/* 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([-58, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([4, 2, 2]) primes_array = [ [2, 2, w],\ [3, 3, w + 1],\ [3, 3, w + 2],\ [7, 7, -2*w + 15],\ [7, 7, -2*w - 15],\ [11, 11, w + 5],\ [11, 11, w + 6],\ [19, 19, w + 1],\ [19, 19, w + 18],\ [23, 23, w + 9],\ [23, 23, -w + 9],\ [25, 5, 5],\ [29, 29, w],\ [37, 37, w + 13],\ [37, 37, w + 24],\ [43, 43, w + 12],\ [43, 43, w + 31],\ [61, 61, w + 27],\ [61, 61, w + 34],\ [71, 71, 12*w - 91],\ [71, 71, 12*w + 91],\ [101, 101, w + 19],\ [101, 101, w + 82],\ [103, 103, -3*w + 25],\ [103, 103, -3*w - 25],\ [131, 131, w + 53],\ [131, 131, w + 78],\ [151, 151, 2*w - 9],\ [151, 151, -2*w - 9],\ [157, 157, w + 23],\ [157, 157, w + 134],\ [163, 163, w + 59],\ [163, 163, w + 104],\ [167, 167, -w - 15],\ [167, 167, w - 15],\ [169, 13, -13],\ [199, 199, -4*w + 27],\ [199, 199, -4*w - 27],\ [211, 211, w + 67],\ [211, 211, w + 144],\ [223, 223, 2*w - 3],\ [223, 223, -2*w - 3],\ [229, 229, w + 79],\ [229, 229, w + 150],\ [233, 233, 3*w - 17],\ [233, 233, -3*w - 17],\ [239, 239, 6*w + 43],\ [239, 239, 6*w - 43],\ [241, 241, -7*w + 51],\ [241, 241, -7*w - 51],\ [251, 251, w + 73],\ [251, 251, w + 178],\ [257, 257, -8*w - 63],\ [257, 257, -8*w + 63],\ [269, 269, w + 70],\ [269, 269, w + 199],\ [281, 281, 15*w + 113],\ [281, 281, 15*w - 113],\ [289, 17, -17],\ [293, 293, w + 75],\ [293, 293, w + 218],\ [307, 307, w + 66],\ [307, 307, w + 241],\ [313, 313, -6*w + 49],\ [313, 313, -6*w - 49],\ [317, 317, w + 117],\ [317, 317, w + 200],\ [331, 331, w + 148],\ [331, 331, w + 183],\ [353, 353, -3*w - 13],\ [353, 353, 3*w - 13],\ [379, 379, w + 171],\ [379, 379, w + 208],\ [383, 383, -w - 21],\ [383, 383, w - 21],\ [389, 389, w + 35],\ [389, 389, w + 354],\ [401, 401, 3*w - 11],\ [401, 401, -3*w - 11],\ [421, 421, w + 30],\ [421, 421, w + 391],\ [431, 431, -12*w - 89],\ [431, 431, -12*w + 89],\ [439, 439, 3*w - 31],\ [439, 439, -3*w - 31],\ [443, 443, w + 175],\ [443, 443, w + 268],\ [457, 457, -11*w - 81],\ [457, 457, -11*w + 81],\ [461, 461, w + 141],\ [461, 461, w + 320],\ [463, 463, 8*w + 57],\ [463, 463, 8*w - 57],\ [467, 467, w + 92],\ [467, 467, w + 375],\ [487, 487, -4*w - 21],\ [487, 487, 4*w - 21],\ [491, 491, w + 135],\ [491, 491, w + 356],\ [521, 521, -3*w - 1],\ [521, 521, 3*w - 1],\ [541, 541, w + 41],\ [541, 541, w + 500],\ [563, 563, w + 109],\ [563, 563, w + 454],\ [593, 593, -4*w - 39],\ [593, 593, 4*w - 39],\ [619, 619, w + 36],\ [619, 619, w + 583],\ [631, 631, -9*w + 73],\ [631, 631, -9*w - 73],\ [647, 647, 24*w - 181],\ [647, 647, 24*w + 181],\ [653, 653, w + 133],\ [653, 653, w + 520],\ [659, 659, w + 306],\ [659, 659, w + 353],\ [673, 673, 12*w + 95],\ [673, 673, 12*w - 95],\ [677, 677, w + 259],\ [677, 677, w + 418],\ [719, 719, -6*w - 37],\ [719, 719, 6*w - 37],\ [733, 733, w + 304],\ [733, 733, w + 429],\ [739, 739, w + 102],\ [739, 739, w + 637],\ [757, 757, w + 315],\ [757, 757, w + 442],\ [761, 761, -10*w + 81],\ [761, 761, -10*w - 81],\ [773, 773, w + 226],\ [773, 773, w + 547],\ [797, 797, w + 281],\ [797, 797, w + 516],\ [827, 827, w + 239],\ [827, 827, w + 588],\ [829, 829, w + 108],\ [829, 829, w + 721],\ [853, 853, w + 42],\ [853, 853, w + 811],\ [857, 857, 2*w - 33],\ [857, 857, -2*w - 33],\ [859, 859, w + 403],\ [859, 859, w + 456],\ [863, 863, 6*w - 35],\ [863, 863, -6*w - 35],\ [907, 907, w + 128],\ [907, 907, w + 779],\ [919, 919, 4*w - 3],\ [919, 919, -4*w - 3],\ [929, 929, 21*w - 157],\ [929, 929, 21*w + 157],\ [937, 937, 6*w - 55],\ [937, 937, -6*w - 55],\ [947, 947, w + 406],\ [947, 947, w + 541],\ [953, 953, 14*w + 111],\ [953, 953, 14*w - 111],\ [961, 31, -31],\ [971, 971, w + 143],\ [971, 971, w + 828],\ [977, 977, 9*w + 61],\ [977, 977, -9*w + 61],\ [991, 991, -21*w + 163],\ [991, 991, -21*w - 163],\ [997, 997, w + 105],\ [997, 997, w + 892]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 + 4*x^3 - 8*x + 1 K. = NumberField(heckePol) hecke_eigenvalues_array = [0, -e - 2, e, e^3 + 2*e^2 - 3*e - 2, -e^3 - 4*e^2 - e + 4, -e^3 - 4*e^2 - 2*e + 4, e^3 + 2*e^2 - 2*e, -e^3 - 2*e^2 + 3*e + 2, e^3 + 4*e^2 + e - 4, -e^3 - 2*e^2 + 3*e - 2, e^3 + 4*e^2 + e - 8, 2*e^2 + 4*e - 5, 2*e^2 + 4*e + 2, -2*e^2 + 12, -2*e^2 - 8*e + 4, e^3 + 6*e^2 + 6*e - 4, -e^3 + 6*e, -2*e^3 - 5*e^2 + 2*e - 3, 2*e^3 + 7*e^2 + 2*e - 11, -2*e^2 - 2*e + 6, -2*e^2 - 6*e + 2, -2*e^3 - 4*e^2 + 10*e + 6, 2*e^3 + 8*e^2 - 2*e - 14, 2*e^3 + 6*e^2 + 2*e - 6, -2*e^3 - 6*e^2 - 2*e - 2, e^3 + 4*e^2 + 3*e - 12, -e^3 - 2*e^2 + e - 10, 5*e^3 + 18*e^2 + 7*e - 22, -5*e^3 - 12*e^2 + 5*e - 4, 4*e^3 + 11*e^2 - 8*e - 17, -4*e^3 - 13*e^2 + 4*e + 11, -2*e^3 - 4*e^2 + 3*e + 6, 2*e^3 + 8*e^2 + 5*e, -4*e^3 - 8*e^2 + 10*e - 4, 4*e^3 + 16*e^2 + 6*e - 24, -2*e^2 - 4*e - 3, -e^3 - 2*e^2 + 5*e + 2, e^3 + 4*e^2 - e - 8, 2*e^3 + 4*e^2 - 15*e - 10, -2*e^3 - 8*e^2 + 7*e + 20, -2*e^3 - 8*e^2 - 4*e + 4, 2*e^3 + 4*e^2 - 4*e - 4, -2*e^3 - 13*e^2 - 14*e + 25, 2*e^3 - e^2 - 14*e + 17, -2*e^3 - 6*e^2 + 4*e + 17, 2*e^3 + 6*e^2 - 4*e + 1, 5*e^3 + 8*e^2 - 25*e - 16, -5*e^3 - 22*e^2 - 3*e + 26, -2*e^3 - 12*e^2 - 8*e + 25, 2*e^3 - 16*e + 9, -2*e^3 + 13*e - 16, 2*e^3 + 12*e^2 + 11*e - 26, -7*e^2 - 10*e + 12, -7*e^2 - 18*e + 4, -4*e^3 - 17*e^2 - 8*e + 23, 4*e^3 + 7*e^2 - 12*e + 3, -2*e^3 - 2*e^2 + 16*e + 13, 2*e^3 + 10*e^2 - 11, -2*e^2 - 4*e - 6, -2*e^3 + 14*e - 6, 2*e^3 + 12*e^2 + 10*e - 18, e^3 + 2*e^2 - 2*e - 20, -e^3 - 4*e^2 - 2*e - 16, -4*e^3 - 15*e^2 - 10*e + 18, 4*e^3 + 9*e^2 - 2*e + 10, -4*e^3 - 13*e^2 + 4*e + 13, 4*e^3 + 11*e^2 - 8*e - 15, -6*e^3 - 12*e^2 + 17*e, 6*e^3 + 24*e^2 + 7*e - 34, 6*e^2 + 12*e - 12, 6*e^2 + 12*e - 12, -7*e^3 - 16*e^2 + 13*e - 4, 7*e^3 + 26*e^2 + 7*e - 38, -e^3 + 6*e^2 + 19*e - 30, e^3 + 12*e^2 + 17*e - 36, -2*e^3 - 12*e^2 - 10*e + 12, 2*e^3 - 14*e, 2*e^3 - 2*e^2 - 24*e + 1, -2*e^3 - 14*e^2 - 8*e + 25, -6*e^3 - 21*e^2 + 6*e + 25, 6*e^3 + 15*e^2 - 18*e - 23, 3*e^3 + 6*e^2 - 15*e - 6, -3*e^3 - 12*e^2 + 3*e + 24, 5*e^3 + 18*e^2 - 3*e - 14, -5*e^3 - 12*e^2 + 15*e + 24, 3*e^3 + 12*e^2 - 5*e - 8, -3*e^3 - 6*e^2 + 17*e + 26, -4*e^3 + 24*e - 26, 4*e^3 + 24*e^2 + 24*e - 42, 6*e^3 + 22*e^2 + 10*e - 30, -6*e^3 - 14*e^2 + 6*e - 10, -e^3 - 4*e^2 + e - 24, e^3 + 2*e^2 - 5*e - 34, 9*e^3 + 34*e^2 - 40, -9*e^3 - 20*e^2 + 28*e + 24, -e^3 + 8*e^2 + 21*e - 16, e^3 + 14*e^2 + 23*e - 18, 7*e^3 + 26*e^2 - 16, -7*e^3 - 16*e^2 + 20*e + 32, -8*e^3 - 25*e^2 + 6*e + 38, 8*e^3 + 23*e^2 - 10*e - 10, 4*e^3 + 2*e^2 - 28*e + 8, -4*e^3 - 22*e^2 - 12*e + 40, -6*e^3 - 24*e^2 - 3*e + 18, 6*e^3 + 12*e^2 - 21*e - 24, -4*e^3 - 9*e^2 + 18*e + 38, 4*e^3 + 15*e^2 - 6*e - 2, -2*e^3 - 8*e^2 + e + 26, 2*e^3 + 4*e^2 - 9*e + 8, -6*e^3 - 12*e^2 + 28*e + 12, 6*e^3 + 24*e^2 - 4*e - 44, e^3 + 6*e^2 - 5*e - 22, -e^3 + 17*e + 4, 2*e^3 - 5*e^2 - 26*e + 5, -2*e^3 - 17*e^2 - 18*e + 21, -12*e^2 - 23*e + 16, -12*e^2 - 25*e + 14, -2*e^3 + 10*e^2 + 40*e - 21, 2*e^3 + 22*e^2 + 24*e - 45, -4*e^3 + 6*e^2 + 40*e - 30, 4*e^3 + 30*e^2 + 32*e - 54, 4*e^3 + 14*e^2 + 4*e - 18, -4*e^3 - 10*e^2 + 4*e - 2, 10*e^3 + 26*e^2 - 26*e - 44, -10*e^3 - 34*e^2 + 10*e + 32, 10*e^3 + 32*e^2 + e - 38, -10*e^3 - 28*e^2 + 7*e + 8, 4*e^3 + 9*e^2 - 16*e - 29, -4*e^3 - 15*e^2 + 4*e + 7, -8*e^3 - 26*e^2 + 12, 8*e^3 + 22*e^2 - 8*e - 28, -12*e^3 - 29*e^2 + 12*e - 1, 12*e^3 + 43*e^2 + 16*e - 45, -4*e^3 - 19*e^2 - 16*e + 35, 4*e^3 + 5*e^2 - 12*e + 23, -8*e^2 - 15*e + 44, -8*e^2 - 17*e + 42, 8*e^3 + 29*e^2 - 47, -8*e^3 - 19*e^2 + 20*e + 5, -16*e^2 - 28*e + 34, -16*e^2 - 36*e + 26, 12*e^3 + 37*e^2 - 10*e - 20, -12*e^3 - 35*e^2 + 14*e + 52, 3*e^3 + 6*e^2 - 14*e - 16, -3*e^3 - 12*e^2 + 2*e + 12, 4*e^2 + 6*e - 40, 4*e^2 + 10*e - 36, -e^3 - 24*e^2 - 45*e + 48, e^3 - 18*e^2 - 39*e + 50, 4*e^3 + 6*e^2 - 18*e - 18, -4*e^3 - 18*e^2 - 6*e + 10, 4*e^3 + 12*e^2 - 8*e - 52, -4*e^3 - 12*e^2 + 8*e - 20, 2*e^2 - 12*e - 36, 2*e^2 + 20*e - 4, 13*e^3 + 40*e^2 - 14*e - 44, -13*e^3 - 38*e^2 + 18*e + 40, -4*e^3 - 4*e^2 + 40*e + 19, 4*e^3 + 20*e^2 - 8*e - 45, -15*e^2 - 30*e + 32, -9*e^3 - 30*e^2 + 11*e + 50, 9*e^3 + 24*e^2 - 23*e - 20, -4*e^3 - 23*e^2 - 16*e + 18, 4*e^3 + e^2 - 28*e - 10, -12*e^3 - 40*e^2 + 8*e + 44, 12*e^3 + 32*e^2 - 24*e - 36, 10*e^3 + 34*e^2 + 2*e - 24, -10*e^3 - 26*e^2 + 14*e + 28] 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])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]