/* 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([19, 14, -13, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([19,19,-4/19*w^3 + 6/19*w^2 + 55/19*w - 2]) primes_array = [ [4, 2, -2/19*w^3 + 3/19*w^2 + 37/19*w - 3],\ [5, 5, 3/19*w^3 + 5/19*w^2 - 27/19*w - 1],\ [5, 5, -3/19*w^3 + 14/19*w^2 + 8/19*w - 2],\ [9, 3, -2/19*w^3 + 3/19*w^2 + 37/19*w - 4],\ [19, 19, -w],\ [19, 19, 4/19*w^3 - 6/19*w^2 - 55/19*w + 1],\ [19, 19, 4/19*w^3 - 6/19*w^2 - 55/19*w + 2],\ [19, 19, w - 1],\ [29, 29, -5/19*w^3 - 2/19*w^2 + 64/19*w + 2],\ [29, 29, 10/19*w^3 - 34/19*w^2 - 71/19*w + 9],\ [29, 29, 4/19*w^3 - 6/19*w^2 - 55/19*w + 4],\ [29, 29, -w + 3],\ [49, 7, 5/19*w^3 - 17/19*w^2 - 64/19*w + 11],\ [49, 7, 6/19*w^3 - 9/19*w^2 - 92/19*w - 1],\ [71, 71, 3/19*w^3 + 5/19*w^2 - 46/19*w - 1],\ [71, 71, 6/19*w^3 - 9/19*w^2 - 73/19*w],\ [71, 71, 6/19*w^3 - 9/19*w^2 - 73/19*w + 4],\ [71, 71, -3/19*w^3 + 14/19*w^2 + 27/19*w - 3],\ [101, 101, 7/19*w^3 - 1/19*w^2 - 63/19*w - 3],\ [101, 101, -9/19*w^3 + 23/19*w^2 + 81/19*w - 5],\ [101, 101, 9/19*w^3 - 4/19*w^2 - 100/19*w],\ [101, 101, 7/19*w^3 - 20/19*w^2 - 44/19*w + 6],\ [121, 11, 6/19*w^3 - 9/19*w^2 - 54/19*w + 1],\ [121, 11, -6/19*w^3 + 9/19*w^2 + 54/19*w - 2],\ [139, 139, 9/19*w^3 - 4/19*w^2 - 100/19*w - 2],\ [139, 139, -7/19*w^3 + 1/19*w^2 + 63/19*w + 1],\ [139, 139, 7/19*w^3 - 20/19*w^2 - 44/19*w + 4],\ [139, 139, w - 5],\ [149, 149, -1/19*w^3 + 11/19*w^2 - 10/19*w - 6],\ [149, 149, -3/19*w^3 - 5/19*w^2 + 46/19*w],\ [149, 149, -3/19*w^3 + 14/19*w^2 + 27/19*w - 2],\ [149, 149, 1/19*w^3 + 8/19*w^2 - 9/19*w - 6],\ [169, 13, 14/19*w^3 - 40/19*w^2 - 126/19*w + 13],\ [169, 13, 10/19*w^3 - 34/19*w^2 - 52/19*w + 7],\ [191, 191, -10/19*w^3 + 15/19*w^2 + 109/19*w - 6],\ [191, 191, 6/19*w^3 - 9/19*w^2 - 35/19*w - 2],\ [191, 191, -6/19*w^3 + 9/19*w^2 + 35/19*w - 4],\ [191, 191, 10/19*w^3 - 15/19*w^2 - 109/19*w],\ [211, 211, 1/19*w^3 + 8/19*w^2 - 28/19*w - 8],\ [211, 211, 1/19*w^3 - 11/19*w^2 - 9/19*w - 1],\ [211, 211, -1/19*w^3 - 8/19*w^2 + 28/19*w - 2],\ [211, 211, -1/19*w^3 + 11/19*w^2 + 9/19*w - 9],\ [239, 239, 4/19*w^3 - 25/19*w^2 - 17/19*w + 12],\ [239, 239, -1/19*w^3 - 8/19*w^2 - 29/19*w + 5],\ [239, 239, 10/19*w^3 - 15/19*w^2 - 128/19*w + 1],\ [239, 239, -4/19*w^3 - 13/19*w^2 + 55/19*w + 10],\ [241, 241, -2/19*w^3 + 3/19*w^2 + 56/19*w - 4],\ [241, 241, -6/19*w^3 + 9/19*w^2 + 92/19*w],\ [241, 241, 6/19*w^3 - 9/19*w^2 - 92/19*w + 5],\ [241, 241, -2/19*w^3 + 3/19*w^2 + 56/19*w + 1],\ [269, 269, -10/19*w^3 + 34/19*w^2 + 71/19*w - 11],\ [269, 269, 8/19*w^3 - 12/19*w^2 - 129/19*w + 11],\ [269, 269, -10/19*w^3 + 34/19*w^2 + 71/19*w - 8],\ [269, 269, -13/19*w^3 + 48/19*w^2 + 60/19*w - 8],\ [289, 17, 30/19*w^3 - 7/19*w^2 - 384/19*w - 14],\ [289, 17, 11/19*w^3 - 45/19*w^2 - 80/19*w + 17],\ [311, 311, -12/19*w^3 + 37/19*w^2 + 108/19*w - 15],\ [311, 311, -24/19*w^3 + 55/19*w^2 + 273/19*w - 22],\ [311, 311, -8/19*w^3 - 7/19*w^2 + 110/19*w + 4],\ [311, 311, 4/19*w^3 + 13/19*w^2 - 36/19*w - 7],\ [331, 331, -1/19*w^3 - 8/19*w^2 + 66/19*w - 2],\ [331, 331, -6/19*w^3 + 28/19*w^2 + 35/19*w - 9],\ [331, 331, 20/19*w^3 - 68/19*w^2 - 161/19*w + 22],\ [331, 331, 16/19*w^3 - 62/19*w^2 - 87/19*w + 15],\ [359, 359, 13/19*w^3 - 48/19*w^2 - 79/19*w + 11],\ [359, 359, -7/19*w^3 + 20/19*w^2 + 101/19*w - 13],\ [359, 359, -15/19*w^3 + 51/19*w^2 + 116/19*w - 17],\ [359, 359, 3/19*w^3 + 5/19*w^2 - 84/19*w + 5],\ [379, 379, 3/19*w^3 + 5/19*w^2 - 65/19*w - 9],\ [379, 379, 25/19*w^3 - 9/19*w^2 - 339/19*w - 11],\ [379, 379, -25/19*w^3 + 66/19*w^2 + 282/19*w - 28],\ [379, 379, 3/19*w^3 - 14/19*w^2 - 46/19*w + 12],\ [389, 389, -13/19*w^3 + 48/19*w^2 + 136/19*w - 24],\ [389, 389, 11/19*w^3 - 7/19*w^2 - 118/19*w + 2],\ [389, 389, -3/19*w^3 + 14/19*w^2 + 46/19*w - 10],\ [389, 389, -13/19*w^3 - 9/19*w^2 + 193/19*w + 15],\ [409, 409, 2/19*w^3 - 3/19*w^2 - 56/19*w],\ [409, 409, 6/19*w^3 - 9/19*w^2 - 92/19*w + 4],\ [409, 409, -6/19*w^3 + 9/19*w^2 + 92/19*w - 1],\ [409, 409, -2/19*w^3 + 3/19*w^2 + 56/19*w - 3],\ [431, 431, -6/19*w^3 + 9/19*w^2 + 130/19*w - 13],\ [431, 431, 12/19*w^3 - 37/19*w^2 - 108/19*w + 11],\ [431, 431, 10/19*w^3 - 15/19*w^2 - 166/19*w + 14],\ [431, 431, -8/19*w^3 + 31/19*w^2 + 34/19*w - 8],\ [461, 461, 5/19*w^3 - 17/19*w^2 - 45/19*w + 1],\ [461, 461, -3/19*w^3 + 14/19*w^2 + 8/19*w - 8],\ [461, 461, 3/19*w^3 + 5/19*w^2 - 27/19*w - 7],\ [461, 461, 5/19*w^3 + 2/19*w^2 - 64/19*w + 2],\ [479, 479, 5/19*w^3 - 17/19*w^2 - 26/19*w + 7],\ [479, 479, 1/19*w^3 - 11/19*w^2 - 28/19*w + 7],\ [479, 479, 7/19*w^3 - 1/19*w^2 - 82/19*w + 1],\ [479, 479, -5/19*w^3 + 17/19*w^2 + 64/19*w - 3],\ [499, 499, -11/19*w^3 + 7/19*w^2 + 118/19*w + 2],\ [499, 499, 9/19*w^3 - 23/19*w^2 - 62/19*w + 4],\ [499, 499, -9/19*w^3 + 4/19*w^2 + 81/19*w],\ [499, 499, -11/19*w^3 + 26/19*w^2 + 99/19*w - 8],\ [509, 509, -6/19*w^3 - 10/19*w^2 + 73/19*w + 9],\ [509, 509, -1/19*w^3 + 11/19*w^2 + 47/19*w - 7],\ [509, 509, -6/19*w^3 - 10/19*w^2 + 73/19*w + 2],\ [509, 509, -9/19*w^3 + 4/19*w^2 + 138/19*w - 2],\ [529, 23, -4/19*w^3 + 6/19*w^2 + 74/19*w - 1],\ [529, 23, 4/19*w^3 - 6/19*w^2 - 74/19*w + 3],\ [571, 571, 1/19*w^3 + 8/19*w^2 - 66/19*w + 6],\ [571, 571, -9/19*w^3 + 23/19*w^2 + 119/19*w - 15],\ [571, 571, -11/19*w^3 + 26/19*w^2 + 156/19*w - 18],\ [571, 571, 3/19*w^3 + 5/19*w^2 - 103/19*w + 9],\ [599, 599, 6/19*w^3 - 9/19*w^2 - 16/19*w - 2],\ [599, 599, -14/19*w^3 + 21/19*w^2 + 164/19*w - 7],\ [599, 599, -2/19*w^3 + 22/19*w^2 + 18/19*w - 13],\ [599, 599, -6/19*w^3 + 9/19*w^2 + 16/19*w - 3],\ [601, 601, 23/19*w^3 - 63/19*w^2 - 264/19*w + 31],\ [601, 601, -36/19*w^3 - 3/19*w^2 + 476/19*w + 23],\ [601, 601, 36/19*w^3 - 111/19*w^2 - 362/19*w + 46],\ [601, 601, -16/19*w^3 + 43/19*w^2 + 125/19*w - 8],\ [619, 619, -5/19*w^3 + 17/19*w^2 + 45/19*w - 11],\ [619, 619, 5/19*w^3 - 17/19*w^2 - 26/19*w - 2],\ [619, 619, -5/19*w^3 - 2/19*w^2 + 45/19*w - 4],\ [619, 619, -7/19*w^3 + 20/19*w^2 + 63/19*w - 12],\ [691, 691, 7/19*w^3 - 1/19*w^2 - 120/19*w],\ [691, 691, 3/19*w^3 - 14/19*w^2 - 65/19*w + 6],\ [691, 691, -3/19*w^3 - 5/19*w^2 + 84/19*w + 2],\ [691, 691, 7/19*w^3 - 20/19*w^2 - 101/19*w + 6],\ [701, 701, -7/19*w^3 + 20/19*w^2 + 63/19*w - 2],\ [701, 701, 5/19*w^3 - 17/19*w^2 - 26/19*w + 8],\ [701, 701, -7/19*w^3 + 20/19*w^2 + 82/19*w - 4],\ [701, 701, -7/19*w^3 + 1/19*w^2 + 82/19*w - 2],\ [719, 719, -8/19*w^3 + 31/19*w^2 + 53/19*w - 12],\ [719, 719, 8/19*w^3 - 31/19*w^2 - 53/19*w + 6],\ [719, 719, -8/19*w^3 - 7/19*w^2 + 91/19*w + 2],\ [719, 719, 8/19*w^3 + 7/19*w^2 - 91/19*w - 8],\ [739, 739, 9/19*w^3 - 23/19*w^2 - 43/19*w + 3],\ [739, 739, -16/19*w^3 + 24/19*w^2 + 163/19*w - 9],\ [739, 739, 16/19*w^3 - 24/19*w^2 - 163/19*w],\ [739, 739, -9/19*w^3 + 4/19*w^2 + 62/19*w],\ [769, 769, 24/19*w^3 - 74/19*w^2 - 254/19*w + 35],\ [769, 769, -3/19*w^3 + 33/19*w^2 - 30/19*w - 11],\ [769, 769, 3/19*w^3 + 24/19*w^2 - 27/19*w - 11],\ [769, 769, 24/19*w^3 + 2/19*w^2 - 330/19*w - 19],\ [811, 811, 5/19*w^3 + 2/19*w^2 - 102/19*w],\ [811, 811, -5/19*w^3 + 17/19*w^2 + 83/19*w - 7],\ [811, 811, 5/19*w^3 + 2/19*w^2 - 102/19*w - 2],\ [811, 811, -5/19*w^3 + 17/19*w^2 + 83/19*w - 5],\ [821, 821, -6/19*w^3 - 10/19*w^2 + 111/19*w + 13],\ [821, 821, 1/19*w^3 + 8/19*w^2 + 10/19*w - 12],\ [821, 821, -1/19*w^3 + 11/19*w^2 - 29/19*w - 11],\ [821, 821, -6/19*w^3 + 28/19*w^2 + 73/19*w - 18],\ [839, 839, -2*w^3 + 26*w + 23],\ [839, 839, 18/19*w^3 - 65/19*w^2 - 143/19*w + 24],\ [839, 839, -18/19*w^3 - 11/19*w^2 + 219/19*w + 14],\ [839, 839, 2*w^3 - 6*w^2 - 20*w + 47],\ [859, 859, 10/19*w^3 - 34/19*w^2 - 109/19*w + 19],\ [859, 859, 3/19*w^3 + 24/19*w^2 - 46/19*w - 14],\ [859, 859, -17/19*w^3 + 35/19*w^2 + 210/19*w - 14],\ [859, 859, -10/19*w^3 - 4/19*w^2 + 147/19*w + 12],\ [911, 911, -1/19*w^3 + 11/19*w^2 + 28/19*w - 9],\ [911, 911, -5/19*w^3 + 17/19*w^2 + 64/19*w - 1],\ [911, 911, 5/19*w^3 + 2/19*w^2 - 83/19*w + 3],\ [911, 911, 1/19*w^3 + 8/19*w^2 - 47/19*w - 7],\ [941, 941, 11/19*w^3 - 7/19*w^2 - 99/19*w - 5],\ [941, 941, -1/19*w^3 + 11/19*w^2 + 28/19*w - 10],\ [941, 941, -13/19*w^3 + 10/19*w^2 + 136/19*w - 4],\ [941, 941, -6/19*w^3 + 28/19*w^2 - 3/19*w - 6],\ [961, 31, 10/19*w^3 - 15/19*w^2 - 90/19*w + 2],\ [961, 31, -10/19*w^3 + 15/19*w^2 + 90/19*w - 3]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^8 - 14*x^6 + 58*x^4 - 63*x^2 + 13 K. = NumberField(heckePol) hecke_eigenvalues_array = [2/13*e^6 - 21/13*e^4 + 62/13*e^2 - 4, -3/13*e^7 + 38/13*e^5 - 132/13*e^3 + 6*e, e, 1/13*e^6 - 4/13*e^4 - 21/13*e^2 + 2, e^4 - 8*e^2 + 7, 5/13*e^6 - 72/13*e^4 + 272/13*e^2 - 13, 1, -5/13*e^6 + 46/13*e^4 - 90/13*e^2 + 1, 11/13*e^7 - 135/13*e^5 + 471/13*e^3 - 28*e, e^3 - 8*e, 5/13*e^7 - 72/13*e^5 + 272/13*e^3 - 10*e, -6/13*e^7 + 63/13*e^5 - 173/13*e^3 + 7*e, -8/13*e^6 + 84/13*e^4 - 235/13*e^2 + 12, -8/13*e^6 + 84/13*e^4 - 209/13*e^2 - 1, -2/13*e^7 + 21/13*e^5 - 75/13*e^3 + 14*e, 4/13*e^7 - 29/13*e^5 + 7/13*e^3 + 7*e, -7/13*e^7 + 93/13*e^5 - 334/13*e^3 + 16*e, 1/13*e^7 - 17/13*e^5 + 57/13*e^3 + 2*e, -2/13*e^7 + 47/13*e^5 - 296/13*e^3 + 33*e, 2/13*e^7 - 8/13*e^5 - 81/13*e^3 + 25*e, 1/13*e^7 + 9/13*e^5 - 164/13*e^3 + 28*e, 4/13*e^7 - 81/13*e^5 + 410/13*e^3 - 29*e, 16/13*e^6 - 155/13*e^4 + 379/13*e^2 - 17, -19/13*e^6 + 193/13*e^4 - 459/13*e^2 + 6, -4/13*e^6 + 42/13*e^4 - 137/13*e^2 + 15, 9/13*e^6 - 114/13*e^4 + 331/13*e^2 + 2, -14/13*e^6 + 134/13*e^4 - 278/13*e^2 + 3, -6/13*e^6 + 115/13*e^4 - 550/13*e^2 + 23, -6/13*e^7 + 76/13*e^5 - 264/13*e^3 + 7*e, 6/13*e^7 - 102/13*e^5 + 472/13*e^3 - 36*e, -14/13*e^7 + 199/13*e^5 - 772/13*e^3 + 39*e, 5/13*e^7 - 85/13*e^5 + 441/13*e^3 - 52*e, -9/13*e^6 + 36/13*e^4 + 150/13*e^2 - 9, -6/13*e^6 + 89/13*e^4 - 316/13*e^2 + 5, 17/13*e^7 - 211/13*e^5 + 761/13*e^3 - 55*e, -11/13*e^7 + 174/13*e^5 - 718/13*e^3 + 30*e, -22/13*e^7 + 296/13*e^5 - 1098/13*e^3 + 55*e, 10/13*e^7 - 144/13*e^5 + 557/13*e^3 - 22*e, -3/13*e^6 + 12/13*e^4 + 76/13*e^2 - 5, -1/13*e^6 + 30/13*e^4 - 174/13*e^2, 2*e^6 - 24*e^4 + 75*e^2 - 43, 9/13*e^6 - 153/13*e^4 + 682/13*e^2 - 27, -1/13*e^7 + 17/13*e^5 - 57/13*e^3 - 3*e, -9/13*e^7 + 153/13*e^5 - 747/13*e^3 + 69*e, 7/13*e^7 - 80/13*e^5 + 308/13*e^3 - 34*e, -14/13*e^7 + 173/13*e^5 - 551/13*e^3 + 16*e, 6/13*e^6 - 11/13*e^4 - 165/13*e^2 - 6, 24/13*e^6 - 213/13*e^4 + 393/13*e^2, 21/13*e^6 - 253/13*e^4 + 729/13*e^2 - 14, -2/13*e^6 + 47/13*e^4 - 244/13*e^2, 22/13*e^7 - 309/13*e^5 + 1215/13*e^3 - 63*e, 29/13*e^7 - 389/13*e^5 + 1471/13*e^3 - 85*e, 33/13*e^7 - 405/13*e^5 + 1387/13*e^3 - 72*e, 23/13*e^7 - 326/13*e^5 + 1272/13*e^3 - 74*e, 3/13*e^6 - 77/13*e^4 + 353/13*e^2 - 4, -29/13*e^6 + 298/13*e^4 - 782/13*e^2 + 33, -10/13*e^7 + 105/13*e^5 - 271/13*e^3 + 12*e, -10/13*e^7 + 118/13*e^5 - 388/13*e^3 + 23*e, 1/13*e^7 + 35/13*e^5 - 398/13*e^3 + 61*e, -e^7 + 15*e^5 - 63*e^3 + 49*e, -28/13*e^6 + 229/13*e^4 - 283/13*e^2 - 26, 9/13*e^6 - 23/13*e^4 - 189/13*e^2 - 5, -1/13*e^6 + 69/13*e^4 - 408/13*e^2 + 22, -5/13*e^6 + 46/13*e^4 - 103/13*e^2 - 1, 5/13*e^7 - 59/13*e^5 + 181/13*e^3 - 14*e, -8/13*e^7 + 110/13*e^5 - 443/13*e^3 + 26*e, e^5 - 4*e^3 - 11*e, 16/13*e^7 - 168/13*e^5 + 457/13*e^3 - 17*e, -31/13*e^6 + 332/13*e^4 - 883/13*e^2 + 25, e^6 - 9*e^4 + 18*e^2 + 5, 7/13*e^6 - 67/13*e^4 + 152/13*e^2 - 23, 29/13*e^6 - 376/13*e^4 + 1380/13*e^2 - 59, -7/13*e^7 + 158/13*e^5 - 828/13*e^3 + 55*e, -21/13*e^7 + 266/13*e^5 - 898/13*e^3 + 40*e, 12/13*e^7 - 178/13*e^5 + 723/13*e^3 - 29*e, 19/13*e^7 - 310/13*e^5 + 1356/13*e^3 - 81*e, 29/13*e^6 - 285/13*e^4 + 652/13*e^2 - 26, 2*e^4 - 12*e^2 - 14, -32/13*e^6 + 284/13*e^4 - 576/13*e^2 + 23, 14/13*e^6 - 95/13*e^4 + 122/13*e^2 - 31, -24/13*e^7 + 330/13*e^5 - 1225/13*e^3 + 52*e, 17/13*e^7 - 211/13*e^5 + 826/13*e^3 - 84*e, -17/13*e^7 + 263/13*e^5 - 1086/13*e^3 + 48*e, 19/13*e^7 - 232/13*e^5 + 745/13*e^3 - 21*e, -2*e^5 + 18*e^3 - 23*e, -8/13*e^7 + 136/13*e^5 - 664/13*e^3 + 72*e, 22/13*e^7 - 244/13*e^5 + 682/13*e^3 - 12*e, 31/13*e^7 - 449/13*e^5 + 1819/13*e^3 - 107*e, 1/13*e^7 + 9/13*e^5 - 99/13*e^3 - 3*e, 8/13*e^7 - 110/13*e^5 + 443/13*e^3 - 36*e, 5/13*e^7 - 59/13*e^5 + 233/13*e^3 - 29*e, -21/13*e^7 + 279/13*e^5 - 1054/13*e^3 + 53*e, -10/13*e^6 + 27/13*e^4 + 197/13*e^2 - 6, -e^4 + 7*e^2 + 6, 2*e^6 - 18*e^4 + 39*e^2 - 33, 37/13*e^6 - 330/13*e^4 + 614/13*e^2 + 1, -e^7 + 16*e^5 - 70*e^3 + 64*e, -16/13*e^7 + 207/13*e^5 - 834/13*e^3 + 76*e, 34/13*e^7 - 435/13*e^5 + 1535/13*e^3 - 73*e, 18/13*e^7 - 254/13*e^5 + 1065/13*e^3 - 97*e, -12/13*e^6 + 178/13*e^4 - 736/13*e^2 + 35, -2*e^6 + 22*e^4 - 62*e^2 + 13, 20/13*e^6 - 262/13*e^4 + 789/13*e^2 - 10, -23/13*e^6 + 235/13*e^4 - 557/13*e^2 + 6, -14/13*e^6 + 264/13*e^4 - 1201/13*e^2 + 48, 37/13*e^6 - 356/13*e^4 + 848/13*e^2 - 52, 17/13*e^7 - 237/13*e^5 + 930/13*e^3 - 56*e, -10/13*e^7 + 105/13*e^5 - 219/13*e^3 - 6*e, -37/13*e^7 + 551/13*e^5 - 2252/13*e^3 + 120*e, -2*e^7 + 26*e^5 - 97*e^3 + 94*e, 12/13*e^6 - 48/13*e^4 - 213/13*e^2 + 7, -27/13*e^6 + 277/13*e^4 - 655/13*e^2 + 7, 18/13*e^6 - 176/13*e^4 + 480/13*e^2 - 17, -12/13*e^6 + 139/13*e^4 - 515/13*e^2 + 38, -2*e^6 + 19*e^4 - 34*e^2 - 33, 42/13*e^6 - 402/13*e^4 + 964/13*e^2 - 58, -4/13*e^6 + 3/13*e^4 + 97/13*e^2 - 4, -2*e^6 + 22*e^4 - 58*e^2 + 19, 5/13*e^6 - 46/13*e^4 + 116/13*e^2 - 17, 8/13*e^6 - 149/13*e^4 + 742/13*e^2 - 42, -45/13*e^6 + 505/13*e^4 - 1486/13*e^2 + 53, -19/13*e^6 + 141/13*e^4 - 108/13*e^2 - 25, 9/13*e^7 - 179/13*e^5 + 916/13*e^3 - 69*e, -44/13*e^7 + 553/13*e^5 - 1949/13*e^3 + 105*e, 3/13*e^7 - 38/13*e^5 + 184/13*e^3 - 31*e, 46/13*e^7 - 548/13*e^5 + 1816/13*e^3 - 97*e, 15/13*e^7 - 229/13*e^5 + 959/13*e^3 - 56*e, 48/13*e^7 - 647/13*e^5 + 2463/13*e^3 - 137*e, -16/13*e^7 + 246/13*e^5 - 1120/13*e^3 + 85*e, -12/13*e^7 + 139/13*e^5 - 346/13*e^3 - 16*e, -20/13*e^6 + 262/13*e^4 - 1023/13*e^2 + 43, 23/13*e^6 - 378/13*e^4 + 1688/13*e^2 - 77, 11/13*e^6 - 174/13*e^4 + 731/13*e^2 - 29, -2/13*e^6 - 44/13*e^4 + 484/13*e^2 - 42, -31/13*e^6 + 254/13*e^4 - 246/13*e^2 - 34, -2/13*e^6 + 151/13*e^4 - 946/13*e^2 + 32, -3*e^6 + 27*e^4 - 60*e^2 + 50, 42/13*e^6 - 441/13*e^4 + 1185/13*e^2 - 53, -11/13*e^6 + 161/13*e^4 - 757/13*e^2 + 55, -17/13*e^6 + 172/13*e^4 - 371/13*e^2 - 12, 25/13*e^6 - 100/13*e^4 - 525/13*e^2 + 67, -40/13*e^6 + 433/13*e^4 - 1162/13*e^2 + 15, -43/13*e^7 + 575/13*e^5 - 2113/13*e^3 + 107*e, -11/13*e^7 + 148/13*e^5 - 588/13*e^3 + 44*e, -12/13*e^7 + 152/13*e^5 - 489/13*e^3, -9/13*e^7 + 75/13*e^5 - 58/13*e^3 - 26*e, -23/13*e^7 + 235/13*e^5 - 648/13*e^3 + 47*e, 23/13*e^7 - 352/13*e^5 + 1597/13*e^3 - 136*e, 3*e^3 - 23*e, -27/13*e^7 + 290/13*e^5 - 889/13*e^3 + 61*e, -20/13*e^6 + 145/13*e^4 - 87/13*e^2 - 36, 47/13*e^6 - 435/13*e^4 + 989/13*e^2 - 44, -12/13*e^6 + 48/13*e^4 + 122/13*e^2 - 3, -11/13*e^6 + 57/13*e^4 + 23/13*e^2 + 32, 3*e^7 - 44*e^5 + 177*e^3 - 122*e, 23/13*e^7 - 378/13*e^5 + 1688/13*e^3 - 115*e, -2/13*e^7 + 73/13*e^5 - 374/13*e^3 - 5*e, -35/13*e^7 + 413/13*e^5 - 1449/13*e^3 + 122*e, 40/13*e^7 - 576/13*e^5 + 2267/13*e^3 - 121*e, 5/13*e^7 - 20/13*e^5 - 40/13*e^3 - 8*e, -35/13*e^7 + 491/13*e^5 - 1956/13*e^3 + 136*e, -29/13*e^7 + 350/13*e^5 - 1094/13*e^3 + 26*e, 28/13*e^6 - 281/13*e^4 + 595/13*e^2 + 2, 2/13*e^6 - 73/13*e^4 + 465/13*e^2 - 37] 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,-4/19*w^3 + 6/19*w^2 + 55/19*w - 2])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]