/*
  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, 7, 11, -2, -7, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {29, 29, w^5 - w^4 - 7*w^3 + 4*w^2 + 11*w}>;

primesArray := [
[5, 5, -2*w^5 + w^4 + 13*w^3 - 2*w^2 - 19*w - 5],
[9, 3, 2*w^5 - w^4 - 14*w^3 + 2*w^2 + 23*w + 6],
[11, 11, w - 1],
[25, 5, w^3 + w^2 - 4*w - 3],
[29, 29, w^5 - w^4 - 7*w^3 + 4*w^2 + 11*w],
[41, 41, w^4 - w^3 - 5*w^2 + 3*w + 3],
[49, 7, w^5 - w^4 - 7*w^3 + 4*w^2 + 11*w + 1],
[59, 59, 2*w^5 - w^4 - 14*w^3 + 2*w^2 + 24*w + 7],
[59, 59, -w^5 + 8*w^3 + 2*w^2 - 15*w - 8],
[61, 61, w^5 - 7*w^3 - 2*w^2 + 12*w + 4],
[61, 61, -w^5 + 7*w^3 - 11*w - 1],
[64, 2, 2],
[71, 71, w^3 + w^2 - 5*w - 3],
[71, 71, -3*w^5 + w^4 + 21*w^3 - w^2 - 33*w - 9],
[79, 79, -2*w^5 + w^4 + 13*w^3 - 3*w^2 - 19*w - 5],
[81, 3, 2*w^5 - w^4 - 13*w^3 + w^2 + 19*w + 8],
[89, 89, 2*w^5 - w^4 - 13*w^3 + 2*w^2 + 20*w + 7],
[89, 89, -w^5 + 8*w^3 + w^2 - 16*w - 5],
[89, 89, -3*w^5 + w^4 + 20*w^3 - 30*w - 11],
[89, 89, -w^5 + 7*w^3 + 2*w^2 - 11*w - 4],
[89, 89, -2*w^5 + 14*w^3 + 3*w^2 - 22*w - 8],
[89, 89, 2*w^5 - w^4 - 14*w^3 + 3*w^2 + 22*w + 6],
[101, 101, 3*w^5 - 2*w^4 - 18*w^3 + 6*w^2 + 22*w + 6],
[101, 101, w^2 - w - 4],
[101, 101, -3*w^5 + w^4 + 20*w^3 - 2*w^2 - 29*w - 7],
[109, 109, w^4 - w^3 - 5*w^2 + 4*w + 5],
[121, 11, w^5 - 7*w^3 - 2*w^2 + 10*w + 6],
[131, 131, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 4*w - 3],
[131, 131, -2*w^5 + w^4 + 13*w^3 - 3*w^2 - 20*w - 6],
[131, 131, 3*w^5 - w^4 - 21*w^3 + w^2 + 35*w + 11],
[131, 131, -w^5 + w^4 + 6*w^3 - 4*w^2 - 7*w + 2],
[139, 139, -w^5 + 6*w^3 + w^2 - 8*w - 1],
[151, 151, -w^3 + w^2 + 4*w - 2],
[179, 179, w^4 - 6*w^2 + 6],
[181, 181, w^3 + w^2 - 3*w],
[191, 191, 3*w^5 - w^4 - 20*w^3 + 2*w^2 + 28*w + 5],
[199, 199, -2*w^5 + w^4 + 12*w^3 - 3*w^2 - 15*w - 2],
[199, 199, 5*w^5 - w^4 - 34*w^3 - w^2 + 52*w + 15],
[211, 211, 3*w^5 - w^4 - 19*w^3 + w^2 + 26*w + 9],
[229, 229, -w^5 + w^4 + 7*w^3 - 3*w^2 - 11*w],
[229, 229, -w^4 + 2*w^3 + 6*w^2 - 7*w - 6],
[239, 239, -w^5 + w^4 + 7*w^3 - 3*w^2 - 13*w - 6],
[239, 239, 3*w^5 - w^4 - 21*w^3 + w^2 + 35*w + 10],
[239, 239, w^3 - w^2 - 4*w + 1],
[239, 239, 5*w^5 - 2*w^4 - 34*w^3 + 2*w^2 + 53*w + 17],
[241, 241, -w^5 + 7*w^3 + w^2 - 13*w - 4],
[241, 241, -3*w^5 + w^4 + 21*w^3 - 35*w - 10],
[241, 241, -w^5 + 7*w^3 - 11*w],
[251, 251, 3*w^5 - w^4 - 21*w^3 + 35*w + 11],
[251, 251, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 8*w - 3],
[269, 269, -w^5 + 8*w^3 + 3*w^2 - 16*w - 8],
[269, 269, -2*w^5 + w^4 + 13*w^3 - w^2 - 21*w - 9],
[269, 269, 3*w^5 - 21*w^3 - 3*w^2 + 32*w + 12],
[271, 271, -5*w^5 + 3*w^4 + 34*w^3 - 8*w^2 - 53*w - 14],
[281, 281, -w^5 + 8*w^3 + 3*w^2 - 16*w - 11],
[281, 281, 4*w^5 - w^4 - 29*w^3 + 49*w + 13],
[281, 281, w^5 - w^4 - 7*w^3 + 3*w^2 + 11*w + 5],
[281, 281, -5*w^5 + 3*w^4 + 33*w^3 - 8*w^2 - 50*w - 12],
[311, 311, 2*w^5 - w^4 - 14*w^3 + w^2 + 21*w + 9],
[311, 311, -3*w^5 + 20*w^3 + 3*w^2 - 30*w - 10],
[311, 311, 5*w^5 - 2*w^4 - 35*w^3 + 2*w^2 + 57*w + 19],
[331, 331, -w^5 - w^4 + 7*w^3 + 6*w^2 - 11*w - 7],
[349, 349, -6*w^5 + 3*w^4 + 41*w^3 - 7*w^2 - 64*w - 17],
[349, 349, 4*w^5 - w^4 - 29*w^3 - w^2 + 50*w + 17],
[349, 349, 5*w^5 - w^4 - 34*w^3 - w^2 + 52*w + 17],
[359, 359, w^5 - 6*w^3 - w^2 + 5*w + 3],
[379, 379, -w^5 + w^4 + 8*w^3 - 3*w^2 - 17*w - 3],
[379, 379, 2*w^5 - 14*w^3 - 4*w^2 + 22*w + 11],
[379, 379, -5*w^5 + 3*w^4 + 34*w^3 - 8*w^2 - 53*w - 13],
[379, 379, 4*w^5 - 2*w^4 - 27*w^3 + 3*w^2 + 43*w + 14],
[389, 389, 4*w^5 - 2*w^4 - 28*w^3 + 4*w^2 + 44*w + 13],
[401, 401, -2*w^5 + 2*w^4 + 13*w^3 - 6*w^2 - 20*w - 6],
[409, 409, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 9*w],
[419, 419, -5*w^5 + w^4 + 34*w^3 + 2*w^2 - 53*w - 19],
[419, 419, 2*w^5 - w^4 - 14*w^3 + 4*w^2 + 23*w + 3],
[421, 421, -w^5 - w^4 + 8*w^3 + 6*w^2 - 13*w - 7],
[421, 421, 2*w^5 - 2*w^4 - 13*w^3 + 8*w^2 + 18*w + 2],
[421, 421, 4*w^5 - w^4 - 26*w^3 + 37*w + 11],
[431, 431, 2*w^5 - w^4 - 13*w^3 + 2*w^2 + 20*w + 9],
[431, 431, w^5 - 8*w^3 - w^2 + 14*w + 2],
[449, 449, 4*w^5 - 2*w^4 - 26*w^3 + 4*w^2 + 38*w + 9],
[449, 449, -3*w^5 + w^4 + 20*w^3 - 31*w - 14],
[461, 461, 3*w^5 - 2*w^4 - 18*w^3 + 6*w^2 + 23*w + 4],
[461, 461, -4*w^5 + 2*w^4 + 26*w^3 - 5*w^2 - 36*w - 9],
[461, 461, -4*w^5 + 2*w^4 + 27*w^3 - 3*w^2 - 42*w - 15],
[491, 491, -3*w^5 + w^4 + 20*w^3 - 29*w - 10],
[491, 491, -5*w^5 + w^4 + 35*w^3 + 2*w^2 - 56*w - 18],
[491, 491, 2*w^5 - 14*w^3 - 4*w^2 + 24*w + 13],
[491, 491, -3*w^5 + w^4 + 21*w^3 - 2*w^2 - 33*w - 10],
[499, 499, 6*w^5 - 3*w^4 - 40*w^3 + 6*w^2 + 61*w + 18],
[499, 499, -4*w^5 + w^4 + 29*w^3 + w^2 - 49*w - 17],
[499, 499, -w^5 - w^4 + 8*w^3 + 5*w^2 - 14*w - 4],
[499, 499, -3*w^5 + w^4 + 19*w^3 - 2*w^2 - 26*w - 8],
[509, 509, -4*w^5 + 2*w^4 + 27*w^3 - 5*w^2 - 43*w - 11],
[509, 509, -2*w^5 + 16*w^3 + 2*w^2 - 30*w - 9],
[521, 521, 3*w^5 - 21*w^3 - 3*w^2 + 34*w + 10],
[521, 521, -3*w^5 + w^4 + 19*w^3 - 26*w - 9],
[521, 521, 4*w^5 - w^4 - 27*w^3 + 40*w + 15],
[521, 521, -3*w^5 + 22*w^3 + 3*w^2 - 37*w - 11],
[541, 541, -4*w^5 + w^4 + 29*w^3 + w^2 - 49*w - 18],
[569, 569, -6*w^5 + 2*w^4 + 40*w^3 - 2*w^2 - 60*w - 19],
[569, 569, -w^5 + w^4 + 7*w^3 - 2*w^2 - 12*w - 6],
[571, 571, 4*w^5 - w^4 - 28*w^3 + 43*w + 14],
[571, 571, -4*w^5 + 2*w^4 + 28*w^3 - 4*w^2 - 45*w - 14],
[599, 599, -w^5 + w^4 + 7*w^3 - 2*w^2 - 13*w - 7],
[601, 601, 4*w^5 - w^4 - 26*w^3 + 38*w + 13],
[601, 601, 4*w^5 - 2*w^4 - 28*w^3 + 5*w^2 + 44*w + 11],
[601, 601, 3*w^5 - w^4 - 21*w^3 + 2*w^2 + 35*w + 9],
[619, 619, 4*w^5 - 2*w^4 - 27*w^3 + 4*w^2 + 40*w + 13],
[619, 619, -4*w^5 + w^4 + 26*w^3 - 36*w - 12],
[619, 619, w^4 - w^3 - 6*w^2 + 4*w + 4],
[631, 631, -3*w^5 + w^4 + 20*w^3 - 30*w - 9],
[631, 631, -5*w^5 + 2*w^4 + 35*w^3 - 3*w^2 - 57*w - 15],
[641, 641, 5*w^5 - w^4 - 34*w^3 - w^2 + 52*w + 16],
[641, 641, -5*w^5 + 2*w^4 + 34*w^3 - 2*w^2 - 52*w - 18],
[659, 659, 3*w^5 - w^4 - 20*w^3 + w^2 + 32*w + 9],
[659, 659, -4*w^5 + w^4 + 27*w^3 - w^2 - 40*w - 11],
[661, 661, 4*w^5 - w^4 - 26*w^3 + 38*w + 12],
[661, 661, -w^5 + 6*w^3 - 8*w + 1],
[691, 691, 3*w^5 - 20*w^3 - 3*w^2 + 29*w + 9],
[691, 691, w^5 + w^4 - 8*w^3 - 7*w^2 + 14*w + 9],
[691, 691, -w^5 + w^4 + 6*w^3 - 3*w^2 - 7*w + 2],
[701, 701, -2*w^5 + w^4 + 13*w^3 - 2*w^2 - 19*w - 9],
[701, 701, 5*w^5 - 2*w^4 - 33*w^3 + 3*w^2 + 48*w + 14],
[701, 701, w^4 + 3*w^3 - 3*w^2 - 10*w - 1],
[701, 701, -2*w^5 + w^4 + 14*w^3 - 3*w^2 - 24*w - 3],
[709, 709, -2*w^5 + 2*w^4 + 14*w^3 - 7*w^2 - 22*w - 5],
[709, 709, -3*w^5 + 21*w^3 + 4*w^2 - 34*w - 11],
[719, 719, 5*w^5 - 2*w^4 - 35*w^3 + 3*w^2 + 58*w + 17],
[719, 719, 4*w^5 - 2*w^4 - 26*w^3 + 4*w^2 + 37*w + 11],
[719, 719, 4*w^5 - 2*w^4 - 27*w^3 + 4*w^2 + 40*w + 12],
[719, 719, 2*w^5 - 16*w^3 - 3*w^2 + 30*w + 10],
[739, 739, -5*w^5 + w^4 + 34*w^3 + w^2 - 51*w - 17],
[751, 751, -w^5 - w^4 + 9*w^3 + 7*w^2 - 18*w - 10],
[761, 761, 4*w^5 - 2*w^4 - 28*w^3 + 3*w^2 + 46*w + 16],
[761, 761, -4*w^5 + 2*w^4 + 27*w^3 - 5*w^2 - 43*w - 12],
[769, 769, -4*w^5 + w^4 + 28*w^3 - 44*w - 12],
[769, 769, -w^4 + w^3 + 3*w^2 - 3*w + 1],
[809, 809, 2*w^5 - 2*w^4 - 13*w^3 + 7*w^2 + 20*w + 1],
[811, 811, -2*w^5 + 15*w^3 + 2*w^2 - 24*w - 8],
[811, 811, -w^5 + 9*w^3 + w^2 - 18*w - 6],
[821, 821, -w^5 + 8*w^3 + 3*w^2 - 17*w - 9],
[821, 821, w^5 + w^4 - 7*w^3 - 6*w^2 + 11*w + 8],
[821, 821, 4*w^5 - 2*w^4 - 28*w^3 + 4*w^2 + 46*w + 11],
[821, 821, 5*w^5 - 2*w^4 - 34*w^3 + 4*w^2 + 51*w + 13],
[829, 829, -2*w^5 + w^4 + 15*w^3 - w^2 - 28*w - 10],
[829, 829, -6*w^5 + 3*w^4 + 42*w^3 - 7*w^2 - 68*w - 21],
[829, 829, 4*w^5 - 2*w^4 - 26*w^3 + 5*w^2 + 38*w + 12],
[829, 829, w^5 - 7*w^3 - 3*w^2 + 11*w + 6],
[839, 839, 3*w^5 - 2*w^4 - 20*w^3 + 5*w^2 + 32*w + 11],
[839, 839, -w^4 + w^3 + 6*w^2 - 3*w - 4],
[841, 29, -3*w^5 + w^4 + 22*w^3 - 38*w - 11],
[911, 911, 3*w^5 - w^4 - 21*w^3 + 36*w + 12],
[919, 919, -w^5 - w^4 + 7*w^3 + 6*w^2 - 12*w - 5],
[941, 941, -w^5 + w^4 + 7*w^3 - 5*w^2 - 11*w],
[941, 941, 4*w^5 - w^4 - 29*w^3 + 48*w + 15],
[971, 971, -4*w^5 + w^4 + 27*w^3 + w^2 - 41*w - 13],
[991, 991, w^5 - w^4 - 8*w^3 + 3*w^2 + 16*w + 2],
[991, 991, -4*w^5 + 3*w^4 + 27*w^3 - 9*w^2 - 42*w - 10]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^12 - x^11 - 41*x^10 + 37*x^9 + 601*x^8 - 440*x^7 - 4061*x^6 + 1788*x^5 + 13740*x^4 - 1460*x^3 - 21000*x^2 - 3360*x + 8208;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -24364/31197*e^11 + 21631/124788*e^10 + 3629441/124788*e^9 - 726895/124788*e^8 - 44872501/124788*e^7 + 6232775/124788*e^6 + 56085119/31197*e^5 + 3576683/41596*e^4 - 37558511/10399*e^3 - 23561650/31197*e^2 + 18388130/10399*e - 1489588/10399, -4141/124788*e^11 + 61499/62394*e^10 + 130709/124788*e^9 - 4568209/124788*e^8 - 1144471/124788*e^7 + 14037047/31197*e^6 + 971912/31197*e^5 - 92268097/41596*e^4 - 15411595/41596*e^3 + 271883863/62394*e^2 + 13448873/10399*e - 20867077/10399, 17231/62394*e^11 - 57811/31197*e^10 - 1241489/124788*e^9 + 8566099/124788*e^8 + 14518783/124788*e^7 - 104527277/124788*e^6 - 69672209/124788*e^5 + 42001080/10399*e^4 + 70953959/41596*e^3 - 486047551/62394*e^2 - 30230652/10399*e + 39069782/10399, -1, -156209/124788*e^11 + 145115/124788*e^10 + 5811061/124788*e^9 - 5297309/124788*e^8 - 71747333/124788*e^7 + 15372478/31197*e^6 + 359944831/124788*e^5 - 40575007/20798*e^4 - 127838159/20798*e^3 + 93765979/31197*e^2 + 43117335/10399*e - 23235640/10399, -1323/41596*e^11 + 165227/41596*e^10 + 15603/41596*e^9 - 6134975/41596*e^8 + 484793/41596*e^7 + 18855366/10399*e^6 - 5225129/41596*e^5 - 93225068/10399*e^4 - 10174985/10399*e^3 + 183761773/10399*e^2 + 51503319/10399*e - 84057404/10399, 59221/124788*e^11 + 166321/62394*e^10 - 1109911/62394*e^9 - 3123890/31197*e^8 + 13805477/62394*e^7 + 157447717/124788*e^6 - 132749783/124788*e^5 - 274185977/41596*e^4 + 20592793/20798*e^3 + 425902816/31197*e^2 + 32750587/10399*e - 64171584/10399, -508003/249576*e^11 - 347897/249576*e^10 + 18951005/249576*e^9 + 13412291/249576*e^8 - 234807121/249576*e^7 - 90574757/124788*e^6 + 1169355305/249576*e^5 + 94253623/20798*e^4 - 361059775/41596*e^3 - 330917801/31197*e^2 + 19731577/10399*e + 38350496/10399, -7685/20798*e^11 + 190239/41596*e^10 + 264281/20798*e^9 - 1760754/10399*e^8 - 2822173/20798*e^7 + 42992571/20798*e^6 + 24032495/41596*e^5 - 417456449/41596*e^4 - 103633057/41596*e^3 + 202334719/10399*e^2 + 64343599/10399*e - 94146755/10399, -102497/124788*e^11 - 108293/62394*e^10 + 1924007/62394*e^9 + 2037085/31197*e^8 - 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186568449/10399, -165785/62394*e^11 + 47179/124788*e^10 + 12357545/124788*e^9 - 1533763/124788*e^8 - 152969029/124788*e^7 + 10521281/124788*e^6 + 382868875/62394*e^5 + 27576305/41596*e^4 - 127530025/10399*e^3 - 99790669/31197*e^2 + 60391406/10399*e - 2505770/10399, -206999/62394*e^11 + 389167/124788*e^10 + 15339155/124788*e^9 - 14112937/124788*e^8 - 187996375/124788*e^7 + 161720795/124788*e^6 + 232664525/31197*e^5 - 206846767/41596*e^4 - 324335267/20798*e^3 + 225970844/31197*e^2 + 105215629/10399*e - 54673746/10399, 245129/41596*e^11 + 123933/41596*e^10 - 4574509/20798*e^9 - 2411495/20798*e^8 + 56737575/20798*e^7 + 66758343/41596*e^6 - 141739822/10399*e^5 - 220251517/20798*e^4 + 1072130955/41596*e^3 + 266244508/10399*e^2 - 75675489/10399*e - 85957631/10399, 512597/124788*e^11 - 1220213/124788*e^10 - 18854317/124788*e^9 + 44906573/124788*e^8 + 228296525/124788*e^7 - 134479834/31197*e^6 - 1119531973/124788*e^5 + 203561263/10399*e^4 + 214834068/10399*e^3 - 1106877340/31197*e^2 - 207119690/10399*e + 193992598/10399, 118367/124788*e^11 + 134309/62394*e^10 - 4609105/124788*e^9 - 9832183/124788*e^8 + 61309343/124788*e^7 + 59518837/62394*e^6 - 169393013/62394*e^5 - 198977565/41596*e^4 + 224352855/41596*e^3 + 591518341/62394*e^2 - 11811249/10399*e - 33015759/10399, -104809/62394*e^11 + 1856027/124788*e^10 + 7537795/124788*e^9 - 68946317/124788*e^8 - 88014611/124788*e^7 + 846247243/124788*e^6 + 107404081/31197*e^5 - 1378350695/41596*e^4 - 248635273/20798*e^3 + 2019277627/31197*e^2 + 241152592/10399*e - 323103486/10399, -1089541/249576*e^11 + 2098027/249576*e^10 + 40206629/249576*e^9 - 77107897/249576*e^8 - 489600865/249576*e^7 + 114894283/31197*e^6 + 2416272611/249576*e^5 - 682357221/41596*e^4 - 903813851/41596*e^3 + 909129241/31197*e^2 + 197594476/10399*e - 166505550/10399, 429419/124788*e^11 - 379039/62394*e^10 - 3947272/31197*e^9 + 13856695/62394*e^8 + 47742401/31197*e^7 - 326633143/124788*e^6 - 930013063/124788*e^5 + 471437881/41596*e^4 + 169251453/10399*e^3 - 606760720/31197*e^2 - 139220262/10399*e + 111311716/10399, 63469/62394*e^11 + 139033/31197*e^10 - 4698799/124788*e^9 - 20998363/124788*e^8 + 57193649/124788*e^7 + 267204941/124788*e^6 - 266485027/124788*e^5 - 118533581/10399*e^4 + 86778945/41596*e^3 + 1496791669/62394*e^2 + 57827957/10399*e - 113814100/10399, 16640/31197*e^11 - 132881/31197*e^10 - 1139405/62394*e^9 + 9782251/62394*e^8 + 11974069/62394*e^7 - 118004147/62394*e^6 - 46143989/62394*e^5 + 92989168/10399*e^4 + 47718727/20798*e^3 - 524721901/31197*e^2 - 52304384/10399*e + 78543472/10399, -7063/31197*e^11 + 1584937/124788*e^10 + 784973/124788*e^9 - 58935379/124788*e^8 - 4431841/124788*e^7 + 726231899/124788*e^6 + 2932381/62394*e^5 - 1200485307/41596*e^4 - 88879651/20798*e^3 + 1780024076/31197*e^2 + 174697929/10399*e - 274347328/10399, 411049/249576*e^11 - 1043959/249576*e^10 - 14853257/249576*e^9 + 37955701/249576*e^8 + 173891173/249576*e^7 - 55585219/31197*e^6 - 800941367/249576*e^5 + 322920613/41596*e^4 + 289244667/41596*e^3 - 416977801/31197*e^2 - 67446660/10399*e + 68274996/10399, 151975/124788*e^11 - 198965/62394*e^10 - 2849779/62394*e^9 + 3711346/31197*e^8 + 35765639/62394*e^7 - 182498225/124788*e^6 - 374517917/124788*e^5 + 291211293/41596*e^4 + 156851923/20798*e^3 - 420735353/31197*e^2 - 83466920/10399*e + 77129192/10399, 179487/41596*e^11 - 44817/10399*e^10 - 6650771/41596*e^9 + 6513153/41596*e^8 + 81535051/41596*e^7 - 37519279/20798*e^6 - 101038350/10399*e^5 + 294142853/41596*e^4 + 851243645/41596*e^3 - 110870188/10399*e^2 - 142006674/10399*e + 77671897/10399, 764437/249576*e^11 + 295745/249576*e^10 - 28560341/249576*e^9 - 11521079/249576*e^8 + 354934825/249576*e^7 + 20209916/31197*e^6 - 1781653943/249576*e^5 - 186232701/41596*e^4 + 572137593/41596*e^3 + 345223058/31197*e^2 - 48645483/10399*e - 32628292/10399, -79729/41596*e^11 - 60562/10399*e^10 + 3001939/41596*e^9 + 9085135/41596*e^8 - 37723927/41596*e^7 - 28638186/10399*e^6 + 94426975/20798*e^5 + 606414697/41596*e^4 - 284733081/41596*e^3 - 316943101/10399*e^2 - 44331078/10399*e + 134817459/10399, -7609/20798*e^11 - 69045/41596*e^10 + 559617/41596*e^9 + 2610861/41596*e^8 - 6736069/41596*e^7 - 33297383/41596*e^6 + 15392997/20798*e^5 + 177635711/41596*e^4 - 6832074/10399*e^3 - 93372992/10399*e^2 - 22153013/10399*e + 42777408/10399, 167321/124788*e^11 + 418855/31197*e^10 - 1610935/31197*e^9 - 31294907/62394*e^8 + 20912312/31197*e^7 + 779734043/124788*e^6 - 420173935/124788*e^5 - 1325399103/41596*e^4 + 37654455/20798*e^3 + 2015338760/31197*e^2 + 164678257/10399*e - 303635800/10399, -386963/41596*e^11 + 945/41596*e^10 + 14402481/41596*e^9 + 302483/41596*e^8 - 177819369/41596*e^7 - 7624783/20798*e^6 + 884910419/41596*e^5 + 123791561/20798*e^4 - 432933767/10399*e^3 - 198849569/10399*e^2 + 179580862/10399*e + 34421776/10399, 452869/83192*e^11 - 97179/83192*e^10 - 16889481/83192*e^9 + 3289653/83192*e^8 + 209335597/83192*e^7 - 7172795/20798*e^6 - 1051062359/83192*e^5 - 22070953/41596*e^4 + 1062059471/41596*e^3 + 51446056/10399*e^2 - 132354132/10399*e + 13645784/10399, 1134157/124788*e^11 + 122753/124788*e^10 - 21149551/62394*e^9 - 1365401/31197*e^8 + 131035114/31197*e^7 + 99001357/124788*e^6 - 1311624937/62394*e^5 - 81424335/10399*e^4 + 1713337037/41596*e^3 + 1392788165/62394*e^2 - 172534977/10399*e - 47364208/10399, -13151/31197*e^11 + 43393/62394*e^10 + 489682/31197*e^9 - 807977/31197*e^8 - 6058616/31197*e^7 + 9833248/31197*e^6 + 61299425/62394*e^5 - 30015147/20798*e^4 - 46101109/20798*e^3 + 81714337/31197*e^2 + 19667361/10399*e - 15214308/10399, 224437/62394*e^11 + 885047/62394*e^10 - 4252600/31197*e^9 - 16560472/31197*e^8 + 54001877/31197*e^7 + 415311265/62394*e^6 - 273427972/31197*e^5 - 361092444/10399*e^4 + 259410341/20798*e^3 + 2237331752/31197*e^2 + 121512607/10399*e - 321211940/10399, -165146/31197*e^11 + 468794/31197*e^10 + 24249673/124788*e^9 - 69115811/124788*e^8 - 292728983/124788*e^7 + 831668863/124788*e^6 + 1434255103/124788*e^5 - 639299903/20798*e^4 - 1139302871/41596*e^3 + 3537672491/62394*e^2 + 301913277/10399*e - 303653619/10399, -245557/31197*e^11 - 2459/31197*e^10 + 9165290/31197*e^9 + 260540/31197*e^8 - 113735062/31197*e^7 - 9526438/31197*e^6 + 571099514/31197*e^5 + 49329529/10399*e^4 - 377775909/10399*e^3 - 469176658/31197*e^2 + 165888981/10399*e + 18797272/10399, 2294801/249576*e^11 - 537359/249576*e^10 - 85400821/249576*e^9 + 18069833/249576*e^8 + 1054429601/249576*e^7 - 39148927/62394*e^6 - 5258531659/249576*e^5 - 36674475/41596*e^4 + 1754255945/41596*e^3 + 274021774/31197*e^2 - 212495143/10399*e + 16628364/10399, 337679/41596*e^11 + 66453/41596*e^10 - 12558507/41596*e^9 - 2807081/41596*e^8 + 154800419/41596*e^7 + 11392033/10399*e^6 - 766666287/41596*e^5 - 96004267/10399*e^4 + 365706910/10399*e^3 + 260804318/10399*e^2 - 124275574/10399*e - 73512700/10399, 763597/124788*e^11 - 41389/124788*e^10 - 28349387/124788*e^9 + 739123/124788*e^8 + 348428479/124788*e^7 + 8264915/62394*e^6 - 1720335923/124788*e^5 - 37092092/10399*e^4 + 555830165/20798*e^3 + 383180098/31197*e^2 - 111809415/10399*e - 24316150/10399, 181735/31197*e^11 - 149273/62394*e^10 - 27009383/124788*e^9 + 10455565/124788*e^8 + 332561089/124788*e^7 - 107022353/124788*e^6 - 1652768963/124788*e^5 + 17302776/10399*e^4 + 1111959351/41596*e^3 + 82184435/62394*e^2 - 144882428/10399*e + 29438692/10399, 173421/41596*e^11 - 262349/20798*e^10 - 1586499/10399*e^9 + 9666435/20798*e^8 + 19041944/10399*e^7 - 232593873/41596*e^6 - 369979889/41596*e^5 + 1074511965/41596*e^4 + 221568287/10399*e^3 - 496127341/10399*e^2 - 242424354/10399*e + 252067426/10399, 103283/20798*e^11 + 33935/20798*e^10 - 3878577/20798*e^9 - 1303439/20798*e^8 + 48644901/20798*e^7 + 9035930/10399*e^6 - 248251031/20798*e^5 - 63328578/10399*e^4 + 247045548/10399*e^3 + 156704142/10399*e^2 - 100391444/10399*e - 33574818/10399, -1089203/124788*e^11 + 395671/62394*e^10 + 20169245/62394*e^9 - 7085666/31197*e^8 - 246971443/62394*e^7 + 314153521/124788*e^6 + 2437745713/124788*e^5 - 351985865/41596*e^4 - 832823513/20798*e^3 + 299607214/31197*e^2 + 245269255/10399*e - 97713720/10399, 1221899/124788*e^11 - 312887/124788*e^10 - 22695791/62394*e^9 + 2623562/31197*e^8 + 139667708/31197*e^7 - 91430677/124788*e^6 - 692855110/31197*e^5 - 16634895/20798*e^4 + 1837929581/41596*e^3 + 588119239/62394*e^2 - 219380711/10399*e + 14332783/10399, -258359/62394*e^11 + 467797/31197*e^10 + 18932465/124788*e^9 - 69180139/124788*e^8 - 227913415/124788*e^7 + 838625765/124788*e^6 + 1119086573/124788*e^5 - 329121937/10399*e^4 - 941513323/41596*e^3 + 3721484251/62394*e^2 + 285427408/10399*e - 311881592/10399, 27803/31197*e^11 - 2127287/124788*e^10 - 3673489/124788*e^9 + 78842867/124788*e^8 + 35924129/124788*e^7 - 965100367/124788*e^6 - 34149520/31197*e^5 + 1572870729/41596*e^4 + 75501440/10399*e^3 - 2301474886/31197*e^2 - 231793790/10399*e + 354319382/10399, 145190/31197*e^11 + 1720675/124788*e^10 - 11035883/62394*e^9 - 16054228/31197*e^8 + 140950981/62394*e^7 + 200743958/31197*e^6 - 1452557275/124788*e^5 - 1399868809/41596*e^4 + 791693757/41596*e^3 + 2169902006/31197*e^2 + 72548127/10399*e - 297082115/10399, 482041/124788*e^11 - 2487961/124788*e^10 - 17523707/124788*e^9 + 92135767/124788*e^8 + 208133863/124788*e^7 - 561207025/62394*e^6 - 1010976515/124788*e^5 + 447732950/10399*e^4 + 467043463/20798*e^3 - 2569978238/31197*e^2 - 343242921/10399*e + 417882764/10399, -59987/62394*e^11 - 564701/124788*e^10 + 4447643/124788*e^9 + 21317051/124788*e^8 - 54284995/124788*e^7 - 271056589/124788*e^6 + 63464300/31197*e^5 + 479815301/41596*e^4 - 40103305/20798*e^3 - 755984500/31197*e^2 - 58359944/10399*e + 115143232/10399, 10729/62394*e^11 + 75251/62394*e^10 - 342107/62394*e^9 - 2909777/62394*e^8 + 2925229/62394*e^7 + 19232258/31197*e^6 - 1666007/62394*e^5 - 35868844/10399*e^4 - 10909331/10399*e^3 + 236809115/31197*e^2 + 32443787/10399*e - 41061600/10399, -192503/62394*e^11 + 318655/31197*e^10 + 3518921/31197*e^9 - 11753662/31197*e^8 - 42173008/31197*e^7 + 283562569/62394*e^6 + 409563853/62394*e^5 - 439968079/20798*e^4 - 166753350/10399*e^3 + 1227926633/31197*e^2 + 191771812/10399*e - 205773642/10399];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {29, 29, w^5 - w^4 - 7*w^3 + 4*w^2 + 11*w}>] := 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;