/*
  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![-8, -14, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {8,8,-1/2*w^2 - 1/2*w + 3}>;

primesArray := [
[2, 2, -1/2*w^2 + 1/2*w + 8],
[2, 2, 1/2*w^2 - 3/2*w - 1],
[2, 2, w + 3],
[11, 11, -w^2 + 3*w + 7],
[11, 11, -2*w - 1],
[11, 11, -w^2 + w + 11],
[27, 3, 3],
[41, 41, 2*w + 5],
[41, 41, -w^2 + 3*w + 3],
[41, 41, -w^2 + w + 15],
[43, 43, -3*w^2 + 11*w + 11],
[47, 47, -w^2 - w + 5],
[47, 47, w^2 - 5*w - 3],
[47, 47, -2*w^2 + 4*w + 23],
[59, 59, -w^2 + w + 13],
[59, 59, w^2 - 3*w - 5],
[59, 59, -2*w - 3],
[97, 97, 7*w^2 - 11*w - 91],
[97, 97, -2*w^2 - 8*w - 5],
[97, 97, 5*w^2 - 19*w - 15],
[107, 107, w^2 + 3*w - 1],
[107, 107, -3*w^2 + 5*w + 37],
[107, 107, w^2 - 5*w - 23],
[113, 113, w^2 - 3*w - 1],
[113, 113, w^2 - w - 17],
[113, 113, -2*w - 7],
[125, 5, -5],
[127, 127, -w^2 - w - 1],
[127, 127, 2*w^2 - 4*w - 29],
[127, 127, -w^2 + 5*w - 3],
[131, 131, -w^2 + 5*w - 1],
[131, 131, 2*w^2 - 4*w - 27],
[131, 131, w^2 + w - 1],
[137, 137, 2*w^2 - 6*w - 5],
[137, 137, -4*w - 11],
[137, 137, 2*w^2 - 2*w - 31],
[151, 151, -2*w^2 + 8*w + 3],
[151, 151, 3*w^2 - 5*w - 41],
[151, 151, -w^2 - 3*w - 3],
[173, 173, 2*w^2 - 4*w - 25],
[173, 173, w^2 + w - 3],
[173, 173, -w^2 + 5*w + 1],
[193, 193, -6*w - 17],
[193, 193, 3*w^2 - 9*w - 7],
[193, 193, 3*w^2 - 3*w - 47],
[199, 199, w^2 + w - 7],
[199, 199, 2*w^2 - 4*w - 21],
[199, 199, -w^2 + 5*w + 5],
[211, 211, 2*w - 7],
[211, 211, w^2 - w - 3],
[211, 211, -w^2 + 3*w + 15],
[223, 223, w^2 - w - 7],
[223, 223, w^2 - 3*w - 11],
[223, 223, 2*w - 3],
[257, 257, -2*w^2 - 4*w - 1],
[257, 257, -3*w^2 + 13*w + 3],
[257, 257, 5*w^2 - 9*w - 67],
[269, 269, 2*w - 5],
[269, 269, w^2 - 3*w - 13],
[269, 269, w^2 - w - 5],
[293, 293, -13*w^2 + 51*w + 35],
[293, 293, -19*w^2 + 31*w + 247],
[293, 293, 6*w^2 + 20*w + 9],
[317, 317, 5*w^2 - 19*w - 17],
[317, 317, 7*w^2 - 11*w - 89],
[317, 317, 2*w^2 + 8*w + 3],
[343, 7, -7],
[379, 379, 3*w^2 - 11*w - 13],
[379, 379, 4*w^2 - 6*w - 49],
[379, 379, w^2 + 5*w + 1],
[383, 383, -4*w - 5],
[383, 383, -2*w^2 + 2*w + 25],
[383, 383, 2*w^2 - 6*w - 11],
[389, 389, 4*w^2 - 18*w - 13],
[389, 389, -7*w^2 + 13*w + 83],
[389, 389, 4*w^2 - 14*w - 11],
[409, 409, 2*w - 11],
[409, 409, w^2 - 3*w - 19],
[409, 409, -w^2 + w - 1],
[419, 419, 10*w^2 - 38*w - 31],
[419, 419, -14*w^2 + 22*w + 181],
[419, 419, 4*w^2 - 14*w - 17],
[431, 431, -4*w^2 + 16*w + 13],
[431, 431, 2*w^2 + 6*w - 1],
[431, 431, -6*w^2 + 10*w + 75],
[457, 457, -2*w^2 + 8*w + 9],
[457, 457, 3*w^2 - 5*w - 35],
[457, 457, w^2 + 3*w - 3],
[557, 557, 5*w^2 - 9*w - 63],
[557, 557, -3*w^2 + 13*w + 7],
[557, 557, 2*w^2 + 4*w - 3],
[563, 563, -9*w^2 + 35*w + 23],
[563, 563, 13*w^2 - 21*w - 171],
[563, 563, -4*w^2 - 14*w - 9],
[601, 601, 5*w^2 - 17*w - 13],
[601, 601, 6*w^2 - 8*w - 85],
[601, 601, -w^2 - 9*w - 17],
[613, 613, -2*w^2 + 15],
[613, 613, 3*w^2 - 7*w - 31],
[613, 613, 22*w^2 - 36*w - 285],
[641, 641, 3*w^2 - 5*w - 43],
[641, 641, -2*w^2 + 8*w + 1],
[641, 641, -w^2 - 3*w - 5],
[643, 643, 2*w^2 + 4*w - 1],
[643, 643, 5*w^2 - 9*w - 65],
[643, 643, -3*w^2 + 13*w + 5],
[647, 647, 9*w^2 - 35*w - 27],
[647, 647, 4*w^2 + 14*w + 5],
[647, 647, 13*w^2 - 21*w - 167],
[653, 653, -4*w^2 + 16*w + 7],
[653, 653, 6*w^2 - 10*w - 81],
[653, 653, -2*w^2 - 6*w - 5],
[661, 661, -w^2 - 7*w - 13],
[661, 661, -4*w^2 + 14*w + 9],
[661, 661, 5*w^2 - 7*w - 71],
[677, 677, -12*w^2 + 20*w + 155],
[677, 677, -4*w^2 - 12*w - 3],
[677, 677, -8*w^2 + 32*w + 21],
[709, 709, -2*w^2 + 10*w - 3],
[709, 709, 19*w^2 - 31*w - 249],
[709, 709, 4*w^2 - 8*w - 55],
[727, 727, -w^2 + 5*w + 21],
[727, 727, -23*w^2 + 37*w + 301],
[727, 727, -w^2 - w + 23],
[733, 733, 2*w^2 + 2*w - 9],
[733, 733, -2*w^2 + 10*w + 5],
[733, 733, 4*w^2 - 8*w - 47],
[739, 739, 2*w^2 - 6*w - 17],
[739, 739, 2*w^2 - 2*w - 19],
[739, 739, 4*w - 1],
[773, 773, 8*w^2 - 12*w - 103],
[773, 773, 6*w^2 - 22*w - 21],
[773, 773, -2*w^2 - 10*w - 7],
[809, 809, 4*w^2 + 16*w + 11],
[809, 809, w^2 - 9*w - 7],
[809, 809, 3*w^2 - w - 29],
[821, 821, 3*w^2 + w - 23],
[821, 821, 2*w^2 - 12*w - 9],
[821, 821, 5*w^2 - 11*w - 51],
[839, 839, 2*w^2 + 4*w - 7],
[839, 839, 5*w^2 - 9*w - 59],
[839, 839, 3*w^2 - 13*w - 11],
[859, 859, w^2 + w - 11],
[859, 859, 2*w^2 - 4*w - 17],
[859, 859, w^2 - 5*w - 9],
[881, 881, 4*w^2 - 14*w - 15],
[881, 881, 5*w^2 - 7*w - 65],
[881, 881, -w^2 - 7*w - 7],
[887, 887, 5*w^2 - 7*w - 67],
[887, 887, 4*w^2 - 14*w - 13],
[887, 887, -w^2 - 7*w - 9],
[907, 907, -3*w^2 - 11*w - 9],
[907, 907, 10*w^2 - 16*w - 133],
[907, 907, -7*w^2 + 27*w + 17],
[911, 911, w^2 - 7*w - 7],
[911, 911, 3*w^2 - 7*w - 27],
[911, 911, 2*w^2 - 19],
[919, 919, -3*w^2 - 5*w - 1],
[919, 919, -4*w^2 + 18*w + 1],
[919, 919, 7*w^2 - 13*w - 95],
[947, 947, 4*w^2 - 6*w - 57],
[947, 947, -3*w^2 + 11*w + 5],
[947, 947, -w^2 - 5*w - 9],
[967, 967, 8*w^2 - 30*w - 25],
[967, 967, -3*w^2 - 13*w - 9],
[967, 967, 11*w^2 - 17*w - 143],
[991, 991, 2*w^2 - 8*w - 11],
[991, 991, w^2 + 3*w - 5],
[991, 991, 3*w^2 - 5*w - 33],
[997, 997, 4*w^2 + 2*w - 27],
[997, 997, 7*w^2 - 15*w - 75],
[997, 997, -3*w^2 + 17*w + 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^11 - 17*x^9 + 2*x^8 + 100*x^7 - 21*x^6 - 240*x^5 + 63*x^4 + 215*x^3 - 57*x^2 - 37*x - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-47/2521*e^10 + 85/2521*e^9 + 538/2521*e^8 - 1657/2521*e^7 - 1006/2521*e^6 + 10423/2521*e^5 - 4620/2521*e^4 - 23624/2521*e^3 + 10413/2521*e^2 + 14689/2521*e - 1279/2521, 0, e, 37/2521*e^10 + 94/2521*e^9 - 799/2521*e^8 - 1002/2521*e^7 + 7014/2521*e^6 + 1235/2521*e^5 - 27205/2521*e^4 + 11571/2521*e^3 + 35035/2521*e^2 - 20414/2521*e + 2026/2521, 540/2521*e^10 + 418/2521*e^9 - 8595/2521*e^8 - 5153/2521*e^7 + 46155/2521*e^6 + 18365/2521*e^5 - 96570/2521*e^4 - 12161/2521*e^3 + 69329/2521*e^2 - 21101/2521*e - 2523/2521, -749/2521*e^10 + 550/2521*e^9 + 12972/2521*e^8 - 9832/2521*e^7 - 77394/2521*e^6 + 55876/2521*e^5 + 186467/2521*e^4 - 111932/2521*e^3 - 161143/2521*e^2 + 66574/2521*e + 17379/2521, -250/2521*e^10 - 567/2521*e^9 + 3559/2521*e^8 + 9155/2521*e^7 - 16186/2521*e^6 - 51406/2521*e^5 + 26221/2521*e^4 + 114640/2521*e^3 - 13376/2521*e^2 - 75058/2521*e + 1028/2521, 1491/2521*e^10 + 790/2521*e^9 - 24362/2521*e^8 - 9172/2521*e^7 + 134792/2521*e^6 + 31030/2521*e^5 - 290590/2521*e^4 - 28858/2521*e^3 + 211411/2521*e^2 - 7324/2521*e - 11294/2521, -12/2521*e^10 + 719/2521*e^9 + 191/2521*e^8 - 11258/2521*e^7 + 655/2521*e^6 + 57855/2521*e^5 - 12980/2521*e^4 - 108749/2521*e^3 + 27815/2521*e^2 + 61197/2521*e + 2409/2521, 1661/2521*e^10 + 268/2521*e^9 - 27488/2521*e^8 - 1784/2521*e^7 + 153664/2521*e^6 - 770/2521*e^5 - 331916/2521*e^4 + 12178/2521*e^3 + 235431/2521*e^2 - 13360/2521*e - 16430/2521, 1324/2521*e^10 + 502/2521*e^9 - 21914/2521*e^8 - 4922/2521*e^7 + 125210/2521*e^6 + 9921/2521*e^5 - 291397/2521*e^4 + 12125/2521*e^3 + 252916/2521*e^2 - 28562/2521*e - 28819/2521, 1453/2521*e^10 + 966/2521*e^9 - 23337/2521*e^8 - 11209/2521*e^7 + 126362/2521*e^6 + 36507/2521*e^5 - 260265/2521*e^4 - 23231/2521*e^3 + 156215/2521*e^2 - 24037/2521*e + 10200/2521, 665/2521*e^10 - 559/2521*e^9 - 11635/2521*e^8 + 9177/2521*e^7 + 69374/2521*e^6 - 49209/2521*e^5 - 163882/2521*e^4 + 96905/2521*e^3 + 136521/2521*e^2 - 59202/2521*e - 18163/2521, 124/2521*e^10 - 707/2521*e^9 - 2814/2521*e^8 + 11291/2521*e^7 + 21803/2521*e^6 - 60862/2521*e^5 - 71755/2521*e^4 + 126264/2521*e^3 + 97451/2521*e^2 - 85312/2521*e - 19851/2521, 65/2521*e^10 + 97/2521*e^9 + 436/2521*e^8 - 1624/2521*e^7 - 16363/2521*e^6 + 12458/2521*e^5 + 87115/2521*e^4 - 48966/2521*e^3 - 118942/2521*e^2 + 68725/2521*e + 11531/2521, 2105/2521*e^10 + 1396/2521*e^9 - 34555/2521*e^8 - 16329/2521*e^7 + 194975/2521*e^6 + 51865/2521*e^5 - 441570/2521*e^4 - 17877/2521*e^3 + 350810/2521*e^2 - 69253/2521*e - 24891/2521, -591/2521*e^10 + 747/2521*e^9 + 10037/2521*e^8 - 13702/2521*e^7 - 57867/2521*e^6 + 81931/2521*e^5 + 132161/2521*e^4 - 179645/2521*e^3 - 109308/2521*e^2 + 121735/2521*e + 9610/2521, -1446/2521*e^10 - 335/2521*e^9 + 21755/2521*e^8 + 2230/2521*e^7 - 103845/2521*e^6 - 298/2521*e^5 + 162795/2521*e^4 - 3878/2521*e^3 - 42819/2521*e^2 - 5989/2521*e - 23580/2521, -2464/2521*e^10 - 264/2521*e^9 + 40059/2521*e^8 - 726/2521*e^7 - 219287/2521*e^6 + 25818/2521*e^5 + 462494/2521*e^4 - 82810/2521*e^3 - 323087/2521*e^2 + 81792/2521*e + 23221/2521, 946/2521*e^10 + 1722/2521*e^9 - 14637/2521*e^8 - 24256/2521*e^7 + 76515/2521*e^6 + 109250/2521*e^5 - 153210/2521*e^4 - 170202/2521*e^3 + 81613/2521*e^2 + 75200/2521*e + 28157/2521, 930/2521*e^10 + 1000/2521*e^9 - 16063/2521*e^8 - 14897/2521*e^7 + 96716/2521*e^6 + 72945/2521*e^5 - 234382/2521*e^4 - 124445/2521*e^3 + 185086/2521*e^2 + 43351/2521*e - 3925/2521, -559/2521*e^10 - 330/2521*e^9 + 7847/2521*e^8 + 5395/2521*e^7 - 32723/2521*e^6 - 37055/2521*e^5 + 32321/2521*e^4 + 117075/2521*e^3 + 21560/2521*e^2 - 107003/2521*e + 8228/2521, 822/2521*e^10 - 92/2521*e^9 - 11823/2521*e^8 + 2268/2521*e^7 + 49670/2521*e^6 - 8879/2521*e^5 - 43640/2521*e^4 - 16635/2521*e^3 - 46090/2521*e^2 + 49588/2521*e + 5151/2521, -116/2521*e^10 + 1068/2521*e^9 + 3527/2521*e^8 - 17231/2521*e^7 - 33164/2521*e^6 + 97922/2521*e^5 + 124946/2521*e^4 - 233596/2521*e^3 - 175658/2521*e^2 + 193253/2521*e + 43455/2521, 1057/2521*e^10 - 517/2521*e^9 - 17034/2521*e^8 + 10553/2521*e^7 + 92515/2521*e^6 - 63515/2521*e^5 - 197010/2521*e^4 + 126695/2521*e^3 + 156466/2521*e^2 - 81840/2521*e - 33832/2521, -920/2521*e^10 - 1179/2521*e^9 + 13803/2521*e^8 + 15035/2521*e^7 - 67430/2521*e^6 - 56872/2521*e^5 + 119989/2521*e^4 + 55826/2521*e^3 - 74232/2521*e^2 + 17836/2521*e + 23346/2521, -420/2521*e^10 - 45/2521*e^9 + 6685/2521*e^8 - 754/2521*e^7 - 37579/2521*e^6 + 10646/2521*e^5 + 92757/2521*e^4 - 22194/2521*e^3 - 100421/2521*e^2 - 26165/2521*e + 36416/2521, -795/2521*e^10 + 365/2521*e^9 + 13284/2521*e^8 - 8450/2521*e^7 - 74463/2521*e^6 + 62108/2521*e^5 + 156038/2521*e^4 - 172922/2521*e^3 - 92754/2521*e^2 + 156205/2521*e + 2664/2521, 1196/2521*e^10 - 232/2521*e^9 - 20717/2521*e^8 + 6925/2521*e^7 + 125474/2521*e^6 - 58671/2521*e^5 - 313044/2521*e^4 + 181543/2521*e^3 + 284064/2521*e^2 - 169909/2521*e - 48501/2521, 162/2521*e^10 + 1638/2521*e^9 - 1318/2521*e^8 - 24487/2521*e^7 - 2540/2521*e^6 + 117694/2521*e^5 + 39096/2521*e^4 - 194488/2521*e^3 - 81806/2521*e^2 + 72577/2521*e + 41848/2521, -422/2521*e^10 + 495/2521*e^9 + 7137/2521*e^8 - 9353/2521*e^7 - 40411/2521*e^6 + 61885/2521*e^5 + 86392/2521*e^4 - 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6657/2521*e^7 + 407102/2521*e^6 - 12197/2521*e^5 - 898329/2521*e^4 + 114863/2521*e^3 + 694898/2521*e^2 - 147324/2521*e - 75273/2521, -3464/2521*e^10 - 11/2521*e^9 + 61858/2521*e^8 - 6963/2521*e^7 - 387392/2521*e^6 + 79857/2521*e^5 + 1000990/2521*e^4 - 277189/2521*e^3 - 933732/2521*e^2 + 316012/2521*e + 110526/2521, -1178/2521*e^10 - 2107/2521*e^9 + 16649/2521*e^8 + 30130/2521*e^7 - 74776/2521*e^6 - 140296/2521*e^5 + 115708/2521*e^4 + 217294/2521*e^3 - 47216/2521*e^2 - 29029/2521*e - 1751/2521, 2255/2521*e^10 + 3753/2521*e^9 - 35682/2521*e^8 - 54595/2521*e^7 + 190569/2521*e^6 + 260187/2521*e^5 - 387723/2521*e^4 - 444643/2521*e^3 + 218668/2521*e^2 + 188050/2521*e + 16845/2521, 3659/2521*e^10 + 302/2521*e^9 - 60550/2521*e^8 + 2091/2521*e^7 + 345866/2521*e^6 - 42483/2521*e^5 - 817796/2521*e^4 + 140375/2521*e^3 + 781107/2521*e^2 - 130005/2521*e - 138958/2521, 7128/2521*e^10 + 1484/2521*e^9 - 115975/2521*e^8 - 8524/2521*e^7 + 636977/2521*e^6 - 29850/2521*e^5 - 1347833/2521*e^4 + 195440/2521*e^3 + 905563/2521*e^2 - 184752/2521*e + 982/2521, -4361/2521*e^10 - 2358/2521*e^9 + 70463/2521*e^8 + 25028/2521*e^7 - 386960/2521*e^6 - 68366/2521*e^5 + 829892/2521*e^4 + 20896/2521*e^3 - 589639/2521*e^2 + 35672/2521*e + 89549/2521, -1909/2521*e^10 - 1375/2521*e^9 + 28074/2521*e^8 + 17017/2521*e^7 - 134245/2521*e^6 - 76665/2521*e^5 + 246015/2521*e^4 + 171427/2521*e^3 - 216048/2521*e^2 - 148788/2521*e + 88905/2521, 2730/2521*e^10 + 1553/2521*e^9 - 47234/2521*e^8 - 20309/2521*e^7 + 280818/2521*e^6 + 87103/2521*e^5 - 657122/2521*e^4 - 143133/2521*e^3 + 495174/2521*e^2 + 67972/2521*e - 14856/2521, -2464/2521*e^10 + 2257/2521*e^9 + 45101/2521*e^8 - 46104/2521*e^7 - 292396/2521*e^6 + 313212/2521*e^5 + 795266/2521*e^4 - 808858/2521*e^3 - 804598/2521*e^2 + 628849/2521*e + 129103/2521, 6568/2521*e^10 + 1424/2521*e^9 - 105381/2521*e^8 - 11210/2521*e^7 + 565023/2521*e^6 + 23000/2521*e^5 - 1148527/2521*e^4 - 66084/2521*e^3 + 768307/2521*e^2 + 109772/2521*e - 98362/2521, 1765/2521*e^10 - 81/2521*e^9 - 23261/2521*e^8 + 6710/2521*e^7 + 79080/2521*e^6 - 58484/2521*e^5 + 11669/2521*e^4 + 134504/2521*e^3 - 224119/2521*e^2 - 57181/2521*e + 43364/2521, 4711/2521*e^10 + 3656/2521*e^9 - 76454/2521*e^8 - 47929/2521*e^7 + 423738/2521*e^6 + 207393/2521*e^5 - 928618/2521*e^4 - 345257/2521*e^3 + 703155/2521*e^2 + 169745/2521*e - 95526/2521];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2,2,1/2*w^2 - 3/2*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;