/*
  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, 10, -7, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {17, 17, -w^2 - w + 3}>;

primesArray := [
[2, 2, -w + 1],
[5, 5, w^3 + w^2 - 4*w + 1],
[7, 7, -w^3 + 6*w - 2],
[8, 2, w^3 - 7*w + 3],
[11, 11, -w^3 + 6*w - 4],
[17, 17, w^3 + w^2 - 6*w - 1],
[17, 17, -w^2 - w + 3],
[31, 31, 2*w^3 + w^2 - 13*w + 3],
[37, 37, -3*w^3 - w^2 + 18*w - 3],
[41, 41, w^2 + 2*w - 4],
[47, 47, 2*w^3 + 2*w^2 - 11*w - 2],
[47, 47, -3*w^3 - w^2 + 19*w - 6],
[59, 59, -2*w^3 + 13*w - 6],
[59, 59, -w^3 - w^2 + 5*w + 2],
[59, 59, w^2 + w - 1],
[59, 59, w^2 + 2*w - 6],
[61, 61, -w^3 + w^2 + 8*w - 7],
[67, 67, w^3 + 2*w^2 - 4*w - 4],
[73, 73, -2*w^3 - w^2 + 12*w - 4],
[81, 3, -3],
[107, 107, 2*w^2 + 2*w - 11],
[109, 109, 2*w^3 + w^2 - 11*w + 3],
[113, 113, w^3 + w^2 - 7*w - 2],
[113, 113, 2*w^3 - 12*w + 9],
[125, 5, -4*w^3 - 3*w^2 + 24*w + 2],
[131, 131, 5*w^3 + 3*w^2 - 31*w + 2],
[151, 151, 4*w^3 + 2*w^2 - 26*w + 1],
[151, 151, w^3 + 2*w^2 - 6*w - 10],
[167, 167, w + 4],
[167, 167, 4*w^3 + w^2 - 26*w + 10],
[173, 173, -2*w^3 + w^2 + 13*w - 11],
[179, 179, 4*w^3 + w^2 - 24*w + 4],
[179, 179, 2*w^3 + 2*w^2 - 12*w - 3],
[179, 179, -w^3 + 8*w - 4],
[179, 179, w^3 + 2*w^2 - 4*w - 6],
[181, 181, 6*w^3 + 3*w^2 - 39*w + 5],
[197, 197, -w^3 + 4*w - 4],
[199, 199, 2*w^3 - 12*w + 3],
[199, 199, 3*w - 2],
[223, 223, w^3 + w^2 - 7*w - 4],
[223, 223, 2*w^3 + w^2 - 10*w + 4],
[227, 227, 7*w^3 + 2*w^2 - 44*w + 12],
[227, 227, -4*w^3 - w^2 + 25*w - 9],
[229, 229, 8*w^3 + 4*w^2 - 51*w + 4],
[229, 229, 3*w^3 + w^2 - 18*w + 5],
[233, 233, -6*w^3 - 3*w^2 + 39*w - 7],
[233, 233, -w^3 - 2*w^2 + 8*w + 2],
[241, 241, 2*w^3 + w^2 - 10*w],
[251, 251, w^3 + w^2 - 7*w - 6],
[257, 257, -6*w^3 - w^2 + 39*w - 17],
[263, 263, -2*w^3 + 13*w - 4],
[263, 263, -w^3 + 6*w - 8],
[269, 269, -8*w^3 - 4*w^2 + 50*w - 7],
[269, 269, -w^3 + w^2 + 7*w - 6],
[277, 277, 2*w^3 + 2*w^2 - 11*w + 4],
[277, 277, w^3 + w^2 - 4*w - 3],
[281, 281, -w^2 - 3*w + 5],
[281, 281, 2*w^3 + 3*w^2 - 9*w - 3],
[283, 283, -3*w^3 - 4*w^2 + 12*w + 2],
[289, 17, -2*w^3 - w^2 + 13*w - 5],
[293, 293, -w^3 - 2*w^2 + 6*w + 6],
[293, 293, w^3 + w^2 - 8*w - 1],
[307, 307, 2*w^3 - 11*w + 12],
[317, 317, 2*w^3 - 14*w + 11],
[317, 317, -2*w^3 - 2*w^2 + 8*w + 1],
[337, 337, 2*w^2 + 4*w - 11],
[343, 7, -5*w^3 - w^2 + 31*w - 10],
[349, 349, 2*w^3 - 11*w + 8],
[349, 349, w^2 + w - 9],
[353, 353, 4*w^3 + 4*w^2 - 23*w - 8],
[373, 373, -2*w - 3],
[373, 373, -4*w^3 + 26*w - 13],
[389, 389, 6*w^3 + 3*w^2 - 37*w + 1],
[389, 389, -2*w^3 - w^2 + 14*w - 6],
[401, 401, 2*w^3 - 12*w + 13],
[421, 421, 2*w^2 + 2*w - 15],
[421, 421, 3*w^3 + 2*w^2 - 20*w + 4],
[431, 431, w^3 + 3*w^2 - 2*w - 9],
[433, 433, 2*w^3 + 2*w^2 - 12*w + 3],
[433, 433, -7*w^3 - 2*w^2 + 44*w - 16],
[433, 433, -3*w^3 + 18*w - 14],
[433, 433, -w^3 + w^2 + 9*w - 10],
[439, 439, 5*w^3 + 3*w^2 - 31*w - 2],
[443, 443, -w^3 - w^2 + 3*w - 4],
[443, 443, -4*w^3 + 27*w - 14],
[449, 449, w^3 + w^2 - 4*w - 5],
[449, 449, w^3 + 3*w^2 - 3*w - 8],
[449, 449, 2*w^3 - 11*w + 6],
[449, 449, -3*w^3 - 2*w^2 + 18*w - 4],
[461, 461, -3*w^3 + w^2 + 20*w - 15],
[463, 463, 2*w^3 - 10*w + 7],
[479, 479, -2*w^3 - 3*w^2 + 11*w + 9],
[491, 491, w^3 + w^2 - 4*w + 5],
[499, 499, -2*w^3 - 2*w^2 + 14*w - 3],
[499, 499, -6*w^3 - 2*w^2 + 39*w - 10],
[499, 499, 4*w^3 + w^2 - 24*w + 14],
[499, 499, -5*w^3 - 3*w^2 + 30*w - 5],
[503, 503, -2*w^3 + w^2 + 14*w - 18],
[509, 509, 4*w^3 + 3*w^2 - 24*w],
[509, 509, 4*w^3 + 2*w^2 - 23*w + 4],
[547, 547, w^3 + 2*w^2 - 2*w - 6],
[547, 547, 6*w^3 + 2*w^2 - 38*w + 7],
[563, 563, -5*w^3 - w^2 + 33*w - 12],
[569, 569, 2*w^3 - w^2 - 15*w + 13],
[569, 569, -4*w^3 - 2*w^2 + 24*w - 7],
[577, 577, -w^2 - 2],
[577, 577, 3*w^3 - 20*w + 8],
[587, 587, 6*w^3 + 4*w^2 - 36*w - 1],
[587, 587, -w^3 + w^2 + 9*w - 8],
[593, 593, 6*w^3 + 3*w^2 - 36*w + 2],
[593, 593, -3*w^3 - w^2 + 21*w - 4],
[599, 599, -2*w^3 - w^2 + 12*w - 10],
[599, 599, -4*w^3 - 2*w^2 + 25*w - 6],
[601, 601, -2*w^3 + 2*w^2 + 15*w - 16],
[607, 607, w^3 - w^2 - 7*w + 4],
[607, 607, -2*w^3 - 2*w^2 + 10*w + 3],
[631, 631, 2*w^2 + 4*w - 7],
[641, 641, -3*w^3 - 2*w^2 + 16*w],
[643, 643, w^3 + w^2 - 3*w - 4],
[643, 643, w^3 + 3*w^2 - 6*w - 13],
[653, 653, -w^3 + 6*w + 2],
[653, 653, -3*w^2 - 5*w + 11],
[659, 659, -2*w^3 + 10*w - 5],
[661, 661, w^3 - w^2 - 8*w + 3],
[673, 673, 3*w^3 + 4*w^2 - 14*w - 2],
[677, 677, 4*w^3 + 2*w^2 - 23*w + 6],
[677, 677, 2*w^3 - 13*w + 14],
[683, 683, 2*w^3 + w^2 - 13*w + 7],
[701, 701, 8*w^3 + 4*w^2 - 51*w + 8],
[701, 701, -4*w^3 - w^2 + 24*w - 12],
[709, 709, -4*w^3 - w^2 + 27*w - 7],
[709, 709, 3*w^3 + 3*w^2 - 18*w - 5],
[709, 709, -3*w^3 + w^2 + 21*w - 14],
[709, 709, w^2 - w - 7],
[719, 719, -9*w^3 - 4*w^2 + 58*w - 8],
[719, 719, 11*w^3 + 5*w^2 - 68*w + 9],
[727, 727, -w^3 + 4*w - 6],
[733, 733, 2*w^3 + w^2 - 13*w + 9],
[739, 739, -2*w^2 - 3*w + 4],
[743, 743, 3*w^3 + 3*w^2 - 19*w - 4],
[751, 751, 2*w^3 + 3*w^2 - 10*w - 2],
[761, 761, -12*w^3 - 5*w^2 + 75*w - 9],
[761, 761, -3*w^3 - 3*w^2 + 17*w],
[769, 769, -6*w^3 - 4*w^2 + 37*w],
[769, 769, -8*w^3 - 3*w^2 + 51*w - 7],
[773, 773, -w^2 - 2*w - 2],
[787, 787, -5*w^3 - 5*w^2 + 28*w + 7],
[811, 811, 7*w^3 + 2*w^2 - 42*w + 6],
[811, 811, -2*w^3 - w^2 + 9*w - 7],
[823, 823, -3*w^3 - 2*w^2 + 14*w - 10],
[829, 829, 4*w^3 + 3*w^2 - 20*w + 6],
[829, 829, -7*w^3 - 2*w^2 + 44*w - 14],
[841, 29, w^3 + 4*w^2 - 2*w - 14],
[841, 29, -w^2 + 10],
[857, 857, -3*w^3 - w^2 + 20*w - 9],
[859, 859, -w^3 + 2*w^2 + 10*w - 18],
[863, 863, 11*w^3 + 5*w^2 - 69*w + 6],
[863, 863, w^3 - 2*w^2 - 10*w + 12],
[877, 877, 3*w^3 - 18*w + 10],
[883, 883, 4*w^3 + 3*w^2 - 22*w],
[887, 887, 3*w^3 + 3*w^2 - 15*w - 2],
[887, 887, w^2 + 4*w - 10],
[887, 887, w^3 + w^2 - 3*w - 8],
[887, 887, w - 6],
[907, 907, 6*w^3 + w^2 - 38*w + 16],
[919, 919, -2*w^3 - 3*w^2 + 11*w + 1],
[919, 919, 5*w^3 - w^2 - 34*w + 27],
[937, 937, -w^3 - w^2 + 8*w + 3],
[941, 941, -4*w^3 + 25*w - 20],
[947, 947, -2*w^3 + w^2 + 8*w - 6],
[967, 967, -7*w^3 - w^2 + 44*w - 23],
[971, 971, 2*w^2 + 2*w - 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^20 - 7*x^19 - 6*x^18 + 138*x^17 - 133*x^16 - 1070*x^15 + 1820*x^14 + 4096*x^13 - 9348*x^12 - 7866*x^11 + 24258*x^10 + 6530*x^9 - 33216*x^8 - 1128*x^7 + 23505*x^6 + 284*x^5 - 8657*x^4 - 821*x^3 + 1359*x^2 + 278*x - 12;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 617511/275090*e^19 - 941187/55018*e^18 - 483048/137545*e^17 + 43442373/137545*e^16 - 133947461/275090*e^15 - 299660091/137545*e^14 + 749094828/137545*e^13 + 888124399/137545*e^12 - 3507145696/137545*e^11 - 123855366/27509*e^10 + 8281518014/137545*e^9 - 2299116692/137545*e^8 - 9873916227/137545*e^7 + 4796733197/137545*e^6 + 11112071623/275090*e^5 - 551288333/27509*e^4 - 3265502907/275090*e^3 + 213148503/55018*e^2 + 441014959/275090*e - 9045642/137545, 56392/137545*e^19 - 94526/27509*e^18 + 214898/137545*e^17 + 8103617/137545*e^16 - 17773177/137545*e^15 - 47875814/137545*e^14 + 174468267/137545*e^13 + 82438981/137545*e^12 - 749261094/137545*e^11 + 49134390/27509*e^10 + 1581281166/137545*e^9 - 1182850613/137545*e^8 - 1527297113/137545*e^7 + 1587107693/137545*e^6 + 508197056/137545*e^5 - 137717526/27509*e^4 - 83510194/137545*e^3 + 19599116/27509*e^2 + 12963198/137545*e + 32177/137545, -776809/275090*e^19 + 1176021/55018*e^18 + 734527/137545*e^17 - 54495127/137545*e^16 + 163682519/275090*e^15 + 378690879/137545*e^14 - 924805152/137545*e^13 - 1143379941/137545*e^12 + 4352547339/137545*e^11 + 181295658/27509*e^10 - 10334371221/137545*e^9 + 2603233068/137545*e^8 + 12420224958/137545*e^7 - 5735324648/137545*e^6 - 14144327927/275090*e^5 + 671156864/27509*e^4 + 4156565223/275090*e^3 - 261196855/55018*e^2 - 552892831/275090*e + 11226108/137545, -1141177/275090*e^19 + 1692399/55018*e^18 + 1703466/137545*e^17 - 79613191/137545*e^16 + 216689417/275090*e^15 + 568871987/137545*e^14 - 1271400091/137545*e^13 - 1835805138/137545*e^12 + 6091855912/137545*e^11 + 412918417/27509*e^10 - 14719526038/137545*e^9 + 2170368094/137545*e^8 + 18129955709/137545*e^7 - 6681816089/137545*e^6 - 21444853761/275090*e^5 + 845865499/27509*e^4 + 6401082919/275090*e^3 - 344033515/55018*e^2 - 832165963/275090*e + 15880599/137545, 635442/137545*e^19 - 959691/27509*e^18 - 1156147/137545*e^17 + 88222902/137545*e^16 - 133108762/137545*e^15 - 604447079/137545*e^14 + 1485965552/137545*e^13 + 1767212796/137545*e^12 - 6875119674/137545*e^11 - 228862233/27509*e^10 + 15863270741/137545*e^9 - 4726673568/137545*e^8 - 18069582918/137545*e^7 + 9382019243/137545*e^6 + 9251008566/137545*e^5 - 999535128/27509*e^4 - 2471898579/137545*e^3 + 177530662/27509*e^2 + 317818748/137545*e - 12834408/137545, -1, 110411/55018*e^19 - 832935/55018*e^18 - 120172/27509*e^17 + 7788698/27509*e^16 - 22916663/55018*e^15 - 54940042/27509*e^14 + 131757032/27509*e^13 + 171266921/27509*e^12 - 630885520/27509*e^11 - 159489754/27509*e^10 + 1536107849/27509*e^9 - 325206883/27509*e^8 - 1925608117/27509*e^7 + 810962577/27509*e^6 + 2354202013/55018*e^5 - 517999328/27509*e^4 - 727058041/55018*e^3 + 215315389/55018*e^2 + 98391185/55018*e - 1888955/27509, -280576/137545*e^19 + 414375/27509*e^18 + 933821/137545*e^17 - 39513796/137545*e^16 + 52633391/137545*e^15 + 288486542/137545*e^14 - 635990146/137545*e^13 - 970164563/137545*e^12 + 3145569082/137545*e^11 + 249624168/27509*e^10 - 7968652888/137545*e^9 + 737145409/137545*e^8 + 10603574094/137545*e^7 - 3288739459/137545*e^6 - 7090145603/137545*e^5 + 488712403/27509*e^4 + 2326825142/137545*e^3 - 111507237/27509*e^2 - 314335094/137545*e + 12637424/137545, 367377/27509*e^19 - 2741326/27509*e^18 - 934379/27509*e^17 + 51107005/27509*e^16 - 72502800/27509*e^15 - 359160959/27509*e^14 + 834337275/27509*e^13 + 1115992189/27509*e^12 - 3945911381/27509*e^11 - 1054425046/27509*e^10 + 9379300505/27509*e^9 - 1956568178/27509*e^8 - 11253846595/27509*e^7 + 4800015598/27509*e^6 + 6371140906/27509*e^5 - 2836889992/27509*e^4 - 1852683226/27509*e^3 + 551669361/27509*e^2 + 242306712/27509*e - 9397143/27509, -2815247/275090*e^19 + 4237991/55018*e^18 + 2945031/137545*e^17 - 196394611/137545*e^16 + 580373437/275090*e^15 + 1365178907/137545*e^14 - 3291406821/137545*e^13 - 4127694873/137545*e^12 + 15472504532/137545*e^11 + 663769781/27509*e^10 - 36586495718/137545*e^9 + 9182728589/137545*e^8 + 43595707659/137545*e^7 - 20269508024/137545*e^6 - 48819563371/275090*e^5 + 2343729380/27509*e^4 + 14128364649/275090*e^3 - 899103107/55018*e^2 - 1869298003/275090*e + 36026709/137545, -2423502/137545*e^19 + 3622460/27509*e^18 + 5845997/137545*e^17 - 336496642/137545*e^16 + 482883752/137545*e^15 + 2350134599/137545*e^14 - 5519090862/137545*e^13 - 7198931416/137545*e^12 + 25950075629/137545*e^11 + 1263307566/27509*e^10 - 61164342991/137545*e^9 + 14108398548/137545*e^8 + 72303902108/137545*e^7 - 32542664643/137545*e^6 - 39794439541/137545*e^5 + 3736148291/27509*e^4 + 11282699659/137545*e^3 - 708635522/27509*e^2 - 1462721888/137545*e + 58641783/137545, -510679/275090*e^19 + 697681/55018*e^18 + 1676172/137545*e^17 - 33982522/137545*e^16 + 59240209/275090*e^15 + 258559309/137545*e^14 - 411906212/137545*e^13 - 956108231/137545*e^12 + 2049662424/137545*e^11 + 338862084/27509*e^10 - 4974485581/137545*e^9 - 1022902122/137545*e^8 + 5982991863/137545*e^7 - 489617048/137545*e^6 - 6536083087/275090*e^5 + 100267169/27509*e^4 + 1607539993/275090*e^3 - 38037343/55018*e^2 - 143705541/275090*e + 2839863/137545, -2147803/137545*e^19 + 3215258/27509*e^18 + 5103123/137545*e^17 - 298887463/137545*e^16 + 430206153/137545*e^15 + 2089982771/137545*e^14 - 4917565093/137545*e^13 - 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11610425283/137545*e^6 - 6830674456/137545*e^5 + 1126067954/27509*e^4 + 1694271289/137545*e^3 - 186492938/27509*e^2 - 237581198/137545*e + 10262018/137545, -3787673/275090*e^19 + 5604535/55018*e^18 + 5270259/137545*e^17 - 260233579/137545*e^16 + 720806373/275090*e^15 + 1818508543/137545*e^14 - 4143009654/137545*e^13 - 5595339637/137545*e^12 + 19367506598/137545*e^11 + 1026006706/27509*e^10 - 44997188477/137545*e^9 + 9913751476/137545*e^8 + 51629915526/137545*e^7 - 23155865176/137545*e^6 - 53365696449/275090*e^5 + 2522820163/27509*e^4 + 14145508541/275090*e^3 - 904711577/55018*e^2 - 1739874357/275090*e + 38365826/137545, -10600947/137545*e^19 + 15863323/27509*e^18 + 24869967/137545*e^17 - 1472254437/137545*e^16 + 2126358517/137545*e^15 + 10264256514/137545*e^14 - 24255099652/137545*e^13 - 31298286851/137545*e^12 + 113975201459/137545*e^11 + 5341045067/27509*e^10 - 268571210196/137545*e^9 + 63922803658/137545*e^8 + 317479981588/137545*e^7 - 144941090818/137545*e^6 - 174812110171/137545*e^5 + 16607171674/27509*e^4 + 49657428619/137545*e^3 - 3150048688/27509*e^2 - 6458106133/137545*e + 261468203/137545, 2213416/137545*e^19 - 3312488/27509*e^18 - 5400051/137545*e^17 + 308866066/137545*e^16 - 442666751/137545*e^15 - 2170940252/137545*e^14 + 5088428751/137545*e^13 + 6741390203/137545*e^12 - 24120363252/137545*e^11 - 1262745739/27509*e^10 + 57623137143/137545*e^9 - 12117725684/137545*e^8 - 69827686924/137545*e^7 + 29656558094/137545*e^6 + 40308945408/137545*e^5 - 3554583213/27509*e^4 - 11969270237/137545*e^3 + 703383928/27509*e^2 + 1583896724/137545*e - 62779014/137545, -7872001/137545*e^19 + 11680153/27509*e^18 + 23015596/137545*e^17 - 1096880596/137545*e^16 + 1501664601/137545*e^15 + 7812556482/137545*e^14 - 17561993361/137545*e^13 - 25036405193/137545*e^12 + 83914397822/137545*e^11 + 5472602347/27509*e^10 - 201988934518/137545*e^9 + 32030581839/137545*e^8 + 247236650024/137545*e^7 - 93703616949/137545*e^6 - 144679405303/137545*e^5 + 11660523218/27509*e^4 + 42854711147/137545*e^3 - 2340631957/27509*e^2 - 5555888549/137545*e + 211718129/137545, -12746637/275090*e^19 + 19130903/55018*e^18 + 14702171/137545*e^17 - 890236591/137545*e^16 + 2578891637/275090*e^15 + 6236557022/137545*e^14 - 14743382531/137545*e^13 - 19220028418/137545*e^12 + 69643022067/137545*e^11 + 3461008859/27509*e^10 - 165620707178/137545*e^9 + 36734001104/137545*e^8 + 199120849164/137545*e^7 - 87032548424/137545*e^6 - 226351192331/275090*e^5 + 10291322237/27509*e^4 + 65972208129/275090*e^3 - 4008135853/55018*e^2 - 8644698223/275090*e + 172744269/137545, -899566/27509*e^19 + 6680399/27509*e^18 + 2371818/27509*e^17 - 124137712/27509*e^16 + 174707496/27509*e^15 + 867573913/27509*e^14 - 2006772404/27509*e^13 - 2663903874/27509*e^12 + 9427655188/27509*e^11 + 2380593361/27509*e^10 - 22134297958/27509*e^9 + 5039139471/27509*e^8 + 25944497242/27509*e^7 - 11713573727/27509*e^6 - 14034826594/27509*e^5 + 6647752453/27509*e^4 + 3917672880/27509*e^3 - 1247011002/27509*e^2 - 505829894/27509*e + 21020334/27509, -7129818/137545*e^19 + 10804355/27509*e^18 + 12538898/137545*e^17 - 996965608/137545*e^16 + 1514252273/137545*e^15 + 6873161986/137545*e^14 - 16985539623/137545*e^13 - 20359511894/137545*e^12 + 79370011181/137545*e^11 + 2848896445/27509*e^10 - 186434351584/137545*e^9 + 52155167932/137545*e^8 + 219862053427/137545*e^7 - 108239391132/137545*e^6 - 120946172124/137545*e^5 + 12235331223/27509*e^4 + 34690013296/137545*e^3 - 2316574930/27509*e^2 - 4619769882/137545*e + 184952372/137545, 7682268/137545*e^19 - 11373718/27509*e^18 - 22855713/137545*e^17 + 1067153358/137545*e^16 - 1454761858/137545*e^15 - 7589137926/137545*e^14 + 17019713263/137545*e^13 + 24240599074/137545*e^12 - 81194825361/137545*e^11 - 5228937773/27509*e^10 + 194817639219/137545*e^9 - 32015977937/137545*e^8 - 237031877827/137545*e^7 + 91603433587/137545*e^6 + 137165163644/137545*e^5 - 11281613585/27509*e^4 - 40194580446/137545*e^3 + 2248216119/27509*e^2 + 5181992437/137545*e - 209624077/137545, -4113951/137545*e^19 + 6099445/27509*e^18 + 11783381/137545*e^17 - 570476566/137545*e^16 + 784824441/137545*e^15 + 4035561077/137545*e^14 - 9120027246/137545*e^13 - 12748946773/137545*e^12 + 43242344432/137545*e^11 + 2630393414/27509*e^10 - 102836541868/137545*e^9 + 18517985219/137545*e^8 + 123228608534/137545*e^7 - 49564399559/137545*e^6 - 69409344673/137545*e^5 + 5897428155/27509*e^4 + 19919047382/137545*e^3 - 1145852602/27509*e^2 - 2546592254/137545*e + 105980124/137545, 4543189/275090*e^19 - 6830395/55018*e^18 - 5108092/137545*e^17 + 318264602/137545*e^16 - 928871359/275090*e^15 - 2233575594/137545*e^14 + 5320988847/137545*e^13 + 6899365361/137545*e^12 - 25272803769/137545*e^11 - 1242805018/27509*e^10 + 60701885686/137545*e^9 - 13451939513/137545*e^8 - 74329162393/137545*e^7 + 32266046188/137545*e^6 + 87382561107/275090*e^5 - 3947256058/27509*e^4 - 26235755913/275090*e^3 + 1586003487/55018*e^2 + 3505319491/275090*e - 66402708/137545, 993425/55018*e^19 - 7596449/55018*e^18 - 565849/27509*e^17 + 69278487/27509*e^16 - 221765733/55018*e^15 - 466970707/27509*e^14 + 1217056348/27509*e^13 + 1303308077/27509*e^12 - 5606069416/27509*e^11 - 498223177/27509*e^10 + 12932463240/27509*e^9 - 4682450013/27509*e^8 - 14796648770/27509*e^7 + 8566305739/27509*e^6 + 15385922755/55018*e^5 - 4603949200/27509*e^4 - 4253746867/55018*e^3 + 1672896141/55018*e^2 + 580089833/55018*e - 10210662/27509, 976802/137545*e^19 - 1378051/27509*e^18 - 5020307/137545*e^17 + 132505477/137545*e^16 - 143381242/137545*e^15 - 984364614/137545*e^14 + 1842733847/137545*e^13 + 3458582631/137545*e^12 - 9102086994/137545*e^11 - 1056591779/27509*e^10 + 22457307716/137545*e^9 + 631546217/137545*e^8 - 28291765668/137545*e^7 + 6336020228/137545*e^6 + 17227864731/137545*e^5 - 950300270/27509*e^4 - 5174555994/137545*e^3 + 205292928/27509*e^2 + 650622188/137545*e - 20586273/137545, -899703/275090*e^19 + 1584661/55018*e^18 - 2421256/137545*e^17 - 69892429/137545*e^16 + 325951633/275090*e^15 + 436775773/137545*e^14 - 1646137569/137545*e^13 - 927667827/137545*e^12 + 7562380413/137545*e^11 - 279417165/27509*e^10 - 18129385952/137545*e^9 + 9559869981/137545*e^8 + 22790237131/137545*e^7 - 15080251806/137545*e^6 - 28821202779/275090*e^5 + 1748313568/27509*e^4 + 9787506471/275090*e^3 - 713493103/55018*e^2 - 1488024187/275090*e + 30891941/137545, 1396229/137545*e^19 - 2018057/27509*e^18 - 6199099/137545*e^17 + 194985889/137545*e^16 - 230194389/137545*e^15 - 1457261723/137545*e^14 + 2903068814/137545*e^13 + 5153493327/137545*e^12 - 14531503808/137545*e^11 - 1572195314/27509*e^10 + 36972477247/137545*e^9 + 344762289/137545*e^8 - 49156084246/137545*e^7 + 11618750546/137545*e^6 + 32572043502/137545*e^5 - 1942828891/27509*e^4 - 10392564693/137545*e^3 + 458652372/27509*e^2 + 1358014536/137545*e - 49237066/137545, 2992112/137545*e^19 - 4474656/27509*e^18 - 7195147/137545*e^17 + 415519147/137545*e^16 - 595564002/137545*e^15 - 2901414014/137545*e^14 + 6795785482/137545*e^13 + 8893541486/137545*e^12 - 31875591489/137545*e^11 - 1578306406/27509*e^10 + 74819693356/137545*e^9 - 16938262233/137545*e^8 - 87755267148/137545*e^7 + 39077999383/137545*e^6 + 47561179631/137545*e^5 - 4395242868/27509*e^4 - 13297621654/137545*e^3 + 813367328/27509*e^2 + 1702258153/137545*e - 61281418/137545, 1520481/275090*e^19 - 2403199/55018*e^18 + 547297/137545*e^17 + 107034918/137545*e^16 - 394725861/275090*e^15 - 686552556/137545*e^14 + 2075137553/137545*e^13 + 1637316074/137545*e^12 - 9384566406/137545*e^11 + 191527332/27509*e^10 + 21326523414/137545*e^9 - 11194784347/137545*e^8 - 23959255292/137545*e^7 + 17650553862/137545*e^6 + 24352661233/275090*e^5 - 1835794629/27509*e^4 - 6998111847/275090*e^3 + 665790521/55018*e^2 + 1063607369/275090*e - 18914797/137545, 2501219/275090*e^19 - 3573087/55018*e^18 - 6143767/137545*e^17 + 172637042/137545*e^16 - 381964139/275090*e^15 - 1293189009/137545*e^14 + 2438124187/137545*e^13 + 4616990546/137545*e^12 - 12092400669/137545*e^11 - 1471628683/27509*e^10 + 30115217056/137545*e^9 + 1771114172/137545*e^8 - 38550954553/137545*e^7 + 7505455383/137545*e^6 + 48126112637/275090*e^5 - 1229438891/27509*e^4 - 14620878983/275090*e^3 + 560473329/55018*e^2 + 1806613191/275090*e - 34784193/137545, 918594/137545*e^19 - 1394720/27509*e^18 - 1493229/137545*e^17 + 127698609/137545*e^16 - 194189764/137545*e^15 - 869736103/137545*e^14 + 2140049964/137545*e^13 + 2516668522/137545*e^12 - 9758669613/137545*e^11 - 311752780/27509*e^10 + 21979063997/137545*e^9 - 6745046941/137545*e^8 - 23840581341/137545*e^7 + 12843485166/137545*e^6 + 10857669152/137545*e^5 - 1253575013/27509*e^4 - 2482271803/137545*e^3 + 197296265/27509*e^2 + 282236816/137545*e - 7345701/137545, -7510149/275090*e^19 + 11181983/55018*e^18 + 9610012/137545*e^17 - 519196612/137545*e^16 + 1468571899/275090*e^15 + 3625809799/137545*e^14 - 8405289612/137545*e^13 - 11120180596/137545*e^12 + 39383967889/137545*e^11 + 1982990478/27509*e^10 - 92108839876/137545*e^9 + 20963520533/137545*e^8 + 107177565578/137545*e^7 - 48440481818/137545*e^6 - 114182663667/275090*e^5 + 5421836544/27509*e^4 + 31290034533/275090*e^3 - 1999783907/55018*e^2 - 3970544131/275090*e + 78475403/137545, 3238906/137545*e^19 - 4927648/27509*e^18 - 5155156/137545*e^17 + 454470211/137545*e^16 - 700350576/137545*e^15 - 3129048977/137545*e^14 + 7835099406/137545*e^13 + 9228154898/137545*e^12 - 36663060942/137545*e^11 - 1239027155/27509*e^10 + 86467414858/137545*e^9 - 24659110074/137545*e^8 - 102812728299/137545*e^7 + 50726532039/137545*e^6 + 57453566038/137545*e^5 - 5783139554/27509*e^4 - 16682161477/137545*e^3 + 1099849106/27509*e^2 + 2227714854/137545*e - 83943219/137545];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {17, 17, -w^2 - 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;