/* 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([4, 0, -9, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([36, 6, -1/2*w^2 - 1/2*w + 5]) primes_array = [ [4, 2, w],\ [4, 2, -1/2*w^3 + 9/2*w],\ [9, 3, 1/4*w^3 - 11/4*w - 1/2],\ [9, 3, 1/4*w^3 - 11/4*w + 1/2],\ [25, 5, 1/2*w^3 - 7/2*w],\ [29, 29, -1/4*w^3 + 1/2*w^2 + 9/4*w - 1/2],\ [29, 29, -1/2*w^2 - 1/2*w + 4],\ [29, 29, -1/2*w^2 + 1/2*w + 4],\ [29, 29, -1/4*w^3 - 1/2*w^2 + 9/4*w + 1/2],\ [49, 7, 3/4*w^3 + 1/2*w^2 - 23/4*w - 3/2],\ [49, 7, 1/2*w^3 + 1/2*w^2 - 3*w - 3],\ [61, 61, 1/2*w^3 + 1/2*w^2 - 5*w - 1],\ [61, 61, -1/4*w^3 - 1/2*w^2 + 13/4*w + 7/2],\ [61, 61, -1/4*w^3 + 1/2*w^2 + 13/4*w - 7/2],\ [61, 61, -1/2*w^3 + 1/2*w^2 + 5*w - 1],\ [79, 79, 3/4*w^3 - 25/4*w - 9/2],\ [79, 79, -1/4*w^3 + 1/2*w^2 - 3/4*w - 3/2],\ [79, 79, 1/4*w^3 - 3/4*w - 9/2],\ [79, 79, 3/2*w^3 + 1/2*w^2 - 13*w - 3],\ [101, 101, 1/4*w^3 + w^2 - 7/4*w - 5/2],\ [101, 101, 5/4*w^3 - 1/2*w^2 - 45/4*w + 9/2],\ [101, 101, 1/4*w^3 + w^2 - 7/4*w - 13/2],\ [101, 101, 1/4*w^3 - w^2 - 7/4*w + 5/2],\ [121, 11, 3/4*w^3 - 21/4*w - 1/2],\ [121, 11, 1/4*w^3 - 7/4*w - 7/2],\ [131, 131, -3/4*w^3 + 29/4*w - 1/2],\ [131, 131, 1/4*w^3 - 15/4*w - 1/2],\ [131, 131, 1/4*w^3 - 15/4*w + 1/2],\ [131, 131, 3/4*w^3 - 29/4*w - 1/2],\ [139, 139, w^2 + w + 1],\ [139, 139, 1/2*w^3 + w^2 - 9/2*w - 10],\ [139, 139, 1/2*w^3 - w^2 - 9/2*w + 10],\ [139, 139, w^2 - w + 1],\ [169, 13, 1/2*w^3 - 11/2*w],\ [179, 179, -3/4*w^3 + w^2 + 21/4*w - 13/2],\ [179, 179, 1/2*w^3 + 1/2*w^2 - 6*w - 5],\ [179, 179, -3/4*w^3 - w^2 + 25/4*w + 7/2],\ [179, 179, -3/2*w^2 + 1/2*w + 2],\ [181, 181, -1/2*w^2 + 1/2*w - 2],\ [181, 181, 1/4*w^3 - 1/2*w^2 - 9/4*w + 13/2],\ [181, 181, 5/4*w^3 - 39/4*w + 3/2],\ [181, 181, 1/2*w^2 + 1/2*w + 2],\ [191, 191, -w^3 + 1/2*w^2 + 19/2*w - 4],\ [191, 191, 1/4*w^3 + 1/2*w^2 - 17/4*w - 1/2],\ [191, 191, 1/2*w^3 + 1/2*w^2 - 6*w - 3],\ [191, 191, -3/4*w^3 + 1/2*w^2 + 31/4*w - 3/2],\ [199, 199, 1/4*w^3 + 1/2*w^2 - 5/4*w - 15/2],\ [199, 199, -1/4*w^3 + w^2 + 11/4*w - 17/2],\ [199, 199, 1/2*w^3 + 1/2*w^2 - 4*w + 3],\ [199, 199, -1/4*w^3 - w^2 + 11/4*w + 1/2],\ [211, 211, -3/4*w^3 + 1/2*w^2 + 19/4*w - 9/2],\ [211, 211, -w^3 - 1/2*w^2 + 15/2*w],\ [211, 211, -1/4*w^3 + 1/2*w^2 + 13/4*w - 11/2],\ [211, 211, 3/4*w^3 + 1/2*w^2 - 19/4*w - 9/2],\ [251, 251, 3/4*w^3 - 1/2*w^2 - 27/4*w + 1/2],\ [251, 251, 1/2*w^2 - 3/2*w - 4],\ [251, 251, 1/2*w^2 + 3/2*w - 4],\ [251, 251, -3/4*w^3 - 1/2*w^2 + 27/4*w + 1/2],\ [269, 269, -3/4*w^3 + 17/4*w - 1/2],\ [269, 269, 1/2*w^3 + 1/2*w^2 - 4*w - 7],\ [269, 269, -5/4*w^3 + 39/4*w + 1/2],\ [269, 269, 3/4*w^3 - 17/4*w - 1/2],\ [289, 17, -1/4*w^3 + 11/4*w - 9/2],\ [289, 17, 1/4*w^3 - 11/4*w - 9/2],\ [311, 311, 3/4*w^3 + w^2 - 29/4*w - 19/2],\ [311, 311, 3/4*w^3 + w^2 - 25/4*w - 21/2],\ [311, 311, 3/4*w^3 - w^2 - 13/4*w + 7/2],\ [311, 311, -3/4*w^3 + w^2 + 29/4*w - 19/2],\ [361, 19, -1/4*w^3 + 7/4*w - 9/2],\ [361, 19, -w^3 + 7*w - 1],\ [389, 389, 5/4*w^3 - 3/2*w^2 - 41/4*w + 13/2],\ [389, 389, 5/4*w^3 - 47/4*w + 9/2],\ [389, 389, 1/2*w^3 + 3/2*w^2 - 2*w - 7],\ [389, 389, 5/4*w^3 + 3/2*w^2 - 41/4*w - 13/2],\ [419, 419, 5/4*w^3 - 1/2*w^2 - 37/4*w + 5/2],\ [419, 419, w^3 - 1/2*w^2 - 13/2*w + 2],\ [419, 419, w^3 + 1/2*w^2 - 13/2*w - 2],\ [419, 419, 5/4*w^3 + 1/2*w^2 - 37/4*w - 5/2],\ [439, 439, -1/4*w^3 - 1/2*w^2 - 7/4*w + 5/2],\ [439, 439, 2*w^3 + 1/2*w^2 - 35/2*w - 2],\ [439, 439, 2*w^3 - 1/2*w^2 - 35/2*w + 2],\ [439, 439, -1/4*w^3 + 1/2*w^2 - 7/4*w - 5/2],\ [491, 491, 1/4*w^3 + 3/2*w^2 - 9/4*w - 7/2],\ [491, 491, -3/2*w^2 + 1/2*w + 10],\ [491, 491, -1/4*w^3 - 5/4*w + 5/2],\ [491, 491, 1/4*w^3 - 3/2*w^2 - 9/4*w + 7/2],\ [521, 521, 1/2*w^2 - 3/2*w - 6],\ [521, 521, -3/4*w^3 - 1/2*w^2 + 27/4*w - 3/2],\ [521, 521, 3/4*w^3 - 1/2*w^2 - 27/4*w - 3/2],\ [521, 521, 1/2*w^2 + 3/2*w - 6],\ [529, 23, 1/2*w^3 - 11/2*w - 6],\ [529, 23, -1/2*w^3 + 11/2*w - 6],\ [569, 569, 3/4*w^3 + w^2 - 33/4*w - 3/2],\ [569, 569, -3/4*w^3 + w^2 + 33/4*w - 15/2],\ [569, 569, 3/4*w^3 + w^2 - 33/4*w - 15/2],\ [569, 569, -3/4*w^3 + w^2 + 33/4*w - 3/2],\ [571, 571, -1/2*w^3 - 3/2*w^2 + 5*w + 9],\ [571, 571, 1/4*w^3 - 19/4*w + 1/2],\ [571, 571, 1/4*w^3 - 19/4*w - 1/2],\ [571, 571, 3/4*w^3 + 3/2*w^2 - 27/4*w - 19/2],\ [599, 599, -3/4*w^3 - w^2 + 21/4*w + 1/2],\ [599, 599, 1/2*w^3 - 3/2*w^2 - 6*w + 9],\ [599, 599, -1/2*w^3 - 3/2*w^2 + 6*w + 9],\ [599, 599, 3/4*w^3 - w^2 - 21/4*w + 1/2],\ [601, 601, 1/2*w^3 - w^2 - 9/2*w],\ [601, 601, -w^2 - w + 9],\ [601, 601, -7/4*w^3 + 57/4*w - 7/2],\ [601, 601, 1/2*w^3 + w^2 - 9/2*w],\ [641, 641, w^3 + 1/2*w^2 - 11/2*w - 2],\ [641, 641, -7/4*w^3 + 1/2*w^2 + 55/4*w - 5/2],\ [641, 641, 7/4*w^3 + 1/2*w^2 - 55/4*w - 5/2],\ [641, 641, -w^3 + 1/2*w^2 + 11/2*w - 2],\ [659, 659, 7/4*w^3 - 1/2*w^2 - 59/4*w + 3/2],\ [659, 659, 5/2*w^3 - w^2 - 41/2*w + 8],\ [659, 659, 5/2*w^3 + w^2 - 41/2*w - 8],\ [659, 659, w^3 + 3/2*w^2 - 19/2*w - 12],\ [701, 701, 1/2*w^3 - 3/2*w^2 - 3*w + 5],\ [701, 701, 3/4*w^3 + 3/2*w^2 - 23/4*w - 17/2],\ [701, 701, 3/4*w^3 - 3/2*w^2 - 23/4*w + 17/2],\ [701, 701, 1/2*w^3 + 3/2*w^2 - 3*w - 5],\ [719, 719, 3/4*w^3 - 3/2*w^2 - 23/4*w + 13/2],\ [719, 719, -1/2*w^3 + 3/2*w^2 + 3*w - 7],\ [719, 719, 1/2*w^3 + 3/2*w^2 - 3*w - 7],\ [719, 719, 3/4*w^3 + 3/2*w^2 - 23/4*w - 13/2],\ [751, 751, 1/2*w^3 - 3/2*w - 4],\ [751, 751, -3/2*w^3 + 25/2*w + 4],\ [751, 751, 3/2*w^3 - 25/2*w + 4],\ [751, 751, -1/2*w^3 + 3/2*w - 4],\ [809, 809, 3/2*w^2 - 1/2*w - 4],\ [809, 809, 1/4*w^3 + 3/2*w^2 - 9/4*w - 19/2],\ [809, 809, 1/4*w^3 - 3/2*w^2 - 9/4*w + 19/2],\ [809, 809, 3/2*w^2 + 1/2*w - 4],\ [829, 829, -3/2*w^3 - 1/2*w^2 + 11*w + 1],\ [829, 829, 5/4*w^3 - 1/2*w^2 - 33/4*w + 7/2],\ [829, 829, 5/4*w^3 + 1/2*w^2 - 33/4*w - 7/2],\ [829, 829, -3/2*w^3 + 1/2*w^2 + 11*w - 1],\ [859, 859, -1/2*w^3 - 1/2*w^2 + 7*w + 3],\ [859, 859, 5/4*w^3 + 1/2*w^2 - 49/4*w - 3/2],\ [859, 859, -5/4*w^3 + 1/2*w^2 + 49/4*w - 3/2],\ [859, 859, 1/2*w^3 - 1/2*w^2 - 7*w + 3],\ [881, 881, 5/4*w^3 + 2*w^2 - 43/4*w - 29/2],\ [881, 881, -1/4*w^3 + 2*w^2 - 1/4*w - 7/2],\ [881, 881, -w^3 + 3/2*w^2 + 15/2*w - 12],\ [881, 881, 5/4*w^3 - 2*w^2 - 43/4*w + 29/2],\ [911, 911, 1/4*w^3 - w^2 - 7/4*w - 5/2],\ [911, 911, -1/4*w^3 + w^2 + 7/4*w - 23/2],\ [911, 911, -9/4*w^3 + 1/2*w^2 + 73/4*w - 9/2],\ [911, 911, 9/4*w^3 + w^2 - 75/4*w - 11/2],\ [919, 919, 5/4*w^3 - 1/2*w^2 - 41/4*w - 1/2],\ [919, 919, -3/4*w^3 - w^2 + 29/4*w + 3/2],\ [919, 919, -3/4*w^3 + w^2 + 29/4*w - 3/2],\ [919, 919, -5/4*w^3 - 1/2*w^2 + 41/4*w - 1/2],\ [961, 31, 1/2*w^3 - 7/2*w - 6],\ [961, 31, 5/4*w^3 - 35/4*w - 1/2],\ [971, 971, -7/4*w^3 + 61/4*w - 7/2],\ [971, 971, -1/4*w^3 - 5/4*w - 7/2],\ [971, 971, 11/4*w^3 + w^2 - 97/4*w - 17/2],\ [971, 971, 1/4*w^3 - 2*w^2 - 7/4*w + 7/2],\ [991, 991, -5/4*w^3 - w^2 + 43/4*w + 7/2],\ [991, 991, w^3 - 1/2*w^2 - 17/2*w - 2],\ [991, 991, -w^3 - 1/2*w^2 + 17/2*w - 2],\ [991, 991, -5/4*w^3 + w^2 + 43/4*w - 7/2]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 - 2*x - 5 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1, e, -1, 0, -e + 1, e - 5, -3*e + 5, 2*e, 2*e, -2*e + 10, e - 5, 2*e + 2, 2, -4*e + 2, e + 7, 10, 2*e - 10, -3*e + 5, -3*e + 5, 2, -6*e + 2, e + 7, 3*e - 13, 4*e - 8, 2*e + 12, -6*e + 2, -4*e - 8, -5*e + 7, 8*e - 8, 10, 4*e, 5*e - 5, 2*e - 20, 2*e - 10, 10, -2*e + 10, 2*e - 10, -10, e + 7, 2, -4*e + 2, -4*e + 2, 2*e - 8, -6*e + 12, -8, -5*e - 3, -4*e, -e + 25, -2*e + 10, -2*e + 20, 3*e + 7, 2*e + 12, 2*e + 2, 2, -6*e + 2, e + 7, 3*e - 3, -6*e + 12, 6*e, -10, e + 5, -8*e + 10, e + 5, 9*e - 15, -e + 17, -6*e + 12, -5*e - 3, -7*e + 7, -5*e - 3, 2*e + 2, -2*e - 30, -8*e, 2*e - 20, -5*e + 5, -8*e + 20, 3*e + 15, -4*e + 10, -10*e + 10, -30, -3*e - 15, 0, -10*e + 10, -10*e - 8, -10*e + 22, 10*e + 2, -e - 23, 2*e - 18, 12*e - 18, 3*e + 17, 8*e - 8, 3*e - 5, -4*e - 20, 2*e, -10, -e - 5, 3*e + 15, -6*e + 32, 4*e - 8, 2*e + 22, 8*e - 8, 10*e - 10, -12*e, -4*e + 20, -2*e - 30, -8*e - 18, 4*e - 28, -6*e + 22, 14*e - 8, -4*e + 22, 10*e + 2, 7*e + 17, -10*e - 8, 2*e + 20, -8*e + 20, 8*e - 10, -14*e + 20, 5*e - 13, 4*e + 12, -4*e + 2, -10*e + 2, -2*e + 20, 6*e - 30, -10*e, 12*e - 10, -2*e - 28, 4*e + 12, -2*e + 32, -19*e + 17, -13*e + 15, 2*e + 20, -8*e + 10, 10, 3*e + 25, -12*e + 10, 12*e + 10, -12*e + 10, -12*e + 30, 8*e, -4*e + 40, -5*e - 5, 10*e + 12, 4*e - 18, 8*e + 2, -4*e - 28, -8*e + 32, 9*e + 27, -3*e + 27, 13*e - 13, 8*e - 40, 5*e - 25, -4*e + 20, -4*e - 40, -14*e - 8, 2, 2*e + 32, 12, 9*e + 7, 12*e - 18, -e - 13, -8, 2*e + 32, 16*e - 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([4,2,w])] = 1 AL_eigenvalues[ZF.ideal([9,3,1/4*w^3-11/4*w-1/2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]