/*
  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 | {19, 19, -w^3 + w^2 + 4*w - 2}>;

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^11 + 5*x^10 - 8*x^9 - 70*x^8 - 33*x^7 + 265*x^6 + 276*x^5 - 323*x^4 - 451*x^3 + 77*x^2 + 204*x + 44;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 6023/11964*e^10 + 23261/11964*e^9 - 12159/1994*e^8 - 165535/5982*e^7 + 163121/11964*e^6 + 437815/3988*e^5 + 23851/1994*e^4 - 1813045/11964*e^3 - 431287/11964*e^2 + 756449/11964*e + 96461/5982, -6941/11964*e^10 - 25835/11964*e^9 + 14677/1994*e^8 + 185533/5982*e^7 - 239663/11964*e^6 - 499861/3988*e^5 - 1117/1994*e^4 + 2115439/11964*e^3 + 389773/11964*e^2 - 895487/11964*e - 125633/5982, 23099/11964*e^10 + 78413/11964*e^9 - 52181/1994*e^8 - 562273/5982*e^7 + 1065569/11964*e^6 + 1514303/3988*e^5 - 150133/1994*e^4 - 6422749/11964*e^3 - 375583/11964*e^2 + 2667749/11964*e + 314741/5982, -2955/3988*e^10 - 10045/3988*e^9 + 19937/1994*e^8 + 71293/1994*e^7 - 136637/3988*e^6 - 561401/3988*e^5 + 70341/1994*e^4 + 747093/3988*e^3 - 35261/3988*e^2 - 265321/3988*e - 21787/1994, -1, -2018/997*e^10 - 6929/997*e^9 + 26923/997*e^8 + 98693/997*e^7 - 87704/997*e^6 - 392758/997*e^5 + 63738/997*e^4 + 541809/997*e^3 + 39127/997*e^2 - 220590/997*e - 52040/997, 3533/5982*e^10 + 12995/5982*e^9 - 7271/997*e^8 - 91528/2991*e^7 + 104627/5982*e^6 + 237499/1994*e^5 + 12089/997*e^4 - 964519/5982*e^3 - 299293/5982*e^2 + 414899/5982*e + 83801/2991, -2801/1994*e^10 - 9535/1994*e^9 + 18761/997*e^8 + 68251/997*e^7 - 121555/1994*e^6 - 550819/1994*e^5 + 34092/997*e^4 + 786655/1994*e^3 + 120367/1994*e^2 - 343873/1994*e - 50902/997, -24209/11964*e^10 - 84575/11964*e^9 + 53375/1994*e^8 + 604165/5982*e^7 - 1014239/11964*e^6 - 1612081/3988*e^5 + 102319/1994*e^4 + 6716683/11964*e^3 + 722857/11964*e^2 - 2705123/11964*e - 406511/5982, -7531/3988*e^10 - 25769/3988*e^9 + 50893/1994*e^8 + 185327/1994*e^7 - 343229/3988*e^6 - 1507081/3988*e^5 + 138775/1994*e^4 + 2154621/3988*e^3 + 136219/3988*e^2 - 909101/3988*e - 102951/1994, 390/997*e^10 + 1680/997*e^9 - 4053/997*e^8 - 23267/997*e^7 + 1474/997*e^6 + 85764/997*e^5 + 41516/997*e^4 - 98343/997*e^3 - 62788/997*e^2 + 25796/997*e + 20134/997, -10513/11964*e^10 - 36007/11964*e^9 + 22863/1994*e^8 + 250757/5982*e^7 - 417019/11964*e^6 - 629153/3988*e^5 + 36085/1994*e^4 + 2287307/11964*e^3 + 243305/11964*e^2 - 657139/11964*e - 98677/5982, 11699/1994*e^10 + 40349/1994*e^9 - 77827/997*e^8 - 287482/997*e^7 + 502509/1994*e^6 + 2290273/1994*e^5 - 173906/997*e^4 - 3165873/1994*e^3 - 245725/1994*e^2 + 1288363/1994*e + 148551/997, -4556/997*e^10 - 15101/997*e^9 + 62241/997*e^8 + 214783/997*e^7 - 217918/997*e^6 - 853009/997*e^5 + 216354/997*e^4 + 1174862/997*e^3 + 4113/997*e^2 - 474910/997*e - 95698/997, 18145/5982*e^10 + 61981/5982*e^9 - 41021/997*e^8 - 446573/2991*e^7 + 837547/5982*e^6 + 1215011/1994*e^5 - 114129/997*e^4 - 5244113/5982*e^3 - 410627/5982*e^2 + 2245057/5982*e + 309301/2991, 1202/997*e^10 + 3644/997*e^9 - 17814/997*e^8 - 53171/997*e^7 + 75831/997*e^6 + 223314/997*e^5 - 115968/997*e^4 - 334496/997*e^3 + 42947/997*e^2 + 142198/997*e + 27430/997, -5481/1994*e^10 - 19239/1994*e^9 + 36598/997*e^8 + 138812/997*e^7 - 238951/1994*e^6 - 1136337/1994*e^5 + 84354/997*e^4 + 1645129/1994*e^3 + 146949/1994*e^2 - 705455/1994*e - 95411/997, -4883/997*e^10 - 16663/997*e^9 + 65785/997*e^8 + 238824/997*e^7 - 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8015509/3988*e - 939515/1994, 35674/2991*e^10 + 120925/2991*e^9 - 159951/997*e^8 - 1722076/2991*e^7 + 1599811/2991*e^6 + 2284641/997*e^5 - 424657/997*e^4 - 9462038/2991*e^3 - 568400/2991*e^2 + 3810313/2991*e + 840020/2991, -46283/3988*e^10 - 161717/3988*e^9 + 306031/1994*e^8 + 1155643/1994*e^7 - 1935945/3988*e^6 - 9258585/3988*e^5 + 588073/1994*e^4 + 12875713/3988*e^3 + 1233415/3988*e^2 - 5207241/3988*e - 662647/1994, 3362/997*e^10 + 12105/997*e^9 - 43099/997*e^8 - 169917/997*e^7 + 126789/997*e^6 + 650612/997*e^5 - 74773/997*e^4 - 818578/997*e^3 - 20887/997*e^2 + 256692/997*e + 41082/997, -13171/2991*e^10 - 50908/2991*e^9 + 54182/997*e^8 + 734203/2991*e^7 - 393535/2991*e^6 - 998819/997*e^5 - 75810/997*e^4 + 4329317/2991*e^3 + 986174/2991*e^2 - 1938637/2991*e - 537104/2991, -4593/1994*e^10 - 13113/1994*e^9 + 33533/997*e^8 + 92138/997*e^7 - 272039/1994*e^6 - 719817/1994*e^5 + 179367/997*e^4 + 1002807/1994*e^3 - 77431/1994*e^2 - 501679/1994*e - 59881/997, 28850/2991*e^10 + 101576/2991*e^9 - 125120/997*e^8 - 1441439/2991*e^7 + 1123703/2991*e^6 + 1896845/997*e^5 - 153350/997*e^4 - 7777195/2991*e^3 - 947008/2991*e^2 + 3183380/2991*e + 719818/2991, 25699/3988*e^10 + 85625/3988*e^9 - 174041/1994*e^8 - 609255/1994*e^7 + 1180413/3988*e^6 + 4859437/3988*e^5 - 492137/1994*e^4 - 6835101/3988*e^3 - 404675/3988*e^2 + 2977973/3988*e + 312729/1994, 184499/11964*e^10 + 625505/11964*e^9 - 418243/1994*e^8 - 4501693/5982*e^7 + 8595065/11964*e^6 + 12227163/3988*e^5 - 1196623/1994*e^4 - 52731817/11964*e^3 - 3767215/11964*e^2 + 22609289/11964*e + 2913953/5982, -17199/997*e^10 - 59133/997*e^9 + 231678/997*e^8 + 849506/997*e^7 - 774668/997*e^6 - 3443013/997*e^5 + 604616/997*e^4 + 4891358/997*e^3 + 383728/997*e^2 - 2042082/997*e - 544742/997, -144943/11964*e^10 - 492229/11964*e^9 + 321543/1994*e^8 + 3484187/5982*e^7 - 6197845/11964*e^6 - 9122235/3988*e^5 + 655593/1994*e^4 + 36892409/11964*e^3 + 3864227/11964*e^2 - 14411881/11964*e - 1909303/5982, -98721/3988*e^10 - 337447/3988*e^9 + 663505/1994*e^8 + 2412487/1994*e^7 - 4413695/3988*e^6 - 19369671/3988*e^5 + 1719051/1994*e^4 + 27147143/3988*e^3 + 1896477/3988*e^2 - 11234635/3988*e - 1405345/1994];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {19, 19, -w^3 + w^2 + 4*w - 2}>] := 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;