/* 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([-5, -9, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([5, 5, w]) primes_array = [ [2, 2, -w - 1],\ [5, 5, w],\ [7, 7, w^2 - 2*w - 8],\ [7, 7, -2*w^2 + 3*w + 16],\ [7, 7, -w^2 + 2*w + 6],\ [11, 11, w^2 - 2*w - 2],\ [19, 19, -w + 2],\ [23, 23, -w^2 + 3*w + 1],\ [25, 5, w^2 - w - 9],\ [27, 3, 3],\ [29, 29, -w^2 - w + 1],\ [29, 29, w^2 - 2*w - 4],\ [29, 29, -w^2 + w + 11],\ [43, 43, 2*w^2 - 3*w - 18],\ [47, 47, -2*w + 7],\ [53, 53, w^2 - w - 3],\ [61, 61, w^2 + 2*w + 2],\ [67, 67, -2*w^2 + 5*w + 8],\ [79, 79, w^2 - 3*w - 9],\ [97, 97, -w^2 - 2*w + 2],\ [109, 109, -2*w^2 + 4*w + 11],\ [109, 109, -3*w^2 + 4*w + 26],\ [109, 109, -3*w^2 + 4*w + 28],\ [113, 113, -4*w^2 + 8*w + 27],\ [121, 11, 5*w^2 - 8*w - 38],\ [127, 127, -3*w^2 + 4*w + 24],\ [131, 131, -4*w^2 + 5*w + 38],\ [139, 139, 2*w - 3],\ [149, 149, -w^2 + 3*w - 1],\ [151, 151, w^2 - 4*w - 6],\ [157, 157, 2*w^2 - 3*w - 12],\ [167, 167, -3*w^2 + 6*w + 22],\ [173, 173, -3*w^2 + 8*w + 8],\ [181, 181, w^2 - 2*w - 12],\ [181, 181, w^2 - 12],\ [181, 181, -4*w - 1],\ [191, 191, 2*w^2 - 6*w - 9],\ [193, 193, -8*w^2 + 12*w + 67],\ [197, 197, 2*w^2 - 3*w - 4],\ [199, 199, 2*w^2 - 2*w - 11],\ [211, 211, 4*w^2 - 6*w - 29],\ [223, 223, 3*w^2 - 5*w - 27],\ [227, 227, -4*w^2 + 7*w + 28],\ [229, 229, 3*w^2 - 8*w - 14],\ [229, 229, w^2 - 4*w - 8],\ [229, 229, -2*w - 7],\ [233, 233, -5*w^2 + 7*w + 43],\ [241, 241, 2*w^2 - 4*w - 7],\ [251, 251, w^2 + 2],\ [251, 251, -2*w^2 + 2*w + 1],\ [251, 251, 2*w^2 - 5*w - 2],\ [263, 263, 2*w^2 - 5*w - 14],\ [269, 269, -w^2 + 5*w - 1],\ [271, 271, -2*w^2 - w + 2],\ [277, 277, -3*w^2 + 7*w + 7],\ [281, 281, -5*w^2 + 10*w + 34],\ [293, 293, -2*w^2 + 5*w + 16],\ [311, 311, w^2 + w + 3],\ [313, 313, 2*w^2 + w - 4],\ [317, 317, -7*w^2 + 13*w + 49],\ [331, 331, w^2 - 5*w - 7],\ [337, 337, -w^2 + 2*w - 2],\ [337, 337, 7*w^2 - 12*w - 52],\ [337, 337, -2*w^2 + 3],\ [347, 347, -10*w^2 + 17*w + 76],\ [349, 349, 2*w^2 - 2*w - 9],\ [353, 353, 2*w^2 - 3*w - 8],\ [359, 359, -4*w^2 + 5*w + 36],\ [361, 19, w^2 + w - 7],\ [367, 367, -4*w^2 + 8*w + 29],\ [373, 373, 2*w^2 - 6*w - 11],\ [373, 373, 8*w^2 - 14*w - 59],\ [379, 379, -2*w^2 + w + 4],\ [383, 383, -3*w^2 + 2*w + 34],\ [383, 383, 5*w^2 - 7*w - 41],\ [383, 383, 3*w^2 - 7*w - 19],\ [397, 397, -7*w^2 + 13*w + 51],\ [431, 431, -8*w^2 + 14*w + 63],\ [433, 433, -2*w^2 + 9*w + 8],\ [443, 443, -6*w^2 + 10*w + 51],\ [443, 443, 3*w - 4],\ [443, 443, w^2 - 7*w - 3],\ [449, 449, -7*w^2 + 10*w + 62],\ [461, 461, -6*w^2 + 11*w + 46],\ [461, 461, 2*w^2 - 4*w - 1],\ [461, 461, w^2 - 5*w - 9],\ [463, 463, 7*w + 6],\ [463, 463, -5*w^2 + 8*w + 36],\ [463, 463, 3*w^2 - 2*w - 16],\ [467, 467, 3*w^2 - 7*w - 9],\ [467, 467, 4*w - 1],\ [467, 467, 6*w^2 - 11*w - 48],\ [479, 479, 2*w^2 - 4*w - 21],\ [479, 479, w^2 + 7*w + 7],\ [479, 479, w^2 - 2*w - 14],\ [487, 487, w^2 - 5*w - 11],\ [487, 487, 3*w^2 - 3*w - 23],\ [487, 487, w^2 + 3*w - 3],\ [491, 491, w^2 - 8*w - 4],\ [499, 499, -6*w^2 + 14*w + 31],\ [509, 509, -w - 8],\ [523, 523, 2*w^2 + 1],\ [529, 23, 2*w^2 - 5*w - 22],\ [541, 541, -w^2 + 5*w - 7],\ [547, 547, 2*w^2 - 6*w - 13],\ [563, 563, 2*w^2 - 9*w - 4],\ [569, 569, 4*w^2 - 7*w - 26],\ [569, 569, -3*w^2 + 5*w + 3],\ [569, 569, 4*w^2 - 6*w - 27],\ [577, 577, 3*w^2 - 7*w - 21],\ [577, 577, -8*w^2 + 12*w + 69],\ [577, 577, -2*w^2 + w + 26],\ [587, 587, w^2 - 6*w - 8],\ [593, 593, -2*w^2 + w + 16],\ [593, 593, 3*w^2 - 5*w - 17],\ [593, 593, 2*w^2 - 8*w - 9],\ [601, 601, -6*w^2 + 8*w + 53],\ [607, 607, -2*w - 9],\ [613, 613, w^2 + 2*w - 6],\ [617, 617, 5*w^2 - 11*w - 31],\ [619, 619, -4*w^2 + 6*w + 37],\ [641, 641, 3*w^2 - 6*w - 14],\ [647, 647, -2*w^2 + 3*w + 22],\ [653, 653, 2*w^2 - 7*w - 22],\ [653, 653, 2*w^2 + w + 2],\ [653, 653, -3*w^2 + 5*w + 29],\ [659, 659, 4*w^2 - 11*w - 18],\ [659, 659, 3*w^2 - 6*w - 8],\ [659, 659, 4*w^2 - 8*w - 23],\ [661, 661, w^2 + w - 11],\ [677, 677, -w^2 - 4],\ [701, 701, -2*w^2 - 5*w + 2],\ [727, 727, -2*w^2 + w + 18],\ [733, 733, -9*w^2 + 13*w + 79],\ [733, 733, -7*w^2 + 10*w + 56],\ [733, 733, 2*w^2 - 13],\ [739, 739, 6*w^2 - 10*w - 43],\ [739, 739, 4*w^2 - 4*w - 29],\ [739, 739, -8*w^2 + 15*w + 58],\ [751, 751, -11*w^2 + 19*w + 87],\ [757, 757, 3*w^2 - 10*w - 12],\ [761, 761, 4*w - 3],\ [769, 769, 3*w^2 - w - 13],\ [811, 811, 5*w^2 - 9*w - 33],\ [823, 823, 2*w^2 - 7*w - 12],\ [853, 853, -7*w - 8],\ [853, 853, -3*w^2 + 4*w + 2],\ [853, 853, 2*w^2 + 5*w + 6],\ [857, 857, 10*w^2 - 15*w - 82],\ [857, 857, 3*w^2 - 8*w - 18],\ [857, 857, -4*w^2 + 3*w + 44],\ [859, 859, -w^2 + 4*w - 6],\ [877, 877, -4*w^2 + 13*w + 14],\ [881, 881, 2*w^2 - 9*w - 2],\ [887, 887, 3*w - 14],\ [907, 907, -11*w^2 + 21*w + 77],\ [919, 919, 12*w^2 - 21*w - 92],\ [929, 929, w^2 - 18],\ [937, 937, 9*w + 4],\ [941, 941, -2*w^2 + 2*w - 1],\ [947, 947, 4*w^2 - 9*w - 18],\ [953, 953, -7*w^2 + 14*w + 48],\ [977, 977, w^2 - 2*w - 16],\ [983, 983, 4*w^2 - 7*w - 24],\ [991, 991, -3*w^2 - 2*w + 4]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^6 - 8*x^4 - 2*x^3 + 14*x^2 + 2*x - 6 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, 1, -e^4 + 6*e^2 + 2*e - 5, -e^5 + 7*e^3 + 2*e^2 - 8*e, -e^4 + e^3 + 6*e^2 - 2*e - 3, e^5 - 6*e^3 - 4*e^2 + 4*e + 6, 2*e^5 - e^4 - 13*e^3 + 2*e^2 + 12*e - 3, 2*e^5 + e^4 - 14*e^3 - 12*e^2 + 12*e + 9, -2*e^4 + 14*e^2 + 2*e - 12, e^5 - 2*e^4 - 5*e^3 + 8*e^2 + 4*e - 2, 2*e^4 - e^3 - 14*e^2 + 2*e + 15, -2*e^5 + 2*e^4 + 13*e^3 - 8*e^2 - 12*e + 9, -2*e^5 + 14*e^3 + 4*e^2 - 16*e - 3, 2*e^2 - 2*e - 2, -2*e^5 + e^4 + 11*e^3 + 2*e^2 - 4*e - 3, -e^3 + 10*e, -3*e^5 + 2*e^4 + 20*e^3 - 4*e^2 - 22*e + 4, 2*e^5 + 2*e^4 - 15*e^3 - 16*e^2 + 14*e + 12, -2*e^5 + 2*e^4 + 11*e^3 - 8*e^2 - 6*e + 10, 2*e^5 - 4*e^4 - 12*e^3 + 18*e^2 + 14*e - 12, 4*e^4 - 3*e^3 - 26*e^2 + 2*e + 27, -2*e^5 + 2*e^4 + 14*e^3 - 8*e^2 - 16*e + 6, e^5 + 4*e^4 - 9*e^3 - 28*e^2 + 4*e + 26, 4*e^4 - 30*e^2 - 6*e + 27, 2*e^4 - 2*e^3 - 10*e^2 + 6*e + 18, -e^5 + 2*e^4 + 5*e^3 - 10*e^2 - 4*e + 6, -e^5 - 4*e^4 + 10*e^3 + 28*e^2 - 18*e - 24, -2*e^5 + e^4 + 13*e^3 + 4*e^2 - 16*e - 15, 2*e^5 + 6*e^4 - 14*e^3 - 44*e^2 + 6*e + 36, 3*e^5 - 23*e^3 - 4*e^2 + 28*e - 4, 6*e^5 - 4*e^4 - 41*e^3 + 10*e^2 + 44*e - 6, -2*e^5 + 3*e^4 + 12*e^3 - 10*e^2 - 10*e + 3, 4*e^5 + 4*e^4 - 31*e^3 - 28*e^2 + 34*e + 18, 3*e^5 - 4*e^4 - 18*e^3 + 16*e^2 + 18*e - 20, -2*e^5 + 2*e^4 + 14*e^3 - 6*e^2 - 28*e - 6, -e^5 + 3*e^3 + 6*e^2 + 8*e - 6, 4*e^4 - 4*e^3 - 20*e^2 + 12*e + 18, -2*e^5 - 4*e^4 + 17*e^3 + 26*e^2 - 16*e - 15, 2*e^4 - 3*e^3 - 14*e^2 + 6*e + 21, -4*e^5 + 7*e^4 + 25*e^3 - 36*e^2 - 34*e + 33, 2*e^5 - 4*e^4 - 14*e^3 + 22*e^2 + 24*e - 12, 3*e^5 + 2*e^4 - 21*e^3 - 24*e^2 + 20*e + 22, 2*e^5 - 4*e^4 - 10*e^3 + 14*e^2 + 2*e, -2*e^4 + e^3 + 6*e^2 + 4*e + 10, -2*e^4 - 2*e^3 + 18*e^2 + 8*e - 29, e^5 + 2*e^4 - 10*e^3 - 10*e^2 + 12*e - 4, -4*e^5 + 4*e^4 + 27*e^3 - 18*e^2 - 36*e + 15, 3*e^5 + 4*e^4 - 23*e^3 - 26*e^2 + 24*e + 6, -4*e^5 - 4*e^4 + 27*e^3 + 40*e^2 - 22*e - 36, 3*e^4 - 20*e^2 - 8*e + 9, 2*e^5 - 15*e^3 - 6*e^2 + 24*e, e^5 - 2*e^4 - 4*e^3 + 2*e + 30, -5*e^5 + 6*e^4 + 29*e^3 - 28*e^2 - 26*e + 36, 2*e^5 - 10*e^4 - 9*e^3 + 56*e^2 + 18*e - 42, -5*e^5 + 28*e^3 + 18*e^2 - 10*e - 24, -3*e^5 + 2*e^4 + 23*e^3 - 12*e^2 - 32*e + 24, -3*e^5 + 2*e^4 + 21*e^3 - 4*e^2 - 28*e + 6, 2*e^5 + 6*e^4 - 11*e^3 - 50*e^2 - 4*e + 48, -2*e^5 + 2*e^4 + 13*e^3 - 10*e^2 - 4*e + 15, 4*e^5 - 4*e^4 - 29*e^3 + 12*e^2 + 46*e + 3, 6*e^5 - 46*e^3 - 10*e^2 + 58*e - 2, 2*e^4 - 4*e^3 - 4*e^2 + 8*e - 10, -2*e^5 - 2*e^4 + 15*e^3 + 10*e^2 - 16*e + 12, -e^5 + 4*e^4 - 2*e^3 - 12*e^2 + 28*e + 12, -8*e^5 + e^4 + 55*e^3 + 16*e^2 - 60*e - 3, 8*e^5 - 4*e^4 - 53*e^3 + 4*e^2 + 52*e + 3, -e^5 - 6*e^4 + 12*e^3 + 44*e^2 - 16*e - 48, 3*e^5 + 6*e^4 - 24*e^3 - 38*e^2 + 22*e + 18, -2*e^5 + 10*e^4 + 12*e^3 - 60*e^2 - 24*e + 54, 2*e^5 - 16*e^3 - 6*e^2 + 14*e + 20, -2*e^5 - 6*e^4 + 17*e^3 + 42*e^2 - 16*e - 38, 6*e^5 + 4*e^4 - 41*e^3 - 44*e^2 + 30*e + 39, 4*e^5 - 10*e^4 - 24*e^3 + 48*e^2 + 30*e - 30, e^5 + 8*e^4 - 7*e^3 - 62*e^2 - 8*e + 60, e^5 - 4*e^4 - 5*e^3 + 20*e^2 + 6*e - 6, 6*e^5 - 4*e^4 - 47*e^3 + 18*e^2 + 72*e - 12, -10*e^5 + 2*e^4 + 61*e^3 + 20*e^2 - 46*e - 26, -8*e^5 - e^4 + 54*e^3 + 26*e^2 - 42*e - 27, -2*e^4 - 2*e^3 + 24*e^2 + 8*e - 49, -2*e^5 - 3*e^4 + 16*e^3 + 24*e^2 - 18*e - 33, 4*e^4 + 3*e^3 - 30*e^2 - 28*e + 36, -8*e^5 - 2*e^4 + 56*e^3 + 36*e^2 - 56*e - 36, 5*e^5 - 33*e^3 - 6*e^2 + 30*e - 12, 2*e^5 + 6*e^4 - 10*e^3 - 50*e^2 - 14*e + 48, -6*e^5 - 4*e^4 + 46*e^3 + 36*e^2 - 54*e - 27, -3*e^5 + 4*e^4 + 21*e^3 - 18*e^2 - 22*e + 18, -7*e^5 + 6*e^4 + 49*e^3 - 20*e^2 - 72*e + 4, 6*e^5 - 2*e^4 - 44*e^3 + 2*e^2 + 60*e - 12, 2*e^5 - 7*e^4 - 15*e^3 + 42*e^2 + 22*e - 21, -2*e^5 - e^4 + 19*e^3 + 10*e^2 - 38*e - 15, -7*e^5 + 2*e^4 + 56*e^3 - 4*e^2 - 80*e + 6, 6*e^5 - 46*e^3 - 16*e^2 + 62*e + 6, 2*e^5 + 3*e^4 - 18*e^3 - 16*e^2 + 26*e - 3, -10*e^5 + 74*e^3 + 16*e^2 - 82*e - 6, -4*e^5 + 14*e^4 + 20*e^3 - 80*e^2 - 30*e + 78, 4*e^5 - 12*e^4 - 23*e^3 + 64*e^2 + 38*e - 40, 4*e^5 - e^4 - 25*e^3 - 4*e^2 + 22*e + 21, -6*e^5 + 6*e^4 + 40*e^3 - 20*e^2 - 46*e + 18, -3*e^5 - 4*e^4 + 25*e^3 + 34*e^2 - 40*e - 30, 2*e^5 + 5*e^4 - 21*e^3 - 30*e^2 + 38*e + 9, 3*e^5 - 6*e^4 - 15*e^3 + 32*e^2 + 4*e - 36, -2*e^5 - 10*e^4 + 16*e^3 + 66*e^2 - 2*e - 50, 9*e^5 - 61*e^3 - 26*e^2 + 58*e + 16, 9*e^5 + 6*e^4 - 58*e^3 - 66*e^2 + 40*e + 62, -4*e^5 + 6*e^4 + 22*e^3 - 18*e^2 - 14*e - 2, -2*e^5 + 10*e^4 + 8*e^3 - 54*e^2 - 14*e + 42, 4*e^5 + 4*e^4 - 37*e^3 - 28*e^2 + 66*e + 21, -5*e^5 + 12*e^4 + 29*e^3 - 60*e^2 - 32*e + 54, 8*e^5 - 2*e^4 - 53*e^3 - 10*e^2 + 56*e - 12, -8*e^5 + 4*e^4 + 49*e^3 - 4*e^2 - 34*e + 10, 6*e^5 - 2*e^4 - 40*e^3 - 6*e^2 + 54*e + 16, -6*e^5 + 44*e^3 + 12*e^2 - 54*e - 16, -8*e^5 + 15*e^4 + 50*e^3 - 80*e^2 - 66*e + 69, -8*e^5 - 2*e^4 + 55*e^3 + 40*e^2 - 58*e - 42, -3*e^5 - 4*e^4 + 25*e^3 + 36*e^2 - 42*e - 42, 9*e^5 - 2*e^4 - 67*e^3 + 2*e^2 + 86*e, 8*e^5 + 4*e^4 - 53*e^3 - 50*e^2 + 34*e + 45, 10*e^5 + 5*e^4 - 74*e^3 - 58*e^2 + 84*e + 41, -3*e^5 - 6*e^4 + 20*e^3 + 44*e^2 + 4*e - 54, -3*e^5 - 12*e^4 + 30*e^3 + 78*e^2 - 32*e - 66, -4*e^5 + 2*e^4 + 29*e^3 - 6*e^2 - 28*e + 20, 6*e^5 - 2*e^4 - 45*e^3 - 6*e^2 + 64*e + 15, -2*e^5 - 12*e^4 + 16*e^3 + 80*e^2 - 66, 2*e^5 + 2*e^4 - 18*e^3 - 10*e^2 + 42*e - 3, 8*e^5 - 4*e^4 - 53*e^3 + 6*e^2 + 52*e, 2*e^5 - 4*e^4 - 12*e^3 + 30*e^2 + 10*e - 45, 8*e^5 - 7*e^4 - 50*e^3 + 24*e^2 + 52*e - 45, 4*e^5 - 4*e^4 - 21*e^3 + 16*e^2 - 2*e - 18, -8*e^5 - 2*e^4 + 62*e^3 + 24*e^2 - 80*e - 6, 3*e^5 + 14*e^4 - 29*e^3 - 92*e^2 + 22*e + 72, -2*e^5 + 8*e^3 + 8*e^2 + 8*e - 15, 12*e^5 - 2*e^4 - 78*e^3 - 16*e^2 + 68*e - 6, -4*e^5 + 4*e^4 + 30*e^3 - 22*e^2 - 48*e + 24, 2*e^5 + 4*e^4 - 24*e^3 - 20*e^2 + 52*e + 1, 2*e^5 - 4*e^4 - 18*e^3 + 18*e^2 + 42*e - 12, 14*e^5 - 4*e^4 - 96*e^3 - 10*e^2 + 92*e - 6, -3*e^5 - 2*e^4 + 16*e^3 + 32*e^2 + 4*e - 36, 8*e^5 + 6*e^4 - 62*e^3 - 46*e^2 + 70*e + 18, -6*e^5 + 7*e^4 + 44*e^3 - 28*e^2 - 78*e + 11, 4*e^5 + 3*e^4 - 24*e^3 - 38*e^2 + 16*e + 37, -4*e^5 - 4*e^4 + 26*e^3 + 46*e^2 - 22*e - 67, -9*e^3 + 12*e^2 + 34*e - 21, 7*e^5 + 6*e^4 - 46*e^3 - 56*e^2 + 20*e + 48, 2*e^5 - 4*e^4 - 4*e^3 + 16*e^2 - 20*e - 6, -6*e^5 + e^4 + 34*e^3 + 14*e^2 - 18*e - 13, -8*e^5 - 8*e^4 + 60*e^3 + 74*e^2 - 54*e - 55, -9*e^5 + 8*e^4 + 61*e^3 - 36*e^2 - 76*e + 40, 6*e^5 - 4*e^4 - 30*e^3 - 2*e^2 + 12*e + 19, 8*e^5 - 2*e^4 - 57*e^3 + 4*e^2 + 68*e - 33, 16*e^5 - 10*e^4 - 104*e^3 + 28*e^2 + 94*e - 48, -5*e^5 + 8*e^4 + 37*e^3 - 36*e^2 - 66*e + 30, 4*e^5 + e^4 - 28*e^3 - 12*e^2 + 24*e - 13, -13*e^5 + 12*e^4 + 78*e^3 - 50*e^2 - 60*e + 76, -4*e^5 - 4*e^4 + 26*e^3 + 34*e^2 - 12*e - 18, 10*e^5 + 7*e^4 - 70*e^3 - 60*e^2 + 60*e + 39, -8*e^5 + 9*e^4 + 48*e^3 - 36*e^2 - 46*e + 31, 12*e^5 - e^4 - 86*e^3 - 14*e^2 + 94*e + 13, 8*e^4 - 12*e^3 - 38*e^2 + 38*e + 18, 2*e^5 - 12*e^4 - 4*e^3 + 66*e^2 + 2*e - 54, 4*e^5 - 12*e^4 - 24*e^3 + 64*e^2 + 32*e - 51, -5*e^5 - 4*e^4 + 41*e^3 + 32*e^2 - 46*e + 6, 6*e^5 - 10*e^4 - 38*e^3 + 54*e^2 + 46*e - 57, -14*e^5 - 2*e^4 + 98*e^3 + 34*e^2 - 98*e - 3, e^4 + 3*e^3 + 4*e^2 - 16*e - 33, 6*e^5 + 5*e^4 - 41*e^3 - 48*e^2 + 26*e + 63] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([5, 5, w])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]