/* 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([7, 6, -5, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([28,14,w^3 - 4*w - 1]) primes_array = [ [4, 2, -w^2 + w + 3],\ [7, 7, w],\ [7, 7, -w^2 + 2*w + 1],\ [7, 7, w^2 - 2],\ [7, 7, w - 1],\ [23, 23, -w^3 + w^2 + 3*w - 1],\ [23, 23, w^3 - 2*w^2 - 2*w + 2],\ [31, 31, w^2 - 5],\ [31, 31, -w^2 + 2*w + 4],\ [41, 41, -w^3 + 2*w^2 + 2*w - 6],\ [41, 41, w^3 - w^2 - 3*w - 3],\ [47, 47, w^2 - 2*w - 5],\ [47, 47, w^2 - 6],\ [71, 71, -w^3 + 2*w^2 + 3*w - 1],\ [71, 71, -w^3 + w^2 + 4*w - 3],\ [79, 79, -w - 3],\ [79, 79, w - 4],\ [81, 3, -3],\ [89, 89, w^2 - 3*w - 2],\ [89, 89, w^2 + w - 4],\ [97, 97, 2*w^3 - 5*w^2 - 4*w + 9],\ [97, 97, -2*w^3 + w^2 + 8*w + 2],\ [113, 113, w^3 - 2*w^2 - 4*w + 2],\ [113, 113, w^3 - w^2 - 5*w + 3],\ [121, 11, 2*w^2 - w - 9],\ [121, 11, -2*w^2 + 3*w + 8],\ [127, 127, w^3 - 3*w^2 - 2*w + 5],\ [127, 127, w^3 - 5*w - 1],\ [137, 137, 2*w - 1],\ [167, 167, 2*w^3 - 2*w^2 - 7*w - 2],\ [167, 167, w^3 - 7*w - 4],\ [167, 167, 2*w^3 - w^2 - 9*w - 2],\ [167, 167, 2*w^3 - 4*w^2 - 5*w + 9],\ [191, 191, 2*w^3 - 4*w^2 - 5*w + 5],\ [191, 191, -2*w^3 + 2*w^2 + 7*w - 2],\ [193, 193, -2*w^3 + 3*w^2 + 6*w - 6],\ [193, 193, -2*w^3 + 2*w^2 + 7*w - 4],\ [193, 193, -2*w^3 + 4*w^2 + 5*w - 3],\ [193, 193, 2*w^3 - 3*w^2 - 6*w + 1],\ [199, 199, -w^3 - 2*w^2 + 6*w + 10],\ [199, 199, -w^3 + 5*w^2 - w - 13],\ [223, 223, -w^3 + 2*w^2 + 3*w - 8],\ [223, 223, w^3 - w^2 - 4*w - 4],\ [233, 233, -w^2 - 1],\ [233, 233, w^2 - 2*w + 2],\ [239, 239, 3*w^2 - 2*w - 13],\ [239, 239, 3*w^2 - 4*w - 12],\ [241, 241, w^3 - w^2 - 6*w + 3],\ [241, 241, w^3 - 2*w^2 - 5*w + 3],\ [257, 257, -w^3 + 3*w^2 + 4*w - 9],\ [257, 257, -2*w^3 + 4*w^2 + 8*w - 11],\ [257, 257, -w^3 + 2*w^2 + 4],\ [257, 257, w^3 - 7*w - 3],\ [263, 263, -w^3 + w^2 + 3*w + 5],\ [263, 263, w^3 - 5*w^2 + 12],\ [263, 263, -w^3 - 2*w^2 + 7*w + 8],\ [263, 263, -w^3 + 7*w + 2],\ [271, 271, -w^3 + 3*w^2 + 2*w - 3],\ [271, 271, w^3 - 5*w + 1],\ [289, 17, 3*w^2 - 3*w - 8],\ [289, 17, 3*w^2 - 3*w - 10],\ [311, 311, -2*w^3 + 2*w^2 + 7*w - 1],\ [311, 311, -w^3 + 5*w^2 - w - 11],\ [311, 311, w^3 + 2*w^2 - 6*w - 8],\ [311, 311, -2*w^3 + 4*w^2 + 5*w - 6],\ [337, 337, w^3 - w^2 - 2*w - 4],\ [337, 337, 2*w^3 - 9*w - 4],\ [337, 337, 2*w^3 - 6*w^2 - 3*w + 11],\ [337, 337, -w^3 + 2*w^2 + w - 6],\ [353, 353, -w^3 + 4*w^2 - 12],\ [353, 353, w^3 + w^2 - 5*w - 9],\ [359, 359, 2*w^3 - 4*w^2 - 6*w + 5],\ [359, 359, 2*w^3 - 2*w^2 - 8*w + 3],\ [361, 19, -w^3 + 5*w^2 - 13],\ [361, 19, w^3 + 2*w^2 - 7*w - 9],\ [367, 367, 2*w^3 - 10*w - 5],\ [367, 367, w^3 - w^2 - 6*w + 2],\ [367, 367, -w^3 + 2*w^2 + 5*w - 4],\ [367, 367, 2*w^3 - 6*w^2 - 4*w + 13],\ [401, 401, -w^3 + 2*w^2 + 5*w - 5],\ [401, 401, -w^3 + w^2 + 6*w - 1],\ [409, 409, 2*w^3 - 3*w^2 - 6*w + 4],\ [409, 409, 3*w^3 - 3*w^2 - 12*w + 1],\ [409, 409, -3*w^3 + 6*w^2 + 9*w - 11],\ [409, 409, -2*w^3 + 3*w^2 + 6*w - 3],\ [431, 431, 3*w^2 - 5*w - 10],\ [431, 431, w^3 + w^2 - 8*w - 3],\ [457, 457, -2*w^3 + 4*w^2 + 4*w - 5],\ [457, 457, -2*w^3 + 2*w^2 + 6*w - 1],\ [463, 463, -2*w^3 + 7*w^2 + w - 16],\ [463, 463, 3*w^3 - 6*w^2 - 8*w + 8],\ [503, 503, -w^3 + 3*w^2 + 2*w - 12],\ [503, 503, -w^3 + 4*w^2 + 4*w - 10],\ [521, 521, w^3 + w^2 - 6*w - 2],\ [521, 521, -w^3 + 4*w^2 + w - 6],\ [529, 23, w^2 - w - 8],\ [569, 569, w^3 - 3*w - 6],\ [569, 569, -w^3 + 3*w^2 - 8],\ [577, 577, -w^3 + 4*w^2 - w - 10],\ [577, 577, w^3 + w^2 - 4*w - 8],\ [593, 593, -w^3 + 3*w^2 - 12],\ [593, 593, w^3 - 3*w - 10],\ [599, 599, w^3 + 3*w^2 - 7*w - 9],\ [599, 599, -w^3 + 3*w^2 + 3*w - 13],\ [601, 601, w^2 + 2*w - 4],\ [601, 601, w^2 - 4*w - 1],\ [607, 607, 2*w^2 - 13],\ [607, 607, w^3 + 2*w^2 - 7*w - 5],\ [607, 607, 2*w^3 - 3*w^2 - 8*w + 8],\ [607, 607, 2*w^2 - 4*w - 11],\ [617, 617, w^3 + w^2 - 9*w - 1],\ [617, 617, w^3 - w^2 - 4*w - 5],\ [617, 617, -w^3 + 2*w^2 + 3*w - 9],\ [617, 617, -w^3 + 4*w^2 + 4*w - 8],\ [625, 5, -5],\ [631, 631, -w^3 + 5*w^2 - 2*w - 13],\ [631, 631, w^3 + 2*w^2 - 5*w - 11],\ [641, 641, w^2 - 2*w + 3],\ [641, 641, -w^3 + 6*w - 2],\ [641, 641, -w^3 + 3*w^2 + 3*w - 3],\ [641, 641, -w^2 - 2],\ [647, 647, -2*w^3 + 5*w^2 + 4*w - 6],\ [647, 647, 2*w^3 - 4*w^2 - 5*w - 1],\ [647, 647, 3*w^3 - 7*w^2 - 7*w + 13],\ [647, 647, -2*w^3 + w^2 + 8*w - 1],\ [673, 673, -w^3 + 7*w - 2],\ [673, 673, -w^3 + 3*w^2 + 4*w - 4],\ [719, 719, -2*w^3 + 5*w^2 + 3*w - 12],\ [719, 719, 4*w^2 - 5*w - 10],\ [719, 719, -4*w^2 + 3*w + 11],\ [719, 719, 2*w^3 - w^2 - 7*w - 6],\ [727, 727, -w - 5],\ [727, 727, w - 6],\ [743, 743, w^2 + 2*w - 11],\ [743, 743, w^3 + 4*w^2 - 11*w - 11],\ [751, 751, -2*w^2 + w + 13],\ [751, 751, 2*w^2 - 3*w - 12],\ [769, 769, 2*w^3 - 6*w^2 - 3*w + 17],\ [769, 769, 3*w^3 - 2*w^2 - 11*w + 2],\ [809, 809, w^3 + 3*w^2 - 6*w - 11],\ [809, 809, -3*w^3 + 5*w^2 + 8*w - 11],\ [823, 823, -w^3 + w^2 + 2*w - 6],\ [823, 823, 4*w^2 - 5*w - 11],\ [823, 823, -4*w^2 + 3*w + 12],\ [823, 823, -w^3 + 4*w^2 + 4*w - 12],\ [839, 839, -2*w^3 + w^2 + 6*w + 5],\ [839, 839, -3*w^3 + 8*w^2 + 7*w - 17],\ [839, 839, 3*w^3 - w^2 - 14*w - 5],\ [839, 839, -2*w^3 + 5*w^2 + 2*w - 10],\ [857, 857, w^3 + w^2 - 8*w - 2],\ [857, 857, -w^3 + 4*w^2 + 3*w - 8],\ [863, 863, -w^3 + w^2 - w - 1],\ [863, 863, w^3 - 2*w^2 + 2*w - 2],\ [911, 911, -w^3 + 3*w^2 - 11],\ [911, 911, w^3 - 3*w - 9],\ [919, 919, -2*w^3 + w^2 + 8*w - 2],\ [919, 919, -2*w^3 + 5*w^2 + 4*w - 5],\ [929, 929, -2*w^3 + 2*w^2 + 9*w - 4],\ [929, 929, 2*w^3 - 9*w - 12],\ [929, 929, 2*w^3 - 6*w^2 - 3*w + 19],\ [929, 929, -2*w^3 + 4*w^2 + 7*w - 5],\ [937, 937, -w^3 + 2*w^2 - 6],\ [937, 937, -w^3 + 3*w^2 - 10],\ [937, 937, w^3 - 3*w - 8],\ [937, 937, w^3 - w^2 - w - 5],\ [953, 953, 2*w^3 - 6*w^2 - 3*w + 18],\ [953, 953, -3*w^3 + 3*w^2 + 11*w - 5],\ [961, 31, 4*w^2 - 4*w - 11],\ [967, 967, -w^3 + 7*w - 3],\ [967, 967, -2*w^3 + 2*w^2 + 11*w + 2],\ [967, 967, -w^3 - 4*w^2 + 9*w + 9],\ [967, 967, -w^3 + 3*w^2 + 4*w - 3],\ [977, 977, 2*w^2 - 5*w - 8],\ [977, 977, -w^3 + 2*w^2 - w + 6],\ [977, 977, -w^3 + w^2 - 6],\ [977, 977, 2*w^2 + w - 11],\ [983, 983, -w^3 + 6*w^2 - 2*w - 16],\ [983, 983, w^3 + 3*w^2 - 7*w - 13]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 - 4*x^3 - 13*x^2 + 34*x + 71 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1, -1/2*e^2 + 1/2*e + 13/2, e^3 - 5/2*e^2 - 15/2*e + 7/2, -1, e, 1/2*e^3 + 3/2*e^2 - 19/2*e - 21, -1/2*e^3 - e^2 + 9*e + 35/2, -e^3 + 3/2*e^2 + 17/2*e + 15/2, 9/2*e^2 - 19/2*e - 75/2, e^3 - 3*e^2 - 6*e + 14, -e^2 + e + 9, -1/2*e^3 + 2*e^2 + 3*e - 3/2, -1/2*e^3 + 13/2*e^2 - 17/2*e - 45, e^3 - 7*e^2 + 3*e + 44, -2*e^3 + 3*e^2 + 19*e + 11, e^3 - 7*e^2 + 2*e + 39, -1/2*e^3 + 23/2*e^2 - 35/2*e - 89, -1/2*e^3 + e^2 + 3*e - 9/2, 3*e^3 - 12*e^2 - 13*e + 52, -1/2*e^3 + e^2 + 4*e + 15/2, 1/2*e^3 - 7/2*e^2 - 1/2*e + 30, e^3 - 13/2*e^2 - 3/2*e + 83/2, -e^3 + 5*e^2 + 2*e - 25, e^3 - 8*e^2 + 4*e + 53, 27/2*e^2 - 51/2*e - 231/2, e^3 - 4*e^2 - 7*e + 27, e^3 - 8*e^2 + 5*e + 47, -23/2*e^2 + 45/2*e + 197/2, 3/2*e^3 - 11*e^2 + 7*e + 139/2, -2*e^3 + 7*e^2 + 15*e - 29, 2*e^3 - 6*e^2 - 16*e + 26, e^3 - 17*e^2 + 19*e + 126, 1/2*e^3 - 10*e^2 + 15*e + 147/2, -9/2*e^3 + 19*e^2 + 18*e - 165/2, -10*e^2 + 21*e + 85, 2*e^3 - 2*e^2 - 24*e - 18, e^3 - 12*e^2 + 12*e + 81, -9/2*e^3 + 15*e^2 + 28*e - 109/2, -6*e^3 + 15*e^2 + 47*e - 33, 4*e^3 - 15/2*e^2 - 71/2*e - 15/2, -2*e^3 + 10*e^2 + 10*e - 56, -1/2*e^3 - 7*e^2 + 18*e + 131/2, -3*e^3 - 5/2*e^2 + 93/2*e + 123/2, -3*e^3 - e^2 + 39*e + 56, e^3 - e^2 - 13*e + 2, 3*e^3 - 13/2*e^2 - 61/2*e + 21/2, 2*e^3 + 6*e^2 - 38*e - 82, -5*e^3 + 23/2*e^2 + 79/2*e - 31/2, -5/2*e^3 + 4*e^2 + 24*e + 25/2, -e^3 + 19/2*e^2 - 17/2*e - 111/2, -3*e^3 + 19*e^2 + 5*e - 104, 2*e^3 - 3*e^2 - 15*e - 25, -3/2*e^3 + 9*e^2 - 2*e - 111/2, 2*e^3 + 7/2*e^2 - 67/2*e - 111/2, 9/2*e^3 - 35/2*e^2 - 45/2*e + 75, -16*e^2 + 29*e + 137, -5/2*e^3 + 15*e^2 + 3*e - 149/2, 1/2*e^3 + 14*e^2 - 30*e - 269/2, 3*e^3 - 3*e^2 - 30*e - 32, 4*e^3 + 1/2*e^2 - 99/2*e - 167/2, 4*e^3 - 45/2*e^2 - 11/2*e + 245/2, -11/2*e^3 + 23*e^2 + 22*e - 211/2, -e^3 - 33/2*e^2 + 89/2*e + 315/2, -2*e^3 + 13*e^2 - 3*e - 77, 1/2*e^3 + 17*e^2 - 37*e - 315/2, -3*e^3 + 8*e^2 + 22*e - 11, -9/2*e^3 + 23*e^2 + 9*e - 237/2, 3*e^3 + 3/2*e^2 - 87/2*e - 133/2, -9/2*e^3 + 17*e^2 + 18*e - 129/2, 3/2*e^2 - 3/2*e + 9/2, 2*e^3 + 5*e^2 - 31*e - 87, -e^3 + 39/2*e^2 - 59/2*e - 289/2, -3*e^3 + e^2 + 35*e + 44, 3*e^3 - 9*e^2 - 19*e + 40, -7/2*e^3 + 8*e^2 + 33*e - 17/2, -e^2 + e + 3, -3*e^3 + 3*e^2 + 27*e + 32, 3*e^3 - 16*e^2 - 4*e + 85, 11/2*e^3 - 11*e^2 - 48*e + 31/2, -5/2*e^3 + e^2 + 31*e + 75/2, 1/2*e^3 + 13/2*e^2 - 53/2*e - 52, 5*e^3 - 61/2*e^2 - 11/2*e + 339/2, -3*e^3 + 63/2*e^2 - 49/2*e - 435/2, 7/2*e^3 - 6*e^2 - 31*e - 31/2, e^3 - 23/2*e^2 + 31/2*e + 149/2, -9/2*e^3 + 23*e^2 + 15*e - 237/2, -5*e^3 + 3*e^2 + 59*e + 56, -8*e^2 + 16*e + 58, e^3 - 23/2*e^2 + 15/2*e + 165/2, -4*e^3 + 23*e^2 + 5*e - 135, -1/2*e^3 + 6*e^2 - 107/2, -e^3 + 13*e^2 - 15*e - 96, 2*e^3 - 20*e^2 + 19*e + 142, 3*e^3 + 5*e^2 - 55*e - 72, 5/2*e^3 - 5/2*e^2 - 47/2*e - 11, 3*e^3 - 34*e - 73, -7/2*e^3 + 39/2*e^2 + 25/2*e - 105, 2*e^3 - 7*e^2 - 19*e + 39, -9/2*e^3 + 13*e^2 + 31*e - 65/2, e^3 + 7/2*e^2 - 49/2*e - 91/2, 2*e^3 - 39*e^2 + 55*e + 301, 3/2*e^3 - 15/2*e^2 - 9/2*e + 28, -2*e^3 - 2*e^2 + 38*e + 28, -3*e^3 - e^2 + 35*e + 70, -2*e^3 + 3*e^2 + 21*e + 9, 4*e^3 - 83/2*e^2 + 61/2*e + 561/2, -5/2*e^3 - 1/2*e^2 + 73/2*e + 54, -e^3 + 6*e^2 + 4*e - 9, -e^3 + 11/2*e^2 + 11/2*e - 111/2, -1/2*e^3 + 13*e^2 - 17*e - 169/2, -4*e^3 - 21/2*e^2 + 137/2*e + 325/2, -15/2*e^3 + 59/2*e^2 + 73/2*e - 136, -9/2*e^3 + 23*e^2 + 15*e - 229/2, 4*e^2 - 10*e - 16, -e^3 + 8*e + 5, 3/2*e^3 - 8*e^2 + 2*e + 101/2, 5/2*e^3 - 4*e^2 - 22*e - 53/2, e^3 - 2*e^2 - 11*e - 16, -3/2*e^3 - 4*e^2 + 33*e + 71/2, 3*e^3 - e^2 - 39*e - 36, -4*e^3 + 33/2*e^2 + 29/2*e - 167/2, e^3 - 37/2*e^2 + 41/2*e + 319/2, -4*e^3 + 13/2*e^2 + 65/2*e + 73/2, -3*e^3 + 5/2*e^2 + 65/2*e + 67/2, e^3 + 7*e^2 - 27*e - 78, -2*e^3 + 39/2*e^2 - 19/2*e - 271/2, -4*e^3 + 25/2*e^2 + 51/2*e - 65/2, 6*e^3 - 85/2*e^2 + 17/2*e + 509/2, 3*e^3 - 1/2*e^2 - 85/2*e - 39/2, 1/2*e^3 - 21*e^2 + 36*e + 309/2, -e^3 - 35/2*e^2 + 83/2*e + 357/2, 7/2*e^3 - 45/2*e^2 + 9/2*e + 156, 5*e^3 - 27*e^2 - 10*e + 147, 13/2*e^3 - 5*e^2 - 75*e - 99/2, e^3 - 7/2*e^2 - 5/2*e + 37/2, 4*e^3 + 3*e^2 - 59*e - 75, 9/2*e^3 - 21/2*e^2 - 91/2*e + 24, 4*e^3 - 17*e^2 - 11*e + 99, 2*e^3 - 12*e^2 + 4*e + 76, -4*e^3 + 22*e^2 + 6*e - 112, -5*e^3 + 12*e^2 + 33*e + 9, 7/2*e^2 - 37/2*e - 21/2, -5/2*e^3 - 5/2*e^2 + 53/2*e + 84, -17/2*e^3 + 53/2*e^2 + 103/2*e - 84, 2*e^3 - 22*e^2 + 12*e + 166, -3*e^3 + 14*e^2 + 16*e - 57, 8*e^3 - 24*e^2 - 56*e + 70, 5/2*e^3 - 8*e^2 - 19*e + 95/2, -2*e^3 + 7*e^2 + 13*e - 21, 15/2*e^3 - 53/2*e^2 - 99/2*e + 108, 15/2*e^3 - 5*e^2 - 94*e - 117/2, 5*e^3 - 43/2*e^2 - 55/2*e + 187/2, 6*e^3 - 15*e^2 - 51*e + 37, -25/2*e^2 + 47/2*e + 255/2, -11/2*e^3 + 49/2*e^2 + 35/2*e - 108, 4*e^3 - 19*e^2 - 9*e + 123, -3*e^3 - 4*e^2 + 50*e + 41, 3*e^3 + 11/2*e^2 - 89/2*e - 287/2, -3*e^3 + 25/2*e^2 + 37/2*e - 117/2, -3*e^3 - 27/2*e^2 + 149/2*e + 303/2, -5*e^3 + 2*e^2 + 54*e + 63, -7*e^3 + 35/2*e^2 + 103/2*e - 91/2, -7*e^3 + 67/2*e^2 + 51/2*e - 271/2, 2*e^3 - 27*e^2 + 31*e + 179, 7/2*e^3 - 8*e^2 - 36*e + 21/2, 2*e^3 - 35/2*e^2 + 27/2*e + 167/2, -9/2*e^3 + 20*e^2 + 27*e - 199/2, 1/2*e^3 + e^2 - 4*e - 67/2, 3*e^3 + 19*e^2 - 77*e - 204, -7/2*e^3 - 43/2*e^2 + 175/2*e + 243, -11/2*e^3 + 24*e^2 + 20*e - 229/2, 2*e^3 - 49/2*e^2 + 61/2*e + 389/2, -7*e^3 + 21*e^2 + 45*e - 48, -6*e^3 + 22*e^2 + 35*e - 92, -6*e^3 + 39/2*e^2 + 65/2*e - 119/2, 4*e^3 - 25*e^2 - 11*e + 149, -13/2*e^3 + 49/2*e^2 + 75/2*e - 97, -17/2*e^3 + 8*e^2 + 88*e + 161/2] 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,-w^2 + w + 3])] = 1 AL_eigenvalues[ZF.ideal([7,7,-w^2 + 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]