/* 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([29, 8, -12, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([25,5,-w + 1]) primes_array = [ [5, 5, -w + 2],\ [5, 5, -3/5*w^3 - w^2 + 26/5*w + 42/5],\ [9, 3, 2/5*w^3 - 19/5*w - 3/5],\ [9, 3, -1/5*w^3 + 2/5*w + 4/5],\ [11, 11, -2/5*w^3 - w^2 + 14/5*w + 28/5],\ [11, 11, 1/5*w^3 - w^2 - 7/5*w + 36/5],\ [16, 2, 2],\ [29, 29, -w],\ [29, 29, 1/5*w^3 - 12/5*w + 1/5],\ [41, 41, -2/5*w^3 - w^2 + 14/5*w + 38/5],\ [41, 41, -w^2 + 10],\ [41, 41, 11/5*w^3 + 3*w^2 - 102/5*w - 149/5],\ [41, 41, -1/5*w^3 + w^2 + 7/5*w - 26/5],\ [49, 7, 1/5*w^3 + w^2 - 12/5*w - 19/5],\ [49, 7, 1/5*w^3 + w^2 - 12/5*w - 44/5],\ [59, 59, -4/5*w^3 - w^2 + 33/5*w + 36/5],\ [59, 59, -2*w^3 - 3*w^2 + 17*w + 26],\ [61, 61, -1/5*w^3 + w^2 + 12/5*w - 51/5],\ [61, 61, 4/5*w^3 + w^2 - 28/5*w - 41/5],\ [71, 71, 1/5*w^3 + w^2 - 7/5*w - 49/5],\ [71, 71, -2*w^3 - 3*w^2 + 18*w + 28],\ [89, 89, -3/5*w^3 + 16/5*w + 7/5],\ [89, 89, 3/5*w^3 + w^2 - 26/5*w - 57/5],\ [121, 11, -3/5*w^3 + 21/5*w + 2/5],\ [131, 131, w^2 - w - 8],\ [131, 131, 2/5*w^3 + w^2 - 19/5*w - 23/5],\ [149, 149, 2/5*w^3 - w^2 - 14/5*w + 22/5],\ [149, 149, 3/5*w^3 + w^2 - 21/5*w - 42/5],\ [179, 179, -6/5*w^3 - w^2 + 57/5*w + 69/5],\ [179, 179, -2/5*w^3 - 2*w^2 + 14/5*w + 73/5],\ [179, 179, 2*w^2 - 11],\ [179, 179, -3/5*w^3 + 2*w^2 + 16/5*w - 48/5],\ [181, 181, -7/5*w^3 - 2*w^2 + 59/5*w + 78/5],\ [181, 181, -2/5*w^3 + 24/5*w - 27/5],\ [191, 191, -3/5*w^3 + 2*w^2 + 21/5*w - 78/5],\ [191, 191, 2/5*w^3 - 2*w^2 - 9/5*w + 57/5],\ [199, 199, -1/5*w^3 + 17/5*w - 16/5],\ [199, 199, 1/5*w^3 - 17/5*w - 14/5],\ [211, 211, 2/5*w^3 - w^2 + 1/5*w + 7/5],\ [211, 211, -1/5*w^3 + w^2 - 3/5*w - 26/5],\ [211, 211, 4/5*w^3 + w^2 - 38/5*w - 36/5],\ [211, 211, -1/5*w^3 - 2*w^2 + 7/5*w + 44/5],\ [239, 239, 3/5*w^3 + w^2 - 16/5*w - 37/5],\ [239, 239, 2/5*w^3 + w^2 - 14/5*w - 53/5],\ [251, 251, 2/5*w^3 - 24/5*w - 13/5],\ [251, 251, 2*w - 3],\ [251, 251, w^2 + w - 10],\ [251, 251, -4/5*w^3 - w^2 + 28/5*w + 46/5],\ [269, 269, 1/5*w^3 - 12/5*w - 24/5],\ [269, 269, w - 5],\ [271, 271, -3/5*w^3 + 31/5*w - 33/5],\ [271, 271, -8/5*w^3 - 2*w^2 + 66/5*w + 77/5],\ [281, 281, 4/5*w^3 - w^2 - 38/5*w + 39/5],\ [281, 281, -14/5*w^3 - 3*w^2 + 128/5*w + 161/5],\ [311, 311, 1/5*w^3 - 2*w^2 - 2/5*w + 51/5],\ [311, 311, 1/5*w^3 - w^2 - 17/5*w + 21/5],\ [331, 331, 1/5*w^3 + 3/5*w - 4/5],\ [331, 331, 3/5*w^3 - 31/5*w - 2/5],\ [359, 359, -w^3 + 6*w - 1],\ [359, 359, -6/5*w^3 + 47/5*w - 6/5],\ [359, 359, 3/5*w^3 + w^2 - 31/5*w - 27/5],\ [359, 359, 3/5*w^3 + w^2 - 26/5*w - 22/5],\ [361, 19, 4/5*w^3 - 28/5*w - 11/5],\ [361, 19, 1/5*w^3 - 7/5*w - 24/5],\ [379, 379, 9/5*w^3 + 3*w^2 - 78/5*w - 121/5],\ [379, 379, -4/5*w^3 + 3*w^2 + 28/5*w - 99/5],\ [389, 389, -3/5*w^3 - 2*w^2 + 26/5*w + 42/5],\ [389, 389, -2*w^2 + w + 17],\ [401, 401, 4/5*w^3 - w^2 - 23/5*w + 24/5],\ [401, 401, 3/5*w^3 + w^2 - 26/5*w - 7/5],\ [401, 401, -4/5*w^3 - w^2 + 38/5*w + 66/5],\ [401, 401, 6/5*w^3 + w^2 - 52/5*w - 39/5],\ [409, 409, -6/5*w^3 + 52/5*w + 19/5],\ [409, 409, 1/5*w^3 + 2*w^2 - 7/5*w - 39/5],\ [421, 421, -1/5*w^3 - w^2 + 17/5*w + 44/5],\ [421, 421, 2/5*w^3 + w^2 - 24/5*w - 18/5],\ [431, 431, -3/5*w^3 - w^2 + 36/5*w + 47/5],\ [431, 431, -13/5*w^3 - 4*w^2 + 116/5*w + 182/5],\ [431, 431, -2/5*w^3 + w^2 + 24/5*w - 47/5],\ [431, 431, 2/5*w^3 - w^2 + 6/5*w - 8/5],\ [439, 439, 6/5*w^3 + w^2 - 47/5*w - 49/5],\ [439, 439, -2/5*w^3 + 2*w^2 + 9/5*w - 42/5],\ [439, 439, 4/5*w^3 - w^2 - 23/5*w + 14/5],\ [439, 439, -w^3 - 2*w^2 + 8*w + 17],\ [461, 461, w^3 + w^2 - 10*w - 9],\ [461, 461, -2*w^3 - 3*w^2 + 17*w + 27],\ [461, 461, -1/5*w^3 + w^2 - 8/5*w - 16/5],\ [461, 461, -3*w^3 - 4*w^2 + 26*w + 35],\ [479, 479, 2*w - 1],\ [479, 479, 2/5*w^3 - 24/5*w - 3/5],\ [491, 491, w^3 - 2*w^2 - 6*w + 11],\ [491, 491, w^3 + w^2 - 10*w - 15],\ [499, 499, -1/5*w^3 + 2*w^2 + 2/5*w - 36/5],\ [499, 499, 4/5*w^3 + 2*w^2 - 33/5*w - 91/5],\ [509, 509, 8/5*w^3 + w^2 - 66/5*w - 42/5],\ [509, 509, 1/5*w^3 - 2*w^2 + 8/5*w + 16/5],\ [521, 521, w^3 - 9*w - 4],\ [521, 521, 3/5*w^3 + w^2 - 16/5*w - 42/5],\ [521, 521, 3/5*w^3 - w^2 - 26/5*w + 23/5],\ [521, 521, -3/5*w^3 + 11/5*w + 22/5],\ [529, 23, 3/5*w^3 + w^2 - 36/5*w - 62/5],\ [529, 23, -4/5*w^3 + 2*w^2 + 23/5*w - 64/5],\ [541, 541, -w^3 - w^2 + 9*w + 7],\ [541, 541, 1/5*w^3 + 2*w^2 - 7/5*w - 64/5],\ [541, 541, 3/5*w^3 + w^2 - 26/5*w - 12/5],\ [571, 571, -w^3 - w^2 + 6*w + 9],\ [571, 571, -w^3 + w^2 + 8*w - 4],\ [619, 619, -w^3 - 2*w^2 + 8*w + 11],\ [619, 619, -2/5*w^3 + 2*w^2 + 9/5*w - 72/5],\ [619, 619, 2/5*w^3 - 2*w^2 - 4/5*w + 57/5],\ [619, 619, -6/5*w^3 - 2*w^2 + 52/5*w + 69/5],\ [631, 631, 1/5*w^3 + w^2 + 3/5*w - 24/5],\ [631, 631, 2/5*w^3 - w^2 - 24/5*w + 42/5],\ [659, 659, -13/5*w^3 - 4*w^2 + 111/5*w + 177/5],\ [659, 659, -11/5*w^3 - 3*w^2 + 92/5*w + 119/5],\ [659, 659, -8/5*w^3 - w^2 + 61/5*w + 27/5],\ [659, 659, -8/5*w^3 + 71/5*w + 2/5],\ [661, 661, -w^3 - w^2 + 7*w + 11],\ [661, 661, 4/5*w^3 - w^2 - 28/5*w + 9/5],\ [691, 691, 1/5*w^3 - w^2 - 17/5*w + 41/5],\ [691, 691, 3/5*w^3 + 2*w^2 - 26/5*w - 67/5],\ [701, 701, 12/5*w^3 + 4*w^2 - 99/5*w - 158/5],\ [701, 701, -17/5*w^3 - 4*w^2 + 149/5*w + 193/5],\ [709, 709, -1/5*w^3 + w^2 + 12/5*w - 6/5],\ [709, 709, 1/5*w^3 + w^2 - 2/5*w - 59/5],\ [719, 719, 2/5*w^3 + w^2 - 29/5*w - 8/5],\ [719, 719, -w^3 + w^2 + 7*w - 4],\ [719, 719, 6/5*w^3 + w^2 - 42/5*w - 44/5],\ [719, 719, -2/5*w^3 - w^2 + 29/5*w + 53/5],\ [739, 739, 14/5*w^3 + 4*w^2 - 123/5*w - 171/5],\ [739, 739, -1/5*w^3 + 2*w^2 + 12/5*w - 56/5],\ [739, 739, -6/5*w^3 + 3*w^2 + 37/5*w - 96/5],\ [739, 739, 2/5*w^3 + 2*w^2 - 9/5*w - 73/5],\ [751, 751, -9/5*w^3 - 2*w^2 + 73/5*w + 86/5],\ [751, 751, -w^3 + 2*w^2 + 5*w - 8],\ [761, 761, 1/5*w^3 + w^2 - 22/5*w + 16/5],\ [761, 761, -16/5*w^3 - 4*w^2 + 147/5*w + 214/5],\ [769, 769, 13/5*w^3 + 3*w^2 - 121/5*w - 172/5],\ [769, 769, 3/5*w^3 - w^2 - 41/5*w + 78/5],\ [769, 769, -12/5*w^3 - 4*w^2 + 99/5*w + 153/5],\ [769, 769, -6/5*w^3 + 3*w^2 + 27/5*w - 61/5],\ [809, 809, 1/5*w^3 + 3*w^2 - 22/5*w - 104/5],\ [809, 809, w^3 + 3*w^2 - 10*w - 17],\ [811, 811, -5*w + 11],\ [811, 811, -16/5*w^3 - 5*w^2 + 137/5*w + 214/5],\ [821, 821, -1/5*w^3 + w^2 + 17/5*w - 36/5],\ [821, 821, w^2 + 2*w - 6],\ [829, 829, 3*w^2 + w - 17],\ [829, 829, 2/5*w^3 + 3*w^2 - 9/5*w - 108/5],\ [841, 29, w^3 - 7*w - 3],\ [859, 859, 6/5*w^3 + w^2 - 37/5*w - 34/5],\ [859, 859, 2/5*w^3 + w^2 - 14/5*w - 68/5],\ [881, 881, 1/5*w^3 + 2*w^2 - 17/5*w - 79/5],\ [881, 881, 3/5*w^3 + 2*w^2 - 31/5*w - 47/5],\ [911, 911, 4/5*w^3 - 3*w^2 - 28/5*w + 89/5],\ [911, 911, 1/5*w^3 - w^2 - 7/5*w + 61/5],\ [911, 911, w^3 - 6*w - 3],\ [911, 911, -11/5*w^3 - 3*w^2 + 92/5*w + 139/5],\ [919, 919, 7/5*w^3 + 2*w^2 - 54/5*w - 88/5],\ [919, 919, 4/5*w^3 - 2*w^2 - 23/5*w + 39/5],\ [929, 929, -7/5*w^3 + w^2 + 54/5*w - 17/5],\ [929, 929, -1/5*w^3 + w^2 + 22/5*w - 71/5],\ [941, 941, 9/5*w^3 + w^2 - 73/5*w - 31/5],\ [941, 941, 1/5*w^3 - w^2 - 2/5*w + 66/5],\ [941, 941, 2/5*w^3 + 2*w^2 - 19/5*w - 103/5],\ [941, 941, -w^3 + w^2 + 5*w - 3],\ [961, 31, w^3 - 7*w - 2],\ [961, 31, 2/5*w^3 - 14/5*w - 33/5],\ [971, 971, -2/5*w^3 - 2*w^2 + 24/5*w + 93/5],\ [971, 971, 2/5*w^3 + 2*w^2 - 24/5*w - 33/5],\ [991, 991, -w^3 + 2*w^2 + 7*w - 10],\ [991, 991, -7/5*w^3 - 2*w^2 + 49/5*w + 78/5]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^3 - 3*x^2 - 13*x + 31 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, 0, -1/4*e^2 - 1/2*e + 19/4, -1/4*e^2 - e + 17/4, 1/4*e^2 + 1/2*e - 19/4, -1/4*e^2 - e + 17/4, -1/4*e^2 + 21/4, -1/2*e^2 + 2*e + 15/2, 1/2*e^2 - e - 3/2, -3/4*e^2 + 3/2*e + 29/4, 1/4*e^2 - 3/2*e + 5/4, 1/4*e^2 - 3/2*e + 5/4, e^2 - 11, 3/4*e^2 - e - 27/4, -5/4*e^2 - 1/2*e + 63/4, -1/4*e^2 + 1/2*e + 23/4, -3/2*e^2 + e + 25/2, -1/2*e^2 + 2*e + 5/2, 5/4*e^2 + 1/2*e - 63/4, 5/4*e^2 - 2*e - 33/4, -e - 11, 1/2*e^2 + e - 19/2, -5/4*e^2 + 7/2*e + 39/4, 1/4*e^2 + 3/2*e - 47/4, 1/2*e^2 - 9/2*e - 5, -3/4*e^2 + 1/2*e + 65/4, -1/2*e^2 + 2*e + 5/2, e^2 - 1/2*e - 23/2, 3/2*e^2 - 3*e - 19/2, -2*e^2 + 27, -3/4*e^2 - 1/2*e + 61/4, -3/4*e^2 - 1/2*e + 61/4, 5*e - 5, -11/4*e^2 + 7/2*e + 125/4, 3/4*e^2 + 3/2*e - 97/4, -1/4*e^2 + 1/2*e + 23/4, -e^2 + 3*e + 19, 11/4*e^2 - 3/2*e - 77/4, 1/4*e^2 + 1/2*e + 49/4, 2*e^2 - 24, e^2 - 2*e, -9/4*e^2 + 7/2*e + 51/4, -1/2*e^2 + 3*e - 3/2, -1/4*e^2 + 5/2*e + 31/4, 1/2*e^2 + 23/2, 1/2*e^2 + 23/2, 3/2*e^2 + 3/2, 7/4*e^2 + 9/2*e - 137/4, -9/4*e^2 + 11/2*e + 51/4, -3*e^2 - e + 42, 11/4*e^2 + 3/2*e - 133/4, -7/4*e^2 + 1/2*e + 61/4, 13/4*e^2 - 5/2*e - 135/4, -9/4*e^2 - 11/2*e + 155/4, -1/2*e^2 + 57/2, -1/2*e^2 + 57/2, 3/2*e^2 + 4*e - 29/2, -1/2*e^2 - 5*e + 7/2, 1/4*e^2 - 5/2*e + 49/4, 1/2*e^2 - 1/2, -1/4*e^2 - 1/2*e + 39/4, 3/2*e^2 - 3/2*e - 13, -17/4*e^2 + 2*e + 165/4, 4*e^2 - 7/2*e - 55/2, 1/2*e^2 - 4*e - 49/2, -1/2*e^2 - 2*e - 23/2, -9/4*e^2 + 11/2*e + 51/4, 1/2*e^2 - 4*e + 11/2, -13/4*e^2 + 153/4, 3/4*e^2 - 3/2*e + 3/4, -7/4*e^2 + 13/2*e + 93/4, -11/4*e^2 - 1/2*e + 101/4, 11/4*e^2 - 1/2*e - 113/4, -3/2*e^2 - 5/2*e + 19, 3*e^2 - 33, 5/4*e^2 + 7/2*e - 11/4, 1/2*e^2 - 3*e - 7/2, 1/2*e^2 - 4*e - 49/2, 3/4*e^2 + 13/2*e - 37/4, 5/4*e^2 - 5/2*e + 45/4, e^2 - 4*e + 15, 5/4*e^2 - 9/2*e - 43/4, 5/4*e^2 + 6*e - 121/4, 5/4*e^2 - 11/2*e - 127/4, 5/2*e^2 + 3*e - 39/2, -1/4*e^2 - 5/2*e + 11/4, 3/4*e^2 + 5/2*e - 53/4, -3/4*e^2 - 4*e + 47/4, -1/4*e^2 + 3/2*e + 147/4, -7/2*e^2 + 107/2, 15/4*e^2 - 5*e - 135/4, -7/4*e^2 + 15/2*e + 97/4, -1/2*e^2 - 8*e + 25/2, 9/2*e - 31/2, 3/2*e^2 - 2*e - 37/2, 9/4*e^2 + 11/2*e - 167/4, -7/4*e^2 + 5/2*e + 77/4, 1/2*e^2 - 7*e - 11/2, 13/4*e^2 - 11/2*e - 167/4, -1/2*e^2 - 1/2*e + 10, -1/4*e^2 - 1/2*e + 79/4, 2*e^2 + 2*e - 20, -5/2*e^2 + 5*e + 75/2, -5/2*e^2 - 3*e + 75/2, -5*e - 5, 13/4*e^2 + 3/2*e - 119/4, -4*e^2 - e + 55, 3/4*e^2 + 1/2*e - 61/4, -7/2*e^2 - 4*e + 79/2, 3/2*e + 43/2, 3*e^2 - 3*e - 31, 7/2*e^2 - 9*e - 69/2, -13/4*e^2 + 15/2*e + 155/4, -1/2*e^2 - 2*e - 43/2, -11/4*e^2 + 21/2*e + 113/4, -2*e^2 + 3*e + 25, 11/2*e^2 - 6*e - 123/2, 3/4*e^2 + 15/2*e - 33/4, 2*e^2 + 9/2*e - 35/2, -23/4*e^2 + 3/2*e + 289/4, -5/4*e^2 + 3/2*e + 71/4, -3/4*e^2 - 11/2*e + 49/4, -4*e^2 + 5*e + 31, -e^2 - e + 40, -2*e^2 - 5*e + 32, e^2 - e - 22, 3*e^2 + 3*e - 60, 6*e^2 - 5*e - 61, 15/4*e^2 + 9/2*e - 297/4, -11/4*e^2 - 8*e + 159/4, -5/4*e^2 + 11/2*e + 7/4, e^2 - 6*e - 7, -1/2*e^2 - 6*e + 9/2, -e^2 + 9*e - 2, -9/4*e^2 - 5/2*e + 111/4, 3/4*e^2 - 7/2*e + 31/4, 4*e + 26, -9/2*e^2 + 5*e + 79/2, 7/2*e^2 - 2*e - 91/2, -7/4*e^2 + 5/2*e + 157/4, -7/4*e^2 + 8*e - 21/4, 5/4*e^2 - 5*e - 85/4, 17/4*e^2 - 19/2*e - 195/4, -e^2 + 6*e + 15, -4*e^2 - 5*e + 57, -27/4*e^2 + e + 331/4, -1/4*e^2 - 13/2*e + 75/4, -5/4*e^2 + 1/2*e + 147/4, 11/4*e^2 - 7/2*e - 5/4, -1/2*e^2 + 2*e + 5/2, 5/2*e^2 - 10*e - 75/2, -5/4*e^2 - 19/2*e + 147/4, -7/2*e^2 + 5/2*e + 36, 9/4*e^2 + 3/2*e - 63/4, 15/4*e^2 - 5/2*e - 177/4, -e^2 - 4*e + 55, -7/2*e^2 - e + 111/2, 3*e^2 + 5*e - 26, 3/4*e^2 + 9/2*e - 205/4, 5/4*e^2 - 9/2*e - 83/4, 5*e^2 - 1/2*e - 91/2, 27/4*e^2 - 19/2*e - 245/4, 7/4*e^2 - 5/2*e - 9/4, 9/4*e^2 - 25/2*e - 47/4, -e^2 - 4*e + 34, e^2 + 2*e - 41, 17/4*e^2 - 19/2*e - 187/4, -1/2*e^2 + 3*e - 49/2, -19/4*e^2 - e + 195/4, -9/4*e^2 + 5/2*e - 81/4, 2*e - 48, 5*e^2 + 19/2*e - 151/2] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([5,5,-1/5*w^3 + 12/5*w + 9/5])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]