441, 24*w - 95], [4463, 4463, 48*w + 49], [4463, 4463, 48*w - 97], [4507, 4507, 44*w - 83], [4507, 4507, 44*w + 39], [4513, 4513, 34*w - 37], [4513, 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177], [8447, 8447, 2*w - 93], [8447, 8447, -2*w - 91], [8467, 8467, -52*w - 29], [8467, 8467, 52*w - 81], [8501, 8501, -20*w - 91], [8501, 8501, 20*w - 111], [8513, 8513, 66*w + 67], [8513, 8513, 66*w - 133], [8521, 8521, 58*w + 47], [8521, 8521, 58*w - 105], [8543, 8543, 34*w - 133], [8543, 8543, 96*w - 223], [8581, 8581, -4*w - 91], [8581, 8581, 4*w - 95], [8623, 8623, 18*w - 109], [8623, 8623, -18*w - 91], [8627, 8627, 68*w + 71], [8627, 8627, 68*w - 139], [8669, 8669, 46*w - 5], [8669, 8669, 46*w - 41], [8689, 8689, 50*w - 69], [8689, 8689, -50*w - 19], [8713, 8713, 16*w - 107], [8713, 8713, -16*w - 91], [8719, 8719, 30*w - 127], [8719, 8719, -30*w - 97], [8737, 8737, 46*w - 7], [8737, 8737, 46*w - 39], [8747, 8747, 92*w - 211], [8747, 8747, 38*w - 141], [8753, 8753, 8*w - 99], [8753, 8753, -8*w - 91], [8807, 8807, 72*w + 79], [8807, 8807, 72*w - 151], [8819, 8819, 2*w - 95], [8819, 8819, -2*w - 93], [8821, 8821, 58*w - 103], [8821, 8821, 58*w + 45], [8831, 8831, -48*w - 7], [8831, 8831, 48*w - 55], [8839, 8839, 56*w + 39], [8839, 8839, 56*w - 95], [8849, 8849, 46*w - 35], [8849, 8849, 46*w - 11], [8861, 8861, -50*w - 17], [8861, 8861, 50*w - 67], [8887, 8887, 42*w - 149], [8887, 8887, 88*w - 199], [8893, 8893, 46*w - 13], [8893, 8893, 46*w - 33], [8923, 8923, 54*w - 173], [8923, 8923, 76*w - 163], [8929, 8929, 46*w - 15], [8929, 8929, 46*w - 31], [8933, 8933, 44*w - 153], [8933, 8933, 86*w - 193], [8941, 8941, 62*w - 117], [8941, 8941, 62*w + 55], [8951, 8951, 48*w - 53], [8951, 8951, -48*w - 5], [8963, 8963, 46*w - 157], [8963, 8963, 84*w - 187], [9001, 9001, -24*w - 95], [9001, 9001, 24*w - 119], [9011, 9011, 100*w - 233], [9011, 9011, 34*w - 135], [9029, 9029, -54*w - 31], [9029, 9029, 54*w - 85], [9043, 9043, 68*w + 69], [9043, 9043, 68*w - 137], [9059, 9059, 60*w - 109], [9059, 9059, 60*w + 49], [9091, 9091, 52*w - 75], [9091, 9091, -52*w - 23], [9103, 9103, 56*w - 93], [9103, 9103, -56*w - 37], [9127, 9127, 64*w - 123], [9127, 9127, -64*w - 59], [9133, 9133, 36*w - 139], [9133, 9133, 98*w - 227], [9137, 9137, -8*w - 93], [9137, 9137, 8*w - 101], [9161, 9161, 70*w - 143], [9161, 9161, 70*w + 73], [9181, 9181, 50*w - 63], [9181, 9181, -50*w - 13], [9199, 9199, 2*w - 97], [9199, 9199, -2*w - 95], [9227, 9227, 26*w - 123], [9227, 9227, -26*w - 97], [9239, 9239, 96*w - 221], [9239, 9239, 38*w - 143], [9257, 9257, 54*w - 83], [9257, 9257, -54*w - 29], [9281, 9281, 62*w - 115], [9281, 9281, 62*w + 53], [9283, 9283, -52*w - 21], [9283, 9283, 52*w - 73], [9337, 9337, 32*w - 133], [9337, 9337, -32*w - 101], [9341, 9341, -4*w - 95], [9341, 9341, 4*w - 99], [9349, 9349, 74*w - 155], [9349, 9349, 74*w + 81], [9371, 9371, 60*w + 47], [9371, 9371, 60*w - 107], [9397, 9397, 58*w + 41], [9397, 9397, 58*w - 99], [9403, 9403, 92*w - 209], [9403, 9403, 42*w - 151], [9409, 97, -97], [9419, 9419, 76*w + 85], [9419, 9419, 76*w - 161], [9431, 9431, 48*w - 5], [9431, 9431, 48*w - 43], [9433, 9433, 24*w - 121], [9433, 9433, -24*w - 97], [9437, 9437, -28*w - 99], [9437, 9437, 28*w - 127], [9439, 9439, -18*w - 95], [9439, 9439, 18*w - 113], [9461, 9461, 90*w - 203], [9461, 9461, 44*w - 155], [9467, 9467, -52*w - 19], [9467, 9467, 52*w - 71], [9473, 9473, 78*w - 167], [9473, 9473, 56*w - 179], [9511, 9511, 54*w - 175], [9511, 9511, 80*w - 173], [9521, 9521, -16*w - 95], [9521, 9521, 16*w - 111], [9533, 9533, 52*w - 171], [9533, 9533, 82*w - 179], [9539, 9539, 50*w - 167], [9539, 9539, 84*w - 185], [9587, 9587, 2*w - 99], [9587, 9587, -2*w - 97], [9601, 9601, 50*w - 57], [9601, 9601, -50*w - 7], [9613, 9613, 62*w - 113], [9613, 9613, 62*w + 51], [9623, 9623, 48*w - 11], [9623, 9623, 48*w - 37], [9631, 9631, 30*w - 131], [9631, 9631, -30*w - 101], [9643, 9643, 52*w - 69], [9643, 9643, -52*w - 17], [9677, 9677, -66*w - 61], [9677, 9677, 66*w - 127], [9689, 9689, -54*w - 25], [9689, 9689, 54*w - 79], [9733, 9733, 4*w - 101], [9733, 9733, -4*w - 97], [9739, 9739, 100*w - 231], [9739, 9739, 38*w - 145], [9743, 9743, 48*w - 31], [9743, 9743, 48*w - 17], [9749, 9749, -20*w - 97], [9749, 9749, 20*w - 117], [9767, 9767, 48*w - 19], [9767, 9767, 48*w - 29], [9791, 9791, 48*w - 25], [9791, 9791, 48*w - 23], [9811, 9811, -52*w - 15], [9811, 9811, 52*w - 67], [9817, 9817, 74*w + 79], [9817, 9817, 74*w - 153], [9839, 9839, 64*w + 55], [9839, 9839, 64*w - 119], [9851, 9851, 68*w - 133], [9851, 9851, -68*w - 65], [9859, 9859, -18*w - 97], [9859, 9859, 18*w - 115], [9907, 9907, 76*w + 83], [9907, 9907, 76*w - 159], [9929, 9929, 8*w - 105], [9929, 9929, -8*w - 97], [9941, 9941, -58*w - 37], [9941, 9941, 58*w - 95], [9949, 9949, -50*w - 1], [9949, 9949, 50*w - 51]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [-1, -1, 1, 2, 2, 6, -4, -4, 1, -4, -4, 0, 0, 2, -6, -6, 0, 0, -4, -4, 12, 12, 18, 18, 18, 18, -4, -4, -22, 20, 20, 6, 6, -6, -6, 8, 8, 2, 2, 24, 24, -24, -24, 20, 20, -10, -10, 24, 24, -24, -24, -18, -18, 0, 0, -16, -16, 18, 18, -6, -6, 20, 20, -28, -28, -10, -10, 6, 6, -24, -24, 26, 26, 0, 0, -6, -6, 26, 26, -10, -10, 26, 26, 12, 12, 26, 26, -30, -30, -4, -4, -36, -36, 24, 24, -6, -6, 20, 20, -46, 18, 18, -12, -12, 18, 18, 2, 2, -12, -12, 30, 30, -24, -24, 2, 2, 32, 32, 24, 24, 48, 48, 14, 14, -6, -6, -28, -28, -22, -22, -52, -52, 2, 2, 18, 18, 2, 2, 42, 42, -30, -30, 38, 38, -22, -4, -4, 24, 24, -4, -4, -16, -16, 26, 26, 6, 6, 2, -4, -4, 24, 24, -42, -42, 48, 48, 14, 14, 50, 50, 56, 56, -28, -28, 14, 14, -52, -52, 6, 6, 0, 0, -6, -6, -52, -52, -28, -28, -30, -30, -24, -24, -22, -22, -46, -46, 0, 0, -30, -30, -40, -40, -12, -12, -28, -28, 42, 42, 60, 60, 20, 20, -6, -6, -70, 66, 66, -10, -10, -30, -30, 12, 12, -58, -58, 20, 20, 2, 2, -16, -16, 18, 18, -28, -28, -24, -24, -48, -48, 20, 20, 68, 68, 38, 38, -28, -28, -24, -24, -22, -22, -48, -48, 66, 66, -48, -48, 2, 2, -12, -12, -46, 44, 44, -30, -30, 42, 42, -6, -6, 20, 20, -46, -46, -40, -40, 26, 26, -52, -52, -36, -36, 38, 38, -22, -22, 0, 0, 72, 72, -10, -10, 24, 24, 8, 8, 66, 66, -66, -66, 56, 56, 42, 42, -28, -28, 50, 50, -78, -78, -12, -12, 48, 48, 74, 74, -28, -28, -72, -72, 26, 26, -72, -72, -30, -30, -54, -54, -4, -4, 2, 2, 60, 60, -34, -34, 20, 20, 74, 74, 78, 78, -40, -40, 18, 18, -4, -4, 20, 20, 42, 42, 14, 14, -66, -66, -48, -48, -48, -48, 0, 0, 92, 92, -22, -22, -52, -52, -24, -24, -78, -78, -40, -40, -46, -46, 42, 42, 26, 26, 6, 6, 8, 8, -94, -94, -24, -24, -48, -48, -52, -52, 24, 24, 32, 32, 42, 42, -54, -54, 30, 30, -28, -28, -30, -30, 74, 74, -6, -6, -4, -4, -54, -54, -60, -60, 2, 2, 96, 96, -78, -78, -72, -72, -78, -78, 20, 20, -22, -22, 24, 24, -78, -78, -34, -34, -16, -16, 86, 86, 72, 72, 102, 102, -52, -52, 14, 14, -52, -52, 66, 66, -22, -22, 90, 90, -10, -10, -96, -96, -24, -24, -100, -100, 26, 26, 80, 80, -36, -36, -4, -4, -12, -12, 56, 56, 42, 42, 74, 74, 90, 90, 12, 12, 62, 62, 80, 80, 26, 26, -48, -48, 90, 90, -76, -76, 20, 20, -94, -94, -42, -42, 48, 48, 50, 50, 24, 24, -24, -24, -22, -22, 20, 20, 26, 26, 72, 72, -22, -4, -4, 68, 68, 90, 90, -94, -94, -6, -6, -28, -28, 30, 30, 96, 96, 24, 24, 2, 2, 2, 2, -24, -24, 32, 32, -84, -84, -54, -54, -100, -100, -76, -76, 68, 68, 18, 18, 62, 62, -28, -28, 72, 72, 98, 98, 68, 68, -48, -48, 50, 50, -102, -102, -48, -48, 66, 66, -70, -70, -42, -42, -78, -78, 80, 80, 18, 18, 0, 0, -120, -120, 108, 108, -22, -22, 68, 68, 54, 54, 92, 92, -6, -6, -102, -102, 74, 74, -96, -96, 116, 116, 50, 50, 108, 108, 60, 60, 80, 80, -124, -124, -78, -78, -16, -16, 60, 60, -6, -6, -22, -22, -114, -114, -24, -24, -48, -48, -48, -48, 32, 32, -66, -66, -46, -46, -58, -58, 0, 0, -126, -126, -58, -58, 48, 48, 48, 48, -16, -16, 114, 114, 80, 80, 44, 44, 20, 20, -142, 96, 96, 42, 42, 72, 72, -48, -48, -10, -10, 74, 74, -64, -64, -84, -84, 78, 78, 44, 44, -30, -30, 44, 44, -102, -102, 90, 90, -24, -24, 48, 48, -4, -4, -142, 68, 68, -96, -96, -6, -6, 12, 12, 102, 102, 44, 44, -100, -100, -88, -88, -102, -102, -118, -118, 36, 36, -108, -108, 50, 50, -52, -52, 122, 122, 44, 44, 48, 48, 44, 44, 26, 26, -42, -42, -28, -28, -6, -6, 66, 66, 120, 120, -94, -94, 116, 116, 42, 42, -54, -54, -100, -100, 80, 80, 90, 90, -142, -142, 18, 18, 12, 12, -130, -130, -70, -70, 78, 78, -144, -144, -70, -70, -4, -4, 90, 90, 92, 92, -142, -142, -102, -102, -22, -22, 98, 98, 44, 44, -126, -126, -54, -54, -12, -12, 18, 18, -94, -124, -124, 126, 126, 18, 18, 8, 8, 98, 98, -96, -96, 120, 120, -108, -108, 26, 26, 92, 92, -144, -144, -100, -100, 74, 74, -28, -28, 20, 20, 44, 44, 14, 14, -66, -66, 2, 2, 50, 50, -4, -4, -70, -70, 108, 108, -76, -76, -46, -46, -78, -78, -46, -46, 152, 152, -78, -78, -120, -120, -94, -94, -24, -24, -34, -34, -72, -72, -42, -42, -54, -54, -28, -28, 24, 24, 90, 90, -10, -10, -142, -142, 24, 24, -144, -144, -136, -136, -102, -102, 120, 120, -16, -16, 26, 26, 90, 90, -72, -72, -16, -16, 132, 132, 54, 54, 68, 68, -126, -126, 20, 20, 98, 98, -10, -10, -168, -168, 26, 26, -94, -94, 20, 20, -42, -42, -124, -124, -78, -78, -94, -94, 24, 24, -12, -12, -6, -6, -108, -108, 134, 134, -70, -70, -94, -94, -24, -24, -40, -40, -100, -100, 72, 72, -126, -126, 110, 110, -28, -28, 72, 72, -94, -94, 72, 72, 98, 98, 138, 138, -78, -78, 114, 114, -100, -100, 2, 2, -136, -136, 114, 114, -132, -132, 54, 54, -54, -54, -52, -52, -48, -48, 20, 20, 66, 66, -154, -154, -96, -96, 138, 138, 12, 12, -46, -46, -76, -76, -94, -94, 144, 144, -78, -78, 180, 180, -4, -4, 140, 140, -114, -114, 8, 8, 2, 2, 74, 74, -108, -108, 26, 26, -154, -154, 140, 140, -48, -48, 32, 32, 48, 48, 68, 68, -30, -30, -90, -90, -118, -118, -168, -168, 14, 14, -4, -4, -108, -108, -102, -102, 2, 2, -46, -46, -112, -112, -118, -118, -132, -132, -114, -114, 72, 72, 168, 168, 86, 86, -72, -72, 32, 32, 162, 162, 90, 90, -28, -28, -70, -70, -4, -4, 146, 146, 42, 42, -130, -130, -48, -48, 12, 12, -70, -70, -96, -96, 138, 138, 140, 140, -48, -48, 164, 164, -76, -76, -52, -52, -46, -46, 174, 174, -6, -6, -178, -178, -112, -112, 108, 108, 96, 96, -42, -42, 90, 90, -148, -148, 122, 122, -30, -30, 62, 62, 96, 96, 50, 50, -28, -28, -190, 120, 120, 72, 72, 74, 74, 162, 162, 152, 152, -6, -6, 84, 84, 126, 126, 80, 80, 114, 114, -30, -30, -168, -168, 108, 108, 146, 146, -70, -70, 96, 96, 80, 80, -172, -172, -126, -126, -6, -6, -142, -142, 44, 44, 168, 168, 42, 42, 48, 48, -48, -48, 140, 140, 170, 170, 192, 192, 0, 0, 140, 140, -52, -52, 90, 90, -6, -6, -130, -130]; 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; 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;