/*
  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![2, 0, -6, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {13, 13, -w^3 + 3*w^2 + 3*w - 1}>;

primesArray := [
[2, 2, w],
[5, 5, -w^3 + 3*w^2 + 2*w - 1],
[5, 5, w - 1],
[11, 11, w^3 - 2*w^2 - 5*w - 1],
[13, 13, -w^3 + 3*w^2 + 3*w - 1],
[17, 17, 2*w^3 - 5*w^2 - 9*w + 5],
[23, 23, -w^3 + 3*w^2 + 4*w - 5],
[25, 5, -w^2 + 3*w + 1],
[29, 29, -2*w^3 + 6*w^2 + 5*w - 1],
[31, 31, -w^2 + 2*w + 1],
[43, 43, -w^3 + 3*w^2 + 2*w - 3],
[43, 43, -w^3 + 2*w^2 + 6*w - 3],
[59, 59, 2*w^3 - 6*w^2 - 7*w + 7],
[73, 73, 3*w^3 - 6*w^2 - 16*w - 5],
[79, 79, -5*w^3 + 14*w^2 + 20*w - 17],
[81, 3, -3],
[83, 83, -w - 3],
[83, 83, w^2 - 3*w - 7],
[89, 89, w^2 - 2*w - 7],
[101, 101, -2*w^3 + 5*w^2 + 8*w - 3],
[101, 101, 5*w^3 - 14*w^2 - 19*w + 17],
[103, 103, -w^3 + w^2 + 8*w + 3],
[103, 103, 2*w^3 - 5*w^2 - 7*w - 1],
[109, 109, 3*w^3 - 7*w^2 - 14*w + 1],
[109, 109, -w^2 + 4*w + 1],
[127, 127, -3*w^3 + 8*w^2 + 11*w - 7],
[127, 127, -w^3 + 2*w^2 + 7*w + 1],
[137, 137, -2*w^3 + 5*w^2 + 9*w - 1],
[139, 139, -w^3 + 3*w^2 + 2*w - 5],
[149, 149, w^3 - 3*w^2 - w - 1],
[149, 149, -2*w^3 + 4*w^2 + 10*w + 1],
[163, 163, w^3 - 4*w^2 - w + 11],
[163, 163, 2*w^3 - 4*w^2 - 10*w + 1],
[173, 173, -4*w^3 + 11*w^2 + 15*w - 13],
[179, 179, 2*w^2 - 6*w - 3],
[179, 179, -w^3 + 2*w^2 + 7*w - 1],
[181, 181, w^3 - 10*w - 9],
[181, 181, -3*w^3 + 8*w^2 + 14*w - 7],
[199, 199, w^3 - 8*w - 9],
[211, 211, 2*w - 3],
[223, 223, -w^3 + 4*w^2 + w - 5],
[227, 227, w^3 - 2*w^2 - 5*w - 5],
[227, 227, -3*w^3 + 7*w^2 + 14*w - 5],
[227, 227, -4*w^3 + 10*w^2 + 19*w - 7],
[227, 227, w^2 - 4*w - 5],
[239, 239, -2*w^3 + 5*w^2 + 7*w - 3],
[251, 251, -w^3 + 4*w^2 + 2*w - 11],
[251, 251, 2*w^3 - 6*w^2 - 6*w + 1],
[257, 257, -3*w^3 + 6*w^2 + 16*w + 3],
[263, 263, -w^3 + 3*w^2 + 3*w - 7],
[263, 263, w^3 - 4*w^2 + 7],
[269, 269, -2*w^3 + 6*w^2 + 6*w - 11],
[269, 269, -w^3 + 4*w^2 - w - 5],
[271, 271, -3*w^3 + 9*w^2 + 11*w - 11],
[271, 271, w^2 - 5*w - 3],
[277, 277, w^3 - 4*w^2 - 2*w + 7],
[277, 277, -2*w^3 + 7*w^2 + 8*w - 9],
[281, 281, 2*w^2 - 4*w - 5],
[281, 281, 2*w^3 - 4*w^2 - 11*w + 1],
[283, 283, -3*w^3 + 8*w^2 + 14*w - 11],
[293, 293, w^3 - 2*w^2 - 6*w + 5],
[307, 307, -2*w^3 + 5*w^2 + 7*w - 1],
[313, 313, 2*w^3 - 4*w^2 - 9*w + 3],
[313, 313, -2*w^3 + 5*w^2 + 6*w - 3],
[317, 317, -2*w^3 + 5*w^2 + 9*w + 1],
[347, 347, 4*w^3 - 9*w^2 - 21*w + 3],
[347, 347, 4*w^3 - 10*w^2 - 18*w + 5],
[349, 349, -2*w^3 + 6*w^2 + 7*w - 3],
[353, 353, -w^3 + 12*w + 5],
[353, 353, w^2 - w - 7],
[359, 359, -2*w^3 + 5*w^2 + 6*w - 7],
[359, 359, -w^3 + 4*w^2 - 11],
[373, 373, -w^3 + 2*w^2 + 6*w + 5],
[379, 379, -3*w^3 + 8*w^2 + 14*w - 5],
[397, 397, 6*w^3 - 14*w^2 - 30*w + 7],
[401, 401, 3*w^3 - 5*w^2 - 20*w - 7],
[409, 409, 3*w^3 - 7*w^2 - 12*w - 3],
[409, 409, 2*w^2 - 4*w - 7],
[421, 421, w^3 - 11*w - 11],
[421, 421, -w^3 + 3*w^2 + w - 7],
[431, 431, 2*w^3 - 3*w^2 - 14*w - 3],
[431, 431, -4*w^3 + 11*w^2 + 15*w - 15],
[433, 433, 2*w^3 - 5*w^2 - 6*w + 1],
[439, 439, -4*w^3 + 11*w^2 + 13*w - 9],
[439, 439, -2*w^3 + 6*w^2 + 8*w - 11],
[443, 443, 2*w - 5],
[443, 443, -6*w^3 + 17*w^2 + 24*w - 21],
[449, 449, w^3 - 2*w^2 - 8*w + 3],
[449, 449, 4*w^3 - 12*w^2 - 15*w + 17],
[457, 457, -2*w^3 + 4*w^2 + 9*w + 5],
[461, 461, -3*w^3 + 6*w^2 + 15*w - 1],
[461, 461, -3*w^3 + 7*w^2 + 17*w + 1],
[463, 463, -4*w^3 + 10*w^2 + 17*w - 7],
[463, 463, -3*w^3 + 10*w^2 + 9*w - 17],
[467, 467, 4*w^3 - 8*w^2 - 21*w - 7],
[467, 467, w^3 - 6*w^2 + 3*w + 19],
[479, 479, -w^3 + w^2 + 8*w + 1],
[487, 487, -4*w^3 + 11*w^2 + 14*w - 15],
[487, 487, 3*w^3 - 6*w^2 - 15*w - 5],
[491, 491, w^2 - 5],
[503, 503, 2*w^3 - 5*w^2 - 11*w + 3],
[523, 523, -3*w^3 + 7*w^2 + 13*w + 1],
[547, 547, 2*w^3 - 3*w^2 - 12*w - 3],
[547, 547, 2*w^3 - 3*w^2 - 12*w - 9],
[557, 557, -3*w^3 + 9*w^2 + 9*w - 11],
[557, 557, -6*w^3 + 15*w^2 + 27*w - 13],
[563, 563, -3*w^3 + 7*w^2 + 14*w - 7],
[571, 571, -5*w^3 + 11*w^2 + 26*w - 3],
[593, 593, -6*w^3 + 15*w^2 + 26*w - 7],
[599, 599, 7*w^3 - 18*w^2 - 29*w + 17],
[599, 599, -2*w^3 + 6*w^2 + 8*w - 3],
[601, 601, -7*w^3 + 20*w^2 + 28*w - 23],
[601, 601, 3*w^3 - 7*w^2 - 12*w + 5],
[607, 607, 5*w^3 - 12*w^2 - 24*w + 9],
[607, 607, w^3 - 5*w^2 + 15],
[613, 613, -w^3 + 5*w^2 - 2*w - 9],
[617, 617, w^3 + w^2 - 13*w - 15],
[617, 617, 2*w^3 - 4*w^2 - 8*w - 1],
[619, 619, 5*w^3 - 10*w^2 - 27*w - 7],
[619, 619, 2*w^3 - 4*w^2 - 10*w + 3],
[643, 643, w^3 - 5*w^2 + 5*w + 5],
[643, 643, 4*w^3 - 9*w^2 - 20*w - 1],
[643, 643, 3*w^3 - 7*w^2 - 14*w - 3],
[643, 643, -3*w^3 + 7*w^2 + 13*w - 3],
[647, 647, -3*w^3 + 8*w^2 + 12*w - 3],
[653, 653, -w^3 + 9*w + 13],
[653, 653, -w^3 + 4*w^2 + 3*w - 9],
[653, 653, -2*w^3 + 7*w^2 + 4*w - 7],
[653, 653, w^2 - 4*w - 9],
[673, 673, -w^3 + 5*w^2 + w - 13],
[683, 683, w^3 - 3*w^2 - 3*w - 3],
[701, 701, 5*w^3 - 13*w^2 - 23*w + 9],
[709, 709, 2*w^3 - w^2 - 19*w - 15],
[719, 719, -5*w^3 + 15*w^2 + 20*w - 19],
[727, 727, -w - 5],
[751, 751, -4*w^3 + 10*w^2 + 15*w - 9],
[751, 751, 6*w^3 - 17*w^2 - 22*w + 21],
[757, 757, 3*w^3 - 6*w^2 - 18*w - 1],
[757, 757, -w^3 + 4*w^2 + 2*w + 1],
[769, 769, w^2 - 2*w + 3],
[773, 773, 5*w^3 - 12*w^2 - 23*w + 9],
[787, 787, 2*w^3 - 4*w^2 - 5*w + 1],
[797, 797, -3*w^3 + 8*w^2 + 10*w - 3],
[809, 809, 2*w^2 - 5*w - 1],
[809, 809, 4*w^3 - 6*w^2 - 28*w - 11],
[827, 827, 4*w^3 - 8*w^2 - 22*w - 1],
[827, 827, 2*w^2 - 6*w - 9],
[829, 829, -w^3 + w^2 + 10*w + 1],
[839, 839, -2*w^3 + 2*w^2 + 16*w + 7],
[877, 877, 6*w^3 - 16*w^2 - 25*w + 13],
[877, 877, w^3 - w^2 - 7*w + 1],
[881, 881, 2*w^3 - 2*w^2 - 15*w - 7],
[881, 881, -4*w^3 + 7*w^2 + 26*w + 9],
[887, 887, 2*w^3 - 4*w^2 - 13*w - 1],
[907, 907, 5*w^3 - 11*w^2 - 26*w - 1],
[907, 907, 5*w^3 - 13*w^2 - 22*w + 9],
[907, 907, w^3 - 5*w^2 + 2*w + 13],
[907, 907, -2*w^3 + 7*w^2 + 5*w - 7],
[911, 911, -3*w^3 + 8*w^2 + 9*w - 5],
[919, 919, -2*w^3 + 7*w^2 + 6*w - 3],
[919, 919, -3*w^3 + 8*w^2 + 15*w - 7],
[947, 947, -2*w^3 + 5*w^2 + 12*w - 9],
[947, 947, -w^2 + w - 3],
[947, 947, 3*w^3 - 6*w^2 - 14*w - 7],
[947, 947, -w^3 + 5*w^2 - w - 17],
[953, 953, 3*w^3 - 9*w^2 - 9*w + 13],
[953, 953, -4*w^3 + 13*w^2 + 11*w - 23],
[967, 967, 8*w^3 - 20*w^2 - 33*w + 19]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^13 + 8*x^12 + 11*x^11 - 66*x^10 - 202*x^9 + 64*x^8 + 804*x^7 + 565*x^6 - 985*x^5 - 1282*x^4 + 21*x^3 + 452*x^2 + 83*x - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 336/97*e^12 + 2218/97*e^11 + 620/97*e^10 - 22907/97*e^9 - 35964/97*e^8 + 70238/97*e^7 + 171078/97*e^6 - 43455/97*e^5 - 264062/97*e^4 - 69501/97*e^3 + 93499/97*e^2 + 22665/97*e + 38/97, -92/97*e^12 - 605/97*e^11 - 179/97*e^10 + 6174/97*e^9 + 9875/97*e^8 - 18375/97*e^7 - 46531/97*e^6 + 9172/97*e^5 + 70947/97*e^4 + 22283/97*e^3 - 24251/97*e^2 - 7214/97*e - 179/97, 312/97*e^12 + 2115/97*e^11 + 839/97*e^10 - 21638/97*e^9 - 36416/97*e^8 + 64833/97*e^7 + 170609/97*e^6 - 34344/97*e^5 - 262993/97*e^4 - 74105/97*e^3 + 94823/97*e^2 + 22702/97*e - 325/97, 1, 1102/97*e^12 + 7405/97*e^11 + 2627/97*e^10 - 76189/97*e^9 - 125069/97*e^8 + 231633/97*e^7 + 590642/97*e^6 - 136780/97*e^5 - 915594/97*e^4 - 237063/97*e^3 + 333770/97*e^2 + 70364/97*e - 1059/97, 1226/97*e^12 + 8212/97*e^11 + 2805/97*e^10 - 84540/97*e^9 - 137704/97*e^8 + 257441/97*e^7 + 651055/97*e^6 - 153929/97*e^5 - 1008595/97*e^4 - 259674/97*e^3 + 366182/97*e^2 + 77173/97*e - 881/97, -466/97*e^12 - 3172/97*e^11 - 1301/97*e^10 + 32521/97*e^9 + 55179/97*e^8 - 97858/97*e^7 - 259132/97*e^6 + 53303/97*e^5 + 401635/97*e^4 + 108615/97*e^3 - 146532/97*e^2 - 31526/97*e - 137/97, -729/97*e^12 - 4911/97*e^11 - 1799/97*e^10 + 50392/97*e^9 + 83222/97*e^8 - 152064/97*e^7 - 391103/97*e^6 + 84723/97*e^5 + 601776/97*e^4 + 166140/97*e^3 - 213875/97*e^2 - 50468/97*e - 247/97, -461/97*e^12 - 3001/97*e^11 - 607/97*e^10 + 31323/97*e^9 + 46624/97*e^8 - 98870/97*e^7 - 224959/97*e^6 + 73727/97*e^5 + 351041/97*e^4 + 72795/97*e^3 - 128976/97*e^2 - 24093/97*e + 945/97, -1148/97*e^12 - 7756/97*e^11 - 2959/97*e^10 + 79567/97*e^9 + 132868/97*e^8 - 239899/97*e^7 - 625402/97*e^6 + 132636/97*e^5 + 968964/97*e^4 + 263967/97*e^3 - 353995/97*e^2 - 79694/97*e + 1309/97, -90/97*e^12 - 556/97*e^11 + 21/97*e^10 + 5947/97*e^9 + 7617/97*e^8 - 19808/97*e^7 - 38837/97*e^6 + 18719/97*e^5 + 62524/97*e^4 + 6791/97*e^3 - 24232/97*e^2 - 1389/97*e + 700/97, -1262/97*e^12 - 8415/97*e^11 - 2719/97*e^10 + 86686/97*e^9 + 139742/97*e^8 - 264142/97*e^7 - 661701/97*e^6 + 157459/97*e^5 + 1022857/97*e^4 + 269258/97*e^3 - 366524/97*e^2 - 81628/97*e + 191/97, -193/97*e^12 - 1285/97*e^11 - 385/97*e^10 + 13321/97*e^9 + 21084/97*e^8 - 41117/97*e^7 - 100331/97*e^6 + 26065/97*e^5 + 155208/97*e^4 + 39978/97*e^3 - 55135/97*e^2 - 12559/97*e + 197/97, 1586/97*e^12 + 10630/97*e^11 + 3594/97*e^10 - 109686/97*e^9 - 177969/97*e^8 + 336091/97*e^7 + 843069/97*e^6 - 209696/97*e^5 - 1310392/97*e^4 - 322437/97*e^3 + 481734/97*e^2 + 95824/97*e - 2517/97, -309/97*e^12 - 1993/97*e^11 - 248/97*e^10 + 21152/97*e^9 + 29634/97*e^8 - 69553/97*e^7 - 145876/97*e^6 + 63263/97*e^5 + 231880/97*e^4 + 28751/97*e^3 - 89508/97*e^2 - 8969/97*e + 528/97, 510/97*e^12 + 3280/97*e^11 + 560/97*e^10 - 34023/97*e^9 - 49953/97*e^8 + 105520/97*e^7 + 240317/97*e^6 - 70249/97*e^5 - 369176/97*e^4 - 95098/97*e^3 + 126677/97*e^2 + 31248/97*e + 560/97, -995/97*e^12 - 6772/97*e^11 - 2791/97*e^10 + 69234/97*e^9 + 117465/97*e^8 - 207176/97*e^7 - 549262/97*e^6 + 109243/97*e^5 + 846222/97*e^4 + 236136/97*e^3 - 304478/97*e^2 - 69854/97*e - 75/97, -916/97*e^12 - 6146/97*e^11 - 2166/97*e^10 + 63129/97*e^9 + 103546/97*e^8 - 191078/97*e^7 - 487946/97*e^6 + 109165/97*e^5 + 752667/97*e^4 + 203001/97*e^3 - 269826/97*e^2 - 60684/97*e + 841/97, 287/97*e^12 + 1842/97*e^11 + 279/97*e^10 - 19140/97*e^9 - 27591/97*e^8 + 59708/97*e^7 + 132537/97*e^6 - 41598/97*e^5 - 201889/97*e^4 - 49623/97*e^3 + 66989/97*e^2 + 16577/97*e + 85/97, 393/97*e^12 + 2596/97*e^11 + 694/97*e^10 - 26903/97*e^9 - 41729/97*e^8 + 83087/97*e^7 + 198685/97*e^6 - 53587/97*e^5 - 305510/97*e^4 - 77336/97*e^3 + 105535/97*e^2 + 22856/97*e + 597/97, 1384/97*e^12 + 9076/97*e^11 + 2212/97*e^10 - 94228/97*e^9 - 144396/97*e^8 + 293323/97*e^7 + 691916/97*e^6 - 202003/97*e^5 - 1075716/97*e^4 - 246307/97*e^3 + 391351/97*e^2 + 76404/97*e - 2250/97, 784/97*e^12 + 5240/97*e^11 + 1673/97*e^10 - 54161/97*e^9 - 86632/97*e^8 + 166734/97*e^7 + 410628/97*e^6 - 107506/97*e^5 - 636741/97*e^4 - 152954/97*e^3 + 232973/97*e^2 + 45804/97*e - 2013/97, 1511/97*e^12 + 10199/97*e^11 + 3854/97*e^10 - 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430865/97*e^9 - 696823/97*e^8 + 1308912/97*e^7 + 3296854/97*e^6 - 762440/97*e^5 - 5093686/97*e^4 - 1373413/97*e^3 + 1824944/97*e^2 + 425035/97*e - 4252/97, 5218/97*e^12 + 35206/97*e^11 + 13132/97*e^10 - 361480/97*e^9 - 599772/97*e^8 + 1092873/97*e^7 + 2823069/97*e^6 - 619658/97*e^5 - 4364772/97*e^4 - 1165832/97*e^3 + 1579940/97*e^2 + 342845/97*e - 4134/97, -4204/97*e^12 - 28211/97*e^11 - 9896/97*e^10 + 290041/97*e^9 + 475309/97*e^8 - 879571/97*e^7 - 2243394/97*e^6 + 508428/97*e^5 + 3468359/97*e^4 + 922784/97*e^3 - 1247297/97*e^2 - 273186/97*e - 1263/97, -172/97*e^12 - 916/97*e^11 + 939/97*e^10 + 11277/97*e^9 + 5232/97*e^8 - 48258/97*e^7 - 43891/97*e^6 + 84162/97*e^5 + 88834/97*e^4 - 53042/97*e^3 - 48485/97*e^2 + 13150/97*e + 1812/97, -2640/97*e^12 - 17344/97*e^11 - 4234/97*e^10 + 180524/97*e^9 + 276200/97*e^8 - 565256/97*e^7 - 1325782/97*e^6 + 401780/97*e^5 + 2065738/97*e^4 + 451338/97*e^3 - 753356/97*e^2 - 139199/97*e + 2071/97, -2002/97*e^12 - 13256/97*e^11 - 3969/97*e^10 + 136338/97*e^9 + 216856/97*e^8 - 413999/97*e^7 - 1025960/97*e^6 + 241617/97*e^5 + 1575754/97*e^4 + 432140/97*e^3 - 552034/97*e^2 - 134015/97*e - 2126/97, -6413/97*e^12 - 42804/97*e^11 - 14098/97*e^10 + 440610/97*e^9 + 713341/97*e^8 - 1340855/97*e^7 - 3376498/97*e^6 + 795417/97*e^5 + 5227824/97*e^4 + 1369971/97*e^3 - 1892041/97*e^2 - 414266/97*e + 6078/97, 3533/97*e^12 + 23654/97*e^11 + 8077/97*e^10 - 243322/97*e^9 - 396556/97*e^8 + 738815/97*e^7 + 1873075/97*e^6 - 429500/97*e^5 - 2893667/97*e^4 - 776299/97*e^3 + 1038726/97*e^2 + 241390/97*e - 1817/97, -27*e^12 - 181*e^11 - 62*e^10 + 1864*e^9 + 3034*e^8 - 5684*e^7 - 14339*e^6 + 3435*e^5 + 22216*e^4 + 5676*e^3 - 8076*e^2 - 1717*e - 20, 3441/97*e^12 + 23631/97*e^11 + 10905/97*e^10 - 240252/97*e^9 - 421213/97*e^8 + 707151/97*e^7 + 1960210/97*e^6 - 319448/97*e^5 - 3019630/97*e^4 - 911059/97*e^3 + 1090814/97*e^2 + 268611/97*e - 2190/97, 5864/97*e^12 + 39102/97*e^11 + 12451/97*e^10 - 403373/97*e^9 - 647390/97*e^8 + 1234130/97*e^7 + 3069180/97*e^6 - 758123/97*e^5 - 4753424/97*e^4 - 1214309/97*e^3 + 1716930/97*e^2 + 369874/97*e - 2778/97, 3983/97*e^12 + 26143/97*e^11 + 6614/97*e^10 - 270729/97*e^9 - 418054/97*e^8 + 836497/97*e^7 + 1998390/97*e^6 - 546278/97*e^5 - 3097647/97*e^4 - 769708/97*e^3 + 1113423/97*e^2 + 251439/97*e - 1728/97, -4172/97*e^12 - 27815/97*e^11 - 8927/97*e^10 + 286700/97*e^9 + 461200/97*e^8 - 875727/97*e^7 - 2185280/97*e^6 + 533431/97*e^5 + 3384225/97*e^4 + 869979/97*e^3 - 1224004/97*e^2 - 268741/97*e + 191/97, 518/97*e^12 + 3476/97*e^11 + 1166/97*e^10 - 35804/97*e^9 - 57627/97*e^8 + 109488/97*e^7 + 271190/97*e^6 - 67757/97*e^5 - 417418/97*e^4 - 109245/97*e^3 + 150518/97*e^2 + 39707/97*e + 196/97, -6491/97*e^12 - 43454/97*e^11 - 14817/97*e^10 + 447038/97*e^9 + 728362/97*e^8 - 1358203/97*e^7 - 3441630/97*e^6 + 795952/97*e^5 + 5322551/97*e^4 + 1407388/97*e^3 - 1917711/97*e^2 - 424452/97*e + 606/97, 3257/97*e^12 + 21354/97*e^11 + 5212/97*e^10 - 221599/97*e^9 - 339771/97*e^8 + 688928/97*e^7 + 1627267/97*e^6 - 470078/97*e^5 - 2526693/97*e^4 - 589073/97*e^3 + 913496/97*e^2 + 185313/97*e - 2742/97, 1058/97*e^12 + 7200/97*e^11 + 3174/97*e^10 - 72844/97*e^9 - 127191/97*e^8 + 210488/97*e^7 + 591512/97*e^6 - 73565/97*e^5 - 902948/97*e^4 - 321196/97*e^3 + 309490/97*e^2 + 104398/97*e + 3271/97, -1275/97*e^12 - 8588/97*e^11 - 3049/97*e^10 + 88695/97*e^9 + 145301/97*e^8 - 272239/97*e^7 - 688432/97*e^6 + 171985/97*e^5 + 1072708/97*e^4 + 253556/97*e^3 - 395214/97*e^2 - 68905/97*e - 818/97, 1829/97*e^12 + 12364/97*e^11 + 4905/97*e^10 - 126160/97*e^9 - 213599/97*e^8 + 373878/97*e^7 + 1000920/97*e^6 - 175275/97*e^5 - 1538047/97*e^4 - 477339/97*e^3 + 541224/97*e^2 + 147696/97*e + 1898/97];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {13, 13, -w^3 + 3*w^2 + 3*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;