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

NN := ideal<ZF | {11, 11, -w^2 + 5}>;

primesArray := [
[2, 2, -w],
[4, 2, w^2 - w - 7],
[5, 5, w - 1],
[7, 7, w + 1],
[11, 11, -w^2 + 5],
[13, 13, w + 3],
[13, 13, -2*w + 1],
[19, 19, 2*w^2 - w - 15],
[25, 5, w^2 - 7],
[27, 3, 3],
[31, 31, w^2 - 2*w - 1],
[37, 37, -3*w^2 + 2*w + 23],
[41, 41, -2*w - 1],
[47, 47, w^2 - 3],
[49, 7, w^2 - 2*w - 5],
[59, 59, 2*w - 3],
[67, 67, w - 5],
[71, 71, w^2 - 2*w - 11],
[73, 73, w^2 + 1],
[83, 83, -4*w^2 + 3*w + 27],
[89, 89, w^2 - 2*w - 7],
[97, 97, w^2 - 6*w + 1],
[109, 109, -w^2 + 4*w + 1],
[113, 113, -w - 5],
[113, 113, -3*w - 1],
[113, 113, 2*w^2 - 3*w - 11],
[137, 137, -2*w^2 + 3*w + 17],
[149, 149, 2*w^2 - 13],
[151, 151, -2*w^2 + 1],
[163, 163, 2*w^2 + w - 7],
[167, 167, 2*w^2 - 5*w - 5],
[191, 191, -6*w^2 + 4*w + 41],
[191, 191, 4*w^2 - 12*w + 1],
[191, 191, 3*w - 7],
[193, 193, w^2 + 2*w - 5],
[197, 197, -4*w^2 + 4*w + 27],
[211, 211, -10*w^2 + 8*w + 69],
[211, 211, 3*w - 5],
[211, 211, 2*w^2 - 4*w - 1],
[223, 223, 2*w^2 - 3*w - 7],
[227, 227, 2*w^2 - 15],
[229, 229, -2*w - 7],
[233, 233, 2*w^2 - 3*w - 13],
[233, 233, -w^2 - 4*w + 3],
[233, 233, 2*w^2 - 2*w - 9],
[239, 239, -5*w^2 + 4*w + 33],
[241, 241, -2*w^2 + 2*w + 1],
[257, 257, 2*w^2 - 6*w - 3],
[257, 257, 2*w^2 - 3*w - 15],
[257, 257, -w^2 - 2*w - 3],
[269, 269, 2*w^2 - w - 9],
[269, 269, 2*w^2 + w - 11],
[269, 269, -2*w^2 + 7*w + 1],
[271, 271, -7*w^2 + 6*w + 49],
[277, 277, -3*w^2 + 2*w + 25],
[277, 277, 3*w^2 - 8*w - 1],
[277, 277, 2*w^2 - 4*w - 9],
[281, 281, w^2 - 4*w - 3],
[283, 283, 4*w^2 - 2*w - 25],
[283, 283, -3*w^2 + 10*w - 5],
[283, 283, 2*w^2 + w - 5],
[293, 293, -5*w + 3],
[307, 307, -4*w^2 + 2*w + 33],
[317, 317, 2*w^2 + w + 1],
[331, 331, 2*w - 9],
[337, 337, -10*w^2 + 6*w + 75],
[337, 337, w^2 + 2*w - 7],
[337, 337, -8*w^2 + 6*w + 55],
[347, 347, w^2 - 13],
[347, 347, 2*w^2 - 4*w - 19],
[347, 347, -5*w^2 + 4*w + 39],
[349, 349, 2*w^2 - w - 19],
[353, 353, -8*w^2 + 7*w + 55],
[359, 359, -4*w^2 + 3*w + 25],
[361, 19, -w^2 - 6*w + 3],
[367, 367, 2*w^2 - 3*w - 3],
[367, 367, 2*w^2 - 2*w - 7],
[367, 367, w^2 - 2*w - 13],
[373, 373, -4*w - 7],
[373, 373, 3*w^2 - 2*w - 17],
[373, 373, -11*w^2 + 8*w + 77],
[379, 379, -7*w + 3],
[383, 383, w^2 + 2*w - 17],
[397, 397, 2*w^2 - w - 7],
[401, 401, -4*w - 3],
[409, 409, 12*w^2 - 7*w - 91],
[419, 419, 5*w + 1],
[419, 419, 2*w^2 - 2*w - 3],
[419, 419, -7*w^2 + 4*w + 51],
[421, 421, w^2 - 4*w - 13],
[431, 431, 3*w^2 - 19],
[449, 449, w^2 + 2*w - 9],
[457, 457, -8*w^2 + 7*w + 59],
[461, 461, 2*w^2 - w - 5],
[463, 463, -4*w - 5],
[479, 479, -4*w^2 + 4*w + 23],
[487, 487, -3*w^2 + 4*w + 23],
[499, 499, 2*w^2 - 4*w - 11],
[509, 509, 4*w - 7],
[541, 541, -w^2 + 4*w - 7],
[557, 557, -2*w^2 - w + 19],
[569, 569, 8*w - 1],
[569, 569, -w^2 + 2*w - 5],
[569, 569, 2*w^2 - 6*w + 5],
[577, 577, -w^2 + 6*w - 11],
[587, 587, w - 9],
[593, 593, 2*w^2 - 8*w - 1],
[599, 599, 4*w^2 - 10*w - 5],
[599, 599, 2*w^2 - 6*w - 5],
[599, 599, -4*w^2 + 3*w + 33],
[613, 613, -3*w^2 - 2*w + 3],
[617, 617, 2*w^2 - 3*w + 3],
[619, 619, -10*w^2 + 7*w + 75],
[631, 631, 3*w^2 + 1],
[643, 643, -10*w^2 + 7*w + 69],
[661, 661, 2*w^2 - 7*w - 3],
[673, 673, 5*w - 13],
[673, 673, -w^2 - 2*w - 5],
[673, 673, 2*w^2 + 2*w - 11],
[677, 677, 6*w + 1],
[677, 677, 2*w^2 + w - 17],
[677, 677, 3*w^2 - 2*w - 15],
[683, 683, 3*w^2 - 25],
[691, 691, -5*w - 9],
[701, 701, -5*w^2 + 2*w + 37],
[701, 701, -3*w^2 + 4*w + 1],
[701, 701, w^2 - 4*w - 9],
[709, 709, -6*w^2 + 4*w + 39],
[727, 727, 2*w^2 - 3*w - 21],
[727, 727, -8*w^2 + 6*w + 61],
[727, 727, 2*w^2 - 4*w - 15],
[733, 733, 2*w^2 + 2*w - 23],
[739, 739, -4*w^2 + w + 31],
[743, 743, 2*w^2 + 4*w - 7],
[751, 751, 4*w^2 + 2*w - 19],
[751, 751, -4*w^2 + w + 29],
[751, 751, -6*w^2 + 6*w + 41],
[761, 761, -10*w^2 + 6*w + 77],
[769, 769, -w^2 - 5],
[773, 773, -3*w^2 + 2*w + 27],
[787, 787, -10*w^2 + 6*w + 73],
[797, 797, -5*w^2 + 6*w + 31],
[797, 797, 3*w^2 - 6*w - 5],
[797, 797, 2*w - 11],
[811, 811, 4*w^2 - 16*w + 9],
[821, 821, 4*w^2 - 8*w - 15],
[827, 827, -2*w^2 + 9*w - 11],
[839, 839, -3*w^2 + 8*w + 7],
[853, 853, -6*w + 5],
[863, 863, -w^2 + 8*w - 13],
[877, 877, 5*w - 7],
[881, 881, -7*w^2 + 6*w + 45],
[881, 881, -2*w^2 + 10*w - 7],
[881, 881, 3*w^2 - 4*w - 11],
[883, 883, 9*w - 1],
[883, 883, 2*w^2 - w + 3],
[883, 883, -2*w^2 + 8*w - 9],
[887, 887, 4*w^2 - 5*w - 27],
[887, 887, -2*w^2 + 5*w - 5],
[887, 887, -5*w - 7],
[929, 929, w^2 + 4*w - 7],
[941, 941, -2*w^2 + 2*w + 21],
[947, 947, -3*w - 11],
[961, 31, -4*w^2 + w + 21],
[967, 967, -2*w^2 + 11*w - 13],
[977, 977, 5*w^2 - 16*w - 1],
[983, 983, 2*w^2 + 2*w - 13],
[997, 997, 3*w^2 - 2*w - 13],
[997, 997, -2*w^2 - w - 3],
[997, 997, -4*w^2 + 4*w + 21]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^17 - 4*x^16 - 22*x^15 + 103*x^14 + 160*x^13 - 1019*x^12 - 309*x^11 + 4873*x^10 - 1166*x^9 - 11821*x^8 + 5385*x^7 + 14542*x^6 - 7055*x^5 - 8208*x^4 + 3394*x^3 + 1198*x^2 - 660*x + 72;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -255787/3801738*e^16 + 851443/3801738*e^15 + 12329099/7603476*e^14 - 11061875/1900869*e^13 - 109337693/7603476*e^12 + 443098219/7603476*e^11 + 142161577/2534492*e^10 - 1076437291/3801738*e^9 - 326206267/3801738*e^8 + 5314232687/7603476*e^7 + 7715311/633623*e^6 - 1644347474/1900869*e^5 + 572846233/7603476*e^4 + 306183885/633623*e^3 - 233685283/3801738*e^2 - 335675329/3801738*e + 11170155/633623, 115045/2534492*e^16 - 673225/2534492*e^15 - 2320211/2534492*e^14 + 8898959/1267246*e^13 + 6480339/1267246*e^12 - 183327895/2534492*e^11 + 8884801/1267246*e^10 + 934648765/2534492*e^9 - 377128665/2534492*e^8 - 1253925607/1267246*e^7 + 1121614119/2534492*e^6 + 3543066637/2534492*e^5 - 580716073/1267246*e^4 - 1162659353/1267246*e^3 + 159036545/1267246*e^2 + 92293610/633623*e - 18727053/633623, 759487/3801738*e^16 - 3125875/3801738*e^15 - 16692673/3801738*e^14 + 80699161/3801738*e^13 + 121910911/3801738*e^12 - 802070303/3801738*e^11 - 83664279/1267246*e^10 + 3869966761/3801738*e^9 - 708890021/3801738*e^8 - 4780012904/1900869*e^7 + 541493353/633623*e^6 + 6090560366/1900869*e^5 - 3518924783/3801738*e^4 - 1210891782/633623*e^3 + 470670923/1900869*e^2 + 558178952/1900869*e - 35477972/633623, 1, -565739/1900869*e^16 + 6534767/7603476*e^15 + 14013917/1900869*e^14 - 170271683/7603476*e^13 - 522032027/7603476*e^12 + 1711757515/7603476*e^11 + 187911959/633623*e^10 - 4182604681/3801738*e^9 - 4557840155/7603476*e^8 + 10417857019/3801738*e^7 + 685254917/1267246*e^6 - 26108131343/7603476*e^5 - 725960161/3801738*e^4 + 1219984928/633623*e^3 - 39507469/3801738*e^2 - 589124459/1900869*e + 31331160/633623, 164093/1900869*e^16 - 1270445/7603476*e^15 - 8462497/3801738*e^14 + 33016253/7603476*e^13 + 168207197/7603476*e^12 - 329714845/7603476*e^11 - 135447473/1267246*e^10 + 396893870/1900869*e^9 + 2009900621/7603476*e^8 - 957898568/1900869*e^7 - 423379749/1267246*e^6 + 4511340587/7603476*e^5 + 726487993/3801738*e^4 - 377229553/1267246*e^3 - 73461233/3801738*e^2 + 82526648/1900869*e - 4679246/633623, 1263305/3801738*e^16 - 4982309/3801738*e^15 - 28739705/3801738*e^14 + 64781134/1900869*e^13 + 226633739/3801738*e^12 - 650538260/1900869*e^11 - 105919951/633623*e^10 + 3185526067/1900869*e^9 - 271806175/3801738*e^8 - 8024839255/1900869*e^7 + 1244924349/1267246*e^6 + 10410084445/1900869*e^5 - 4747457857/3801738*e^4 - 4172863605/1267246*e^3 + 778201408/1900869*e^2 + 1008030067/1900869*e - 64824966/633623, 2627987/7603476*e^16 - 2416097/1900869*e^15 - 61245899/7603476*e^14 + 253295417/7603476*e^13 + 507099281/7603476*e^12 - 1286224073/3801738*e^11 - 274216291/1267246*e^10 + 12809904155/7603476*e^9 + 380934631/3801738*e^8 - 16540895893/3801738*e^7 + 1782770407/2534492*e^6 + 11077474097/1900869*e^5 - 4040229275/3801738*e^4 - 2299438215/633623*e^3 + 702585170/1900869*e^2 + 1157264915/1900869*e - 73341305/633623, -153154/633623*e^16 + 2524909/2534492*e^15 + 3397583/633623*e^14 - 65690745/2534492*e^13 - 100973557/2534492*e^12 + 660425325/2534492*e^11 + 55087297/633623*e^10 - 1621493309/1267246*e^9 + 541081755/2534492*e^8 + 4113327181/1267246*e^7 - 1358713879/1267246*e^6 - 10841585137/2534492*e^5 + 1561625923/1267246*e^4 + 1671344225/633623*e^3 - 445601491/1267246*e^2 - 262759003/633623*e + 50355965/633623, 80273/3801738*e^16 + 1575533/7603476*e^15 - 4098923/3801738*e^14 - 39887069/7603476*e^13 + 143060761/7603476*e^12 + 388229323/7603476*e^11 - 193944771/1267246*e^10 - 923304439/3801738*e^9 + 4732505731/7603476*e^8 + 2329995985/3801738*e^7 - 1537227699/1267246*e^6 - 6703724561/7603476*e^5 + 3676368281/3801738*e^4 + 831223685/1267246*e^3 - 574044679/3801738*e^2 - 166589141/1900869*e + 10359285/633623, 2971831/7603476*e^16 - 9330349/7603476*e^15 - 72296839/7603476*e^14 + 121563329/3801738*e^13 + 162958285/1900869*e^12 - 2445826541/7603476*e^11 - 438645589/1267246*e^10 + 11979392575/7603476*e^9 + 4450823515/7603476*e^8 - 7506515662/1900869*e^7 - 625775055/2534492*e^6 + 38238749491/7603476*e^5 - 981506779/3801738*e^4 - 3712284801/1267246*e^3 + 940741751/3801738*e^2 + 982605781/1900869*e - 62389380/633623, -325233/2534492*e^16 + 246170/633623*e^15 + 7649671/2534492*e^14 - 24719413/2534492*e^13 - 64957451/2534492*e^12 + 58639054/633623*e^11 + 115046855/1267246*e^10 - 1040979741/2534492*e^9 - 121886455/1267246*e^8 + 545879267/633623*e^7 - 286456353/2534492*e^6 - 1006892167/1267246*e^5 + 374417087/1267246*e^4 + 163213507/633623*e^3 - 133007176/633623*e^2 - 24990350/633623*e + 14734428/633623, -182411/633623*e^16 + 2187691/2534492*e^15 + 8919421/1267246*e^14 - 56898873/2534492*e^13 - 162374679/2534492*e^12 + 570497921/2534492*e^11 + 167480252/633623*e^10 - 1388781717/1267246*e^9 - 1208648379/2534492*e^8 + 1721579085/633623*e^7 + 184513192/633623*e^6 - 8605365649/2534492*e^5 + 91491815/1267246*e^4 + 1214161193/633623*e^3 - 171261021/1267246*e^2 - 206096760/633623*e + 38435925/633623, -659671/1900869*e^16 + 11528965/7603476*e^15 + 29455409/3801738*e^14 - 300989929/7603476*e^13 - 445284475/7603476*e^12 + 3041559839/7603476*e^11 + 177258769/1267246*e^10 - 7523786081/3801738*e^9 + 1697451743/7603476*e^8 + 9636579019/1900869*e^7 - 1654509131/1267246*e^6 - 51197874193/7603476*e^5 + 2820207929/1900869*e^4 + 2630015667/633623*e^3 - 1604456597/3801738*e^2 - 1272490294/1900869*e + 80498533/633623, 34177/1267246*e^16 - 117059/633623*e^15 - 195798/633623*e^14 + 5573179/1267246*e^13 - 3496015/1267246*e^12 - 24254939/633623*e^11 + 73432975/1267246*e^10 + 91952381/633623*e^9 - 203395223/633623*e^8 - 273280073/1267246*e^7 + 946943633/1267246*e^6 + 86369425/1267246*e^5 - 492091799/633623*e^4 + 41668927/1267246*e^3 + 210029898/633623*e^2 + 10787330/633623*e - 18738699/633623, -631243/1900869*e^16 + 7615813/7603476*e^15 + 15651085/1900869*e^14 - 200667991/7603476*e^13 - 582742567/7603476*e^12 + 2050987433/7603476*e^11 + 416993911/1267246*e^10 - 5140451591/3801738*e^9 - 4912912225/7603476*e^8 + 13316011073/3801738*e^7 + 324580105/633623*e^6 - 35271207559/7603476*e^5 - 178368658/1900869*e^4 + 1769738040/633623*e^3 - 156434177/3801738*e^2 - 901844935/1900869*e + 48434146/633623, -635089/1267246*e^16 + 4981855/2534492*e^15 + 14354601/1267246*e^14 - 130095149/2534492*e^13 - 222395679/2534492*e^12 + 1314719829/2534492*e^11 + 143757630/633623*e^10 - 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1766847027/633623*e + 334576876/633623, -667513/1267246*e^16 + 2024009/1267246*e^15 + 7703987/633623*e^14 - 51812597/1267246*e^13 - 125175423/1267246*e^12 + 507552387/1267246*e^11 + 191395519/633623*e^10 - 1193677171/633623*e^9 - 15261187/1267246*e^8 + 5644848331/1267246*e^7 - 948518496/633623*e^6 - 6771422743/1267246*e^5 + 3035191907/1267246*e^4 + 3908308211/1267246*e^3 - 754742278/633623*e^2 - 377926110/633623*e + 110181792/633623, -15039/1267246*e^16 - 624547/2534492*e^15 + 433295/1267246*e^14 + 16902625/2534492*e^13 - 8859369/2534492*e^12 - 180518937/2534492*e^11 + 7016463/633623*e^10 + 485410533/1267246*e^9 + 124049239/2534492*e^8 - 1396126137/1267246*e^7 - 278957448/633623*e^6 + 4193254681/2534492*e^5 + 663474404/633623*e^4 - 1370261921/1267246*e^3 - 1041314961/1267246*e^2 + 72037995/633623*e + 49522462/633623, -4867709/2534492*e^16 + 8962369/1267246*e^15 + 112656335/2534492*e^14 - 462623745/2534492*e^13 - 925152259/2534492*e^12 + 2296154505/1267246*e^11 + 748794334/633623*e^10 - 22071497153/2534492*e^9 - 440138518/633623*e^8 + 13478661416/633623*e^7 - 7183408359/2534492*e^6 - 33374818891/1267246*e^5 + 2484189232/633623*e^4 + 9414185648/633623*e^3 - 696868837/633623*e^2 - 1397180831/633623*e + 238457673/633623, -8614739/2534492*e^16 + 31588445/2534492*e^15 + 199393237/2534492*e^14 - 410276823/1267246*e^13 - 408122991/633623*e^12 + 8225151345/2534492*e^11 + 1294077834/633623*e^10 - 40148977985/2534492*e^9 - 1988103203/2534492*e^8 + 50288827035/1267246*e^7 - 17091738475/2534492*e^6 - 129317064113/2534492*e^5 + 12128645735/1267246*e^4 + 19207457676/633623*e^3 - 3978533437/1267246*e^2 - 3052371899/633623*e + 574924236/633623, -935073/1267246*e^16 + 3548861/1267246*e^15 + 21913511/1267246*e^14 - 46620261/633623*e^13 - 91919455/633623*e^12 + 950598997/1267246*e^11 + 312036302/633623*e^10 - 4762548537/1267246*e^9 - 498736323/1267246*e^8 + 6205720223/633623*e^7 - 1167179725/1267246*e^6 - 16795185861/1267246*e^5 + 814077103/633623*e^4 + 5237487490/633623*e^3 - 40575690/633623*e^2 - 804087738/633623*e + 109044450/633623, -930294/633623*e^16 + 14113459/2534492*e^15 + 42767861/1267246*e^14 - 365465951/2534492*e^13 - 690493929/2534492*e^12 + 3645980357/2534492*e^11 + 1049757413/1267246*e^10 - 8829988645/1267246*e^9 - 259432895/2534492*e^8 + 10916629531/633623*e^7 - 4374755761/1267246*e^6 - 55011823955/2534492*e^5 + 3112215604/633623*e^4 + 7981438775/633623*e^3 - 2570751365/1267246*e^2 - 1320501579/633623*e + 283625317/633623, -3657595/3801738*e^16 + 8196469/3801738*e^15 + 94021903/3801738*e^14 - 106420814/1900869*e^13 - 929817457/3801738*e^12 + 1062292717/1900869*e^11 + 743302481/633623*e^10 - 5116211741/1900869*e^9 - 10935990967/3801738*e^8 + 12367241801/1900869*e^7 + 4586582909/1267246*e^6 - 14633575859/1900869*e^5 - 7715333677/3801738*e^4 + 5030853857/1267246*e^3 + 96349330/1900869*e^2 - 1360909976/1900869*e + 88609488/633623, -2072282/1900869*e^16 + 32157563/7603476*e^15 + 93489967/3801738*e^14 - 838931669/7603476*e^13 - 1442663147/7603476*e^12 + 8465588617/7603476*e^11 + 306824781/633623*e^10 - 20895403597/3801738*e^9 + 4453970737/7603476*e^8 + 26709740954/1900869*e^7 - 2663903662/633623*e^6 - 142338043661/7603476*e^5 + 20384139131/3801738*e^4 + 7453740078/633623*e^3 - 7265516251/3801738*e^2 - 3843110126/1900869*e + 266147013/633623, -669861/633623*e^16 + 3318296/633623*e^15 + 28319893/1267246*e^14 - 173263281/1267246*e^13 - 92546386/633623*e^12 + 1752159819/1267246*e^11 + 68085145/633623*e^10 - 4345483751/633623*e^9 + 1378940296/633623*e^8 + 22422585301/1267246*e^7 - 9373403369/1267246*e^6 - 30292102773/1267246*e^5 + 5096896793/633623*e^4 + 9647698782/633623*e^3 - 1601354301/633623*e^2 - 1628898740/633623*e + 352765392/633623, -1076817/633623*e^16 + 5149968/633623*e^15 + 45189935/1267246*e^14 - 266894673/1267246*e^13 - 290265651/1267246*e^12 + 1334561887/633623*e^11 + 82128675/633623*e^10 - 13017009927/1267246*e^9 + 2293728063/633623*e^8 + 32760178867/1267246*e^7 - 15143519115/1267246*e^6 - 21490350495/633623*e^5 + 16283250335/1267246*e^4 + 26749544451/1267246*e^3 - 2611978053/633623*e^2 - 2268426637/633623*e + 493104528/633623, -385733/633623*e^16 + 1985833/633623*e^15 + 7807546/633623*e^14 - 51232372/633623*e^13 - 90366487/1267246*e^12 + 1018468225/1267246*e^11 - 29242429/633623*e^10 - 4924446937/1267246*e^9 + 1035258618/633623*e^8 + 6127375819/633623*e^7 - 2966745767/633623*e^6 - 15903286137/1267246*e^5 + 5546338953/1267246*e^4 + 9766003539/1267246*e^3 - 593435692/633623*e^2 - 733291599/633623*e + 152239734/633623, -1120951/2534492*e^16 + 5509821/2534492*e^15 + 23406597/2534492*e^14 - 35407614/633623*e^13 - 37342153/633623*e^12 + 1398188675/2534492*e^11 + 43352275/1267246*e^10 - 6669324987/2534492*e^9 + 2255493661/2534492*e^8 + 8066888233/1267246*e^7 - 7168958865/2534492*e^6 - 19799336213/2534492*e^5 + 3640357675/1267246*e^4 + 5561704795/1267246*e^3 - 1086040165/1267246*e^2 - 415923592/633623*e + 81246315/633623, 829573/1267246*e^16 - 9748575/2534492*e^15 - 16197429/1267246*e^14 + 251440875/2534492*e^13 + 164823381/2534492*e^12 - 2498225081/2534492*e^11 + 199310621/1267246*e^10 + 6032346813/1267246*e^9 - 5591744201/2534492*e^8 - 14946628885/1267246*e^7 + 7415910297/1267246*e^6 + 38244598231/2534492*e^5 - 6639329435/1267246*e^4 - 11324457427/1267246*e^3 + 1303866407/1267246*e^2 + 772790177/633623*e - 132933360/633623, -2671671/1267246*e^16 + 5288291/633623*e^15 + 60474915/1267246*e^14 - 275174957/1267246*e^13 - 235732857/633623*e^12 + 2766123681/1267246*e^11 + 633770295/633623*e^10 - 6787576725/633623*e^9 + 463695867/633623*e^8 + 17186784745/633623*e^7 - 8671593989/1267246*e^6 - 45088117209/1267246*e^5 + 10762851695/1267246*e^4 + 27634136777/1267246*e^3 - 1744779127/633623*e^2 - 2278607707/633623*e + 447764229/633623, -300512/1900869*e^16 - 937541/3801738*e^15 + 8942738/1900869*e^14 + 13767355/1900869*e^13 - 215676967/3801738*e^12 - 163536953/1900869*e^11 + 449808331/1267246*e^10 + 2012253179/3801738*e^9 - 4631993431/3801738*e^8 - 3393478714/1900869*e^7 + 2779448163/1267246*e^6 + 12059061281/3801738*e^5 - 3186966503/1900869*e^4 - 3126530413/1267246*e^3 + 271947106/1900869*e^2 + 630574534/1900869*e - 9684081/633623, 5575939/3801738*e^16 - 15788281/3801738*e^15 - 138741853/3801738*e^14 + 205211222/1900869*e^13 + 652066169/1900869*e^12 - 4110848111/3801738*e^11 - 959737653/633623*e^10 + 19966331053/3801738*e^9 + 12372195013/3801738*e^8 - 24601431281/1900869*e^7 - 4487889691/1267246*e^6 + 60555423103/3801738*e^5 + 4135144880/1900869*e^4 - 5500630565/633623*e^3 - 1172526715/1900869*e^2 + 2581897568/1900869*e - 95315894/633623, 10578167/2534492*e^16 - 18377399/1267246*e^15 - 247702669/2534492*e^14 + 953682929/2534492*e^13 + 2077258615/2534492*e^12 - 4770417585/1267246*e^11 - 3526806647/1267246*e^10 + 46407372151/2534492*e^9 + 1390031835/633623*e^8 - 28881404445/633623*e^7 + 14536177087/2534492*e^6 + 36762869341/633623*e^5 - 12257007379/1267246*e^4 - 21576914113/633623*e^3 + 2254181991/633623*e^2 + 3464982285/633623*e - 654858105/633623, 964336/633623*e^16 - 11657553/2534492*e^15 - 23106568/633623*e^14 + 298886229/2534492*e^13 + 814571633/2534492*e^12 - 2934886565/2534492*e^11 - 793334556/633623*e^10 + 6923126719/1267246*e^9 + 5098740529/2534492*e^8 - 16377617757/1267246*e^7 - 1237090261/1267246*e^6 + 38589870457/2534492*e^5 - 60740623/1267246*e^4 - 5115111403/633623*e^3 + 21196879/1267246*e^2 + 776254224/633623*e - 116002731/633623, 1374617/2534492*e^16 - 1379675/1267246*e^15 - 37303929/2534492*e^14 + 71659387/2534492*e^13 + 401178939/2534492*e^12 - 178478749/633623*e^11 - 550304005/633623*e^10 + 3408747925/2534492*e^9 + 1672348235/633623*e^8 - 3994953939/1267246*e^7 - 11461961295/2534492*e^6 + 2132381177/633623*e^5 + 2473667411/633623*e^4 - 801368556/633623*e^3 - 670987682/633623*e^2 + 133864275/633623*e + 19309108/633623, 1875761/1900869*e^16 - 29364131/7603476*e^15 - 84224605/3801738*e^14 + 768683531/7603476*e^13 + 1280380511/7603476*e^12 - 7795947787/7603476*e^11 - 502335493/1267246*e^10 + 9695351048/1900869*e^9 - 5994423325/7603476*e^8 - 25076405306/1900869*e^7 + 5834704395/1267246*e^6 + 135808354697/7603476*e^5 - 22719061733/3801738*e^4 - 14568953911/1267246*e^3 + 8782079281/3801738*e^2 + 3998455493/1900869*e - 298888934/633623, 8585213/7603476*e^16 - 18361357/3801738*e^15 - 185227241/7603476*e^14 + 951029597/7603476*e^13 + 1286333111/7603476*e^12 - 4750730405/3801738*e^11 - 315996873/1267246*e^10 + 46265872451/7603476*e^9 - 3180167173/1900869*e^8 - 58118381929/3801738*e^7 + 16201165129/2534492*e^6 + 38140602125/1900869*e^5 - 13495432825/1900869*e^4 - 15923652131/1267246*e^3 + 4303219088/1900869*e^2 + 4094314343/1900869*e - 301867562/633623];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {11, 11, -w^2 + 5}>] := -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;