/*
  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<x> := PolynomialRing(Rationals());
g := P![-1, 3, 4, -5, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {17, 17, w^4 - w^3 - 5*w^2 + 3*w + 1}>;

primesArray := [
[3, 3, w^4 - w^3 - 5*w^2 + 3*w + 3],
[9, 3, -w^4 + 5*w^2 - 3],
[9, 3, -w^4 + w^3 + 5*w^2 - 3*w - 2],
[13, 13, -w^4 + w^3 + 4*w^2 - 3*w - 1],
[17, 17, w^4 - w^3 - 5*w^2 + 3*w + 1],
[19, 19, -w^3 + w^2 + 4*w - 2],
[23, 23, -w^2 + 3],
[31, 31, w^3 - 4*w + 2],
[32, 2, 2],
[37, 37, w^3 - 3*w - 1],
[53, 53, -2*w^4 + w^3 + 9*w^2 - 3*w - 2],
[59, 59, -w^4 + 5*w^2 + w - 4],
[61, 61, -w^4 + w^3 + 5*w^2 - 4*w],
[67, 67, -w^4 + 6*w^2 + 2*w - 4],
[71, 71, 2*w^4 - w^3 - 9*w^2 + 4*w + 5],
[79, 79, 2*w^4 - w^3 - 10*w^2 + 2*w + 7],
[83, 83, -w^4 + 2*w^3 + 5*w^2 - 7*w - 2],
[83, 83, -w^4 + w^3 + 4*w^2 - 3*w + 3],
[83, 83, w^4 - w^3 - 5*w^2 + 4*w - 1],
[83, 83, -w^4 + w^3 + 4*w^2 - 4*w - 2],
[83, 83, -2*w^4 + w^3 + 9*w^2 - 3*w - 5],
[97, 97, -w^4 - w^3 + 6*w^2 + 4*w - 4],
[101, 101, -w^4 + 4*w^2 + 2*w - 2],
[107, 107, -2*w^4 + w^3 + 11*w^2 - 3*w - 7],
[127, 127, w^3 - w^2 - 4*w],
[131, 131, w^4 - w^3 - 4*w^2 + 2*w - 1],
[137, 137, w^4 - w^3 - 6*w^2 + 3*w + 3],
[139, 139, -2*w^4 + w^3 + 9*w^2 - 3*w + 1],
[157, 157, 2*w^4 - w^3 - 9*w^2 + 4*w + 2],
[163, 163, -w^2 - 2*w + 3],
[167, 167, w^4 - 5*w^2 + 2*w + 1],
[169, 13, -2*w^4 + 2*w^3 + 9*w^2 - 7*w - 5],
[169, 13, w^4 - 2*w^3 - 5*w^2 + 8*w + 2],
[191, 191, 2*w^4 - w^3 - 9*w^2 + 3*w + 4],
[193, 193, 2*w^4 - 2*w^3 - 8*w^2 + 7*w],
[199, 199, -w^4 + 2*w^3 + 5*w^2 - 9*w - 3],
[211, 211, -3*w^4 + w^3 + 13*w^2 - 2*w - 1],
[227, 227, 2*w^3 - w^2 - 9*w + 1],
[233, 233, -w^3 + w^2 + 4*w + 1],
[251, 251, -3*w^4 + 14*w^2 + w - 2],
[257, 257, -2*w^4 + 2*w^3 + 10*w^2 - 6*w - 5],
[257, 257, -w^4 + w^3 + 5*w^2 - 6*w - 1],
[257, 257, 2*w^4 - 2*w^3 - 11*w^2 + 9*w + 9],
[257, 257, -2*w^4 + 2*w^3 + 9*w^2 - 6*w - 3],
[257, 257, 2*w^4 - w^3 - 10*w^2 + 5*w + 5],
[269, 269, -w^4 + 2*w^3 + 4*w^2 - 6*w - 2],
[271, 271, -w^4 + 4*w^2 + w - 3],
[277, 277, -2*w^4 + 9*w^2 + w - 4],
[281, 281, w^4 - w^3 - 3*w^2 + 3*w - 4],
[283, 283, 2*w^4 - 3*w^3 - 8*w^2 + 13*w],
[289, 17, -w^4 + w^3 + 7*w^2 - 5*w - 5],
[289, 17, 4*w^4 - 3*w^3 - 19*w^2 + 11*w + 6],
[293, 293, -2*w^4 + 11*w^2 - 7],
[317, 317, w^3 - 6*w],
[337, 337, -w^4 + 2*w^3 + 5*w^2 - 9*w - 1],
[337, 337, w^4 - 2*w^3 - 5*w^2 + 6*w + 3],
[337, 337, -w^4 + w^3 + 5*w^2 - 4*w - 6],
[337, 337, -2*w^3 + 8*w + 1],
[337, 337, w^2 + w - 5],
[347, 347, -2*w^4 + w^3 + 9*w^2 - 5*w - 3],
[349, 349, 3*w^4 - 2*w^3 - 13*w^2 + 8*w + 2],
[353, 353, w^4 - 3*w^3 - 4*w^2 + 12*w + 2],
[359, 359, 2*w^3 - w^2 - 7*w + 3],
[361, 19, w^4 - 2*w^3 - 4*w^2 + 7*w + 3],
[361, 19, w^4 - w^3 - 3*w^2 + 3*w - 3],
[367, 367, -2*w^4 + w^3 + 9*w^2 - w - 2],
[379, 379, -2*w^3 + w^2 + 7*w + 2],
[379, 379, -2*w^4 + w^3 + 10*w^2 - 4*w],
[379, 379, 2*w^3 - w^2 - 6*w - 2],
[379, 379, -2*w^3 + 2*w^2 + 6*w - 3],
[379, 379, -3*w^4 + 2*w^3 + 14*w^2 - 7*w - 6],
[383, 383, 2*w^4 + w^3 - 10*w^2 - 7*w + 4],
[383, 383, -2*w^3 + w^2 + 10*w],
[383, 383, w^3 + w^2 - 5*w - 1],
[383, 383, 3*w^4 + 3*w^3 - 15*w^2 - 15*w + 4],
[383, 383, -2*w^4 + 3*w^3 + 10*w^2 - 10*w - 2],
[389, 389, 3*w^4 - w^3 - 14*w^2 + 3*w + 3],
[397, 397, w^4 - 6*w^2 - 2*w + 5],
[397, 397, w^4 - 2*w^3 - 2*w^2 + 6*w - 5],
[397, 397, 2*w^4 - 2*w^3 - 11*w^2 + 10*w + 6],
[397, 397, w^4 - w^3 - 3*w^2 + 4*w - 6],
[397, 397, w^3 + w^2 - 3*w - 4],
[401, 401, w^3 - w^2 - 6*w + 3],
[401, 401, -w^4 + 4*w^2 + 2*w - 3],
[401, 401, -w^4 + 6*w^2 - w - 7],
[421, 421, -w^4 - w^3 + 5*w^2 + 5*w - 4],
[421, 421, -w^4 + w^3 + 6*w^2 - 3*w - 2],
[421, 421, -w^4 - w^3 + 6*w^2 + 6*w - 5],
[421, 421, 3*w^4 - w^3 - 15*w^2 + w + 10],
[421, 421, 2*w^4 - 9*w^2 + 2*w + 4],
[431, 431, 2*w^4 - 11*w^2 - 2*w + 11],
[439, 439, 2*w^4 - 3*w^3 - 9*w^2 + 10*w + 6],
[443, 443, -3*w^4 + 2*w^3 + 14*w^2 - 7*w - 2],
[449, 449, -w^4 + 2*w^3 + 5*w^2 - 6*w - 4],
[461, 461, 3*w^4 - 4*w^3 - 14*w^2 + 14*w + 5],
[463, 463, w^3 - 6*w - 1],
[467, 467, 3*w^4 - 3*w^3 - 13*w^2 + 10*w],
[487, 487, 3*w^4 - w^3 - 14*w^2 + w + 3],
[487, 487, 2*w^2 - w - 4],
[487, 487, -2*w^4 + 2*w^3 + 9*w^2 - 8*w - 5],
[487, 487, 2*w^4 - 10*w^2 - 3*w + 4],
[487, 487, -2*w^4 + w^3 + 8*w^2 - 3*w - 1],
[499, 499, -w^4 + 2*w^3 + 4*w^2 - 8*w - 2],
[499, 499, -2*w^4 + w^3 + 10*w^2 - w - 9],
[499, 499, -w^3 - 2*w^2 + 2*w + 5],
[499, 499, -w^4 + 4*w^2 + 4*w],
[499, 499, -2*w^4 + w^3 + 12*w^2 - 4*w - 9],
[509, 509, 3*w^4 - 2*w^3 - 16*w^2 + 6*w + 9],
[521, 521, -2*w^4 + w^3 + 10*w^2 - 5*w - 4],
[523, 523, w - 4],
[529, 23, -w^3 + 2*w^2 + 4*w - 4],
[529, 23, w^4 + 2*w^3 - 6*w^2 - 11*w + 4],
[569, 569, -2*w^4 + 11*w^2 - 10],
[571, 571, -2*w^4 + w^3 + 10*w^2 - 3*w - 2],
[593, 593, -2*w^4 + 3*w^3 + 10*w^2 - 11*w - 8],
[613, 613, -2*w^3 + 2*w^2 + 9*w - 5],
[617, 617, -2*w^4 + 10*w^2 - w - 7],
[631, 631, -2*w^4 + 2*w^3 + 7*w^2 - 8*w + 3],
[641, 641, -2*w^4 + 2*w^3 + 8*w^2 - 9*w + 3],
[643, 643, -3*w^4 + w^3 + 13*w^2 - w + 1],
[643, 643, w^4 + w^3 - 3*w^2 - 6*w - 2],
[643, 643, -w^4 - w^3 + 6*w^2 + 3*w - 4],
[643, 643, 2*w^4 - w^3 - 10*w^2 + 4*w + 1],
[643, 643, 2*w^4 - 2*w^3 - 8*w^2 + 6*w + 5],
[647, 647, -2*w^3 + w^2 + 6*w - 4],
[659, 659, 4*w^4 - 3*w^3 - 19*w^2 + 11*w + 10],
[661, 661, 2*w^4 + 3*w^3 - 11*w^2 - 14*w + 6],
[673, 673, w^4 - 3*w^3 - 3*w^2 + 10*w - 1],
[683, 683, w^4 - 2*w^3 - 4*w^2 + 6*w - 4],
[701, 701, w^3 - w^2 - 2*w + 4],
[727, 727, -w^4 + 6*w^2 + 2*w - 7],
[743, 743, w^4 - w^3 - 7*w^2 + 5*w + 9],
[769, 769, w^4 + 2*w^3 - 5*w^2 - 9*w + 4],
[787, 787, w^4 + w^3 - 7*w^2 - 3*w + 5],
[797, 797, -2*w^4 - w^3 + 10*w^2 + 4*w - 4],
[797, 797, w^4 - 3*w^3 - 5*w^2 + 12*w + 5],
[797, 797, -3*w^4 + 2*w^3 + 13*w^2 - 8*w],
[797, 797, -w^4 + w^3 + 3*w^2 - 5*w - 1],
[797, 797, 2*w^4 - 11*w^2 + 2*w + 7],
[809, 809, -4*w^4 + 2*w^3 + 18*w^2 - 6*w + 1],
[809, 809, 3*w^4 - 3*w^3 - 14*w^2 + 11*w + 8],
[809, 809, w^4 - 2*w^3 - 2*w^2 + 8*w - 4],
[809, 809, 2*w^4 - 8*w^2 - 2*w + 3],
[809, 809, 2*w^4 - 3*w^3 - 9*w^2 + 13*w + 4],
[821, 821, -2*w^4 + 2*w^3 + 8*w^2 - 6*w + 3],
[823, 823, -2*w^4 + 3*w^3 + 8*w^2 - 12*w - 1],
[829, 829, 2*w^4 - 3*w^3 - 10*w^2 + 13*w + 4],
[839, 839, -2*w^4 + 12*w^2 - w - 10],
[863, 863, 2*w^4 - w^3 - 9*w^2 + 5],
[877, 877, -w^4 + w^3 + 5*w^2 - 5*w - 7],
[881, 881, 2*w^4 - w^3 - 9*w^2 + w - 1],
[887, 887, 3*w^4 - 3*w^3 - 15*w^2 + 11*w + 6],
[907, 907, -3*w^4 + w^3 + 13*w^2 - w - 5],
[919, 919, w^4 + w^3 - 3*w^2 - 5*w - 3],
[929, 929, 2*w^4 - 3*w^3 - 10*w^2 + 9*w + 4],
[937, 937, -2*w^4 + 10*w^2 + 3*w - 7],
[941, 941, -w^4 + 2*w^3 + 3*w^2 - 6*w - 2],
[961, 31, 2*w^4 + 2*w^3 - 10*w^2 - 12*w + 5],
[961, 31, -w^4 + 5*w^2 - w - 7],
[967, 967, -2*w^4 + w^3 + 9*w^2 - 3*w + 2],
[991, 991, -3*w^4 + w^3 + 15*w^2 - 4*w - 5],
[997, 997, -3*w^4 + 15*w^2 - 5],
[997, 997, 2*w^4 - w^3 - 9*w^2 + 6*w + 2],
[997, 997, 2*w^4 - w^3 - 11*w^2 + w + 7],
[997, 997, 2*w^3 - 5*w - 1],
[997, 997, 2*w^4 - 3*w^3 - 6*w^2 + 5*w + 1]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^12 + 5*x^11 - 11*x^10 - 76*x^9 + 22*x^8 + 363*x^7 - 22*x^6 - 754*x^5 + 108*x^4 + 648*x^3 - 158*x^2 - 162*x + 39;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 217/927*e^11 + 1387/927*e^10 - 269/309*e^9 - 18619/927*e^8 - 16375/927*e^7 + 68554/927*e^6 + 72298/927*e^5 - 100811/927*e^4 - 86725/927*e^3 + 54292/927*e^2 + 8345/309*e - 1850/309, 1841/927*e^11 + 11498/927*e^10 - 2023/309*e^9 - 147824/927*e^8 - 141824/927*e^7 + 495482/927*e^6 + 570695/927*e^5 - 684550/927*e^4 - 646682/927*e^3 + 386237/927*e^2 + 63352/309*e - 19144/309, 3364/927*e^11 + 20878/927*e^10 - 3854/309*e^9 - 268897/927*e^8 - 254098/927*e^7 + 905407/927*e^6 + 1035241/927*e^5 - 1258106/927*e^4 - 1191760/927*e^3 + 714682/927*e^2 + 120410/309*e - 35426/309, -1, -1016/309*e^11 - 6353/309*e^10 + 1109/103*e^9 + 81644/309*e^8 + 78320/309*e^7 - 273830/309*e^6 - 313589/309*e^5 + 381826/309*e^4 + 355679/309*e^3 - 218861/309*e^2 - 35392/103*e + 10586/103, -222/103*e^11 - 1362/103*e^10 + 818/103*e^9 + 17550/103*e^8 + 16037/103*e^7 - 59161/103*e^6 - 65693/103*e^5 + 82306/103*e^4 + 75020/103*e^3 - 46881/103*e^2 - 22135/103*e + 6921/103, 740/927*e^11 + 4952/927*e^10 - 268/309*e^9 - 61178/927*e^8 - 74915/927*e^7 + 185942/927*e^6 + 264125/927*e^5 - 238990/927*e^4 - 273791/927*e^3 + 133919/927*e^2 + 24274/309*e - 6454/309, 1270/309*e^11 + 7864/309*e^10 - 1515/103*e^9 - 101746/309*e^8 - 93883/309*e^7 + 346768/309*e^6 + 386347/309*e^5 - 490724/309*e^4 - 448384/309*e^3 + 284005/309*e^2 + 46197/103*e - 14108/103, -1879/927*e^11 - 11617/927*e^10 + 2147/309*e^9 + 148846/927*e^8 + 141274/927*e^7 - 494197/927*e^6 - 569113/927*e^5 + 671117/927*e^4 + 645892/927*e^3 - 364645/927*e^2 - 65447/309*e + 16190/309, -568/309*e^11 - 3649/309*e^10 + 444/103*e^9 + 46225/309*e^8 + 50554/309*e^7 - 149035/309*e^6 - 197581/309*e^5 + 195659/309*e^4 + 226642/309*e^3 - 106246/309*e^2 - 23297/103*e + 5112/103, -601/309*e^11 - 3793/309*e^10 + 633/103*e^9 + 49243/309*e^8 + 48385/309*e^7 - 169297/309*e^6 - 201574/309*e^5 + 242858/309*e^4 + 242398/309*e^3 - 146758/309*e^2 - 26187/103*e + 8190/103, -1184/927*e^11 - 7367/927*e^10 + 1294/309*e^9 + 94733/927*e^8 + 92981/927*e^7 - 316418/927*e^6 - 385520/927*e^5 + 425953/927*e^4 + 460499/927*e^3 - 227990/927*e^2 - 49159/309*e + 10450/309, -2077/309*e^11 - 12790/309*e^10 + 2663/103*e^9 + 166645/309*e^8 + 147109/309*e^7 - 577453/309*e^6 - 615628/309*e^5 + 830402/309*e^4 + 714472/309*e^3 - 484735/309*e^2 - 71766/103*e + 23328/103, -2318/309*e^11 - 14366/309*e^10 + 2826/103*e^9 + 186560/309*e^8 + 169154/309*e^7 - 641585/309*e^6 - 698555/309*e^5 + 916858/309*e^4 + 807431/309*e^3 - 533489/309*e^2 - 81960/103*e + 26115/103, 262/927*e^11 + 1162/927*e^10 - 1229/309*e^9 - 20122/927*e^8 + 11696/927*e^7 + 111016/927*e^6 + 10999/927*e^5 - 223490/927*e^4 - 50905/927*e^3 + 156307/927*e^2 + 8915/309*e - 6890/309, -1598/309*e^11 - 9932/309*e^10 + 1886/103*e^9 + 128831/309*e^8 + 119564/309*e^7 - 441143/309*e^6 - 495251/309*e^5 + 622594/309*e^4 + 575297/309*e^3 - 356639/309*e^2 - 58111/103*e + 17575/103, -107/103*e^11 - 701/103*e^10 + 256/103*e^9 + 9074/103*e^8 + 9597/103*e^7 - 30987/103*e^6 - 38491/103*e^5 + 44318/103*e^4 + 45123/103*e^3 - 27402/103*e^2 - 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2791213/927*e^2 - 403997/309*e + 139400/309, 5024/309*e^11 + 31118/309*e^10 - 6273/103*e^9 - 405608/309*e^8 - 361787/309*e^7 + 1408514/309*e^6 + 1504931/309*e^5 - 2042638/309*e^4 - 1748090/309*e^3 + 1209506/309*e^2 + 179890/103*e - 61078/103, 2812/309*e^11 + 17458/309*e^10 - 2996/103*e^9 - 221167/309*e^8 - 214534/309*e^7 + 716962/309*e^6 + 821365/309*e^5 - 974288/309*e^4 - 884608/309*e^3 + 555613/309*e^2 + 83033/103*e - 28604/103, -3845/927*e^11 - 23726/927*e^10 + 4090/309*e^9 + 302081/927*e^8 + 303488/927*e^7 - 989282/927*e^6 - 1239599/927*e^5 + 1339753/927*e^4 + 1496306/927*e^3 - 746063/927*e^2 - 173608/309*e + 35017/309, 1756/103*e^11 + 10893/103*e^10 - 6294/103*e^9 - 140968/103*e^8 - 129030/103*e^7 + 481093/103*e^6 + 526214/103*e^5 - 684470/103*e^4 - 601351/103*e^3 + 398821/103*e^2 + 180543/103*e - 59360/103, 4663/927*e^11 + 29215/927*e^10 - 4889/309*e^9 - 373651/927*e^8 - 366394/927*e^7 + 1235476/927*e^6 + 1456615/927*e^5 - 1674461/927*e^4 - 1654552/927*e^3 + 913432/927*e^2 + 171266/309*e - 42791/309, -878/103*e^11 - 5498/103*e^10 + 2529/103*e^9 + 69557/103*e^8 + 71519/103*e^7 - 224118/103*e^6 - 279175/103*e^5 + 298666/103*e^4 + 314735/103*e^3 - 168871/103*e^2 - 96194/103*e + 27105/103, 26068/927*e^11 + 162283/927*e^10 - 29444/309*e^9 - 2089738/927*e^8 - 1978207/927*e^7 + 7043413/927*e^6 + 7999348/927*e^5 - 9871892/927*e^4 - 9158188/927*e^3 + 5694982/927*e^2 + 923612/309*e - 283949/309, -9620/309*e^11 - 59741/309*e^10 + 11106/103*e^9 + 770285/309*e^8 + 722060/309*e^7 - 2603264/309*e^6 - 2935517/309*e^5 + 3649165/309*e^4 + 3367703/309*e^3 - 2084864/309*e^2 - 338428/103*e + 97704/103, -18983/927*e^11 - 118604/927*e^10 + 21742/309*e^9 + 1536683/927*e^8 + 1439513/927*e^7 - 5254112/927*e^6 - 5902355/927*e^5 + 7455460/927*e^4 + 6849029/927*e^3 - 4340348/927*e^2 - 711745/309*e + 219154/309, 451/103*e^11 + 2895/103*e^10 - 1260/103*e^9 - 37332/103*e^8 - 37822/103*e^7 + 126225/103*e^6 + 151597/103*e^5 - 177330/103*e^4 - 177428/103*e^3 + 102936/103*e^2 + 57514/103*e - 17224/103];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {17, 17, w^4 - w^3 - 5*w^2 + 3*w + 1}>] := 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;