/* 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([31, 14, -14, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 2, 2]) primes_array = [ [5, 5, 4*w^3 + 7*w^2 - 37*w - 45],\ [11, 11, 2*w^3 + 3*w^2 - 18*w - 16],\ [11, 11, -w + 2],\ [16, 2, 2],\ [19, 19, 4*w^3 + 7*w^2 - 36*w - 45],\ [19, 19, -w^3 - w^2 + 9*w + 3],\ [19, 19, -2*w^3 - 4*w^2 + 17*w + 23],\ [19, 19, w^3 + 2*w^2 - 10*w - 10],\ [29, 29, -w - 2],\ [29, 29, -w^3 - 2*w^2 + 8*w + 15],\ [29, 29, 2*w^3 + 3*w^2 - 18*w - 20],\ [29, 29, -3*w^3 - 5*w^2 + 27*w + 28],\ [31, 31, 3*w^3 + 5*w^2 - 27*w - 31],\ [31, 31, -3*w^3 - 5*w^2 + 27*w + 30],\ [31, 31, -2*w^3 - 3*w^2 + 18*w + 17],\ [31, 31, 2*w^3 + 3*w^2 - 18*w - 18],\ [71, 71, -w^3 - 2*w^2 + 10*w + 8],\ [71, 71, 4*w^3 + 8*w^2 - 35*w - 50],\ [71, 71, -4*w^3 - 7*w^2 + 36*w + 47],\ [71, 71, -w^3 - 2*w^2 + 8*w + 17],\ [79, 79, w^3 + 2*w^2 - 10*w - 14],\ [79, 79, -w^3 - w^2 + 9*w + 7],\ [79, 79, -2*w^3 - 4*w^2 + 17*w + 27],\ [79, 79, -4*w^3 - 7*w^2 + 36*w + 41],\ [81, 3, -3],\ [109, 109, -2*w^3 - 3*w^2 + 19*w + 13],\ [109, 109, 3*w^3 + 5*w^2 - 26*w - 26],\ [109, 109, 2*w^3 + 3*w^2 - 19*w - 22],\ [109, 109, 3*w^3 + 5*w^2 - 26*w - 35],\ [139, 139, 4*w^3 + 7*w^2 - 36*w - 40],\ [139, 139, -w^3 - w^2 + 9*w + 8],\ [139, 139, -w^3 - 2*w^2 + 10*w + 15],\ [139, 139, 2*w^3 + 4*w^2 - 17*w - 28],\ [149, 149, 4*w^3 + 8*w^2 - 35*w - 52],\ [149, 149, -3*w^3 - 5*w^2 + 26*w + 34],\ [149, 149, -2*w^3 - 3*w^2 + 19*w + 14],\ [149, 149, -6*w^3 - 11*w^2 + 54*w + 66],\ [181, 181, w^3 + 2*w^2 - 11*w - 13],\ [181, 181, -3*w^3 - 4*w^2 + 27*w + 24],\ [181, 181, -7*w^3 - 12*w^2 + 63*w + 72],\ [181, 181, 3*w^3 + 6*w^2 - 25*w - 39],\ [191, 191, -7*w^3 - 12*w^2 + 66*w + 80],\ [191, 191, 5*w^3 + 9*w^2 - 47*w - 57],\ [191, 191, -2*w^3 - 2*w^2 + 18*w + 9],\ [191, 191, 2*w^3 + 2*w^2 - 17*w - 13],\ [239, 239, 7*w^3 + 13*w^2 - 62*w - 81],\ [239, 239, -2*w^3 - 5*w^2 + 17*w + 32],\ [239, 239, 6*w^3 + 11*w^2 - 55*w - 67],\ [239, 239, -w^3 - 3*w^2 + 10*w + 20],\ [251, 251, 7*w^3 + 13*w^2 - 63*w - 82],\ [251, 251, -5*w^3 - 10*w^2 + 44*w + 61],\ [251, 251, 4*w^3 + 8*w^2 - 37*w - 48],\ [251, 251, -2*w^3 - 5*w^2 + 18*w + 34],\ [311, 311, 5*w^3 + 9*w^2 - 44*w - 50],\ [311, 311, -4*w^3 - 7*w^2 + 37*w + 48],\ [311, 311, 2*w^3 + 4*w^2 - 19*w - 19],\ [311, 311, w^3 + w^2 - 8*w - 11],\ [331, 331, 4*w^3 + 6*w^2 - 36*w - 37],\ [331, 331, -6*w^3 - 10*w^2 + 54*w + 59],\ [331, 331, -2*w - 1],\ [331, 331, -2*w^3 - 4*w^2 + 16*w + 27],\ [349, 349, -3*w^3 - 5*w^2 + 28*w + 27],\ [349, 349, 4*w^3 + 8*w^2 - 35*w - 53],\ [349, 349, -3*w^3 - 6*w^2 + 28*w + 40],\ [349, 349, 6*w^3 + 11*w^2 - 54*w - 65],\ [359, 359, -w^3 - 3*w^2 + 8*w + 19],\ [359, 359, -6*w^3 - 11*w^2 + 53*w + 69],\ [359, 359, 5*w^3 + 9*w^2 - 46*w - 54],\ [359, 359, w^2 - w - 8],\ [401, 401, 5*w^3 + 8*w^2 - 45*w - 50],\ [401, 401, w^3 + 2*w^2 - 7*w - 15],\ [401, 401, w^3 + 2*w^2 - 7*w - 11],\ [401, 401, 5*w^3 + 8*w^2 - 45*w - 46],\ [409, 409, -9*w^3 - 16*w^2 + 81*w + 97],\ [409, 409, w^3 + w^2 - 11*w - 10],\ [409, 409, 6*w^3 + 10*w^2 - 53*w - 66],\ [409, 409, -3*w^3 - 6*w^2 + 29*w + 38],\ [421, 421, -3*w^3 - 4*w^2 + 26*w + 23],\ [421, 421, -4*w^3 - 6*w^2 + 35*w + 39],\ [421, 421, 8*w^3 + 14*w^2 - 71*w - 86],\ [421, 421, -7*w^3 - 12*w^2 + 62*w + 70],\ [439, 439, w^3 + w^2 - 10*w - 9],\ [439, 439, -4*w^3 - 7*w^2 + 35*w + 47],\ [439, 439, 3*w^3 + 5*w^2 - 28*w - 26],\ [439, 439, 2*w^3 + 3*w^2 - 17*w - 14],\ [479, 479, 3*w^3 + 6*w^2 - 29*w - 33],\ [479, 479, 5*w^3 + 10*w^2 - 43*w - 59],\ [479, 479, w^3 - 9*w + 6],\ [479, 479, 9*w^3 + 16*w^2 - 81*w - 102],\ [521, 521, 4*w^3 + 7*w^2 - 38*w - 44],\ [521, 521, 3*w^3 + 4*w^2 - 26*w - 25],\ [521, 521, 5*w^3 + 8*w^2 - 44*w - 46],\ [521, 521, -8*w^3 - 14*w^2 + 71*w + 84],\ [569, 569, 21*w^3 + 37*w^2 - 191*w - 236],\ [569, 569, -13*w^3 - 23*w^2 + 120*w + 149],\ [569, 569, 10*w^3 + 17*w^2 - 92*w - 111],\ [569, 569, -18*w^3 - 32*w^2 + 164*w + 201],\ [631, 631, -2*w^3 - 3*w^2 + 20*w + 25],\ [631, 631, 7*w^3 + 12*w^2 - 64*w - 80],\ [631, 631, -9*w^3 - 17*w^2 + 80*w + 106],\ [631, 631, 19*w^3 + 33*w^2 - 175*w - 215],\ [641, 641, 4*w^3 + 6*w^2 - 35*w - 35],\ [641, 641, 4*w^3 + 6*w^2 - 35*w - 38],\ [641, 641, 7*w^3 + 12*w^2 - 62*w - 71],\ [641, 641, 7*w^3 + 12*w^2 - 62*w - 74],\ [659, 659, 6*w^3 + 11*w^2 - 54*w - 63],\ [659, 659, 4*w^3 + 6*w^2 - 37*w - 38],\ [659, 659, 4*w^3 + 8*w^2 - 35*w - 55],\ [659, 659, 3*w^3 + 6*w^2 - 28*w - 42],\ [661, 661, -27*w^3 - 48*w^2 + 247*w + 306],\ [661, 661, 14*w^3 + 24*w^2 - 129*w - 158],\ [661, 661, 11*w^3 + 19*w^2 - 102*w - 126],\ [661, 661, 2*w^3 + 3*w^2 - 19*w - 24],\ [691, 691, 6*w^3 + 12*w^2 - 53*w - 73],\ [691, 691, 8*w^3 + 15*w^2 - 72*w - 95],\ [691, 691, -5*w^3 - 10*w^2 + 46*w + 60],\ [691, 691, -3*w^3 - 7*w^2 + 27*w + 47],\ [739, 739, -2*w^3 - 5*w^2 + 19*w + 34],\ [739, 739, 7*w^3 + 13*w^2 - 64*w - 78],\ [739, 739, -3*w^3 - 7*w^2 + 26*w + 43],\ [739, 739, 8*w^3 + 15*w^2 - 71*w - 95],\ [751, 751, -4*w^3 - 9*w^2 + 37*w + 59],\ [751, 751, -5*w^3 - 9*w^2 + 43*w + 48],\ [751, 751, -6*w^3 - 10*w^2 + 55*w + 68],\ [751, 751, -10*w^3 - 17*w^2 + 93*w + 114],\ [769, 769, 6*w^3 + 11*w^2 - 52*w - 69],\ [769, 769, w^3 + w^2 - 7*w - 5],\ [769, 769, 2*w^3 + 2*w^2 - 19*w - 11],\ [769, 769, 7*w^3 + 12*w^2 - 64*w - 72],\ [809, 809, -3*w^3 - 4*w^2 + 29*w + 17],\ [809, 809, -5*w^3 - 8*w^2 + 43*w + 54],\ [809, 809, w^3 - 9*w + 9],\ [809, 809, 10*w^3 + 18*w^2 - 90*w - 109],\ [881, 881, -10*w^3 - 17*w^2 + 92*w + 113],\ [881, 881, -6*w^3 - 10*w^2 + 56*w + 69],\ [881, 881, -31*w^3 - 55*w^2 + 283*w + 350],\ [881, 881, -11*w^3 - 19*w^2 + 101*w + 125],\ [911, 911, 5*w^3 + 8*w^2 - 44*w - 49],\ [911, 911, -6*w^3 - 10*w^2 + 53*w + 60],\ [911, 911, 5*w^3 + 8*w^2 - 44*w - 48],\ [911, 911, 6*w^3 + 10*w^2 - 53*w - 61],\ [919, 919, 4*w^3 + 7*w^2 - 37*w - 38],\ [919, 919, 5*w^3 + 9*w^2 - 44*w - 60],\ [919, 919, 8*w^3 + 14*w^2 - 73*w - 85],\ [919, 919, -7*w^3 - 13*w^2 + 61*w + 81],\ [971, 971, 28*w^3 + 49*w^2 - 255*w - 312],\ [971, 971, 16*w^3 + 28*w^2 - 145*w - 176],\ [971, 971, -25*w^3 - 44*w^2 + 227*w + 278],\ [971, 971, 15*w^3 + 26*w^2 - 136*w - 165],\ [991, 991, 5*w^3 + 10*w^2 - 42*w - 64],\ [991, 991, -4*w^3 - 5*w^2 + 36*w + 29],\ [991, 991, -11*w^3 - 19*w^2 + 99*w + 115],\ [991, 991, 10*w^3 + 17*w^2 - 90*w - 108]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 - 2*x - 20 K. = NumberField(heckePol) hecke_eigenvalues_array = [4, -e + 2, e, -1, 4, 4, -4, -4, e + 4, e + 4, e - 6, e - 6, e, -e + 2, e, -e + 2, -2*e, 2*e - 4, 2*e - 4, -2*e, -e + 12, -e - 10, -e + 12, -e - 10, 14, -4, 4, -4, 4, -3*e - 2, -3*e - 2, -3*e + 8, -3*e + 8, -2*e - 4, -2*e - 4, -2*e + 8, -2*e + 8, 2, 2, 2, 2, e - 2, e - 2, -e, -e, -3*e, -3*e + 6, -3*e + 6, -3*e, -12, -12, -12, -12, 16, 16, 16, 16, -3*e + 18, -3*e + 18, 3*e + 12, 3*e + 12, 3*e - 10, 3*e + 4, 3*e + 4, 3*e - 10, -e - 2, -e + 4, -e - 2, -e + 4, 12, 12, 12, 12, 3*e + 2, 3*e + 2, 3*e - 8, 3*e - 8, 4*e - 24, -4*e - 16, 4*e - 24, -4*e - 16, 4*e, 4*e - 8, 4*e, 4*e - 8, -e + 26, -e + 26, -e - 24, -e - 24, -4*e + 8, 4*e, -4*e + 8, 4*e, e - 40, e - 40, e + 38, e + 38, 2*e - 12, -2*e - 8, 2*e - 12, -2*e - 8, -34, -34, -34, -34, -9*e + 14, -9*e + 14, -9*e + 4, -9*e + 4, -6*e - 6, 6*e - 18, -6*e - 6, 6*e - 18, -2*e - 8, 2*e - 12, -2*e - 8, 2*e - 12, 3*e + 20, 3*e - 26, 3*e - 26, 3*e + 20, 8*e - 16, 8*e - 16, -8*e, -8*e, -4*e - 4, -4*e - 4, -4*e + 12, -4*e + 12, 4*e - 4, 4*e - 4, 4*e - 4, 4*e - 4, -46, -46, -46, -46, -e + 22, -e + 22, e + 20, e + 20, -4*e - 32, -4*e + 40, -4*e - 32, -4*e + 40, -5*e + 26, -5*e + 26, 5*e + 16, 5*e + 16, 7*e - 12, -7*e + 2, -7*e + 2, 7*e - 12] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([16, 2, 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]