/* 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, -5, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [4, 2, -w^3 + 4*w + 1], [5, 5, w - 1], [7, 7, -w^2 + w + 2], [13, 13, -w^3 + w^2 + 4*w], [13, 13, -w^2 + 3], [17, 17, -w^2 + 2], [19, 19, -w^3 + 5*w], [31, 31, -w^2 + 2*w + 3], [37, 37, -2*w^3 + w^2 + 8*w - 1], [43, 43, -w - 3], [53, 53, -w^3 + 2*w^2 + 3*w - 2], [53, 53, w^3 - 6*w - 2], [59, 59, 2*w^3 - w^2 - 10*w - 2], [61, 61, 2*w^3 - w^2 - 10*w], [61, 61, 2*w^3 - w^2 - 8*w], [71, 71, 2*w^3 - 9*w - 2], [73, 73, -w^3 - w^2 + 6*w + 3], [81, 3, -3], [83, 83, -w^3 + 2*w^2 + 3*w - 3], [101, 101, 2*w^3 - 8*w - 3], [125, 5, w^3 + w^2 - 4*w - 6], [127, 127, 2*w^3 - w^2 - 7*w - 2], [127, 127, w^3 - 7*w - 7], [131, 131, w^3 - 3*w - 4], [131, 131, 3*w^3 - 13*w - 7], [137, 137, w^2 + w - 4], [137, 137, -w^3 + 2*w^2 + 4*w - 4], [149, 149, 3*w + 5], [157, 157, -2*w^3 - w^2 + 7*w + 2], [157, 157, -2*w^3 + 2*w^2 + 9*w - 1], [163, 163, -2*w^3 + w^2 + 7*w], [163, 163, -3*w^3 + w^2 + 15*w + 3], [163, 163, 2*w^3 - 7*w - 1], [163, 163, 2*w^3 - 11*w - 5], [167, 167, -2*w^3 + w^2 + 9*w - 4], [167, 167, w^3 - w^2 - 7*w + 1], [167, 167, w^3 - w^2 - 7*w - 1], [167, 167, w^2 - 2*w - 5], [169, 13, w^3 + w^2 - 6*w - 2], [173, 173, -3*w^3 + 13*w + 3], [181, 181, 2*w^2 + w - 4], [191, 191, 2*w^3 - 10*w - 1], [191, 191, -2*w^3 + 2*w^2 + 6*w - 3], [193, 193, 3*w^3 - 13*w - 6], [197, 197, -w^3 + w^2 + 2*w - 3], [197, 197, w^3 + w^2 - 7*w - 7], [199, 199, -w^2 + 2*w + 6], [223, 223, -2*w^3 + w^2 + 11*w + 2], [227, 227, -3*w^3 + 2*w^2 + 15*w - 1], [229, 229, -w^3 + w^2 + 6*w - 5], [233, 233, 2*w^3 - 11*w - 4], [239, 239, -2*w^3 + 11*w + 3], [251, 251, 2*w^3 - w^2 - 10*w - 4], [257, 257, -2*w^3 + w^2 + 6*w + 1], [257, 257, w^3 - 7*w - 2], [263, 263, 2*w^3 - 9*w], [269, 269, w^3 - 4*w - 6], [269, 269, -2*w^3 + 8*w - 3], [271, 271, 2*w^2 - w - 5], [277, 277, -3*w^3 + 2*w^2 + 11*w - 2], [277, 277, 3*w^3 - w^2 - 15*w - 5], [281, 281, 2*w^2 - 5], [283, 283, -3*w^3 + 2*w^2 + 12*w + 2], [283, 283, -2*w^3 + w^2 + 6*w + 6], [293, 293, -2*w^3 - w^2 + 12*w + 7], [293, 293, -2*w^3 + 9*w - 1], [311, 311, -3*w^3 + w^2 + 12*w + 4], [313, 313, -w^3 + w^2 + 2*w - 4], [313, 313, -4*w^3 + 2*w^2 + 16*w - 3], [331, 331, 3*w^3 - w^2 - 16*w - 5], [331, 331, 2*w^3 - 2*w^2 - 8*w - 1], [343, 7, 2*w^3 - 3*w^2 - 9*w + 6], [359, 359, -2*w^3 + 10*w - 1], [367, 367, -w^3 + 2*w^2 + w - 4], [367, 367, 3*w^3 - 11*w - 1], [373, 373, -w^3 + w^2 + 2*w - 5], [373, 373, 2*w^3 - 2*w^2 - 11*w - 1], [389, 389, -2*w^3 + 3*w^2 + 8*w - 6], [409, 409, 3*w^3 - 2*w^2 - 13*w - 1], [419, 419, 3*w^3 - w^2 - 12*w - 1], [419, 419, w^3 - w^2 - 4*w - 4], [439, 439, w^2 + 2*w - 4], [439, 439, 3*w^3 + w^2 - 11*w - 5], [443, 443, 2*w^3 + 2*w^2 - 7*w - 8], [449, 449, 4*w^3 - w^2 - 16*w - 8], [457, 457, 2*w^3 + w^2 - 11*w - 4], [461, 461, -5*w^3 + 2*w^2 + 24*w + 2], [463, 463, -w^3 + 3*w^2 + 6*w - 5], [467, 467, w^3 - 3*w^2 - 6*w + 4], [467, 467, 3*w^3 - w^2 - 12*w - 2], [479, 479, -w^3 + w^2 + 4*w - 6], [479, 479, -w^3 + 2*w + 8], [491, 491, -w^3 + 2*w^2 + 2*w - 6], [491, 491, w - 5], [499, 499, 2*w^3 + w^2 - 10*w - 4], [509, 509, -4*w^3 + w^2 + 20*w + 5], [521, 521, w^3 - w + 6], [523, 523, -4*w^3 + 18*w + 5], [523, 523, -2*w^3 - 2*w^2 + 9*w + 12], [557, 557, w^2 - 3*w - 6], [571, 571, 3*w^3 - 11*w - 8], [577, 577, 2*w^3 + 2*w^2 - 11*w - 7], [577, 577, -3*w^3 + 4*w^2 + 11*w - 6], [587, 587, -w^3 - w^2 + 7*w + 11], [593, 593, w^3 + w^2 - 8*w - 8], [593, 593, -2*w^3 + 4*w^2 + 6*w - 9], [599, 599, 4*w^3 - 17*w - 9], [599, 599, w^3 + w^2 - 6*w - 10], [601, 601, 3*w^2 - 10], [607, 607, 4*w^3 - w^2 - 18*w - 8], [619, 619, w^2 - w - 8], [641, 641, -w^3 + 3*w^2 + 2*w - 10], [643, 643, 3*w^3 - w^2 - 16*w + 1], [653, 653, -2*w^3 + 2*w^2 + 9*w + 3], [677, 677, w^3 + w^2 - 8*w - 11], [677, 677, 2*w^3 - w^2 - 12*w - 3], [683, 683, 3*w^3 - 2*w^2 - 13*w - 2], [683, 683, 3*w^3 + 2*w^2 - 16*w - 10], [691, 691, -2*w^3 - 2*w^2 + 12*w + 15], [691, 691, 3*w^3 - 2*w^2 - 15*w - 2], [691, 691, w^3 - 2*w^2 - 6*w + 6], [691, 691, -4*w^3 + 3*w^2 + 15*w - 2], [719, 719, 2*w^3 - 7*w - 8], [719, 719, w^2 + 2*w - 5], [727, 727, -w^3 + 3*w^2 + 4*w - 8], [727, 727, 2*w^3 - 9*w - 9], [733, 733, w^3 - 8*w - 2], [739, 739, -w^3 - 2*w^2 + 4*w + 10], [743, 743, w^3 - w^2 - 2*w + 8], [743, 743, 4*w^3 - w^2 - 17*w - 4], [751, 751, -2*w^3 + 2*w^2 + 6*w - 5], [751, 751, w^3 - 5*w - 7], [769, 769, 3*w^3 - 2*w^2 - 10*w], [787, 787, -2*w^3 + 3*w^2 + 8*w], [787, 787, 3*w^3 - w^2 - 10*w - 4], [809, 809, 3*w^3 + w^2 - 16*w - 6], [811, 811, 3*w^3 - 14*w - 2], [811, 811, -w^3 + 3*w^2 + 5*w - 5], [811, 811, w^3 - 2*w - 6], [811, 811, 3*w^3 - w^2 - 11*w - 3], [823, 823, w^3 + 2*w^2 - 3*w - 8], [827, 827, -w^3 + 3*w^2 + 3*w - 7], [829, 829, w^3 + w^2 - 3*w - 7], [853, 853, 3*w^3 + w^2 - 14*w - 6], [857, 857, 3*w^3 + w^2 - 13*w - 5], [857, 857, 3*w^3 - 3*w^2 - 11*w - 1], [859, 859, 4*w^3 - 2*w^2 - 18*w - 3], [859, 859, 3*w^3 - 11*w - 3], [863, 863, 3*w^3 + w^2 - 17*w - 9], [877, 877, -w^3 - 2*w^2 + 9*w + 1], [877, 877, 5*w^3 + w^2 - 23*w - 11], [911, 911, w^2 - 4*w - 6], [911, 911, -2*w^3 + 2*w^2 + 10*w + 3], [919, 919, -4*w^3 + 3*w^2 + 18*w - 1], [929, 929, 2*w^3 - 13*w - 12], [929, 929, -w^3 + 2*w^2 + 7*w - 5], [953, 953, -3*w^3 + 3*w^2 + 13*w - 1], [967, 967, -3*w^3 + 4*w^2 + 13*w - 7], [971, 971, 2*w^2 - 3*w - 12], [971, 971, 3*w^3 - w^2 - 16*w - 3]]; primes := [ideal : I in primesArray]; heckePol := x^7 - 8*x^6 + 9*x^5 + 62*x^4 - 136*x^3 - 62*x^2 + 285*x - 139; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -21/103*e^6 + e^5 + 120/103*e^4 - 960/103*e^3 + 223/103*e^2 + 1953/103*e - 1073/103, 11/309*e^6 - 70/103*e^4 - 71/309*e^3 + 342/103*e^2 + 110/309*e - 929/309, -17/103*e^6 + e^5 + 53/103*e^4 - 1042/103*e^3 + 568/103*e^2 + 2405/103*e - 1673/103, 98/309*e^6 - 2*e^5 - 15/103*e^4 + 5407/309*e^3 - 1457/103*e^2 - 10144/309*e + 9196/309, 65/309*e^6 - e^5 - 114/103*e^4 + 2839/309*e^3 - 423/103*e^2 - 6148/309*e + 5494/309, -43/309*e^6 + e^5 - 26/103*e^4 - 2981/309*e^3 + 1004/103*e^2 + 6677/309*e - 6116/309, 43/309*e^6 - e^5 + 26/103*e^4 + 2672/309*e^3 - 798/103*e^2 - 4205/309*e + 3644/309, -1, -29/309*e^6 + 222/103*e^4 + 131/309*e^3 - 1426/103*e^2 - 599/309*e + 6101/309, -52/309*e^6 + e^5 - 53/103*e^4 - 2024/309*e^3 + 1183/103*e^2 + 2879/309*e - 4148/309, 217/309*e^6 - 4*e^5 - 173/103*e^4 + 10847/309*e^3 - 2439/103*e^2 - 20078/309*e + 18023/309, -8/103*e^6 + 134/103*e^4 + 61/103*e^3 - 381/103*e^2 - 389/103*e - 139/103, -16/309*e^6 + e^5 - 151/103*e^4 - 3071/309*e^3 + 982/103*e^2 + 6329/309*e - 2750/309, -169/309*e^6 + 3*e^5 + 214/103*e^4 - 8123/309*e^3 + 935/103*e^2 + 14996/309*e - 7301/309, 37/309*e^6 - e^5 + 8/103*e^4 + 3104/309*e^3 - 404/103*e^2 - 6737/309*e + 1763/309, 29/103*e^6 - e^5 - 254/103*e^4 + 796/103*e^3 + 364/103*e^2 - 1049/103*e + 1212/103, -5/309*e^6 - 15/103*e^4 + 566/309*e^3 - 15/103*e^2 - 2522/309*e + 2810/309, 64/309*e^6 - 2*e^5 + 192/103*e^4 + 5795/309*e^3 - 2486/103*e^2 - 12029/309*e + 14090/309, -59/103*e^6 + 3*e^5 + 293/103*e^4 - 2859/103*e^3 + 705/103*e^2 + 5796/103*e - 2480/103, -113/309*e^6 + e^5 + 485/103*e^4 - 2473/309*e^3 - 1987/103*e^2 + 2887/309*e + 6650/309, 41/103*e^6 - 2*e^5 - 249/103*e^4 + 1786/103*e^3 + 266/103*e^2 - 3092/103*e + 339/103, -1/309*e^6 + e^5 - 312/103*e^4 - 3224/309*e^3 + 2469/103*e^2 + 9569/309*e - 13652/309, -18/103*e^6 + e^5 + 44/103*e^4 - 970/103*e^3 + 765/103*e^2 + 2189/103*e - 1729/103, -49/309*e^6 + e^5 - 44/103*e^4 - 2858/309*e^3 + 1707/103*e^2 + 6308/309*e - 10469/309, 70/309*e^6 - 2*e^5 + 107/103*e^4 + 5981/309*e^3 - 1747/103*e^2 - 12896/309*e + 10718/309, 143/309*e^6 - 4*e^5 + 223/103*e^4 + 12364/309*e^3 - 4412/103*e^2 - 27616/309*e + 28093/309, 9/103*e^6 - e^5 + 81/103*e^4 + 1103/103*e^3 - 846/103*e^2 - 3206/103*e + 1946/103, -32/309*e^6 + e^5 + 7/103*e^4 - 3670/309*e^3 + 213/103*e^2 + 11113/309*e - 1792/309, 113/309*e^6 - 3*e^5 + 133/103*e^4 + 9271/309*e^3 - 3266/103*e^2 - 20191/309*e + 20542/309, -140/309*e^6 + 2*e^5 + 301/103*e^4 - 4855/309*e^3 - 420/103*e^2 + 7870/309*e - 424/309, 142/309*e^6 - 2*e^5 - 398/103*e^4 + 6050/309*e^3 + 323/103*e^2 - 12485/309*e + 8261/309, -1/3*e^6 + 2*e^5 - 50/3*e^3 + 15*e^2 + 98/3*e - 50/3, 1/103*e^6 - e^5 + 318/103*e^4 + 855/103*e^3 - 2154/103*e^2 - 1226/103*e + 2219/103, 15/103*e^6 - 380/103*e^4 + 53/103*e^3 + 2401/103*e^2 - 468/103*e - 2765/103, -47/309*e^6 + 374/103*e^4 + 191/309*e^3 - 2510/103*e^2 - 1397/309*e + 11582/309, -62/309*e^6 + 2*e^5 - 289/103*e^4 - 5527/309*e^3 + 3316/103*e^2 + 12667/309*e - 15523/309, -380/309*e^6 + 7*e^5 + 302/103*e^4 - 19093/309*e^3 + 4319/103*e^2 + 35443/309*e - 31786/309, 386/309*e^6 - 7*e^5 - 387/103*e^4 + 19897/309*e^3 - 4198/103*e^2 - 41254/309*e + 33667/309, -151/309*e^6 + 3*e^5 + 62/103*e^4 - 8801/309*e^3 + 2637/103*e^2 + 20738/309*e - 21743/309, 7/103*e^6 - 246/103*e^4 + 320/103*e^3 + 1505/103*e^2 - 1578/103*e - 1256/103, 64/103*e^6 - 4*e^5 - 42/103*e^4 + 3838/103*e^3 - 3132/103*e^2 - 8218/103*e + 6777/103, 281/309*e^6 - 6*e^5 + 19/103*e^4 + 16024/309*e^3 - 4307/103*e^2 - 29326/309*e + 24697/309, -12/103*e^6 + 2*e^5 - 417/103*e^4 - 1917/103*e^3 + 3703/103*e^2 + 3691/103*e - 4792/103, 188/309*e^6 - 4*e^5 + 49/103*e^4 + 10978/309*e^3 - 3659/103*e^2 - 22840/309*e + 21034/309, -19/309*e^6 + 46/103*e^4 + 1162/309*e^3 - 160/103*e^2 - 6370/309*e - 137/309, -167/309*e^6 + 4*e^5 - 192/103*e^4 - 10945/309*e^3 + 4134/103*e^2 + 21814/309*e - 23566/309, -25/103*e^6 + 2*e^5 - 122/103*e^4 - 1702/103*e^3 + 1526/103*e^2 + 2428/103*e - 1812/103, 59/309*e^6 - e^5 - 132/103*e^4 + 3271/309*e^3 + 74/103*e^2 - 7753/309*e + 2686/309, -13/309*e^6 - e^5 + 476/103*e^4 + 2584/309*e^3 - 3129/103*e^2 - 6619/309*e + 10396/309, 224/309*e^6 - 4*e^5 - 358/103*e^4 + 12094/309*e^3 - 1079/103*e^2 - 25879/309*e + 11308/309, -5/309*e^6 - e^5 + 294/103*e^4 + 3656/309*e^3 - 2075/103*e^2 - 9320/309*e + 5282/309, 319/309*e^6 - 6*e^5 - 176/103*e^4 + 16790/309*e^3 - 4708/103*e^2 - 36053/309*e + 36095/309, 5/309*e^6 + e^5 - 294/103*e^4 - 3347/309*e^3 + 1972/103*e^2 + 7157/309*e - 6518/309, -33/103*e^6 + 2*e^5 - 91/103*e^4 - 1538/103*e^3 + 2278/103*e^2 + 2554/103*e - 3599/103, 44/103*e^6 - e^5 - 428/103*e^4 + 334/103*e^3 + 1117/103*e^2 + 1058/103*e - 2068/103, 96/103*e^6 - 5*e^5 - 578/103*e^4 + 5242/103*e^3 - 578/103*e^2 - 11709/103*e + 4758/103, -316/309*e^6 + 6*e^5 + 185/103*e^4 - 16388/309*e^3 + 4099/103*e^2 + 31139/309*e - 26966/309, -43/103*e^6 + 2*e^5 + 128/103*e^4 - 1436/103*e^3 + 849/103*e^2 + 1939/103*e - 451/103, 81/103*e^6 - 3*e^5 - 713/103*e^4 + 2717/103*e^3 + 1141/103*e^2 - 5576/103*e + 2785/103, -151/309*e^6 + 4*e^5 - 144/103*e^4 - 13127/309*e^3 + 4388/103*e^2 + 31244/309*e - 30086/309, 118/309*e^6 - 2*e^5 - 58/103*e^4 + 4997/309*e^3 - 1809/103*e^2 - 10253/309*e + 14024/309, -53/103*e^6 + 4*e^5 - 168/103*e^4 - 4012/103*e^3 + 4364/103*e^2 + 9358/103*e - 9457/103, -4/103*e^6 - 36/103*e^4 + 82/103*e^3 + 1200/103*e^2 + 63/103*e - 2902/103, 2/3*e^6 - 4*e^5 - e^4 + 106/3*e^3 - 24*e^2 - 205/3*e + 121/3, -153/103*e^6 + 8*e^5 + 580/103*e^4 - 7627/103*e^3 + 4185/103*e^2 + 15465/103*e - 12688/103, -104/309*e^6 + e^5 + 409/103*e^4 - 2812/309*e^3 - 1136/103*e^2 + 4522/309*e + 4682/309, -358/309*e^6 + 6*e^5 + 471/103*e^4 - 15836/309*e^3 + 2531/103*e^2 + 26084/309*e - 21902/309, -218/309*e^6 + 4*e^5 + 376/103*e^4 - 13144/309*e^3 + 1303/103*e^2 + 31501/309*e - 17770/309, -1/309*e^6 - 3/103*e^4 + 175/309*e^3 + 512/103*e^2 - 4645/309*e - 6236/309, 115/309*e^6 - 3*e^5 + 139/103*e^4 + 8612/309*e^3 - 2848/103*e^2 - 18008/309*e + 17255/309, 559/309*e^6 - 10*e^5 - 589/103*e^4 + 27938/309*e^3 - 5636/103*e^2 - 56210/309*e + 49535/309, 217/309*e^6 - 4*e^5 - 276/103*e^4 + 11774/309*e^3 - 1306/103*e^2 - 25640/309*e + 9371/309, -52/103*e^6 + 4*e^5 - 262/103*e^4 - 3466/103*e^3 + 3858/103*e^2 + 6381/103*e - 5178/103, 185/309*e^6 - 4*e^5 - 63/103*e^4 + 11503/309*e^3 - 2020/103*e^2 - 23488/309*e + 12832/309, 182/309*e^6 - 2*e^5 - 587/103*e^4 + 5848/309*e^3 + 855/103*e^2 - 16411/309*e + 5866/309, 43/309*e^6 - e^5 + 26/103*e^4 + 3290/309*e^3 - 1416/103*e^2 - 7604/309*e + 7352/309, 50/103*e^6 - 2*e^5 - 477/103*e^4 + 2168/103*e^3 + 656/103*e^2 - 5062/103*e + 2697/103, 175/103*e^6 - 9*e^5 - 691/103*e^4 + 8515/103*e^3 - 4502/103*e^2 - 18026/103*e + 14538/103, -143/309*e^6 + 2*e^5 + 395/103*e^4 - 5875/309*e^3 - 944/103*e^2 + 13402/309*e + 3116/309, 49/309*e^6 - 2*e^5 + 250/103*e^4 + 6257/309*e^3 - 2840/103*e^2 - 12488/309*e + 16958/309, 83/309*e^6 - 369/103*e^4 - 1238/309*e^3 + 352/103*e^2 + 5774/309*e + 6811/309, -311/309*e^6 + 6*e^5 + 303/103*e^4 - 17263/309*e^3 + 2569/103*e^2 + 35206/309*e - 21742/309, -7/309*e^6 - e^5 + 494/103*e^4 + 1225/309*e^3 - 2802/103*e^2 + 2711/309*e + 7024/309, 256/309*e^6 - 4*e^5 - 365/103*e^4 + 9893/309*e^3 - 1395/103*e^2 - 16907/309*e + 9701/309, 413/309*e^6 - 7*e^5 - 615/103*e^4 + 19807/309*e^3 - 2778/103*e^2 - 39130/309*e + 33016/309, -135/103*e^6 + 7*e^5 + 330/103*e^4 - 6245/103*e^3 + 5274/103*e^2 + 12864/103*e - 13637/103, -295/309*e^6 + 5*e^5 + 248/103*e^4 - 12647/309*e^3 + 3544/103*e^2 + 22079/309*e - 29498/309, -176/309*e^6 + 4*e^5 - 219/103*e^4 - 10297/309*e^3 + 4519/103*e^2 + 18016/309*e - 20053/309, 154/309*e^6 - 3*e^5 - 156/103*e^4 + 9821/309*e^3 - 2113/103*e^2 - 24725/309*e + 17894/309, 151/309*e^6 - 4*e^5 + 350/103*e^4 + 10964/309*e^3 - 5933/103*e^2 - 19502/309*e + 29468/309, -54/103*e^6 + 4*e^5 - 74/103*e^4 - 4146/103*e^3 + 3737/103*e^2 + 8833/103*e - 8174/103, -557/309*e^6 + 10*e^5 + 698/103*e^4 - 29215/309*e^3 + 4612/103*e^2 + 63028/309*e - 43552/309, -2*e^5 + 4*e^4 + 24*e^3 - 24*e^2 - 52*e + 12, -48/103*e^6 + 4*e^5 - 226/103*e^4 - 4063/103*e^3 + 4512/103*e^2 + 9408/103*e - 11649/103, 40/309*e^6 - 189/103*e^4 - 511/309*e^3 + 326/103*e^2 - 527/309*e + 9965/309, -404/309*e^6 + 6*e^5 + 848/103*e^4 - 17674/309*e^3 + 1878/103*e^2 + 38602/309*e - 34057/309, -7/103*e^6 + 246/103*e^4 - 114/103*e^3 - 2020/103*e^2 + 1166/103*e + 2492/103, -8/103*e^6 + 134/103*e^4 + 164/103*e^3 - 896/103*e^2 - 389/103*e + 685/103, 17/309*e^6 - 2*e^5 + 566/103*e^4 + 5677/309*e^3 - 4584/103*e^2 - 12190/309*e + 21964/309, -14/103*e^6 + e^5 + 80/103*e^4 - 1155/103*e^3 - 23/103*e^2 + 2126/103*e + 1997/103, -148/309*e^6 + 2*e^5 + 380/103*e^4 - 5000/309*e^3 - 650/103*e^2 + 6245/309*e - 3344/309, -36/103*e^6 + 2*e^5 - 15/103*e^4 - 1631/103*e^3 + 2354/103*e^2 + 2730/103*e - 4694/103, 581/309*e^6 - 11*e^5 - 420/103*e^4 + 29959/309*e^3 - 6394/103*e^2 - 55063/309*e + 43969/309, 15/103*e^6 - e^5 - 174/103*e^4 + 1495/103*e^3 + 547/103*e^2 - 4073/103*e + 943/103, 644/309*e^6 - 11*e^5 - 746/103*e^4 + 28513/309*e^3 - 4660/103*e^2 - 50107/309*e + 40390/309, 122/103*e^6 - 7*e^5 - 138/103*e^4 + 6666/103*e^3 - 6730/103*e^2 - 14951/103*e + 16514/103, -169/309*e^6 + 3*e^5 + 214/103*e^4 - 8741/309*e^3 + 1553/103*e^2 + 16850/309*e - 19043/309, -139/309*e^6 + 2*e^5 + 98/103*e^4 - 4103/309*e^3 + 2055/103*e^2 + 8189/309*e - 17054/309, 31/309*e^6 - 3*e^5 + 917/103*e^4 + 7553/309*e^3 - 7323/103*e^2 - 14213/309*e + 29855/309, 152/309*e^6 - 3*e^5 - 59/103*e^4 + 8317/309*e^3 - 2222/103*e^2 - 17020/309*e + 15001/309, -211/309*e^6 + 4*e^5 + 191/103*e^4 - 11897/309*e^3 + 2354/103*e^2 + 26936/309*e - 14597/309, 310/309*e^6 - 4*e^5 - 924/103*e^4 + 12494/309*e^3 + 827/103*e^2 - 30272/309*e + 11180/309, -140/103*e^6 + 8*e^5 + 491/103*e^4 - 8048/103*e^3 + 3684/103*e^2 + 17243/103*e - 10209/103, -87/103*e^6 + 5*e^5 + 453/103*e^4 - 5066/103*e^3 - 268/103*e^2 + 11078/103*e - 958/103, 79/309*e^6 - 3*e^5 + 443/103*e^4 + 8423/309*e^3 - 4501/103*e^2 - 18677/309*e + 24509/309, -211/309*e^6 + 5*e^5 - 15/103*e^4 - 15605/309*e^3 + 3693/103*e^2 + 34352/309*e - 20159/309, 160/309*e^6 - 4*e^5 + 274/103*e^4 + 10007/309*e^3 - 3949/103*e^2 - 14777/309*e + 15758/309, 679/309*e^6 - 12*e^5 - 847/103*e^4 + 33203/309*e^3 - 4761/103*e^2 - 63662/309*e + 36170/309, 416/309*e^6 - 5*e^5 - 1224/103*e^4 + 12484/309*e^3 + 2690/103*e^2 - 21178/309*e + 4756/309, -329/309*e^6 + 6*e^5 + 249/103*e^4 - 16894/309*e^3 + 4163/103*e^2 + 37189/309*e - 25840/309, 358/309*e^6 - 6*e^5 - 471/103*e^4 + 16454/309*e^3 - 2840/103*e^2 - 32573/309*e + 23447/309, 445/309*e^6 - 6*e^5 - 1137/103*e^4 + 15752/309*e^3 + 1335/103*e^2 - 23978/309*e + 2363/309, 874/309*e^6 - 16*e^5 - 983/103*e^4 + 46664/309*e^3 - 8605/103*e^2 - 98174/309*e + 79844/309, -87/103*e^6 + 4*e^5 + 453/103*e^4 - 3315/103*e^3 + 1277/103*e^2 + 3559/103*e - 5387/103, -184/103*e^6 + 12*e^5 - 8/103*e^4 - 10957/103*e^3 + 7820/103*e^2 + 20923/103*e - 14630/103, 29/103*e^6 - 3*e^5 + 261/103*e^4 + 3577/103*e^3 - 4271/103*e^2 - 9495/103*e + 7804/103, 245/309*e^6 - 5*e^5 - 89/103*e^4 + 14908/309*e^3 - 3694/103*e^2 - 34012/309*e + 19900/309, 18/103*e^6 - 44/103*e^4 - 987/103*e^3 - 662/103*e^2 + 4506/103*e + 184/103, -62/309*e^6 - e^5 + 844/103*e^4 + 3125/309*e^3 - 5954/103*e^2 - 6491/309*e + 22175/309, 401/309*e^6 - 9*e^5 + 173/103*e^4 + 24070/309*e^3 - 7037/103*e^2 - 43576/309*e + 32962/309, -746/309*e^6 + 15*e^5 + 337/103*e^4 - 43108/309*e^3 + 10740/103*e^2 + 90493/309*e - 78238/309, -214/309*e^6 + 4*e^5 + 285/103*e^4 - 11681/309*e^3 + 1521/103*e^2 + 23507/309*e - 19709/309, -337/309*e^6 + 6*e^5 + 328/103*e^4 - 15803/309*e^3 + 3109/103*e^2 + 29693/309*e - 31850/309, -30/103*e^6 + 2*e^5 - 167/103*e^4 - 1239/103*e^3 + 1790/103*e^2 + 1245/103*e + 895/103, -275/309*e^6 + 5*e^5 + 308/103*e^4 - 13675/309*e^3 + 2471/103*e^2 + 21970/309*e - 15400/309, 42/103*e^6 - 2*e^5 - 240/103*e^4 + 1920/103*e^3 - 549/103*e^2 - 3803/103*e + 2970/103, 145/103*e^6 - 8*e^5 - 240/103*e^4 + 6967/103*e^3 - 5905/103*e^2 - 11631/103*e + 13785/103, 114/103*e^6 - 5*e^5 - 725/103*e^4 + 4873/103*e^3 - 931/103*e^2 - 11426/103*e + 4221/103, 544/309*e^6 - 8*e^5 - 943/103*e^4 + 20984/309*e^3 - 2900/103*e^2 - 40910/309*e + 26447/309, 152/103*e^6 - 7*e^5 - 898/103*e^4 + 6463/103*e^3 - 1104/103*e^2 - 13003/103*e + 6967/103, -98/309*e^6 + 530/103*e^4 + 773/309*e^3 - 1942/103*e^2 + 1492/309*e + 2546/309, -179/309*e^6 + 6*e^5 - 434/103*e^4 - 19351/309*e^3 + 5746/103*e^2 + 47341/309*e - 35671/309, -169/103*e^6 + 10*e^5 + 230/103*e^4 - 9256/103*e^3 + 7028/103*e^2 + 17571/103*e - 12966/103, 295/309*e^6 - 9*e^5 + 988/103*e^4 + 23771/309*e^3 - 11269/103*e^2 - 42473/309*e + 52673/309, -340/309*e^6 + 5*e^5 + 937/103*e^4 - 14351/309*e^3 - 2462/103*e^2 + 30281/309*e + 2590/309, 6/103*e^6 - 255/103*e^4 + 495/103*e^3 + 981/103*e^2 - 2927/103*e + 645/103, -533/309*e^6 + 9*e^5 + 564/103*e^4 - 23218/309*e^3 + 4993/103*e^2 + 41020/309*e - 36337/309, -51/103*e^6 + 4*e^5 - 356/103*e^4 - 3744/103*e^3 + 5721/103*e^2 + 8760/103*e - 9963/103, -179/103*e^6 + 11*e^5 + 243/103*e^4 - 10596/103*e^3 + 7968/103*e^2 + 20767/103*e - 19706/103, -46/309*e^6 + 171/103*e^4 + 1561/309*e^3 + 171/103*e^2 - 11584/309*e - 2267/309, 182/309*e^6 - 3*e^5 + 31/103*e^4 + 5848/309*e^3 - 3471/103*e^2 - 6214/309*e + 17608/309, 36/103*e^6 - 1118/103*e^4 + 807/103*e^3 + 6401/103*e^2 - 3039/103*e - 5297/103, 55/309*e^6 - 247/103*e^4 - 1282/309*e^3 + 989/103*e^2 + 3022/309*e - 8971/309, -116/103*e^6 + 7*e^5 + 295/103*e^4 - 6892/103*e^3 + 3488/103*e^2 + 14084/103*e - 9174/103, -166/309*e^6 + 2*e^5 + 429/103*e^4 - 4631/309*e^3 - 910/103*e^2 + 7610/309*e + 13570/309, 254/309*e^6 - 4*e^5 - 268/103*e^4 + 8698/309*e^3 - 1916/103*e^2 - 6730/309*e + 13297/309, -181/309*e^6 + 3*e^5 + 281/103*e^4 - 8495/309*e^3 + 796/103*e^2 + 17657/309*e - 4265/309, 191/309*e^6 - 5*e^5 + 470/103*e^4 + 13234/309*e^3 - 6946/103*e^2 - 27445/309*e + 33871/309, -380/309*e^6 + 8*e^5 - 213/103*e^4 - 20329/309*e^3 + 8233/103*e^2 + 36061/309*e - 42910/309]; 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;