/*
  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![3, -4, -8, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {9, 3, w^3 - 2*w^2 - 5*w + 1}>;

primesArray := [
[3, 3, w],
[3, 3, w^3 - 2*w^2 - 6*w + 1],
[9, 3, w^3 - 2*w^2 - 5*w + 1],
[16, 2, 2],
[17, 17, w^3 - w^2 - 8*w - 5],
[17, 17, -2*w^3 + 4*w^2 + 11*w - 2],
[17, 17, -w^3 + 3*w^2 + 2*w - 2],
[17, 17, w^3 - 2*w^2 - 4*w + 1],
[25, 5, w^3 - 3*w^2 - 3*w + 1],
[25, 5, w^2 - 2*w - 7],
[29, 29, w^3 - 2*w^2 - 6*w - 1],
[29, 29, w - 2],
[53, 53, w^3 - 3*w^2 - 4*w + 4],
[53, 53, -w^3 + 3*w^2 + 4*w - 5],
[61, 61, -w^3 + 2*w^2 + 7*w - 4],
[61, 61, -4*w^3 + 11*w^2 + 14*w - 8],
[101, 101, -w^3 + 2*w^2 + 7*w - 1],
[103, 103, -w^3 + 3*w^2 + 2*w - 5],
[103, 103, -w^3 + w^2 + 8*w + 2],
[113, 113, w^3 - 3*w^2 - 3*w + 7],
[113, 113, w^2 - 2*w - 1],
[127, 127, w^3 - 2*w^2 - 4*w - 2],
[127, 127, -2*w^3 + 4*w^2 + 11*w + 1],
[131, 131, -3*w^3 + 7*w^2 + 14*w - 8],
[131, 131, -w^2 + w + 8],
[131, 131, 2*w^3 - 5*w^2 - 9*w + 1],
[131, 131, -3*w^3 + 6*w^2 + 16*w + 1],
[139, 139, -2*w^3 + 5*w^2 + 9*w - 2],
[139, 139, -w^2 + w + 7],
[157, 157, -2*w^3 + 5*w^2 + 7*w - 5],
[157, 157, -2*w^3 + 3*w^2 + 13*w + 2],
[157, 157, -2*w^3 + 5*w^2 + 6*w - 2],
[157, 157, -3*w^3 + 5*w^2 + 19*w + 4],
[169, 13, -2*w^3 + 4*w^2 + 10*w - 1],
[173, 173, 3*w^3 - 8*w^2 - 10*w + 7],
[173, 173, 2*w^3 - 2*w^2 - 15*w - 8],
[179, 179, -3*w^2 + 6*w + 17],
[179, 179, -2*w^3 + 3*w^2 + 12*w + 1],
[179, 179, -3*w^3 + 7*w^2 + 13*w - 7],
[179, 179, -2*w^3 + 3*w^2 + 14*w + 4],
[191, 191, w^3 - 3*w^2 - w + 4],
[191, 191, -2*w^3 + 3*w^2 + 14*w + 2],
[199, 199, -w^3 + 8*w + 10],
[199, 199, 5*w^3 - 15*w^2 - 14*w + 14],
[211, 211, -w - 4],
[211, 211, -w^3 + 2*w^2 + 6*w - 5],
[211, 211, 2*w^3 - 3*w^2 - 12*w - 4],
[211, 211, -3*w^3 + 7*w^2 + 13*w - 4],
[257, 257, -3*w^3 + 8*w^2 + 10*w - 2],
[257, 257, 2*w^3 - 2*w^2 - 16*w - 7],
[263, 263, 3*w^3 - 6*w^2 - 17*w + 1],
[263, 263, w^3 - 2*w^2 - 3*w - 1],
[269, 269, 5*w^3 - 13*w^2 - 19*w + 7],
[269, 269, -3*w^3 + 6*w^2 + 18*w - 1],
[277, 277, w^3 - 4*w^2 - w + 5],
[277, 277, -w^3 + 4*w^2 + w - 11],
[283, 283, -w^3 + 5*w^2 - w - 14],
[283, 283, 2*w^3 - 7*w^2 - 4*w + 10],
[283, 283, -w^3 + w^2 + 9*w + 1],
[283, 283, w^2 - 4*w - 5],
[311, 311, -w^3 + w^2 + 9*w + 5],
[311, 311, w^2 - 4*w - 1],
[313, 313, -4*w^3 + 9*w^2 + 19*w - 10],
[313, 313, -2*w^2 + 5*w + 11],
[337, 337, -3*w^3 + 6*w^2 + 14*w + 2],
[337, 337, -4*w^3 + 8*w^2 + 21*w + 1],
[347, 347, -2*w^3 + 6*w^2 + 7*w - 10],
[347, 347, -w^3 + 4*w^2 + 2*w - 7],
[361, 19, -4*w^3 + 11*w^2 + 13*w - 10],
[361, 19, -2*w^3 + 5*w^2 + 8*w - 10],
[373, 373, 2*w^3 - 5*w^2 - 10*w + 2],
[373, 373, -w^3 + 2*w^2 + 8*w - 2],
[373, 373, -2*w^3 + 4*w^2 + 13*w - 1],
[373, 373, -w^3 + 3*w^2 + 5*w - 8],
[389, 389, -w^3 + 4*w^2 + 2*w - 10],
[389, 389, -2*w^3 + 6*w^2 + 7*w - 7],
[419, 419, -2*w^3 + 5*w^2 + 6*w - 1],
[419, 419, -3*w^3 + 5*w^2 + 19*w + 5],
[433, 433, w^2 - 4*w - 2],
[433, 433, w^3 - w^2 - 9*w - 4],
[439, 439, -w^3 + 4*w^2 + w - 2],
[439, 439, w^3 - 4*w^2 - w + 14],
[443, 443, -2*w^3 + 5*w^2 + 9*w + 1],
[443, 443, -w^2 + w + 10],
[467, 467, w^3 - w^2 - 8*w + 4],
[467, 467, w^3 - 4*w^2 + 2],
[491, 491, -4*w^3 + 12*w^2 + 10*w - 11],
[491, 491, 2*w^3 - w^2 - 18*w - 11],
[491, 491, 3*w^3 - 9*w^2 - 7*w + 11],
[491, 491, 2*w^3 - 20*w - 19],
[503, 503, w^3 - 3*w^2 - 4*w + 10],
[503, 503, -w^3 + w^2 + 10*w - 2],
[523, 523, -4*w^3 + 9*w^2 + 22*w - 7],
[523, 523, w^3 - 3*w^2 - 7*w + 5],
[529, 23, -2*w^3 + 4*w^2 + 10*w + 5],
[529, 23, -2*w^3 + 4*w^2 + 10*w - 7],
[547, 547, 2*w^3 - w^2 - 18*w - 14],
[547, 547, 3*w^3 - 8*w^2 - 14*w + 8],
[571, 571, 5*w^3 - 11*w^2 - 25*w + 5],
[571, 571, -4*w^3 + 9*w^2 + 19*w - 11],
[599, 599, -5*w^3 + 12*w^2 + 20*w - 4],
[599, 599, 4*w^3 - 6*w^2 - 25*w - 11],
[601, 601, -4*w^3 + 9*w^2 + 20*w - 5],
[601, 601, w^3 - 9*w - 14],
[601, 601, 3*w^3 - 8*w^2 - 11*w + 2],
[601, 601, -2*w^3 + 3*w^2 + 14*w - 1],
[641, 641, -2*w^3 + 3*w^2 + 15*w - 5],
[641, 641, -4*w^3 + 8*w^2 + 23*w - 2],
[647, 647, -3*w^3 + 8*w^2 + 13*w - 11],
[647, 647, -w^3 + 4*w^2 + 3*w - 7],
[659, 659, 3*w^3 - 7*w^2 - 14*w - 2],
[659, 659, w^3 - w^2 - 6*w - 11],
[673, 673, -3*w^3 + 7*w^2 + 11*w - 1],
[673, 673, -4*w^3 + 7*w^2 + 24*w + 5],
[677, 677, 5*w^3 - 12*w^2 - 21*w + 11],
[677, 677, -2*w^3 + 6*w^2 + 5*w - 1],
[677, 677, -2*w^3 + 6*w^2 + 8*w - 7],
[677, 677, 2*w^3 - 6*w^2 - 8*w + 11],
[719, 719, -2*w^3 + 6*w^2 + 10*w - 7],
[719, 719, -4*w^3 + 10*w^2 + 20*w - 13],
[727, 727, 3*w^3 - 10*w^2 - 8*w + 10],
[727, 727, 3*w^3 - 8*w^2 - 13*w + 10],
[727, 727, 2*w^3 - 8*w^2 - 3*w + 23],
[727, 727, -w^3 + 4*w^2 + 3*w - 8],
[751, 751, -7*w^3 + 18*w^2 + 25*w - 7],
[751, 751, w^3 - 7*w^2 + 5*w + 26],
[757, 757, -4*w^3 + 9*w^2 + 18*w - 7],
[757, 757, -3*w^3 + 5*w^2 + 17*w + 1],
[797, 797, 2*w^2 - 4*w - 5],
[797, 797, 2*w^3 - 6*w^2 - 6*w + 11],
[823, 823, -2*w^3 + 2*w^2 + 15*w + 14],
[823, 823, -3*w^3 + 8*w^2 + 10*w - 1],
[829, 829, -6*w^3 + 16*w^2 + 22*w - 7],
[829, 829, 5*w^3 - 13*w^2 - 20*w + 8],
[829, 829, -w^3 + 7*w + 4],
[829, 829, -5*w^3 + 12*w^2 + 23*w - 14],
[841, 29, -3*w^3 + 6*w^2 + 15*w - 2],
[859, 859, -4*w^3 + 7*w^2 + 24*w - 1],
[859, 859, -5*w^3 + 15*w^2 + 14*w - 13],
[881, 881, 3*w^3 - 7*w^2 - 16*w + 4],
[881, 881, -w^3 + 3*w^2 + 6*w - 7],
[883, 883, -2*w^3 + 3*w^2 + 12*w + 10],
[883, 883, -3*w^3 + 7*w^2 + 13*w + 2],
[887, 887, 4*w^3 - 11*w^2 - 11*w + 11],
[887, 887, 2*w^3 - 7*w^2 - 3*w + 7],
[887, 887, 4*w^3 - 5*w^2 - 29*w - 10],
[887, 887, 3*w^2 - 7*w - 16],
[907, 907, -w^3 + 3*w^2 + 8*w - 5],
[907, 907, -5*w^3 + 11*w^2 + 28*w - 8],
[911, 911, -4*w^3 + 10*w^2 + 18*w - 17],
[911, 911, 2*w^2 - 2*w - 1],
[919, 919, 4*w^3 - 7*w^2 - 25*w - 4],
[919, 919, 2*w^3 - 5*w^2 - 5*w + 1],
[937, 937, -w^3 + 3*w^2 + w - 8],
[937, 937, -2*w^3 + 3*w^2 + 14*w - 2],
[953, 953, -2*w^3 + w^2 + 17*w + 19],
[953, 953, 4*w^3 - 11*w^2 - 13*w + 4],
[961, 31, 2*w^3 - 3*w^2 - 11*w - 8],
[961, 31, w^3 - 4*w^2 + 2*w + 11],
[971, 971, 2*w^2 - 5*w - 4],
[971, 971, -w^3 + 4*w^2 - 11],
[991, 991, -2*w^3 + 6*w^2 + 9*w - 8],
[991, 991, -2*w^3 + 3*w^2 + 13*w - 1],
[991, 991, 3*w^3 - 8*w^2 - 14*w + 11],
[991, 991, -2*w^3 + 5*w^2 + 7*w - 8],
[997, 997, -2*w^3 + 5*w^2 + 6*w + 1],
[997, 997, -3*w^3 + 5*w^2 + 19*w + 7],
[997, 997, -6*w^3 + 12*w^2 + 33*w - 4],
[997, 997, 4*w^3 - 11*w^2 - 15*w + 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^9 - 18*x^7 - 2*x^6 + 94*x^5 + 26*x^4 - 140*x^3 - 72*x^2 + 20*x + 10;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, e, -1, -425/747*e^8 + 268/747*e^7 + 7423/747*e^6 - 3634/747*e^5 - 12155/249*e^4 + 9554/747*e^3 + 15614/249*e^2 + 1964/249*e - 3211/747, -256/747*e^8 + 5/747*e^7 + 4640/747*e^6 + 328/747*e^5 - 8218/249*e^4 - 4733/747*e^3 + 12706/249*e^2 + 4147/249*e - 5576/747, 136/747*e^8 - 26/747*e^7 - 2465/747*e^6 + 386/747*e^5 + 4288/249*e^4 - 1384/747*e^3 - 6112/249*e^2 + 248/249*e + 2402/747, -256/747*e^8 + 5/747*e^7 + 4640/747*e^6 + 328/747*e^5 - 8218/249*e^4 - 4733/747*e^3 + 12706/249*e^2 + 4147/249*e - 5576/747, 136/747*e^8 - 26/747*e^7 - 2465/747*e^6 + 386/747*e^5 + 4288/249*e^4 - 1384/747*e^3 - 6112/249*e^2 + 248/249*e + 2402/747, -196/249*e^8 + 140/249*e^7 + 3428/249*e^6 - 2021/249*e^5 - 5672/83*e^4 + 6418/249*e^3 + 7666/83*e^2 + 82/83*e - 3242/249, -196/249*e^8 + 140/249*e^7 + 3428/249*e^6 - 2021/249*e^5 - 5672/83*e^4 + 6418/249*e^3 + 7666/83*e^2 + 82/83*e - 3242/249, 65/249*e^8 - 82/249*e^7 - 1147/249*e^6 + 1294/249*e^5 + 1942/83*e^4 - 5246/249*e^3 - 2804/83*e^2 + 1344/83*e + 1906/249, 65/249*e^8 - 82/249*e^7 - 1147/249*e^6 + 1294/249*e^5 + 1942/83*e^4 - 5246/249*e^3 - 2804/83*e^2 + 1344/83*e + 1906/249, 146/747*e^8 + 38/747*e^7 - 2833/747*e^6 - 794/747*e^5 + 5570/249*e^4 + 4666/747*e^3 - 10355/249*e^2 - 1550/249*e + 10510/747, 146/747*e^8 + 38/747*e^7 - 2833/747*e^6 - 794/747*e^5 + 5570/249*e^4 + 4666/747*e^3 - 10355/249*e^2 - 1550/249*e + 10510/747, -278/249*e^8 + 163/249*e^7 + 4852/249*e^6 - 2305/249*e^5 - 7984/83*e^4 + 7106/249*e^3 + 10736/83*e^2 - 48/83*e - 6382/249, -278/249*e^8 + 163/249*e^7 + 4852/249*e^6 - 2305/249*e^5 - 7984/83*e^4 + 7106/249*e^3 + 10736/83*e^2 - 48/83*e - 6382/249, -1100/747*e^8 + 430/747*e^7 + 19564/747*e^6 - 6154/747*e^5 - 33452/249*e^4 + 20246/747*e^3 + 48908/249*e^2 - 1420/249*e - 31336/747, 70/249*e^8 - 50/249*e^7 - 1331/249*e^6 + 704/249*e^5 + 2500/83*e^4 - 1972/249*e^3 - 4137/83*e^2 - 468/83*e + 1976/249, 70/249*e^8 - 50/249*e^7 - 1331/249*e^6 + 704/249*e^5 + 2500/83*e^4 - 1972/249*e^3 - 4137/83*e^2 - 468/83*e + 1976/249, 754/747*e^8 - 254/747*e^7 - 13106/747*e^6 + 2909/747*e^5 + 21166/249*e^4 - 2488/747*e^3 - 25654/249*e^2 - 8131/249*e + 5078/747, 754/747*e^8 - 254/747*e^7 - 13106/747*e^6 + 2909/747*e^5 + 21166/249*e^4 - 2488/747*e^3 - 25654/249*e^2 - 8131/249*e + 5078/747, -42/83*e^8 + 30/83*e^7 + 782/83*e^6 - 439/83*e^5 - 4334/83*e^4 + 1482/83*e^3 + 7264/83*e^2 - 386/83*e - 1584/83, -42/83*e^8 + 30/83*e^7 + 782/83*e^6 - 439/83*e^5 - 4334/83*e^4 + 1482/83*e^3 + 7264/83*e^2 - 386/83*e - 1584/83, -430/747*e^8 + 236/747*e^7 + 7607/747*e^6 - 3044/747*e^5 - 12796/249*e^4 + 5782/747*e^3 + 17362/249*e^2 + 4606/249*e - 5024/747, 98/249*e^8 - 70/249*e^7 - 1714/249*e^6 + 1135/249*e^5 + 2836/83*e^4 - 4952/249*e^3 - 3916/83*e^2 + 1702/83*e + 2368/249, 98/249*e^8 - 70/249*e^7 - 1714/249*e^6 + 1135/249*e^5 + 2836/83*e^4 - 4952/249*e^3 - 3916/83*e^2 + 1702/83*e + 2368/249, -430/747*e^8 + 236/747*e^7 + 7607/747*e^6 - 3044/747*e^5 - 12796/249*e^4 + 5782/747*e^3 + 17362/249*e^2 + 4606/249*e - 5024/747, 29/249*e^8 + 86/249*e^7 - 619/249*e^6 - 1430/249*e^5 + 1344/83*e^4 + 6340/249*e^3 - 2602/83*e^2 - 1944/83*e - 92/249, 29/249*e^8 + 86/249*e^7 - 619/249*e^6 - 1430/249*e^5 + 1344/83*e^4 + 6340/249*e^3 - 2602/83*e^2 - 1944/83*e - 92/249, 13/249*e^8 - 116/249*e^7 - 80/249*e^6 + 1952/249*e^5 - 342/83*e^4 - 9316/249*e^3 + 1896/83*e^2 + 4286/83*e - 1810/249, 13/249*e^8 - 116/249*e^7 - 80/249*e^6 + 1952/249*e^5 - 342/83*e^4 - 9316/249*e^3 + 1896/83*e^2 + 4286/83*e - 1810/249, 164/249*e^8 - 46/249*e^7 - 2848/249*e^6 + 568/249*e^5 + 4624/83*e^4 - 878/249*e^3 - 5974/83*e^2 - 902/83*e + 2794/249, 164/249*e^8 - 46/249*e^7 - 2848/249*e^6 + 568/249*e^5 + 4624/83*e^4 - 878/249*e^3 - 5974/83*e^2 - 902/83*e + 2794/249, -1639/747*e^8 + 566/747*e^7 + 29240/747*e^6 - 6794/747*e^5 - 50212/249*e^4 + 10132/747*e^3 + 73600/249*e^2 + 11380/249*e - 27926/747, -571/747*e^8 + 230/747*e^7 + 10256/747*e^6 - 2840/747*e^5 - 17725/249*e^4 + 4888/747*e^3 + 25720/249*e^2 + 4510/249*e - 12974/747, -571/747*e^8 + 230/747*e^7 + 10256/747*e^6 - 2840/747*e^5 - 17725/249*e^4 + 4888/747*e^3 + 25720/249*e^2 + 4510/249*e - 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23056/249*e - 17008/747, -290/747*e^8 + 385/747*e^7 + 4696/747*e^6 - 6865/747*e^5 - 6800/249*e^4 + 38192/747*e^3 + 8258/249*e^2 - 23056/249*e - 17008/747, -3920/747*e^8 + 1804/747*e^7 + 69556/747*e^6 - 24484/747*e^5 - 118088/249*e^4 + 63620/747*e^3 + 167513/249*e^2 + 19070/249*e - 64840/747, 190/83*e^8 - 112/83*e^7 - 3340/83*e^6 + 1650/83*e^5 + 16634/83*e^4 - 5815/83*e^3 - 22150/83*e^2 + 2011/83*e + 3324/83, -3920/747*e^8 + 1804/747*e^7 + 69556/747*e^6 - 24484/747*e^5 - 118088/249*e^4 + 63620/747*e^3 + 167513/249*e^2 + 19070/249*e - 64840/747, 190/83*e^8 - 112/83*e^7 - 3340/83*e^6 + 1650/83*e^5 + 16634/83*e^4 - 5815/83*e^3 - 22150/83*e^2 + 2011/83*e + 3324/83, -416/249*e^8 + 475/249*e^7 + 7042/249*e^6 - 7435/249*e^5 - 10968/83*e^4 + 29690/249*e^3 + 13364/83*e^2 - 6427/83*e - 9808/249, -416/249*e^8 + 475/249*e^7 + 7042/249*e^6 - 7435/249*e^5 - 10968/83*e^4 + 29690/249*e^3 + 13364/83*e^2 - 6427/83*e - 9808/249, 1463/747*e^8 - 796/747*e^7 - 26050/747*e^6 + 11128/747*e^5 + 44282/249*e^4 - 31454/747*e^3 - 61472/249*e^2 - 5432/249*e + 14008/747, 1463/747*e^8 - 796/747*e^7 - 26050/747*e^6 + 11128/747*e^5 + 44282/249*e^4 - 31454/747*e^3 - 61472/249*e^2 - 5432/249*e + 14008/747, 3328/747*e^8 - 1559/747*e^7 - 58826/747*e^6 + 20387/747*e^5 + 98866/249*e^4 - 45292/747*e^3 - 136294/249*e^2 - 20794/249*e + 60536/747, 3328/747*e^8 - 1559/747*e^7 - 58826/747*e^6 + 20387/747*e^5 + 98866/249*e^4 - 45292/747*e^3 - 136294/249*e^2 - 20794/249*e + 60536/747, 550/249*e^8 - 464/249*e^7 - 9782/249*e^6 + 6812/249*e^5 + 16726/83*e^4 - 23320/249*e^3 - 24786/83*e^2 + 2370/83*e + 17162/249, 550/249*e^8 - 464/249*e^7 - 9782/249*e^6 + 6812/249*e^5 + 16726/83*e^4 - 23320/249*e^3 - 24786/83*e^2 + 2370/83*e + 17162/249, 5015/747*e^8 - 2266/747*e^7 - 88189/747*e^6 + 29734/747*e^5 + 146666/249*e^4 - 68216/747*e^3 - 197741/249*e^2 - 28454/249*e + 77182/747, 5015/747*e^8 - 2266/747*e^7 - 88189/747*e^6 + 29734/747*e^5 + 146666/249*e^4 - 68216/747*e^3 - 197741/249*e^2 - 28454/249*e + 77182/747, -1099/249*e^8 + 536/249*e^7 + 19328/249*e^6 - 7268/249*e^5 - 32278/83*e^4 + 19606/249*e^3 + 44508/83*e^2 + 3596/83*e - 23354/249, -1099/249*e^8 + 536/249*e^7 + 19328/249*e^6 - 7268/249*e^5 - 32278/83*e^4 + 19606/249*e^3 + 44508/83*e^2 + 3596/83*e - 23354/249, -59/83*e^8 + 54/83*e^7 + 1059/83*e^6 - 674/83*e^5 - 5610/83*e^4 + 1406/83*e^3 + 9307/83*e^2 + 1596/83*e - 2984/83, -59/83*e^8 + 54/83*e^7 + 1059/83*e^6 - 674/83*e^5 - 5610/83*e^4 + 1406/83*e^3 + 9307/83*e^2 + 1596/83*e - 2984/83, -773/249*e^8 + 232/249*e^7 + 13606/249*e^6 - 2908/249*e^5 - 22639/83*e^4 + 5186/249*e^3 + 30238/83*e^2 + 5372/83*e - 12316/249, -1712/747*e^8 + 1294/747*e^7 + 30283/747*e^6 - 19096/747*e^5 - 51254/249*e^4 + 65318/747*e^3 + 72926/249*e^2 - 3532/249*e - 33928/747, -773/249*e^8 + 232/249*e^7 + 13606/249*e^6 - 2908/249*e^5 - 22639/83*e^4 + 5186/249*e^3 + 30238/83*e^2 + 5372/83*e - 12316/249, -1712/747*e^8 + 1294/747*e^7 + 30283/747*e^6 - 19096/747*e^5 - 51254/249*e^4 + 65318/747*e^3 + 72926/249*e^2 - 3532/249*e - 33928/747, 647/747*e^8 - 640/747*e^7 - 11260/747*e^6 + 8812/747*e^5 + 18056/249*e^4 - 23150/747*e^3 - 21563/249*e^2 - 6422/249*e + 7066/747, 647/747*e^8 - 640/747*e^7 - 11260/747*e^6 + 8812/747*e^5 + 18056/249*e^4 - 23150/747*e^3 - 21563/249*e^2 - 6422/249*e + 7066/747, -1517/747*e^8 + 1048/747*e^7 + 26842/747*e^6 - 15214/747*e^5 - 45179/249*e^4 + 48086/747*e^3 + 62024/249*e^2 + 2/249*e - 16258/747, -1517/747*e^8 + 1048/747*e^7 + 26842/747*e^6 - 15214/747*e^5 - 45179/249*e^4 + 48086/747*e^3 + 62024/249*e^2 + 2/249*e - 16258/747];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {9, 3, w^3 - 2*w^2 - 5*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;