/* 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([19, 19, 4*w^3 + 7*w^2 - 36*w - 45]) 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^6 - 28*x^4 + 207*x^2 - 162 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -1/27*e^5 + 19/27*e^3 - 10/3*e, 2/27*e^5 - 29/27*e^3 + 1/3*e, -1/9*e^4 + 10/9*e^2 + 1, -1, -4/9*e^4 + 58/9*e^2 - 4, 1/27*e^5 - 19/27*e^3 + 10/3*e, 1/3*e^3 - 13/3*e, 5/9*e^4 - 77/9*e^2 + 14, 1/9*e^4 - 19/9*e^2 + 12, 2/9*e^5 - 35/9*e^3 + 35/3*e, -1/27*e^5 + 10/27*e^3 + 3*e, -2/27*e^5 + 20/27*e^3 + 5*e, 5/27*e^5 - 95/27*e^3 + 41/3*e, 1/9*e^5 - 13/9*e^3 - 5/3*e, -2/27*e^5 + 29/27*e^3 - 4/3*e, -2/9*e^4 + 11/9*e^2 + 14, -8/9*e^4 + 125/9*e^2 - 22, 11/9*e^4 - 164/9*e^2 + 22, -2/3*e^4 + 29/3*e^2 - 2, -4/9*e^4 + 76/9*e^2 - 20, -2/9*e^5 + 35/9*e^3 - 29/3*e, -7/9*e^4 + 97/9*e^2 - 4, 2/9*e^5 - 23/9*e^3 - 23/3*e, -2/9*e^4 + 29/9*e^2 - 8, 1/27*e^5 + 8/27*e^3 - 29/3*e, -e^4 + 15*e^2 - 10, 5/27*e^5 - 104/27*e^3 + 15*e, -5/9*e^4 + 59/9*e^2 + 12, -1/3*e^5 + 19/3*e^3 - 26*e, -4/3*e^3 + 52/3*e, e^4 - 15*e^2 + 28, 1/9*e^4 - 1/9*e^2 - 2, 2/9*e^4 - 20/9*e^2 + 6, -5/9*e^4 + 59/9*e^2 + 14, -5/27*e^5 + 68/27*e^3 + 1/3*e, 1/27*e^5 + 17/27*e^3 - 12*e, -2/27*e^5 + 92/27*e^3 - 89/3*e, 7/27*e^5 - 115/27*e^3 + 23/3*e, 7/27*e^5 - 142/27*e^3 + 80/3*e, -4/9*e^5 + 58/9*e^3 - 3*e, -4/27*e^5 + 103/27*e^3 - 58/3*e, 1/27*e^5 - 1/27*e^3 - 37/3*e, -1/27*e^5 + 10/27*e^3 + 4*e, 2/3*e^3 - 23/3*e, 10/9*e^4 - 136/9*e^2 - 8, 4/9*e^5 - 70/9*e^3 + 64/3*e, -2/9*e^5 + 17/9*e^3 + 37/3*e, 11/9*e^4 - 137/9*e^2 - 6, -5/9*e^4 + 86/9*e^2 - 34, -2*e^4 + 27*e^2 + 2, -2/9*e^4 + 47/9*e^2 - 24, -1/3*e^4 + 10/3*e^2 + 12, 2/3*e^4 - 32/3*e^2 + 20, 2*e^2 - 4, 14/9*e^4 - 194/9*e^2 + 4, 4/3*e^4 - 64/3*e^2 + 36, 1/27*e^5 + 53/27*e^3 - 103/3*e, -16/27*e^5 + 259/27*e^3 - 68/3*e, 4/27*e^5 - 103/27*e^3 + 82/3*e, 1/9*e^5 - 34/9*e^3 + 80/3*e, -5/27*e^5 + 95/27*e^3 - 14/3*e, 22/9*e^4 - 346/9*e^2 + 66, -2/3*e^4 + 20/3*e^2 + 16, -10/27*e^5 + 190/27*e^3 - 88/3*e, -1/3*e^5 + 7/3*e^3 + 28*e, -11/9*e^4 + 137/9*e^2 + 6, -22/27*e^5 + 400/27*e^3 - 140/3*e, -10/9*e^4 + 154/9*e^2 - 12, -5/27*e^5 + 95/27*e^3 - 56/3*e, 1/27*e^5 + 35/27*e^3 - 74/3*e, -2/9*e^5 + 23/9*e^3 + 17/3*e, -17/27*e^5 + 323/27*e^3 - 122/3*e, -1/27*e^5 + 64/27*e^3 - 19*e, -2/27*e^5 - 7/27*e^3 + 19*e, -4/9*e^4 + 76/9*e^2 - 12, -2/9*e^4 + 38/9*e^2 + 2, -2/27*e^5 + 83/27*e^3 - 73/3*e, 5/9*e^5 - 53/9*e^3 - 68/3*e, -2/27*e^5 + 65/27*e^3 - 71/3*e, -8/27*e^5 + 89/27*e^3 + 23/3*e, -11/27*e^5 + 110/27*e^3 + 17*e, -1/3*e^4 + 19/3*e^2 - 4, 2/3*e^3 - 8/3*e, -2/9*e^4 + 56/9*e^2 - 26, 14/27*e^5 - 176/27*e^3 - 14/3*e, 17/27*e^5 - 233/27*e^3 - 2/3*e, e^4 - 13*e^2 - 10, -1/9*e^4 + 1/9*e^2 + 42, 8/9*e^5 - 143/9*e^3 + 50*e, -7/9*e^5 + 124/9*e^3 - 44*e, 10/27*e^5 - 118/27*e^3 - 19/3*e, -11/27*e^5 + 92/27*e^3 + 80/3*e, 7/3*e^4 - 97/3*e^2 + 12, -2*e^4 + 26*e^2 + 6, 10/3*e^3 - 136/3*e, -7/27*e^5 + 196/27*e^3 - 161/3*e, 8/9*e^4 - 125/9*e^2 - 8, 32/9*e^4 - 455/9*e^2 + 26, 5/3*e^4 - 62/3*e^2 - 6, 1/9*e^4 - 46/9*e^2 + 24, 14/27*e^5 - 275/27*e^3 + 42*e, -2/27*e^5 - 52/27*e^3 + 125/3*e, 4/27*e^5 - 67/27*e^3 + 10*e, -26/27*e^5 + 413/27*e^3 - 80/3*e, 11/3*e^3 - 167/3*e, 7/27*e^5 - 133/27*e^3 + 34/3*e, -2/9*e^4 + 20/9*e^2 + 28, -10/9*e^4 + 154/9*e^2 - 32, -e^4 + 18*e^2 - 44, -4/9*e^4 + 49/9*e^2 - 28, -2/3*e^4 + 35/3*e^2 - 20, -16/9*e^4 + 223/9*e^2 - 10, 2/9*e^4 + 25/9*e^2 - 36, -23/9*e^4 + 338/9*e^2 - 52, 5/9*e^4 - 50/9*e^2 - 4, 14/9*e^4 - 185/9*e^2 - 2, -7/3*e^4 + 97/3*e^2 - 10, 5/27*e^5 - 122/27*e^3 + 53/3*e, -19/27*e^5 + 235/27*e^3 + 22/3*e, -1/9*e^4 - 17/9*e^2 + 14, -5/9*e^4 + 104/9*e^2 - 44, 20/9*e^4 - 317/9*e^2 + 74, 26/9*e^4 - 341/9*e^2 - 14, 14/9*e^4 - 221/9*e^2 + 52, -11/3*e^4 + 161/3*e^2 - 52, 28/9*e^4 - 424/9*e^2 + 50, 2/3*e^5 - 9*e^3 + 13/3*e, -4/27*e^5 + 130/27*e^3 - 142/3*e, -4*e^3 + 58*e, 10/9*e^4 - 190/9*e^2 + 34, -2/9*e^4 + 56/9*e^2 + 2, 10/3*e^3 - 100/3*e, -1/3*e^4 + 10/3*e^2 + 38, 20/9*e^4 - 281/9*e^2 + 38, -e^2 + 24, 1/3*e^4 - 28/3*e^2 + 44, 13/3*e^3 - 157/3*e, -11/3*e^3 + 137/3*e, -4/27*e^5 + 103/27*e^3 - 55/3*e, 1/3*e^5 - 29/3*e^3 + 172/3*e, -1/27*e^5 + 37/27*e^3 - 8*e, 4/9*e^4 - 76/9*e^2 + 64, -2/27*e^5 - 25/27*e^3 + 53/3*e, -1/3*e^4 + 7/3*e^2 + 46, -e^5 + 40/3*e^3 + 8/3*e, 2/27*e^5 - 11/27*e^3 - 4/3*e, 20/27*e^5 - 362/27*e^3 + 53*e, -5/27*e^5 + 158/27*e^3 - 48*e, 5/9*e^5 - 119/9*e^3 + 206/3*e, 29/27*e^5 - 389/27*e^3 - 16/3*e, 23/27*e^5 - 293/27*e^3 - 38/3*e, -8/27*e^5 + 215/27*e^3 - 63*e] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([19, 19, 4*w^3 + 7*w^2 - 36*w - 45])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]