/*
  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![-49, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {9, 3, 3}>;

primesArray := [
[4, 2, 2],
[7, 7, w - 7],
[7, 7, w + 6],
[9, 3, 3],
[19, 19, w + 5],
[19, 19, w - 6],
[23, 23, w + 8],
[23, 23, -w + 9],
[25, 5, 5],
[29, 29, -w - 4],
[29, 29, w - 5],
[37, 37, -w - 3],
[37, 37, w - 4],
[41, 41, -w - 9],
[41, 41, w - 10],
[43, 43, -w - 2],
[43, 43, w - 3],
[47, 47, -w - 1],
[47, 47, w - 2],
[53, 53, 2*w - 13],
[53, 53, -2*w - 11],
[59, 59, -4*w - 25],
[59, 59, -4*w + 29],
[61, 61, -w - 10],
[61, 61, w - 11],
[83, 83, -w - 11],
[83, 83, w - 12],
[97, 97, 2*w - 11],
[97, 97, -2*w - 9],
[101, 101, 3*w - 20],
[101, 101, -3*w - 17],
[107, 107, -w - 12],
[107, 107, w - 13],
[109, 109, -5*w - 31],
[109, 109, -5*w + 36],
[121, 11, -11],
[127, 127, 2*w - 19],
[127, 127, -2*w - 17],
[137, 137, -3*w - 16],
[137, 137, 3*w - 19],
[157, 157, -3*w - 23],
[157, 157, 3*w - 26],
[163, 163, -4*w + 27],
[163, 163, 4*w + 23],
[169, 13, -13],
[173, 173, -6*w - 37],
[173, 173, -6*w + 43],
[181, 181, 2*w - 5],
[181, 181, -2*w - 3],
[191, 191, -w - 15],
[191, 191, w - 16],
[193, 193, 2*w - 3],
[193, 193, -2*w - 1],
[197, 197, 2*w - 1],
[223, 223, -w - 16],
[223, 223, w - 17],
[233, 233, -3*w - 13],
[233, 233, 3*w - 16],
[239, 239, -5*w + 34],
[239, 239, -5*w - 29],
[251, 251, 5*w - 41],
[251, 251, 5*w + 36],
[257, 257, -w - 17],
[257, 257, w - 18],
[289, 17, -17],
[293, 293, -w - 18],
[293, 293, w - 19],
[311, 311, -3*w - 10],
[311, 311, 3*w - 13],
[313, 313, -3*w - 26],
[313, 313, 3*w - 29],
[331, 331, -w - 19],
[331, 331, w - 20],
[347, 347, 4*w - 23],
[347, 347, -4*w - 19],
[353, 353, 3*w - 11],
[353, 353, -3*w - 8],
[379, 379, 2*w - 25],
[379, 379, -2*w - 23],
[401, 401, 3*w - 8],
[401, 401, -3*w - 5],
[409, 409, 5*w - 43],
[409, 409, 5*w + 38],
[419, 419, -5*w - 26],
[419, 419, 5*w - 31],
[431, 431, 3*w - 5],
[431, 431, -3*w - 2],
[433, 433, -7*w + 48],
[433, 433, -7*w - 41],
[443, 443, 3*w - 2],
[443, 443, 3*w - 1],
[449, 449, -9*w - 55],
[449, 449, -9*w + 64],
[457, 457, -w - 22],
[457, 457, w - 23],
[479, 479, 2*w - 27],
[479, 479, -2*w - 25],
[487, 487, -3*w - 29],
[487, 487, 3*w - 32],
[491, 491, -5*w - 39],
[491, 491, 5*w - 44],
[499, 499, 4*w - 19],
[499, 499, -4*w - 15],
[503, 503, -w - 23],
[503, 503, w - 24],
[521, 521, 11*w - 86],
[521, 521, -7*w + 47],
[557, 557, 7*w + 51],
[557, 557, 7*w - 58],
[563, 563, -4*w - 13],
[563, 563, 4*w - 17],
[569, 569, -10*w - 61],
[569, 569, -10*w + 71],
[587, 587, 2*w - 29],
[587, 587, -2*w - 27],
[601, 601, -w - 25],
[601, 601, w - 26],
[607, 607, -7*w - 39],
[607, 607, 7*w - 46],
[613, 613, 3*w - 34],
[613, 613, -3*w - 31],
[617, 617, -6*w - 31],
[617, 617, 6*w - 37],
[619, 619, 4*w - 15],
[619, 619, -4*w - 11],
[631, 631, 5*w - 27],
[631, 631, -5*w - 22],
[653, 653, -w - 26],
[653, 653, w - 27],
[661, 661, 5*w - 46],
[661, 661, -5*w - 41],
[683, 683, -9*w + 62],
[683, 683, -9*w - 53],
[691, 691, 7*w - 45],
[691, 691, -7*w - 38],
[727, 727, -6*w - 47],
[727, 727, 6*w - 53],
[733, 733, 4*w - 41],
[733, 733, -4*w - 37],
[739, 739, -4*w - 5],
[739, 739, 4*w - 9],
[751, 751, 8*w - 53],
[751, 751, -8*w - 45],
[769, 769, 5*w - 24],
[769, 769, -5*w - 19],
[773, 773, 7*w - 44],
[773, 773, -7*w - 37],
[787, 787, 4*w - 3],
[787, 787, 4*w - 1],
[797, 797, -9*w - 52],
[797, 797, -9*w + 61],
[811, 811, -5*w - 18],
[811, 811, 5*w - 23],
[821, 821, -w - 29],
[821, 821, w - 30],
[827, 827, 2*w - 33],
[827, 827, -2*w - 31],
[829, 829, -10*w + 69],
[829, 829, -10*w - 59],
[839, 839, 5*w - 48],
[839, 839, -5*w - 43],
[853, 853, -7*w - 36],
[853, 853, 7*w - 43],
[881, 881, -w - 30],
[881, 881, w - 31],
[961, 31, -31],
[991, 991, -5*w - 13],
[991, 991, 5*w - 18]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 + 3*x^5 - 28*x^4 - 72*x^3 + 226*x^2 + 383*x - 577;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-3/49*e^5 + 4/49*e^4 + 83/49*e^3 - 111/49*e^2 - 540/49*e + 701/49, -1/49*e^5 - 1/49*e^4 + 30/49*e^3 + 12/49*e^2 - 250/49*e + 19/49, e, 1, -1/112*e^5 + 1/28*e^4 + 2/7*e^3 - 9/14*e^2 - 85/56*e + 535/112, -57/784*e^5 + 5/196*e^4 + 99/49*e^3 - 135/98*e^2 - 5333/392*e + 12479/784, -73/1568*e^5 - 27/392*e^4 + 87/98*e^3 + 183/196*e^2 - 2349/784*e - 3009/1568, 361/1568*e^5 - 13/392*e^4 - 599/98*e^3 + 561/196*e^2 + 29277/784*e - 56223/1568, 1/49*e^5 - 6/49*e^4 - 23/49*e^3 + 135/49*e^2 + 138/49*e - 663/49, -47/1568*e^5 + 11/392*e^4 + 61/98*e^3 - 199/196*e^2 - 1227/784*e + 8873/1568, 111/1568*e^5 + 5/392*e^4 - 181/98*e^3 + 103/196*e^2 + 7659/784*e - 19497/1568, 33/784*e^5 - 25/196*e^4 - 75/49*e^3 + 367/98*e^2 + 5021/392*e - 19527/784, 143/784*e^5 - 15/196*e^4 - 234/49*e^3 + 251/98*e^2 + 11155/392*e - 17865/784, -361/1568*e^5 + 13/392*e^4 + 599/98*e^3 - 561/196*e^2 - 29277/784*e + 49951/1568, 73/1568*e^5 + 27/392*e^4 - 87/98*e^3 - 183/196*e^2 + 2349/784*e - 3263/1568, -15/98*e^5 + 3/49*e^4 + 190/49*e^3 - 155/49*e^2 - 1070/49*e + 3001/98, 13/98*e^5 + 3/49*e^4 - 167/49*e^3 + 20/49*e^2 + 932/49*e - 1675/98, 3/1568*e^5 - 15/392*e^4 - 17/98*e^3 + 363/196*e^2 + 2223/784*e - 28981/1568, -67/1568*e^5 - 1/392*e^4 + 137/98*e^3 - 267/196*e^2 - 8655/784*e + 20789/1568, 57/784*e^5 + 23/196*e^4 - 106/49*e^3 - 159/98*e^2 + 5837/392*e - 4527/784, 1/112*e^5 + 3/28*e^4 - 3/7*e^3 - 33/14*e^2 + 157/56*e + 601/112, 23/196*e^5 - 3/49*e^4 - 176/49*e^3 + 127/49*e^2 + 2595/98*e - 4637/196, 37/196*e^5 - 3/49*e^4 - 253/49*e^3 + 134/49*e^2 + 3057/98*e - 5407/196, 97/784*e^5 - 9/196*e^4 - 146/49*e^3 + 173/98*e^2 + 6357/392*e - 13687/784, -97/784*e^5 - 19/196*e^4 + 153/49*e^3 + 121/98*e^2 - 6861/392*e + 1815/784, 849/1568*e^5 - 101/392*e^4 - 1437/98*e^3 + 1985/196*e^2 + 71797/784*e - 142775/1568, 271/1568*e^5 - 123/392*e^4 - 537/98*e^3 + 1627/196*e^2 + 31691/784*e - 82793/1568, 345/784*e^5 - 45/196*e^4 - 562/49*e^3 + 977/98*e^2 + 27949/392*e - 68575/784, -201/784*e^5 - 59/196*e^4 + 327/49*e^3 + 277/98*e^2 - 15997/392*e + 15887/784, 241/1568*e^5 + 27/392*e^4 - 367/98*e^3 - 43/196*e^2 + 14165/784*e - 25047/1568, -39/224*e^5 + 3/56*e^4 + 59/14*e^3 - 71/28*e^2 - 2339/112*e + 4817/224, 81/1568*e^5 + 43/392*e^4 - 81/98*e^3 - 195/196*e^2 + 2229/784*e - 12375/1568, -103/224*e^5 + 3/56*e^4 + 25/2*e^3 - 171/28*e^2 - 8931/112*e + 2247/32, -225/784*e^5 + 33/196*e^4 + 372/49*e^3 - 667/98*e^2 - 18213/392*e + 48487/784, 1/16*e^5 + 5/28*e^4 - 10/7*e^3 - 5/2*e^2 + 363/56*e + 335/112, 10/49*e^5 - 4/49*e^4 - 286/49*e^3 + 174/49*e^2 + 1884/49*e - 1870/49, -167/392*e^5 - 5/98*e^4 + 565/49*e^3 - 166/49*e^2 - 14407/196*e + 22185/392, 47/392*e^5 + 3/98*e^4 - 129/49*e^3 + 52/49*e^2 + 2851/196*e - 7249/392, 81/1568*e^5 - 13/392*e^4 - 165/98*e^3 + 197/196*e^2 + 10629/784*e - 11031/1568, 239/1568*e^5 - 19/392*e^4 - 407/98*e^3 + 499/196*e^2 + 19515/784*e - 39401/1568, -59/392*e^5 + 1/98*e^4 + 195/49*e^3 - 83/49*e^2 - 5107/196*e + 12013/392, 5/56*e^5 + 1/14*e^4 - 16/7*e^3 - 4/7*e^2 + 421/28*e + 205/56, -311/784*e^5 + 15/196*e^4 + 514/49*e^3 - 489/98*e^2 - 25323/392*e + 42785/784, 55/784*e^5 + 33/196*e^4 - 62/49*e^3 - 303/98*e^2 + 1611/392*e + 6431/784, -12/49*e^5 + 2/49*e^4 + 346/49*e^3 - 150/49*e^2 - 2286/49*e + 1712/49, -297/784*e^5 + 29/196*e^4 + 528/49*e^3 - 657/98*e^2 - 28333/392*e + 49519/784, -33/112*e^5 + 5/28*e^4 + 59/7*e^3 - 87/14*e^2 - 3029/56*e + 5783/112, 18/49*e^5 - 10/49*e^4 - 512/49*e^3 + 372/49*e^2 + 3464/49*e - 3212/49, 16/49*e^5 - 12/49*e^4 - 452/49*e^3 + 396/49*e^2 + 2866/49*e - 3174/49, -151/1568*e^5 + 27/392*e^4 + 319/98*e^3 - 631/196*e^2 - 19939/784*e + 34145/1568, -393/1568*e^5 + 5/392*e^4 + 659/98*e^3 - 317/196*e^2 - 31709/784*e + 25471/1568, 23/98*e^5 + 1/49*e^4 - 310/49*e^3 + 107/49*e^2 + 1972/49*e - 3643/98, -9/98*e^5 - 1/49*e^4 + 107/49*e^3 - 44/49*e^2 - 628/49*e + 1109/98, -8/49*e^5 - 1/49*e^4 + 233/49*e^3 - 51/49*e^2 - 1545/49*e + 1090/49, -17/784*e^5 - 55/196*e^4 - 32/49*e^3 + 589/98*e^2 + 6051/392*e - 21881/784, 529/784*e^5 - 41/196*e^4 - 872/49*e^3 + 995/98*e^2 + 41373/392*e - 81255/784, -499/1568*e^5 + 87/392*e^4 + 849/98*e^3 - 1775/196*e^2 - 45407/784*e + 104261/1568, 115/1568*e^5 + 97/392*e^4 - 199/98*e^3 - 589/196*e^2 + 11855/784*e - 22149/1568, -79/224*e^5 - 13/56*e^4 + 125/14*e^3 + 45/28*e^2 - 5627/112*e + 7401/224, 649/1568*e^5 - 53/392*e^4 - 1013/98*e^3 + 1305/196*e^2 + 46013/784*e - 90367/1568, -51/224*e^5 + 23/56*e^4 + 95/14*e^3 - 291/28*e^2 - 5007/112*e + 14181/224, -955/1568*e^5 + 127/392*e^4 + 1641/98*e^3 - 2463/196*e^2 - 85719/784*e + 171165/1568, -181/1568*e^5 - 103/392*e^4 + 265/98*e^3 + 639/196*e^2 - 10361/784*e + 7667/1568, 725/1568*e^5 - 97/392*e^4 - 1201/98*e^3 + 2073/196*e^2 + 58985/784*e - 147923/1568, 25/49*e^5 - 3/49*e^4 - 722/49*e^3 + 288/49*e^2 + 4773/49*e - 4031/49, -1237/1568*e^5 + 137/392*e^4 + 2119/98*e^3 - 3265/196*e^2 - 108985/784*e + 230451/1568, -107/1568*e^5 + 143/392*e^4 + 247/98*e^3 - 1187/196*e^2 - 14999/784*e + 12365/1568, 25/1568*e^5 - 181/392*e^4 + 3/98*e^3 + 2045/196*e^2 - 515/784*e - 61935/1568, -121/1568*e^5 - 179/392*e^4 + 261/98*e^3 + 1627/196*e^2 - 15181/784*e - 22257/1568, -681/784*e^5 + 45/196*e^4 + 1122/49*e^3 - 1355/98*e^2 - 54325/392*e + 125471/784, 31/112*e^5 + 9/28*e^4 - 44/7*e^3 - 43/14*e^2 + 1675/56*e - 409/112, -215/784*e^5 - 45/196*e^4 + 306/49*e^3 + 347/98*e^2 - 11307/392*e + 3777/784, 423/784*e^5 - 15/196*e^4 - 668/49*e^3 + 517/98*e^2 + 30195/392*e - 45809/784, -129/98*e^5 + 30/49*e^4 + 1718/49*e^3 - 1333/49*e^2 - 10693/49*e + 24445/98, 23/98*e^5 + 36/49*e^4 - 247/49*e^3 - 530/49*e^2 + 1111/49*e + 837/98, -355/1568*e^5 + 95/392*e^4 + 635/98*e^3 - 1011/196*e^2 - 34575/784*e + 45749/1568, -765/1568*e^5 + 129/392*e^4 + 1339/98*e^3 - 2601/196*e^2 - 68913/784*e + 179819/1568, 293/784*e^5 + 75/196*e^4 - 461/49*e^3 - 513/98*e^2 + 20217/392*e - 19427/784, -51/112*e^5 + 1/4*e^4 + 78/7*e^3 - 123/14*e^2 - 3447/56*e + 6277/112, -47/1568*e^5 + 67/392*e^4 + 145/98*e^3 - 591/196*e^2 - 8843/784*e + 18505/1568, -881/1568*e^5 + 37/392*e^4 + 1511/98*e^3 - 1545/196*e^2 - 78373/784*e + 141591/1568, 113/392*e^5 - 5/98*e^4 - 303/49*e^3 + 149/49*e^2 + 4437/196*e - 11639/392, -233/392*e^5 - 11/98*e^4 + 746/49*e^3 - 116/49*e^2 - 16245/196*e + 25343/392, -565/1568*e^5 + 25/392*e^4 + 929/98*e^3 - 549/196*e^2 - 44921/784*e + 48787/1568, -139/1568*e^5 + 79/392*e^4 + 321/98*e^3 - 1335/196*e^2 - 20791/784*e + 77037/1568, 773/1568*e^5 - 113/392*e^4 - 1333/98*e^3 + 1805/196*e^2 + 66441/784*e - 124643/1568, 731/1568*e^5 - 127/392*e^4 - 1333/98*e^3 + 2407/196*e^2 + 73623/784*e - 168253/1568, -43/392*e^5 - 23/98*e^4 + 100/49*e^3 + 236/49*e^2 - 475/196*e - 4867/392, 163/392*e^5 - 17/98*e^4 - 515/49*e^3 + 319/49*e^2 + 11275/196*e - 21605/392, 55/1568*e^5 + 173/392*e^4 - 97/98*e^3 - 1773/196*e^2 + 4131/784*e + 54815/1568, -311/1568*e^5 + 155/392*e^4 + 479/98*e^3 - 1959/196*e^2 - 22803/784*e + 91169/1568, 55/1568*e^5 + 5/392*e^4 - 153/98*e^3 + 579/196*e^2 + 12083/784*e - 66593/1568, -55/1568*e^5 + 51/392*e^4 + 139/98*e^3 - 1167/196*e^2 - 11075/784*e + 65249/1568, -27/196*e^5 + 23/49*e^4 + 185/49*e^3 - 507/49*e^2 - 2619/98*e + 7345/196, -61/196*e^5 + 25/49*e^4 + 405/49*e^3 - 699/49*e^2 - 4965/98*e + 15775/196, -5/8*e^5 + 5/14*e^4 + 120/7*e^3 - 17*e^2 - 3123/28*e + 7573/56, 53/392*e^5 + 29/98*e^4 - 176/49*e^3 - 55/49*e^2 + 4581/196*e - 20691/392, 533/784*e^5 + 135/196*e^4 - 813/49*e^3 - 677/98*e^2 + 35713/392*e - 32611/784, -1109/784*e^5 + 85/196*e^4 + 1802/49*e^3 - 2281/98*e^2 - 87049/392*e + 182211/784, -715/1568*e^5 + 47/392*e^4 + 1275/98*e^3 - 863/196*e^2 - 70391/784*e + 83725/1568, -853/1568*e^5 + 121/392*e^4 + 1525/98*e^3 - 2665/196*e^2 - 77113/784*e + 208595/1568, 55/392*e^5 - 37/98*e^4 - 285/49*e^3 + 530/49*e^2 + 10347/196*e - 32265/392, 55/56*e^5 - 3/14*e^4 - 183/7*e^3 + 82/7*e^2 + 4407/28*e - 7769/56, 393/1568*e^5 - 5/392*e^4 - 561/98*e^3 + 1297/196*e^2 + 23085/784*e - 111711/1568, -1481/1568*e^5 + 13/392*e^4 + 2531/98*e^3 - 2605/196*e^2 - 127389/784*e + 276191/1568, 789/1568*e^5 + 31/392*e^4 - 1349/98*e^3 + 605/196*e^2 + 73705/784*e - 90675/1568, -213/1568*e^5 - 55/392*e^4 + 311/98*e^3 + 295/196*e^2 - 18841/784*e + 8499/1568, 719/1568*e^5 - 67/392*e^4 - 1069/98*e^3 + 1347/196*e^2 + 46699/784*e - 82121/1568, -911/1568*e^5 - 149/392*e^4 + 1471/98*e^3 + 705/196*e^2 - 69019/784*e + 64713/1568, 127/196*e^5 - 5/49*e^4 - 830/49*e^3 + 305/49*e^2 + 9505/98*e - 14761/196, 9/196*e^5 - 10/49*e^4 - 141/49*e^3 + 316/49*e^2 + 3281/98*e - 8991/196, 101/112*e^5 - 9/28*e^4 - 169/7*e^3 + 181/14*e^2 + 1199/8*e - 13123/112, 27/112*e^5 - 15/28*e^4 - 57/7*e^3 + 215/14*e^2 + 3463/56*e - 10813/112, -117/98*e^5 + 29/49*e^4 + 1594/49*e^3 - 1356/49*e^2 - 10187/49*e + 23909/98, 1/14*e^5 + 4/7*e^4 - 8/7*e^3 - 48/7*e^2 + 20/7*e - 257/14, 1/784*e^5 - 5/196*e^4 + 27/49*e^3 - 369/98*e^2 - 3963/392*e + 25881/784, 25/112*e^5 - 5/28*e^4 - 48/7*e^3 + 141/14*e^2 + 411/8*e - 11727/112, 5/98*e^5 + 13/49*e^4 - 61/49*e^3 - 373/49*e^2 + 107/49*e + 3797/98, 13/98*e^5 + 10/49*e^4 - 223/49*e^3 - 29/49*e^2 + 1828/49*e - 3131/98, -411/784*e^5 - 17/196*e^4 + 642/49*e^3 - 437/98*e^2 - 29423/392*e + 75037/784, 587/784*e^5 + 5/196*e^4 - 958/49*e^3 + 761/98*e^2 + 46103/392*e - 70765/784, -31/112*e^5 - 5/28*e^4 + 50/7*e^3 - 13/14*e^2 - 349/8*e + 5913/112, 409/784*e^5 - 29/196*e^4 - 682/49*e^3 + 979/98*e^2 + 34381/392*e - 73711/784, -267/392*e^5 + 33/98*e^4 + 933/49*e^3 - 800/49*e^2 - 25087/196*e + 57461/392, -29/392*e^5 + 33/98*e^4 + 107/49*e^3 - 226/49*e^2 - 2141/196*e + 467/392, -849/784*e^5 + 157/196*e^4 + 1423/49*e^3 - 2867/98*e^2 - 72749/392*e + 167303/784, 1/16*e^5 + 25/28*e^4 - 8/7*e^3 - 31/2*e^2 + 275/56*e + 1983/112, 647/1568*e^5 - 211/392*e^4 - 1025/98*e^3 + 2435/196*e^2 + 43075/784*e - 100465/1568, 537/1568*e^5 - 221/392*e^4 - 1013/98*e^3 + 3433/196*e^2 + 62813/784*e - 177391/1568, 87/392*e^5 + 27/98*e^4 - 237/49*e^3 - 253/49*e^2 + 3791/196*e - 1681/392, -207/392*e^5 + 13/98*e^4 + 652/49*e^3 - 302/49*e^2 - 14591/196*e + 18745/392, 893/1568*e^5 - 209/392*e^4 - 1649/98*e^3 + 3977/196*e^2 + 93089/784*e - 270507/1568, 803/1568*e^5 - 151/392*e^4 - 1335/98*e^3 + 1711/196*e^2 + 63247/784*e - 100405/1568, 225/392*e^5 - 19/98*e^4 - 702/49*e^3 + 520/49*e^2 + 15329/196*e - 40983/392, -31/56*e^5 - 3/14*e^4 + 99/7*e^3 - 13/7*e^2 - 2183/28*e + 3849/56, 239/784*e^5 + 121/196*e^4 - 344/49*e^3 - 1069/98*e^2 + 14587/392*e + 13687/784, -719/784*e^5 + 67/196*e^4 + 1167/49*e^3 - 1445/98*e^2 - 57283/392*e + 102505/784, 67/392*e^5 + 99/98*e^4 - 127/49*e^3 - 909/49*e^2 + 227/196*e + 23507/392, -93/56*e^5 + 11/14*e^4 + 44*e^3 - 228/7*e^2 - 7589/28*e + 2373/8, 1/8*e^5 + 3/14*e^4 - 12/7*e^3 - 4*e^2 + 75/28*e + 607/56, -369/392*e^5 - 3/98*e^4 + 1235/49*e^3 - 353/49*e^2 - 30921/196*e + 42599/392, -239/784*e^5 - 37/196*e^4 + 421/49*e^3 - 303/98*e^2 - 21307/392*e + 61129/784, 271/784*e^5 - 67/196*e^4 - 453/49*e^3 + 1431/98*e^2 + 22507/392*e - 90409/784, 457/784*e^5 + 95/196*e^4 - 702/49*e^3 - 269/98*e^2 + 30637/392*e - 33407/784, -921/784*e^5 + 69/196*e^4 + 1502/49*e^3 - 1975/98*e^2 - 72229/392*e + 170351/784, 1277/1568*e^5 - 337/392*e^4 - 2215/98*e^3 + 4577/196*e^2 + 111969/784*e - 255179/1568, 229/224*e^5 - 7/8*e^4 - 405/14*e^3 + 829/28*e^2 + 21753/112*e - 52579/224, -339/784*e^5 + 15/196*e^4 + 430/49*e^3 - 447/98*e^2 - 12527/392*e + 24725/784, 963/784*e^5 + 57/196*e^4 - 1579/49*e^3 + 393/98*e^2 + 73727/392*e - 109829/784, -341/1568*e^5 + 305/392*e^4 + 649/98*e^3 - 2845/196*e^2 - 39545/784*e + 81491/1568, -1739/1568*e^5 + 351/392*e^4 + 2957/98*e^3 - 6383/196*e^2 - 148327/784*e + 409613/1568, 533/392*e^5 - 19/98*e^4 - 1794/49*e^3 + 1038/49*e^2 + 44309/196*e - 93987/392, -173/392*e^5 - 17/98*e^4 + 507/49*e^3 - 255/49*e^2 - 10397/196*e + 39211/392, -647/1568*e^5 - 293/392*e^4 + 759/98*e^3 + 2661/196*e^2 - 20003/784*e - 50511/1568, 3047/1568*e^5 - 115/392*e^4 - 5007/98*e^3 + 4323/196*e^2 + 243059/784*e - 397393/1568, 521/784*e^5 - 57/196*e^4 - 829/49*e^3 + 1301/98*e^2 + 37181/392*e - 111343/784, -313/784*e^5 - 59/196*e^4 + 481/49*e^3 + 151/98*e^2 - 19301/392*e + 15775/784, 1011/784*e^5 - 43/196*e^4 - 1690/49*e^3 + 1497/98*e^2 + 82807/392*e - 129221/784, 45/784*e^5 - 85/196*e^4 - 192/49*e^3 + 1035/98*e^2 + 16265/392*e - 38235/784, 2665/1568*e^5 - 333/392*e^4 - 4401/98*e^3 + 6905/196*e^2 + 221181/784*e - 478335/1568, -681/1568*e^5 - 459/392*e^4 + 1003/98*e^3 + 3643/196*e^2 - 44973/784*e - 30209/1568, -421/392*e^5 + 75/98*e^4 + 1507/49*e^3 - 1157/49*e^2 - 40193/196*e + 70411/392, -571/392*e^5 + 83/98*e^4 + 2010/49*e^3 - 1618/49*e^2 - 52195/196*e + 111173/392, 463/1568*e^5 + 93/392*e^4 - 841/98*e^3 + 751/196*e^2 + 44603/784*e - 151625/1568, -719/1568*e^5 + 179/392*e^4 + 1237/98*e^3 - 3895/196*e^2 - 64283/784*e + 245641/1568, -30/49*e^5 + 26/49*e^4 + 844/49*e^3 - 816/49*e^2 - 5526/49*e + 4546/49, 27/28*e^5 - 5/7*e^4 - 25*e^3 + 150/7*e^2 + 2075/14*e - 635/4, 23/196*e^5 - 45/49*e^4 - 232/49*e^3 + 1107/49*e^2 + 4387/98*e - 23425/196];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {9, 3, 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;