/*
  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 | {28, 14, w^2 - w - 5}>;

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^8 - 3*x^7 - 26*x^6 + 102*x^5 + 53*x^4 - 507*x^3 + 320*x^2 + 296*x - 172;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, 1, 11/312*e^7 - 9/104*e^6 - 103/104*e^5 + 931/312*e^4 + 49/12*e^3 - 29/2*e^2 + 35/78*e + 283/39, e, -9/208*e^7 + 2/39*e^6 + 355/312*e^5 - 94/39*e^4 - 73/16*e^3 + 155/12*e^2 - 379/156*e - 1043/156, 3/104*e^7 - 1/156*e^6 - 127/156*e^5 + 125/156*e^4 + 39/8*e^3 - 29/6*e^2 - 437/78*e + 365/78, -3/208*e^7 + 7/156*e^6 + 101/312*e^5 - 251/156*e^4 + 5/16*e^3 + 97/12*e^2 - 1097/156*e - 937/156, 23/312*e^7 - 17/156*e^6 - 163/78*e^5 + 223/52*e^4 + 245/24*e^3 - 127/6*e^2 - 305/78*e + 283/26, 7/208*e^7 - 37/312*e^6 - 73/78*e^5 + 1193/312*e^4 + 49/16*e^3 - 227/12*e^2 + 445/156*e + 1883/156, 53/624*e^7 - 2/39*e^6 - 745/312*e^5 + 253/78*e^4 + 653/48*e^3 - 227/12*e^2 - 801/52*e + 2317/156, -5/39*e^7 + 19/312*e^6 + 1087/312*e^5 - 1387/312*e^4 - 439/24*e^3 + 68/3*e^2 + 238/13*e - 731/78, 55/624*e^7 - 3/104*e^6 - 119/52*e^5 + 865/312*e^4 + 493/48*e^3 - 53/4*e^2 - 605/156*e + 997/156, -33/208*e^7 + 21/104*e^6 + 111/26*e^5 - 909/104*e^4 - 295/16*e^3 + 169/4*e^2 + 103/52*e - 775/52, 11/78*e^7 - 43/312*e^6 - 1171/312*e^5 + 717/104*e^4 + 407/24*e^3 - 211/6*e^2 - 346/39*e + 577/26, -5/52*e^7 + 21/104*e^6 + 275/104*e^5 - 753/104*e^4 - 81/8*e^3 + 71/2*e^2 - 114/13*e - 381/26, -1/624*e^7 + 2/39*e^6 + 43/312*e^5 - 14/13*e^4 - 37/48*e^3 + 41/12*e^2 - 197/156*e + 25/52, 11/312*e^7 + 25/312*e^6 - 257/312*e^5 - 421/312*e^4 + 43/12*e^3 + 29/3*e^2 + 3/26*e - 523/39, 5/104*e^7 - 19/156*e^6 - 229/156*e^5 + 581/156*e^4 + 57/8*e^3 - 49/3*e^2 - 35/78*e + 695/78, 7/52*e^7 - 11/78*e^6 - 133/39*e^5 + 278/39*e^4 + 49/4*e^3 - 215/6*e^2 + 224/39*e + 661/39, -73/624*e^7 + 11/312*e^6 + 121/39*e^5 - 1115/312*e^4 - 727/48*e^3 + 205/12*e^2 + 337/52*e + 769/156, -5/26*e^7 + 37/156*e^6 + 419/78*e^5 - 1583/156*e^4 - 27*e^3 + 299/6*e^2 + 889/39*e - 1000/39, 7/52*e^7 + 17/156*e^6 - 133/39*e^5 + 59/156*e^4 + 69/4*e^3 - 1/3*e^2 - 478/39*e - 509/39, 61/624*e^7 - 7/156*e^6 - 881/312*e^5 + 511/156*e^4 + 853/48*e^3 - 241/12*e^2 - 1281/52*e + 2237/156, 1/78*e^7 + 1/156*e^6 - 29/156*e^5 + 83/156*e^4 - 13/12*e^3 - 17/3*e^2 + 88/13*e + 487/39, 1/39*e^7 - 11/156*e^6 - 55/78*e^5 + 129/52*e^4 + 17/6*e^3 - 38/3*e^2 - 31/39*e + 82/13, 1/156*e^7 + 7/156*e^6 - 2/39*e^5 - 49/52*e^4 - 5/12*e^3 + 22/3*e^2 + 2/39*e - 155/13, -83/624*e^7 + 41/312*e^6 + 595/156*e^5 - 599/104*e^4 - 1001/48*e^3 + 301/12*e^2 + 2993/156*e + 125/52, -1/312*e^7 + 4/39*e^6 + 43/156*e^5 - 28/13*e^4 - 37/24*e^3 + 47/6*e^2 + 193/78*e - 235/26, -5/104*e^7 + 1/26*e^6 + 18/13*e^5 - 47/26*e^4 - 59/8*e^3 + 15/2*e^2 + 3/26*e + 193/26, 97/624*e^7 - 11/78*e^6 - 1337/312*e^5 + 569/78*e^4 + 1057/48*e^3 - 481/12*e^2 - 737/52*e + 3593/156, -47/312*e^7 + 61/312*e^6 + 1351/312*e^5 - 843/104*e^4 - 70/3*e^3 + 125/3*e^2 + 1583/78*e - 316/13, 35/624*e^7 - 1/312*e^6 - 50/39*e^5 + 541/312*e^4 + 173/48*e^3 - 113/12*e^2 + 253/52*e + 1093/156, -89/624*e^7 + 7/312*e^6 + 581/156*e^5 - 1291/312*e^4 - 875/48*e^3 + 263/12*e^2 + 517/52*e + 149/156, -83/624*e^7 + 10/39*e^6 + 1229/312*e^5 - 231/26*e^4 - 983/48*e^3 + 493/12*e^2 + 2057/156*e - 421/52, 19/624*e^7 - 5/312*e^6 - 71/78*e^5 + 365/312*e^4 + 325/48*e^3 - 103/12*e^2 - 971/52*e + 2501/156, 137/624*e^7 - 73/312*e^6 - 911/156*e^5 + 3457/312*e^4 + 1187/48*e^3 - 665/12*e^2 + 347/52*e + 4675/156, -29/624*e^7 + 25/78*e^6 + 493/312*e^5 - 655/78*e^4 - 269/48*e^3 + 425/12*e^2 - 327/52*e - 2245/156, -1/24*e^7 + 5/24*e^6 + 35/24*e^5 - 47/8*e^4 - 113/12*e^3 + 167/6*e^2 + 115/6*e - 17, 25/624*e^7 - 11/39*e^6 - 373/312*e^5 + 103/13*e^4 + 97/48*e^3 - 413/12*e^2 + 1649/156*e + 259/52, 19/208*e^7 - 5/104*e^6 - 129/52*e^5 + 313/104*e^4 + 193/16*e^3 - 45/4*e^2 - 209/52*e - 463/52, 37/624*e^7 - 7/312*e^6 - 115/78*e^5 + 257/104*e^4 + 307/48*e^3 - 203/12*e^2 + 269/156*e + 609/52, -11/312*e^7 - 77/312*e^6 + 205/312*e^5 + 539/104*e^4 - 61/12*e^3 - 79/3*e^2 + 1343/78*e + 287/13, -17/312*e^7 - 5/104*e^6 + 171/104*e^5 - 49/312*e^4 - 43/3*e^3 + 13/2*e^2 + 2839/78*e - 370/39, -7/624*e^7 + 47/312*e^6 + 10/39*e^5 - 1403/312*e^4 + 71/48*e^3 + 277/12*e^2 - 113/52*e - 2465/156, -5/39*e^7 + 1/52*e^6 + 179/52*e^5 - 505/156*e^4 - 209/12*e^3 + 29/2*e^2 + 337/39*e + 577/39, 8/39*e^7 - 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4523/312*e^4 - 787/24*e^3 + 71*e^2 + 367/39*e - 2383/78, 127/624*e^7 - 277/312*e^6 - 239/39*e^5 + 2823/104*e^4 + 1153/48*e^3 - 1643/12*e^2 + 2243/156*e + 3923/52, 1/52*e^7 - 19/52*e^6 - 21/52*e^5 + 529/52*e^4 - 8*e^3 - 103/2*e^2 + 561/13*e + 562/13, -175/624*e^7 + 11/78*e^6 + 2585/312*e^5 - 343/39*e^4 - 2563/48*e^3 + 553/12*e^2 + 4299/52*e - 4451/156, 157/624*e^7 - 107/312*e^6 - 1081/156*e^5 + 4379/312*e^4 + 1543/48*e^3 - 805/12*e^2 - 1113/52*e + 5567/156, -31/78*e^7 + 185/312*e^6 + 3293/312*e^5 - 2595/104*e^4 - 1033/24*e^3 + 373/3*e^2 + 266/39*e - 1515/26, 41/208*e^7 - 11/78*e^6 - 1571/312*e^5 + 317/39*e^4 + 281/16*e^3 - 445/12*e^2 + 5771/156*e - 125/156, -1/208*e^7 - 95/312*e^6 - 7/156*e^5 + 2359/312*e^4 - 75/16*e^3 - 481/12*e^2 + 4349/156*e + 5815/156, -25/104*e^7 + 59/104*e^6 + 733/104*e^5 - 1967/104*e^4 - 67/2*e^3 + 87*e^2 + 249/26*e - 44/13, 15/104*e^7 - 31/156*e^6 - 661/156*e^5 + 1223/156*e^4 + 199/8*e^3 - 239/6*e^2 - 3329/78*e + 2657/78, -341/624*e^7 + 55/104*e^6 + 189/13*e^5 - 8405/312*e^4 - 3131/48*e^3 + 567/4*e^2 + 3127/156*e - 14075/156, -71/624*e^7 + 35/156*e^6 + 973/312*e^5 - 1229/156*e^4 - 515/48*e^3 + 425/12*e^2 - 693/52*e - 2683/156, -11/39*e^7 + 73/312*e^6 + 2251/312*e^5 - 4237/312*e^4 - 691/24*e^3 + 457/6*e^2 - 103/13*e - 3839/78, -19/624*e^7 + 35/156*e^6 + 193/312*e^5 - 349/52*e^4 + 257/48*e^3 + 383/12*e^2 - 6395/156*e - 877/52, 323/624*e^7 - 241/312*e^6 - 1129/78*e^5 + 9637/312*e^4 + 3365/48*e^3 - 1865/12*e^2 - 2779/52*e + 13813/156, -59/156*e^7 + 227/312*e^6 + 3245/312*e^5 - 2837/104*e^4 - 1039/24*e^3 + 821/6*e^2 - 118/39*e - 2185/26, 161/624*e^7 + 11/312*e^6 - 1115/156*e^5 + 1381/312*e^4 + 2027/48*e^3 - 293/12*e^2 - 3017/52*e + 2563/156, 77/312*e^7 - 5/78*e^6 - 1075/156*e^5 + 241/39*e^4 + 929/24*e^3 - 67/3*e^2 - 967/26*e - 1511/78, 7/104*e^7 - 25/78*e^6 - 305/156*e^5 + 863/78*e^4 + 71/8*e^3 - 433/6*e^2 - 439/78*e + 6121/78, 20/39*e^7 - 103/156*e^6 - 2161/156*e^5 + 1449/52*e^4 + 701/12*e^3 - 397/3*e^2 + 82/39*e + 795/13, -185/312*e^7 + 37/78*e^6 + 653/39*e^5 - 649/26*e^4 - 2255/24*e^3 + 793/6*e^2 + 8249/78*e - 2109/26, -147/208*e^7 + 205/312*e^6 + 2975/156*e^5 - 10337/312*e^4 - 1441/16*e^3 + 2003/12*e^2 + 7919/156*e - 12971/156, -181/208*e^7 + 101/104*e^6 + 1259/52*e^5 - 4565/104*e^4 - 1943/16*e^3 + 897/4*e^2 + 4279/52*e - 6783/52, 167/312*e^7 - 131/312*e^6 - 4625/312*e^5 + 6755/312*e^4 + 913/12*e^3 - 599/6*e^2 - 1661/26*e + 959/39, -29/104*e^7 - 37/312*e^6 + 2243/312*e^5 - 991/312*e^4 - 69/2*e^3 + 43/3*e^2 + 359/78*e + 812/39, 77/156*e^7 - 131/312*e^6 - 4079/312*e^5 + 2269/104*e^4 + 1381/24*e^3 - 605/6*e^2 - 509/39*e + 895/26, -1/6*e^7 + 1/24*e^6 + 121/24*e^5 - 109/24*e^4 - 889/24*e^3 + 187/6*e^2 + 71*e - 197/6, -31/312*e^7 + 20/39*e^6 + 122/39*e^5 - 602/39*e^4 - 313/24*e^3 + 487/6*e^2 - 155/26*e - 2849/78, 33/104*e^7 - 21/52*e^6 - 111/13*e^5 + 909/52*e^4 + 295/8*e^3 - 169/2*e^2 - 415/26*e + 1139/26, 49/104*e^7 - 115/312*e^6 - 3685/312*e^5 + 6887/312*e^4 + 171/4*e^3 - 350/3*e^2 + 2543/78*e + 2255/39, -151/156*e^7 + 113/156*e^6 + 2047/78*e^5 - 2143/52*e^4 - 1567/12*e^3 + 659/3*e^2 + 3481/39*e - 1867/13, 11/104*e^7 - 1/104*e^6 - 283/104*e^5 + 229/104*e^4 + 12*e^3 - 3*e^2 - 251/26*e + 10/13, -95/312*e^7 + 47/104*e^6 + 873/104*e^5 - 5665/312*e^4 - 221/6*e^3 + 88*e^2 - 905/78*e - 760/39, 397/624*e^7 - 125/312*e^6 - 1307/78*e^5 + 7721/312*e^4 + 3595/48*e^3 - 1399/12*e^2 - 1213/52*e + 6131/156, 11/156*e^7 - 53/156*e^6 - 335/156*e^5 + 1529/156*e^4 + 23/3*e^3 - 140/3*e^2 + 3/13*e + 904/39, 1/16*e^7 + 5/12*e^6 - 35/24*e^5 - 109/12*e^4 + 221/16*e^3 + 509/12*e^2 - 373/12*e - 341/12, -191/624*e^7 + 11/52*e^6 + 831/104*e^5 - 1915/156*e^4 - 1595/48*e^3 + 223/4*e^2 + 73/156*e - 3563/156, -43/156*e^7 + 3/104*e^6 + 823/104*e^5 - 1801/312*e^4 - 1223/24*e^3 + 22*e^2 + 3515/39*e + 925/78, 53/104*e^7 - 11/78*e^6 - 1085/78*e^5 + 1141/78*e^4 + 599/8*e^3 - 455/6*e^2 - 5987/78*e + 3701/78, 33/104*e^7 + 23/104*e^6 - 849/104*e^5 + 141/104*e^4 + 42*e^3 + e^2 - 935/26*e - 490/13, -5/12*e^7 + 3/4*e^6 + 47/4*e^5 - 337/12*e^4 - 167/3*e^3 + 140*e^2 + 127/3*e - 284/3];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

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