/*
  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, 10, -7, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {11, 11, -w^3 + 6*w - 4}>;

primesArray := [
[2, 2, -w + 1],
[5, 5, w^3 + w^2 - 4*w + 1],
[7, 7, -w^3 + 6*w - 2],
[8, 2, w^3 - 7*w + 3],
[11, 11, -w^3 + 6*w - 4],
[17, 17, w^3 + w^2 - 6*w - 1],
[17, 17, -w^2 - w + 3],
[31, 31, 2*w^3 + w^2 - 13*w + 3],
[37, 37, -3*w^3 - w^2 + 18*w - 3],
[41, 41, w^2 + 2*w - 4],
[47, 47, 2*w^3 + 2*w^2 - 11*w - 2],
[47, 47, -3*w^3 - w^2 + 19*w - 6],
[59, 59, -2*w^3 + 13*w - 6],
[59, 59, -w^3 - w^2 + 5*w + 2],
[59, 59, w^2 + w - 1],
[59, 59, w^2 + 2*w - 6],
[61, 61, -w^3 + w^2 + 8*w - 7],
[67, 67, w^3 + 2*w^2 - 4*w - 4],
[73, 73, -2*w^3 - w^2 + 12*w - 4],
[81, 3, -3],
[107, 107, 2*w^2 + 2*w - 11],
[109, 109, 2*w^3 + w^2 - 11*w + 3],
[113, 113, w^3 + w^2 - 7*w - 2],
[113, 113, 2*w^3 - 12*w + 9],
[125, 5, -4*w^3 - 3*w^2 + 24*w + 2],
[131, 131, 5*w^3 + 3*w^2 - 31*w + 2],
[151, 151, 4*w^3 + 2*w^2 - 26*w + 1],
[151, 151, w^3 + 2*w^2 - 6*w - 10],
[167, 167, w + 4],
[167, 167, 4*w^3 + w^2 - 26*w + 10],
[173, 173, -2*w^3 + w^2 + 13*w - 11],
[179, 179, 4*w^3 + w^2 - 24*w + 4],
[179, 179, 2*w^3 + 2*w^2 - 12*w - 3],
[179, 179, -w^3 + 8*w - 4],
[179, 179, w^3 + 2*w^2 - 4*w - 6],
[181, 181, 6*w^3 + 3*w^2 - 39*w + 5],
[197, 197, -w^3 + 4*w - 4],
[199, 199, 2*w^3 - 12*w + 3],
[199, 199, 3*w - 2],
[223, 223, w^3 + w^2 - 7*w - 4],
[223, 223, 2*w^3 + w^2 - 10*w + 4],
[227, 227, 7*w^3 + 2*w^2 - 44*w + 12],
[227, 227, -4*w^3 - w^2 + 25*w - 9],
[229, 229, 8*w^3 + 4*w^2 - 51*w + 4],
[229, 229, 3*w^3 + w^2 - 18*w + 5],
[233, 233, -6*w^3 - 3*w^2 + 39*w - 7],
[233, 233, -w^3 - 2*w^2 + 8*w + 2],
[241, 241, 2*w^3 + w^2 - 10*w],
[251, 251, w^3 + w^2 - 7*w - 6],
[257, 257, -6*w^3 - w^2 + 39*w - 17],
[263, 263, -2*w^3 + 13*w - 4],
[263, 263, -w^3 + 6*w - 8],
[269, 269, -8*w^3 - 4*w^2 + 50*w - 7],
[269, 269, -w^3 + w^2 + 7*w - 6],
[277, 277, 2*w^3 + 2*w^2 - 11*w + 4],
[277, 277, w^3 + w^2 - 4*w - 3],
[281, 281, -w^2 - 3*w + 5],
[281, 281, 2*w^3 + 3*w^2 - 9*w - 3],
[283, 283, -3*w^3 - 4*w^2 + 12*w + 2],
[289, 17, -2*w^3 - w^2 + 13*w - 5],
[293, 293, -w^3 - 2*w^2 + 6*w + 6],
[293, 293, w^3 + w^2 - 8*w - 1],
[307, 307, 2*w^3 - 11*w + 12],
[317, 317, 2*w^3 - 14*w + 11],
[317, 317, -2*w^3 - 2*w^2 + 8*w + 1],
[337, 337, 2*w^2 + 4*w - 11],
[343, 7, -5*w^3 - w^2 + 31*w - 10],
[349, 349, 2*w^3 - 11*w + 8],
[349, 349, w^2 + w - 9],
[353, 353, 4*w^3 + 4*w^2 - 23*w - 8],
[373, 373, -2*w - 3],
[373, 373, -4*w^3 + 26*w - 13],
[389, 389, 6*w^3 + 3*w^2 - 37*w + 1],
[389, 389, -2*w^3 - w^2 + 14*w - 6],
[401, 401, 2*w^3 - 12*w + 13],
[421, 421, 2*w^2 + 2*w - 15],
[421, 421, 3*w^3 + 2*w^2 - 20*w + 4],
[431, 431, w^3 + 3*w^2 - 2*w - 9],
[433, 433, 2*w^3 + 2*w^2 - 12*w + 3],
[433, 433, -7*w^3 - 2*w^2 + 44*w - 16],
[433, 433, -3*w^3 + 18*w - 14],
[433, 433, -w^3 + w^2 + 9*w - 10],
[439, 439, 5*w^3 + 3*w^2 - 31*w - 2],
[443, 443, -w^3 - w^2 + 3*w - 4],
[443, 443, -4*w^3 + 27*w - 14],
[449, 449, w^3 + w^2 - 4*w - 5],
[449, 449, w^3 + 3*w^2 - 3*w - 8],
[449, 449, 2*w^3 - 11*w + 6],
[449, 449, -3*w^3 - 2*w^2 + 18*w - 4],
[461, 461, -3*w^3 + w^2 + 20*w - 15],
[463, 463, 2*w^3 - 10*w + 7],
[479, 479, -2*w^3 - 3*w^2 + 11*w + 9],
[491, 491, w^3 + w^2 - 4*w + 5],
[499, 499, -2*w^3 - 2*w^2 + 14*w - 3],
[499, 499, -6*w^3 - 2*w^2 + 39*w - 10],
[499, 499, 4*w^3 + w^2 - 24*w + 14],
[499, 499, -5*w^3 - 3*w^2 + 30*w - 5],
[503, 503, -2*w^3 + w^2 + 14*w - 18],
[509, 509, 4*w^3 + 3*w^2 - 24*w],
[509, 509, 4*w^3 + 2*w^2 - 23*w + 4],
[547, 547, w^3 + 2*w^2 - 2*w - 6],
[547, 547, 6*w^3 + 2*w^2 - 38*w + 7],
[563, 563, -5*w^3 - w^2 + 33*w - 12],
[569, 569, 2*w^3 - w^2 - 15*w + 13],
[569, 569, -4*w^3 - 2*w^2 + 24*w - 7],
[577, 577, -w^2 - 2],
[577, 577, 3*w^3 - 20*w + 8],
[587, 587, 6*w^3 + 4*w^2 - 36*w - 1],
[587, 587, -w^3 + w^2 + 9*w - 8],
[593, 593, 6*w^3 + 3*w^2 - 36*w + 2],
[593, 593, -3*w^3 - w^2 + 21*w - 4],
[599, 599, -2*w^3 - w^2 + 12*w - 10],
[599, 599, -4*w^3 - 2*w^2 + 25*w - 6],
[601, 601, -2*w^3 + 2*w^2 + 15*w - 16],
[607, 607, w^3 - w^2 - 7*w + 4],
[607, 607, -2*w^3 - 2*w^2 + 10*w + 3],
[631, 631, 2*w^2 + 4*w - 7],
[641, 641, -3*w^3 - 2*w^2 + 16*w],
[643, 643, w^3 + w^2 - 3*w - 4],
[643, 643, w^3 + 3*w^2 - 6*w - 13],
[653, 653, -w^3 + 6*w + 2],
[653, 653, -3*w^2 - 5*w + 11],
[659, 659, -2*w^3 + 10*w - 5],
[661, 661, w^3 - w^2 - 8*w + 3],
[673, 673, 3*w^3 + 4*w^2 - 14*w - 2],
[677, 677, 4*w^3 + 2*w^2 - 23*w + 6],
[677, 677, 2*w^3 - 13*w + 14],
[683, 683, 2*w^3 + w^2 - 13*w + 7],
[701, 701, 8*w^3 + 4*w^2 - 51*w + 8],
[701, 701, -4*w^3 - w^2 + 24*w - 12],
[709, 709, -4*w^3 - w^2 + 27*w - 7],
[709, 709, 3*w^3 + 3*w^2 - 18*w - 5],
[709, 709, -3*w^3 + w^2 + 21*w - 14],
[709, 709, w^2 - w - 7],
[719, 719, -9*w^3 - 4*w^2 + 58*w - 8],
[719, 719, 11*w^3 + 5*w^2 - 68*w + 9],
[727, 727, -w^3 + 4*w - 6],
[733, 733, 2*w^3 + w^2 - 13*w + 9],
[739, 739, -2*w^2 - 3*w + 4],
[743, 743, 3*w^3 + 3*w^2 - 19*w - 4],
[751, 751, 2*w^3 + 3*w^2 - 10*w - 2],
[761, 761, -12*w^3 - 5*w^2 + 75*w - 9],
[761, 761, -3*w^3 - 3*w^2 + 17*w],
[769, 769, -6*w^3 - 4*w^2 + 37*w],
[769, 769, -8*w^3 - 3*w^2 + 51*w - 7],
[773, 773, -w^2 - 2*w - 2],
[787, 787, -5*w^3 - 5*w^2 + 28*w + 7],
[811, 811, 7*w^3 + 2*w^2 - 42*w + 6],
[811, 811, -2*w^3 - w^2 + 9*w - 7],
[823, 823, -3*w^3 - 2*w^2 + 14*w - 10],
[829, 829, 4*w^3 + 3*w^2 - 20*w + 6],
[829, 829, -7*w^3 - 2*w^2 + 44*w - 14],
[841, 29, w^3 + 4*w^2 - 2*w - 14],
[841, 29, -w^2 + 10],
[857, 857, -3*w^3 - w^2 + 20*w - 9],
[859, 859, -w^3 + 2*w^2 + 10*w - 18],
[863, 863, 11*w^3 + 5*w^2 - 69*w + 6],
[863, 863, w^3 - 2*w^2 - 10*w + 12],
[877, 877, 3*w^3 - 18*w + 10],
[883, 883, 4*w^3 + 3*w^2 - 22*w],
[887, 887, 3*w^3 + 3*w^2 - 15*w - 2],
[887, 887, w^2 + 4*w - 10],
[887, 887, w^3 + w^2 - 3*w - 8],
[887, 887, w - 6],
[907, 907, 6*w^3 + w^2 - 38*w + 16],
[919, 919, -2*w^3 - 3*w^2 + 11*w + 1],
[919, 919, 5*w^3 - w^2 - 34*w + 27],
[937, 937, -w^3 - w^2 + 8*w + 3],
[941, 941, -4*w^3 + 25*w - 20],
[947, 947, -2*w^3 + w^2 + 8*w - 6],
[967, 967, -7*w^3 - w^2 + 44*w - 23],
[971, 971, 2*w^2 + 2*w - 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^13 - 4*x^12 - 11*x^11 + 56*x^10 + 31*x^9 - 282*x^8 + 27*x^7 + 622*x^6 - 214*x^5 - 579*x^4 + 250*x^3 + 175*x^2 - 81*x + 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -503/3809*e^12 + 1516/3809*e^11 + 6725/3809*e^10 - 21423/3809*e^9 - 2380/293*e^8 + 109398/3809*e^7 + 55311/3809*e^6 - 247708/3809*e^5 - 24825/3809*e^4 + 244532/3809*e^3 - 12976/3809*e^2 - 81836/3809*e + 5663/3809, -154/3809*e^12 - 490/3809*e^11 + 3369/3809*e^10 + 7261/3809*e^9 - 1843/293*e^8 - 38264/3809*e^7 + 65232/3809*e^6 + 85290/3809*e^5 - 51900/3809*e^4 - 76153/3809*e^3 - 10258/3809*e^2 + 30270/3809*e + 6868/3809, 131/3809*e^12 - 622/3809*e^11 - 714/3809*e^10 + 7056/3809*e^9 - 371/293*e^8 - 23448/3809*e^7 + 41170/3809*e^6 + 12186/3809*e^5 - 94954/3809*e^4 + 43830/3809*e^3 + 75705/3809*e^2 - 35890/3809*e - 6238/3809, -1, -721/3809*e^12 + 2900/3809*e^11 + 9367/3809*e^10 - 43051/3809*e^9 - 3288/293*e^8 + 233612/3809*e^7 + 82751/3809*e^6 - 564711/3809*e^5 - 58423/3809*e^4 + 586193/3809*e^3 - 19978/3809*e^2 - 203169/3809*e + 28692/3809, -131/3809*e^12 + 622/3809*e^11 + 714/3809*e^10 - 7056/3809*e^9 + 371/293*e^8 + 23448/3809*e^7 - 41170/3809*e^6 - 12186/3809*e^5 + 94954/3809*e^4 - 43830/3809*e^3 - 75705/3809*e^2 + 35890/3809*e + 17665/3809, -503/3809*e^12 + 1516/3809*e^11 + 6725/3809*e^10 - 21423/3809*e^9 - 2380/293*e^8 + 109398/3809*e^7 + 55311/3809*e^6 - 251517/3809*e^5 - 21016/3809*e^4 + 271195/3809*e^3 - 32021/3809*e^2 - 116117/3809*e + 24708/3809, -912/3809*e^12 + 1946/3809*e^11 + 11542/3809*e^10 - 22198/3809*e^9 - 3597/293*e^8 + 73715/3809*e^7 + 55668/3809*e^6 - 37268/3809*e^5 + 30359/3809*e^4 - 134837/3809*e^3 - 37400/3809*e^2 + 116487/3809*e - 3007/3809, 1210/3809*e^12 - 3768/3809*e^11 - 16132/3809*e^10 + 52322/3809*e^9 + 5900/293*e^8 - 258189/3809*e^7 - 161565/3809*e^6 + 540038/3809*e^5 + 161289/3809*e^4 - 433894/3809*e^3 - 92983/3809*e^2 + 84841/3809*e + 36909/3809, -1821/3809*e^12 + 3209/3809*e^11 + 28534/3809*e^10 - 45601/3809*e^9 - 12727/293*e^8 + 239443/3809*e^7 + 434968/3809*e^6 - 581261/3809*e^5 - 504229/3809*e^4 + 647874/3809*e^3 + 194058/3809*e^2 - 231819/3809*e + 12452/3809, -882/3809*e^12 + 2734/3809*e^11 + 8907/3809*e^10 - 34248/3809*e^9 - 1206/293*e^8 + 148247/3809*e^7 - 82786/3809*e^6 - 270897/3809*e^5 + 300075/3809*e^4 + 218999/3809*e^3 - 258896/3809*e^2 - 78722/3809*e + 52147/3809, -2378/3809*e^12 + 5592/3809*e^11 + 36193/3809*e^10 - 79615/3809*e^9 - 16012/293*e^8 + 410466/3809*e^7 + 568853/3809*e^6 - 937126/3809*e^5 - 743342/3809*e^4 + 922838/3809*e^3 + 373327/3809*e^2 - 308136/3809*e - 20980/3809, 972/3809*e^12 - 370/3809*e^11 - 16812/3809*e^10 + 1907/3809*e^9 + 8379/293*e^8 + 10596/3809*e^7 - 321149/3809*e^6 - 71944/3809*e^5 + 406230/3809*e^4 + 103563/3809*e^3 - 158007/3809*e^2 - 38398/3809*e + 14281/3809, 48/293*e^12 - 87/293*e^11 - 700/293*e^10 + 1230/293*e^9 + 3556/293*e^8 - 6563/293*e^7 - 6816/293*e^6 + 16704/293*e^5 + 777/293*e^4 - 19628/293*e^3 + 9571/293*e^2 + 7270/293*e - 3990/293, -1822/3809*e^12 + 2167/3809*e^11 + 31796/3809*e^10 - 31233/3809*e^9 - 16059/293*e^8 + 163064/3809*e^7 + 634350/3809*e^6 - 376802/3809*e^5 - 877106/3809*e^4 + 375036/3809*e^3 + 453597/3809*e^2 - 126405/3809*e - 33063/3809, 2438/3809*e^12 - 7825/3809*e^11 - 30036/3809*e^10 + 105032/3809*e^9 + 9367/293*e^8 - 493751/3809*e^7 - 179186/3809*e^6 + 965038/3809*e^5 + 75321/3809*e^4 - 706527/3809*e^3 - 35474/3809*e^2 + 108168/3809*e + 13209/3809, -242/3809*e^12 + 3039/3809*e^11 + 941/3809*e^10 - 46269/3809*e^9 + 871/293*e^8 + 252753/3809*e^7 - 70530/3809*e^6 - 582609/3809*e^5 + 124673/3809*e^4 + 485962/3809*e^3 - 81961/3809*e^2 - 59629/3809*e + 12425/3809, -834/3809*e^12 + 3233/3809*e^11 + 12309/3809*e^10 - 49719/3809*e^9 - 5350/293*e^8 + 279687/3809*e^7 + 189334/3809*e^6 - 695744/3809*e^5 - 237389/3809*e^4 + 726478/3809*e^3 + 66236/3809*e^2 - 247838/3809*e + 58119/3809, 120/293*e^12 - 364/293*e^11 - 1750/293*e^10 + 5126/293*e^9 + 10062/293*e^8 - 25930/293*e^7 - 30225/293*e^6 + 56996/293*e^5 + 49555/293*e^4 - 51707/293*e^3 - 35405/293*e^2 + 15538/293*e + 4382/293, -894/3809*e^12 + 5466/3809*e^11 + 6152/3809*e^10 - 75136/3809*e^9 + 1295/293*e^8 + 362972/3809*e^7 - 200333/3809*e^6 - 731274/3809*e^5 + 442059/3809*e^4 + 555875/3809*e^3 - 313516/3809*e^2 - 93578/3809*e + 58272/3809, -3043/3809*e^12 + 13518/3809*e^11 + 34293/3809*e^10 - 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8287/3809*e^11 - 84321/3809*e^10 + 110028/3809*e^9 + 38545/293*e^8 - 525693/3809*e^7 - 1394676/3809*e^6 + 1143400/3809*e^5 + 1832941/3809*e^4 - 1206460/3809*e^3 - 1020141/3809*e^2 + 524138/3809*e + 173024/3809, 16570/3809*e^12 - 45965/3809*e^11 - 230170/3809*e^10 + 639888/3809*e^9 + 88857/293*e^8 - 3193106/3809*e^7 - 2571811/3809*e^6 + 6935137/3809*e^5 + 2461222/3809*e^4 - 6253672/3809*e^3 - 699209/3809*e^2 + 1689582/3809*e - 63203/3809, -10233/3809*e^12 + 25268/3809*e^11 + 142712/3809*e^10 - 346275/3809*e^9 - 54444/293*e^8 + 1696453/3809*e^7 + 1489711/3809*e^6 - 3612359/3809*e^5 - 1188235/3809*e^4 + 3199598/3809*e^3 + 136371/3809*e^2 - 891026/3809*e + 154267/3809, -12046/3809*e^12 + 36813/3809*e^11 + 156577/3809*e^10 - 506820/3809*e^9 - 53993/293*e^8 + 2493602/3809*e^7 + 1274255/3809*e^6 - 5334122/3809*e^5 - 796780/3809*e^4 + 4799869/3809*e^3 - 66114/3809*e^2 - 1387189/3809*e + 184220/3809, -5059/3809*e^12 + 19223/3809*e^11 + 70432/3809*e^10 - 285110/3809*e^9 - 28719/293*e^8 + 1541480/3809*e^7 + 994970/3809*e^6 - 3697262/3809*e^5 - 1409677/3809*e^4 + 3769843/3809*e^3 + 780638/3809*e^2 - 1193263/3809*e - 2526/3809, -1114/293*e^12 + 3008/293*e^11 + 14732/293*e^10 - 40486/293*e^9 - 67244/293*e^8 + 193788/293*e^7 + 121563/293*e^6 - 403494/293*e^5 - 61580/293*e^4 + 360064/293*e^3 - 26464/293*e^2 - 107512/293*e + 14883/293, -7519/3809*e^12 + 19360/3809*e^11 + 107479/3809*e^10 - 268305/3809*e^9 - 43707/293*e^8 + 1333806/3809*e^7 + 1393514/3809*e^6 - 2887637/3809*e^5 - 1590358/3809*e^4 + 2587654/3809*e^3 + 637256/3809*e^2 - 688755/3809*e - 53388/3809, 599/3809*e^12 - 8136/3809*e^11 + 7697/3809*e^10 + 116178/3809*e^9 - 13233/293*e^8 - 608318/3809*e^7 + 888874/3809*e^6 + 1458683/3809*e^5 - 1720487/3809*e^4 - 1636862/3809*e^3 + 1146983/3809*e^2 + 714899/3809*e - 168933/3809, 16013/3809*e^12 - 39773/3809*e^11 - 222511/3809*e^10 + 533503/3809*e^9 + 85572/293*e^8 - 2515486/3809*e^7 - 2456971/3809*e^6 + 4994728/3809*e^5 + 2393514/3809*e^4 - 3898994/3809*e^3 - 942739/3809*e^2 + 851465/3809*e + 131905/3809, -102/293*e^12 + 75/293*e^11 + 2220/293*e^10 - 2101/293*e^9 - 17079/293*e^8 + 18085/293*e^7 + 57848/293*e^6 - 63624/293*e^5 - 84973/293*e^4 + 92545/293*e^3 + 42620/293*e^2 - 47752/293*e - 3168/293, 84/3809*e^12 - 79/3809*e^11 + 240/3809*e^10 - 3268/3809*e^9 - 806/293*e^8 + 54806/3809*e^7 + 30557/3809*e^6 - 281641/3809*e^5 + 80250/3809*e^4 + 578607/3809*e^3 - 288044/3809*e^2 - 375942/3809*e + 82822/3809, -3817/3809*e^12 + 18327/3809*e^11 + 41332/3809*e^10 - 262147/3809*e^9 - 9369/293*e^8 + 1342985/3809*e^7 - 915/3809*e^6 - 2954173/3809*e^5 + 361286/3809*e^4 + 2582355/3809*e^3 - 350021/3809*e^2 - 646007/3809*e + 161522/3809, -4969/3809*e^12 + 6351/3809*e^11 + 81572/3809*e^10 - 88911/3809*e^9 - 37661/293*e^8 + 454780/3809*e^7 + 1272846/3809*e^6 - 1042420/3809*e^5 - 1232823/3809*e^4 + 1022351/3809*e^3 + 78060/3809*e^2 - 342897/3809*e + 113419/3809, -206/3809*e^12 - 5157/3809*e^11 + 16824/3809*e^10 + 64968/3809*e^9 - 14250/293*e^8 - 280961/3809*e^7 + 735382/3809*e^6 + 493802/3809*e^5 - 1158304/3809*e^4 - 303744/3809*e^3 + 626586/3809*e^2 - 11252/3809*e - 135456/3809, 1026/3809*e^12 + 2572/3809*e^11 - 25364/3809*e^10 - 38828/3809*e^9 + 16609/293*e^8 + 211792/3809*e^7 - 807286/3809*e^6 - 489429/3809*e^5 + 1344228/3809*e^4 + 381660/3809*e^3 - 872085/3809*e^2 + 62735/3809*e + 198118/3809, -750/3809*e^12 + 3154/3809*e^11 + 16358/3809*e^10 - 56796/3809*e^9 - 11137/293*e^8 + 384010/3809*e^7 + 619836/3809*e^6 - 1194498/3809*e^5 - 1212232/3809*e^4 + 1647895/3809*e^3 + 825667/3809*e^2 - 699960/3809*e - 57127/3809, -2538/3809*e^12 + 10277/3809*e^11 + 28662/3809*e^10 - 146124/3809*e^9 - 6789/293*e^8 + 744290/3809*e^7 - 13179/3809*e^6 - 1620998/3809*e^5 + 327034/3809*e^4 + 1350315/3809*e^3 - 174647/3809*e^2 - 220541/3809*e - 58662/3809, 4539/3809*e^12 - 8758/3809*e^11 - 65388/3809*e^10 + 111807/3809*e^9 + 25951/293*e^8 - 498451/3809*e^7 - 726871/3809*e^6 + 971890/3809*e^5 + 496622/3809*e^4 - 922648/3809*e^3 + 114269/3809*e^2 + 389525/3809*e + 43/3809, 5387/3809*e^12 - 20257/3809*e^11 - 66230/3809*e^10 + 291757/3809*e^9 + 20228/293*e^8 - 1525725/3809*e^7 - 336588/3809*e^6 + 3528368/3809*e^5 - 52504/3809*e^4 - 3466714/3809*e^3 + 234914/3809*e^2 + 1153674/3809*e - 144576/3809, -9614/3809*e^12 + 22736/3809*e^11 + 130877/3809*e^10 - 296535/3809*e^9 - 48210/293*e^8 + 1332082/3809*e^7 + 1270549/3809*e^6 - 2407193/3809*e^5 - 1088502/3809*e^4 + 1496446/3809*e^3 + 514823/3809*e^2 - 136131/3809*e - 170886/3809, 10266/3809*e^12 - 25163/3809*e^11 - 143706/3809*e^10 + 344447/3809*e^9 + 55111/293*e^8 - 1682268/3809*e^7 - 1521646/3809*e^6 + 3557625/3809*e^5 + 1266286/3809*e^4 - 3112813/3809*e^3 - 294695/3809*e^2 + 809992/3809*e - 31130/3809, 51/293*e^12 + 109/293*e^11 - 1989/293*e^10 - 268/293*e^9 + 20699/293*e^8 - 6845/293*e^7 - 87817/293*e^6 + 36207/293*e^5 + 160419/293*e^4 - 49935/293*e^3 - 106573/293*e^2 + 14207/293*e + 8909/293, 13305/3809*e^12 - 35231/3809*e^11 - 180187/3809*e^10 + 472168/3809*e^9 + 66992/293*e^8 - 2220207/3809*e^7 - 1858252/3809*e^6 + 4372593/3809*e^5 + 1791440/3809*e^4 - 3325982/3809*e^3 - 643087/3809*e^2 + 690903/3809*e - 15835/3809, -5105/3809*e^12 + 16999/3809*e^11 + 68124/3809*e^10 - 241240/3809*e^9 - 25236/293*e^8 + 1235224/3809*e^7 + 724031/3809*e^6 - 2774791/3809*e^5 - 785416/3809*e^4 + 2642486/3809*e^3 + 332564/3809*e^2 - 911210/3809*e + 74914/3809, -5446/3809*e^12 + 19723/3809*e^11 + 68238/3809*e^10 - 274407/3809*e^9 - 22850/293*e^8 + 1361624/3809*e^7 + 555047/3809*e^6 - 2892287/3809*e^5 - 434007/3809*e^4 + 2430058/3809*e^3 - 28607/3809*e^2 - 556332/3809*e + 121682/3809, 5221/3809*e^12 - 21824/3809*e^11 - 61807/3809*e^10 + 315265/3809*e^9 + 17370/293*e^8 - 1646366/3809*e^7 - 194644/3809*e^6 + 3758912/3809*e^5 - 339511/3809*e^4 - 3537374/3809*e^3 + 505228/3809*e^2 + 993874/3809*e - 143010/3809, -816/3809*e^12 + 6753/3809*e^11 - 699/3809*e^10 - 83612/3809*e^9 + 6867/293*e^8 + 340404/3809*e^7 - 462803/3809*e^6 - 475590/3809*e^5 + 734234/3809*e^4 + 11669/3809*e^3 - 141318/3809*e^2 + 227717/3809*e - 170086/3809, 505/3809*e^12 + 568/3809*e^11 - 5631/3809*e^10 - 14931/3809*e^9 - 39/293*e^8 + 127158/3809*e^7 + 197264/3809*e^6 - 431649/3809*e^5 - 718740/3809*e^4 + 503021/3809*e^3 + 712778/3809*e^2 - 79475/3809*e - 112701/3809, -1961/3809*e^12 + 17307/3809*e^11 + 5280/3809*e^10 - 239492/3809*e^9 + 9810/293*e^8 + 1167642/3809*e^7 - 737076/3809*e^6 - 2371866/3809*e^5 + 1201768/3809*e^4 + 1770861/3809*e^3 - 486344/3809*e^2 - 290869/3809*e - 6236/3809, 1786/3809*e^12 - 13016/3809*e^11 - 13398/3809*e^10 + 194244/3809*e^9 - 1343/293*e^8 - 1061534/3809*e^7 + 315687/3809*e^6 + 2595176/3809*e^5 - 696667/3809*e^4 - 2735373/3809*e^3 + 468108/3809*e^2 + 938862/3809*e - 123776/3809, 1034/293*e^12 - 3156/293*e^11 - 12784/293*e^10 + 42245/293*e^9 + 51160/293*e^8 - 198867/293*e^7 - 65667/293*e^6 + 394992/293*e^5 - 14430/293*e^4 - 309966/293*e^3 + 47528/293*e^2 + 70295/293*e - 7940/293, 2375/3809*e^12 - 1100/3809*e^11 - 49261/3809*e^10 + 23685/3809*e^9 + 28870/293*e^8 - 174905/3809*e^7 - 1296239/3809*e^6 + 563972/3809*e^5 + 1993909/3809*e^4 - 827192/3809*e^3 - 1118310/3809*e^2 + 475827/3809*e + 105357/3809, -5316/3809*e^12 + 10441/3809*e^11 + 70786/3809*e^10 - 123477/3809*e^9 - 24209/293*e^8 + 461907/3809*e^7 + 479452/3809*e^6 - 546411/3809*e^5 + 19940/3809*e^4 + 3257/3809*e^3 - 264713/3809*e^2 + 103587/3809*e - 86240/3809, -2559/3809*e^12 + 11249/3809*e^11 + 28602/3809*e^10 - 160543/3809*e^9 - 7613/293*e^8 + 839145/3809*e^7 + 105831/3809*e^6 - 1966721/3809*e^5 + 23201/3809*e^4 + 1962702/3809*e^3 - 201670/3809*e^2 - 581731/3809*e + 189167/3809, 5469/3809*e^12 - 7184/3809*e^11 - 86129/3809*e^10 + 88685/3809*e^9 + 38542/293*e^8 - 370516/3809*e^7 - 1305170/3809*e^6 + 623681/3809*e^5 + 1445504/3809*e^4 - 436100/3809*e^3 - 528201/3809*e^2 + 139153/3809*e - 15660/3809, -4582/3809*e^12 + 17278/3809*e^11 + 53294/3809*e^10 - 232929/3809*e^9 - 15695/293*e^8 + 1114570/3809*e^7 + 326293/3809*e^6 - 2304475/3809*e^5 - 380173/3809*e^4 + 1981236/3809*e^3 + 487360/3809*e^2 - 473231/3809*e - 128868/3809, 2152/3809*e^12 - 1117/3809*e^11 - 34118/3809*e^10 + 5335/3809*e^9 + 15404/293*e^8 + 39730/3809*e^7 - 538519/3809*e^6 - 277190/3809*e^5 + 690130/3809*e^4 + 511859/3809*e^3 - 456102/3809*e^2 - 333507/3809*e + 83643/3809, 66/3809*e^12 - 3599/3809*e^11 + 9439/3809*e^10 + 38243/3809*e^9 - 9214/293*e^8 - 112563/3809*e^7 + 496053/3809*e^6 + 42892/3809*e^5 - 826620/3809*e^4 + 70727/3809*e^3 + 532759/3809*e^2 + 127416/3809*e - 92727/3809, 5517/3809*e^12 - 14303/3809*e^11 - 71300/3809*e^10 + 187484/3809*e^9 + 23264/293*e^8 - 848516/3809*e^7 - 412183/3809*e^6 + 1554838/3809*e^5 - 150862/3809*e^4 - 1037040/3809*e^3 + 478742/3809*e^2 + 236667/3809*e - 70632/3809];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {11, 11, -w^3 + 6*w - 4}>] := 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;