/*
  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, -5, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {19, 19, -w^2 + w + 1}>;

primesArray := [
[2, 2, w - 1],
[3, 3, w],
[3, 3, w - 2],
[13, 13, w^2 - 2*w - 2],
[17, 17, -w^2 + 2],
[19, 19, -w^2 + w + 1],
[31, 31, -w + 4],
[41, 41, -w^2 + 2*w - 2],
[41, 41, -2*w^2 - 3*w + 4],
[41, 41, 2*w + 1],
[43, 43, -w^2 - w + 5],
[47, 47, -w^2 + 8],
[47, 47, 2*w^2 - w - 8],
[53, 53, w^2 + w - 7],
[59, 59, w^2 - 2*w - 4],
[61, 61, -3*w^2 + 14],
[61, 61, 4*w^2 - 2*w - 19],
[61, 61, -2*w^2 + 7],
[67, 67, -2*w^2 - w + 8],
[71, 71, w^2 + 2*w - 4],
[73, 73, -2*w^2 + 2*w + 11],
[83, 83, -3*w + 4],
[89, 89, -3*w^2 - 3*w + 5],
[101, 101, w^2 + 2*w + 2],
[109, 109, 4*w - 1],
[113, 113, 2*w^2 - 2*w - 5],
[113, 113, -w^2 - 2*w + 10],
[113, 113, -5*w^2 + w + 25],
[125, 5, -5],
[127, 127, -2*w^2 - 4*w - 1],
[131, 131, -3*w^2 + 16],
[137, 137, -3*w^2 + 3*w + 13],
[137, 137, 2*w^2 - w - 2],
[137, 137, 2*w^2 + 2*w - 7],
[139, 139, w^2 - 2*w - 10],
[149, 149, -3*w - 4],
[151, 151, -3*w - 2],
[163, 163, 2*w^2 - w - 4],
[169, 13, w^2 + 4*w + 2],
[173, 173, 2*w^2 - w - 14],
[181, 181, w^2 + 3*w - 5],
[191, 191, w^2 + w - 11],
[193, 193, 2*w^2 - 4*w - 5],
[197, 197, 3*w^2 + w - 13],
[199, 199, 6*w^2 - 3*w - 34],
[211, 211, -w^2 + 2*w - 4],
[223, 223, w^2 - 3*w - 5],
[229, 229, -3*w^2 + 3*w + 17],
[239, 239, 3*w^2 - w - 11],
[251, 251, 4*w^2 - 2*w - 17],
[263, 263, 3*w + 8],
[269, 269, -2*w^2 + 3*w + 10],
[277, 277, -2*w - 7],
[277, 277, 4*w^2 - 23],
[277, 277, 7*w^2 - 4*w - 40],
[289, 17, 3*w^2 - 2*w - 10],
[313, 313, 3*w^2 + w - 7],
[331, 331, -3*w^2 + 3*w + 1],
[331, 331, -3*w^2 + 4*w + 20],
[331, 331, -w^2 - 3*w + 11],
[343, 7, -7],
[347, 347, -3*w^2 - w + 19],
[349, 349, 4*w^2 + 4*w - 11],
[353, 353, w^2 - 5*w - 1],
[361, 19, 4*w^2 - w - 16],
[367, 367, -4*w - 5],
[373, 373, -w^2 - 4],
[383, 383, 2*w^2 - 4*w - 7],
[389, 389, -5*w^2 + 4*w + 28],
[389, 389, 3*w^2 - 8],
[389, 389, -9*w^2 + 4*w + 50],
[397, 397, -8*w^2 + w + 38],
[397, 397, 6*w^2 - w - 32],
[397, 397, 2*w^2 + 2*w - 11],
[401, 401, w^2 - 4*w - 4],
[409, 409, 5*w^2 - w - 31],
[421, 421, 3*w^2 - 3*w - 7],
[431, 431, -2*w^2 - 3*w - 2],
[433, 433, 5*w^2 + 2*w - 16],
[443, 443, w^2 - 4*w - 10],
[443, 443, 2*w^2 + 2*w - 13],
[443, 443, -3*w^2 + 4*w + 2],
[463, 463, 6*w - 1],
[479, 479, -w^2 + w - 5],
[487, 487, 5*w^2 - 4*w - 26],
[487, 487, -w^2 - 4*w + 14],
[487, 487, 2*w^2 - w - 16],
[491, 491, -4*w^2 - 5*w + 10],
[499, 499, 2*w^2 - 8*w + 1],
[509, 509, -4*w^2 + 3*w + 26],
[509, 509, -6*w^2 + w + 26],
[509, 509, 4*w^2 - 4*w - 19],
[521, 521, 4*w^2 - 3*w - 14],
[541, 541, w^2 + 4*w - 8],
[541, 541, w^2 + w - 13],
[541, 541, 3*w^2 + 3*w - 11],
[547, 547, 3*w^2 - 2*w - 4],
[563, 563, 2*w^2 - 4*w + 5],
[563, 563, -3*w^2 + 2*w - 2],
[563, 563, -6*w + 7],
[571, 571, -3*w^2 - 5*w - 1],
[571, 571, -w^2 + 3*w - 7],
[571, 571, 3*w^2 - 22],
[577, 577, 3*w^2 - 4*w - 14],
[587, 587, -3*w^2 - 6*w - 2],
[587, 587, 2*w^2 + 3*w - 10],
[587, 587, -7*w^2 + 2*w + 32],
[593, 593, -w^2 - w - 5],
[593, 593, 5*w^2 - 28],
[593, 593, w^2 + 5*w - 7],
[607, 607, 3*w^2 + 2*w - 14],
[617, 617, -2*w^2 + 4*w + 13],
[631, 631, w^2 - 8*w + 8],
[643, 643, -9*w^2 + 3*w + 43],
[643, 643, -2*w^2 + 4*w + 11],
[643, 643, -5*w^2 + 3*w + 31],
[647, 647, -4*w^2 + 7*w + 10],
[653, 653, 6*w^2 - 5*w - 28],
[659, 659, w^2 + 4*w - 10],
[661, 661, -8*w^2 + 3*w + 46],
[661, 661, 4*w^2 + w - 20],
[661, 661, 2*w^2 - 4*w - 19],
[673, 673, -4*w + 13],
[677, 677, -5*w^2 - w + 17],
[677, 677, -3*w^2 - 4*w + 10],
[677, 677, 2*w^2 - 2*w - 17],
[683, 683, 4*w^2 - 5*w - 16],
[691, 691, 5*w^2 + 2*w - 20],
[719, 719, 6*w^2 - 3*w - 26],
[727, 727, -6*w^2 - w + 22],
[743, 743, -7*w^2 + w + 37],
[743, 743, 7*w^2 - 3*w - 41],
[743, 743, 4*w + 11],
[751, 751, 2*w^2 + 5*w - 8],
[757, 757, 3*w^2 - 6*w - 8],
[761, 761, 5*w^2 - 2*w - 20],
[769, 769, 4*w^2 + w - 10],
[769, 769, -5*w^2 - w + 23],
[769, 769, -3*w - 10],
[773, 773, 2*w^2 + 3*w - 16],
[787, 787, 5*w^2 + 7*w - 11],
[821, 821, -6*w - 1],
[821, 821, -11*w^2 + 7*w + 61],
[821, 821, -w^2 - 6*w - 4],
[827, 827, -4*w^2 - 4*w + 13],
[829, 829, w^2 - 5*w - 5],
[853, 853, -2*w^2 + w - 4],
[853, 853, -3*w^2 + 2*w + 22],
[853, 853, w - 10],
[859, 859, 6*w^2 - 35],
[863, 863, 3*w^2 + 3*w - 13],
[877, 877, 5*w^2 - w - 19],
[883, 883, 5*w^2 - 5*w - 23],
[883, 883, 3*w^2 - 4*w + 4],
[883, 883, w^2 - 5*w - 11],
[887, 887, 4*w^2 - 11],
[887, 887, -6*w^2 + 6*w + 37],
[887, 887, 4*w^2 + w - 8],
[911, 911, 5*w^2 + 2*w - 14],
[919, 919, w^2 + 6*w - 8],
[929, 929, 6*w^2 - w - 38],
[937, 937, 12*w^2 - 6*w - 67],
[941, 941, 7*w^2 - 5*w - 29],
[947, 947, -8*w^2 + w + 40],
[961, 31, w^2 + 3*w + 7],
[977, 977, 6*w^2 - 23],
[983, 983, -4*w^2 + 5*w + 2],
[983, 983, w^2 - 5*w - 7],
[983, 983, 6*w^2 - 31],
[997, 997, -6*w^2 + 2*w + 37]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^10 - x^9 - 14*x^8 + 12*x^7 + 67*x^6 - 44*x^5 - 126*x^4 + 50*x^3 + 80*x^2 - 20*x - 12;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, e^4 - 6*e^2 + 4, -1/2*e^9 + 13/2*e^7 - 55/2*e^5 - 1/2*e^4 + 41*e^3 + 3*e^2 - 16*e - 2, -e^7 + 11*e^5 + e^4 - 34*e^3 - 6*e^2 + 22*e + 2, -1/2*e^9 - e^8 + 13/2*e^7 + 12*e^6 - 57/2*e^5 - 89/2*e^4 + 48*e^3 + 51*e^2 - 26*e - 12, 1, -1/2*e^9 + 15/2*e^7 + e^6 - 75/2*e^5 - 21/2*e^4 + 68*e^3 + 29*e^2 - 28*e - 10, 1/2*e^9 - 1/2*e^8 - 6*e^7 + 6*e^6 + 43/2*e^5 - 22*e^4 - 22*e^3 + 23*e^2 + 8*e, e^8 - e^7 - 12*e^6 + 12*e^5 + 46*e^4 - 40*e^3 - 60*e^2 + 26*e + 18, -1/2*e^9 + 13/2*e^7 + e^6 - 55/2*e^5 - 21/2*e^4 + 39*e^3 + 29*e^2 - 6*e - 12, e^8 + 2*e^7 - 10*e^6 - 22*e^5 + 24*e^4 + 66*e^3 + 4*e^2 - 32*e - 4, 1/2*e^9 + 1/2*e^8 - 6*e^7 - 5*e^6 + 45/2*e^5 + 13*e^4 - 28*e^3 - e^2 + 12*e - 6, e^9 - e^8 - 14*e^7 + 10*e^6 + 66*e^5 - 23*e^4 - 114*e^3 - 10*e^2 + 42*e + 12, -e^9 + e^8 + 12*e^7 - 12*e^6 - 44*e^5 + 42*e^4 + 46*e^3 - 36*e^2 - 2*e + 6, -3/2*e^9 + 41/2*e^7 + e^6 - 187/2*e^5 - 29/2*e^4 + 156*e^3 + 51*e^2 - 64*e - 18, e^9 - 14*e^7 - 2*e^6 + 65*e^5 + 21*e^4 - 110*e^3 - 60*e^2 + 48*e + 26, -1/2*e^9 - e^8 + 13/2*e^7 + 13*e^6 - 59/2*e^5 - 111/2*e^4 + 54*e^3 + 83*e^2 - 30*e - 28, 2*e^9 - 26*e^7 - e^6 + 111*e^5 + 13*e^4 - 170*e^3 - 42*e^2 + 70*e + 14, e^5 - 2*e^4 - 8*e^3 + 10*e^2 + 12*e - 4, 1/2*e^9 - 1/2*e^8 - 5*e^7 + 6*e^6 + 23/2*e^5 - 21*e^4 + 8*e^3 + 17*e^2 - 20*e + 6, e^7 + 2*e^6 - 11*e^5 - 20*e^4 + 34*e^3 + 54*e^2 - 22*e - 22, -1/2*e^9 + 2*e^8 + 15/2*e^7 - 23*e^6 - 73/2*e^5 + 157/2*e^4 + 61*e^3 - 73*e^2 - 20*e + 18, -2*e^9 + e^8 + 26*e^7 - 11*e^6 - 111*e^5 + 32*e^4 + 170*e^3 - 10*e^2 - 64*e - 6, e^8 - 11*e^6 - 2*e^5 + 35*e^4 + 12*e^3 - 30*e^2 - 8*e + 18, -1/2*e^9 + 11/2*e^7 - 2*e^6 - 35/2*e^5 + 39/2*e^4 + 15*e^3 - 51*e^2 - 6*e + 20, 2*e^9 - e^8 - 26*e^7 + 11*e^6 + 111*e^5 - 33*e^4 - 168*e^3 + 18*e^2 + 58*e - 6, -e^9 - e^8 + 12*e^7 + 12*e^6 - 45*e^5 - 45*e^4 + 56*e^3 + 52*e^2 - 26*e - 6, 1/2*e^9 - e^8 - 15/2*e^7 + 10*e^6 + 73/2*e^5 - 45/2*e^4 - 59*e^3 - 13*e^2 + 10*e + 12, 3/2*e^9 + 3/2*e^8 - 20*e^7 - 19*e^6 + 181/2*e^5 + 80*e^4 - 156*e^3 - 119*e^2 + 72*e + 36, 1/2*e^9 - 1/2*e^8 - 4*e^7 + 7*e^6 + 7/2*e^5 - 31*e^4 + 20*e^3 + 45*e^2 - 8*e - 10, -e^9 + e^8 + 12*e^7 - 11*e^6 - 45*e^5 + 34*e^4 + 56*e^3 - 18*e^2 - 24*e - 12, -e^8 - 2*e^7 + 10*e^6 + 23*e^5 - 26*e^4 - 74*e^3 + 6*e^2 + 44*e + 6, 2*e^9 - e^8 - 27*e^7 + 10*e^6 + 123*e^5 - 23*e^4 - 212*e^3 - 8*e^2 + 104*e + 6, 3/2*e^9 - 1/2*e^8 - 19*e^7 + 6*e^6 + 157/2*e^5 - 22*e^4 - 116*e^3 + 25*e^2 + 52*e, 3/2*e^9 - 5/2*e^8 - 19*e^7 + 29*e^6 + 153/2*e^5 - 99*e^4 - 102*e^3 + 91*e^2 + 24*e - 22, e^9 - 2*e^8 - 13*e^7 + 23*e^6 + 53*e^5 - 77*e^4 - 70*e^3 + 64*e^2 + 22*e - 6, 5/2*e^9 + 1/2*e^8 - 31*e^7 - 6*e^6 + 245/2*e^5 + 25*e^4 - 162*e^3 - 41*e^2 + 52*e + 14, -2*e^9 + e^8 + 28*e^7 - 10*e^6 - 131*e^5 + 17*e^4 + 220*e^3 + 48*e^2 - 76*e - 40, 1/2*e^9 + e^8 - 21/2*e^7 - 16*e^6 + 141/2*e^5 + 165/2*e^4 - 170*e^3 - 145*e^2 + 90*e + 44, -e^9 + 2*e^8 + 16*e^7 - 19*e^6 - 87*e^5 + 37*e^4 + 174*e^3 + 46*e^2 - 66*e - 42, 1/2*e^9 - e^8 - 13/2*e^7 + 11*e^6 + 57/2*e^5 - 65/2*e^4 - 45*e^3 + 13*e^2 + 10*e + 8, -3/2*e^9 - e^8 + 33/2*e^7 + 10*e^6 - 99/2*e^5 - 53/2*e^4 + 19*e^3 + 11*e^2 + 24*e - 6, 3/2*e^9 - 33/2*e^7 + 2*e^6 + 101/2*e^5 - 39/2*e^4 - 26*e^3 + 49*e^2 - 14*e - 16, 1/2*e^9 + e^8 - 19/2*e^7 - 14*e^6 + 121/2*e^5 + 135/2*e^4 - 141*e^3 - 125*e^2 + 74*e + 48, -1/2*e^9 + 3*e^8 + 17/2*e^7 - 33*e^6 - 97/2*e^5 + 201/2*e^4 + 98*e^3 - 57*e^2 - 24*e + 2, e^9 + e^8 - 14*e^7 - 11*e^6 + 66*e^5 + 41*e^4 - 112*e^3 - 62*e^2 + 34*e + 20, -e^9 + e^8 + 10*e^7 - 12*e^6 - 22*e^5 + 45*e^4 - 20*e^3 - 54*e^2 + 36*e + 8, -3*e^8 - 6*e^7 + 33*e^6 + 62*e^5 - 99*e^4 - 178*e^3 + 46*e^2 + 100*e + 2, 3*e^9 + 2*e^8 - 39*e^7 - 26*e^6 + 167*e^5 + 113*e^4 - 262*e^3 - 176*e^2 + 120*e + 60, 2*e^9 - e^8 - 27*e^7 + 10*e^6 + 122*e^5 - 22*e^4 - 206*e^3 - 16*e^2 + 92*e + 12, -3/2*e^9 + 3*e^8 + 49/2*e^7 - 31*e^6 - 269/2*e^5 + 161/2*e^4 + 268*e^3 - 3*e^2 - 104*e - 18, -3*e^8 - 4*e^7 + 31*e^6 + 42*e^5 - 86*e^4 - 122*e^3 + 32*e^2 + 60*e + 18, -2*e^8 - 2*e^7 + 25*e^6 + 20*e^5 - 94*e^4 - 52*e^3 + 100*e^2 + 16*e - 22, -e^9 + 2*e^8 + 16*e^7 - 21*e^6 - 88*e^5 + 59*e^4 + 180*e^3 - 22*e^2 - 76*e + 2, -3/2*e^9 - 7/2*e^8 + 18*e^7 + 38*e^6 - 133/2*e^5 - 118*e^4 + 76*e^3 + 87*e^2 - 28*e - 16, -1/2*e^9 + 4*e^8 + 17/2*e^7 - 45*e^6 - 95/2*e^5 + 281/2*e^4 + 90*e^3 - 77*e^2 - 18*e - 16, -e^8 + 4*e^7 + 15*e^6 - 42*e^5 - 78*e^4 + 124*e^3 + 152*e^2 - 80*e - 58, 2*e^9 + e^8 - 23*e^7 - 8*e^6 + 77*e^5 + 10*e^4 - 66*e^3 + 26*e^2 + 20*e - 16, 3/2*e^9 - e^8 - 31/2*e^7 + 13*e^6 + 75/2*e^5 - 103/2*e^4 + 19*e^3 + 53*e^2 - 44*e + 14, -1/2*e^9 - 3*e^8 + 15/2*e^7 + 36*e^6 - 79/2*e^5 - 273/2*e^4 + 83*e^3 + 169*e^2 - 48*e - 46, -3/2*e^9 + 1/2*e^8 + 14*e^7 - 8*e^6 - 43/2*e^5 + 41*e^4 - 64*e^3 - 69*e^2 + 60*e + 2, -e^9 + 2*e^8 + 15*e^7 - 23*e^6 - 73*e^5 + 74*e^4 + 124*e^3 - 46*e^2 - 52*e, 3*e^8 + e^7 - 35*e^6 - 7*e^5 + 117*e^4 + 6*e^3 - 88*e^2 + 16*e + 14, -3*e^9 + 5*e^8 + 41*e^7 - 57*e^6 - 185*e^5 + 187*e^4 + 300*e^3 - 140*e^2 - 116*e + 18, -e^9 - e^8 + 9*e^7 + 7*e^6 - 12*e^5 + 3*e^4 - 48*e^3 - 70*e^2 + 48*e + 38, 3/2*e^9 - e^8 - 41/2*e^7 + 7*e^6 + 189/2*e^5 + 3/2*e^4 - 164*e^3 - 55*e^2 + 76*e + 14, -e^9 + 16*e^7 + 4*e^6 - 89*e^5 - 45*e^4 + 192*e^3 + 136*e^2 - 102*e - 58, e^9 - 3*e^8 - 14*e^7 + 35*e^6 + 64*e^5 - 112*e^4 - 98*e^3 + 60*e^2 + 16*e + 12, 1/2*e^9 - 2*e^8 - 21/2*e^7 + 21*e^6 + 143/2*e^5 - 115/2*e^4 - 173*e^3 + 9*e^2 + 86*e + 24, 1/2*e^9 + e^8 - 17/2*e^7 - 13*e^6 + 93/2*e^5 + 101/2*e^4 - 88*e^3 - 55*e^2 + 34*e, -1/2*e^9 - 1/2*e^8 + 11*e^7 + 8*e^6 - 151/2*e^5 - 46*e^4 + 182*e^3 + 101*e^2 - 84*e - 24, 1/2*e^9 - 4*e^8 - 9/2*e^7 + 46*e^6 + 11/2*e^5 - 319/2*e^4 + 27*e^3 + 155*e^2 - 30*e - 16, e^9 - 11*e^7 + 32*e^5 - 4*e^3 + 4*e^2 - 38*e - 10, -e^9 + 12*e^7 - e^6 - 45*e^5 + 12*e^4 + 62*e^3 - 42*e^2 - 52*e + 14, 9/2*e^9 - 5*e^8 - 119/2*e^7 + 56*e^6 + 515/2*e^5 - 347/2*e^4 - 395*e^3 + 101*e^2 + 146*e, e^9 - 3*e^8 - 13*e^7 + 31*e^6 + 53*e^5 - 83*e^4 - 62*e^3 + 14*e^2 - 22*e + 26, 2*e^9 + 2*e^8 - 21*e^7 - 21*e^6 + 56*e^5 + 60*e^4 + 6*e^3 - 28*e^2 - 64*e + 2, 5/2*e^9 + 2*e^8 - 61/2*e^7 - 24*e^6 + 235/2*e^5 + 169/2*e^4 - 152*e^3 - 79*e^2 + 60*e + 18, 3/2*e^9 - 3*e^8 - 41/2*e^7 + 34*e^6 + 185/2*e^5 - 221/2*e^4 - 151*e^3 + 79*e^2 + 62*e - 4, -e^9 + e^8 + 16*e^7 - 8*e^6 - 85*e^5 + e^4 + 160*e^3 + 78*e^2 - 52*e - 48, 2*e^9 + e^8 - 23*e^7 - 9*e^6 + 78*e^5 + 14*e^4 - 70*e^3 + 36*e^2 + 12*e - 24, 2*e^9 + e^8 - 27*e^7 - 14*e^6 + 120*e^5 + 69*e^4 - 196*e^3 - 132*e^2 + 96*e + 48, 3/2*e^9 - 3*e^8 - 43/2*e^7 + 31*e^6 + 205/2*e^5 - 177/2*e^4 - 181*e^3 + 49*e^2 + 88*e + 2, 2*e^9 + 2*e^8 - 21*e^7 - 22*e^6 + 58*e^5 + 71*e^4 - 12*e^3 - 64*e^2 - 20*e + 36, e^9 + e^8 - 11*e^7 - 11*e^6 + 35*e^5 + 35*e^4 - 36*e^3 - 26*e^2 + 40*e + 8, -7/2*e^9 + 5/2*e^8 + 45*e^7 - 27*e^6 - 373/2*e^5 + 73*e^4 + 262*e^3 + e^2 - 76*e - 22, 3/2*e^9 - 45/2*e^7 - 2*e^6 + 225/2*e^5 + 43/2*e^4 - 207*e^3 - 63*e^2 + 96*e + 38, -1/2*e^9 - e^8 + 17/2*e^7 + 14*e^6 - 101/2*e^5 - 131/2*e^4 + 112*e^3 + 113*e^2 - 44*e - 42, -4*e^9 + e^8 + 51*e^7 - 12*e^6 - 209*e^5 + 42*e^4 + 294*e^3 - 34*e^2 - 108*e - 4, -e^9 + 3*e^8 + 19*e^7 - 29*e^6 - 120*e^5 + 65*e^4 + 276*e^3 + 24*e^2 - 130*e - 18, 9/2*e^9 - 1/2*e^8 - 58*e^7 + 5*e^6 + 487/2*e^5 - 8*e^4 - 356*e^3 - 25*e^2 + 120*e + 24, 3/2*e^9 + e^8 - 37/2*e^7 - 10*e^6 + 147/2*e^5 + 67/2*e^4 - 104*e^3 - 47*e^2 + 46*e + 12, -3*e^9 - e^8 + 34*e^7 + 10*e^6 - 113*e^5 - 25*e^4 + 96*e^3 - 8*e^2 - 14*e + 42, 11/2*e^9 + 1/2*e^8 - 71*e^7 - 8*e^6 + 607/2*e^5 + 50*e^4 - 472*e^3 - 117*e^2 + 188*e + 32, -3*e^9 + e^8 + 38*e^7 - 12*e^6 - 156*e^5 + 42*e^4 + 224*e^3 - 36*e^2 - 86*e - 10, 5*e^9 + e^8 - 63*e^7 - 12*e^6 + 257*e^5 + 54*e^4 - 368*e^3 - 98*e^2 + 146*e + 14, 7/2*e^9 - 95/2*e^7 - 5*e^6 + 431/2*e^5 + 105/2*e^4 - 363*e^3 - 145*e^2 + 160*e + 50, -e^9 + 2*e^8 + 11*e^7 - 20*e^6 - 33*e^5 + 47*e^4 + 6*e^3 + 12*e^2 + 52*e - 24, -5*e^9 - 3*e^8 + 67*e^7 + 41*e^6 - 300*e^5 - 185*e^4 + 500*e^3 + 294*e^2 - 234*e - 84, -5/2*e^9 - 5/2*e^8 + 37*e^7 + 30*e^6 - 375/2*e^5 - 119*e^4 + 362*e^3 + 171*e^2 - 180*e - 30, 3/2*e^9 - 4*e^8 - 45/2*e^7 + 42*e^6 + 229/2*e^5 - 233/2*e^4 - 214*e^3 + 31*e^2 + 80*e + 26, e^9 + e^8 - 15*e^7 - 15*e^6 + 76*e^5 + 75*e^4 - 140*e^3 - 124*e^2 + 52*e + 20, 4*e^9 - 3*e^8 - 54*e^7 + 31*e^6 + 242*e^5 - 70*e^4 - 388*e^3 - 56*e^2 + 124*e + 56, -2*e^9 - 3*e^8 + 28*e^7 + 36*e^6 - 139*e^5 - 132*e^4 + 282*e^3 + 138*e^2 - 178*e - 22, 3/2*e^9 + 3/2*e^8 - 14*e^7 - 13*e^6 + 55/2*e^5 + 13*e^4 + 28*e^3 + 85*e^2 - 32*e - 54, 5/2*e^9 - e^8 - 69/2*e^7 + 6*e^6 + 325/2*e^5 + 27/2*e^4 - 290*e^3 - 99*e^2 + 124*e + 54, 1/2*e^9 - 2*e^8 - 13/2*e^7 + 20*e^6 + 47/2*e^5 - 99/2*e^4 - 13*e^3 - 5*e^2 - 24*e + 30, e^9 - 12*e^7 - e^6 + 44*e^5 + 6*e^4 - 50*e^3 - 8*e^2 + 22*e + 6, 2*e^9 + 2*e^8 - 29*e^7 - 26*e^6 + 140*e^5 + 114*e^4 - 244*e^3 - 188*e^2 + 92*e + 66, 3/2*e^9 + e^8 - 41/2*e^7 - 14*e^6 + 187/2*e^5 + 147/2*e^4 - 154*e^3 - 155*e^2 + 62*e + 48, -e^9 + 18*e^7 + 7*e^6 - 108*e^5 - 69*e^4 + 232*e^3 + 178*e^2 - 94*e - 64, 1/2*e^9 - 2*e^8 - 15/2*e^7 + 24*e^6 + 77/2*e^5 - 183/2*e^4 - 78*e^3 + 115*e^2 + 62*e - 36, -3*e^8 + 33*e^6 - 6*e^5 - 99*e^4 + 48*e^3 + 50*e^2 - 86*e + 8, -e^9 + e^8 + 22*e^7 - 4*e^6 - 153*e^5 - 39*e^4 + 376*e^3 + 176*e^2 - 184*e - 64, -3*e^9 - e^8 + 37*e^7 + 9*e^6 - 141*e^5 - 24*e^4 + 172*e^3 + 34*e^2 - 72*e - 40, -1/2*e^9 + 7/2*e^7 - 3*e^6 + 13/2*e^5 + 55/2*e^4 - 59*e^3 - 63*e^2 + 12*e + 14, -2*e^9 - 2*e^8 + 27*e^7 + 25*e^6 - 123*e^5 - 103*e^4 + 204*e^3 + 148*e^2 - 68*e - 48, -5*e^7 - 2*e^6 + 47*e^5 + 23*e^4 - 116*e^3 - 68*e^2 + 42*e + 6, -e^8 - 3*e^7 + 9*e^6 + 34*e^5 - 14*e^4 - 106*e^3 - 32*e^2 + 56*e + 24, e^9 + 5*e^8 - 18*e^7 - 64*e^6 + 107*e^5 + 263*e^4 - 234*e^3 - 362*e^2 + 122*e + 98, -5/2*e^9 + 3*e^8 + 67/2*e^7 - 33*e^6 - 301/2*e^5 + 201/2*e^4 + 253*e^3 - 59*e^2 - 118*e + 8, -4*e^9 - e^8 + 57*e^7 + 18*e^6 - 277*e^5 - 115*e^4 + 504*e^3 + 272*e^2 - 218*e - 106, -2*e^8 + 7*e^7 + 29*e^6 - 78*e^5 - 143*e^4 + 240*e^3 + 258*e^2 - 144*e - 82, e^9 - 4*e^8 - 14*e^7 + 49*e^6 + 66*e^5 - 185*e^4 - 118*e^3 + 214*e^2 + 52*e - 54, -e^9 - e^8 + 15*e^7 + 12*e^6 - 78*e^5 - 51*e^4 + 160*e^3 + 82*e^2 - 104*e - 18, 2*e^7 - 3*e^6 - 23*e^5 + 24*e^4 + 74*e^3 - 50*e^2 - 48*e + 30, 3*e^9 - 5*e^8 - 42*e^7 + 53*e^6 + 201*e^5 - 145*e^4 - 358*e^3 + 32*e^2 + 128*e + 12, e^9 - 4*e^8 - 11*e^7 + 48*e^6 + 33*e^5 - 176*e^4 - 10*e^3 + 198*e^2 - 36*e - 64, -2*e^9 + 4*e^8 + 30*e^7 - 45*e^6 - 152*e^5 + 140*e^4 + 286*e^3 - 80*e^2 - 136*e + 12, e^8 + e^7 - 9*e^6 - 7*e^5 + 13*e^4 + 12*e^3 + 40*e^2 - 8*e - 40, -7/2*e^9 - 2*e^8 + 93/2*e^7 + 26*e^6 - 415/2*e^5 - 237/2*e^4 + 349*e^3 + 203*e^2 - 172*e - 54, -1/2*e^9 + e^8 + 3/2*e^7 - 17*e^6 + 61/2*e^5 + 171/2*e^4 - 149*e^3 - 135*e^2 + 108*e + 42, -3*e^9 - 4*e^8 + 35*e^7 + 41*e^6 - 123*e^5 - 110*e^4 + 132*e^3 + 26*e^2 - 72*e + 24, -e^9 - 2*e^8 + 10*e^7 + 21*e^6 - 21*e^5 - 66*e^4 - 30*e^3 + 66*e^2 + 44*e - 28, 3/2*e^9 - 37/2*e^7 + e^6 + 147/2*e^5 - 15/2*e^4 - 97*e^3 + 7*e^2 + 18*e + 20, -2*e^9 + e^8 + 27*e^7 - 8*e^6 - 124*e^5 + 4*e^4 + 224*e^3 + 56*e^2 - 138*e - 30, -2*e^9 + 31*e^7 + 6*e^6 - 166*e^5 - 61*e^4 + 340*e^3 + 162*e^2 - 160*e - 46, -1/2*e^9 + 4*e^8 + 15/2*e^7 - 47*e^6 - 75/2*e^5 + 329/2*e^4 + 64*e^3 - 159*e^2 - 10*e + 44, -2*e^9 + 3*e^8 + 20*e^7 - 36*e^6 - 50*e^5 + 137*e^4 - 172*e^2 + 22*e + 38, 5*e^9 + 4*e^8 - 62*e^7 - 44*e^6 + 245*e^5 + 145*e^4 - 328*e^3 - 142*e^2 + 122*e + 66, 3/2*e^9 - 3*e^8 - 45/2*e^7 + 31*e^6 + 231/2*e^5 - 155/2*e^4 - 216*e^3 - 17*e^2 + 76*e + 38, 1/2*e^9 - e^8 - 15/2*e^7 + 10*e^6 + 71/2*e^5 - 35/2*e^4 - 49*e^3 - 49*e^2 - 18*e + 48, -1/2*e^9 + 1/2*e^8 + 6*e^7 - 12*e^6 - 43/2*e^5 + 76*e^4 + 24*e^3 - 147*e^2 - 16*e + 60, -1/2*e^9 + 5*e^8 + 29/2*e^7 - 55*e^6 - 221/2*e^5 + 323/2*e^4 + 282*e^3 - 57*e^2 - 154*e - 12, 11/2*e^9 - 1/2*e^8 - 74*e^7 + 669/2*e^5 + 47*e^4 - 558*e^3 - 189*e^2 + 228*e + 90, -4*e^9 + 2*e^8 + 49*e^7 - 20*e^6 - 188*e^5 + 52*e^4 + 228*e^3 - 12*e^2 - 44*e - 10, -3/2*e^9 + e^8 + 43/2*e^7 - 7*e^6 - 207/2*e^5 - 3/2*e^4 + 188*e^3 + 63*e^2 - 102*e - 28, -2*e^9 + 4*e^8 + 24*e^7 - 48*e^6 - 87*e^5 + 173*e^4 + 88*e^3 - 176*e^2 - 10*e + 38, -3/2*e^9 - 3*e^8 + 37/2*e^7 + 37*e^6 - 147/2*e^5 - 283/2*e^4 + 102*e^3 + 161*e^2 - 22*e - 16, -3*e^9 + 2*e^8 + 44*e^7 - 18*e^6 - 222*e^5 + 23*e^4 + 430*e^3 + 90*e^2 - 222*e - 40, e^8 - e^7 - 11*e^6 + 7*e^5 + 33*e^4 - 8*e^3 - 22*e^2 + 4*e + 24, -2*e^9 + e^8 + 28*e^7 - 6*e^6 - 128*e^5 - 19*e^4 + 202*e^3 + 130*e^2 - 60*e - 58, e^9 + 3*e^8 - 18*e^7 - 37*e^6 + 114*e^5 + 146*e^4 - 280*e^3 - 192*e^2 + 176*e + 32, 11/2*e^9 - 3/2*e^8 - 70*e^7 + 15*e^6 + 575/2*e^5 - 25*e^4 - 410*e^3 - 67*e^2 + 160*e + 62, 2*e^9 + 2*e^8 - 29*e^7 - 28*e^6 + 146*e^5 + 126*e^4 - 284*e^3 - 192*e^2 + 136*e + 44, -5/2*e^9 + 3*e^8 + 65/2*e^7 - 34*e^6 - 275/2*e^5 + 231/2*e^4 + 208*e^3 - 117*e^2 - 92*e + 30, -5/2*e^9 + 5*e^8 + 71/2*e^7 - 54*e^6 - 347/2*e^5 + 323/2*e^4 + 327*e^3 - 89*e^2 - 164*e - 6, -3*e^9 + 2*e^8 + 41*e^7 - 22*e^6 - 188*e^5 + 62*e^4 + 320*e^3 - 8*e^2 - 148*e, -5*e^9 + 62*e^7 + 2*e^6 - 246*e^5 - 21*e^4 + 338*e^3 + 56*e^2 - 140*e - 36, 3*e^9 - 3*e^8 - 40*e^7 + 31*e^6 + 172*e^5 - 79*e^4 - 240*e^3 - 14*e^2 + 28*e + 44, 5/2*e^9 + 2*e^8 - 53/2*e^7 - 15*e^6 + 147/2*e^5 + 9/2*e^4 - 15*e^3 + 107*e^2 - 22*e - 48, -e^9 + 2*e^8 + 12*e^7 - 19*e^6 - 43*e^5 + 41*e^4 + 34*e^3 + 20*e^2 + 44*e - 22, e^8 - 8*e^7 - 20*e^6 + 89*e^5 + 123*e^4 - 274*e^3 - 252*e^2 + 148*e + 66, e^8 + e^7 - 13*e^6 - 7*e^5 + 50*e^4 - 50*e^2 + 52*e, 3*e^9 - 4*e^8 - 41*e^7 + 45*e^6 + 185*e^5 - 144*e^4 - 290*e^3 + 106*e^2 + 78*e - 10, 5*e^9 - 3*e^8 - 71*e^7 + 31*e^6 + 339*e^5 - 79*e^4 - 602*e^3 + 2*e^2 + 278*e + 18, 1/2*e^9 - 3/2*e^8 - 6*e^7 + 23*e^6 + 41/2*e^5 - 107*e^4 - 8*e^3 + 159*e^2 - 32*e - 54, -e^9 + 14*e^7 + 2*e^6 - 68*e^5 - 25*e^4 + 126*e^3 + 78*e^2 - 46*e - 12, -2*e^8 + e^7 + 23*e^6 - 9*e^5 - 74*e^4 + 22*e^3 + 50*e^2 - 20*e, -5/2*e^9 + 1/2*e^8 + 32*e^7 - 5*e^6 - 273/2*e^5 + 216*e^3 + 67*e^2 - 80*e - 40];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {19, 19, -w^2 + 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;