/* 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, -1, -6, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([7, 7, -w^2 + 2]) primes_array = [ [5, 5, w],\ [5, 5, -w^2 + w + 2],\ [7, 7, -w^2 + 2],\ [13, 13, -w^2 + 3],\ [13, 13, w^2 - w - 4],\ [16, 2, 2],\ [19, 19, -w^2 + w + 1],\ [23, 23, -w^3 + 4*w + 2],\ [25, 5, -w^3 + w^2 + 3*w - 1],\ [29, 29, w^3 - w^2 - 4*w + 1],\ [29, 29, -w + 3],\ [31, 31, -w^3 + w^2 + 5*w - 2],\ [37, 37, w^3 - 4*w - 1],\ [37, 37, w^3 - 3*w + 1],\ [53, 53, -w^3 + 2*w^2 + 4*w - 6],\ [59, 59, w^2 + w - 4],\ [61, 61, -2*w^2 + w + 8],\ [73, 73, -w^3 + 5*w - 1],\ [79, 79, 2*w^2 + w - 6],\ [79, 79, 2*w^3 - w^2 - 9*w + 3],\ [81, 3, -3],\ [83, 83, -w^3 + w^2 + 3*w - 4],\ [97, 97, -w^3 + 6*w + 2],\ [97, 97, -2*w^3 + w^2 + 10*w - 1],\ [97, 97, 3*w^3 - 4*w^2 - 15*w + 13],\ [97, 97, w^3 - w^2 - 6*w + 2],\ [103, 103, -2*w^3 + 3*w^2 + 11*w - 9],\ [109, 109, w^2 + 3*w - 3],\ [137, 137, -w^2 + 7],\ [149, 149, w^3 - 2*w^2 - 5*w + 4],\ [149, 149, w^3 + 2*w^2 - 6*w - 7],\ [151, 151, w^3 + w^2 - 5*w - 3],\ [163, 163, w^3 - 3*w - 4],\ [163, 163, -w^3 + 6*w - 3],\ [167, 167, w^3 - 2*w^2 - 6*w + 4],\ [169, 13, w^2 + 2*w - 4],\ [191, 191, w^3 - w^2 - 5*w - 2],\ [191, 191, 2*w^3 - 9*w - 2],\ [197, 197, w^3 - 4*w + 4],\ [211, 211, -w^3 + 2*w^2 + 6*w - 8],\ [223, 223, -w^3 + 3*w^2 + 4*w - 6],\ [229, 229, w^3 - w^2 - 5*w - 1],\ [251, 251, 2*w^3 - 2*w^2 - 7*w + 3],\ [257, 257, 3*w^2 - 14],\ [263, 263, w^2 - 3*w - 1],\ [263, 263, w^2 - 2*w - 7],\ [269, 269, -w^3 + w^2 + 2*w - 4],\ [269, 269, 2*w^3 - 11*w - 1],\ [271, 271, -w^3 + w^2 + 4*w - 7],\ [277, 277, w^3 - w^2 - 3*w + 8],\ [311, 311, w^3 + 2*w^2 - 7*w - 11],\ [311, 311, -3*w^2 - 2*w + 8],\ [317, 317, -2*w^3 + 2*w^2 + 7*w - 4],\ [331, 331, w^2 - w - 8],\ [331, 331, 3*w^3 - 2*w^2 - 12*w + 9],\ [337, 337, -2*w^3 + 2*w^2 + 11*w - 8],\ [343, 7, w^3 - 3*w^2 - 4*w + 11],\ [347, 347, -2*w^3 + 2*w^2 + 9*w - 4],\ [349, 349, -w^3 + 2*w^2 + 2*w - 6],\ [353, 353, 2*w^2 - w - 12],\ [353, 353, 2*w^3 - w^2 - 9*w - 1],\ [367, 367, -2*w^3 + w^2 + 10*w - 6],\ [367, 367, 2*w^3 - 11*w - 3],\ [379, 379, -3*w - 1],\ [379, 379, w^3 - 2*w - 3],\ [383, 383, -w^3 + 2*w^2 + 4*w - 11],\ [383, 383, -w^3 - w^2 + 7*w + 4],\ [389, 389, 2*w^3 - w^2 - 11*w + 4],\ [401, 401, w^3 + 2*w^2 - 7*w - 6],\ [401, 401, 2*w^3 - 11*w - 6],\ [419, 419, -2*w^3 + w^2 + 8*w - 1],\ [421, 421, -w^3 + 3*w^2 + 3*w - 12],\ [421, 421, -w^3 - 2*w^2 + 2*w + 6],\ [433, 433, w^3 - 2*w^2 - 6*w + 11],\ [433, 433, w^2 - 8],\ [439, 439, -w^3 + 3*w^2 + 3*w - 7],\ [443, 443, w^3 - w^2 - 2*w - 4],\ [443, 443, 2*w^3 + w^2 - 7*w - 2],\ [443, 443, 2*w^3 - w^2 - 8*w + 2],\ [443, 443, -w^2 - 2],\ [467, 467, 3*w^3 - 5*w^2 - 15*w + 19],\ [467, 467, 2*w^3 - 3*w^2 - 8*w + 13],\ [467, 467, -w^3 + 2*w^2 + 3*w - 9],\ [467, 467, -w^3 + 3*w^2 + 3*w - 8],\ [479, 479, -w^3 - 3*w^2 + 5*w + 13],\ [479, 479, -w^3 + w^2 + w - 4],\ [487, 487, 3*w^3 - w^2 - 13*w - 2],\ [491, 491, w^2 + w - 9],\ [499, 499, -3*w^3 + 3*w^2 + 13*w - 7],\ [503, 503, 2*w^2 + w - 9],\ [509, 509, -2*w^2 - 3*w + 6],\ [509, 509, -w^3 + w^2 - 4],\ [521, 521, -3*w^2 + w + 12],\ [523, 523, 2*w^3 - 8*w - 3],\ [529, 23, 2*w^3 - w^2 - 6*w - 1],\ [541, 541, -w^3 + w^2 + 7*w - 4],\ [547, 547, 2*w^3 - 3*w^2 - 9*w + 7],\ [557, 557, 3*w^2 - 2*w - 9],\ [557, 557, w^3 - 6*w - 7],\ [563, 563, -2*w^3 + w^2 + 6*w - 6],\ [577, 577, -w^3 + 4*w^2 + 2*w - 13],\ [587, 587, w^2 - 2*w + 3],\ [587, 587, w^2 + 2*w - 6],\ [593, 593, -2*w^3 + 2*w^2 + 11*w - 3],\ [599, 599, w^3 + 3*w^2 - 6*w - 12],\ [601, 601, -2*w^3 + w^2 + 9*w + 2],\ [607, 607, -w^3 + 3*w^2 + 4*w - 9],\ [617, 617, -2*w^3 + w^2 + 7*w - 3],\ [619, 619, -2*w^3 + 4*w^2 + 9*w - 11],\ [631, 631, 3*w^2 - w - 7],\ [631, 631, -w^3 - w^2 + 6*w + 1],\ [641, 641, -w^3 + 2*w^2 + 5*w - 1],\ [641, 641, -w^3 + 2*w^2 + 2*w - 7],\ [643, 643, -3*w^3 + w^2 + 14*w - 4],\ [647, 647, -3*w^3 + 5*w^2 + 14*w - 18],\ [647, 647, -w^3 + 7*w + 2],\ [647, 647, -2*w^3 + 7*w - 1],\ [647, 647, 2*w^3 - w^2 - 10*w - 1],\ [653, 653, -2*w^3 - w^2 + 11*w + 4],\ [659, 659, -w^3 - w^2 + 5*w - 1],\ [673, 673, -w^3 + w^2 + 7*w - 2],\ [673, 673, -w^3 + w^2 + 2*w - 8],\ [683, 683, 2*w^3 - 11*w - 8],\ [683, 683, 2*w^3 - 7*w - 3],\ [701, 701, 2*w^3 - 4*w^2 - 10*w + 17],\ [727, 727, -w^3 + w^2 + 4*w - 8],\ [727, 727, 3*w^2 - 11],\ [739, 739, 2*w^3 + 3*w^2 - 8*w - 13],\ [743, 743, 2*w^2 + w - 12],\ [743, 743, 3*w^2 - 2*w - 14],\ [751, 751, w^3 + w^2 - 3*w - 7],\ [757, 757, -2*w^3 + 10*w - 1],\ [761, 761, -2*w^3 + w^2 + 7*w - 1],\ [761, 761, 3*w^2 + w - 7],\ [773, 773, -2*w^3 + 2*w^2 + 8*w - 1],\ [797, 797, -2*w^3 + w^2 + 9*w + 3],\ [797, 797, 3*w^3 - 14*w - 8],\ [809, 809, 5*w^3 - 5*w^2 - 24*w + 16],\ [821, 821, -4*w^3 + 3*w^2 + 18*w - 9],\ [821, 821, w^3 - w - 3],\ [823, 823, 3*w^2 + w - 8],\ [829, 829, -2*w^3 + 3*w^2 + 10*w - 7],\ [829, 829, 3*w^2 - 2*w - 16],\ [841, 29, 2*w^2 + w - 11],\ [853, 853, 3*w^3 - 2*w^2 - 14*w + 3],\ [853, 853, 3*w^3 - 5*w^2 - 14*w + 17],\ [857, 857, 3*w^2 - w - 8],\ [857, 857, 3*w^3 - w^2 - 13*w + 3],\ [859, 859, w^3 - 3*w^2 - 6*w + 12],\ [877, 877, w^3 + 2*w^2 - 5*w - 4],\ [877, 877, w^2 - w - 9],\ [881, 881, w^3 + 2*w^2 - 4*w - 12],\ [881, 881, w^2 - 4*w - 4],\ [883, 883, -3*w^3 + 2*w^2 + 11*w - 8],\ [887, 887, -2*w^3 + w^2 + 6*w - 1],\ [911, 911, 3*w^2 - w - 17],\ [911, 911, 2*w^3 - 2*w^2 - 12*w + 3],\ [919, 919, 5*w^3 - 6*w^2 - 22*w + 24],\ [919, 919, -2*w^3 + w^2 + 6*w - 7],\ [929, 929, 2*w^3 - 7*w - 1],\ [937, 937, -w^3 - w^2 + 8*w - 2],\ [947, 947, 3*w^3 + w^2 - 15*w - 6],\ [947, 947, -2*w^3 + 9*w + 9],\ [953, 953, w^3 - 2*w^2 - 6*w + 1],\ [977, 977, w^3 - w^2 - 3*w + 9],\ [977, 977, w^3 + w^2 - 3*w - 8],\ [983, 983, w^3 - 5*w - 7],\ [991, 991, -w^3 + 2*w^2 + 2*w - 8],\ [991, 991, w^2 + 4*w - 1],\ [997, 997, 2*w^3 + w^2 - 12*w - 8]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^5 - 4*x^4 - 16*x^3 + 56*x^2 + 72*x - 168 K. = NumberField(heckePol) hecke_eigenvalues_array = [1/2*e^3 - 3/2*e^2 - 4*e + 10, e, -1, 1/2*e^3 - 3/2*e^2 - 4*e + 10, 1/2*e^3 - 2*e^2 - 4*e + 14, -1/4*e^4 + 1/2*e^3 + 3*e^2 - 2*e - 5, 1/2*e^3 - e^2 - 4*e + 4, 1/2*e^3 - 3/2*e^2 - 5*e + 6, -1/2*e^3 + 1/2*e^2 + 6*e - 2, -1/2*e^4 + 2*e^3 + 3*e^2 - 14*e + 6, -e^3 + 2*e^2 + 9*e - 8, 1/2*e^3 - 3/2*e^2 - 5*e + 14, -1/4*e^4 + 3/2*e^3 - 12*e + 14, -1/2*e^4 + e^3 + 6*e^2 - 6*e - 8, 3/2*e^3 - 9/2*e^2 - 14*e + 32, -1/2*e^4 + 3/2*e^3 + 9/2*e^2 - 9*e - 2, 1/2*e^4 - 5/2*e^3 - 5/2*e^2 + 19*e - 6, -1/2*e^3 + 5/2*e^2 + 3*e - 16, -1/4*e^4 + e^3 + e^2 - 4*e + 4, e^2 + e - 12, 1/2*e^3 - 1/2*e^2 - 4*e - 4, -1/4*e^4 + 1/2*e^3 + 4*e^2 - 2*e - 12, -e^3 + 3*e^2 + 6*e - 16, e^3 - 5*e^2 - 6*e + 34, 2*e^3 - 4*e^2 - 19*e + 20, -1/2*e^4 + 3*e^3 - 23*e + 18, e^2 - 2*e - 8, 1/2*e^3 - 3/2*e^2 - 3*e + 10, -1/2*e^4 + 2*e^3 + 2*e^2 - 11*e + 18, -1/2*e^4 - e^3 + 10*e^2 + 14*e - 26, 1/2*e^3 - 3*e^2 + 22, 1/2*e^3 - e^2 - 2*e + 8, e^4 - 3/2*e^3 - 14*e^2 + 10*e + 28, 1/4*e^4 - 3*e^3 + 2*e^2 + 28*e - 20, 1/4*e^4 - 2*e^3 + 16*e - 20, 1/4*e^4 - 1/2*e^3 - 3*e^2 + 4*e - 6, -1/2*e^3 + e^2 + 2*e + 4, e^4 - 5/2*e^3 - 11*e^2 + 14*e + 12, -1/2*e^4 + 2*e^3 + 3*e^2 - 14*e + 12, -5/2*e^3 + 7*e^2 + 24*e - 52, 1/2*e^4 - 3/2*e^3 - 6*e^2 + 12*e + 20, -1/2*e^4 + 3/2*e^3 + 4*e^2 - 10*e + 6, -3/2*e^3 + 7/2*e^2 + 14*e - 26, 3/2*e^3 - 4*e^2 - 14*e + 18, -1/2*e^4 + 3/2*e^3 + 15/2*e^2 - 15*e - 18, -1/2*e^4 + 4*e^3 - 4*e^2 - 32*e + 60, -e^4 + 7/2*e^3 + 15/2*e^2 - 23*e - 2, -e^3 + e^2 + 10*e, -1/2*e^4 + 3/2*e^3 + 4*e^2 - 8*e - 4, 1/2*e^4 - 3/2*e^3 - e^2 + 8*e - 30, 1/2*e^4 - 3/2*e^3 - 7*e^2 + 16*e + 20, 1/2*e^4 + 1/2*e^3 - 23/2*e^2 - 3*e + 50, 1/2*e^3 - 7/2*e^2 - 4*e + 24, e^4 - 3/2*e^3 - 27/2*e^2 + 7*e + 16, 1/2*e^4 - 2*e^3 - 6*e^2 + 17*e + 12, 3/2*e^4 - 11/2*e^3 - 9*e^2 + 36*e - 26, -1/2*e^4 + 9/2*e^3 - 7/2*e^2 - 38*e + 44, 1/2*e^4 - 9/2*e^3 + 7/2*e^2 + 40*e - 42, -3*e^2 + 5*e + 28, 1/2*e^4 - 3*e^3 + e^2 + 23*e - 26, -3/2*e^3 + 4*e^2 + 8*e - 10, -5/2*e^3 + 7*e^2 + 22*e - 52, 7/2*e^3 - 10*e^2 - 32*e + 72, e^4 - 7/2*e^3 - 9*e^2 + 22*e, -e^4 - 1/2*e^3 + 37/2*e^2 + 14*e - 64, -3/2*e^4 + 6*e^3 + 10*e^2 - 41*e + 14, -e^4 + 13/2*e^3 - 1/2*e^2 - 48*e + 48, e^4 - 11/2*e^3 - 5*e^2 + 40*e - 14, 3*e^3 - 8*e^2 - 31*e + 60, 1/2*e^4 + e^3 - 11*e^2 - 8*e + 36, 3/2*e^4 - 7*e^3 - 3*e^2 + 44*e - 76, -1/2*e^4 + 3/2*e^3 + 11/2*e^2 - 11*e + 14, -3/4*e^4 + 6*e^3 - 2*e^2 - 50*e + 34, 1/2*e^4 - 13/2*e^3 + 17/2*e^2 + 57*e - 66, -1/2*e^4 + 15/2*e^3 - 21/2*e^2 - 63*e + 82, -1/2*e^4 + 7*e^2 + 9*e - 18, -e^4 + 4*e^3 + 5*e^2 - 29*e + 34, -e^4 + 9/2*e^3 + 9/2*e^2 - 34*e + 16, 2*e^4 - 9/2*e^3 - 23*e^2 + 26*e + 44, -5/4*e^4 + 7/2*e^3 + 15*e^2 - 22*e - 48, -1/2*e^4 + 1/2*e^3 + 4*e^2 + 6*e + 8, -3/4*e^4 - 3/2*e^3 + 17*e^2 + 16*e - 48, 2*e^3 - 26*e - 6, -1/4*e^4 + 7/2*e^3 - 5*e^2 - 28*e + 40, -1/4*e^4 - 3*e^3 + 14*e^2 + 26*e - 68, -2*e^4 + 3*e^3 + 28*e^2 - 16*e - 64, -e^4 + 3/2*e^3 + 11*e^2 - 16, 3*e^3 - 8*e^2 - 26*e + 42, -1/2*e^3 + 17/2*e^2 - 4*e - 54, -5/4*e^4 + 7/2*e^3 + 13*e^2 - 20*e - 28, e^4 - 13/2*e^3 - e^2 + 46*e - 42, -1/4*e^4 + 1/2*e^3 + 7*e^2 - 4*e - 34, 1/2*e^4 + 5/2*e^3 - 17*e^2 - 26*e + 74, 1/2*e^4 - 7*e^3 + 12*e^2 + 58*e - 88, 3*e^3 - 14*e^2 - 22*e + 82, 1/2*e^4 + 2*e^3 - 16*e^2 - 22*e + 80, 3/2*e^4 - 3/2*e^3 - 43/2*e^2 - 6*e + 68, -1/2*e^3 + 5/2*e^2 + 3*e - 24, 1/4*e^4 - 9/2*e^3 + 11*e^2 + 36*e - 86, 6*e - 8, 3/4*e^4 - 14*e^2 + 30, -1/4*e^4 - 3/2*e^3 + 10*e^2 + 12*e - 52, -2*e^4 + 7*e^3 + 14*e^2 - 44*e + 30, -e^4 + 22*e^2 - 90, 1/2*e^4 - 5/2*e^3 + 3/2*e^2 + 6*e - 46, e^4 - 7/2*e^3 - 5/2*e^2 + 21*e - 50, -3/2*e^4 + 9/2*e^3 + 33/2*e^2 - 30*e - 44, -13/2*e^3 + 19*e^2 + 54*e - 102, 9/2*e^3 - 25/2*e^2 - 39*e + 92, -1/4*e^4 + 11/2*e^3 - 12*e^2 - 50*e + 80, 3*e^3 - 11*e^2 - 21*e + 82, -4*e^3 + 9*e^2 + 40*e - 54, 1/4*e^4 - 3*e^3 + 32*e + 6, 3/2*e^4 - 10*e^3 + 6*e^2 + 75*e - 120, -1/2*e^4 - 1/2*e^3 + 13/2*e^2 + 14*e - 18, -3/2*e^4 + 4*e^3 + 10*e^2 - 21*e + 38, 5/4*e^4 - 13/2*e^3 - 5*e^2 + 44*e - 24, 2*e^3 - 6*e^2 - 17*e + 62, 3/2*e^4 - 5*e^3 - 10*e^2 + 32*e - 34, -e^4 - e^3 + 22*e^2 + 13*e - 82, 7/2*e^3 - 10*e^2 - 28*e + 50, 1/2*e^4 + 3*e^3 - 17*e^2 - 36*e + 94, -e^4 + 7/2*e^3 + 5*e^2 - 26*e + 40, 5/4*e^4 - 11/2*e^3 - e^2 + 36*e - 72, -e^4 + 3*e^3 + 11*e^2 - 22*e - 6, 3/2*e^4 - 10*e^3 + 77*e - 64, 2*e^4 - 11/2*e^3 - 19*e^2 + 32*e + 12, 1/2*e^4 - e^3 - 10*e^2 + 14*e + 52, e^4 - 4*e^3 - 10*e^2 + 24*e + 32, -e^4 - e^3 + 17*e^2 + 19*e - 18, -3/4*e^4 + 5/2*e^3 + 9*e^2 - 24*e - 28, -e^4 + 7*e^3 - 59*e + 34, e^4 - 15/2*e^3 + 11/2*e^2 + 53*e - 92, -5/2*e^3 + 7/2*e^2 + 23*e - 12, e^4 - 11/2*e^3 - 11/2*e^2 + 44*e - 16, -2*e^4 + 3*e^3 + 29*e^2 - 20*e - 58, 2*e^4 - 13/2*e^3 - 35/2*e^2 + 37*e + 10, e^4 - 19*e^2 - 4*e + 70, -1/2*e^4 - 7/2*e^3 + 19*e^2 + 28*e - 74, e^4 - 9*e^3 + 6*e^2 + 74*e - 60, -1/2*e^4 + 6*e^3 - 6*e^2 - 52*e + 56, 5*e^3 - 11*e^2 - 54*e + 46, -1/2*e^4 - 4*e^3 + 20*e^2 + 44*e - 102, -1/2*e^4 + 1/2*e^3 + 17/2*e^2 + 6*e - 16, 3/2*e^4 - 1/2*e^3 - 53/2*e^2 + e + 84, -1/2*e^4 - 3*e^3 + 15*e^2 + 26*e - 46, -1/4*e^4 - 4*e^3 + 19*e^2 + 42*e - 122, 1/2*e^4 + 2*e^3 - 20*e^2 - 16*e + 90, -4*e^3 + 7*e^2 + 40*e - 32, -e^4 + 2*e^3 + 8*e^2 - 8*e + 2, e^4 - 7/2*e^3 - 6*e^2 + 28*e - 14, 2*e^4 - 7*e^3 - 10*e^2 + 40*e - 78, 5/2*e^4 - 9*e^3 - 19*e^2 + 63*e - 12, 2*e^4 - 13/2*e^3 - 23*e^2 + 46*e + 44, 1/2*e^4 - 11/2*e^3 + 12*e^2 + 34*e - 108, e^4 + 2*e^3 - 26*e^2 - 18*e + 96, e^4 - 15/2*e^3 + e^2 + 70*e - 48, e^4 - 11/2*e^3 - 3*e^2 + 46*e - 8, 3/2*e^4 - 2*e^3 - 15*e^2 - 4*e + 16, 1/2*e^4 + 5/2*e^3 - 33/2*e^2 - 24*e + 50, -3/2*e^4 - 2*e^3 + 34*e^2 + 24*e - 130, -1/2*e^4 + 19/2*e^3 - 23*e^2 - 78*e + 168, -1/2*e^4 + 4*e^3 + 2*e^2 - 44*e + 4, -e^4 + 6*e^3 - 48*e + 60, -1/2*e^4 + 2*e^3 + 8*e^2 - 28*e - 24, 7/2*e^3 - 27/2*e^2 - 21*e + 78, 5/2*e^4 - 17/2*e^3 - 43/2*e^2 + 62*e + 28, -e^4 + 3*e^3 + 6*e^2 - 21*e + 32, 1/4*e^4 - 7/2*e^3 + 7*e^2 + 20*e - 56, -5/4*e^4 + 10*e^3 - 2*e^2 - 84*e + 58] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([7, 7, -w^2 + 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]