/*
  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 | {16, 2, 2}>;

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^9 - 4*x^8 - 23*x^7 + 84*x^6 + 188*x^5 - 580*x^4 - 636*x^3 + 1522*x^2 + 764*x - 1176;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, e, 149/6304*e^8 - 609/6304*e^7 - 1285/3152*e^6 + 4847/3152*e^5 + 6031/3152*e^4 - 20001/3152*e^3 - 557/3152*e^2 + 9809/1576*e - 5671/788, 1, -383/12608*e^8 + 275/12608*e^7 + 5831/6304*e^6 - 253/6304*e^5 - 57917/6304*e^4 - 22861/6304*e^3 + 197575/6304*e^2 + 47589/3152*e - 41355/1576, 247/12608*e^8 - 523/12608*e^7 - 3135/6304*e^6 + 3381/6304*e^5 + 28677/6304*e^4 - 7115/6304*e^3 - 100095/6304*e^2 - 1837/3152*e + 23547/1576, 247/12608*e^8 - 523/12608*e^7 - 3135/6304*e^6 + 3381/6304*e^5 + 28677/6304*e^4 - 7115/6304*e^3 - 100095/6304*e^2 - 1837/3152*e + 23547/1576, -585/12608*e^8 + 741/12608*e^7 + 9001/6304*e^6 - 5851/6304*e^5 - 90315/6304*e^4 + 17349/6304*e^3 + 306577/6304*e^2 + 5595/3152*e - 57677/1576, 149/6304*e^8 - 609/6304*e^7 - 1285/3152*e^6 + 4847/3152*e^5 + 6031/3152*e^4 - 20001/3152*e^3 + 2595/3152*e^2 + 9809/1576*e - 10399/788, 111/12608*e^8 - 771/12608*e^7 - 439/6304*e^6 + 6509/6304*e^5 - 563/6304*e^4 - 37091/6304*e^3 - 2615/6304*e^2 + 40763/3152*e + 5739/1576, -883/12608*e^8 + 1959/12608*e^7 + 11571/6304*e^6 - 15545/6304*e^5 - 102377/6304*e^4 + 57351/6304*e^3 + 313995/6304*e^2 - 17175/3152*e - 55791/1576, -209/6304*e^8 + 685/6304*e^7 + 2289/3152*e^6 - 5043/3152*e^5 - 18931/3152*e^4 + 14749/3152*e^3 + 67481/3152*e^2 + 3979/1576*e - 19197/788, -159/3152*e^8 + 359/3152*e^7 + 2109/1576*e^6 - 3041/1576*e^5 - 18425/1576*e^4 + 12559/1576*e^3 + 53475/1576*e^2 - 5541/788*e - 8061/394, -193/3152*e^8 + 297/3152*e^7 + 2783/1576*e^6 - 2259/1576*e^5 - 25735/1576*e^4 + 5065/1576*e^3 + 76269/1576*e^2 + 5897/788*e - 10149/394, -383/12608*e^8 + 275/12608*e^7 + 5831/6304*e^6 - 253/6304*e^5 - 57917/6304*e^4 - 22861/6304*e^3 + 210183/6304*e^2 + 41285/3152*e - 60267/1576, 639/12608*e^8 - 179/12608*e^7 - 10535/6304*e^6 - 2483/6304*e^5 + 112957/6304*e^4 + 57037/6304*e^3 - 409991/6304*e^2 - 98853/3152*e + 86835/1576, 37/394*e^8 - 30/197*e^7 - 1015/394*e^6 + 331/197*e^5 + 4803/197*e^4 + 310/197*e^3 - 16172/197*e^2 - 4870/197*e + 13562/197, 1383/12608*e^8 - 3643/12608*e^7 - 17311/6304*e^6 + 30837/6304*e^5 + 146837/6304*e^4 - 131259/6304*e^3 - 430415/6304*e^2 + 47267/3152*e + 73379/1576, 1545/12608*e^8 - 5109/12608*e^7 - 17185/6304*e^6 + 43659/6304*e^5 + 129659/6304*e^4 - 201237/6304*e^3 - 340353/6304*e^2 + 112637/3152*e + 50533/1576, 117/12608*e^8 - 1409/12608*e^7 + 91/6304*e^6 + 15039/6304*e^5 - 13457/6304*e^4 - 96769/6304*e^3 + 68547/6304*e^2 + 87137/3152*e - 11159/1576, 237/6304*e^8 + 15/6304*e^7 - 4281/3152*e^6 - 329/3152*e^5 + 45439/3152*e^4 + 9871/3152*e^3 - 158285/3152*e^2 - 21603/1576*e + 33849/788, 763/12608*e^8 - 1807/12608*e^7 - 9563/6304*e^6 + 15153/6304*e^5 + 76577/6304*e^4 - 61551/6304*e^3 - 192755/6304*e^2 + 19535/3152*e + 15511/1576, 5/3152*e^8 - 269/3152*e^7 + 573/1576*e^6 + 1855/1576*e^5 - 9957/1576*e^4 - 5341/1576*e^3 + 48007/1576*e^2 + 427/788*e - 14607/394, -1757/12608*e^8 + 4537/12608*e^7 + 21573/6304*e^6 - 37207/6304*e^5 - 178391/6304*e^4 + 155993/6304*e^3 + 503061/6304*e^2 - 71169/3152*e - 70737/1576, -1019/6304*e^8 + 1711/6304*e^7 + 14267/3152*e^6 - 12417/3152*e^5 - 131617/3152*e^4 + 27375/3152*e^3 + 408323/3152*e^2 + 28577/1576*e - 68871/788, 375/12608*e^8 - 3627/12608*e^7 + 817/6304*e^6 + 31957/6304*e^5 - 50971/6304*e^4 - 166539/6304*e^3 + 272801/6304*e^2 + 123827/3152*e - 74277/1576, -99/1576*e^8 + 283/1576*e^7 + 1105/788*e^6 - 2057/788*e^5 - 8677/788*e^4 + 6779/788*e^3 + 23587/788*e^2 - 1205/394*e - 923/197, -17/788*e^8 - 31/788*e^7 + 337/394*e^6 + 391/394*e^5 - 3655/394*e^4 - 3353/394*e^3 + 11397/394*e^2 + 3552/197*e - 1694/197, 465/3152*e^8 - 589/3152*e^7 - 6993/1576*e^6 + 3883/1576*e^5 + 69243/1576*e^4 - 2637/1576*e^3 - 235769/1576*e^2 - 17419/788*e + 48129/394, 643/6304*e^8 - 1655/6304*e^7 - 7555/3152*e^6 + 11609/3152*e^5 + 60233/3152*e^4 - 24775/3152*e^3 - 166075/3152*e^2 - 23809/1576*e + 24087/788, -99/1576*e^8 + 283/1576*e^7 + 1105/788*e^6 - 2057/788*e^5 - 8677/788*e^4 + 6779/788*e^3 + 24375/788*e^2 - 1205/394*e - 3681/197, 57/3152*e^8 - 545/3152*e^7 - 87/1576*e^6 + 5387/1576*e^5 - 3505/1576*e^4 - 31889/1576*e^3 + 11755/1576*e^2 + 24095/788*e + 1797/394, -135/1576*e^8 + 565/1576*e^7 + 637/394*e^6 - 4775/788*e^5 - 8143/788*e^4 + 21673/788*e^3 + 20165/788*e^2 - 5075/197*e - 3354/197, 921/6304*e^8 - 1797/6304*e^7 - 12417/3152*e^6 + 13883/3152*e^5 + 112123/3152*e^4 - 43413/3152*e^3 - 359217/3152*e^2 - 12203/1576*e + 79053/788, 23/3152*e^8 - 607/3152*e^7 + 587/1576*e^6 + 6169/1576*e^5 - 10815/1576*e^4 - 40959/1576*e^3 + 44005/1576*e^2 + 39473/788*e - 12111/394, 133/3152*e^8 - 1009/3152*e^7 - 597/1576*e^6 + 9943/1576*e^5 - 3713/1576*e^4 - 58385/1576*e^3 + 35571/1576*e^2 + 46897/788*e - 13931/394, 1743/12608*e^8 - 947/12608*e^7 - 28063/6304*e^6 + 493/6304*e^5 + 290429/6304*e^4 + 92525/6304*e^3 - 1024231/6304*e^2 - 178877/3152*e + 219435/1576, -125/1576*e^8 + 421/1576*e^7 + 1435/788*e^6 - 3823/788*e^5 - 11115/788*e^4 + 19265/788*e^3 + 30681/788*e^2 - 13433/394*e - 5185/197, -2129/12608*e^8 + 3117/12608*e^7 + 31265/6304*e^6 - 23923/6304*e^5 - 296195/6304*e^4 + 57869/6304*e^3 + 929337/6304*e^2 + 46467/3152*e - 157781/1576, 597/3152*e^8 - 2017/3152*e^7 - 6365/1576*e^6 + 16607/1576*e^5 + 45615/1576*e^4 - 70513/1576*e^3 - 113821/1576*e^2 + 32781/788*e + 16001/394, -2*e - 2, 663/6304*e^8 - 1155/6304*e^7 - 9203/3152*e^6 + 7997/3152*e^5 + 88173/3152*e^4 - 19347/3152*e^3 - 306583/3152*e^2 - 5553/1576*e + 67311/788, 57/3152*e^8 - 545/3152*e^7 - 87/1576*e^6 + 5387/1576*e^5 - 3505/1576*e^4 - 31889/1576*e^3 + 13331/1576*e^2 + 26459/788*e - 567/394, 2765/12608*e^8 - 7705/12608*e^7 - 31821/6304*e^6 + 58151/6304*e^5 + 253271/6304*e^4 - 193209/6304*e^3 - 749237/6304*e^2 + 15097/3152*e + 166385/1576, 1275/12608*e^8 - 1615/12608*e^7 - 18971/6304*e^6 + 9681/6304*e^5 + 186657/6304*e^4 + 6801/6304*e^3 - 630195/6304*e^2 - 64081/3152*e + 125383/1576, -153/12608*e^8 + 2085/12608*e^7 - 1695/6304*e^6 - 18939/6304*e^5 + 43541/6304*e^4 + 111269/6304*e^3 - 214991/6304*e^2 - 108493/3152*e + 57387/1576, -1791/12608*e^8 + 2899/12608*e^7 + 25399/6304*e^6 - 21453/6304*e^5 - 234557/6304*e^4 + 53939/6304*e^3 + 697639/6304*e^2 + 39557/3152*e - 95283/1576, -333/12608*e^8 + 2313/12608*e^7 + 1317/6304*e^6 - 19527/6304*e^5 + 7993/6304*e^4 + 92361/6304*e^3 - 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1807/12608*e^7 - 9563/6304*e^6 + 15153/6304*e^5 + 76577/6304*e^4 - 61551/6304*e^3 - 186451/6304*e^2 + 623/3152*e + 37575/1576, -705/6304*e^8 + 2469/6304*e^7 + 7069/3152*e^6 - 20427/3152*e^5 - 45195/3152*e^4 + 93525/3152*e^3 + 92129/3152*e^2 - 59025/1576*e - 10489/788, -1163/12608*e^8 + 4415/12608*e^7 + 11003/6304*e^6 - 31169/6304*e^5 - 80625/6304*e^4 + 91679/6304*e^3 + 273283/6304*e^2 + 6193/3152*e - 82535/1576, 97/6304*e^8 + 2819/6304*e^7 - 6929/3152*e^6 - 27053/3152*e^5 + 102019/3152*e^4 + 167299/3152*e^3 - 435529/3152*e^2 - 155699/1576*e + 127645/788, 41/12608*e^8 - 1733/12608*e^7 + 1783/6304*e^6 + 18363/6304*e^5 - 26645/6304*e^4 - 124645/6304*e^3 + 32911/6304*e^2 + 120677/3152*e + 36877/1576, 697/6304*e^8 - 2669/6304*e^7 - 6725/3152*e^6 + 22187/3152*e^5 + 43475/3152*e^4 - 100109/3152*e^3 - 119769/3152*e^2 + 58657/1576*e + 32757/788, -1273/6304*e^8 + 2453/6304*e^7 + 16521/3152*e^6 - 15243/3152*e^5 - 146827/3152*e^4 + 12181/3152*e^3 + 457441/3152*e^2 + 59051/1576*e - 73229/788, 1141/12608*e^8 + 2127/12608*e^7 - 23453/6304*e^6 - 27425/6304*e^5 + 279199/6304*e^4 + 238511/6304*e^3 - 1118381/6304*e^2 - 302311/3152*e + 299993/1576, 1487/12608*e^8 - 7347/12608*e^7 - 10751/6304*e^6 + 63117/6304*e^5 + 27357/6304*e^4 - 319891/6304*e^3 + 108569/6304*e^2 + 260083/3152*e - 76629/1576, 31/6304*e^8 + 381/6304*e^7 - 1727/3152*e^6 - 1107/3152*e^5 + 23213/3152*e^4 - 14675/3152*e^3 - 80439/3152*e^2 + 30779/1576*e + 7779/788, -1139/6304*e^8 + 1863/6304*e^7 + 16275/3152*e^6 - 12809/3152*e^5 - 157417/3152*e^4 + 23175/3152*e^3 + 526411/3152*e^2 + 30937/1576*e - 96543/788, -2537/12608*e^8 + 8677/12608*e^7 + 26745/6304*e^6 - 74427/6304*e^5 - 182187/6304*e^4 + 358789/6304*e^3 + 377041/6304*e^2 - 238645/3152*e - 7901/1576, -365/3152*e^8 + 725/3152*e^7 + 4663/1576*e^6 - 5395/1576*e^5 - 37499/1576*e^4 + 16381/1576*e^3 + 84041/1576*e^2 + 2713/788*e + 2117/394, 29/12608*e^8 - 457/12608*e^7 + 723/6304*e^6 + 1303/6304*e^5 - 7161/6304*e^4 + 7319/6304*e^3 + 4059/6304*e^2 - 16199/3152*e - 8127/1576, -849/12608*e^8 - 2707/12608*e^7 + 20353/6304*e^6 + 28589/6304*e^5 - 254243/6304*e^4 - 199923/6304*e^3 + 983065/6304*e^2 + 187299/3152*e - 225093/1576, 533/3152*e^8 - 1253/3152*e^7 - 6371/1576*e^6 + 7835/1576*e^5 + 53131/1576*e^4 - 5773/1576*e^3 - 159217/1576*e^2 - 39901/788*e + 25119/394, 377/12608*e^8 + 363/12608*e^7 - 6361/6304*e^6 - 8277/6304*e^5 + 70811/6304*e^4 + 88843/6304*e^3 - 306561/6304*e^2 - 100267/3152*e + 140205/1576, -1883/12608*e^8 + 5327/12608*e^7 + 23051/6304*e^6 - 46129/6304*e^5 - 191297/6304*e^4 + 217775/6304*e^3 + 521619/6304*e^2 - 149855/3152*e - 56295/1576, 19/3152*e^8 + 81/3152*e^7 - 423/1576*e^6 - 831/1576*e^5 + 3297/1576*e^4 + 8545/1576*e^3 + 5757/1576*e^2 - 17053/788*e - 15161/394, -925/6304*e^8 + 3273/6304*e^7 + 9437/3152*e^6 - 27975/3152*e^5 - 59399/3152*e^4 + 131529/3152*e^3 + 99541/3152*e^2 - 78601/1576*e + 8911/788, -4023/12608*e^8 + 3835/12608*e^7 + 59911/6304*e^6 - 14245/6304*e^5 - 578901/6304*e^4 - 159717/6304*e^3 + 1896783/6304*e^2 + 415989/3152*e - 370115/1576, -317/3152*e^8 + 1137/3152*e^7 + 2993/1576*e^6 - 8863/1576*e^5 - 16935/1576*e^4 + 34609/1576*e^3 + 17421/1576*e^2 - 10445/788*e + 13895/394, -25/788*e^8 + 163/788*e^7 + 45/197*e^6 - 1395/394*e^5 + 1323/394*e^4 + 6611/394*e^3 - 11909/394*e^2 - 4893/197*e + 7776/197, -2217/12608*e^8 + 4069/12608*e^7 + 31897/6304*e^6 - 37659/6304*e^5 - 289899/6304*e^4 + 174565/6304*e^3 + 827025/6304*e^2 - 100997/3152*e - 107469/1576, -4465/12608*e^8 + 7757/12608*e^7 + 62369/6304*e^6 - 58451/6304*e^5 - 577795/6304*e^4 + 177837/6304*e^3 + 1797529/6304*e^2 + 2051/3152*e - 290485/1576, -581/12608*e^8 + 2417/12608*e^7 + 4101/6304*e^6 - 16975/6304*e^5 - 10655/6304*e^4 + 55313/6304*e^3 - 20019/6304*e^2 - 4465/3152*e - 6953/1576, -1539/6304*e^8 + 2895/6304*e^7 + 20079/3152*e^6 - 16217/3152*e^5 - 185105/3152*e^4 - 8161/3152*e^3 + 624275/3152*e^2 + 101581/1576*e - 139927/788, -889/12608*e^8 - 555/12608*e^7 + 15769/6304*e^6 + 13749/6304*e^5 - 174587/6304*e^4 - 182411/6304*e^3 + 643137/6304*e^2 + 284747/3152*e - 108237/1576, -883/12608*e^8 + 1959/12608*e^7 + 11571/6304*e^6 - 15545/6304*e^5 - 102377/6304*e^4 + 57351/6304*e^3 + 313995/6304*e^2 - 23479/3152*e - 52639/1576, -1879/6304*e^8 + 5427/6304*e^7 + 22091/3152*e^6 - 43069/3152*e^5 - 182557/3152*e^4 + 162755/3152*e^3 + 555927/3152*e^2 - 42503/1576*e - 102495/788, 3155/12608*e^8 - 11351/12608*e^7 - 32043/6304*e^6 + 92521/6304*e^5 + 222073/6304*e^4 - 403351/6304*e^3 - 555419/6304*e^2 + 195759/3152*e + 59319/1576, -1093/6304*e^8 + 3801/6304*e^7 + 11145/3152*e^6 - 30415/3152*e^5 - 75031/3152*e^4 + 120921/3152*e^3 + 179445/3152*e^2 - 46141/1576*e - 26993/788, 799/3152*e^8 - 907/3152*e^7 - 11899/1576*e^6 + 4869/1576*e^5 + 115837/1576*e^4 + 15357/1576*e^3 - 394607/1576*e^2 - 64701/788*e + 91423/394, 635/6304*e^8 + 1297/6304*e^7 - 13515/3152*e^6 - 13423/3152*e^5 + 156225/3152*e^4 + 97873/3152*e^3 - 584563/3152*e^2 - 102977/1576*e + 140127/788, 1837/6304*e^8 - 4113/6304*e^7 - 24225/3152*e^6 + 33791/3152*e^5 + 219231/3152*e^4 - 145313/3152*e^3 - 704189/3152*e^2 + 75637/1576*e + 129373/788, 903/6304*e^8 - 1459/6304*e^7 - 13219/3152*e^6 + 11933/3152*e^5 + 127165/3152*e^4 - 36163/3152*e^3 - 426135/3152*e^2 - 19729/1576*e + 97439/788, 5/197*e^8 - 91/788*e^7 - 341/788*e^6 + 919/394*e^5 + 557/394*e^4 - 5801/394*e^3 + 1529/394*e^2 + 9523/394*e + 2329/197, -1745/6304*e^8 + 6413/6304*e^7 + 17905/3152*e^6 - 54819/3152*e^5 - 119075/3152*e^4 + 247821/3152*e^3 + 254537/3152*e^2 - 128141/1576*e - 11549/788, 535/3152*e^8 - 1203/3152*e^7 - 6851/1576*e^6 + 9365/1576*e^5 + 59077/1576*e^4 - 34859/1576*e^3 - 165703/1576*e^2 + 11411/788*e + 13051/394, -1743/12608*e^8 + 947/12608*e^7 + 28063/6304*e^6 - 493/6304*e^5 - 290429/6304*e^4 - 92525/6304*e^3 + 1036839/6304*e^2 + 191485/3152*e - 250955/1576, 327/12608*e^8 - 4827/12608*e^7 + 2881/6304*e^6 + 48821/6304*e^5 - 86507/6304*e^4 - 313211/6304*e^3 + 472593/6304*e^2 + 310739/3152*e - 107725/1576, 3681/12608*e^8 - 5293/12608*e^7 - 52297/6304*e^6 + 32355/6304*e^5 + 488035/6304*e^4 + 12211/6304*e^3 - 1527609/6304*e^2 - 222507/3152*e + 264773/1576, -299/12608*e^8 - 5505/12608*e^7 + 14827/6304*e^6 + 56127/6304*e^5 - 216369/6304*e^4 - 350881/6304*e^3 + 919619/6304*e^2 + 288641/3152*e - 243255/1576, -57/1576*e^8 + 151/1576*e^7 + 339/394*e^6 - 1447/788*e^5 - 4769/788*e^4 + 9431/788*e^3 + 5187/788*e^2 - 4660/197*e + 4507/197, -661/1576*e^8 + 1993/1576*e^7 + 7541/788*e^6 - 16711/788*e^5 - 57011/788*e^4 + 73001/788*e^3 + 143285/788*e^2 - 34937/394*e - 15945/197, -939/12608*e^8 + 3711/12608*e^7 + 10827/6304*e^6 - 36321/6304*e^5 - 89201/6304*e^4 + 206687/6304*e^3 + 259203/6304*e^2 - 147407/3152*e - 28359/1576, 721/3152*e^8 - 493/3152*e^7 - 11697/1576*e^6 + 2723/1576*e^5 + 119555/1576*e^4 + 11051/1576*e^3 - 404057/1576*e^2 - 31647/788*e + 76273/394, 2589/6304*e^8 - 7377/6304*e^7 - 29769/3152*e^6 + 57471/3152*e^5 + 232767/3152*e^4 - 210401/3152*e^3 - 630781/3152*e^2 + 42461/1576*e + 86557/788, 601/12608*e^8 - 6645/12608*e^7 + 2919/6304*e^6 + 55915/6304*e^5 - 101669/6304*e^4 - 268949/6304*e^3 + 543007/6304*e^2 + 162197/3152*e - 196467/1576, 1267/6304*e^8 - 1815/6304*e^7 - 17051/3152*e^6 + 6713/3152*e^5 + 159721/3152*e^4 + 53801/3152*e^3 - 541211/3152*e^2 - 140097/1576*e + 112191/788, -2657/12608*e^8 + 8829/12608*e^7 + 28753/6304*e^6 - 74819/6304*e^5 - 201683/6304*e^4 + 341981/6304*e^3 + 422633/6304*e^2 - 189005/3152*e - 13509/1576, -77/12608*e^8 - 743/12608*e^7 + 4493/6304*e^6 - 199/6304*e^5 - 59895/6304*e^4 + 60345/6304*e^3 + 189429/6304*e^2 - 137305/3152*e - 23745/1576, 1517/12608*e^8 - 7385/12608*e^7 - 12829/6304*e^6 + 67943/6304*e^5 + 60599/6304*e^4 - 369273/6304*e^3 - 49397/6304*e^2 + 298105/3152*e - 25583/1576, -2965/12608*e^8 + 5857/12608*e^7 + 40421/6304*e^6 - 44095/6304*e^5 - 378223/6304*e^4 + 129473/6304*e^3 + 1268605/6304*e^2 + 18255/3152*e - 256633/1576, 2687/12608*e^8 - 2563/12608*e^7 - 40287/6304*e^6 + 10301/6304*e^5 + 386221/6304*e^4 + 100349/6304*e^3 - 1210999/6304*e^2 - 252077/3152*e + 196939/1576, 2251/12608*e^8 - 5583/12608*e^7 - 27843/6304*e^6 + 43969/6304*e^5 + 229441/6304*e^4 - 157615/6304*e^3 - 614995/6304*e^2 + 54887/3152*e + 73703/1576, 3193/12608*e^8 - 4885/12608*e^7 - 45497/6304*e^6 + 34123/6304*e^5 + 430395/6304*e^4 - 64757/6304*e^3 - 1407553/6304*e^2 - 109419/3152*e + 266973/1576, -335/6304*e^8 + 3051/6304*e^7 - 173/3152*e^6 - 28149/3152*e^5 + 35931/3152*e^4 + 160059/3152*e^3 - 199217/3152*e^2 - 130655/1576*e + 54345/788, 681/6304*e^8 - 1493/6304*e^7 - 8401/3152*e^6 + 9947/3152*e^5 + 69979/3152*e^4 - 23445/3152*e^3 - 192385/3152*e^2 + 13005/1576*e + 22133/788, 211/6304*e^8 + 1729/6304*e^7 - 7103/3152*e^6 - 16279/3152*e^5 + 91857/3152*e^4 + 103521/3152*e^3 - 333219/3152*e^2 - 116965/1576*e + 54015/788];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2, 2, -w + 1}>] := -1;
ALEigenvalues[ideal<ZF | {8, 2, w^3 - 7*w + 3}>] := -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;