/*
  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, 2, -5, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {19, 19, -w^2 + w + 4}>;

primesArray := [
[3, 3, w],
[4, 2, -w^3 + 3*w^2 + w - 2],
[11, 11, -w^3 + 3*w^2 + 2*w - 5],
[13, 13, w^3 - 2*w^2 - 4*w + 2],
[13, 13, -w + 2],
[17, 17, w^3 - 3*w^2 - 2*w + 2],
[19, 19, -w^2 + w + 4],
[19, 19, w^3 - 2*w^2 - 3*w + 2],
[23, 23, w^2 - 2*w - 1],
[27, 3, w^3 - 2*w^2 - 5*w - 1],
[29, 29, -w^3 + 2*w^2 + 5*w - 1],
[37, 37, -w^3 + 2*w^2 + 5*w - 4],
[41, 41, 2*w^3 - 5*w^2 - 6*w + 4],
[53, 53, w^3 - 2*w^2 - 3*w - 2],
[59, 59, w - 4],
[67, 67, 2*w^2 - 3*w - 8],
[73, 73, 2*w^3 - 5*w^2 - 6*w + 5],
[89, 89, -2*w^3 + 6*w^2 + 5*w - 7],
[97, 97, -2*w^3 + 6*w^2 + 3*w - 7],
[97, 97, -w^3 + 3*w^2 + 3*w - 1],
[103, 103, w^2 - 4*w - 4],
[113, 113, 2*w^3 - 5*w^2 - 4*w + 1],
[113, 113, -w^3 + 3*w^2 + w - 5],
[139, 139, 2*w^2 - 3*w - 4],
[139, 139, w^3 - w^2 - 6*w - 1],
[149, 149, 2*w^3 - 5*w^2 - 5*w + 2],
[149, 149, w^3 - 4*w^2 + 10],
[151, 151, -3*w - 1],
[157, 157, w^2 - w - 8],
[157, 157, w^3 - 7*w - 8],
[163, 163, -3*w^3 + 8*w^2 + 8*w - 10],
[163, 163, w^3 - 3*w^2 - 4*w + 7],
[163, 163, -w^3 + 4*w^2 + w - 8],
[163, 163, -2*w^3 + 5*w^2 + 7*w - 4],
[167, 167, -2*w^3 + 5*w^2 + 8*w - 4],
[169, 13, -2*w^3 + 4*w^2 + 7*w - 5],
[173, 173, 2*w^3 - 4*w^2 - 9*w + 4],
[173, 173, -w^3 + 5*w^2 - 2*w - 5],
[181, 181, 3*w^3 - 7*w^2 - 12*w + 7],
[191, 191, w^3 - 3*w^2 - 3*w - 1],
[199, 199, 2*w^3 - 3*w^2 - 10*w - 1],
[199, 199, 2*w^3 - 6*w^2 - 6*w + 13],
[223, 223, -2*w^3 + 6*w^2 + 4*w - 11],
[223, 223, 3*w^3 - 7*w^2 - 12*w + 8],
[223, 223, -2*w^3 + 5*w^2 + 6*w - 1],
[229, 229, -w^3 + 3*w^2 + 2*w + 1],
[229, 229, 3*w^3 - 4*w^2 - 15*w - 5],
[233, 233, -3*w^3 + 7*w^2 + 11*w - 5],
[233, 233, 2*w^3 - 5*w^2 - 4*w + 5],
[241, 241, w^3 - 3*w^2 - 5*w + 7],
[257, 257, w^3 - 3*w^2 - 4*w + 8],
[263, 263, w^2 - 4*w - 5],
[271, 271, -2*w^3 + 6*w^2 + 5*w - 13],
[277, 277, -2*w^3 + 6*w^2 + 3*w - 8],
[277, 277, w^3 - 2*w^2 - 2*w - 2],
[281, 281, 2*w^3 - 5*w^2 - 4*w + 2],
[283, 283, -3*w^2 + 7*w + 10],
[283, 283, -w^3 + 4*w^2 - w - 7],
[293, 293, 2*w^3 - 5*w^2 - 3*w + 4],
[293, 293, w^3 - 8*w - 4],
[307, 307, -w^3 + 2*w^2 + 6*w - 2],
[311, 311, w^2 - 5],
[317, 317, -w^3 + 5*w^2 - 2*w - 11],
[331, 331, -w^3 + 3*w^2 - 5],
[349, 349, -2*w^2 + 5*w + 2],
[353, 353, -2*w^3 + 5*w^2 + 4*w - 4],
[353, 353, 3*w^3 - 7*w^2 - 10*w + 2],
[353, 353, -2*w^3 + 7*w^2 + 2*w - 10],
[353, 353, w^3 + w^2 - 9*w - 13],
[361, 19, -w^3 + w^2 + 5*w - 1],
[373, 373, -w^3 + 3*w^2 + 4*w - 11],
[373, 373, 2*w^3 - 5*w^2 - 5*w - 2],
[383, 383, w^3 - 5*w^2 + w + 13],
[389, 389, 3*w^3 - 10*w^2 - 4*w + 16],
[397, 397, -w^3 + 2*w^2 + 3*w + 5],
[397, 397, -3*w^3 + 7*w^2 + 10*w - 7],
[419, 419, -w^3 + 2*w^2 + 4*w + 4],
[419, 419, 3*w^3 - 7*w^2 - 12*w + 5],
[421, 421, -w^2 + 5*w - 2],
[421, 421, 3*w^3 - 7*w^2 - 10*w + 5],
[431, 431, w^2 - 10],
[439, 439, w^3 - w^2 - 7*w - 7],
[461, 461, -2*w^3 + 7*w^2 + 5*w - 10],
[467, 467, -3*w^3 + 6*w^2 + 14*w - 4],
[479, 479, 3*w^2 - 6*w - 5],
[479, 479, -4*w^3 + 11*w^2 + 10*w - 10],
[491, 491, -w^3 + w^2 + 6*w - 1],
[499, 499, -4*w^3 + 9*w^2 + 15*w - 10],
[499, 499, 2*w^3 - 5*w^2 - 7*w + 2],
[503, 503, -2*w^3 + 7*w^2 - 8],
[521, 521, 2*w^2 - w - 7],
[521, 521, -3*w^3 + 7*w^2 + 11*w - 11],
[569, 569, 2*w^2 - 5*w - 1],
[569, 569, -4*w - 1],
[571, 571, -w^3 + 3*w^2 - 7],
[577, 577, -w^3 + w^2 + 8*w - 4],
[587, 587, -3*w^3 + 8*w^2 + 7*w - 8],
[593, 593, 3*w^2 - 4*w - 11],
[593, 593, -3*w^3 + 9*w^2 + 6*w - 14],
[601, 601, -3*w^3 + 9*w^2 + 4*w - 10],
[601, 601, 5*w^3 - 9*w^2 - 24*w - 2],
[607, 607, 2*w^2 - 7*w - 5],
[613, 613, w^3 - 6*w - 2],
[613, 613, -w^2 + 3*w + 8],
[617, 617, -w^2 + 4*w - 5],
[619, 619, 3*w^3 - 6*w^2 - 13*w + 5],
[625, 5, -5],
[631, 631, -2*w^3 + 4*w^2 + 8*w - 7],
[647, 647, -2*w^3 + 4*w^2 + 5*w - 1],
[647, 647, w^3 - 3*w^2 - 2*w - 2],
[653, 653, w^3 - 4*w^2 - w + 13],
[653, 653, w^3 - 2*w^2 - 7*w + 5],
[659, 659, -4*w^3 + 10*w^2 + 15*w - 14],
[659, 659, 5*w^3 - 12*w^2 - 17*w + 14],
[661, 661, w^2 - 5*w - 2],
[661, 661, -w^3 + 5*w^2 - 2*w - 7],
[673, 673, -w^3 + w^2 + 6*w - 2],
[683, 683, 2*w^3 - 8*w^2 + w + 8],
[683, 683, 5*w^3 - 12*w^2 - 19*w + 14],
[701, 701, -2*w^3 + 6*w^2 + 5*w - 4],
[709, 709, -w^2 + 3*w - 4],
[709, 709, -2*w^3 + 5*w^2 + 9*w - 8],
[719, 719, w^2 - 7],
[719, 719, -2*w^3 + 3*w^2 + 8*w + 5],
[733, 733, -2*w^3 + 7*w^2 + 3*w - 10],
[739, 739, 4*w^3 - 11*w^2 - 13*w + 14],
[739, 739, -5*w^3 + 10*w^2 + 21*w - 4],
[739, 739, 3*w^3 - 8*w^2 - 6*w + 8],
[739, 739, -w^3 + w^2 + 8*w - 5],
[743, 743, -w - 5],
[743, 743, w^3 - w^2 - 8*w - 8],
[751, 751, -2*w^3 + 5*w^2 + 8*w - 2],
[751, 751, -2*w^3 + 6*w^2 - 1],
[757, 757, -2*w^3 + 6*w^2 + 3*w - 10],
[757, 757, 5*w^2 - 9*w - 22],
[761, 761, -2*w^3 + 4*w^2 + 9*w - 7],
[761, 761, 2*w^3 - 3*w^2 - 12*w - 1],
[769, 769, 5*w^3 - 13*w^2 - 14*w + 10],
[773, 773, -3*w^3 + 5*w^2 + 16*w + 1],
[773, 773, w^3 - 11*w + 2],
[787, 787, -w^3 + 4*w^2 - w - 11],
[797, 797, -5*w^3 + 12*w^2 + 17*w - 11],
[809, 809, -2*w^3 + 7*w^2 + 4*w - 11],
[809, 809, 4*w^3 - 9*w^2 - 14*w + 4],
[821, 821, -w^3 + 5*w^2 - 2*w - 17],
[827, 827, -3*w^3 + 8*w^2 + 9*w - 7],
[829, 829, 2*w^3 - 6*w^2 - 5*w + 2],
[829, 829, -w^3 + 4*w^2 - w - 10],
[839, 839, w^2 - 8],
[853, 853, -5*w^3 + 12*w^2 + 18*w - 16],
[853, 853, w^3 - 6*w - 8],
[863, 863, 5*w^3 - 11*w^2 - 19*w + 7],
[881, 881, -3*w^3 + 9*w^2 + 9*w - 11],
[907, 907, -w^3 + 2*w^2 + 7*w - 4],
[907, 907, -2*w^3 + 3*w^2 + 7*w + 4],
[929, 929, -w^3 + 5*w^2 - 3*w - 11],
[937, 937, 2*w^3 - 5*w^2 - 6*w + 11],
[937, 937, -2*w^3 + 6*w^2 + 3*w - 11],
[941, 941, -3*w^3 + 8*w^2 + 11*w - 8],
[953, 953, -4*w^3 + 10*w^2 + 10*w - 13],
[953, 953, w^3 - 5*w^2 - w + 13],
[967, 967, 3*w^3 - 7*w^2 - 9*w + 7],
[967, 967, -2*w^3 + 5*w^2 + 8*w + 1],
[967, 967, 4*w^3 - 13*w^2 - 6*w + 20],
[967, 967, -2*w^2 + 3*w + 13],
[971, 971, -6*w^3 + 13*w^2 + 25*w - 8],
[977, 977, w^3 - w^2 - 4*w - 10],
[977, 977, 2*w^3 - 6*w^2 - 7*w + 13],
[983, 983, -4*w^3 + 10*w^2 + 11*w - 13],
[983, 983, -w^3 + 5*w^2 - w - 7],
[983, 983, 3*w^3 - 9*w^2 - 7*w + 19],
[983, 983, -2*w^3 + 3*w^2 + 10*w - 1],
[997, 997, 5*w^3 - 12*w^2 - 17*w + 13],
[997, 997, -2*w^3 + 3*w^2 + 13*w - 4],
[997, 997, w^3 - 7*w - 2],
[997, 997, w^3 - 9*w - 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^10 - 20*x^8 + 129*x^6 - 274*x^4 + 80*x^2 - 4;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1/14*e^9 + 9/7*e^7 - 107/14*e^5 + 114/7*e^3 - 50/7*e, -1/14*e^8 + 25/14*e^6 - 92/7*e^4 + 191/7*e^2 - 22/7, 1/14*e^9 - 9/7*e^7 + 107/14*e^5 - 114/7*e^3 + 50/7*e, 1/7*e^8 - 18/7*e^6 + 100/7*e^4 - 165/7*e^2 + 2/7, 3/14*e^9 - 61/14*e^7 + 199/7*e^5 - 419/7*e^3 + 87/7*e, 1, 1/2*e^9 - 21/2*e^7 + 70*e^5 - 147*e^3 + 25*e, -5/14*e^9 + 97/14*e^7 - 299/7*e^5 + 577/7*e^3 - 68/7*e, -3/14*e^9 + 61/14*e^7 - 199/7*e^5 + 412/7*e^3 - 45/7*e, -1/7*e^9 + 43/14*e^7 - 291/14*e^5 + 312/7*e^3 - 100/7*e, 3/7*e^9 - 115/14*e^7 + 705/14*e^5 - 698/7*e^3 + 160/7*e, -1/7*e^9 + 25/7*e^7 - 191/7*e^5 + 466/7*e^3 - 205/7*e, -1/7*e^8 + 18/7*e^6 - 100/7*e^4 + 165/7*e^2 - 58/7, e^6 - 13*e^4 + 40*e^2 - 8, 1/14*e^8 - 39/14*e^6 + 176/7*e^4 - 429/7*e^2 + 78/7, -12/7*e^9 + 237/7*e^7 - 1501/7*e^5 + 3051/7*e^3 - 521/7*e, -1/7*e^9 + 25/7*e^7 - 184/7*e^5 + 382/7*e^3 + 5/7*e, 4/7*e^8 - 72/7*e^6 + 421/7*e^4 - 835/7*e^2 + 134/7, -2*e^9 + 39*e^7 - 244*e^5 + 492*e^3 - 95*e, 3/7*e^8 - 68/7*e^6 + 482/7*e^4 - 1048/7*e^2 + 132/7, 1/7*e^9 - 11/7*e^7 + 16/7*e^5 + 80/7*e^3 - 131/7*e, 5/14*e^8 - 83/14*e^6 + 222/7*e^4 - 409/7*e^2 + 68/7, -6/7*e^8 + 87/7*e^6 - 369/7*e^4 + 479/7*e^2 - 124/7, 3/14*e^9 - 75/14*e^7 + 290/7*e^5 - 741/7*e^3 + 423/7*e, 13/7*e^9 - 503/14*e^7 + 3125/14*e^5 - 3146/7*e^3 + 698/7*e, 3/7*e^8 - 75/7*e^6 + 559/7*e^4 - 1237/7*e^2 + 216/7, -29/14*e^9 + 599/14*e^7 - 1975/7*e^5 + 4153/7*e^3 - 792/7*e, 1/2*e^8 - 21/2*e^6 + 70*e^4 - 147*e^2 + 16, -e^8 + 18*e^6 - 102*e^4 + 179*e^2 - 18, 3/2*e^9 - 59/2*e^7 + 186*e^5 - 375*e^3 + 53*e, -3/14*e^8 + 61/14*e^6 - 192/7*e^4 + 356/7*e^2 - 66/7, 4/7*e^8 - 65/7*e^6 + 330/7*e^4 - 562/7*e^2 + 148/7, -3/7*e^9 + 54/7*e^7 - 314/7*e^5 + 614/7*e^3 - 153/7*e, -11/7*e^9 + 219/7*e^7 - 1394/7*e^5 + 2844/7*e^3 - 554/7*e, -3/7*e^9 + 61/7*e^7 - 391/7*e^5 + 754/7*e^3 + 1/7*e, 17/14*e^9 - 167/7*e^7 + 2099/14*e^5 - 2099/7*e^3 + 262/7*e, -5/14*e^8 + 55/14*e^6 - 47/7*e^4 - 109/7*e^2 + 44/7, 12/7*e^9 - 481/14*e^7 + 3107/14*e^5 - 3296/7*e^3 + 906/7*e, 3/7*e^8 - 61/7*e^6 + 398/7*e^4 - 838/7*e^2 + 76/7, -9/14*e^9 + 183/14*e^7 - 611/7*e^5 + 1425/7*e^3 - 674/7*e, -3/2*e^8 + 55/2*e^6 - 162*e^4 + 310*e^2 - 46, -15/14*e^8 + 235/14*e^6 - 568/7*e^4 + 884/7*e^2 - 134/7, e^7 - 10*e^5 + 11*e^3 + 34*e, -3/2*e^9 + 59/2*e^7 - 187*e^5 + 387*e^3 - 92*e, -17/14*e^8 + 257/14*e^6 - 605/7*e^4 + 1007/7*e^2 - 248/7, 27/14*e^9 - 278/7*e^7 + 3659/14*e^5 - 3869/7*e^3 + 930/7*e, 17/7*e^9 - 341/7*e^7 + 2183/7*e^5 - 4450/7*e^3 + 839/7*e, 25/14*e^9 - 485/14*e^7 + 1516/7*e^5 - 3081/7*e^3 + 571/7*e, 5/14*e^8 - 97/14*e^6 + 285/7*e^4 - 437/7*e^2 - 72/7, 9/14*e^8 - 141/14*e^6 + 331/7*e^4 - 424/7*e^2 - 96/7, 3/14*e^8 + 23/14*e^6 - 305/7*e^4 + 960/7*e^2 - 200/7, 1/14*e^8 - 39/14*e^6 + 176/7*e^4 - 408/7*e^2 + 8/7, -3/7*e^8 + 33/7*e^6 - 62/7*e^4 - 93/7*e^2 + 64/7, -1/14*e^8 + 39/14*e^6 - 162/7*e^4 + 289/7*e^2 + 34/7, 27/14*e^9 - 535/14*e^7 + 1714/7*e^5 - 3631/7*e^3 + 1049/7*e, 1/2*e^8 - 9/2*e^6 - 2*e^4 + 46*e^2 + 2, 6/7*e^8 - 115/7*e^6 + 705/7*e^4 - 1382/7*e^2 + 152/7, 5/14*e^9 - 52/7*e^7 + 689/14*e^5 - 745/7*e^3 + 278/7*e, 3/7*e^9 - 115/14*e^7 + 705/14*e^5 - 684/7*e^3 + 118/7*e, -1/7*e^9 + 18/7*e^7 - 86/7*e^5 + 11/7*e^3 + 341/7*e, -9/14*e^8 + 211/14*e^6 - 758/7*e^4 + 1656/7*e^2 - 324/7, -3/7*e^8 + 54/7*e^6 - 314/7*e^4 + 607/7*e^2 - 188/7, 4/7*e^8 - 58/7*e^6 + 267/7*e^4 - 499/7*e^2 + 176/7, 29/14*e^9 - 296/7*e^7 + 3831/14*e^5 - 3880/7*e^3 + 596/7*e, -41/14*e^9 + 829/14*e^7 - 2680/7*e^5 + 5549/7*e^3 - 1119/7*e, 3/2*e^9 - 59/2*e^7 + 186*e^5 - 381*e^3 + 95*e, 2/7*e^8 - 43/7*e^6 + 291/7*e^4 - 575/7*e^2 - 10/7, -5/7*e^8 + 90/7*e^6 - 507/7*e^4 + 860/7*e^2 - 122/7, 15/7*e^9 - 284/7*e^7 + 1738/7*e^5 - 3504/7*e^3 + 751/7*e, 3/14*e^8 - 61/14*e^6 + 185/7*e^4 - 307/7*e^2 - 46/7, -3/14*e^8 + 131/14*e^6 - 619/7*e^4 + 1553/7*e^2 - 290/7, -19/14*e^8 + 335/14*e^6 - 929/7*e^4 + 1592/7*e^2 - 138/7, 6/7*e^8 - 101/7*e^6 + 530/7*e^4 - 843/7*e^2 - 44/7, 1/2*e^8 - 19/2*e^6 + 60*e^4 - 129*e^2 + 18, 45/14*e^9 - 454/7*e^7 + 5865/14*e^5 - 6061/7*e^3 + 1158/7*e, -19/14*e^8 + 265/14*e^6 - 530/7*e^4 + 647/7*e^2 - 82/7, -29/14*e^9 + 557/14*e^7 - 1702/7*e^5 + 3264/7*e^3 - 337/7*e, -45/14*e^9 + 440/7*e^7 - 5529/14*e^5 + 5599/7*e^3 - 1018/7*e, 3*e^9 - 119/2*e^7 + 759/2*e^5 - 787*e^3 + 180*e, -11/14*e^8 + 205/14*e^6 - 606/7*e^4 + 1142/7*e^2 - 172/7, -1/7*e^8 + 18/7*e^6 - 107/7*e^4 + 242/7*e^2 - 198/7, e^8 - 20*e^6 + 126*e^4 - 245*e^2 + 42, -5/7*e^9 + 111/7*e^7 - 780/7*e^5 + 1770/7*e^3 - 619/7*e, 2/7*e^9 - 50/7*e^7 + 375/7*e^5 - 841/7*e^3 + 130/7*e, 9/14*e^9 - 197/14*e^7 + 674/7*e^5 - 1453/7*e^3 + 478/7*e, -3*e^9 + 57*e^7 - 350*e^5 + 707*e^3 - 149*e, -5/7*e^9 + 97/7*e^7 - 619/7*e^5 + 1364/7*e^3 - 493/7*e, 1/7*e^9 - 11/7*e^7 + 9/7*e^5 + 192/7*e^3 - 551/7*e, -3/7*e^8 + 61/7*e^6 - 370/7*e^4 + 628/7*e^2 - 62/7, -8/7*e^8 + 172/7*e^6 - 1185/7*e^4 + 2601/7*e^2 - 366/7, 3/14*e^9 - 75/14*e^7 + 269/7*e^5 - 496/7*e^3 - 249/7*e, 4*e^9 - 76*e^7 + 462*e^5 - 899*e^3 + 143*e, 13/14*e^9 - 269/14*e^7 + 895/7*e^5 - 1958/7*e^3 + 629/7*e, -9/7*e^8 + 120/7*e^6 - 431/7*e^4 + 393/7*e^2 - 4/7, -3/14*e^9 + 75/14*e^7 - 297/7*e^5 + 804/7*e^3 - 563/7*e, -2/7*e^9 + 29/7*e^7 - 109/7*e^5 - 27/7*e^3 + 563/7*e, 1/14*e^8 - 39/14*e^6 + 176/7*e^4 - 415/7*e^2 + 36/7, 1/7*e^8 - 46/7*e^6 + 429/7*e^4 - 984/7*e^2 - 54/7, -9/14*e^8 + 85/14*e^6 - 9/7*e^4 - 346/7*e^2 + 68/7, -2*e^9 + 38*e^7 - 235*e^5 + 491*e^3 - 153*e, 8/7*e^9 - 172/7*e^7 + 1164/7*e^5 - 2468/7*e^3 + 534/7*e, 1/14*e^9 - 2/7*e^7 - 75/14*e^5 + 201/7*e^3 - 230/7*e, 11/14*e^8 - 135/14*e^6 + 200/7*e^4 - 99/7*e^2 - 164/7, -3/14*e^8 + 19/14*e^6 + 46/7*e^4 - 176/7*e^2 - 108/7, -1/14*e^9 + 39/14*e^7 - 204/7*e^5 + 723/7*e^3 - 729/7*e, 4/7*e^8 - 72/7*e^6 + 428/7*e^4 - 849/7*e^2 - 62/7, -4/7*e^9 + 86/7*e^7 - 603/7*e^5 + 1458/7*e^3 - 708/7*e, -17/7*e^9 + 341/7*e^7 - 2190/7*e^5 + 4506/7*e^3 - 846/7*e, 6/7*e^8 - 122/7*e^6 + 810/7*e^4 - 1802/7*e^2 + 418/7, -11/7*e^8 + 198/7*e^6 - 1142/7*e^4 + 2151/7*e^2 - 288/7, -11/14*e^9 + 113/7*e^7 - 1513/14*e^5 + 1737/7*e^3 - 858/7*e, -13/14*e^9 + 297/14*e^7 - 1035/7*e^5 + 2126/7*e^3 - 181/7*e, -23/7*e^9 + 456/7*e^7 - 2895/7*e^5 + 5881/7*e^3 - 1047/7*e, -7/2*e^9 + 68*e^7 - 849/2*e^5 + 856*e^3 - 160*e, -4/7*e^8 + 107/7*e^6 - 841/7*e^4 + 1955/7*e^2 - 316/7, 1/7*e^9 + 3/7*e^7 - 166/7*e^5 + 689/7*e^3 - 537/7*e, 15/7*e^8 - 235/7*e^6 + 1157/7*e^4 - 1957/7*e^2 + 436/7, 9/14*e^9 - 169/14*e^7 + 492/7*e^5 - 802/7*e^3 - 271/7*e, -12/7*e^9 + 481/14*e^7 - 3121/14*e^5 + 3429/7*e^3 - 1480/7*e, -17/14*e^8 + 411/14*e^6 - 1508/7*e^4 + 3331/7*e^2 - 472/7, 5/14*e^9 - 38/7*e^7 + 395/14*e^5 - 514/7*e^3 + 656/7*e, 5/7*e^8 - 83/7*e^6 + 437/7*e^4 - 734/7*e^2 + 52/7, -2/7*e^8 + 36/7*e^6 - 214/7*e^4 + 484/7*e^2 - 102/7, 25/14*e^8 - 457/14*e^6 + 1348/7*e^4 - 2605/7*e^2 + 312/7, 24/7*e^9 - 481/7*e^7 + 3065/7*e^5 - 6137/7*e^3 + 937/7*e, -4/7*e^9 + 93/7*e^7 - 652/7*e^5 + 1304/7*e^3 + 27/7*e, 15/7*e^9 - 298/7*e^7 + 1871/7*e^5 - 3609/7*e^3 + 219/7*e, 11/7*e^9 - 205/7*e^7 + 1240/7*e^5 - 2508/7*e^3 + 589/7*e, -e^8 + 23*e^6 - 162*e^4 + 340*e^2 - 58, e^8 - 12*e^6 + 30*e^4 + 20*e^2 - 42, 45/14*e^9 - 901/14*e^7 + 2887/7*e^5 - 5970/7*e^3 + 1452/7*e, -2/7*e^8 - 13/7*e^6 + 353/7*e^4 - 986/7*e^2 + 248/7, 5/7*e^8 - 97/7*e^6 + 591/7*e^4 - 1147/7*e^2 + 388/7, -5*e^6 + 62*e^4 - 177*e^2 + 36, -3*e^9 + 60*e^7 - 382*e^5 + 770*e^3 - 135*e, 11/7*e^9 - 219/7*e^7 + 1415/7*e^5 - 3033/7*e^3 + 785/7*e, e^9 - 21*e^7 + 139*e^5 - 282*e^3 + 19*e, -8/7*e^9 + 309/14*e^7 - 1929/14*e^5 + 2041/7*e^3 - 828/7*e, -4/7*e^9 + 137/14*e^7 - 779/14*e^5 + 842/7*e^3 - 456/7*e, 10/7*e^8 - 222/7*e^6 + 1560/7*e^4 - 3442/7*e^2 + 552/7, -73/14*e^9 + 734/7*e^7 - 9449/14*e^5 + 9764/7*e^3 - 2096/7*e, -19/14*e^8 + 251/14*e^6 - 474/7*e^4 + 661/7*e^2 - 264/7, 13/7*e^9 - 283/7*e^7 + 1965/7*e^5 - 4441/7*e^3 + 1433/7*e, 11/7*e^8 - 219/7*e^6 + 1394/7*e^4 - 2795/7*e^2 + 288/7, 4/7*e^9 - 65/7*e^7 + 337/7*e^5 - 611/7*e^3 + 225/7*e, 11/2*e^9 - 107*e^7 + 1335/2*e^5 - 1338*e^3 + 224*e, -8/7*e^8 + 130/7*e^6 - 660/7*e^4 + 1033/7*e^2 + 40/7, 4/7*e^8 - 16/7*e^6 - 258/7*e^4 + 1062/7*e^2 - 342/7, -11/7*e^9 + 487/14*e^7 - 3341/14*e^5 + 3439/7*e^3 - 246/7*e, 4/7*e^8 - 65/7*e^6 + 344/7*e^4 - 653/7*e^2 + 190/7, 20/7*e^9 - 402/7*e^7 + 2574/7*e^5 - 5190/7*e^3 + 740/7*e, 16/7*e^9 - 337/7*e^7 + 2258/7*e^5 - 4831/7*e^3 + 1145/7*e, -2/7*e^9 + 29/7*e^7 - 123/7*e^5 + 141/7*e^3 + 115/7*e, -3*e^8 + 55*e^6 - 325*e^4 + 625*e^2 - 92, -1/14*e^8 + 151/14*e^6 - 834/7*e^4 + 2109/7*e^2 - 204/7, 33/14*e^8 - 573/14*e^6 + 1601/7*e^4 - 2985/7*e^2 + 432/7, -1/14*e^8 - 59/14*e^6 + 384/7*e^4 - 943/7*e^2 + 104/7, 8/7*e^9 - 351/14*e^7 + 2433/14*e^5 - 2720/7*e^3 + 1052/7*e, 10/7*e^9 - 180/7*e^7 + 1049/7*e^5 - 2084/7*e^3 + 699/7*e, 1/14*e^8 + 73/14*e^6 - 475/7*e^4 + 1223/7*e^2 - 216/7, -4/7*e^9 + 72/7*e^7 - 400/7*e^5 + 618/7*e^3 + 160/7*e, -27/7*e^9 + 507/7*e^7 - 3043/7*e^5 + 5827/7*e^3 - 796/7*e, -1/7*e^8 + 74/7*e^6 - 765/7*e^4 + 1922/7*e^2 - 366/7, 11/14*e^8 - 247/14*e^6 + 858/7*e^4 - 1786/7*e^2 + 256/7, 20/7*e^9 - 381/7*e^7 + 2343/7*e^5 - 4735/7*e^3 + 1111/7*e, -6/7*e^8 + 122/7*e^6 - 761/7*e^4 + 1354/7*e^2 - 82/7, 3/7*e^8 - 19/7*e^6 - 78/7*e^4 + 275/7*e^2 + 34/7, 1/7*e^9 - 25/7*e^7 + 156/7*e^5 - 81/7*e^3 - 558/7*e, 1/7*e^8 - 46/7*e^6 + 415/7*e^4 - 844/7*e^2 - 166/7, 4/7*e^8 - 79/7*e^6 + 491/7*e^4 - 940/7*e^2 + 22/7, 25/7*e^9 - 506/7*e^7 + 3256/7*e^5 - 6624/7*e^3 + 1226/7*e, 75/14*e^9 - 745/7*e^7 + 9467/14*e^5 - 9614/7*e^3 + 1706/7*e, -25/14*e^9 + 239/7*e^7 - 2941/14*e^5 + 2941/7*e^3 - 634/7*e, -5/7*e^9 + 159/14*e^7 - 755/14*e^5 + 475/7*e^3 + 186/7*e, -5/14*e^9 + 38/7*e^7 - 367/14*e^5 + 346/7*e^3 - 306/7*e];
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 + 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;