/*
  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![9, 6, -7, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {23, 23, -1/3*w^3 + 2/3*w^2 + 1/3*w - 2}>;

primesArray := [
[4, 2, -1/3*w^3 + 2/3*w^2 + 4/3*w - 1],
[7, 7, 1/3*w^3 - 2/3*w^2 - 7/3*w + 1],
[7, 7, w - 2],
[7, 7, w - 1],
[9, 3, w],
[9, 3, -1/3*w^3 + 2/3*w^2 + 7/3*w - 2],
[23, 23, -1/3*w^3 + 2/3*w^2 + 1/3*w - 2],
[23, 23, -2/3*w^3 + 4/3*w^2 + 11/3*w - 4],
[31, 31, -2/3*w^3 + 1/3*w^2 + 14/3*w + 2],
[41, 41, -1/3*w^3 + 5/3*w^2 + 4/3*w - 7],
[41, 41, -1/3*w^3 + 5/3*w^2 + 4/3*w - 4],
[47, 47, 1/3*w^3 - 5/3*w^2 - 1/3*w + 5],
[47, 47, w^2 - w - 4],
[73, 73, -w^3 + 3*w^2 + 4*w - 8],
[73, 73, -2/3*w^3 + 7/3*w^2 - 1/3*w - 4],
[73, 73, -w^3 + w^2 + 7*w + 1],
[73, 73, 2/3*w^3 - 4/3*w^2 - 14/3*w + 5],
[79, 79, w^2 - 5],
[79, 79, -2/3*w^3 + 7/3*w^2 + 8/3*w - 6],
[97, 97, -2/3*w^3 + 7/3*w^2 + 5/3*w - 4],
[97, 97, 1/3*w^3 + 1/3*w^2 - 7/3*w - 5],
[103, 103, w^3 - 3*w^2 - 2*w + 5],
[103, 103, -4/3*w^3 + 11/3*w^2 + 13/3*w - 10],
[113, 113, -1/3*w^3 + 5/3*w^2 + 4/3*w - 3],
[113, 113, w^3 - 2*w^2 - 5*w + 1],
[113, 113, -1/3*w^3 + 5/3*w^2 + 4/3*w - 1],
[113, 113, -1/3*w^3 + 5/3*w^2 + 4/3*w - 10],
[127, 127, -2/3*w^3 + 1/3*w^2 + 14/3*w + 1],
[127, 127, -2/3*w^3 + 7/3*w^2 + 2/3*w - 6],
[137, 137, -1/3*w^3 + 5/3*w^2 - 2/3*w - 6],
[137, 137, 1/3*w^3 + 1/3*w^2 - 10/3*w - 1],
[151, 151, -w^3 + 2*w^2 + 3*w - 5],
[151, 151, 1/3*w^3 + 1/3*w^2 - 10/3*w - 7],
[169, 13, -2/3*w^3 + 7/3*w^2 + 5/3*w - 2],
[169, 13, 1/3*w^3 + 1/3*w^2 - 7/3*w - 7],
[191, 191, 1/3*w^3 - 2/3*w^2 + 2/3*w - 2],
[191, 191, 2*w - 5],
[191, 191, -w^3 + 2*w^2 + 6*w - 2],
[191, 191, 2/3*w^3 - 4/3*w^2 - 14/3*w - 1],
[239, 239, 1/3*w^3 - 2/3*w^2 - 7/3*w - 3],
[239, 239, w - 5],
[241, 241, 1/3*w^3 + 1/3*w^2 - 10/3*w],
[241, 241, 4/3*w^3 - 17/3*w^2 + 2/3*w + 9],
[241, 241, -1/3*w^3 + 5/3*w^2 - 2/3*w - 7],
[241, 241, -4/3*w^3 + 14/3*w^2 + 13/3*w - 14],
[257, 257, -w - 4],
[257, 257, -1/3*w^3 + 2/3*w^2 + 7/3*w - 6],
[263, 263, -4/3*w^3 + 11/3*w^2 + 16/3*w - 8],
[263, 263, 2/3*w^3 - 1/3*w^2 - 8/3*w - 3],
[281, 281, -1/3*w^3 - 1/3*w^2 + 10/3*w - 2],
[281, 281, -1/3*w^3 + 5/3*w^2 - 5/3*w - 5],
[281, 281, -1/3*w^3 + 5/3*w^2 - 2/3*w - 9],
[281, 281, -2/3*w^3 + 1/3*w^2 + 17/3*w],
[289, 17, -w^3 + 2*w^2 + 4*w - 4],
[289, 17, w^3 - 2*w^2 - 4*w + 2],
[311, 311, -2/3*w^3 + 10/3*w^2 - 4/3*w - 9],
[311, 311, -2/3*w^3 - 2/3*w^2 + 20/3*w + 5],
[313, 313, -1/3*w^3 + 5/3*w^2 - 2/3*w - 8],
[313, 313, -1/3*w^3 - 1/3*w^2 + 10/3*w - 1],
[313, 313, w^3 - 4*w^2 - 2*w + 10],
[313, 313, 1/3*w^3 + 4/3*w^2 - 10/3*w - 8],
[337, 337, -4/3*w^3 + 5/3*w^2 + 19/3*w + 4],
[337, 337, 5/3*w^3 - 13/3*w^2 - 17/3*w + 5],
[353, 353, -4/3*w^3 + 8/3*w^2 + 25/3*w - 3],
[353, 353, -1/3*w^3 - 1/3*w^2 + 16/3*w + 6],
[353, 353, -2/3*w^3 + 10/3*w^2 + 2/3*w - 13],
[353, 353, -5/3*w^3 + 7/3*w^2 + 29/3*w - 3],
[367, 367, -5/3*w^3 + 4/3*w^2 + 35/3*w + 4],
[367, 367, 2/3*w^3 - 7/3*w^2 - 8/3*w + 4],
[367, 367, -4/3*w^3 + 14/3*w^2 + 1/3*w - 8],
[367, 367, w^2 - 7],
[383, 383, 4/3*w^3 - 14/3*w^2 - 10/3*w + 15],
[383, 383, -5/3*w^3 + 13/3*w^2 + 20/3*w - 12],
[383, 383, 2*w^2 - 2*w - 13],
[383, 383, w^3 - w^2 - 8*w - 1],
[401, 401, -4/3*w^3 + 11/3*w^2 + 22/3*w - 11],
[401, 401, 5/3*w^3 - 10/3*w^2 - 29/3*w + 5],
[409, 409, -w^3 + 2*w^2 + 3*w - 2],
[409, 409, -1/3*w^3 + 8/3*w^2 - 2/3*w - 6],
[409, 409, 1/3*w^3 - 8/3*w^2 + 2/3*w + 12],
[409, 409, 4/3*w^3 - 8/3*w^2 - 19/3*w + 4],
[433, 433, -w^3 + 3*w^2 - 1],
[433, 433, -1/3*w^3 + 5/3*w^2 - 8/3*w - 5],
[433, 433, -w^3 + w^2 + 8*w - 2],
[433, 433, 5/3*w^3 - 7/3*w^2 - 32/3*w - 2],
[439, 439, w^2 - 10],
[439, 439, -2/3*w^3 + 4/3*w^2 + 17/3*w - 7],
[449, 449, -2/3*w^3 + 4/3*w^2 + 11/3*w - 8],
[449, 449, -4/3*w^3 + 5/3*w^2 + 22/3*w + 2],
[457, 457, 2/3*w^3 - 7/3*w^2 - 14/3*w + 6],
[457, 457, 2/3*w^3 - 7/3*w^2 - 14/3*w + 9],
[463, 463, w^3 - 3*w^2 - 6*w + 11],
[463, 463, -1/3*w^3 + 5/3*w^2 + 10/3*w - 4],
[479, 479, -1/3*w^3 + 5/3*w^2 - 2/3*w + 1],
[479, 479, 5/3*w^3 - 10/3*w^2 - 23/3*w + 9],
[487, 487, -2/3*w^3 + 7/3*w^2 + 2/3*w - 8],
[487, 487, -2/3*w^3 + 1/3*w^2 + 14/3*w - 1],
[503, 503, w^3 - 4*w^2 - 2*w + 16],
[503, 503, -1/3*w^3 + 5/3*w^2 - 2/3*w - 11],
[521, 521, -4/3*w^3 + 11/3*w^2 + 10/3*w - 7],
[521, 521, 4/3*w^3 - 5/3*w^2 - 22/3*w],
[529, 23, -1/3*w^3 + 2/3*w^2 + 4/3*w - 6],
[569, 569, -1/3*w^3 + 8/3*w^2 + 1/3*w - 10],
[569, 569, -2/3*w^3 + 10/3*w^2 + 5/3*w - 10],
[577, 577, -2/3*w^3 + 10/3*w^2 + 5/3*w - 12],
[577, 577, -1/3*w^3 + 8/3*w^2 + 1/3*w - 8],
[593, 593, w^3 - w^2 - 5*w - 5],
[593, 593, 4/3*w^3 - 11/3*w^2 - 13/3*w + 4],
[599, 599, 2*w^2 - 3*w - 7],
[599, 599, 2/3*w^3 - 1/3*w^2 - 8/3*w - 4],
[599, 599, -1/3*w^3 + 8/3*w^2 - 5/3*w - 9],
[599, 599, -4/3*w^3 + 11/3*w^2 + 16/3*w - 7],
[607, 607, 4/3*w^3 - 8/3*w^2 - 22/3*w + 9],
[607, 607, -2/3*w^3 + 4/3*w^2 + 2/3*w - 5],
[625, 5, -5],
[631, 631, -1/3*w^3 + 2/3*w^2 + 13/3*w - 5],
[631, 631, -2/3*w^3 + 4/3*w^2 + 17/3*w - 1],
[641, 641, -2/3*w^3 + 7/3*w^2 + 14/3*w - 7],
[641, 641, 2/3*w^3 - 7/3*w^2 - 14/3*w + 8],
[647, 647, -2/3*w^3 - 5/3*w^2 + 26/3*w + 8],
[647, 647, -1/3*w^3 + 8/3*w^2 + 1/3*w - 9],
[647, 647, -2/3*w^3 + 10/3*w^2 + 5/3*w - 11],
[647, 647, -2/3*w^3 + 13/3*w^2 - 10/3*w - 13],
[673, 673, -4/3*w^3 + 5/3*w^2 + 25/3*w - 2],
[673, 673, -w^3 + 3*w^2 + w - 7],
[727, 727, -2/3*w^3 + 7/3*w^2 + 2/3*w - 11],
[727, 727, -2/3*w^3 + 1/3*w^2 + 14/3*w - 4],
[743, 743, w^2 - w - 10],
[743, 743, 1/3*w^3 - 5/3*w^2 - 1/3*w - 1],
[769, 769, -w^3 + 2*w^2 + 7*w - 4],
[769, 769, 3*w - 2],
[809, 809, -1/3*w^3 + 8/3*w^2 - 2/3*w - 8],
[809, 809, -1/3*w^3 + 8/3*w^2 - 2/3*w - 10],
[823, 823, -w^3 + 2*w^2 + 7*w - 5],
[823, 823, 3*w - 1],
[839, 839, 4/3*w^3 - 2/3*w^2 - 25/3*w - 8],
[839, 839, 2/3*w^3 - 7/3*w^2 - 14/3*w + 12],
[857, 857, -1/3*w^3 + 5/3*w^2 - 5/3*w - 7],
[857, 857, -2/3*w^3 + 1/3*w^2 + 17/3*w - 2],
[863, 863, -2/3*w^3 + 7/3*w^2 - 7/3*w - 2],
[863, 863, 5/3*w^3 - 7/3*w^2 - 35/3*w + 1],
[881, 881, -2/3*w^3 + 1/3*w^2 + 14/3*w + 8],
[881, 881, -2/3*w^3 + 13/3*w^2 - 4/3*w - 18],
[881, 881, -5/3*w^3 + 7/3*w^2 + 35/3*w + 3],
[881, 881, -2/3*w^3 + 7/3*w^2 + 2/3*w + 1],
[887, 887, -1/3*w^3 + 11/3*w^2 - 8/3*w - 7],
[887, 887, -1/3*w^3 + 11/3*w^2 - 8/3*w - 18],
[911, 911, 5/3*w^3 - 4/3*w^2 - 29/3*w - 3],
[911, 911, 2*w^3 - 6*w^2 - 5*w + 13],
[929, 929, 2*w^3 - 4*w^2 - 9*w + 2],
[929, 929, 5/3*w^3 - 10/3*w^2 - 17/3*w],
[929, 929, -4/3*w^3 + 14/3*w^2 + 13/3*w - 11],
[929, 929, 1/3*w^3 + 4/3*w^2 - 7/3*w - 9],
[937, 937, -w^3 + 4*w^2 + w - 13],
[937, 937, 2/3*w^3 + 2/3*w^2 - 17/3*w - 3],
[953, 953, -2*w^3 + w^2 + 15*w + 8],
[953, 953, -5/3*w^3 + 19/3*w^2 - 1/3*w - 11],
[961, 31, 4/3*w^3 - 8/3*w^2 - 16/3*w + 5],
[967, 967, -4/3*w^3 + 2/3*w^2 + 31/3*w + 11],
[967, 967, 7/3*w^3 - 5/3*w^2 - 46/3*w - 9],
[977, 977, 2*w^3 - 5*w^2 - 8*w + 10],
[977, 977, -w^3 + 3*w^2 + 2*w - 14],
[977, 977, -4/3*w^3 + 8/3*w^2 + 25/3*w - 11],
[977, 977, -w^3 + w^2 + 6*w - 7],
[983, 983, -5/3*w^3 + 16/3*w^2 + 8/3*w - 12],
[983, 983, -1/3*w^3 + 11/3*w^2 - 11/3*w - 15],
[983, 983, 3*w^2 - 2*w - 16],
[983, 983, -5/3*w^3 + 16/3*w^2 + 11/3*w - 14],
[991, 991, 4/3*w^3 - 8/3*w^2 - 22/3*w + 1],
[991, 991, 2/3*w^3 - 4/3*w^2 - 2/3*w - 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^10 - 34*x^8 + 385*x^6 - 1640*x^4 + 1928*x^2 - 576;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1/1012*e^8 - 3/88*e^6 + 931/2024*e^4 - 67/23*e^2 + 1230/253, 109/6072*e^9 - 79/132*e^7 + 39481/6072*e^5 - 3439/138*e^3 + 13399/759*e, 7/1012*e^9 - 21/88*e^7 + 5505/2024*e^5 - 501/46*e^3 + 1526/253*e, e, 1/1104*e^9 - 1/24*e^7 + 661/1104*e^5 - 167/69*e^3 - 305/138*e, 73/12144*e^9 - 13/66*e^7 + 25759/12144*e^5 - 1169/138*e^3 + 14029/1518*e, -1, -15/2024*e^8 + 17/88*e^6 - 351/253*e^4 + 54/23*e^2 + 136/253, -27/1012*e^8 + 35/44*e^6 - 3691/506*e^4 + 498/23*e^2 - 2344/253, -475/12144*e^9 + 337/264*e^7 - 163555/12144*e^5 + 6839/138*e^3 - 48739/1518*e, -73/4048*e^9 + 13/22*e^7 - 25759/4048*e^5 + 1169/46*e^3 - 13017/506*e, -3/506*e^9 + 9/44*e^7 - 2287/1012*e^5 + 367/46*e^3 - 43/253*e, -7/759*e^9 + 73/264*e^7 - 15695/6072*e^5 + 565/69*e^3 - 5345/759*e, 67/4048*e^9 - 53/88*e^7 + 29291/4048*e^5 - 691/23*e^3 + 8821/506*e, -51/2024*e^8 + 71/88*e^6 - 8771/1012*e^4 + 800/23*e^2 - 5812/253, -14/253*e^8 + 73/44*e^6 - 15695/1012*e^4 + 1130/23*e^2 - 9172/253, -397/12144*e^9 + 295/264*e^7 - 152293/12144*e^5 + 6917/138*e^3 - 61489/1518*e, 13/759*e^9 - 145/264*e^7 + 35003/6072*e^5 - 3035/138*e^3 + 13613/759*e, 13/1012*e^9 - 39/88*e^7 + 10079/2024*e^5 - 434/23*e^3 + 2075/253*e, -179/4048*e^9 + 63/44*e^7 - 60681/4048*e^5 + 1279/23*e^3 - 20523/506*e, -9/368*e^9 + 7/8*e^7 - 3833/368*e^5 + 2063/46*e^3 - 1809/46*e, 9/184*e^8 - 11/8*e^6 + 527/46*e^4 - 706/23*e^2 + 636/23, e^2 - 4, 145/12144*e^9 - 53/132*e^7 + 53203/12144*e^5 - 1135/69*e^3 + 15805/1518*e, 403/12144*e^9 - 283/264*e^7 + 136111/12144*e^5 - 2800/69*e^3 + 27229/1518*e, -171/4048*e^9 + 15/11*e^7 - 56957/4048*e^5 + 1122/23*e^3 - 10543/506*e, 523/12144*e^9 - 373/264*e^7 + 179827/12144*e^5 - 3568/69*e^3 + 35755/1518*e, -31/1012*e^8 + 41/44*e^6 - 2311/253*e^4 + 720/23*e^2 - 4228/253, 13/253*e^8 - 39/22*e^6 + 10079/506*e^4 - 1782/23*e^2 + 12348/253, 41/2024*e^8 - 45/88*e^6 + 3281/1012*e^4 - 51/23*e^2 - 1856/253, 13/2024*e^8 - 25/88*e^6 + 4101/1012*e^4 - 447/23*e^2 + 6224/253, 1/253*e^8 - 3/22*e^6 + 931/506*e^4 - 222/23*e^2 - 140/253, -3/1012*e^8 - 1/44*e^6 + 883/506*e^4 - 282/23*e^2 + 5924/253, 85/12144*e^9 - 61/264*e^7 + 31345/12144*e^5 - 1571/138*e^3 + 22927/1518*e, 601/12144*e^9 - 415/264*e^7 + 197161/12144*e^5 - 8507/138*e^3 + 94351/1518*e, -89/3036*e^9 + 139/132*e^7 - 19513/1518*e^5 + 8131/138*e^3 - 47485/759*e, -21/2024*e^9 + 13/44*e^7 - 4589/2024*e^5 + 73/46*e^3 + 5048/253*e, -97/3036*e^9 + 35/33*e^7 - 33895/3036*e^5 + 2635/69*e^3 - 9761/759*e, -37/1518*e^9 + 25/33*e^7 - 5639/759*e^5 + 3361/138*e^3 - 3988/759*e, 14/253*e^8 - 21/11*e^6 + 5505/253*e^4 - 2004/23*e^2 + 15244/253, -15/253*e^8 + 45/22*e^6 - 11435/506*e^4 + 1904/23*e^2 - 12068/253, 31/506*e^8 - 93/44*e^6 + 23801/1012*e^4 - 2038/23*e^2 + 13516/253, 151/12144*e^9 - 127/264*e^7 + 74971/12144*e^5 - 3851/138*e^3 + 34675/1518*e, -39/1012*e^8 + 21/22*e^6 - 5631/1012*e^4 - 9/23*e^2 + 4148/253, -1/16*e^9 + 17/8*e^7 - 377/16*e^5 + 91*e^3 - 107/2*e, 167/2024*e^8 - 223/88*e^6 + 24385/1012*e^4 - 1719/23*e^2 + 8336/253, 49/2024*e^8 - 57/88*e^6 + 5143/1012*e^4 - 319/23*e^2 + 5088/253, 15/506*e^9 - 45/44*e^7 + 11941/1012*e^5 - 1159/23*e^3 + 13624/253*e, -29/552*e^9 + 5/3*e^7 - 9371/552*e^5 + 4235/69*e^3 - 3644/69*e, -2/253*e^8 + 23/44*e^6 - 9037/1012*e^4 + 1065/23*e^2 - 10852/253, -7/2024*e^8 + 5/88*e^6 + 229/506*e^4 - 237/23*e^2 + 8598/253, 39/2024*e^8 - 75/88*e^6 + 12303/1012*e^4 - 1341/23*e^2 + 9564/253, 7/253*e^8 - 21/22*e^6 + 5505/506*e^4 - 1002/23*e^2 + 9646/253, -59/1012*e^8 + 83/44*e^6 - 9621/506*e^4 + 1423/23*e^2 - 6790/253, -255/2024*e^8 + 333/88*e^6 - 17753/506*e^4 + 2505/23*e^2 - 17422/253, 17/2024*e^8 + 13/88*e^6 - 3809/506*e^4 + 1190/23*e^2 - 8520/253, -71/2024*e^8 + 101/88*e^6 - 6207/506*e^4 + 1056/23*e^2 - 7992/253, 79/1012*e^8 - 51/22*e^6 + 21215/1012*e^4 - 1360/23*e^2 + 3560/253, 1/88*e^8 - 51/88*e^6 + 417/44*e^4 - 53*e^2 + 560/11, -61/12144*e^9 + 43/264*e^7 - 20173/12144*e^5 + 767/138*e^3 - 15757/1518*e, -31/4048*e^9 + 13/44*e^7 - 14557/4048*e^5 + 659/46*e^3 - 4391/506*e, -367/4048*e^9 + 32/11*e^7 - 121377/4048*e^5 + 4969/46*e^3 - 36461/506*e, -155/4048*e^9 + 119/88*e^7 - 64183/4048*e^5 + 3065/46*e^3 - 24485/506*e, -985/12144*e^9 + 703/264*e^7 - 345553/12144*e^5 + 7339/69*e^3 - 86113/1518*e, -129/1012*e^8 + 177/44*e^6 - 20727/506*e^4 + 3376/23*e^2 - 22050/253, -269/12144*e^9 + 155/264*e^7 - 52229/12144*e^5 + 521/69*e^3 + 1039/1518*e, -1/2024*e^8 + 7/88*e^6 - 580/253*e^4 + 413/23*e^2 - 4410/253, -29/1012*e^8 + 27/44*e^6 - 497/253*e^4 - 288/23*e^2 + 7340/253, -329/3036*e^9 + 233/66*e^7 - 113555/3036*e^5 + 19591/138*e^3 - 83866/759*e, 51/1012*e^8 - 71/44*e^6 + 8265/506*e^4 - 1278/23*e^2 + 12636/253, -59/2024*e^9 + 47/44*e^7 - 26579/2024*e^5 + 2803/46*e^3 - 20346/253*e, 31/253*e^8 - 41/11*e^6 + 8991/253*e^4 - 2604/23*e^2 + 14888/253, -37/1012*e^8 + 61/44*e^6 - 8675/506*e^4 + 1697/23*e^2 - 13632/253, -13/1518*e^9 + 23/264*e^7 + 12479/6072*e^5 - 3209/138*e^3 + 28487/759*e, 127/2024*e^9 - 185/88*e^7 + 5856/253*e^5 - 2127/23*e^3 + 20421/253*e, 485/12144*e^9 - 175/132*e^7 + 172511/12144*e^5 - 7243/138*e^3 + 51347/1518*e, 25/12144*e^9 - 2/33*e^7 + 9487/12144*e^5 - 505/69*e^3 + 46747/1518*e, -35/1012*e^8 + 29/22*e^6 - 17431/1012*e^4 + 1816/23*e^2 - 10160/253, 223/12144*e^9 - 37/66*e^7 + 64465/12144*e^5 - 1165/69*e^3 + 15199/1518*e, 995/12144*e^9 - 179/66*e^7 + 354509/12144*e^5 - 7679/69*e^3 + 126671/1518*e, -3/506*e^8 + 9/44*e^6 - 1781/1012*e^4 + 11/23*e^2 + 3752/253, -439/12144*e^9 + 277/264*e^7 - 114919/12144*e^5 + 2161/69*e^3 - 63031/1518*e, 13/1012*e^8 - 3/44*e^6 - 1112/253*e^4 + 877/23*e^2 - 6274/253, 31/1012*e^8 - 41/44*e^6 + 2058/253*e^4 - 352/23*e^2 - 5386/253, -169/4048*e^9 + 113/88*e^7 - 49701/4048*e^5 + 1717/46*e^3 - 6277/506*e, -403/6072*e^9 + 283/132*e^7 - 136111/6072*e^5 + 5807/69*e^3 - 60625/759*e, -119/6072*e^9 + 151/264*e^7 - 14731/3036*e^5 + 1349/138*e^3 + 9799/759*e, -119/2024*e^8 + 151/88*e^6 - 15237/1012*e^4 + 985/23*e^2 - 10188/253, -65/1012*e^8 + 23/11*e^6 - 23047/1012*e^4 + 2124/23*e^2 - 18724/253, 191/12144*e^9 - 179/264*e^7 + 116867/12144*e^5 - 3389/69*e^3 + 89129/1518*e, 313/12144*e^9 - 83/132*e^7 + 43363/12144*e^5 + 122/69*e^3 - 29639/1518*e, 1/88*e^9 - 5/11*e^7 + 515/88*e^5 - 47/2*e^3 + 32/11*e, -5/44*e^9 + 167/44*e^7 - 454/11*e^5 + 315/2*e^3 - 1156/11*e, 327/2024*e^8 - 441/88*e^6 + 12560/253*e^4 - 3951/23*e^2 + 25776/253, 17/1012*e^8 - 31/44*e^6 + 2263/253*e^4 - 817/23*e^2 + 7248/253, 1/46*e^8 - e^6 + 661/46*e^4 - 1543/23*e^2 + 1172/23, -113/1012*e^8 + 153/44*e^6 - 17509/506*e^4 + 2718/23*e^2 - 19068/253, 61/1012*e^8 - 97/44*e^6 + 13249/506*e^4 - 2454/23*e^2 + 19876/253, 191/2024*e^8 - 281/88*e^6 + 8821/253*e^4 - 2983/23*e^2 + 20060/253, -1/253*e^8 + 3/22*e^6 - 931/506*e^4 + 222/23*e^2 + 2670/253, -47/506*e^8 + 65/22*e^6 - 7587/253*e^4 + 2434/23*e^2 - 19986/253, 37/2024*e^8 - 83/88*e^6 + 3497/253*e^4 - 1366/23*e^2 + 2262/253, 1063/12144*e^9 - 745/264*e^7 + 356815/12144*e^5 - 7438/69*e^3 + 115867/1518*e, -243/4048*e^9 + 87/44*e^7 - 86425/4048*e^5 + 1914/23*e^3 - 36607/506*e, -75/4048*e^9 + 37/88*e^7 - 5691/4048*e^5 - 627/46*e^3 + 25221/506*e, -395/4048*e^9 + 36/11*e^7 - 147061/4048*e^5 + 6781/46*e^3 - 64307/506*e, 157/4048*e^9 - 115/88*e^7 + 58789/4048*e^5 - 1359/23*e^3 + 33811/506*e, -127/1104*e^9 + 11/3*e^7 - 41305/1104*e^5 + 9065/69*e^3 - 10669/138*e, -6/253*e^9 + 9/11*e^7 - 4827/506*e^5 + 1859/46*e^3 - 6497/253*e, -179/3036*e^9 + 137/66*e^7 - 73331/3036*e^5 + 6956/69*e^3 - 58756/759*e, 335/3036*e^9 - 475/132*e^7 + 57415/1518*e^5 - 9404/69*e^3 + 52537/759*e, 223/3036*e^9 - 329/132*e^7 + 20860/759*e^5 - 7144/69*e^3 + 31916/759*e, -9/184*e^8 + 11/8*e^6 - 527/46*e^4 + 683/23*e^2 - 452/23, 4/253*e^8 - 1/22*e^6 - 3107/506*e^4 + 1090/23*e^2 - 5620/253, 101/2024*e^8 - 135/88*e^6 + 7105/506*e^4 - 865/23*e^2 + 130/253, -5/264*e^9 + 167/264*e^7 - 919/132*e^5 + 173/6*e^3 - 1040/33*e, -43/759*e^9 + 505/264*e^7 - 128507/6072*e^5 + 11249/138*e^3 - 28388/759*e, -235/12144*e^9 + 223/264*e^7 - 145951/12144*e^5 + 4057/69*e^3 - 72673/1518*e, -1231/12144*e^9 + 871/264*e^7 - 416803/12144*e^5 + 16709/138*e^3 - 105337/1518*e, -3/46*e^8 + 2*e^6 - 879/46*e^4 + 1340/23*e^2 - 572/23, -233/3036*e^9 + 677/264*e^7 - 171889/6072*e^5 + 7952/69*e^3 - 85708/759*e, 127/2024*e^9 - 87/44*e^7 + 40523/2024*e^5 - 1690/23*e^3 + 17132/253*e, -1/184*e^8 + 3/8*e^6 - 309/46*e^4 + 748/23*e^2 - 316/23, 127/2024*e^8 - 207/88*e^6 + 28737/1012*e^4 - 2633/23*e^2 + 19156/253, 25/184*e^8 - 33/8*e^6 + 3651/92*e^4 - 3060/23*e^2 + 1828/23, -41/2024*e^8 + 67/88*e^6 - 4803/506*e^4 + 856/23*e^2 + 1856/253, -93/1012*e^8 + 145/44*e^6 - 19685/506*e^4 + 3678/23*e^2 - 21792/253, -3/184*e^8 + 9/8*e^6 - 927/46*e^4 + 2543/23*e^2 - 1960/23, -19/2024*e^8 + 45/88*e^6 - 2165/253*e^4 + 993/23*e^2 + 712/253, -7/12144*e^9 - 47/264*e^7 + 55817/12144*e^5 - 4009/138*e^3 + 44777/1518*e, -185/4048*e^9 + 125/88*e^7 - 59173/4048*e^5 + 1391/23*e^3 - 48501/506*e, 1657/12144*e^9 - 1207/264*e^7 + 609793/12144*e^5 - 13558/69*e^3 + 214009/1518*e, -83/12144*e^9 + 131/264*e^7 - 116687/12144*e^5 + 4202/69*e^3 - 142631/1518*e, 269/2024*e^9 - 365/88*e^7 + 41421/1012*e^5 - 6293/46*e^3 + 18442/253*e, 1/23*e^9 - 5/4*e^7 + 965/92*e^5 - 1089/46*e^3 - 186/23*e, -135/1012*e^9 + 197/44*e^7 - 25033/506*e^5 + 9097/46*e^3 - 40056/253*e, -103/6072*e^9 + 29/66*e^7 - 17713/6072*e^5 + 277/138*e^3 + 7748/759*e, -117/2024*e^8 + 137/88*e^6 - 11609/1012*e^4 + 412/23*e^2 + 4704/253, 139/2024*e^8 - 203/88*e^6 + 25205/1012*e^4 - 2023/23*e^2 + 10344/253, 29/759*e^9 - 293/264*e^7 + 59167/6072*e^5 - 3745/138*e^3 + 10867/759*e, 211/2024*e^9 - 75/22*e^7 + 73553/2024*e^5 - 3193/23*e^3 + 26541/253*e, -103/1012*e^8 + 29/11*e^6 - 18725/1012*e^4 + 622/23*e^2 + 5376/253, 7/528*e^9 - 37/132*e^7 + 349/528*e^5 + 61/6*e^3 - 1679/66*e, 1517/12144*e^9 - 281/66*e^7 + 580043/12144*e^5 - 26635/138*e^3 + 239735/1518*e, -45/2024*e^8 + 73/88*e^6 - 9525/1012*e^4 + 622/23*e^2 + 6480/253, -1099/6072*e^9 + 1577/264*e^7 - 48689/759*e^5 + 33337/138*e^3 - 119032/759*e, -47/552*e^9 + 35/12*e^7 - 18095/552*e^5 + 17941/138*e^3 - 6020/69*e, -145/2024*e^8 + 179/88*e^6 - 8557/506*e^4 + 890/23*e^2 - 372/253, 73/1012*e^8 - 115/44*e^6 + 8021/253*e^4 - 3097/23*e^2 + 24516/253, -163/2024*e^8 + 261/88*e^6 - 9026/253*e^4 + 3425/23*e^2 - 22574/253, 315/2024*e^8 - 401/88*e^6 + 20561/506*e^4 - 2629/23*e^2 + 9794/253, 1499/12144*e^9 - 547/132*e^7 + 552689/12144*e^5 - 12326/69*e^3 + 207413/1518*e, -157/1104*e^9 + 59/12*e^7 - 61687/1104*e^5 + 30703/138*e^3 - 19873/138*e, 81/1012*e^8 - 47/22*e^6 + 15821/1012*e^4 - 689/23*e^2 + 12092/253, -91/2024*e^8 + 131/88*e^6 - 17069/1012*e^4 + 1680/23*e^2 - 10172/253, -19/253*e^8 + 23/11*e^6 - 4417/253*e^4 + 1090/23*e^2 - 4930/253, 97/1012*e^8 - 107/44*e^6 + 3983/253*e^4 - 266/23*e^2 - 6178/253, 31/2024*e^8 - 19/88*e^6 - 1197/1012*e^4 + 422/23*e^2 - 8512/253, -52/253*e^8 + 145/22*e^6 - 34497/506*e^4 + 5656/23*e^2 - 39272/253, -21/253*e^8 + 26/11*e^6 - 5095/253*e^4 + 1304/23*e^2 - 14264/253, -75/506*e^8 + 203/44*e^6 - 46043/1012*e^4 + 3610/23*e^2 - 22580/253, 18/253*e^8 - 27/11*e^6 + 7114/253*e^4 - 2662/23*e^2 + 23286/253, -93/2024*e^8 + 101/88*e^6 - 7035/1012*e^4 - 47/23*e^2 + 6308/253, 15/1012*e^8 - 17/44*e^6 + 449/253*e^4 + 168/23*e^2 + 3270/253, -13/92*e^8 + 17/4*e^6 - 1847/46*e^4 + 3026/23*e^2 - 1776/23, 311/6072*e^9 - 203/132*e^7 + 87719/6072*e^5 - 6485/138*e^3 + 24032/759*e, 565/6072*e^9 - 421/132*e^7 + 221389/6072*e^5 - 21295/138*e^3 + 110920/759*e, 389/2024*e^8 - 523/88*e^6 + 14618/253*e^4 - 4280/23*e^2 + 24944/253, -181/2024*e^9 + 61/22*e^7 - 55287/2024*e^5 + 2119/23*e^3 - 13404/253*e, -683/6072*e^9 + 241/66*e^7 - 233561/6072*e^5 + 9943/69*e^3 - 88868/759*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {23, 23, -1/3*w^3 + 2/3*w^2 + 1/3*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;