/*
  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![-1, -8, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {7, 7, w - 3}>;

primesArray := [
[3, 3, -w - 2],
[5, 5, w + 1],
[7, 7, w - 3],
[8, 2, 2],
[9, 3, w^2 - 3*w - 2],
[13, 13, w + 3],
[13, 13, -w + 2],
[19, 19, -w^2 + 2*w + 4],
[23, 23, w^2 - 4*w + 1],
[25, 5, w^2 - 2*w - 1],
[31, 31, w^2 - 2*w - 9],
[37, 37, -w^2 + 3*w + 3],
[41, 41, w^2 - w - 1],
[41, 41, w^2 - w - 5],
[41, 41, w^2 - w - 10],
[43, 43, -2*w - 3],
[47, 47, -w^2 - w + 4],
[49, 7, -w^2 + 5*w - 5],
[59, 59, w^2 - w - 4],
[59, 59, w^2 - 3],
[59, 59, w - 5],
[67, 67, 2*w^2 - 4*w - 9],
[73, 73, 2*w^2 - 3*w - 12],
[79, 79, -3*w^2 + 9*w + 5],
[97, 97, 2*w^2 - w - 14],
[97, 97, 2*w^2 - 2*w - 13],
[97, 97, -2*w^2 + 5*w + 5],
[101, 101, w^2 - w - 11],
[131, 131, w - 6],
[139, 139, -2*w^2 + 5*w + 4],
[149, 149, -3*w + 11],
[149, 149, -3*w - 4],
[157, 157, w^2 - 3*w - 6],
[173, 173, w^2 - 5*w + 3],
[173, 173, -2*w^2 + 5*w + 8],
[173, 173, 3*w^2 - 5*w - 17],
[179, 179, 2*w^2 - w - 11],
[179, 179, -3*w^2 + 7*w + 12],
[179, 179, w^2 - 2*w - 12],
[181, 181, w^2 + w - 16],
[199, 199, 2*w^2 - 9],
[211, 211, -w^2 + w - 2],
[229, 229, -w^2 + 3*w - 3],
[233, 233, w^2 + w - 7],
[257, 257, 3*w^2 - 3*w - 20],
[271, 271, w^2 - 4*w - 13],
[271, 271, 3*w^2 - 4*w - 19],
[271, 271, 2*w - 9],
[277, 277, 3*w^2 - 5*w - 20],
[281, 281, -w^2 - w + 13],
[283, 283, -3*w + 8],
[293, 293, -4*w - 7],
[307, 307, w^2 + 2*w - 5],
[311, 311, 2*w^2 - 4*w - 5],
[313, 313, -w^2 - w - 3],
[317, 317, -3*w^2 + 8*w + 5],
[331, 331, -w^2 + 2*w - 3],
[337, 337, -w - 7],
[337, 337, w^2 + w - 9],
[337, 337, 3*w - 5],
[359, 359, 2*w^2 - 4*w - 15],
[361, 19, 2*w^2 + w - 5],
[367, 367, -w^2 - 3],
[367, 367, 3*w^2 - 11*w + 4],
[367, 367, -2*w^2 + 9*w - 6],
[373, 373, 2*w^2 - w - 9],
[379, 379, 3*w^2 - 5*w - 21],
[383, 383, w - 8],
[389, 389, -4*w^2 + 9*w + 15],
[397, 397, 2*w^2 + w - 4],
[397, 397, -4*w^2 + 13*w + 5],
[397, 397, -w^2 - 2*w - 4],
[401, 401, w^2 - 5*w - 4],
[421, 421, 3*w^2 - 4*w - 18],
[431, 431, w^2 - 4*w - 16],
[433, 433, w^2 - 4*w - 7],
[439, 439, w^2 + 2*w - 17],
[457, 457, 2*w^2 - 3],
[457, 457, 4*w - 3],
[457, 457, 2*w^2 - 3*w - 6],
[463, 463, 3*w^2 - 12*w + 7],
[479, 479, w^2 - 6*w + 6],
[479, 479, -w^2 + 15],
[479, 479, -6*w^2 + 19*w + 1],
[487, 487, 2*w^2 - 7*w - 5],
[487, 487, w^2 + 5*w + 8],
[487, 487, 3*w^2 - 6*w - 17],
[491, 491, -4*w + 15],
[503, 503, 8*w^2 - 25*w - 5],
[509, 509, 3*w^2 - 12*w + 4],
[509, 509, -3*w^2 - 5*w + 6],
[509, 509, w^2 - w - 14],
[521, 521, -4*w^2 + 10*w + 11],
[523, 523, 2*w^2 - 7*w + 4],
[523, 523, 2*w^2 - 2*w - 7],
[523, 523, -7*w^2 + 22*w + 8],
[529, 23, 4*w^2 - 14*w + 3],
[541, 541, w^2 + 4*w - 3],
[541, 541, -5*w - 9],
[541, 541, -5*w^2 + 11*w + 20],
[557, 557, -w^2 + 5*w - 8],
[563, 563, 2*w^2 - 15],
[563, 563, -w^2 - 2*w + 20],
[563, 563, w^2 - 3*w - 15],
[569, 569, 2*w^2 - w - 5],
[577, 577, 2*w^2 - 3*w - 21],
[587, 587, -4*w^2 + 11*w + 6],
[599, 599, 2*w^2 + 2*w - 9],
[607, 607, -3*w - 10],
[613, 613, 4*w^2 - 7*w - 25],
[631, 631, -3*w^2 + 7*w + 7],
[641, 641, 3*w^2 - 2*w - 17],
[641, 641, -3*w^2 + 14*w - 12],
[641, 641, -5*w^2 + 15*w + 9],
[643, 643, -5*w - 8],
[647, 647, 3*w^2 - w - 21],
[653, 653, w^2 + 7*w + 9],
[673, 673, -3*w^2 - 2*w + 9],
[683, 683, 5*w^2 - 17*w + 2],
[683, 683, 3*w^2 - 6*w - 11],
[683, 683, 2*w^2 - 7*w - 6],
[691, 691, w^2 + 3*w + 6],
[701, 701, 2*w^2 - 5*w - 19],
[709, 709, -3*w^2 + 9*w + 8],
[727, 727, 3*w - 13],
[739, 739, -w - 9],
[743, 743, 5*w - 3],
[743, 743, 2*w^2 - 5*w - 13],
[743, 743, 3*w^2 - 3*w - 17],
[757, 757, 3*w^2 - 4*w - 16],
[761, 761, w^2 + 4*w - 4],
[769, 769, 3*w^2 - w - 15],
[773, 773, -w^2 + 2*w - 5],
[787, 787, -2*w^2 + 10*w - 9],
[797, 797, w^2 - 4*w - 17],
[811, 811, 6*w^2 - 20*w + 1],
[811, 811, -w^2 + 4*w - 7],
[811, 811, 4*w - 9],
[821, 821, 2*w^2 - 5*w - 14],
[823, 823, -5*w^2 + 14*w + 8],
[827, 827, 3*w^2 - w - 22],
[829, 829, -3*w^2 + 6*w + 10],
[859, 859, -9*w^2 + 29*w + 2],
[859, 859, 4*w^2 - 13*w + 2],
[859, 859, -w^2 + w - 5],
[863, 863, 2*w^2 + w - 13],
[877, 877, w^2 + 2*w - 11],
[881, 881, 2*w^2 - 8*w + 7],
[883, 883, 3*w^2 + 3*w - 4],
[907, 907, w^2 + 2*w - 13],
[929, 929, -3*w - 11],
[937, 937, w^2 - 5*w - 12],
[941, 941, 5*w - 4],
[947, 947, 4*w^2 - 7*w - 20],
[953, 953, 2*w^2 + 6*w - 3],
[961, 31, 5*w^2 - 8*w - 29],
[967, 967, -5*w^2 + 14*w + 7],
[971, 971, w^2 - 5*w - 10],
[977, 977, 2*w^2 - 4*w - 23],
[977, 977, w^2 - 8*w + 17],
[977, 977, 3*w^2 - 19],
[991, 991, 3*w^2 - 4*w - 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^12 - 31*x^10 + 358*x^8 - 1867*x^6 + 4116*x^4 - 2528*x^2 + 128;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -2/535*e^11 + 509/4280*e^9 - 1235/856*e^7 + 17361/2140*e^5 - 83167/4280*e^3 + 5858/535*e, 1, 1/4280*e^10 - 199/4280*e^8 + 383/428*e^6 - 24707/4280*e^4 + 13663/1070*e^2 - 2573/535, 99/17120*e^11 - 2581/17120*e^9 + 2393/1712*e^7 - 96273/17120*e^5 + 45097/4280*e^3 - 6303/535*e, -11/8560*e^10 + 49/8560*e^8 + 281/856*e^6 - 27823/8560*e^4 + 4023/535*e^2 - 1096/535, -11/8560*e^10 + 49/8560*e^8 + 281/856*e^6 - 27823/8560*e^4 + 4023/535*e^2 - 1096/535, 99/17120*e^11 - 2581/17120*e^9 + 2393/1712*e^7 - 96273/17120*e^5 + 40817/4280*e^3 - 2023/535*e, 111/4280*e^10 - 2829/4280*e^8 + 2495/428*e^6 - 88877/4280*e^4 + 13844/535*e^2 - 2588/535, 3/160*e^11 - 97/160*e^9 + 115/16*e^7 - 5961/160*e^5 + 1517/20*e^3 - 166/5*e, -29/17120*e^11 + 1491/17120*e^9 - 2547/1712*e^7 + 181503/17120*e^5 - 125517/4280*e^3 + 9693/535*e, -1/856*e^11 + 23/214*e^9 - 1797/856*e^7 + 13365/856*e^5 - 38923/856*e^3 + 3643/107*e, 19/1070*e^11 - 571/1070*e^9 + 643/107*e^7 - 32873/1070*e^5 + 35019/535*e^3 - 16858/535*e, 13/1712*e^11 - 233/1712*e^9 + 49/107*e^7 + 4517/1712*e^5 - 10381/856*e^3 + 21/107*e, 211/8560*e^10 - 5609/8560*e^8 + 5271/856*e^6 - 205577/8560*e^4 + 17867/535*e^2 - 1544/535, -209/17120*e^11 + 7351/17120*e^9 - 9427/1712*e^7 + 528523/17120*e^5 - 298267/4280*e^3 + 22758/535*e, -11/428*e^10 + 263/428*e^8 - 528/107*e^6 + 6417/428*e^4 - 3233/214*e^2 + 1180/107, -161/8560*e^10 + 4219/8560*e^8 - 4097/856*e^6 + 177187/8560*e^4 - 36559/1070*e^2 + 3024/535, -583/17120*e^11 + 17577/17120*e^9 - 19561/1712*e^7 + 969261/17120*e^5 - 493719/4280*e^3 + 29806/535*e, 11/8560*e^11 - 49/8560*e^9 - 281/856*e^7 + 27823/8560*e^5 - 3488/535*e^3 - 3184/535*e, -123/4280*e^10 + 3077/4280*e^8 - 2597/428*e^6 + 81481/4280*e^4 - 14559/1070*e^2 - 776/535, -697/17120*e^11 + 18863/17120*e^9 - 18283/1712*e^7 + 755619/17120*e^5 - 297571/4280*e^3 + 10849/535*e, 629/17120*e^11 - 16031/17120*e^9 + 14281/1712*e^7 - 517903/17120*e^5 + 19969/535*e^3 - 1883/535*e, 77/17120*e^11 - 2483/17120*e^9 + 2955/1712*e^7 - 151919/17120*e^5 + 73001/4280*e^3 - 2584/535*e, -221/4280*e^10 + 5459/4280*e^8 - 4607/428*e^6 + 153047/4280*e^4 - 44923/1070*e^2 + 11698/535, 25/856*e^11 - 401/428*e^9 + 9615/856*e^7 - 52073/856*e^5 + 116861/856*e^3 - 7508/107*e, -467/17120*e^11 + 13753/17120*e^9 - 15151/1712*e^7 + 761129/17120*e^5 - 51587/535*e^3 + 34369/535*e, 11/8560*e^10 - 49/8560*e^8 - 281/856*e^6 + 36383/8560*e^4 - 8838/535*e^2 + 3236/535, 87/4280*e^10 - 2333/4280*e^8 + 2291/428*e^6 - 99389/4280*e^4 + 21623/535*e^2 - 2896/535, -649/17120*e^11 + 17871/17120*e^9 - 17875/1712*e^7 + 785203/17120*e^5 - 354367/4280*e^3 + 21168/535*e, 147/2140*e^10 - 3573/2140*e^8 + 3015/214*e^6 - 105209/2140*e^4 + 35381/535*e^2 - 5322/535, -29/8560*e^11 + 1491/8560*e^9 - 2547/856*e^7 + 181503/8560*e^5 - 127657/2140*e^3 + 22596/535*e, -199/17120*e^11 + 7501/17120*e^9 - 10519/1712*e^7 + 670933/17120*e^5 - 231421/2140*e^3 + 47088/535*e, -177/8560*e^10 + 3123/8560*e^8 - 1237/856*e^6 - 26701/8560*e^4 + 9191/535*e^2 + 2208/535, 111/8560*e^11 - 3899/8560*e^9 + 1239/214*e^7 - 279337/8560*e^5 + 339461/4280*e^3 - 34999/535*e, -87/2140*e^11 + 2333/2140*e^9 - 2291/214*e^7 + 101529/2140*e^5 - 49666/535*e^3 + 29332/535*e, -191/8560*e^11 + 3769/8560*e^9 - 1677/856*e^7 - 53163/8560*e^5 + 95327/2140*e^3 - 15091/535*e, -93/2140*e^11 + 748/535*e^9 - 7145/428*e^7 + 190921/2140*e^5 - 424391/2140*e^3 + 63418/535*e, 239/4280*e^10 - 4761/4280*e^8 + 2513/428*e^6 + 5707/4280*e^4 - 51543/1070*e^2 + 6188/535, -721/8560*e^10 + 17219/8560*e^8 - 13565/856*e^6 + 377027/8560*e^4 - 12322/535*e^2 + 4424/535, 799/17120*e^11 - 22041/17120*e^9 + 22253/1712*e^7 - 1007333/17120*e^5 + 488517/4280*e^3 - 33393/535*e, 29/2140*e^10 - 421/2140*e^8 - 64/107*e^6 + 38917/2140*e^4 - 70501/1070*e^2 + 14476/535, -597/8560*e^10 + 13943/8560*e^8 - 11013/856*e^6 + 343599/8560*e^4 - 47983/1070*e^2 + 4328/535, 451/8560*e^10 - 10569/8560*e^8 + 8167/856*e^6 - 220297/8560*e^4 + 5722/535*e^2 + 4276/535, 103/8560*e^11 - 3377/8560*e^9 + 3925/856*e^7 - 186541/8560*e^5 + 74069/2140*e^3 - 1702/535*e, -219/17120*e^11 + 5061/17120*e^9 - 4269/1712*e^7 + 193513/17120*e^5 - 149207/4280*e^3 + 24643/535*e, -25/1712*e^11 + 267/1712*e^9 + 1559/856*e^7 - 48293/1712*e^5 + 10505/107*e^3 - 7481/107*e, 121/4280*e^10 - 2679/4280*e^8 + 1831/428*e^6 - 32067/4280*e^4 - 12767/1070*e^2 + 13412/535, 33/2140*e^11 - 341/1070*e^9 + 775/428*e^7 - 1061/2140*e^5 - 33139/2140*e^3 + 13687/535*e, -673/8560*e^10 + 16227/8560*e^8 - 13157/856*e^6 + 406611/8560*e^4 - 24916/535*e^2 + 11152/535, 43/1070*e^10 - 1067/1070*e^8 + 954/107*e^6 - 38411/1070*e^4 + 33863/535*e^2 - 1716/535, 21/856*e^11 - 81/107*e^9 + 7349/856*e^7 - 36705/856*e^5 + 70523/856*e^3 - 1175/107*e, 403/4280*e^10 - 9577/4280*e^8 + 7331/428*e^6 - 181401/4280*e^4 + 787/535*e^2 + 5796/535, -1027/17120*e^11 + 28893/17120*e^9 - 29541/1712*e^7 + 1307649/17120*e^5 - 548831/4280*e^3 + 14739/535*e, 3/80*e^10 - 57/80*e^8 + 23/8*e^6 + 839/80*e^4 - 294/5*e^2 + 108/5, 699/17120*e^11 - 17121/17120*e^9 + 14127/1712*e^7 - 432673/17120*e^5 + 19833/1070*e^3 + 437/535*e, -67/1070*e^10 + 1563/1070*e^8 - 1158/107*e^6 + 24689/1070*e^4 + 10628/535*e^2 - 18776/535, -739/8560*e^10 + 16521/8560*e^8 - 11471/856*e^6 + 213993/8560*e^4 + 13667/535*e^2 - 10404/535, 17/428*e^10 - 387/428*e^8 + 1525/214*e^6 - 10423/428*e^4 + 4040/107*e^2 - 598/107, 407/8560*e^10 - 10373/8560*e^8 + 9291/856*e^6 - 340149/8560*e^4 + 26094/535*e^2 + 4172/535, 433/17120*e^11 - 13407/17120*e^9 + 15183/1712*e^7 - 764251/17120*e^5 + 406649/4280*e^3 - 30421/535*e, 581/17120*e^11 - 15039/17120*e^9 + 13873/1712*e^7 - 547487/17120*e^5 + 55207/1070*e^3 - 17552/535*e, -31/2140*e^10 + 819/2140*e^8 - 426/107*e^6 + 42597/2140*e^4 - 46211/1070*e^2 + 15076/535, 22/535*e^10 - 633/535*e^8 + 1337/107*e^6 - 31024/535*e^4 + 54769/535*e^2 - 20568/535, 97/2140*e^10 - 2183/2140*e^8 + 1627/214*e^6 - 44719/2140*e^4 + 8141/535*e^2 + 2668/535, -65/1712*e^11 + 1807/1712*e^9 - 8915/856*e^7 + 72859/1712*e^5 - 25033/428*e^3 - 1068/107*e, 23/428*e^11 - 725/428*e^9 + 4241/214*e^7 - 44817/428*e^5 + 25349/107*e^3 - 16922/107*e, 119/4280*e^10 - 2281/4280*e^8 + 1065/428*e^6 + 8787/4280*e^4 - 17623/1070*e^2 + 2508/535, 1333/17120*e^11 - 40567/17120*e^9 + 45517/1712*e^7 - 2255391/17120*e^5 + 555217/2140*e^3 - 56691/535*e, 233/17120*e^11 - 7847/17120*e^9 + 10487/1712*e^7 - 667811/17120*e^5 + 473169/4280*e^3 - 53176/535*e, 159/8560*e^11 - 4891/8560*e^9 + 1341/214*e^7 - 249753/8560*e^5 + 225869/4280*e^3 - 14896/535*e, 151/1712*e^10 - 3941/1712*e^8 + 17599/856*e^6 - 121861/1712*e^4 + 8090/107*e^2 - 1180/107, -641/17120*e^11 + 18419/17120*e^9 - 18877/1712*e^7 + 814387/17120*e^5 - 80927/1070*e^3 + 9067/535*e, 9/2140*e^11 - 907/4280*e^9 + 2767/856*e^7 - 9982/535*e^5 + 141111/4280*e^3 + 2371/535*e, -67/1070*e^10 + 1563/1070*e^8 - 1158/107*e^6 + 26829/1070*e^4 - 2212/535*e^2 + 11184/535, -401/17120*e^11 + 13459/17120*e^9 - 16837/1712*e^7 + 928067/17120*e^5 - 60981/535*e^3 + 17327/535*e, -3/2140*e^10 + 597/2140*e^8 - 1149/214*e^6 + 71981/2140*e^4 - 35639/535*e^2 + 16508/535, 99/3424*e^11 - 2581/3424*e^9 + 11965/1712*e^7 - 92849/3424*e^5 + 31401/856*e^3 + 117/107*e, -927/8560*e^10 + 23973/8560*e^8 - 21415/856*e^6 + 767229/8560*e^4 - 120113/1070*e^2 + 20668/535, 257/4280*e^11 - 976/535*e^9 + 17637/856*e^7 - 446509/4280*e^5 + 912919/4280*e^3 - 40126/535*e, 281/4280*e^10 - 6699/4280*e^8 + 5545/428*e^6 - 193107/4280*e^4 + 35114/535*e^2 - 13068/535, -33/1070*e^10 + 341/535*e^8 - 989/214*e^6 + 18181/1070*e^4 - 44971/1070*e^2 + 18636/535, -201/4280*e^10 + 5759/4280*e^8 - 5507/428*e^6 + 193907/4280*e^4 - 44513/1070*e^2 + 8388/535, -37/4280*e^10 + 943/4280*e^8 - 1117/428*e^6 + 79559/4280*e^4 - 31543/535*e^2 + 16556/535, 383/8560*e^11 - 12017/8560*e^9 + 14009/856*e^7 - 735861/8560*e^5 + 410439/2140*e^3 - 63927/535*e, -337/2140*e^10 + 8213/2140*e^8 - 3385/107*e^6 + 215659/2140*e^4 - 111187/1070*e^2 + 16852/535, 11/1712*e^11 - 905/1712*e^9 + 8439/856*e^7 - 119409/1712*e^5 + 41067/214*e^3 - 16880/107*e, -61/856*e^10 + 1439/856*e^8 - 5749/428*e^6 + 35663/856*e^4 - 4825/107*e^2 + 2552/107, -273/8560*e^11 + 9387/8560*e^9 - 11897/856*e^7 + 667411/8560*e^5 - 188577/1070*e^3 + 49207/535*e, 557/8560*e^10 - 10263/8560*e^8 + 3825/856*e^6 + 173881/8560*e^4 - 56196/535*e^2 + 17172/535, 501/8560*e^10 - 11959/8560*e^8 + 9341/856*e^6 - 240127/8560*e^4 - 7571/1070*e^2 + 22876/535, 343/8560*e^10 - 6197/8560*e^8 + 1899/856*e^6 + 153979/8560*e^4 - 43249/535*e^2 + 11608/535, 493/8560*e^11 - 11437/8560*e^9 + 4155/428*e^7 - 155891/8560*e^5 - 161687/4280*e^3 + 44403/535*e, -351/4280*e^10 + 7789/4280*e^8 - 5391/428*e^6 + 99317/4280*e^4 + 17401/535*e^2 - 24032/535, -2091/17120*e^11 + 60869/17120*e^9 - 64693/1712*e^7 + 3011577/17120*e^5 - 1405243/4280*e^3 + 78557/535*e, -121/1712*e^11 + 3963/1712*e^9 - 23921/856*e^7 + 255483/1712*e^5 - 34713/107*e^3 + 19509/107*e, 97/8560*e^10 - 4323/8560*e^8 + 6549/856*e^6 - 393539/8560*e^4 + 48179/535*e^2 - 4148/535, -373/8560*e^10 + 7887/8560*e^8 - 5257/856*e^6 + 133551/8560*e^4 - 15086/535*e^2 + 17892/535, 843/17120*e^11 - 28657/17120*e^9 + 35039/1712*e^7 - 1833361/17120*e^5 + 228151/1070*e^3 - 43506/535*e, -1021/17120*e^11 + 31979/17120*e^9 - 36231/1712*e^7 + 1758607/17120*e^5 - 803903/4280*e^3 + 30407/535*e, 799/8560*e^10 - 19901/8560*e^8 + 16475/856*e^6 - 480893/8560*e^4 + 13123/535*e^2 + 19884/535, -283/4280*e^10 + 7097/4280*e^8 - 6311/428*e^6 + 242521/4280*e^4 - 46637/535*e^2 + 4304/535, -71/428*e^10 + 1717/428*e^8 - 7137/214*e^6 + 48189/428*e^4 - 15205/107*e^2 + 2636/107, -19/214*e^10 + 232/107*e^8 - 3755/214*e^6 + 10617/214*e^4 - 3805/214*e^2 - 3044/107, 83/8560*e^11 - 3677/8560*e^9 + 5253/856*e^7 - 300161/8560*e^5 + 152839/2140*e^3 - 14492/535*e, -729/8560*e^10 + 18811/8560*e^8 - 16629/856*e^6 + 566123/8560*e^4 - 32158/535*e^2 - 6684/535, 409/3424*e^11 - 12055/3424*e^9 + 65907/1712*e^7 - 650643/3424*e^5 + 339817/856*e^3 - 22901/107*e, -123/2140*e^10 + 3077/2140*e^8 - 2597/214*e^6 + 79341/2140*e^4 - 12954/535*e^2 + 7008/535, -51/4280*e^10 + 1589/4280*e^8 - 1985/428*e^6 + 117297/4280*e^4 - 73003/1070*e^2 + 19408/535, -25/428*e^11 + 1497/856*e^9 - 16341/856*e^7 + 19563/214*e^5 - 150369/856*e^3 + 7205/107*e, -59/535*e^11 + 1576/535*e^9 - 3036/107*e^7 + 64038/535*e^5 - 113443/535*e^3 + 59926/535*e, -467/17120*e^11 + 9473/17120*e^9 - 5307/1712*e^7 + 33529/17120*e^5 + 13537/2140*e^3 + 22599/535*e, -199/8560*e^10 + 3221/8560*e^8 - 675/856*e^6 - 82347/8560*e^4 + 18842/535*e^2 - 14964/535, 31/17120*e^11 - 6169/17120*e^9 + 14013/1712*e^7 - 1121157/17120*e^5 + 823733/4280*e^3 - 67957/535*e, 1363/17120*e^11 - 40117/17120*e^9 + 43953/1712*e^7 - 2187681/17120*e^5 + 1164549/4280*e^3 - 80536/535*e, 25/428*e^10 - 695/428*e^8 + 3577/214*e^6 - 32599/428*e^4 + 14583/107*e^2 - 5172/107, 1401/17120*e^11 - 41259/17120*e^9 + 45453/1712*e^7 - 2283387/17120*e^5 + 153156/535*e^3 - 85452/535*e, -393/8560*e^11 + 10797/8560*e^9 - 2721/214*e^7 + 497151/8560*e^5 - 511523/4280*e^3 + 44157/535*e, -43/1070*e^10 + 1067/1070*e^8 - 954/107*e^6 + 36271/1070*e^4 - 23163/535*e^2 - 6844/535, -1331/17120*e^11 + 33749/17120*e^9 - 29985/1712*e^7 + 1123137/17120*e^5 - 457693/4280*e^3 + 52997/535*e, 189/8560*e^11 - 7651/8560*e^9 + 10755/856*e^7 - 650703/8560*e^5 + 409137/2140*e^3 - 78101/535*e, 59/1070*e^10 - 1041/1070*e^8 + 341/107*e^6 + 21027/1070*e^4 - 51081/535*e^2 + 17652/535, 171/3424*e^11 - 6209/3424*e^9 + 41347/1712*e^7 - 487601/3424*e^5 + 36758/107*e^3 - 21694/107*e, 119/8560*e^11 - 4421/8560*e^9 + 5987/856*e^7 - 355013/8560*e^5 + 50833/535*e^3 - 11051/535*e, -263/4280*e^10 + 7397/4280*e^8 - 7639/428*e^6 + 347581/4280*e^4 - 73717/535*e^2 + 8484/535, -211/4280*e^10 + 3469/4280*e^8 - 777/428*e^6 - 89743/4280*e^4 + 83147/1070*e^2 - 11892/535, -37/1070*e^10 + 943/1070*e^8 - 796/107*e^6 + 23919/1070*e^4 - 9542/535*e^2 + 6304/535, -317/4280*e^11 + 8513/4280*e^9 - 2078/107*e^7 + 361379/4280*e^5 - 345767/2140*e^3 + 58616/535*e, 163/8560*e^11 - 6757/8560*e^9 + 9785/856*e^7 - 607521/8560*e^5 + 373829/2140*e^3 - 54817/535*e, 1183/17120*e^11 - 34257/17120*e^9 + 37073/1712*e^7 - 1840661/17120*e^5 + 1004639/4280*e^3 - 85126/535*e, -27/428*e^10 + 665/428*e^8 - 2699/214*e^6 + 15673/428*e^4 - 2105/107*e^2 - 3582/107, -17/856*e^11 + 247/428*e^9 - 5083/856*e^7 + 22193/856*e^5 - 40449/856*e^3 + 4472/107*e, 11/856*e^10 - 477/856*e^8 + 3517/428*e^6 - 43225/856*e^4 + 24033/214*e^2 - 2730/107, 12/535*e^10 - 248/535*e^8 + 204/107*e^6 + 5256/535*e^4 - 30581/535*e^2 + 16212/535, -97/1712*e^10 + 2611/1712*e^8 - 12201/856*e^6 + 95651/1712*e^4 - 9766/107*e^2 + 2864/107, 129/1070*e^10 - 3201/1070*e^8 + 2648/107*e^6 - 78853/1070*e^4 + 23479/535*e^2 + 5552/535, 1071/4280*e^10 - 26949/4280*e^8 + 23495/428*e^6 - 823997/4280*e^4 + 123194/535*e^2 - 22368/535, 111/4280*e^10 - 2829/4280*e^8 + 2495/428*e^6 - 93157/4280*e^4 + 20799/535*e^2 - 4728/535, 143/8560*e^11 - 4917/8560*e^9 + 6191/856*e^7 - 323101/8560*e^5 + 64997/1070*e^3 + 11573/535*e, 811/8560*e^11 - 24429/8560*e^9 + 26849/856*e^7 - 1303817/8560*e^5 + 669593/2140*e^3 - 111649/535*e, -17/856*e^10 + 387/856*e^8 - 1525/428*e^6 + 8283/856*e^4 + 1083/107*e^2 - 4516/107, 1269/17120*e^11 - 36391/17120*e^9 + 38981/1712*e^7 - 1906783/17120*e^5 + 499991/2140*e^3 - 69558/535*e, -7/428*e^10 + 109/428*e^8 - 137/214*e^6 - 1247/428*e^4 + 873/107*e^2 - 3140/107, -629/4280*e^10 + 13891/4280*e^8 - 9359/428*e^6 + 145543/4280*e^4 + 81533/1070*e^2 - 30988/535, -13/535*e^10 + 359/1070*e^8 + 307/214*e^6 - 18534/535*e^4 + 123593/1070*e^2 - 23448/535, 59/4280*e^10 - 1041/4280*e^8 + 127/428*e^6 + 57407/4280*e^4 - 32699/535*e^2 + 17788/535, -779/8560*e^10 + 20201/8560*e^8 - 18659/856*e^6 + 731473/8560*e^4 - 129831/1070*e^2 + 13236/535, -1/80*e^10 + 19/80*e^8 - 5/8*e^6 - 573/80*e^4 + 123/5*e^2 - 16/5, -537/4280*e^11 + 15913/4280*e^9 - 8729/214*e^7 + 857799/4280*e^5 - 878017/2140*e^3 + 113751/535*e, 77/2140*e^10 - 2483/2140*e^8 + 2741/214*e^6 - 117679/2140*e^4 + 33946/535*e^2 + 15708/535, 539/4280*e^10 - 13101/4280*e^8 + 10841/428*e^6 - 348673/4280*e^4 + 94777/1070*e^2 - 32762/535, 65/428*e^11 - 3721/856*e^9 + 39405/856*e^7 - 46969/214*e^5 + 376065/856*e^3 - 22371/107*e, 1371/8560*e^10 - 31009/8560*e^8 + 22407/856*e^6 - 532097/8560*e^4 + 9367/535*e^2 + 1976/535, -359/17120*e^11 + 15801/17120*e^9 - 23649/1712*e^7 + 1478253/17120*e^5 - 872187/4280*e^3 + 38728/535*e, 559/8560*e^10 - 14941/8560*e^8 + 13579/856*e^6 - 474733/8560*e^4 + 33293/535*e^2 - 22316/535, -711/17120*e^11 + 23789/17120*e^9 - 30279/1712*e^7 + 1799157/17120*e^5 - 591909/2140*e^3 + 113212/535*e, -747/17120*e^11 + 20253/17120*e^9 - 20313/1712*e^7 + 938089/17120*e^5 - 482101/4280*e^3 + 30974/535*e, -69/535*e^11 + 1961/535*e^9 - 4062/107*e^7 + 92293/535*e^5 - 170973/535*e^3 + 88146/535*e, -25/1712*e^10 + 1123/1712*e^8 - 7429/856*e^6 + 69835/1712*e^4 - 10555/214*e^2 - 312/107, 27/535*e^10 - 558/535*e^8 + 673/107*e^6 - 3154/535*e^4 - 17046/535*e^2 + 10262/535, -75/1712*e^10 + 2513/1712*e^8 - 15011/856*e^6 + 149585/1712*e^4 - 16100/107*e^2 + 3344/107, -299/8560*e^11 + 8141/8560*e^9 - 7945/856*e^7 + 329673/8560*e^5 - 130517/2140*e^3 + 14711/535*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {7, 7, 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;