/* 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![1, -2, -6, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [5, 5, -w^3 + 2*w^2 + 3*w], [7, 7, w - 1], [11, 11, -w^3 + 2*w^2 + 4*w], [13, 13, -2*w^3 + 3*w^2 + 10*w - 2], [16, 2, 2], [17, 17, -w^3 + 2*w^2 + 5*w - 3], [17, 17, -w^3 + w^2 + 5*w], [17, 17, -w^2 + 2*w + 1], [19, 19, w^2 - w - 2], [29, 29, w^3 - 2*w^2 - 5*w], [29, 29, w^2 - w - 3], [31, 31, -w^3 + 2*w^2 + 3*w - 2], [43, 43, 2*w^3 - 3*w^2 - 11*w], [47, 47, w^3 - 7*w - 4], [53, 53, -2*w^3 + 3*w^2 + 9*w - 1], [61, 61, -w - 3], [73, 73, -w^3 + 2*w^2 + 3*w - 3], [73, 73, w^3 - w^2 - 7*w - 1], [81, 3, -3], [83, 83, -2*w^3 + 3*w^2 + 9*w + 1], [83, 83, -w^3 + 3*w^2 + 2*w - 3], [89, 89, w - 4], [103, 103, w^3 - 2*w^2 - 6*w + 3], [113, 113, 2*w^3 - w^2 - 14*w - 6], [121, 11, -w^2 + 2*w + 7], [125, 5, -3*w^3 + 4*w^2 + 15*w + 1], [139, 139, w^3 - 2*w^2 - 5*w - 3], [139, 139, w^3 - 5*w - 5], [151, 151, w^3 - 9*w - 7], [157, 157, -3*w^3 + 6*w^2 + 13*w - 6], [163, 163, w^3 - 3*w^2 - 2*w + 9], [167, 167, -w^3 + 7*w + 3], [167, 167, -3*w^3 + 4*w^2 + 15*w - 1], [173, 173, 2*w^3 - 3*w^2 - 7*w - 2], [181, 181, -w^3 + 3*w^2 + 4*w - 5], [191, 191, w^3 - w^2 - 4*w - 4], [193, 193, -w^3 + w^2 + 7*w + 6], [193, 193, 2*w^3 - 2*w^2 - 13*w - 5], [199, 199, -2*w^3 + 5*w^2 + 5*w - 5], [211, 211, w^3 - w^2 - 8*w - 3], [211, 211, -2*w^2 + 3*w + 8], [227, 227, -w^3 + 3*w^2 + 4*w - 4], [227, 227, w^2 - 3*w - 6], [229, 229, w^2 - 5], [229, 229, 3*w^3 - 4*w^2 - 17*w - 2], [233, 233, 3*w^3 - 6*w^2 - 13*w + 3], [233, 233, 3*w^3 - 4*w^2 - 16*w + 1], [241, 241, -2*w^3 + 4*w^2 + 8*w + 1], [251, 251, 2*w^3 - 2*w^2 - 9*w - 3], [277, 277, 3*w^3 - 4*w^2 - 17*w - 1], [281, 281, -2*w^2 + 5*w + 1], [283, 283, -4*w^3 + 6*w^2 + 19*w - 2], [293, 293, -3*w^3 + 6*w^2 + 12*w - 2], [307, 307, w^3 - w^2 - 4*w - 5], [307, 307, w^3 - w^2 - 8*w - 2], [311, 311, 2*w^2 - 3*w - 7], [313, 313, -2*w^3 + w^2 + 15*w + 3], [317, 317, -w^3 + 3*w^2 + w - 5], [331, 331, 2*w^3 - 3*w^2 - 12*w], [337, 337, w^3 - w^2 - 8*w - 1], [343, 7, -w^3 + 6*w + 8], [347, 347, 2*w^3 - 4*w^2 - 7*w + 5], [349, 349, 2*w^3 - 4*w^2 - 9*w], [353, 353, -3*w^3 + 4*w^2 + 16*w + 5], [353, 353, -3*w^3 + 4*w^2 + 14*w], [353, 353, -w^3 + 3*w^2 + 4*w - 8], [353, 353, -w^3 + 2*w^2 + 8*w - 3], [359, 359, -4*w^3 + 7*w^2 + 19*w - 3], [367, 367, -w^3 + 2*w^2 + 6*w - 5], [373, 373, 5*w^3 - 8*w^2 - 26*w + 3], [379, 379, -3*w^3 + 5*w^2 + 17*w - 4], [379, 379, -w^3 + 2*w^2 + 4*w - 6], [383, 383, w^2 - 6], [389, 389, w^3 - 2*w^2 - 3*w - 4], [389, 389, w^2 + w - 4], [397, 397, -4*w^3 + 5*w^2 + 23*w + 5], [397, 397, -3*w^3 + 5*w^2 + 12*w - 2], [419, 419, -3*w^3 + 4*w^2 + 17*w + 5], [419, 419, -3*w^3 + 5*w^2 + 15*w], [419, 419, 3*w^3 - 5*w^2 - 14*w - 1], [419, 419, 2*w^3 - 3*w^2 - 10*w - 6], [431, 431, 2*w^3 - 2*w^2 - 11*w], [433, 433, w^3 - 8*w - 1], [433, 433, 2*w^2 - 2*w - 5], [439, 439, 2*w^3 - 2*w^2 - 9*w - 6], [443, 443, 2*w^3 - 5*w^2 - 7*w], [443, 443, 2*w^3 - 5*w^2 - 9*w + 4], [457, 457, -2*w^3 + 5*w^2 + 7*w - 4], [457, 457, -w^3 + 2*w^2 - 3], [461, 461, -4*w^3 + 7*w^2 + 17*w - 7], [479, 479, 3*w^3 - 5*w^2 - 12*w + 6], [487, 487, -2*w^3 + 5*w^2 + 9*w - 9], [487, 487, w^3 - 10*w - 8], [499, 499, 2*w^3 - 2*w^2 - 12*w + 1], [503, 503, w^2 - 7], [503, 503, w^3 + w^2 - 7*w - 8], [529, 23, -3*w^3 + 5*w^2 + 12*w + 1], [529, 23, 3*w^3 - 4*w^2 - 14*w - 1], [541, 541, w^3 - 4*w^2 - 2*w + 11], [563, 563, -2*w^3 + w^2 + 15*w + 6], [569, 569, w^3 - 2*w^2 - 4*w - 4], [601, 601, 3*w^3 - 5*w^2 - 17*w + 1], [607, 607, -2*w^3 + 5*w^2 + 7*w - 5], [607, 607, -w^3 + 3*w^2 + 6*w - 5], [613, 613, -4*w^3 + 5*w^2 + 21*w + 1], [613, 613, -5*w^3 + 8*w^2 + 23*w - 4], [619, 619, w^3 + w^2 - 9*w - 5], [619, 619, -3*w^3 + 4*w^2 + 14*w + 2], [641, 641, -6*w^3 + 8*w^2 + 31*w + 3], [643, 643, 3*w^3 - 4*w^2 - 17*w - 6], [643, 643, -2*w^3 + 5*w^2 + 7*w - 6], [647, 647, 3*w^3 - 4*w^2 - 13*w - 10], [653, 653, -3*w^3 + 5*w^2 + 14*w + 2], [653, 653, w^3 - 2*w^2 - w - 3], [661, 661, 5*w^3 - 9*w^2 - 24*w + 5], [683, 683, 2*w^3 - 3*w^2 - 13*w - 2], [683, 683, 4*w^3 - 7*w^2 - 18*w + 1], [719, 719, -w^3 + 2*w^2 + 3*w - 7], [719, 719, 3*w^3 - 5*w^2 - 16*w - 1], [727, 727, -2*w^3 + 5*w^2 + 8*w - 7], [727, 727, 2*w^3 - 5*w^2 - 7*w + 13], [733, 733, -2*w^2 + 7], [733, 733, -3*w^3 + 5*w^2 + 11*w + 2], [739, 739, -6*w^3 + 9*w^2 + 29*w - 3], [739, 739, -2*w^3 + 3*w^2 + 7*w + 5], [751, 751, 4*w^3 - 5*w^2 - 23*w - 3], [757, 757, -w^3 + 10*w + 4], [757, 757, 2*w^3 - w^2 - 14*w - 4], [769, 769, -w^3 + 4*w^2 + 4*w - 8], [769, 769, 4*w^3 - 6*w^2 - 21*w - 2], [787, 787, 3*w^3 - 3*w^2 - 19*w - 6], [787, 787, -2*w^2 + 6*w + 7], [839, 839, -3*w^3 + 3*w^2 + 19*w + 5], [841, 29, 3*w^3 - 4*w^2 - 13*w - 4], [853, 853, -3*w^3 + 5*w^2 + 14*w + 3], [853, 853, 2*w^2 - w - 9], [853, 853, 3*w^3 - 5*w^2 - 11*w + 1], [853, 853, w - 6], [857, 857, 2*w^3 - 2*w^2 - 9*w - 7], [857, 857, -4*w^3 + 8*w^2 + 13*w - 1], [859, 859, 2*w^3 - 4*w^2 - 13*w + 2], [859, 859, w^3 - w^2 - 9*w - 2], [863, 863, -2*w^3 - w^2 + 17*w + 11], [877, 877, 5*w^3 - 6*w^2 - 28*w - 9], [877, 877, -4*w^3 + 5*w^2 + 20*w + 6], [881, 881, 4*w^3 - 5*w^2 - 18*w - 8], [911, 911, -w^3 + 4*w^2 - w - 8], [911, 911, 2*w^3 - 3*w^2 - 13*w - 1], [919, 919, w^3 - w^2 - 7*w + 5], [929, 929, 2*w^3 - 3*w^2 - 13*w + 1], [929, 929, -2*w^3 + 5*w^2 + 6*w - 11], [941, 941, -3*w^3 + 7*w^2 + 12*w - 5], [953, 953, -2*w^3 + 5*w^2 + 5*w - 7], [971, 971, w^3 - w^2 - 3*w - 5], [971, 971, 3*w^3 - 7*w^2 - 12*w + 11], [977, 977, 3*w^3 - 5*w^2 - 11*w], [997, 997, -2*w^3 + 3*w^2 + 6*w - 5], [997, 997, -4*w^3 + 7*w^2 + 16*w - 2]]; primes := [ideal : I in primesArray]; heckePol := x^5 - x^4 - 27*x^3 + 36*x^2 + 170*x - 289; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -14/17*e^4 - 20/17*e^3 + 344/17*e^2 + 295/17*e - 110, 1/17*e^4 - 1/17*e^3 - 10/17*e^2 + 19/17*e - 2, -1/17*e^4 + 1/17*e^3 + 27/17*e^2 - 19/17*e - 10, -23/17*e^4 - 28/17*e^3 + 553/17*e^2 + 413/17*e - 168, 22/17*e^4 + 29/17*e^3 - 526/17*e^2 - 432/17*e + 163, -8/17*e^4 - 9/17*e^3 + 182/17*e^2 + 137/17*e - 54, 9/17*e^4 + 8/17*e^3 - 209/17*e^2 - 101/17*e + 63, -1, -12/17*e^4 - 22/17*e^3 + 290/17*e^2 + 316/17*e - 90, 15/17*e^4 + 19/17*e^3 - 371/17*e^2 - 310/17*e + 121, -15/17*e^4 - 19/17*e^3 + 354/17*e^2 + 276/17*e - 108, 6/17*e^4 + 11/17*e^3 - 162/17*e^2 - 175/17*e + 60, -33/17*e^4 - 35/17*e^3 + 772/17*e^2 + 512/17*e - 226, -9/17*e^4 - 8/17*e^3 + 209/17*e^2 + 101/17*e - 61, 10/17*e^4 + 7/17*e^3 - 253/17*e^2 - 82/17*e + 77, -2/17*e^4 + 2/17*e^3 + 54/17*e^2 - 4/17*e - 22, 22/17*e^4 + 29/17*e^3 - 543/17*e^2 - 449/17*e + 169, e^4 + e^3 - 22*e^2 - 14*e + 98, 8/17*e^4 + 9/17*e^3 - 199/17*e^2 - 154/17*e + 73, -3/17*e^4 + 3/17*e^3 + 81/17*e^2 - 74/17*e - 25, 33/17*e^4 + 52/17*e^3 - 789/17*e^2 - 733/17*e + 245, -4/17*e^4 - 13/17*e^3 + 91/17*e^2 + 196/17*e - 29, -41/17*e^4 - 61/17*e^3 + 988/17*e^2 + 921/17*e - 312, 52/17*e^4 + 67/17*e^3 - 1251/17*e^2 - 950/17*e + 387, -19/17*e^4 - 32/17*e^3 + 496/17*e^2 + 455/17*e - 173, -14/17*e^4 - 37/17*e^3 + 327/17*e^2 + 499/17*e - 99, 71/17*e^4 + 99/17*e^3 - 1713/17*e^2 - 1456/17*e + 535, -25/17*e^4 - 26/17*e^3 + 556/17*e^2 + 358/17*e - 152, 28/17*e^4 + 23/17*e^3 - 654/17*e^2 - 335/17*e + 196, 22/17*e^4 + 12/17*e^3 - 509/17*e^2 - 194/17*e + 140, 43/17*e^4 + 59/17*e^3 - 1008/17*e^2 - 849/17*e + 312, -9/17*e^4 - 25/17*e^3 + 209/17*e^2 + 322/17*e - 63, 15/17*e^4 + 19/17*e^3 - 371/17*e^2 - 259/17*e + 124, -12/17*e^4 - 22/17*e^3 + 290/17*e^2 + 316/17*e - 96, 28/17*e^4 + 40/17*e^3 - 671/17*e^2 - 573/17*e + 215, 36/17*e^4 + 49/17*e^3 - 853/17*e^2 - 676/17*e + 253, -6/17*e^4 - 11/17*e^3 + 128/17*e^2 + 124/17*e - 29, 6/17*e^4 - 6/17*e^3 - 128/17*e^2 + 80/17*e + 28, -81/17*e^4 - 106/17*e^3 + 1949/17*e^2 + 1521/17*e - 605, -22/17*e^4 - 29/17*e^3 + 543/17*e^2 + 466/17*e - 161, 23/17*e^4 + 45/17*e^3 - 587/17*e^2 - 651/17*e + 198, 13/17*e^4 + 21/17*e^3 - 317/17*e^2 - 314/17*e + 105, -45/17*e^4 - 57/17*e^3 + 1130/17*e^2 + 913/17*e - 370, 37/17*e^4 + 31/17*e^3 - 863/17*e^2 - 453/17*e + 254, 6/17*e^4 - 6/17*e^3 - 128/17*e^2 + 114/17*e + 20, 13/17*e^4 + 21/17*e^3 - 317/17*e^2 - 263/17*e + 94, -16/17*e^4 - 18/17*e^3 + 381/17*e^2 + 240/17*e - 104, 33/17*e^4 + 52/17*e^3 - 789/17*e^2 - 852/17*e + 249, 11/17*e^4 + 23/17*e^3 - 280/17*e^2 - 369/17*e + 98, -2*e^4 - 3*e^3 + 49*e^2 + 44*e - 263, -1/17*e^4 + 1/17*e^3 - 41/17*e^2 - 70/17*e + 45, -70/17*e^4 - 83/17*e^3 + 1669/17*e^2 + 1271/17*e - 503, -22/17*e^4 - 12/17*e^3 + 492/17*e^2 + 194/17*e - 132, -43/17*e^4 - 59/17*e^3 + 1059/17*e^2 + 951/17*e - 348, -99/17*e^4 - 122/17*e^3 + 2350/17*e^2 + 1740/17*e - 714, 76/17*e^4 + 111/17*e^3 - 1831/17*e^2 - 1650/17*e + 589, -48/17*e^4 - 54/17*e^3 + 1126/17*e^2 + 805/17*e - 334, 54/17*e^4 + 82/17*e^3 - 1305/17*e^2 - 1133/17*e + 417, -37/17*e^4 - 31/17*e^3 + 846/17*e^2 + 453/17*e - 230, -46/17*e^4 - 56/17*e^3 + 1106/17*e^2 + 826/17*e - 332, 59/17*e^4 + 60/17*e^3 - 1389/17*e^2 - 885/17*e + 405, -114/17*e^4 - 124/17*e^3 + 2704/17*e^2 + 1846/17*e - 816, 5/17*e^4 + 12/17*e^3 - 135/17*e^2 - 245/17*e + 61, 41/17*e^4 + 61/17*e^3 - 1039/17*e^2 - 921/17*e + 344, 48/17*e^4 + 71/17*e^3 - 1177/17*e^2 - 1179/17*e + 375, 40/17*e^4 + 45/17*e^3 - 961/17*e^2 - 668/17*e + 301, 33/17*e^4 + 35/17*e^3 - 806/17*e^2 - 529/17*e + 258, 80/17*e^4 + 107/17*e^3 - 1905/17*e^2 - 1557/17*e + 577, -54/17*e^4 - 65/17*e^3 + 1305/17*e^2 + 946/17*e - 401, 9/17*e^4 + 25/17*e^3 - 192/17*e^2 - 339/17*e + 72, -116/17*e^4 - 139/17*e^3 + 2758/17*e^2 + 2080/17*e - 841, 52/17*e^4 + 50/17*e^3 - 1183/17*e^2 - 695/17*e + 335, -8/17*e^4 + 8/17*e^3 + 114/17*e^2 - 186/17*e - 2, 35/17*e^4 + 67/17*e^3 - 860/17*e^2 - 933/17*e + 294, 128/17*e^4 + 178/17*e^3 - 3048/17*e^2 - 2634/17*e + 954, -45/17*e^4 - 57/17*e^3 + 1113/17*e^2 + 879/17*e - 346, 57/17*e^4 + 96/17*e^3 - 1420/17*e^2 - 1433/17*e + 459, 22/17*e^4 + 12/17*e^3 - 526/17*e^2 - 194/17*e + 158, -2*e^4 - 3*e^3 + 52*e^2 + 45*e - 308, 140/17*e^4 + 183/17*e^3 - 3389/17*e^2 - 2746/17*e + 1059, -19/17*e^4 - 32/17*e^3 + 479/17*e^2 + 387/17*e - 169, -38/17*e^4 - 30/17*e^3 + 856/17*e^2 + 400/17*e - 236, 37/17*e^4 + 48/17*e^3 - 897/17*e^2 - 827/17*e + 277, 59/17*e^4 + 77/17*e^3 - 1474/17*e^2 - 1208/17*e + 488, -27/17*e^4 - 7/17*e^3 + 610/17*e^2 + 99/17*e - 160, 42/17*e^4 + 43/17*e^3 - 1049/17*e^2 - 766/17*e + 349, 72/17*e^4 + 81/17*e^3 - 1740/17*e^2 - 1199/17*e + 530, -137/17*e^4 - 186/17*e^3 + 3308/17*e^2 + 2701/17*e - 1029, 98/17*e^4 + 140/17*e^3 - 2408/17*e^2 - 2082/17*e + 754, 75/17*e^4 + 95/17*e^3 - 1804/17*e^2 - 1465/17*e + 548, -48/17*e^4 - 71/17*e^3 + 1194/17*e^2 + 1043/17*e - 388, e^4 + e^3 - 27*e^2 - 15*e + 158, 59/17*e^4 + 60/17*e^3 - 1355/17*e^2 - 817/17*e + 389, -21/17*e^4 - 47/17*e^3 + 618/17*e^2 + 723/17*e - 248, 46/17*e^4 + 22/17*e^3 - 1106/17*e^2 - 384/17*e + 328, -161/17*e^4 - 196/17*e^3 + 3837/17*e^2 + 2891/17*e - 1169, -54/17*e^4 - 65/17*e^3 + 1322/17*e^2 + 929/17*e - 400, -45/17*e^4 - 74/17*e^3 + 1130/17*e^2 + 1117/17*e - 369, -30/17*e^4 - 55/17*e^3 + 691/17*e^2 + 756/17*e - 193, 56/17*e^4 + 97/17*e^3 - 1376/17*e^2 - 1435/17*e + 428, 24/17*e^4 + 27/17*e^3 - 597/17*e^2 - 428/17*e + 205, 43/17*e^4 + 59/17*e^3 - 1008/17*e^2 - 849/17*e + 332, 8/17*e^4 + 26/17*e^3 - 250/17*e^2 - 426/17*e + 112, 15/17*e^4 + 2/17*e^3 - 354/17*e^2 - 72/17*e + 80, 135/17*e^4 + 188/17*e^3 - 3237/17*e^2 - 2773/17*e + 1011, 109/17*e^4 + 163/17*e^3 - 2620/17*e^2 - 2315/17*e + 826, -20/17*e^4 - 31/17*e^3 + 421/17*e^2 + 470/17*e - 103, 41/17*e^4 + 44/17*e^3 - 920/17*e^2 - 581/17*e + 255, -30/17*e^4 - 38/17*e^3 + 742/17*e^2 + 603/17*e - 220, -27/17*e^4 - 24/17*e^3 + 644/17*e^2 + 439/17*e - 177, 89/17*e^4 + 98/17*e^3 - 2131/17*e^2 - 1573/17*e + 639, -7/17*e^4 - 10/17*e^3 + 172/17*e^2 + 122/17*e - 30, 132/17*e^4 + 157/17*e^3 - 3173/17*e^2 - 2235/17*e + 959, 77/17*e^4 + 93/17*e^3 - 1858/17*e^2 - 1291/17*e + 572, 82/17*e^4 + 105/17*e^3 - 1976/17*e^2 - 1655/17*e + 612, -76/17*e^4 - 77/17*e^3 + 1797/17*e^2 + 1174/17*e - 525, 135/17*e^4 + 205/17*e^3 - 3271/17*e^2 - 2960/17*e + 1039, -50/17*e^4 - 69/17*e^3 + 1282/17*e^2 + 1141/17*e - 444, -28/17*e^4 - 40/17*e^3 + 620/17*e^2 + 692/17*e - 172, 4*e^4 + 4*e^3 - 92*e^2 - 56*e + 476, 16/17*e^4 + 18/17*e^3 - 432/17*e^2 - 274/17*e + 158, 14/17*e^4 + 54/17*e^3 - 310/17*e^2 - 720/17*e + 100, e^4 + 3*e^3 - 27*e^2 - 45*e + 174, -71/17*e^4 - 99/17*e^3 + 1730/17*e^2 + 1439/17*e - 556, -78/17*e^4 - 109/17*e^3 + 1885/17*e^2 + 1697/17*e - 605, -89/17*e^4 - 98/17*e^3 + 2131/17*e^2 + 1505/17*e - 627, 121/17*e^4 + 168/17*e^3 - 2995/17*e^2 - 2495/17*e + 947, -33/17*e^4 - 52/17*e^3 + 755/17*e^2 + 699/17*e - 237, -2*e^4 - 2*e^3 + 45*e^2 + 22*e - 218, -7/17*e^4 - 10/17*e^3 + 291/17*e^2 + 173/17*e - 125, 4/17*e^4 + 13/17*e^3 - 91/17*e^2 - 162/17*e + 7, 7*e^4 + 10*e^3 - 171*e^2 - 149*e + 925, -5*e^4 - 7*e^3 + 121*e^2 + 104*e - 679, 139/17*e^4 + 184/17*e^3 - 3311/17*e^2 - 2629/17*e + 1005, 101/17*e^4 + 120/17*e^3 - 2387/17*e^2 - 1617/17*e + 715, -53/17*e^4 - 83/17*e^3 + 1312/17*e^2 + 1271/17*e - 432, 96/17*e^4 + 142/17*e^3 - 2354/17*e^2 - 2086/17*e + 742, 80/17*e^4 + 90/17*e^3 - 1905/17*e^2 - 1200/17*e + 572, 15/17*e^4 + 19/17*e^3 - 388/17*e^2 - 259/17*e + 114, 129/17*e^4 + 160/17*e^3 - 3160/17*e^2 - 2292/17*e + 980, -2*e^4 - 3*e^3 + 45*e^2 + 48*e - 223, -14/17*e^4 - 3/17*e^3 + 361/17*e^2 + 40/17*e - 133, -116/17*e^4 - 156/17*e^3 + 2809/17*e^2 + 2403/17*e - 887, 74/17*e^4 + 113/17*e^3 - 1760/17*e^2 - 1671/17*e + 564, 82/17*e^4 + 88/17*e^3 - 1942/17*e^2 - 1230/17*e + 576, 57/17*e^4 + 96/17*e^3 - 1403/17*e^2 - 1501/17*e + 435, 10/17*e^4 + 7/17*e^3 - 287/17*e^2 - 14/17*e + 85, 1/17*e^4 - 1/17*e^3 - 44/17*e^2 + 121/17*e + 38, -39/17*e^4 - 63/17*e^3 + 968/17*e^2 + 993/17*e - 300, 18/17*e^4 - 18/17*e^3 - 367/17*e^2 + 257/17*e + 65, -32/17*e^4 - 19/17*e^3 + 745/17*e^2 + 344/17*e - 217, 160/17*e^4 + 214/17*e^3 - 3844/17*e^2 - 3080/17*e + 1178, 49/17*e^4 + 70/17*e^3 - 1221/17*e^2 - 1177/17*e + 389, -82/17*e^4 - 122/17*e^3 + 1942/17*e^2 + 1604/17*e - 592, 13/17*e^4 + 21/17*e^3 - 470/17*e^2 - 382/17*e + 214, -101/17*e^4 - 103/17*e^3 + 2438/17*e^2 + 1634/17*e - 762, 98/17*e^4 + 123/17*e^3 - 2323/17*e^2 - 1878/17*e + 707]; 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;