/*
  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![-49, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[4, 2, 2],
[7, 7, w - 7],
[7, 7, w + 6],
[9, 3, 3],
[19, 19, w + 5],
[19, 19, w - 6],
[23, 23, w + 8],
[23, 23, -w + 9],
[25, 5, 5],
[29, 29, -w - 4],
[29, 29, w - 5],
[37, 37, -w - 3],
[37, 37, w - 4],
[41, 41, -w - 9],
[41, 41, w - 10],
[43, 43, -w - 2],
[43, 43, w - 3],
[47, 47, -w - 1],
[47, 47, w - 2],
[53, 53, 2*w - 13],
[53, 53, -2*w - 11],
[59, 59, -4*w - 25],
[59, 59, -4*w + 29],
[61, 61, -w - 10],
[61, 61, w - 11],
[83, 83, -w - 11],
[83, 83, w - 12],
[97, 97, 2*w - 11],
[97, 97, -2*w - 9],
[101, 101, 3*w - 20],
[101, 101, -3*w - 17],
[107, 107, -w - 12],
[107, 107, w - 13],
[109, 109, -5*w - 31],
[109, 109, -5*w + 36],
[121, 11, -11],
[127, 127, 2*w - 19],
[127, 127, -2*w - 17],
[137, 137, -3*w - 16],
[137, 137, 3*w - 19],
[157, 157, -3*w - 23],
[157, 157, 3*w - 26],
[163, 163, -4*w + 27],
[163, 163, 4*w + 23],
[169, 13, -13],
[173, 173, -6*w - 37],
[173, 173, -6*w + 43],
[181, 181, 2*w - 5],
[181, 181, -2*w - 3],
[191, 191, -w - 15],
[191, 191, w - 16],
[193, 193, 2*w - 3],
[193, 193, -2*w - 1],
[197, 197, 2*w - 1],
[223, 223, -w - 16],
[223, 223, w - 17],
[233, 233, -3*w - 13],
[233, 233, 3*w - 16],
[239, 239, -5*w + 34],
[239, 239, -5*w - 29],
[251, 251, 5*w - 41],
[251, 251, 5*w + 36],
[257, 257, -w - 17],
[257, 257, w - 18],
[289, 17, -17],
[293, 293, -w - 18],
[293, 293, w - 19],
[311, 311, -3*w - 10],
[311, 311, 3*w - 13],
[313, 313, -3*w - 26],
[313, 313, 3*w - 29],
[331, 331, -w - 19],
[331, 331, w - 20],
[347, 347, 4*w - 23],
[347, 347, -4*w - 19],
[353, 353, 3*w - 11],
[353, 353, -3*w - 8],
[379, 379, 2*w - 25],
[379, 379, -2*w - 23],
[401, 401, 3*w - 8],
[401, 401, -3*w - 5],
[409, 409, 5*w - 43],
[409, 409, 5*w + 38],
[419, 419, -5*w - 26],
[419, 419, 5*w - 31],
[431, 431, 3*w - 5],
[431, 431, -3*w - 2],
[433, 433, -7*w + 48],
[433, 433, -7*w - 41],
[443, 443, 3*w - 2],
[443, 443, 3*w - 1],
[449, 449, -9*w - 55],
[449, 449, -9*w + 64],
[457, 457, -w - 22],
[457, 457, w - 23],
[479, 479, 2*w - 27],
[479, 479, -2*w - 25],
[487, 487, -3*w - 29],
[487, 487, 3*w - 32],
[491, 491, -5*w - 39],
[491, 491, 5*w - 44],
[499, 499, 4*w - 19],
[499, 499, -4*w - 15],
[503, 503, -w - 23],
[503, 503, w - 24],
[521, 521, 11*w - 86],
[521, 521, -7*w + 47],
[557, 557, 7*w + 51],
[557, 557, 7*w - 58],
[563, 563, -4*w - 13],
[563, 563, 4*w - 17],
[569, 569, -10*w - 61],
[569, 569, -10*w + 71],
[587, 587, 2*w - 29],
[587, 587, -2*w - 27],
[601, 601, -w - 25],
[601, 601, w - 26],
[607, 607, -7*w - 39],
[607, 607, 7*w - 46],
[613, 613, 3*w - 34],
[613, 613, -3*w - 31],
[617, 617, -6*w - 31],
[617, 617, 6*w - 37],
[619, 619, 4*w - 15],
[619, 619, -4*w - 11],
[631, 631, 5*w - 27],
[631, 631, -5*w - 22],
[653, 653, -w - 26],
[653, 653, w - 27],
[661, 661, 5*w - 46],
[661, 661, -5*w - 41],
[683, 683, -9*w + 62],
[683, 683, -9*w - 53],
[691, 691, 7*w - 45],
[691, 691, -7*w - 38],
[727, 727, -6*w - 47],
[727, 727, 6*w - 53],
[733, 733, 4*w - 41],
[733, 733, -4*w - 37],
[739, 739, -4*w - 5],
[739, 739, 4*w - 9],
[751, 751, 8*w - 53],
[751, 751, -8*w - 45],
[769, 769, 5*w - 24],
[769, 769, -5*w - 19],
[773, 773, 7*w - 44],
[773, 773, -7*w - 37],
[787, 787, 4*w - 3],
[787, 787, 4*w - 1],
[797, 797, -9*w - 52],
[797, 797, -9*w + 61],
[811, 811, -5*w - 18],
[811, 811, 5*w - 23],
[821, 821, -w - 29],
[821, 821, w - 30],
[827, 827, 2*w - 33],
[827, 827, -2*w - 31],
[829, 829, -10*w + 69],
[829, 829, -10*w - 59],
[839, 839, 5*w - 48],
[839, 839, -5*w - 43],
[853, 853, -7*w - 36],
[853, 853, 7*w - 43],
[881, 881, -w - 30],
[881, 881, w - 31],
[961, 31, -31],
[991, 991, -5*w - 13],
[991, 991, 5*w - 18]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - 8*x^7 + 4*x^6 + 103*x^5 - 186*x^4 - 346*x^3 + 867*x^2 + 101*x - 724;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -4/23*e^7 + 26/23*e^6 + e^5 - 366/23*e^4 + 149/23*e^3 + 1527/23*e^2 - 798/23*e - 1555/23, -4/23*e^7 + 26/23*e^6 + e^5 - 366/23*e^4 + 149/23*e^3 + 1527/23*e^2 - 798/23*e - 1555/23, -1/4*e^7 + 5/4*e^6 + 11/4*e^5 - 37/2*e^4 - 4*e^3 + 157/2*e^2 - 81/4*e - 73, 1/23*e^7 - 18/23*e^6 + e^5 + 287/23*e^4 - 388/23*e^3 - 1434/23*e^2 + 1154/23*e + 1809/23, 1/23*e^7 - 18/23*e^6 + e^5 + 287/23*e^4 - 388/23*e^3 - 1434/23*e^2 + 1154/23*e + 1809/23, 9/23*e^7 - 47/23*e^6 - 4*e^5 + 674/23*e^4 + 96/23*e^3 - 2763/23*e^2 + 772/23*e + 2619/23, 9/23*e^7 - 47/23*e^6 - 4*e^5 + 674/23*e^4 + 96/23*e^3 - 2763/23*e^2 + 772/23*e + 2619/23, 16/23*e^7 - 81/23*e^6 - 7*e^5 + 1142/23*e^4 + 163/23*e^3 - 4682/23*e^2 + 1329/23*e + 4702/23, 9/92*e^7 - 1/92*e^6 - 9/4*e^5 - 77/46*e^4 + 392/23*e^3 + 999/46*e^2 - 3391/92*e - 1076/23, 9/92*e^7 - 1/92*e^6 - 9/4*e^5 - 77/46*e^4 + 392/23*e^3 + 999/46*e^2 - 3391/92*e - 1076/23, 31/92*e^7 - 167/92*e^6 - 11/4*e^5 + 1125/46*e^4 - 63/23*e^3 - 4471/46*e^2 + 3827/92*e + 2238/23, 31/92*e^7 - 167/92*e^6 - 11/4*e^5 + 1125/46*e^4 - 63/23*e^3 - 4471/46*e^2 + 3827/92*e + 2238/23, 49/92*e^7 - 353/92*e^6 - 9/4*e^5 + 2627/46*e^4 - 751/23*e^3 - 11857/46*e^2 + 13329/92*e + 6664/23, 49/92*e^7 - 353/92*e^6 - 9/4*e^5 + 2627/46*e^4 - 751/23*e^3 - 11857/46*e^2 + 13329/92*e + 6664/23, 8/23*e^7 - 52/23*e^6 - 2*e^5 + 755/23*e^4 - 321/23*e^3 - 3307/23*e^2 + 1642/23*e + 3593/23, 8/23*e^7 - 52/23*e^6 - 2*e^5 + 755/23*e^4 - 321/23*e^3 - 3307/23*e^2 + 1642/23*e + 3593/23, -24/23*e^7 + 133/23*e^6 + 9*e^5 - 1874/23*e^4 + 112/23*e^3 + 7782/23*e^2 - 2741/23*e - 7927/23, -24/23*e^7 + 133/23*e^6 + 9*e^5 - 1874/23*e^4 + 112/23*e^3 + 7782/23*e^2 - 2741/23*e - 7927/23, 113/92*e^7 - 585/92*e^6 - 49/4*e^5 + 4175/46*e^4 + 217/23*e^3 - 17219/46*e^2 + 10825/92*e + 8261/23, 113/92*e^7 - 585/92*e^6 - 49/4*e^5 + 4175/46*e^4 + 217/23*e^3 - 17219/46*e^2 + 10825/92*e + 8261/23, -21/46*e^7 + 79/46*e^6 + 13/2*e^5 - 564/23*e^4 - 572/23*e^3 + 2223/23*e^2 + 307/46*e - 1894/23, -21/46*e^7 + 79/46*e^6 + 13/2*e^5 - 564/23*e^4 - 572/23*e^3 + 2223/23*e^2 + 307/46*e - 1894/23, 25/92*e^7 - 105/92*e^6 - 13/4*e^5 + 701/46*e^4 + 197/23*e^3 - 2515/46*e^2 + 629/92*e + 870/23, 25/92*e^7 - 105/92*e^6 - 13/4*e^5 + 701/46*e^4 + 197/23*e^3 - 2515/46*e^2 + 629/92*e + 870/23, -13/46*e^7 + 119/46*e^6 - 1/2*e^5 - 888/23*e^4 + 843/23*e^3 + 4123/23*e^2 - 5871/46*e - 4893/23, -13/46*e^7 + 119/46*e^6 - 1/2*e^5 - 888/23*e^4 + 843/23*e^3 + 4123/23*e^2 - 5871/46*e - 4893/23, -113/92*e^7 + 585/92*e^6 + 49/4*e^5 - 4175/46*e^4 - 217/23*e^3 + 17219/46*e^2 - 10733/92*e - 8261/23, -113/92*e^7 + 585/92*e^6 + 49/4*e^5 - 4175/46*e^4 - 217/23*e^3 + 17219/46*e^2 - 10733/92*e - 8261/23, -13/92*e^7 - 19/92*e^6 + 17/4*e^5 + 147/46*e^4 - 717/23*e^3 - 1121/46*e^2 + 4755/92*e + 1176/23, -13/92*e^7 - 19/92*e^6 + 17/4*e^5 + 147/46*e^4 - 717/23*e^3 - 1121/46*e^2 + 4755/92*e + 1176/23, -13/23*e^7 + 73/23*e^6 + 5*e^5 - 1063/23*e^4 + 122/23*e^3 + 4451/23*e^2 - 1984/23*e - 4289/23, -13/23*e^7 + 73/23*e^6 + 5*e^5 - 1063/23*e^4 + 122/23*e^3 + 4451/23*e^2 - 1984/23*e - 4289/23, -19/92*e^7 + 135/92*e^6 + 3/4*e^5 - 967/46*e^4 + 279/23*e^3 + 4285/46*e^2 - 4147/92*e - 2584/23, -19/92*e^7 + 135/92*e^6 + 3/4*e^5 - 967/46*e^4 + 279/23*e^3 + 4285/46*e^2 - 4147/92*e - 2584/23, -9/92*e^7 - 91/92*e^6 + 21/4*e^5 + 767/46*e^4 - 1174/23*e^3 - 4357/46*e^2 + 10751/92*e + 3491/23, -77/46*e^7 + 443/46*e^6 + 27/2*e^5 - 3149/23*e^4 + 471/23*e^3 + 13234/23*e^2 - 10589/46*e - 13630/23, -77/46*e^7 + 443/46*e^6 + 27/2*e^5 - 3149/23*e^4 + 471/23*e^3 + 13234/23*e^2 - 10589/46*e - 13630/23, -73/92*e^7 + 417/92*e^6 + 25/4*e^5 - 2897/46*e^4 + 204/23*e^3 + 11815/46*e^2 - 9101/92*e - 5604/23, -73/92*e^7 + 417/92*e^6 + 25/4*e^5 - 2897/46*e^4 + 204/23*e^3 + 11815/46*e^2 - 9101/92*e - 5604/23, 3/92*e^7 - 31/92*e^6 + 1/4*e^5 + 281/46*e^4 - 199/23*e^3 - 1599/46*e^2 + 3071/92*e + 960/23, 3/92*e^7 - 31/92*e^6 + 1/4*e^5 + 281/46*e^4 - 199/23*e^3 - 1599/46*e^2 + 3071/92*e + 960/23, 12/23*e^7 - 55/23*e^6 - 6*e^5 + 776/23*e^4 + 289/23*e^3 - 3132/23*e^2 + 692/23*e + 3124/23, 12/23*e^7 - 55/23*e^6 - 6*e^5 + 776/23*e^4 + 289/23*e^3 - 3132/23*e^2 + 692/23*e + 3124/23, -16/23*e^7 + 81/23*e^6 + 7*e^5 - 1165/23*e^4 - 117/23*e^3 + 4889/23*e^2 - 1559/23*e - 4610/23, 75/92*e^7 - 407/92*e^6 - 31/4*e^5 + 2977/46*e^4 + 39/23*e^3 - 12789/46*e^2 + 7867/92*e + 6543/23, 75/92*e^7 - 407/92*e^6 - 31/4*e^5 + 2977/46*e^4 + 39/23*e^3 - 12789/46*e^2 + 7867/92*e + 6543/23, -37/92*e^7 + 321/92*e^6 + 1/4*e^5 - 2515/46*e^4 + 1059/23*e^3 + 11901/46*e^2 - 16133/92*e - 6665/23, -37/92*e^7 + 321/92*e^6 + 1/4*e^5 - 2515/46*e^4 + 1059/23*e^3 + 11901/46*e^2 - 16133/92*e - 6665/23, -7/46*e^7 + 103/46*e^6 - 5/2*e^5 - 786/23*e^4 + 1013/23*e^3 + 3823/23*e^2 - 5617/46*e - 5009/23, -7/46*e^7 + 103/46*e^6 - 5/2*e^5 - 786/23*e^4 + 1013/23*e^3 + 3823/23*e^2 - 5617/46*e - 5009/23, 21/92*e^7 - 125/92*e^6 - 9/4*e^5 + 1001/46*e^4 - 36/23*e^3 - 4569/46*e^2 + 3925/92*e + 2189/23, 21/92*e^7 - 125/92*e^6 - 9/4*e^5 + 1001/46*e^4 - 36/23*e^3 - 4569/46*e^2 + 3925/92*e + 2189/23, 56/23*e^7 - 318/23*e^6 - 20*e^5 + 4526/23*e^4 - 660/23*e^3 - 18940/23*e^2 + 7768/23*e + 19194/23, -11/46*e^7 - 9/46*e^6 + 11/2*e^5 + 204/23*e^4 - 971/23*e^3 - 1819/23*e^2 + 4303/46*e + 3241/23, -11/46*e^7 - 9/46*e^6 + 11/2*e^5 + 204/23*e^4 - 971/23*e^3 - 1819/23*e^2 + 4303/46*e + 3241/23, -15/23*e^7 + 132/23*e^6 + e^5 - 2074/23*e^4 + 1519/23*e^3 + 9780/23*e^2 - 6132/23*e - 11173/23, -15/23*e^7 + 132/23*e^6 + e^5 - 2074/23*e^4 + 1519/23*e^3 + 9780/23*e^2 - 6132/23*e - 11173/23, -129/46*e^7 + 735/46*e^6 + 47/2*e^5 - 5275/23*e^4 + 646/23*e^3 + 22320/23*e^2 - 17697/46*e - 22645/23, -129/46*e^7 + 735/46*e^6 + 47/2*e^5 - 5275/23*e^4 + 646/23*e^3 + 22320/23*e^2 - 17697/46*e - 22645/23, 13/46*e^7 - 27/46*e^6 - 9/2*e^5 + 129/23*e^4 + 491/23*e^3 - 121/23*e^2 - 937/46*e - 443/23, 13/46*e^7 - 27/46*e^6 - 9/2*e^5 + 129/23*e^4 + 491/23*e^3 - 121/23*e^2 - 937/46*e - 443/23, -9/92*e^7 + 1/92*e^6 + 9/4*e^5 - 61/46*e^4 - 254/23*e^3 + 381/46*e^2 + 355/92*e - 166/23, -9/92*e^7 + 1/92*e^6 + 9/4*e^5 - 61/46*e^4 - 254/23*e^3 + 381/46*e^2 + 355/92*e - 166/23, -64/23*e^7 + 370/23*e^6 + 22*e^5 - 5235/23*e^4 + 889/23*e^3 + 21856/23*e^2 - 9088/23*e - 21890/23, 123/92*e^7 - 535/92*e^6 - 63/4*e^5 + 3655/46*e^4 + 926/23*e^3 - 14085/46*e^2 + 4379/92*e + 5872/23, 123/92*e^7 - 535/92*e^6 - 63/4*e^5 + 3655/46*e^4 + 926/23*e^3 - 14085/46*e^2 + 4379/92*e + 5872/23, 30/23*e^7 - 149/23*e^6 - 14*e^5 + 2124/23*e^4 + 527/23*e^3 - 8796/23*e^2 + 2167/23*e + 8707/23, 30/23*e^7 - 149/23*e^6 - 14*e^5 + 2124/23*e^4 + 527/23*e^3 - 8796/23*e^2 + 2167/23*e + 8707/23, -279/92*e^7 + 1595/92*e^6 + 103/4*e^5 - 11505/46*e^4 + 636/23*e^3 + 48887/46*e^2 - 37663/92*e - 25340/23, -279/92*e^7 + 1595/92*e^6 + 103/4*e^5 - 11505/46*e^4 + 636/23*e^3 + 48887/46*e^2 - 37663/92*e - 25340/23, 167/46*e^7 - 959/46*e^6 - 61/2*e^5 + 6910/23*e^4 - 888/23*e^3 - 29165/23*e^2 + 23645/46*e + 29255/23, 167/46*e^7 - 959/46*e^6 - 61/2*e^5 + 6910/23*e^4 - 888/23*e^3 - 29165/23*e^2 + 23645/46*e + 29255/23, 87/46*e^7 - 485/46*e^6 - 31/2*e^5 + 3388/23*e^4 - 387/23*e^3 - 14010/23*e^2 + 10399/46*e + 14142/23, 87/46*e^7 - 485/46*e^6 - 31/2*e^5 + 3388/23*e^4 - 387/23*e^3 - 14010/23*e^2 + 10399/46*e + 14142/23, 37/92*e^7 - 413/92*e^6 + 11/4*e^5 + 3067/46*e^4 - 1703/23*e^3 - 14155/46*e^2 + 21837/92*e + 8390/23, 37/92*e^7 - 413/92*e^6 + 11/4*e^5 + 3067/46*e^4 - 1703/23*e^3 - 14155/46*e^2 + 21837/92*e + 8390/23, 10/23*e^7 - 65/23*e^6 - 3*e^5 + 984/23*e^4 - 292/23*e^3 - 4450/23*e^2 + 1788/23*e + 4796/23, 10/23*e^7 - 65/23*e^6 - 3*e^5 + 984/23*e^4 - 292/23*e^3 - 4450/23*e^2 + 1788/23*e + 4796/23, 39/92*e^7 - 219/92*e^6 - 19/4*e^5 + 1721/46*e^4 + 173/23*e^3 - 7815/46*e^2 + 2847/92*e + 4108/23, 39/92*e^7 - 219/92*e^6 - 19/4*e^5 + 1721/46*e^4 + 173/23*e^3 - 7815/46*e^2 + 2847/92*e + 4108/23, -27/92*e^7 + 95/92*e^6 + 15/4*e^5 - 413/46*e^4 - 463/23*e^3 + 177/46*e^2 + 3733/92*e + 1066/23, -27/92*e^7 + 95/92*e^6 + 15/4*e^5 - 413/46*e^4 - 463/23*e^3 + 177/46*e^2 + 3733/92*e + 1066/23, 11/23*e^7 - 106/23*e^6 + 2*e^5 + 1524/23*e^4 - 1623/23*e^3 - 6988/23*e^2 + 4989/23*e + 8721/23, 11/23*e^7 - 106/23*e^6 + 2*e^5 + 1524/23*e^4 - 1623/23*e^3 - 6988/23*e^2 + 4989/23*e + 8721/23, 21/23*e^7 - 148/23*e^6 - 4*e^5 + 2163/23*e^4 - 1202/23*e^3 - 9552/23*e^2 + 5351/23*e + 10895/23, 21/23*e^7 - 148/23*e^6 - 4*e^5 + 2163/23*e^4 - 1202/23*e^3 - 9552/23*e^2 + 5351/23*e + 10895/23, -121/92*e^7 + 729/92*e^6 + 41/4*e^5 - 5369/46*e^4 + 605/23*e^3 + 23461/46*e^2 - 20701/92*e - 12408/23, -121/92*e^7 + 729/92*e^6 + 41/4*e^5 - 5369/46*e^4 + 605/23*e^3 + 23461/46*e^2 - 20701/92*e - 12408/23, -70/23*e^7 + 386/23*e^6 + 27*e^5 - 5485/23*e^4 + 250/23*e^3 + 22778/23*e^2 - 8445/23*e - 22417/23, -70/23*e^7 + 386/23*e^6 + 27*e^5 - 5485/23*e^4 + 250/23*e^3 + 22778/23*e^2 - 8445/23*e - 22417/23, 111/92*e^7 - 503/92*e^6 - 55/4*e^5 + 3451/46*e^4 + 756/23*e^3 - 13623/46*e^2 + 4147/92*e + 6678/23, 111/92*e^7 - 503/92*e^6 - 55/4*e^5 + 3451/46*e^4 + 756/23*e^3 - 13623/46*e^2 + 4147/92*e + 6678/23, 41/92*e^7 - 117/92*e^6 - 29/4*e^5 + 697/46*e^4 + 945/23*e^3 - 1981/46*e^2 - 6575/92*e + 194/23, 41/92*e^7 - 117/92*e^6 - 29/4*e^5 + 697/46*e^4 + 945/23*e^3 - 1981/46*e^2 - 6575/92*e + 194/23, 7/46*e^7 + 35/46*e^6 - 13/2*e^5 - 249/23*e^4 + 1333/23*e^3 + 1352/23*e^2 - 5837/46*e - 2374/23, 7/46*e^7 + 35/46*e^6 - 13/2*e^5 - 249/23*e^4 + 1333/23*e^3 + 1352/23*e^2 - 5837/46*e - 2374/23, -103/46*e^7 + 405/46*e^6 + 61/2*e^5 - 2786/23*e^4 - 2673/23*e^3 + 10509/23*e^2 + 2371/46*e - 8443/23, -103/46*e^7 + 405/46*e^6 + 61/2*e^5 - 2786/23*e^4 - 2673/23*e^3 + 10509/23*e^2 + 2371/46*e - 8443/23, 39/23*e^7 - 196/23*e^6 - 17*e^5 + 2683/23*e^4 + 462/23*e^3 - 10455/23*e^2 + 2939/23*e + 9693/23, 39/23*e^7 - 196/23*e^6 - 17*e^5 + 2683/23*e^4 + 462/23*e^3 - 10455/23*e^2 + 2939/23*e + 9693/23, -61/46*e^7 + 247/46*e^6 + 33/2*e^5 - 1543/23*e^4 - 1368/23*e^3 + 4959/23*e^2 + 1481/46*e - 2470/23, -61/46*e^7 + 247/46*e^6 + 33/2*e^5 - 1543/23*e^4 - 1368/23*e^3 + 4959/23*e^2 + 1481/46*e - 2470/23, -12/23*e^7 + 55/23*e^6 + 7*e^5 - 822/23*e^4 - 496/23*e^3 + 3408/23*e^2 - 623/23*e - 3239/23, -12/23*e^7 + 55/23*e^6 + 7*e^5 - 822/23*e^4 - 496/23*e^3 + 3408/23*e^2 - 623/23*e - 3239/23, 40/23*e^7 - 260/23*e^6 - 10*e^5 + 3683/23*e^4 - 1490/23*e^3 - 15569/23*e^2 + 7911/23*e + 16079/23, 40/23*e^7 - 260/23*e^6 - 10*e^5 + 3683/23*e^4 - 1490/23*e^3 - 15569/23*e^2 + 7911/23*e + 16079/23, -117/92*e^7 + 565/92*e^6 + 53/4*e^5 - 3875/46*e^4 - 450/23*e^3 + 15303/46*e^2 - 8081/92*e - 7080/23, -117/92*e^7 + 565/92*e^6 + 53/4*e^5 - 3875/46*e^4 - 450/23*e^3 + 15303/46*e^2 - 8081/92*e - 7080/23, 5/23*e^7 - 44/23*e^6 - e^5 + 722/23*e^4 - 307/23*e^3 - 3559/23*e^2 + 1400/23*e + 4468/23, 5/23*e^7 - 44/23*e^6 - e^5 + 722/23*e^4 - 307/23*e^3 - 3559/23*e^2 + 1400/23*e + 4468/23, -269/92*e^7 + 1277/92*e^6 + 129/4*e^5 - 8897/46*e^4 - 1530/23*e^3 + 35369/46*e^2 - 11633/92*e - 16137/23, -269/92*e^7 + 1277/92*e^6 + 129/4*e^5 - 8897/46*e^4 - 1530/23*e^3 + 35369/46*e^2 - 11633/92*e - 16137/23, 39/46*e^7 - 173/46*e^6 - 21/2*e^5 + 1238/23*e^4 + 668/23*e^3 - 4917/23*e^2 + 961/46*e + 4306/23, 39/46*e^7 - 173/46*e^6 - 21/2*e^5 + 1238/23*e^4 + 668/23*e^3 - 4917/23*e^2 + 961/46*e + 4306/23, -105/92*e^7 + 533/92*e^6 + 45/4*e^5 - 3809/46*e^4 - 142/23*e^3 + 15945/46*e^2 - 10517/92*e - 8162/23, -105/92*e^7 + 533/92*e^6 + 45/4*e^5 - 3809/46*e^4 - 142/23*e^3 + 15945/46*e^2 - 10517/92*e - 8162/23, -67/46*e^7 + 493/46*e^6 + 11/2*e^5 - 3692/23*e^4 + 2257/23*e^3 + 16828/23*e^2 - 19657/46*e - 19167/23, -67/46*e^7 + 493/46*e^6 + 11/2*e^5 - 3692/23*e^4 + 2257/23*e^3 + 16828/23*e^2 - 19657/46*e - 19167/23, -71/92*e^7 + 519/92*e^6 + 11/4*e^5 - 3829/46*e^4 + 1206/23*e^3 + 17051/46*e^2 - 19535/92*e - 9426/23, -71/92*e^7 + 519/92*e^6 + 11/4*e^5 - 3829/46*e^4 + 1206/23*e^3 + 17051/46*e^2 - 19535/92*e - 9426/23, -19/92*e^7 + 411/92*e^6 - 29/4*e^5 - 3221/46*e^4 + 2487/23*e^3 + 16153/46*e^2 - 27791/92*e - 10887/23, -19/92*e^7 + 411/92*e^6 - 29/4*e^5 - 3221/46*e^4 + 2487/23*e^3 + 16153/46*e^2 - 27791/92*e - 10887/23, -27/23*e^7 + 233/23*e^6 + e^5 - 3517/23*e^4 + 2725/23*e^3 + 16086/23*e^2 - 10113/23*e - 18368/23, -27/23*e^7 + 233/23*e^6 + e^5 - 3517/23*e^4 + 2725/23*e^3 + 16086/23*e^2 - 10113/23*e - 18368/23, -15/23*e^7 + 109/23*e^6 + 4*e^5 - 1752/23*e^4 + 760/23*e^3 + 8400/23*e^2 - 4223/23*e - 9563/23, -15/23*e^7 + 109/23*e^6 + 4*e^5 - 1752/23*e^4 + 760/23*e^3 + 8400/23*e^2 - 4223/23*e - 9563/23, 137/92*e^7 - 833/92*e^6 - 49/4*e^5 + 6331/46*e^4 - 616/23*e^3 - 28493/46*e^2 + 23525/92*e + 15550/23, 137/92*e^7 - 833/92*e^6 - 49/4*e^5 + 6331/46*e^4 - 616/23*e^3 - 28493/46*e^2 + 23525/92*e + 15550/23, -55/92*e^7 + 323/92*e^6 + 23/4*e^5 - 2545/46*e^4 + 91/23*e^3 + 11605/46*e^2 - 8339/92*e - 6008/23, -55/92*e^7 + 323/92*e^6 + 23/4*e^5 - 2545/46*e^4 + 91/23*e^3 + 11605/46*e^2 - 8339/92*e - 6008/23, 72/23*e^7 - 468/23*e^6 - 19*e^5 + 6772/23*e^4 - 2590/23*e^3 - 29418/23*e^2 + 14364/23*e + 31693/23, 72/23*e^7 - 468/23*e^6 - 19*e^5 + 6772/23*e^4 - 2590/23*e^3 - 29418/23*e^2 + 14364/23*e + 31693/23, 123/46*e^7 - 719/46*e^6 - 41/2*e^5 + 5127/23*e^4 - 1115/23*e^3 - 21606/23*e^2 + 19099/46*e + 21933/23, 123/46*e^7 - 719/46*e^6 - 41/2*e^5 + 5127/23*e^4 - 1115/23*e^3 - 21606/23*e^2 + 19099/46*e + 21933/23, 58/23*e^7 - 262/23*e^6 - 29*e^5 + 3628/23*e^4 + 1531/23*e^3 - 14310/23*e^2 + 2716/23*e + 13336/23, 58/23*e^7 - 262/23*e^6 - 29*e^5 + 3628/23*e^4 + 1531/23*e^3 - 14310/23*e^2 + 2716/23*e + 13336/23, -203/92*e^7 + 1331/92*e^6 + 47/4*e^5 - 9339/46*e^4 + 2050/23*e^3 + 39383/46*e^2 - 39935/92*e - 20639/23, -203/92*e^7 + 1331/92*e^6 + 47/4*e^5 - 9339/46*e^4 + 2050/23*e^3 + 39383/46*e^2 - 39935/92*e - 20639/23, -e^5 + 3*e^4 + 7*e^3 - 16*e^2 - 9*e - 8, -e^5 + 3*e^4 + 7*e^3 - 16*e^2 - 9*e - 8, 9/23*e^7 - 93/23*e^6 + e^5 + 1479/23*e^4 - 1330/23*e^3 - 7179/23*e^2 + 4429/23*e + 8484/23, 9/23*e^7 - 93/23*e^6 + e^5 + 1479/23*e^4 - 1330/23*e^3 - 7179/23*e^2 + 4429/23*e + 8484/23, -173/92*e^7 + 1205/92*e^6 + 37/4*e^5 - 8829/46*e^4 + 2245/23*e^3 + 38665/46*e^2 - 42989/92*e - 21090/23, -173/92*e^7 + 1205/92*e^6 + 37/4*e^5 - 8829/46*e^4 + 2245/23*e^3 + 38665/46*e^2 - 42989/92*e - 21090/23, -84/23*e^7 + 477/23*e^6 + 30*e^5 - 6720/23*e^4 + 783/23*e^3 + 27927/23*e^2 - 10548/23*e - 28377/23, -84/23*e^7 + 477/23*e^6 + 30*e^5 - 6720/23*e^4 + 783/23*e^3 + 27927/23*e^2 - 10548/23*e - 28377/23, -1/2*e^7 + 5/2*e^6 + 11/2*e^5 - 37*e^4 - 5*e^3 + 154*e^2 - 129/2*e - 146, -1/2*e^7 + 5/2*e^6 + 11/2*e^5 - 37*e^4 - 5*e^3 + 154*e^2 - 129/2*e - 146, -70/23*e^7 + 363/23*e^6 + 31*e^5 - 5209/23*e^4 - 762/23*e^3 + 21743/23*e^2 - 6053/23*e - 21819/23, -70/23*e^7 + 363/23*e^6 + 31*e^5 - 5209/23*e^4 - 762/23*e^3 + 21743/23*e^2 - 6053/23*e - 21819/23, 33/23*e^7 - 180/23*e^6 - 13*e^5 + 2617/23*e^4 - 200/23*e^3 - 11028/23*e^2 + 4410/23*e + 10477/23, 33/23*e^7 - 180/23*e^6 - 13*e^5 + 2617/23*e^4 - 200/23*e^3 - 11028/23*e^2 + 4410/23*e + 10477/23, -329/92*e^7 + 1989/92*e^6 + 105/4*e^5 - 14241/46*e^4 + 1714/23*e^3 + 60311/46*e^2 - 52537/92*e - 30898/23, -329/92*e^7 + 1989/92*e^6 + 105/4*e^5 - 14241/46*e^4 + 1714/23*e^3 + 60311/46*e^2 - 52537/92*e - 30898/23, 107/46*e^7 - 707/46*e^6 - 29/2*e^5 + 5292/23*e^4 - 2128/23*e^3 - 23681/23*e^2 + 24371/46*e + 25654/23, 107/46*e^7 - 707/46*e^6 - 29/2*e^5 + 5292/23*e^4 - 2128/23*e^3 - 23681/23*e^2 + 24371/46*e + 25654/23, 109/92*e^7 - 237/92*e^6 - 85/4*e^5 + 1209/46*e^4 + 2928/23*e^3 - 1103/46*e^2 - 20011/92*e - 3047/23, 109/92*e^7 - 237/92*e^6 - 85/4*e^5 + 1209/46*e^4 + 2928/23*e^3 - 1103/46*e^2 - 20011/92*e - 3047/23, 48/23*e^7 - 312/23*e^6 - 14*e^5 + 4622/23*e^4 - 1466/23*e^3 - 20394/23*e^2 + 9231/23*e + 21857/23, 48/23*e^7 - 312/23*e^6 - 14*e^5 + 4622/23*e^4 - 1466/23*e^3 - 20394/23*e^2 + 9231/23*e + 21857/23, -101/92*e^7 + 369/92*e^6 + 61/4*e^5 - 2499/46*e^4 - 1404/23*e^3 + 9443/46*e^2 + 4035/92*e - 3616/23, -101/92*e^7 + 369/92*e^6 + 61/4*e^5 - 2499/46*e^4 - 1404/23*e^3 + 9443/46*e^2 + 4035/92*e - 3616/23, 19/92*e^7 + 49/92*e^6 - 27/4*e^5 - 505/46*e^4 + 1331/23*e^3 + 3719/46*e^2 - 11769/92*e - 3488/23, 19/92*e^7 + 49/92*e^6 - 27/4*e^5 - 505/46*e^4 + 1331/23*e^3 + 3719/46*e^2 - 11769/92*e - 3488/23, -53/92*e^7 + 241/92*e^6 + 21/4*e^5 - 1315/46*e^4 - 310/23*e^3 + 3731/46*e^2 + 2019/92*e - 469/23, 89/46*e^7 - 567/46*e^6 - 27/2*e^5 + 4227/23*e^4 - 1373/23*e^3 - 18641/23*e^2 + 17905/46*e + 19815/23, 89/46*e^7 - 567/46*e^6 - 27/2*e^5 + 4227/23*e^4 - 1373/23*e^3 - 18641/23*e^2 + 17905/46*e + 19815/23];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();

// 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;