/* 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([44, 2, -15, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 2, 2]) primes_array = [ [4, 2, w + 2],\ [4, 2, -1/2*w^3 + 3/2*w^2 + 9/2*w - 10],\ [9, 3, 1/2*w^3 - 5/2*w^2 - 5/2*w + 17],\ [9, 3, 3/2*w^3 - 11/2*w^2 - 19/2*w + 28],\ [11, 11, 1/2*w^3 - 3/2*w^2 - 9/2*w + 11],\ [11, 11, -w - 3],\ [25, 5, w^3 - 3*w^2 - 7*w + 15],\ [29, 29, w + 1],\ [29, 29, 1/2*w^3 - 3/2*w^2 - 9/2*w + 9],\ [31, 31, -1/2*w^3 + 5/2*w^2 + 5/2*w - 16],\ [31, 31, 1/2*w^3 - 3/2*w^2 - 9/2*w + 5],\ [31, 31, -w + 3],\ [31, 31, 3/2*w^3 - 11/2*w^2 - 19/2*w + 29],\ [59, 59, 2*w^2 - w - 13],\ [59, 59, 9/2*w^3 - 31/2*w^2 - 61/2*w + 85],\ [61, 61, 2*w^3 - 6*w^2 - 15*w + 31],\ [61, 61, -3/2*w^3 + 11/2*w^2 + 21/2*w - 34],\ [71, 71, 3/2*w^3 - 11/2*w^2 - 19/2*w + 32],\ [71, 71, -1/2*w^3 + 5/2*w^2 + 5/2*w - 13],\ [79, 79, -3*w^3 + 10*w^2 + 19*w - 51],\ [79, 79, 15/2*w^3 - 51/2*w^2 - 103/2*w + 139],\ [101, 101, -2*w^3 + 7*w^2 + 14*w - 37],\ [101, 101, -1/2*w^3 + 5/2*w^2 + 7/2*w - 16],\ [109, 109, w^3 - 3*w^2 - 6*w + 19],\ [109, 109, -w^3 + 4*w^2 + 5*w - 19],\ [121, 11, -3/2*w^3 + 9/2*w^2 + 21/2*w - 23],\ [139, 139, -w^3 + 4*w^2 + 7*w - 25],\ [139, 139, -5/2*w^3 + 17/2*w^2 + 33/2*w - 41],\ [149, 149, 1/2*w^3 + 1/2*w^2 - 11/2*w - 8],\ [149, 149, 5/2*w^3 - 19/2*w^2 - 35/2*w + 50],\ [149, 149, -9/2*w^3 + 31/2*w^2 + 59/2*w - 82],\ [149, 149, -5/2*w^3 + 19/2*w^2 + 35/2*w - 56],\ [151, 151, -1/2*w^3 + 3/2*w^2 + 3/2*w - 12],\ [151, 151, 3/2*w^3 - 9/2*w^2 - 25/2*w + 28],\ [179, 179, -1/2*w^3 + 3/2*w^2 + 9/2*w - 3],\ [179, 179, w - 5],\ [229, 229, -15/2*w^3 + 51/2*w^2 + 105/2*w - 142],\ [229, 229, 1/2*w^3 - 7/2*w^2 - 7/2*w + 19],\ [229, 229, -17/2*w^3 + 57/2*w^2 + 119/2*w - 156],\ [229, 229, -9/2*w^3 + 33/2*w^2 + 55/2*w - 82],\ [239, 239, -9*w^3 + 31*w^2 + 59*w - 161],\ [239, 239, w^3 + w^2 - 11*w - 19],\ [241, 241, 5/2*w^3 - 15/2*w^2 - 37/2*w + 37],\ [241, 241, -2*w^3 + 6*w^2 + 13*w - 29],\ [251, 251, -7/2*w^3 + 25/2*w^2 + 45/2*w - 67],\ [251, 251, 1/2*w^3 - 5/2*w^2 - 13/2*w + 7],\ [251, 251, -1/2*w^3 + 1/2*w^2 + 13/2*w + 4],\ [251, 251, -1/2*w^3 + 7/2*w^2 + 3/2*w - 23],\ [271, 271, -w^3 + 4*w^2 + 5*w - 25],\ [271, 271, -5/2*w^3 + 17/2*w^2 + 33/2*w - 47],\ [281, 281, 5/2*w^3 - 15/2*w^2 - 39/2*w + 41],\ [281, 281, 3/2*w^3 - 7/2*w^2 - 27/2*w + 14],\ [289, 17, -7/2*w^3 + 25/2*w^2 + 45/2*w - 62],\ [289, 17, 5/2*w^3 - 15/2*w^2 - 39/2*w + 37],\ [311, 311, -19/2*w^3 + 65/2*w^2 + 129/2*w - 175],\ [311, 311, -7/2*w^3 + 25/2*w^2 + 47/2*w - 71],\ [311, 311, 7/2*w^3 - 25/2*w^2 - 49/2*w + 67],\ [311, 311, -3/2*w^3 + 11/2*w^2 + 15/2*w - 31],\ [331, 331, 19/2*w^3 - 65/2*w^2 - 127/2*w + 173],\ [331, 331, -w^3 - w^2 + 10*w + 15],\ [349, 349, -5/2*w^3 + 15/2*w^2 + 33/2*w - 35],\ [349, 349, 3/2*w^3 - 11/2*w^2 - 21/2*w + 36],\ [361, 19, 2*w^3 - 6*w^2 - 14*w + 29],\ [361, 19, 2*w^3 - 6*w^2 - 14*w + 31],\ [379, 379, 1/2*w^3 - 5/2*w^2 - 1/2*w + 20],\ [379, 379, -1/2*w^3 + 5/2*w^2 + 1/2*w - 9],\ [401, 401, -7/2*w^3 + 25/2*w^2 + 45/2*w - 69],\ [401, 401, w^3 - 3*w^2 - 5*w + 19],\ [409, 409, 3/2*w^3 - 7/2*w^2 - 23/2*w + 13],\ [409, 409, -11/2*w^3 + 37/2*w^2 + 77/2*w - 100],\ [409, 409, 7/2*w^3 - 23/2*w^2 - 47/2*w + 58],\ [409, 409, -w^3 + 3*w^2 + 9*w - 9],\ [419, 419, 3/2*w^3 - 13/2*w^2 - 17/2*w + 36],\ [419, 419, -5/2*w^3 + 19/2*w^2 + 31/2*w - 54],\ [421, 421, -w^3 + 4*w^2 + 5*w - 27],\ [421, 421, 7/2*w^3 - 23/2*w^2 - 47/2*w + 63],\ [439, 439, 2*w^3 - 7*w^2 - 14*w + 35],\ [439, 439, -1/2*w^3 + 5/2*w^2 + 7/2*w - 18],\ [461, 461, -5/2*w^3 + 17/2*w^2 + 33/2*w - 48],\ [461, 461, -13/2*w^3 + 45/2*w^2 + 85/2*w - 119],\ [461, 461, -1/2*w^3 + 3/2*w^2 + 1/2*w - 15],\ [461, 461, 3/2*w^3 - 11/2*w^2 - 17/2*w + 32],\ [479, 479, -3*w^3 + 10*w^2 + 19*w - 53],\ [479, 479, 5/2*w^3 - 17/2*w^2 - 33/2*w + 51],\ [491, 491, 3*w^2 - 2*w - 23],\ [491, 491, 13/2*w^3 - 45/2*w^2 - 87/2*w + 120],\ [499, 499, -3*w^3 + 11*w^2 + 21*w - 63],\ [499, 499, -5/2*w^3 + 19/2*w^2 + 31/2*w - 50],\ [499, 499, 11/2*w^3 - 37/2*w^2 - 75/2*w + 101],\ [499, 499, 9*w^3 - 31*w^2 - 58*w + 159],\ [509, 509, 3/2*w^3 - 9/2*w^2 - 23/2*w + 29],\ [509, 509, w^3 - 3*w^2 - 6*w + 21],\ [521, 521, 2*w^3 - 8*w^2 - 12*w + 43],\ [521, 521, -3/2*w^3 + 11/2*w^2 + 19/2*w - 36],\ [521, 521, 2*w^3 - 8*w^2 - 12*w + 47],\ [521, 521, -7/2*w^3 + 23/2*w^2 + 45/2*w - 58],\ [541, 541, -1/2*w^3 + 5/2*w^2 + 13/2*w - 6],\ [541, 541, 9/2*w^3 - 33/2*w^2 - 55/2*w + 80],\ [541, 541, -15/2*w^3 + 51/2*w^2 + 101/2*w - 134],\ [541, 541, w^3 - w^2 - 7*w + 1],\ [571, 571, 13/2*w^3 - 47/2*w^2 - 79/2*w + 116],\ [571, 571, -5/2*w^3 + 11/2*w^2 + 43/2*w - 21],\ [599, 599, -3*w^3 + 9*w^2 + 19*w - 47],\ [599, 599, -15/2*w^3 + 51/2*w^2 + 103/2*w - 137],\ [599, 599, -w^3 + 2*w^2 + 11*w + 1],\ [599, 599, 5/2*w^3 - 17/2*w^2 - 35/2*w + 38],\ [601, 601, -1/2*w^3 + 11/2*w^2 - 5/2*w - 47],\ [601, 601, -3/2*w^3 + 13/2*w^2 + 17/2*w - 30],\ [619, 619, -w^3 + 5*w^2 + 5*w - 27],\ [619, 619, -1/2*w^3 + 1/2*w^2 + 11/2*w - 2],\ [619, 619, -2*w^3 + 7*w^2 + 12*w - 39],\ [619, 619, -3*w^3 + 11*w^2 + 19*w - 63],\ [631, 631, -3/2*w^3 + 13/2*w^2 + 17/2*w - 37],\ [631, 631, 5/2*w^3 - 19/2*w^2 - 31/2*w + 53],\ [641, 641, 5*w^3 - 16*w^2 - 35*w + 79],\ [641, 641, -2*w^3 + 8*w^2 + 12*w - 53],\ [641, 641, 3/2*w^3 - 11/2*w^2 - 23/2*w + 34],\ [641, 641, 3/2*w^3 - 11/2*w^2 - 23/2*w + 27],\ [659, 659, -5/2*w^3 + 19/2*w^2 + 31/2*w - 51],\ [659, 659, -3/2*w^3 + 13/2*w^2 + 17/2*w - 39],\ [661, 661, -5/2*w^3 + 17/2*w^2 + 33/2*w - 50],\ [661, 661, 5*w^3 - 17*w^2 - 30*w + 83],\ [661, 661, 5/2*w^3 - 17/2*w^2 - 31/2*w + 46],\ [661, 661, 3/2*w^3 - 5/2*w^2 - 21/2*w + 9],\ [691, 691, -3/2*w^3 + 13/2*w^2 + 19/2*w - 35],\ [691, 691, 3*w^3 - 11*w^2 - 20*w + 63],\ [709, 709, 19/2*w^3 - 65/2*w^2 - 127/2*w + 171],\ [709, 709, 4*w^3 - 14*w^2 - 28*w + 75],\ [709, 709, 1/2*w^3 - 5/2*w^2 - 15/2*w + 4],\ [709, 709, -w^3 + 5*w^2 + 7*w - 31],\ [739, 739, 11/2*w^3 - 37/2*w^2 - 73/2*w + 95],\ [739, 739, -3/2*w^3 + 5/2*w^2 + 25/2*w - 5],\ [739, 739, 27/2*w^3 - 91/2*w^2 - 187/2*w + 247],\ [739, 739, 4*w^3 - 15*w^2 - 24*w + 73],\ [751, 751, -5/2*w^3 + 19/2*w^2 + 35/2*w - 49],\ [751, 751, -2*w^2 + 2*w + 21],\ [751, 751, -9*w^3 + 31*w^2 + 61*w - 169],\ [751, 751, 17/2*w^3 - 57/2*w^2 - 117/2*w + 152],\ [761, 761, 5/2*w^3 - 15/2*w^2 - 33/2*w + 39],\ [761, 761, -3*w^3 + 9*w^2 + 22*w - 47],\ [761, 761, 9/2*w^3 - 29/2*w^2 - 61/2*w + 80],\ [761, 761, 25/2*w^3 - 85/2*w^2 - 169/2*w + 227],\ [769, 769, 2*w^3 - 8*w^2 - 12*w + 45],\ [811, 811, 5/2*w^3 - 19/2*w^2 - 27/2*w + 45],\ [811, 811, 1/2*w^3 - 7/2*w^2 + 1/2*w + 29],\ [821, 821, 25/2*w^3 - 85/2*w^2 - 173/2*w + 233],\ [821, 821, 19/2*w^3 - 63/2*w^2 - 135/2*w + 173],\ [829, 829, -7/2*w^3 + 27/2*w^2 + 43/2*w - 79],\ [829, 829, 10*w^3 - 34*w^2 - 67*w + 179],\ [829, 829, -3/2*w^3 + 1/2*w^2 + 27/2*w + 9],\ [829, 829, 5/2*w^3 - 13/2*w^2 - 41/2*w + 31],\ [839, 839, -3/2*w^3 + 11/2*w^2 + 25/2*w - 26],\ [839, 839, 2*w^3 - 7*w^2 - 16*w + 43],\ [841, 29, 5/2*w^3 - 15/2*w^2 - 35/2*w + 39],\ [881, 881, -8*w^3 + 27*w^2 + 56*w - 151],\ [881, 881, -w^3 + 3*w^2 + 4*w - 21],\ [911, 911, 5/2*w^3 - 19/2*w^2 - 33/2*w + 51],\ [911, 911, 2*w^3 - 8*w^2 - 13*w + 47],\ [919, 919, 3/2*w^3 - 9/2*w^2 - 27/2*w + 29],\ [919, 919, -3*w - 5],\ [919, 919, 1/2*w^3 - 7/2*w^2 - 5/2*w + 17],\ [919, 919, -4*w^3 + 14*w^2 + 27*w - 81],\ [929, 929, 7/2*w^3 - 23/2*w^2 - 49/2*w + 58],\ [929, 929, 13*w^3 - 44*w^2 - 89*w + 237],\ [929, 929, 9/2*w^3 - 29/2*w^2 - 59/2*w + 78],\ [929, 929, -w^3 + 2*w^2 + 7*w - 5],\ [941, 941, 5/2*w^3 - 19/2*w^2 - 31/2*w + 59],\ [941, 941, -3/2*w^3 + 13/2*w^2 + 17/2*w - 31],\ [941, 941, w^3 - 3*w^2 - 10*w + 7],\ [941, 941, -8*w^3 + 27*w^2 + 56*w - 147],\ [971, 971, -9/2*w^3 + 29/2*w^2 + 73/2*w - 92],\ [971, 971, -1/2*w^3 + 5/2*w^2 + 17/2*w - 1],\ [991, 991, w^3 - 4*w^2 - 3*w + 15],\ [991, 991, -1/2*w^3 + 1/2*w^2 + 15/2*w + 6],\ [1019, 1019, 3*w^3 - 9*w^2 - 19*w + 45],\ [1019, 1019, 3*w^3 - 11*w^2 - 20*w + 53],\ [1019, 1019, -1/2*w^3 + 1/2*w^2 + 17/2*w + 9],\ [1019, 1019, -1/2*w^3 + 5/2*w^2 - 3/2*w - 4],\ [1021, 1021, -3/2*w^3 + 13/2*w^2 + 19/2*w - 37],\ [1021, 1021, 3*w^3 - 11*w^2 - 20*w + 61],\ [1049, 1049, w^3 - 4*w^2 - 9*w + 15],\ [1049, 1049, 5/2*w^3 - 17/2*w^2 - 39/2*w + 54],\ [1051, 1051, 3/2*w^3 - 11/2*w^2 - 15/2*w + 21],\ [1051, 1051, -1/2*w^3 + 7/2*w^2 + 7/2*w - 23],\ [1051, 1051, 9/2*w^3 - 31/2*w^2 - 63/2*w + 83],\ [1051, 1051, -1/2*w^3 + 1/2*w^2 + 13/2*w + 8],\ [1061, 1061, 2*w^3 - 5*w^2 - 14*w + 19],\ [1061, 1061, 9/2*w^3 - 29/2*w^2 - 63/2*w + 72],\ [1091, 1091, 3/2*w^3 - 5/2*w^2 - 21/2*w + 10],\ [1091, 1091, 3/2*w^3 - 3/2*w^2 - 19/2*w + 7],\ [1091, 1091, -13/2*w^3 + 43/2*w^2 + 91/2*w - 116],\ [1091, 1091, 19/2*w^3 - 63/2*w^2 - 135/2*w + 174],\ [1109, 1109, -5/2*w^3 + 17/2*w^2 + 33/2*w - 53],\ [1109, 1109, 4*w^3 - 13*w^2 - 26*w + 67],\ [1109, 1109, -3/2*w^3 + 11/2*w^2 + 23/2*w - 25],\ [1109, 1109, 4*w^3 - 13*w^2 - 28*w + 65],\ [1171, 1171, 1/2*w^3 - 1/2*w^2 + 3/2*w + 12],\ [1171, 1171, 11/2*w^3 - 35/2*w^2 - 87/2*w + 105],\ [1181, 1181, 2*w^3 - 8*w^2 - 13*w + 45],\ [1181, 1181, -5/2*w^3 + 19/2*w^2 + 33/2*w - 53],\ [1201, 1201, -5/2*w^3 + 15/2*w^2 + 39/2*w - 45],\ [1201, 1201, -1/2*w^3 + 3/2*w^2 + 13/2*w - 3],\ [1201, 1201, -w^3 + 3*w^2 + 10*w - 21],\ [1201, 1201, 3/2*w^3 - 9/2*w^2 - 17/2*w + 29],\ [1229, 1229, -5*w^3 + 17*w^2 + 35*w - 91],\ [1229, 1229, 3/2*w^3 - 9/2*w^2 - 25/2*w + 17],\ [1249, 1249, 5/2*w^3 - 17/2*w^2 - 35/2*w + 40],\ [1249, 1249, 3/2*w^3 - 3/2*w^2 - 21/2*w + 4],\ [1249, 1249, w^2 - 13],\ [1249, 1249, 9*w^3 - 30*w^2 - 63*w + 163],\ [1289, 1289, 3/2*w^3 - 1/2*w^2 - 23/2*w - 3],\ [1289, 1289, 11*w^3 - 37*w^2 - 76*w + 201],\ [1291, 1291, -1/2*w^3 - 7/2*w^2 + 13/2*w + 29],\ [1291, 1291, 23/2*w^3 - 79/2*w^2 - 155/2*w + 212],\ [1301, 1301, 11*w^3 - 38*w^2 - 73*w + 201],\ [1301, 1301, 1/2*w^3 - 5/2*w^2 - 17/2*w + 3],\ [1319, 1319, 12*w^3 - 40*w^2 - 85*w + 219],\ [1319, 1319, 3/2*w^3 - 1/2*w^2 - 19/2*w - 1],\ [1321, 1321, -1/2*w^3 - 1/2*w^2 - 1/2*w - 7],\ [1321, 1321, -3*w^3 + 11*w^2 + 15*w - 45],\ [1361, 1361, -3*w^3 + 11*w^2 + 21*w - 57],\ [1361, 1361, 5/2*w^3 - 15/2*w^2 - 39/2*w + 35],\ [1381, 1381, -3*w^3 + 9*w^2 + 23*w - 43],\ [1381, 1381, 2*w^3 - 6*w^2 - 12*w + 27],\ [1381, 1381, -1/2*w^3 + 3/2*w^2 + 9/2*w - 1],\ [1381, 1381, w - 7],\ [1399, 1399, 9/2*w^3 - 33/2*w^2 - 57/2*w + 85],\ [1399, 1399, -5/2*w^3 + 15/2*w^2 + 41/2*w - 37],\ [1409, 1409, 3/2*w^3 - 1/2*w^2 - 21/2*w - 1],\ [1409, 1409, -23/2*w^3 + 77/2*w^2 + 161/2*w - 211],\ [1409, 1409, 11/2*w^3 - 39/2*w^2 - 71/2*w + 98],\ [1409, 1409, 1/2*w^3 - 9/2*w^2 - 1/2*w + 37],\ [1429, 1429, 3/2*w^3 - 5/2*w^2 - 33/2*w - 5],\ [1429, 1429, 7/2*w^3 - 25/2*w^2 - 37/2*w + 53],\ [1439, 1439, -7*w^3 + 23*w^2 + 52*w - 135],\ [1439, 1439, -1/2*w^3 - 1/2*w^2 + 1/2*w - 5],\ [1451, 1451, 1/2*w^3 + 3/2*w^2 - 13/2*w - 15],\ [1451, 1451, -13/2*w^3 + 45/2*w^2 + 85/2*w - 120],\ [1459, 1459, -7/2*w^3 + 27/2*w^2 + 43/2*w - 70],\ [1459, 1459, 5/2*w^3 - 21/2*w^2 - 29/2*w + 65],\ [1471, 1471, 5/2*w^3 - 17/2*w^2 - 37/2*w + 43],\ [1471, 1471, -3*w^3 + 10*w^2 + 21*w - 49],\ [1481, 1481, -6*w^3 + 21*w^2 + 40*w - 113],\ [1481, 1481, -16*w^3 + 54*w^2 + 112*w - 299],\ [1481, 1481, 9*w^3 - 30*w^2 - 63*w + 161],\ [1481, 1481, 1/2*w^3 - 9/2*w^2 - 3/2*w + 30],\ [1489, 1489, -8*w^3 + 26*w^2 + 60*w - 151],\ [1489, 1489, -w^3 + 5*w^2 + 6*w - 35],\ [1489, 1489, 7/2*w^3 - 25/2*w^2 - 47/2*w + 63],\ [1489, 1489, -w^3 + w^2 + 3*w - 13],\ [1499, 1499, 3*w^3 - 10*w^2 - 21*w + 45],\ [1499, 1499, 9/2*w^3 - 29/2*w^2 - 65/2*w + 75],\ [1511, 1511, -11/2*w^3 + 35/2*w^2 + 79/2*w - 91],\ [1511, 1511, -5/2*w^3 + 21/2*w^2 + 33/2*w - 54],\ [1511, 1511, 7/2*w^3 - 19/2*w^2 - 57/2*w + 42],\ [1511, 1511, 5/2*w^3 - 15/2*w^2 - 29/2*w + 41],\ [1531, 1531, -4*w^2 + 2*w + 29],\ [1531, 1531, 9*w^3 - 31*w^2 - 61*w + 167],\ [1549, 1549, -1/2*w^3 + 1/2*w^2 + 13/2*w - 5],\ [1549, 1549, -3/2*w^3 + 11/2*w^2 + 15/2*w - 34],\ [1559, 1559, 9/2*w^3 - 29/2*w^2 - 63/2*w + 76],\ [1559, 1559, 2*w^3 - 5*w^2 - 14*w + 23],\ [1571, 1571, 3*w^3 - 9*w^2 - 20*w + 43],\ [1571, 1571, 2*w^3 - 7*w^2 - 14*w + 45],\ [1601, 1601, -27/2*w^3 + 93/2*w^2 + 179/2*w - 247],\ [1601, 1601, 1/2*w^3 - 1/2*w^2 + 5/2*w + 16],\ [1609, 1609, -1/2*w^3 + 7/2*w^2 + 1/2*w - 31],\ [1609, 1609, 3/2*w^3 - 7/2*w^2 - 23/2*w + 25],\ [1621, 1621, 7/2*w^3 - 25/2*w^2 - 45/2*w + 71],\ [1621, 1621, -1/2*w^3 + 7/2*w^2 + 3/2*w - 19],\ [1681, 41, -3*w^3 + 9*w^2 + 21*w - 43],\ [1681, 41, 3*w^3 - 9*w^2 - 21*w + 47],\ [1721, 1721, -1/2*w^3 - 3/2*w^2 + 21/2*w + 25],\ [1721, 1721, -9/2*w^3 + 33/2*w^2 + 49/2*w - 78],\ [1759, 1759, 4*w^3 - 14*w^2 - 25*w + 75],\ [1759, 1759, 2*w^3 - 7*w^2 - 12*w + 43],\ [1831, 1831, -3/2*w^3 + 7/2*w^2 + 29/2*w - 6],\ [1831, 1831, 15/2*w^3 - 51/2*w^2 - 103/2*w + 136],\ [1849, 43, -1/2*w^3 + 7/2*w^2 - 3/2*w - 31],\ [1849, 43, 15/2*w^3 - 51/2*w^2 - 101/2*w + 138],\ [1871, 1871, 3*w^3 - 9*w^2 - 18*w + 43],\ [1871, 1871, -7/2*w^3 + 21/2*w^2 + 45/2*w - 51],\ [1879, 1879, 21/2*w^3 - 71/2*w^2 - 145/2*w + 195],\ [1879, 1879, -w^3 - w^2 + 8*w + 9],\ [1889, 1889, 2*w^3 - 4*w^2 - 16*w + 15],\ [1889, 1889, -6*w^3 + 20*w^2 + 40*w - 105],\ [1901, 1901, 2*w^2 - 7*w - 29],\ [1901, 1901, -19/2*w^3 + 65/2*w^2 + 133/2*w - 182],\ [1901, 1901, 1/2*w^3 - 11/2*w^2 - 7/2*w + 30],\ [1901, 1901, -1/2*w^3 + 5/2*w^2 + 1/2*w - 24],\ [1931, 1931, -1/2*w^3 + 9/2*w^2 + 3/2*w - 34],\ [1931, 1931, -5/2*w^3 + 19/2*w^2 + 35/2*w - 59],\ [1931, 1931, -5/2*w^3 + 19/2*w^2 + 35/2*w - 47],\ [1931, 1931, 6*w^3 - 21*w^2 - 40*w + 109],\ [1949, 1949, -3/2*w^3 + 15/2*w^2 + 19/2*w - 46],\ [1949, 1949, -11/2*w^3 + 39/2*w^2 + 75/2*w - 105],\ [1979, 1979, -7/2*w^3 + 21/2*w^2 + 55/2*w - 51],\ [1979, 1979, 3/2*w^3 - 5/2*w^2 - 29/2*w + 7],\ [1979, 1979, 9/2*w^3 - 31/2*w^2 - 55/2*w + 81],\ [1979, 1979, -2*w^3 + 6*w^2 + 11*w - 27]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^7 - 5*x^6 - 39*x^5 + 171*x^4 + 315*x^3 - 663*x^2 - 1105*x - 250 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1, -1, e, e, 1/135*e^6 - 1/9*e^5 - 8/45*e^4 + 167/45*e^3 - 7/9*e^2 - 172/15*e - 122/27, 1/135*e^6 - 1/9*e^5 - 8/45*e^4 + 167/45*e^3 - 7/9*e^2 - 172/15*e - 122/27, -2/135*e^6 + 1/45*e^5 + 22/45*e^4 - 22/45*e^3 - 47/45*e^2 - 4*e - 8/27, 4/135*e^6 - 7/45*e^5 - 13/15*e^4 + 26/5*e^3 - 7/15*e^2 - 154/9*e - 80/27, 4/135*e^6 - 7/45*e^5 - 13/15*e^4 + 26/5*e^3 - 7/15*e^2 - 154/9*e - 80/27, -2/135*e^6 + 1/5*e^5 + 4/9*e^4 - 311/45*e^3 - 43/45*e^2 + 1199/45*e + 358/27, 4/135*e^6 - 4/45*e^5 - 17/15*e^4 + 3*e^3 + 116/15*e^2 - 521/45*e - 332/27, 4/135*e^6 - 4/45*e^5 - 17/15*e^4 + 3*e^3 + 116/15*e^2 - 521/45*e - 332/27, -2/135*e^6 + 1/5*e^5 + 4/9*e^4 - 311/45*e^3 - 43/45*e^2 + 1199/45*e + 358/27, 1/135*e^6 - 1/15*e^5 - 16/45*e^4 + 106/45*e^3 + 206/45*e^2 - 95/9*e - 440/27, 1/135*e^6 - 1/15*e^5 - 16/45*e^4 + 106/45*e^3 + 206/45*e^2 - 95/9*e - 440/27, -8/135*e^6 + 4/15*e^5 + 86/45*e^4 - 392/45*e^3 - 214/45*e^2 + 1214/45*e + 298/27, -8/135*e^6 + 4/15*e^5 + 86/45*e^4 - 392/45*e^3 - 214/45*e^2 + 1214/45*e + 298/27, 1/135*e^6 + 1/9*e^5 - 2/5*e^4 - 61/15*e^3 + 14/3*e^2 + 904/45*e + 88/27, 1/135*e^6 + 1/9*e^5 - 2/5*e^4 - 61/15*e^3 + 14/3*e^2 + 904/45*e + 88/27, 1/45*e^6 - 4/45*e^5 - 38/45*e^4 + 128/45*e^3 + 253/45*e^2 - 59/9*e - 10, 1/45*e^6 - 4/45*e^5 - 38/45*e^4 + 128/45*e^3 + 253/45*e^2 - 59/9*e - 10, 8/135*e^6 - 11/45*e^5 - 2*e^4 + 41/5*e^3 + 109/15*e^2 - 1381/45*e - 412/27, 8/135*e^6 - 11/45*e^5 - 2*e^4 + 41/5*e^3 + 109/15*e^2 - 1381/45*e - 412/27, 1/15*e^6 - 4/15*e^5 - 11/5*e^4 + 133/15*e^3 + 103/15*e^2 - 30*e - 70/3, 1/15*e^6 - 4/15*e^5 - 11/5*e^4 + 133/15*e^3 + 103/15*e^2 - 30*e - 70/3, 4/135*e^6 - 7/45*e^5 - 13/15*e^4 + 26/5*e^3 - 7/15*e^2 - 172/9*e + 244/27, -1/45*e^6 - 1/45*e^5 + 43/45*e^4 + 32/45*e^3 - 383/45*e^2 + 1/9*e + 70/9, -1/45*e^6 - 1/45*e^5 + 43/45*e^4 + 32/45*e^3 - 383/45*e^2 + 1/9*e + 70/9, 2/135*e^6 - 2/15*e^5 - 17/45*e^4 + 197/45*e^3 - 98/45*e^2 - 97/9*e + 200/27, -11/135*e^6 + 2/5*e^5 + 116/45*e^4 - 596/45*e^3 - 211/45*e^2 + 367/9*e + 160/27, 2/135*e^6 - 2/15*e^5 - 17/45*e^4 + 197/45*e^3 - 98/45*e^2 - 97/9*e + 200/27, -11/135*e^6 + 2/5*e^5 + 116/45*e^4 - 596/45*e^3 - 211/45*e^2 + 367/9*e + 160/27, -1/15*e^6 + 2/9*e^5 + 107/45*e^4 - 323/45*e^3 - 113/9*e^2 + 934/45*e + 256/9, -1/15*e^6 + 2/9*e^5 + 107/45*e^4 - 323/45*e^3 - 113/9*e^2 + 934/45*e + 256/9, 8/135*e^6 - 14/45*e^5 - 26/15*e^4 + 151/15*e^3 - 8/5*e^2 - 233/9*e + 290/27, 8/135*e^6 - 14/45*e^5 - 26/15*e^4 + 151/15*e^3 - 8/5*e^2 - 233/9*e + 290/27, -1/27*e^6 + 1/3*e^5 + 10/9*e^4 - 103/9*e^3 - 8/9*e^2 + 400/9*e + 490/27, 11/135*e^6 - 23/45*e^5 - 37/15*e^4 + 84/5*e^3 + 14/5*e^2 - 461/9*e - 760/27, -1/27*e^6 + 1/3*e^5 + 10/9*e^4 - 103/9*e^3 - 8/9*e^2 + 400/9*e + 490/27, 11/135*e^6 - 23/45*e^5 - 37/15*e^4 + 84/5*e^3 + 14/5*e^2 - 461/9*e - 760/27, -11/135*e^6 + 2/5*e^5 + 116/45*e^4 - 596/45*e^3 - 211/45*e^2 + 385/9*e + 430/27, -11/135*e^6 + 2/5*e^5 + 116/45*e^4 - 596/45*e^3 - 211/45*e^2 + 385/9*e + 430/27, 1/135*e^6 - 1/3*e^5 + 2/45*e^4 + 502/45*e^3 - 62/9*e^2 - 1471/45*e - 152/27, 1/135*e^6 - 1/3*e^5 + 2/45*e^4 + 502/45*e^3 - 62/9*e^2 - 1471/45*e - 152/27, 2/45*e^6 - 4/9*e^5 - 58/45*e^4 + 697/45*e^3 - 2/9*e^2 - 2891/45*e - 58/3, -8/135*e^6 + 14/45*e^5 + 31/15*e^4 - 161/15*e^3 - 52/5*e^2 + 428/9*e + 754/27, -8/135*e^6 + 14/45*e^5 + 31/15*e^4 - 161/15*e^3 - 52/5*e^2 + 428/9*e + 754/27, 2/45*e^6 - 4/9*e^5 - 58/45*e^4 + 697/45*e^3 - 2/9*e^2 - 2891/45*e - 58/3, 4/27*e^6 - 2/3*e^5 - 43/9*e^4 + 199/9*e^3 + 104/9*e^2 - 655/9*e - 1006/27, 4/27*e^6 - 2/3*e^5 - 43/9*e^4 + 199/9*e^3 + 104/9*e^2 - 655/9*e - 1006/27, 11/135*e^6 - 5/9*e^5 - 133/45*e^4 + 862/45*e^3 + 169/9*e^2 - 1157/15*e - 1432/27, 11/135*e^6 - 5/9*e^5 - 133/45*e^4 + 862/45*e^3 + 169/9*e^2 - 1157/15*e - 1432/27, 4/135*e^6 - 2/45*e^5 - 44/45*e^4 + 44/45*e^3 + 94/45*e^2 + 9*e + 340/27, 4/135*e^6 - 2/45*e^5 - 44/45*e^4 + 44/45*e^3 + 94/45*e^2 + 9*e + 340/27, 1/135*e^6 + 2/9*e^5 - 8/45*e^4 - 358/45*e^3 - 25/9*e^2 + 523/15*e + 688/27, 1/27*e^6 - 22/45*e^5 - 22/45*e^4 + 736/45*e^3 - 811/45*e^2 - 259/5*e + 398/27, 1/135*e^6 + 2/9*e^5 - 8/45*e^4 - 358/45*e^3 - 25/9*e^2 + 523/15*e + 688/27, 1/27*e^6 - 22/45*e^5 - 22/45*e^4 + 736/45*e^3 - 811/45*e^2 - 259/5*e + 398/27, 11/135*e^6 - 4/9*e^5 - 12/5*e^4 + 214/15*e^3 - 4/3*e^2 - 1681/45*e + 338/27, 11/135*e^6 - 4/9*e^5 - 12/5*e^4 + 214/15*e^3 - 4/3*e^2 - 1681/45*e + 338/27, 13/135*e^6 - 14/45*e^5 - 143/45*e^4 + 443/45*e^3 + 403/45*e^2 - 24*e - 20/27, 13/135*e^6 - 14/45*e^5 - 143/45*e^4 + 443/45*e^3 + 403/45*e^2 - 24*e - 20/27, -1/135*e^6 + 1/45*e^5 + 8/15*e^4 - 1/3*e^3 - 48/5*e^2 - 496/45*e + 938/27, 17/135*e^6 - 2/5*e^5 - 179/45*e^4 + 563/45*e^3 + 241/45*e^2 - 1226/45*e + 728/27, -2/27*e^6 + 5/9*e^5 + 7/3*e^4 - 19*e^3 - 5*e^2 + 649/9*e + 1010/27, -2/27*e^6 + 5/9*e^5 + 7/3*e^4 - 19*e^3 - 5*e^2 + 649/9*e + 1010/27, -1/45*e^6 + 14/45*e^5 + 28/45*e^4 - 478/45*e^3 - 53/45*e^2 + 370/9*e + 268/9, -1/45*e^6 + 14/45*e^5 + 28/45*e^4 - 478/45*e^3 - 53/45*e^2 + 370/9*e + 268/9, -7/135*e^6 + 1/45*e^5 + 8/5*e^4 + 1/15*e^3 - 4/15*e^2 - 119/9*e - 400/27, 1/27*e^6 - 1/9*e^5 - 4/3*e^4 + 11/3*e^3 + 19/3*e^2 - 134/9*e - 10/27, -7/135*e^6 + 1/45*e^5 + 8/5*e^4 + 1/15*e^3 - 4/15*e^2 - 119/9*e - 400/27, 1/27*e^6 - 1/9*e^5 - 4/3*e^4 + 11/3*e^3 + 19/3*e^2 - 134/9*e - 10/27, 7/135*e^6 - 11/45*e^5 - 77/45*e^4 + 377/45*e^3 + 262/45*e^2 - 38*e - 530/27, 7/135*e^6 - 11/45*e^5 - 77/45*e^4 + 377/45*e^3 + 262/45*e^2 - 38*e - 530/27, 7/135*e^6 - 1/45*e^5 - 29/15*e^4 + 4/15*e^3 + 58/5*e^2 + 17/9*e - 356/27, 7/135*e^6 - 1/45*e^5 - 29/15*e^4 + 4/15*e^3 + 58/5*e^2 + 17/9*e - 356/27, -1/45*e^6 + 14/45*e^5 + 28/45*e^4 - 478/45*e^3 + 37/45*e^2 + 325/9*e - 20/9, -1/45*e^6 + 14/45*e^5 + 28/45*e^4 - 478/45*e^3 + 37/45*e^2 + 325/9*e - 20/9, -1/15*e^6 + 1/9*e^5 + 112/45*e^4 - 163/45*e^3 - 130/9*e^2 + 554/45*e + 236/9, 8/135*e^6 + 4/45*e^5 - 7/3*e^4 - 19/5*e^3 + 244/15*e^2 + 1079/45*e + 218/27, 8/135*e^6 + 4/45*e^5 - 7/3*e^4 - 19/5*e^3 + 244/15*e^2 + 1079/45*e + 218/27, -1/15*e^6 + 1/9*e^5 + 112/45*e^4 - 163/45*e^3 - 130/9*e^2 + 554/45*e + 236/9, -16/135*e^6 + 38/45*e^5 + 161/45*e^4 - 1271/45*e^3 - 166/45*e^2 + 269/3*e + 1250/27, -16/135*e^6 + 38/45*e^5 + 161/45*e^4 - 1271/45*e^3 - 166/45*e^2 + 269/3*e + 1250/27, -17/135*e^6 + 19/45*e^5 + 38/9*e^4 - 631/45*e^3 - 608/45*e^2 + 713/15*e + 598/27, -17/135*e^6 + 19/45*e^5 + 38/9*e^4 - 631/45*e^3 - 608/45*e^2 + 713/15*e + 598/27, -14/135*e^6 + 22/45*e^5 + 154/45*e^4 - 724/45*e^3 - 464/45*e^2 + 157/3*e + 700/27, -14/135*e^6 + 22/45*e^5 + 154/45*e^4 - 724/45*e^3 - 464/45*e^2 + 157/3*e + 700/27, -1/27*e^6 + 2/9*e^5 + 8/9*e^4 - 62/9*e^3 + 53/9*e^2 + 10*e - 200/27, -1/27*e^6 + 2/9*e^5 + 8/9*e^4 - 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3896/45*e - 1562/27, -4/15*e^6 + 73/45*e^5 + 371/45*e^4 - 2396/45*e^3 - 541/45*e^2 + 1397/9*e + 400/9, 1/9*e^6 - 8/9*e^5 - 25/9*e^4 + 268/9*e^3 - 133/9*e^2 - 899/9*e - 140/9, 1/9*e^6 - 8/9*e^5 - 25/9*e^4 + 268/9*e^3 - 133/9*e^2 - 899/9*e - 140/9, -4/15*e^6 + 73/45*e^5 + 371/45*e^4 - 2396/45*e^3 - 541/45*e^2 + 1397/9*e + 400/9, -26/135*e^6 + 23/45*e^5 + 97/15*e^4 - 242/15*e^3 - 99/5*e^2 + 329/9*e - 290/27, -26/135*e^6 + 23/45*e^5 + 97/15*e^4 - 242/15*e^3 - 99/5*e^2 + 329/9*e - 290/27, 26/135*e^6 - 23/45*e^5 - 34/5*e^4 + 79/5*e^3 + 169/5*e^2 - 317/9*e - 1726/27, -1/45*e^6 - 16/45*e^5 + 43/45*e^4 + 557/45*e^3 - 293/45*e^2 - 398/9*e - 92/9, 26/135*e^6 - 23/45*e^5 - 34/5*e^4 + 79/5*e^3 + 169/5*e^2 - 317/9*e - 1726/27, -1/45*e^6 - 16/45*e^5 + 43/45*e^4 + 557/45*e^3 - 293/45*e^2 - 398/9*e - 92/9, 1/135*e^6 + 4/9*e^5 - 1/15*e^4 - 251/15*e^3 - 10*e^2 + 4654/45*e + 2068/27, 1/135*e^6 + 4/9*e^5 - 1/15*e^4 - 251/15*e^3 - 10*e^2 + 4654/45*e + 2068/27, -1/45*e^6 - 1/45*e^5 + 58/45*e^4 + 17/45*e^3 - 938/45*e^2 + 94/9*e + 520/9, -1/45*e^6 - 1/45*e^5 + 58/45*e^4 + 17/45*e^3 - 938/45*e^2 + 94/9*e + 520/9, -1/45*e^6 + 29/45*e^5 + 43/45*e^4 - 1018/45*e^3 - 608/45*e^2 + 835/9*e + 610/9, -1/45*e^6 + 29/45*e^5 + 43/45*e^4 - 1018/45*e^3 - 608/45*e^2 + 835/9*e + 610/9, 7/45*e^6 - 11/9*e^5 - 218/45*e^4 + 1832/45*e^3 + 110/9*e^2 - 5791/45*e - 238/3, 7/45*e^6 - 11/9*e^5 - 218/45*e^4 + 1832/45*e^3 + 110/9*e^2 - 5791/45*e - 238/3, -11/45*e^6 + 82/45*e^5 + 326/45*e^4 - 2732/45*e^3 - 94/45*e^2 + 8419/45*e + 476/9, -11/45*e^6 + 82/45*e^5 + 326/45*e^4 - 2732/45*e^3 - 94/45*e^2 + 8419/45*e + 476/9, 4/9*e^5 - 4/9*e^4 - 134/9*e^3 + 110/9*e^2 + 391/9*e + 20/9, 4/9*e^5 - 4/9*e^4 - 134/9*e^3 + 110/9*e^2 + 391/9*e + 20/9, -31/135*e^6 + 11/15*e^5 + 331/45*e^4 - 1036/45*e^3 - 626/45*e^2 + 491/9*e - 706/27, -31/135*e^6 + 11/15*e^5 + 331/45*e^4 - 1036/45*e^3 - 626/45*e^2 + 491/9*e - 706/27, 32/135*e^6 - 16/45*e^5 - 367/45*e^4 + 457/45*e^3 + 1352/45*e^2 - 37/3*e + 614/27, -23/135*e^6 + 13/15*e^5 + 278/45*e^4 - 1313/45*e^3 - 1618/45*e^2 + 994/9*e + 3064/27, 38/135*e^6 - 41/45*e^5 - 31/3*e^4 + 458/15*e^3 + 303/5*e^2 - 5326/45*e - 2632/27, 38/135*e^6 - 41/45*e^5 - 31/3*e^4 + 458/15*e^3 + 303/5*e^2 - 5326/45*e - 2632/27, 16/135*e^6 - 58/45*e^5 - 14/5*e^4 + 652/15*e^3 - 278/15*e^2 - 1318/9*e - 590/27, 16/135*e^6 - 58/45*e^5 - 14/5*e^4 + 652/15*e^3 - 278/15*e^2 - 1318/9*e - 590/27, 14/135*e^6 - 37/45*e^5 - 109/45*e^4 + 1174/45*e^3 - 901/45*e^2 - 50*e + 1064/27, 14/135*e^6 - 37/45*e^5 - 109/45*e^4 + 1174/45*e^3 - 901/45*e^2 - 50*e + 1064/27, 19/135*e^6 - 14/15*e^5 - 184/45*e^4 + 1384/45*e^3 - 46/45*e^2 - 809/9*e - 1520/27, 19/135*e^6 - 14/15*e^5 - 184/45*e^4 + 1384/45*e^3 - 46/45*e^2 - 809/9*e - 1520/27, 37/135*e^6 - 22/15*e^5 - 382/45*e^4 + 2182/45*e^3 + 437/45*e^2 - 1331/9*e - 1106/27, 37/135*e^6 - 22/15*e^5 - 382/45*e^4 + 2182/45*e^3 + 437/45*e^2 - 1331/9*e - 1106/27, -2/45*e^6 + 13/45*e^5 + 71/45*e^4 - 416/45*e^3 - 331/45*e^2 + 122/9*e + 110/9, -2/45*e^6 + 13/45*e^5 + 71/45*e^4 - 416/45*e^3 - 331/45*e^2 + 122/9*e + 110/9, 7/45*e^6 - 38/45*e^5 - 256/45*e^4 + 1336/45*e^3 + 1526/45*e^2 - 1201/9*e - 880/9, 7/45*e^6 - 38/45*e^5 - 256/45*e^4 + 1336/45*e^3 + 1526/45*e^2 - 1201/9*e - 880/9, 1/135*e^6 - 4/9*e^5 + 7/45*e^4 + 707/45*e^3 - 79/9*e^2 - 1022/15*e - 482/27, -2/45*e^6 + 7/15*e^5 + 8/15*e^4 - 15*e^3 + 391/15*e^2 + 563/15*e - 644/9, -2/45*e^6 + 7/15*e^5 + 8/15*e^4 - 15*e^3 + 391/15*e^2 + 563/15*e - 644/9, 1/135*e^6 - 4/9*e^5 + 7/45*e^4 + 707/45*e^3 - 79/9*e^2 - 1022/15*e - 482/27, 10/27*e^6 - 77/45*e^5 - 58/5*e^4 + 847/15*e^3 + 238/15*e^2 - 7861/45*e - 982/27, 2/15*e^6 - 74/45*e^5 - 193/45*e^4 + 2548/45*e^3 + 788/45*e^2 - 1978/9*e - 1112/9, 2/15*e^6 - 74/45*e^5 - 193/45*e^4 + 2548/45*e^3 + 788/45*e^2 - 1978/9*e - 1112/9, 10/27*e^6 - 77/45*e^5 - 58/5*e^4 + 847/15*e^3 + 238/15*e^2 - 7861/45*e - 982/27, 14/45*e^6 - 22/15*e^5 - 154/15*e^4 + 724/15*e^3 + 494/15*e^2 - 154*e - 610/9, 14/45*e^6 - 22/15*e^5 - 154/15*e^4 + 724/15*e^3 + 494/15*e^2 - 154*e - 610/9, -16/135*e^6 + 28/45*e^5 + 19/5*e^4 - 317/15*e^3 - 142/15*e^2 + 790/9*e + 1400/27, -7/27*e^6 + 14/9*e^5 + 74/9*e^4 - 464/9*e^3 - 175/9*e^2 + 457/3*e + 2290/27, -7/27*e^6 + 14/9*e^5 + 74/9*e^4 - 464/9*e^3 - 175/9*e^2 + 457/3*e + 2290/27, -16/135*e^6 + 28/45*e^5 + 19/5*e^4 - 317/15*e^3 - 142/15*e^2 + 790/9*e + 1400/27] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([4, 2, w + 2])] = 1 AL_eigenvalues[ZF.ideal([4, 2, -1/2*w^3 + 3/2*w^2 + 9/2*w - 10])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]