/* 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([1, -5, 4, 7, -6, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([59, 59, -3*w^5 + 2*w^4 + 18*w^3 - 15*w^2 - 14*w + 7]) primes_array = [ [4, 2, -w^5 + 6*w^3 - w^2 - 6*w],\ [19, 19, 3*w^5 - 2*w^4 - 19*w^3 + 14*w^2 + 18*w - 9],\ [29, 29, -w^2 - w + 2],\ [29, 29, 2*w^5 - w^4 - 13*w^3 + 7*w^2 + 14*w - 3],\ [29, 29, -2*w^5 + w^4 + 13*w^3 - 8*w^2 - 15*w + 6],\ [49, 7, -3*w^5 + 2*w^4 + 19*w^3 - 15*w^2 - 19*w + 9],\ [49, 7, w^5 - w^4 - 7*w^3 + 6*w^2 + 9*w - 4],\ [49, 7, w^2 - 3],\ [59, 59, -3*w^5 + 2*w^4 + 18*w^3 - 15*w^2 - 14*w + 7],\ [59, 59, 2*w^5 - 12*w^3 + 3*w^2 + 12*w - 3],\ [59, 59, -2*w^5 + w^4 + 12*w^3 - 8*w^2 - 11*w + 3],\ [61, 61, -4*w^5 + 2*w^4 + 24*w^3 - 16*w^2 - 19*w + 8],\ [61, 61, 5*w^5 - 3*w^4 - 31*w^3 + 22*w^2 + 27*w - 13],\ [61, 61, 3*w^5 - w^4 - 18*w^3 + 9*w^2 + 16*w - 4],\ [71, 71, 2*w^5 - 12*w^3 + 2*w^2 + 11*w - 1],\ [71, 71, 3*w^5 - 2*w^4 - 18*w^3 + 15*w^2 + 13*w - 9],\ [71, 71, -w^5 + w^4 + 6*w^3 - 7*w^2 - 6*w + 5],\ [79, 79, -w^5 + 7*w^3 - w^2 - 10*w],\ [79, 79, 4*w^5 - 2*w^4 - 25*w^3 + 15*w^2 + 23*w - 9],\ [79, 79, -w^5 + 6*w^3 - 2*w^2 - 7*w + 4],\ [89, 89, -w^5 + w^4 + 6*w^3 - 6*w^2 - 4*w + 4],\ [89, 89, w^5 - w^4 - 7*w^3 + 6*w^2 + 8*w - 1],\ [89, 89, 4*w^5 - 2*w^4 - 25*w^3 + 16*w^2 + 24*w - 12],\ [101, 101, -2*w^5 + 2*w^4 + 13*w^3 - 13*w^2 - 12*w + 6],\ [101, 101, w^3 + w^2 - 4*w - 1],\ [101, 101, -4*w^5 + 2*w^4 + 25*w^3 - 16*w^2 - 25*w + 10],\ [109, 109, -w^3 + 5*w],\ [109, 109, -2*w^5 + w^4 + 12*w^3 - 9*w^2 - 11*w + 8],\ [109, 109, 4*w^5 - 2*w^4 - 25*w^3 + 15*w^2 + 24*w - 9],\ [125, 5, 2*w^5 - w^4 - 12*w^3 + 8*w^2 + 10*w - 7],\ [149, 149, 3*w^5 - 2*w^4 - 19*w^3 + 14*w^2 + 20*w - 9],\ [149, 149, 5*w^5 - 2*w^4 - 31*w^3 + 16*w^2 + 30*w - 9],\ [149, 149, -2*w^5 + 12*w^3 - 3*w^2 - 10*w + 1],\ [199, 199, -3*w^5 + 3*w^4 + 20*w^3 - 19*w^2 - 20*w + 10],\ [199, 199, 2*w^5 - w^4 - 13*w^3 + 7*w^2 + 15*w - 2],\ [199, 199, 3*w^5 - w^4 - 17*w^3 + 10*w^2 + 10*w - 5],\ [211, 211, 9*w^5 - 5*w^4 - 57*w^3 + 38*w^2 + 56*w - 21],\ [211, 211, -5*w^5 + 4*w^4 + 31*w^3 - 27*w^2 - 27*w + 12],\ [211, 211, -7*w^5 + 3*w^4 + 43*w^3 - 26*w^2 - 39*w + 15],\ [241, 241, 5*w^5 - 3*w^4 - 30*w^3 + 22*w^2 + 24*w - 12],\ [241, 241, -5*w^5 + 3*w^4 + 32*w^3 - 21*w^2 - 32*w + 12],\ [241, 241, -3*w^5 + w^4 + 17*w^3 - 10*w^2 - 12*w + 7],\ [251, 251, -3*w^5 + 17*w^3 - 4*w^2 - 12*w + 2],\ [251, 251, -8*w^5 + 5*w^4 + 49*w^3 - 38*w^2 - 42*w + 23],\ [251, 251, -3*w^5 + 2*w^4 + 18*w^3 - 14*w^2 - 16*w + 7],\ [269, 269, -w^4 + 6*w^2 - w - 3],\ [269, 269, -w^5 + 6*w^3 - w^2 - 4*w - 1],\ [269, 269, w^5 - w^4 - 6*w^3 + 7*w^2 + 3*w - 5],\ [271, 271, -6*w^5 + 4*w^4 + 37*w^3 - 28*w^2 - 33*w + 14],\ [271, 271, -3*w^5 + 2*w^4 + 19*w^3 - 14*w^2 - 20*w + 8],\ [271, 271, 8*w^5 - 4*w^4 - 49*w^3 + 33*w^2 + 44*w - 21],\ [311, 311, 6*w^5 - 4*w^4 - 37*w^3 + 29*w^2 + 34*w - 17],\ [311, 311, -3*w^5 + w^4 + 18*w^3 - 8*w^2 - 15*w + 1],\ [311, 311, -5*w^5 + 3*w^4 + 31*w^3 - 23*w^2 - 30*w + 13],\ [311, 311, -6*w^5 + 3*w^4 + 36*w^3 - 25*w^2 - 29*w + 16],\ [311, 311, 4*w^5 - 3*w^4 - 25*w^3 + 21*w^2 + 22*w - 9],\ [311, 311, w^5 - 7*w^3 + 2*w^2 + 9*w - 1],\ [331, 331, 6*w^5 - 3*w^4 - 37*w^3 + 24*w^2 + 32*w - 13],\ [331, 331, 8*w^5 - 4*w^4 - 49*w^3 + 31*w^2 + 44*w - 17],\ [331, 331, 7*w^5 - 4*w^4 - 44*w^3 + 30*w^2 + 42*w - 19],\ [331, 331, -7*w^5 + 4*w^4 + 43*w^3 - 31*w^2 - 37*w + 18],\ [331, 331, 6*w^5 - 3*w^4 - 38*w^3 + 24*w^2 + 38*w - 14],\ [331, 331, 5*w^5 - 2*w^4 - 30*w^3 + 17*w^2 + 24*w - 11],\ [349, 349, 3*w^5 - 3*w^4 - 19*w^3 + 19*w^2 + 17*w - 10],\ [349, 349, 5*w^5 - 2*w^4 - 31*w^3 + 18*w^2 + 29*w - 13],\ [349, 349, -12*w^5 + 6*w^4 + 74*w^3 - 46*w^2 - 67*w + 23],\ [349, 349, -w^5 + w^4 + 6*w^3 - 8*w^2 - 4*w + 4],\ [349, 349, -w^5 + w^4 + 7*w^3 - 6*w^2 - 10*w + 1],\ [349, 349, 2*w^5 - 2*w^4 - 13*w^3 + 14*w^2 + 14*w - 12],\ [389, 389, 8*w^5 - 4*w^4 - 49*w^3 + 33*w^2 + 43*w - 21],\ [389, 389, -5*w^5 + 4*w^4 + 31*w^3 - 27*w^2 - 28*w + 14],\ [389, 389, -2*w^5 + 2*w^4 + 12*w^3 - 14*w^2 - 9*w + 6],\ [401, 401, w^3 + w^2 - 2*w - 3],\ [401, 401, -6*w^5 + 2*w^4 + 37*w^3 - 18*w^2 - 35*w + 12],\ [401, 401, 4*w^5 - 3*w^4 - 24*w^3 + 21*w^2 + 17*w - 12],\ [409, 409, 8*w^5 - 4*w^4 - 49*w^3 + 31*w^2 + 44*w - 16],\ [409, 409, 4*w^5 - w^4 - 25*w^3 + 10*w^2 + 26*w - 6],\ [409, 409, -2*w^5 + 12*w^3 - 3*w^2 - 11*w + 6],\ [421, 421, -7*w^5 + 5*w^4 + 43*w^3 - 36*w^2 - 37*w + 20],\ [421, 421, -2*w^5 + 2*w^4 + 13*w^3 - 13*w^2 - 11*w + 5],\ [421, 421, w^5 - 2*w^4 - 6*w^3 + 12*w^2 + 3*w - 8],\ [431, 431, -4*w^5 + 4*w^4 + 26*w^3 - 26*w^2 - 25*w + 14],\ [431, 431, 7*w^5 - 4*w^4 - 44*w^3 + 31*w^2 + 41*w - 20],\ [431, 431, -6*w^5 + 4*w^4 + 37*w^3 - 29*w^2 - 32*w + 14],\ [439, 439, 3*w^5 - w^4 - 19*w^3 + 9*w^2 + 18*w - 7],\ [439, 439, 4*w^5 - 2*w^4 - 26*w^3 + 15*w^2 + 28*w - 11],\ [439, 439, w^5 - w^4 - 7*w^3 + 8*w^2 + 8*w - 7],\ [461, 461, 5*w^5 - 3*w^4 - 31*w^3 + 22*w^2 + 30*w - 13],\ [461, 461, -3*w^5 + 3*w^4 + 19*w^3 - 19*w^2 - 16*w + 7],\ [461, 461, 5*w^5 - 3*w^4 - 30*w^3 + 24*w^2 + 24*w - 16],\ [479, 479, 5*w^5 - 3*w^4 - 30*w^3 + 22*w^2 + 23*w - 11],\ [479, 479, -4*w^5 + 3*w^4 + 25*w^3 - 21*w^2 - 21*w + 9],\ [479, 479, -3*w^5 + w^4 + 19*w^3 - 9*w^2 - 18*w + 8],\ [491, 491, 6*w^5 - 4*w^4 - 38*w^3 + 28*w^2 + 35*w - 14],\ [491, 491, 5*w^5 - 2*w^4 - 31*w^3 + 18*w^2 + 28*w - 13],\ [491, 491, w^5 + w^4 - 6*w^3 - 3*w^2 + 8*w],\ [541, 541, -w^4 - w^3 + 6*w^2 + 2*w - 5],\ [541, 541, -5*w^5 + 4*w^4 + 31*w^3 - 28*w^2 - 28*w + 16],\ [541, 541, 5*w^5 - 2*w^4 - 30*w^3 + 18*w^2 + 24*w - 11],\ [619, 619, w^3 - 6*w],\ [619, 619, -w^5 + w^4 + 6*w^3 - 8*w^2 - 6*w + 8],\ [619, 619, 3*w^5 - w^4 - 19*w^3 + 8*w^2 + 20*w - 5],\ [619, 619, -2*w^5 + w^4 + 13*w^3 - 8*w^2 - 14*w + 2],\ [619, 619, -4*w^5 + 3*w^4 + 26*w^3 - 21*w^2 - 25*w + 11],\ [619, 619, -6*w^5 + 2*w^4 + 37*w^3 - 19*w^2 - 34*w + 14],\ [631, 631, 6*w^5 - 4*w^4 - 38*w^3 + 29*w^2 + 35*w - 16],\ [631, 631, -4*w^5 + w^4 + 25*w^3 - 11*w^2 - 24*w + 9],\ [631, 631, -2*w^5 + 3*w^4 + 13*w^3 - 18*w^2 - 11*w + 10],\ [641, 641, -3*w^5 + 2*w^4 + 20*w^3 - 14*w^2 - 23*w + 10],\ [641, 641, -6*w^5 + 3*w^4 + 37*w^3 - 24*w^2 - 32*w + 14],\ [641, 641, -4*w^5 + 2*w^4 + 25*w^3 - 17*w^2 - 25*w + 11],\ [661, 661, 2*w^5 - w^4 - 13*w^3 + 9*w^2 + 15*w - 6],\ [661, 661, -4*w^5 + 3*w^4 + 25*w^3 - 22*w^2 - 22*w + 12],\ [661, 661, -5*w^5 + 3*w^4 + 32*w^3 - 22*w^2 - 33*w + 15],\ [709, 709, 2*w^5 - 2*w^4 - 12*w^3 + 13*w^2 + 10*w - 5],\ [709, 709, 5*w^5 - 3*w^4 - 31*w^3 + 23*w^2 + 26*w - 14],\ [709, 709, 7*w^5 - 4*w^4 - 43*w^3 + 31*w^2 + 37*w - 17],\ [719, 719, 6*w^5 - 4*w^4 - 38*w^3 + 29*w^2 + 38*w - 16],\ [719, 719, -6*w^5 + 4*w^4 + 37*w^3 - 28*w^2 - 31*w + 14],\ [719, 719, -6*w^5 + 2*w^4 + 37*w^3 - 19*w^2 - 36*w + 13],\ [729, 3, -3],\ [739, 739, 11*w^5 - 5*w^4 - 69*w^3 + 41*w^2 + 67*w - 24],\ [739, 739, -9*w^5 + 5*w^4 + 56*w^3 - 38*w^2 - 51*w + 18],\ [739, 739, 8*w^5 - 4*w^4 - 49*w^3 + 31*w^2 + 43*w - 17],\ [751, 751, -7*w^5 + 4*w^4 + 44*w^3 - 31*w^2 - 43*w + 19],\ [751, 751, -7*w^5 + 5*w^4 + 44*w^3 - 35*w^2 - 40*w + 18],\ [751, 751, -7*w^5 + 5*w^4 + 44*w^3 - 36*w^2 - 41*w + 20],\ [761, 761, -3*w^5 + 2*w^4 + 19*w^3 - 16*w^2 - 20*w + 11],\ [761, 761, 7*w^5 - 4*w^4 - 42*w^3 + 31*w^2 + 33*w - 17],\ [761, 761, 3*w^5 - 2*w^4 - 20*w^3 + 14*w^2 + 24*w - 10],\ [761, 761, -6*w^5 + 2*w^4 + 36*w^3 - 18*w^2 - 31*w + 9],\ [761, 761, -8*w^5 + 5*w^4 + 50*w^3 - 36*w^2 - 45*w + 21],\ [761, 761, 3*w^5 - 2*w^4 - 20*w^3 + 15*w^2 + 23*w - 10],\ [769, 769, 5*w^5 - 3*w^4 - 30*w^3 + 22*w^2 + 22*w - 10],\ [769, 769, w^5 - w^4 - 5*w^3 + 8*w^2 + 2*w - 5],\ [769, 769, -6*w^5 + 2*w^4 + 37*w^3 - 18*w^2 - 36*w + 10],\ [811, 811, 5*w^5 - w^4 - 30*w^3 + 13*w^2 + 26*w - 10],\ [811, 811, -3*w^5 + 2*w^4 + 20*w^3 - 15*w^2 - 21*w + 10],\ [811, 811, -8*w^5 + 6*w^4 + 50*w^3 - 42*w^2 - 45*w + 24],\ [821, 821, 8*w^5 - 4*w^4 - 50*w^3 + 32*w^2 + 47*w - 21],\ [821, 821, -2*w^5 + 13*w^3 - 3*w^2 - 16*w + 2],\ [821, 821, -6*w^5 + 3*w^4 + 38*w^3 - 23*w^2 - 37*w + 15],\ [829, 829, -9*w^5 + 5*w^4 + 56*w^3 - 38*w^2 - 51*w + 24],\ [829, 829, -5*w^5 + 3*w^4 + 30*w^3 - 22*w^2 - 23*w + 13],\ [829, 829, -5*w^5 + 2*w^4 + 32*w^3 - 17*w^2 - 33*w + 13],\ [829, 829, 6*w^5 - 4*w^4 - 38*w^3 + 28*w^2 + 35*w - 13],\ [829, 829, -2*w^5 + w^4 + 11*w^3 - 9*w^2 - 7*w + 8],\ [829, 829, 7*w^5 - 3*w^4 - 43*w^3 + 25*w^2 + 40*w - 17],\ [859, 859, 3*w^5 - w^4 - 19*w^3 + 9*w^2 + 20*w - 9],\ [859, 859, -11*w^5 + 6*w^4 + 68*w^3 - 45*w^2 - 61*w + 23],\ [859, 859, -9*w^5 + 4*w^4 + 55*w^3 - 32*w^2 - 49*w + 15],\ [919, 919, -2*w^5 + 2*w^4 + 11*w^3 - 14*w^2 - 5*w + 8],\ [919, 919, -5*w^5 + 3*w^4 + 32*w^3 - 22*w^2 - 32*w + 16],\ [919, 919, 4*w^5 - 3*w^4 - 24*w^3 + 21*w^2 + 19*w - 9],\ [919, 919, -8*w^5 + 4*w^4 + 49*w^3 - 32*w^2 - 43*w + 17],\ [919, 919, 3*w^5 - w^4 - 18*w^3 + 11*w^2 + 17*w - 11],\ [919, 919, -7*w^5 + 4*w^4 + 44*w^3 - 29*w^2 - 42*w + 17],\ [929, 929, 7*w^5 - 4*w^4 - 43*w^3 + 30*w^2 + 39*w - 15],\ [929, 929, 6*w^5 - 3*w^4 - 36*w^3 + 25*w^2 + 30*w - 14],\ [929, 929, -3*w^5 + w^4 + 17*w^3 - 9*w^2 - 11*w + 2],\ [941, 941, 7*w^5 - 4*w^4 - 42*w^3 + 31*w^2 + 34*w - 19],\ [941, 941, 6*w^5 - 3*w^4 - 36*w^3 + 24*w^2 + 31*w - 15],\ [941, 941, 7*w^5 - 3*w^4 - 42*w^3 + 25*w^2 + 35*w - 15]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [-2, 4, 2, 2, 2, 6, 2, -6, 1, -12, 8, -10, -10, 6, -12, 8, 0, -4, -16, -8, -10, -2, -6, -10, 10, -6, -14, -6, -18, 2, -2, -18, 22, -8, 24, 16, -8, 4, -4, -10, -30, 2, 0, -16, -4, 2, 10, 2, 8, 4, 0, 0, -32, 28, -4, 0, -16, -16, -20, 8, -20, 12, -20, -18, -6, -2, 2, -26, -34, 18, 26, -30, 2, 6, 22, -34, 26, -14, 10, 26, 22, 20, -32, 24, -8, 24, -16, 10, 34, 2, 0, 28, 16, -24, 28, -36, -42, 38, -14, 40, 32, -8, -20, 28, -4, -16, 40, 8, 6, 30, 46, 22, 14, -10, -26, -30, 10, -12, 28, 48, 22, -4, 20, 20, 8, -32, -40, -46, -42, -30, 18, 22, -38, 14, 10, -2, -52, -28, -36, -54, -10, -30, -2, -14, 6, -2, -18, 14, -4, 36, -36, -56, -52, -48, 36, 44, 56, -6, 6, 50, 46, 14, -46] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([59, 59, -3*w^5 + 2*w^4 + 18*w^3 - 15*w^2 - 14*w + 7])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]