/* 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([-28, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([4, 4, -5*w - 24]) primes_array = [ [2, 2, -w + 6],\ [2, 2, w + 5],\ [7, 7, 6*w - 35],\ [7, 7, -6*w - 29],\ [9, 3, 3],\ [11, 11, 4*w + 19],\ [11, 11, 4*w - 23],\ [13, 13, -2*w + 11],\ [13, 13, 2*w + 9],\ [25, 5, -5],\ [31, 31, 2*w - 13],\ [31, 31, -2*w - 11],\ [41, 41, -8*w - 39],\ [41, 41, 8*w - 47],\ [53, 53, -26*w - 125],\ [53, 53, 26*w - 151],\ [61, 61, -14*w + 81],\ [61, 61, -14*w - 67],\ [83, 83, 2*w - 15],\ [83, 83, -2*w - 13],\ [97, 97, 2*w - 5],\ [97, 97, -2*w - 3],\ [109, 109, 2*w - 3],\ [109, 109, -2*w - 1],\ [113, 113, 2*w - 1],\ [127, 127, 8*w - 45],\ [127, 127, 8*w + 37],\ [131, 131, -50*w + 291],\ [131, 131, 50*w + 241],\ [139, 139, 6*w - 37],\ [139, 139, -6*w - 31],\ [149, 149, 20*w + 97],\ [149, 149, 20*w - 117],\ [157, 157, -12*w - 59],\ [157, 157, 12*w - 71],\ [163, 163, 4*w - 19],\ [163, 163, 4*w + 15],\ [173, 173, 4*w + 23],\ [173, 173, 4*w - 27],\ [211, 211, 2*w - 19],\ [211, 211, -2*w - 17],\ [227, 227, -4*w - 13],\ [227, 227, 4*w - 17],\ [233, 233, 6*w + 25],\ [233, 233, -6*w + 31],\ [239, 239, -14*w - 69],\ [239, 239, 14*w - 83],\ [241, 241, 46*w + 221],\ [241, 241, -46*w + 267],\ [251, 251, 22*w - 129],\ [251, 251, 22*w + 107],\ [257, 257, -34*w + 197],\ [257, 257, -34*w - 163],\ [277, 277, 4*w - 29],\ [277, 277, -4*w - 25],\ [283, 283, 4*w - 15],\ [283, 283, -4*w - 11],\ [289, 17, -17],\ [307, 307, 68*w - 395],\ [307, 307, 68*w + 327],\ [311, 311, 10*w - 61],\ [311, 311, -10*w - 51],\ [313, 313, -32*w - 155],\ [313, 313, 32*w - 187],\ [317, 317, -18*w - 85],\ [317, 317, 18*w - 103],\ [331, 331, -4*w - 9],\ [331, 331, 4*w - 13],\ [337, 337, -16*w - 79],\ [337, 337, 16*w - 95],\ [347, 347, -12*w + 67],\ [347, 347, 12*w + 55],\ [353, 353, 14*w - 79],\ [353, 353, 14*w + 65],\ [361, 19, -19],\ [367, 367, -32*w - 153],\ [367, 367, -32*w + 185],\ [383, 383, 56*w + 269],\ [383, 383, 56*w - 325],\ [389, 389, -4*w - 27],\ [389, 389, 4*w - 31],\ [401, 401, 8*w - 51],\ [401, 401, -8*w - 43],\ [421, 421, 12*w - 73],\ [421, 421, -12*w - 61],\ [439, 439, -8*w - 33],\ [439, 439, 8*w - 41],\ [443, 443, -4*w - 1],\ [443, 443, 4*w - 5],\ [461, 461, -30*w + 173],\ [461, 461, 30*w + 143],\ [463, 463, 2*w - 25],\ [463, 463, -2*w - 23],\ [467, 467, 34*w + 165],\ [467, 467, 34*w - 199],\ [503, 503, -26*w - 127],\ [503, 503, 26*w - 153],\ [509, 509, 4*w - 33],\ [509, 509, -4*w - 29],\ [521, 521, -10*w + 53],\ [521, 521, 10*w + 43],\ [529, 23, -23],\ [547, 547, 14*w - 85],\ [547, 547, -14*w - 71],\ [557, 557, 66*w + 317],\ [557, 557, 66*w - 383],\ [563, 563, 2*w - 27],\ [563, 563, -2*w - 25],\ [569, 569, -64*w + 373],\ [569, 569, 64*w + 309],\ [587, 587, 12*w + 53],\ [587, 587, -12*w + 65],\ [593, 593, 8*w + 45],\ [593, 593, 8*w - 53],\ [601, 601, -26*w - 123],\ [601, 601, 26*w - 149],\ [617, 617, 6*w - 23],\ [617, 617, -6*w - 17],\ [647, 647, 24*w - 137],\ [647, 647, -24*w - 113],\ [653, 653, 28*w - 165],\ [653, 653, -28*w - 137],\ [677, 677, -22*w - 103],\ [677, 677, 22*w - 125],\ [691, 691, 20*w - 113],\ [691, 691, 20*w + 93],\ [709, 709, -10*w - 41],\ [709, 709, 10*w - 51],\ [719, 719, -8*w + 37],\ [719, 719, -8*w - 29],\ [727, 727, -22*w - 109],\ [727, 727, 22*w - 131],\ [739, 739, 46*w - 269],\ [739, 739, -46*w - 223],\ [761, 761, -6*w - 13],\ [761, 761, 6*w - 19],\ [769, 769, 110*w - 639],\ [769, 769, 110*w + 529],\ [773, 773, 4*w - 37],\ [773, 773, -4*w - 33],\ [787, 787, 2*w - 31],\ [787, 787, -2*w - 29],\ [809, 809, 56*w + 271],\ [809, 809, -56*w + 327],\ [821, 821, 6*w - 17],\ [821, 821, -6*w - 11],\ [823, 823, 38*w - 223],\ [823, 823, -38*w - 185],\ [827, 827, 132*w - 767],\ [827, 827, 132*w + 635],\ [841, 29, -29],\ [853, 853, -76*w + 443],\ [853, 853, 76*w + 367],\ [863, 863, 14*w - 87],\ [863, 863, -14*w - 73],\ [911, 911, 2*w - 33],\ [911, 911, -2*w - 31],\ [919, 919, -6*w - 41],\ [919, 919, 6*w - 47],\ [929, 929, 50*w + 239],\ [929, 929, 50*w - 289],\ [953, 953, 6*w - 11],\ [953, 953, -6*w - 5],\ [967, 967, -8*w - 25],\ [967, 967, 8*w - 33],\ [991, 991, 16*w + 71],\ [991, 991, -16*w + 87]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^3 + 2*x^2 - 3*x - 3 K. = NumberField(heckePol) hecke_eigenvalues_array = [0, e, e^2 + e - 4, -e^2 - e + 2, -e^2 - e + 1, e^2 - 6, e^2 + 2*e, -e^2 + 2, -e^2 - 4*e + 2, 2*e^2 + 2*e - 7, -e^2 + e - 1, e^2 + e - 7, -e + 3, 3*e + 3, 2*e^2 - e - 9, -2*e^2 - e + 3, -e^2 - 6*e + 5, -e^2 - 2*e - 7, 2*e^2 + 7*e - 6, -4*e^2 - 5*e + 6, e^2 - 3*e - 10, 5*e^2 + 9*e - 10, 2*e^2 + 2*e - 7, -2*e^2 + 2*e + 17, 6*e^2 + 8*e - 18, 2*e^2 + 3*e - 10, 3*e + 14, 4*e^2 + e - 9, 2*e^2 - e - 21, e^2 + 7*e - 4, 3*e^2 + 9*e + 2, -6*e^2 - 3*e + 21, 2*e^2 + 9*e - 3, 5*e^2 + 12*e - 13, -3*e^2 + 11, -e^2 + 3*e + 5, e^2 + 7*e - 1, 2*e^2 + 2*e - 3, -6*e^2 - 2*e + 21, -7*e^2 - 17*e + 14, -e^2 + 5*e - 4, -2*e^2 - 12*e, 6*e^2 + 4*e - 12, 4*e^2 + 3*e - 27, -4*e^2 - 17*e + 9, e^2 - 10*e - 15, -3*e^2 - 2*e - 9, 6*e^2 + 12*e - 10, -2*e^2 - 10, -3*e^2 + e + 9, 3*e^2 - 3*e - 21, -e^2 + 4*e + 15, -e^2 - 8*e - 9, -7*e^2 - 9*e + 29, -3*e^2 - 9*e + 17, 2*e^2 + 12*e + 2, 6*e^2 + 12*e - 10, -2*e^2 - 5*e - 16, -7*e^2 - 7*e + 17, -13*e^2 - 19*e + 23, e^2 + 2*e + 3, -11*e^2 - 18*e + 21, -2*e^2 - 14*e - 10, 2*e^2 - 6*e + 2, -6*e^2 - 9*e + 24, 2*e^2 - 9*e - 12, 11*e^2 + 17*e - 25, -11*e^2 - 11*e + 17, -9*e^2 - 12*e + 26, 11*e^2 + 8*e - 34, -e - 15, 6*e^2 + 9*e - 3, -3*e^2 - e + 21, -11*e^2 - 5*e + 33, 3*e - 4, 2*e^2 + e + 2, -3*e - 22, -7*e^2 - 2*e + 36, -7*e^2 - 12*e + 6, -13*e^2 - 17*e + 30, -5*e^2 - 17*e + 18, 8*e^2 + 11*e - 6, -4*e^2 - 5*e + 30, 6*e^2 + 6*e - 19, -2*e^2 + 2*e + 5, -8*e^2 - 18*e + 17, 12*e + 5, -e^2 - 16*e - 3, -e^2 - 16*e - 9, -7*e^2 - 5*e + 39, -3*e^2 - 17*e + 3, e^2 + 10*e - 1, 9*e^2 + 6*e - 43, 5*e^2 + 7*e, 3*e^2 + e - 6, 5*e^2 + 10*e - 12, -11*e^2 - 24*e + 18, 4*e^2 + 6*e, -4*e^2 + 6*e + 24, 2*e^2 + 12*e + 3, -6*e^2 - 20*e + 3, 5*e^2 + 18*e - 10, 3*e + 23, -14*e^2 - 15*e + 35, -e^2 - 2*e - 9, 3*e^2 + 6*e - 33, 5*e^2 + 8*e - 27, 5*e^2 + 4*e - 9, 3*e^2 + 15*e + 15, -5*e^2 - e + 27, 10*e^2 + 5*e - 33, 3*e + 3, 8*e^2 + 19*e - 15, 4*e^2 - 5*e - 15, -e^2 - 13, -9*e^2 + 35, 7*e^2 + 22*e - 21, 3*e^2 - 2*e - 21, 6*e^2 + 2*e - 33, 6*e^2 + 4*e - 21, 2*e^2 + 17*e - 9, 6*e^2 + e - 33, -11*e^2 - 18*e + 3, -11*e^2 - 14*e + 27, 15*e + 5, 6*e^2 - 3*e - 31, 3*e^2 + 5*e + 32, 11*e^2 + 13*e - 16, -4*e^2 - 8*e - 9, -4*e^2 - 14*e + 3, 10*e^2 + 19*e - 22, -4*e^2 - 5*e + 26, 7*e^2 + 3*e - 43, e^2 + 3*e + 11, 9*e^2 + e - 18, -11*e^2 - 3*e + 42, -e^2 - 15*e + 14, 7*e^2 + 9*e + 14, -5*e^2 - 19*e + 12, 15*e^2 + 21*e - 24, -13*e^2 - 24*e + 26, 7*e^2 - 6*e - 28, 5*e^2 + 22*e + 3, 9*e^2 + 6*e - 33, 4*e^2 - 2*e - 12, -12*e^2 - 22*e + 36, 8*e^2 + 3*e - 43, 2*e^2 - 3*e - 19, -5*e^2 - 14*e + 18, -9*e^2 - 8*e + 24, -7*e^2 - 21*e + 11, 14*e^2 + 18*e - 43, 2*e^2 - 18*e - 19, -11*e^2 - 4*e + 30, 9*e^2 + 6*e - 24, -5*e - 12, -6*e^2 + 15*e + 48, 5*e^2 + 15*e - 19, -e^2 - 9*e - 1, 2*e^2 - 3, 10*e^2 + 8*e - 15, 14*e^2 + 11*e - 24, -2*e^2 - e + 12, -e^2 - 4*e - 22, 11*e^2 + 6*e - 52, -6*e^2 + 17, 2*e^2 + 2*e + 5] 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 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]