/* 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, -6, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([13, 13, -w^2 + w + 4]) primes_array = [ [3, 3, w + 1],\ [7, 7, w - 1],\ [8, 2, 2],\ [9, 3, -w^2 + 2*w + 4],\ [11, 11, -w^2 + 2*w + 2],\ [13, 13, -w^2 + w + 4],\ [19, 19, w + 3],\ [19, 19, -w^2 + 2*w + 5],\ [19, 19, -w^2 + 3*w + 2],\ [23, 23, w^2 - w - 3],\ [23, 23, -w^2 + 2],\ [23, 23, -w + 4],\ [31, 31, w^2 - 5],\ [43, 43, w^2 - 3*w - 3],\ [49, 7, w^2 - 6],\ [53, 53, 2*w - 5],\ [61, 61, 2*w^2 - 2*w - 9],\ [71, 71, 2*w - 3],\ [73, 73, 2*w^2 - 5*w - 5],\ [83, 83, w^2 - w - 9],\ [97, 97, 2*w^2 - 3*w - 7],\ [103, 103, w^2 - 3*w - 6],\ [109, 109, w^2 - 4*w - 3],\ [121, 11, 2*w^2 - w - 7],\ [125, 5, -5],\ [127, 127, w^2 + w - 5],\ [131, 131, w^2 - 11],\ [137, 137, 3*w^2 - 7*w - 8],\ [137, 137, 2*w^2 - 5*w - 6],\ [137, 137, -w^2 + w - 2],\ [139, 139, 3*w^2 - 2*w - 16],\ [139, 139, 3*w - 2],\ [139, 139, -w^2 + 3*w - 3],\ [151, 151, w^2 + w - 11],\ [157, 157, 2*w^2 - 3*w - 3],\ [163, 163, w^2 - w - 10],\ [169, 13, 2*w^2 - 3*w - 4],\ [173, 173, -3*w^2 + 6*w + 8],\ [191, 191, -4*w^2 + 7*w + 15],\ [193, 193, 2*w^2 - 3],\ [197, 197, w^2 - 3*w - 11],\ [199, 199, 3*w^2 - 5*w - 16],\ [211, 211, w^2 - 4*w - 9],\ [223, 223, 2*w^2 - w - 4],\ [227, 227, 3*w - 4],\ [229, 229, w^2 + w - 8],\ [241, 241, w^2 - 4*w - 6],\ [251, 251, w - 7],\ [257, 257, w^2 - 4*w - 7],\ [263, 263, 3*w^2 - 5*w - 10],\ [263, 263, 2*w^2 - 5*w - 15],\ [263, 263, -w^2 + 6*w - 1],\ [269, 269, -w^2 - w + 14],\ [277, 277, 2*w^2 - 5*w - 8],\ [277, 277, w^2 + 3*w - 3],\ [277, 277, 3*w^2 - 4*w - 12],\ [281, 281, -2*w - 7],\ [293, 293, 3*w^2 - 2*w - 12],\ [307, 307, 2*w^2 + w - 8],\ [313, 313, w^2 + 4*w - 2],\ [317, 317, 3*w^2 - 6*w - 5],\ [331, 331, 4*w^2 - 4*w - 19],\ [337, 337, -5*w^2 + 8*w + 21],\ [347, 347, 3*w^2 - 6*w - 13],\ [347, 347, 3*w^2 - 5*w - 22],\ [347, 347, 3*w^2 - 5*w - 9],\ [349, 349, w^2 - 5*w - 11],\ [353, 353, -w^2 + 6*w - 7],\ [359, 359, -4*w^2 + 8*w + 11],\ [367, 367, 3*w^2 - 4*w - 11],\ [373, 373, w^2 + 2*w - 6],\ [379, 379, 5*w^2 - 9*w - 18],\ [409, 409, -w^2 - 4],\ [439, 439, 3*w^2 - 6*w - 14],\ [443, 443, -w^2 - 2*w + 13],\ [443, 443, 2*w^2 - 15],\ [443, 443, 3*w^2 - 13],\ [449, 449, w^2 + 2*w - 7],\ [449, 449, 2*w^2 - 6*w - 7],\ [449, 449, 3*w^2 - 3*w - 11],\ [461, 461, -w^2 - 2*w - 5],\ [467, 467, 3*w^2 - 5*w - 7],\ [479, 479, -w^2 + 6*w - 4],\ [491, 491, 3*w^2 - 5*w - 6],\ [499, 499, w^2 - 5*w - 8],\ [509, 509, 3*w^2 - 6*w - 19],\ [521, 521, 3*w^2 - 2*w - 10],\ [523, 523, 4*w - 9],\ [541, 541, 3*w^2 - 3*w - 10],\ [547, 547, 2*w^2 + w - 20],\ [547, 547, w^2 - 6*w - 5],\ [547, 547, 6*w^2 - 11*w - 24],\ [557, 557, -5*w^2 + 11*w + 11],\ [563, 563, 3*w^2 - 6*w - 16],\ [569, 569, 4*w^2 - 5*w - 17],\ [587, 587, 5*w^2 - 8*w - 20],\ [593, 593, 5*w - 3],\ [593, 593, 3*w^2 - w - 17],\ [593, 593, w - 9],\ [613, 613, 2*w^2 - w - 19],\ [613, 613, 3*w^2 - 4*w - 5],\ [613, 613, 3*w^2 - w - 8],\ [617, 617, 3*w^2 - 2*w - 9],\ [617, 617, -w^2 + w - 5],\ [617, 617, 4*w^2 - 6*w - 15],\ [619, 619, -w^2 + 3*w - 6],\ [631, 631, 4*w^2 - 7*w - 21],\ [641, 641, -w^2 - 3*w + 20],\ [643, 643, 3*w^2 - 14],\ [643, 643, w^2 - w - 13],\ [643, 643, 2*w^2 - 9*w + 2],\ [647, 647, 2*w^2 - 6*w - 9],\ [653, 653, 3*w^2 - 5],\ [659, 659, 6*w^2 - 11*w - 21],\ [661, 661, w^2 - 6*w - 6],\ [661, 661, w^2 + 3*w - 6],\ [661, 661, 5*w^2 - 7*w - 22],\ [673, 673, w^2 - 6*w - 12],\ [677, 677, 4*w^2 - 7*w - 22],\ [677, 677, 3*w^2 - 7*w - 12],\ [677, 677, -2*w^2 - 3],\ [683, 683, -2*w - 9],\ [683, 683, 4*w^2 - 7*w - 12],\ [683, 683, 3*w^2 - 3*w - 8],\ [691, 691, 2*w^2 + w - 11],\ [701, 701, 3*w^2 - 3*w - 5],\ [709, 709, -w^2 - 2*w - 6],\ [727, 727, 3*w^2 - w - 6],\ [727, 727, 3*w^2 - w - 27],\ [727, 727, 3*w^2 - w - 5],\ [733, 733, -2*w^2 + 9*w - 5],\ [739, 739, 3*w^2 - 4*w - 23],\ [743, 743, 2*w^2 - 7*w - 7],\ [743, 743, 3*w^2 - w - 20],\ [743, 743, 3*w^2 - 2*w - 7],\ [757, 757, -w - 9],\ [757, 757, 4*w^2 - 2*w - 29],\ [757, 757, 2*w^2 - 3*w - 18],\ [761, 761, 3*w^2 - 2*w - 6],\ [761, 761, 3*w^2 - 2*w - 25],\ [773, 773, -5*w - 12],\ [787, 787, -3*w - 10],\ [809, 809, 4*w^2 - w - 19],\ [821, 821, 5*w^2 - 10*w - 19],\ [823, 823, -4*w - 11],\ [827, 827, -7*w - 5],\ [827, 827, 2*w^2 - 6*w - 13],\ [827, 827, -w^2 - 2*w + 18],\ [839, 839, 6*w^2 - 9*w - 26],\ [839, 839, 5*w^2 - 12*w - 13],\ [839, 839, w - 10],\ [857, 857, -w^2 + 4*w - 8],\ [859, 859, 3*w^2 - 8*w - 10],\ [859, 859, 4*w^2 - 9*w - 14],\ [859, 859, 3*w^2 - 26],\ [863, 863, 2*w^2 + 3*w - 7],\ [877, 877, -5*w^2 + 11*w + 9],\ [877, 877, -5*w^2 + 10*w + 13],\ [877, 877, -w^2 + w - 6],\ [881, 881, w^2 - 3*w - 14],\ [883, 883, -w^2 + 3*w - 7],\ [887, 887, -3*w^2 + 6*w - 2],\ [907, 907, w^2 + 3*w - 8],\ [911, 911, 4*w^2 - 6*w - 29],\ [937, 937, -6*w^2 + 11*w + 25],\ [953, 953, 6*w^2 - 6*w - 29],\ [961, 31, -w^2 + 7*w - 5],\ [967, 967, 2*w^2 + w - 14],\ [971, 971, 5*w^2 - 3*w - 27],\ [971, 971, 4*w - 15],\ [971, 971, w^2 - 2*w - 14],\ [977, 977, 3*w^2 - 5*w - 24],\ [983, 983, w^2 - 4*w - 15],\ [991, 991, 6*w^2 - 10*w - 23],\ [991, 991, 5*w^2 - 6*w - 22],\ [991, 991, 3*w^2 + w - 13]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^5 - 5*x^4 + 3*x^3 + 13*x^2 - 9*x - 8 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -e^3 + e^2 + 4*e, -e^4 + 3*e^3 + 2*e^2 - 7*e - 1, e^4 - 2*e^3 - 5*e^2 + 6*e + 8, -e^3 + 3*e^2 - 2, 1, e^4 - 3*e^3 + e^2 + e - 4, -e^4 + 2*e^3 + 5*e^2 - 7*e - 4, -2*e^3 + 4*e^2 + 4*e - 2, -2*e^4 + 7*e^3 + 3*e^2 - 17*e - 2, e^4 - 2*e^3 - 3*e^2 + 2*e + 6, 2*e^4 - 5*e^3 - 8*e^2 + 12*e + 12, -e^4 + 5*e^3 - 2*e^2 - 10*e, e^3 - 3*e^2 - e + 2, -2*e^4 + 5*e^3 + 10*e^2 - 15*e - 18, -e^4 + 3*e^3 - e + 2, 2*e^4 - 3*e^3 - 14*e^2 + 9*e + 20, 2*e^4 - 9*e^3 + 7*e^2 + 13*e - 8, -3*e^4 + 9*e^3 + 3*e^2 - 13*e, -4*e^4 + 9*e^3 + 13*e^2 - 16*e - 12, 5*e^4 - 13*e^3 - 15*e^2 + 31*e + 16, 3*e^4 - 12*e^3 + e^2 + 23*e + 6, -3*e^4 + 6*e^3 + 19*e^2 - 20*e - 30, -3*e^4 + 9*e^3 - e^2 - 10*e + 10, e^4 + 3*e^3 - 20*e^2 + 3*e + 26, -2*e^4 + 5*e^3 + 7*e^2 - 10*e - 12, 4*e^4 - 10*e^3 - 14*e^2 + 16*e + 28, -4*e^4 + 12*e^3 + e^2 - 14*e + 14, -2*e^2 + 3*e - 2, e^4 - 2*e^3 - 7*e^2 + 9*e + 6, -6*e^4 + 20*e^3 + 6*e^2 - 43*e - 8, 2*e^4 - 5*e^3 - 4*e^2 + 8*e, -e^4 + 5*e^3 - 3*e^2 - 14*e, 4*e^4 - 10*e^3 - 12*e^2 + 25*e, 2*e^3 - 13*e^2 + 10*e + 28, 3*e^4 - 6*e^3 - 9*e^2 + 12*e - 4, -e^3 - e^2 + 13*e - 2, -e^4 + 7*e^3 - 6*e^2 - 12*e - 4, -e^4 - 2*e^3 + 14*e^2 + 6*e - 12, 2*e^4 - 8*e^3 + 7*e^2 + 7*e - 6, 6*e^4 - 21*e^3 - 4*e^2 + 38*e - 2, -4*e^4 + 12*e^3 + 3*e^2 - 12*e - 4, 6*e^4 - 19*e^3 - 8*e^2 + 38*e + 18, 3*e^4 - 10*e^3 - 2*e^2 + 17*e + 10, -5*e^4 + 19*e^3 - 42*e, -3*e^4 + 10*e^3 - e^2 - 7*e + 2, e^4 - 8*e^2 - 8*e + 6, -7*e^4 + 27*e^3 - 3*e^2 - 50*e + 4, -10*e^4 + 33*e^3 + 13*e^2 - 66*e - 18, 2*e^4 - 12*e^3 + 14*e^2 + 12*e - 22, -e^4 + 4*e^3 + 7*e^2 - 28*e - 6, -4*e^4 + 11*e^3 + 14*e^2 - 32*e - 12, 4*e^4 - 16*e^3 + e^2 + 23*e + 2, -3*e^4 + 17*e^3 - 14*e^2 - 39*e + 18, -6*e^4 + 13*e^3 + 29*e^2 - 39*e - 26, -10*e^4 + 30*e^3 + 16*e^2 - 51*e - 24, -e^4 + 2*e^3 + 11*e^2 - 12*e - 38, 5*e^4 - 19*e^3 + 6*e^2 + 30*e - 16, -2*e^4 + 2*e^3 + 15*e^2 - 14*e - 12, 5*e^4 - 13*e^3 - 22*e^2 + 36*e + 34, 3*e^4 - 6*e^3 - 21*e^2 + 34*e + 24, -e^4 + 8*e^3 - 10*e^2 - 19*e + 12, 8*e^4 - 17*e^3 - 34*e^2 + 40*e + 42, -5*e^4 + 8*e^3 + 30*e^2 - 15*e - 44, -4*e^4 + 17*e^3 - 4*e^2 - 37*e - 4, 6*e^4 - 15*e^3 - 19*e^2 + 32*e + 38, 2*e^3 + 5*e^2 - 16*e - 14, 4*e^4 - 7*e^3 - 12*e^2 - 2*e + 18, 5*e^4 - 15*e^3 + 21*e - 24, 2*e^3 - 13*e^2 + 13*e + 26, -3*e^4 + 9*e^3 + 6, -16*e^4 + 50*e^3 + 23*e^2 - 99*e - 16, -2*e^4 + e^3 + 19*e^2 - 12*e - 18, 3*e^4 - 17*e^3 + 9*e^2 + 34*e - 10, 3*e^4 - 6*e^3 - 27*e^2 + 32*e + 44, -e^4 - e^3 + 11*e^2 + 9*e - 28, e^4 + e^3 - 25*e^2 + 20*e + 44, 3*e^4 - 5*e^3 - 22*e^2 + 26*e + 30, 3*e^4 - 4*e^3 - 18*e^2 + 11*e + 12, -2*e^4 + 3*e^3 + 10*e + 20, -11*e^4 + 34*e^3 + 17*e^2 - 65*e - 30, -9*e^4 + 22*e^3 + 41*e^2 - 70*e - 52, -5*e^4 + 23*e^3 - 8*e^2 - 45*e - 4, 9*e^4 - 25*e^3 - 28*e^2 + 56*e + 42, -3*e^4 + 9*e^3 + 2*e^2 - 21*e + 14, 9*e^4 - 19*e^3 - 39*e^2 + 33*e + 64, -16*e^4 + 41*e^3 + 52*e^2 - 92*e - 70, -2*e^4 - 5*e^3 + 24*e^2 + 12*e - 16, 8*e^4 - 27*e^3 + e^2 + 41*e - 8, -6*e^4 + 8*e^3 + 35*e^2 - 24*e - 12, 6*e^4 - 15*e^3 - 21*e^2 + 38*e + 16, -8*e^4 + 16*e^3 + 41*e^2 - 43*e - 62, 3*e^4 - 8*e^3 - 7*e^2 + 20*e - 6, 2*e^4 - 9*e^3 + 9*e^2 + 24*e - 32, -4*e^3 + 17*e^2 - e - 32, -4*e^4 + 16*e^3 - 14*e^2 - 11*e + 36, -e^4 + 3*e^3 - 9*e^2 + 4*e + 26, -e^4 + 5*e^3 - 16*e^2 + 5*e + 46, -e^4 - 2*e^3 + 27*e^2 - 17*e - 26, -8*e^4 + 16*e^3 + 43*e^2 - 34*e - 74, 3*e^4 - 7*e^3 - 22*e^2 + 29*e + 30, -e^4 + 5*e^3 - 12*e^2 - 2*e + 40, -8*e^4 + 34*e^3 - 5*e^2 - 63*e - 8, e^4 - 10*e^2 + 4*e + 6, 9*e^4 - 19*e^3 - 39*e^2 + 56*e + 30, e^4 + 3*e^3 - 11*e^2 - 10*e - 12, 2*e^4 - 6*e^3 + 3*e^2 + 5*e - 14, 2*e^4 - 12*e^3 + 2*e^2 + 38*e - 6, 7*e^4 - 31*e^3 + 24*e^2 + 42*e - 36, -4*e^4 + 21*e^3 - 20*e^2 - 28*e + 16, 3*e^4 - 5*e^3 - 15*e^2 + 2*e + 28, 3*e^4 - 23*e^3 + 32*e^2 + 30*e - 40, 5*e^4 - 18*e^3 - 13*e^2 + 46*e + 30, -7*e^4 + 24*e^3 + 12*e^2 - 52*e - 18, -7*e^4 + 28*e^3 - 5*e^2 - 43*e + 14, -3*e^4 - 3*e^3 + 27*e^2 + 25*e - 30, -3*e^4 + 5*e^3 + 11*e^2 - 21*e + 18, -e^4 + 4*e^3 + e^2 - 20*e + 8, 9*e^4 - 32*e^3 - e^2 + 52*e - 20, -2*e^3 + 5*e^2 + 6*e - 38, 7*e^4 - 25*e^3 - 2*e^2 + 54*e + 2, -8*e^4 + 31*e^3 - 2*e^2 - 58*e + 12, 7*e^4 - 22*e^3 - 13*e^2 + 40*e + 28, -5*e^4 + 4*e^3 + 43*e^2 - 28*e - 58, -5*e^4 + 8*e^3 + 31*e^2 - 35*e - 36, e^4 + 4*e^3 - 17*e^2 - 14*e - 2, 9*e^4 - 22*e^3 - 34*e^2 + 47*e + 70, -e^4 - 5*e^3 + 25*e^2 + 2*e - 14, 7*e^4 - 17*e^3 - 18*e^2 + 24*e + 16, -10*e^4 + 32*e^3 + 10*e^2 - 59*e + 6, -7*e^4 + 14*e^3 + 27*e^2 - 28*e - 30, 9*e^4 - 22*e^3 - 17*e^2 + 21*e + 12, 2*e^4 - 15*e^3 + 19*e^2 + 30*e - 12, -9*e^4 + 32*e^3 + 9*e^2 - 64*e - 28, -e^4 + 9*e^3 - 5*e^2 - 22*e - 8, -6*e^4 + 14*e^3 + 4*e^2 + 3*e + 22, 11*e^4 - 20*e^3 - 59*e^2 + 55*e + 86, 9*e^4 - 25*e^3 - 35*e^2 + 67*e + 66, -e^4 + 6*e^3 + 2*e^2 - 37*e + 10, 6*e^4 - 33*e^3 + 22*e^2 + 63*e - 26, -10*e^4 + 30*e^3 + 10*e^2 - 40*e - 6, -7*e^4 + 15*e^3 + 41*e^2 - 59*e - 40, -14*e^4 + 41*e^3 + 37*e^2 - 89*e - 66, -4*e^4 + 17*e^3 - e^2 - 49*e + 20, -11*e^4 + 30*e^3 + 31*e^2 - 67*e - 48, 8*e^4 - 21*e^3 - 16*e^2 + 21*e + 26, -7*e^4 + 9*e^3 + 31*e^2 + 5*e - 32, 3*e^4 - e^3 - 16*e^2 - 6*e + 4, 5*e^4 - 9*e^3 - 23*e^2 + 27*e + 6, 14*e^4 - 39*e^3 - 23*e^2 + 56*e + 26, -4*e^3 + 7*e^2 + 20*e - 36, 2*e^4 - 20*e^3 + 24*e^2 + 42*e - 6, 6*e^4 - 18*e^3 - 8*e^2 + 28*e + 2, 12*e^4 - 36*e^3 - 24*e^2 + 80*e + 16, -e^4 - 4*e^3 + 12*e^2 + 24*e - 36, e^4 - e^3 - 14*e^2 + 13*e + 24, 8*e^4 - 27*e^3 + 11*e^2 + 24*e - 44, -8*e^4 + 20*e^3 + 29*e^2 - 48*e - 56, -15*e^3 + 39*e^2 + 19*e - 42, 4*e^4 - 22*e^3 + 10*e^2 + 54*e - 10, 6*e^4 - 15*e^3 + e - 16, -5*e^4 + 17*e^3 - e^2 - 29*e + 32, -13*e^4 + 26*e^3 + 55*e^2 - 46*e - 84, 6*e^4 - 13*e^3 - 30*e^2 + 36*e + 24, -5*e^4 + 26*e^3 - 10*e^2 - 59*e + 10, -12*e^4 + 43*e^3 - 4*e^2 - 72*e - 4, -5*e^4 + 13*e^3 + 8*e^2 - 21*e + 6, 4*e^4 - 8*e^3 - 4*e^2 + 3*e - 32, 10*e^4 - 20*e^3 - 45*e^2 + 43*e + 60, 3*e^4 - 42*e^2 + 28*e + 64, 15*e^4 - 36*e^3 - 49*e^2 + 69*e + 84, -5*e^4 + 15*e^3 + 8*e^2 - 16*e - 6, -2*e^3 + 6*e^2 - 6*e, 3*e^4 - 12*e^3 + 4*e^2 + 21*e + 4, e^4 - 6*e^3 + 23*e^2 - 11*e - 54, -2*e^4 - 3*e^3 + 24*e^2 + 9*e - 24] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([13, 13, -w^2 + w + 4])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]