/* 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([36, 6, -13, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 2, 2]) primes_array = [ [4, 2, 1/3*w^3 - 4/3*w^2 - 4/3*w + 6],\ [4, 2, -1/3*w^3 - 2/3*w^2 + 10/3*w + 7],\ [9, 3, -1/2*w^3 + 3/2*w^2 + 7/2*w - 9],\ [9, 3, 1/3*w^3 + 2/3*w^2 - 7/3*w - 5],\ [11, 11, -1/6*w^3 + 1/6*w^2 + 1/6*w],\ [19, 19, w + 1],\ [19, 19, 1/6*w^3 - 1/6*w^2 - 13/6*w + 2],\ [25, 5, -1/3*w^3 + 1/3*w^2 + 7/3*w - 1],\ [29, 29, -1/6*w^3 + 1/6*w^2 + 13/6*w],\ [29, 29, w - 1],\ [31, 31, 1/3*w^3 - 1/3*w^2 - 10/3*w + 3],\ [31, 31, 1/6*w^3 - 1/6*w^2 - 1/6*w + 2],\ [41, 41, -1/2*w^3 - 1/2*w^2 + 9/2*w + 5],\ [41, 41, -1/2*w^3 + 3/2*w^2 + 5/2*w - 8],\ [59, 59, 1/3*w^3 - 1/3*w^2 - 10/3*w - 1],\ [59, 59, -1/3*w^3 + 4/3*w^2 + 7/3*w - 7],\ [59, 59, 1/2*w^3 - 1/2*w^2 - 5/2*w + 2],\ [79, 79, w^2 - 11],\ [79, 79, 1/6*w^3 - 7/6*w^2 - 7/6*w + 3],\ [89, 89, -1/6*w^3 + 1/6*w^2 + 19/6*w - 5],\ [89, 89, 1/6*w^3 - 1/6*w^2 - 19/6*w - 3],\ [109, 109, -1/6*w^3 + 7/6*w^2 + 1/6*w - 9],\ [109, 109, -7/6*w^3 + 19/6*w^2 + 49/6*w - 20],\ [109, 109, -5/6*w^3 + 17/6*w^2 + 23/6*w - 12],\ [109, 109, 1/6*w^3 + 5/6*w^2 - 13/6*w - 4],\ [121, 11, 1/6*w^3 - 1/6*w^2 - 7/6*w - 3],\ [139, 139, 1/3*w^3 - 4/3*w^2 + 5/3*w - 1],\ [139, 139, -5/3*w^3 - 7/3*w^2 + 50/3*w + 29],\ [169, 13, w^3 + w^2 - 10*w - 13],\ [169, 13, -5/6*w^3 + 17/6*w^2 + 17/6*w - 12],\ [179, 179, -1/3*w^3 + 4/3*w^2 + 1/3*w - 7],\ [179, 179, -2/3*w^3 + 2/3*w^2 + 17/3*w - 5],\ [181, 181, -w^3 + 3*w^2 + 4*w - 13],\ [181, 181, -1/6*w^3 + 7/6*w^2 + 1/6*w - 11],\ [181, 181, 7/6*w^3 + 11/6*w^2 - 79/6*w - 25],\ [181, 181, 1/6*w^3 + 5/6*w^2 - 13/6*w - 2],\ [191, 191, -5/6*w^3 - 1/6*w^2 + 53/6*w + 7],\ [191, 191, 1/6*w^3 + 11/6*w^2 - 19/6*w - 16],\ [211, 211, -1/6*w^3 + 1/6*w^2 - 5/6*w - 2],\ [211, 211, -1/2*w^3 + 1/2*w^2 + 9/2*w + 2],\ [211, 211, 1/3*w^3 - 1/3*w^2 - 4/3*w - 3],\ [211, 211, -1/2*w^3 + 1/2*w^2 + 11/2*w - 4],\ [229, 229, 5/6*w^3 + 1/6*w^2 - 53/6*w - 9],\ [229, 229, 1/2*w^3 - 3/2*w^2 - 1/2*w + 2],\ [239, 239, 5/6*w^3 + 7/6*w^2 - 59/6*w - 19],\ [239, 239, 1/2*w^3 - 5/2*w^2 + 1/2*w + 5],\ [241, 241, 1/6*w^3 - 1/6*w^2 - 13/6*w - 4],\ [241, 241, w - 5],\ [269, 269, -1/6*w^3 + 7/6*w^2 + 1/6*w - 2],\ [269, 269, 1/3*w^3 - 7/3*w^2 - 1/3*w + 7],\ [271, 271, 5/3*w^3 + 7/3*w^2 - 47/3*w - 25],\ [271, 271, -5/3*w^3 + 17/3*w^2 + 23/3*w - 27],\ [281, 281, 1/6*w^3 + 5/6*w^2 - 1/6*w - 7],\ [281, 281, -1/2*w^3 + 3/2*w^2 + 9/2*w - 8],\ [311, 311, 11/6*w^3 + 7/6*w^2 - 113/6*w - 23],\ [311, 311, -4/3*w^3 + 13/3*w^2 + 10/3*w - 13],\ [311, 311, -2/3*w^3 + 5/3*w^2 + 20/3*w - 17],\ [311, 311, 1/6*w^3 + 5/6*w^2 + 5/6*w + 1],\ [349, 349, -1/2*w^3 + 1/2*w^2 + 11/2*w - 1],\ [349, 349, -5/6*w^3 + 17/6*w^2 + 29/6*w - 14],\ [349, 349, 1/6*w^3 - 1/6*w^2 + 5/6*w - 1],\ [349, 349, -2/3*w^3 - 4/3*w^2 + 17/3*w + 13],\ [359, 359, -1/6*w^3 + 13/6*w^2 + 7/6*w - 18],\ [359, 359, -1/6*w^3 - 11/6*w^2 + 13/6*w + 18],\ [361, 19, 2/3*w^3 - 2/3*w^2 - 14/3*w + 1],\ [379, 379, 1/2*w^3 + 1/2*w^2 - 7/2*w - 7],\ [379, 379, -5/6*w^3 + 11/6*w^2 + 23/6*w - 7],\ [379, 379, w^3 - 9*w - 5],\ [379, 379, -2/3*w^3 + 5/3*w^2 + 14/3*w - 7],\ [401, 401, -1/3*w^3 + 7/3*w^2 + 4/3*w - 17],\ [401, 401, -4/3*w^3 - 8/3*w^2 + 43/3*w + 31],\ [419, 419, 1/6*w^3 - 1/6*w^2 + 5/6*w],\ [419, 419, 1/2*w^3 - 1/2*w^2 - 11/2*w + 2],\ [421, 421, -5/6*w^3 - 7/6*w^2 + 47/6*w + 11],\ [421, 421, 1/3*w^3 - 1/3*w^2 - 19/3*w - 7],\ [431, 431, 5/6*w^3 + 1/6*w^2 - 41/6*w - 6],\ [431, 431, 2/3*w^3 - 8/3*w^2 - 8/3*w + 15],\ [431, 431, 5/6*w^3 - 11/6*w^2 - 29/6*w + 7],\ [431, 431, 1/2*w^3 - 5/2*w^2 + 5/2*w + 2],\ [439, 439, 5/6*w^3 + 1/6*w^2 - 41/6*w - 4],\ [439, 439, 1/2*w^3 - 1/2*w^2 - 9/2*w + 8],\ [449, 449, 5/6*w^3 - 17/6*w^2 - 29/6*w + 18],\ [449, 449, 11/6*w^3 - 41/6*w^2 - 41/6*w + 29],\ [449, 449, -7/6*w^3 + 31/6*w^2 + 7/6*w - 16],\ [449, 449, 5/3*w^3 + 7/3*w^2 - 56/3*w - 33],\ [461, 461, -1/6*w^3 + 1/6*w^2 + 19/6*w - 3],\ [461, 461, 1/6*w^3 - 1/6*w^2 - 19/6*w - 1],\ [479, 479, -2*w - 1],\ [479, 479, 1/3*w^3 - 1/3*w^2 - 13/3*w + 3],\ [491, 491, 2/3*w^3 + 1/3*w^2 - 14/3*w - 7],\ [491, 491, 5/6*w^3 - 11/6*w^2 - 35/6*w + 7],\ [509, 509, 5/6*w^3 - 11/6*w^2 - 29/6*w + 8],\ [509, 509, 5/6*w^3 + 1/6*w^2 - 41/6*w - 5],\ [521, 521, -1/6*w^3 + 7/6*w^2 - 17/6*w + 1],\ [521, 521, -1/6*w^3 + 7/6*w^2 - 11/6*w - 3],\ [521, 521, -1/6*w^3 + 7/6*w^2 + 19/6*w - 5],\ [521, 521, 2/3*w^3 + 1/3*w^2 - 26/3*w - 11],\ [541, 541, 1/6*w^3 + 5/6*w^2 - 31/6*w - 17],\ [541, 541, 5/6*w^3 - 17/6*w^2 - 47/6*w + 23],\ [541, 541, 3*w^3 + 3*w^2 - 32*w - 47],\ [541, 541, 2/3*w^3 + 4/3*w^2 - 14/3*w - 11],\ [571, 571, -1/3*w^3 + 1/3*w^2 + 13/3*w - 1],\ [571, 571, 2*w - 1],\ [571, 571, -1/3*w^3 + 4/3*w^2 + 7/3*w - 1],\ [571, 571, 1/6*w^3 + 5/6*w^2 - 7/6*w - 13],\ [599, 599, 1/2*w^3 + 1/2*w^2 - 9/2*w - 2],\ [599, 599, -1/6*w^3 + 1/6*w^2 + 19/6*w - 2],\ [599, 599, 1/6*w^3 - 1/6*w^2 - 19/6*w],\ [599, 599, 1/2*w^3 - 3/2*w^2 - 5/2*w + 11],\ [601, 601, 1/6*w^3 + 11/6*w^2 - 7/6*w - 15],\ [601, 601, -1/3*w^3 + 7/3*w^2 - 5/3*w - 5],\ [601, 601, 2/3*w^3 + 4/3*w^2 - 26/3*w - 19],\ [601, 601, 1/2*w^3 - 5/2*w^2 - 7/2*w + 13],\ [659, 659, w^2 - 13],\ [659, 659, -5/6*w^3 + 11/6*w^2 + 29/6*w - 12],\ [659, 659, -1/6*w^3 + 7/6*w^2 + 7/6*w - 1],\ [659, 659, 5/6*w^3 + 1/6*w^2 - 41/6*w - 1],\ [691, 691, 1/2*w^3 - 1/2*w^2 - 11/2*w + 10],\ [691, 691, -1/6*w^3 + 1/6*w^2 - 5/6*w - 8],\ [701, 701, -2*w^2 + w + 11],\ [701, 701, -1/6*w^3 + 13/6*w^2 + 7/6*w - 9],\ [701, 701, -1/6*w^3 + 13/6*w^2 + 1/6*w - 16],\ [701, 701, -w^3 + w^2 + 9*w - 1],\ [709, 709, -2/3*w^3 + 11/3*w^2 + 14/3*w - 21],\ [709, 709, 1/3*w^3 + 5/3*w^2 + 2/3*w - 5],\ [709, 709, 7/6*w^3 - 25/6*w^2 - 43/6*w + 25],\ [709, 709, 2/3*w^3 - 2/3*w^2 - 23/3*w - 1],\ [719, 719, 2*w^2 - 2*w - 19],\ [719, 719, -2*w^2 + 2*w + 7],\ [739, 739, 5/6*w^3 - 5/6*w^2 - 29/6*w + 4],\ [739, 739, -5/6*w^3 + 11/6*w^2 + 53/6*w - 17],\ [739, 739, -1/6*w^3 - 5/6*w^2 - 11/6*w],\ [739, 739, -w^3 + w^2 + 8*w - 5],\ [751, 751, -1/6*w^3 + 1/6*w^2 + 25/6*w + 8],\ [751, 751, -1/3*w^3 + 1/3*w^2 + 16/3*w - 11],\ [761, 761, -2/3*w^3 + 5/3*w^2 + 8/3*w - 9],\ [761, 761, 5/6*w^3 + 1/6*w^2 - 47/6*w - 3],\ [769, 769, 1/3*w^3 - 7/3*w^2 - 7/3*w + 11],\ [769, 769, -4/3*w^3 - 8/3*w^2 + 46/3*w + 33],\ [769, 769, -1/2*w^3 + 5/2*w^2 + 5/2*w - 20],\ [769, 769, 2*w^2 - 17],\ [809, 809, 1/6*w^3 + 11/6*w^2 - 31/6*w - 15],\ [809, 809, -5/6*w^3 - 1/6*w^2 + 35/6*w + 3],\ [809, 809, 1/6*w^3 + 11/6*w^2 - 31/6*w - 9],\ [809, 809, w^3 - 2*w^2 - 7*w + 11],\ [811, 811, -1/6*w^3 - 5/6*w^2 + 25/6*w + 7],\ [811, 811, 1/6*w^3 + 5/6*w^2 - 25/6*w - 15],\ [811, 811, -1/6*w^3 - 5/6*w^2 + 25/6*w - 4],\ [811, 811, 1/6*w^3 + 5/6*w^2 - 25/6*w - 4],\ [821, 821, -1/2*w^3 - 5/2*w^2 + 13/2*w + 16],\ [821, 821, 1/6*w^3 - 13/6*w^2 - 7/6*w + 21],\ [821, 821, 3/2*w^3 + 1/2*w^2 - 35/2*w - 20],\ [821, 821, -1/6*w^3 + 13/6*w^2 + 7/6*w - 7],\ [829, 829, -1/6*w^3 + 19/6*w^2 - 11/6*w - 23],\ [829, 829, 5/6*w^3 - 17/6*w^2 - 5/6*w + 8],\ [829, 829, -4/3*w^3 - 2/3*w^2 + 43/3*w + 15],\ [829, 829, -1/6*w^3 - 17/6*w^2 + 25/6*w + 16],\ [839, 839, 1/3*w^3 - 7/3*w^2 + 2/3*w + 5],\ [839, 839, 1/2*w^3 + 3/2*w^2 - 13/2*w - 20],\ [841, 29, -5/6*w^3 + 5/6*w^2 + 35/6*w - 4],\ [859, 859, 1/6*w^3 + 5/6*w^2 - 37/6*w + 7],\ [859, 859, 13/6*w^3 + 17/6*w^2 - 127/6*w - 33],\ [881, 881, -2/3*w^3 + 2/3*w^2 + 20/3*w + 5],\ [881, 881, 17/6*w^3 + 19/6*w^2 - 167/6*w - 42],\ [911, 911, -1/6*w^3 + 13/6*w^2 - 11/6*w - 6],\ [911, 911, 3/2*w^3 + 1/2*w^2 - 33/2*w - 17],\ [911, 911, 5/6*w^3 - 17/6*w^2 + 1/6*w + 5],\ [911, 911, 7/6*w^3 - 1/6*w^2 - 61/6*w - 1],\ [929, 929, -7/6*w^3 - 11/6*w^2 + 67/6*w + 22],\ [929, 929, -1/6*w^3 + 1/6*w^2 - 11/6*w + 6],\ [941, 941, 11/6*w^3 + 19/6*w^2 - 107/6*w - 34],\ [941, 941, 4/3*w^3 - 10/3*w^2 - 25/3*w + 19],\ [941, 941, -11/6*w^3 + 41/6*w^2 + 47/6*w - 31],\ [941, 941, 1/2*w^3 + 1/2*w^2 - 13/2*w - 13],\ [961, 31, -1/3*w^3 + 1/3*w^2 + 7/3*w + 5],\ [991, 991, -1/6*w^3 - 5/6*w^2 + 25/6*w + 6],\ [991, 991, 1/6*w^3 + 5/6*w^2 - 25/6*w - 5]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [1, -1, -1, -3, 2, -2, 4, -2, -7, -1, -2, -4, 9, 3, -10, 10, -6, 6, -2, -7, -11, 9, 9, 1, 13, -15, -14, -14, -1, -21, -10, 6, 13, -2, 7, 2, -12, -22, -26, -4, 14, 8, 5, -25, 26, 8, -13, -3, -1, -21, 12, -2, -23, 7, 12, -10, -6, 10, 10, -13, 2, 33, 0, 2, -13, 18, -20, 22, -22, 30, -30, -16, -4, 3, -29, -28, -2, -30, 0, -24, -2, 15, 41, -6, 6, 25, -13, 6, -18, 22, -20, -27, 27, 3, 22, -14, 11, -37, 15, -19, 25, 30, 28, -34, 10, 0, 30, 12, -14, -33, -10, -14, 5, -20, 34, -6, -14, 4, 38, -15, 14, -11, -38, 9, 27, -35, 39, 0, 18, -50, 6, 38, 6, 52, -14, 18, 18, -23, -7, 37, 9, 26, -9, -10, -9, -32, -20, -46, 2, 38, 18, 30, -54, -2, 41, 53, -42, 34, -4, -26, 52, 0, -21, 5, 38, 30, -6, -42, 58, -50, -55, 15, -13, 33, 22, -20, 0] 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, 1/3*w^3 - 4/3*w^2 - 4/3*w + 6])] = -1 AL_eigenvalues[ZF.ideal([4, 2, -1/3*w^3 - 2/3*w^2 + 10/3*w + 7])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]