/*
  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,-2/3*w^3 + 4/3*w^2 + 11/3*w - 4}>;

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 - 18*x^6 + 71*x^4 - 80*x^2 + 8;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-5/66*e^6 + 38/33*e^4 - 43/22*e^2 - 7/33, e, -13/66*e^7 + 112/33*e^5 - 257/22*e^3 + 391/33*e, -2/11*e^7 + 37/11*e^5 - 155/11*e^3 + 166/11*e, 29/132*e^7 - 227/66*e^5 + 333/44*e^3 + 17/33*e, 0, -1/66*e^6 + 1/33*e^4 + 31/22*e^2 + 58/33, -1, -19/66*e^6 + 151/33*e^4 - 247/22*e^2 + 112/33, -8/33*e^7 + 148/33*e^5 - 203/11*e^3 + 532/33*e, -79/132*e^7 + 673/66*e^5 - 1467/44*e^3 + 971/33*e, 31/132*e^7 - 229/66*e^5 + 227/44*e^3 + 157/33*e, -13/44*e^7 + 123/22*e^5 - 1123/44*e^3 + 377/11*e, 8/33*e^7 - 115/33*e^5 + 38/11*e^3 + 425/33*e, -1/11*e^6 + 13/11*e^4 - 17/11*e^2 + 116/11, 2/11*e^6 - 37/11*e^4 + 166/11*e^2 - 144/11, -13/33*e^7 + 224/33*e^5 - 257/11*e^3 + 650/33*e, 35/132*e^7 - 299/66*e^5 + 631/44*e^3 - 124/33*e, -1/11*e^7 + 13/11*e^5 + 5/11*e^3 - 60/11*e, -1/2*e^7 + 8*e^5 - 39/2*e^3 + 3*e, 15/22*e^7 - 125/11*e^5 + 761/22*e^3 - 320/11*e, -5/6*e^6 + 38/3*e^4 - 45/2*e^2 - 22/3, e^6 - 16*e^4 + 39*e^2 - 10, 95/132*e^7 - 887/66*e^5 + 2555/44*e^3 - 2194/33*e, 13/132*e^7 - 79/66*e^5 - 95/44*e^3 + 415/33*e, -4/33*e^7 + 74/33*e^5 - 107/11*e^3 + 365/33*e, -73/132*e^7 + 667/66*e^5 - 1829/44*e^3 + 1523/33*e, -3/11*e^6 + 50/11*e^4 - 150/11*e^2 + 172/11, -16/33*e^6 + 230/33*e^4 - 98/11*e^2 - 58/33, 1/11*e^6 - 24/11*e^4 + 171/11*e^2 - 226/11, 23/33*e^6 - 343/33*e^4 + 189/11*e^2 + 368/33, -3/11*e^6 + 50/11*e^4 - 128/11*e^2 - 48/11, -7/11*e^6 + 102/11*e^4 - 119/11*e^2 - 134/11, 3/22*e^7 - 36/11*e^5 + 469/22*e^3 - 284/11*e, 5/22*e^7 - 38/11*e^5 + 129/22*e^3 + 106/11*e, 27/22*e^7 - 225/11*e^5 + 1339/22*e^3 - 389/11*e, -65/132*e^7 + 461/66*e^5 - 229/44*e^3 - 887/33*e, 29/132*e^7 - 227/66*e^5 + 289/44*e^3 + 413/33*e, -41/66*e^7 + 338/33*e^5 - 665/22*e^3 + 695/33*e, 7/33*e^6 - 80/33*e^4 - 74/11*e^2 + 838/33, 1/11*e^6 - 24/11*e^4 + 116/11*e^2 + 60/11, 5/11*e^6 - 87/11*e^4 + 305/11*e^2 - 316/11, -1/3*e^7 + 20/3*e^5 - 33*e^3 + 116/3*e, -41/22*e^6 + 327/11*e^4 - 1599/22*e^2 + 222/11, -29/66*e^7 + 260/33*e^5 - 685/22*e^3 + 1517/33*e, -3/11*e^6 + 61/11*e^4 - 293/11*e^2 + 304/11, 1/11*e^6 - 24/11*e^4 + 149/11*e^2 - 72/11, 7/12*e^7 - 61/6*e^5 + 139/4*e^3 - 56/3*e, -95/66*e^7 + 821/33*e^5 - 1895/22*e^3 + 2540/33*e, -5/22*e^6 + 49/11*e^4 - 459/22*e^2 + 180/11, 113/66*e^6 - 905/33*e^4 + 1469/22*e^2 - 812/33, 31/33*e^6 - 524/33*e^4 + 535/11*e^2 - 296/33, 13/11*e^6 - 213/11*e^4 + 595/11*e^2 - 254/11, 7/11*e^6 - 124/11*e^4 + 449/11*e^2 - 328/11, -61/66*e^6 + 523/33*e^4 - 1167/22*e^2 + 700/33, 53/33*e^6 - 832/33*e^4 + 645/11*e^2 - 340/33, -13/33*e^6 + 158/33*e^4 + 40/11*e^2 - 604/33, 6/11*e^6 - 100/11*e^4 + 278/11*e^2 - 14/11, 2/11*e^6 - 26/11*e^4 + 12/11*e^2 + 10/11, -43/22*e^7 + 373/11*e^5 - 2579/22*e^3 + 976/11*e, 53/44*e^7 - 449/22*e^5 + 2859/44*e^3 - 448/11*e, 27/22*e^7 - 236/11*e^5 + 1691/22*e^3 - 818/11*e, 29/44*e^7 - 249/22*e^5 + 1659/44*e^3 - 280/11*e, 29/33*e^7 - 520/33*e^5 + 696/11*e^3 - 3001/33*e, -40/33*e^6 + 608/33*e^4 - 366/11*e^2 - 178/33, -40/33*e^7 + 674/33*e^5 - 696/11*e^3 + 1373/33*e, 53/33*e^6 - 832/33*e^4 + 645/11*e^2 - 868/33, -4/11*e^6 + 52/11*e^4 - 2/11*e^2 - 152/11, -13/11*e^7 + 235/11*e^5 - 925/11*e^3 + 793/11*e, 41/22*e^6 - 316/11*e^4 + 1247/22*e^2 + 64/11, -161/132*e^7 + 1415/66*e^5 - 3457/44*e^3 + 2392/33*e, -4/11*e^6 + 63/11*e^4 - 156/11*e^2 + 178/11, -46/33*e^6 + 752/33*e^4 - 697/11*e^2 + 782/33, -2/33*e^7 + 37/33*e^5 - 59/11*e^3 + 529/33*e, 7/33*e^7 - 146/33*e^5 + 256/11*e^3 - 1010/33*e, -41/132*e^7 + 437/66*e^5 - 1633/44*e^3 + 1288/33*e, -5/66*e^7 + 38/33*e^5 - 65/22*e^3 + 323/33*e, 59/33*e^6 - 976/33*e^4 + 921/11*e^2 - 1102/33, -79/44*e^7 + 651/22*e^5 - 3697/44*e^3 + 432/11*e, 45/44*e^7 - 397/22*e^5 + 2899/44*e^3 - 590/11*e, -2/11*e^6 + 26/11*e^4 + 10/11*e^2 - 164/11, 9/22*e^7 - 64/11*e^5 + 131/22*e^3 + 105/11*e, -2/11*e^6 + 26/11*e^4 + 21/11*e^2 - 10/11, -20/11*e^6 + 326/11*e^4 - 879/11*e^2 + 450/11, 199/132*e^7 - 1651/66*e^5 + 3203/44*e^3 - 1349/33*e, 103/132*e^7 - 961/66*e^5 + 2835/44*e^3 - 3185/33*e, -45/44*e^7 + 397/22*e^5 - 2899/44*e^3 + 700/11*e, 13/22*e^6 - 123/11*e^4 + 1079/22*e^2 - 270/11, 61/66*e^6 - 523/33*e^4 + 1057/22*e^2 - 172/33, 75/44*e^7 - 625/22*e^5 + 3717/44*e^3 - 503/11*e, 5/44*e^7 - 93/22*e^5 + 1779/44*e^3 - 772/11*e, -181/132*e^7 + 1435/66*e^5 - 2221/44*e^3 + 167/33*e, 45/22*e^7 - 386/11*e^5 + 2591/22*e^3 - 1015/11*e, -4/33*e^6 + 74/33*e^4 - 74/11*e^2 + 68/33, -25/22*e^6 + 179/11*e^4 - 403/22*e^2 - 310/11, 8/11*e^6 - 126/11*e^4 + 290/11*e^2 + 18/11, 2*e^2 - 6, -71/33*e^6 + 1132/33*e^4 - 923/11*e^2 + 976/33, -19/66*e^6 + 151/33*e^4 - 247/22*e^2 + 244/33, 92/33*e^6 - 1438/33*e^4 + 1086/11*e^2 - 244/33, -13/66*e^6 + 145/33*e^4 - 565/22*e^2 + 490/33, 13/11*e^6 - 202/11*e^4 + 419/11*e^2 + 32/11, 239/132*e^7 - 2153/66*e^5 + 5571/44*e^3 - 3961/33*e, 53/66*e^7 - 416/33*e^5 + 623/22*e^3 - 71/33*e, -8/11*e^7 + 126/11*e^5 - 279/11*e^3 - 106/11*e, -247/132*e^7 + 2161/66*e^5 - 5147/44*e^3 + 3401/33*e, -7/6*e^7 + 55/3*e^5 - 83/2*e^3 + 13/3*e, 383/132*e^7 - 3287/66*e^5 + 7355/44*e^3 - 4144/33*e, -67/66*e^7 + 529/33*e^5 - 849/22*e^3 + 685/33*e, 17/66*e^7 - 83/33*e^5 - 307/22*e^3 + 1588/33*e, 163/132*e^7 - 1351/66*e^5 + 2647/44*e^3 - 1130/33*e, 5/66*e^7 - 5/33*e^5 - 331/22*e^3 + 1360/33*e, -23/22*e^6 + 188/11*e^4 - 1029/22*e^2 + 344/11, -2/3*e^6 + 34/3*e^4 - 32*e^2 - 50/3, 47/33*e^6 - 820/33*e^4 + 941/11*e^2 - 1360/33, 41/66*e^7 - 272/33*e^5 - 39/22*e^3 + 1813/33*e, 155/132*e^7 - 1343/66*e^5 + 3027/44*e^3 - 1195/33*e, -307/132*e^7 + 2683/66*e^5 - 6455/44*e^3 + 4811/33*e, 7/11*e^7 - 124/11*e^5 + 493/11*e^3 - 724/11*e, -4/3*e^6 + 62/3*e^4 - 46*e^2 + 32/3, -239/132*e^7 + 1889/66*e^5 - 2887/44*e^3 + 1/33*e, -107/132*e^7 + 1031/66*e^5 - 3195/44*e^3 + 2641/33*e, 5/6*e^6 - 35/3*e^4 + 25/2*e^2 + 58/3, e^6 - 16*e^4 + 37*e^2 + 20, 14/11*e^6 - 237/11*e^4 + 700/11*e^2 - 150/11, -65/22*e^6 + 527/11*e^4 - 2865/22*e^2 + 668/11, 18/11*e^6 - 289/11*e^4 + 702/11*e^2 - 218/11, 10/33*e^6 - 152/33*e^4 + 152/11*e^2 - 830/33, 2*e^6 - 34*e^4 + 110*e^2 - 54, -9/44*e^7 + 75/22*e^5 - 307/44*e^3 - 267/11*e, -7/11*e^7 + 135/11*e^5 - 625/11*e^3 + 801/11*e, -73/33*e^7 + 1268/33*e^5 - 1466/11*e^3 + 3683/33*e, -79/66*e^7 + 706/33*e^5 - 1841/22*e^3 + 3097/33*e, -1/4*e^7 + 5/2*e^5 + 53/4*e^3 - 48*e, 21/44*e^7 - 131/22*e^5 - 281/44*e^3 + 326/11*e, 20/33*e^7 - 337/33*e^5 + 348/11*e^3 - 571/33*e, -179/132*e^7 + 1433/66*e^5 - 2415/44*e^3 + 538/33*e, 9/11*e^6 - 128/11*e^4 + 87/11*e^2 + 320/11, -5/33*e^6 + 142/33*e^4 - 351/11*e^2 + 1174/33, 23/132*e^7 - 155/66*e^5 + 123/44*e^3 - 502/33*e, 17/33*e^7 - 331/33*e^5 + 551/11*e^3 - 2830/33*e, -40/33*e^6 + 608/33*e^4 - 366/11*e^2 - 1102/33, 329/132*e^7 - 2903/66*e^5 + 7181/44*e^3 - 5251/33*e, 205/132*e^7 - 1855/66*e^5 + 4953/44*e^3 - 3899/33*e, -13/66*e^6 + 211/33*e^4 - 1225/22*e^2 + 1876/33, -7/6*e^7 + 58/3*e^5 - 111/2*e^3 + 37/3*e, 233/132*e^7 - 1883/66*e^5 + 3337/44*e^3 - 1147/33*e, 109/33*e^6 - 1736/33*e^4 + 1406/11*e^2 - 962/33, 43/11*e^6 - 702/11*e^4 + 1897/11*e^2 - 764/11, -89/66*e^6 + 749/33*e^4 - 1553/22*e^2 + 1202/33, 79/22*e^6 - 629/11*e^4 + 3037/22*e^2 - 512/11, -7/4*e^7 + 61/2*e^5 - 437/4*e^3 + 112*e, -109/44*e^7 + 923/22*e^5 - 5923/44*e^3 + 1126/11*e, 19/22*e^6 - 151/11*e^4 + 829/22*e^2 - 222/11, -29/11*e^6 + 476/11*e^4 - 1318/11*e^2 + 416/11, -155/66*e^6 + 1277/33*e^4 - 2521/22*e^2 + 2192/33, 17/66*e^6 - 215/33*e^4 + 881/22*e^2 - 1448/33, -12/11*e^6 + 200/11*e^4 - 578/11*e^2 + 138/11, 63/22*e^6 - 503/11*e^4 + 2523/22*e^2 - 376/11, -21/11*e^6 + 328/11*e^4 - 731/11*e^2 + 38/11, -35/33*e^6 + 565/33*e^4 - 411/11*e^2 - 758/33, 34/33*e^6 - 563/33*e^4 + 530/11*e^2 - 644/33, 19/22*e^6 - 151/11*e^4 + 829/22*e^2 - 288/11, -26/33*e^6 + 448/33*e^4 - 536/11*e^2 + 574/33, -38/33*e^6 + 604/33*e^4 - 538/11*e^2 + 1966/33, 134/33*e^7 - 2281/33*e^5 + 2490/11*e^3 - 5941/33*e, -70/33*e^7 + 1229/33*e^5 - 1482/11*e^3 + 4127/33*e, -e^6 + 18*e^4 - 65*e^2 + 36, -101/132*e^7 + 959/66*e^5 - 2941/44*e^3 + 2731/33*e, 1/44*e^7 + 43/22*e^5 - 1545/44*e^3 + 939/11*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {23,23,-2/3*w^3 + 4/3*w^2 + 11/3*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;