/* 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, 8, -7, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([59, 59, 3/7*w^3 - 1/7*w^2 - 32/7*w + 15/7]) primes_array = [ [4, 2, -2/7*w^3 + 3/7*w^2 + 19/7*w - 17/7],\ [9, 3, -2/7*w^3 + 3/7*w^2 + 19/7*w - 10/7],\ [11, 11, -w - 1],\ [11, 11, 4/7*w^3 - 6/7*w^2 - 31/7*w + 27/7],\ [11, 11, 4/7*w^3 - 6/7*w^2 - 31/7*w + 6/7],\ [11, 11, -w + 2],\ [25, 5, 4/7*w^3 - 6/7*w^2 - 24/7*w + 13/7],\ [49, 7, 5/7*w^3 - 11/7*w^2 - 23/7*w + 25/7],\ [49, 7, w^3 - w^2 - 7*w + 1],\ [59, 59, 3/7*w^3 - 1/7*w^2 - 32/7*w + 15/7],\ [59, 59, -3/7*w^3 + 8/7*w^2 + 25/7*w - 43/7],\ [59, 59, -3/7*w^3 + 1/7*w^2 + 32/7*w + 13/7],\ [59, 59, 3/7*w^3 - 8/7*w^2 - 25/7*w + 15/7],\ [61, 61, -8/7*w^3 + 12/7*w^2 + 55/7*w - 26/7],\ [61, 61, -4/7*w^3 + 6/7*w^2 + 17/7*w - 6/7],\ [61, 61, 6/7*w^3 - 9/7*w^2 - 50/7*w + 16/7],\ [61, 61, 8/7*w^3 - 12/7*w^2 - 55/7*w + 33/7],\ [71, 71, -3/7*w^3 + 1/7*w^2 + 25/7*w - 15/7],\ [71, 71, -w^3 + w^2 + 8*w - 2],\ [71, 71, w^3 - 2*w^2 - 7*w + 6],\ [71, 71, -3/7*w^3 + 8/7*w^2 + 18/7*w - 8/7],\ [109, 109, -6/7*w^3 + 9/7*w^2 + 50/7*w - 30/7],\ [109, 109, -2/7*w^3 + 3/7*w^2 + 26/7*w - 10/7],\ [109, 109, 2/7*w^3 - 3/7*w^2 - 26/7*w + 17/7],\ [109, 109, -6/7*w^3 + 9/7*w^2 + 50/7*w - 23/7],\ [131, 131, 5/7*w^3 - 4/7*w^2 - 37/7*w + 18/7],\ [131, 131, 3/7*w^3 - 1/7*w^2 - 18/7*w - 20/7],\ [131, 131, 5/7*w^3 - 4/7*w^2 - 44/7*w + 18/7],\ [131, 131, 1/7*w^3 - 5/7*w^2 - 13/7*w + 33/7],\ [169, 13, w^3 - 2*w^2 - 5*w + 3],\ [169, 13, 9/7*w^3 - 10/7*w^2 - 61/7*w + 3/7],\ [179, 179, 9/7*w^3 - 10/7*w^2 - 75/7*w + 31/7],\ [179, 179, 1/7*w^3 - 5/7*w^2 - 20/7*w + 33/7],\ [179, 179, 3/7*w^3 - 1/7*w^2 - 18/7*w - 34/7],\ [179, 179, 4/7*w^3 + 1/7*w^2 - 31/7*w - 36/7],\ [181, 181, -2/7*w^3 + 10/7*w^2 + 5/7*w - 38/7],\ [181, 181, -2/7*w^3 + 10/7*w^2 + 5/7*w - 31/7],\ [181, 181, 2/7*w^3 + 4/7*w^2 - 19/7*w - 18/7],\ [181, 181, 2/7*w^3 + 4/7*w^2 - 19/7*w - 25/7],\ [191, 191, w^3 - 2*w^2 - 7*w + 5],\ [191, 191, -1/7*w^3 + 5/7*w^2 - 8/7*w - 19/7],\ [191, 191, 1/7*w^3 + 2/7*w^2 + 1/7*w - 23/7],\ [191, 191, -w^3 + w^2 + 8*w - 3],\ [229, 229, 6/7*w^3 - 9/7*w^2 - 43/7*w + 58/7],\ [229, 229, -9/7*w^3 + 17/7*w^2 + 54/7*w - 45/7],\ [229, 229, 9/7*w^3 - 10/7*w^2 - 61/7*w + 17/7],\ [229, 229, 6/7*w^3 - 9/7*w^2 - 43/7*w - 12/7],\ [239, 239, -4/7*w^3 - 1/7*w^2 + 24/7*w + 22/7],\ [239, 239, 2*w - 5],\ [239, 239, -2*w - 3],\ [239, 239, -5/7*w^3 + 4/7*w^2 + 51/7*w + 17/7],\ [241, 241, 9/7*w^3 - 10/7*w^2 - 61/7*w + 10/7],\ [241, 241, w^3 - w^2 - 6*w + 2],\ [241, 241, w^3 - 2*w^2 - 5*w + 4],\ [241, 241, -9/7*w^3 + 17/7*w^2 + 54/7*w - 52/7],\ [251, 251, -5/7*w^3 + 4/7*w^2 + 23/7*w + 10/7],\ [251, 251, 11/7*w^3 - 13/7*w^2 - 80/7*w + 34/7],\ [251, 251, -11/7*w^3 + 20/7*w^2 + 73/7*w - 48/7],\ [251, 251, 10/7*w^3 - 15/7*w^2 - 74/7*w + 22/7],\ [289, 17, -10/7*w^3 + 15/7*w^2 + 67/7*w - 36/7],\ [289, 17, 6/7*w^3 - 9/7*w^2 - 29/7*w + 16/7],\ [311, 311, 5/7*w^3 - 4/7*w^2 - 44/7*w + 32/7],\ [311, 311, 1/7*w^3 + 2/7*w^2 - 20/7*w - 30/7],\ [311, 311, -1/7*w^3 + 5/7*w^2 + 13/7*w - 47/7],\ [311, 311, -5/7*w^3 + 11/7*w^2 + 37/7*w - 11/7],\ [349, 349, -6/7*w^3 + 16/7*w^2 + 36/7*w - 93/7],\ [349, 349, 11/7*w^3 - 13/7*w^2 - 94/7*w + 6/7],\ [349, 349, 9/7*w^3 - 3/7*w^2 - 75/7*w - 25/7],\ [349, 349, 2*w^3 - 3*w^2 - 14*w + 8],\ [359, 359, w^3 - 2*w^2 - 6*w + 2],\ [359, 359, 2/7*w^3 - 10/7*w^2 - 19/7*w + 52/7],\ [359, 359, 2/7*w^3 + 4/7*w^2 - 33/7*w - 25/7],\ [359, 359, -w^3 + w^2 + 7*w - 5],\ [361, 19, -8/7*w^3 + 12/7*w^2 + 48/7*w - 33/7],\ [361, 19, 8/7*w^3 - 12/7*w^2 - 48/7*w + 19/7],\ [409, 409, 1/7*w^3 + 2/7*w^2 - 13/7*w - 44/7],\ [409, 409, -1/7*w^3 - 2/7*w^2 + 13/7*w - 26/7],\ [409, 409, 1/7*w^3 - 5/7*w^2 - 6/7*w - 16/7],\ [409, 409, -1/7*w^3 + 5/7*w^2 + 6/7*w - 54/7],\ [419, 419, -6/7*w^3 + 16/7*w^2 + 36/7*w - 37/7],\ [419, 419, -2/7*w^3 + 10/7*w^2 - 2/7*w - 45/7],\ [419, 419, 2/7*w^3 + 4/7*w^2 - 12/7*w - 39/7],\ [419, 419, 6/7*w^3 - 2/7*w^2 - 50/7*w + 9/7],\ [421, 421, 13/7*w^3 - 16/7*w^2 - 92/7*w + 30/7],\ [421, 421, -3/7*w^3 + 15/7*w^2 + 18/7*w - 71/7],\ [421, 421, -w^3 + 3*w^2 + 6*w - 13],\ [421, 421, -13/7*w^3 + 23/7*w^2 + 85/7*w - 65/7],\ [431, 431, w^2 - 2*w - 7],\ [431, 431, -4/7*w^3 - 1/7*w^2 + 38/7*w - 13/7],\ [431, 431, 4/7*w^3 - 13/7*w^2 - 24/7*w + 20/7],\ [431, 431, w^2 - 8],\ [479, 479, 1/7*w^3 + 2/7*w^2 - 27/7*w - 16/7],\ [479, 479, 9/7*w^3 - 17/7*w^2 - 68/7*w + 38/7],\ [479, 479, 9/7*w^3 - 10/7*w^2 - 75/7*w + 38/7],\ [479, 479, -1/7*w^3 + 5/7*w^2 + 20/7*w - 40/7],\ [491, 491, 10/7*w^3 - 8/7*w^2 - 74/7*w + 15/7],\ [491, 491, 2/7*w^3 - 3/7*w^2 - 33/7*w + 52/7],\ [491, 491, 11/7*w^3 - 6/7*w^2 - 87/7*w - 1/7],\ [491, 491, 5/7*w^3 + 3/7*w^2 - 30/7*w - 31/7],\ [529, 23, -6/7*w^3 + 9/7*w^2 + 57/7*w - 16/7],\ [529, 23, 6/7*w^3 - 9/7*w^2 - 57/7*w + 44/7],\ [541, 541, 4/7*w^3 - 6/7*w^2 - 45/7*w + 13/7],\ [541, 541, -8/7*w^3 + 12/7*w^2 + 69/7*w - 47/7],\ [541, 541, 8/7*w^3 - 12/7*w^2 - 69/7*w + 26/7],\ [541, 541, 4/7*w^3 - 6/7*w^2 - 45/7*w + 34/7],\ [599, 599, -3/7*w^3 + 15/7*w^2 + 4/7*w - 78/7],\ [599, 599, w^3 - 2*w^2 - 5*w + 11],\ [599, 599, -4/7*w^3 + 13/7*w^2 + 31/7*w - 76/7],\ [599, 599, 3/7*w^3 + 6/7*w^2 - 25/7*w - 62/7],\ [601, 601, 11/7*w^3 - 20/7*w^2 - 87/7*w + 83/7],\ [601, 601, -3/7*w^3 + 8/7*w^2 + 39/7*w - 15/7],\ [601, 601, -2/7*w^3 + 10/7*w^2 + 5/7*w - 80/7],\ [601, 601, -11/7*w^3 + 13/7*w^2 + 94/7*w - 13/7],\ [659, 659, -3/7*w^3 + 8/7*w^2 - 3/7*w - 22/7],\ [659, 659, 13/7*w^3 - 23/7*w^2 - 92/7*w + 58/7],\ [659, 659, -13/7*w^3 + 16/7*w^2 + 99/7*w - 44/7],\ [659, 659, 3/7*w^3 - 1/7*w^2 - 4/7*w - 20/7],\ [661, 661, -11/7*w^3 + 13/7*w^2 + 73/7*w - 13/7],\ [661, 661, 9/7*w^3 - 10/7*w^2 - 54/7*w + 24/7],\ [661, 661, 9/7*w^3 - 17/7*w^2 - 47/7*w + 31/7],\ [661, 661, 11/7*w^3 - 20/7*w^2 - 66/7*w + 62/7],\ [709, 709, 11/7*w^3 - 13/7*w^2 - 94/7*w + 34/7],\ [709, 709, -3/7*w^3 + 1/7*w^2 + 46/7*w - 8/7],\ [709, 709, 3/7*w^3 - 8/7*w^2 - 39/7*w + 36/7],\ [709, 709, -11/7*w^3 + 20/7*w^2 + 87/7*w - 62/7],\ [719, 719, 3/7*w^3 - 1/7*w^2 - 39/7*w - 27/7],\ [719, 719, w^3 - 2*w^2 - 8*w + 2],\ [719, 719, -w^3 + w^2 + 9*w - 7],\ [719, 719, -3/7*w^3 + 8/7*w^2 + 32/7*w - 64/7],\ [769, 769, -5/7*w^3 + 11/7*w^2 + 51/7*w - 25/7],\ [769, 769, 9/7*w^3 - 10/7*w^2 - 82/7*w + 10/7],\ [769, 769, -9/7*w^3 + 17/7*w^2 + 75/7*w - 73/7],\ [769, 769, 5/7*w^3 - 4/7*w^2 - 58/7*w + 32/7],\ [829, 829, -11/7*w^3 + 20/7*w^2 + 87/7*w - 69/7],\ [829, 829, -3/7*w^3 + 8/7*w^2 + 39/7*w - 29/7],\ [829, 829, 3/7*w^3 - 1/7*w^2 - 46/7*w + 15/7],\ [829, 829, -11/7*w^3 + 13/7*w^2 + 94/7*w - 27/7],\ [839, 839, 1/7*w^3 + 2/7*w^2 - 27/7*w - 23/7],\ [839, 839, -9/7*w^3 + 10/7*w^2 + 75/7*w - 45/7],\ [839, 839, 9/7*w^3 - 17/7*w^2 - 68/7*w + 31/7],\ [839, 839, -1/7*w^3 + 5/7*w^2 + 20/7*w - 47/7],\ [841, 29, -10/7*w^3 + 15/7*w^2 + 60/7*w - 43/7],\ [841, 29, 10/7*w^3 - 15/7*w^2 - 60/7*w + 22/7],\ [911, 911, 8/7*w^3 - 5/7*w^2 - 76/7*w + 26/7],\ [911, 911, 4/7*w^3 - 13/7*w^2 - 38/7*w + 69/7],\ [911, 911, 4/7*w^3 + 1/7*w^2 - 52/7*w - 22/7],\ [911, 911, 8/7*w^3 - 19/7*w^2 - 62/7*w + 47/7],\ [961, 31, 10/7*w^3 - 15/7*w^2 - 60/7*w + 29/7],\ [961, 31, -10/7*w^3 + 15/7*w^2 + 60/7*w - 36/7],\ [971, 971, -2/7*w^3 + 10/7*w^2 - 2/7*w - 59/7],\ [971, 971, -6/7*w^3 + 2/7*w^2 + 50/7*w - 23/7],\ [971, 971, 6/7*w^3 - 16/7*w^2 - 36/7*w + 23/7],\ [971, 971, 2/7*w^3 + 4/7*w^2 - 12/7*w - 53/7]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 + 3*x + 1 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, 3*e + 6, -2*e - 3, 5, 3*e + 7, 2*e + 3, 0, -5*e - 9, -3*e - 6, -1, 2*e + 3, -10*e - 15, 9*e + 11, -3*e - 9, 3, -4*e - 3, -12, -8*e - 12, 6*e + 9, -3*e + 3, e - 6, 9, -3*e + 7, -15*e - 21, -6*e - 20, 4*e + 6, -12*e - 18, 15, 4*e + 6, 15*e + 16, -15*e - 24, -4*e - 6, -15*e - 15, 15, 5*e + 15, -16*e - 27, 11*e + 6, 4*e + 18, e + 6, -2*e - 18, 15, -12*e - 18, 15, -2*e - 24, -6*e - 15, 2*e - 3, -3*e - 3, -9*e - 6, -6*e + 1, -15*e - 20, 5*e + 30, -3*e - 5, 6*e + 11, 18*e + 19, -15*e - 18, -3*e - 2, 16*e + 24, 16*e + 24, -6*e - 24, 6, -3*e + 9, 15, -3*e - 7, -15*e - 5, -10*e - 15, -12*e - 17, 6*e + 25, -12*e - 27, 3*e + 23, -12*e - 8, 11*e + 9, 26*e + 39, 9*e + 26, 3*e + 25, -15*e - 37, -12*e - 42, 15*e + 11, -3*e + 14, 15*e + 26, 5*e + 15, -3*e - 12, -9*e - 36, 14*e + 21, -6*e - 11, 12*e + 6, 18*e + 10, 18*e + 30, -12*e - 33, 4*e - 9, -26*e - 39, 5, -15, -10*e - 45, 3*e + 27, -12*e + 2, -13*e - 42, -6*e - 29, 12*e + 33, -7*e - 18, 11*e + 3, -16*e - 15, 30*e + 33, 6*e - 18, -5*e - 42, -24*e - 33, 3*e - 23, 5*e + 30, -17*e - 33, 3*e + 27, -3, -2*e + 39, 27, -7*e + 9, -9*e - 11, 6*e + 24, -4*e + 24, 6*e + 39, -32*e - 51, e - 24, 10*e - 3, -18, 30*e + 39, 11*e + 3, -10*e - 6, -6*e - 30, -17*e - 18, 21*e + 14, -6*e + 11, 19*e + 36, 30*e + 49, 27*e + 27, 12*e + 42, -12*e - 14, -30*e - 29, 12*e + 44, -12*e - 2, 24*e + 17, 28*e + 42, -9*e + 14, 3*e - 8, -24*e - 36, -27, 23*e + 30, -17*e + 12, 27*e + 23, 24*e + 26, 20*e + 15, 6*e + 1, -9*e - 9, -3*e - 42, -45, -20*e - 45, 3*e + 52] 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, 3/7*w^3 - 1/7*w^2 - 32/7*w + 15/7])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]