/* 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([-30, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([7,7,-w + 3]) primes_array = [ [2, 2, w],\ [3, 3, w],\ [5, 5, -w + 5],\ [7, 7, w + 3],\ [7, 7, w + 4],\ [13, 13, w + 2],\ [13, 13, w + 11],\ [17, 17, w + 8],\ [17, 17, w + 9],\ [19, 19, w + 7],\ [19, 19, -w + 7],\ [29, 29, -w - 1],\ [29, 29, w - 1],\ [37, 37, w + 17],\ [37, 37, w + 20],\ [71, 71, 2*w - 7],\ [71, 71, -2*w - 7],\ [83, 83, w + 14],\ [83, 83, w + 69],\ [101, 101, -7*w + 37],\ [101, 101, -3*w + 13],\ [103, 103, w + 37],\ [103, 103, w + 66],\ [107, 107, w + 43],\ [107, 107, w + 64],\ [113, 113, w + 16],\ [113, 113, w + 97],\ [121, 11, -11],\ [127, 127, w + 41],\ [127, 127, w + 86],\ [137, 137, w + 21],\ [137, 137, w + 116],\ [139, 139, -w - 13],\ [139, 139, w - 13],\ [149, 149, 3*w - 11],\ [149, 149, -3*w - 11],\ [157, 157, w + 40],\ [157, 157, w + 117],\ [191, 191, -4*w - 17],\ [191, 191, 4*w - 17],\ [211, 211, -5*w + 31],\ [211, 211, -7*w + 41],\ [223, 223, w + 91],\ [223, 223, w + 132],\ [227, 227, w + 22],\ [227, 227, w + 205],\ [233, 233, w + 27],\ [233, 233, w + 206],\ [239, 239, 6*w - 29],\ [239, 239, 8*w - 41],\ [241, 241, 2*w - 19],\ [241, 241, -2*w - 19],\ [257, 257, w + 95],\ [257, 257, w + 162],\ [269, 269, -3*w - 1],\ [269, 269, 3*w - 1],\ [277, 277, w + 108],\ [277, 277, w + 169],\ [311, 311, -4*w - 13],\ [311, 311, 4*w - 13],\ [331, 331, -w - 19],\ [331, 331, w - 19],\ [347, 347, w + 77],\ [347, 347, w + 270],\ [353, 353, w + 33],\ [353, 353, w + 320],\ [359, 359, 4*w - 11],\ [359, 359, -4*w - 11],\ [367, 367, w + 146],\ [367, 367, w + 221],\ [373, 373, w + 75],\ [373, 373, w + 298],\ [379, 379, -7*w + 43],\ [379, 379, -9*w + 53],\ [389, 389, 5*w - 19],\ [389, 389, -5*w - 19],\ [397, 397, w + 53],\ [397, 397, w + 344],\ [409, 409, 2*w - 23],\ [409, 409, -2*w - 23],\ [431, 431, 4*w - 7],\ [431, 431, -4*w - 7],\ [443, 443, w + 159],\ [443, 443, w + 284],\ [461, 461, 5*w - 17],\ [461, 461, -5*w - 17],\ [463, 463, w + 131],\ [463, 463, w + 332],\ [467, 467, w + 214],\ [467, 467, w + 253],\ [479, 479, -4*w - 1],\ [479, 479, 4*w - 1],\ [487, 487, w + 70],\ [487, 487, w + 417],\ [499, 499, -w - 23],\ [499, 499, w - 23],\ [509, 509, -7*w + 31],\ [509, 509, -15*w + 79],\ [529, 23, -23],\ [563, 563, w + 34],\ [563, 563, w + 529],\ [571, 571, -3*w - 29],\ [571, 571, 3*w - 29],\ [587, 587, w + 183],\ [587, 587, w + 404],\ [593, 593, w + 246],\ [593, 593, w + 347],\ [599, 599, 10*w - 49],\ [599, 599, 12*w - 61],\ [601, 601, -16*w + 91],\ [601, 601, 6*w - 41],\ [607, 607, w + 89],\ [607, 607, w + 518],\ [613, 613, w + 231],\ [613, 613, w + 382],\ [617, 617, w + 257],\ [617, 617, w + 360],\ [619, 619, -5*w - 37],\ [619, 619, 5*w - 37],\ [683, 683, w + 264],\ [683, 683, w + 419],\ [691, 691, 3*w - 31],\ [691, 691, -3*w - 31],\ [701, 701, 5*w - 7],\ [701, 701, -5*w - 7],\ [719, 719, -6*w - 19],\ [719, 719, 6*w - 19],\ [727, 727, w + 311],\ [727, 727, w + 416],\ [733, 733, w + 105],\ [733, 733, w + 628],\ [739, 739, -7*w + 47],\ [739, 739, 17*w - 97],\ [757, 757, w + 73],\ [757, 757, w + 684],\ [769, 769, 6*w - 43],\ [769, 769, -6*w - 43],\ [811, 811, -w - 29],\ [811, 811, w - 29],\ [821, 821, 15*w - 77],\ [821, 821, 11*w - 53],\ [823, 823, w + 186],\ [823, 823, w + 637],\ [827, 827, w + 388],\ [827, 827, w + 439],\ [839, 839, 12*w - 59],\ [839, 839, 14*w - 71],\ [853, 853, w + 258],\ [853, 853, w + 595],\ [857, 857, w + 51],\ [857, 857, w + 806],\ [859, 859, -11*w + 67],\ [859, 859, -13*w + 77],\ [877, 877, w + 192],\ [877, 877, w + 685],\ [911, 911, -6*w - 13],\ [911, 911, 6*w - 13],\ [941, 941, 7*w - 23],\ [941, 941, -7*w - 23],\ [947, 947, w + 127],\ [947, 947, w + 820],\ [953, 953, w + 44],\ [953, 953, w + 909],\ [961, 31, -31],\ [967, 967, w + 442],\ [967, 967, w + 525],\ [977, 977, w + 185],\ [977, 977, w + 792],\ [997, 997, w + 100],\ [997, 997, w + 897]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^6 - 6*x^4 + 8*x^2 - 1 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -e^5 + 6*e^3 - 7*e, e^3 - 3*e, e^2 + 1, -1, e^4 - 5*e^2 + 6, -2*e^4 + 8*e^2 - 2, -e^5 + 4*e^3 - 3*e, 2*e^5 - 9*e^3 + 5*e, -e^4 + e^2 + 6, -2*e^4 + 11*e^2 - 7, -3*e^5 + 15*e^3 - 10*e, -4*e^5 + 24*e^3 - 32*e, 2*e^4 - 9*e^2 + 9, -4*e^4 + 17*e^2 - 7, e^5 - 4*e^3 + 3*e, 3*e^5 - 23*e^3 + 38*e, -9*e^5 + 51*e^3 - 54*e, 6*e^5 - 30*e^3 + 26*e, 7*e^5 - 42*e^3 + 59*e, -5*e^5 + 25*e^3 - 22*e, 5*e^4 - 22*e^2 + 21, 5*e^4 - 23*e^2 + 12, -10*e^5 + 50*e^3 - 42*e, -e^5 + 10*e^3 - 27*e, 5*e^5 - 33*e^3 + 50*e, 4*e^5 - 24*e^3 + 28*e, 2*e^4 - 6*e^2 - 2, -3*e^4 + 13*e^2, -2*e^4 + 3*e^2 + 13, 5*e^5 - 33*e^3 + 58*e, -3*e^5 + 11*e^3 - 2*e, -2*e^4 + 14, 6*e^4 - 30*e^2 + 16, -9*e^5 + 52*e^3 - 63*e, 6*e^5 - 29*e^3 + 17*e, -3*e^4 + 15*e^2 - 10, 6*e^4 - 24*e^2 + 14, -11*e^5 + 56*e^3 - 53*e, 14*e^5 - 73*e^3 + 73*e, 4*e^4 - 18*e^2 + 14, e^4 - 6*e^2 - 3, e^4 - 11*e^2 + 20, 3*e^4 - 13*e^2 + 24, 6*e^5 - 34*e^3 + 38*e, -4*e^5 + 30*e^3 - 48*e, -3*e^5 + 21*e^3 - 32*e, -2*e^5 + 12*e^3 - 10*e, 11*e^5 - 60*e^3 + 49*e, 15*e^5 - 82*e^3 + 79*e, -2*e^4 + 10*e^2 - 6, 3*e^4 - 7*e^2 - 12, 3*e^5 - 16*e^3 + 21*e, -2*e^5 + 15*e^3 - 31*e, 14*e^5 - 67*e^3 + 51*e, 6*e^5 - 38*e^3 + 40*e, 3*e^4 - 6*e^2 - 13, e^4 - e^2 + 10, -5*e^5 + 30*e^3 - 29*e, 11*e^5 - 60*e^3 + 73*e, e^4 + 2*e^2 - 11, 7*e^4 - 24*e^2 + 5, 5*e^5 - 26*e^3 + 23*e, 9*e^5 - 49*e^3 + 44*e, 9*e^5 - 58*e^3 + 93*e, 7*e^5 - 41*e^3 + 40*e, -3*e^5 + 11*e^3 + 6*e, 7*e^5 - 39*e^3 + 34*e, 9*e^4 - 33*e^2 + 14, 6*e^4 - 20*e^2 + 6, -9*e^4 + 40*e^2 - 11, 4*e^4 - 25*e^2 + 7, -3*e^4 + 12*e^2 + 11, -e^4 + 5*e^2 - 30, -7*e^5 + 34*e^3 - 11*e, -11*e^5 + 70*e^3 - 99*e, 3*e^4 - 7*e^2 + 2, 7*e^4 - 31*e^2 + 14, 7*e^4 - 39*e^2 + 28, 7*e^4 - 22*e^2 - 3, 12*e^5 - 70*e^3 + 86*e, 14*e^5 - 74*e^3 + 72*e, 11*e^5 - 70*e^3 + 89*e, e^5 - 6*e^3 + 3*e, 8*e^5 - 54*e^3 + 86*e, -10*e^3 + 26*e, 7*e^2 - 1, -20*e^4 + 87*e^2 - 53, -19*e^5 + 105*e^3 - 114*e, e^5 - 23*e^3 + 58*e, 8*e^5 - 40*e^3 + 16*e, -2*e^5 + 9*e^3 - 21*e, e^4 + 11*e^2 - 30, 6*e^4 - 20*e^2 + 22, 5*e^4 - 22*e^2 + 17, -11*e^4 + 53*e^2 - 36, -3*e^5 + 20*e^3 - 45*e, -5*e^5 + 10*e^3 + 27*e, -4*e^4 + 15*e^2 - 15, 10*e^5 - 69*e^3 + 99*e, -e^5 + 19*e^3 - 58*e, 7*e^4 - 28*e^2 + 17, 3*e^4 - 7*e^2 - 22, -14*e^5 + 74*e^3 - 78*e, -4*e^5 + 27*e^3 - 23*e, -3*e^5 + 30*e^3 - 63*e, 17*e^3 - 55*e, 3*e^5 - 24*e^3 + 37*e, -6*e^5 + 30*e^3 - 36*e, -7*e^4 + 24*e^2 - 7, -5*e^4 + 23*e^2 + 6, -4*e^4 + 8*e^2 + 28, -8*e^4 + 47*e^2 - 33, -6*e^4 + 31*e^2 - 47, 7*e^4 - 28*e^2 + 25, 14*e^5 - 75*e^3 + 71*e, 7*e^5 - 40*e^3 + 33*e, 5*e^4 - 23*e^2 + 8, -6*e^4 + 23*e^2 + 25, 5*e^5 - 34*e^3 + 63*e, -26*e^5 + 135*e^3 - 137*e, -7*e^4 + 29*e^2, -2*e^4 - 6*e^2 + 16, 3*e^5 - 19*e^3 + 22*e, e^5 - 14*e^3 + 45*e, -2*e^5 + 9*e^3 - 13*e, -14*e^5 + 78*e^3 - 76*e, -9*e^4 + 39*e^2, -10*e^4 + 30*e^2, 9*e^4 - 38*e^2 + 5, -10*e^4 + 52*e^2 - 38, 18*e^4 - 76*e^2 + 46, 4*e^4 - 21*e^2 + 19, 3*e^4 - 10*e^2 - 9, -19*e^4 + 93*e^2 - 60, 7*e^4 - 30*e^2 + 25, 2*e^4 - 17*e^2 - 5, -5*e^4 + 3*e^2 + 28, 8*e^4 - 44*e^2 + 24, 4*e^5 - 21*e^3 + 43*e, 7*e^5 - 50*e^3 + 91*e, 3*e^4 + 3*e^2 - 8, 9*e^4 - 41*e^2 + 30, 11*e^5 - 51*e^3 + 40*e, 17*e^5 - 92*e^3 + 105*e, 8*e^5 - 33*e^3 + 23*e, -6*e^5 + 39*e^3 - 35*e, 13*e^4 - 59*e^2 + 28, 4*e^4 - 3*e^2 - 27, -7*e^5 + 19*e^3 + 18*e, -2*e^5 + 16*e^3 - 46*e, -16*e^4 + 89*e^2 - 51, 11*e^4 - 47*e^2 + 34, -7*e^4 + 9*e^2 + 28, -8*e^4 + 37*e^2 - 7, -30*e^5 + 156*e^3 - 142*e, 6*e^5 - 33*e^3 + 25*e, 5*e^5 - 20*e^3 + 3*e, -23*e^5 + 122*e^3 - 131*e, e^5 - e^3 - 20*e, 12*e^5 - 72*e^3 + 106*e, 17*e^5 - 100*e^3 + 131*e, -12*e^5 + 68*e^3 - 56*e, 9*e^4 - 28*e^2 - 35, -12*e^4 + 55*e^2 - 21, -2*e^4 - 10*e^2 + 56, 14*e^5 - 88*e^3 + 122*e, 5*e^5 - 33*e^3 + 58*e, -2*e^4 - 5*e^2 + 33, 3*e^4 - 23*e^2 + 18] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([7,7,-w + 3])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]