/*
  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![29, 16, -15, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {16, 2, 2}>;

primesArray := [
[4, 2, -2/23*w^3 + 3/23*w^2 - 3/23*w + 1/23],
[9, 3, -2/23*w^3 + 3/23*w^2 + 43/23*w + 24/23],
[9, 3, -2/23*w^3 + 3/23*w^2 + 43/23*w - 68/23],
[19, 19, -2/23*w^3 + 3/23*w^2 - 3/23*w + 24/23],
[19, 19, -6/23*w^3 + 9/23*w^2 + 83/23*w - 20/23],
[19, 19, 6/23*w^3 - 9/23*w^2 - 83/23*w + 66/23],
[19, 19, -2/23*w^3 + 3/23*w^2 - 3/23*w - 22/23],
[25, 5, -4/23*w^3 + 6/23*w^2 + 40/23*w - 21/23],
[29, 29, w],
[29, 29, -4/23*w^3 + 6/23*w^2 + 63/23*w - 21/23],
[29, 29, 4/23*w^3 - 6/23*w^2 - 63/23*w + 44/23],
[29, 29, -w + 1],
[31, 31, w + 2],
[31, 31, -4/23*w^3 + 6/23*w^2 + 63/23*w + 25/23],
[31, 31, 4/23*w^3 - 6/23*w^2 - 63/23*w + 90/23],
[31, 31, -w + 3],
[49, 7, -2/23*w^3 + 3/23*w^2 + 43/23*w - 22/23],
[59, 59, -20/23*w^3 + 30/23*w^2 + 269/23*w - 128/23],
[59, 59, -8/23*w^3 + 12/23*w^2 + 103/23*w - 88/23],
[59, 59, -8/23*w^3 + 12/23*w^2 + 103/23*w - 19/23],
[59, 59, -4/23*w^3 + 6/23*w^2 + 17/23*w - 44/23],
[109, 109, -9/23*w^3 + 2/23*w^2 + 113/23*w + 62/23],
[109, 109, 4/23*w^3 - 6/23*w^2 - 63/23*w - 71/23],
[109, 109, 7/23*w^3 - 22/23*w^2 - 47/23*w + 100/23],
[109, 109, 9/23*w^3 - 25/23*w^2 - 90/23*w + 168/23],
[121, 11, -2/23*w^3 + 3/23*w^2 + 20/23*w - 91/23],
[121, 11, 2/23*w^3 - 3/23*w^2 - 20/23*w - 70/23],
[131, 131, -5/23*w^3 + 19/23*w^2 + 50/23*w - 101/23],
[131, 131, -3/23*w^3 + 16/23*w^2 + 7/23*w - 125/23],
[131, 131, 3/23*w^3 + 7/23*w^2 - 30/23*w - 105/23],
[131, 131, 5/23*w^3 + 4/23*w^2 - 73/23*w - 37/23],
[139, 139, -2/23*w^3 - 20/23*w^2 + 43/23*w + 323/23],
[139, 139, 17/23*w^3 - 37/23*w^2 - 216/23*w + 302/23],
[139, 139, 17/23*w^3 - 14/23*w^2 - 239/23*w - 66/23],
[139, 139, -2/23*w^3 + 26/23*w^2 - 3/23*w - 344/23],
[149, 149, -1/23*w^3 - 10/23*w^2 + 33/23*w - 34/23],
[149, 149, 1/23*w^3 + 10/23*w^2 - 33/23*w - 196/23],
[149, 149, -1/23*w^3 + 13/23*w^2 + 10/23*w - 218/23],
[149, 149, 1/23*w^3 - 13/23*w^2 - 10/23*w - 12/23],
[169, 13, 6/23*w^3 - 9/23*w^2 - 37/23*w + 20/23],
[169, 13, -10/23*w^3 + 15/23*w^2 + 123/23*w - 64/23],
[199, 199, -1/23*w^3 + 13/23*w^2 - 13/23*w - 195/23],
[199, 199, 3/23*w^3 + 7/23*w^2 - 53/23*w + 33/23],
[199, 199, 3/23*w^3 - 16/23*w^2 - 30/23*w + 10/23],
[199, 199, 1/23*w^3 + 10/23*w^2 - 10/23*w - 196/23],
[251, 251, 6/23*w^3 - 9/23*w^2 - 14/23*w + 43/23],
[251, 251, 14/23*w^3 - 21/23*w^2 - 186/23*w + 62/23],
[251, 251, 4/23*w^3 - 6/23*w^2 - 109/23*w + 228/23],
[251, 251, 6/23*w^3 - 9/23*w^2 - 14/23*w - 26/23],
[271, 271, -8/23*w^3 + 12/23*w^2 + 126/23*w + 27/23],
[271, 271, 8/23*w^3 - 35/23*w^2 - 80/23*w + 226/23],
[271, 271, 13/23*w^3 - 31/23*w^2 - 153/23*w + 189/23],
[271, 271, 8/23*w^3 - 12/23*w^2 - 126/23*w + 157/23],
[281, 281, -w^2 + 2*w + 9],
[281, 281, 13/23*w^3 - 8/23*w^2 - 153/23*w + 28/23],
[281, 281, -13/23*w^3 + 31/23*w^2 + 130/23*w - 120/23],
[281, 281, 11/23*w^3 - 28/23*w^2 - 87/23*w + 190/23],
[289, 17, -11/23*w^3 + 51/23*w^2 + 110/23*w - 581/23],
[289, 17, -8/23*w^3 + 35/23*w^2 + 80/23*w - 410/23],
[311, 311, 5/23*w^3 + 4/23*w^2 - 50/23*w - 106/23],
[311, 311, -7/23*w^3 + 22/23*w^2 + 70/23*w - 100/23],
[311, 311, 7/23*w^3 + 1/23*w^2 - 93/23*w - 15/23],
[311, 311, -5/23*w^3 + 19/23*w^2 + 27/23*w - 147/23],
[389, 389, -8/23*w^3 + 35/23*w^2 + 80/23*w - 295/23],
[389, 389, -4/23*w^3 + 29/23*w^2 - 6/23*w - 136/23],
[389, 389, -4/23*w^3 - 17/23*w^2 + 40/23*w + 117/23],
[389, 389, 2/23*w^3 + 20/23*w^2 - 20/23*w - 185/23],
[401, 401, 11/23*w^3 - 5/23*w^2 - 179/23*w - 17/23],
[401, 401, 2/23*w^3 + 20/23*w^2 - 43/23*w - 93/23],
[401, 401, 1/23*w^3 + 10/23*w^2 + 36/23*w - 104/23],
[401, 401, 11/23*w^3 - 28/23*w^2 - 156/23*w + 190/23],
[419, 419, -6/23*w^3 - 14/23*w^2 + 83/23*w + 72/23],
[419, 419, 3/23*w^3 + 7/23*w^2 - 76/23*w + 79/23],
[419, 419, -6/23*w^3 + 32/23*w^2 + 37/23*w - 296/23],
[419, 419, -1/23*w^3 + 13/23*w^2 + 56/23*w - 172/23],
[421, 421, -8/23*w^3 + 12/23*w^2 + 57/23*w - 65/23],
[421, 421, -12/23*w^3 + 18/23*w^2 + 143/23*w - 109/23],
[421, 421, 12/23*w^3 - 18/23*w^2 - 143/23*w + 40/23],
[421, 421, 8/23*w^3 - 12/23*w^2 - 57/23*w - 4/23],
[439, 439, 7/23*w^3 - 22/23*w^2 - 93/23*w + 123/23],
[439, 439, 1/23*w^3 + 10/23*w^2 + 13/23*w - 127/23],
[439, 439, 5/23*w^3 + 4/23*w^2 - 73/23*w + 9/23],
[439, 439, 7/23*w^3 + 1/23*w^2 - 116/23*w - 15/23],
[449, 449, -11/23*w^3 + 5/23*w^2 + 133/23*w + 63/23],
[449, 449, -9/23*w^3 + 2/23*w^2 + 90/23*w + 16/23],
[449, 449, 9/23*w^3 - 25/23*w^2 - 67/23*w + 99/23],
[449, 449, -11/23*w^3 + 28/23*w^2 + 110/23*w - 190/23],
[479, 479, 10/23*w^3 - 15/23*w^2 - 123/23*w - 74/23],
[479, 479, 6/23*w^3 - 9/23*w^2 - 37/23*w + 158/23],
[479, 479, -6/23*w^3 + 9/23*w^2 + 37/23*w + 118/23],
[479, 479, 10/23*w^3 - 15/23*w^2 - 123/23*w + 202/23],
[529, 23, -4/23*w^3 + 29/23*w^2 + 40/23*w - 274/23],
[529, 23, w^2 - 6],
[541, 541, 6/23*w^3 + 14/23*w^2 - 83/23*w - 141/23],
[541, 541, -6/23*w^3 + 32/23*w^2 + 37/23*w - 227/23],
[541, 541, 6/23*w^3 + 14/23*w^2 - 83/23*w - 164/23],
[541, 541, -6/23*w^3 + 32/23*w^2 + 37/23*w - 204/23],
[569, 569, -5/23*w^3 - 4/23*w^2 + 50/23*w - 78/23],
[569, 569, -7/23*w^3 + 22/23*w^2 + 70/23*w - 284/23],
[569, 569, -5/23*w^3 + 19/23*w^2 + 50/23*w - 262/23],
[569, 569, 5/23*w^3 - 19/23*w^2 - 27/23*w - 37/23],
[619, 619, -2/23*w^3 - 43/23*w^2 + 66/23*w + 622/23],
[619, 619, 17/23*w^3 - 37/23*w^2 - 216/23*w + 233/23],
[619, 619, 17/23*w^3 - 14/23*w^2 - 239/23*w + 3/23],
[619, 619, -2/23*w^3 + 49/23*w^2 - 26/23*w - 643/23],
[641, 641, -1/23*w^3 + 13/23*w^2 - 59/23*w + 127/23],
[641, 641, -14/23*w^3 + 44/23*w^2 + 117/23*w - 246/23],
[641, 641, -11/23*w^3 + 5/23*w^2 + 179/23*w + 201/23],
[641, 641, -1/23*w^3 - 10/23*w^2 - 36/23*w - 80/23],
[691, 691, 5/23*w^3 - 65/23*w^2 - 4/23*w + 837/23],
[691, 691, -2/23*w^3 + 26/23*w^2 + 20/23*w - 321/23],
[691, 691, -2/23*w^3 - 20/23*w^2 + 66/23*w + 277/23],
[691, 691, -5/23*w^3 - 50/23*w^2 + 119/23*w + 773/23],
[701, 701, 2/23*w^3 + 20/23*w^2 - 66/23*w - 185/23],
[701, 701, w^2 - 7],
[701, 701, -2/23*w^3 + 26/23*w^2 + 20/23*w - 183/23],
[701, 701, -2/23*w^3 + 26/23*w^2 + 20/23*w - 229/23],
[709, 709, -18/23*w^3 + 27/23*w^2 + 226/23*w - 152/23],
[709, 709, -33/23*w^3 + 38/23*w^2 + 445/23*w - 41/23],
[709, 709, -33/23*w^3 + 61/23*w^2 + 422/23*w - 409/23],
[709, 709, 18/23*w^3 - 27/23*w^2 - 226/23*w + 83/23],
[719, 719, -4/23*w^3 + 29/23*w^2 + 40/23*w - 136/23],
[719, 719, 12/23*w^3 - 18/23*w^2 - 166/23*w + 155/23],
[719, 719, w^2 - 12],
[719, 719, 4/23*w^3 + 17/23*w^2 - 86/23*w - 71/23],
[809, 809, -10/23*w^3 + 15/23*w^2 + 123/23*w - 248/23],
[809, 809, 7/23*w^3 - 22/23*w^2 - 93/23*w + 307/23],
[809, 809, 7/23*w^3 + 1/23*w^2 - 116/23*w - 199/23],
[809, 809, 10/23*w^3 - 15/23*w^2 - 123/23*w - 120/23],
[811, 811, -7/23*w^3 - 1/23*w^2 + 93/23*w - 31/23],
[811, 811, -7/23*w^3 + 22/23*w^2 + 93/23*w - 100/23],
[811, 811, 7/23*w^3 + 1/23*w^2 - 116/23*w + 8/23],
[811, 811, 1/23*w^3 + 10/23*w^2 + 13/23*w - 150/23],
[821, 821, -4/23*w^3 + 29/23*w^2 + 17/23*w - 182/23],
[821, 821, 4/23*w^3 + 17/23*w^2 - 63/23*w - 186/23],
[821, 821, -4/23*w^3 + 29/23*w^2 + 17/23*w - 228/23],
[821, 821, 4/23*w^3 + 17/23*w^2 - 63/23*w - 140/23],
[839, 839, -16/23*w^3 + 70/23*w^2 + 160/23*w - 751/23],
[839, 839, 22/23*w^3 - 33/23*w^2 - 289/23*w + 196/23],
[839, 839, -22/23*w^3 + 33/23*w^2 + 289/23*w - 104/23],
[839, 839, -16/23*w^3 - 22/23*w^2 + 252/23*w + 537/23],
[859, 859, -7/23*w^3 + 22/23*w^2 + 47/23*w - 192/23],
[859, 859, -9/23*w^3 + 2/23*w^2 + 113/23*w - 30/23],
[859, 859, -9/23*w^3 + 25/23*w^2 + 90/23*w - 76/23],
[859, 859, 7/23*w^3 + 1/23*w^2 - 70/23*w - 130/23],
[971, 971, -10/23*w^3 + 15/23*w^2 + 146/23*w + 97/23],
[971, 971, -14/23*w^3 - 2/23*w^2 + 186/23*w + 99/23],
[971, 971, 14/23*w^3 - 44/23*w^2 - 140/23*w + 269/23],
[971, 971, -10/23*w^3 + 15/23*w^2 + 146/23*w - 248/23]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - 148*x^6 + 7716*x^4 - 168784*x^2 + 1317904;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, -1/1368*e^6 + 11/114*e^4 - 224/57*e^2 + 8066/171, -1/1368*e^6 + 11/114*e^4 - 224/57*e^2 + 8066/171, e, e, 115/130872*e^7 - 1227/10906*e^5 + 48689/10906*e^3 - 126194/2337*e, 115/130872*e^7 - 1227/10906*e^5 + 48689/10906*e^3 - 126194/2337*e, 1/684*e^6 - 11/57*e^4 + 448/57*e^2 - 15790/171, -13/4788*e^6 + 89/266*e^4 - 1650/133*e^2 + 23323/171, -1/4788*e^6 + 41/798*e^4 - 1322/399*e^2 + 9625/171, -13/4788*e^6 + 89/266*e^4 - 1650/133*e^2 + 23323/171, -1/4788*e^6 + 41/798*e^4 - 1322/399*e^2 + 9625/171, -11/10332*e^7 + 149/1148*e^5 - 1401/287*e^3 + 21044/369*e, 317/392616*e^7 - 540/5453*e^5 + 20148/5453*e^3 - 291484/7011*e, -11/10332*e^7 + 149/1148*e^5 - 1401/287*e^3 + 21044/369*e, 317/392616*e^7 - 540/5453*e^5 + 20148/5453*e^3 - 291484/7011*e, -1/684*e^6 + 11/57*e^4 - 448/57*e^2 + 17500/171, -43/196308*e^7 + 965/32718*e^5 - 21335/16359*e^3 + 134974/7011*e, -29/98154*e^7 + 524/16359*e^5 - 17491/16359*e^3 + 81730/7011*e, -43/196308*e^7 + 965/32718*e^5 - 21335/16359*e^3 + 134974/7011*e, -29/98154*e^7 + 524/16359*e^5 - 17491/16359*e^3 + 81730/7011*e, 1/504*e^6 - 5/21*e^4 + 199/21*e^2 - 1118/9, -47/9576*e^6 + 83/133*e^4 - 3351/133*e^2 + 54532/171, 1/504*e^6 - 5/21*e^4 + 199/21*e^2 - 1118/9, -47/9576*e^6 + 83/133*e^4 - 3351/133*e^2 + 54532/171, 47/4788*e^6 - 166/133*e^4 + 6569/133*e^2 - 100685/171, 1/2394*e^6 - 41/399*e^4 + 2245/399*e^2 - 11213/171, 1/779*e^7 - 1489/9348*e^5 + 14456/2337*e^3 - 173596/2337*e, 163/130872*e^7 - 2585/16359*e^5 + 103114/16359*e^3 - 182470/2337*e, 1/779*e^7 - 1489/9348*e^5 + 14456/2337*e^3 - 173596/2337*e, 163/130872*e^7 - 2585/16359*e^5 + 103114/16359*e^3 - 182470/2337*e, 101/49077*e^7 - 1342/5453*e^5 + 98083/10906*e^3 - 698552/7011*e, -181/196308*e^7 + 2243/21812*e^5 - 18226/5453*e^3 + 218629/7011*e, 101/49077*e^7 - 1342/5453*e^5 + 98083/10906*e^3 - 698552/7011*e, -181/196308*e^7 + 2243/21812*e^5 - 18226/5453*e^3 + 218629/7011*e, 1/76*e^6 - 179/114*e^4 + 3272/57*e^2 - 36616/57, 1/76*e^6 - 179/114*e^4 + 3272/57*e^2 - 36616/57, -1/76*e^6 + 179/114*e^4 - 3272/57*e^2 + 36730/57, -1/76*e^6 + 179/114*e^4 - 3272/57*e^2 + 36730/57, -1/171*e^6 + 44/57*e^4 - 1792/57*e^2 + 67777/171, -1/171*e^6 + 44/57*e^4 - 1792/57*e^2 + 67777/171, 181/196308*e^7 - 2243/21812*e^5 + 18226/5453*e^3 - 204607/7011*e, 181/196308*e^7 - 2243/21812*e^5 + 18226/5453*e^3 - 204607/7011*e, -59/196308*e^7 + 115/5453*e^5 - 705/10906*e^3 - 58612/7011*e, -59/196308*e^7 + 115/5453*e^5 - 705/10906*e^3 - 58612/7011*e, 101/49077*e^7 - 1342/5453*e^5 + 98083/10906*e^3 - 712574/7011*e, -263/98154*e^7 + 7151/21812*e^5 - 66915/5453*e^3 + 975793/7011*e, 101/49077*e^7 - 1342/5453*e^5 + 98083/10906*e^3 - 712574/7011*e, -263/98154*e^7 + 7151/21812*e^5 - 66915/5453*e^3 + 975793/7011*e, -11/5166*e^7 + 149/574*e^5 - 2802/287*e^3 + 40981/369*e, -11/5166*e^7 + 149/574*e^5 - 2802/287*e^3 + 40981/369*e, -401/392616*e^7 + 1521/10906*e^5 - 65475/10906*e^3 + 552778/7011*e, -401/392616*e^7 + 1521/10906*e^5 - 65475/10906*e^3 + 552778/7011*e, 13/3192*e^6 - 167/399*e^4 + 5164/399*e^2 - 7396/57, -55/3192*e^6 + 860/399*e^4 - 33388/399*e^2 + 56818/57, 13/3192*e^6 - 167/399*e^4 + 5164/399*e^2 - 7396/57, -55/3192*e^6 + 860/399*e^4 - 33388/399*e^2 + 56818/57, 1/684*e^6 - 11/57*e^4 + 448/57*e^2 - 18013/171, 1/684*e^6 - 11/57*e^4 + 448/57*e^2 - 18013/171, -29/56088*e^7 + 131/2337*e^5 - 7577/4674*e^3 + 55390/7011*e, -29/56088*e^7 + 131/2337*e^5 - 7577/4674*e^3 + 55390/7011*e, 131/196308*e^7 - 2179/32718*e^5 + 31138/16359*e^3 - 117227/7011*e, 131/196308*e^7 - 2179/32718*e^5 + 31138/16359*e^3 - 117227/7011*e, 47/9576*e^6 - 83/133*e^4 + 3351/133*e^2 - 56242/171, -1/504*e^6 + 5/21*e^4 - 199/21*e^2 + 1028/9, 47/9576*e^6 - 83/133*e^4 + 3351/133*e^2 - 56242/171, -1/504*e^6 + 5/21*e^4 - 199/21*e^2 + 1028/9, 121/4788*e^6 - 1217/399*e^4 + 44917/399*e^2 - 217171/171, 121/4788*e^6 - 1217/399*e^4 + 44917/399*e^2 - 217171/171, -43/2394*e^6 + 832/399*e^4 - 29237/399*e^2 + 133433/171, -43/2394*e^6 + 832/399*e^4 - 29237/399*e^2 + 133433/171, 11/3444*e^7 - 447/1148*e^5 + 4203/287*e^3 - 20921/123*e, -101/65436*e^7 + 2013/10906*e^5 - 72199/10906*e^3 + 165290/2337*e, 11/3444*e^7 - 447/1148*e^5 + 4203/287*e^3 - 20921/123*e, -101/65436*e^7 + 2013/10906*e^5 - 72199/10906*e^3 + 165290/2337*e, -125/9576*e^6 + 433/266*e^4 - 8301/133*e^2 + 122278/171, -125/9576*e^6 + 433/266*e^4 - 8301/133*e^2 + 122278/171, 13/9576*e^6 - 67/798*e^4 - 185/399*e^2 + 5410/171, 13/9576*e^6 - 67/798*e^4 - 185/399*e^2 + 5410/171, -83/98154*e^7 + 6563/65436*e^5 - 58522/16359*e^3 + 271873/7011*e, -83/98154*e^7 + 6563/65436*e^5 - 58522/16359*e^3 + 271873/7011*e, 389/196308*e^7 - 7969/32718*e^5 + 301937/32718*e^3 - 751796/7011*e, 389/196308*e^7 - 7969/32718*e^5 + 301937/32718*e^3 - 751796/7011*e, -23/3192*e^6 + 683/798*e^4 - 12758/399*e^2 + 22214/57, 37/3192*e^6 - 1145/798*e^4 + 22166/399*e^2 - 12478/19, -23/3192*e^6 + 683/798*e^4 - 12758/399*e^2 + 22214/57, 37/3192*e^6 - 1145/798*e^4 + 22166/399*e^2 - 12478/19, -1/779*e^7 + 1489/9348*e^5 - 14456/2337*e^3 + 173596/2337*e, -163/130872*e^7 + 2585/16359*e^5 - 103114/16359*e^3 + 182470/2337*e, -1/779*e^7 + 1489/9348*e^5 - 14456/2337*e^3 + 173596/2337*e, -163/130872*e^7 + 2585/16359*e^5 - 103114/16359*e^3 + 182470/2337*e, -5/684*e^6 + 55/57*e^4 - 2240/57*e^2 + 81002/171, -5/684*e^6 + 55/57*e^4 - 2240/57*e^2 + 81002/171, 11/4788*e^6 - 185/798*e^4 + 2705/399*e^2 - 13307/171, 11/4788*e^6 - 185/798*e^4 + 2705/399*e^2 - 13307/171, -23/2394*e^6 + 955/798*e^4 - 18385/399*e^2 + 89863/171, -23/2394*e^6 + 955/798*e^4 - 18385/399*e^2 + 89863/171, -1/684*e^6 + 11/57*e^4 - 448/57*e^2 + 18526/171, -1/684*e^6 + 11/57*e^4 - 448/57*e^2 + 18526/171, -1/684*e^6 + 11/57*e^4 - 448/57*e^2 + 18526/171, -1/684*e^6 + 11/57*e^4 - 448/57*e^2 + 18526/171, -137/98154*e^7 + 5515/32718*e^5 - 99553/16359*e^3 + 440983/7011*e, -137/98154*e^7 + 5515/32718*e^5 - 99553/16359*e^3 + 440983/7011*e, 607/392616*e^7 - 2930/16359*e^5 + 208343/32718*e^3 - 502820/7011*e, 607/392616*e^7 - 2930/16359*e^5 + 208343/32718*e^3 - 502820/7011*e, 7/684*e^6 - 45/38*e^4 + 754/19*e^2 - 62536/171, -5/684*e^6 + 91/114*e^4 - 1366/57*e^2 + 33350/171, 7/684*e^6 - 45/38*e^4 + 754/19*e^2 - 62536/171, -5/684*e^6 + 91/114*e^4 - 1366/57*e^2 + 33350/171, 139/65436*e^7 - 8851/32718*e^5 + 352295/32718*e^3 - 104446/779*e, -31/65436*e^7 + 4301/65436*e^5 - 44875/16359*e^3 + 27043/779*e, 139/65436*e^7 - 8851/32718*e^5 + 352295/32718*e^3 - 104446/779*e, -31/65436*e^7 + 4301/65436*e^5 - 44875/16359*e^3 + 27043/779*e, -115/9576*e^6 + 409/266*e^4 - 8137/133*e^2 + 128816/171, -115/9576*e^6 + 409/266*e^4 - 8137/133*e^2 + 128816/171, -25/9576*e^6 + 313/798*e^4 - 6949/399*e^2 + 39344/171, -25/9576*e^6 + 313/798*e^4 - 6949/399*e^2 + 39344/171, 31/1197*e^6 - 2557/798*e^4 + 48484/399*e^2 - 238351/171, 31/1197*e^6 - 2557/798*e^4 + 48484/399*e^2 - 238351/171, -13/2394*e^6 + 401/798*e^4 - 4580/399*e^2 + 9083/171, -13/2394*e^6 + 401/798*e^4 - 4580/399*e^2 + 9083/171, 851/196308*e^7 - 17069/32718*e^5 + 315584/16359*e^3 - 1539089/7011*e, -953/392616*e^7 + 4681/16359*e^5 - 329797/32718*e^3 + 734110/7011*e, 851/196308*e^7 - 17069/32718*e^5 + 315584/16359*e^3 - 1539089/7011*e, -953/392616*e^7 + 4681/16359*e^5 - 329797/32718*e^3 + 734110/7011*e, -1/266*e^6 + 103/399*e^4 - 122/399*e^2 - 6299/57, 4/133*e^6 - 1489/399*e^4 + 56570/399*e^2 - 93343/57, -1/266*e^6 + 103/399*e^4 - 122/399*e^2 - 6299/57, 4/133*e^6 - 1489/399*e^4 + 56570/399*e^2 - 93343/57, -431/196308*e^7 + 4646/16359*e^5 - 188737/16359*e^3 + 985046/7011*e, -1151/196308*e^7 + 12091/16359*e^5 - 473183/16359*e^3 + 2420930/7011*e, -431/196308*e^7 + 4646/16359*e^5 - 188737/16359*e^3 + 985046/7011*e, -1151/196308*e^7 + 12091/16359*e^5 - 473183/16359*e^3 + 2420930/7011*e, 263/9576*e^6 - 2665/798*e^4 + 50419/399*e^2 - 253168/171, 263/9576*e^6 - 2665/798*e^4 + 50419/399*e^2 - 253168/171, -235/9576*e^6 + 2357/798*e^4 - 44147/399*e^2 + 221588/171, -235/9576*e^6 + 2357/798*e^4 - 44147/399*e^2 + 221588/171, -1069/392616*e^7 + 1735/5453*e^5 - 121593/10906*e^3 + 815840/7011*e, -1069/392616*e^7 + 1735/5453*e^5 - 121593/10906*e^3 + 815840/7011*e, 202/49077*e^7 - 2684/5453*e^5 + 98083/5453*e^3 - 1404115/7011*e, 202/49077*e^7 - 2684/5453*e^5 + 98083/5453*e^3 - 1404115/7011*e, -1/7011*e^7 + 21/779*e^5 - 1199/779*e^3 + 202240/7011*e, -1/7011*e^7 + 21/779*e^5 - 1199/779*e^3 + 202240/7011*e, 68/49077*e^7 - 2077/10906*e^5 + 44140/5453*e^3 - 728678/7011*e, 68/49077*e^7 - 2077/10906*e^5 + 44140/5453*e^3 - 728678/7011*e, 55/10332*e^7 - 745/1148*e^5 + 7005/287*e^3 - 103006/369*e, 485/392616*e^7 - 981/5453*e^5 + 45327/5453*e^3 - 814072/7011*e, 55/10332*e^7 - 745/1148*e^5 + 7005/287*e^3 - 103006/369*e, 485/392616*e^7 - 981/5453*e^5 + 45327/5453*e^3 - 814072/7011*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {4,2,-2/23*w^3+3/23*w^2-3/23*w+1/23}>] := 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;