/* 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([20, 10, -11, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([19,19,w - 1]) primes_array = [ [4, 2, -w + 2],\ [4, 2, 1/2*w^3 + 1/2*w^2 - 9/2*w - 4],\ [5, 5, -1/2*w^3 - 3/2*w^2 + 7/2*w + 10],\ [5, 5, -1/2*w^3 - 1/2*w^2 + 5/2*w + 3],\ [19, 19, 1/2*w^3 + 1/2*w^2 - 9/2*w - 3],\ [19, 19, w - 1],\ [31, 31, 1/2*w^3 + 3/2*w^2 - 7/2*w - 6],\ [31, 31, -w^3 - 2*w^2 + 7*w + 13],\ [49, 7, -2*w^3 - 2*w^2 + 15*w + 9],\ [49, 7, 1/2*w^3 + 3/2*w^2 - 9/2*w - 4],\ [59, 59, -3/2*w^3 - 3/2*w^2 + 17/2*w + 8],\ [59, 59, -1/2*w^3 - 5/2*w^2 + 11/2*w + 9],\ [61, 61, -1/2*w^3 - 3/2*w^2 + 9/2*w + 9],\ [61, 61, -3/2*w^3 - 5/2*w^2 + 23/2*w + 12],\ [71, 71, -3/2*w^3 - 1/2*w^2 + 23/2*w - 1],\ [71, 71, 3/2*w^3 + 3/2*w^2 - 19/2*w - 9],\ [79, 79, -1/2*w^3 - 3/2*w^2 + 11/2*w + 4],\ [79, 79, -5/2*w^3 - 7/2*w^2 + 37/2*w + 18],\ [81, 3, -3],\ [89, 89, w^3 - 7*w + 1],\ [89, 89, -2*w^3 - 3*w^2 + 14*w + 17],\ [89, 89, 1/2*w^3 - 1/2*w^2 - 7/2*w + 2],\ [89, 89, -5/2*w^3 - 7/2*w^2 + 35/2*w + 18],\ [101, 101, -2*w^2 + 11],\ [101, 101, 3*w^3 + 5*w^2 - 21*w - 27],\ [121, 11, -3/2*w^3 - 3/2*w^2 + 21/2*w + 7],\ [121, 11, 3/2*w^3 + 3/2*w^2 - 21/2*w - 8],\ [131, 131, -3/2*w^3 - 5/2*w^2 + 21/2*w + 16],\ [131, 131, -3/2*w^3 - 5/2*w^2 + 19/2*w + 14],\ [139, 139, -2*w^3 - w^2 + 16*w + 1],\ [139, 139, 1/2*w^3 + 1/2*w^2 - 3/2*w - 8],\ [149, 149, -3/2*w^3 - 7/2*w^2 + 21/2*w + 24],\ [149, 149, -w^3 + w^2 + 5*w - 3],\ [149, 149, -5*w^3 - 7*w^2 + 37*w + 39],\ [149, 149, -3/2*w^3 - 7/2*w^2 + 21/2*w + 14],\ [181, 181, -5/2*w^3 - 9/2*w^2 + 37/2*w + 27],\ [181, 181, -5/2*w^3 - 7/2*w^2 + 33/2*w + 19],\ [181, 181, 3/2*w^3 + 1/2*w^2 - 23/2*w - 2],\ [181, 181, w^3 + 3*w^2 - 8*w - 13],\ [191, 191, 1/2*w^3 + 5/2*w^2 - 7/2*w - 14],\ [191, 191, -5/2*w^3 - 9/2*w^2 + 35/2*w + 24],\ [199, 199, -1/2*w^3 - 5/2*w^2 + 7/2*w + 16],\ [199, 199, 5/2*w^3 + 9/2*w^2 - 35/2*w - 22],\ [211, 211, -2*w^3 - 4*w^2 + 14*w + 19],\ [211, 211, -w^3 - 3*w^2 + 7*w + 19],\ [229, 229, 3/2*w^3 + 7/2*w^2 - 21/2*w - 17],\ [229, 229, -3/2*w^3 - 7/2*w^2 + 21/2*w + 21],\ [229, 229, -1/2*w^3 - 3/2*w^2 + 13/2*w + 1],\ [229, 229, 5/2*w^3 + 7/2*w^2 - 41/2*w - 24],\ [241, 241, -1/2*w^3 - 5/2*w^2 + 7/2*w + 12],\ [241, 241, 3/2*w^3 + 3/2*w^2 - 23/2*w - 3],\ [241, 241, w^3 + w^2 - 6*w - 1],\ [241, 241, -5/2*w^3 - 9/2*w^2 + 35/2*w + 26],\ [251, 251, 2*w^3 + 2*w^2 - 13*w - 9],\ [251, 251, -5/2*w^3 - 5/2*w^2 + 37/2*w + 11],\ [269, 269, -2*w^3 - 3*w^2 + 12*w + 17],\ [269, 269, 5/2*w^3 + 11/2*w^2 - 37/2*w - 21],\ [271, 271, 3/2*w^3 + 5/2*w^2 - 23/2*w - 9],\ [271, 271, 1/2*w^3 + 3/2*w^2 - 9/2*w - 12],\ [281, 281, 5/2*w^3 + 9/2*w^2 - 39/2*w - 28],\ [281, 281, 3/2*w^3 + 7/2*w^2 - 25/2*w - 14],\ [311, 311, 1/2*w^3 + 7/2*w^2 - 5/2*w - 24],\ [311, 311, 3*w^3 + 5*w^2 - 22*w - 27],\ [311, 311, 3/2*w^3 + 5/2*w^2 - 17/2*w - 12],\ [311, 311, 1/2*w^3 + 5/2*w^2 - 9/2*w - 13],\ [349, 349, 3*w^3 + 4*w^2 - 21*w - 21],\ [349, 349, -3/2*w^3 - 1/2*w^2 + 21/2*w + 2],\ [359, 359, -w^3 - 3*w^2 + 7*w + 17],\ [359, 359, -2*w^3 - 4*w^2 + 14*w + 21],\ [361, 19, -2*w^3 - 2*w^2 + 14*w + 11],\ [389, 389, -2*w^3 - 2*w^2 + 13*w + 11],\ [389, 389, 5/2*w^3 + 5/2*w^2 - 37/2*w - 13],\ [401, 401, 3/2*w^3 + 7/2*w^2 - 21/2*w - 19],\ [401, 401, 5/2*w^3 + 9/2*w^2 - 37/2*w - 23],\ [401, 401, -w^3 - 3*w^2 + 8*w + 17],\ [431, 431, 5/2*w^3 + 5/2*w^2 - 29/2*w - 13],\ [431, 431, -15/2*w^3 - 23/2*w^2 + 107/2*w + 61],\ [439, 439, -3/2*w^3 + 1/2*w^2 + 21/2*w - 6],\ [439, 439, -9/2*w^3 - 13/2*w^2 + 63/2*w + 32],\ [449, 449, -w^2 - 2*w + 1],\ [449, 449, -1/2*w^3 - 3/2*w^2 + 3/2*w + 14],\ [461, 461, -2*w^3 - 2*w^2 + 12*w + 9],\ [461, 461, -3*w^3 - 3*w^2 + 23*w + 13],\ [479, 479, 7/2*w^3 + 9/2*w^2 - 47/2*w - 26],\ [479, 479, 3/2*w^3 + 5/2*w^2 - 27/2*w - 16],\ [479, 479, -7/2*w^3 - 9/2*w^2 + 51/2*w + 24],\ [479, 479, 3/2*w^3 + 1/2*w^2 - 19/2*w - 3],\ [491, 491, 2*w^3 + 4*w^2 - 15*w - 23],\ [491, 491, -7/2*w^3 - 7/2*w^2 + 53/2*w + 17],\ [491, 491, -1/2*w^3 - 5/2*w^2 + 11/2*w + 6],\ [491, 491, -3/2*w^3 - 7/2*w^2 + 23/2*w + 17],\ [499, 499, 3*w^2 - 19],\ [499, 499, 3/2*w^3 + 5/2*w^2 - 19/2*w - 16],\ [499, 499, -9/2*w^3 - 15/2*w^2 + 63/2*w + 38],\ [499, 499, -1/2*w^3 + 1/2*w^2 + 9/2*w - 1],\ [509, 509, 2*w^2 - w - 7],\ [509, 509, 7/2*w^3 + 11/2*w^2 - 51/2*w - 33],\ [521, 521, 4*w^3 + 5*w^2 - 32*w - 31],\ [521, 521, -1/2*w^3 + 1/2*w^2 - 1/2*w + 4],\ [541, 541, 1/2*w^3 - 7/2*w^2 - 3/2*w + 18],\ [541, 541, -15/2*w^3 - 23/2*w^2 + 109/2*w + 62],\ [541, 541, -w^3 - 3*w^2 + 7*w + 11],\ [541, 541, 2*w^3 + 4*w^2 - 14*w - 27],\ [569, 569, 1/2*w^3 + 5/2*w^2 - 7/2*w - 21],\ [569, 569, -5/2*w^3 - 5/2*w^2 + 31/2*w + 14],\ [571, 571, 2*w^2 + 2*w - 11],\ [571, 571, 2*w^3 + 4*w^2 - 12*w - 23],\ [601, 601, w^3 + 3*w^2 - 8*w - 9],\ [601, 601, 3*w^3 + 2*w^2 - 23*w - 7],\ [601, 601, -3/2*w^3 - 7/2*w^2 + 23/2*w + 19],\ [601, 601, -2*w^3 - 4*w^2 + 15*w + 21],\ [631, 631, -5/2*w^3 - 3/2*w^2 + 33/2*w + 3],\ [631, 631, -13/2*w^3 - 19/2*w^2 + 101/2*w + 58],\ [641, 641, 13/2*w^3 + 19/2*w^2 - 97/2*w - 52],\ [641, 641, -3/2*w^3 - 9/2*w^2 + 23/2*w + 16],\ [659, 659, 1/2*w^3 - 1/2*w^2 - 1/2*w + 2],\ [659, 659, 3/2*w^3 + 7/2*w^2 - 15/2*w - 11],\ [661, 661, -3/2*w^3 - 3/2*w^2 + 25/2*w + 3],\ [661, 661, -1/2*w^3 - 1/2*w^2 + 3/2*w - 1],\ [691, 691, -2*w^3 - 7*w^2 + 14*w + 49],\ [691, 691, -1/2*w^3 - 7/2*w^2 + 13/2*w + 14],\ [691, 691, -13/2*w^3 - 19/2*w^2 + 101/2*w + 59],\ [691, 691, -7/2*w^3 - 13/2*w^2 + 49/2*w + 36],\ [701, 701, 3/2*w^3 + 11/2*w^2 - 21/2*w - 38],\ [701, 701, -2*w^3 - 2*w^2 + 10*w + 11],\ [709, 709, 1/2*w^3 + 5/2*w^2 - 9/2*w - 17],\ [709, 709, -3*w^3 - 3*w^2 + 22*w + 13],\ [709, 709, -3*w^3 - 5*w^2 + 22*w + 23],\ [709, 709, -5/2*w^3 - 5/2*w^2 + 33/2*w + 11],\ [751, 751, -3/2*w^3 + 3/2*w^2 + 15/2*w - 4],\ [751, 751, 15/2*w^3 + 21/2*w^2 - 111/2*w - 59],\ [751, 751, 1/2*w^3 - 1/2*w^2 - 9/2*w - 2],\ [751, 751, -3/2*w^3 - 5/2*w^2 + 19/2*w + 19],\ [769, 769, -9/2*w^3 - 15/2*w^2 + 65/2*w + 38],\ [769, 769, -1/2*w^3 - 7/2*w^2 + 9/2*w + 21],\ [809, 809, -7/2*w^3 - 13/2*w^2 + 51/2*w + 42],\ [809, 809, -5/2*w^3 - 3/2*w^2 + 27/2*w + 14],\ [809, 809, -3*w^3 - 5*w^2 + 21*w + 29],\ [809, 809, 2*w^2 - 9],\ [811, 811, 3*w^3 + 5*w^2 - 20*w - 27],\ [811, 811, 1/2*w^3 - 7/2*w^2 - 9/2*w + 31],\ [811, 811, w^3 - w^2 - 2*w + 1],\ [811, 811, 3/2*w^3 + 5/2*w^2 - 19/2*w - 18],\ [821, 821, 9/2*w^3 + 13/2*w^2 - 67/2*w - 34],\ [821, 821, 3/2*w^3 + 3/2*w^2 - 25/2*w - 1],\ [829, 829, 1/2*w^3 + 5/2*w^2 - 1/2*w - 21],\ [829, 829, 4*w^3 + 2*w^2 - 32*w - 3],\ [839, 839, -1/2*w^3 - 1/2*w^2 + 3/2*w - 2],\ [839, 839, -3/2*w^3 - 3/2*w^2 + 25/2*w + 2],\ [841, 29, -5/2*w^3 - 5/2*w^2 + 35/2*w + 11],\ [841, 29, 5/2*w^3 + 5/2*w^2 - 35/2*w - 14],\ [881, 881, -9/2*w^3 - 13/2*w^2 + 63/2*w + 36],\ [881, 881, 1/2*w^3 + 5/2*w^2 - 3/2*w - 21],\ [911, 911, -1/2*w^3 + 3/2*w^2 + 7/2*w - 7],\ [911, 911, 7/2*w^3 + 13/2*w^2 - 49/2*w - 38],\ [911, 911, 7*w^3 + 11*w^2 - 51*w - 61],\ [911, 911, -7/2*w^3 - 11/2*w^2 + 49/2*w + 31],\ [919, 919, -1/2*w^3 + 7/2*w^2 + 3/2*w - 16],\ [919, 919, 15/2*w^3 + 23/2*w^2 - 109/2*w - 64],\ [929, 929, -3/2*w^3 - 7/2*w^2 + 19/2*w + 23],\ [929, 929, -5/2*w^3 - 7/2*w^2 + 33/2*w + 21],\ [941, 941, -1/2*w^3 - 7/2*w^2 + 11/2*w + 18],\ [941, 941, -5*w^3 - 8*w^2 + 37*w + 43],\ [961, 31, 5/2*w^3 + 5/2*w^2 - 35/2*w - 13],\ [971, 971, 2*w^3 + w^2 - 14*w - 3],\ [971, 971, -5/2*w^3 - 9/2*w^2 + 35/2*w + 31],\ [971, 971, 7/2*w^3 + 9/2*w^2 - 49/2*w - 22],\ [971, 971, 3/2*w^3 + 1/2*w^2 - 27/2*w + 6],\ [991, 991, 5/2*w^3 + 7/2*w^2 - 39/2*w - 16],\ [991, 991, 5/2*w^3 + 7/2*w^2 - 37/2*w - 14],\ [1009, 1009, 1/2*w^3 + 7/2*w^2 - 9/2*w - 18],\ [1009, 1009, 9/2*w^3 + 15/2*w^2 - 65/2*w - 41],\ [1019, 1019, 2*w^3 + 5*w^2 - 14*w - 21],\ [1019, 1019, 5/2*w^3 + 11/2*w^2 - 35/2*w - 36],\ [1021, 1021, 1/2*w^3 + 3/2*w^2 - 7/2*w - 2],\ [1021, 1021, w^3 + 2*w^2 - 7*w - 17],\ [1039, 1039, -3/2*w^3 - 5/2*w^2 + 27/2*w + 8],\ [1039, 1039, -7/2*w^3 - 13/2*w^2 + 51/2*w + 31],\ [1039, 1039, -3*w^3 - 4*w^2 + 25*w + 19],\ [1039, 1039, -4*w^3 - 6*w^2 + 29*w + 29],\ [1051, 1051, 3/2*w^3 + 7/2*w^2 - 27/2*w - 13],\ [1051, 1051, -7/2*w^3 - 9/2*w^2 + 53/2*w + 22],\ [1061, 1061, 1/2*w^3 + 3/2*w^2 - 1/2*w - 17],\ [1061, 1061, -3*w^3 - w^2 + 24*w - 3],\ [1069, 1069, -7/2*w^3 - 9/2*w^2 + 47/2*w + 23],\ [1069, 1069, 2*w^3 + 3*w^2 - 14*w - 23],\ [1129, 1129, 3*w^3 + 4*w^2 - 23*w - 19],\ [1129, 1129, 5/2*w^3 + 7/2*w^2 - 37/2*w - 13],\ [1151, 1151, w^3 + w^2 - 4*w - 13],\ [1151, 1151, -5*w^3 - 8*w^2 + 35*w + 43],\ [1229, 1229, -3*w^3 - 4*w^2 + 19*w + 23],\ [1229, 1229, -3/2*w^3 - 9/2*w^2 + 19/2*w + 31],\ [1249, 1249, 3*w^3 + 6*w^2 - 21*w - 23],\ [1249, 1249, 7/2*w^3 + 1/2*w^2 - 55/2*w + 7],\ [1279, 1279, 2*w^3 + 2*w^2 - 17*w - 9],\ [1279, 1279, -3*w^3 - 6*w^2 + 21*w + 29],\ [1319, 1319, 3/2*w^3 + 9/2*w^2 - 19/2*w - 22],\ [1319, 1319, -w^3 + w^2 + 9],\ [1319, 1319, 5/2*w^3 + 11/2*w^2 - 33/2*w - 33],\ [1319, 1319, 9*w^3 + 14*w^2 - 65*w - 77],\ [1321, 1321, 3*w^3 + 3*w^2 - 16*w - 19],\ [1321, 1321, -3*w^3 - 8*w^2 + 21*w + 53],\ [1361, 1361, -11*w^3 - 17*w^2 + 79*w + 89],\ [1361, 1361, -9/2*w^3 - 11/2*w^2 + 59/2*w + 24],\ [1409, 1409, -3/2*w^3 - 1/2*w^2 + 15/2*w + 3],\ [1409, 1409, -5*w^3 - 6*w^2 + 39*w + 31],\ [1409, 1409, 7/2*w^3 + 13/2*w^2 - 51/2*w - 27],\ [1409, 1409, 1/2*w^3 + 5/2*w^2 - 9/2*w - 21],\ [1429, 1429, -1/2*w^3 - 1/2*w^2 + 13/2*w - 1],\ [1429, 1429, 7/2*w^3 + 13/2*w^2 - 49/2*w - 34],\ [1429, 1429, 1/2*w^3 - 5/2*w^2 - 5/2*w + 17],\ [1429, 1429, 11/2*w^3 + 17/2*w^2 - 79/2*w - 42],\ [1439, 1439, -17/2*w^3 - 25/2*w^2 + 125/2*w + 71],\ [1439, 1439, 9/2*w^3 + 17/2*w^2 - 65/2*w - 51],\ [1451, 1451, 3/2*w^3 - 3/2*w^2 - 15/2*w + 7],\ [1451, 1451, -7/2*w^3 - 5/2*w^2 + 47/2*w + 7],\ [1459, 1459, 3/2*w^3 + 11/2*w^2 - 25/2*w - 18],\ [1459, 1459, -7/2*w^3 - 9/2*w^2 + 55/2*w + 22],\ [1459, 1459, 3*w^3 + 5*w^2 - 22*w - 21],\ [1459, 1459, w^3 + 7*w^2 - 5*w - 53],\ [1471, 1471, -1/2*w^3 - 9/2*w^2 + 7/2*w + 37],\ [1471, 1471, -21/2*w^3 - 31/2*w^2 + 151/2*w + 82],\ [1481, 1481, -11/2*w^3 - 15/2*w^2 + 81/2*w + 38],\ [1481, 1481, 3/2*w^3 - 1/2*w^2 - 17/2*w + 4],\ [1489, 1489, -7/2*w^3 - 7/2*w^2 + 45/2*w + 22],\ [1489, 1489, 9/2*w^3 + 9/2*w^2 - 67/2*w - 26],\ [1499, 1499, 7/2*w^3 + 9/2*w^2 - 45/2*w - 26],\ [1499, 1499, 7*w^3 + 11*w^2 - 49*w - 53],\ [1511, 1511, -7/2*w^3 - 9/2*w^2 + 51/2*w + 19],\ [1511, 1511, -3/2*w^3 - 1/2*w^2 + 19/2*w - 2],\ [1511, 1511, -5*w^3 - 3*w^2 + 39*w + 9],\ [1511, 1511, 1/2*w^3 - 9/2*w^2 - 5/2*w + 26],\ [1559, 1559, -9/2*w^3 - 13/2*w^2 + 65/2*w + 31],\ [1559, 1559, -3/2*w^3 - 5/2*w^2 + 27/2*w + 4],\ [1559, 1559, -6*w^3 - 8*w^2 + 43*w + 37],\ [1559, 1559, w^3 - w^2 - 6*w + 9],\ [1571, 1571, 9/2*w^3 + 17/2*w^2 - 65/2*w - 39],\ [1571, 1571, -2*w^3 - 6*w^2 + 15*w + 39],\ [1601, 1601, -3*w^3 - 5*w^2 + 24*w + 33],\ [1601, 1601, -7/2*w^3 - 9/2*w^2 + 51/2*w + 14],\ [1609, 1609, 4*w^3 + 5*w^2 - 28*w - 23],\ [1609, 1609, -3/2*w^3 - 3/2*w^2 + 13/2*w + 8],\ [1609, 1609, -13/2*w^3 - 19/2*w^2 + 91/2*w + 48],\ [1609, 1609, -5/2*w^3 - 3/2*w^2 + 35/2*w + 4],\ [1619, 1619, 5/2*w^3 + 7/2*w^2 - 29/2*w - 14],\ [1619, 1619, -11/2*w^3 - 17/2*w^2 + 75/2*w + 44],\ [1621, 1621, 2*w^3 + 2*w^2 - 16*w - 3],\ [1621, 1621, w^3 + w^2 - 5*w + 1],\ [1669, 1669, -6*w^3 - 6*w^2 + 46*w + 31],\ [1669, 1669, -1/2*w^3 - 9/2*w^2 + 15/2*w + 11],\ [1681, 41, 3*w^3 + 3*w^2 - 21*w - 13],\ [1681, 41, 3*w^3 + 3*w^2 - 21*w - 17],\ [1699, 1699, 7/2*w^3 + 7/2*w^2 - 39/2*w - 23],\ [1699, 1699, -7/2*w^3 - 17/2*w^2 + 49/2*w + 56],\ [1709, 1709, -1/2*w^3 + 1/2*w^2 + 13/2*w - 1],\ [1709, 1709, 1/2*w^3 + 3/2*w^2 - 1/2*w - 12],\ [1721, 1721, 5/2*w^3 + 9/2*w^2 - 39/2*w - 19],\ [1721, 1721, -3*w^3 - 3*w^2 + 23*w + 11],\ [1741, 1741, 5/2*w^3 + 9/2*w^2 - 31/2*w - 26],\ [1741, 1741, -3*w^3 - 5*w^2 + 21*w + 31],\ [1759, 1759, w^3 + 5*w^2 - 6*w - 33],\ [1759, 1759, 9/2*w^3 + 17/2*w^2 - 61/2*w - 41],\ [1789, 1789, 5/2*w^3 + 11/2*w^2 - 35/2*w - 28],\ [1789, 1789, 2*w^3 + 5*w^2 - 14*w - 29],\ [1801, 1801, w^3 + 3*w^2 - 4*w - 23],\ [1801, 1801, 3/2*w^3 + 9/2*w^2 - 21/2*w - 26],\ [1801, 1801, 3*w^3 + 6*w^2 - 21*w - 31],\ [1801, 1801, 11/2*w^3 + 17/2*w^2 - 79/2*w - 48],\ [1811, 1811, 1/2*w^3 + 3/2*w^2 - 5/2*w - 1],\ [1811, 1811, 1/2*w^3 + 3/2*w^2 - 5/2*w - 16],\ [1831, 1831, -9/2*w^3 - 13/2*w^2 + 63/2*w + 37],\ [1831, 1831, -w^3 - 3*w^2 + 5*w + 23],\ [1849, 43, 6*w^3 + 8*w^2 - 44*w - 43],\ [1849, 43, 1/2*w^3 - 3/2*w^2 + 1/2*w - 3],\ [1861, 1861, 5/2*w^3 + 7/2*w^2 - 33/2*w - 23],\ [1861, 1861, 5/2*w^3 + 9/2*w^2 - 33/2*w - 27],\ [1861, 1861, -2*w^3 + 13*w - 1],\ [1861, 1861, 2*w^3 + 3*w^2 - 12*w - 7],\ [1871, 1871, 3/2*w^3 + 5/2*w^2 - 15/2*w - 14],\ [1871, 1871, 3/2*w^3 + 1/2*w^2 - 27/2*w - 1],\ [1901, 1901, -4*w^3 - 5*w^2 + 28*w + 27],\ [1901, 1901, 5/2*w^3 + 3/2*w^2 - 35/2*w - 8],\ [1931, 1931, -2*w^3 + w^2 + 10*w + 1],\ [1931, 1931, 17/2*w^3 + 23/2*w^2 - 127/2*w - 66],\ [1949, 1949, -w^3 - 2*w^2 + 3*w + 17],\ [1949, 1949, -4*w^2 + 2*w + 17],\ [1949, 1949, -w^3 - 5*w^2 + 7*w + 27],\ [1949, 1949, -5*w^3 - 9*w^2 + 35*w + 49],\ [1951, 1951, -13/2*w^3 - 17/2*w^2 + 103/2*w + 52],\ [1951, 1951, 11/2*w^3 + 13/2*w^2 - 87/2*w - 41],\ [1979, 1979, -15/2*w^3 - 19/2*w^2 + 119/2*w + 59],\ [1979, 1979, 9/2*w^3 + 17/2*w^2 - 61/2*w - 47],\ [1999, 1999, 3*w^3 + 3*w^2 - 20*w - 17],\ [1999, 1999, 7/2*w^3 + 7/2*w^2 - 51/2*w - 19]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 - 2*x^3 - 6*x^2 + 2*x + 1 K. = NumberField(heckePol) hecke_eigenvalues_array = [2, e, -1/2*e^2 + e + 5/2, -e^3 + 2*e^2 + 7*e - 2, -3*e^3 + 13/2*e^2 + 15*e - 9/2, -1, 1/2*e^3 + e^2 - 17/2*e - 6, -2*e^2 + 4*e + 4, 2*e^3 - 5*e^2 - 10*e + 9, -3*e^3 + 11/2*e^2 + 16*e + 5/2, -2*e^3 + 3*e^2 + 14*e + 5, 3*e^3 - 6*e^2 - 15*e + 2, e^3 - 2*e^2 - 7*e - 6, 2*e^3 - 4*e^2 - 12*e + 4, -3/2*e^3 + 4*e^2 + 23/2*e - 12, 2*e^3 - 3*e^2 - 16*e + 5, -e^2 + 9, 5/2*e^3 - 7*e^2 - 17/2*e + 11, 4*e^3 - 17/2*e^2 - 22*e + 9/2, 3*e^3 - 9*e^2 - 9*e + 13, -7/2*e^3 + 6*e^2 + 35/2*e + 1, e^3 + e^2 - 15*e - 7, 2*e^3 - 3*e^2 - 16*e + 11, -3*e^3 + 6*e^2 + 17*e - 8, e^3 - 7/2*e^2 - 4*e + 23/2, 6*e^3 - 21/2*e^2 - 36*e + 1/2, -6*e^3 + 12*e^2 + 28*e - 4, e^3 - 9/2*e^2 - 4*e + 17/2, 5*e^3 - 25/2*e^2 - 23*e + 17/2, -13/2*e^3 + 12*e^2 + 77/2*e - 1, -2*e^3 + 2*e^2 + 20*e + 2, 3/2*e^3 - 4*e^2 - 23/2*e + 18, 13/2*e^3 - 11*e^2 - 73/2*e + 1, -e^3 + 1/2*e^2 + 9*e + 19/2, -1/2*e^2 + 4*e - 11/2, -e^3 + 5/2*e^2 + 3*e - 9/2, -11/2*e^3 + 14*e^2 + 47/2*e - 10, 17/2*e^3 - 14*e^2 - 105/2*e + 9, -2*e^3 + 3*e^2 + 12*e - 7, -6*e^3 + 14*e^2 + 32*e - 12, 5/2*e^3 - 6*e^2 - 25/2*e + 13, 11/2*e^3 - 12*e^2 - 67/2*e + 1, -2*e^3 + 3*e^2 + 16*e - 3, 5*e^3 - 5*e^2 - 41*e - 3, 2*e^2 - 6*e - 4, -e^3 + 7/2*e^2 + 3*e + 33/2, 4*e^3 - 8*e^2 - 22*e + 6, 5/2*e^3 - 2*e^2 - 41/2*e + 9, 6*e^3 - 11*e^2 - 40*e - 1, -7/2*e^2 + 47/2, -13/2*e^3 + 13*e^2 + 69/2*e - 15, -5*e^3 + 8*e^2 + 31*e, 7*e^3 - 15*e^2 - 33*e + 13, 7*e^3 - 12*e^2 - 45*e + 6, -5*e^3 + 9*e^2 + 35*e - 9, -4*e^3 + 7*e^2 + 24*e + 5, 3/2*e^3 - 8*e^2 + 1/2*e + 11, 5*e^3 - 10*e^2 - 29*e + 16, -4*e^2 + 16*e + 18, -3*e^3 + 10*e^2 + 11*e - 20, -5*e^3 + 13*e^2 + 23*e - 21, e^3 + e^2 - 15*e - 15, 4*e^3 - 5/2*e^2 - 37*e - 15/2, 6*e^3 - 16*e^2 - 30*e + 24, -13/2*e^3 + 18*e^2 + 45/2*e - 20, 5*e^3 - 11/2*e^2 - 48*e - 17/2, 9*e^3 - 18*e^2 - 51*e + 6, e^3 - 8*e^2 - 3*e + 30, 7/2*e^2 - 7*e - 19/2, -1/2*e^3 + 3*e^2 - 7/2*e - 11, 3*e^3 + 1/2*e^2 - 28*e - 13/2, -2*e^3 + e^2 + 20*e - 17, -8*e^3 + 35/2*e^2 + 46*e - 19/2, 8*e^3 - 14*e^2 - 46*e - 2, -7*e^3 + 20*e^2 + 33*e - 24, -4*e^3 + 9*e^2 + 12*e - 9, 6*e^3 - 9*e^2 - 34*e - 1, 4*e^3 - 5*e^2 - 40*e - 7, -15*e^3 + 63/2*e^2 + 85*e - 27/2, 4*e^3 - 9*e^2 - 16*e - 13, 11/2*e^3 - 11*e^2 - 59/2*e + 10, 2*e^3 - 11*e^2 + 19, 9*e^3 - 15*e^2 - 45*e - 7, 1/2*e^2 + 11*e - 33/2, -4*e^3 + 7*e^2 + 36*e - 13, -21/2*e^3 + 22*e^2 + 125/2*e - 16, -9/2*e^2 + 53/2, 9*e^3 - 17*e^2 - 59*e + 23, 7/2*e^3 - 13*e^2 - 31/2*e + 27, -11*e^3 + 21*e^2 + 65*e - 21, -4*e^2 + 12*e + 34, -7/2*e^3 + 14*e^2 + 3/2*e - 25, -2*e^3 + 21/2*e^2 - 6*e - 69/2, -10*e^3 + 18*e^2 + 70*e - 12, -1/2*e^3 + 2*e^2 + 5/2*e - 20, -e^3 + 4*e^2 - 3*e + 4, -14*e^3 + 61/2*e^2 + 71*e - 45/2, 12*e^3 - 45/2*e^2 - 65*e + 9/2, 4*e^3 - 1/2*e^2 - 34*e - 11/2, 3*e^3 - 14*e^2 + e + 26, -7*e^3 + 10*e^2 + 41*e + 6, -2*e^3 - 3/2*e^2 + 11*e + 59/2, -3*e^3 + 5*e^2 + 19*e + 5, 5*e^3 - 8*e^2 - 33*e - 22, 4*e^3 - 11*e^2 - 6*e + 1, -5*e^3 + 7*e^2 + 39*e + 3, -5*e^3 + 8*e^2 + 37*e - 16, 15/2*e^3 - 14*e^2 - 111/2*e, 2*e^3 + 1/2*e^2 - 18*e + 31/2, 4*e^3 - 14*e^2 - 4*e + 28, -2*e^2 - 12*e + 28, 7*e^3 - 29/2*e^2 - 40*e + 33/2, 15/2*e^3 - 18*e^2 - 51/2*e + 7, -25/2*e^3 + 25*e^2 + 141/2*e - 6, 3*e^3 + 3/2*e^2 - 30*e - 31/2, 5*e^3 - 13*e^2 - 23*e + 25, -25/2*e^3 + 28*e^2 + 137/2*e - 30, 14*e^3 - 27*e^2 - 76*e + 21, e^3 - 5*e^2 - e + 1, -7*e^3 + 20*e^2 + 37*e - 38, -2*e^3 + 2*e^2 + 22*e + 18, -e^3 + 19/2*e^2 - 18*e - 39/2, 12*e^3 - 49/2*e^2 - 60*e + 29/2, -11*e^3 + 16*e^2 + 83*e - 6, e^3 + 2*e^2 - 9*e - 14, -5*e^3 + 6*e^2 + 31*e + 16, 5*e^3 - e^2 - 47*e - 11, -6*e^3 + 15*e^2 + 30*e - 29, 8*e^3 - 10*e^2 - 52*e - 8, 17/2*e^3 - 15*e^2 - 101/2*e - 19, -e^3 + 1/2*e^2 + 9*e - 1/2, -6*e^3 + 17/2*e^2 + 44*e + 39/2, -2*e^3 + 25/2*e^2 - 11*e - 65/2, 2*e^3 + 8*e^2 - 30*e - 28, -10*e^3 + 43/2*e^2 + 57*e - 35/2, -9*e^3 + 27/2*e^2 + 48*e + 17/2, 11*e^3 - 31/2*e^2 - 74*e - 1/2, -19/2*e^3 + 21*e^2 + 99/2*e - 2, 31/2*e^3 - 25*e^2 - 211/2*e + 26, 11*e^3 - 23*e^2 - 55*e + 17, 8*e^3 - 31/2*e^2 - 43*e + 55/2, -5*e^3 + 17*e^2 + 21*e - 41, 5*e^3 - 10*e^2 - 33*e + 26, 12*e^3 - 23*e^2 - 78*e + 33, -4*e^3 + 13/2*e^2 + 35*e - 5/2, 10*e^3 - 22*e^2 - 48*e + 18, -8*e^3 + 12*e^2 + 44*e - 6, -9/2*e^3 + 17*e^2 + 9/2*e - 31, 8*e^3 - 17*e^2 - 52*e + 23, -7/2*e^3 + 11*e^2 - 5/2*e - 31, 19*e^3 - 37*e^2 - 105*e + 15, 3*e^3 - 6*e^2 - 9*e + 18, -3*e^3 + 16*e^2 - e - 38, 10*e^3 - 26*e^2 - 50*e + 36, 2*e^3 + 3*e^2 - 32*e - 15, -15*e^3 + 34*e^2 + 77*e - 12, -7*e^3 + 13*e^2 + 37*e - 25, -4*e^3 + 39/2*e^2 + e - 67/2, 11*e^3 - 39/2*e^2 - 60*e + 59/2, 6*e^3 - 15*e^2 - 16*e + 23, -11*e^3 + 57/2*e^2 + 46*e - 69/2, -4*e^3 + 9*e^2 + 12*e - 3, -6*e^3 + 7*e^2 + 60*e + 1, 12*e^3 - 28*e^2 - 66*e + 24, 7*e^3 - 21*e^2 - 31*e + 21, -6*e^3 + 39/2*e^2 + 20*e - 75/2, -3/2*e^3 + 3*e^2 + 19/2*e + 7, 9*e^3 - 16*e^2 - 49*e + 18, -6*e^3 + 10*e^2 + 36*e - 18, 3*e^3 - 7*e^2 - 7*e + 7, 5*e^3 - 13*e^2 - 25*e + 37, -4*e^3 + 15/2*e^2 + 15*e - 11/2, -5/2*e^3 + 4*e^2 + 25/2*e + 19, -25/2*e^3 + 24*e^2 + 169/2*e + 8, 2*e^3 - 10*e^2 - 14*e + 36, -9*e^3 + 18*e^2 + 55*e + 26, -3/2*e^3 - 6*e^2 + 35/2*e + 31, 5*e^3 - 14*e^2 - 29*e + 42, 7/2*e^3 - 16*e^2 + 1/2*e + 41, -27/2*e^3 + 25*e^2 + 167/2*e + 4, -3/2*e^3 + 8*e^2 + 7/2*e - 42, 13*e^3 - 27*e^2 - 59*e + 31, -11/2*e^3 + 9*e^2 + 67/2*e + 30, -14*e^3 + 20*e^2 + 110*e - 4, -13*e^3 + 18*e^2 + 99*e - 12, -16*e^3 + 39*e^2 + 76*e - 45, 4*e^3 - 21*e^2 - 2*e + 29, 12*e^3 - 69/2*e^2 - 51*e + 101/2, 11*e^3 - 24*e^2 - 71*e + 8, e^3 - 13*e^2 + 17*e + 39, -7*e^3 + 20*e^2 + 31*e - 24, -3*e^3 + 3*e^2 + 29*e - 3, 17*e^3 - 31*e^2 - 99*e + 27, -2*e^3 - 7*e^2 + 42*e + 29, -3*e^3 + 6*e^2 + 21*e + 22, 5*e^3 - 24*e^2 - 5*e + 54, -3/2*e^3 + 12*e^2 - 5/2*e - 66, -37/2*e^3 + 37*e^2 + 233/2*e - 33, 3*e^3 - 17/2*e^2 + 9*e + 9/2, 17/2*e^3 - 24*e^2 - 69/2*e + 12, -17*e^3 + 30*e^2 + 107*e - 2, -5/2*e^3 - 4*e^2 + 73/2*e + 26, 8*e^3 - 18*e^2 - 30*e + 24, 3*e^3 - 7*e^2 - 7*e - 7, 3*e^3 - 12*e^2 - 25*e + 36, -12*e^3 + 20*e^2 + 90*e - 4, 13*e^3 - 26*e^2 - 71*e - 2, 19*e^3 - 38*e^2 - 125*e + 22, -24*e^3 + 45*e^2 + 136*e - 17, -1/2*e^3 + 2*e^2 + 1/2*e + 29, -6*e^3 + 57/2*e^2 + 14*e - 141/2, -20*e^3 + 49*e^2 + 92*e - 35, -3*e^3 + 6*e^2 + 3*e - 26, -9/2*e^3 + 5*e^2 + 73/2*e - 39, 6*e^3 - 8*e^2 - 30*e - 24, -19*e^3 + 41*e^2 + 85*e - 41, -3*e^3 - 2*e^2 + 35*e - 6, 9/2*e^3 - 9*e^2 - 73/2*e - 12, 9*e^3 - 31/2*e^2 - 57*e + 95/2, 12*e^3 - 25*e^2 - 74*e + 13, 7/2*e^3 - 13*e^2 - 11/2*e - 4, -5/2*e^3 + 14*e^2 + 1/2*e - 19, -6*e^3 + 19*e^2 + 24*e - 19, 19/2*e^3 - 22*e^2 - 95/2*e + 12, 10*e^3 - 23*e^2 - 50*e + 1, -3*e^3 + 8*e^2 + 11*e + 30, 9*e^3 - 13*e^2 - 57*e + 3, 3/2*e^3 + e^2 - 63/2*e - 6, 14*e^3 - 33*e^2 - 76*e + 37, 16*e^3 - 27*e^2 - 104*e + 15, 39/2*e^3 - 34*e^2 - 215/2*e + 7, -2*e^3 + 9/2*e^2 + 18*e + 23/2, 14*e^3 - 55/2*e^2 - 74*e + 27/2, -21*e^3 + 36*e^2 + 121*e - 10, -17*e^3 + 30*e^2 + 109*e - 32, e^3 + e^2 - 5*e - 11, 7/2*e^3 - 59/2*e - 18, -9*e^3 + 37/2*e^2 + 54*e + 43/2, -14*e^3 + 39*e^2 + 72*e - 53, -22*e^3 + 46*e^2 + 112*e - 40, 6*e^3 + 1/2*e^2 - 63*e - 49/2, -12*e^3 + 67/2*e^2 + 41*e - 91/2, -3*e^3 + 35*e, 13/2*e^3 - 11*e^2 - 101/2*e + 21, -6*e^3 + 12*e^2 + 46*e - 32, 25/2*e^3 - 15*e^2 - 169/2*e - 20, -6*e^3 + 11*e^2 + 60*e - 19, 18*e^3 - 27*e^2 - 128*e + 13, -7/2*e^3 + 18*e^2 + 11/2*e - 30, -3*e^3 + 3*e^2 + 19*e - 39, -13*e^3 + 51/2*e^2 + 88*e - 67/2, e^3 + 3*e^2 - 25*e - 43, 17*e^3 - 37*e^2 - 91*e + 7, 20*e^3 - 67/2*e^2 - 118*e + 27/2, 17/2*e^3 - 18*e^2 - 89/2*e + 37, 6*e^3 - 18*e^2 - 10*e + 64, -2*e^3 + 8*e^2 + 6, -4*e^3 + 23/2*e^2 + 3*e - 15/2, -2*e^3 + 5*e^2 + 20*e - 35, 10*e^3 - 27*e^2 - 38*e + 37, -5*e^3 + 10*e^2 + 37*e + 18, -e^3 - 2*e^2 + 27*e - 22, -24*e^3 + 47*e^2 + 128*e - 9, -11*e^3 + 13*e^2 + 77*e - 27, -35/2*e^3 + 46*e^2 + 131/2*e - 53, -e^3 - 14*e^2 + 33*e + 62, 15*e^3 - 18*e^2 - 103*e - 6, -14*e^3 + 27*e^2 + 100*e - 17, e^3 - 19/2*e^2 + 2*e + 99/2, 3*e^3 - 15*e^2 + 3*e + 25, 3*e^3 + 1/2*e^2 - 49*e - 53/2, -22*e^3 + 52*e^2 + 104*e - 44, 7*e^3 - 27/2*e^2 - 55*e + 27/2, 5*e^3 - 12*e^2 - 35*e + 12, 7/2*e^3 - 10*e^2 - 3/2*e + 50, -10*e^3 + 21*e^2 + 42*e - 15, -11/2*e^3 + 9*e^2 + 79/2*e - 17, -27/2*e^3 + 27*e^2 + 103/2*e - 11, -7*e^3 + 25/2*e^2 + 35*e - 17/2, 9/2*e^3 - 19*e^2 + 3/2*e + 26, -5*e^3 + 17*e^2 + 11*e - 21, 25/2*e^3 - 26*e^2 - 153/2*e + 15, 29/2*e^3 - 27*e^2 - 169/2*e + 30, 21*e^3 - 54*e^2 - 101*e + 62, 5/2*e^3 - 11*e^2 - 1/2*e - 20, 19*e^3 - 36*e^2 - 105*e + 24, -9*e^3 + 15*e^2 + 47*e + 51, -4*e^3 + 9*e^2 + 16*e + 21, -1/2*e^3 - 15*e^2 + 41/2*e + 58, -6*e^3 + 11*e^2 + 26*e + 25, 2*e^3 + 3*e^2 - 42*e - 13, e^3 - e - 18, -23/2*e^3 + 25*e^2 + 83/2*e - 23, -3*e^3 + 4*e^2 + 11*e + 16] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([19,19,w - 1])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]