/*
  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![-98, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {6, 6, -5*w - 47}>;

primesArray := [
[2, 2, -17*w - 160],
[2, 2, -17*w + 177],
[3, 3, -842*w + 8767],
[7, 7, -2*w + 21],
[7, 7, 2*w + 19],
[13, 13, -12*w - 113],
[13, 13, 12*w - 125],
[17, 17, 182*w - 1895],
[17, 17, 182*w + 1713],
[23, 23, -512*w - 4819],
[23, 23, 512*w - 5331],
[25, 5, -5],
[29, 29, 22*w - 229],
[29, 29, 22*w + 207],
[43, 43, 114*w + 1073],
[43, 43, 114*w - 1187],
[47, 47, 8*w - 83],
[47, 47, -8*w - 75],
[61, 61, -1172*w - 11031],
[61, 61, 1172*w - 12203],
[71, 71, 216*w + 2033],
[71, 71, 216*w - 2249],
[83, 83, 8932*w - 93001],
[83, 83, -2196*w + 22865],
[109, 109, -4*w - 39],
[109, 109, 4*w - 43],
[121, 11, -11],
[131, 131, 5564*w - 57933],
[137, 137, 2*w - 17],
[137, 137, -2*w - 15],
[149, 149, 18*w - 187],
[149, 149, 18*w + 169],
[151, 151, -1502*w - 14137],
[151, 151, -1502*w + 15639],
[173, 173, -6*w - 55],
[173, 173, 6*w - 61],
[193, 193, 808*w - 8413],
[193, 193, -808*w - 7605],
[197, 197, 2*w - 15],
[197, 197, -2*w - 13],
[211, 211, -70*w + 729],
[211, 211, -70*w - 659],
[227, 227, 284*w + 2673],
[227, 227, 284*w - 2957],
[251, 251, 124*w - 1291],
[251, 251, 124*w + 1167],
[257, 257, 42*w + 395],
[257, 257, 42*w - 437],
[271, 271, 410*w + 3859],
[271, 271, 410*w - 4269],
[277, 277, 4*w - 45],
[277, 277, -4*w - 41],
[281, 281, -3550*w - 33413],
[281, 281, 3550*w - 36963],
[283, 283, 2*w - 27],
[283, 283, -2*w - 25],
[293, 293, 2*w - 11],
[293, 293, -2*w - 9],
[307, 307, -6*w - 59],
[307, 307, 6*w - 65],
[337, 337, 376*w - 3915],
[337, 337, 376*w + 3539],
[347, 347, -4*w - 33],
[347, 347, -4*w + 37],
[359, 359, -8*w - 73],
[359, 359, -8*w + 81],
[361, 19, -19],
[367, 367, -774*w + 8059],
[367, 367, -774*w - 7285],
[379, 379, -30*w + 313],
[379, 379, -30*w - 283],
[389, 389, 2*w - 3],
[389, 389, -2*w - 1],
[397, 397, -172*w + 1791],
[397, 397, -172*w - 1619],
[401, 401, 6*w - 59],
[401, 401, 6*w + 53],
[409, 409, -40*w - 377],
[409, 409, 40*w - 417],
[419, 419, 100*w - 1041],
[419, 419, -100*w - 941],
[421, 421, 308*w + 2899],
[421, 421, 308*w - 3207],
[439, 439, 274*w - 2853],
[439, 439, 274*w + 2579],
[443, 443, -52*w - 489],
[443, 443, 52*w - 541],
[449, 449, 15338*w - 159701],
[449, 449, -6918*w + 72031],
[457, 457, 17352*w - 180671],
[457, 457, 6224*w - 64805],
[461, 461, 614*w - 6393],
[461, 461, 614*w + 5779],
[479, 479, 192*w - 1999],
[479, 479, 192*w + 1807],
[487, 487, -94*w + 979],
[487, 487, -94*w - 885],
[491, 491, -20*w - 187],
[491, 491, 20*w - 207],
[503, 503, -16*w + 165],
[503, 503, 16*w + 149],
[509, 509, 2890*w - 30091],
[509, 509, -2890*w - 27201],
[521, 521, 11970*w - 124633],
[521, 521, -10286*w + 107099],
[577, 577, -1104*w - 10391],
[577, 577, 1104*w - 11495],
[593, 593, 134*w + 1261],
[593, 593, 134*w - 1395],
[601, 601, -8*w - 79],
[601, 601, 8*w - 87],
[613, 613, -2492*w - 23455],
[613, 613, 2492*w - 25947],
[617, 617, 226*w - 2353],
[617, 617, 226*w + 2127],
[631, 631, 2*w - 33],
[631, 631, -2*w - 31],
[677, 677, -62*w - 583],
[677, 677, 62*w - 645],
[691, 691, 1798*w - 18721],
[691, 691, -1798*w - 16923],
[739, 739, 74*w + 697],
[739, 739, 74*w - 771],
[743, 743, 48*w - 499],
[743, 743, 48*w + 451],
[757, 757, -44*w + 459],
[757, 757, -44*w - 415],
[761, 761, 14*w + 129],
[761, 761, 14*w - 143],
[769, 769, 16*w + 153],
[769, 769, 16*w - 169],
[773, 773, 4574*w - 47625],
[773, 773, -39938*w + 415839],
[787, 787, 3846*w + 36199],
[787, 787, 3846*w - 40045],
[809, 809, 30*w + 281],
[809, 809, -30*w + 311],
[811, 811, 1070*w - 11141],
[811, 811, -1070*w - 10071],
[829, 829, -4*w - 47],
[829, 829, 4*w - 51],
[857, 857, -18*w + 185],
[857, 857, 18*w + 167],
[877, 877, -196*w - 1845],
[877, 877, -196*w + 2041],
[881, 881, -26*w - 243],
[881, 881, 26*w - 269],
[907, 907, -14*w - 135],
[907, 907, -14*w + 149],
[937, 937, 8*w - 89],
[937, 937, -8*w - 81],
[941, 941, 294*w - 3061],
[941, 941, 294*w + 2767],
[947, 947, 4*w - 27],
[947, 947, -4*w - 23],
[961, 31, -31],
[971, 971, -1604*w + 16701],
[971, 971, 1604*w + 15097],
[983, 983, 144*w - 1499],
[983, 983, 144*w + 1355],
[991, 991, -10*w - 99],
[991, 991, 10*w - 109],
[997, 997, 604*w - 6289],
[997, 997, 604*w + 5685]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^10 + 2*x^9 - 14*x^8 - 24*x^7 + 71*x^6 + 89*x^5 - 162*x^4 - 96*x^3 + 154*x^2 - 12*x - 8;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1, 1, -3/4*e^9 - 1/2*e^8 + 19/2*e^7 + 4*e^6 - 149/4*e^5 - 19/4*e^4 + 85/2*e^3 - 11*e^2 + 7/2*e - 2, 3/2*e^9 + 5/2*e^8 - 18*e^7 - 27*e^6 + 135/2*e^5 + 85*e^4 - 167/2*e^3 - 70*e^2 + 28*e + 5, -3/4*e^9 + 1/2*e^8 + 21/2*e^7 - 8*e^6 - 185/4*e^5 + 157/4*e^4 + 123/2*e^3 - 58*e^2 + 11/2*e - 1, 3/4*e^9 - 5/2*e^8 - 23/2*e^7 + 34*e^6 + 213/4*e^5 - 585/4*e^4 - 125/2*e^3 + 197*e^2 - 91/2*e - 9, 1/4*e^9 + e^8 - 5/2*e^7 - 12*e^6 + 27/4*e^5 + 179/4*e^4 - 5*e^3 - 52*e^2 + 15/2*e + 4, 11/4*e^9 + 5/2*e^8 - 69/2*e^7 - 23*e^6 + 541/4*e^5 + 199/4*e^4 - 329/2*e^3 + 4*e^2 + 31/2*e - 1, -3*e^9 - 5/2*e^8 + 37*e^7 + 21*e^6 - 140*e^5 - 65/2*e^4 + 301/2*e^3 - 39*e^2 + 19*e + 1, -7/4*e^9 - 9/2*e^8 + 39/2*e^7 + 51*e^6 - 261/4*e^5 - 695/4*e^4 + 141/2*e^3 + 167*e^2 - 83/2*e - 5, 3/2*e^8 + e^7 - 19*e^6 - 8*e^5 + 151/2*e^4 + 21/2*e^3 - 93*e^2 + 19*e + 7, 7/4*e^9 + 3*e^8 - 41/2*e^7 - 32*e^6 + 289/4*e^5 + 389/4*e^4 - 72*e^3 - 69*e^2 + 11/2*e - 2, -e^9 - 5/2*e^8 + 11*e^7 + 28*e^6 - 35*e^5 - 187/2*e^4 + 57/2*e^3 + 89*e^2 - 3*e - 14, 1/2*e^9 + 5*e^8 - 2*e^7 - 61*e^6 - 29/2*e^5 + 463/2*e^4 + 52*e^3 - 271*e^2 + 18*e + 23, -2*e^9 - 9/2*e^8 + 22*e^7 + 50*e^6 - 70*e^5 - 331/2*e^4 + 117/2*e^3 + 153*e^2 - 15*e - 11, 3/4*e^9 + 1/2*e^8 - 19/2*e^7 - 4*e^6 + 153/4*e^5 + 23/4*e^4 - 99/2*e^3 + 6*e^2 + 7/2*e + 1, -11/4*e^9 - 3/2*e^8 + 71/2*e^7 + 10*e^6 - 581/4*e^5 + 13/4*e^4 + 377/2*e^3 - 72*e^2 - 29/2*e + 8, -7/4*e^9 - 6*e^8 + 35/2*e^7 + 69*e^6 - 177/4*e^5 - 957/4*e^4 + 9*e^3 + 239*e^2 - 3/2*e - 20, -2*e^9 + 3/2*e^8 + 28*e^7 - 24*e^6 - 125*e^5 + 235/2*e^4 + 353/2*e^3 - 177*e^2 - 5*e + 15, 2*e^9 + 11/2*e^8 - 21*e^7 - 61*e^6 + 59*e^5 + 399/2*e^4 - 41/2*e^3 - 178*e^2 - 27*e + 14, -15/4*e^9 - 11/2*e^8 + 91/2*e^7 + 58*e^6 - 689/4*e^5 - 699/4*e^4 + 421/2*e^3 + 128*e^2 - 111/2*e - 6, 7/4*e^9 + 5/2*e^8 - 43/2*e^7 - 26*e^6 + 333/4*e^5 + 307/4*e^4 - 211/2*e^3 - 55*e^2 + 49/2*e + 2, -3*e^8 - 2*e^7 + 38*e^6 + 16*e^5 - 150*e^4 - 19*e^3 + 182*e^2 - 45*e - 15, 3/2*e^9 - 4*e^8 - 24*e^7 + 54*e^6 + 243/2*e^5 - 455/2*e^4 - 193*e^3 + 292*e^2 + e - 9, -3*e^9 + e^8 + 41*e^7 - 21*e^6 - 178*e^5 + 123*e^4 + 246*e^3 - 210*e^2 - 25*e + 5, 3/2*e^9 - 19*e^7 + 5*e^6 + 143/2*e^5 - 89/2*e^4 - 62*e^3 + 94*e^2 - 45*e - 1, e^9 + 5/2*e^8 - 10*e^7 - 27*e^6 + 22*e^5 + 165/2*e^4 + 51/2*e^3 - 57*e^2 - 66*e + 5, 1/2*e^9 + 15/2*e^8 - 93*e^6 - 65/2*e^5 + 360*e^4 + 177/2*e^3 - 431*e^2 + 29*e + 42, -7/2*e^8 - 2*e^7 + 45*e^6 + 15*e^5 - 361/2*e^4 - 25/2*e^3 + 220*e^2 - 55*e - 11, 1/2*e^9 + 5/2*e^8 - 4*e^7 - 29*e^6 + 9/2*e^5 + 103*e^4 + 13/2*e^3 - 106*e^2 + 27*e - 13, -9/2*e^9 - 3*e^8 + 57*e^7 + 23*e^6 - 453/2*e^5 - 39/2*e^4 + 278*e^3 - 86*e^2 - 7*e - 7, -3/2*e^9 - 11/2*e^8 + 16*e^7 + 66*e^6 - 103/2*e^5 - 244*e^4 + 135/2*e^3 + 267*e^2 - 89*e - 7, 1/2*e^9 - 5*e^8 - 11*e^7 + 64*e^6 + 131/2*e^5 - 519/2*e^4 - 105*e^3 + 333*e^2 - 32*e - 24, 5/2*e^9 + 1/2*e^8 - 33*e^7 + 2*e^6 + 277/2*e^5 - 48*e^4 - 373/2*e^3 + 122*e^2 + 21*e - 15, 5/2*e^9 - 34*e^7 + 8*e^6 + 301/2*e^5 - 141/2*e^4 - 228*e^3 + 148*e^2 + 51*e - 15, 5*e^9 + 11/2*e^8 - 60*e^7 - 52*e^6 + 217*e^5 + 247/2*e^4 - 419/2*e^3 - 28*e^2 - 46*e + 11, 1/4*e^9 + 4*e^8 + 1/2*e^7 - 48*e^6 - 89/4*e^5 + 711/4*e^4 + 70*e^3 - 199*e^2 - 55/2*e + 8, 25/4*e^9 + 7/2*e^8 - 161/2*e^7 - 26*e^6 + 1311/4*e^5 + 73/4*e^4 - 845/2*e^3 + 104*e^2 + 81/2*e + 4, 5/2*e^9 + 6*e^8 - 27*e^7 - 65*e^6 + 163/2*e^5 + 407/2*e^4 - 49*e^3 - 156*e^2 - 11*e - 6, -13/4*e^9 - 8*e^8 + 69/2*e^7 + 89*e^6 - 395/4*e^5 - 1175/4*e^4 + 43*e^3 + 264*e^2 + 33/2*e - 14, 13/4*e^9 - 4*e^8 - 93/2*e^7 + 59*e^6 + 847/4*e^5 - 1081/4*e^4 - 301*e^3 + 379*e^2 - 21/2*e - 19, 23/4*e^9 - 4*e^8 - 161/2*e^7 + 67*e^6 + 1445/4*e^5 - 1371/4*e^4 - 521*e^3 + 539*e^2 + 67/2*e - 36, 3/2*e^9 - 9*e^8 - 28*e^7 + 119*e^6 + 315/2*e^5 - 991/2*e^4 - 262*e^3 + 644*e^2 - 40*e - 50, 7/4*e^9 + 4*e^8 - 39/2*e^7 - 46*e^6 + 261/4*e^5 + 649/4*e^4 - 74*e^3 - 172*e^2 + 113/2*e + 12, -11/4*e^9 - 29/2*e^8 + 45/2*e^7 + 172*e^6 - 97/4*e^5 - 2511/4*e^4 - 159/2*e^3 + 687*e^2 - 51/2*e - 49, 3/2*e^9 + 7/2*e^8 - 17*e^7 - 39*e^6 + 119/2*e^5 + 132*e^4 - 147/2*e^3 - 133*e^2 + 52*e + 2, -9/2*e^9 - 1/2*e^8 + 58*e^7 - 10*e^6 - 463/2*e^5 + 118*e^4 + 531/2*e^3 - 272*e^2 + 39*e + 31, 3/2*e^9 - 15/2*e^8 - 28*e^7 + 97*e^6 + 313/2*e^5 - 396*e^4 - 517/2*e^3 + 502*e^2 - 28*e - 11, -1/4*e^9 - 13*e^8 - 15/2*e^7 + 162*e^6 + 329/4*e^5 - 2503/4*e^4 - 167*e^3 + 724*e^2 - 167/2*e - 17, 9/4*e^9 - 61/2*e^7 + 5*e^6 + 531/4*e^5 - 173/4*e^4 - 189*e^3 + 92*e^2 + 37/2*e - 15, e^9 + 11/2*e^8 - 8*e^7 - 66*e^6 + 8*e^5 + 487/2*e^4 + 43/2*e^3 - 265*e^2 + 44*e + 5, 1/2*e^9 - 4*e^8 - 9*e^7 + 54*e^6 + 93/2*e^5 - 467/2*e^4 - 58*e^3 + 326*e^2 - 53*e - 35, 1/2*e^9 + 35/2*e^8 + 8*e^7 - 218*e^6 - 205/2*e^5 + 844*e^4 + 429/2*e^3 - 994*e^2 + 112*e + 57, 9/4*e^9 + e^8 - 61/2*e^7 - 7*e^6 + 543/4*e^5 + 7/4*e^4 - 213*e^3 + 35*e^2 + 129/2*e - 4, -5*e^9 - e^8 + 65*e^7 - 4*e^6 - 266*e^5 + 98*e^4 + 334*e^3 - 259*e^2 + 7*e + 35, 2*e^9 + 12*e^8 - 16*e^7 - 144*e^6 + 15*e^5 + 533*e^4 + 56*e^3 - 593*e^2 + 51*e + 34, -13/2*e^9 - 3*e^8 + 84*e^7 + 19*e^6 - 683/2*e^5 + 33/2*e^4 + 433*e^3 - 168*e^2 - 27*e + 19, -1/4*e^9 + 14*e^8 + 29/2*e^7 - 175*e^6 - 443/4*e^5 + 2713/4*e^4 + 193*e^3 - 795*e^2 + 203/2*e + 36, -7/4*e^9 - e^8 + 39/2*e^7 + 4*e^6 - 229/4*e^5 + 99/4*e^4 + 2*e^3 - 89*e^2 + 181/2*e - 2, 9*e^9 + 11/2*e^8 - 115*e^7 - 41*e^6 + 462*e^5 + 57/2*e^4 - 1145/2*e^3 + 162*e^2 + 11*e + 23, 1/4*e^9 - 5/2*e^8 - 9/2*e^7 + 33*e^6 + 75/4*e^5 - 555/4*e^4 + 11/2*e^3 + 185*e^2 - 157/2*e - 11, -3/4*e^9 - 11*e^8 + 1/2*e^7 + 135*e^6 + 163/4*e^5 - 2061/4*e^4 - 96*e^3 + 610*e^2 - 175/2*e - 53, 17/2*e^9 + 15/2*e^8 - 105*e^7 - 67*e^6 + 799/2*e^5 + 131*e^4 - 891/2*e^3 + 57*e^2 - 7*e - 16, -11/2*e^9 + 7/2*e^8 + 77*e^7 - 58*e^6 - 687/2*e^5 + 291*e^4 + 971/2*e^3 - 439*e^2 - 26*e + 6, -5*e^9 - 1/2*e^8 + 64*e^7 - 11*e^6 - 251*e^5 + 261/2*e^4 + 549/2*e^3 - 309*e^2 + 56*e + 40, 9/2*e^9 + 7/2*e^8 - 55*e^7 - 30*e^6 + 403/2*e^5 + 53*e^4 - 373/2*e^3 + 28*e^2 - 72*e + 17, 25/4*e^9 + 8*e^8 - 147/2*e^7 - 80*e^6 + 1023/4*e^5 + 867/4*e^4 - 223*e^3 - 112*e^2 - 119/2*e + 15, 3/2*e^9 + 2*e^8 - 19*e^7 - 20*e^6 + 155/2*e^5 + 107/2*e^4 - 112*e^3 - 21*e^2 + 58*e - 20, -4*e^9 + 7/2*e^8 + 55*e^7 - 57*e^6 - 237*e^5 + 569/2*e^4 + 599/2*e^3 - 438*e^2 + 60*e + 29, 13/4*e^9 + 9*e^8 - 69/2*e^7 - 101*e^6 + 403/4*e^5 + 1339/4*e^4 - 56*e^3 - 296*e^2 - 9/2*e + 17, 3*e^9 - 3/2*e^8 - 43*e^7 + 28*e^6 + 201*e^5 - 305/2*e^4 - 641/2*e^3 + 244*e^2 + 76*e - 1, 11/4*e^9 + 23*e^8 - 35/2*e^7 - 280*e^6 - 39/4*e^5 + 4229/4*e^4 + 77*e^3 - 1211*e^2 + 461/2*e + 54, 7/4*e^9 + 4*e^8 - 41/2*e^7 - 45*e^6 + 293/4*e^5 + 597/4*e^4 - 86*e^3 - 135*e^2 + 95/2*e + 16, 4*e^9 + 21/2*e^8 - 45*e^7 - 120*e^6 + 153*e^5 + 827/2*e^4 - 329/2*e^3 - 412*e^2 + 70*e + 37, 1/4*e^9 + 7*e^8 + 5/2*e^7 - 85*e^6 - 149/4*e^5 + 1263/4*e^4 + 74*e^3 - 341*e^2 + 137/2*e - 9, 5*e^9 + 21/2*e^8 - 58*e^7 - 116*e^6 + 207*e^5 + 755/2*e^4 - 473/2*e^3 - 326*e^2 + 77*e + 11, 1/2*e^9 - 5/2*e^8 - 10*e^7 + 31*e^6 + 125/2*e^5 - 118*e^4 - 269/2*e^3 + 138*e^2 + 57*e - 8, 17/4*e^9 - 10*e^8 - 129/2*e^7 + 136*e^6 + 1223/4*e^5 - 2317/4*e^4 - 426*e^3 + 761*e^2 - 167/2*e - 33, 33/4*e^9 + 1/2*e^8 - 213/2*e^7 + 21*e^6 + 1695/4*e^5 - 847/4*e^4 - 935/2*e^3 + 452*e^2 - 293/2*e - 17, 7*e^9 - 1/2*e^8 - 95*e^7 + 26*e^6 + 409*e^5 - 399/2*e^4 - 1103/2*e^3 + 403*e^2 + 15*e - 33, -9*e^9 - 37/2*e^8 + 105*e^7 + 204*e^6 - 376*e^5 - 1325/2*e^4 + 857/2*e^3 + 576*e^2 - 150*e - 27, 3/2*e^9 + 18*e^8 - 4*e^7 - 221*e^6 - 119/2*e^5 + 1685/2*e^4 + 180*e^3 - 983*e^2 + 81*e + 81, 25/4*e^9 + 18*e^8 - 135/2*e^7 - 206*e^6 + 855/4*e^5 + 2863/4*e^4 - 212*e^3 - 726*e^2 + 319/2*e + 38, -7/2*e^9 - 7*e^8 + 41*e^7 + 76*e^6 - 295/2*e^5 - 479/2*e^4 + 164*e^3 + 189*e^2 - 35*e + 4, 33/4*e^9 - 3/2*e^8 - 221/2*e^7 + 45*e^6 + 1879/4*e^5 - 1207/4*e^4 - 1255/2*e^3 + 561*e^2 + 45/2*e - 48, -13/4*e^9 + 6*e^8 + 103/2*e^7 - 82*e^6 - 1055/4*e^5 + 1409/4*e^4 + 451*e^3 - 471*e^2 - 173/2*e + 20, -1/2*e^9 + e^8 + 9*e^7 - 13*e^6 - 107/2*e^5 + 107/2*e^4 + 119*e^3 - 73*e^2 - 74*e + 19, -9/4*e^9 - 7/2*e^8 + 53/2*e^7 + 36*e^6 - 367/4*e^5 - 381/4*e^4 + 153/2*e^3 + 21*e^2 + 47/2*e + 25, -11/2*e^9 + 74*e^7 - 16*e^6 - 639/2*e^5 + 273/2*e^4 + 448*e^3 - 270*e^2 - 38*e - 1, 4*e^9 + 15*e^8 - 42*e^7 - 177*e^6 + 131*e^5 + 638*e^4 - 155*e^3 - 672*e^2 + 188*e + 20, -11/4*e^9 + 1/2*e^8 + 73/2*e^7 - 13*e^6 - 601/4*e^5 + 317/4*e^4 + 349/2*e^3 - 127*e^2 + 89/2*e - 17, 19/2*e^9 - 9/2*e^8 - 130*e^7 + 84*e^6 + 1131/2*e^5 - 456*e^4 - 1539/2*e^3 + 737*e^2 + 5*e - 38, -19/2*e^9 - 6*e^8 + 122*e^7 + 46*e^6 - 991/2*e^5 - 83/2*e^4 + 640*e^3 - 161*e^2 - 73*e - 6, -3/2*e^9 + 5*e^8 + 27*e^7 - 63*e^6 - 309/2*e^5 + 495/2*e^4 + 300*e^3 - 303*e^2 - 99*e + 29, -17/2*e^9 - 33/2*e^8 + 97*e^7 + 180*e^6 - 653/2*e^5 - 577*e^4 + 585/2*e^3 + 504*e^2 - 16*e - 51, -15/2*e^9 + 13/2*e^8 + 104*e^7 - 107*e^6 - 915/2*e^5 + 543*e^4 + 1237/2*e^3 - 855*e^2 + 48*e + 60, -12*e^9 - 21/2*e^8 + 150*e^7 + 95*e^6 - 585*e^5 - 399/2*e^4 + 1419/2*e^3 - 14*e^2 - 81*e - 27, -21/4*e^9 - 22*e^8 + 107/2*e^7 + 262*e^6 - 627/4*e^5 - 3835/4*e^4 + 175*e^3 + 1047*e^2 - 533/2*e - 64, -9/4*e^9 + 4*e^8 + 69/2*e^7 - 53*e^6 - 675/4*e^5 + 881/4*e^4 + 267*e^3 - 284*e^2 - 63/2*e + 16, -27/2*e^9 - 18*e^8 + 164*e^7 + 184*e^6 - 1233/2*e^5 - 1039/2*e^4 + 713*e^3 + 308*e^2 - 99*e - 28, 11/4*e^9 - 14*e^8 - 97/2*e^7 + 183*e^6 + 1033/4*e^5 - 3007/4*e^4 - 391*e^3 + 955*e^2 - 249/2*e - 42, -19/2*e^9 - 25/2*e^8 + 115*e^7 + 126*e^6 - 849/2*e^5 - 344*e^4 + 889/2*e^3 + 178*e^2 + 25*e - 29, 15/2*e^9 + 15/2*e^8 - 91*e^7 - 69*e^6 + 671/2*e^5 + 150*e^4 - 691/2*e^3 + 12*e^2 - 33*e - 12, 7*e^9 + 49/2*e^8 - 74*e^7 - 286*e^6 + 228*e^5 + 2035/2*e^4 - 465/2*e^3 - 1062*e^2 + 230*e + 63, 7/2*e^9 - 3/2*e^8 - 47*e^7 + 30*e^6 + 395/2*e^5 - 166*e^4 - 483/2*e^3 + 256*e^2 - 52*e + 14, 3*e^9 - e^8 - 39*e^7 + 24*e^6 + 157*e^5 - 146*e^4 - 178*e^3 + 252*e^2 - 45*e - 25, -21/4*e^9 + 15/2*e^8 + 157/2*e^7 - 108*e^6 - 1507/4*e^5 + 1955/4*e^4 + 1169/2*e^3 - 689*e^2 - 85/2*e + 41, 6*e^9 + 23*e^8 - 62*e^7 - 271*e^6 + 185*e^5 + 977*e^4 - 189*e^3 - 1045*e^2 + 222*e + 83, -9/2*e^9 - 13/2*e^8 + 55*e^7 + 67*e^6 - 417/2*e^5 - 188*e^4 + 491/2*e^3 + 96*e^2 - 45*e + 7, -23/4*e^9 - 15/2*e^8 + 145/2*e^7 + 77*e^6 - 1169/4*e^5 - 891/4*e^4 + 791/2*e^3 + 156*e^2 - 217/2*e - 34, -10*e^9 - 43/2*e^8 + 116*e^7 + 238*e^6 - 412*e^5 - 1561/2*e^4 + 921/2*e^3 + 706*e^2 - 152*e - 66, -35/4*e^9 - 6*e^8 + 221/2*e^7 + 49*e^6 - 1741/4*e^5 - 265/4*e^4 + 522*e^3 - 111*e^2 - 21/2*e - 21, -e^9 - 5/2*e^8 + 12*e^7 + 32*e^6 - 45*e^5 - 255/2*e^4 + 117/2*e^3 + 150*e^2 - 27*e - 1, -43/4*e^9 - 8*e^8 + 271/2*e^7 + 66*e^6 - 2121/4*e^5 - 353/4*e^4 + 619*e^3 - 174*e^2 + 21/2*e - 1, 13/2*e^9 + 37/2*e^8 - 72*e^7 - 214*e^6 + 473/2*e^5 + 749*e^4 - 479/2*e^3 - 760*e^2 + 139*e + 58, 1/4*e^9 + 13/2*e^8 + 5/2*e^7 - 82*e^6 - 165/4*e^5 + 1297/4*e^4 + 221/2*e^3 - 402*e^2 - 29/2*e + 45, -17/2*e^9 + 10*e^8 + 123*e^7 - 150*e^6 - 1133/2*e^5 + 1409/2*e^4 + 813*e^3 - 1024*e^2 + 32*e + 23, 10*e^9 - 2*e^8 - 133*e^7 + 57*e^6 + 553*e^5 - 373*e^4 - 677*e^3 + 670*e^2 - 97*e - 14, -17/2*e^9 - 15*e^8 + 100*e^7 + 161*e^6 - 719/2*e^5 - 1003/2*e^4 + 396*e^3 + 408*e^2 - 93*e - 48, -11/2*e^9 - 8*e^8 + 69*e^7 + 83*e^6 - 563/2*e^5 - 491/2*e^4 + 415*e^3 + 176*e^2 - 187*e + 10, -15/2*e^9 - e^8 + 100*e^7 - 11*e^6 - 853/2*e^5 + 303/2*e^4 + 590*e^3 - 339*e^2 - 78*e + 5, 19/2*e^9 + 43/2*e^8 - 106*e^7 - 238*e^6 + 695/2*e^5 + 785*e^4 - 631/2*e^3 - 723*e^2 + 82*e + 43, 1/2*e^9 + 4*e^8 - 2*e^7 - 46*e^6 - 35/2*e^5 + 313/2*e^4 + 81*e^3 - 143*e^2 - 47*e - 13, 17/2*e^8 + 6*e^7 - 106*e^6 - 43*e^5 + 833/2*e^4 + 37/2*e^3 - 509*e^2 + 201*e + 4, 9*e^9 + 21/2*e^8 - 111*e^7 - 104*e^6 + 427*e^5 + 543/2*e^4 - 1035/2*e^3 - 103*e^2 + 93*e + 4, -8*e^9 - 49/2*e^8 + 86*e^7 + 282*e^6 - 268*e^5 - 1963/2*e^4 + 509/2*e^3 + 978*e^2 - 208*e - 22, 5/4*e^9 + 30*e^8 + 13/2*e^7 - 374*e^6 - 521/4*e^5 + 5799/4*e^4 + 255*e^3 - 1707*e^2 + 569/2*e + 74, 13/4*e^9 + 19*e^8 - 61/2*e^7 - 230*e^6 + 323/4*e^5 + 3435/4*e^4 - 121*e^3 - 961*e^2 + 597/2*e + 38, -1/4*e^9 + 19*e^8 + 31/2*e^7 - 243*e^6 - 431/4*e^5 + 3889/4*e^4 + 110*e^3 - 1191*e^2 + 597/2*e + 44, 3*e^9 + 16*e^8 - 28*e^7 - 194*e^6 + 68*e^5 + 732*e^4 - 67*e^3 - 848*e^2 + 194*e + 46, 4*e^9 + e^8 - 48*e^7 + e^6 + 167*e^5 - 66*e^4 - 119*e^3 + 170*e^2 - 110*e + 10, -5/2*e^9 + 43/2*e^8 + 52*e^7 - 276*e^6 - 623/2*e^5 + 1106*e^4 + 1091/2*e^3 - 1362*e^2 + 59*e + 87, -9/4*e^9 + 19/2*e^8 + 77/2*e^7 - 127*e^6 - 819/4*e^5 + 2155/4*e^4 + 669/2*e^3 - 727*e^2 + 59/2*e + 72, -1/2*e^9 + 11/2*e^8 + 11*e^7 - 71*e^6 - 133/2*e^5 + 281*e^4 + 203/2*e^3 - 320*e^2 + 68*e - 11, 83/4*e^9 + 20*e^8 - 519/2*e^7 - 189*e^6 + 4065/4*e^5 + 1781/4*e^4 - 1257*e^3 - 76*e^2 + 389/2*e - 6, -7/2*e^9 + 17/2*e^8 + 54*e^7 - 118*e^6 - 529/2*e^5 + 515*e^4 + 797/2*e^3 - 694*e^2 + 40*e + 22, -13/4*e^9 - 6*e^8 + 67/2*e^7 + 62*e^6 - 335/4*e^5 - 735/4*e^4 - 33*e^3 + 139*e^2 + 291/2*e - 28, 19/4*e^9 - 37/2*e^8 - 155/2*e^7 + 246*e^6 + 1553/4*e^5 - 4109/4*e^4 - 1115/2*e^3 + 1328*e^2 - 301/2*e - 63, 37/2*e^9 + 19/2*e^8 - 238*e^7 - 62*e^6 + 1927/2*e^5 - 32*e^4 - 2425/2*e^3 + 474*e^2 + 64*e - 23, 63/4*e^9 + 18*e^8 - 387/2*e^7 - 176*e^6 + 2957/4*e^5 + 1825/4*e^4 - 874*e^3 - 204*e^2 + 209/2*e + 32, 25/4*e^9 + 23/2*e^8 - 147/2*e^7 - 127*e^6 + 1059/4*e^5 + 1653/4*e^4 - 603/2*e^3 - 355*e^2 + 221/2*e + 11, 5/4*e^9 + 21/2*e^8 - 21/2*e^7 - 130*e^6 + 115/4*e^5 + 1997/4*e^4 - 193/2*e^3 - 579*e^2 + 489/2*e + 20, -3*e^9 + 1/2*e^8 + 45*e^7 - 10*e^6 - 226*e^5 + 113/2*e^4 + 839/2*e^3 - 98*e^2 - 192*e + 25, -16*e^9 - 11*e^8 + 204*e^7 + 89*e^6 - 820*e^5 - 117*e^4 + 1041*e^3 - 204*e^2 - 100*e - 26, 3/2*e^9 - 39/2*e^8 - 34*e^7 + 249*e^6 + 399/2*e^5 - 994*e^4 - 551/2*e^3 + 1220*e^2 - 239*e - 61, 3/4*e^9 + 7*e^8 - 9/2*e^7 - 90*e^6 - 15/4*e^5 + 1465/4*e^4 + 10*e^3 - 467*e^2 + 217/2*e + 39, 9*e^9 + 23*e^8 - 100*e^7 - 263*e^6 + 330*e^5 + 907*e^4 - 341*e^3 - 890*e^2 + 206*e + 45, -11/4*e^9 - 69/2*e^8 + 11/2*e^7 + 422*e^6 + 495/4*e^5 - 6391/4*e^4 - 699/2*e^3 + 1823*e^2 - 357/2*e - 83, 23/4*e^9 + 29/2*e^8 - 123/2*e^7 - 159*e^6 + 709/4*e^5 + 2027/4*e^4 - 129/2*e^3 - 416*e^2 - 189/2*e + 18, -1/2*e^9 - 7/2*e^8 + e^7 + 41*e^6 + 55/2*e^5 - 150*e^4 - 215/2*e^3 + 169*e^2 + 59*e - 2, 27/4*e^9 + 19/2*e^8 - 159/2*e^7 - 96*e^6 + 1121/4*e^5 + 1043/4*e^4 - 541/2*e^3 - 120*e^2 - 5/2*e - 27, -11/2*e^9 + 15/2*e^8 + 76*e^7 - 114*e^6 - 657/2*e^5 + 537*e^4 + 821/2*e^3 - 777*e^2 + 106*e + 45, e^9 - 13/2*e^8 - 15*e^7 + 87*e^6 + 57*e^5 - 739/2*e^4 + 35/2*e^3 + 492*e^2 - 254*e - 19, -10*e^9 + 47/2*e^8 + 153*e^7 - 326*e^6 - 733*e^5 + 2859/2*e^4 + 2087/2*e^3 - 1962*e^2 + 193*e + 90, -7/2*e^9 - 11/2*e^8 + 42*e^7 + 60*e^6 - 313/2*e^5 - 200*e^4 + 375/2*e^3 + 212*e^2 - 47*e - 51, -13*e^9 + 1/2*e^8 + 173*e^7 - 46*e^6 - 726*e^5 + 751/2*e^4 + 1885/2*e^3 - 775*e^2 - 12*e + 75, 8*e^9 + 10*e^8 - 93*e^7 - 97*e^6 + 316*e^5 + 246*e^4 - 246*e^3 - 95*e^2 - 122*e + 12, 73/4*e^9 + 26*e^8 - 445/2*e^7 - 270*e^6 + 3399/4*e^5 + 3159/4*e^4 - 1052*e^3 - 539*e^2 + 531/2*e + 54, -29/4*e^9 - 25/2*e^8 + 169/2*e^7 + 133*e^6 - 1203/4*e^5 - 1629/4*e^4 + 673/2*e^3 + 313*e^2 - 221/2*e - 21, -21/4*e^9 - 23/2*e^8 + 125/2*e^7 + 132*e^6 - 943/4*e^5 - 1849/4*e^4 + 635/2*e^3 + 489*e^2 - 327/2*e - 49, 75/4*e^9 + 27/2*e^8 - 477/2*e^7 - 111*e^6 + 3833/4*e^5 + 635/4*e^4 - 2449/2*e^3 + 235*e^2 + 275/2*e - 21, 31/4*e^9 - 29/2*e^8 - 233/2*e^7 + 203*e^6 + 2201/4*e^5 - 3589/4*e^4 - 1557/2*e^3 + 1243*e^2 - 229/2*e - 71, -25/4*e^9 + 3*e^8 + 171/2*e^7 - 59*e^6 - 1487/4*e^5 + 1321/4*e^4 + 496*e^3 - 541*e^2 + 101/2*e + 18];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2, 2, -17*w + 177}>] := 1;
ALEigenvalues[ideal<ZF | {3, 3, -842*w + 8767}>] := -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;