/* 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, 4, w^3 - 3*w^2 - 8*w + 16]) 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^5 - 8*x^3 - 4*x^2 + 6*x - 1 K. = NumberField(heckePol) hecke_eigenvalues_array = [0, e, -e^4 + 7*e^2 + 5*e - 1, 2*e^4 - 15*e^2 - 8*e + 7, -e^4 + 7*e^2 + 6*e, -2*e^4 - 2*e^3 + 18*e^2 + 21*e - 11, e^4 + e^3 - 10*e^2 - 9*e + 9, e^4 - 8*e^2 - 7*e + 8, e^4 - e^3 - 6*e^2 + e + 5, -e^4 - e^3 + 7*e^2 + 11*e, 4*e^4 - 30*e^2 - 19*e + 11, -3*e^4 + e^3 + 22*e^2 + 10*e - 10, -2*e^4 + 15*e^2 + 8*e - 5, e + 5, -e^3 + e^2 + 5*e - 11, -4*e^4 + 30*e^2 + 22*e - 10, 2*e^4 + 2*e^3 - 16*e^2 - 24*e + 2, 2*e^4 - 16*e^2 - 10*e + 12, 3*e^4 - 22*e^2 - 14*e + 9, -3*e^4 + 22*e^2 + 16*e - 9, 2*e^4 + e^3 - 17*e^2 - 13*e + 11, 3*e^4 - 19*e^2 - 17*e - 3, 3*e^4 + e^3 - 22*e^2 - 17*e + 5, -4*e^4 - e^3 + 30*e^2 + 24*e - 17, -2*e^4 + e^3 + 15*e^2 + 4*e + 2, -5*e^4 + e^3 + 39*e^2 + 16*e - 23, 5*e^4 - 2*e^3 - 37*e^2 - 9*e + 17, 11*e^4 + 2*e^3 - 82*e^2 - 66*e + 37, -e^4 + 6*e^2 + 4*e - 3, -2*e^4 - e^3 + 14*e^2 + 11*e + 2, 5*e^4 + e^3 - 40*e^2 - 21*e + 25, -8*e^4 - 3*e^3 + 65*e^2 + 50*e - 32, 5*e^4 - e^3 - 37*e^2 - 21*e + 22, -5*e^4 + 2*e^3 + 35*e^2 + 13*e - 17, -e^4 - 4*e^3 + 12*e^2 + 33*e - 10, -13*e^4 - 3*e^3 + 98*e^2 + 77*e - 39, -5*e^4 - 9*e^3 + 49*e^2 + 75*e - 28, 4*e^4 - 2*e^3 - 28*e^2 - 8*e - 4, 7*e^4 + 6*e^3 - 60*e^2 - 67*e + 22, 4*e^4 + 3*e^3 - 36*e^2 - 43*e + 34, 2*e^4 - 2*e^3 - 15*e^2 + 7*e + 14, 2*e^4 - 3*e^3 - 15*e^2 + 7*e + 21, -5*e^4 + 2*e^3 + 37*e^2 + 3*e - 21, -3*e^4 + e^3 + 17*e^2 + 11*e + 8, -3*e^4 + 3*e^3 + 23*e^2 - 6*e - 5, 7*e^4 + 4*e^3 - 54*e^2 - 62*e + 27, 8*e^4 - e^3 - 60*e^2 - 24*e + 35, -10*e^4 - 8*e^3 + 86*e^2 + 85*e - 51, e^4 - 2*e^3 - 3*e^2 - e - 9, -5*e^3 + 10*e^2 + 25*e - 20, -7*e^4 + e^3 + 49*e^2 + 27*e - 28, 4*e^4 - 35*e^2 - 11*e + 36, 6*e^4 + 8*e^3 - 58*e^2 - 73*e + 41, e^4 + 3*e^3 - 9*e^2 - 28*e + 5, -2*e^4 + 17*e^2 + 12*e - 3, e^4 + 2*e^3 - 16*e^2 - 7*e + 26, 14*e^4 + e^3 - 108*e^2 - 72*e + 45, 12*e^4 - 2*e^3 - 85*e^2 - 48*e + 33, 7*e^4 - 4*e^3 - 53*e^2 - 7*e + 31, -12*e^4 - 5*e^3 + 100*e^2 + 78*e - 55, -5*e^4 + 7*e^3 + 30*e^2 - 19*e - 23, 4*e^4 - 34*e^2 - 14*e + 22, 7*e^4 + 7*e^3 - 59*e^2 - 79*e + 18, 10*e^4 + 2*e^3 - 76*e^2 - 60*e + 30, -7*e^4 - e^3 + 51*e^2 + 49*e - 16, -3*e^4 + 4*e^3 + 16*e^2 - 11*e + 4, 2*e^4 + 2*e^3 - 20*e^2 - 8*e + 18, 3*e^4 + 4*e^3 - 24*e^2 - 34*e + 9, 2*e^4 + 7*e^3 - 23*e^2 - 48*e + 18, 7*e^4 - 2*e^3 - 52*e^2 - 23*e + 26, -4*e^4 - 5*e^3 + 34*e^2 + 48*e - 15, -9*e^4 - e^3 + 66*e^2 + 60*e - 16, 12*e^4 - 2*e^3 - 87*e^2 - 52*e + 39, 6*e^4 + 5*e^3 - 51*e^2 - 55*e + 47, 7*e^4 + 4*e^3 - 59*e^2 - 45*e + 31, e^4 - 10*e^2 - 11*e + 12, -9*e^4 - 5*e^3 + 74*e^2 + 65*e - 55, 8*e^4 + 6*e^3 - 70*e^2 - 72*e + 48, 17*e^4 - e^3 - 126*e^2 - 72*e + 62, 8*e^4 + 7*e^3 - 66*e^2 - 75*e + 18, -4*e^4 - 3*e^3 + 35*e^2 + 26*e - 12, 15*e^4 - 109*e^2 - 74*e + 42, -e^4 - 7*e^3 + 9*e^2 + 55*e + 4, -4*e^4 - 9*e^3 + 38*e^2 + 66*e - 7, 5*e^4 + 4*e^3 - 41*e^2 - 45*e + 17, -14*e^4 + 99*e^2 + 69*e - 28, 12*e^4 - 3*e^3 - 90*e^2 - 32*e + 45, -e^4 - 9*e^3 + 16*e^2 + 60*e - 8, 5*e^4 + 2*e^3 - 46*e^2 - 31*e + 36, 5*e^4 - 7*e^3 - 24*e^2 + 15*e - 5, -3*e^4 - 4*e^3 + 20*e^2 + 41*e + 6, -2*e^4 - 3*e^3 + 25*e^2 + 21*e - 35, -3*e^4 + 2*e^3 + 24*e^2 + 8*e - 31, 13*e^4 + 7*e^3 - 101*e^2 - 102*e + 51, 17*e^4 - 122*e^2 - 80*e + 53, -4*e^4 + 2*e^3 + 26*e^2 + 18*e + 12, 17*e^4 + 8*e^3 - 137*e^2 - 127*e + 67, 3*e^4 - e^3 - 20*e^2 - 18*e + 4, -3*e^4 - 5*e^3 + 25*e^2 + 52*e - 1, -3*e^4 + e^3 + 19*e^2 + 18*e + 5, 9*e^4 + e^3 - 62*e^2 - 57*e + 3, -12*e^4 + 93*e^2 + 53*e - 60, -5*e^4 + 4*e^3 + 39*e^2 - 10*e - 28, -3*e^4 - 14*e^3 + 36*e^2 + 92*e - 15, -e^4 + 3*e^3 + 2*e^2 - 16*e + 12, -10*e^4 + 5*e^3 + 64*e^2 + 14*e - 13, -10*e^4 + 3*e^3 + 67*e^2 + 27*e - 29, 4*e^4 + 10*e^3 - 33*e^2 - 81*e + 2, -4*e^4 - e^3 + 34*e^2 + 23*e + 4, 4*e^4 + 5*e^3 - 29*e^2 - 44*e - 12, 3*e^4 + 4*e^3 - 20*e^2 - 46*e - 5, 2*e^4 - 7*e^3 - 14*e^2 + 41*e + 10, 10*e^4 - 76*e^2 - 50*e + 36, -14*e^4 - 4*e^3 + 112*e^2 + 84*e - 42, 4*e^4 + 7*e^3 - 42*e^2 - 60*e + 43, 3*e^4 + 8*e^3 - 35*e^2 - 64*e + 34, -8*e^4 + 4*e^3 + 61*e^2 + 12*e - 27, 15*e^4 + 3*e^3 - 115*e^2 - 88*e + 65, 6*e^4 - 47*e^2 - 33*e + 20, 11*e^4 + 8*e^3 - 94*e^2 - 109*e + 64, -6*e^3 + e^2 + 42*e + 21, -11*e^4 - 10*e^3 + 98*e^2 + 106*e - 53, -3*e^4 - 2*e^3 + 25*e^2 + 18*e - 4, 2*e^4 - 3*e^3 - 19*e^2 + 12*e + 36, 5*e^4 - 4*e^3 - 36*e^2 - 6*e + 37, 6*e^4 - 4*e^3 - 44*e^2 - 15*e + 41, -11*e^4 + e^3 + 85*e^2 + 34*e - 41, 10*e^4 + e^3 - 70*e^2 - 52*e + 11, -16*e^4 - e^3 + 124*e^2 + 78*e - 45, 8*e^4 + 9*e^3 - 64*e^2 - 101*e + 30, -27*e^4 - 3*e^3 + 204*e^2 + 148*e - 90, 9*e^4 + 10*e^3 - 79*e^2 - 102*e + 46, -13*e^4 + 5*e^3 + 95*e^2 + 33*e - 32, -7*e^4 - 3*e^3 + 58*e^2 + 32*e - 34, -3*e^3 + 6*e^2 + 18*e - 11, -e^4 + 9*e^3 + 3*e^2 - 50*e - 11, -25*e^4 - 5*e^3 + 189*e^2 + 148*e - 89, 17*e^4 + 11*e^3 - 139*e^2 - 138*e + 65, 13*e^4 - e^3 - 96*e^2 - 50*e + 50, 15*e^4 + 2*e^3 - 111*e^2 - 85*e + 35, -10*e^4 - 3*e^3 + 85*e^2 + 64*e - 64, 8*e^4 - 4*e^3 - 59*e^2 - 8*e + 21, 4*e^4 - e^3 - 23*e^2 - 22*e - 22, -e^4 - 2*e^3 + 9*e^2 + 3*e + 1, 21*e^4 + 3*e^3 - 156*e^2 - 121*e + 53, -7*e^4 - 9*e^3 + 64*e^2 + 87*e - 39, -4*e^4 + 32*e^2 + e - 29, 7*e^4 + e^3 - 53*e^2 - 44*e + 37, -6*e^4 - 5*e^3 + 51*e^2 + 50*e - 2, -3*e^4 - e^3 + 30*e^2 + 5*e - 45, 24*e^4 + e^3 - 182*e^2 - 115*e + 88, 16*e^4 - e^3 - 123*e^2 - 73*e + 49, -20*e^4 + 7*e^3 + 143*e^2 + 55*e - 59, 5*e^4 + 5*e^3 - 40*e^2 - 55*e + 23, 10*e^4 - 8*e^3 - 63*e^2 - 9*e - 2, 11*e^4 + e^3 - 75*e^2 - 61*e + 10, -14*e^4 - 10*e^3 + 108*e^2 + 134*e - 50, -30*e^4 - 7*e^3 + 237*e^2 + 180*e - 104, -5*e^4 + 7*e^3 + 37*e^2 - 20*e - 41, -12*e^4 + e^3 + 85*e^2 + 41*e - 29, -3*e^4 - 8*e^3 + 28*e^2 + 56*e + 19, -28*e^4 - 12*e^3 + 220*e^2 + 198*e - 94, -8*e^4 + 3*e^3 + 57*e^2 + 19*e - 17, -18*e^4 + 136*e^2 + 73*e - 59, -e^4 - 8*e^3 + 13*e^2 + 47*e - 3, -5*e^4 + 2*e^3 + 39*e^2 + 13*e - 19, -10*e^4 + 2*e^3 + 71*e^2 + 47*e - 14, -3*e^4 + 3*e^3 + 14*e^2 - 8*e + 42, -8*e^4 + 6*e^3 + 62*e^2 + 6*e - 60, 6*e^4 + 8*e^3 - 57*e^2 - 70*e + 23, -6*e^4 + 7*e^3 + 35*e^2 - 2*e + 14, 18*e^4 - 10*e^3 - 124*e^2 - 29*e + 67, 16*e^4 + 14*e^3 - 136*e^2 - 149*e + 61, 18*e^4 + 18*e^3 - 157*e^2 - 190*e + 75, -23*e^4 - 2*e^3 + 175*e^2 + 131*e - 83, 5*e^4 - 5*e^3 - 31*e^2 - e - 16, -e^4 - 4*e^3 + 19*e^2 + 19*e - 51, 18*e^4 + 10*e^3 - 145*e^2 - 137*e + 80, -9*e^4 - 8*e^3 + 79*e^2 + 85*e - 47, 14*e^4 - 3*e^3 - 101*e^2 - 61*e + 41, -20*e^4 - 11*e^3 + 160*e^2 + 142*e - 61, 24*e^4 + 10*e^3 - 199*e^2 - 165*e + 96, -19*e^4 + 3*e^3 + 137*e^2 + 80*e - 49, 16*e^4 - e^3 - 123*e^2 - 49*e + 53, 18*e^4 - 130*e^2 - 85*e + 27, -30*e^4 - 15*e^3 + 244*e^2 + 234*e - 119, -12*e^4 - 3*e^3 + 85*e^2 + 82*e - 20, 10*e^4 - 9*e^3 - 64*e^2 + e + 32, -e^4 + 12*e^3 - e^2 - 55*e - 9, 4*e^4 + 10*e^3 - 31*e^2 - 80*e + 1, -10*e^4 - 8*e^3 + 82*e^2 + 94*e - 62, -19*e^4 - 4*e^3 + 143*e^2 + 99*e - 45, -4*e^4 - 7*e^3 + 41*e^2 + 52*e - 34, -8*e^4 - 5*e^3 + 68*e^2 + 59*e - 72, -e^4 - e^3 + 6*e^2 + 11*e - 33, 9*e^4 - 4*e^3 - 64*e^2 - 30*e + 11, 4*e^4 + 4*e^3 - 33*e^2 - 24*e + 5, -27*e^4 - 14*e^3 + 219*e^2 + 205*e - 97, -27*e^4 - 8*e^3 + 205*e^2 + 174*e - 98, -19*e^4 - 5*e^3 + 153*e^2 + 126*e - 93, 5*e^3 - 14*e^2 - 29*e + 16, -13*e^4 - 2*e^3 + 106*e^2 + 53*e - 70, -2*e^4 - 14*e^3 + 37*e^2 + 93*e - 34, -2*e^4 - 6*e^3 + 14*e^2 + 58*e + 4, -15*e^4 - 11*e^3 + 119*e^2 + 122*e - 41, -9*e^4 + 2*e^3 + 63*e^2 + 48*e + 4, -16*e^4 - 5*e^3 + 121*e^2 + 109*e - 51, 7*e^4 - 5*e^3 - 45*e^2 - 16*e + 9, 6*e^4 + 3*e^3 - 51*e^2 - 52*e + 42, -12*e^4 - 7*e^3 + 101*e^2 + 107*e - 41, 16*e^4 + e^3 - 119*e^2 - 72*e + 54, -4*e^4 - 4*e^3 + 38*e^2 + 45*e - 51, -e^4 - 10*e^3 + 20*e^2 + 55*e - 14, 15*e^4 + 23*e^3 - 139*e^2 - 189*e + 82, 4*e^4 + 7*e^3 - 42*e^2 - 68*e + 39, -26*e^4 + 5*e^3 + 186*e^2 + 112*e - 43, -7*e^4 - 6*e^3 + 54*e^2 + 77*e - 40, -15*e^4 - 8*e^3 + 124*e^2 + 125*e - 70, -5*e^4 - 8*e^3 + 48*e^2 + 83*e - 14, 19*e^4 - 2*e^3 - 133*e^2 - 77*e + 63, 14*e^4 + 13*e^3 - 116*e^2 - 156*e + 41, -25*e^4 - 17*e^3 + 206*e^2 + 205*e - 87, 3*e^4 - 22*e^2 - 39*e + 18, -17*e^4 - 14*e^3 + 134*e^2 + 167*e - 58, -24*e^4 + 8*e^3 + 166*e^2 + 75*e - 41, 6*e^4 + 18*e^3 - 68*e^2 - 132*e + 48, 13*e^4 + e^3 - 97*e^2 - 65*e + 38, 40*e^4 + 10*e^3 - 310*e^2 - 251*e + 137, -20*e^4 - 9*e^3 + 152*e^2 + 155*e - 68, 8*e^4 + 2*e^3 - 51*e^2 - 67*e - 24, 8*e^4 - 54*e^2 - 45*e + 7, 9*e^4 + 7*e^3 - 84*e^2 - 91*e + 69, -21*e^4 + 153*e^2 + 99*e - 65, -26*e^4 - 7*e^3 + 199*e^2 + 162*e - 60, -8*e^4 - 14*e^3 + 75*e^2 + 119*e - 34, 7*e^4 - 7*e^3 - 46*e^2 + 4*e + 6, 25*e^4 - 3*e^3 - 180*e^2 - 114*e + 50, -23*e^4 + e^3 + 169*e^2 + 97*e - 70, -9*e^3 + 10*e^2 + 48*e - 23, e^4 + 4*e^3 - 13*e^2 - 21*e + 61, -31*e^4 - 13*e^3 + 246*e^2 + 240*e - 108, -3*e^4 - 8*e^3 + 30*e^2 + 71*e - 32, 2*e^4 - 11*e^3 - 5*e^2 + 48*e - 22, -16*e^4 + 4*e^3 + 115*e^2 + 54*e - 79, -6*e^4 + 11*e^3 + 40*e^2 - 34*e - 35, 13*e^4 + 16*e^3 - 124*e^2 - 157*e + 96, -10*e^4 + 4*e^3 + 72*e^2 + 31*e - 11, -14*e^4 - 6*e^3 + 123*e^2 + 96*e - 81, -4*e^4 + 11*e^3 + 20*e^2 - 50*e - 9, -6*e^4 + 6*e^3 + 43*e^2 - 5*e - 42, 15*e^4 + 12*e^3 - 120*e^2 - 143*e + 62, 18*e^4 + 6*e^3 - 143*e^2 - 112*e + 63, -7*e^4 + 46*e^2 + 43*e + 32, 2*e^4 - 10*e^3 - e^2 + 46*e - 13, -13*e^4 + 12*e^3 + 98*e^2 - 19*e - 72, -32*e^4 + e^3 + 241*e^2 + 136*e - 100, 20*e^4 + 19*e^3 - 172*e^2 - 209*e + 70, -20*e^4 - 4*e^3 + 149*e^2 + 137*e - 48, -15*e^4 + 3*e^3 + 119*e^2 + 56*e - 81, -11*e^4 + 6*e^3 + 73*e^2 + 24*e - 28, -15*e^4 - 2*e^3 + 117*e^2 + 84*e - 28, 8*e^4 + 3*e^3 - 67*e^2 - 63*e + 47, 7*e^4 - 3*e^3 - 50*e^2 - 27*e + 61, 19*e^4 + 13*e^3 - 148*e^2 - 173*e + 61, -11*e^4 + 12*e^3 + 74*e^2 - 11*e - 40, 23*e^4 + 9*e^3 - 188*e^2 - 142*e + 106, -3*e^4 + 6*e^3 + 27*e^2 - 34*e - 30, -19*e^4 + 6*e^3 + 128*e^2 + 59*e - 22, -10*e^4 - e^3 + 79*e^2 + 31*e - 31, -e^4 + 12*e^3 - 14*e^2 - 44*e + 67, 17*e^4 + 2*e^3 - 129*e^2 - 81*e + 33, -32*e^4 - 18*e^3 + 260*e^2 + 246*e - 140, 17*e^4 + 7*e^3 - 139*e^2 - 111*e + 40, 27*e^4 + 26*e^3 - 232*e^2 - 265*e + 120, -2*e^4 + 4*e^3 - 2*e^2 + 2*e + 36, e^4 + e^3 - 12*e^2 - 32*e + 44, -18*e^4 - 9*e^3 + 159*e^2 + 120*e - 106, 30*e^4 + 15*e^3 - 240*e^2 - 237*e + 128, 3*e^4 + 14*e^3 - 39*e^2 - 88*e + 24, 31*e^4 + 9*e^3 - 246*e^2 - 206*e + 116, -29*e^4 + 4*e^3 + 213*e^2 + 111*e - 89, 10*e^4 - 10*e^3 - 62*e^2 + 19*e + 1, -10*e^4 - 14*e^3 + 101*e^2 + 117*e - 72, -4*e^4 + 17*e^3 + 17*e^2 - 84*e - 46, 12*e^4 + 19*e^3 - 124*e^2 - 156*e + 99, 3*e^4 + 13*e^3 - 40*e^2 - 90*e + 42, -7*e^4 - e^3 + 40*e^2 + 34*e + 30, 23*e^4 + 12*e^3 - 178*e^2 - 183*e + 86, e^4 + 10*e^3 - 31*e^2 - 62*e + 76, 15*e^4 - 2*e^3 - 122*e^2 - 36*e + 97, -26*e^4 - 7*e^3 + 198*e^2 + 173*e - 92, 26*e^4 - 9*e^3 - 188*e^2 - 63*e + 102, 18*e^4 + e^3 - 133*e^2 - 78*e + 18, 11*e^4 - 9*e^3 - 83*e^2 - e + 58, 15*e^4 + 5*e^3 - 120*e^2 - 82*e + 26, -9*e^4 + 5*e^3 + 52*e^2 + 12*e + 12, -12*e^4 - 6*e^3 + 84*e^2 + 101*e + 7, 3*e^4 - 6*e^3 - 25*e^2 + 45*e + 37, 25*e^4 + 15*e^3 - 199*e^2 - 208*e + 109, 3*e^4 + 10*e^3 - 39*e^2 - 76*e + 54] 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 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]