/*
  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, -2, -6, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[2, 2, w + 1],
[3, 3, -w^3 + w^2 + 5*w - 2],
[11, 11, -w^3 + 5*w + 1],
[11, 11, -w + 2],
[13, 13, -w^3 + w^2 + 4*w + 1],
[17, 17, -w^3 + w^2 + 6*w - 1],
[17, 17, -w^2 - w + 3],
[23, 23, w^3 - 6*w],
[27, 3, w^3 + w^2 - 5*w - 4],
[41, 41, -w^3 + w^2 + 5*w],
[41, 41, 2*w^3 - 11*w - 4],
[53, 53, 2*w^3 - 2*w^2 - 11*w + 6],
[59, 59, 2*w^3 - 11*w - 2],
[67, 67, w^3 - 7*w - 1],
[71, 71, 3*w^3 - 2*w^2 - 15*w + 1],
[79, 79, -3*w^3 + w^2 + 16*w + 3],
[89, 89, -4*w^3 + 2*w^2 + 22*w - 3],
[101, 101, -2*w^3 + 13*w + 6],
[101, 101, w^3 + w^2 - 6*w - 3],
[101, 101, -3*w^3 + 2*w^2 + 17*w - 3],
[101, 101, w^3 - w^2 - 8*w - 3],
[107, 107, 2*w^3 - 3*w^2 - 6*w],
[109, 109, -w^3 + w^2 + 4*w - 3],
[113, 113, w^3 - 8*w - 4],
[113, 113, 7*w^3 - 4*w^2 - 39*w + 7],
[121, 11, w^3 - w^2 - 6*w - 1],
[127, 127, -2*w^3 + w^2 + 9*w + 3],
[127, 127, 2*w^3 - 13*w],
[131, 131, 6*w^3 - 3*w^2 - 34*w + 6],
[139, 139, w^2 - 2*w - 4],
[149, 149, 3*w^3 - w^2 - 18*w - 3],
[151, 151, -2*w^3 + 2*w^2 + 9*w - 4],
[163, 163, 3*w^3 - 2*w^2 - 16*w + 2],
[163, 163, 2*w^3 - 10*w + 1],
[167, 167, 3*w^3 - 18*w - 4],
[173, 173, -3*w^3 + w^2 + 18*w - 3],
[193, 193, 4*w^3 - 3*w^2 - 23*w + 9],
[197, 197, 3*w^3 - w^2 - 17*w + 2],
[197, 197, -3*w^3 - 2*w^2 + 15*w + 9],
[199, 199, 2*w^3 - w^2 - 8*w - 6],
[223, 223, -6*w^3 + 3*w^2 + 35*w - 7],
[227, 227, 2*w^3 - 12*w - 1],
[227, 227, -w^3 + w^2 + 7*w - 8],
[227, 227, 3*w^3 - w^2 - 18*w - 1],
[227, 227, -4*w^3 - 2*w^2 + 21*w + 14],
[229, 229, -w^3 + 2*w^2 + 3*w - 5],
[233, 233, w^3 - 3*w - 3],
[233, 233, -3*w^3 + 3*w^2 + 15*w - 8],
[241, 241, 4*w^2 - 7*w - 8],
[241, 241, w^3 + w^2 - 4*w - 5],
[251, 251, 2*w^3 - w^2 - 13*w - 1],
[251, 251, 2*w^3 - 9*w - 6],
[257, 257, w^3 - 8*w],
[263, 263, -w^3 + 2*w^2 + 5*w - 7],
[269, 269, 3*w^3 - 3*w^2 - 12*w - 1],
[271, 271, 2*w^3 - w^2 - 12*w - 2],
[271, 271, 9*w^3 - 5*w^2 - 50*w + 9],
[277, 277, w^3 - 2*w^2 - 6*w + 6],
[277, 277, w^3 + w^2 - 7*w - 2],
[281, 281, 4*w^3 - 2*w^2 - 24*w + 5],
[281, 281, -7*w^3 + 4*w^2 + 40*w - 12],
[289, 17, -5*w^3 + 5*w^2 + 26*w - 15],
[293, 293, -w^3 + 2*w^2 + 4*w - 4],
[293, 293, -9*w^3 + 5*w^2 + 51*w - 12],
[307, 307, -2*w^3 + 2*w^2 + 12*w - 7],
[311, 311, -4*w^3 + 24*w + 11],
[313, 313, 2*w^3 - 9*w - 4],
[317, 317, 2*w^3 - 4*w^2 - 2*w - 3],
[317, 317, 2*w^3 - 14*w - 7],
[331, 331, -3*w^3 + 19*w + 3],
[349, 349, -w^3 - 2*w^2 + 8*w + 8],
[349, 349, w^3 + w^2 - 5*w - 8],
[361, 19, -w^3 + w^2 + 7*w - 6],
[361, 19, w^3 + w^2 - 3*w - 4],
[373, 373, 2*w^3 - w^2 - 11*w - 3],
[379, 379, -4*w^3 + 25*w + 14],
[383, 383, -3*w^3 + w^2 + 15*w - 2],
[383, 383, 3*w^3 - 4*w^2 - 10*w],
[401, 401, 4*w^3 - w^2 - 24*w - 4],
[409, 409, w^2 + 5*w + 5],
[419, 419, w^2 - 2*w - 6],
[419, 419, -6*w^3 + 3*w^2 + 34*w - 8],
[433, 433, 3*w^3 - 16*w - 6],
[433, 433, 3*w^3 - w^2 - 16*w - 5],
[443, 443, -w^3 + 2*w^2 + w + 3],
[449, 449, 2*w^2 - 2*w - 5],
[457, 457, -4*w^3 + 4*w^2 + 20*w - 9],
[457, 457, w^2 + w - 7],
[463, 463, w^3 - 7*w - 7],
[463, 463, w^3 + 2*w^2 - 5*w - 9],
[487, 487, w^3 + 2*w^2 - 5*w - 5],
[487, 487, -w^3 + w^2 + 6*w - 7],
[509, 509, -2*w^3 - 2*w^2 + 13*w + 10],
[509, 509, 3*w^3 - 17*w - 3],
[509, 509, -5*w^3 + 3*w^2 + 30*w - 11],
[509, 509, -w^3 - 2*w^2 + 4*w + 6],
[521, 521, -2*w^3 - w^2 + 9*w + 7],
[541, 541, -2*w^3 - 2*w^2 + 12*w + 9],
[541, 541, -2*w^3 + 12*w + 9],
[547, 547, 2*w^3 + w^2 - 7*w - 3],
[547, 547, 5*w^3 - 3*w^2 - 30*w + 3],
[557, 557, w^3 + 2*w^2 - 5*w - 11],
[557, 557, -w^3 + w^2 + 3*w - 4],
[563, 563, 3*w^3 - w^2 - 15*w - 4],
[569, 569, 2*w^2 - w - 8],
[577, 577, 4*w^3 - 22*w - 7],
[593, 593, -3*w^3 + w^2 + 17*w + 4],
[607, 607, -w^3 - 3*w^2 + w + 4],
[613, 613, 3*w^3 - 2*w^2 - 14*w],
[613, 613, w^2 - 3*w - 5],
[619, 619, -w^3 + 2*w^2 + 4*w - 10],
[625, 5, -5],
[641, 641, -3*w^3 - 3*w^2 + 12*w + 7],
[643, 643, 2*w^3 + 2*w^2 - 14*w - 13],
[643, 643, 3*w^3 - 19*w - 7],
[647, 647, w^3 - 5*w - 7],
[653, 653, -3*w^3 + 2*w^2 + 14*w - 2],
[659, 659, -6*w^3 + 2*w^2 + 34*w - 1],
[659, 659, -2*w^3 + w^2 + 17*w + 9],
[683, 683, w^3 + 2*w^2 - 7*w - 7],
[691, 691, 3*w^3 - 15*w - 7],
[701, 701, w^3 + 2*w^2 - 6*w - 10],
[701, 701, -w^3 + 3*w^2 + 6*w - 13],
[701, 701, 5*w^3 - 2*w^2 - 30*w + 4],
[701, 701, 4*w^3 - 2*w^2 - 21*w + 2],
[727, 727, -2*w^3 - 3*w^2 + 9*w + 7],
[743, 743, 3*w - 4],
[751, 751, w^3 - 9*w - 3],
[757, 757, w^3 - 5*w^2 + 4*w + 7],
[757, 757, 4*w^3 - 5*w^2 - 11*w - 5],
[761, 761, 4*w^3 - 5*w^2 - 20*w + 16],
[769, 769, -3*w^3 + 16*w + 12],
[773, 773, 2*w^3 + 2*w^2 - 11*w - 10],
[773, 773, w^3 - 3*w - 5],
[773, 773, -w^3 + 2*w^2 + 9*w + 3],
[773, 773, 5*w^3 - w^2 - 27*w],
[787, 787, -4*w^3 + 3*w^2 + 22*w - 4],
[787, 787, -2*w^3 + w^2 + 14*w - 6],
[797, 797, 2*w^3 - w^2 - 14*w + 2],
[797, 797, -3*w^3 + 2*w^2 + 19*w - 7],
[811, 811, 2*w^3 + w^2 - 14*w - 12],
[811, 811, 2*w^3 + 2*w^2 - 8*w - 3],
[829, 829, -2*w^3 + 14*w + 11],
[857, 857, -3*w^3 + 3*w^2 + 14*w - 7],
[859, 859, 3*w^3 - w^2 - 15*w],
[859, 859, -11*w^3 + 7*w^2 + 60*w - 15],
[863, 863, 2*w^2 - 2*w - 13],
[863, 863, 2*w^3 + w^2 - 12*w - 4],
[877, 877, -6*w^3 - 2*w^2 + 32*w + 19],
[887, 887, 3*w^3 + w^2 - 14*w - 7],
[887, 887, 2*w^3 - w^2 - 15*w - 5],
[911, 911, 3*w^3 - 18*w - 2],
[919, 919, w^3 - w^2 - 6*w - 5],
[919, 919, -4*w^3 + w^2 + 24*w - 2],
[941, 941, 3*w^3 - 3*w^2 - 13*w],
[947, 947, 2*w^3 - 2*w^2 - 10*w - 1],
[947, 947, -3*w^3 + 2*w^2 + 14*w + 6],
[953, 953, -4*w^3 + 3*w^2 + 21*w - 3],
[953, 953, 4*w^3 - w^2 - 23*w + 1],
[961, 31, -2*w^3 + 3*w^2 + 12*w - 8],
[961, 31, 3*w^2 - 4*w - 6],
[967, 967, -5*w^3 - 3*w^2 + 23*w + 14],
[971, 971, 5*w^3 - w^2 - 27*w - 8],
[971, 971, -9*w^3 + 6*w^2 + 50*w - 12],
[983, 983, -w^3 + w^2 + 10*w + 3],
[997, 997, 2*w^3 + w^2 - 11*w - 3],
[997, 997, w^3 + 2*w^2 - 6*w - 8],
[997, 997, -5*w^3 + 4*w^2 + 29*w - 15]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^12 - 15*x^10 + 82*x^8 - 203*x^6 + 220*x^4 - 76*x^2 + 4;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1/2*e^11 + 13/2*e^9 - 29*e^7 + 111/2*e^5 - 45*e^3 + 13*e, -1/2*e^11 + 13/2*e^9 - 28*e^7 + 91/2*e^5 - 18*e^3 - 7*e, 2*e^10 - 25*e^8 + 102*e^6 - 157*e^4 + 70*e^2 - 8, 1/2*e^11 - 11/2*e^9 + 17*e^7 - 19/2*e^5 - 20*e^3 + 11*e, e^11 - 13*e^9 + 57*e^7 - 102*e^5 + 71*e^3 - 20*e, 1, -2*e^10 + 25*e^8 - 101*e^6 + 148*e^4 - 51*e^2, -e^9 + 12*e^7 - 46*e^5 + 64*e^3 - 20*e, -3/2*e^11 + 39/2*e^9 - 85*e^7 + 295/2*e^5 - 91*e^3 + 19*e, 2*e^11 - 28*e^9 + 138*e^7 - 293*e^5 + 249*e^3 - 54*e, -e^11 + 12*e^9 - 43*e^7 + 37*e^5 + 39*e^3 - 24*e, -2*e^10 + 26*e^8 - 115*e^6 + 211*e^4 - 146*e^2 + 20, -3*e^10 + 37*e^8 - 150*e^6 + 240*e^4 - 132*e^2 + 12, 5/2*e^11 - 61/2*e^9 + 120*e^7 - 349/2*e^5 + 60*e^3 + 19*e, 3*e^11 - 38*e^9 + 158*e^7 - 248*e^5 + 108*e^3 - 2*e, e^9 - 13*e^7 + 56*e^5 - 93*e^3 + 48*e, -3*e^10 + 37*e^8 - 144*e^6 + 185*e^4 - 19*e^2 - 6, 3*e^10 - 38*e^8 + 159*e^6 - 256*e^4 + 122*e^2 - 10, -3/2*e^11 + 31/2*e^9 - 39*e^7 - 33/2*e^5 + 116*e^3 - 41*e, -3*e^11 + 38*e^9 - 160*e^7 + 269*e^5 - 169*e^3 + 48*e, -3/2*e^11 + 45/2*e^9 - 122*e^7 + 587/2*e^5 - 291*e^3 + 63*e, e^11 - 11*e^9 + 31*e^7 + 9*e^5 - 103*e^3 + 50*e, -2*e^10 + 23*e^8 - 78*e^6 + 67*e^4 + 40*e^2 - 10, 1/2*e^11 - 7/2*e^9 - 7*e^7 + 161/2*e^5 - 131*e^3 + 23*e, 1/2*e^11 - 11/2*e^9 + 17*e^7 - 19/2*e^5 - 19*e^3 + 9*e, -e^11 + 14*e^9 - 69*e^7 + 146*e^5 - 120*e^3 + 16*e, 5*e^10 - 63*e^8 + 262*e^6 - 422*e^4 + 208*e^2 - 12, 4*e^10 - 47*e^8 + 171*e^6 - 207*e^4 + 30*e^2 + 4, -3*e^11 + 31*e^9 - 75*e^7 - 62*e^5 + 300*e^3 - 110*e, -e^11 + 12*e^9 - 46*e^7 + 65*e^5 - 27*e^3 + 8*e, -2*e^9 + 22*e^7 - 70*e^5 + 60*e^3 + 10*e, 2*e^11 - 29*e^9 + 148*e^7 - 319*e^5 + 258*e^3 - 32*e, 2*e^6 - 21*e^4 + 54*e^2 - 24, -4*e^10 + 49*e^8 - 191*e^6 + 260*e^4 - 67*e^2 - 4, 2*e^10 - 23*e^8 + 80*e^6 - 85*e^4 - 5*e^2 + 6, e^10 - 12*e^8 + 47*e^6 - 73*e^4 + 44*e^2 - 26, 4*e^10 - 47*e^8 + 170*e^6 - 196*e^4 + 14, -8*e^10 + 98*e^8 - 390*e^6 + 591*e^4 - 268*e^2 + 6, -e^10 + 10*e^8 - 21*e^6 - 33*e^4 + 102*e^2 - 24, -8*e^10 + 98*e^8 - 386*e^6 + 552*e^4 - 181*e^2 - 8, e^10 - 12*e^8 + 45*e^6 - 60*e^4 + 31*e^2 - 20, 5*e^10 - 62*e^8 + 251*e^6 - 387*e^4 + 170*e^2, -6*e^10 + 75*e^8 - 305*e^6 + 462*e^4 - 187*e^2, -3*e^10 + 38*e^8 - 159*e^6 + 259*e^4 - 142*e^2 + 28, 8*e^11 - 103*e^9 + 446*e^7 - 782*e^5 + 495*e^3 - 80*e, -3*e^10 + 34*e^8 - 117*e^6 + 131*e^4 - 16*e^2 + 2, -9/2*e^11 + 105/2*e^9 - 185*e^7 + 371/2*e^5 + 73*e^3 - 65*e, e^11 - 11*e^9 + 31*e^7 + 8*e^5 - 91*e^3 + 14*e, -e^10 + 10*e^8 - 24*e^6 - 3*e^4 + 26*e^2 + 6, -e^10 + 15*e^8 - 80*e^6 + 182*e^4 - 155*e^2 + 20, 9*e^10 - 111*e^8 + 445*e^6 - 674*e^4 + 298*e^2 - 24, -7*e^10 + 86*e^8 - 340*e^6 + 487*e^4 - 152*e^2 - 14, -11/2*e^11 + 137/2*e^9 - 280*e^7 + 889/2*e^5 - 237*e^3 + 47*e, -3/2*e^11 + 41/2*e^9 - 95*e^7 + 347/2*e^5 - 101*e^3 + 3*e, -3/2*e^11 + 33/2*e^9 - 51*e^7 + 59/2*e^5 + 52*e^3 - 27*e, 3/2*e^11 - 31/2*e^9 + 43*e^7 - 45/2*e^5 - 15*e^3 - 19*e, e^11 - 15*e^9 + 84*e^7 - 223*e^5 + 271*e^3 - 102*e, 4*e^10 - 48*e^8 + 182*e^6 - 243*e^4 + 70*e^2 - 6, e^8 - 15*e^6 + 73*e^4 - 122*e^2 + 18, -7*e^10 + 90*e^8 - 385*e^6 + 645*e^4 - 351*e^2 + 26, 3*e^11 - 33*e^9 + 99*e^7 - 32*e^5 - 157*e^3 + 38*e, -2*e^11 + 26*e^9 - 117*e^7 + 228*e^5 - 169*e^3 - 4*e, -10*e^10 + 126*e^8 - 516*e^6 + 775*e^4 - 292*e^2 + 10, -2*e^10 + 24*e^8 - 90*e^6 + 110*e^4 - 6*e^2, 3*e^11 - 44*e^9 + 232*e^7 - 542*e^5 + 526*e^3 - 116*e, -5*e^11 + 69*e^9 - 332*e^7 + 678*e^5 - 537*e^3 + 94*e, -5*e^8 + 54*e^6 - 171*e^4 + 169*e^2 - 18, 7*e^10 - 87*e^8 + 356*e^6 - 572*e^4 + 304*e^2 - 22, -5*e^10 + 58*e^8 - 207*e^6 + 244*e^4 - 26*e^2 - 20, 3*e^11 - 44*e^9 + 227*e^7 - 494*e^5 + 413*e^3 - 78*e, 2*e^8 - 21*e^6 + 61*e^4 - 42*e^2 - 6, 7*e^8 - 76*e^6 + 243*e^4 - 250*e^2 + 42, -13*e^10 + 163*e^8 - 673*e^6 + 1077*e^4 - 542*e^2 + 46, -1/2*e^11 + 21/2*e^9 - 78*e^7 + 493/2*e^5 - 310*e^3 + 101*e, 1/2*e^11 - 19/2*e^9 + 65*e^7 - 387/2*e^5 + 239*e^3 - 83*e, -5/2*e^11 + 71/2*e^9 - 179*e^7 + 791/2*e^5 - 364*e^3 + 87*e, 3/2*e^11 - 31/2*e^9 + 36*e^7 + 89/2*e^5 - 180*e^3 + 75*e, 4*e^10 - 53*e^8 + 237*e^6 - 421*e^4 + 252*e^2 - 18, 7/2*e^11 - 79/2*e^9 + 124*e^7 - 93/2*e^5 - 226*e^3 + 115*e, -2*e^11 + 25*e^9 - 100*e^7 + 138*e^5 - 26*e^3 - 10*e, 4*e^10 - 51*e^8 + 219*e^6 - 384*e^4 + 242*e^2 - 40, -12*e^11 + 154*e^9 - 663*e^7 + 1151*e^5 - 723*e^3 + 136*e, 11/2*e^11 - 141/2*e^9 + 305*e^7 - 1093/2*e^5 + 398*e^3 - 125*e, 11*e^10 - 135*e^8 + 532*e^6 - 756*e^4 + 237*e^2 + 12, -13/2*e^11 + 165/2*e^9 - 347*e^7 + 1137/2*e^5 - 287*e^3 - 5*e, 11/2*e^11 - 147/2*e^9 + 335*e^7 - 1247/2*e^5 + 418*e^3 - 57*e, 12*e^10 - 146*e^8 + 566*e^6 - 777*e^4 + 216*e^2 + 6, 6*e^10 - 77*e^8 + 328*e^6 - 542*e^4 + 290*e^2 - 52, 3/2*e^11 - 31/2*e^9 + 36*e^7 + 93/2*e^5 - 188*e^3 + 57*e, 9*e^11 - 112*e^9 + 453*e^7 - 684*e^5 + 290*e^3 - 40*e, -3*e^10 + 39*e^8 - 167*e^6 + 264*e^4 - 94*e^2 - 20, -11/2*e^11 + 155/2*e^9 - 388*e^7 + 1707/2*e^5 - 773*e^3 + 165*e, 19*e^10 - 237*e^8 + 967*e^6 - 1505*e^4 + 704*e^2 - 58, -6*e^10 + 74*e^8 - 294*e^6 + 423*e^4 - 136*e^2 - 10, 8*e^11 - 98*e^9 + 387*e^7 - 564*e^5 + 221*e^3 - 34*e, 4*e^10 - 44*e^8 + 137*e^6 - 89*e^4 - 112*e^2 + 34, 1/2*e^11 + 5/2*e^9 - 82*e^7 + 761/2*e^5 - 549*e^3 + 133*e, 11*e^10 - 136*e^8 + 547*e^6 - 830*e^4 + 370*e^2 - 58, -2*e^11 + 27*e^9 - 123*e^7 + 224*e^5 - 162*e^3 + 88*e, -9*e^10 + 110*e^8 - 429*e^6 + 592*e^4 - 165*e^2 - 8, -3*e^10 + 35*e^8 - 126*e^6 + 151*e^4 - 31*e^2 + 14, 7/2*e^11 - 103/2*e^9 + 275*e^7 - 1325/2*e^5 + 684*e^3 - 181*e, 4*e^11 - 49*e^9 + 197*e^7 - 314*e^5 + 180*e^3 - 36*e, -4*e^9 + 53*e^7 - 234*e^5 + 391*e^3 - 174*e, 3/2*e^11 - 39/2*e^9 + 85*e^7 - 301/2*e^5 + 113*e^3 - 53*e, 3*e^11 - 34*e^9 + 110*e^7 - 69*e^5 - 111*e^3 + 30*e, -23/2*e^11 + 303/2*e^9 - 679*e^7 + 2501/2*e^5 - 859*e^3 + 157*e, 11/2*e^11 - 149/2*e^9 + 346*e^7 - 1319/2*e^5 + 456*e^3 - 61*e, -2*e^9 + 24*e^7 - 91*e^5 + 115*e^3 + 8*e, e^11 - 17*e^9 + 105*e^7 - 282*e^5 + 292*e^3 - 22*e, -9*e^10 + 111*e^8 - 443*e^6 + 654*e^4 - 236*e^2 - 34, 2*e^10 - 21*e^8 + 64*e^6 - 68*e^4 + 36*e^2 - 18, 9/2*e^11 - 109/2*e^9 + 209*e^7 - 567/2*e^5 + 105*e^3 - 53*e, -3*e^10 + 40*e^8 - 181*e^6 + 331*e^4 - 220*e^2 + 28, 16*e^10 - 199*e^8 + 803*e^6 - 1201*e^4 + 480*e^2 - 32, -7/2*e^11 + 73/2*e^9 - 91*e^7 - 119/2*e^5 + 323*e^3 - 93*e, 5*e^10 - 63*e^8 + 255*e^6 - 366*e^4 + 122*e^2 - 28, -2*e^10 + 26*e^8 - 116*e^6 + 222*e^4 - 172*e^2 + 28, 6*e^11 - 87*e^9 + 449*e^7 - 1008*e^5 + 914*e^3 - 190*e, 3*e^11 - 39*e^9 + 171*e^7 - 298*e^5 + 150*e^3 + 42*e, e^11 - 12*e^9 + 47*e^7 - 73*e^5 + 33*e^3 + 32*e, 17*e^10 - 207*e^8 + 807*e^6 - 1139*e^4 + 387*e^2 - 34, 17*e^10 - 213*e^8 + 875*e^6 - 1372*e^4 + 636*e^2 - 46, 2*e^11 - 26*e^9 + 113*e^7 - 195*e^5 + 117*e^3 - 2*e, 7*e^11 - 98*e^9 + 482*e^7 - 1021*e^5 + 886*e^3 - 230*e, -3*e^10 + 34*e^8 - 121*e^6 + 167*e^4 - 86*e^2, 5*e^10 - 62*e^8 + 245*e^6 - 330*e^4 + 53*e^2 - 4, -13/2*e^11 + 183/2*e^9 - 458*e^7 + 2017/2*e^5 - 913*e^3 + 195*e, -4*e^10 + 47*e^8 - 174*e^6 + 231*e^4 - 72*e^2 - 2, 5*e^7 - 42*e^5 + 71*e^3 + 18*e, 20*e^10 - 251*e^8 + 1029*e^6 - 1590*e^4 + 718*e^2 - 90, 1/2*e^11 - 13/2*e^9 + 33*e^7 - 185/2*e^5 + 126*e^3 - 35*e, -10*e^10 + 117*e^8 - 423*e^6 + 506*e^4 - 62*e^2 - 14, 1/2*e^11 - 11/2*e^9 + 15*e^7 + 25/2*e^5 - 82*e^3 + 41*e, -8*e^10 + 99*e^8 - 402*e^6 + 634*e^4 - 312*e^2 - 2, 17/2*e^11 - 195/2*e^9 + 326*e^7 - 491/2*e^5 - 290*e^3 + 141*e, 6*e^10 - 70*e^8 + 252*e^6 - 300*e^4 + 52*e^2 - 20, -11*e^10 + 144*e^8 - 631*e^6 + 1079*e^4 - 580*e^2 + 42, -6*e^10 + 74*e^8 - 294*e^6 + 429*e^4 - 172*e^2 + 14, 6*e^11 - 75*e^9 + 308*e^7 - 491*e^5 + 272*e^3 - 86*e, -13*e^10 + 157*e^8 - 605*e^6 + 846*e^4 - 293*e^2 + 8, -12*e^10 + 148*e^8 - 588*e^6 + 845*e^4 - 264*e^2 - 26, 11/2*e^11 - 153/2*e^9 + 375*e^7 - 1593/2*e^5 + 684*e^3 - 151*e, -5*e^11 + 64*e^9 - 274*e^7 + 465*e^5 - 240*e^3 - 40*e, 7*e^11 - 83*e^9 + 305*e^7 - 364*e^5 + 20*e^3 + 26*e, 1/2*e^11 - 9/2*e^9 + 169/2*e^5 - 199*e^3 + 89*e, 12*e^10 - 152*e^8 + 632*e^6 - 992*e^4 + 429*e^2 + 4, 7*e^11 - 90*e^9 + 388*e^7 - 673*e^5 + 423*e^3 - 86*e, -19/2*e^11 + 217/2*e^9 - 361*e^7 + 543/2*e^5 + 323*e^3 - 169*e, -4*e^10 + 42*e^8 - 116*e^6 + 31*e^4 + 130*e^2 - 4, 11*e^10 - 136*e^8 + 541*e^6 - 777*e^4 + 268*e^2 - 44, 9*e^10 - 109*e^8 + 416*e^6 - 536*e^4 + 69*e^2 + 20, 6*e^10 - 68*e^8 + 232*e^6 - 244*e^4 - 12*e^2 + 32, 3/2*e^11 - 53/2*e^9 + 166*e^7 - 863/2*e^5 + 404*e^3 - 49*e, -6*e^9 + 70*e^7 - 261*e^5 + 382*e^3 - 194*e, -9*e^11 + 120*e^9 - 553*e^7 + 1087*e^5 - 872*e^3 + 222*e, -e^11 + 2*e^9 + 77*e^7 - 415*e^5 + 621*e^3 - 156*e, 4*e^10 - 43*e^8 + 126*e^6 - 52*e^4 - 152*e^2 + 38, -10*e^11 + 133*e^9 - 605*e^7 + 1144*e^5 - 853*e^3 + 232*e, -5/2*e^11 + 69/2*e^9 - 178*e^7 + 915/2*e^5 - 584*e^3 + 253*e, 10*e^11 - 128*e^9 + 544*e^7 - 902*e^5 + 480*e^3 - 54*e, -27/2*e^11 + 351/2*e^9 - 768*e^7 + 2709/2*e^5 - 871*e^3 + 187*e, -5/2*e^11 + 69/2*e^9 - 169*e^7 + 735/2*e^5 - 327*e^3 + 35*e, 8*e^10 - 98*e^8 + 374*e^6 - 452*e^4 + 2*e^2 + 16, -11*e^10 + 136*e^8 - 537*e^6 + 743*e^4 - 183*e^2 - 38, -5/2*e^11 + 41/2*e^9 + e^7 - 581/2*e^5 + 563*e^3 - 147*e, 3*e^10 - 37*e^8 + 146*e^6 - 200*e^4 + 28*e^2 + 26];
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;