/* 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([-34, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([6, 6, w + 2]) primes_array = [ [2, 2, w - 6],\ [3, 3, w + 1],\ [3, 3, w + 2],\ [5, 5, w + 2],\ [5, 5, w + 3],\ [11, 11, w + 1],\ [11, 11, w + 10],\ [17, 17, -3*w + 17],\ [29, 29, w + 11],\ [29, 29, w + 18],\ [37, 37, w + 16],\ [37, 37, w + 21],\ [47, 47, -w - 9],\ [47, 47, w - 9],\ [49, 7, -7],\ [61, 61, w + 20],\ [61, 61, w + 41],\ [89, 89, 2*w - 15],\ [89, 89, -2*w - 15],\ [103, 103, -14*w + 81],\ [103, 103, 4*w - 21],\ [107, 107, w + 26],\ [107, 107, w + 81],\ [109, 109, w + 19],\ [109, 109, w + 90],\ [127, 127, 2*w - 3],\ [127, 127, -2*w - 3],\ [131, 131, w + 54],\ [131, 131, w + 77],\ [137, 137, -3*w - 13],\ [137, 137, 3*w - 13],\ [139, 139, w + 27],\ [139, 139, w + 112],\ [151, 151, 8*w - 45],\ [151, 151, -10*w + 57],\ [163, 163, w + 69],\ [163, 163, w + 94],\ [169, 13, -13],\ [173, 173, w + 42],\ [173, 173, w + 131],\ [181, 181, w + 45],\ [181, 181, w + 136],\ [191, 191, -w - 15],\ [191, 191, w - 15],\ [197, 197, w + 25],\ [197, 197, w + 172],\ [211, 211, w + 33],\ [211, 211, w + 178],\ [223, 223, -3*w - 23],\ [223, 223, 3*w - 23],\ [227, 227, w + 48],\ [227, 227, w + 179],\ [239, 239, -5*w + 33],\ [239, 239, -23*w + 135],\ [257, 257, -3*w - 7],\ [257, 257, 3*w - 7],\ [263, 263, -24*w + 139],\ [263, 263, 6*w - 31],\ [269, 269, w + 29],\ [269, 269, w + 240],\ [271, 271, -9*w + 55],\ [271, 271, -15*w + 89],\ [277, 277, w + 119],\ [277, 277, w + 158],\ [281, 281, 3*w - 5],\ [281, 281, -3*w - 5],\ [283, 283, w + 113],\ [283, 283, w + 170],\ [317, 317, w + 44],\ [317, 317, w + 273],\ [347, 347, w + 46],\ [347, 347, w + 301],\ [353, 353, 9*w - 49],\ [353, 353, -21*w + 121],\ [359, 359, -25*w + 147],\ [359, 359, -7*w + 45],\ [361, 19, -19],\ [379, 379, w + 105],\ [379, 379, w + 274],\ [383, 383, 6*w - 29],\ [383, 383, -36*w + 209],\ [397, 397, w + 35],\ [397, 397, w + 362],\ [409, 409, -5*w - 21],\ [409, 409, 5*w - 21],\ [419, 419, w + 156],\ [419, 419, w + 263],\ [433, 433, -12*w + 73],\ [433, 433, -18*w + 107],\ [457, 457, 6*w + 41],\ [457, 457, -6*w + 41],\ [463, 463, -4*w - 9],\ [463, 463, 4*w - 9],\ [499, 499, w + 212],\ [499, 499, w + 287],\ [529, 23, -23],\ [541, 541, w + 121],\ [541, 541, w + 420],\ [547, 547, w + 157],\ [547, 547, w + 390],\ [569, 569, -28*w + 165],\ [569, 569, -10*w + 63],\ [571, 571, w + 264],\ [571, 571, w + 307],\ [577, 577, -47*w + 273],\ [577, 577, 7*w - 33],\ [593, 593, 2*w - 27],\ [593, 593, -2*w - 27],\ [599, 599, -6*w - 25],\ [599, 599, 6*w - 25],\ [619, 619, w + 167],\ [619, 619, w + 452],\ [631, 631, -15*w + 91],\ [631, 631, -21*w + 125],\ [643, 643, w + 57],\ [643, 643, w + 586],\ [647, 647, -11*w + 69],\ [647, 647, -29*w + 171],\ [653, 653, w + 120],\ [653, 653, w + 533],\ [677, 677, w + 64],\ [677, 677, w + 613],\ [683, 683, w + 248],\ [683, 683, w + 435],\ [691, 691, w + 322],\ [691, 691, w + 369],\ [709, 709, w + 265],\ [709, 709, w + 444],\ [727, 727, -9*w + 59],\ [727, 727, -39*w + 229],\ [761, 761, -27*w + 155],\ [761, 761, 15*w - 83],\ [769, 769, 5*w - 9],\ [769, 769, -5*w - 9],\ [787, 787, w + 63],\ [787, 787, w + 724],\ [811, 811, w + 70],\ [811, 811, w + 741],\ [821, 821, w + 149],\ [821, 821, w + 672],\ [827, 827, w + 193],\ [827, 827, w + 634],\ [853, 853, w + 253],\ [853, 853, w + 600],\ [863, 863, -6*w - 19],\ [863, 863, 6*w - 19],\ [877, 877, w + 339],\ [877, 877, w + 538],\ [907, 907, w + 74],\ [907, 907, w + 833],\ [919, 919, -3*w - 35],\ [919, 919, 3*w - 35],\ [937, 937, 7*w - 27],\ [937, 937, -7*w - 27],\ [941, 941, w + 144],\ [941, 941, w + 797],\ [947, 947, w + 292],\ [947, 947, w + 655],\ [953, 953, 2*w - 33],\ [953, 953, -2*w - 33],\ [961, 31, -31],\ [967, 967, 9*w - 61],\ [967, 967, -9*w - 61],\ [977, 977, -4*w - 39],\ [977, 977, 4*w - 39],\ [997, 997, w + 55],\ [997, 997, w + 942]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^6 + 8*x^4 + 16*x^2 + 4 K. = NumberField(heckePol) hecke_eigenvalues_array = [1, e, 1/2*e^3 + 2*e, e^3 + 3*e, 1/2*e^5 + 3*e^3 + 2*e, -3/2*e^5 - 9*e^3 - 10*e, -1/2*e^5 - 2*e^3 + e, -2*e^2 - 4, 3/2*e^5 + 9*e^3 + 9*e, 4*e, 5/2*e^5 + 17*e^3 + 25*e, -2*e, 3*e^4 + 17*e^2 + 12, -e^4 - 5*e^2 - 6, -e^4 - 5*e^2 - 4, -1/2*e^5 - 4*e^3 - e, 4*e^3 + 14*e, -e^4 - 8*e^2 - 12, -e^4 - 8*e^2 - 12, -e^4 - 3*e^2 + 12, -e^4 - 4*e^2 - 4, -1/2*e^5 - 7*e^3 - 26*e, 3/2*e^5 + 13*e^3 + 27*e, -3/2*e^5 - 9*e^3 - 6*e, -3*e^3 - 9*e, 6*e^2 + 14, -e^4 - 12*e^2 - 18, 7/2*e^5 + 21*e^3 + 25*e, -1/2*e^5 - 9*e^3 - 26*e, -e^4 - 11*e^2 - 12, -2*e^4 - 12*e^2 - 2, 1/2*e^5 + e^3 - 8*e, -7/2*e^5 - 25*e^3 - 45*e, -e^4 - 3*e^2 + 10, -5*e^4 - 26*e^2 - 24, -2*e^5 - 16*e^3 - 38*e, -e^5 - 4*e^3 + 3*e, -3*e^4 - 16*e^2 - 8, -5/2*e^5 - 20*e^3 - 39*e, -5*e^5 - 32*e^3 - 46*e, 3*e^5 + 18*e^3 + 15*e, -7/2*e^5 - 19*e^3 - 18*e, e^4 + 4*e^2, e^4 + 4*e^2, 2*e^5 + 10*e^3 + e, -1/2*e^5 - 13*e^3 - 42*e, -9/2*e^5 - 29*e^3 - 32*e, -1/2*e^5 - 2*e^3 + 11*e, 3*e^4 + 21*e^2 + 22, -2*e^4 - 19*e^2 - 28, e^5 - 4*e^3 - 32*e, -4*e^5 - 30*e^3 - 43*e, 3*e^4 + 19*e^2 + 6, -6*e^4 - 25*e^2 - 4, 4*e^4 + 20*e^2 + 4, 18, 6*e^4 + 34*e^2 + 32, 7*e^4 + 35*e^2 + 22, -5/2*e^5 - 17*e^3 - 13*e, 4*e^3 + 8*e, -e^4 - 4*e^2 + 4, 2*e^4 + 16*e^2 + 16, -4*e^5 - 31*e^3 - 59*e, 9/2*e^5 + 27*e^3 + 28*e, -5*e^4 - 33*e^2 - 36, 5*e^4 + 30*e^2 + 22, -3/2*e^5 - 7*e^3 - 2*e, -3/2*e^5 - 12*e^3 - 17*e, -1/2*e^5 - 6*e^3 - 15*e, 3*e^5 + 22*e^3 + 22*e, 8*e^5 + 52*e^3 + 77*e, -e^5 - 2*e^3 + 12*e, e^4 - 12, -2*e^2 - 18, -10*e^2 - 30, e^4 + 8*e^2 + 2, -10*e^2 - 28, 7*e^5 + 48*e^3 + 60*e, e^5 - 7*e^3 - 49*e, e^4 + 2*e^2 - 10, -4*e^4 - 22*e^2 - 34, 15/2*e^5 + 55*e^3 + 83*e, 6*e^5 + 46*e^3 + 78*e, -8*e^4 - 48*e^2 - 50, e^4 + 15*e^2 + 34, -e^5 + 2*e^3 + 22*e, 4*e^5 + 17*e^3 - 3*e, 7*e^4 + 28*e^2 + 10, -2*e^4 + 26, 10*e^4 + 46*e^2 + 14, 4*e^4 + 24*e^2 + 48, -e^4 + 6*e^2 + 26, -10*e^2 - 26, -3/2*e^5 - 7*e^3 - 14*e, 15/2*e^5 + 46*e^3 + 45*e, -9*e^4 - 51*e^2 - 52, -7*e^5 - 48*e^3 - 71*e, 21/2*e^5 + 69*e^3 + 86*e, 1/2*e^5 - 7*e^3 - 32*e, 1/2*e^5 + 13*e^3 + 33*e, -4*e^4 - 14*e^2 - 10, -11*e^4 - 56*e^2 - 40, e^5 - 31*e, -e^5 - 10*e^3 - 14*e, -4*e^4 - 16*e^2 + 6, -2*e^4 - 14*e^2 - 14, -6*e^4 - 17*e^2 + 18, 11*e^4 + 55*e^2 + 38, 12*e^2 + 52, 11*e^2 + 36, -5*e^5 - 43*e^3 - 69*e, -3*e^5 - 22*e^3 - 50*e, -7*e^4 - 20*e^2 + 34, -4*e^4 - 16*e^2 + 20, e^5 + 7*e^3 + 17*e, -7*e^5 - 46*e^3 - 48*e, -12*e^4 - 64*e^2 - 64, 2*e^4 + 4*e^2 - 30, 4*e^5 + 22*e^3 + 9*e, 3/2*e^5 + 13*e^3 + 40*e, -13/2*e^5 - 53*e^3 - 101*e, 5*e^5 + 28*e^3 + 16*e, -e^5 - 6*e^3 - 14*e, e^5 + 7*e^3 + 7*e, -11/2*e^5 - 36*e^3 - 53*e, -13/2*e^5 - 41*e^3 - 62*e, 11/2*e^5 + 27*e^3 + 18*e, -3*e^5 - 16*e^3 + e, -4*e^2 - 26, -2*e^4 - 22*e^2 - 32, e^4 - 6*e^2 - 10, 11*e^4 + 57*e^2 + 48, 4*e^4 + 28*e^2 + 34, -e^4 - 12*e^2 - 16, 13/2*e^5 + 41*e^3 + 60*e, 21/2*e^5 + 73*e^3 + 103*e, -8*e^5 - 57*e^3 - 79*e, 8*e^5 + 48*e^3 + 50*e, -27/2*e^5 - 85*e^3 - 99*e, -7*e^5 - 44*e^3 - 52*e, -11/2*e^5 - 25*e^3 - 4*e, 11/2*e^5 + 30*e^3 + 15*e, -13/2*e^5 - 27*e^3 + 8*e, 5*e^5 + 36*e^3 + 37*e, 11*e^4 + 50*e^2 + 18, -2*e^4 + 16, -e^5 + 11*e^3 + 65*e, 5/2*e^5 + 27*e^3 + 90*e, -9/2*e^5 - 27*e^3 - 42*e, 11/2*e^5 + 32*e^3 + 27*e, -6*e^4 - 26*e^2 - 30, -5*e^4 - 40*e^2 - 50, -3*e^4 - 23*e^2 - 32, -7*e^4 - 46*e^2 - 66, 5*e^5 + 46*e^3 + 106*e, 5/2*e^5 - 2*e^3 - 57*e, -9/2*e^5 - 23*e^3 - 28*e, 11/2*e^5 + 48*e^3 + 103*e, -4*e^2 - 30, 10*e^4 + 42*e^2 - 14, 9*e^4 + 58*e^2 + 46, 8*e^4 + 40*e^2 + 20, 8*e^4 + 40*e^2 + 20, 7*e^4 + 39*e^2 + 40, 5*e^4 + e^2 - 56, 10*e^5 + 71*e^3 + 109*e, -13/2*e^5 - 53*e^3 - 96*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([2, 2, w - 6])] = -1 AL_eigenvalues[ZF.ideal([3, 3, w + 2])] = -1/2*e^3 - 2*e # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]