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

NN := ideal<ZF | {29, 29, w^3 - 2*w^2 - 5*w}>;

primesArray := [
[5, 5, -w^3 + 2*w^2 + 3*w],
[7, 7, w - 1],
[11, 11, -w^3 + 2*w^2 + 4*w],
[13, 13, -2*w^3 + 3*w^2 + 10*w - 2],
[16, 2, 2],
[17, 17, -w^3 + 2*w^2 + 5*w - 3],
[17, 17, -w^3 + w^2 + 5*w],
[17, 17, -w^2 + 2*w + 1],
[19, 19, w^2 - w - 2],
[29, 29, w^3 - 2*w^2 - 5*w],
[29, 29, w^2 - w - 3],
[31, 31, -w^3 + 2*w^2 + 3*w - 2],
[43, 43, 2*w^3 - 3*w^2 - 11*w],
[47, 47, w^3 - 7*w - 4],
[53, 53, -2*w^3 + 3*w^2 + 9*w - 1],
[61, 61, -w - 3],
[73, 73, -w^3 + 2*w^2 + 3*w - 3],
[73, 73, w^3 - w^2 - 7*w - 1],
[81, 3, -3],
[83, 83, -2*w^3 + 3*w^2 + 9*w + 1],
[83, 83, -w^3 + 3*w^2 + 2*w - 3],
[89, 89, w - 4],
[103, 103, w^3 - 2*w^2 - 6*w + 3],
[113, 113, 2*w^3 - w^2 - 14*w - 6],
[121, 11, -w^2 + 2*w + 7],
[125, 5, -3*w^3 + 4*w^2 + 15*w + 1],
[139, 139, w^3 - 2*w^2 - 5*w - 3],
[139, 139, w^3 - 5*w - 5],
[151, 151, w^3 - 9*w - 7],
[157, 157, -3*w^3 + 6*w^2 + 13*w - 6],
[163, 163, w^3 - 3*w^2 - 2*w + 9],
[167, 167, -w^3 + 7*w + 3],
[167, 167, -3*w^3 + 4*w^2 + 15*w - 1],
[173, 173, 2*w^3 - 3*w^2 - 7*w - 2],
[181, 181, -w^3 + 3*w^2 + 4*w - 5],
[191, 191, w^3 - w^2 - 4*w - 4],
[193, 193, -w^3 + w^2 + 7*w + 6],
[193, 193, 2*w^3 - 2*w^2 - 13*w - 5],
[199, 199, -2*w^3 + 5*w^2 + 5*w - 5],
[211, 211, w^3 - w^2 - 8*w - 3],
[211, 211, -2*w^2 + 3*w + 8],
[227, 227, -w^3 + 3*w^2 + 4*w - 4],
[227, 227, w^2 - 3*w - 6],
[229, 229, w^2 - 5],
[229, 229, 3*w^3 - 4*w^2 - 17*w - 2],
[233, 233, 3*w^3 - 6*w^2 - 13*w + 3],
[233, 233, 3*w^3 - 4*w^2 - 16*w + 1],
[241, 241, -2*w^3 + 4*w^2 + 8*w + 1],
[251, 251, 2*w^3 - 2*w^2 - 9*w - 3],
[277, 277, 3*w^3 - 4*w^2 - 17*w - 1],
[281, 281, -2*w^2 + 5*w + 1],
[283, 283, -4*w^3 + 6*w^2 + 19*w - 2],
[293, 293, -3*w^3 + 6*w^2 + 12*w - 2],
[307, 307, w^3 - w^2 - 4*w - 5],
[307, 307, w^3 - w^2 - 8*w - 2],
[311, 311, 2*w^2 - 3*w - 7],
[313, 313, -2*w^3 + w^2 + 15*w + 3],
[317, 317, -w^3 + 3*w^2 + w - 5],
[331, 331, 2*w^3 - 3*w^2 - 12*w],
[337, 337, w^3 - w^2 - 8*w - 1],
[343, 7, -w^3 + 6*w + 8],
[347, 347, 2*w^3 - 4*w^2 - 7*w + 5],
[349, 349, 2*w^3 - 4*w^2 - 9*w],
[353, 353, -3*w^3 + 4*w^2 + 16*w + 5],
[353, 353, -3*w^3 + 4*w^2 + 14*w],
[353, 353, -w^3 + 3*w^2 + 4*w - 8],
[353, 353, -w^3 + 2*w^2 + 8*w - 3],
[359, 359, -4*w^3 + 7*w^2 + 19*w - 3],
[367, 367, -w^3 + 2*w^2 + 6*w - 5],
[373, 373, 5*w^3 - 8*w^2 - 26*w + 3],
[379, 379, -3*w^3 + 5*w^2 + 17*w - 4],
[379, 379, -w^3 + 2*w^2 + 4*w - 6],
[383, 383, w^2 - 6],
[389, 389, w^3 - 2*w^2 - 3*w - 4],
[389, 389, w^2 + w - 4],
[397, 397, -4*w^3 + 5*w^2 + 23*w + 5],
[397, 397, -3*w^3 + 5*w^2 + 12*w - 2],
[419, 419, -3*w^3 + 4*w^2 + 17*w + 5],
[419, 419, -3*w^3 + 5*w^2 + 15*w],
[419, 419, 3*w^3 - 5*w^2 - 14*w - 1],
[419, 419, 2*w^3 - 3*w^2 - 10*w - 6],
[431, 431, 2*w^3 - 2*w^2 - 11*w],
[433, 433, w^3 - 8*w - 1],
[433, 433, 2*w^2 - 2*w - 5],
[439, 439, 2*w^3 - 2*w^2 - 9*w - 6],
[443, 443, 2*w^3 - 5*w^2 - 7*w],
[443, 443, 2*w^3 - 5*w^2 - 9*w + 4],
[457, 457, -2*w^3 + 5*w^2 + 7*w - 4],
[457, 457, -w^3 + 2*w^2 - 3],
[461, 461, -4*w^3 + 7*w^2 + 17*w - 7],
[479, 479, 3*w^3 - 5*w^2 - 12*w + 6],
[487, 487, -2*w^3 + 5*w^2 + 9*w - 9],
[487, 487, w^3 - 10*w - 8],
[499, 499, 2*w^3 - 2*w^2 - 12*w + 1],
[503, 503, w^2 - 7],
[503, 503, w^3 + w^2 - 7*w - 8],
[529, 23, -3*w^3 + 5*w^2 + 12*w + 1],
[529, 23, 3*w^3 - 4*w^2 - 14*w - 1],
[541, 541, w^3 - 4*w^2 - 2*w + 11],
[563, 563, -2*w^3 + w^2 + 15*w + 6],
[569, 569, w^3 - 2*w^2 - 4*w - 4],
[601, 601, 3*w^3 - 5*w^2 - 17*w + 1],
[607, 607, -2*w^3 + 5*w^2 + 7*w - 5],
[607, 607, -w^3 + 3*w^2 + 6*w - 5],
[613, 613, -4*w^3 + 5*w^2 + 21*w + 1],
[613, 613, -5*w^3 + 8*w^2 + 23*w - 4],
[619, 619, w^3 + w^2 - 9*w - 5],
[619, 619, -3*w^3 + 4*w^2 + 14*w + 2],
[641, 641, -6*w^3 + 8*w^2 + 31*w + 3],
[643, 643, 3*w^3 - 4*w^2 - 17*w - 6],
[643, 643, -2*w^3 + 5*w^2 + 7*w - 6],
[647, 647, 3*w^3 - 4*w^2 - 13*w - 10],
[653, 653, -3*w^3 + 5*w^2 + 14*w + 2],
[653, 653, w^3 - 2*w^2 - w - 3],
[661, 661, 5*w^3 - 9*w^2 - 24*w + 5],
[683, 683, 2*w^3 - 3*w^2 - 13*w - 2],
[683, 683, 4*w^3 - 7*w^2 - 18*w + 1],
[719, 719, -w^3 + 2*w^2 + 3*w - 7],
[719, 719, 3*w^3 - 5*w^2 - 16*w - 1],
[727, 727, -2*w^3 + 5*w^2 + 8*w - 7],
[727, 727, 2*w^3 - 5*w^2 - 7*w + 13],
[733, 733, -2*w^2 + 7],
[733, 733, -3*w^3 + 5*w^2 + 11*w + 2],
[739, 739, -6*w^3 + 9*w^2 + 29*w - 3],
[739, 739, -2*w^3 + 3*w^2 + 7*w + 5],
[751, 751, 4*w^3 - 5*w^2 - 23*w - 3],
[757, 757, -w^3 + 10*w + 4],
[757, 757, 2*w^3 - w^2 - 14*w - 4],
[769, 769, -w^3 + 4*w^2 + 4*w - 8],
[769, 769, 4*w^3 - 6*w^2 - 21*w - 2],
[787, 787, 3*w^3 - 3*w^2 - 19*w - 6],
[787, 787, -2*w^2 + 6*w + 7],
[839, 839, -3*w^3 + 3*w^2 + 19*w + 5],
[841, 29, 3*w^3 - 4*w^2 - 13*w - 4],
[853, 853, -3*w^3 + 5*w^2 + 14*w + 3],
[853, 853, 2*w^2 - w - 9],
[853, 853, 3*w^3 - 5*w^2 - 11*w + 1],
[853, 853, w - 6],
[857, 857, 2*w^3 - 2*w^2 - 9*w - 7],
[857, 857, -4*w^3 + 8*w^2 + 13*w - 1],
[859, 859, 2*w^3 - 4*w^2 - 13*w + 2],
[859, 859, w^3 - w^2 - 9*w - 2],
[863, 863, -2*w^3 - w^2 + 17*w + 11],
[877, 877, 5*w^3 - 6*w^2 - 28*w - 9],
[877, 877, -4*w^3 + 5*w^2 + 20*w + 6],
[881, 881, 4*w^3 - 5*w^2 - 18*w - 8],
[911, 911, -w^3 + 4*w^2 - w - 8],
[911, 911, 2*w^3 - 3*w^2 - 13*w - 1],
[919, 919, w^3 - w^2 - 7*w + 5],
[929, 929, 2*w^3 - 3*w^2 - 13*w + 1],
[929, 929, -2*w^3 + 5*w^2 + 6*w - 11],
[941, 941, -3*w^3 + 7*w^2 + 12*w - 5],
[953, 953, -2*w^3 + 5*w^2 + 5*w - 7],
[971, 971, w^3 - w^2 - 3*w - 5],
[971, 971, 3*w^3 - 7*w^2 - 12*w + 11],
[977, 977, 3*w^3 - 5*w^2 - 11*w],
[997, 997, -2*w^3 + 3*w^2 + 6*w - 5],
[997, 997, -4*w^3 + 7*w^2 + 16*w - 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^7 + 6*x^6 - 50*x^4 - 43*x^3 + 119*x^2 + 81*x - 115;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -e^6 - 5*e^5 + 2*e^4 + 34*e^3 + 20*e^2 - 40*e - 17, -2*e^6 - 8*e^5 + 14*e^4 + 63*e^3 - 32*e^2 - 110*e + 70, -e^6 - 3*e^5 + 13*e^4 + 31*e^3 - 61*e^2 - 80*e + 104, -e^5 - 4*e^4 + 5*e^3 + 26*e^2 - 29, e^6 + 4*e^5 - 7*e^4 - 31*e^3 + 17*e^2 + 51*e - 38, e^6 + 5*e^5 - 2*e^4 - 34*e^3 - 20*e^2 + 40*e + 15, e^6 + 4*e^5 - 7*e^4 - 32*e^3 + 16*e^2 + 59*e - 37, -2*e^6 - 6*e^5 + 24*e^4 + 57*e^3 - 105*e^2 - 133*e + 175, 1, 4*e^6 + 17*e^5 - 24*e^4 - 131*e^3 + 37*e^2 + 218*e - 110, 3*e^6 + 12*e^5 - 23*e^4 - 99*e^3 + 66*e^2 + 186*e - 147, e^6 + 4*e^5 - 9*e^4 - 36*e^3 + 36*e^2 + 80*e - 83, -2*e^6 - 7*e^5 + 19*e^4 + 60*e^3 - 67*e^2 - 118*e + 118, e^6 + 3*e^5 - 12*e^4 - 28*e^3 + 50*e^2 + 61*e - 72, -7*e^6 - 25*e^5 + 65*e^4 + 214*e^3 - 227*e^2 - 430*e + 408, 2*e^6 + 6*e^5 - 23*e^4 - 54*e^3 + 97*e^2 + 119*e - 163, 2*e^6 + 7*e^5 - 18*e^4 - 57*e^3 + 60*e^2 + 105*e - 105, -e^6 - 3*e^5 + 13*e^4 + 31*e^3 - 63*e^2 - 82*e + 109, 4*e^6 + 10*e^5 - 59*e^4 - 111*e^3 + 287*e^2 + 301*e - 457, 2*e^5 + 6*e^4 - 15*e^3 - 36*e^2 + 24*e + 19, -e^6 - 7*e^5 - 9*e^4 + 37*e^3 + 101*e^2 - 2*e - 144, e^6 - e^5 - 35*e^4 - 24*e^3 + 223*e^2 + 153*e - 345, 3*e^6 + 9*e^5 - 33*e^4 - 78*e^3 + 128*e^2 + 156*e - 195, 4*e^6 + 12*e^5 - 46*e^4 - 108*e^3 + 194*e^2 + 237*e - 318, 2*e^6 + 2*e^5 - 44*e^4 - 46*e^3 + 249*e^2 + 186*e - 382, 7*e^6 + 34*e^5 - 24*e^4 - 247*e^3 - 58*e^2 + 357*e - 39, 7*e^6 + 29*e^5 - 46*e^4 - 227*e^3 + 93*e^2 + 391*e - 230, -5*e^6 - 24*e^5 + 18*e^4 + 175*e^3 + 36*e^2 - 259*e + 32, 4*e^6 + 18*e^5 - 15*e^4 - 124*e^3 - 36*e^2 + 151*e + 27, -e^6 - 4*e^5 + 6*e^4 + 28*e^3 - 7*e^2 - 35*e + 14, -e^6 - 4*e^5 + 9*e^4 + 34*e^3 - 41*e^2 - 71*e + 97, 9*e^6 + 36*e^5 - 66*e^4 - 289*e^3 + 174*e^2 + 520*e - 389, 11*e^6 + 44*e^5 - 80*e^4 - 351*e^3 + 202*e^2 + 622*e - 439, 2*e^5 + 10*e^4 - 7*e^3 - 72*e^2 - 15*e + 100, -e^6 - 5*e^5 + 4*e^4 + 38*e^3 + 3*e^2 - 62*e + 7, e^6 + 9*e^5 + 18*e^4 - 46*e^3 - 167*e^2 - 10*e + 225, -2*e^6 - 4*e^5 + 35*e^4 + 55*e^3 - 183*e^2 - 177*e + 287, -8*e^6 - 30*e^5 + 64*e^4 + 238*e^3 - 188*e^2 - 421*e + 361, -e^5 - 8*e^4 - 5*e^3 + 64*e^2 + 55*e - 114, 8*e^6 + 34*e^5 - 50*e^4 - 265*e^3 + 95*e^2 + 454*e - 260, -3*e^6 - 16*e^5 + e^4 + 104*e^3 + 98*e^2 - 99*e - 119, -3*e^6 - 17*e^5 + e^4 + 122*e^3 + 92*e^2 - 167*e - 68, e^6 + 6*e^5 - 2*e^4 - 50*e^3 - 14*e^2 + 100*e - 16, 10*e^6 + 45*e^5 - 46*e^4 - 330*e^3 - 12*e^2 + 485*e - 115, 4*e^6 + 14*e^5 - 39*e^4 - 123*e^3 + 146*e^2 + 259*e - 280, 6*e^6 + 25*e^5 - 41*e^4 - 199*e^3 + 102*e^2 + 356*e - 256, 10*e^6 + 36*e^5 - 91*e^4 - 305*e^3 + 308*e^2 + 598*e - 551, -3*e^6 - 6*e^5 + 45*e^4 + 61*e^3 - 215*e^2 - 155*e + 302, 3*e^6 + 9*e^5 - 28*e^4 - 66*e^3 + 88*e^2 + 97*e - 108, 4*e^6 + 10*e^5 - 57*e^4 - 107*e^3 + 267*e^2 + 279*e - 403, -3*e^6 - 8*e^5 + 46*e^4 + 95*e^3 - 230*e^2 - 270*e + 376, 11*e^6 + 46*e^5 - 70*e^4 - 357*e^3 + 133*e^2 + 610*e - 344, e^3 - e^2 - 13*e - 3, -8*e^6 - 33*e^5 + 51*e^4 + 254*e^3 - 91*e^2 - 414*e + 228, -e^5 - 3*e^4 + 10*e^3 + 22*e^2 - 25*e - 24, e^6 + 3*e^5 - 7*e^4 - 17*e^3 + 7*e^2 + 8*e + 13, -4*e^6 - 17*e^5 + 27*e^4 + 137*e^3 - 64*e^2 - 254*e + 157, 8*e^6 + 27*e^5 - 76*e^4 - 226*e^3 + 265*e^2 + 441*e - 443, -4*e^6 - 15*e^5 + 33*e^4 + 121*e^3 - 108*e^2 - 226*e + 211, 4*e^6 + 18*e^5 - 14*e^4 - 120*e^3 - 40*e^2 + 136*e + 39, -4*e^6 - 11*e^5 + 57*e^4 + 123*e^3 - 276*e^2 - 334*e + 459, -5*e^6 - 18*e^5 + 46*e^4 + 154*e^3 - 159*e^2 - 308*e + 293, -11*e^6 - 38*e^5 + 108*e^4 + 330*e^3 - 403*e^2 - 676*e + 716, -13*e^6 - 38*e^5 + 157*e^4 + 360*e^3 - 676*e^2 - 832*e + 1090, -14*e^6 - 57*e^5 + 96*e^4 + 449*e^3 - 214*e^2 - 776*e + 505, 7*e^6 + 36*e^5 - 10*e^4 - 246*e^3 - 170*e^2 + 291*e + 147, -6*e^6 - 18*e^5 + 77*e^4 + 183*e^3 - 353*e^2 - 455*e + 585, -10*e^6 - 35*e^5 + 91*e^4 + 291*e^3 - 301*e^2 - 550*e + 520, -7*e^6 - 25*e^5 + 71*e^4 + 226*e^3 - 285*e^2 - 493*e + 542, -4*e^6 - 19*e^5 + 16*e^4 + 139*e^3 + 12*e^2 - 205*e + 49, -3*e^6 - 4*e^5 + 56*e^4 + 62*e^3 - 288*e^2 - 209*e + 412, -6*e^5 - 28*e^4 + 21*e^3 + 195*e^2 + 54*e - 256, -5*e^6 - 13*e^5 + 62*e^4 + 116*e^3 - 263*e^2 - 250*e + 372, -5*e^6 - 14*e^5 + 67*e^4 + 146*e^3 - 311*e^2 - 374*e + 511, -11*e^6 - 47*e^5 + 61*e^4 + 351*e^3 - 60*e^2 - 543*e + 207, -2*e^6 - 7*e^5 + 20*e^4 + 63*e^3 - 72*e^2 - 130*e + 115, -4*e^6 - 16*e^5 + 29*e^4 + 128*e^3 - 80*e^2 - 240*e + 188, -2*e^6 - 22*e^5 - 46*e^4 + 128*e^3 + 386*e^2 - 62*e - 444, -6*e^6 - 27*e^5 + 26*e^4 + 195*e^3 + 21*e^2 - 284*e + 36, 7*e^6 + 30*e^5 - 38*e^4 - 221*e^3 + 39*e^2 + 339*e - 149, -3*e^6 - 23*e^5 - 29*e^4 + 136*e^3 + 307*e^2 - 76*e - 369, -4*e^6 - 13*e^5 + 44*e^4 + 121*e^3 - 172*e^2 - 270*e + 276, 4*e^6 + 22*e^5 - 4*e^4 - 156*e^3 - 101*e^2 + 211*e + 59, 7*e^6 + 24*e^5 - 71*e^4 - 212*e^3 + 275*e^2 + 446*e - 492, -5*e^6 - 21*e^5 + 34*e^4 + 168*e^3 - 80*e^2 - 303*e + 209, -3*e^6 - 8*e^5 + 38*e^4 + 74*e^3 - 162*e^2 - 158*e + 234, -10*e^6 - 49*e^5 + 28*e^4 + 344*e^3 + 135*e^2 - 450*e - 51, 7*e^5 + 34*e^4 - 24*e^3 - 248*e^2 - 76*e + 346, -12*e^6 - 46*e^5 + 91*e^4 + 365*e^3 - 236*e^2 - 640*e + 460, -10*e^6 - 49*e^5 + 29*e^4 + 348*e^3 + 126*e^2 - 473*e - 33, 16*e^6 + 48*e^5 - 190*e^4 - 452*e^3 + 818*e^2 + 1045*e - 1346, -3*e^6 - 11*e^5 + 21*e^4 + 80*e^3 - 29*e^2 - 110*e + 11, -11*e^6 - 41*e^5 + 94*e^4 + 339*e^3 - 312*e^2 - 647*e + 608, -4*e^6 - 10*e^5 + 63*e^4 + 119*e^3 - 330*e^2 - 346*e + 548, 11*e^6 + 38*e^5 - 110*e^4 - 336*e^3 + 414*e^2 + 707*e - 730, -15*e^6 - 61*e^5 + 106*e^4 + 489*e^3 - 257*e^2 - 878*e + 595, -16*e^6 - 59*e^5 + 141*e^4 + 497*e^3 - 470*e^2 - 972*e + 890, -8*e^6 - 34*e^5 + 47*e^4 + 259*e^3 - 73*e^2 - 429*e + 243, -16*e^6 - 72*e^5 + 74*e^4 + 530*e^3 + 10*e^2 - 800*e + 213, -11*e^6 - 32*e^5 + 133*e^4 + 302*e^3 - 575*e^2 - 690*e + 937, 6*e^6 + 31*e^5 - 4*e^4 - 203*e^3 - 181*e^2 + 216*e + 189, 3*e^6 + 13*e^5 - 15*e^4 - 94*e^3 + 5*e^2 + 137*e - 44, 10*e^6 + 42*e^5 - 69*e^4 - 337*e^3 + 173*e^2 + 596*e - 442, 9*e^5 + 34*e^4 - 53*e^3 - 232*e^2 + 30*e + 265, -14*e^6 - 57*e^5 + 99*e^4 + 452*e^3 - 250*e^2 - 796*e + 579, -7*e^6 - 28*e^5 + 49*e^4 + 218*e^3 - 112*e^2 - 372*e + 224, 6*e^6 + 19*e^5 - 68*e^4 - 178*e^3 + 279*e^2 + 402*e - 456, 9*e^6 + 42*e^5 - 37*e^4 - 309*e^3 - 29*e^2 + 470*e - 118, 15*e^6 + 57*e^5 - 120*e^4 - 464*e^3 + 346*e^2 + 861*e - 669, 27*e^6 + 104*e^5 - 213*e^4 - 845*e^3 + 612*e^2 + 1563*e - 1213, 6*e^6 + 23*e^5 - 54*e^4 - 205*e^3 + 183*e^2 + 433*e - 358, 10*e^6 + 36*e^5 - 99*e^4 - 322*e^3 + 394*e^2 + 698*e - 761, -7*e^5 - 32*e^4 + 28*e^3 + 228*e^2 + 39*e - 303, -4*e^6 - 11*e^5 + 47*e^4 + 99*e^3 - 194*e^2 - 223*e + 289, -12*e^6 - 37*e^5 + 143*e^4 + 356*e^3 - 623*e^2 - 842*e + 1058, -5*e^6 - 23*e^5 + 16*e^4 + 152*e^3 + 57*e^2 - 166*e - 45, -12*e^6 - 52*e^5 + 68*e^4 + 397*e^3 - 86*e^2 - 648*e + 292, -6*e^6 - 26*e^5 + 44*e^4 + 221*e^3 - 130*e^2 - 433*e + 339, 7*e^6 + 35*e^5 - 13*e^4 - 235*e^3 - 143*e^2 + 279*e + 119, -e^6 - e^5 + 29*e^4 + 38*e^3 - 195*e^2 - 177*e + 348, -10*e^6 - 32*e^5 + 111*e^4 + 295*e^3 - 449*e^2 - 654*e + 746, 2*e^6 + 9*e^5 - 7*e^4 - 62*e^3 - 21*e^2 + 88*e + 41, 11*e^6 + 26*e^5 - 167*e^4 - 296*e^3 + 816*e^2 + 825*e - 1285, -16*e^6 - 54*e^5 + 158*e^4 + 463*e^3 - 588*e^2 - 940*e + 1005, -13*e^6 - 60*e^5 + 52*e^4 + 431*e^3 + 71*e^2 - 612*e + 51, -8*e^6 - 32*e^5 + 52*e^4 + 239*e^3 - 106*e^2 - 382*e + 258, 9*e^6 + 23*e^5 - 125*e^4 - 237*e^3 + 592*e^2 + 611*e - 942, -17*e^6 - 79*e^5 + 65*e^4 + 569*e^3 + 116*e^2 - 807*e + 64, 12*e^6 + 48*e^5 - 93*e^4 - 397*e^3 + 271*e^2 + 747*e - 598, -6*e^6 - 31*e^5 + 8*e^4 + 210*e^3 + 140*e^2 - 249*e - 98, 20*e^6 + 73*e^5 - 177*e^4 - 612*e^3 + 596*e^2 + 1198*e - 1111, 11*e^6 + 25*e^5 - 169*e^4 - 286*e^3 + 830*e^2 + 800*e - 1302, -7*e^6 - 28*e^5 + 50*e^4 + 222*e^3 - 125*e^2 - 384*e + 310, -14*e^6 - 54*e^5 + 108*e^4 + 429*e^3 - 307*e^2 - 760*e + 631, 16*e^6 + 52*e^5 - 173*e^4 - 467*e^3 + 705*e^2 + 1015*e - 1210, -12*e^6 - 44*e^5 + 112*e^4 + 381*e^3 - 415*e^2 - 789*e + 780, -5*e^6 - 15*e^5 + 55*e^4 + 132*e^3 - 219*e^2 - 283*e + 341, 14*e^6 + 29*e^5 - 225*e^4 - 344*e^3 + 1133*e^2 + 1009*e - 1736, 12*e^6 + 51*e^5 - 79*e^4 - 407*e^3 + 170*e^2 + 726*e - 428, 9*e^6 + 45*e^5 - 22*e^4 - 314*e^3 - 140*e^2 + 411*e + 78, e^6 + e^5 - 16*e^4 - 10*e^3 + 65*e^2 + 24*e - 69, 15*e^6 + 53*e^5 - 143*e^4 - 457*e^3 + 518*e^2 + 924*e - 934, -22*e^6 - 93*e^5 + 136*e^4 + 719*e^3 - 235*e^2 - 1201*e + 660, -8*e^6 - 22*e^5 + 112*e^4 + 236*e^3 - 538*e^2 - 625*e + 869, 3*e^6 + 10*e^5 - 27*e^4 - 76*e^3 + 96*e^2 + 129*e - 178, -8*e^6 - 23*e^5 + 98*e^4 + 216*e^3 - 421*e^2 - 485*e + 655, 12*e^6 + 42*e^5 - 114*e^4 - 360*e^3 + 406*e^2 + 717*e - 742, 10*e^6 + 44*e^5 - 49*e^4 - 320*e^3 + 13*e^2 + 475*e - 139, -2*e^6 - e^5 + 41*e^4 + 26*e^3 - 208*e^2 - 96*e + 262, 27*e^6 + 106*e^5 - 206*e^4 - 855*e^3 + 569*e^2 + 1552*e - 1192, 11*e^6 + 53*e^5 - 40*e^4 - 390*e^3 - 73*e^2 + 590*e - 98, 6*e^6 + 9*e^5 - 113*e^4 - 137*e^3 + 601*e^2 + 477*e - 901, -3*e^6 - 18*e^5 - 6*e^4 + 116*e^3 + 142*e^2 - 104*e - 135, -12*e^6 - 32*e^5 + 163*e^4 + 325*e^3 - 767*e^2 - 825*e + 1228, e^6 + 10*e^5 + 17*e^4 - 63*e^3 - 144*e^2 + 47*e + 139, 25*e^6 + 114*e^5 - 119*e^4 - 856*e^3 + 26*e^2 + 1350*e - 443, -2*e^6 - 10*e^5 + 8*e^4 + 71*e^3 - 5*e^2 - 92*e + 58];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {29, 29, w^3 - 2*w^2 - 5*w}>] := -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;