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

NN := ideal<ZF | {11, 11, -w^2 + 2*w + 2}>;

primesArray := [
[3, 3, w + 1],
[7, 7, w - 1],
[8, 2, 2],
[9, 3, -w^2 + 2*w + 4],
[11, 11, -w^2 + 2*w + 2],
[13, 13, -w^2 + w + 4],
[19, 19, w + 3],
[19, 19, -w^2 + 2*w + 5],
[19, 19, -w^2 + 3*w + 2],
[23, 23, w^2 - w - 3],
[23, 23, -w^2 + 2],
[23, 23, -w + 4],
[31, 31, w^2 - 5],
[43, 43, w^2 - 3*w - 3],
[49, 7, w^2 - 6],
[53, 53, 2*w - 5],
[61, 61, 2*w^2 - 2*w - 9],
[71, 71, 2*w - 3],
[73, 73, 2*w^2 - 5*w - 5],
[83, 83, w^2 - w - 9],
[97, 97, 2*w^2 - 3*w - 7],
[103, 103, w^2 - 3*w - 6],
[109, 109, w^2 - 4*w - 3],
[121, 11, 2*w^2 - w - 7],
[125, 5, -5],
[127, 127, w^2 + w - 5],
[131, 131, w^2 - 11],
[137, 137, 3*w^2 - 7*w - 8],
[137, 137, 2*w^2 - 5*w - 6],
[137, 137, -w^2 + w - 2],
[139, 139, 3*w^2 - 2*w - 16],
[139, 139, 3*w - 2],
[139, 139, -w^2 + 3*w - 3],
[151, 151, w^2 + w - 11],
[157, 157, 2*w^2 - 3*w - 3],
[163, 163, w^2 - w - 10],
[169, 13, 2*w^2 - 3*w - 4],
[173, 173, -3*w^2 + 6*w + 8],
[191, 191, -4*w^2 + 7*w + 15],
[193, 193, 2*w^2 - 3],
[197, 197, w^2 - 3*w - 11],
[199, 199, 3*w^2 - 5*w - 16],
[211, 211, w^2 - 4*w - 9],
[223, 223, 2*w^2 - w - 4],
[227, 227, 3*w - 4],
[229, 229, w^2 + w - 8],
[241, 241, w^2 - 4*w - 6],
[251, 251, w - 7],
[257, 257, w^2 - 4*w - 7],
[263, 263, 3*w^2 - 5*w - 10],
[263, 263, 2*w^2 - 5*w - 15],
[263, 263, -w^2 + 6*w - 1],
[269, 269, -w^2 - w + 14],
[277, 277, 2*w^2 - 5*w - 8],
[277, 277, w^2 + 3*w - 3],
[277, 277, 3*w^2 - 4*w - 12],
[281, 281, -2*w - 7],
[293, 293, 3*w^2 - 2*w - 12],
[307, 307, 2*w^2 + w - 8],
[313, 313, w^2 + 4*w - 2],
[317, 317, 3*w^2 - 6*w - 5],
[331, 331, 4*w^2 - 4*w - 19],
[337, 337, -5*w^2 + 8*w + 21],
[347, 347, 3*w^2 - 6*w - 13],
[347, 347, 3*w^2 - 5*w - 22],
[347, 347, 3*w^2 - 5*w - 9],
[349, 349, w^2 - 5*w - 11],
[353, 353, -w^2 + 6*w - 7],
[359, 359, -4*w^2 + 8*w + 11],
[367, 367, 3*w^2 - 4*w - 11],
[373, 373, w^2 + 2*w - 6],
[379, 379, 5*w^2 - 9*w - 18],
[409, 409, -w^2 - 4],
[439, 439, 3*w^2 - 6*w - 14],
[443, 443, -w^2 - 2*w + 13],
[443, 443, 2*w^2 - 15],
[443, 443, 3*w^2 - 13],
[449, 449, w^2 + 2*w - 7],
[449, 449, 2*w^2 - 6*w - 7],
[449, 449, 3*w^2 - 3*w - 11],
[461, 461, -w^2 - 2*w - 5],
[467, 467, 3*w^2 - 5*w - 7],
[479, 479, -w^2 + 6*w - 4],
[491, 491, 3*w^2 - 5*w - 6],
[499, 499, w^2 - 5*w - 8],
[509, 509, 3*w^2 - 6*w - 19],
[521, 521, 3*w^2 - 2*w - 10],
[523, 523, 4*w - 9],
[541, 541, 3*w^2 - 3*w - 10],
[547, 547, 2*w^2 + w - 20],
[547, 547, w^2 - 6*w - 5],
[547, 547, 6*w^2 - 11*w - 24],
[557, 557, -5*w^2 + 11*w + 11],
[563, 563, 3*w^2 - 6*w - 16],
[569, 569, 4*w^2 - 5*w - 17],
[587, 587, 5*w^2 - 8*w - 20],
[593, 593, 5*w - 3],
[593, 593, 3*w^2 - w - 17],
[593, 593, w - 9],
[613, 613, 2*w^2 - w - 19],
[613, 613, 3*w^2 - 4*w - 5],
[613, 613, 3*w^2 - w - 8],
[617, 617, 3*w^2 - 2*w - 9],
[617, 617, -w^2 + w - 5],
[617, 617, 4*w^2 - 6*w - 15],
[619, 619, -w^2 + 3*w - 6],
[631, 631, 4*w^2 - 7*w - 21],
[641, 641, -w^2 - 3*w + 20],
[643, 643, 3*w^2 - 14],
[643, 643, w^2 - w - 13],
[643, 643, 2*w^2 - 9*w + 2],
[647, 647, 2*w^2 - 6*w - 9],
[653, 653, 3*w^2 - 5],
[659, 659, 6*w^2 - 11*w - 21],
[661, 661, w^2 - 6*w - 6],
[661, 661, w^2 + 3*w - 6],
[661, 661, 5*w^2 - 7*w - 22],
[673, 673, w^2 - 6*w - 12],
[677, 677, 4*w^2 - 7*w - 22],
[677, 677, 3*w^2 - 7*w - 12],
[677, 677, -2*w^2 - 3],
[683, 683, -2*w - 9],
[683, 683, 4*w^2 - 7*w - 12],
[683, 683, 3*w^2 - 3*w - 8],
[691, 691, 2*w^2 + w - 11],
[701, 701, 3*w^2 - 3*w - 5],
[709, 709, -w^2 - 2*w - 6],
[727, 727, 3*w^2 - w - 6],
[727, 727, 3*w^2 - w - 27],
[727, 727, 3*w^2 - w - 5],
[733, 733, -2*w^2 + 9*w - 5],
[739, 739, 3*w^2 - 4*w - 23],
[743, 743, 2*w^2 - 7*w - 7],
[743, 743, 3*w^2 - w - 20],
[743, 743, 3*w^2 - 2*w - 7],
[757, 757, -w - 9],
[757, 757, 4*w^2 - 2*w - 29],
[757, 757, 2*w^2 - 3*w - 18],
[761, 761, 3*w^2 - 2*w - 6],
[761, 761, 3*w^2 - 2*w - 25],
[773, 773, -5*w - 12],
[787, 787, -3*w - 10],
[809, 809, 4*w^2 - w - 19],
[821, 821, 5*w^2 - 10*w - 19],
[823, 823, -4*w - 11],
[827, 827, -7*w - 5],
[827, 827, 2*w^2 - 6*w - 13],
[827, 827, -w^2 - 2*w + 18],
[839, 839, 6*w^2 - 9*w - 26],
[839, 839, 5*w^2 - 12*w - 13],
[839, 839, w - 10],
[857, 857, -w^2 + 4*w - 8],
[859, 859, 3*w^2 - 8*w - 10],
[859, 859, 4*w^2 - 9*w - 14],
[859, 859, 3*w^2 - 26],
[863, 863, 2*w^2 + 3*w - 7],
[877, 877, -5*w^2 + 11*w + 9],
[877, 877, -5*w^2 + 10*w + 13],
[877, 877, -w^2 + w - 6],
[881, 881, w^2 - 3*w - 14],
[883, 883, -w^2 + 3*w - 7],
[887, 887, -3*w^2 + 6*w - 2],
[907, 907, w^2 + 3*w - 8],
[911, 911, 4*w^2 - 6*w - 29],
[937, 937, -6*w^2 + 11*w + 25],
[953, 953, 6*w^2 - 6*w - 29],
[961, 31, -w^2 + 7*w - 5],
[967, 967, 2*w^2 + w - 14],
[971, 971, 5*w^2 - 3*w - 27],
[971, 971, 4*w - 15],
[971, 971, w^2 - 2*w - 14],
[977, 977, 3*w^2 - 5*w - 24],
[983, 983, w^2 - 4*w - 15],
[991, 991, 6*w^2 - 10*w - 23],
[991, 991, 5*w^2 - 6*w - 22],
[991, 991, 3*w^2 + w - 13]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^5 + 2*x^4 - 7*x^3 - 8*x^2 + 8*x - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, e^4 + 2*e^3 - 7*e^2 - 9*e + 6, -4/3*e^4 - 10/3*e^3 + 23/3*e^2 + 14*e - 14/3, e^4 + 2*e^3 - 6*e^2 - 8*e + 1, -1, -e^4 - 2*e^3 + 6*e^2 + 7*e - 4, 1/3*e^4 + 4/3*e^3 - 5/3*e^2 - 5*e + 5/3, e^2 + 2*e - 3, -4/3*e^4 - 7/3*e^3 + 26/3*e^2 + 8*e - 29/3, -5/3*e^4 - 11/3*e^3 + 31/3*e^2 + 16*e - 25/3, 2*e^4 + 4*e^3 - 13*e^2 - 18*e + 6, 1/3*e^4 + 4/3*e^3 + 1/3*e^2 - 6*e - 16/3, 4/3*e^4 + 13/3*e^3 - 23/3*e^2 - 21*e + 20/3, e^4 + 3*e^3 - 4*e^2 - 14*e, -7/3*e^4 - 16/3*e^3 + 41/3*e^2 + 19*e - 35/3, -10/3*e^4 - 25/3*e^3 + 59/3*e^2 + 38*e - 41/3, 2*e^4 + 4*e^3 - 13*e^2 - 13*e + 13, 4/3*e^4 + 7/3*e^3 - 32/3*e^2 - 11*e + 20/3, -5*e^4 - 13*e^3 + 29*e^2 + 54*e - 24, -1/3*e^4 + 2/3*e^3 + 8/3*e^2 - 5*e - 2/3, e^4 + e^3 - 9*e^2 - 5*e + 9, 4*e^4 + 9*e^3 - 26*e^2 - 43*e + 17, 2*e^3 + 4*e^2 - 11*e - 15, e^4 + 2*e^3 - 10*e^2 - 11*e + 16, 4/3*e^4 + 7/3*e^3 - 32/3*e^2 - 4*e + 41/3, 2*e^4 + 6*e^3 - 9*e^2 - 29*e + 3, -11/3*e^4 - 29/3*e^3 + 58/3*e^2 + 37*e - 28/3, 11/3*e^4 + 26/3*e^3 - 70/3*e^2 - 35*e + 34/3, -2/3*e^4 - 11/3*e^3 + 7/3*e^2 + 21*e - 16/3, -5/3*e^4 - 11/3*e^3 + 37/3*e^2 + 20*e - 37/3, 20/3*e^4 + 47/3*e^3 - 121/3*e^2 - 64*e + 76/3, e^4 - 7*e^2 + 9*e + 7, 4/3*e^4 + 4/3*e^3 - 26/3*e^2 - 5*e + 8/3, -2*e^4 - 5*e^3 + 14*e^2 + 28*e - 18, 10/3*e^4 + 25/3*e^3 - 59/3*e^2 - 32*e + 26/3, 2*e^4 + 6*e^3 - 9*e^2 - 20*e, -10/3*e^4 - 16/3*e^3 + 74/3*e^2 + 21*e - 56/3, 22/3*e^4 + 55/3*e^3 - 134/3*e^2 - 81*e + 113/3, 11/3*e^4 + 23/3*e^3 - 79/3*e^2 - 33*e + 73/3, -5/3*e^4 - 11/3*e^3 + 43/3*e^2 + 15*e - 64/3, 3*e^4 + 7*e^3 - 15*e^2 - 29*e + 2, -17/3*e^4 - 47/3*e^3 + 94/3*e^2 + 70*e - 70/3, -13/3*e^4 - 37/3*e^3 + 62/3*e^2 + 53*e - 32/3, -5*e^4 - 13*e^3 + 32*e^2 + 63*e - 36, 1/3*e^4 - 5/3*e^3 - 17/3*e^2 + 10*e + 50/3, -7/3*e^4 - 13/3*e^3 + 41/3*e^2 + 13*e - 50/3, 7/3*e^4 + 13/3*e^3 - 44/3*e^2 - 10*e + 23/3, -14/3*e^4 - 35/3*e^3 + 94/3*e^2 + 58*e - 58/3, -1/3*e^4 + 2/3*e^3 + 11/3*e^2 - 9*e + 1/3, -e^4 - 4*e^3 + 2*e^2 + 21*e + 13, -4*e^4 - 11*e^3 + 19*e^2 + 42*e - 5, -11/3*e^4 - 23/3*e^3 + 79/3*e^2 + 33*e - 82/3, 8/3*e^4 + 17/3*e^3 - 64/3*e^2 - 28*e + 88/3, -14/3*e^4 - 32/3*e^3 + 79/3*e^2 + 41*e - 70/3, 5*e^4 + 12*e^3 - 27*e^2 - 41*e + 9, -4*e^4 - 9*e^3 + 22*e^2 + 38*e - 5, -3*e^4 - 6*e^3 + 18*e^2 + 18*e, 14/3*e^4 + 41/3*e^3 - 76/3*e^2 - 56*e + 52/3, 3*e^4 + 10*e^3 - 16*e^2 - 49*e + 15, 2*e^4 + 4*e^3 - 9*e^2 - 8*e - 5, -2/3*e^4 + 4/3*e^3 + 34/3*e^2 - 4*e - 103/3, -e^4 - 4*e^3 + 8*e + 25, -8/3*e^4 - 20/3*e^3 + 49/3*e^2 + 22*e - 16/3, 23/3*e^4 + 59/3*e^3 - 139/3*e^2 - 85*e + 73/3, -7/3*e^4 - 1/3*e^3 + 65/3*e^2 + e - 56/3, -10/3*e^4 - 22/3*e^3 + 80/3*e^2 + 34*e - 89/3, -26/3*e^4 - 53/3*e^3 + 160/3*e^2 + 73*e - 70/3, 28/3*e^4 + 58/3*e^3 - 179/3*e^2 - 84*e + 137/3, 19/3*e^4 + 49/3*e^3 - 110/3*e^2 - 76*e + 65/3, 25/3*e^4 + 55/3*e^3 - 152/3*e^2 - 68*e + 107/3, 13/3*e^4 + 28/3*e^3 - 83/3*e^2 - 32*e + 80/3, -5/3*e^4 - 20/3*e^3 + 34/3*e^2 + 43*e - 43/3, -8/3*e^4 - 14/3*e^3 + 67/3*e^2 + 23*e - 64/3, -26/3*e^4 - 74/3*e^3 + 145/3*e^2 + 110*e - 100/3, 19/3*e^4 + 37/3*e^3 - 134/3*e^2 - 51*e + 143/3, 1/3*e^4 + 1/3*e^3 - 23/3*e^2 - 7*e + 74/3, -e^4 - e^3 + 9*e^2 + 2*e - 25, -17/3*e^4 - 50/3*e^3 + 118/3*e^2 + 89*e - 118/3, -8/3*e^4 - 32/3*e^3 + 40/3*e^2 + 48*e - 64/3, -1/3*e^4 + 2/3*e^3 + 17/3*e^2 - 8*e + 16/3, 8/3*e^4 + 20/3*e^3 - 46/3*e^2 - 23*e - 47/3, -22/3*e^4 - 49/3*e^3 + 143/3*e^2 + 74*e - 83/3, -22/3*e^4 - 46/3*e^3 + 131/3*e^2 + 49*e - 80/3, -4*e^4 - 10*e^3 + 26*e^2 + 52*e - 16, -4/3*e^4 - 10/3*e^3 + 29/3*e^2 + 17*e + 49/3, -32/3*e^4 - 68/3*e^3 + 211/3*e^2 + 98*e - 115/3, 6*e^4 + 17*e^3 - 35*e^2 - 80*e + 20, -22/3*e^4 - 58/3*e^3 + 143/3*e^2 + 98*e - 95/3, 13*e^4 + 29*e^3 - 79*e^2 - 118*e + 53, -6*e^4 - 10*e^3 + 46*e^2 + 40*e - 32, -16/3*e^4 - 40/3*e^3 + 104/3*e^2 + 60*e - 152/3, 5*e^4 + 11*e^3 - 39*e^2 - 53*e + 38, 5/3*e^4 - 10/3*e^3 - 76/3*e^2 + 19*e + 100/3, -4*e^4 - 10*e^3 + 32*e^2 + 52*e - 34, 11*e^4 + 26*e^3 - 71*e^2 - 122*e + 51, -10*e^4 - 19*e^3 + 65*e^2 + 76*e - 55, e^4 + 4*e^3 - 5*e^2 - 20*e - 6, -19/3*e^4 - 46/3*e^3 + 116/3*e^2 + 60*e - 134/3, 5/3*e^4 - 4/3*e^3 - 67/3*e^2 + 12*e + 112/3, 3*e^4 + 9*e^3 - 17*e^2 - 43*e - 3, 11/3*e^4 + 26/3*e^3 - 76/3*e^2 - 34*e + 64/3, 31/3*e^4 + 79/3*e^3 - 182/3*e^2 - 121*e + 113/3, -8/3*e^4 + 1/3*e^3 + 64/3*e^2 - 15*e - 46/3, 8/3*e^4 + 11/3*e^3 - 67/3*e^2 - 13*e + 76/3, 40/3*e^4 + 103/3*e^3 - 248/3*e^2 - 156*e + 194/3, 16/3*e^4 + 40/3*e^3 - 119/3*e^2 - 61*e + 146/3, -10*e^4 - 25*e^3 + 57*e^2 + 103*e - 22, 25/3*e^4 + 61/3*e^3 - 131/3*e^2 - 86*e + 44/3, -13/3*e^4 - 40/3*e^3 + 80/3*e^2 + 67*e - 101/3, 5/3*e^4 + 5/3*e^3 - 64/3*e^2 - 19*e + 118/3, -6*e^4 - 18*e^3 + 34*e^2 + 83*e - 9, 1/3*e^4 - 2/3*e^3 + 1/3*e^2 + 3*e - 58/3, 23/3*e^4 + 47/3*e^3 - 148/3*e^2 - 67*e + 97/3, -e^4 + 3*e^3 + 8*e^2 - 26*e + 6, -11*e^4 - 23*e^3 + 70*e^2 + 90*e - 53, -7*e^4 - 13*e^3 + 51*e^2 + 58*e - 37, 11/3*e^4 + 17/3*e^3 - 85/3*e^2 - 26*e + 34/3, -17/3*e^4 - 35/3*e^3 + 100/3*e^2 + 45*e + 5/3, 35/3*e^4 + 83/3*e^3 - 190/3*e^2 - 110*e + 88/3, -1/3*e^4 + 5/3*e^3 + 17/3*e^2 - 13*e - 86/3, -e^3 - 9*e^2 + 6*e + 32, -37/3*e^4 - 82/3*e^3 + 236/3*e^2 + 116*e - 158/3, 13/3*e^4 + 22/3*e^3 - 95/3*e^2 - 34*e + 65/3, 8/3*e^4 + 5/3*e^3 - 70/3*e^2 + 4*e + 88/3, 7/3*e^4 + 16/3*e^3 - 47/3*e^2 - 21*e + 8/3, -7*e^4 - 11*e^3 + 49*e^2 + 33*e - 41, -5*e^3 - 5*e^2 + 20*e, -5/3*e^4 - 14/3*e^3 - 11/3*e^2 + 8*e + 122/3, -10/3*e^4 - 16/3*e^3 + 74/3*e^2 + 9*e - 131/3, 10*e^4 + 24*e^3 - 67*e^2 - 116*e + 56, 17/3*e^4 + 32/3*e^3 - 97/3*e^2 - 47*e - 14/3, 16/3*e^4 + 31/3*e^3 - 95/3*e^2 - 39*e + 35/3, -4*e^4 - 3*e^3 + 38*e^2 + 6*e - 46, 6*e^4 + 17*e^3 - 31*e^2 - 81*e + 18, 29/3*e^4 + 65/3*e^3 - 196/3*e^2 - 87*e + 196/3, 25/3*e^4 + 58/3*e^3 - 170/3*e^2 - 83*e + 110/3, 4/3*e^4 + 10/3*e^3 - 20/3*e^2 - 26*e + 23/3, 2*e^4 - 22*e^2 + 7*e + 26, -31/3*e^4 - 67/3*e^3 + 185/3*e^2 + 90*e - 68/3, 8/3*e^4 + 17/3*e^3 - 64/3*e^2 - 22*e + 61/3, -6*e^4 - 14*e^3 + 36*e^2 + 64*e - 31, 12*e^4 + 30*e^3 - 80*e^2 - 136*e + 72, -32/3*e^4 - 59/3*e^3 + 220/3*e^2 + 87*e - 130/3, 56/3*e^4 + 140/3*e^3 - 331/3*e^2 - 199*e + 241/3, -13*e^4 - 27*e^3 + 81*e^2 + 102*e - 39, -10/3*e^4 - 16/3*e^3 + 89/3*e^2 + 32*e - 188/3, 16/3*e^4 + 28/3*e^3 - 128/3*e^2 - 44*e + 107/3, 46/3*e^4 + 106/3*e^3 - 278/3*e^2 - 145*e + 200/3, -22/3*e^4 - 52/3*e^3 + 137/3*e^2 + 78*e - 122/3, -14/3*e^4 - 41/3*e^3 + 82/3*e^2 + 61*e - 115/3, -3*e^4 - 4*e^3 + 21*e^2 - 2*e - 42, -8/3*e^4 - 14/3*e^3 + 61/3*e^2 + 32*e - 16/3, -10*e^4 - 23*e^3 + 64*e^2 + 108*e - 39, -8/3*e^4 - 20/3*e^3 + 67/3*e^2 + 26*e - 97/3, -31/3*e^4 - 79/3*e^3 + 188/3*e^2 + 109*e - 140/3, -8/3*e^4 - 17/3*e^3 + 46/3*e^2 + 22*e - 70/3, -2/3*e^4 - 8/3*e^3 + 4/3*e^2 + 17*e + 56/3, -29/3*e^4 - 77/3*e^3 + 154/3*e^2 + 112*e - 133/3, -19*e^4 - 41*e^3 + 117*e^2 + 161*e - 85, -11*e^4 - 22*e^3 + 70*e^2 + 67*e - 62, -22/3*e^4 - 52/3*e^3 + 134/3*e^2 + 85*e - 68/3, -17/3*e^4 - 44/3*e^3 + 127/3*e^2 + 68*e - 166/3, -3*e^4 - 4*e^3 + 19*e^2 + e - 5, 10/3*e^4 + 1/3*e^3 - 86/3*e^2 + 5*e + 101/3, 10*e^4 + 26*e^3 - 52*e^2 - 105*e + 28, 44/3*e^4 + 95/3*e^3 - 274/3*e^2 - 129*e + 190/3, 17*e^4 + 36*e^3 - 100*e^2 - 137*e + 41, -14*e^4 - 35*e^3 + 87*e^2 + 163*e - 78, 31/3*e^4 + 70/3*e^3 - 194/3*e^2 - 95*e + 74/3, 26/3*e^4 + 59/3*e^3 - 145/3*e^2 - 61*e + 49/3, 8/3*e^4 + 32/3*e^3 - 28/3*e^2 - 59*e - 44/3, 11/3*e^4 + 26/3*e^3 - 25/3*e^2 - 22*e - 107/3, 25/3*e^4 + 61/3*e^3 - 143/3*e^2 - 94*e + 41/3, -4/3*e^4 - 19/3*e^3 + 17/3*e^2 + 21*e - 38/3, 1/3*e^4 - 5/3*e^3 - 23/3*e^2 + 24*e + 104/3, 2*e^4 + 10*e^3 - 9*e^2 - 44*e + 13];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {11, 11, -w^2 + 2*w + 2}>] := 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;