/*
  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, 1, -7, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {13, 13, -w^3 + w^2 + 6*w - 3}>;

primesArray := [
[2, 2, w],
[2, 2, -1/2*w^3 + w^2 + 3/2*w],
[11, 11, 1/2*w^3 - w^2 - 5/2*w + 4],
[11, 11, -w^3 + w^2 + 6*w + 1],
[13, 13, -w^3 + w^2 + 6*w - 3],
[19, 19, -1/2*w^3 + w^2 + 5/2*w],
[23, 23, -1/2*w^3 + 2*w^2 - 1/2*w - 2],
[31, 31, -w^3 + w^2 + 6*w - 1],
[31, 31, 1/2*w^3 - 7/2*w],
[43, 43, 1/2*w^3 - w^2 - 9/2*w + 2],
[67, 67, w^2 - w - 5],
[67, 67, -1/2*w^3 + 11/2*w + 2],
[73, 73, w^2 + w + 1],
[79, 79, 1/2*w^3 + w^2 - 5/2*w - 2],
[81, 3, -3],
[83, 83, 2*w^3 - 2*w^2 - 12*w + 5],
[83, 83, -1/2*w^3 - w^2 + 1/2*w + 2],
[89, 89, -3/2*w^3 + 2*w^2 + 17/2*w - 2],
[89, 89, 3/2*w^3 - 2*w^2 - 21/2*w + 2],
[97, 97, 1/2*w^3 - w^2 - 1/2*w - 2],
[97, 97, -w^3 + w^2 + 8*w + 1],
[103, 103, 2*w^3 - 2*w^2 - 14*w + 1],
[103, 103, w^2 - w - 3],
[107, 107, -3/2*w^3 + 2*w^2 + 21/2*w - 4],
[109, 109, 5/2*w^3 - 3*w^2 - 33/2*w + 4],
[109, 109, -1/2*w^3 + w^2 + 9/2*w - 4],
[113, 113, -w^3 + w^2 + 8*w - 3],
[113, 113, 1/2*w^3 - w^2 - 1/2*w + 2],
[121, 11, 5/2*w^3 - 2*w^2 - 31/2*w + 2],
[127, 127, -3/2*w^3 + w^2 + 23/2*w],
[127, 127, -1/2*w^3 + 11/2*w],
[131, 131, w^2 - 3*w - 3],
[137, 137, 1/2*w^3 - w^2 - 5/2*w + 6],
[139, 139, -3/2*w^3 + 2*w^2 + 13/2*w + 2],
[151, 151, -1/2*w^3 + 2*w^2 + 3/2*w - 10],
[157, 157, 7/2*w^3 - 5*w^2 - 47/2*w + 10],
[157, 157, 7/2*w^3 - 4*w^2 - 45/2*w + 4],
[167, 167, 1/2*w^3 + w^2 - 5/2*w - 4],
[167, 167, 2*w + 5],
[173, 173, -w^3 + w^2 + 4*w - 7],
[181, 181, w^3 - 2*w^2 - 9*w + 1],
[193, 193, -1/2*w^3 + 2*w^2 + 3/2*w - 8],
[197, 197, -1/2*w^3 - w^2 + 13/2*w + 2],
[197, 197, -w^3 + w^2 + 4*w - 1],
[197, 197, w^3 - 2*w^2 - 7*w + 7],
[197, 197, -2*w^3 + 3*w^2 + 13*w - 11],
[199, 199, -2*w^3 + 3*w^2 + 11*w - 5],
[239, 239, -w^3 + 2*w^2 + 5*w - 1],
[239, 239, 7/2*w^3 - 4*w^2 - 45/2*w + 6],
[241, 241, 1/2*w^3 - 2*w^2 - 7/2*w + 8],
[257, 257, 3/2*w^3 - 2*w^2 - 17/2*w - 4],
[257, 257, 1/2*w^3 - 7/2*w - 6],
[257, 257, 7/2*w^3 - 5*w^2 - 43/2*w + 10],
[257, 257, 3*w^3 - 4*w^2 - 19*w + 7],
[269, 269, 1/2*w^3 + w^2 + 3/2*w + 2],
[269, 269, -2*w^3 + 3*w^2 + 11*w - 11],
[277, 277, 5/2*w^3 - 4*w^2 - 35/2*w + 8],
[281, 281, -3/2*w^3 + w^2 + 23/2*w - 2],
[283, 283, -w^3 - w^2 + 8*w + 11],
[283, 283, -3*w^2 + 3*w + 19],
[311, 311, 3/2*w^3 - 2*w^2 - 17/2*w - 2],
[331, 331, -4*w^3 + 7*w^2 + 25*w - 21],
[337, 337, -5/2*w^3 + 2*w^2 + 35/2*w + 4],
[337, 337, -2*w^3 + 3*w^2 + 11*w - 9],
[347, 347, 2*w^3 - 4*w^2 - 12*w + 13],
[347, 347, 5/2*w^3 - 5*w^2 - 29/2*w + 20],
[347, 347, -1/2*w^3 - 1/2*w - 6],
[347, 347, -1/2*w^3 + 3*w^2 + 1/2*w - 16],
[349, 349, 1/2*w^3 + w^2 - 9/2*w - 4],
[353, 353, -1/2*w^3 + 2*w^2 + 7/2*w],
[359, 359, 3/2*w^3 - w^2 - 19/2*w + 2],
[373, 373, -2*w^3 + 2*w^2 + 12*w + 1],
[373, 373, -1/2*w^3 + w^2 + 9/2*w - 6],
[379, 379, w^2 - w - 9],
[379, 379, -2*w^3 + 3*w^2 + 13*w - 5],
[383, 383, w^2 + w - 5],
[389, 389, 9/2*w^3 - 6*w^2 - 59/2*w + 14],
[397, 397, 2*w^2 - 2*w - 15],
[401, 401, -1/2*w^3 + 2*w^2 + 3/2*w - 4],
[419, 419, -3/2*w^3 + w^2 + 15/2*w - 2],
[419, 419, w^3 + w^2 - 8*w - 15],
[431, 431, -5/2*w^3 + 4*w^2 + 27/2*w - 16],
[431, 431, 2*w^3 - 4*w^2 - 10*w + 11],
[443, 443, 1/2*w^3 - 3*w^2 - 5/2*w + 14],
[449, 449, 1/2*w^3 + w^2 - 9/2*w - 6],
[457, 457, -w^3 + 2*w^2 + 7*w - 9],
[461, 461, 4*w^3 - 5*w^2 - 27*w + 7],
[487, 487, 2*w^2 - 9],
[499, 499, 3/2*w^3 - w^2 - 15/2*w - 2],
[499, 499, 3/2*w^3 - 4*w^2 - 17/2*w + 16],
[503, 503, -5/2*w^3 + 3*w^2 + 37/2*w],
[509, 509, -3*w^3 + 3*w^2 + 18*w + 7],
[521, 521, -1/2*w^3 - 2*w^2 + 11/2*w + 18],
[521, 521, -5/2*w^3 + 4*w^2 + 31/2*w - 8],
[523, 523, -1/2*w^3 + 2*w^2 + 11/2*w],
[523, 523, -1/2*w^3 - w^2 + 9/2*w],
[547, 547, -7/2*w^3 + 6*w^2 + 45/2*w - 18],
[547, 547, -w^3 - w^2 + 12*w + 5],
[547, 547, 3/2*w^3 - w^2 - 19/2*w + 4],
[547, 547, w^3 - w^2 - 6*w - 5],
[557, 557, 1/2*w^3 + w^2 - 9/2*w - 12],
[557, 557, 3/2*w^3 - 2*w^2 - 21/2*w],
[557, 557, 2*w^3 - 2*w^2 - 12*w + 1],
[557, 557, w^3 - 3*w^2 - 6*w + 11],
[569, 569, w^3 - 2*w^2 - 7*w + 1],
[577, 577, w^3 + w^2 - 6*w - 5],
[599, 599, -3/2*w^3 + 3*w^2 + 15/2*w - 10],
[607, 607, -5/2*w^3 + 3*w^2 + 25/2*w + 4],
[613, 613, w^3 - 9*w + 1],
[619, 619, 3*w^2 - w - 17],
[619, 619, w^3 - 3*w^2 - 6*w + 13],
[625, 5, -5],
[631, 631, 11/2*w^3 - 7*w^2 - 71/2*w + 16],
[631, 631, -3/2*w^3 + 2*w^2 + 13/2*w],
[643, 643, w^2 - 3*w - 5],
[653, 653, 2*w^2 - 2*w - 11],
[659, 659, 5*w^3 - 3*w^2 - 36*w - 9],
[659, 659, -w^3 + 7*w - 1],
[673, 673, 5/2*w^3 - 2*w^2 - 31/2*w - 4],
[677, 677, 13/2*w^3 - 6*w^2 - 87/2*w + 2],
[691, 691, -3/2*w^3 + 2*w^2 + 13/2*w - 2],
[701, 701, -w^3 + 3*w^2 + 4*w - 13],
[701, 701, w^3 - 2*w^2 - 9*w + 3],
[709, 709, -w^2 - 5*w - 1],
[739, 739, 3/2*w^3 - w^2 - 15/2*w],
[743, 743, -2*w^3 + w^2 + 15*w + 1],
[751, 751, -2*w^3 + 4*w^2 + 12*w - 11],
[757, 757, -6*w^3 + 9*w^2 + 39*w - 23],
[757, 757, -w^3 - w^2 + 8*w + 3],
[757, 757, 13/2*w^3 - 8*w^2 - 83/2*w + 16],
[757, 757, 3*w^3 - 4*w^2 - 17*w + 3],
[761, 761, 7/2*w^3 - 5*w^2 - 47/2*w + 12],
[769, 769, -3*w^2 + 7*w + 3],
[773, 773, -3/2*w^3 - w^2 + 19/2*w + 4],
[787, 787, -4*w^3 + 5*w^2 + 23*w - 17],
[787, 787, 1/2*w^3 - w^2 - 9/2*w - 4],
[809, 809, -2*w^3 + 2*w^2 + 16*w + 3],
[809, 809, -4*w^3 + 5*w^2 + 25*w - 9],
[811, 811, 3*w^3 - 3*w^2 - 20*w + 5],
[811, 811, w^3 - 3*w^2 - 6*w + 17],
[823, 823, 5/2*w^3 - w^2 - 33/2*w - 10],
[829, 829, 5/2*w^3 - 3*w^2 - 33/2*w + 2],
[829, 829, -3*w^3 + 2*w^2 + 23*w - 1],
[839, 839, -11/2*w^3 + 7*w^2 + 67/2*w - 18],
[853, 853, -w^3 + 2*w^2 + 3*w - 5],
[853, 853, -5*w^3 + 7*w^2 + 32*w - 15],
[857, 857, 3/2*w^3 - 5*w^2 - 15/2*w + 26],
[859, 859, -w^3 + 3*w^2 + 4*w - 17],
[863, 863, 4*w^3 - 4*w^2 - 26*w + 3],
[877, 877, w^3 - 5*w - 5],
[887, 887, 3/2*w^3 - w^2 - 11/2*w + 8],
[907, 907, -9/2*w^3 + 7*w^2 + 53/2*w - 24],
[919, 919, 7/2*w^3 - 6*w^2 - 41/2*w + 22],
[929, 929, -2*w^3 + 3*w^2 + 11*w - 1],
[937, 937, 2*w^3 - 4*w^2 - 12*w + 7],
[947, 947, 9/2*w^3 - 5*w^2 - 61/2*w + 10],
[953, 953, -1/2*w^3 + 2*w^2 + 3/2*w + 2],
[961, 31, -1/2*w^3 + 2*w^2 - 1/2*w - 6],
[977, 977, 5/2*w^3 - 2*w^2 - 31/2*w],
[983, 983, -5/2*w^3 + 6*w^2 + 27/2*w - 24],
[983, 983, -3/2*w^3 + 2*w^2 + 21/2*w - 10],
[983, 983, -w^3 + 4*w^2 + 7*w - 15],
[983, 983, -1/2*w^3 + 3*w^2 - 3/2*w - 4],
[997, 997, 1/2*w^3 - w^2 - 5/2*w - 4],
[997, 997, 1/2*w^3 + w^2 - 17/2*w - 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^18 - 4*x^17 - 19*x^16 + 90*x^15 + 124*x^14 - 814*x^13 - 222*x^12 + 3802*x^11 - 1023*x^10 - 9774*x^9 + 5785*x^8 + 13540*x^7 - 11139*x^6 - 8912*x^5 + 9483*x^4 + 1598*x^3 - 2988*x^2 + 436*x + 40;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-17892337/19430084*e^17 + 49752117/19430084*e^16 + 100283421/4857521*e^15 - 560781335/9715042*e^14 - 899579169/4857521*e^13 + 5090393611/9715042*e^12 + 4126726013/4857521*e^11 - 23950068733/9715042*e^10 - 40771065907/19430084*e^9 + 125059468565/19430084*e^8 + 12790775120/4857521*e^7 - 2362512209/255659*e^6 - 23737116371/19430084*e^5 + 130467874855/19430084*e^4 - 1644009338/4857521*e^3 - 18657141899/9715042*e^2 + 1617822337/4857521*e + 150118723/4857521, e, -9041561/9715042*e^17 + 24394923/9715042*e^16 + 101429639/4857521*e^15 - 273823492/4857521*e^14 - 910155190/4857521*e^13 + 2473967285/4857521*e^12 + 4173813415/4857521*e^11 - 11581612074/4857521*e^10 - 20601373283/9715042*e^9 + 60166868135/9715042*e^8 + 12941673877/4857521*e^7 - 2261783112/255659*e^6 - 12294724513/9715042*e^5 + 62113632311/9715042*e^4 - 1396029250/4857521*e^3 - 8800353852/4857521*e^2 + 1508705483/4857521*e + 117212480/4857521, 19791728/4857521*e^17 - 53120522/4857521*e^16 - 445985714/4857521*e^15 + 1194768496/4857521*e^14 + 4026273003/4857521*e^13 - 10819937141/4857521*e^12 - 18621738169/4857521*e^11 + 50799358593/4857521*e^10 + 46515077964/4857521*e^9 - 132432128813/4857521*e^8 - 59469809434/4857521*e^7 + 10004516786/255659*e^6 + 29138356766/4857521*e^5 - 138358767373/4857521*e^4 + 6114970300/4857521*e^3 + 39734529269/4857521*e^2 - 7071866945/4857521*e - 605089910/4857521, -1, -18899797/4857521*e^17 + 53723936/4857521*e^16 + 422993153/4857521*e^15 - 1213756943/4857521*e^14 - 3787801760/4857521*e^13 + 11046476590/4857521*e^12 + 17344305869/4857521*e^11 - 52133850000/4857521*e^10 - 42724887602/4857521*e^9 + 136618570450/4857521*e^8 + 53220580585/4857521*e^7 - 10371934163/255659*e^6 - 23662097739/4857521*e^5 + 144087574192/4857521*e^4 - 8378261980/4857521*e^3 - 41519335691/4857521*e^2 + 7397362898/4857521*e + 648867954/4857521, -50717567/9715042*e^17 + 136530567/9715042*e^16 + 570927638/4857521*e^15 - 1536017185/4857521*e^14 - 5147637353/4857521*e^13 + 13919576082/4857521*e^12 + 23762350328/4857521*e^11 - 65420120412/4857521*e^10 - 118347187809/9715042*e^9 + 341616538775/9715042*e^8 + 75236831548/4857521*e^7 - 12930100655/255659*e^6 - 72619738611/9715042*e^5 + 358507641269/9715042*e^4 - 8191525246/4857521*e^3 - 51618860860/4857521*e^2 + 9012068158/4857521*e + 809699324/4857521, -12983168/4857521*e^17 + 35213504/4857521*e^16 + 292842727/4857521*e^15 - 793168005/4857521*e^14 - 2649113389/4857521*e^13 + 7194494000/4857521*e^12 + 12299080998/4857521*e^11 - 33832584848/4857521*e^10 - 30929927620/4857521*e^9 + 88327485974/4857521*e^8 + 40034754341/4857521*e^7 - 6679320639/255659*e^6 - 20259898376/4857521*e^5 + 92382009564/4857521*e^4 - 3559251140/4857521*e^3 - 26456668523/4857521*e^2 + 4625656656/4857521*e + 381107924/4857521, -11783539/4857521*e^17 + 30460793/4857521*e^16 + 266994464/4857521*e^15 - 683745399/4857521*e^14 - 2425989843/4857521*e^13 + 6178166158/4857521*e^12 + 11306526328/4857521*e^11 - 28938122802/4857521*e^10 - 28515340405/4857521*e^9 + 75271985655/4857521*e^8 + 36987148408/4857521*e^7 - 5675794711/255659*e^6 - 18834504313/4857521*e^5 + 78386808089/4857521*e^4 - 3040158512/4857521*e^3 - 22472015903/4857521*e^2 + 4098695180/4857521*e + 332966518/4857521, -27238387/9715042*e^17 + 70249387/9715042*e^16 + 307825204/4857521*e^15 - 786799165/4857521*e^14 - 2788118166/4857521*e^13 + 7093260210/4857521*e^12 + 12942332118/4857521*e^11 - 33144053076/4857521*e^10 - 64977700215/9715042*e^9 + 171962856001/9715042*e^8 + 41985760714/4857521*e^7 - 6461290837/255659*e^6 - 43139144233/9715042*e^5 + 177526007327/9715042*e^4 - 2898445399/4857521*e^3 - 25205636384/4857521*e^2 + 4368161548/4857521*e + 348260252/4857521, -11662167/9715042*e^17 + 33832343/9715042*e^16 + 130194272/4857521*e^15 - 382714748/4857521*e^14 - 1163147006/4857521*e^13 + 3489530803/4857521*e^12 + 5314906737/4857521*e^11 - 16505296588/4857521*e^10 - 26141071905/9715042*e^9 + 86695544819/9715042*e^8 + 16267902747/4857521*e^7 - 3294467854/255659*e^6 - 14510215273/9715042*e^5 + 91304343029/9715042*e^4 - 2524738418/4857521*e^3 - 13033569974/4857521*e^2 + 2250878899/4857521*e + 219292600/4857521, -2873343/255659*e^17 + 7662380/255659*e^16 + 64730712/255659*e^15 - 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28649331376/4857521*e - 2215538944/4857521, 152878993/4857521*e^17 - 429022145/4857521*e^16 - 3425042783/4857521*e^15 + 9676360315/4857521*e^14 + 30706728644/4857521*e^13 - 87888696030/4857521*e^12 - 140796362362/4857521*e^11 + 413812189528/4857521*e^10 + 347388023975/4857521*e^9 - 1081401238342/4857521*e^8 - 433767648739/4857521*e^7 + 81830584989/255659*e^6 + 194481461981/4857521*e^5 - 1132556643868/4857521*e^4 + 65970924161/4857521*e^3 + 325217395215/4857521*e^2 - 58955031198/4857521*e - 5095566656/4857521, 12779025/4857521*e^17 - 45667481/4857521*e^16 - 279733299/4857521*e^15 + 1045419500/4857521*e^14 + 2452358860/4857521*e^13 - 9633272895/4857521*e^12 - 11041009194/4857521*e^11 + 45903565334/4857521*e^10 + 26940378611/4857521*e^9 - 120814107266/4857521*e^8 - 33398114167/4857521*e^7 + 9143931367/255659*e^6 + 14089830659/4857521*e^5 - 125674146846/4857521*e^4 + 7006820603/4857521*e^3 + 35635163159/4857521*e^2 - 5494858348/4857521*e - 625439750/4857521, 96810096/4857521*e^17 - 261392958/4857521*e^16 - 2176822122/4857521*e^15 + 5879773994/4857521*e^14 + 19592082050/4857521*e^13 - 53263850114/4857521*e^12 - 90203919981/4857521*e^11 + 250222551230/4857521*e^10 + 223660154041/4857521*e^9 - 652965868092/4857521*e^8 - 281829691741/4857521*e^7 + 49394957026/255659*e^6 + 131386499612/4857521*e^5 - 684046958840/4857521*e^4 + 36497126121/4857521*e^3 + 196564337430/4857521*e^2 - 35926779834/4857521*e - 3050962188/4857521];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {13, 13, -w^3 + w^2 + 6*w - 3}>] := 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;