/* 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([5, -6, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([23, 23, w^2 - 3]) primes_array = [ [3, 3, w - 2],\ [5, 5, w],\ [5, 5, -w + 3],\ [8, 2, 2],\ [9, 3, w^2 + w - 4],\ [13, 13, w + 3],\ [17, 17, -w^2 + w + 3],\ [23, 23, w^2 - 2],\ [23, 23, w^2 - 3],\ [23, 23, -w^2 + 8],\ [29, 29, w - 4],\ [37, 37, w^2 + w - 8],\ [41, 41, w^2 + 2*w - 4],\ [47, 47, 2*w^2 + w - 8],\ [59, 59, -2*w^2 - 3*w + 6],\ [61, 61, -2*w - 1],\ [67, 67, -2*w - 3],\ [79, 79, 2*w^2 - 9],\ [109, 109, w^2 + 2*w - 6],\ [109, 109, 2*w^2 + w - 14],\ [109, 109, -w^2 - w - 1],\ [113, 113, 2*w^2 + w - 12],\ [127, 127, w^2 - 3*w - 2],\ [131, 131, -4*w^2 - 5*w + 9],\ [137, 137, w^2 + 2*w - 7],\ [137, 137, 3*w + 8],\ [137, 137, 2*w^2 + w - 17],\ [139, 139, 3*w^2 + 2*w - 11],\ [149, 149, w - 6],\ [151, 151, w^2 + 2*w - 9],\ [157, 157, -w^2 - 2*w + 13],\ [157, 157, 2*w^2 - 7],\ [163, 163, w^2 + 3*w + 3],\ [163, 163, 2*w^2 - 3*w - 3],\ [163, 163, 2*w^2 - w - 7],\ [167, 167, 3*w^2 + w - 13],\ [169, 13, 2*w^2 + 5*w - 1],\ [173, 173, w^2 + 2*w + 2],\ [179, 179, 4*w^2 + 2*w - 19],\ [181, 181, w^2 + 3*w - 6],\ [191, 191, 3*w^2 + 2*w - 14],\ [193, 193, w^2 - 3*w - 3],\ [197, 197, 3*w^2 - 2*w - 13],\ [211, 211, -w - 6],\ [223, 223, -w^2 + w - 3],\ [223, 223, -3*w - 2],\ [223, 223, 2*w^2 - 2*w - 3],\ [229, 229, 2*w^2 - w - 4],\ [233, 233, 2*w + 7],\ [239, 239, -3*w - 4],\ [239, 239, w^2 - 11],\ [239, 239, 4*w^2 - w - 21],\ [241, 241, 3*w^2 + w - 19],\ [251, 251, -w^2 - 4*w - 1],\ [257, 257, w - 7],\ [263, 263, 3*w^2 + w - 12],\ [269, 269, 3*w^2 - 4*w - 11],\ [271, 271, 4*w^2 + 6*w - 11],\ [277, 277, -4*w^2 - 3*w + 17],\ [283, 283, 5*w^2 + 4*w - 18],\ [289, 17, 3*w^2 + 2*w - 9],\ [307, 307, -2*w^2 - w + 18],\ [307, 307, w^2 + 2*w + 3],\ [307, 307, -w^2 - 3],\ [313, 313, 3*w^2 - 13],\ [317, 317, 3*w^2 - 2*w - 12],\ [331, 331, w^2 + 3*w - 9],\ [331, 331, w^2 + 3*w - 11],\ [331, 331, w^2 + 3*w + 4],\ [343, 7, -7],\ [347, 347, 2*w^2 + 2*w - 13],\ [353, 353, 4*w^2 + 5*w - 13],\ [359, 359, 3*w^2 + 6*w - 1],\ [379, 379, w^2 - 3*w - 6],\ [383, 383, 3*w^2 - 23],\ [383, 383, w^2 - 12],\ [383, 383, 3*w^2 + 3*w - 13],\ [401, 401, -6*w^2 - 7*w + 16],\ [409, 409, -4*w^2 + 3*w + 21],\ [421, 421, w^2 - 3*w - 9],\ [433, 433, w^2 - 3*w - 8],\ [433, 433, w^2 + 4*w - 8],\ [433, 433, 3*w^2 - w - 12],\ [439, 439, -4*w - 11],\ [449, 449, 2*w^2 + 3*w - 19],\ [457, 457, w^2 - 4*w - 3],\ [461, 461, 5*w^2 + 6*w - 16],\ [461, 461, -2*w^2 + 3*w + 16],\ [461, 461, 2*w^2 + 4*w - 9],\ [467, 467, 3*w^2 - 4*w - 12],\ [479, 479, 3*w^2 + w - 9],\ [487, 487, 5*w^2 + 2*w - 22],\ [491, 491, 3*w^2 - 11],\ [499, 499, 5*w^2 + w - 24],\ [503, 503, 3*w^2 + 2*w - 17],\ [503, 503, 3*w^2 + w - 7],\ [503, 503, 2*w^2 - 4*w - 7],\ [509, 509, 3*w - 11],\ [521, 521, 3*w^2 - 3*w - 16],\ [523, 523, -w - 8],\ [541, 541, 6*w^2 + 7*w - 19],\ [547, 547, 3*w^2 + 4*w - 12],\ [557, 557, 2*w^2 - 3*w - 12],\ [563, 563, 4*w^2 + 3*w - 12],\ [563, 563, 4*w^2 + w - 17],\ [563, 563, -w^2 - 5*w - 3],\ [569, 569, w^2 - 4*w - 4],\ [577, 577, w^2 - w - 13],\ [587, 587, 4*w^2 + 3*w - 8],\ [587, 587, 2*w^2 - 3*w - 13],\ [587, 587, 2*w^2 + 3*w - 18],\ [593, 593, 4*w - 13],\ [599, 599, w - 9],\ [613, 613, -w^2 + 3*w - 7],\ [613, 613, w^2 + 5*w + 2],\ [613, 613, 3*w^2 - 3*w - 17],\ [617, 617, w^2 + 5*w + 8],\ [641, 641, 4*w^2 + 3*w - 11],\ [641, 641, w^2 + w - 14],\ [641, 641, 3*w^2 - 2*w - 9],\ [647, 647, 2*w^2 + 3*w - 13],\ [653, 653, 3*w^2 - 8],\ [659, 659, 5*w^2 + 4*w - 21],\ [659, 659, 2*w^2 - 4*w - 19],\ [659, 659, -2*w^2 - 6*w + 9],\ [677, 677, -6*w - 13],\ [691, 691, 6*w - 11],\ [701, 701, 3*w^2 - 3*w - 19],\ [719, 719, 2*w^2 + 3*w - 16],\ [727, 727, 3*w^2 + 4*w - 13],\ [733, 733, 3*w^2 - w - 8],\ [739, 739, 2*w^2 + 4*w - 11],\ [739, 739, 4*w^2 - 3*w - 24],\ [739, 739, 3*w^2 - w - 6],\ [743, 743, 4*w^2 - 2*w - 17],\ [751, 751, -w - 9],\ [761, 761, 5*w^2 + 5*w - 19],\ [769, 769, w^2 - 4*w - 6],\ [769, 769, -5*w - 1],\ [769, 769, 2*w^2 - 19],\ [773, 773, 2*w^2 - 6*w - 3],\ [797, 797, 4*w^2 - 17],\ [809, 809, -7*w^2 - 4*w + 29],\ [811, 811, 6*w^2 + 3*w - 29],\ [823, 823, 4*w^2 + 2*w - 13],\ [823, 823, -w^2 - 4*w - 7],\ [823, 823, 4*w^2 + 4*w - 17],\ [827, 827, -2*w^2 - 3*w - 2],\ [829, 829, -w^2 + w - 6],\ [829, 829, -w^2 + 4*w - 9],\ [829, 829, -2*w^2 + 3*w + 21],\ [839, 839, 6*w^2 - 2*w - 31],\ [841, 29, w^2 + 3*w + 6],\ [853, 853, -7*w^2 - 6*w + 27],\ [859, 859, 6*w - 1],\ [881, 881, w^2 + 5*w - 16],\ [907, 907, 2*w^2 + 7*w + 7],\ [919, 919, w^2 - 4*w - 9],\ [953, 953, -4*w^2 - w + 32],\ [967, 967, 5*w^2 - 5*w - 18],\ [971, 971, 3*w^2 - 4*w - 24],\ [977, 977, 2*w^2 + 4*w + 3],\ [997, 997, w^2 + 5*w - 12],\ [997, 997, 5*w^2 - 3*w - 22],\ [997, 997, 3*w^2 + 8*w - 2]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^5 - 12*x^3 + 4*x^2 + 20*x - 4 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, 0, -1/6*e^4 - 1/6*e^3 + 4/3*e^2 + 2/3*e - 2/3, 1/3*e^4 + 1/3*e^3 - 11/3*e^2 - 7/3*e + 13/3, -1/3*e^4 - 1/3*e^3 + 11/3*e^2 + 4/3*e - 10/3, 1/6*e^4 + 1/6*e^3 - 4/3*e^2 - 2/3*e - 4/3, 1/2*e^4 - 6*e^2 + 2*e + 6, e^2 - e - 8, -1, 1/2*e^3 - 4*e - 2, 1/3*e^4 - 1/6*e^3 - 14/3*e^2 + 2/3*e + 28/3, e^3 + e^2 - 10*e - 2, -5/6*e^4 - 1/3*e^3 + 26/3*e^2 + 4/3*e - 16/3, -1/6*e^4 - 1/6*e^3 + 4/3*e^2 - 4/3*e - 8/3, -5/6*e^4 - 5/6*e^3 + 26/3*e^2 + 10/3*e - 34/3, -1/3*e^4 + 2/3*e^3 + 14/3*e^2 - 23/3*e - 22/3, 1/3*e^4 - 7/6*e^3 - 14/3*e^2 + 26/3*e + 10/3, -1/2*e^3 - 2*e^2 + 2*e + 6, 4/3*e^4 + 5/6*e^3 - 44/3*e^2 - 10/3*e + 58/3, 2/3*e^4 + 2/3*e^3 - 16/3*e^2 - 8/3*e + 2/3, -1/3*e^4 + 1/6*e^3 + 14/3*e^2 - 2/3*e - 22/3, 1/3*e^4 + 1/3*e^3 - 11/3*e^2 + 2/3*e + 22/3, -2/3*e^4 - 7/6*e^3 + 16/3*e^2 + 20/3*e - 14/3, e^4 + e^3 - 10*e^2 - 2*e + 2, 2/3*e^4 - 1/3*e^3 - 25/3*e^2 + 34/3*e + 38/3, -1/2*e^4 - 1/2*e^3 + 6*e^2 + 6*e - 6, -2/3*e^4 + 1/3*e^3 + 25/3*e^2 - 16/3*e - 38/3, -7/6*e^4 - 2/3*e^3 + 34/3*e^2 - 10/3*e - 26/3, -1/3*e^4 - 1/3*e^3 + 11/3*e^2 + 16/3*e + 14/3, -1/3*e^4 + 1/6*e^3 + 14/3*e^2 + 10/3*e - 22/3, -4/3*e^4 - 1/3*e^3 + 38/3*e^2 - 2/3*e - 34/3, -1/3*e^4 + 2/3*e^3 + 8/3*e^2 - 20/3*e - 10/3, -4/3*e^4 + 2/3*e^3 + 47/3*e^2 - 26/3*e - 52/3, e^4 - 11*e^2 + 4*e, -e^3 + 8*e - 4, -2/3*e^4 + 1/3*e^3 + 28/3*e^2 - 22/3*e - 20/3, 1/3*e^4 - 1/6*e^3 - 8/3*e^2 + 8/3*e - 14/3, -1/2*e^4 - e^3 + 6*e^2 + 14*e - 12, -1/2*e^4 + 1/2*e^3 + 6*e^2 - 4*e - 6, -1/3*e^4 - 7/3*e^3 + 8/3*e^2 + 61/3*e - 10/3, 1/3*e^4 + 1/3*e^3 - 8/3*e^2 + 8/3*e - 20/3, 2/3*e^4 + 5/3*e^3 - 10/3*e^2 - 26/3*e - 28/3, -1/2*e^4 + 2*e^2 - 4*e + 8, -7/6*e^4 + 5/6*e^3 + 40/3*e^2 - 34/3*e - 56/3, -5/3*e^4 - 5/3*e^3 + 52/3*e^2 + 14/3*e - 44/3, -2*e^3 - 2*e^2 + 14*e + 6, 1/2*e^4 - 2*e^3 - 6*e^2 + 22*e + 8, -4/3*e^4 - 1/3*e^3 + 53/3*e^2 + 1/3*e - 82/3, 4/3*e^4 + 1/3*e^3 - 38/3*e^2 - 4/3*e + 22/3, 19/6*e^4 + 13/6*e^3 - 100/3*e^2 - 2/3*e + 86/3, e^3 + 4*e^2 - 7*e - 16, -1/6*e^4 + 1/3*e^3 + 16/3*e^2 - 4/3*e - 50/3, 3/2*e^4 + 3/2*e^3 - 16*e^2 - 10*e + 20, -5/3*e^4 + 4/3*e^3 + 55/3*e^2 - 46/3*e - 56/3, -e^4 - 7/2*e^3 + 8*e^2 + 26*e - 16, 2/3*e^4 - 4/3*e^3 - 22/3*e^2 + 28/3*e + 8/3, -e^4 + 10*e^2 - 8*e - 2, 8/3*e^4 + 5/3*e^3 - 79/3*e^2 + 4/3*e + 56/3, 1/3*e^4 + 1/3*e^3 - 17/3*e^2 - 16/3*e + 58/3, -2/3*e^4 - 2/3*e^3 + 22/3*e^2 + 14/3*e - 74/3, e^3 - 2*e^2 - 13*e - 2, 5/6*e^4 - 2/3*e^3 - 26/3*e^2 + 20/3*e - 14/3, e^4 - 2*e^3 - 10*e^2 + 26*e + 10, -1/3*e^4 - 4/3*e^3 + 5/3*e^2 + 34/3*e - 4/3, -2/3*e^4 + 11/6*e^3 + 28/3*e^2 - 64/3*e - 50/3, e^4 - 1/2*e^3 - 10*e^2 + 4*e + 8, e^4 + 3*e^3 - 13*e^2 - 22*e + 28, 4/3*e^4 - 5/3*e^3 - 41/3*e^2 + 62/3*e + 52/3, -1/3*e^4 - 4/3*e^3 - 4/3*e^2 + 28/3*e + 68/3, -1/3*e^4 - 5/6*e^3 + 2/3*e^2 + 10/3*e + 62/3, -7/6*e^4 - 13/6*e^3 + 34/3*e^2 + 32/3*e - 68/3, 4/3*e^4 + 1/3*e^3 - 32/3*e^2 + 8/3*e + 10/3, e^4 + e^3 - 7*e^2 - 8*e - 4, 8/3*e^4 + 5/3*e^3 - 82/3*e^2 - 2/3*e + 44/3, -1/6*e^4 + 11/6*e^3 + 10/3*e^2 - 34/3*e - 38/3, -2*e^4 - e^3 + 23*e^2 + 6*e - 28, e^4 + e^3 - 11*e^2 - 4*e + 4, 1/3*e^4 + 4/3*e^3 + 4/3*e^2 - 16/3*e - 62/3, -4/3*e^4 - 7/3*e^3 + 44/3*e^2 + 43/3*e - 118/3, -5/2*e^4 - 3*e^3 + 24*e^2 + 18*e - 28, 4/3*e^4 + 4/3*e^3 - 41/3*e^2 - 16/3*e + 22/3, 11/6*e^4 + 11/6*e^3 - 62/3*e^2 - 40/3*e + 100/3, 2/3*e^4 - 7/3*e^3 - 28/3*e^2 + 79/3*e + 50/3, -1/3*e^4 - 1/3*e^3 + 11/3*e^2 + 16/3*e - 16/3, -2/3*e^4 + 5/6*e^3 + 10/3*e^2 - 34/3*e + 64/3, 11/6*e^4 + 7/3*e^3 - 50/3*e^2 - 22/3*e + 4/3, 1/6*e^4 + 11/3*e^3 + 2/3*e^2 - 80/3*e - 22/3, 23/6*e^4 + 5/6*e^3 - 116/3*e^2 + 8/3*e + 88/3, -5/3*e^4 - 11/3*e^3 + 46/3*e^2 + 56/3*e - 50/3, 7/3*e^4 + 4/3*e^3 - 77/3*e^2 - 10/3*e + 100/3, e^4 - 3*e^3 - 14*e^2 + 24*e + 28, -1/3*e^4 + 5/3*e^3 + 11/3*e^2 - 50/3*e + 20/3, -17/6*e^4 - 17/6*e^3 + 86/3*e^2 + 34/3*e - 52/3, 5/3*e^4 + 13/6*e^3 - 58/3*e^2 - 32/3*e + 92/3, -11/6*e^4 + 5/3*e^3 + 50/3*e^2 - 74/3*e - 46/3, e^4 - 7*e^2 + 2*e - 16, 7/3*e^4 + 7/3*e^3 - 77/3*e^2 - 22/3*e + 64/3, 7/3*e^4 + 4/3*e^3 - 68/3*e^2 - 10/3*e + 46/3, -3/2*e^4 + 3/2*e^3 + 14*e^2 - 22*e - 12, -5/3*e^4 + 1/3*e^3 + 49/3*e^2 - 37/3*e - 20/3, 4/3*e^4 + 7/3*e^3 - 53/3*e^2 - 70/3*e + 82/3, -3*e^4 + 30*e^2 - 11*e - 28, 7/6*e^4 - 17/6*e^3 - 46/3*e^2 + 64/3*e + 62/3, -2/3*e^4 - 5/3*e^3 + 25/3*e^2 + 50/3*e - 32/3, -8/3*e^4 - 11/3*e^3 + 88/3*e^2 + 62/3*e - 110/3, -2*e^4 - e^3 + 20*e^2 - 6*e - 4, -e^4 + 3*e^3 + 14*e^2 - 24*e - 34, e^3 + 2*e^2 - 5*e - 26, e^4 - 3*e^3 - 14*e^2 + 33*e + 16, -1/6*e^4 - 8/3*e^3 + 10/3*e^2 + 74/3*e - 38/3, 17/6*e^4 + 11/6*e^3 - 92/3*e^2 - 40/3*e + 94/3, 1/3*e^4 + 4/3*e^3 - 11/3*e^2 - 19/3*e + 58/3, -2*e^4 - e^3 + 22*e^2 - 9*e - 36, -e^4 + 13*e^2 - 10*e - 42, 5/2*e^4 + 2*e^3 - 26*e^2 + 14, 1/3*e^4 + 4/3*e^3 - 8/3*e^2 - 46/3*e - 50/3, 2/3*e^4 + 8/3*e^3 - 28/3*e^2 - 50/3*e + 92/3, -e^4 + 15*e^2 - 4*e - 18, 2/3*e^4 - 7/3*e^3 - 43/3*e^2 + 70/3*e + 86/3, -1/2*e^4 + 4*e^3 + 8*e^2 - 32*e + 2, 25/6*e^4 - 1/3*e^3 - 142/3*e^2 + 34/3*e + 140/3, e^4 - e^3 - 9*e^2 + 8*e - 6, 2*e^4 + 2*e^3 - 22*e^2 - 2*e + 38, -1/6*e^4 - 2/3*e^3 + 10/3*e^2 - 16/3*e - 32/3, -2/3*e^4 + 11/6*e^3 + 16/3*e^2 - 64/3*e + 22/3, -13/3*e^4 - 1/3*e^3 + 146/3*e^2 - 11/3*e - 154/3, 2*e^4 + 7/2*e^3 - 18*e^2 - 16*e + 16, -13/6*e^4 - 19/6*e^3 + 76/3*e^2 + 38/3*e - 140/3, 2/3*e^4 - 1/3*e^3 - 19/3*e^2 + 49/3*e + 56/3, -e^4 + 10*e^2 - 8*e, -2*e^2 + 6, -e^4 + e^3 + 14*e^2 - 24*e - 26, 1/3*e^4 + 7/3*e^3 - 20/3*e^2 - 46/3*e + 100/3, 1/3*e^4 + 1/3*e^3 - 32/3*e^2 - 10/3*e + 100/3, 2/3*e^4 - 1/3*e^3 - 13/3*e^2 + 19/3*e + 56/3, -2/3*e^4 + 1/3*e^3 + 25/3*e^2 + 11/3*e - 80/3, -4/3*e^4 - 7/3*e^3 + 41/3*e^2 + 52/3*e - 82/3, -7/3*e^4 - 10/3*e^3 + 74/3*e^2 + 52/3*e - 46/3, -2/3*e^4 - 8/3*e^3 + 16/3*e^2 + 74/3*e + 10/3, -5/3*e^4 - 2/3*e^3 + 46/3*e^2 - 28/3*e - 14/3, 11/3*e^4 + 14/3*e^3 - 118/3*e^2 - 98/3*e + 176/3, e^4 - e^3 - 14*e^2 + 7*e + 34, 3*e^4 + 3/2*e^3 - 34*e^2 - 4*e + 36, 3*e^3 + 2*e^2 - 13*e - 4, 5/3*e^4 - 4/3*e^3 - 61/3*e^2 + 49/3*e + 116/3, -5/2*e^4 - 3/2*e^3 + 30*e^2 - 30, -7/6*e^4 - 2/3*e^3 + 34/3*e^2 - 10/3*e + 40/3, 11/6*e^4 - 5/3*e^3 - 62/3*e^2 + 98/3*e + 46/3, 5/3*e^4 + 2/3*e^3 - 49/3*e^2 - 8/3*e + 14/3, 5/6*e^4 + 5/6*e^3 - 20/3*e^2 + 14/3*e - 26/3, -13/6*e^4 - 2/3*e^3 + 70/3*e^2 + 8/3*e - 2/3, 2/3*e^4 + 5/3*e^3 - 31/3*e^2 - 38/3*e + 140/3, -11/6*e^4 - 17/6*e^3 + 56/3*e^2 + 22/3*e - 94/3, -7/3*e^4 - 10/3*e^3 + 62/3*e^2 + 49/3*e - 10/3, -e^4 - 5/2*e^3 + 12*e^2 + 16*e + 2, e^3 + e^2 - 10*e - 22, 1/2*e^4 - 7/2*e^3 - 10*e^2 + 24*e + 34, -1/2*e^4 + 4*e^3 + 14*e^2 - 36*e - 48, 8/3*e^4 + 2/3*e^3 - 70/3*e^2 + 22/3*e - 22/3, -13/3*e^4 + 2/3*e^3 + 152/3*e^2 - 92/3*e - 160/3, -8/3*e^4 - 5/3*e^3 + 79/3*e^2 + 17/3*e - 140/3, 1/3*e^4 - 5/3*e^3 - 17/3*e^2 + 44/3*e + 10/3, -4/3*e^4 - 23/6*e^3 + 32/3*e^2 + 64/3*e - 22/3, -25/6*e^4 - 13/6*e^3 + 124/3*e^2 + 8/3*e - 176/3, -2/3*e^4 + 4/3*e^3 + 10/3*e^2 + 5/3*e + 130/3] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([23, 23, w^2 - 3])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]