/*
  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![41, 13, -12, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[9, 3, -w^3 + 3*w^2 + 5*w - 15],
[9, 3, -w^3 + 8*w + 8],
[16, 2, 2],
[19, 19, w + 1],
[19, 19, -w^2 + 6],
[19, 19, -w^2 + 2*w + 5],
[19, 19, -w + 2],
[25, 5, 2*w^2 - 2*w - 13],
[29, 29, -w^2 + 9],
[29, 29, -w^2 + 2*w + 6],
[29, 29, w^2 - 7],
[29, 29, -w^2 + 2*w + 8],
[31, 31, -2*w^2 + w + 12],
[31, 31, 2*w^2 - 3*w - 11],
[41, 41, -w],
[41, 41, -w + 1],
[49, 7, w^3 + 2*w^2 - 10*w - 20],
[49, 7, w^3 - 5*w^2 - 3*w + 27],
[61, 61, 2*w^2 - 3*w - 14],
[61, 61, 2*w^2 - w - 15],
[71, 71, -w^3 + w^2 + 7*w - 5],
[71, 71, w^3 - 2*w^2 - 6*w + 2],
[79, 79, w^3 - 2*w^2 - 6*w + 4],
[79, 79, 3*w^2 - 2*w - 18],
[89, 89, 2*w^3 - 6*w^2 - 11*w + 33],
[89, 89, -4*w^2 + 3*w + 25],
[101, 101, w^3 + w^2 - 9*w - 12],
[101, 101, w^3 - 4*w^2 - 4*w + 19],
[109, 109, -w^3 + 3*w^2 + 6*w - 19],
[109, 109, -2*w^2 + w + 18],
[121, 11, -3*w^2 + 3*w + 19],
[121, 11, 3*w^2 - 3*w - 20],
[131, 131, 2*w^2 - 3*w - 15],
[131, 131, 2*w^2 - w - 16],
[139, 139, 3*w^2 - 4*w - 21],
[139, 139, -w^3 - w^2 + 7*w + 13],
[151, 151, 4*w^2 - 3*w - 27],
[151, 151, 2*w^3 - w^2 - 13*w - 8],
[151, 151, w^3 + 3*w^2 - 13*w - 27],
[151, 151, 4*w^2 - 5*w - 26],
[179, 179, w^3 - 3*w^2 - 5*w + 11],
[179, 179, 3*w^3 - 11*w^2 - 14*w + 61],
[179, 179, 2*w^3 - 2*w^2 - 12*w + 3],
[179, 179, w^3 - 8*w - 4],
[181, 181, -w - 4],
[181, 181, w^3 - 5*w^2 - 3*w + 25],
[181, 181, w^3 + 2*w^2 - 10*w - 18],
[181, 181, w - 5],
[229, 229, w^2 - 2*w - 11],
[229, 229, w^2 - 12],
[239, 239, 6*w^2 - 7*w - 42],
[239, 239, 2*w^3 - 2*w^2 - 14*w - 3],
[241, 241, w^3 - 4*w^2 - 4*w + 18],
[241, 241, 3*w^2 - 2*w - 16],
[269, 269, 3*w^2 - 4*w - 22],
[269, 269, -w^3 + 6*w + 8],
[281, 281, w^3 - w^2 - 9*w + 4],
[281, 281, 2*w^3 - 9*w^2 - 8*w + 49],
[311, 311, -2*w^3 - w^2 + 17*w + 19],
[311, 311, 2*w^3 - 7*w^2 - 9*w + 33],
[331, 331, w^3 + 2*w^2 - 9*w - 21],
[331, 331, w^3 - 5*w^2 - 2*w + 27],
[349, 349, 2*w^3 - w^2 - 15*w - 4],
[349, 349, -2*w^3 + 5*w^2 + 11*w - 18],
[359, 359, -w^3 + 6*w^2 - 31],
[359, 359, -w^3 - 3*w^2 + 9*w + 26],
[379, 379, 2*w^3 - 5*w^2 - 11*w + 20],
[379, 379, 2*w^3 - 14*w - 11],
[401, 401, w^3 - 3*w^2 - 4*w + 16],
[401, 401, -2*w^3 + 4*w^2 + 11*w - 16],
[401, 401, 2*w^3 - 2*w^2 - 13*w - 3],
[401, 401, w^3 - 7*w - 10],
[409, 409, w^3 - 4*w^2 - 4*w + 17],
[409, 409, 2*w^3 - 6*w^2 - 9*w + 22],
[409, 409, -2*w^3 + 15*w + 9],
[409, 409, -w^3 + 3*w^2 + 6*w - 12],
[419, 419, -w^3 + 5*w^2 + 3*w - 24],
[419, 419, w^3 + 2*w^2 - 10*w - 17],
[421, 421, w^3 + 3*w^2 - 12*w - 25],
[421, 421, -w^3 - w^2 + 9*w + 7],
[431, 431, -w^3 - 2*w^2 + 12*w + 20],
[431, 431, w^3 + 3*w^2 - 10*w - 28],
[431, 431, -w^3 + 6*w^2 + w - 34],
[431, 431, -w^3 + 5*w^2 + 5*w - 29],
[439, 439, w^3 - 7*w^2 - 2*w + 39],
[439, 439, w^3 + 4*w^2 - 13*w - 31],
[449, 449, -w^3 - 4*w^2 + 13*w + 34],
[449, 449, 6*w^2 - 5*w - 36],
[461, 461, -w^3 + 5*w^2 + w - 25],
[461, 461, 2*w^3 - 15*w - 12],
[461, 461, -2*w^3 + 6*w^2 + 9*w - 25],
[461, 461, w^3 + 2*w^2 - 8*w - 20],
[479, 479, -w^3 - 4*w^2 + 12*w + 31],
[479, 479, 2*w^3 - 15*w - 16],
[479, 479, -2*w^3 + 6*w^2 + 9*w - 29],
[479, 479, w^3 - 7*w^2 - w + 38],
[491, 491, w^2 - 3*w - 7],
[491, 491, -2*w^3 + 3*w^2 + 11*w - 2],
[491, 491, 4*w^2 - 2*w - 29],
[491, 491, w^2 + w - 9],
[499, 499, 5*w^2 - 6*w - 33],
[499, 499, 5*w^2 - 4*w - 34],
[509, 509, 2*w - 3],
[509, 509, -2*w - 1],
[541, 541, -w^3 - 4*w^2 + 12*w + 29],
[541, 541, -w^3 + 7*w^2 + w - 36],
[599, 599, -w^3 - w^2 + 8*w + 7],
[599, 599, w^3 - 4*w^2 - 3*w + 13],
[619, 619, w^2 - 3*w - 6],
[619, 619, w^2 + w - 8],
[641, 641, -2*w^3 + 16*w + 11],
[641, 641, -2*w^3 + 6*w^2 + 10*w - 25],
[659, 659, 2*w^3 + w^2 - 16*w - 16],
[659, 659, 2*w^3 - 7*w^2 - 8*w + 29],
[691, 691, -w^3 + 5*w^2 + 3*w - 23],
[691, 691, -w^3 + 3*w^2 + 6*w - 11],
[701, 701, w^3 - w^2 - 9*w + 3],
[701, 701, 2*w^3 - w^2 - 14*w - 5],
[709, 709, 2*w^3 - 3*w^2 - 11*w + 6],
[719, 719, w^2 - 2*w - 12],
[719, 719, -w^2 + 13],
[739, 739, -2*w^3 + 8*w^2 + 7*w - 36],
[739, 739, w^2 + 2*w - 13],
[751, 751, -2*w^3 + 5*w^2 + 10*w - 20],
[751, 751, 2*w^3 - w^2 - 14*w - 7],
[761, 761, w^3 + 4*w^2 - 12*w - 32],
[761, 761, -w^3 + 7*w^2 + w - 39],
[769, 769, -2*w^3 - 4*w^2 + 23*w + 45],
[769, 769, 2*w^3 - 5*w^2 - 9*w + 21],
[769, 769, 2*w^3 + 2*w^2 - 18*w - 31],
[769, 769, 2*w^3 - 10*w^2 - 9*w + 62],
[811, 811, -w^3 + 10*w + 5],
[811, 811, -5*w^2 + 8*w + 30],
[811, 811, w^3 - 8*w^2 + 2*w + 43],
[811, 811, 2*w^3 - 2*w^2 - 13*w + 2],
[829, 829, -2*w^3 - 2*w^2 + 18*w + 25],
[829, 829, -2*w^3 + 8*w^2 + 8*w - 39],
[859, 859, -w^3 - 4*w^2 + 12*w + 33],
[859, 859, -w^3 + 7*w^2 + w - 40],
[919, 919, w^3 - 4*w^2 - 3*w + 24],
[919, 919, -w^3 + 5*w^2 + w - 26],
[929, 929, 5*w^2 - 7*w - 28],
[929, 929, -w^3 + 2*w^2 + 6*w - 13],
[929, 929, w^3 - w^2 - 7*w - 6],
[929, 929, -2*w^3 + 2*w^2 + 14*w - 5],
[941, 941, -2*w^3 + 3*w^2 + 11*w - 1],
[941, 941, 2*w^3 - 3*w^2 - 11*w + 11],
[961, 31, 5*w^2 - 5*w - 33],
[971, 971, -w^3 - 3*w^2 + 11*w + 22],
[971, 971, 7*w^2 - 5*w - 45],
[971, 971, 7*w^2 - 9*w - 43],
[971, 971, w^3 - 6*w^2 - 2*w + 29],
[991, 991, w^3 - 3*w^2 - 3*w + 15],
[991, 991, -w^3 + 5*w^2 + 2*w - 29]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^7 + 7*x^6 - 4*x^5 - 97*x^4 - 72*x^3 + 332*x^2 + 336*x - 64;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-37/1616*e^6 - 265/1616*e^5 - 13/808*e^4 + 3017/1616*e^3 + 1795/808*e^2 - 1157/202*e - 374/101, e, 29/808*e^6 + 235/808*e^5 + 1/202*e^4 - 2725/808*e^3 - 205/101*e^2 + 1669/202*e + 185/101, -1, 115/3232*e^6 + 507/3232*e^5 - 915/1616*e^4 - 6975/3232*e^3 + 3773/1616*e^2 + 2799/404*e + 543/101, -11/404*e^6 - 133/808*e^5 + 293/808*e^4 + 529/202*e^3 - 849/808*e^2 - 3497/404*e - 412/101, -1/16*e^6 - 5/16*e^5 + 7/8*e^4 + 69/16*e^3 - 33/8*e^2 - 25/2*e + 6, 29/1616*e^6 + 33/1616*e^5 - 503/808*e^4 - 1513/1616*e^3 + 2917/808*e^2 + 493/101*e + 244/101, -47/808*e^6 - 63/202*e^5 + 401/808*e^4 + 2877/808*e^3 + 119/808*e^2 - 2655/404*e - 366/101, 47/1616*e^6 + 353/1616*e^5 - 75/404*e^4 - 4695/1616*e^3 + 62/101*e^2 + 3903/404*e - 120/101, 77/3232*e^6 + 617/3232*e^5 - 235/1616*e^4 - 10537/3232*e^3 - 2327/1616*e^2 + 1311/101*e + 714/101, -29/404*e^6 - 67/202*e^5 + 501/404*e^4 + 2119/404*e^3 - 2501/404*e^2 - 3641/202*e + 438/101, -63/1616*e^6 - 413/1616*e^5 + 16/101*e^4 + 5279/1616*e^3 + 1131/404*e^2 - 3981/404*e - 847/101, 69/808*e^6 + 385/808*e^5 - 347/404*e^4 - 4993/808*e^3 + 769/404*e^2 + 1740/101*e + 172/101, 57/3232*e^6 + 441/3232*e^5 + 91/1616*e^4 - 3949/3232*e^3 - 3677/1616*e^2 + 19/404*e + 1107/101, -151/1616*e^6 - 743/1616*e^5 + 815/808*e^4 + 8491/1616*e^3 - 2365/808*e^2 - 1254/101*e - 156/101, -309/3232*e^6 - 1285/3232*e^5 + 2441/1616*e^4 + 16985/3232*e^3 - 13535/1616*e^2 - 6567/404*e + 1037/101, 297/3232*e^6 + 1341/3232*e^5 - 2811/1616*e^4 - 22405/3232*e^3 + 15553/1616*e^2 + 5049/202*e - 781/101, -37/1616*e^6 - 63/1616*e^5 + 123/202*e^4 + 1805/1616*e^3 - 971/404*e^2 - 2213/404*e - 71/101, -113/3232*e^6 - 853/3232*e^5 + 539/1616*e^4 + 14477/3232*e^3 - 305/1616*e^2 - 4143/202*e - 653/101, -71/3232*e^6 - 39/3232*e^5 + 1531/1616*e^4 + 3955/3232*e^3 - 12317/1616*e^2 - 2313/404*e + 875/101, -17/808*e^6 - 89/808*e^5 + 65/404*e^4 + 469/808*e^3 - 895/404*e^2 + 129/101*e + 1316/101, -1/404*e^6 + 18/101*e^5 + 275/404*e^4 - 1125/404*e^3 - 2963/404*e^2 + 2187/202*e + 844/101, -99/3232*e^6 - 447/3232*e^5 + 937/1616*e^4 + 9623/3232*e^3 - 2491/1616*e^2 - 3097/202*e - 110/101, 121/3232*e^6 + 681/3232*e^5 - 629/1616*e^4 - 11133/3232*e^3 - 165/1616*e^2 + 6437/404*e + 1021/101, 167/1616*e^6 + 803/1616*e^5 - 1197/808*e^4 - 11499/1616*e^3 + 4455/808*e^2 + 2132/101*e - 392/101, 325/3232*e^6 + 1749/3232*e^5 - 1409/1616*e^4 - 19993/3232*e^3 + 879/1616*e^2 + 5697/404*e + 911/101, 93/3232*e^6 + 273/3232*e^5 - 1223/1616*e^4 - 5465/3232*e^3 + 8045/1616*e^2 + 639/101*e - 267/101, 189/1616*e^6 + 1037/1616*e^5 - 889/808*e^4 - 13009/1616*e^3 + 991/808*e^2 + 4305/202*e + 824/101, 39/3232*e^6 + 323/3232*e^5 - 565/1616*e^4 - 8443/3232*e^3 + 663/1616*e^2 + 887/101*e + 1592/101, -1/1616*e^6 - 29/1616*e^5 + 87/808*e^4 + 1501/1616*e^3 - 2037/808*e^2 - 926/101*e + 918/101, 63/1616*e^6 + 413/1616*e^5 - 16/101*e^4 - 3663/1616*e^3 + 485/404*e^2 + 1961/404*e - 769/101, -11/3232*e^6 - 319/3232*e^5 - 659/1616*e^4 + 3583/3232*e^3 + 8297/1616*e^2 - 1399/202*e - 910/101, -41/404*e^6 - 70/101*e^5 + 165/404*e^4 + 2961/404*e^3 - 283/404*e^2 - 1475/101*e + 1678/101, -323/3232*e^6 - 2499/3232*e^5 + 23/1616*e^4 + 29919/3232*e^3 + 11679/1616*e^2 - 10477/404*e - 1425/101, 53/3232*e^6 + 325/3232*e^5 + 439/1616*e^4 + 2055/3232*e^3 - 3745/1616*e^2 - 3349/404*e - 289/101, -27/3232*e^6 - 379/3232*e^5 - 681/1616*e^4 + 935/3232*e^3 + 10247/1616*e^2 + 3829/404*e - 1747/101, -49/1616*e^6 - 411/1616*e^5 + 81/202*e^4 + 6081/1616*e^3 - 1477/404*e^2 - 4443/404*e + 138/101, 89/3232*e^6 + 965/3232*e^5 + 1145/1616*e^4 - 4309/3232*e^3 - 6971/1616*e^2 - 911/202*e - 1158/101, 1/808*e^6 - 173/808*e^5 - 47/101*e^4 + 3751/808*e^3 + 1271/202*e^2 - 4679/202*e - 1432/101, 653/3232*e^6 + 3181/3232*e^5 - 4089/1616*e^4 - 42065/3232*e^3 + 13727/1616*e^2 + 16647/404*e + 647/101, -11/1616*e^6 - 117/1616*e^5 - 77/404*e^4 + 755/1616*e^3 + 368/101*e^2 + 969/404*e - 911/101, 207/3232*e^6 + 751/3232*e^5 - 1243/1616*e^4 - 2859/3232*e^3 + 9781/1616*e^2 - 2153/404*e - 1689/101, -53/3232*e^6 + 887/3232*e^5 + 3399/1616*e^4 - 4479/3232*e^3 - 29989/1616*e^2 - 295/202*e + 2511/101, -371/3232*e^6 - 1467/3232*e^5 + 2987/1616*e^4 + 17935/3232*e^3 - 17821/1616*e^2 - 5241/404*e + 1821/101, 9/202*e^6 + 337/808*e^5 + 301/808*e^4 - 1667/404*e^3 - 4937/808*e^2 + 3179/404*e + 2180/101, 459/1616*e^6 + 2201/1616*e^5 - 767/202*e^4 - 29227/1616*e^3 + 5871/404*e^2 + 21617/404*e - 495/101, -35/404*e^6 - 313/808*e^5 + 969/808*e^4 + 967/202*e^3 - 2885/808*e^2 - 4773/404*e - 1669/101, -13/101*e^6 - 491/808*e^5 + 1633/808*e^4 + 3413/404*e^3 - 11313/808*e^2 - 11619/404*e + 2478/101, -413/3232*e^6 - 2685/3232*e^5 + 1793/1616*e^4 + 38961/3232*e^3 - 1567/1616*e^2 - 17981/404*e - 717/101, -27/3232*e^6 + 25/3232*e^5 + 329/1616*e^4 - 1489/3232*e^3 - 2075/1616*e^2 - 55/202*e - 333/101, 157/1616*e^6 + 513/1616*e^5 - 1943/808*e^4 - 9417/1616*e^3 + 13173/808*e^2 + 2265/101*e - 706/101, -249/3232*e^6 - 1161/3232*e^5 + 1261/1616*e^4 + 10957/3232*e^3 - 3627/1616*e^2 - 487/404*e + 969/101, -1043/3232*e^6 - 6007/3232*e^5 + 4285/1616*e^4 + 72359/3232*e^3 - 5207/1616*e^2 - 11671/202*e - 710/101, -361/1616*e^6 - 1985/1616*e^5 + 1713/808*e^4 + 23933/1616*e^3 - 1895/808*e^2 - 6921/202*e - 1498/101, -99/3232*e^6 - 1255/3232*e^5 - 275/1616*e^4 + 19319/3232*e^3 + 5993/1616*e^2 - 3905/202*e + 294/101, -53/1616*e^6 - 123/1616*e^5 + 437/404*e^4 + 4813/1616*e^3 - 504/101*e^2 - 5321/404*e - 230/101, -51/404*e^6 - 267/404*e^5 + 195/202*e^4 + 3023/404*e^3 + 547/202*e^2 - 1347/101*e - 1194/101, -37/3232*e^6 - 669/3232*e^5 - 1427/1616*e^4 - 215/3232*e^3 + 12501/1616*e^2 + 6317/404*e - 1601/101, 269/1616*e^6 + 1539/1616*e^5 - 339/404*e^4 - 13101/1616*e^3 - 148/101*e^2 + 1627/404*e + 912/101, 671/3232*e^6 + 3703/3232*e^5 - 4847/1616*e^4 - 57771/3232*e^3 + 21305/1616*e^2 + 25945/404*e - 141/101, 121/3232*e^6 + 1489/3232*e^5 + 1391/1616*e^4 - 9517/3232*e^3 - 8649/1616*e^2 - 2653/404*e - 797/101, 355/3232*e^6 + 1003/3232*e^5 - 4827/1616*e^4 - 26239/3232*e^3 + 24013/1616*e^2 + 14393/404*e + 1079/101, -907/3232*e^6 - 4891/3232*e^5 + 3967/1616*e^4 + 54871/3232*e^3 - 8753/1616*e^2 - 15991/404*e + 799/101, 77/404*e^6 + 213/404*e^5 - 471/101*e^4 - 3871/404*e^3 + 5955/202*e^2 + 6533/202*e - 3378/101, -107/808*e^6 - 275/808*e^5 + 1431/404*e^4 + 5471/808*e^3 - 9293/404*e^2 - 2426/101*e + 1926/101, -29/3232*e^6 - 437/3232*e^5 - 507/1616*e^4 + 3937/3232*e^3 + 2941/1616*e^2 - 279/404*e + 2403/101, 40/101*e^6 + 401/202*e^5 - 891/202*e^4 - 2369/101*e^3 + 3427/202*e^2 + 5931/101*e - 2124/101, 71/808*e^6 + 443/808*e^5 - 117/404*e^4 - 3955/808*e^3 - 409/404*e^2 + 899/101*e - 470/101, 153/1616*e^6 + 801/1616*e^5 - 585/808*e^4 - 9069/1616*e^3 - 2449/808*e^2 + 2677/202*e + 1552/101, 457/3232*e^6 + 1941/3232*e^5 - 5015/1616*e^4 - 39557/3232*e^3 + 25141/1616*e^2 + 10193/202*e + 115/101, 555/3232*e^6 + 2763/3232*e^5 - 4855/1616*e^4 - 46871/3232*e^3 + 22969/1616*e^2 + 20991/404*e - 1235/101, 335/1616*e^6 + 1029/1616*e^5 - 999/202*e^4 - 20863/1616*e^3 + 11887/404*e^2 + 21669/404*e - 3318/101, 399/3232*e^6 + 1471/3232*e^5 - 2595/1616*e^4 - 13099/3232*e^3 + 10621/1616*e^2 + 1143/404*e - 735/101, 549/3232*e^6 + 2589/3232*e^5 - 3525/1616*e^4 - 33017/3232*e^3 + 10747/1616*e^2 + 10889/404*e + 1519/101, 119/3232*e^6 + 219/3232*e^5 - 1465/1616*e^4 - 5707/3232*e^3 + 3235/1616*e^2 + 860/101*e + 3353/101, -34/101*e^6 - 178/101*e^5 + 823/202*e^4 + 4805/202*e^3 - 1156/101*e^2 - 14355/202*e - 2174/101, 337/3232*e^6 + 2501/3232*e^5 + 173/1616*e^4 - 27501/3232*e^3 - 16087/1616*e^2 + 3507/202*e + 1665/101, -209/1616*e^6 - 1415/1616*e^5 + 355/404*e^4 + 17577/1616*e^3 - 257/202*e^2 - 11081/404*e + 164/101, 95/808*e^6 + 331/808*e^5 - 791/404*e^4 - 3215/808*e^3 + 4443/404*e^2 - 711/202*e - 1710/101, -455/3232*e^6 - 1479/3232*e^5 + 5043/1616*e^4 + 29283/3232*e^3 - 26925/1616*e^2 - 17187/404*e + 1593/101, -363/3232*e^6 - 1235/3232*e^5 + 3907/1616*e^4 + 28551/3232*e^3 - 11221/1616*e^2 - 17897/404*e - 2457/101, -219/1616*e^6 - 1099/1616*e^5 + 671/808*e^4 + 7943/1616*e^3 - 289/808*e^2 + 533/202*e + 254/101, 271/3232*e^6 + 991/3232*e^5 - 2771/1616*e^4 - 21355/3232*e^3 + 11677/1616*e^2 + 12951/404*e - 361/101, 271/3232*e^6 + 1799/3232*e^5 - 1559/1616*e^4 - 24587/3232*e^3 + 9657/1616*e^2 + 10123/404*e + 43/101, 61/808*e^6 + 355/808*e^5 - 179/202*e^4 - 4701/808*e^3 + 436/101*e^2 + 2431/202*e - 896/101, 29/808*e^6 + 33/808*e^5 - 99/404*e^4 + 3335/808*e^3 + 2109/404*e^2 - 3862/101*e - 2138/101, 429/1616*e^6 + 1937/1616*e^5 - 2983/808*e^4 - 25001/1616*e^3 + 12949/808*e^2 + 4566/101*e - 326/101, -981/3232*e^6 - 6229/3232*e^5 + 2729/1616*e^4 + 70601/3232*e^3 + 3321/1616*e^2 - 19921/404*e - 181/101, -255/3232*e^6 - 123/3232*e^5 + 4005/1616*e^4 + 4611/3232*e^3 - 28575/1616*e^2 - 901/202*e + 3824/101, 667/3232*e^6 + 3991/3232*e^5 - 2681/1616*e^4 - 52575/3232*e^3 - 2397/1616*e^2 + 4306/101*e + 281/101, -111/1616*e^6 - 391/1616*e^5 + 567/808*e^4 + 971/1616*e^3 - 3705/808*e^2 + 1983/202*e + 1706/101, 3/16*e^6 + 13/16*e^5 - 15/4*e^4 - 235/16*e^3 + 21*e^2 + 223/4*e - 13, -77/808*e^6 - 415/808*e^5 + 84/101*e^4 + 5285/808*e^3 + 507/202*e^2 - 2913/202*e - 3462/101, 41/3232*e^6 - 23/3232*e^5 - 133/1616*e^4 + 3907/3232*e^3 - 2333/1616*e^2 - 4161/404*e - 841/101, 535/3232*e^6 + 2183/3232*e^5 - 4731/1616*e^4 - 39475/3232*e^3 + 16165/1616*e^2 + 22331/404*e + 1885/101, -53/1616*e^6 - 729/1616*e^5 - 641/808*e^4 + 8449/1616*e^3 + 7987/808*e^2 - 1305/101*e - 1442/101, 945/3232*e^6 + 4377/3232*e^5 - 7273/1616*e^4 - 65045/3232*e^3 + 32023/1616*e^2 + 28393/404*e - 667/101, -71/808*e^6 - 35/202*e^5 + 2153/808*e^4 + 2541/808*e^3 - 17665/808*e^2 - 3495/404*e + 2187/101, 769/3232*e^6 + 4929/3232*e^5 - 1253/1616*e^4 - 52965/3232*e^3 - 17429/1616*e^2 + 12713/404*e + 3963/101, -213/3232*e^6 - 1733/3232*e^5 - 255/1616*e^4 + 23177/3232*e^3 + 12337/1616*e^2 - 10373/404*e - 2425/101, 553/3232*e^6 + 3109/3232*e^5 - 2863/1616*e^4 - 38213/3232*e^3 + 6573/1616*e^2 + 4755/202*e + 188/101, -81/808*e^6 - 215/404*e^5 + 661/808*e^4 + 3815/808*e^3 - 1037/808*e^2 + 2013/404*e + 650/101, -149/1616*e^6 - 483/1616*e^5 + 573/404*e^4 + 5893/1616*e^3 - 1113/202*e^2 - 3567/404*e + 2351/101, -41/808*e^6 + 23/808*e^5 + 941/404*e^4 + 941/808*e^3 - 8171/404*e^2 - 182/101*e + 2960/101, 311/808*e^6 + 823/404*e^5 - 2907/808*e^4 - 20189/808*e^3 + 1383/808*e^2 + 26843/404*e + 2987/101, -201/808*e^6 - 541/404*e^5 + 2553/808*e^4 + 15063/808*e^3 - 8753/808*e^2 - 21175/404*e - 523/101, 25/404*e^6 + 119/404*e^5 - 128/101*e^4 - 2781/404*e^3 + 566/101*e^2 + 4124/101*e - 294/101, -165/808*e^6 - 221/404*e^5 + 3965/808*e^4 + 7083/808*e^3 - 24901/808*e^2 - 9411/404*e + 3677/101, 49/404*e^6 + 317/808*e^5 - 1801/808*e^4 - 485/101*e^3 + 12725/808*e^2 + 5753/404*e - 4289/101, -9/404*e^6 + 185/808*e^5 + 2223/808*e^4 + 341/202*e^3 - 17075/808*e^2 - 6387/404*e - 585/101, 485/3232*e^6 + 1541/3232*e^5 - 4017/1616*e^4 - 12905/3232*e^3 + 22183/1616*e^2 + 835/404*e + 999/101, -811/3232*e^6 - 3723/3232*e^5 + 6119/1616*e^4 + 52983/3232*e^3 - 22473/1616*e^2 - 17979/404*e + 165/101, 307/3232*e^6 + 1631/3232*e^5 - 449/1616*e^4 - 5095/3232*e^3 + 5219/1616*e^2 - 2659/202*e - 3048/101, -787/3232*e^6 - 4643/3232*e^5 + 3223/1616*e^4 + 58975/3232*e^3 + 4599/1616*e^2 - 20799/404*e - 3781/101, -19/808*e^6 - 23/404*e^5 + 579/808*e^4 + 845/808*e^3 - 5595/808*e^2 - 2685/404*e + 443/101, -487/3232*e^6 - 4023/3232*e^5 - 1869/1616*e^4 + 40147/3232*e^3 + 36363/1616*e^2 - 12417/404*e - 2303/101, 407/1616*e^6 + 2107/1616*e^5 - 1877/808*e^4 - 25107/1616*e^3 + 2475/808*e^2 + 8687/202*e + 882/101, -123/1616*e^6 - 537/1616*e^5 + 125/202*e^4 + 4843/1616*e^3 + 1429/404*e^2 + 1231/404*e - 2630/101, 387/1616*e^6 + 2133/1616*e^5 - 1331/404*e^4 - 34275/1616*e^3 + 1973/202*e^2 + 32881/404*e + 1365/101, 195/1616*e^6 + 1615/1616*e^5 + 407/808*e^4 - 19591/1616*e^3 - 12845/808*e^2 + 4022/101*e + 3598/101, -223/808*e^6 - 557/404*e^5 + 2543/808*e^4 + 14149/808*e^3 - 6067/808*e^2 - 20127/404*e + 1085/101, 381/808*e^6 + 929/404*e^5 - 4785/808*e^4 - 24663/808*e^3 + 13661/808*e^2 + 35757/404*e + 1222/101, -131/808*e^6 - 971/808*e^5 + 287/404*e^4 + 12407/808*e^3 + 803/404*e^2 - 4656/101*e - 672/101, 347/3232*e^6 + 771/3232*e^5 - 4131/1616*e^4 - 14231/3232*e^3 + 23877/1616*e^2 + 7253/404*e - 299/101, 37/808*e^6 + 467/808*e^5 + 57/202*e^4 - 6653/808*e^3 - 373/101*e^2 + 6455/202*e + 748/101, -693/3232*e^6 - 3937/3232*e^5 + 4539/1616*e^4 + 65745/3232*e^3 - 5721/1616*e^2 - 8365/101*e - 3598/101, 107/808*e^6 + 289/404*e^5 - 1347/808*e^4 - 8905/808*e^3 + 2527/808*e^2 + 14855/404*e + 2316/101, -205/3232*e^6 - 1501/3232*e^5 + 665/1616*e^4 + 20865/3232*e^3 + 1161/1616*e^2 - 6465/404*e - 1451/101, -463/1616*e^6 - 1509/1616*e^5 + 1359/202*e^4 + 27151/1616*e^3 - 18227/404*e^2 - 26333/404*e + 5985/101, 247/808*e^6 + 1103/808*e^5 - 1895/404*e^4 - 16035/808*e^3 + 8845/404*e^2 + 6479/101*e - 3638/101, 259/3232*e^6 + 239/3232*e^5 - 1929/1616*e^4 + 13625/3232*e^3 + 17331/1616*e^2 - 3606/101*e - 2428/101, -3/808*e^6 + 115/808*e^5 + 40/101*e^4 - 3577/808*e^3 - 644/101*e^2 + 2928/101*e + 1064/101, 419/1616*e^6 + 2455/1616*e^5 - 1709/808*e^4 - 32615/1616*e^3 - 2977/808*e^2 + 5608/101*e + 3804/101, -757/1616*e^6 - 3773/1616*e^5 + 4653/808*e^4 + 47881/1616*e^3 - 16707/808*e^2 - 15941/202*e + 652/101, 63/404*e^6 + 927/808*e^5 - 815/808*e^4 - 3397/202*e^3 - 665/808*e^2 + 20065/404*e - 2066/101, 557/1616*e^6 + 3225/1616*e^5 - 2807/808*e^4 - 41793/1616*e^3 + 8257/808*e^2 + 16211/202*e + 946/101, 7/101*e^6 + 513/808*e^5 + 1265/808*e^4 - 517/404*e^3 - 11933/808*e^2 - 12281/404*e + 2123/101, -143/1616*e^6 - 511/1616*e^5 + 927/808*e^4 + 4563/1616*e^3 - 4653/808*e^2 - 7/101*e + 1388/101, 327/1616*e^6 + 1807/1616*e^5 - 2795/808*e^4 - 30267/1616*e^3 + 16265/808*e^2 + 6468/101*e - 2438/101, -259/1616*e^6 - 2057/1616*e^5 - 24/101*e^4 + 27987/1616*e^3 + 7747/404*e^2 - 25793/404*e - 2820/101, -5/1616*e^6 - 347/1616*e^5 - 641/404*e^4 - 6635/1616*e^3 + 406/101*e^2 + 17941/404*e + 3883/101, -239/808*e^6 - 1477/808*e^5 + 347/202*e^4 + 16147/808*e^3 + 645/202*e^2 - 4288/101*e - 1758/101, 139/808*e^6 + 349/404*e^5 - 1663/808*e^4 - 9669/808*e^3 + 1191/808*e^2 + 15773/404*e + 3255/101, 49/808*e^6 + 7/808*e^5 - 61/202*e^4 + 5231/808*e^3 + 385/101*e^2 - 10101/202*e - 3508/101, -585/3232*e^6 - 2825/3232*e^5 + 3021/1616*e^4 + 28877/3232*e^3 - 16611/1616*e^2 - 10497/404*e + 4703/101, 165/3232*e^6 + 1553/3232*e^5 + 189/1616*e^4 - 14961/3232*e^3 + 1593/1616*e^2 + 179/202*e - 3015/101, -1409/3232*e^6 - 7329/3232*e^5 + 6837/1616*e^4 + 89253/3232*e^3 - 6379/1616*e^2 - 29625/404*e - 3305/101, 263/1616*e^6 + 1971/1616*e^5 + 147/808*e^4 - 24699/1616*e^3 - 14113/808*e^2 + 5582/101*e + 2986/101, 45/202*e^6 + 1079/808*e^5 - 717/808*e^4 - 6517/404*e^3 - 10343/808*e^2 + 23773/404*e + 4133/101, -1151/3232*e^6 - 7119/3232*e^5 + 4187/1616*e^4 + 89835/3232*e^3 + 2451/1616*e^2 - 28933/404*e - 1537/101, -53/202*e^6 - 997/808*e^5 + 2851/808*e^4 + 7101/404*e^3 - 4683/808*e^2 - 21257/404*e - 4264/101, 59/1616*e^6 + 95/1616*e^5 - 1093/808*e^4 - 7759/1616*e^3 + 6255/808*e^2 + 4033/101*e - 632/101, 675/3232*e^6 + 3011/3232*e^5 - 3175/1616*e^4 - 28223/3232*e^3 - 847/1616*e^2 + 3861/404*e + 4285/101];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {19, 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;