/* This code can be loaded, or copied and pasted, into Magma. It will load the data associated to the HMF, including the field, level, and Hecke and Atkin-Lehner eigenvalue data. At the *bottom* of the file, there is code to recreate the Hilbert modular form in Magma, by creating the HMF space and cutting out the corresponding Hecke irreducible subspace. From there, you can ask for more eigenvalues or modify as desired. It is commented out, as this computation may be lengthy. */ P := PolynomialRing(Rationals()); g := P![11, 6, -8, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [9, 3, 1/3*w^3 - 8/3*w - 2/3], [9, 3, -w + 1], [11, 11, w], [11, 11, -1/3*w^3 + 2/3*w + 5/3], [11, 11, 1/3*w^3 - 8/3*w + 1/3], [11, 11, -2/3*w^3 + 13/3*w + 4/3], [16, 2, 2], [25, 5, 2/3*w^3 - 10/3*w - 1/3], [29, 29, -w - 3], [29, 29, -1/3*w^3 + 8/3*w - 10/3], [31, 31, w^3 - 6*w + 1], [31, 31, w^3 - 6*w - 2], [41, 41, 2/3*w^3 + w^2 - 13/3*w - 10/3], [41, 41, 2/3*w^3 - 13/3*w + 5/3], [59, 59, 2/3*w^3 - 13/3*w + 8/3], [59, 59, w^3 + w^2 - 6*w - 4], [79, 79, 2/3*w^3 + w^2 - 10/3*w - 19/3], [79, 79, 1/3*w^3 + w^2 - 2/3*w - 17/3], [89, 89, 4/3*w^3 - 23/3*w - 5/3], [89, 89, 1/3*w^3 + 2*w^2 - 5/3*w - 32/3], [101, 101, 5/3*w^3 + w^2 - 31/3*w - 16/3], [101, 101, 2/3*w^3 - w^2 - 4/3*w + 8/3], [109, 109, -w^3 - 2*w^2 + 7*w + 7], [109, 109, -1/3*w^3 + 2*w^2 + 2/3*w - 16/3], [131, 131, w^3 - w^2 - 7*w + 5], [131, 131, -2/3*w^3 - w^2 + 4/3*w + 13/3], [139, 139, -2/3*w^3 - w^2 + 13/3*w + 4/3], [139, 139, w^2 - w - 7], [151, 151, -5/3*w^3 + 28/3*w - 5/3], [151, 151, -1/3*w^3 + w^2 + 2/3*w - 16/3], [151, 151, w^3 + w^2 - 6*w - 3], [151, 151, -w^3 + w^2 + 7*w - 6], [179, 179, 5/3*w^3 + w^2 - 28/3*w - 19/3], [179, 179, -1/3*w^3 + w^2 + 11/3*w - 4/3], [181, 181, -4/3*w^3 - 2*w^2 + 20/3*w + 29/3], [181, 181, 1/3*w^3 + w^2 - 2/3*w - 20/3], [181, 181, w^3 + w^2 - 5*w - 7], [181, 181, 5/3*w^3 + w^2 - 28/3*w - 16/3], [191, 191, 2*w - 1], [191, 191, 2/3*w^3 - 16/3*w - 1/3], [199, 199, 1/3*w^3 - 11/3*w + 4/3], [199, 199, 1/3*w^3 - 11/3*w - 2/3], [211, 211, -2/3*w^3 - w^2 + 19/3*w + 16/3], [211, 211, 2/3*w^3 + w^2 - 19/3*w - 7/3], [229, 229, 1/3*w^3 - 11/3*w + 1/3], [239, 239, -5/3*w^3 - w^2 + 25/3*w + 19/3], [239, 239, -4/3*w^3 + w^2 + 20/3*w - 7/3], [251, 251, -2/3*w^3 + w^2 + 16/3*w - 14/3], [251, 251, -1/3*w^3 - w^2 - 1/3*w + 14/3], [269, 269, 2/3*w^3 + w^2 - 7/3*w - 19/3], [269, 269, 1/3*w^3 + 2*w^2 - 2/3*w - 32/3], [281, 281, w^3 - w^2 - 5*w + 1], [281, 281, 4/3*w^3 + w^2 - 20/3*w - 23/3], [289, 17, 2/3*w^3 + w^2 - 16/3*w - 4/3], [289, 17, 1/3*w^3 + w^2 - 11/3*w - 20/3], [331, 331, 1/3*w^3 + 2*w^2 - 8/3*w - 23/3], [331, 331, 2/3*w^3 + 2*w^2 - 13/3*w - 28/3], [349, 349, 4/3*w^3 - 17/3*w + 1/3], [349, 349, -5/3*w^3 + 28/3*w - 2/3], [361, 19, -1/3*w^3 + 5/3*w + 14/3], [361, 19, 4/3*w^3 - 20/3*w - 5/3], [379, 379, 4/3*w^3 + 2*w^2 - 23/3*w - 41/3], [379, 379, -5/3*w^3 + 31/3*w - 5/3], [389, 389, 5/3*w^3 + w^2 - 25/3*w - 16/3], [389, 389, -4/3*w^3 + w^2 + 20/3*w - 10/3], [401, 401, -1/3*w^3 - 3*w^2 + 5/3*w + 26/3], [401, 401, 1/3*w^3 + w^2 - 11/3*w - 23/3], [401, 401, -2/3*w^3 - 3*w^2 + 10/3*w + 52/3], [401, 401, 2/3*w^3 + w^2 - 16/3*w - 1/3], [409, 409, -2*w^3 - w^2 + 11*w + 2], [409, 409, -5/3*w^3 - 2*w^2 + 28/3*w + 37/3], [409, 409, 1/3*w^3 - 11/3*w - 17/3], [409, 409, 5/3*w^3 + 3*w^2 - 25/3*w - 37/3], [419, 419, -1/3*w^3 + 3*w^2 - 1/3*w - 37/3], [419, 419, 2*w^3 + 3*w^2 - 12*w - 13], [421, 421, -1/3*w^3 + 3*w^2 + 2/3*w - 31/3], [421, 421, -5/3*w^3 - w^2 + 25/3*w + 13/3], [421, 421, -w^3 + 7*w - 4], [421, 421, -4/3*w^3 + w^2 + 20/3*w - 13/3], [439, 439, -4/3*w^3 + w^2 + 14/3*w - 7/3], [439, 439, 4/3*w^3 - 17/3*w - 5/3], [439, 439, -7/3*w^3 - w^2 + 41/3*w + 17/3], [439, 439, 5/3*w^3 - 28/3*w - 4/3], [449, 449, w^3 + 3*w^2 - 6*w - 17], [449, 449, 1/3*w^3 + 3*w^2 - 8/3*w - 26/3], [449, 449, -4/3*w^3 + 26/3*w + 17/3], [449, 449, 7/3*w^3 + w^2 - 38/3*w - 8/3], [461, 461, -5/3*w^3 + w^2 + 28/3*w - 14/3], [461, 461, -2/3*w^3 - 2*w^2 + 10/3*w + 43/3], [461, 461, -2*w^3 - w^2 + 12*w + 4], [461, 461, -w^3 + 8*w - 5], [479, 479, w^3 + w^2 - 7*w - 2], [479, 479, -4/3*w^3 - w^2 + 23/3*w + 5/3], [499, 499, -4/3*w^3 - 2*w^2 + 23/3*w + 17/3], [499, 499, -4/3*w^3 - w^2 + 29/3*w + 8/3], [521, 521, 2*w^3 - 11*w + 2], [521, 521, 4/3*w^3 - w^2 - 14/3*w + 1/3], [569, 569, w^3 + 3*w^2 - 6*w - 9], [569, 569, 4/3*w^3 + w^2 - 26/3*w - 32/3], [569, 569, -w^3 + 6*w - 6], [569, 569, -1/3*w^3 - 3*w^2 + 8/3*w + 50/3], [571, 571, 2/3*w^3 + 3*w^2 - 10/3*w - 46/3], [571, 571, 2*w^2 - 15], [599, 599, -w^3 + 2*w^2 + 4*w - 10], [599, 599, -w^2 - 3*w + 8], [601, 601, 5/3*w^3 + 3*w^2 - 28/3*w - 34/3], [601, 601, 1/3*w^3 - 3*w^2 - 2/3*w + 43/3], [619, 619, 2/3*w^3 - w^2 - 13/3*w + 2/3], [619, 619, 2/3*w^3 + w^2 - 7/3*w - 25/3], [619, 619, 2*w^3 + w^2 - 11*w - 6], [619, 619, -2/3*w^3 + 13/3*w - 17/3], [631, 631, 2/3*w^3 + 2*w^2 - 19/3*w - 31/3], [631, 631, w^3 + 2*w^2 - 8*w - 6], [631, 631, 2/3*w^3 + 2*w^2 - 4/3*w - 31/3], [631, 631, -2/3*w^3 + 2*w^2 + 16/3*w - 23/3], [659, 659, -2/3*w^3 + 1/3*w + 16/3], [659, 659, 5/3*w^3 - 34/3*w - 13/3], [661, 661, w^3 - 8*w + 2], [661, 661, 4/3*w^3 + w^2 - 23/3*w + 1/3], [661, 661, 5/3*w^3 + w^2 - 31/3*w - 10/3], [661, 661, -3*w - 1], [691, 691, -4/3*w^3 + 29/3*w + 2/3], [691, 691, 1/3*w^3 + 4/3*w - 5/3], [691, 691, -5/3*w^3 + w^2 + 25/3*w - 5/3], [691, 691, 2*w^3 + w^2 - 10*w - 7], [701, 701, -2/3*w^3 - 3*w^2 + 19/3*w + 46/3], [701, 701, w^3 - 2*w^2 - 6*w + 8], [701, 701, 4/3*w^3 + 2*w^2 - 17/3*w - 29/3], [701, 701, 5/3*w^3 + 3*w^2 - 28/3*w - 52/3], [709, 709, 1/3*w^3 - 14/3*w + 10/3], [709, 709, -2/3*w^3 + 19/3*w + 7/3], [719, 719, 3*w^2 - w - 10], [719, 719, -4/3*w^3 - 3*w^2 + 23/3*w + 47/3], [739, 739, 5/3*w^3 + w^2 - 34/3*w - 10/3], [739, 739, 4/3*w^3 + 2*w^2 - 23/3*w - 14/3], [761, 761, 7/3*w^3 + 2*w^2 - 41/3*w - 26/3], [761, 761, 2*w^3 + 3*w^2 - 12*w - 15], [769, 769, -2/3*w^3 + 19/3*w - 8/3], [769, 769, -4/3*w^3 + 2*w^2 + 17/3*w - 31/3], [769, 769, 1/3*w^3 - 14/3*w - 5/3], [769, 769, -1/3*w^3 + 3*w^2 + 2/3*w - 34/3], [809, 809, -4/3*w^3 + w^2 + 26/3*w - 7/3], [809, 809, w^3 + w^2 - 3*w - 7], [811, 811, 4/3*w^3 + w^2 - 17/3*w - 23/3], [811, 811, -1/3*w^3 - 2*w^2 + 2/3*w + 38/3], [821, 821, 8/3*w^3 + 3*w^2 - 43/3*w - 34/3], [821, 821, 2/3*w^3 + 2*w^2 - 7/3*w - 34/3], [821, 821, -w^3 - w^2 + 4*w + 8], [821, 821, 2/3*w^3 + 4*w^2 - 7/3*w - 58/3], [829, 829, 4/3*w^3 + 3*w^2 - 26/3*w - 50/3], [829, 829, 7/3*w^3 - 41/3*w - 5/3], [839, 839, 2*w^3 + w^2 - 10*w - 3], [839, 839, 5/3*w^3 - w^2 - 25/3*w + 17/3], [841, 29, 5/3*w^3 - 25/3*w - 7/3], [859, 859, -7/3*w^3 - w^2 + 32/3*w + 29/3], [859, 859, 2*w^3 + w^2 - 14*w - 6], [859, 859, 1/3*w^3 + 2*w^2 - 2/3*w - 47/3], [859, 859, -5/3*w^3 - 2*w^2 + 31/3*w + 49/3], [881, 881, w^3 + 4*w^2 - 4*w - 20], [881, 881, 8/3*w^3 + 2*w^2 - 43/3*w - 34/3], [919, 919, -w^3 + 3*w - 7], [919, 919, 1/3*w^3 + 3*w^2 - 5/3*w - 35/3], [919, 919, -5/3*w^3 + 31/3*w - 23/3], [919, 919, 2/3*w^3 + 3*w^2 - 10/3*w - 43/3], [929, 929, w^2 - 2*w - 9], [929, 929, w^3 + w^2 - 7*w + 1], [961, 31, 2/3*w^3 - 10/3*w - 19/3], [971, 971, -w^3 - 3*w^2 + 7*w + 16], [971, 971, -5/3*w^3 + w^2 + 25/3*w - 8/3], [971, 971, 2*w^3 + w^2 - 10*w - 6], [971, 971, 1/3*w^3 + 2*w^2 - 8/3*w - 47/3]]; primes := [ideal : I in primesArray]; heckePol := x^6 + 5*x^5 - 11*x^4 - 111*x^3 - 248*x^2 - 225*x - 72; K := NumberField(heckePol); heckeEigenvaluesArray := [e, e^5 + 3*e^4 - 17*e^3 - 77*e^2 - 94*e - 38, 5/3*e^5 + 19/3*e^4 - 79/3*e^3 - 154*e^2 - 661/3*e - 92, -5/3*e^5 - 16/3*e^4 + 85/3*e^3 + 135*e^2 + 487/3*e + 53, -11/3*e^5 - 40/3*e^4 + 175/3*e^3 + 327*e^2 + 1402/3*e + 201, 10/3*e^5 + 35/3*e^4 - 161/3*e^3 - 289*e^2 - 1208/3*e - 168, e^5 + 3*e^4 - 17*e^3 - 77*e^2 - 93*e - 36, 1/3*e^5 + 2/3*e^4 - 20/3*e^3 - 20*e^2 - 14/3*e + 9, -4*e^5 - 15*e^4 + 63*e^3 + 365*e^2 + 531*e + 232, -17/3*e^5 - 58/3*e^4 + 277/3*e^3 + 482*e^2 + 1957/3*e + 261, e^5 + 4*e^4 - 15*e^3 - 96*e^2 - 153*e - 74, 11/3*e^5 + 43/3*e^4 - 172/3*e^3 - 345*e^2 - 1528/3*e - 228, 8/3*e^5 + 31/3*e^4 - 124/3*e^3 - 250*e^2 - 1123/3*e - 163, e^5 + 3*e^4 - 17*e^3 - 78*e^2 - 94*e - 25, -5*e^5 - 19*e^4 + 77*e^3 + 462*e^2 + 698*e + 317, 2*e^5 + 8*e^4 - 30*e^3 - 192*e^2 - 306*e - 146, 1, 20/3*e^5 + 73/3*e^4 - 325/3*e^3 - 596*e^2 - 2428/3*e - 311, 4*e^5 + 15*e^4 - 65*e^3 - 364*e^2 - 496*e - 194, -8*e^5 - 28*e^4 + 130*e^3 + 692*e^2 + 946*e + 392, 5*e^5 + 17*e^4 - 81*e^3 - 424*e^2 - 583*e - 238, -14/3*e^5 - 52/3*e^4 + 223/3*e^3 + 422*e^2 + 1804/3*e + 254, -32/3*e^5 - 115/3*e^4 + 517/3*e^3 + 942*e^2 + 3904/3*e + 525, -5/3*e^5 - 10/3*e^4 + 94/3*e^3 + 99*e^2 + 184/3*e - 14, -11/3*e^5 - 34/3*e^4 + 187/3*e^3 + 289*e^2 + 1060/3*e + 124, -2/3*e^5 - 4/3*e^4 + 37/3*e^3 + 38*e^2 + 91/3*e + 9, -23/3*e^5 - 88/3*e^4 + 361/3*e^3 + 711*e^2 + 3118/3*e + 451, 11*e^5 + 40*e^4 - 178*e^3 - 981*e^2 - 1348*e - 534, 4/3*e^5 + 8/3*e^4 - 74/3*e^3 - 79*e^2 - 164/3*e + 11, 14/3*e^5 + 46/3*e^4 - 229/3*e^3 - 384*e^2 - 1567/3*e - 232, 2/3*e^5 + 13/3*e^4 - 19/3*e^3 - 95*e^2 - 616/3*e - 122, -4/3*e^5 - 11/3*e^4 + 65/3*e^3 + 99*e^2 + 392/3*e + 44, 2/3*e^5 + 1/3*e^4 - 43/3*e^3 - 22*e^2 + 92/3*e + 46, -14/3*e^5 - 46/3*e^4 + 232/3*e^3 + 387*e^2 + 1501/3*e + 178, -29/3*e^5 - 109/3*e^4 + 451/3*e^3 + 885*e^2 + 3949/3*e + 588, 44/3*e^5 + 160/3*e^4 - 700/3*e^3 - 1308*e^2 - 5593/3*e - 798, 17/3*e^5 + 58/3*e^4 - 274/3*e^3 - 482*e^2 - 2014/3*e - 294, 5/3*e^5 + 22/3*e^4 - 73/3*e^3 - 173*e^2 - 835/3*e - 141, -2/3*e^5 - 7/3*e^4 + 40/3*e^3 + 57*e^2 + 115/3*e - 11, -4/3*e^5 - 17/3*e^4 + 59/3*e^3 + 133*e^2 + 650/3*e + 104, 34/3*e^5 + 119/3*e^4 - 548/3*e^3 - 983*e^2 - 4100/3*e - 570, -19/3*e^5 - 74/3*e^4 + 287/3*e^3 + 598*e^2 + 2795/3*e + 433, 10/3*e^5 + 29/3*e^4 - 176/3*e^3 - 252*e^2 - 806/3*e - 70, 6*e^5 + 22*e^4 - 94*e^3 - 539*e^2 - 789*e - 348, -40/3*e^5 - 140/3*e^4 + 641/3*e^3 + 1157*e^2 + 4868/3*e + 678, -12*e^5 - 46*e^4 + 188*e^3 + 1115*e^2 + 1633*e + 710, 16/3*e^5 + 53/3*e^4 - 263/3*e^3 - 442*e^2 - 1763/3*e - 258, 5/3*e^5 + 13/3*e^4 - 85/3*e^3 - 114*e^2 - 430/3*e - 80, -5*e^5 - 19*e^4 + 77*e^3 + 461*e^2 + 701*e + 336, -41/3*e^5 - 148/3*e^4 + 664/3*e^3 + 1211*e^2 + 4987/3*e + 665, -7/3*e^5 - 20/3*e^4 + 125/3*e^3 + 173*e^2 + 530/3*e + 38, -37/3*e^5 - 128/3*e^4 + 602/3*e^3 + 1059*e^2 + 4328/3*e + 583, 34/3*e^5 + 122/3*e^4 - 539/3*e^3 - 1001*e^2 - 4313/3*e - 633, 3*e^5 + 10*e^4 - 52*e^3 - 249*e^2 - 289*e - 101, -e^5 - e^4 + 19*e^3 + 42*e^2 + 11*e - 12, 9*e^5 + 29*e^4 - 151*e^3 - 734*e^2 - 923*e - 330, 74/3*e^5 + 268/3*e^4 - 1177/3*e^3 - 2194*e^2 - 9397/3*e - 1351, -20/3*e^5 - 82/3*e^4 + 298/3*e^3 + 654*e^2 + 3097/3*e + 474, 22/3*e^5 + 80/3*e^4 - 350/3*e^3 - 652*e^2 - 2819/3*e - 414, 19/3*e^5 + 68/3*e^4 - 308/3*e^3 - 559*e^2 - 2297/3*e - 294, -10*e^5 - 36*e^4 + 163*e^3 + 887*e^2 + 1193*e + 454, -16/3*e^5 - 47/3*e^4 + 278/3*e^3 + 407*e^2 + 1358/3*e + 123, -20*e^5 - 71*e^4 + 324*e^3 + 1752*e^2 + 2405*e + 971, -e^5 - e^4 + 22*e^3 + 38*e^2 - 36*e - 46, 20/3*e^5 + 67/3*e^4 - 319/3*e^3 - 559*e^2 - 2398/3*e - 358, -41/3*e^5 - 136/3*e^4 + 676/3*e^3 + 1138*e^2 + 4492/3*e + 580, -55/3*e^5 - 197/3*e^4 + 887/3*e^3 + 1618*e^2 + 6716/3*e + 915, 34/3*e^5 + 119/3*e^4 - 539/3*e^3 - 982*e^2 - 4262/3*e - 646, -29/3*e^5 - 97/3*e^4 + 484/3*e^3 + 807*e^2 + 3115/3*e + 401, -6*e^5 - 20*e^4 + 95*e^3 + 503*e^2 + 729*e + 331, -25/3*e^5 - 86/3*e^4 + 407/3*e^3 + 715*e^2 + 2891/3*e + 353, 16/3*e^5 + 59/3*e^4 - 263/3*e^3 - 479*e^2 - 1907/3*e - 251, 4*e^5 + 17*e^4 - 61*e^3 - 405*e^2 - 604*e - 228, 18*e^5 + 63*e^4 - 291*e^3 - 1560*e^2 - 2145*e - 892, -29*e^5 - 103*e^4 + 465*e^3 + 2540*e^2 + 3573*e + 1517, -9*e^5 - 31*e^4 + 146*e^3 + 769*e^2 + 1062*e + 447, -4/3*e^5 - 5/3*e^4 + 77/3*e^3 + 56*e^2 + 77/3*e + 22, 23/3*e^5 + 67/3*e^4 - 394/3*e^3 - 582*e^2 - 2035/3*e - 226, -26/3*e^5 - 85/3*e^4 + 439/3*e^3 + 713*e^2 + 2650/3*e + 300, 27*e^5 + 95*e^4 - 439*e^3 - 2349*e^2 - 3187*e - 1262, 17*e^5 + 61*e^4 - 274*e^3 - 1502*e^2 - 2076*e - 850, -5*e^5 - 15*e^4 + 86*e^3 + 386*e^2 + 450*e + 160, 5*e^5 + 14*e^4 - 86*e^3 - 370*e^2 - 425*e - 131, -1/3*e^5 - 8/3*e^4 - 7/3*e^3 + 60*e^2 + 629/3*e + 156, 11*e^5 + 36*e^4 - 186*e^3 - 906*e^2 - 1112*e - 378, 5/3*e^5 + 7/3*e^4 - 100/3*e^3 - 80*e^2 - 7/3*e + 43, -13/3*e^5 - 44/3*e^4 + 212/3*e^3 + 365*e^2 + 1487/3*e + 218, -11/3*e^5 - 52/3*e^4 + 157/3*e^3 + 401*e^2 + 1981/3*e + 319, 12*e^5 + 39*e^4 - 197*e^3 - 982*e^2 - 1311*e - 552, -44/3*e^5 - 163/3*e^4 + 697/3*e^3 + 1328*e^2 + 5719/3*e + 808, 27*e^5 + 95*e^4 - 437*e^3 - 2351*e^2 - 3225*e - 1300, 41/3*e^5 + 142/3*e^4 - 664/3*e^3 - 1174*e^2 - 4870/3*e - 676, -29/3*e^5 - 112/3*e^4 + 445/3*e^3 + 904*e^2 + 4120/3*e + 629, 11/3*e^5 + 37/3*e^4 - 184/3*e^3 - 309*e^2 - 1153/3*e - 126, 11/3*e^5 + 31/3*e^4 - 187/3*e^3 - 267*e^2 - 1000/3*e - 159, -25/3*e^5 - 83/3*e^4 + 422/3*e^3 + 694*e^2 + 2558/3*e + 292, 2*e^5 + 9*e^4 - 28*e^3 - 210*e^2 - 365*e - 199, 25/3*e^5 + 101/3*e^4 - 386/3*e^3 - 807*e^2 - 3620/3*e - 516, 67/3*e^5 + 236/3*e^4 - 1085/3*e^3 - 1945*e^2 - 8006/3*e - 1104, 3*e^4 + 4*e^3 - 55*e^2 - 142*e - 98, 14*e^5 + 52*e^4 - 225*e^3 - 1267*e^2 - 1770*e - 721, -4/3*e^5 - 17/3*e^4 + 68/3*e^3 + 134*e^2 + 482/3*e + 34, -49/3*e^5 - 173/3*e^4 + 797/3*e^3 + 1427*e^2 + 5762/3*e + 736, -4*e^5 - 11*e^4 + 73*e^3 + 290*e^2 + 260*e + 36, 38/3*e^5 + 127/3*e^4 - 625/3*e^3 - 1060*e^2 - 4216/3*e - 552, 4/3*e^5 - 4/3*e^4 - 92/3*e^3 - 5*e^2 + 430/3*e + 122, -91/3*e^5 - 320/3*e^4 + 1469/3*e^3 + 2639*e^2 + 10919/3*e + 1490, 44/3*e^5 + 160/3*e^4 - 712/3*e^3 - 1304*e^2 - 5422/3*e - 771, 10*e^5 + 35*e^4 - 161*e^3 - 870*e^2 - 1202*e - 477, -24*e^5 - 82*e^4 + 392*e^3 + 2043*e^2 + 2745*e + 1088, 71/3*e^5 + 232/3*e^4 - 1174/3*e^3 - 1949*e^2 - 7639/3*e - 991, -61/3*e^5 - 215/3*e^4 + 998/3*e^3 + 1769*e^2 + 7118/3*e + 933, -17/3*e^5 - 52/3*e^4 + 277/3*e^3 + 443*e^2 + 1825/3*e + 296, -4/3*e^5 + 4/3*e^4 + 92/3*e^3 + 4*e^2 - 415/3*e - 118, -4*e^5 - 13*e^4 + 68*e^3 + 325*e^2 + 405*e + 158, -14/3*e^5 - 43/3*e^4 + 247/3*e^3 + 364*e^2 + 1171/3*e + 100, -95/3*e^5 - 334/3*e^4 + 1537/3*e^3 + 2751*e^2 + 11383/3*e + 1560, 58/3*e^5 + 215/3*e^4 - 914/3*e^3 - 1753*e^2 - 7625/3*e - 1098, 73/3*e^5 + 266/3*e^4 - 1163/3*e^3 - 2175*e^2 - 9257/3*e - 1315, -4/3*e^5 - 29/3*e^4 + 35/3*e^3 + 210*e^2 + 1352/3*e + 270, -9*e^5 - 31*e^4 + 149*e^3 + 770*e^2 + 1011*e + 372, -77/3*e^5 - 277/3*e^4 + 1228/3*e^3 + 2271*e^2 + 9691/3*e + 1373, -3*e^5 - 13*e^4 + 44*e^3 + 309*e^2 + 493*e + 201, 98/3*e^5 + 361/3*e^4 - 1564/3*e^3 - 2940*e^2 - 12511/3*e - 1774, 13/3*e^5 + 38/3*e^4 - 221/3*e^3 - 330*e^2 - 1175/3*e - 144, 10*e^5 + 39*e^4 - 156*e^3 - 941*e^2 - 1383*e - 612, 38/3*e^5 + 136/3*e^4 - 613/3*e^3 - 1113*e^2 - 4648/3*e - 671, 46/3*e^5 + 182/3*e^4 - 713/3*e^3 - 1460*e^2 - 6533/3*e - 960, 83/3*e^5 + 301/3*e^4 - 1318/3*e^3 - 2465*e^2 - 10573/3*e - 1512, 19/3*e^5 + 77/3*e^4 - 290/3*e^3 - 619*e^2 - 2795/3*e - 387, -4*e^5 - 16*e^4 + 65*e^3 + 387*e^2 + 513*e + 156, -22*e^5 - 76*e^4 + 354*e^3 + 1890*e^2 + 2629*e + 1083, -4/3*e^5 - 17/3*e^4 + 53/3*e^3 + 134*e^2 + 749/3*e + 159, 11/3*e^5 + 34/3*e^4 - 187/3*e^3 - 291*e^2 - 1039/3*e - 103, 44/3*e^5 + 154/3*e^4 - 718/3*e^3 - 1272*e^2 - 5134/3*e - 676, -2/3*e^5 + 11/3*e^4 + 70/3*e^3 - 53*e^2 - 842/3*e - 214, 4*e^5 + 19*e^4 - 55*e^3 - 441*e^2 - 764*e - 397, -16/3*e^5 - 59/3*e^4 + 251/3*e^3 + 478*e^2 + 2135/3*e + 336, -64/3*e^5 - 227/3*e^4 + 1043/3*e^3 + 1865*e^2 + 7586/3*e + 1010, 89/3*e^5 + 316/3*e^4 - 1435/3*e^3 - 2597*e^2 - 10819/3*e - 1489, 4*e^5 + 13*e^4 - 67*e^3 - 324*e^2 - 419*e - 196, -21*e^5 - 71*e^4 + 342*e^3 + 1776*e^2 + 2401*e + 950, -85/3*e^5 - 308/3*e^4 + 1355/3*e^3 + 2521*e^2 + 10757/3*e + 1529, 31/3*e^5 + 110/3*e^4 - 497/3*e^3 - 907*e^2 - 3800/3*e - 528, 25*e^5 + 87*e^4 - 405*e^3 - 2155*e^2 - 2970*e - 1247, -25*e^5 - 85*e^4 + 410*e^3 + 2119*e^2 + 2827*e + 1110, 31*e^5 + 113*e^4 - 493*e^3 - 2768*e^2 - 3961*e - 1714, 1/3*e^5 - 1/3*e^4 - 32/3*e^3 + e^2 + 238/3*e + 47, 50/3*e^5 + 187/3*e^4 - 790/3*e^3 - 1520*e^2 - 6568/3*e - 957, -26*e^5 - 92*e^4 + 419*e^3 + 2272*e^2 + 3158*e + 1302, -4/3*e^5 - 20/3*e^4 + 41/3*e^3 + 152*e^2 + 1058/3*e + 234, -25*e^5 - 83*e^4 + 411*e^3 + 2084*e^2 + 2759*e + 1084, 35/3*e^5 + 115/3*e^4 - 583/3*e^3 - 964*e^2 - 3718/3*e - 461, -20/3*e^5 - 61/3*e^4 + 331/3*e^3 + 524*e^2 + 2014/3*e + 239, -22/3*e^5 - 74/3*e^4 + 356/3*e^3 + 615*e^2 + 2582/3*e + 384, 9*e^5 + 32*e^4 - 144*e^3 - 786*e^2 - 1126*e - 520, -6*e^5 - 27*e^4 + 87*e^3 + 630*e^2 + 1040*e + 515, -97/3*e^5 - 344/3*e^4 + 1559/3*e^3 + 2827*e^2 + 11882/3*e + 1673, -27*e^5 - 95*e^4 + 436*e^3 + 2345*e^2 + 3262*e + 1388, -20/3*e^5 - 79/3*e^4 + 319/3*e^3 + 632*e^2 + 2680/3*e + 352, 59/3*e^5 + 211/3*e^4 - 946/3*e^3 - 1731*e^2 - 7291/3*e - 1025, -43/3*e^5 - 143/3*e^4 + 713/3*e^3 + 1192*e^2 + 4673/3*e + 597, 101/3*e^5 + 379/3*e^4 - 1597/3*e^3 - 3075*e^2 - 13297/3*e - 1907, 21*e^5 + 77*e^4 - 330*e^3 - 1885*e^2 - 2764*e - 1224, 16*e^5 + 61*e^4 - 250*e^3 - 1479*e^2 - 2178*e - 946, -139/3*e^5 - 479/3*e^4 + 2252/3*e^3 + 3970*e^2 + 16355/3*e + 2256, 23/3*e^5 + 85/3*e^4 - 358/3*e^3 - 695*e^2 - 3100/3*e - 455, -47*e^5 - 170*e^4 + 753*e^3 + 4175*e^2 + 5858*e + 2456, -25*e^5 - 87*e^4 + 404*e^3 + 2159*e^2 + 2976*e + 1232, 24*e^5 + 87*e^4 - 390*e^3 - 2133*e^2 - 2910*e - 1168, -16*e^5 - 55*e^4 + 257*e^3 + 1368*e^2 + 1921*e + 820]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := -1; // EXAMPLE: // pp := Factorization(2*ZF)[1][1]; // heckeEigenvalues[pp]; print "To reconstruct the Hilbert newform f, type f, iso := Explode(make_newform());"; function make_newform(); M := HilbertCuspForms(F, NN); S := NewSubspace(M); // SetVerbose("ModFrmHil", 1); NFD := NewformDecomposition(S); newforms := [* Eigenform(U) : U in NFD *]; if #newforms eq 0 then; print "No Hilbert newforms at this level"; return 0; end if; print "Testing ", #newforms, " possible newforms"; newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *]; print #newforms, " newforms have the correct Hecke field"; if #newforms eq 0 then; print "No Hilbert newform found with the correct Hecke field"; return 0; end if; autos := Automorphisms(K); xnewforms := [* *]; for f in newforms do; if K eq RationalField() then; Append(~xnewforms, [* f, autos[1] *]); else; flag, iso := IsIsomorphic(K,BaseField(f)); for a in autos do; Append(~xnewforms, [* f, a*iso *]); end for; end if; end for; newforms := xnewforms; for P in primes do; xnewforms := [* *]; for f_iso in newforms do; f, iso := Explode(f_iso); if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then; Append(~xnewforms, f_iso); end if; end for; newforms := xnewforms; if #newforms eq 0 then; print "No Hilbert newform found which matches the Hecke eigenvalues"; return 0; else if #newforms eq 1 then; print "success: unique match"; return newforms[1]; end if; end if; end for; print #newforms, "Hilbert newforms found which match the Hecke eigenvalues"; return newforms[1]; end function;