/* 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, 0, 9, -2, -3, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 2, w^4 - 2*w^3 - 3*w^2 + 4*w + 1]) primes_array = [ [4, 2, w^3 - 2*w^2 - 2*w + 3],\ [25, 5, w^4 - 2*w^3 - 2*w^2 + 3*w + 1],\ [25, 5, -w^5 + 3*w^4 + 2*w^3 - 8*w^2 - 2*w + 2],\ [25, 5, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 5*w + 4],\ [27, 3, -w^3 + 2*w^2 + 3*w - 2],\ [27, 3, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 6*w - 4],\ [37, 37, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 3*w + 2],\ [37, 37, -2*w^5 + 5*w^4 + 6*w^3 - 14*w^2 - 5*w + 5],\ [37, 37, w^5 - 3*w^4 - 2*w^3 + 8*w^2 - 2],\ [67, 67, w^5 - w^4 - 6*w^3 + 4*w^2 + 7*w - 2],\ [67, 67, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 7*w - 5],\ [67, 67, -2*w^5 + 6*w^4 + 4*w^3 - 16*w^2 - 3*w + 5],\ [67, 67, -2*w^5 + 4*w^4 + 8*w^3 - 12*w^2 - 9*w + 6],\ [67, 67, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 6*w + 4],\ [67, 67, w^5 - 4*w^4 + 10*w^2 - 2*w - 3],\ [107, 107, w^5 - 3*w^4 - w^3 + 7*w^2 - 3*w - 2],\ [107, 107, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 5*w - 5],\ [107, 107, 2*w^5 - 5*w^4 - 6*w^3 + 13*w^2 + 6*w - 4],\ [107, 107, 2*w^5 - 5*w^4 - 6*w^3 + 15*w^2 + 4*w - 6],\ [107, 107, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - 3*w + 2],\ [107, 107, -w^5 + 2*w^4 + 3*w^3 - 4*w^2 - w - 1],\ [137, 137, w^4 - w^3 - 4*w^2 + w + 4],\ [137, 137, -w^5 + 3*w^4 + 2*w^3 - 9*w^2 - w + 3],\ [137, 137, w^5 - 3*w^4 - w^3 + 6*w^2 + 1],\ [137, 137, -w^5 + 2*w^4 + 3*w^3 - 5*w^2 - 2*w + 4],\ [137, 137, w^5 - 2*w^4 - 4*w^3 + 5*w^2 + 6*w - 3],\ [137, 137, w^4 - 3*w^3 - w^2 + 6*w + 1],\ [139, 139, 2*w^5 - 4*w^4 - 8*w^3 + 12*w^2 + 9*w - 5],\ [139, 139, -w^5 + 2*w^4 + 3*w^3 - 3*w^2 - 3*w - 2],\ [139, 139, w^5 - w^4 - 6*w^3 + 4*w^2 + 7*w - 3],\ [139, 139, -w^5 + 4*w^4 - 10*w^2 + 2*w + 2],\ [139, 139, -w^5 + 3*w^4 + 2*w^3 - 8*w^2 - 3*w + 4],\ [139, 139, -2*w^5 + 6*w^4 + 4*w^3 - 16*w^2 - 3*w + 6],\ [151, 151, -3*w^5 + 8*w^4 + 8*w^3 - 23*w^2 - 7*w + 10],\ [151, 151, 2*w^4 - 3*w^3 - 6*w^2 + 3*w + 3],\ [151, 151, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 9*w + 4],\ [151, 151, 2*w^5 - 7*w^4 - 2*w^3 + 19*w^2 - w - 5],\ [151, 151, -2*w^5 + 4*w^4 + 9*w^3 - 13*w^2 - 12*w + 4],\ [151, 151, w^4 - 2*w^3 - 3*w^2 + 5*w + 4],\ [169, 13, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - 5*w + 4],\ [169, 13, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 7*w - 5],\ [169, 13, w^4 - 2*w^3 - w^2 + 2*w - 2],\ [233, 233, -w^5 + 2*w^4 + 2*w^3 - 4*w^2 + 3*w],\ [233, 233, -2*w^5 + 6*w^4 + 3*w^3 - 16*w^2 + 2*w + 4],\ [233, 233, 2*w^5 - 5*w^4 - 7*w^3 + 15*w^2 + 7*w - 7],\ [233, 233, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + 4*w - 3],\ [233, 233, 2*w^5 - 5*w^4 - 6*w^3 + 13*w^2 + 5*w - 1],\ [233, 233, w^5 - 3*w^4 - 4*w^3 + 12*w^2 + 5*w - 7],\ [269, 269, -2*w^5 + 6*w^4 + 2*w^3 - 14*w^2 + 4*w + 5],\ [269, 269, -w^5 + 2*w^4 + 3*w^3 - 4*w^2 - 2*w - 2],\ [269, 269, 3*w^5 - 8*w^4 - 5*w^3 + 18*w^2 - 2*w - 4],\ [269, 269, -3*w^5 + 7*w^4 + 7*w^3 - 15*w^2 - 2*w + 2],\ [269, 269, w^5 - 3*w^4 - w^3 + 7*w^2 - 2*w - 4],\ [269, 269, -2*w^5 + 4*w^4 + 6*w^3 - 8*w^2 - 4*w - 1],\ [289, 17, 2*w^5 - 5*w^4 - 4*w^3 + 12*w^2 - 2],\ [289, 17, w^4 - 6*w^2 + 2],\ [289, 17, -2*w^3 + 2*w^2 + 6*w - 3],\ [293, 293, 2*w^3 - 2*w^2 - 5*w - 1],\ [293, 293, w^5 - 3*w^4 + 6*w^2 - 5*w - 3],\ [293, 293, -3*w^5 + 7*w^4 + 9*w^3 - 18*w^2 - 9*w + 7],\ [293, 293, -3*w^5 + 8*w^4 + 7*w^3 - 21*w^2 - 5*w + 7],\ [293, 293, -w^5 + 2*w^4 + 2*w^3 - 2*w^2 - 4],\ [293, 293, w^4 - 3*w^3 - w^2 + 8*w - 1],\ [317, 317, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 8*w - 7],\ [317, 317, -w^4 + 2*w^3 + w^2 - 4*w + 4],\ [317, 317, 2*w^5 - 4*w^4 - 9*w^3 + 13*w^2 + 12*w - 8],\ [317, 317, 2*w^5 - 6*w^4 - 5*w^3 + 18*w^2 + 5*w - 6],\ [317, 317, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 6*w + 5],\ [317, 317, w^5 - 3*w^4 - 3*w^3 + 9*w^2 + 6*w - 3],\ [343, 7, w^5 - 3*w^4 - 2*w^3 + 10*w^2 - 4],\ [343, 7, w^5 - 2*w^4 - 4*w^3 + 4*w^2 + 7*w - 2],\ [349, 349, -w^5 + 4*w^4 + w^3 - 11*w^2 - 2*w + 4],\ [349, 349, -2*w^5 + 6*w^4 + 2*w^3 - 13*w^2 + w + 2],\ [349, 349, 2*w^5 - 5*w^4 - 7*w^3 + 16*w^2 + 9*w - 8],\ [349, 349, -2*w^5 + 5*w^4 + 7*w^3 - 15*w^2 - 10*w + 7],\ [349, 349, w^5 - 4*w^4 + 12*w^2 - 5*w - 6],\ [349, 349, w^5 - w^4 - 7*w^3 + 6*w^2 + 10*w - 5],\ [361, 19, -2*w^3 + 2*w^2 + 6*w + 1],\ [361, 19, w^5 - 3*w^4 - 2*w^3 + 9*w^2 + 2*w - 7],\ [361, 19, -2*w^3 + 4*w^2 + 4*w - 7],\ [367, 367, -w^5 + w^4 + 5*w^3 - w^2 - 7*w],\ [367, 367, -2*w^5 + 5*w^4 + 4*w^3 - 11*w^2 - 2*w + 2],\ [367, 367, w^5 - 4*w^4 + w^3 + 10*w^2 - 5*w - 5],\ [367, 367, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 8*w + 6],\ [367, 367, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 6*w + 6],\ [367, 367, w^5 - 4*w^4 + w^3 + 10*w^2 - 5*w - 3],\ [491, 491, 3*w^5 - 8*w^4 - 7*w^3 + 20*w^2 + 6*w - 5],\ [491, 491, 2*w^4 - 5*w^3 - 3*w^2 + 8*w + 1],\ [491, 491, -w^5 + 4*w^4 - 11*w^2 + w + 4],\ [491, 491, -w^5 + w^4 + 6*w^3 - 3*w^2 - 10*w + 3],\ [491, 491, 2*w^4 - 3*w^3 - 6*w^2 + 5*w + 3],\ [491, 491, 3*w^5 - 7*w^4 - 9*w^3 + 19*w^2 + 8*w - 9],\ [529, 23, -2*w^5 + 4*w^4 + 9*w^3 - 13*w^2 - 10*w + 5],\ [529, 23, -w^5 + 2*w^4 + 4*w^3 - 5*w^2 - 3*w],\ [529, 23, -3*w^5 + 8*w^4 + 7*w^3 - 20*w^2 - 4*w + 5],\ [601, 601, -w^5 + w^4 + 7*w^3 - 6*w^2 - 9*w + 5],\ [601, 601, -2*w^5 + 4*w^4 + 8*w^3 - 13*w^2 - 8*w + 7],\ [601, 601, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 9*w - 4],\ [601, 601, -w^5 + 3*w^4 + 3*w^3 - 10*w^2 - 5*w + 6],\ [601, 601, 2*w^5 - 6*w^4 - 4*w^3 + 15*w^2 + 4*w - 4],\ [601, 601, -w^5 + 4*w^4 + w^3 - 11*w^2 - w + 3],\ [691, 691, w^5 - 3*w^4 - 2*w^3 + 7*w^2 + 1],\ [691, 691, 3*w^5 - 8*w^4 - 9*w^3 + 24*w^2 + 8*w - 9],\ [691, 691, 2*w^5 - 4*w^4 - 7*w^3 + 11*w^2 + 4*w - 3],\ [691, 691, 2*w^5 - 6*w^4 - 3*w^3 + 14*w^2 - w - 3],\ [691, 691, -3*w^5 + 7*w^4 + 11*w^3 - 21*w^2 - 12*w + 9],\ [691, 691, -w^5 + 3*w^4 + 4*w^3 - 11*w^2 - 7*w + 7],\ [823, 823, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 8*w - 4],\ [823, 823, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 6*w + 8],\ [823, 823, w^5 - 2*w^4 - 6*w^3 + 10*w^2 + 8*w - 6],\ [823, 823, -w^5 + 3*w^4 + 4*w^3 - 10*w^2 - 7*w + 5],\ [823, 823, -2*w^5 + 4*w^4 + 9*w^3 - 14*w^2 - 11*w + 6],\ [823, 823, w^5 - 3*w^4 - 3*w^3 + 9*w^2 + 6*w - 6],\ [839, 839, -3*w^5 + 6*w^4 + 11*w^3 - 17*w^2 - 10*w + 8],\ [839, 839, -w^5 + w^4 + 8*w^3 - 7*w^2 - 12*w + 7],\ [839, 839, 2*w^5 - 3*w^4 - 11*w^3 + 11*w^2 + 14*w - 5],\ [839, 839, -w^5 + 2*w^4 + 5*w^3 - 6*w^2 - 10*w + 1],\ [839, 839, -w^5 + 2*w^4 + 4*w^3 - 5*w^2 - 8*w + 3],\ [839, 839, -3*w^5 + 9*w^4 + 5*w^3 - 22*w^2 - 2*w + 5],\ [841, 29, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 7*w - 5],\ [841, 29, -2*w^5 + 4*w^4 + 7*w^3 - 9*w^2 - 6*w],\ [841, 29, w^5 - 2*w^4 - 6*w^3 + 10*w^2 + 9*w - 8],\ [863, 863, 3*w^5 - 7*w^4 - 10*w^3 + 20*w^2 + 11*w - 10],\ [863, 863, w^4 - 4*w^3 + 2*w^2 + 8*w - 5],\ [863, 863, -w^5 + w^4 + 6*w^3 - 4*w^2 - 9*w + 5],\ [863, 863, 2*w^5 - 5*w^4 - 8*w^3 + 18*w^2 + 10*w - 10],\ [863, 863, 2*w^4 - 4*w^3 - 4*w^2 + 7*w + 1],\ [863, 863, 3*w^5 - 8*w^4 - 8*w^3 + 22*w^2 + 8*w - 7],\ [877, 877, 2*w^5 - 5*w^4 - 4*w^3 + 12*w^2 - 6],\ [877, 877, w^4 - 4*w^3 + 8*w + 1],\ [877, 877, -w^4 + 2*w^3 + 4*w^2 - 6*w - 1],\ [877, 877, -w^4 + 2*w^3 + 4*w^2 - 4*w - 2],\ [877, 877, -w^4 + 6*w^2 - 6],\ [877, 877, 2*w^5 - 5*w^4 - 4*w^3 + 10*w^2 + 2*w + 1],\ [881, 881, 2*w^5 - 4*w^4 - 10*w^3 + 15*w^2 + 14*w - 8],\ [881, 881, -3*w^5 + 7*w^4 + 8*w^3 - 17*w^2 - 6*w + 7],\ [881, 881, -2*w^5 + 6*w^4 + 4*w^3 - 17*w^2 - 3*w + 6],\ [881, 881, w^5 - 4*w^4 + w^3 + 10*w^2 - 4*w - 3],\ [881, 881, -w^5 + 2*w^4 + 5*w^3 - 9*w^2 - 7*w + 7],\ [881, 881, 2*w^5 - 6*w^4 - 6*w^3 + 19*w^2 + 8*w - 9],\ [929, 929, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 11*w - 6],\ [929, 929, 2*w^5 - 6*w^4 - 3*w^3 + 15*w^2 + 2*w - 6],\ [929, 929, 3*w^5 - 6*w^4 - 11*w^3 + 16*w^2 + 12*w - 6],\ [929, 929, -3*w^5 + 9*w^4 + 5*w^3 - 23*w^2 - 2*w + 8],\ [929, 929, 2*w^5 - 4*w^4 - 7*w^3 + 10*w^2 + 9*w - 4],\ [929, 929, -w^5 + 3*w^4 + 4*w^3 - 12*w^2 - 6*w + 6],\ [941, 941, -3*w^5 + 7*w^4 + 11*w^3 - 22*w^2 - 11*w + 11],\ [941, 941, w^5 - w^4 - 6*w^3 + 2*w^2 + 9*w - 1],\ [941, 941, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 8*w + 7],\ [941, 941, -w^5 + 3*w^4 + w^3 - 6*w^2 + 3*w - 2],\ [941, 941, 2*w^5 - 6*w^4 - 4*w^3 + 15*w^2 + 2*w - 4],\ [941, 941, -3*w^5 + 8*w^4 + 9*w^3 - 23*w^2 - 9*w + 7],\ [961, 31, 2*w^5 - 4*w^4 - 8*w^3 + 12*w^2 + 10*w - 7],\ [961, 31, -2*w^4 + 4*w^3 + 4*w^2 - 6*w - 1],\ [961, 31, 2*w^5 - 6*w^4 - 4*w^3 + 16*w^2 + 4*w - 5],\ [971, 971, -w^4 + 4*w^2 + 2*w + 1],\ [971, 971, 2*w^5 - 5*w^4 - 6*w^3 + 12*w^2 + 8*w - 4],\ [971, 971, w^4 - 6*w^2 - 2*w + 7],\ [971, 971, -w^4 + 4*w^3 - 10*w],\ [971, 971, 2*w^5 - 5*w^4 - 6*w^3 + 16*w^2 + 4*w - 7],\ [971, 971, w^4 - 4*w^3 + 2*w^2 + 6*w - 6]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [0, -2, -2, 10, 4, 4, -10, 2, 2, 4, 4, 4, 4, 4, 4, 12, 12, -12, 12, -12, 12, -6, 6, -6, -18, -18, 6, 4, 4, 4, 4, 4, 4, -16, 8, 8, 8, -16, 8, -10, 26, -10, -6, -18, 6, -6, 6, -18, -30, 18, -30, -18, -18, 18, -14, -2, -2, 6, 6, 6, 6, 6, 6, -18, -18, 18, -6, 18, -6, 16, 16, -14, -14, -2, -14, -14, -2, -10, -10, 26, 8, 8, -16, 8, 8, -16, -36, -12, -12, -36, -36, -36, -14, -14, 34, 46, 22, 22, 46, -38, -38, 4, -44, 28, -44, 4, 28, -16, 32, 32, 32, -16, 32, 24, 0, 0, -24, 24, -24, -38, -38, 10, 24, 0, 24, 0, 48, 48, -14, 10, -14, 10, -2, -2, -30, 6, 6, -18, -30, -18, -18, -18, -30, -18, -30, -18, -30, -18, -18, -30, 42, 42, 50, -46, -46, 12, 12, -12, 12, 12, -12] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([4, 2, w^3 - 2*w^2 - 2*w + 3])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]