/* 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([36,6,-w^2 + w + 4]) 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^2 - 3*x - 2 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1, e, 2, -1, 2, -2*e + 2, -2, 2, 3*e - 5, e + 1, -3*e + 7, e + 3, e + 3, -e + 7, -e + 1, -2*e + 4, 4*e, 8, -e - 1, -4*e + 6, 3*e + 1, -e + 15, 2*e - 12, -4*e + 8, -6*e + 6, -2*e + 6, -4*e, -4*e + 4, 18, 4*e - 6, -4*e - 12, 0, 5*e - 13, 2*e - 4, -2*e + 10, -4*e + 18, -2*e + 2, 4*e + 10, -6*e + 6, 2*e + 12, 2*e + 14, -4*e - 8, -e - 17, 4*e - 6, 2*e - 2, 3*e - 19, -2*e - 2, 2*e - 12, -5*e + 9, 2*e - 16, 7*e - 23, 9*e - 15, 3*e - 5, 3*e + 5, 12, 3*e - 15, -4, -2*e + 18, -13*e + 11, 5*e - 9, -6*e + 8, 2*e - 2, e - 1, 8*e - 16, 6*e - 20, -5*e + 21, 3*e + 5, -3*e + 21, 14, -11*e + 13, -2*e - 18, 10*e - 22, 2*e + 18, -8*e + 2, 6*e + 2, -6, -16, e + 17, 5*e + 3, -12*e + 32, 4*e - 34, 22, -e - 17, 3*e - 27, 8*e + 2, -10*e + 28, 14*e - 24, -2*e + 22, -4*e + 8, -7*e - 15, 15*e - 29, -e + 15, -2*e + 10, 12, -17*e + 33, 4*e - 30, 12*e - 6, -11*e + 29, 9*e - 5, -14*e + 16, -6*e - 14, 2*e + 14, -4*e, 5*e - 5, -18*e + 22, -7*e + 19, 6*e - 4, 10*e + 12, 3*e + 11, -11*e + 21, -11*e + 33, 10*e, -10*e + 18, 9*e - 39, -2*e + 34, -12, 12*e, -3*e + 23, -17*e + 39, -4*e + 18, -6*e - 8, -46, 3*e - 9, e - 41, -5*e + 7, -10*e + 8, -2*e - 6, -11*e + 25, -11*e + 39, -e + 1, 7*e - 33, 4*e - 24, 4*e - 36, 7*e - 3, 12*e - 44, 11*e - 19, 2*e + 12, -15*e + 37, -18*e + 40, 2*e + 44, -9*e - 1, 12*e - 14, 6*e, -18*e + 16, -13*e - 1, 3*e + 37, 7*e - 7, -6*e - 24, 8*e - 12, -22, 10*e - 28, -6*e + 18, -e + 23, 16*e - 46, 11*e + 7, -9*e + 41, -4*e - 10, -14*e + 44, -2*e - 28, -3*e + 25, 12*e, -12*e - 8, -17*e + 11, 24*e - 38, 18*e - 34, -10*e + 18, -3*e - 41, -16*e + 28, -7*e - 3, 6*e - 42, -8*e + 34, -11*e + 25, 4*e - 14, 16*e - 38, -12*e + 18, 20*e - 36, -22*e + 26] 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([9,3,-1/3*w^3 - 2/3*w^2 + 7/3*w + 5])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]