/*
  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, 10, -7, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {17, 17, w^3 + w^2 - 6*w - 1}>;

primesArray := [
[2, 2, -w + 1],
[5, 5, w^3 + w^2 - 4*w + 1],
[7, 7, -w^3 + 6*w - 2],
[8, 2, w^3 - 7*w + 3],
[11, 11, -w^3 + 6*w - 4],
[17, 17, w^3 + w^2 - 6*w - 1],
[17, 17, -w^2 - w + 3],
[31, 31, 2*w^3 + w^2 - 13*w + 3],
[37, 37, -3*w^3 - w^2 + 18*w - 3],
[41, 41, w^2 + 2*w - 4],
[47, 47, 2*w^3 + 2*w^2 - 11*w - 2],
[47, 47, -3*w^3 - w^2 + 19*w - 6],
[59, 59, -2*w^3 + 13*w - 6],
[59, 59, -w^3 - w^2 + 5*w + 2],
[59, 59, w^2 + w - 1],
[59, 59, w^2 + 2*w - 6],
[61, 61, -w^3 + w^2 + 8*w - 7],
[67, 67, w^3 + 2*w^2 - 4*w - 4],
[73, 73, -2*w^3 - w^2 + 12*w - 4],
[81, 3, -3],
[107, 107, 2*w^2 + 2*w - 11],
[109, 109, 2*w^3 + w^2 - 11*w + 3],
[113, 113, w^3 + w^2 - 7*w - 2],
[113, 113, 2*w^3 - 12*w + 9],
[125, 5, -4*w^3 - 3*w^2 + 24*w + 2],
[131, 131, 5*w^3 + 3*w^2 - 31*w + 2],
[151, 151, 4*w^3 + 2*w^2 - 26*w + 1],
[151, 151, w^3 + 2*w^2 - 6*w - 10],
[167, 167, w + 4],
[167, 167, 4*w^3 + w^2 - 26*w + 10],
[173, 173, -2*w^3 + w^2 + 13*w - 11],
[179, 179, 4*w^3 + w^2 - 24*w + 4],
[179, 179, 2*w^3 + 2*w^2 - 12*w - 3],
[179, 179, -w^3 + 8*w - 4],
[179, 179, w^3 + 2*w^2 - 4*w - 6],
[181, 181, 6*w^3 + 3*w^2 - 39*w + 5],
[197, 197, -w^3 + 4*w - 4],
[199, 199, 2*w^3 - 12*w + 3],
[199, 199, 3*w - 2],
[223, 223, w^3 + w^2 - 7*w - 4],
[223, 223, 2*w^3 + w^2 - 10*w + 4],
[227, 227, 7*w^3 + 2*w^2 - 44*w + 12],
[227, 227, -4*w^3 - w^2 + 25*w - 9],
[229, 229, 8*w^3 + 4*w^2 - 51*w + 4],
[229, 229, 3*w^3 + w^2 - 18*w + 5],
[233, 233, -6*w^3 - 3*w^2 + 39*w - 7],
[233, 233, -w^3 - 2*w^2 + 8*w + 2],
[241, 241, 2*w^3 + w^2 - 10*w],
[251, 251, w^3 + w^2 - 7*w - 6],
[257, 257, -6*w^3 - w^2 + 39*w - 17],
[263, 263, -2*w^3 + 13*w - 4],
[263, 263, -w^3 + 6*w - 8],
[269, 269, -8*w^3 - 4*w^2 + 50*w - 7],
[269, 269, -w^3 + w^2 + 7*w - 6],
[277, 277, 2*w^3 + 2*w^2 - 11*w + 4],
[277, 277, w^3 + w^2 - 4*w - 3],
[281, 281, -w^2 - 3*w + 5],
[281, 281, 2*w^3 + 3*w^2 - 9*w - 3],
[283, 283, -3*w^3 - 4*w^2 + 12*w + 2],
[289, 17, -2*w^3 - w^2 + 13*w - 5],
[293, 293, -w^3 - 2*w^2 + 6*w + 6],
[293, 293, w^3 + w^2 - 8*w - 1],
[307, 307, 2*w^3 - 11*w + 12],
[317, 317, 2*w^3 - 14*w + 11],
[317, 317, -2*w^3 - 2*w^2 + 8*w + 1],
[337, 337, 2*w^2 + 4*w - 11],
[343, 7, -5*w^3 - w^2 + 31*w - 10],
[349, 349, 2*w^3 - 11*w + 8],
[349, 349, w^2 + w - 9],
[353, 353, 4*w^3 + 4*w^2 - 23*w - 8],
[373, 373, -2*w - 3],
[373, 373, -4*w^3 + 26*w - 13],
[389, 389, 6*w^3 + 3*w^2 - 37*w + 1],
[389, 389, -2*w^3 - w^2 + 14*w - 6],
[401, 401, 2*w^3 - 12*w + 13],
[421, 421, 2*w^2 + 2*w - 15],
[421, 421, 3*w^3 + 2*w^2 - 20*w + 4],
[431, 431, w^3 + 3*w^2 - 2*w - 9],
[433, 433, 2*w^3 + 2*w^2 - 12*w + 3],
[433, 433, -7*w^3 - 2*w^2 + 44*w - 16],
[433, 433, -3*w^3 + 18*w - 14],
[433, 433, -w^3 + w^2 + 9*w - 10],
[439, 439, 5*w^3 + 3*w^2 - 31*w - 2],
[443, 443, -w^3 - w^2 + 3*w - 4],
[443, 443, -4*w^3 + 27*w - 14],
[449, 449, w^3 + w^2 - 4*w - 5],
[449, 449, w^3 + 3*w^2 - 3*w - 8],
[449, 449, 2*w^3 - 11*w + 6],
[449, 449, -3*w^3 - 2*w^2 + 18*w - 4],
[461, 461, -3*w^3 + w^2 + 20*w - 15],
[463, 463, 2*w^3 - 10*w + 7],
[479, 479, -2*w^3 - 3*w^2 + 11*w + 9],
[491, 491, w^3 + w^2 - 4*w + 5],
[499, 499, -2*w^3 - 2*w^2 + 14*w - 3],
[499, 499, -6*w^3 - 2*w^2 + 39*w - 10],
[499, 499, 4*w^3 + w^2 - 24*w + 14],
[499, 499, -5*w^3 - 3*w^2 + 30*w - 5],
[503, 503, -2*w^3 + w^2 + 14*w - 18],
[509, 509, 4*w^3 + 3*w^2 - 24*w],
[509, 509, 4*w^3 + 2*w^2 - 23*w + 4],
[547, 547, w^3 + 2*w^2 - 2*w - 6],
[547, 547, 6*w^3 + 2*w^2 - 38*w + 7],
[563, 563, -5*w^3 - w^2 + 33*w - 12],
[569, 569, 2*w^3 - w^2 - 15*w + 13],
[569, 569, -4*w^3 - 2*w^2 + 24*w - 7],
[577, 577, -w^2 - 2],
[577, 577, 3*w^3 - 20*w + 8],
[587, 587, 6*w^3 + 4*w^2 - 36*w - 1],
[587, 587, -w^3 + w^2 + 9*w - 8],
[593, 593, 6*w^3 + 3*w^2 - 36*w + 2],
[593, 593, -3*w^3 - w^2 + 21*w - 4],
[599, 599, -2*w^3 - w^2 + 12*w - 10],
[599, 599, -4*w^3 - 2*w^2 + 25*w - 6],
[601, 601, -2*w^3 + 2*w^2 + 15*w - 16],
[607, 607, w^3 - w^2 - 7*w + 4],
[607, 607, -2*w^3 - 2*w^2 + 10*w + 3],
[631, 631, 2*w^2 + 4*w - 7],
[641, 641, -3*w^3 - 2*w^2 + 16*w],
[643, 643, w^3 + w^2 - 3*w - 4],
[643, 643, w^3 + 3*w^2 - 6*w - 13],
[653, 653, -w^3 + 6*w + 2],
[653, 653, -3*w^2 - 5*w + 11],
[659, 659, -2*w^3 + 10*w - 5],
[661, 661, w^3 - w^2 - 8*w + 3],
[673, 673, 3*w^3 + 4*w^2 - 14*w - 2],
[677, 677, 4*w^3 + 2*w^2 - 23*w + 6],
[677, 677, 2*w^3 - 13*w + 14],
[683, 683, 2*w^3 + w^2 - 13*w + 7],
[701, 701, 8*w^3 + 4*w^2 - 51*w + 8],
[701, 701, -4*w^3 - w^2 + 24*w - 12],
[709, 709, -4*w^3 - w^2 + 27*w - 7],
[709, 709, 3*w^3 + 3*w^2 - 18*w - 5],
[709, 709, -3*w^3 + w^2 + 21*w - 14],
[709, 709, w^2 - w - 7],
[719, 719, -9*w^3 - 4*w^2 + 58*w - 8],
[719, 719, 11*w^3 + 5*w^2 - 68*w + 9],
[727, 727, -w^3 + 4*w - 6],
[733, 733, 2*w^3 + w^2 - 13*w + 9],
[739, 739, -2*w^2 - 3*w + 4],
[743, 743, 3*w^3 + 3*w^2 - 19*w - 4],
[751, 751, 2*w^3 + 3*w^2 - 10*w - 2],
[761, 761, -12*w^3 - 5*w^2 + 75*w - 9],
[761, 761, -3*w^3 - 3*w^2 + 17*w],
[769, 769, -6*w^3 - 4*w^2 + 37*w],
[769, 769, -8*w^3 - 3*w^2 + 51*w - 7],
[773, 773, -w^2 - 2*w - 2],
[787, 787, -5*w^3 - 5*w^2 + 28*w + 7],
[811, 811, 7*w^3 + 2*w^2 - 42*w + 6],
[811, 811, -2*w^3 - w^2 + 9*w - 7],
[823, 823, -3*w^3 - 2*w^2 + 14*w - 10],
[829, 829, 4*w^3 + 3*w^2 - 20*w + 6],
[829, 829, -7*w^3 - 2*w^2 + 44*w - 14],
[841, 29, w^3 + 4*w^2 - 2*w - 14],
[841, 29, -w^2 + 10],
[857, 857, -3*w^3 - w^2 + 20*w - 9],
[859, 859, -w^3 + 2*w^2 + 10*w - 18],
[863, 863, 11*w^3 + 5*w^2 - 69*w + 6],
[863, 863, w^3 - 2*w^2 - 10*w + 12],
[877, 877, 3*w^3 - 18*w + 10],
[883, 883, 4*w^3 + 3*w^2 - 22*w],
[887, 887, 3*w^3 + 3*w^2 - 15*w - 2],
[887, 887, w^2 + 4*w - 10],
[887, 887, w^3 + w^2 - 3*w - 8],
[887, 887, w - 6],
[907, 907, 6*w^3 + w^2 - 38*w + 16],
[919, 919, -2*w^3 - 3*w^2 + 11*w + 1],
[919, 919, 5*w^3 - w^2 - 34*w + 27],
[937, 937, -w^3 - w^2 + 8*w + 3],
[941, 941, -4*w^3 + 25*w - 20],
[947, 947, -2*w^3 + w^2 + 8*w - 6],
[967, 967, -7*w^3 - w^2 + 44*w - 23],
[971, 971, 2*w^2 + 2*w - 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^14 - 16*x^12 + 101*x^10 - 318*x^8 + 519*x^6 - 411*x^4 + 124*x^2 - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -e^10 + 11*e^8 - 42*e^6 + 65*e^4 - 36*e^2 + 2, -e^11 + 12*e^9 - 51*e^7 + 90*e^5 - 57*e^3 + 5*e, e^11 - 12*e^9 + 52*e^7 - 98*e^5 + 77*e^3 - 21*e, e^13 - 15*e^11 + 87*e^9 - 244*e^7 + 338*e^5 - 212*e^3 + 47*e, 1, -e^13 + 15*e^11 - 87*e^9 + 243*e^7 - 330*e^5 + 192*e^3 - 32*e, -e^11 + 11*e^9 - 43*e^7 + 73*e^5 - 55*e^3 + 16*e, -e^13 + 15*e^11 - 87*e^9 + 243*e^7 - 326*e^5 + 171*e^3 - 12*e, e^13 - 12*e^11 + 49*e^9 - 63*e^7 - 68*e^5 + 223*e^3 - 130*e, -e^12 + 16*e^10 - 96*e^8 + 267*e^6 - 338*e^4 + 151*e^2 - 2, -e^13 + 13*e^11 - 62*e^9 + 128*e^7 - 91*e^5 - 32*e^3 + 48*e, e^13 - 13*e^11 + 64*e^9 - 151*e^7 + 181*e^5 - 108*e^3 + 27*e, e^12 - 15*e^10 + 86*e^8 - 232*e^6 + 284*e^4 - 111*e^2 - 12, -2*e^11 + 26*e^9 - 127*e^7 + 282*e^5 - 265*e^3 + 71*e, 2*e^12 - 23*e^10 + 94*e^8 - 165*e^6 + 127*e^4 - 51*e^2 + 10, 2*e^13 - 29*e^11 + 163*e^9 - 446*e^7 + 608*e^5 - 372*e^3 + 75*e, -e^12 + 10*e^10 - 29*e^8 + 7*e^6 + 60*e^4 - 28*e^2 - 12, -e^13 + 16*e^11 - 95*e^9 + 253*e^7 - 275*e^5 + 45*e^3 + 55*e, 2*e^10 - 22*e^8 + 86*e^6 - 141*e^4 + 82*e^2 - 5, -e^12 + 14*e^10 - 77*e^8 + 204*e^6 - 251*e^4 + 114*e^2 - 14, -e^11 + 6*e^9 + 13*e^7 - 144*e^5 + 282*e^3 - 161*e, 4*e^12 - 50*e^10 + 233*e^8 - 501*e^6 + 497*e^4 - 185*e^2 - 2, -e^13 + 18*e^11 - 130*e^9 + 471*e^7 - 875*e^5 + 757*e^3 - 225*e, e^12 - 20*e^10 + 142*e^8 - 448*e^6 + 616*e^4 - 283*e^2 - 7, 4*e^12 - 52*e^10 + 252*e^8 - 557*e^6 + 541*e^4 - 170*e^2, e^12 - 12*e^10 + 49*e^8 - 68*e^6 - 21*e^4 + 83*e^2 - 17, -e^12 + 13*e^10 - 65*e^8 + 158*e^6 - 187*e^4 + 83*e^2 - 8, 2*e^12 - 29*e^10 + 160*e^8 - 418*e^6 + 514*e^4 - 233*e^2 + 4, e^13 - 17*e^11 + 113*e^9 - 367*e^7 + 591*e^5 - 422*e^3 + 105*e, 4*e^13 - 59*e^11 + 338*e^9 - 946*e^7 + 1334*e^5 - 874*e^3 + 203*e, 2*e^13 - 29*e^11 + 163*e^9 - 443*e^7 + 590*e^5 - 345*e^3 + 64*e, -5*e^12 + 68*e^10 - 352*e^8 + 856*e^6 - 953*e^4 + 357*e^2 + 18, -7*e^13 + 102*e^11 - 573*e^9 + 1548*e^7 - 2032*e^5 + 1137*e^3 - 185*e, 2*e^12 - 30*e^10 + 170*e^8 - 448*e^6 + 537*e^4 - 224*e^2 - 10, 3*e^11 - 35*e^9 + 145*e^7 - 255*e^5 + 176*e^3 - 19*e, 2*e^12 - 31*e^10 + 183*e^8 - 510*e^6 + 665*e^4 - 328*e^2 + 13, -2*e^13 + 28*e^11 - 149*e^9 + 373*e^7 - 433*e^5 + 181*e^3 - 3*e, -e^13 + 16*e^11 - 104*e^9 + 353*e^7 - 656*e^5 + 617*e^3 - 224*e, -4*e^12 + 54*e^10 - 277*e^8 + 667*e^6 - 741*e^4 + 295*e^2 - 6, -6*e^13 + 91*e^11 - 532*e^9 + 1488*e^7 - 1987*e^5 + 1069*e^3 - 128*e, -5*e^13 + 68*e^11 - 348*e^9 + 823*e^7 - 870*e^5 + 290*e^3 + 38*e, -e^13 + 9*e^11 - 16*e^9 - 58*e^7 + 220*e^5 - 227*e^3 + 90*e, 3*e^10 - 39*e^8 + 181*e^6 - 349*e^4 + 229*e^2 - 10, 6*e^13 - 85*e^11 + 464*e^9 - 1221*e^7 + 1578*e^5 - 907*e^3 + 185*e, e^13 - 15*e^11 + 93*e^9 - 314*e^7 + 621*e^5 - 668*e^3 + 289*e, -3*e^10 + 37*e^8 - 161*e^6 + 282*e^4 - 154*e^2 - 1, 2*e^12 - 26*e^10 + 129*e^8 - 305*e^6 + 340*e^4 - 134*e^2 + 5, -e^12 + 13*e^10 - 66*e^8 + 165*e^6 - 203*e^4 + 89*e^2 + 4, -e^13 + 22*e^11 - 169*e^9 + 589*e^7 - 966*e^5 + 684*e^3 - 151*e, 2*e^13 - 28*e^11 + 149*e^9 - 376*e^7 + 454*e^5 - 217*e^3 + 2*e, e^10 - 7*e^8 + 4*e^6 + 46*e^4 - 56*e^2 - 3, -6*e^13 + 88*e^11 - 498*e^9 + 1360*e^7 - 1823*e^5 + 1063*e^3 - 170*e, e^13 - 11*e^11 + 35*e^9 + 19*e^7 - 310*e^5 + 555*e^3 - 281*e, 4*e^13 - 65*e^11 + 411*e^9 - 1265*e^7 + 1930*e^5 - 1309*e^3 + 281*e, -6*e^12 + 81*e^10 - 417*e^8 + 1022*e^6 - 1200*e^4 + 558*e^2 - 28, 3*e^13 - 39*e^11 + 190*e^9 - 423*e^7 + 398*e^5 - 59*e^3 - 63*e, -e^12 + 13*e^10 - 62*e^8 + 125*e^6 - 76*e^4 - 27*e^2 + 4, -3*e^12 + 40*e^10 - 198*e^8 + 444*e^6 - 433*e^4 + 137*e^2 - 6, 5*e^13 - 76*e^11 + 448*e^9 - 1275*e^7 + 1756*e^5 - 997*e^3 + 144*e, -e^12 + 16*e^10 - 93*e^8 + 244*e^6 - 288*e^4 + 124*e^2 + 3, e^12 - 20*e^10 + 140*e^8 - 433*e^6 + 589*e^4 - 291*e^2 + 7, 2*e^12 - 28*e^10 + 147*e^8 - 353*e^6 + 367*e^4 - 107*e^2 - 10, -2*e^13 + 31*e^11 - 182*e^9 + 496*e^7 - 605*e^5 + 245*e^3 - 3*e, -6*e^12 + 89*e^10 - 500*e^8 + 1306*e^6 - 1545*e^4 + 641*e^2 - 1, -9*e^12 + 124*e^10 - 650*e^8 + 1601*e^6 - 1821*e^4 + 747*e^2 - 4, -e^13 + 12*e^11 - 58*e^9 + 160*e^7 - 288*e^5 + 279*e^3 - 79*e, e^9 - 13*e^7 + 54*e^5 - 80*e^3 + 26*e, 4*e^10 - 47*e^8 + 195*e^6 - 331*e^4 + 195*e^2 - 23, 3*e^12 - 46*e^10 + 266*e^8 - 716*e^6 + 886*e^4 - 415*e^2 + 27, -2*e^12 + 36*e^10 - 239*e^8 + 728*e^6 - 996*e^4 + 458*e^2 - 3, e^11 - 5*e^9 - 26*e^7 + 206*e^5 - 405*e^3 + 231*e, -5*e^12 + 63*e^10 - 294*e^8 + 624*e^6 - 595*e^4 + 208*e^2 - 14, -4*e^13 + 57*e^11 - 309*e^9 + 773*e^7 - 819*e^5 + 158*e^3 + 135*e, 6*e^12 - 80*e^10 + 409*e^8 - 1001*e^6 + 1175*e^4 - 542*e^2 + 33, -6*e^12 + 82*e^10 - 422*e^8 + 1007*e^6 - 1089*e^4 + 410*e^2 - 10, -6*e^13 + 83*e^11 - 451*e^9 + 1233*e^7 - 1790*e^5 + 1282*e^3 - 332*e, -5*e^12 + 66*e^10 - 323*e^8 + 712*e^6 - 667*e^4 + 172*e^2 + 13, e^13 - 18*e^11 + 118*e^9 - 347*e^7 + 439*e^5 - 157*e^3 - 42*e, -6*e^13 + 86*e^11 - 477*e^9 + 1279*e^7 - 1668*e^5 + 898*e^3 - 122*e, -4*e^13 + 57*e^11 - 310*e^9 + 796*e^7 - 944*e^5 + 376*e^3 + 43*e, 8*e^11 - 100*e^9 + 461*e^7 - 955*e^5 + 845*e^3 - 216*e, -2*e^12 + 29*e^10 - 156*e^8 + 373*e^6 - 359*e^4 + 73*e^2 + 17, -7*e^12 + 94*e^10 - 478*e^8 + 1143*e^6 - 1276*e^4 + 516*e^2 + 4, 2*e^13 - 22*e^11 + 71*e^9 + 3*e^7 - 379*e^5 + 560*e^3 - 211*e, 4*e^12 - 59*e^10 + 330*e^8 - 852*e^6 + 968*e^4 - 348*e^2 - 28, -e^12 + 7*e^10 + 7*e^8 - 142*e^6 + 306*e^4 - 160*e^2 - 19, -e^13 + 11*e^11 - 37*e^9 + 13*e^7 + 143*e^5 - 225*e^3 + 79*e, -e^13 + 12*e^11 - 61*e^9 + 196*e^7 - 444*e^5 + 564*e^3 - 255*e, -4*e^13 + 50*e^11 - 228*e^9 + 455*e^7 - 364*e^5 + 59*e^3 + 29*e, -e^13 + 21*e^11 - 160*e^9 + 571*e^7 - 1003*e^5 + 824*e^3 - 241*e, -5*e^12 + 78*e^10 - 456*e^8 + 1226*e^6 - 1476*e^4 + 617*e^2, -9*e^12 + 116*e^10 - 564*e^8 + 1286*e^6 - 1366*e^4 + 519*e^2 - 4, -5*e^13 + 73*e^11 - 415*e^9 + 1156*e^7 - 1629*e^5 + 1083*e^3 - 272*e, e^13 - 11*e^11 + 42*e^9 - 69*e^7 + 81*e^5 - 143*e^3 + 103*e, 3*e^13 - 43*e^11 + 229*e^9 - 543*e^7 + 499*e^5 + 11*e^3 - 159*e, 4*e^13 - 65*e^11 + 415*e^9 - 1302*e^7 + 2040*e^5 - 1419*e^3 + 302*e, 10*e^12 - 136*e^10 + 701*e^8 - 1687*e^6 + 1850*e^4 - 695*e^2 - 23, 2*e^12 - 23*e^10 + 97*e^8 - 193*e^6 + 204*e^4 - 100*e^2 + 2, 4*e^13 - 53*e^11 + 262*e^9 - 580*e^7 + 499*e^5 + 24*e^3 - 167*e, 7*e^12 - 97*e^10 + 506*e^8 - 1222*e^6 + 1336*e^4 - 513*e^2 - 8, 6*e^13 - 93*e^11 + 560*e^9 - 1639*e^7 + 2373*e^5 - 1521*e^3 + 301*e, 3*e^13 - 37*e^11 + 175*e^9 - 423*e^7 + 615*e^5 - 546*e^3 + 193*e, -6*e^13 + 90*e^11 - 526*e^9 + 1504*e^7 - 2154*e^5 + 1398*e^3 - 312*e, -5*e^13 + 82*e^11 - 518*e^9 + 1580*e^7 - 2381*e^5 + 1617*e^3 - 389*e, -12*e^12 + 157*e^10 - 778*e^8 + 1818*e^6 - 2012*e^4 + 871*e^2 - 40, -6*e^13 + 83*e^11 - 426*e^9 + 966*e^7 - 811*e^5 - 129*e^3 + 315*e, -9*e^12 + 128*e^10 - 700*e^8 + 1823*e^6 - 2225*e^4 + 996*e^2 - 29, 7*e^13 - 105*e^11 + 604*e^9 - 1650*e^7 + 2125*e^5 - 1068*e^3 + 92*e, 5*e^12 - 59*e^10 + 251*e^8 - 469*e^6 + 382*e^4 - 111*e^2 - 27, -2*e^13 + 22*e^11 - 86*e^9 + 159*e^7 - 212*e^5 + 255*e^3 - 110*e, -7*e^12 + 90*e^10 - 433*e^8 + 962*e^6 - 978*e^4 + 372*e^2 - 18, -6*e^13 + 85*e^11 - 461*e^9 + 1191*e^7 - 1474*e^5 + 758*e^3 - 102*e, -e^13 + 10*e^11 - 13*e^9 - 171*e^7 + 753*e^5 - 1110*e^3 + 520*e, -6*e^13 + 76*e^11 - 354*e^9 + 734*e^7 - 635*e^5 + 135*e^3 + 47*e, -4*e^12 + 53*e^10 - 262*e^8 + 590*e^6 - 582*e^4 + 187*e^2 - 14, 5*e^13 - 80*e^11 + 493*e^9 - 1452*e^7 + 2040*e^5 - 1163*e^3 + 168*e, -2*e^12 + 22*e^10 - 83*e^8 + 124*e^6 - 70*e^4 + 49*e^2 - 48, 5*e^12 - 53*e^10 + 184*e^8 - 208*e^6 - 27*e^4 + 96*e^2 + 1, 3*e^12 - 51*e^10 + 329*e^8 - 990*e^6 + 1362*e^4 - 685*e^2 + 26, e^12 - 8*e^10 + 3*e^8 + 106*e^6 - 244*e^4 + 101*e^2 + 20, -2*e^13 + 26*e^11 - 129*e^9 + 308*e^7 - 363*e^5 + 185*e^3 - 30*e, 5*e^13 - 74*e^11 + 436*e^9 - 1300*e^7 + 2035*e^5 - 1528*e^3 + 412*e, -3*e^13 + 48*e^11 - 296*e^9 + 881*e^7 - 1307*e^5 + 931*e^3 - 293*e, 5*e^12 - 63*e^10 + 293*e^8 - 622*e^6 + 621*e^4 - 280*e^2 + 12, -4*e^11 + 64*e^9 - 384*e^7 + 1068*e^5 - 1344*e^3 + 576*e, 5*e^12 - 74*e^10 + 415*e^8 - 1075*e^6 + 1233*e^4 - 477*e^2 + 16, 2*e^13 - 36*e^11 + 239*e^9 - 735*e^7 + 1068*e^5 - 677*e^3 + 163*e, -3*e^13 + 42*e^11 - 234*e^9 + 671*e^7 - 1051*e^5 + 830*e^3 - 235*e, 2*e^13 - 32*e^11 + 191*e^9 - 525*e^7 + 656*e^5 - 294*e^3 - 19*e, -2*e^13 + 26*e^11 - 125*e^9 + 279*e^7 - 317*e^5 + 217*e^3 - 79*e, -7*e^12 + 89*e^10 - 425*e^8 + 957*e^6 - 1049*e^4 + 484*e^2 - 22, 8*e^13 - 129*e^11 + 801*e^9 - 2384*e^7 + 3430*e^5 - 2104*e^3 + 386*e, 8*e^12 - 109*e^10 + 567*e^8 - 1392*e^6 + 1572*e^4 - 595*e^2 - 31, 3*e^13 - 31*e^11 + 102*e^9 - 107*e^7 + 51*e^5 - 214*e^3 + 204*e, 3*e^11 - 52*e^9 + 334*e^7 - 974*e^5 + 1252*e^3 - 547*e, -2*e^12 + 37*e^10 - 256*e^8 + 818*e^6 - 1182*e^4 + 606*e^2 - 32, -6*e^13 + 89*e^11 - 517*e^9 + 1492*e^7 - 2239*e^5 + 1650*e^3 - 455*e, -2*e^13 + 19*e^11 - 40*e^9 - 115*e^7 + 552*e^5 - 680*e^3 + 242*e, 7*e^12 - 98*e^10 + 525*e^8 - 1341*e^6 + 1640*e^4 - 805*e^2 + 53, -4*e^12 + 50*e^10 - 234*e^8 + 504*e^6 - 492*e^4 + 192*e^2 - 19, -e^13 + 4*e^11 + 54*e^9 - 439*e^7 + 1211*e^5 - 1410*e^3 + 560*e, -3*e^12 + 38*e^10 - 182*e^8 + 422*e^6 - 504*e^4 + 274*e^2 - 41, e^12 - 13*e^10 + 65*e^8 - 159*e^6 + 197*e^4 - 119*e^2 + 49, 3*e^13 - 40*e^11 + 222*e^9 - 695*e^7 + 1327*e^5 - 1362*e^3 + 517*e, 4*e^12 - 53*e^10 + 260*e^8 - 579*e^6 + 580*e^4 - 218*e^2 - 14, -11*e^12 + 148*e^10 - 753*e^8 + 1789*e^6 - 1950*e^4 + 759*e^2 - 6, -2*e^13 + 24*e^11 - 106*e^9 + 219*e^7 - 256*e^5 + 243*e^3 - 131*e, -11*e^12 + 147*e^10 - 739*e^8 + 1719*e^6 - 1803*e^4 + 632*e^2 + 37, -e^12 - 3*e^10 + 118*e^8 - 573*e^6 + 989*e^4 - 552*e^2 + 34, -2*e^13 + 35*e^11 - 228*e^9 + 683*e^7 - 924*e^5 + 453*e^3 - 23*e, 3*e^13 - 34*e^11 + 139*e^9 - 257*e^7 + 240*e^5 - 129*e^3 + 44*e, -3*e^12 + 33*e^10 - 131*e^8 + 244*e^6 - 257*e^4 + 148*e^2 - 24, 6*e^12 - 81*e^10 + 411*e^8 - 957*e^6 + 986*e^4 - 350*e^2 + 11, e^13 - 17*e^11 + 120*e^9 - 441*e^7 + 865*e^5 - 837*e^3 + 315*e, -4*e^12 + 46*e^10 - 188*e^8 + 338*e^6 - 305*e^4 + 179*e^2 - 31, -9*e^12 + 129*e^10 - 710*e^8 + 1860*e^6 - 2287*e^4 + 1037*e^2 - 49, e^13 - 16*e^11 + 117*e^9 - 486*e^7 + 1125*e^5 - 1271*e^3 + 498*e, 20*e^11 - 239*e^9 + 1035*e^7 - 1972*e^5 + 1616*e^3 - 479*e, 13*e^12 - 171*e^10 + 841*e^8 - 1898*e^6 + 1908*e^4 - 626*e^2 - 43, 9*e^12 - 118*e^10 + 578*e^8 - 1299*e^6 + 1309*e^4 - 476*e^2 + 31, 12*e^12 - 160*e^10 + 807*e^8 - 1900*e^6 + 2056*e^4 - 814*e^2 + 5, -20*e^10 + 222*e^8 - 855*e^6 + 1315*e^4 - 672*e^2 + 40, -16*e^12 + 223*e^10 - 1176*e^8 + 2900*e^6 - 3304*e^4 + 1398*e^2 - 52, 7*e^13 - 104*e^11 + 589*e^9 - 1562*e^7 + 1873*e^5 - 723*e^3 - 81*e, -5*e^12 + 80*e^10 - 483*e^8 + 1344*e^6 - 1657*e^4 + 666*e^2 + 22, 5*e^13 - 85*e^11 + 555*e^9 - 1758*e^7 + 2809*e^5 - 2125*e^3 + 572*e, -7*e^12 + 95*e^10 - 482*e^8 + 1122*e^6 - 1163*e^4 + 398*e^2 + 17, -9*e^13 + 120*e^11 - 602*e^9 + 1390*e^7 - 1394*e^5 + 322*e^3 + 180*e, e^13 - 13*e^11 + 64*e^9 - 152*e^7 + 199*e^5 - 176*e^3 + 78*e, e^13 - 7*e^11 - 3*e^9 + 99*e^7 - 141*e^5 - 104*e^3 + 158*e, e^13 - 78*e^9 + 387*e^7 - 645*e^5 + 364*e^3 - 68*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {17, 17, w^3 + w^2 - 6*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;