/* 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([-39, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([9, 3, 3]) primes_array = [ [3, 3, w + 6],\ [3, 3, -w + 7],\ [4, 2, 2],\ [11, 11, -3*w - 17],\ [11, 11, 3*w - 20],\ [13, 13, 2*w - 13],\ [13, 13, 2*w + 11],\ [17, 17, w + 7],\ [17, 17, -w + 8],\ [19, 19, -w - 4],\ [19, 19, -w + 5],\ [25, 5, 5],\ [31, 31, -6*w - 35],\ [31, 31, -6*w + 41],\ [37, 37, -w - 1],\ [37, 37, w - 2],\ [47, 47, 3*w + 16],\ [47, 47, -3*w + 19],\ [49, 7, -7],\ [67, 67, 3*w - 22],\ [67, 67, -3*w - 19],\ [71, 71, -w - 10],\ [71, 71, w - 11],\ [89, 89, -8*w - 47],\ [89, 89, 8*w - 55],\ [101, 101, 4*w - 29],\ [101, 101, -4*w - 25],\ [109, 109, -3*w - 20],\ [109, 109, 3*w - 23],\ [113, 113, 3*w - 17],\ [113, 113, -3*w - 14],\ [127, 127, -9*w - 53],\ [127, 127, 9*w - 62],\ [157, 157, 2*w - 1],\ [167, 167, 2*w - 19],\ [167, 167, -2*w - 17],\ [173, 173, 13*w - 89],\ [173, 173, -13*w - 76],\ [193, 193, 11*w - 73],\ [193, 193, -11*w - 62],\ [197, 197, 3*w - 14],\ [197, 197, -3*w - 11],\ [199, 199, 3*w - 25],\ [199, 199, -3*w - 22],\ [233, 233, -w - 16],\ [233, 233, w - 17],\ [239, 239, 7*w - 50],\ [239, 239, -7*w - 43],\ [257, 257, 6*w + 31],\ [257, 257, -6*w + 37],\ [263, 263, 3*w - 11],\ [263, 263, -3*w - 8],\ [277, 277, -12*w - 71],\ [277, 277, 12*w - 83],\ [281, 281, -3*w - 7],\ [281, 281, 3*w - 10],\ [283, 283, -7*w + 44],\ [283, 283, 7*w + 37],\ [311, 311, 3*w - 8],\ [311, 311, -3*w - 5],\ [313, 313, -13*w - 73],\ [313, 313, 13*w - 86],\ [317, 317, -9*w + 58],\ [317, 317, 9*w + 49],\ [331, 331, -5*w - 23],\ [331, 331, 5*w - 28],\ [347, 347, -3*w - 1],\ [347, 347, 3*w - 4],\ [349, 349, 3*w - 28],\ [349, 349, -3*w - 25],\ [353, 353, 3*w - 2],\ [353, 353, 3*w - 1],\ [389, 389, 6*w - 35],\ [389, 389, 6*w + 29],\ [419, 419, 2*w - 25],\ [419, 419, -2*w - 23],\ [431, 431, 10*w - 71],\ [431, 431, -10*w - 61],\ [457, 457, 27*w + 157],\ [457, 457, 27*w - 184],\ [461, 461, -4*w - 31],\ [461, 461, 4*w - 35],\ [467, 467, -w - 22],\ [467, 467, w - 23],\ [487, 487, 8*w + 41],\ [487, 487, -8*w + 49],\ [523, 523, -6*w - 41],\ [523, 523, 6*w - 47],\ [529, 23, -23],\ [547, 547, 4*w - 11],\ [547, 547, -4*w - 7],\ [557, 557, 28*w - 191],\ [557, 557, -28*w - 163],\ [571, 571, 20*w - 133],\ [571, 571, -20*w - 113],\ [577, 577, -3*w - 29],\ [577, 577, 3*w - 32],\ [593, 593, 19*w - 131],\ [593, 593, -19*w - 112],\ [601, 601, 5*w - 22],\ [601, 601, -5*w - 17],\ [617, 617, -18*w - 101],\ [617, 617, 18*w - 119],\ [619, 619, -4*w - 1],\ [619, 619, 4*w - 5],\ [631, 631, 17*w - 112],\ [631, 631, -17*w - 95],\ [641, 641, 15*w + 83],\ [641, 641, 15*w - 98],\ [647, 647, -17*w - 101],\ [647, 647, 17*w - 118],\ [653, 653, -11*w - 68],\ [653, 653, 11*w - 79],\ [659, 659, 5*w - 43],\ [659, 659, -5*w - 38],\ [661, 661, -25*w - 142],\ [661, 661, 25*w - 167],\ [677, 677, 13*w - 92],\ [677, 677, -13*w - 79],\ [709, 709, 5*w - 19],\ [709, 709, -5*w - 14],\ [727, 727, -9*w - 58],\ [727, 727, 9*w - 67],\ [733, 733, 7*w - 38],\ [733, 733, -7*w - 31],\ [739, 739, -18*w - 107],\ [739, 739, 18*w - 125],\ [743, 743, 2*w - 31],\ [743, 743, -2*w - 29],\ [769, 769, -3*w - 32],\ [769, 769, 3*w - 35],\ [773, 773, -w - 28],\ [773, 773, w - 29],\ [797, 797, 21*w - 139],\ [797, 797, -21*w - 118],\ [821, 821, -15*w + 97],\ [821, 821, 15*w + 82],\ [827, 827, 9*w - 53],\ [827, 827, -9*w - 44],\ [829, 829, -19*w - 106],\ [829, 829, 19*w - 125],\ [841, 29, -29],\ [853, 853, 9*w - 68],\ [853, 853, -9*w - 59],\ [907, 907, 3*w - 37],\ [907, 907, -3*w - 34],\ [911, 911, 5*w - 46],\ [911, 911, -5*w - 41],\ [929, 929, -6*w - 19],\ [929, 929, 6*w - 25],\ [941, 941, -20*w - 119],\ [941, 941, 20*w - 139],\ [953, 953, -w - 31],\ [953, 953, w - 32],\ [967, 967, 11*w + 56],\ [967, 967, -11*w + 67],\ [977, 977, 16*w - 113],\ [977, 977, -16*w - 97],\ [991, 991, -8*w - 35],\ [991, 991, 8*w - 43]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^6 - 9*x^5 + 17*x^4 + 46*x^3 - 153*x^2 + 61*x + 29 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1, -1, e, -e^5 + 5*e^4 + 3*e^3 - 33*e^2 + 16*e + 4, -e^5 + 5*e^4 + 3*e^3 - 33*e^2 + 16*e + 4, -1/2*e^5 + 3*e^4 - 1/2*e^3 - 39/2*e^2 + 21*e + 3/2, -1/2*e^5 + 3*e^4 - 1/2*e^3 - 39/2*e^2 + 21*e + 3/2, -3/2*e^5 + 8*e^4 + 7/2*e^3 - 111/2*e^2 + 32*e + 29/2, -3/2*e^5 + 8*e^4 + 7/2*e^3 - 111/2*e^2 + 32*e + 29/2, e^5 - 5*e^4 - 4*e^3 + 35*e^2 - 9*e - 8, e^5 - 5*e^4 - 4*e^3 + 35*e^2 - 9*e - 8, -e^5 + 5*e^4 + 3*e^3 - 34*e^2 + 18*e + 15, 2*e^5 - 10*e^4 - 7*e^3 + 70*e^2 - 29*e - 22, 2*e^5 - 10*e^4 - 7*e^3 + 70*e^2 - 29*e - 22, -5/2*e^5 + 13*e^4 + 13/2*e^3 - 175/2*e^2 + 46*e + 27/2, -5/2*e^5 + 13*e^4 + 13/2*e^3 - 175/2*e^2 + 46*e + 27/2, e^5 - 5*e^4 - 2*e^3 + 32*e^2 - 27*e + 1, e^5 - 5*e^4 - 2*e^3 + 32*e^2 - 27*e + 1, 3/2*e^5 - 8*e^4 - 7/2*e^3 + 111/2*e^2 - 32*e - 13/2, 2*e^5 - 11*e^4 - 4*e^3 + 76*e^2 - 42*e - 21, 2*e^5 - 11*e^4 - 4*e^3 + 76*e^2 - 42*e - 21, 3*e^5 - 15*e^4 - 11*e^3 + 103*e^2 - 32*e - 26, 3*e^5 - 15*e^4 - 11*e^3 + 103*e^2 - 32*e - 26, 11/2*e^5 - 29*e^4 - 25/2*e^3 + 393/2*e^2 - 117*e - 77/2, 11/2*e^5 - 29*e^4 - 25/2*e^3 + 393/2*e^2 - 117*e - 77/2, 3/2*e^5 - 7*e^4 - 15/2*e^3 + 99/2*e^2 - 6*e - 31/2, 3/2*e^5 - 7*e^4 - 15/2*e^3 + 99/2*e^2 - 6*e - 31/2, 1/2*e^5 - 2*e^4 - 7/2*e^3 + 29/2*e^2 + e - 3/2, 1/2*e^5 - 2*e^4 - 7/2*e^3 + 29/2*e^2 + e - 3/2, 1/2*e^5 - 4*e^4 + 7/2*e^3 + 53/2*e^2 - 36*e - 11/2, 1/2*e^5 - 4*e^4 + 7/2*e^3 + 53/2*e^2 - 36*e - 11/2, -2*e^5 + 10*e^4 + 8*e^3 - 71*e^2 + 18*e + 27, -2*e^5 + 10*e^4 + 8*e^3 - 71*e^2 + 18*e + 27, 6*e^5 - 32*e^4 - 12*e^3 + 216*e^2 - 138*e - 42, -e^4 + 3*e^3 + 5*e^2 - 15*e + 18, -e^4 + 3*e^3 + 5*e^2 - 15*e + 18, -e^4 + 4*e^3 + 6*e^2 - 20*e - 11, -e^4 + 4*e^3 + 6*e^2 - 20*e - 11, -e^5 + 6*e^4 - 40*e^2 + 37*e - 4, -e^5 + 6*e^4 - 40*e^2 + 37*e - 4, -9/2*e^5 + 24*e^4 + 21/2*e^3 - 329/2*e^2 + 90*e + 55/2, -9/2*e^5 + 24*e^4 + 21/2*e^3 - 329/2*e^2 + 90*e + 55/2, -3*e^5 + 14*e^4 + 14*e^3 - 96*e^2 + 17*e + 22, -3*e^5 + 14*e^4 + 14*e^3 - 96*e^2 + 17*e + 22, 9/2*e^5 - 23*e^4 - 25/2*e^3 + 309/2*e^2 - 86*e - 41/2, 9/2*e^5 - 23*e^4 - 25/2*e^3 + 309/2*e^2 - 86*e - 41/2, 2*e^5 - 10*e^4 - 5*e^3 + 67*e^2 - 43*e - 25, 2*e^5 - 10*e^4 - 5*e^3 + 67*e^2 - 43*e - 25, 4*e^5 - 20*e^4 - 14*e^3 + 140*e^2 - 60*e - 44, 4*e^5 - 20*e^4 - 14*e^3 + 140*e^2 - 60*e - 44, 2*e^5 - 10*e^4 - 7*e^3 + 71*e^2 - 27*e - 39, 2*e^5 - 10*e^4 - 7*e^3 + 71*e^2 - 27*e - 39, -15/2*e^5 + 39*e^4 + 37/2*e^3 - 531/2*e^2 + 157*e + 135/2, -15/2*e^5 + 39*e^4 + 37/2*e^3 - 531/2*e^2 + 157*e + 135/2, -5/2*e^5 + 13*e^4 + 11/2*e^3 - 175/2*e^2 + 65*e + 15/2, -5/2*e^5 + 13*e^4 + 11/2*e^3 - 175/2*e^2 + 65*e + 15/2, e^5 - 5*e^4 - 3*e^3 + 32*e^2 - 12*e - 5, e^5 - 5*e^4 - 3*e^3 + 32*e^2 - 12*e - 5, -e^5 + 6*e^4 - 40*e^2 + 31*e + 8, -e^5 + 6*e^4 - 40*e^2 + 31*e + 8, 1/2*e^5 - 4*e^4 + 7/2*e^3 + 55/2*e^2 - 40*e - 9/2, 1/2*e^5 - 4*e^4 + 7/2*e^3 + 55/2*e^2 - 40*e - 9/2, 9/2*e^5 - 24*e^4 - 23/2*e^3 + 331/2*e^2 - 85*e - 53/2, 9/2*e^5 - 24*e^4 - 23/2*e^3 + 331/2*e^2 - 85*e - 53/2, -e^4 + 3*e^3 + 11*e^2 - 21*e - 18, -e^4 + 3*e^3 + 11*e^2 - 21*e - 18, 2*e^5 - 11*e^4 - 2*e^3 + 74*e^2 - 64*e + 1, 2*e^5 - 11*e^4 - 2*e^3 + 74*e^2 - 64*e + 1, 9/2*e^5 - 20*e^4 - 49/2*e^3 + 281/2*e^2 - 14*e - 91/2, 9/2*e^5 - 20*e^4 - 49/2*e^3 + 281/2*e^2 - 14*e - 91/2, 5*e^5 - 27*e^4 - 10*e^3 + 184*e^2 - 105*e - 53, 5*e^5 - 27*e^4 - 10*e^3 + 184*e^2 - 105*e - 53, 3/2*e^5 - 7*e^4 - 15/2*e^3 + 91/2*e^2 - 6*e + 33/2, 3/2*e^5 - 7*e^4 - 15/2*e^3 + 91/2*e^2 - 6*e + 33/2, -2*e^5 + 10*e^4 + 7*e^3 - 67*e^2 + 27*e - 5, -2*e^5 + 10*e^4 + 7*e^3 - 67*e^2 + 27*e - 5, -3*e^5 + 16*e^4 + 4*e^3 - 104*e^2 + 91*e, -3*e^5 + 16*e^4 + 4*e^3 - 104*e^2 + 91*e, 21/2*e^5 - 55*e^4 - 55/2*e^3 + 755/2*e^2 - 197*e - 171/2, 21/2*e^5 - 55*e^4 - 55/2*e^3 + 755/2*e^2 - 197*e - 171/2, 5/2*e^5 - 13*e^4 - 11/2*e^3 + 175/2*e^2 - 65*e - 15/2, 5/2*e^5 - 13*e^4 - 11/2*e^3 + 175/2*e^2 - 65*e - 15/2, -2*e^5 + 10*e^4 + 8*e^3 - 70*e^2 + 16*e + 10, -2*e^5 + 10*e^4 + 8*e^3 - 70*e^2 + 16*e + 10, -e^5 + 4*e^4 + 4*e^3 - 22*e^2 + 23*e - 24, -e^5 + 4*e^4 + 4*e^3 - 22*e^2 + 23*e - 24, -4*e^5 + 23*e^4 + 2*e^3 - 155*e^2 + 128*e + 32, -4*e^5 + 23*e^4 + 2*e^3 - 155*e^2 + 128*e + 32, 11/2*e^5 - 29*e^4 - 25/2*e^3 + 391/2*e^2 - 125*e - 7/2, -3*e^5 + 16*e^4 + 7*e^3 - 110*e^2 + 56*e + 30, -3*e^5 + 16*e^4 + 7*e^3 - 110*e^2 + 56*e + 30, -5*e^5 + 28*e^4 + 6*e^3 - 192*e^2 + 143*e + 38, -5*e^5 + 28*e^4 + 6*e^3 - 192*e^2 + 143*e + 38, -2*e^5 + 8*e^4 + 17*e^3 - 63*e^2 - 41*e + 39, -2*e^5 + 8*e^4 + 17*e^3 - 63*e^2 - 41*e + 39, -19/2*e^5 + 48*e^4 + 63/2*e^3 - 661/2*e^2 + 142*e + 131/2, -19/2*e^5 + 48*e^4 + 63/2*e^3 - 661/2*e^2 + 142*e + 131/2, -9/2*e^5 + 22*e^4 + 35/2*e^3 - 303/2*e^2 + 43*e + 85/2, -9/2*e^5 + 22*e^4 + 35/2*e^3 - 303/2*e^2 + 43*e + 85/2, -3/2*e^5 + 10*e^4 - 3/2*e^3 - 149/2*e^2 + 57*e + 83/2, -3/2*e^5 + 10*e^4 - 3/2*e^3 - 149/2*e^2 + 57*e + 83/2, -5/2*e^5 + 13*e^4 + 15/2*e^3 - 183/2*e^2 + 49*e + 43/2, -5/2*e^5 + 13*e^4 + 15/2*e^3 - 183/2*e^2 + 49*e + 43/2, 9*e^5 - 46*e^4 - 28*e^3 + 315*e^2 - 131*e - 77, 9*e^5 - 46*e^4 - 28*e^3 + 315*e^2 - 131*e - 77, -11*e^5 + 58*e^4 + 26*e^3 - 396*e^2 + 223*e + 96, -11*e^5 + 58*e^4 + 26*e^3 - 396*e^2 + 223*e + 96, -3*e^5 + 16*e^4 + 6*e^3 - 102*e^2 + 57*e - 4, -3*e^5 + 16*e^4 + 6*e^3 - 102*e^2 + 57*e - 4, 4*e^5 - 20*e^4 - 17*e^3 + 141*e^2 - 33*e - 13, 4*e^5 - 20*e^4 - 17*e^3 + 141*e^2 - 33*e - 13, 3*e^5 - 16*e^4 - 8*e^3 + 112*e^2 - 53*e - 16, 3*e^5 - 16*e^4 - 8*e^3 + 112*e^2 - 53*e - 16, -3*e^5 + 15*e^4 + 6*e^3 - 92*e^2 + 69*e - 11, -3*e^5 + 15*e^4 + 6*e^3 - 92*e^2 + 69*e - 11, -3/2*e^5 + 7*e^4 + 13/2*e^3 - 99/2*e^2 + 19*e + 55/2, -3/2*e^5 + 7*e^4 + 13/2*e^3 - 99/2*e^2 + 19*e + 55/2, -3*e^5 + 16*e^4 + 8*e^3 - 112*e^2 + 53*e + 40, -3*e^5 + 16*e^4 + 8*e^3 - 112*e^2 + 53*e + 40, 2*e^2 + 6*e - 22, 2*e^2 + 6*e - 22, -6*e^5 + 33*e^4 + 7*e^3 - 217*e^2 + 163*e + 10, -6*e^5 + 33*e^4 + 7*e^3 - 217*e^2 + 163*e + 10, 6*e^5 - 30*e^4 - 17*e^3 + 198*e^2 - 105*e - 30, 6*e^5 - 30*e^4 - 17*e^3 + 198*e^2 - 105*e - 30, -8*e^5 + 40*e^4 + 30*e^3 - 281*e^2 + 90*e + 99, -8*e^5 + 40*e^4 + 30*e^3 - 281*e^2 + 90*e + 99, -5*e^5 + 27*e^4 + 11*e^3 - 185*e^2 + 100*e + 34, -5*e^5 + 27*e^4 + 11*e^3 - 185*e^2 + 100*e + 34, 3/2*e^5 - 9*e^4 + 11/2*e^3 + 103/2*e^2 - 91*e + 1/2, 3/2*e^5 - 9*e^4 + 11/2*e^3 + 103/2*e^2 - 91*e + 1/2, 15/2*e^5 - 38*e^4 - 47/2*e^3 + 527/2*e^2 - 124*e - 165/2, 15/2*e^5 - 38*e^4 - 47/2*e^3 + 527/2*e^2 - 124*e - 165/2, -3*e^5 + 15*e^4 + 16*e^3 - 114*e^2 - 5*e + 57, -3*e^5 + 15*e^4 + 16*e^3 - 114*e^2 - 5*e + 57, -15/2*e^5 + 38*e^4 + 49/2*e^3 - 533/2*e^2 + 113*e + 195/2, -15/2*e^5 + 38*e^4 + 49/2*e^3 - 533/2*e^2 + 113*e + 195/2, -14*e^5 + 74*e^4 + 31*e^3 - 503*e^2 + 309*e + 101, -14*e^5 + 74*e^4 + 31*e^3 - 503*e^2 + 309*e + 101, 3/2*e^5 - 10*e^4 + 5/2*e^3 + 145/2*e^2 - 64*e - 75/2, 3/2*e^5 - 10*e^4 + 5/2*e^3 + 145/2*e^2 - 64*e - 75/2, 6*e^5 - 30*e^4 - 17*e^3 + 196*e^2 - 109*e + 4, 6*e^5 - 32*e^4 - 11*e^3 + 212*e^2 - 139*e - 14, 6*e^5 - 32*e^4 - 11*e^3 + 212*e^2 - 139*e - 14, -e^4 + 4*e^3 + 8*e^2 - 32*e - 27, -e^4 + 4*e^3 + 8*e^2 - 32*e - 27, -6*e^5 + 32*e^4 + 14*e^3 - 216*e^2 + 116*e + 28, -6*e^5 + 32*e^4 + 14*e^3 - 216*e^2 + 116*e + 28, 19/2*e^5 - 51*e^4 - 41/2*e^3 + 705/2*e^2 - 213*e - 177/2, 19/2*e^5 - 51*e^4 - 41/2*e^3 + 705/2*e^2 - 213*e - 177/2, -7*e^5 + 36*e^4 + 24*e^3 - 258*e^2 + 103*e + 96, -7*e^5 + 36*e^4 + 24*e^3 - 258*e^2 + 103*e + 96, 1/2*e^5 - 6*e^4 + 31/2*e^3 + 57/2*e^2 - 114*e + 57/2, 1/2*e^5 - 6*e^4 + 31/2*e^3 + 57/2*e^2 - 114*e + 57/2, -6*e^5 + 27*e^4 + 32*e^3 - 184*e^2 + 2*e + 53, -6*e^5 + 27*e^4 + 32*e^3 - 184*e^2 + 2*e + 53, 11/2*e^5 - 30*e^4 - 25/2*e^3 + 413/2*e^2 - 105*e - 103/2, 11/2*e^5 - 30*e^4 - 25/2*e^3 + 413/2*e^2 - 105*e - 103/2, 3*e^4 - 9*e^3 - 25*e^2 + 57*e + 12, 3*e^4 - 9*e^3 - 25*e^2 + 57*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([3, 3, w + 6])] = 1 AL_eigenvalues[ZF.ideal([3, 3, -w + 7])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]