/*
  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![25, 5, -11, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {25, 5, 2/5*w^3 - 2/5*w^2 - 12/5*w + 1}>;

primesArray := [
[5, 5, 1/5*w^3 + 4/5*w^2 - 11/5*w - 7],
[5, 5, 1/5*w^3 - 6/5*w^2 - 1/5*w + 4],
[11, 11, w + 1],
[11, 11, 1/5*w^3 - 1/5*w^2 - 11/5*w + 2],
[16, 2, 2],
[19, 19, -1/5*w^3 + 1/5*w^2 + 1/5*w + 1],
[19, 19, 1/5*w^3 - 1/5*w^2 - 11/5*w],
[19, 19, -2/5*w^3 + 2/5*w^2 + 17/5*w],
[19, 19, w - 1],
[29, 29, w^2 - w - 8],
[29, 29, 2/5*w^3 - 2/5*w^2 - 7/5*w + 2],
[31, 31, 1/5*w^3 - 1/5*w^2 - 1/5*w + 2],
[31, 31, 2/5*w^3 - 2/5*w^2 - 17/5*w + 3],
[61, 61, -2/5*w^3 - 3/5*w^2 + 12/5*w + 5],
[61, 61, -4/5*w^3 - 6/5*w^2 + 39/5*w + 15],
[61, 61, 3/5*w^3 + 2/5*w^2 - 18/5*w - 3],
[61, 61, 7/5*w^3 + 3/5*w^2 - 62/5*w - 13],
[71, 71, 1/5*w^3 - 6/5*w^2 - 6/5*w + 4],
[71, 71, w^2 - 8],
[79, 79, 3/5*w^3 + 12/5*w^2 - 33/5*w - 22],
[79, 79, 1/5*w^3 + 4/5*w^2 - 11/5*w - 9],
[81, 3, -3],
[101, 101, 3/5*w^3 + 7/5*w^2 - 28/5*w - 15],
[101, 101, 1/5*w^3 + 9/5*w^2 - 16/5*w - 16],
[121, 11, 3/5*w^3 - 3/5*w^2 - 18/5*w + 1],
[131, 131, 1/5*w^3 + 4/5*w^2 - 16/5*w - 8],
[131, 131, w^2 - 2*w - 2],
[149, 149, 4/5*w^3 + 1/5*w^2 - 29/5*w - 5],
[149, 149, 4/5*w^3 - 9/5*w^2 - 19/5*w + 6],
[151, 151, 2/5*w^3 + 3/5*w^2 - 17/5*w - 3],
[151, 151, -2/5*w^3 + 7/5*w^2 + 7/5*w - 8],
[151, 151, 4/5*w^3 + 1/5*w^2 - 29/5*w - 2],
[151, 151, -4/5*w^3 + 9/5*w^2 + 19/5*w - 9],
[179, 179, -3/5*w^3 + 8/5*w^2 + 18/5*w - 6],
[179, 179, 2/5*w^3 + 3/5*w^2 - 12/5*w - 6],
[211, 211, 1/5*w^3 + 4/5*w^2 - 6/5*w - 10],
[211, 211, 2/5*w^3 - 7/5*w^2 - 12/5*w + 2],
[229, 229, 6/5*w^3 - 11/5*w^2 - 41/5*w + 11],
[229, 229, 2*w^2 - 2*w - 9],
[229, 229, -4/5*w^3 - 1/5*w^2 + 19/5*w + 2],
[229, 229, -3/5*w^3 + 3/5*w^2 + 28/5*w - 3],
[239, 239, 11/5*w^3 + 4/5*w^2 - 96/5*w - 19],
[239, 239, -6/5*w^3 + 1/5*w^2 + 51/5*w + 3],
[241, 241, 1/5*w^3 + 4/5*w^2 - 1/5*w - 6],
[241, 241, 4/5*w^3 - 4/5*w^2 - 29/5*w - 1],
[251, 251, -1/5*w^3 + 6/5*w^2 + 11/5*w - 12],
[251, 251, -1/5*w^3 + 11/5*w^2 + 1/5*w - 15],
[269, 269, 2/5*w^3 + 3/5*w^2 - 17/5*w - 2],
[269, 269, -2/5*w^3 + 7/5*w^2 + 7/5*w - 9],
[271, 271, 1/5*w^3 - 1/5*w^2 + 4/5*w],
[271, 271, 3/5*w^3 - 3/5*w^2 - 28/5*w + 2],
[281, 281, w^3 - 3*w^2 - 4*w + 13],
[281, 281, -3/5*w^3 + 3/5*w^2 + 28/5*w - 6],
[281, 281, -6/5*w^3 + 26/5*w^2 + 11/5*w - 16],
[281, 281, w^3 + w^2 - 8*w - 9],
[289, 17, 2/5*w^3 + 3/5*w^2 - 7/5*w - 5],
[289, 17, 4/5*w^3 - 9/5*w^2 - 29/5*w + 8],
[311, 311, -w^2 + 2*w + 11],
[311, 311, -1/5*w^3 + 6/5*w^2 + 1/5*w - 1],
[311, 311, 1/5*w^3 + 4/5*w^2 - 16/5*w + 1],
[311, 311, -1/5*w^3 - 4/5*w^2 + 11/5*w + 10],
[349, 349, -1/5*w^3 - 9/5*w^2 + 11/5*w + 16],
[349, 349, 2/5*w^3 - 12/5*w^2 - 7/5*w + 7],
[359, 359, -4/5*w^3 + 4/5*w^2 + 19/5*w - 1],
[359, 359, -w^3 + 3*w^2 + 5*w - 12],
[359, 359, w^3 - w^2 - 7*w + 2],
[359, 359, 4/5*w^3 + 6/5*w^2 - 29/5*w - 11],
[379, 379, 4/5*w^3 + 6/5*w^2 - 34/5*w - 9],
[379, 379, 3/5*w^3 - 3/5*w^2 - 28/5*w + 7],
[389, 389, w^2 - 3*w - 8],
[389, 389, -4/5*w^3 + 14/5*w^2 + 19/5*w - 16],
[401, 401, -3*w^2 + 2*w + 23],
[401, 401, 1/5*w^3 - 16/5*w^2 + 4/5*w + 11],
[409, 409, -w^3 + 2*w^2 + 3*w - 7],
[409, 409, 7/5*w^3 - 2/5*w^2 - 57/5*w - 2],
[419, 419, -1/5*w^3 + 1/5*w^2 + 16/5*w - 2],
[419, 419, 1/5*w^3 - 1/5*w^2 - 16/5*w],
[421, 421, 2/5*w^3 + 3/5*w^2 - 27/5*w - 3],
[421, 421, 1/5*w^3 + 4/5*w^2 + 4/5*w - 3],
[439, 439, -w^3 + 3*w^2 + 5*w - 16],
[439, 439, -4/5*w^3 - 6/5*w^2 + 29/5*w + 7],
[439, 439, -3/5*w^3 + 13/5*w^2 + 3/5*w - 11],
[439, 439, 4/5*w^3 + 6/5*w^2 - 39/5*w - 10],
[449, 449, 2/5*w^3 - 12/5*w^2 - 2/5*w + 5],
[449, 449, 4/5*w^3 - 24/5*w^2 - 4/5*w + 15],
[461, 461, -1/5*w^3 - 4/5*w^2 - 4/5*w + 4],
[461, 461, 2*w^2 - 2*w - 11],
[461, 461, 3/5*w^3 + 2/5*w^2 - 33/5*w - 4],
[479, 479, -w^3 + 3*w^2 + 6*w - 14],
[479, 479, -3/5*w^3 + 3/5*w^2 + 28/5*w + 3],
[499, 499, 2/5*w^3 + 8/5*w^2 - 17/5*w - 6],
[499, 499, -3/5*w^3 + 13/5*w^2 + 13/5*w - 17],
[509, 509, -w^3 + 2*w^2 + 4*w - 6],
[509, 509, 6/5*w^3 - 1/5*w^2 - 46/5*w - 4],
[521, 521, -2/5*w^3 + 2/5*w^2 + 7/5*w - 7],
[521, 521, -3/5*w^3 + 3/5*w^2 + 23/5*w - 8],
[529, 23, -7/5*w^3 + 17/5*w^2 + 42/5*w - 15],
[529, 23, 2/5*w^3 + 13/5*w^2 - 27/5*w - 21],
[541, 541, 1/5*w^3 - 11/5*w^2 - 1/5*w + 10],
[541, 541, 3/5*w^3 - 8/5*w^2 - 13/5*w + 13],
[541, 541, -2*w^2 + w + 13],
[541, 541, 3/5*w^3 + 2/5*w^2 - 23/5*w + 2],
[569, 569, w^3 - 2*w^2 - 5*w + 7],
[569, 569, -1/5*w^3 + 1/5*w^2 + 1/5*w - 6],
[571, 571, w^2 + w - 9],
[571, 571, -2/5*w^3 + 12/5*w^2 + 7/5*w - 12],
[571, 571, 1/5*w^3 + 9/5*w^2 - 11/5*w - 11],
[571, 571, -2/5*w^3 + 7/5*w^2 + 17/5*w - 4],
[601, 601, -3/5*w^3 + 13/5*w^2 + 18/5*w - 11],
[601, 601, -4/5*w^3 + 4/5*w^2 + 34/5*w + 1],
[631, 631, 2/5*w^3 + 18/5*w^2 - 27/5*w - 30],
[631, 631, 3/5*w^3 - 23/5*w^2 - 3/5*w + 15],
[661, 661, -2*w^3 + 17*w + 12],
[661, 661, -9/5*w^3 + 4/5*w^2 + 74/5*w + 1],
[691, 691, -7/5*w^3 - 18/5*w^2 + 72/5*w + 35],
[691, 691, 6/5*w^3 - 31/5*w^2 - 6/5*w + 19],
[701, 701, -1/5*w^3 + 6/5*w^2 + 16/5*w - 6],
[701, 701, 2/5*w^3 - 7/5*w^2 - 22/5*w + 8],
[709, 709, -3/5*w^3 + 13/5*w^2 + 13/5*w - 18],
[709, 709, 1/5*w^3 + 14/5*w^2 - 11/5*w - 20],
[709, 709, -2/5*w^3 - 8/5*w^2 + 17/5*w + 5],
[709, 709, 1/5*w^3 + 4/5*w^2 - 21/5*w - 10],
[719, 719, -3/5*w^3 + 8/5*w^2 + 13/5*w - 12],
[719, 719, 3/5*w^3 + 2/5*w^2 - 23/5*w + 1],
[719, 719, -7/5*w^3 + 17/5*w^2 + 32/5*w - 14],
[719, 719, 7/5*w^3 + 3/5*w^2 - 52/5*w - 8],
[739, 739, 1/5*w^3 - 1/5*w^2 - 11/5*w - 5],
[739, 739, w - 6],
[761, 761, -4/5*w^3 + 9/5*w^2 + 19/5*w - 11],
[761, 761, 4/5*w^3 + 1/5*w^2 - 29/5*w],
[769, 769, 3/5*w^3 - 13/5*w^2 - 13/5*w + 8],
[769, 769, -2/5*w^3 - 8/5*w^2 + 17/5*w + 15],
[769, 769, 7/5*w^3 - 17/5*w^2 - 37/5*w + 18],
[769, 769, -6/5*w^3 - 4/5*w^2 + 41/5*w + 5],
[809, 809, -1/5*w^3 + 6/5*w^2 - 4/5*w - 14],
[809, 809, -2/5*w^3 - 3/5*w^2 + 22/5*w - 4],
[809, 809, -6/5*w^3 + 6/5*w^2 + 41/5*w - 5],
[809, 809, w^3 - w^2 - 5*w + 4],
[821, 821, -3/5*w^3 - 17/5*w^2 + 33/5*w + 30],
[821, 821, -4/5*w^3 + 24/5*w^2 + 9/5*w - 15],
[829, 829, 7/5*w^3 - 12/5*w^2 - 52/5*w + 10],
[829, 829, -1/5*w^3 + 11/5*w^2 - 4/5*w - 19],
[829, 829, -13/5*w^3 + 3/5*w^2 + 108/5*w + 10],
[829, 829, -w^3 + 4*w^2 + 5*w - 23],
[839, 839, 4/5*w^3 - 9/5*w^2 - 24/5*w + 4],
[839, 839, 3/5*w^3 + 2/5*w^2 - 18/5*w - 8],
[841, 29, -w^3 + w^2 + 6*w - 4],
[859, 859, 6/5*w^3 - 11/5*w^2 - 36/5*w + 9],
[859, 859, w^3 - 6*w - 3],
[881, 881, 4/5*w^3 - 4/5*w^2 - 9/5*w - 1],
[881, 881, 9/5*w^3 - 24/5*w^2 - 19/5*w + 14],
[881, 881, -w^3 + 6*w + 2],
[881, 881, 1/5*w^3 - 1/5*w^2 - 1/5*w - 6],
[911, 911, w^3 - w^2 - 5*w + 1],
[911, 911, 6/5*w^3 - 6/5*w^2 - 41/5*w + 2],
[919, 919, -7/5*w^3 + 12/5*w^2 + 42/5*w - 7],
[919, 919, -6/5*w^3 + 1/5*w^2 + 36/5*w + 5],
[929, 929, -2/5*w^3 - 8/5*w^2 + 17/5*w + 16],
[929, 929, -3/5*w^3 + 13/5*w^2 + 13/5*w - 7],
[961, 31, w^3 - w^2 - 6*w + 2],
[971, 971, 8/5*w^3 - 13/5*w^2 - 48/5*w + 6],
[971, 971, -2/5*w^3 - 3/5*w^2 + 22/5*w],
[971, 971, 7/5*w^3 - 7/5*w^2 - 57/5*w + 7],
[971, 971, -1/5*w^3 + 6/5*w^2 - 4/5*w - 10],
[991, 991, 2/5*w^3 + 8/5*w^2 - 22/5*w - 7],
[991, 991, -2/5*w^3 + 12/5*w^2 + 2/5*w - 15]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - 51*x^6 + 25*x^5 + 711*x^4 - 363*x^3 - 2619*x^2 + 432*x + 1620;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, -1, e, e, 23/6232*e^7 - 181/28044*e^6 - 10225/56088*e^5 + 22093/56088*e^4 + 42041/18696*e^3 - 26343/6232*e^2 - 36387/6232*e + 12805/3116, 71/14022*e^7 + 32/7011*e^6 - 2905/14022*e^5 + 421/4674*e^4 + 3073/1558*e^3 - 15049/4674*e^2 - 1981/1558*e + 5935/779, -13/28044*e^7 + 49/14022*e^6 + 883/28044*e^5 - 5783/28044*e^4 - 1989/3116*e^3 + 27859/9348*e^2 + 12717/3116*e - 9535/1558, 71/14022*e^7 + 32/7011*e^6 - 2905/14022*e^5 + 421/4674*e^4 + 3073/1558*e^3 - 15049/4674*e^2 - 1981/1558*e + 5935/779, -13/28044*e^7 + 49/14022*e^6 + 883/28044*e^5 - 5783/28044*e^4 - 1989/3116*e^3 + 27859/9348*e^2 + 12717/3116*e - 9535/1558, 65/28044*e^7 - 245/14022*e^6 - 4415/28044*e^5 + 19567/28044*e^4 + 6829/3116*e^3 - 20465/3116*e^2 - 10613/3116*e + 16515/1558, 65/28044*e^7 - 245/14022*e^6 - 4415/28044*e^5 + 19567/28044*e^4 + 6829/3116*e^3 - 20465/3116*e^2 - 10613/3116*e + 16515/1558, -16/7011*e^7 - 179/7011*e^6 + 128/7011*e^5 + 634/779*e^4 + 668/779*e^3 - 14789/2337*e^2 - 4842/779*e + 7090/779, -16/7011*e^7 - 179/7011*e^6 + 128/7011*e^5 + 634/779*e^4 + 668/779*e^3 - 14789/2337*e^2 - 4842/779*e + 7090/779, 1/76*e^7 + 7/342*e^6 - 395/684*e^5 - 229/684*e^4 + 1547/228*e^3 + 17/228*e^2 - 1321/76*e - 5/38, 1/76*e^7 + 7/342*e^6 - 395/684*e^5 - 229/684*e^4 + 1547/228*e^3 + 17/228*e^2 - 1321/76*e - 5/38, -49/14022*e^7 - 55/7011*e^6 + 2729/14022*e^5 + 481/1558*e^4 - 14701/4674*e^3 - 14959/4674*e^2 + 20129/1558*e + 5647/779, -49/14022*e^7 - 55/7011*e^6 + 2729/14022*e^5 + 481/1558*e^4 - 14701/4674*e^3 - 14959/4674*e^2 + 20129/1558*e + 5647/779, 71/14022*e^7 + 32/7011*e^6 - 2905/14022*e^5 + 421/4674*e^4 + 10777/4674*e^3 - 4497/1558*e^2 - 12887/1558*e + 4377/779, 71/14022*e^7 + 32/7011*e^6 - 2905/14022*e^5 + 421/4674*e^4 + 10777/4674*e^3 - 4497/1558*e^2 - 12887/1558*e + 4377/779, -421/28044*e^7 - 91/14022*e^6 + 19727/28044*e^5 - 1465/9348*e^4 - 77947/9348*e^3 + 32839/9348*e^2 + 61405/3116*e - 6775/1558, -421/28044*e^7 - 91/14022*e^6 + 19727/28044*e^5 - 1465/9348*e^4 - 77947/9348*e^3 + 32839/9348*e^2 + 61405/3116*e - 6775/1558, 227/28044*e^7 + 223/14022*e^6 - 10385/28044*e^5 - 11915/28044*e^4 + 44989/9348*e^3 + 10265/3116*e^2 - 53315/3116*e + 3505/1558, 65/28044*e^7 - 245/14022*e^6 - 4415/28044*e^5 + 19567/28044*e^4 + 6829/3116*e^3 - 20465/3116*e^2 - 7497/3116*e + 16515/1558, 65/28044*e^7 - 245/14022*e^6 - 4415/28044*e^5 + 19567/28044*e^4 + 6829/3116*e^3 - 20465/3116*e^2 - 7497/3116*e + 16515/1558, 107/14022*e^7 + 136/7011*e^6 - 4751/14022*e^5 - 8849/14022*e^4 + 19511/4674*e^3 + 29327/4674*e^2 - 20299/1558*e - 1457/779, 25/4674*e^7 - 43/7011*e^6 - 4495/14022*e^5 + 2563/14022*e^4 + 21961/4674*e^3 + 955/1558*e^2 - 17519/1558*e - 11505/779, 25/4674*e^7 - 43/7011*e^6 - 4495/14022*e^5 + 2563/14022*e^4 + 21961/4674*e^3 + 955/1558*e^2 - 17519/1558*e - 11505/779, -227/28044*e^7 - 223/14022*e^6 + 10385/28044*e^5 + 11915/28044*e^4 - 44989/9348*e^3 - 10265/3116*e^2 + 47083/3116*e + 12075/1558, -227/28044*e^7 - 223/14022*e^6 + 10385/28044*e^5 + 11915/28044*e^4 - 44989/9348*e^3 - 10265/3116*e^2 + 47083/3116*e + 12075/1558, -2/7011*e^7 + 25/2337*e^6 + 265/2337*e^5 - 650/7011*e^4 - 6371/2337*e^3 - 6620/2337*e^2 + 6990/779*e + 8092/779, -2/7011*e^7 + 25/2337*e^6 + 265/2337*e^5 - 650/7011*e^4 - 6371/2337*e^3 - 6620/2337*e^2 + 6990/779*e + 8092/779, 1/2337*e^7 + 277/7011*e^6 + 755/7011*e^5 - 9152/7011*e^4 - 1878/779*e^3 + 20836/2337*e^2 + 4316/779*e - 1232/779, 1/2337*e^7 + 277/7011*e^6 + 755/7011*e^5 - 9152/7011*e^4 - 1878/779*e^3 + 20836/2337*e^2 + 4316/779*e - 1232/779, -67/9348*e^7 - 107/4674*e^6 + 2873/9348*e^5 + 4711/9348*e^4 - 42403/9348*e^3 - 2871/3116*e^2 + 74621/3116*e - 15/1558, -67/9348*e^7 - 107/4674*e^6 + 2873/9348*e^5 + 4711/9348*e^4 - 42403/9348*e^3 - 2871/3116*e^2 + 74621/3116*e - 15/1558, -71/7011*e^7 - 64/7011*e^6 + 2905/7011*e^5 - 421/2337*e^4 - 3073/779*e^3 + 12712/2337*e^2 + 423/779*e - 2522/779, -71/7011*e^7 - 64/7011*e^6 + 2905/7011*e^5 - 421/2337*e^4 - 3073/779*e^3 + 12712/2337*e^2 + 423/779*e - 2522/779, -251/28044*e^7 + 227/14022*e^6 + 10577/28044*e^5 - 33277/28044*e^4 - 10359/3116*e^3 + 148393/9348*e^2 + 91/3116*e - 39085/1558, 439/28044*e^7 + 143/14022*e^6 - 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3888/779, 23/7011*e^7 - 473/7011*e^6 - 2521/7011*e^5 + 5348/2337*e^4 + 4298/779*e^3 - 35267/2337*e^2 - 6380/779*e - 5810/779, -7/492*e^7 + 35/738*e^6 + 1439/1476*e^5 - 2315/1476*e^4 - 2557/164*e^3 + 1553/164*e^2 + 7385/164*e - 895/82, 23/7011*e^7 - 473/7011*e^6 - 2521/7011*e^5 + 5348/2337*e^4 + 4298/779*e^3 - 35267/2337*e^2 - 6380/779*e - 5810/779, -7/492*e^7 + 35/738*e^6 + 1439/1476*e^5 - 2315/1476*e^4 - 2557/164*e^3 + 1553/164*e^2 + 7385/164*e - 895/82, 259/28044*e^7 - 377/14022*e^6 - 13757/28044*e^5 + 11959/9348*e^4 + 50329/9348*e^3 - 39599/3116*e^2 + 3109/3116*e + 19785/1558, 259/28044*e^7 - 377/14022*e^6 - 13757/28044*e^5 + 11959/9348*e^4 + 50329/9348*e^3 - 39599/3116*e^2 + 3109/3116*e + 19785/1558, -11/1476*e^7 + 73/738*e^6 + 949/1476*e^5 - 625/164*e^4 - 1683/164*e^3 + 5507/164*e^2 + 4251/164*e - 2265/82, -11/1476*e^7 + 73/738*e^6 + 949/1476*e^5 - 625/164*e^4 - 1683/164*e^3 + 5507/164*e^2 + 4251/164*e - 2265/82, 53/3116*e^7 + 599/14022*e^6 - 17059/28044*e^5 - 21029/28044*e^4 + 47563/9348*e^3 + 2953/9348*e^2 - 17193/3116*e - 3685/1558, 53/3116*e^7 + 599/14022*e^6 - 17059/28044*e^5 - 21029/28044*e^4 + 47563/9348*e^3 + 2953/9348*e^2 - 17193/3116*e - 3685/1558, -145/9348*e^7 + 187/4674*e^6 + 8171/9348*e^5 - 5841/3116*e^4 - 109301/9348*e^3 + 58339/3116*e^2 + 82291/3116*e - 31425/1558, -145/9348*e^7 + 187/4674*e^6 + 8171/9348*e^5 - 5841/3116*e^4 - 109301/9348*e^3 + 58339/3116*e^2 + 82291/3116*e - 31425/1558, 17/684*e^7 + 1/38*e^6 - 229/228*e^5 + 167/684*e^4 + 2255/228*e^3 - 2839/228*e^2 - 1693/76*e + 835/38, 17/684*e^7 + 1/38*e^6 - 229/228*e^5 + 167/684*e^4 + 2255/228*e^3 - 2839/228*e^2 - 1693/76*e + 835/38, -517/14022*e^7 - 628/7011*e^6 + 20495/14022*e^5 + 9947/4674*e^4 - 67481/4674*e^3 - 20709/1558*e^2 + 37027/1558*e + 22285/779, -517/14022*e^7 - 628/7011*e^6 + 20495/14022*e^5 + 9947/4674*e^4 - 67481/4674*e^3 - 20709/1558*e^2 + 37027/1558*e + 22285/779, -149/28044*e^7 - 517/14022*e^6 + 5087/28044*e^5 + 37265/28044*e^4 - 4101/3116*e^3 - 37939/3116*e^2 - 19871/3116*e + 22545/1558, -149/28044*e^7 - 517/14022*e^6 + 5087/28044*e^5 + 37265/28044*e^4 - 4101/3116*e^3 - 37939/3116*e^2 - 19871/3116*e + 22545/1558, -31/1558*e^7 - 3/779*e^6 + 4639/4674*e^5 - 727/1558*e^4 - 64183/4674*e^3 + 13897/1558*e^2 + 83929/1558*e - 10485/779, -31/1558*e^7 - 3/779*e^6 + 4639/4674*e^5 - 727/1558*e^4 - 64183/4674*e^3 + 13897/1558*e^2 + 83929/1558*e - 10485/779, -87/3116*e^7 + 25/4674*e^6 + 12215/9348*e^5 - 11599/9348*e^4 - 141277/9348*e^3 + 60763/3116*e^2 + 83311/3116*e - 56565/1558, -87/3116*e^7 + 25/4674*e^6 + 12215/9348*e^5 - 11599/9348*e^4 - 141277/9348*e^3 + 60763/3116*e^2 + 83311/3116*e - 56565/1558, -299/28044*e^7 - 431/14022*e^6 + 14077/28044*e^5 + 7597/9348*e^4 - 68689/9348*e^3 - 11805/3116*e^2 + 86835/3116*e - 16765/1558, 103/14022*e^7 + 211/7011*e^6 - 3161/14022*e^5 - 1647/1558*e^4 + 2095/4674*e^3 + 59711/4674*e^2 + 17051/1558*e - 32315/779, 103/14022*e^7 + 211/7011*e^6 - 3161/14022*e^5 - 1647/1558*e^4 + 2095/4674*e^3 + 59711/4674*e^2 + 17051/1558*e - 32315/779, -299/28044*e^7 - 431/14022*e^6 + 14077/28044*e^5 + 7597/9348*e^4 - 68689/9348*e^3 - 11805/3116*e^2 + 86835/3116*e - 16765/1558, -23/3116*e^7 + 181/14022*e^6 + 10225/28044*e^5 - 22093/28044*e^4 - 42041/9348*e^3 + 26343/3116*e^2 + 48851/3116*e + 2775/1558, -23/3116*e^7 + 181/14022*e^6 + 10225/28044*e^5 - 22093/28044*e^4 - 42041/9348*e^3 + 26343/3116*e^2 + 48851/3116*e + 2775/1558, -1309/28044*e^7 - 193/4674*e^6 + 20369/9348*e^5 - 91/28044*e^4 - 258071/9348*e^3 + 122311/9348*e^2 + 257737/3116*e - 14515/1558, 611/28044*e^7 + 271/4674*e^6 - 6563/9348*e^5 - 21103/28044*e^4 + 12467/3116*e^3 - 56741/9348*e^2 + 22385/3116*e + 27485/1558, 611/28044*e^7 + 271/4674*e^6 - 6563/9348*e^5 - 21103/28044*e^4 + 12467/3116*e^3 - 56741/9348*e^2 + 22385/3116*e + 27485/1558, 142/7011*e^7 + 128/7011*e^6 - 5810/7011*e^5 + 21/779*e^4 + 6146/779*e^3 - 2762/779*e^2 - 10194/779*e - 5862/779, 142/7011*e^7 + 128/7011*e^6 - 5810/7011*e^5 + 21/779*e^4 + 6146/779*e^3 - 2762/779*e^2 - 10194/779*e - 5862/779, -679/14022*e^7 - 317/7011*e^6 + 29581/14022*e^5 - 2555/14022*e^4 - 109121/4674*e^3 + 21787/1558*e^2 + 78171/1558*e + 2577/779, -679/14022*e^7 - 317/7011*e^6 + 29581/14022*e^5 - 2555/14022*e^4 - 109121/4674*e^3 + 21787/1558*e^2 + 78171/1558*e + 2577/779, -44/7011*e^7 + 92/7011*e^6 + 2689/7011*e^5 - 1391/2337*e^4 - 14743/2337*e^3 + 4685/779*e^2 + 20571/779*e - 5820/779, -44/7011*e^7 + 92/7011*e^6 + 2689/7011*e^5 - 1391/2337*e^4 - 14743/2337*e^3 + 4685/779*e^2 + 20571/779*e - 5820/779, -32/2337*e^7 + 484/7011*e^6 + 6221/7011*e^5 - 20294/7011*e^4 - 30821/2337*e^3 + 23394/779*e^2 + 27036/779*e - 35360/779, -32/2337*e^7 + 484/7011*e^6 + 6221/7011*e^5 - 20294/7011*e^4 - 30821/2337*e^3 + 23394/779*e^2 + 27036/779*e - 35360/779, 46/2337*e^7 - 167/2337*e^6 - 2705/2337*e^5 + 7939/2337*e^4 + 13324/779*e^3 - 26910/779*e^2 - 42175/779*e + 11880/779, 46/2337*e^7 - 167/2337*e^6 - 2705/2337*e^5 + 7939/2337*e^4 + 13324/779*e^3 - 26910/779*e^2 - 42175/779*e + 11880/779, 305/14022*e^7 - 71/7011*e^6 - 15683/14022*e^5 + 13435/14022*e^4 + 73001/4674*e^3 - 17409/1558*e^2 - 84435/1558*e + 23323/779, -67/4674*e^7 - 107/2337*e^6 + 2873/4674*e^5 + 6269/4674*e^4 - 12057/1558*e^3 - 15335/1558*e^2 + 37229/1558*e + 9333/779, 25/2337*e^7 - 86/7011*e^6 - 4495/7011*e^5 + 2563/7011*e^4 + 21961/2337*e^3 - 1382/779*e^2 - 22193/779*e + 14382/779, -67/4674*e^7 - 107/2337*e^6 + 2873/4674*e^5 + 6269/4674*e^4 - 12057/1558*e^3 - 15335/1558*e^2 + 37229/1558*e + 9333/779, 25/2337*e^7 - 86/7011*e^6 - 4495/7011*e^5 + 2563/7011*e^4 + 21961/2337*e^3 - 1382/779*e^2 - 22193/779*e + 14382/779, -248/7011*e^7 - 16/2337*e^6 + 4037/2337*e^5 - 3479/7011*e^4 - 16911/779*e^3 + 3957/779*e^2 + 46473/779*e + 12520/779, -248/7011*e^7 - 16/2337*e^6 + 4037/2337*e^5 - 3479/7011*e^4 - 16911/779*e^3 + 3957/779*e^2 + 46473/779*e + 12520/779];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {5, 5, 1/5*w^3 + 4/5*w^2 - 11/5*w - 7}>] := 1;
ALEigenvalues[ideal<ZF | {5, 5, 1/5*w^3 - 6/5*w^2 - 1/5*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;