/*
  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![19, 14, -13, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {19,19,4/19*w^3 - 6/19*w^2 - 55/19*w + 1}>;

primesArray := [
[4, 2, -2/19*w^3 + 3/19*w^2 + 37/19*w - 3],
[5, 5, 3/19*w^3 + 5/19*w^2 - 27/19*w - 1],
[5, 5, -3/19*w^3 + 14/19*w^2 + 8/19*w - 2],
[9, 3, -2/19*w^3 + 3/19*w^2 + 37/19*w - 4],
[19, 19, -w],
[19, 19, 4/19*w^3 - 6/19*w^2 - 55/19*w + 1],
[19, 19, 4/19*w^3 - 6/19*w^2 - 55/19*w + 2],
[19, 19, w - 1],
[29, 29, -5/19*w^3 - 2/19*w^2 + 64/19*w + 2],
[29, 29, 10/19*w^3 - 34/19*w^2 - 71/19*w + 9],
[29, 29, 4/19*w^3 - 6/19*w^2 - 55/19*w + 4],
[29, 29, -w + 3],
[49, 7, 5/19*w^3 - 17/19*w^2 - 64/19*w + 11],
[49, 7, 6/19*w^3 - 9/19*w^2 - 92/19*w - 1],
[71, 71, 3/19*w^3 + 5/19*w^2 - 46/19*w - 1],
[71, 71, 6/19*w^3 - 9/19*w^2 - 73/19*w],
[71, 71, 6/19*w^3 - 9/19*w^2 - 73/19*w + 4],
[71, 71, -3/19*w^3 + 14/19*w^2 + 27/19*w - 3],
[101, 101, 7/19*w^3 - 1/19*w^2 - 63/19*w - 3],
[101, 101, -9/19*w^3 + 23/19*w^2 + 81/19*w - 5],
[101, 101, 9/19*w^3 - 4/19*w^2 - 100/19*w],
[101, 101, 7/19*w^3 - 20/19*w^2 - 44/19*w + 6],
[121, 11, 6/19*w^3 - 9/19*w^2 - 54/19*w + 1],
[121, 11, -6/19*w^3 + 9/19*w^2 + 54/19*w - 2],
[139, 139, 9/19*w^3 - 4/19*w^2 - 100/19*w - 2],
[139, 139, -7/19*w^3 + 1/19*w^2 + 63/19*w + 1],
[139, 139, 7/19*w^3 - 20/19*w^2 - 44/19*w + 4],
[139, 139, w - 5],
[149, 149, -1/19*w^3 + 11/19*w^2 - 10/19*w - 6],
[149, 149, -3/19*w^3 - 5/19*w^2 + 46/19*w],
[149, 149, -3/19*w^3 + 14/19*w^2 + 27/19*w - 2],
[149, 149, 1/19*w^3 + 8/19*w^2 - 9/19*w - 6],
[169, 13, 14/19*w^3 - 40/19*w^2 - 126/19*w + 13],
[169, 13, 10/19*w^3 - 34/19*w^2 - 52/19*w + 7],
[191, 191, -10/19*w^3 + 15/19*w^2 + 109/19*w - 6],
[191, 191, 6/19*w^3 - 9/19*w^2 - 35/19*w - 2],
[191, 191, -6/19*w^3 + 9/19*w^2 + 35/19*w - 4],
[191, 191, 10/19*w^3 - 15/19*w^2 - 109/19*w],
[211, 211, 1/19*w^3 + 8/19*w^2 - 28/19*w - 8],
[211, 211, 1/19*w^3 - 11/19*w^2 - 9/19*w - 1],
[211, 211, -1/19*w^3 - 8/19*w^2 + 28/19*w - 2],
[211, 211, -1/19*w^3 + 11/19*w^2 + 9/19*w - 9],
[239, 239, 4/19*w^3 - 25/19*w^2 - 17/19*w + 12],
[239, 239, -1/19*w^3 - 8/19*w^2 - 29/19*w + 5],
[239, 239, 10/19*w^3 - 15/19*w^2 - 128/19*w + 1],
[239, 239, -4/19*w^3 - 13/19*w^2 + 55/19*w + 10],
[241, 241, -2/19*w^3 + 3/19*w^2 + 56/19*w - 4],
[241, 241, -6/19*w^3 + 9/19*w^2 + 92/19*w],
[241, 241, 6/19*w^3 - 9/19*w^2 - 92/19*w + 5],
[241, 241, -2/19*w^3 + 3/19*w^2 + 56/19*w + 1],
[269, 269, -10/19*w^3 + 34/19*w^2 + 71/19*w - 11],
[269, 269, 8/19*w^3 - 12/19*w^2 - 129/19*w + 11],
[269, 269, -10/19*w^3 + 34/19*w^2 + 71/19*w - 8],
[269, 269, -13/19*w^3 + 48/19*w^2 + 60/19*w - 8],
[289, 17, 30/19*w^3 - 7/19*w^2 - 384/19*w - 14],
[289, 17, 11/19*w^3 - 45/19*w^2 - 80/19*w + 17],
[311, 311, -12/19*w^3 + 37/19*w^2 + 108/19*w - 15],
[311, 311, -24/19*w^3 + 55/19*w^2 + 273/19*w - 22],
[311, 311, -8/19*w^3 - 7/19*w^2 + 110/19*w + 4],
[311, 311, 4/19*w^3 + 13/19*w^2 - 36/19*w - 7],
[331, 331, -1/19*w^3 - 8/19*w^2 + 66/19*w - 2],
[331, 331, -6/19*w^3 + 28/19*w^2 + 35/19*w - 9],
[331, 331, 20/19*w^3 - 68/19*w^2 - 161/19*w + 22],
[331, 331, 16/19*w^3 - 62/19*w^2 - 87/19*w + 15],
[359, 359, 13/19*w^3 - 48/19*w^2 - 79/19*w + 11],
[359, 359, -7/19*w^3 + 20/19*w^2 + 101/19*w - 13],
[359, 359, -15/19*w^3 + 51/19*w^2 + 116/19*w - 17],
[359, 359, 3/19*w^3 + 5/19*w^2 - 84/19*w + 5],
[379, 379, 3/19*w^3 + 5/19*w^2 - 65/19*w - 9],
[379, 379, 25/19*w^3 - 9/19*w^2 - 339/19*w - 11],
[379, 379, -25/19*w^3 + 66/19*w^2 + 282/19*w - 28],
[379, 379, 3/19*w^3 - 14/19*w^2 - 46/19*w + 12],
[389, 389, -13/19*w^3 + 48/19*w^2 + 136/19*w - 24],
[389, 389, 11/19*w^3 - 7/19*w^2 - 118/19*w + 2],
[389, 389, -3/19*w^3 + 14/19*w^2 + 46/19*w - 10],
[389, 389, -13/19*w^3 - 9/19*w^2 + 193/19*w + 15],
[409, 409, 2/19*w^3 - 3/19*w^2 - 56/19*w],
[409, 409, 6/19*w^3 - 9/19*w^2 - 92/19*w + 4],
[409, 409, -6/19*w^3 + 9/19*w^2 + 92/19*w - 1],
[409, 409, -2/19*w^3 + 3/19*w^2 + 56/19*w - 3],
[431, 431, -6/19*w^3 + 9/19*w^2 + 130/19*w - 13],
[431, 431, 12/19*w^3 - 37/19*w^2 - 108/19*w + 11],
[431, 431, 10/19*w^3 - 15/19*w^2 - 166/19*w + 14],
[431, 431, -8/19*w^3 + 31/19*w^2 + 34/19*w - 8],
[461, 461, 5/19*w^3 - 17/19*w^2 - 45/19*w + 1],
[461, 461, -3/19*w^3 + 14/19*w^2 + 8/19*w - 8],
[461, 461, 3/19*w^3 + 5/19*w^2 - 27/19*w - 7],
[461, 461, 5/19*w^3 + 2/19*w^2 - 64/19*w + 2],
[479, 479, 5/19*w^3 - 17/19*w^2 - 26/19*w + 7],
[479, 479, 1/19*w^3 - 11/19*w^2 - 28/19*w + 7],
[479, 479, 7/19*w^3 - 1/19*w^2 - 82/19*w + 1],
[479, 479, -5/19*w^3 + 17/19*w^2 + 64/19*w - 3],
[499, 499, -11/19*w^3 + 7/19*w^2 + 118/19*w + 2],
[499, 499, 9/19*w^3 - 23/19*w^2 - 62/19*w + 4],
[499, 499, -9/19*w^3 + 4/19*w^2 + 81/19*w],
[499, 499, -11/19*w^3 + 26/19*w^2 + 99/19*w - 8],
[509, 509, -6/19*w^3 - 10/19*w^2 + 73/19*w + 9],
[509, 509, -1/19*w^3 + 11/19*w^2 + 47/19*w - 7],
[509, 509, -6/19*w^3 - 10/19*w^2 + 73/19*w + 2],
[509, 509, -9/19*w^3 + 4/19*w^2 + 138/19*w - 2],
[529, 23, -4/19*w^3 + 6/19*w^2 + 74/19*w - 1],
[529, 23, 4/19*w^3 - 6/19*w^2 - 74/19*w + 3],
[571, 571, 1/19*w^3 + 8/19*w^2 - 66/19*w + 6],
[571, 571, -9/19*w^3 + 23/19*w^2 + 119/19*w - 15],
[571, 571, -11/19*w^3 + 26/19*w^2 + 156/19*w - 18],
[571, 571, 3/19*w^3 + 5/19*w^2 - 103/19*w + 9],
[599, 599, 6/19*w^3 - 9/19*w^2 - 16/19*w - 2],
[599, 599, -14/19*w^3 + 21/19*w^2 + 164/19*w - 7],
[599, 599, -2/19*w^3 + 22/19*w^2 + 18/19*w - 13],
[599, 599, -6/19*w^3 + 9/19*w^2 + 16/19*w - 3],
[601, 601, 23/19*w^3 - 63/19*w^2 - 264/19*w + 31],
[601, 601, -36/19*w^3 - 3/19*w^2 + 476/19*w + 23],
[601, 601, 36/19*w^3 - 111/19*w^2 - 362/19*w + 46],
[601, 601, -16/19*w^3 + 43/19*w^2 + 125/19*w - 8],
[619, 619, -5/19*w^3 + 17/19*w^2 + 45/19*w - 11],
[619, 619, 5/19*w^3 - 17/19*w^2 - 26/19*w - 2],
[619, 619, -5/19*w^3 - 2/19*w^2 + 45/19*w - 4],
[619, 619, -7/19*w^3 + 20/19*w^2 + 63/19*w - 12],
[691, 691, 7/19*w^3 - 1/19*w^2 - 120/19*w],
[691, 691, 3/19*w^3 - 14/19*w^2 - 65/19*w + 6],
[691, 691, -3/19*w^3 - 5/19*w^2 + 84/19*w + 2],
[691, 691, 7/19*w^3 - 20/19*w^2 - 101/19*w + 6],
[701, 701, -7/19*w^3 + 20/19*w^2 + 63/19*w - 2],
[701, 701, 5/19*w^3 - 17/19*w^2 - 26/19*w + 8],
[701, 701, -7/19*w^3 + 20/19*w^2 + 82/19*w - 4],
[701, 701, -7/19*w^3 + 1/19*w^2 + 82/19*w - 2],
[719, 719, -8/19*w^3 + 31/19*w^2 + 53/19*w - 12],
[719, 719, 8/19*w^3 - 31/19*w^2 - 53/19*w + 6],
[719, 719, -8/19*w^3 - 7/19*w^2 + 91/19*w + 2],
[719, 719, 8/19*w^3 + 7/19*w^2 - 91/19*w - 8],
[739, 739, 9/19*w^3 - 23/19*w^2 - 43/19*w + 3],
[739, 739, -16/19*w^3 + 24/19*w^2 + 163/19*w - 9],
[739, 739, 16/19*w^3 - 24/19*w^2 - 163/19*w],
[739, 739, -9/19*w^3 + 4/19*w^2 + 62/19*w],
[769, 769, 24/19*w^3 - 74/19*w^2 - 254/19*w + 35],
[769, 769, -3/19*w^3 + 33/19*w^2 - 30/19*w - 11],
[769, 769, 3/19*w^3 + 24/19*w^2 - 27/19*w - 11],
[769, 769, 24/19*w^3 + 2/19*w^2 - 330/19*w - 19],
[811, 811, 5/19*w^3 + 2/19*w^2 - 102/19*w],
[811, 811, -5/19*w^3 + 17/19*w^2 + 83/19*w - 7],
[811, 811, 5/19*w^3 + 2/19*w^2 - 102/19*w - 2],
[811, 811, -5/19*w^3 + 17/19*w^2 + 83/19*w - 5],
[821, 821, -6/19*w^3 - 10/19*w^2 + 111/19*w + 13],
[821, 821, 1/19*w^3 + 8/19*w^2 + 10/19*w - 12],
[821, 821, -1/19*w^3 + 11/19*w^2 - 29/19*w - 11],
[821, 821, -6/19*w^3 + 28/19*w^2 + 73/19*w - 18],
[839, 839, -2*w^3 + 26*w + 23],
[839, 839, 18/19*w^3 - 65/19*w^2 - 143/19*w + 24],
[839, 839, -18/19*w^3 - 11/19*w^2 + 219/19*w + 14],
[839, 839, 2*w^3 - 6*w^2 - 20*w + 47],
[859, 859, 10/19*w^3 - 34/19*w^2 - 109/19*w + 19],
[859, 859, 3/19*w^3 + 24/19*w^2 - 46/19*w - 14],
[859, 859, -17/19*w^3 + 35/19*w^2 + 210/19*w - 14],
[859, 859, -10/19*w^3 - 4/19*w^2 + 147/19*w + 12],
[911, 911, -1/19*w^3 + 11/19*w^2 + 28/19*w - 9],
[911, 911, -5/19*w^3 + 17/19*w^2 + 64/19*w - 1],
[911, 911, 5/19*w^3 + 2/19*w^2 - 83/19*w + 3],
[911, 911, 1/19*w^3 + 8/19*w^2 - 47/19*w - 7],
[941, 941, 11/19*w^3 - 7/19*w^2 - 99/19*w - 5],
[941, 941, -1/19*w^3 + 11/19*w^2 + 28/19*w - 10],
[941, 941, -13/19*w^3 + 10/19*w^2 + 136/19*w - 4],
[941, 941, -6/19*w^3 + 28/19*w^2 - 3/19*w - 6],
[961, 31, 10/19*w^3 - 15/19*w^2 - 90/19*w + 2],
[961, 31, -10/19*w^3 + 15/19*w^2 + 90/19*w - 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - 32*x^6 + 334*x^4 - 1143*x^2 + 11;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [4/39*e^6 - 31/13*e^4 + 532/39*e^2 - 112/39, e, -7/351*e^7 + 10/13*e^5 - 3310/351*e^3 + 12754/351*e, -19/351*e^6 + 16/13*e^4 - 2215/351*e^2 - 482/351, 61/351*e^6 - 50/13*e^4 + 6982/351*e^2 + 2231/351, -1, 61/351*e^6 - 50/13*e^4 + 6982/351*e^2 + 2231/351, -44/351*e^6 + 35/13*e^4 - 4760/351*e^2 - 211/351, -8/117*e^7 + 25/13*e^5 - 1961/117*e^3 + 5333/117*e, 23/351*e^7 - 18/13*e^5 + 1850/351*e^3 + 5830/351*e, 4/27*e^7 - 4*e^5 + 877/27*e^3 - 2050/27*e, -31/351*e^7 + 35/13*e^5 - 9193/351*e^3 + 28402/351*e, 110/351*e^6 - 94/13*e^4 + 14357/351*e^2 - 2456/351, -148/351*e^6 + 126/13*e^4 - 18787/351*e^2 + 1141/351, -7/351*e^7 - 3/13*e^5 + 4763/351*e^3 - 31472/351*e, 7/351*e^7 + 3/13*e^5 - 4412/351*e^3 + 27260/351*e, -58/351*e^7 + 55/13*e^5 - 11029/351*e^3 + 21085/351*e, 4/39*e^7 - 31/13*e^5 + 571/39*e^3 - 580/39*e, -38/351*e^7 + 45/13*e^5 - 12854/351*e^3 + 45719/351*e, 43/351*e^7 - 41/13*e^5 + 8800/351*e^3 - 22186/351*e, -62/351*e^7 + 70/13*e^5 - 18035/351*e^3 + 52943/351*e, -8/117*e^7 + 25/13*e^5 - 2078/117*e^3 + 6971/117*e, 7/117*e^6 - 17/13*e^4 + 619/117*e^2 + 1286/117, -44/117*e^6 + 105/13*e^4 - 4643/117*e^2 - 1849/117, 10/351*e^6 - 5/13*e^4 + 316/351*e^2 - 319/351, 4/117*e^6 - 6/13*e^4 - 14/117*e^2 + 2189/117, -103/351*e^6 + 84/13*e^4 - 12451/351*e^2 + 4444/351, -46/351*e^6 + 36/13*e^4 - 4753/351*e^2 - 77/351, -43/351*e^7 + 41/13*e^5 - 8449/351*e^3 + 18676/351*e, -34/351*e^7 + 17/13*e^5 + 3278/351*e^3 - 43001/351*e, -46/351*e^7 + 36/13*e^5 - 4402/351*e^3 - 5342/351*e, -10/351*e^7 + 18/13*e^5 - 7336/351*e^3 + 33313/351*e, 227/351*e^6 - 198/13*e^4 + 30620/351*e^2 - 4679/351, -70/351*e^6 + 61/13*e^4 - 8530/351*e^2 - 4085/351, -8/39*e^7 + 62/13*e^5 - 1103/39*e^3 + 614/39*e, 8/351*e^7 - 30/13*e^5 + 15416/351*e^3 - 79511/351*e, -37/351*e^7 + 38/13*e^5 - 8470/351*e^3 + 21082/351*e, 17/351*e^7 - 15/13*e^5 + 1871/351*e^3 + 4477/351*e, -109/351*e^6 + 87/13*e^4 - 12430/351*e^2 + 8005/351, 50/351*e^6 - 38/13*e^4 + 4739/351*e^2 + 3319/351, -257/351*e^6 + 226/13*e^4 - 35078/351*e^2 + 4232/351, -37/117*e^6 + 88/13*e^4 - 4024/117*e^2 + 373/117, -2/351*e^7 + 27/13*e^5 - 15437/351*e^3 + 80864/351*e, 31/351*e^7 - 48/13*e^5 + 16564/351*e^3 - 65608/351*e, -139/351*e^7 + 141/13*e^5 - 31981/351*e^3 + 80917/351*e, 1/27*e^7 - e^5 + 199/27*e^3 - 337/27*e, -92/351*e^6 + 85/13*e^4 - 13718/351*e^2 + 1250/351, 13/27*e^6 - 11*e^4 + 1669/27*e^2 + 20/27, -146/351*e^6 + 125/13*e^4 - 19145/351*e^2 + 8378/351, -4/27*e^6 + 3*e^4 - 391/27*e^2 - 2/27, -1/117*e^7 - 18/13*e^5 + 4040/117*e^3 - 22748/117*e, 19/117*e^7 - 61/13*e^5 + 5023/117*e^3 - 14260/117*e, -7/117*e^7 + 17/13*e^5 - 619/117*e^3 - 2339/117*e, -34/117*e^7 + 103/13*e^5 - 7603/117*e^3 + 18307/117*e, 59/351*e^6 - 49/13*e^4 + 7691/351*e^2 - 8516/351, -157/351*e^6 + 124/13*e^4 - 17176/351*e^2 + 691/351, -23/351*e^7 + 31/13*e^5 - 9923/351*e^3 + 37343/351*e, -23/351*e^7 + 5/13*e^5 + 5872/351*e^3 - 46195/351*e, 46/351*e^7 - 62/13*e^5 + 19846/351*e^3 - 76441/351*e, 4/39*e^7 - 57/13*e^5 + 2287/39*e^3 - 9784/39*e, -25/117*e^6 + 70/13*e^4 - 3481/117*e^2 - 3239/117, -3/13*e^6 + 73/13*e^4 - 438/13*e^2 + 162/13, 41/117*e^6 - 107/13*e^4 + 5297/117*e^2 + 2635/117, -242/351*e^6 + 199/13*e^4 - 28637/351*e^2 + 2174/351, -20/351*e^7 + 10/13*e^5 + 2527/351*e^3 - 32707/351*e, 64/117*e^7 - 187/13*e^5 + 13348/117*e^3 - 30847/117*e, -188/351*e^7 + 172/13*e^5 - 31985/351*e^3 + 49100/351*e, 47/117*e^7 - 155/13*e^5 + 13349/117*e^3 - 41174/117*e, -67/117*e^6 + 172/13*e^4 - 8248/117*e^2 - 2297/117, -101/351*e^6 + 83/13*e^4 - 11756/351*e^2 + 5363/351, 7/351*e^6 + 3/13*e^4 - 902/351*e^2 - 8893/351, 85/117*e^6 - 212/13*e^4 + 10525/117*e^2 - 4057/117, -23/351*e^7 + 18/13*e^5 - 1850/351*e^3 - 4075/351*e, 80/351*e^7 - 92/13*e^5 + 24641/351*e^3 - 80123/351*e, -155/351*e^7 + 162/13*e^5 - 39998/351*e^3 + 122354/351*e, 1/27*e^7 - 422/27*e^3 + 3119/27*e, -128/351*e^6 + 103/13*e^4 - 14294/351*e^2 + 3311/351, 80/351*e^6 - 66/13*e^4 + 9548/351*e^2 + 1309/351, 548/351*e^6 - 456/13*e^4 + 66176/351*e^2 + 2596/351, -283/351*e^6 + 239/13*e^4 - 35338/351*e^2 + 2464/351, -113/351*e^7 + 102/13*e^5 - 18383/351*e^3 + 23717/351*e, 35/351*e^7 - 37/13*e^5 + 8828/351*e^3 - 25160/351*e, -1/117*e^7 + 21/13*e^5 - 3916/117*e^3 + 21010/117*e, 70/117*e^7 - 196/13*e^5 + 12625/117*e^3 - 21070/117*e, 113/351*e^7 - 89/13*e^5 + 9959/351*e^3 + 23317/351*e, -16/117*e^7 + 50/13*e^5 - 3922/117*e^3 + 10198/117*e, 44/351*e^7 - 48/13*e^5 + 11078/351*e^3 - 24008/351*e, -62/351*e^7 + 18/13*e^5 + 12502/351*e^3 - 105709/351*e, 43/351*e^7 - 41/13*e^5 + 9502/351*e^3 - 30961/351*e, -121/351*e^7 + 145/13*e^5 - 42223/351*e^3 + 154123/351*e, 166/351*e^7 - 174/13*e^5 + 40837/351*e^3 - 105190/351*e, 19/351*e^7 - 29/13*e^5 + 9937/351*e^3 - 39181/351*e, 49/351*e^6 - 44/13*e^4 + 6322/351*e^2 + 3737/351, 422/351*e^6 - 354/13*e^4 + 51875/351*e^2 + 2263/351, -16/39*e^6 + 137/13*e^4 - 2635/39*e^2 + 1228/39, -86/117*e^6 + 207/13*e^4 - 9293/117*e^2 - 3598/117, -40/117*e^7 + 138/13*e^5 - 12379/117*e^3 + 39301/117*e, 34/117*e^7 - 103/13*e^5 + 7837/117*e^3 - 21349/117*e, 62/351*e^7 - 31/13*e^5 - 6184/351*e^3 + 79384/351*e, 13/27*e^7 - 12*e^5 + 2290/27*e^3 - 3922/27*e, 164/351*e^6 - 134/13*e^4 + 18380/351*e^2 + 4105/351, -352/351*e^6 + 306/13*e^4 - 47908/351*e^2 + 11299/351, -11/117*e^6 + 36/13*e^4 - 2594/117*e^2 + 2024/117, -188/351*e^6 + 146/13*e^4 - 20051/351*e^2 + 9086/351, 79/117*e^6 - 203/13*e^4 + 10429/117*e^2 - 2134/117, 224/351*e^6 - 190/13*e^4 + 28349/351*e^2 - 8690/351, 2/9*e^7 - 8*e^5 + 821/9*e^3 - 2978/9*e, -251/351*e^7 + 249/13*e^5 - 54404/351*e^3 + 128084/351*e, 76/351*e^7 - 103/13*e^5 + 33781/351*e^3 - 134962/351*e, 37/351*e^7 + 1/13*e^5 - 14696/351*e^3 + 103172/351*e, -20/39*e^6 + 155/13*e^4 - 2543/39*e^2 + 560/39, 16/351*e^6 - 8/13*e^4 + 646/351*e^2 - 3529/351, -5/39*e^6 + 29/13*e^4 - 353/39*e^2 + 725/39, 40/39*e^6 - 310/13*e^4 + 5203/39*e^2 - 925/39, 212/351*e^6 - 184/13*e^4 + 27338/351*e^2 + 1591/351, -406/351*e^6 + 333/13*e^4 - 48421/351*e^2 + 8950/351, 392/351*e^6 - 326/13*e^4 + 48470/351*e^2 - 8012/351, -94/351*e^6 + 73/13*e^4 - 9850/351*e^2 - 1775/351, 205/351*e^6 - 161/13*e^4 + 21220/351*e^2 + 2411/351, -97/351*e^6 + 81/13*e^4 - 11068/351*e^2 - 2627/351, -560/351*e^6 + 475/13*e^4 - 68942/351*e^2 - 13726/351, 307/351*e^6 - 264/13*e^4 + 39466/351*e^2 + 6107/351, -98/351*e^7 + 62/13*e^5 + 694/351*e^3 - 71005/351*e, -11/39*e^7 + 82/13*e^5 - 1268/39*e^3 - 511/39*e, -140/351*e^7 + 135/13*e^5 - 27941/351*e^3 + 59573/351*e, -239/351*e^7 + 217/13*e^5 - 39002/351*e^3 + 49007/351*e, 14/117*e^7 - 21/13*e^5 - 1102/117*e^3 + 15208/117*e, -6/13*e^7 + 172/13*e^5 - 1526/13*e^3 + 4081/13*e, -2/39*e^7 + 61/13*e^5 - 3503/39*e^3 + 18425/39*e, -125/351*e^7 + 147/13*e^5 - 42209/351*e^3 + 150530/351*e, 28/39*e^6 - 230/13*e^4 + 4270/39*e^2 - 1876/39, -35/27*e^6 + 30*e^4 - 4481/27*e^2 + 347/27, -139/117*e^6 + 358/13*e^4 - 18292/117*e^2 + 3931/117, -4/9*e^6 + 10*e^4 - 499/9*e^2 + 277/9, -34/117*e^6 + 77/13*e^4 - 3391/117*e^2 + 2629/117, -107/351*e^6 + 99/13*e^4 - 17702/351*e^2 + 13838/351, 68/39*e^6 - 527/13*e^4 + 8810/39*e^2 - 110/39, -415/351*e^6 + 344/13*e^4 - 48916/351*e^2 - 11858/351, -478/351*e^6 + 395/13*e^4 - 56242/351*e^2 - 6935/351, -1/39*e^6 - 2/13*e^4 + 335/39*e^2 - 1103/39, -161/117*e^6 + 404/13*e^4 - 19385/117*e^2 - 5476/117, 67/117*e^6 - 159/13*e^4 + 6961/117*e^2 + 3467/117, -5/9*e^7 + 13*e^5 - 680/9*e^3 + 272/9*e, -92/351*e^7 + 98/13*e^5 - 24599/351*e^3 + 74258/351*e, 11/117*e^7 - 10/13*e^5 - 1852/117*e^3 + 14590/117*e, -275/351*e^7 + 261/13*e^5 - 51863/351*e^3 + 95645/351*e, 23/117*e^7 - 106/13*e^5 + 12731/117*e^3 - 53021/117*e, 64/351*e^7 - 84/13*e^5 + 26803/351*e^3 - 106429/351*e, 149/351*e^7 - 172/13*e^5 + 46337/351*e^3 - 148628/351*e, -19/117*e^7 + 35/13*e^5 + 8/117*e^3 - 11597/117*e, 37/117*e^6 - 101/13*e^4 + 5779/117*e^2 - 1660/117, 74/117*e^6 - 176/13*e^4 + 8282/117*e^2 - 395/117, 605/351*e^6 - 517/13*e^4 + 77033/351*e^2 - 2978/351, -512/351*e^6 + 425/13*e^4 - 60335/351*e^2 - 7114/351, -11/351*e^7 - 53/13*e^5 + 37069/351*e^3 - 210916/351*e, 202/351*e^7 - 179/13*e^5 + 29830/351*e^3 - 24415/351*e, -43/351*e^7 + 80/13*e^5 - 32668/351*e^3 + 154513/351*e, 71/117*e^7 - 204/13*e^5 + 13733/117*e^3 - 26402/117*e, -7/117*e^7 + 56/13*e^5 - 8926/117*e^3 + 45280/117*e, -23/117*e^7 + 93/13*e^5 - 10508/117*e^3 + 44948/117*e, -19/117*e^7 + 74/13*e^5 - 8182/117*e^3 + 34384/117*e, 82/351*e^7 - 54/13*e^5 + 1819/351*e^3 + 41891/351*e, -344/351*e^6 + 289/13*e^4 - 41969/351*e^2 - 8191/351, -184/351*e^6 + 157/13*e^4 - 22171/351*e^2 - 17156/351];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {19,19,4/19*w^3 - 6/19*w^2 - 55/19*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;