/* 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([20, 0, -10, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([11, 11, -1/2*w^3 - 1/2*w^2 + 3*w + 3]) primes_array = [ [4, 2, 1/2*w^3 - 3*w + 2],\ [5, 5, w^2 - w - 5],\ [11, 11, -1/2*w^3 - 1/2*w^2 + 3*w + 3],\ [11, 11, -1/2*w^2 + w + 2],\ [11, 11, -1/2*w^2 - w + 2],\ [11, 11, 1/2*w^3 - 1/2*w^2 - 3*w + 3],\ [29, 29, -1/2*w^2 - w + 4],\ [29, 29, -1/2*w^3 + 1/2*w^2 + 3*w - 1],\ [29, 29, -1/2*w^3 - 1/2*w^2 + 3*w + 1],\ [29, 29, -1/2*w^2 + w + 4],\ [41, 41, w^3 - 1/2*w^2 - 6*w + 6],\ [41, 41, -1/2*w^3 + 5/2*w^2 + 5*w - 14],\ [41, 41, 1/2*w^3 - 1/2*w^2 - 3*w + 6],\ [41, 41, -3/2*w^2 - 2*w + 6],\ [79, 79, -w^3 - w^2 + 4*w - 1],\ [79, 79, -3/2*w^2 - w + 9],\ [79, 79, -w^3 + 7/2*w^2 + 9*w - 21],\ [79, 79, w^3 - w^2 - 6*w + 9],\ [81, 3, -3],\ [109, 109, w^2 - w - 7],\ [109, 109, -1/2*w^3 + 1/2*w^2 + 4*w - 1],\ [109, 109, 1/2*w^3 + 1/2*w^2 - 4*w - 1],\ [109, 109, w^2 + w - 7],\ [131, 131, -1/2*w^3 - 1/2*w^2 + 4*w + 7],\ [131, 131, -1/2*w^3 + 7/2*w^2 + 6*w - 18],\ [131, 131, 1/2*w^3 + 1/2*w^2 - 2*w + 2],\ [131, 131, 1/2*w^3 - 1/2*w^2 - 4*w + 7],\ [149, 149, w^3 + w^2 - 5*w - 7],\ [149, 149, 1/2*w^3 - w^2 - 5*w + 3],\ [149, 149, -1/2*w^3 - w^2 + 5*w + 3],\ [149, 149, -w^3 + w^2 + 5*w - 7],\ [199, 199, -w^3 - w^2 + 6*w + 3],\ [199, 199, -w^2 + 2*w + 7],\ [199, 199, -w^2 - 2*w + 7],\ [199, 199, w^3 - w^2 - 6*w + 3],\ [211, 211, 1/2*w^3 - 2*w^2 - 3*w + 11],\ [211, 211, -1/2*w^3 - 1/2*w^2 + 3*w + 7],\ [211, 211, 1/2*w^3 - 1/2*w^2 - 3*w + 7],\ [211, 211, 1/2*w^3 + 2*w^2 - 3*w - 11],\ [229, 229, -1/2*w^3 - 1/2*w^2 + 4*w - 2],\ [229, 229, 1/2*w^3 + 1/2*w^2 - 2*w - 7],\ [229, 229, 1/2*w^3 - 1/2*w^2 - 2*w + 7],\ [229, 229, 1/2*w^3 - 1/2*w^2 - 4*w - 2],\ [239, 239, 1/2*w^2 + 2*w - 6],\ [239, 239, -w^3 - 1/2*w^2 + 6*w - 1],\ [239, 239, w^3 - 1/2*w^2 - 6*w - 1],\ [239, 239, 1/2*w^2 - 2*w - 6],\ [241, 241, -w^3 + 1/2*w^2 + 6*w - 2],\ [241, 241, -w^3 - 1/2*w^2 + 6*w + 4],\ [241, 241, w^3 - 1/2*w^2 - 6*w + 4],\ [241, 241, w^3 + 1/2*w^2 - 6*w - 2],\ [251, 251, 1/2*w^3 + w^2 - 5*w - 7],\ [251, 251, w^3 + w^2 - 5*w - 3],\ [251, 251, -w^3 + w^2 + 5*w - 3],\ [251, 251, -1/2*w^3 + w^2 + 5*w - 7],\ [269, 269, 1/2*w^3 + 1/2*w^2 - 2*w - 6],\ [269, 269, 1/2*w^3 - 1/2*w^2 - 4*w - 1],\ [269, 269, -1/2*w^3 - 1/2*w^2 + 4*w - 1],\ [269, 269, -1/2*w^3 + 1/2*w^2 + 2*w - 6],\ [281, 281, w^3 - 6*w + 1],\ [281, 281, w^3 - 1/2*w^2 - 6*w + 3],\ [281, 281, -w^3 - 1/2*w^2 + 6*w + 3],\ [281, 281, -w^3 + 6*w + 1],\ [331, 331, 3/2*w^3 + w^2 - 7*w + 1],\ [331, 331, -1/2*w^3 + 3*w^2 + 5*w - 17],\ [331, 331, -3/2*w^3 + w^2 + 9*w - 13],\ [331, 331, 1/2*w^3 + 3*w^2 + w - 11],\ [349, 349, 1/2*w^3 + 3/2*w^2 - 2*w - 9],\ [349, 349, -1/2*w^3 + 3/2*w^2 + 4*w - 6],\ [349, 349, 1/2*w^3 + 3/2*w^2 - 4*w - 6],\ [349, 349, -1/2*w^3 + 3/2*w^2 + 2*w - 9],\ [359, 359, -w^3 + 7/2*w^2 + 9*w - 19],\ [359, 359, w^3 - 2*w^2 - 8*w + 11],\ [359, 359, w^3 - 1/2*w^2 - 6*w - 2],\ [359, 359, -5/2*w^2 - 3*w + 9],\ [361, 19, 2*w^2 - 11],\ [361, 19, -1/2*w^2 + 7],\ [389, 389, -1/2*w^3 + 5/2*w^2 + 6*w - 11],\ [389, 389, 5/2*w^3 - 3/2*w^2 - 16*w + 17],\ [389, 389, 3/2*w^3 - 1/2*w^2 - 10*w + 7],\ [389, 389, -3/2*w^3 + 7/2*w^2 + 12*w - 21],\ [401, 401, -w^3 + 1/2*w^2 + 5*w - 1],\ [401, 401, 1/2*w^3 + 1/2*w^2 - 5*w - 4],\ [401, 401, -1/2*w^3 + 1/2*w^2 + 5*w - 4],\ [401, 401, w^3 + 1/2*w^2 - 5*w - 1],\ [439, 439, -2*w^3 + 3/2*w^2 + 12*w - 16],\ [439, 439, -w^3 + 1/2*w^2 + 5*w - 7],\ [439, 439, w^3 + 1/2*w^2 - 5*w - 7],\ [439, 439, w^3 + 1/2*w^2 - 4*w + 4],\ [479, 479, -w^3 + 9/2*w^2 + 10*w - 26],\ [479, 479, -3/2*w^3 + 3/2*w^2 + 9*w - 14],\ [479, 479, -3/2*w^3 - 1/2*w^2 + 7*w - 6],\ [479, 479, -5/2*w^2 - 2*w + 14],\ [491, 491, 1/2*w^3 + 5/2*w^2 - 4*w - 14],\ [491, 491, 1/2*w^3 + 5/2*w^2 - 2*w - 11],\ [491, 491, -1/2*w^3 + 5/2*w^2 + 2*w - 11],\ [491, 491, -1/2*w^3 + 5/2*w^2 + 4*w - 14],\ [509, 509, 1/2*w^3 - 1/2*w^2 - 5*w - 3],\ [509, 509, -w^3 + 1/2*w^2 + 5*w - 8],\ [509, 509, w^3 + 1/2*w^2 - 5*w - 8],\ [509, 509, -7/2*w^2 - 3*w + 18],\ [521, 521, -9/2*w^2 - 4*w + 21],\ [521, 521, -5/2*w^3 + 2*w^2 + 16*w - 19],\ [521, 521, -1/2*w^3 + 2*w^2 + 6*w - 9],\ [521, 521, w^3 - 7/2*w^2 - 8*w + 22],\ [571, 571, 1/2*w^3 - 1/2*w^2 - 4*w - 4],\ [571, 571, 3/2*w^3 - 3/2*w^2 - 9*w + 7],\ [571, 571, -3/2*w^3 - 3/2*w^2 + 9*w + 7],\ [571, 571, -1/2*w^3 - 1/2*w^2 + 4*w - 4],\ [599, 599, 1/2*w^3 + w^2 - 2*w - 9],\ [599, 599, -1/2*w^3 + w^2 + 4*w - 1],\ [599, 599, 1/2*w^3 + w^2 - 4*w - 1],\ [599, 599, -1/2*w^3 + w^2 + 2*w - 9],\ [601, 601, 5/2*w^2 + w - 11],\ [601, 601, 1/2*w^3 - 5/2*w^2 - 3*w + 14],\ [601, 601, -1/2*w^3 - 5/2*w^2 + 3*w + 14],\ [601, 601, 5/2*w^2 - w - 11],\ [641, 641, 5/2*w^2 + w - 13],\ [641, 641, -1/2*w^3 + 5/2*w^2 + 3*w - 12],\ [641, 641, 1/2*w^3 + 5/2*w^2 - 3*w - 12],\ [641, 641, 5/2*w^2 - w - 13],\ [691, 691, 1/2*w^3 + 5/2*w^2 - 3*w - 13],\ [691, 691, -1/2*w^3 - w^2 + 3*w + 11],\ [691, 691, 1/2*w^3 - w^2 - 3*w + 11],\ [691, 691, -1/2*w^3 + 5/2*w^2 + 3*w - 13],\ [709, 709, 3/2*w^3 - 3/2*w^2 - 8*w + 2],\ [709, 709, -1/2*w^3 + 3/2*w^2 + 6*w - 13],\ [709, 709, 1/2*w^3 + 3/2*w^2 - 6*w - 13],\ [709, 709, -3/2*w^3 - 3/2*w^2 + 8*w + 2],\ [719, 719, w^2 + 2*w - 9],\ [719, 719, w^3 + w^2 - 6*w - 1],\ [719, 719, -w^3 + w^2 + 6*w - 1],\ [719, 719, w^2 - 2*w - 9],\ [761, 761, 1/2*w^3 + w^2 - 6*w - 9],\ [761, 761, 3/2*w^3 + w^2 - 8*w - 1],\ [761, 761, 3/2*w^3 - 5*w^2 - 14*w + 29],\ [761, 761, -1/2*w^3 + w^2 + 6*w - 9],\ [811, 811, 3/2*w^3 - 1/2*w^2 - 8*w + 12],\ [811, 811, w^3 - 2*w^2 - 9*w + 11],\ [811, 811, -w^3 - 2*w^2 + 9*w + 11],\ [811, 811, -3/2*w^3 + 9/2*w^2 + 12*w - 28],\ [829, 829, w^3 + w^2 - 5*w - 9],\ [829, 829, 1/2*w^3 - w^2 - 5*w + 1],\ [829, 829, -1/2*w^3 - w^2 + 5*w + 1],\ [829, 829, -w^3 + w^2 + 5*w - 9],\ [839, 839, w^3 - 5/2*w^2 - 8*w + 13],\ [839, 839, 2*w^3 - 1/2*w^2 - 13*w - 1],\ [839, 839, -2*w^3 - 1/2*w^2 + 13*w - 1],\ [839, 839, w^3 + 5/2*w^2 - 8*w - 13],\ [881, 881, -w^3 + 9/2*w^2 + 9*w - 27],\ [881, 881, -2*w^3 + 1/2*w^2 + 12*w - 9],\ [881, 881, w^3 - 7/2*w^2 - 10*w + 19],\ [881, 881, 2*w^3 - 5/2*w^2 - 13*w + 23],\ [919, 919, 1/2*w^3 + 2*w^2 - 4*w - 7],\ [919, 919, 1/2*w^3 + 2*w^2 - 2*w - 13],\ [919, 919, -1/2*w^3 + 2*w^2 + 2*w - 13],\ [919, 919, -1/2*w^3 + 2*w^2 + 4*w - 7],\ [961, 31, -w^2 + 11],\ [961, 31, 5/2*w^2 - 13],\ [971, 971, 1/2*w^3 + 7/2*w^2 - 5*w - 17],\ [971, 971, 2*w^3 - 13*w - 3],\ [971, 971, -2*w^3 + 13*w - 3],\ [971, 971, -1/2*w^3 + 7/2*w^2 + 5*w - 17]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 - 7*x^2 + 4 K. = NumberField(heckePol) hecke_eigenvalues_array = [1/2*e^3 - 5/2*e, e, -1, e^2, -e^2 + 5, -e^2 + 5, -e^3 + 10*e, 2*e^3 - 12*e, -1/2*e^3 + 3/2*e, -1/2*e^3 + 3/2*e, 3, -2*e^2 + 8, 2*e^2 - 2, -2*e^2 + 8, 7/2*e^3 - 41/2*e, e^3 - 7*e, -2*e^3 + 15*e, -4*e^3 + 20*e, -4, -7/2*e^3 + 33/2*e, -7/2*e^3 + 33/2*e, 3/2*e^3 - 21/2*e, -e^3 + 3*e, -e^2 - 1, 2*e^2 + 6, e^2 - 6, 3*e^2 - 11, 6*e, -3*e^3 + 28*e, 5/2*e^3 - 15/2*e, 7/2*e^3 - 49/2*e, -5/2*e^3 + 49/2*e, -3/2*e^3 + 15/2*e, -e^3 - e, 3/2*e^3 - 29/2*e, 9*e^2 - 32, 3*e^2 - 17, -6*e^2 + 20, e^2 - 12, 11/2*e^3 - 73/2*e, 11/2*e^3 - 73/2*e, -2*e^3 + 4*e, 11/2*e^3 - 73/2*e, 5/2*e^3 - 33/2*e, 3*e^3 - 25*e, 5/2*e^3 - 33/2*e, -2*e^3 + 2*e, 8*e^2 - 29, 20, -2*e^2 - 4, 20, -2*e^2 + 20, 3*e^2 - 7, e^2 - 2, -5*e^2 + 13, 8*e^3 - 43*e, -4*e^3 + 16*e, -4*e^3 + 16*e, -4*e^3 + 16*e, -e^2 + 10, -7, 10*e^2 - 32, -3*e^2 + 15, -e^2 - 11, -5*e^2 + 28, -e^2 + 18, e^2 - 16, -8*e^3 + 49*e, 3/2*e^3 + 7/2*e, -3*e^3 + 22*e, -7*e^3 + 32*e, 5/2*e^3 - 33/2*e, -2*e^3 + 2*e, -13/2*e^3 + 99/2*e, 2*e^3 - 8*e, -24, -6*e^2 + 20, -6*e^3 + 51*e, 6*e^3 - 37*e, 3/2*e^3 - 37/2*e, 1/2*e^3 - 3/2*e, -14, -12*e^2 + 45, -4*e^2 - 4, 15, 6*e^3 - 36*e, -11/2*e^3 + 87/2*e, -9/2*e^3 + 53/2*e, 6*e^3 - 36*e, -2*e, -3/2*e^3 - 11/2*e, -e^3 + 15*e, -4*e^3 + 37*e, 5*e^2 + 7, -3*e^2 + 27, e^2 - 12, -2*e^2 + 10, -2*e^3 + 25*e, -6*e^3 + 35*e, -8*e^3 + 40*e, 2*e^3 - 14*e, -18, 4*e^2 - 28, -4*e^2 + 21, 8*e^2 - 9, -6*e^2 + 12, 13*e^2 - 50, 2*e^2 - 8, 3*e^2 + 4, -5*e^3 + 21*e, -2*e^3 + 28*e, e^3 - 23*e, 5*e^3 - 33*e, -3*e^2 + 7, 12*e^2 - 45, -2*e^2 - 10, -3*e^2 + 7, 3*e^2 - 6, 10*e^2 - 38, 6*e^2 - 28, 3*e^2 - 6, 3*e^2 - 21, 2*e^2 - 4, 6*e^2 - 14, 11*e^2 - 41, -2*e^3 + 2*e, -8*e^3 + 46*e, 5*e^3 - 30*e, -2*e^3 + 2*e, 13/2*e^3 - 49/2*e, -2*e^3 + 4*e, -11/2*e^3 + 69/2*e, 4*e^3 - 11*e, 7*e^2 - 23, -8*e^2 + 58, 4*e^2 - 30, 6*e^2 - 6, -5*e^2 - 7, -e^2 + 41, 9*e^2 - 42, -2*e^2 + 58, -11/2*e^3 + 89/2*e, -20*e, -9/2*e^3 + 55/2*e, 21/2*e^3 - 107/2*e, -5*e^3 + 51*e, -9*e^3 + 61*e, 3*e^3 - 27*e, 2*e^3 - 10*e, 8*e^2 - 34, 4*e^2 - 53, -4*e^2 + 54, e^2 - 2, -15/2*e^3 + 91/2*e, 12*e^3 - 54*e, 3*e^3 - 17*e, -10*e^3 + 59*e, 16*e^2 - 51, -2*e^2 - 6, -17*e^2 + 61, -14*e^2 + 68, -e^2 - 37, 6*e^2 + 18] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([11, 11, -1/2*w^3 - 1/2*w^2 + 3*w + 3])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]