/* 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([5, 5, 3/19*w^3 + 5/19*w^2 - 27/19*w - 1]) 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 K = QQ e = 1 hecke_eigenvalues_array = [0, 1, -2, -2, -2, -2, 6, 6, -6, -6, -6, -6, 2, 2, 0, 0, 8, 8, 6, -10, 6, -10, -6, -6, -2, -2, -10, -10, -14, -6, -14, -6, 6, 6, 12, 12, 20, 20, 10, 10, -22, -22, 20, 20, -28, -28, 26, 10, 26, 10, -6, -14, -6, -14, -18, -18, 0, 0, -24, -24, -34, -34, -18, -18, 4, -12, 4, -12, -30, -30, -6, -6, -18, 22, -18, 22, -26, 38, -26, 38, 0, 8, 0, 8, -30, -30, 18, 18, -4, -4, -36, -36, -14, -6, -14, -6, -10, -26, -10, -26, 18, -30, -2, -2, -18, -18, -28, -28, 20, 20, 22, 22, 22, 22, -26, 30, -26, 30, 38, 38, -26, -26, 22, 22, -18, -18, 40, 40, 8, 8, 10, 2, 10, 2, -2, -2, 46, 46, -6, -6, 10, 10, -38, -46, -38, -46, -44, 20, -44, 20, 18, 26, 18, 26, -24, 0, -24, 0, 46, 46, 46, 46, -34, -34] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([5, 5, 3/19*w^3 + 5/19*w^2 - 27/19*w - 1])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]