/* 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([11, 6, -8, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([79,79,1/3*w^3 - w^2 - 5/3*w + 7/3]) primes_array = [ [9, 3, 1/3*w^3 - 8/3*w - 2/3],\ [9, 3, -w + 1],\ [11, 11, w],\ [11, 11, -1/3*w^3 + 2/3*w + 5/3],\ [11, 11, 1/3*w^3 - 8/3*w + 1/3],\ [11, 11, -2/3*w^3 + 13/3*w + 4/3],\ [16, 2, 2],\ [25, 5, 2/3*w^3 - 10/3*w - 1/3],\ [29, 29, -w - 3],\ [29, 29, -1/3*w^3 + 8/3*w - 10/3],\ [31, 31, w^3 - 6*w + 1],\ [31, 31, w^3 - 6*w - 2],\ [41, 41, 2/3*w^3 + w^2 - 13/3*w - 10/3],\ [41, 41, 2/3*w^3 - 13/3*w + 5/3],\ [59, 59, 2/3*w^3 - 13/3*w + 8/3],\ [59, 59, w^3 + w^2 - 6*w - 4],\ [79, 79, 2/3*w^3 + w^2 - 10/3*w - 19/3],\ [79, 79, 1/3*w^3 + w^2 - 2/3*w - 17/3],\ [89, 89, 4/3*w^3 - 23/3*w - 5/3],\ [89, 89, 1/3*w^3 + 2*w^2 - 5/3*w - 32/3],\ [101, 101, 5/3*w^3 + w^2 - 31/3*w - 16/3],\ [101, 101, 2/3*w^3 - w^2 - 4/3*w + 8/3],\ [109, 109, -w^3 - 2*w^2 + 7*w + 7],\ [109, 109, -1/3*w^3 + 2*w^2 + 2/3*w - 16/3],\ [131, 131, w^3 - w^2 - 7*w + 5],\ [131, 131, -2/3*w^3 - w^2 + 4/3*w + 13/3],\ [139, 139, -2/3*w^3 - w^2 + 13/3*w + 4/3],\ [139, 139, w^2 - w - 7],\ [151, 151, -5/3*w^3 + 28/3*w - 5/3],\ [151, 151, -1/3*w^3 + w^2 + 2/3*w - 16/3],\ [151, 151, w^3 + w^2 - 6*w - 3],\ [151, 151, -w^3 + w^2 + 7*w - 6],\ [179, 179, 5/3*w^3 + w^2 - 28/3*w - 19/3],\ [179, 179, -1/3*w^3 + w^2 + 11/3*w - 4/3],\ [181, 181, -4/3*w^3 - 2*w^2 + 20/3*w + 29/3],\ [181, 181, 1/3*w^3 + w^2 - 2/3*w - 20/3],\ [181, 181, w^3 + w^2 - 5*w - 7],\ [181, 181, 5/3*w^3 + w^2 - 28/3*w - 16/3],\ [191, 191, 2*w - 1],\ [191, 191, 2/3*w^3 - 16/3*w - 1/3],\ [199, 199, 1/3*w^3 - 11/3*w + 4/3],\ [199, 199, 1/3*w^3 - 11/3*w - 2/3],\ [211, 211, -2/3*w^3 - w^2 + 19/3*w + 16/3],\ [211, 211, 2/3*w^3 + w^2 - 19/3*w - 7/3],\ [229, 229, 1/3*w^3 - 11/3*w + 1/3],\ [239, 239, -5/3*w^3 - w^2 + 25/3*w + 19/3],\ [239, 239, -4/3*w^3 + w^2 + 20/3*w - 7/3],\ [251, 251, -2/3*w^3 + w^2 + 16/3*w - 14/3],\ [251, 251, -1/3*w^3 - w^2 - 1/3*w + 14/3],\ [269, 269, 2/3*w^3 + w^2 - 7/3*w - 19/3],\ [269, 269, 1/3*w^3 + 2*w^2 - 2/3*w - 32/3],\ [281, 281, w^3 - w^2 - 5*w + 1],\ [281, 281, 4/3*w^3 + w^2 - 20/3*w - 23/3],\ [289, 17, 2/3*w^3 + w^2 - 16/3*w - 4/3],\ [289, 17, 1/3*w^3 + w^2 - 11/3*w - 20/3],\ [331, 331, 1/3*w^3 + 2*w^2 - 8/3*w - 23/3],\ [331, 331, 2/3*w^3 + 2*w^2 - 13/3*w - 28/3],\ [349, 349, 4/3*w^3 - 17/3*w + 1/3],\ [349, 349, -5/3*w^3 + 28/3*w - 2/3],\ [361, 19, -1/3*w^3 + 5/3*w + 14/3],\ [361, 19, 4/3*w^3 - 20/3*w - 5/3],\ [379, 379, 4/3*w^3 + 2*w^2 - 23/3*w - 41/3],\ [379, 379, -5/3*w^3 + 31/3*w - 5/3],\ [389, 389, 5/3*w^3 + w^2 - 25/3*w - 16/3],\ [389, 389, -4/3*w^3 + w^2 + 20/3*w - 10/3],\ [401, 401, -1/3*w^3 - 3*w^2 + 5/3*w + 26/3],\ [401, 401, 1/3*w^3 + w^2 - 11/3*w - 23/3],\ [401, 401, -2/3*w^3 - 3*w^2 + 10/3*w + 52/3],\ [401, 401, 2/3*w^3 + w^2 - 16/3*w - 1/3],\ [409, 409, -2*w^3 - w^2 + 11*w + 2],\ [409, 409, -5/3*w^3 - 2*w^2 + 28/3*w + 37/3],\ [409, 409, 1/3*w^3 - 11/3*w - 17/3],\ [409, 409, 5/3*w^3 + 3*w^2 - 25/3*w - 37/3],\ [419, 419, -1/3*w^3 + 3*w^2 - 1/3*w - 37/3],\ [419, 419, 2*w^3 + 3*w^2 - 12*w - 13],\ [421, 421, -1/3*w^3 + 3*w^2 + 2/3*w - 31/3],\ [421, 421, -5/3*w^3 - w^2 + 25/3*w + 13/3],\ [421, 421, -w^3 + 7*w - 4],\ [421, 421, -4/3*w^3 + w^2 + 20/3*w - 13/3],\ [439, 439, -4/3*w^3 + w^2 + 14/3*w - 7/3],\ [439, 439, 4/3*w^3 - 17/3*w - 5/3],\ [439, 439, -7/3*w^3 - w^2 + 41/3*w + 17/3],\ [439, 439, 5/3*w^3 - 28/3*w - 4/3],\ [449, 449, w^3 + 3*w^2 - 6*w - 17],\ [449, 449, 1/3*w^3 + 3*w^2 - 8/3*w - 26/3],\ [449, 449, -4/3*w^3 + 26/3*w + 17/3],\ [449, 449, 7/3*w^3 + w^2 - 38/3*w - 8/3],\ [461, 461, -5/3*w^3 + w^2 + 28/3*w - 14/3],\ [461, 461, -2/3*w^3 - 2*w^2 + 10/3*w + 43/3],\ [461, 461, -2*w^3 - w^2 + 12*w + 4],\ [461, 461, -w^3 + 8*w - 5],\ [479, 479, w^3 + w^2 - 7*w - 2],\ [479, 479, -4/3*w^3 - w^2 + 23/3*w + 5/3],\ [499, 499, -4/3*w^3 - 2*w^2 + 23/3*w + 17/3],\ [499, 499, -4/3*w^3 - w^2 + 29/3*w + 8/3],\ [521, 521, 2*w^3 - 11*w + 2],\ [521, 521, 4/3*w^3 - w^2 - 14/3*w + 1/3],\ [569, 569, w^3 + 3*w^2 - 6*w - 9],\ [569, 569, 4/3*w^3 + w^2 - 26/3*w - 32/3],\ [569, 569, -w^3 + 6*w - 6],\ [569, 569, -1/3*w^3 - 3*w^2 + 8/3*w + 50/3],\ [571, 571, 2/3*w^3 + 3*w^2 - 10/3*w - 46/3],\ [571, 571, 2*w^2 - 15],\ [599, 599, -w^3 + 2*w^2 + 4*w - 10],\ [599, 599, -w^2 - 3*w + 8],\ [601, 601, 5/3*w^3 + 3*w^2 - 28/3*w - 34/3],\ [601, 601, 1/3*w^3 - 3*w^2 - 2/3*w + 43/3],\ [619, 619, 2/3*w^3 - w^2 - 13/3*w + 2/3],\ [619, 619, 2/3*w^3 + w^2 - 7/3*w - 25/3],\ [619, 619, 2*w^3 + w^2 - 11*w - 6],\ [619, 619, -2/3*w^3 + 13/3*w - 17/3],\ [631, 631, 2/3*w^3 + 2*w^2 - 19/3*w - 31/3],\ [631, 631, w^3 + 2*w^2 - 8*w - 6],\ [631, 631, 2/3*w^3 + 2*w^2 - 4/3*w - 31/3],\ [631, 631, -2/3*w^3 + 2*w^2 + 16/3*w - 23/3],\ [659, 659, -2/3*w^3 + 1/3*w + 16/3],\ [659, 659, 5/3*w^3 - 34/3*w - 13/3],\ [661, 661, w^3 - 8*w + 2],\ [661, 661, 4/3*w^3 + w^2 - 23/3*w + 1/3],\ [661, 661, 5/3*w^3 + w^2 - 31/3*w - 10/3],\ [661, 661, -3*w - 1],\ [691, 691, -4/3*w^3 + 29/3*w + 2/3],\ [691, 691, 1/3*w^3 + 4/3*w - 5/3],\ [691, 691, -5/3*w^3 + w^2 + 25/3*w - 5/3],\ [691, 691, 2*w^3 + w^2 - 10*w - 7],\ [701, 701, -2/3*w^3 - 3*w^2 + 19/3*w + 46/3],\ [701, 701, w^3 - 2*w^2 - 6*w + 8],\ [701, 701, 4/3*w^3 + 2*w^2 - 17/3*w - 29/3],\ [701, 701, 5/3*w^3 + 3*w^2 - 28/3*w - 52/3],\ [709, 709, 1/3*w^3 - 14/3*w + 10/3],\ [709, 709, -2/3*w^3 + 19/3*w + 7/3],\ [719, 719, 3*w^2 - w - 10],\ [719, 719, -4/3*w^3 - 3*w^2 + 23/3*w + 47/3],\ [739, 739, 5/3*w^3 + w^2 - 34/3*w - 10/3],\ [739, 739, 4/3*w^3 + 2*w^2 - 23/3*w - 14/3],\ [761, 761, 7/3*w^3 + 2*w^2 - 41/3*w - 26/3],\ [761, 761, 2*w^3 + 3*w^2 - 12*w - 15],\ [769, 769, -2/3*w^3 + 19/3*w - 8/3],\ [769, 769, -4/3*w^3 + 2*w^2 + 17/3*w - 31/3],\ [769, 769, 1/3*w^3 - 14/3*w - 5/3],\ [769, 769, -1/3*w^3 + 3*w^2 + 2/3*w - 34/3],\ [809, 809, -4/3*w^3 + w^2 + 26/3*w - 7/3],\ [809, 809, w^3 + w^2 - 3*w - 7],\ [811, 811, 4/3*w^3 + w^2 - 17/3*w - 23/3],\ [811, 811, -1/3*w^3 - 2*w^2 + 2/3*w + 38/3],\ [821, 821, 8/3*w^3 + 3*w^2 - 43/3*w - 34/3],\ [821, 821, 2/3*w^3 + 2*w^2 - 7/3*w - 34/3],\ [821, 821, -w^3 - w^2 + 4*w + 8],\ [821, 821, 2/3*w^3 + 4*w^2 - 7/3*w - 58/3],\ [829, 829, 4/3*w^3 + 3*w^2 - 26/3*w - 50/3],\ [829, 829, 7/3*w^3 - 41/3*w - 5/3],\ [839, 839, 2*w^3 + w^2 - 10*w - 3],\ [839, 839, 5/3*w^3 - w^2 - 25/3*w + 17/3],\ [841, 29, 5/3*w^3 - 25/3*w - 7/3],\ [859, 859, -7/3*w^3 - w^2 + 32/3*w + 29/3],\ [859, 859, 2*w^3 + w^2 - 14*w - 6],\ [859, 859, 1/3*w^3 + 2*w^2 - 2/3*w - 47/3],\ [859, 859, -5/3*w^3 - 2*w^2 + 31/3*w + 49/3],\ [881, 881, w^3 + 4*w^2 - 4*w - 20],\ [881, 881, 8/3*w^3 + 2*w^2 - 43/3*w - 34/3],\ [919, 919, -w^3 + 3*w - 7],\ [919, 919, 1/3*w^3 + 3*w^2 - 5/3*w - 35/3],\ [919, 919, -5/3*w^3 + 31/3*w - 23/3],\ [919, 919, 2/3*w^3 + 3*w^2 - 10/3*w - 43/3],\ [929, 929, w^2 - 2*w - 9],\ [929, 929, w^3 + w^2 - 7*w + 1],\ [961, 31, 2/3*w^3 - 10/3*w - 19/3],\ [971, 971, -w^3 - 3*w^2 + 7*w + 16],\ [971, 971, -5/3*w^3 + w^2 + 25/3*w - 8/3],\ [971, 971, 2*w^3 + w^2 - 10*w - 6],\ [971, 971, 1/3*w^3 + 2*w^2 - 8/3*w - 47/3]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 - 4*x - 1 K. = NumberField(heckePol) hecke_eigenvalues_array = [0, e, -4, e - 4, -e + 4, 1/2*e - 9/2, -5/2*e + 7/2, 5, 1/2*e - 11/2, -1/2*e - 1/2, 2*e - 5, 2*e + 1, -e - 7, -7/2*e + 19/2, -2*e + 11, -3, -5, 1, -9/2*e + 19/2, 5*e - 17, -2*e + 1, 1/2*e - 17/2, -13/2*e + 35/2, 1/2*e + 7/2, 6*e - 11, -3/2*e + 3/2, 4*e - 20, 3*e - 9, 3/2*e + 3/2, -7/2*e + 5/2, -9/2*e + 23/2, 9/2*e - 3/2, -e, 2*e + 13, -3/2*e - 13/2, -2*e - 6, 3*e - 17, 5*e - 7, 5*e, -15/2*e + 13/2, -e - 2, 4*e - 7, -1/2*e - 39/2, -7*e + 19, e + 12, -3/2*e - 17/2, 3*e - 17, -15/2*e + 9/2, 19/2*e - 19/2, -5*e - 8, 9/2*e - 31/2, 3*e - 14, -2*e + 13, -e + 9, 1/2*e - 5/2, 8*e - 6, 7/2*e - 31/2, -23/2*e + 35/2, -12*e + 26, 9*e - 24, 5/2*e - 9/2, -7/2*e - 17/2, -6*e, 15, 21/2*e - 63/2, 7*e, -19/2*e + 19/2, 15*e - 31, -2*e + 33, -1/2*e - 51/2, -3*e - 4, -6*e + 13, 14*e - 33, -3*e - 5, -11/2*e + 5/2, -11*e + 13, -3*e - 5, 4*e + 21, 3*e + 20, -2*e + 25, -e - 15, -5/2*e - 45/2, -2*e - 5, 5*e + 8, 15/2*e - 69/2, 3/2*e - 3/2, 4*e - 12, -4*e + 3, -7/2*e + 63/2, -5/2*e + 67/2, -9/2*e + 15/2, -25/2*e + 41/2, -e + 20, 9/2*e - 17/2, -6*e + 9, 7/2*e + 19/2, 4*e - 26, -25/2*e + 35/2, e - 18, -8*e + 23, 12*e - 6, 6*e - 40, -2*e + 12, 10*e - 10, -11/2*e - 21/2, -9*e + 37, 14*e - 23, -5*e + 4, -9/2*e - 11/2, -7*e + 6, 7*e - 44, 14, -1/2*e + 69/2, -e - 36, 11/2*e - 17/2, 6*e - 43, -3*e + 15, 9/2*e - 17/2, 9*e - 34, 15/2*e - 79/2, -27/2*e + 65/2, 16*e - 21, 21/2*e - 59/2, 31/2*e - 65/2, -2*e - 5, -1/2*e - 93/2, -23/2*e - 1/2, -9*e + 4, 5*e - 16, 10*e - 41, 14*e - 9, 11*e - 2, -23/2*e + 91/2, 8*e - 39, 21/2*e - 9/2, 7/2*e + 27/2, 7/2*e + 23/2, 4*e + 16, 23/2*e + 3/2, -3*e - 30, -1/2*e + 3/2, 7/2*e + 3/2, -9/2*e - 19/2, 8*e - 45, -21*e + 43, -9*e + 46, 21/2*e - 109/2, 23/2*e - 55/2, -6*e + 11, -12*e + 12, -2*e + 30, 2*e - 6, 2*e + 22, -21/2*e + 33/2, 11/2*e - 37/2, 15/2*e - 37/2, -3*e - 13, -19/2*e + 23/2, -10*e + 17, -16*e + 42, 11/2*e + 23/2, -9/2*e - 21/2, -9*e + 56, -21/2*e + 29/2, 9/2*e - 43/2, -3*e + 4, -17*e + 16, -13*e + 22, 5/2*e + 35/2, -15/2*e + 55/2, 5*e - 34] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([79,79,1/3*w^3 - w^2 - 5/3*w + 7/3])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]