/* This code can be loaded, or copied and paste using cpaste, into Sage. It will load the data associated to the HMF, including the field, level, and Hecke and Atkin-Lehner eigenvalue data. */ P. = PolynomialRing(QQ) g = P([-2, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([576, 24, 24]) primes_array = [ [2, 2, -w],\ [7, 7, -2*w + 1],\ [7, 7, -2*w - 1],\ [9, 3, 3],\ [17, 17, 3*w + 1],\ [17, 17, 3*w - 1],\ [23, 23, w + 5],\ [23, 23, -w + 5],\ [25, 5, 5],\ [31, 31, 4*w + 1],\ [31, 31, -4*w + 1],\ [41, 41, 2*w - 7],\ [41, 41, -2*w - 7],\ [47, 47, -w - 7],\ [47, 47, w - 7],\ [71, 71, -6*w - 1],\ [71, 71, 6*w - 1],\ [73, 73, -7*w - 5],\ [73, 73, 7*w - 5],\ [79, 79, -w - 9],\ [79, 79, w - 9],\ [89, 89, -4*w - 11],\ [89, 89, 4*w - 11],\ [97, 97, -7*w - 1],\ [97, 97, 7*w - 1],\ [103, 103, -3*w - 11],\ [103, 103, 3*w - 11],\ [113, 113, 2*w - 11],\ [113, 113, -2*w - 11],\ [121, 11, -11],\ [127, 127, -8*w - 1],\ [127, 127, 8*w - 1],\ [137, 137, 4*w - 13],\ [137, 137, -4*w - 13],\ [151, 151, 3*w - 13],\ [151, 151, -3*w - 13],\ [167, 167, -w - 13],\ [167, 167, w - 13],\ [169, 13, -13],\ [191, 191, 10*w - 3],\ [191, 191, -10*w - 3],\ [193, 193, -4*w - 15],\ [193, 193, 4*w - 15],\ [199, 199, -10*w - 1],\ [199, 199, 10*w - 1],\ [223, 223, -w - 15],\ [223, 223, w - 15],\ [233, 233, -11*w - 3],\ [233, 233, 11*w - 3],\ [239, 239, -5*w - 17],\ [239, 239, 5*w - 17],\ [241, 241, -11*w - 1],\ [241, 241, 11*w - 1],\ [257, 257, 4*w - 17],\ [257, 257, -4*w - 17],\ [263, 263, 7*w - 19],\ [263, 263, -7*w - 19],\ [271, 271, -3*w - 17],\ [271, 271, 3*w - 17],\ [281, 281, 2*w - 17],\ [281, 281, -2*w - 17],\ [311, 311, -5*w - 19],\ [311, 311, 5*w - 19],\ [313, 313, -8*w - 21],\ [313, 313, 8*w - 21],\ [337, 337, -13*w - 1],\ [337, 337, 13*w - 1],\ [353, 353, 2*w - 19],\ [353, 353, -2*w - 19],\ [359, 359, -w - 19],\ [359, 359, w - 19],\ [361, 19, -19],\ [367, 367, 9*w - 23],\ [367, 367, -9*w - 23],\ [383, 383, -14*w - 3],\ [383, 383, 14*w - 3],\ [401, 401, -8*w - 23],\ [401, 401, 8*w - 23],\ [409, 409, -17*w - 13],\ [409, 409, 17*w - 13],\ [431, 431, -7*w - 23],\ [431, 431, 7*w - 23],\ [433, 433, 2*w - 21],\ [433, 433, -2*w - 21],\ [439, 439, -w - 21],\ [439, 439, w - 21],\ [449, 449, -15*w - 1],\ [449, 449, 15*w - 1],\ [457, 457, -6*w - 23],\ [457, 457, 6*w - 23],\ [463, 463, 9*w - 25],\ [463, 463, -9*w - 25],\ [479, 479, -18*w - 13],\ [479, 479, 18*w - 13],\ [487, 487, 16*w - 5],\ [487, 487, -16*w - 5],\ [503, 503, 16*w - 3],\ [503, 503, -16*w - 3],\ [521, 521, 2*w - 23],\ [521, 521, -2*w - 23],\ [569, 569, -17*w - 3],\ [569, 569, 17*w - 3],\ [577, 577, -17*w - 1],\ [577, 577, 17*w - 1],\ [593, 593, 4*w - 25],\ [593, 593, -4*w - 25],\ [599, 599, -11*w - 29],\ [599, 599, 11*w - 29],\ [601, 601, 8*w - 27],\ [601, 601, -8*w - 27],\ [607, 607, 3*w - 25],\ [607, 607, -3*w - 25],\ [617, 617, 2*w - 25],\ [617, 617, -2*w - 25],\ [631, 631, -7*w - 27],\ [631, 631, 7*w - 27],\ [641, 641, -10*w - 29],\ [641, 641, 10*w - 29],\ [647, 647, -18*w - 1],\ [647, 647, 18*w - 1],\ [673, 673, -12*w - 31],\ [673, 673, 12*w - 31],\ [719, 719, 11*w - 31],\ [719, 719, -11*w - 31],\ [727, 727, -w - 27],\ [727, 727, w - 27],\ [743, 743, 7*w - 29],\ [743, 743, -7*w - 29],\ [751, 751, 13*w - 33],\ [751, 751, -13*w - 33],\ [761, 761, 10*w - 31],\ [761, 761, -10*w - 31],\ [769, 769, 23*w - 17],\ [769, 769, -23*w - 17],\ [809, 809, 4*w - 29],\ [809, 809, -4*w - 29],\ [823, 823, -3*w - 29],\ [823, 823, 3*w - 29],\ [839, 839, -w - 29],\ [839, 839, w - 29],\ [841, 29, -29],\ [857, 857, 21*w - 5],\ [857, 857, -21*w - 5],\ [863, 863, 24*w - 17],\ [863, 863, -24*w - 17],\ [881, 881, -21*w - 1],\ [881, 881, 21*w - 1],\ [887, 887, 13*w - 35],\ [887, 887, -13*w - 35],\ [911, 911, 26*w + 21],\ [911, 911, -26*w + 21],\ [919, 919, 22*w - 7],\ [919, 919, -22*w - 7],\ [929, 929, -4*w - 31],\ [929, 929, 4*w - 31],\ [937, 937, -12*w - 35],\ [937, 937, 12*w - 35],\ [953, 953, 2*w - 31],\ [953, 953, -2*w - 31],\ [967, 967, -22*w - 1],\ [967, 967, 22*w - 1],\ [977, 977, -14*w - 37],\ [977, 977, 14*w - 37],\ [983, 983, -11*w - 35],\ [983, 983, 11*w - 35],\ [991, 991, -26*w - 19],\ [991, 991, 26*w - 19],\ [1009, 1009, -23*w - 7],\ [1009, 1009, 23*w - 7],\ [1031, 1031, 13*w - 37],\ [1031, 1031, -13*w - 37],\ [1033, 1033, 23*w - 5],\ [1033, 1033, -23*w - 5],\ [1039, 1039, 5*w - 33],\ [1039, 1039, -5*w - 33],\ [1049, 1049, -23*w - 3],\ [1049, 1049, 23*w - 3],\ [1063, 1063, -9*w - 35],\ [1063, 1063, 9*w - 35],\ [1087, 1087, -w - 33],\ [1087, 1087, w - 33],\ [1097, 1097, -27*w - 19],\ [1097, 1097, 27*w - 19],\ [1103, 1103, -24*w - 7],\ [1103, 1103, 24*w - 7],\ [1129, 1129, -14*w - 39],\ [1129, 1129, 14*w - 39],\ [1151, 1151, -24*w - 1],\ [1151, 1151, 24*w - 1],\ [1153, 1153, -6*w - 35],\ [1153, 1153, 6*w - 35],\ [1193, 1193, -4*w - 35],\ [1193, 1193, 4*w - 35],\ [1201, 1201, 25*w - 7],\ [1201, 1201, -25*w - 7],\ [1217, 1217, 2*w - 35],\ [1217, 1217, -2*w - 35],\ [1223, 1223, -w - 35],\ [1223, 1223, w - 35],\ [1231, 1231, -15*w - 41],\ [1231, 1231, 15*w - 41],\ [1249, 1249, -25*w - 1],\ [1249, 1249, 25*w - 1],\ [1279, 1279, 11*w - 39],\ [1279, 1279, -11*w - 39],\ [1289, 1289, -14*w - 41],\ [1289, 1289, 14*w - 41],\ [1297, 1297, 31*w + 25],\ [1297, 1297, -31*w + 25],\ [1303, 1303, 26*w - 7],\ [1303, 1303, -26*w - 7],\ [1319, 1319, -5*w - 37],\ [1319, 1319, 5*w - 37],\ [1321, 1321, -10*w - 39],\ [1321, 1321, 10*w - 39],\ [1327, 1327, 26*w - 5],\ [1327, 1327, -26*w - 5],\ [1361, 1361, 2*w - 37],\ [1361, 1361, -2*w - 37],\ [1367, 1367, -w - 37],\ [1367, 1367, w - 37],\ [1369, 37, -37],\ [1399, 1399, 15*w - 43],\ [1399, 1399, -15*w - 43],\ [1409, 1409, -27*w - 7],\ [1409, 1409, 27*w - 7],\ [1423, 1423, -32*w - 25],\ [1423, 1423, 32*w - 25],\ [1433, 1433, -27*w - 5],\ [1433, 1433, 27*w - 5],\ [1439, 1439, -11*w - 41],\ [1439, 1439, 11*w - 41],\ [1447, 1447, -17*w - 45],\ [1447, 1447, 17*w - 45],\ [1471, 1471, -5*w - 39],\ [1471, 1471, 5*w - 39],\ [1481, 1481, 10*w - 41],\ [1481, 1481, -10*w - 41],\ [1487, 1487, 28*w - 9],\ [1487, 1487, -28*w - 9],\ [1489, 1489, -4*w - 39],\ [1489, 1489, 4*w - 39],\ [1511, 1511, -13*w - 43],\ [1511, 1511, 13*w - 43],\ [1543, 1543, 28*w - 5],\ [1543, 1543, -28*w - 5],\ [1553, 1553, -33*w - 25],\ [1553, 1553, 33*w - 25],\ [1559, 1559, 28*w - 3],\ [1559, 1559, -28*w - 3],\ [1567, 1567, -28*w - 1],\ [1567, 1567, 28*w - 1],\ [1583, 1583, -34*w + 27],\ [1583, 1583, 34*w + 27],\ [1601, 1601, -29*w - 9],\ [1601, 1601, 29*w - 9],\ [1607, 1607, -11*w - 43],\ [1607, 1607, 11*w - 43],\ [1609, 1609, -6*w - 41],\ [1609, 1609, 6*w - 41],\ [1657, 1657, -29*w - 5],\ [1657, 1657, 29*w - 5],\ [1663, 1663, -3*w - 41],\ [1663, 1663, 3*w - 41],\ [1697, 1697, -16*w - 47],\ [1697, 1697, 16*w - 47],\ [1721, 1721, -35*w - 27],\ [1721, 1721, 35*w - 27],\ [1753, 1753, 18*w - 49],\ [1753, 1753, -18*w - 49],\ [1759, 1759, -15*w - 47],\ [1759, 1759, 15*w - 47],\ [1777, 1777, 6*w - 43],\ [1777, 1777, -6*w - 43],\ [1783, 1783, 11*w - 45],\ [1783, 1783, -11*w - 45],\ [1801, 1801, -20*w - 51],\ [1801, 1801, 20*w - 51],\ [1823, 1823, 17*w - 49],\ [1823, 1823, -17*w - 49],\ [1831, 1831, 3*w - 43],\ [1831, 1831, -3*w - 43],\ [1847, 1847, -w - 43],\ [1847, 1847, w - 43],\ [1849, 43, -43],\ [1871, 1871, 13*w - 47],\ [1871, 1871, -13*w - 47],\ [1873, 1873, -31*w - 7],\ [1873, 1873, 31*w - 7],\ [1879, 1879, 19*w - 51],\ [1879, 1879, -19*w - 51],\ [1889, 1889, 16*w - 49],\ [1889, 1889, -16*w - 49],\ [1913, 1913, 31*w - 3],\ [1913, 1913, -31*w - 3],\ [1951, 1951, 15*w - 49],\ [1951, 1951, -15*w - 49],\ [1993, 1993, 4*w - 45],\ [1993, 1993, -4*w - 45],\ [1999, 1999, 32*w - 7],\ [1999, 1999, -32*w - 7],\ [2017, 2017, 2*w - 45],\ [2017, 2017, -2*w - 45],\ [2039, 2039, -32*w - 3],\ [2039, 2039, 32*w - 3],\ [2063, 2063, 13*w - 49],\ [2063, 2063, -13*w - 49],\ [2081, 2081, -39*w + 31],\ [2081, 2081, 39*w + 31],\ [2087, 2087, -19*w - 53],\ [2087, 2087, 19*w - 53],\ [2089, 2089, -16*w - 51],\ [2089, 2089, 16*w - 51],\ [2111, 2111, 7*w - 47],\ [2111, 2111, -7*w - 47],\ [2113, 2113, 12*w - 49],\ [2113, 2113, -12*w - 49],\ [2129, 2129, 33*w - 7],\ [2129, 2129, -33*w - 7],\ [2137, 2137, -6*w - 47],\ [2137, 2137, 6*w - 47],\ [2143, 2143, -21*w - 55],\ [2143, 2143, 21*w - 55],\ [2153, 2153, 33*w - 5],\ [2153, 2153, -33*w - 5],\ [2161, 2161, -18*w - 53],\ [2161, 2161, 18*w - 53],\ [2207, 2207, -w - 47],\ [2207, 2207, w - 47],\ [2239, 2239, 40*w - 31],\ [2239, 2239, -40*w - 31],\ [2273, 2273, 8*w - 49],\ [2273, 2273, -8*w - 49],\ [2281, 2281, -22*w - 57],\ [2281, 2281, 22*w - 57],\ [2287, 2287, -34*w - 5],\ [2287, 2287, 34*w - 5],\ [2297, 2297, -16*w - 53],\ [2297, 2297, 16*w - 53],\ [2311, 2311, -34*w - 1],\ [2311, 2311, 34*w - 1],\ [2351, 2351, -5*w - 49],\ [2351, 2351, 5*w - 49],\ [2377, 2377, 18*w - 55],\ [2377, 2377, -18*w - 55],\ [2383, 2383, 3*w - 49],\ [2383, 2383, -3*w - 49],\ [2393, 2393, 2*w - 49],\ [2393, 2393, -2*w - 49],\ [2399, 2399, -w - 49],\ [2399, 2399, w - 49],\ [2417, 2417, -14*w - 53],\ [2417, 2417, 14*w - 53],\ [2423, 2423, -23*w - 59],\ [2423, 2423, 23*w - 59],\ [2441, 2441, -35*w - 3],\ [2441, 2441, 35*w - 3],\ [2447, 2447, -17*w - 55],\ [2447, 2447, 17*w - 55],\ [2473, 2473, 8*w - 51],\ [2473, 2473, -8*w - 51],\ [2503, 2503, -7*w - 51],\ [2503, 2503, 7*w - 51],\ [2521, 2521, -41*w - 29],\ [2521, 2521, 41*w - 29],\ [2543, 2543, 36*w - 7],\ [2543, 2543, -36*w - 7],\ [2551, 2551, 5*w - 51],\ [2551, 2551, -5*w - 51],\ [2591, 2591, -36*w - 1],\ [2591, 2591, 36*w - 1],\ [2593, 2593, 2*w - 51],\ [2593, 2593, -2*w - 51],\ [2609, 2609, 43*w - 33],\ [2609, 2609, -43*w - 33],\ [2617, 2617, 37*w - 11],\ [2617, 2617, -37*w - 11],\ [2633, 2633, -14*w - 55],\ [2633, 2633, 14*w - 55],\ [2647, 2647, -44*w + 35],\ [2647, 2647, 44*w + 35],\ [2657, 2657, 37*w - 9],\ [2657, 2657, -37*w - 9],\ [2663, 2663, -23*w - 61],\ [2663, 2663, 23*w - 61],\ [2671, 2671, 17*w - 57],\ [2671, 2671, -17*w - 57],\ [2687, 2687, -13*w - 55],\ [2687, 2687, 13*w - 55],\ [2689, 2689, -37*w - 7],\ [2689, 2689, 37*w - 7],\ [2711, 2711, 7*w - 53],\ [2711, 2711, -7*w - 53],\ [2713, 2713, -37*w - 5],\ [2713, 2713, 37*w - 5],\ [2719, 2719, 25*w - 63],\ [2719, 2719, -25*w - 63],\ [2729, 2729, 37*w - 3],\ [2729, 2729, -37*w - 3],\ [2753, 2753, -22*w - 61],\ [2753, 2753, 22*w - 61],\ [2767, 2767, -38*w - 11],\ [2767, 2767, 38*w - 11],\ [2777, 2777, 4*w - 53],\ [2777, 2777, -4*w - 53],\ [2791, 2791, -3*w - 53],\ [2791, 2791, 3*w - 53],\ [2801, 2801, 2*w - 53],\ [2801, 2801, -2*w - 53],\ [2809, 53, -53],\ [2833, 2833, -18*w - 59],\ [2833, 2833, 18*w - 59],\ [2857, 2857, 14*w - 57],\ [2857, 2857, -14*w - 57],\ [2879, 2879, -38*w - 3],\ [2879, 2879, 38*w - 3],\ [2887, 2887, -38*w - 1],\ [2887, 2887, 38*w - 1],\ [2897, 2897, -8*w - 55],\ [2897, 2897, 8*w - 55],\ [2903, 2903, -17*w - 59],\ [2903, 2903, 17*w - 59],\ [2927, 2927, -7*w - 55],\ [2927, 2927, 7*w - 55],\ [2953, 2953, 6*w - 55],\ [2953, 2953, -6*w - 55],\ [2969, 2969, 16*w - 59],\ [2969, 2969, -16*w - 59],\ [2999, 2999, 19*w - 61],\ [2999, 2999, -19*w - 61],\ [3001, 3001, 22*w - 63],\ [3001, 3001, -22*w - 63],\ [3023, 3023, -w - 55],\ [3023, 3023, w - 55],\ [3041, 3041, -39*w - 1],\ [3041, 3041, 39*w - 1],\ [3049, 3049, -47*w - 37],\ [3049, 3049, 47*w - 37],\ [3079, 3079, -40*w - 11],\ [3079, 3079, 40*w - 11],\ [3089, 3089, 14*w - 59],\ [3089, 3089, -14*w - 59],\ [3119, 3119, -40*w - 9],\ [3119, 3119, 40*w - 9],\ [3121, 3121, 8*w - 57],\ [3121, 3121, -8*w - 57],\ [3137, 3137, 26*w - 67],\ [3137, 3137, -26*w - 67],\ [3167, 3167, -23*w - 65],\ [3167, 3167, 23*w - 65],\ [3169, 3169, 20*w - 63],\ [3169, 3169, -20*w - 63],\ [3191, 3191, 40*w - 3],\ [3191, 3191, -40*w - 3],\ [3209, 3209, 16*w - 61],\ [3209, 3209, -16*w - 61],\ [3217, 3217, 4*w - 57],\ [3217, 3217, -4*w - 57],\ [3257, 3257, -22*w - 65],\ [3257, 3257, 22*w - 65],\ [3271, 3271, 15*w - 61],\ [3271, 3271, -15*w - 61],\ [3313, 3313, -41*w - 7],\ [3313, 3313, 41*w - 7],\ [3319, 3319, 9*w - 59],\ [3319, 3319, -9*w - 59],\ [3329, 3329, -47*w - 33],\ [3329, 3329, 47*w - 33],\ [3343, 3343, -21*w - 65],\ [3343, 3343, 21*w - 65],\ [3359, 3359, -42*w - 13],\ [3359, 3359, 42*w - 13],\ [3361, 3361, -41*w - 1],\ [3361, 3361, 41*w - 1],\ [3391, 3391, -17*w - 63],\ [3391, 3391, 17*w - 63],\ [3407, 3407, 42*w - 11],\ [3407, 3407, -42*w - 11],\ [3433, 3433, -49*w - 37],\ [3433, 3433, 49*w - 37],\ [3449, 3449, -4*w - 59],\ [3449, 3449, 4*w - 59],\ [3457, 3457, -16*w - 63],\ [3457, 3457, 16*w - 63],\ [3463, 3463, -3*w - 59],\ [3463, 3463, 3*w - 59],\ [3481, 59, -59],\ [3511, 3511, 25*w - 69],\ [3511, 3511, -25*w - 69],\ [3527, 3527, -42*w - 1],\ [3527, 3527, 42*w - 1],\ [3529, 3529, -43*w - 13],\ [3529, 3529, 43*w - 13],\ [3559, 3559, 9*w - 61],\ [3559, 3559, -9*w - 61],\ [3583, 3583, 27*w - 71],\ [3583, 3583, -27*w - 71],\ [3593, 3593, -8*w - 61],\ [3593, 3593, 8*w - 61],\ [3607, 3607, -21*w - 67],\ [3607, 3607, 21*w - 67],\ [3617, 3617, 43*w - 9],\ [3617, 3617, -43*w - 9],\ [3623, 3623, 7*w - 61],\ [3623, 3623, -7*w - 61],\ [3631, 3631, -50*w - 37],\ [3631, 3631, 50*w - 37],\ [3671, 3671, 5*w - 61],\ [3671, 3671, -5*w - 61],\ [3673, 3673, 43*w - 5],\ [3673, 3673, -43*w - 5],\ [3697, 3697, -43*w - 1],\ [3697, 3697, 43*w - 1],\ [3719, 3719, -w - 61],\ [3719, 3719, w - 61],\ [3721, 61, -61],\ [3727, 3727, 52*w - 41],\ [3727, 3727, -52*w - 41],\ [3761, 3761, 28*w - 73],\ [3761, 3761, -28*w - 73],\ [3767, 3767, 19*w - 67],\ [3767, 3767, -19*w - 67],\ [3769, 3769, -10*w - 63],\ [3769, 3769, 10*w - 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19],\ [8887, 8887, -68*w - 19],\ [8929, 8929, 67*w - 7],\ [8929, 8929, -67*w - 7],\ [8951, 8951, 25*w - 101],\ [8951, 8951, -25*w - 101],\ [8969, 8969, 67*w - 3],\ [8969, 8969, -67*w - 3],\ [8999, 8999, -35*w - 107],\ [8999, 8999, 35*w - 107],\ [9001, 9001, 79*w - 59],\ [9001, 9001, -79*w - 59],\ [9007, 9007, -3*w - 95],\ [9007, 9007, 3*w - 95],\ [9041, 9041, 28*w - 103],\ [9041, 9041, -28*w - 103],\ [9049, 9049, 24*w - 101],\ [9049, 9049, -24*w - 101],\ [9103, 9103, 31*w - 105],\ [9103, 9103, -31*w - 105],\ [9127, 9127, -68*w - 11],\ [9127, 9127, 68*w - 11],\ [9137, 9137, 34*w - 107],\ [9137, 9137, -34*w - 107],\ [9151, 9151, -27*w - 103],\ [9151, 9151, 27*w - 103],\ [9161, 9161, 69*w - 19],\ [9161, 9161, -69*w - 19],\ [9199, 9199, 68*w - 7],\ [9199, 9199, -68*w - 7],\ [9209, 9209, 10*w - 97],\ [9209, 9209, -10*w - 97],\ [9239, 9239, -68*w - 3],\ [9239, 9239, 68*w - 3],\ [9241, 9241, -42*w - 113],\ [9241, 9241, 42*w - 113],\ [9257, 9257, -26*w - 103],\ [9257, 9257, 26*w - 103],\ [9281, 9281, 8*w - 97],\ [9281, 9281, -8*w - 97],\ [9311, 9311, -7*w - 97],\ [9311, 9311, 7*w - 97],\ [9319, 9319, -80*w - 59],\ [9319, 9319, 80*w - 59],\ [9337, 9337, 6*w - 97],\ [9337, 9337, -6*w - 97],\ [9343, 9343, -29*w - 105],\ [9343, 9343, 29*w - 105],\ [9377, 9377, 4*w - 97],\ [9377, 9377, -4*w - 97],\ [9391, 9391, 3*w - 97],\ [9391, 9391, -3*w - 97],\ [9431, 9431, 35*w - 109],\ [9431, 9431, -35*w - 109],\ [9433, 9433, -38*w - 111],\ [9433, 9433, 38*w - 111],\ [9439, 9439, 70*w - 19],\ [9439, 9439, -70*w - 19],\ [9463, 9463, 13*w - 99],\ [9463, 9463, -13*w - 99],\ [9473, 9473, -69*w - 7],\ [9473, 9473, 69*w - 7],\ [9479, 9479, 82*w - 63],\ [9479, 9479, -82*w - 63],\ [9497, 9497, -69*w - 5],\ [9497, 9497, 69*w - 5],\ [9511, 9511, -70*w - 17],\ [9511, 9511, 70*w - 17],\ [9521, 9521, -69*w - 1],\ [9521, 9521, 69*w - 1],\ [9551, 9551, 80*w - 57],\ [9551, 9551, -80*w - 57],\ [9601, 9601, -10*w - 99],\ [9601, 9601, 10*w - 99],\ [9623, 9623, -84*w + 67],\ [9623, 9623, 84*w + 67],\ [9631, 9631, -70*w - 13],\ [9631, 9631, 70*w - 13],\ [9649, 9649, -30*w - 107],\ [9649, 9649, 30*w - 107],\ [9679, 9679, 70*w - 11],\ [9679, 9679, -70*w - 11],\ [9689, 9689, -16*w - 101],\ [9689, 9689, 16*w - 101],\ [9697, 9697, 42*w - 115],\ [9697, 9697, -42*w - 115],\ [9719, 9719, 70*w - 9],\ [9719, 9719, -70*w - 9],\ [9721, 9721, -71*w - 19],\ [9721, 9721, 71*w - 19],\ [9743, 9743, -47*w - 119],\ [9743, 9743, 47*w - 119],\ [9767, 9767, -29*w - 107],\ [9767, 9767, 29*w - 107],\ [9769, 9769, -4*w - 99],\ [9769, 9769, 4*w - 99],\ [9791, 9791, 70*w - 3],\ [9791, 9791, -70*w - 3],\ [9817, 9817, -44*w - 117],\ [9817, 9817, 44*w - 117],\ [9833, 9833, 32*w - 109],\ [9833, 9833, -32*w - 109],\ [9839, 9839, -49*w - 121],\ [9839, 9839, 49*w - 121],\ [9857, 9857, -71*w - 15],\ [9857, 9857, 71*w - 15],\ [9871, 9871, 35*w - 111],\ [9871, 9871, -35*w - 111],\ [9887, 9887, 84*w - 65],\ [9887, 9887, -84*w - 65],\ [9929, 9929, -46*w - 119],\ [9929, 9929, 46*w - 119],\ [9967, 9967, 82*w - 59],\ [9967, 9967, -82*w - 59]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 - 4*x - 4 K. = NumberField(heckePol) hecke_eigenvalues_array = [0, -e + 4, e, -1, -2*e + 2, 2*e - 6, 4, 4, -2, e + 4, -e + 8, -2*e + 6, 2*e - 2, -2*e, 2*e - 8, -4, -4, 2, 2, 5*e - 12, -5*e + 8, -10, -10, 4*e - 10, -4*e + 6, -e + 8, e + 4, 10, 10, -10, 3*e - 12, -3*e, 2*e - 18, -2*e - 10, -e + 20, e + 16, 2*e - 4, -2*e + 4, -6, -4*e, 4*e - 16, 2, 2, -5*e, 5*e - 20, 3*e - 12, -3*e, -4*e + 6, 4*e - 10, -4*e + 8, 4*e - 8, -4*e + 2, 4*e - 14, 4*e + 10, -4*e + 26, -2*e - 20, 2*e - 28, e + 8, -e + 12, 4*e - 10, -4*e + 6, 6*e - 28, -6*e - 4, 4*e - 18, -4*e - 2, -4*e - 2, 4*e - 18, 10, 10, 8*e - 20, -8*e + 12, -10, 11*e - 16, -11*e + 28, -4*e, 4*e - 16, -6*e + 26, 6*e + 2, 4*e - 14, -4*e + 2, 6*e - 8, -6*e + 16, -18, -18, -3*e - 12, 3*e - 24, -6*e + 10, 6*e - 14, 4*e - 14, -4*e + 2, 5*e - 8, -5*e + 12, -2*e + 32, 2*e + 24, e - 8, -e - 4, 12*e - 28, -12*e + 20, 6*e - 18, -6*e + 6, -2*e - 26, 2*e - 34, -4*e + 6, 4*e - 10, -4*e + 18, 4*e + 2, 4*e + 20, -4*e + 36, 8*e - 22, -8*e + 10, 3*e + 20, -3*e + 32, -8*e + 30, 8*e - 2, 5*e + 16, -5*e + 36, -2*e - 22, 2*e - 30, 4*e - 4, -4*e + 12, 8*e - 34, -8*e - 2, 2*e + 16, -2*e + 24, -e - 20, e - 24, -2*e + 28, 2*e + 20, 3*e + 32, -3*e + 44, 6*e - 2, -6*e + 22, 18, 18, 2*e + 30, -2*e + 38, e, -e + 4, 12*e - 36, -12*e + 12, 14, 2*e - 50, -2*e - 42, 4*e - 16, -4*e, -8*e + 18, 8*e - 14, 6*e - 20, -6*e + 4, -4*e + 24, 4*e + 8, e - 16, -e - 12, 14*e - 14, -14*e + 42, -16*e + 38, 16*e - 26, 14*e - 18, -14*e + 38, -5*e + 16, 5*e - 4, 6*e + 18, -6*e + 42, 6*e + 12, -6*e + 36, -7*e + 4, 7*e - 24, -4*e + 38, 4*e + 22, -6*e - 12, 6*e - 36, 38, 38, -e + 32, e + 28, -18, -18, 9*e + 12, -9*e + 48, -7*e + 12, 7*e - 16, -12*e + 38, 12*e - 10, 40, 40, 8*e - 2, -8*e + 30, 8*e, -8*e + 32, 18, 18, 4*e - 26, -4*e - 10, -4*e - 10, 4*e - 26, -16*e + 42, 16*e - 22, 2*e - 4, -2*e + 4, -3*e + 48, 3*e + 36, 4*e - 50, -4*e - 34, -e - 12, e - 16, 16*e - 50, -16*e + 14, -8*e - 14, 8*e - 46, 3*e - 40, -3*e - 28, 4*e + 12, -4*e + 28, 16*e - 6, -16*e + 58, 9*e - 4, -9*e + 32, -14*e + 34, 14*e - 22, 14*e - 44, -14*e + 12, 26, e - 36, -e - 32, -2*e + 10, 2*e + 2, 7*e + 4, -7*e + 32, -8*e + 38, 8*e + 6, -10*e + 48, 10*e + 8, -15*e + 12, 15*e - 48, -9*e + 20, 9*e - 16, -2, -2, 18*e - 56, -18*e + 16, 4*e - 26, -4*e - 10, 12*e - 52, -12*e - 4, -3*e, 3*e - 12, -16*e + 42, 16*e - 22, 36, 36, -3*e - 44, 3*e - 56, -16*e + 24, 16*e - 40, -16*e + 26, 16*e - 38, -4*e + 76, 4*e + 60, -8*e + 70, 8*e + 38, 8*e + 10, -8*e + 42, -3*e - 8, 3*e - 20, -18*e + 42, 18*e - 30, 4*e - 58, -4*e - 42, -8*e + 22, 8*e - 10, 3*e - 32, -3*e - 20, -8*e + 50, 8*e + 18, 9*e - 52, -9*e - 16, 8*e + 10, -8*e + 42, -2*e + 32, 2*e + 24, 9*e - 20, -9*e + 16, 4*e + 4, -4*e + 20, 38, 16*e - 8, -16*e + 56, 20*e - 14, -20*e + 66, -5*e - 4, 5*e - 24, 12*e - 14, -12*e + 34, -12*e + 30, 12*e - 18, -17*e + 12, 17*e - 56, -4*e - 34, 4*e - 50, 17*e - 20, -17*e + 48, -4*e + 2, 4*e - 14, -16*e + 52, 16*e - 12, -8*e - 8, 8*e - 40, 10*e - 30, -10*e + 10, -10*e + 4, 10*e - 36, 16*e - 26, -16*e + 38, 6*e - 40, -6*e - 16, 4*e - 2, -4*e + 14, 4*e + 10, -4*e + 26, -12*e - 2, 12*e - 50, -7*e - 28, 7*e - 56, -2*e - 10, 2*e - 18, -20*e + 38, 20*e - 42, 16*e + 16, -16*e + 80, 3*e - 12, -3*e, -16*e + 18, 16*e - 46, -30, -30, 13*e - 48, -13*e + 4, -6*e + 38, 6*e + 14, -e - 16, e - 20, 2*e - 16, -2*e - 8, -20*e + 50, 20*e - 30, 3*e + 44, -3*e + 56, 20*e - 18, -20*e + 62, -16*e + 32, 16*e - 32, 6*e - 38, -6*e - 14, 18*e - 4, -18*e + 68, -4*e - 34, 4*e - 50, 8*e - 88, -8*e - 56, -16*e + 26, 16*e - 38, 5*e + 16, -5*e + 36, 8*e - 70, -8*e - 38, -4*e - 8, 4*e - 24, 9*e + 8, -9*e + 44, -4*e + 48, 4*e + 32, 50, 50, -2*e - 38, 2*e - 46, 14, 14, -6*e + 86, 6*e + 62, -e + 4, e, -4*e + 58, 4*e + 42, -2*e + 4, 2*e - 4, -13*e + 16, 13*e - 36, -26*e + 40, 26*e - 64, 24*e - 30, -24*e + 66, -22*e + 68, 22*e - 20, -4*e + 18, 4*e + 2, 5*e - 56, -5*e - 36, -2*e + 14, 2*e + 6, 10*e - 46, -10*e - 6, 19*e - 12, -19*e + 64, -6*e + 62, 6*e + 38, -31*e + 72, 31*e - 52, 26*e - 38, -26*e + 66, -66, 20*e - 26, -20*e + 54, -16*e - 14, 16*e - 78, 20*e, -20*e + 80, 13*e - 64, -13*e - 12, -10*e + 50, 10*e + 10, -8*e - 12, 8*e - 44, 8*e - 24, -8*e + 8, 20*e - 82, -20*e - 2, -4*e + 94, 4*e + 78, -8*e + 4, 8*e - 28, 16*e - 18, -16*e + 46, -18*e + 64, 18*e - 8, -2*e + 82, 2*e + 74, -8*e + 2, 8*e - 30, -e, e - 4, 24*e - 46, -24*e + 50, -6*e - 32, 6*e - 56, 16*e - 62, -16*e + 2, 8*e - 14, -8*e + 18, 10*e - 48, -10*e - 8, 8*e - 46, -8*e - 14, -12*e + 4, 12*e - 44, -6*e + 62, 6*e + 38, 16*e - 46, -16*e + 18, -8*e - 10, 8*e - 42, -e - 4, e - 8, 12*e - 2, -12*e + 46, -e + 8, e + 4, 2*e + 50, -2*e + 58, 7*e + 4, -7*e + 32, -18*e + 32, 18*e - 40, -78, -78, 3*e + 60, -3*e + 72, -6*e - 24, 6*e - 48, -24*e + 30, 24*e - 66, -4*e - 18, 4*e - 34, 28*e - 42, -28*e + 70, -29*e + 72, 29*e - 44, 38, -e + 8, e + 4, -4*e - 20, 4*e - 36, -6, -6, 5*e - 56, -5*e - 36, -27*e + 20, 27*e - 88, 6*e - 26, -6*e - 2, -39*e + 76, 39*e - 80, 22*e - 30, -22*e + 58, 44, 44, 17*e - 48, -17*e + 20, -16*e - 28, 16*e - 92, 8*e - 10, -8*e + 22, 24*e - 78, -24*e + 18, 10*e - 84, -10*e - 44, 10, 25*e - 72, -25*e + 28, -12*e + 2, 12*e - 46, 6*e + 68, -6*e + 92, -6, -6, -12*e - 30, 12*e - 78, -25*e + 40, 25*e - 60, 12*e - 66, -12*e - 18, 33*e - 84, -33*e + 48, 12*e + 20, -12*e + 68, 6*e + 86, -6*e + 110, -8*e + 110, 8*e + 78, -18*e + 44, 18*e - 28, 25*e - 64, -25*e + 36, -2*e - 42, 2*e - 50, 5*e - 88, -5*e - 68, -33*e + 68, 33*e - 64, -20*e + 50, 20*e - 30, 18*e - 76, -18*e - 4, 24*e - 102, -24*e - 6, -86, -86, 26*e - 58, -26*e + 46, 28*e - 24, -28*e + 88, -7*e + 108, 7*e + 80, 20*e - 16, -20*e + 64, 8*e - 94, -8*e - 62, -16*e + 58, 16*e - 6, -e - 96, e - 100, 20*e - 34, -20*e + 46, 24*e - 14, -24*e + 82, 8*e + 6, -8*e + 38, 3*e + 32, -3*e + 44, 14*e - 110, -14*e - 54, -4*e + 40, 4*e + 24, 20*e - 66, -20*e + 14, -14, -14, -8*e + 22, 8*e - 10, -23*e + 68, 23*e - 24, 22*e - 86, -22*e + 2, 6*e - 4, -6*e + 20, 4*e + 6, -4*e + 22, 13*e - 36, -13*e + 16, 12*e - 50, -12*e - 2, -21*e + 28, 21*e - 56, -28*e + 22, 28*e - 90, 6*e - 120, -6*e - 96, -8*e + 26, 8*e - 6, 54, 4*e - 42, -4*e - 26, 37*e - 76, -37*e + 72, 30, 30, -13*e + 100, 13*e + 48, 30*e - 76, -30*e + 44, 23*e - 56, -23*e + 36, -5*e + 80, 5*e + 60, 30*e - 18, -30*e + 102, -12*e - 2, 12*e - 50, 19*e + 32, -19*e + 108, 6*e + 42, -6*e + 66, -20*e + 44, 20*e - 36, -20*e + 80, 20*e, 12*e - 78, -12*e - 30, -12*e - 30, 12*e - 78, 18*e - 24, -18*e + 48, -19*e - 36, 19*e - 112, -15*e + 84, 15*e + 24, -10*e + 62, 10*e + 22, -20*e + 48, 20*e - 32, 66, 66, 16*e + 26, -16*e + 90, -17*e + 92, 17*e + 24, -28*e + 44, 28*e - 68, -42*e + 70, 42*e - 98, -11*e + 128, 11*e + 84, -14*e + 36, 14*e - 20, -4*e - 34, 4*e - 50, -14*e + 96, 14*e + 40, -33*e + 48, 33*e - 84, -20*e + 60, 20*e - 20, 20*e - 2, -20*e + 78, -24*e + 98, 24*e + 2, -7*e + 72, 7*e + 44, 42*e - 86, -42*e + 82, -5*e - 24, 5*e - 44, -12*e + 8, 12*e - 40, -16*e - 34, 16*e - 98, -12*e + 16, 12*e - 32, 8*e - 54, -8*e - 22, -35*e + 64, 35*e - 76, 10*e + 18, -10*e + 58, -31*e + 56, 31*e - 68, 24*e - 10, -24*e + 86, 2*e + 24, -2*e + 32, 16*e - 62, -16*e + 2, -2*e + 86, 2*e + 78, 10*e, -10*e + 40, -8*e + 18, 8*e - 14, -4*e + 50, 4*e + 34, 4*e - 76, -4*e - 60, 10*e - 12, -10*e + 28, -24*e - 22, 24*e - 118, 16*e - 44, -16*e + 20, -11*e - 44, 11*e - 88, 18*e - 34, -18*e + 38, -e + 80, e + 76, 6*e - 94, -6*e - 70, -12*e + 46, 12*e - 2, -34*e + 96, 34*e - 40, e - 32, -e - 28, 13*e - 108, -13*e - 56, -4*e - 88, 4*e - 104, 8*e + 2, -8*e + 34, 11*e - 124, -11*e - 80, -28*e + 46, 28*e - 66, 20*e + 4, -20*e + 84, -e - 76, e - 80, -6*e - 20, 6*e - 44, 16*e - 98, -16*e - 34, 7*e + 4, -7*e + 32, 18*e - 66, -18*e + 6, 82, 82, -6*e + 112, 6*e + 88, 32*e - 94, -32*e + 34, 3*e - 52, -3*e - 40, 8*e - 44, -8*e - 12, -3*e + 100, 3*e + 88, 22*e - 50, -22*e + 38, 18*e - 24, -18*e + 48, -7*e + 4, 7*e - 24, 6*e + 54, -6*e + 78, 20*e - 26, -20*e + 54, 6*e - 92, -6*e - 68, 16*e - 42, -16*e + 22, 10*e - 26, -10*e + 14, 6*e + 24, -6*e + 48, -4*e - 4, 4*e - 20, -12*e - 14, 12*e - 62, -3*e - 84, 3*e - 96, 30*e - 96, -30*e + 24, -40*e + 106, 40*e - 54, -13*e - 16, 13*e - 68, -20*e + 14, 20*e - 66, -10*e + 26, 10*e - 14, 24*e - 42, -24*e + 54, 28*e - 64, -28*e + 48, 27*e - 52, -27*e + 56, -e + 48, e + 44, 8*e + 46, -8*e + 78, -17*e + 128, 17*e + 60, 6*e - 22, -6*e + 2, 24*e - 92, -24*e + 4, -9*e + 20, 9*e - 16, 24*e - 8, -24*e + 88, -12*e - 36, 12*e - 84, 42*e - 98, -42*e + 70, -16*e + 110, 16*e + 46, 17*e - 64, -17*e + 4, 4*e - 86, -4*e - 70, 10*e + 52, -10*e + 92, -16*e - 62, 16*e - 126, -35*e + 32, 35*e - 108, -28*e + 66, 28*e - 46, -10*e + 110, 10*e + 70, 8*e - 94, -8*e - 62, 16*e - 114, -16*e - 50, 52*e - 90, -52*e + 118, -6*e - 108, 6*e - 132, 32*e - 106, -32*e + 22, -4*e + 62, 4*e + 46, -4*e + 102, 4*e + 86, 10*e - 20, -10*e + 20, -13*e + 16, 13*e - 36, -20*e + 6, 20*e - 74, -9*e + 44, 9*e + 8, 98, 98, -25*e + 100, 25*e, -20*e + 144, 20*e + 64, 2*e - 126, -2*e - 118, -2, -2, 12*e + 60, -12*e + 108, -8*e - 90, 8*e - 122, -9*e - 40, 9*e - 76, 18*e - 126, -18*e - 54, -16*e + 114, 16*e + 50, -16*e - 50, 16*e - 114, 6*e + 120, -6*e + 144, 7*e - 56, -7*e - 28, 6, 32*e, -32*e + 128, 16*e - 8, -16*e + 56, 12*e - 94, -12*e - 46, -21*e + 72, 21*e - 12, -6*e - 102, 6*e - 126, 18*e + 20, -18*e + 92, 15*e - 4, -15*e + 56, 38*e - 90, -38*e + 62, -33*e + 60, 33*e - 72, 16*e - 30, -16*e + 34, -2*e + 20, 2*e + 12, -6*e - 112, 6*e - 136, -16*e + 10, 16*e - 54, -28*e + 36, 28*e - 76, 16*e + 2, -16*e + 66, 36*e - 104, -36*e + 40, -e + 92, e + 88, -40*e + 42, 40*e - 118, -46*e + 86, 46*e - 98, -5*e + 156, 5*e + 136, -120, -120, 4*e + 114, -4*e + 130, 40*e - 38, -40*e + 122, -11*e + 24, 11*e - 20, 24*e - 86, -24*e + 10, 28*e + 6, -28*e + 118, 4*e - 98, -4*e - 82, 10*e - 90, -10*e - 50, -30*e + 26, 30*e - 94, -16*e - 26, 16*e - 90, 18*e - 40, -18*e + 32, 4*e + 126, -4*e + 142, 36*e - 114, -36*e + 30, -16*e + 146, 16*e + 82, -18*e + 4, 18*e - 68, -34, -34, 8*e + 14, -8*e + 46, 22*e + 32, -22*e + 120, 49*e - 108, -49*e + 88, -38*e + 60, 38*e - 92, -27*e - 20, 27*e - 128, -2*e - 38, 2*e - 46, 14*e - 66, -14*e - 10, 8*e + 62, -8*e + 94, -15*e + 16, 15*e - 44, 14*e - 52, -14*e + 4, -2*e - 24, 2*e - 32, 24*e - 10, -24*e + 86, 7*e + 108, -7*e + 136, 4*e + 58, -4*e + 74, -24*e + 150, 24*e + 54, -16*e + 104, 16*e + 40, 44*e - 86, -44*e + 90, -28*e + 114, 28*e + 2, 33*e - 48, -33*e + 84, -46*e + 64, 46*e - 120, 17*e - 44, -17*e + 24, -26*e + 66, 26*e - 38, 45*e - 104, -45*e + 76, -24*e + 102, 24*e + 6, -30*e + 14, 30*e - 106, 8*e - 82, -8*e - 50, 20*e - 68, -20*e + 12, -12*e - 22, 12*e - 70, -8*e - 108, 8*e - 140, 24*e + 42, -24*e + 138, -2*e - 56, 2*e - 64, 4*e + 126, -4*e + 142, -e + 28, e + 24, -43*e + 124, 43*e - 48, -36*e + 102, 36*e - 42, 6*e - 108, -6*e - 84, 32*e - 86, -32*e + 42, -19*e + 36, 19*e - 40, -28*e + 10, 28*e - 102, 15*e + 48, -15*e + 108, -2*e - 2, 2*e - 10, 9*e + 68, -9*e + 104, 4*e - 46, -4*e - 30, -28*e + 102, 28*e - 10, 32*e + 18, -32*e + 146, 28*e - 110, -28*e + 2, -32*e + 92, 32*e - 36, -23*e + 68, 23*e - 24, -26*e + 40, 26*e - 64, 34*e - 94, -34*e + 42, 12*e - 18, -12*e + 30, 31*e - 124, -31*e, -2, -2, 18*e - 24, -18*e + 48, -29*e - 8, 29*e - 124, 16*e - 38, -16*e + 26, 9*e - 112, -9*e - 76, -66, -66, -19*e + 148, 19*e + 72, 4*e + 100, -4*e + 116, -6*e + 46, 6*e + 22, 4*e + 2, -4*e + 18, 24*e + 34, -24*e + 130, 29*e - 96, -29*e + 20, -8*e - 146, 8*e - 178, 18*e + 10, -18*e + 82, -16*e + 102, 16*e + 38, -56*e + 120, 56*e - 104, 16*e - 4, -16*e + 60, -30*e + 104, 30*e - 16, -7*e + 56, 7*e + 28, -18*e + 90, 18*e + 18, 7*e + 96, -7*e + 124, -3*e - 156, 3*e - 168, -32*e + 158, 32*e + 30, -24*e + 164, 24*e + 68, -4*e + 142, 4*e + 126, 42*e - 60, -42*e + 108, -24*e - 10, 24*e - 106, 35*e, -35*e + 140, 30*e - 30, -30*e + 90, -24*e + 142, 24*e + 46, 21*e + 20, -21*e + 104, -11*e - 32, 11*e - 76, 48*e - 86, -48*e + 106, -5*e + 16, 5*e - 4, 34*e - 154, -34*e - 18, 33*e - 44, -33*e + 88, -28*e + 110, 28*e - 2, -12*e + 4, 12*e - 44, -16*e + 26, 16*e - 38, -22*e + 22, 22*e - 66, -16*e + 18, 16*e - 46, -2*e - 112, 2*e - 120, 49*e - 48, -49*e + 148, -40*e + 26, 40*e - 134, 41*e - 76, -41*e + 88, -4*e + 154, 4*e + 138, -37*e + 76, 37*e - 72, -38*e + 84, 38*e - 68, 8*e - 146, -8*e - 114, -27*e + 72, 27*e - 36, 5*e - 24, -5*e - 4, 44*e - 110, -44*e + 66, 6*e - 108, -6*e - 84, 18*e - 58, -18*e + 14, 17*e - 64, -17*e + 4, -20*e + 34, 20*e - 46, 2*e - 40, -2*e - 32, 48*e - 110, -48*e + 82, 42*e - 92, -42*e + 76, -37*e + 136, 37*e - 12, -24*e + 2, 24*e - 94, -e - 48, e - 52, 26*e - 146, -26*e - 42, -8*e + 82, 8*e + 50, -22*e - 44, 22*e - 132, -24*e + 118, 24*e + 22, 28*e - 40, -28*e + 72, 8*e + 76, -8*e + 108, -16*e - 10, 16*e - 74, -24*e + 176, 24*e + 80, 4*e + 178, -4*e + 194, -22*e - 74, 22*e - 162, 28*e - 72, -28*e + 40, 2*e + 106, -2*e + 114, -e + 164, e + 160, 16*e - 128, -16*e - 64, 54*e - 114, -54*e + 102, 19*e - 96, -19*e - 20] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([2, 2, -w])] = 1 AL_eigenvalues[ZF.ideal([9, 3, 3])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]