/*
  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![7, 0, -6, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {23,23,w^3 - w^2 - 3*w + 4}>;

primesArray := [
[2, 2, w + 1],
[7, 7, w],
[9, 3, w^2 + w - 1],
[9, 3, w^2 - w - 1],
[17, 17, -w^3 + 3*w - 1],
[17, 17, w^3 - 3*w - 1],
[23, 23, -w^3 - w^2 + 3*w + 4],
[23, 23, w^3 - w^2 - 3*w + 4],
[41, 41, w^3 - w^2 - 2*w + 3],
[41, 41, -w^3 - w^2 + 2*w + 3],
[49, 7, w^2 - 6],
[71, 71, w^3 + 3*w^2 - 3*w - 6],
[71, 71, -2*w^2 - 2*w + 1],
[73, 73, w^2 - 2*w - 2],
[73, 73, w^2 + 2*w - 2],
[79, 79, w^3 - w^2 - 5*w + 2],
[79, 79, -w^3 - w^2 + 5*w + 2],
[89, 89, 2*w - 1],
[89, 89, -2*w - 1],
[97, 97, -w^3 + w^2 + 4*w - 1],
[97, 97, w^3 + w^2 - 4*w - 1],
[103, 103, -2*w^3 + w^2 + 8*w - 2],
[103, 103, w^3 - w^2 - 3*w + 6],
[103, 103, -w^3 - w^2 + 3*w + 6],
[103, 103, 2*w^3 + w^2 - 8*w - 2],
[127, 127, w^3 + 3*w^2 - 4*w - 11],
[127, 127, -w^3 + 3*w^2 + 4*w - 11],
[137, 137, -w^3 - 3*w^2 + 2*w + 9],
[137, 137, -2*w^2 + w + 10],
[137, 137, -2*w^2 - w + 10],
[137, 137, w^3 - 3*w^2 - 2*w + 9],
[151, 151, 3*w^2 - 2*w - 10],
[151, 151, w^3 + 3*w^2 - 4*w - 9],
[167, 167, w^3 + 4*w^2 - 2*w - 8],
[167, 167, 2*w^3 - 8*w - 3],
[167, 167, 2*w^3 - 8*w + 3],
[167, 167, 2*w^3 + 5*w^2 - 7*w - 13],
[191, 191, -w^3 + 6*w - 2],
[191, 191, w^3 - 6*w - 2],
[193, 193, -2*w^3 + 2*w^2 + 6*w - 5],
[193, 193, w^3 + 2*w^2 - 4*w - 4],
[193, 193, -w^3 + 2*w^2 + 4*w - 4],
[193, 193, -2*w^3 - 2*w^2 + 6*w + 5],
[199, 199, 2*w^2 - 3*w - 6],
[199, 199, w^3 + w^2 + w + 2],
[199, 199, 2*w^3 + 2*w^2 - 4*w - 5],
[199, 199, 2*w^2 + 3*w - 6],
[223, 223, 2*w^3 + w^2 - 6*w - 6],
[223, 223, w^3 - w^2 - 5*w - 2],
[223, 223, 2*w^3 + 4*w^2 - 6*w - 9],
[223, 223, -2*w^3 + w^2 + 6*w - 6],
[239, 239, -2*w^3 - 5*w^2 + 5*w + 11],
[239, 239, w^3 + 4*w^2 - 4*w - 10],
[241, 241, -3*w^3 - 4*w^2 + 9*w + 11],
[241, 241, -2*w^3 - w^2 + 6*w + 4],
[257, 257, 2*w^3 + 5*w^2 - 6*w - 12],
[257, 257, -w^3 - 4*w^2 + 3*w + 9],
[263, 263, -w^3 - w^2 + 6*w + 3],
[263, 263, w^3 - w^2 - 6*w + 3],
[271, 271, -2*w^3 - w^2 + 7*w + 3],
[271, 271, -w^3 + 2*w^2 + 3*w - 9],
[271, 271, w^3 + 2*w^2 - 3*w - 9],
[271, 271, 2*w^3 - w^2 - 7*w + 3],
[289, 17, 3*w^2 - 8],
[313, 313, -w^3 + 2*w^2 - 6],
[313, 313, w^3 + 2*w^2 - 6],
[353, 353, -w^3 + 3*w - 5],
[353, 353, w^3 - 3*w - 5],
[359, 359, w^3 - w + 3],
[359, 359, 4*w^3 + 5*w^2 - 14*w - 18],
[361, 19, -w^3 + 2*w^2 + 4*w - 2],
[361, 19, w^3 + 2*w^2 - 4*w - 2],
[367, 367, -2*w^3 - 3*w^2 + 6*w + 10],
[367, 367, -3*w - 2],
[367, 367, 3*w - 2],
[367, 367, w^3 + 2*w^2 - 7*w - 11],
[401, 401, -2*w^3 + 6*w - 1],
[401, 401, -3*w^3 + 3*w^2 + 11*w - 10],
[401, 401, 3*w^3 + 3*w^2 - 11*w - 10],
[401, 401, 2*w^3 - 6*w - 1],
[409, 409, w^3 - 3*w^2 - 6*w + 13],
[409, 409, -w^3 - 3*w^2 + 6*w + 13],
[431, 431, w^3 - 4*w^2 - 2*w + 10],
[431, 431, -w^3 - 4*w^2 + 2*w + 10],
[433, 433, w^3 - w^2 - 6*w - 3],
[433, 433, 2*w^3 + 4*w^2 - 7*w - 10],
[463, 463, w^2 + 3*w - 3],
[463, 463, w^2 - 3*w - 3],
[487, 487, 2*w^2 + 2*w - 9],
[487, 487, 2*w^2 - 2*w - 9],
[521, 521, -w^3 - 4*w^2 + 5*w + 13],
[521, 521, w^3 - 4*w^2 - 5*w + 13],
[529, 23, -w^2 - 2],
[577, 577, w^3 + 2*w^2 - 2*w - 8],
[577, 577, -w^3 + 2*w^2 + 2*w - 8],
[593, 593, -2*w^3 - 3*w^2 + 5*w + 9],
[593, 593, w^3 + 2*w^2 - 8*w - 12],
[599, 599, w^3 + 2*w^2 - 5*w - 3],
[599, 599, -w^3 + 2*w^2 + 5*w - 3],
[601, 601, -w^3 + 4*w^2 + 4*w - 8],
[601, 601, w^3 + 4*w^2 - 4*w - 8],
[607, 607, -w^3 + 8*w + 6],
[607, 607, -3*w^3 - 5*w^2 + 7*w + 12],
[607, 607, 2*w^2 - 4*w - 9],
[607, 607, -2*w^3 - 3*w^2 + 11*w + 13],
[617, 617, 3*w^3 + 2*w^2 - 11*w - 9],
[617, 617, -2*w^3 + w^2 + 9*w - 1],
[617, 617, 2*w^3 + w^2 - 9*w - 1],
[617, 617, 3*w^3 - 12*w - 4],
[625, 5, -5],
[631, 631, w^3 - w^2 - 7*w - 4],
[631, 631, 2*w^3 + 4*w^2 - 8*w - 11],
[641, 641, -2*w^3 + 3*w^2 + 9*w - 9],
[641, 641, w^3 - 2*w^2 - 7*w + 5],
[641, 641, w^3 + 2*w^2 - 7*w - 5],
[641, 641, 2*w^3 + 3*w^2 - 9*w - 9],
[647, 647, -w^3 + 3*w^2 - 1],
[647, 647, 2*w^3 - 3*w^2 - 8*w + 8],
[647, 647, -2*w^3 - 3*w^2 + 8*w + 8],
[647, 647, w^3 + 3*w^2 - 1],
[727, 727, -3*w^3 - 2*w^2 + 11*w + 3],
[727, 727, -2*w^3 + 5*w^2 + 9*w - 17],
[727, 727, 2*w^3 + 5*w^2 - 9*w - 17],
[727, 727, 3*w^3 - 2*w^2 - 11*w + 3],
[743, 743, 3*w^2 - 2*w - 12],
[743, 743, 3*w^2 + 2*w - 12],
[751, 751, 2*w^3 - 2*w^2 - 6*w + 9],
[751, 751, -2*w^3 - 2*w^2 + 6*w + 9],
[761, 761, -3*w^3 + 5*w^2 + 6*w - 11],
[761, 761, 3*w^3 + 5*w^2 - 6*w - 11],
[769, 769, w^3 - 7*w - 3],
[769, 769, -w^3 + 7*w - 3],
[823, 823, w^3 + 2*w^2 - 5*w - 1],
[823, 823, w^3 - 2*w^2 - 5*w + 1],
[839, 839, 4*w^2 + w - 12],
[839, 839, -3*w^3 + w^2 + 13*w - 8],
[839, 839, 3*w^3 + w^2 - 13*w - 8],
[839, 839, 4*w^2 - w - 12],
[841, 29, 2*w^3 + 4*w^2 - 7*w - 16],
[841, 29, -2*w^3 + 4*w^2 + 7*w - 16],
[857, 857, 3*w^2 - 3*w - 11],
[857, 857, 3*w^2 + 3*w - 11],
[863, 863, w^3 + 3*w^2 - 5*w - 6],
[863, 863, -w^3 + 3*w^2 + 5*w - 6],
[881, 881, w^3 - w^2 - 5*w - 4],
[881, 881, 2*w^2 + 4*w + 3],
[887, 887, w^2 - w - 9],
[887, 887, 3*w^3 + 4*w^2 - 11*w - 11],
[887, 887, -3*w^3 + 4*w^2 + 11*w - 11],
[887, 887, w^2 + w - 9],
[911, 911, w^2 - 4*w - 8],
[911, 911, -3*w^3 - 4*w^2 + 9*w + 13],
[919, 919, 2*w^3 - w^2 - 8*w - 2],
[919, 919, -2*w^3 - w^2 + 8*w - 2],
[929, 929, 4*w^3 + w^2 - 16*w - 2],
[929, 929, 2*w^3 - 11*w - 6],
[937, 937, 3*w^3 - 3*w^2 - 10*w + 5],
[937, 937, -3*w^3 - 3*w^2 + 10*w + 5],
[953, 953, w^3 - 9*w - 11],
[953, 953, 5*w^2 + 3*w - 17],
[953, 953, 5*w^2 - 3*w - 17],
[953, 953, -2*w^3 - w^2 + 8*w + 10],
[961, 31, 4*w^2 - 11],
[961, 31, 4*w^2 - 13],
[967, 967, w^3 - w^2 - 6*w - 5],
[967, 967, 2*w^2 + 3*w + 2],
[983, 983, -w^2 + w - 3],
[983, 983, 2*w^3 - 2*w^2 - 11*w + 10],
[983, 983, 2*w^3 + 2*w^2 - 11*w - 10],
[983, 983, -w^2 - w - 3],
[991, 991, w^3 + 3*w^2 - 10*w - 15],
[991, 991, -2*w^3 - 4*w^2 + w + 6]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - 13*x^6 + 51*x^4 - 63*x^2 + 8;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1/4*e^7 + 5/2*e^5 - 21/4*e^3 - e, -1/4*e^7 + 3*e^5 - 39/4*e^3 + 8*e, -1/2*e^5 + 9/2*e^3 - 8*e, 1/2*e^7 - 6*e^5 + 39/2*e^3 - 14*e, e^3 - 5*e, 1/4*e^6 - 5/2*e^4 + 25/4*e^2 + 2, -1, 1/2*e^4 - 5/2*e^2 + 2, -1/4*e^6 + 2*e^4 - 11/4*e^2, -1/4*e^6 + 7/2*e^4 - 57/4*e^2 + 12, e^7 - 25/2*e^5 + 91/2*e^3 - 46*e, -3/4*e^7 + 17/2*e^5 - 111/4*e^3 + 28*e, -1/4*e^7 + 3/2*e^5 + 15/4*e^3 - 21*e, -1/4*e^7 + 5/2*e^5 - 25/4*e^3 + 2*e, -1/4*e^7 + 3*e^5 - 39/4*e^3 + 8*e, e^7 - 11*e^5 + 31*e^3 - 21*e, -1/2*e^7 + 15/2*e^5 - 36*e^3 + 49*e, -1/4*e^7 + 3*e^5 - 43/4*e^3 + 11*e, 1/2*e^5 - 11/2*e^3 + 9*e, e^3 - e, 3/4*e^7 - 9*e^5 + 109/4*e^3 - 12*e, -3/4*e^6 + 8*e^4 - 73/4*e^2 + 6, 3/2*e^6 - 29/2*e^4 + 27*e^2 + 8, -1/2*e^7 + 6*e^5 - 37/2*e^3 + 11*e, -2*e^6 + 20*e^4 - 48*e^2 + 24, -3/4*e^6 + 15/2*e^4 - 63/4*e^2 + 2, -1/4*e^6 + 5/2*e^4 - 33/4*e^2 + 8, 1/2*e^6 - 7/2*e^4 - e^2 + 14, -5/4*e^6 + 29/2*e^4 - 173/4*e^2 + 20, 3/4*e^6 - 11*e^4 + 173/4*e^2 - 28, 5/4*e^6 - 12*e^4 + 103/4*e^2 - 10, 3/4*e^6 - 8*e^4 + 65/4*e^2 + 10, 5/4*e^6 - 23/2*e^4 + 89/4*e^2 + 2, 1/4*e^7 - 3*e^5 + 43/4*e^3 - 11*e, -3/4*e^7 + 10*e^5 - 165/4*e^3 + 59*e, -9/4*e^6 + 24*e^4 - 235/4*e^2 + 18, 3/4*e^7 - 17/2*e^5 + 115/4*e^3 - 34*e, 3/2*e^7 - 18*e^5 + 123/2*e^3 - 59*e, -3/2*e^7 + 18*e^5 - 123/2*e^3 + 65*e, 5/4*e^7 - 29/2*e^5 + 189/4*e^3 - 40*e, -2*e^5 + 18*e^3 - 32*e, 1/4*e^7 + 1/2*e^5 - 91/4*e^3 + 55*e, -1/2*e^7 + 8*e^5 - 85/2*e^3 + 71*e, 1/2*e^6 - 7/2*e^4 - e^2 + 24, -3/4*e^6 + 13/2*e^4 - 47/4*e^2 + 2, -1/4*e^7 + 2*e^5 - 7/4*e^3 - e, e^6 - 21/2*e^4 + 41/2*e^2 + 12, 1/2*e^7 - 6*e^5 + 41/2*e^3 - 17*e, -3/4*e^7 + 19/2*e^5 - 139/4*e^3 + 30*e, 1/4*e^6 - 7/2*e^4 + 37/4*e^2 + 10, -1/2*e^6 + 4*e^4 - 7/2*e^2 - 12, e^6 - 9*e^4 + 10*e^2 + 16, 1/2*e^7 - 6*e^5 + 39/2*e^3 - 24*e, -2*e^7 + 49/2*e^5 - 171/2*e^3 + 76*e, 3/2*e^6 - 31/2*e^4 + 39*e^2 - 18, 7/2*e^6 - 35*e^4 + 163/2*e^2 - 22, 3/2*e^7 - 18*e^5 + 129/2*e^3 - 76*e, -e^7 + 21/2*e^5 - 57/2*e^3 + 21*e, -7/4*e^7 + 43/2*e^5 - 303/4*e^3 + 71*e, 1/4*e^6 - 9/2*e^4 + 53/4*e^2 + 18, -1/2*e^6 + 5*e^4 - 13/2*e^2 - 12, 2*e^7 - 43/2*e^5 + 117/2*e^3 - 37*e, -e^6 + 13*e^4 - 46*e^2 + 26, 15/4*e^6 - 77/2*e^4 + 363/4*e^2 - 24, -1/4*e^6 - e^4 + 65/4*e^2 - 4, -2*e^6 + 47/2*e^4 - 137/2*e^2 + 42, 13/4*e^6 - 67/2*e^4 + 337/4*e^2 - 32, 1/4*e^6 - 5*e^4 + 95/4*e^2 - 10, -1/4*e^6 + 29/4*e^2 + 2, 1/2*e^7 - 4*e^5 - 7/2*e^3 + 43*e, -3/4*e^7 + 23/2*e^5 - 211/4*e^3 + 62*e, 5/2*e^6 - 24*e^4 + 95/2*e^2 + 12, 1/2*e^7 - 5/2*e^5 - 11*e^3 + 44*e, 1/2*e^7 - 7*e^5 + 63/2*e^3 - 45*e, 7/4*e^6 - 20*e^4 + 245/4*e^2 - 30, 1/4*e^7 + 1/2*e^5 - 79/4*e^3 + 42*e, -3/2*e^5 + 19/2*e^3 + 3*e, -5/2*e^7 + 32*e^5 - 233/2*e^3 + 113*e, 1/2*e^7 - 13/2*e^5 + 22*e^3 - 11*e, 1/2*e^6 - 5/2*e^4 - 4*e^2 + 18, -15/4*e^6 + 73/2*e^4 - 307/4*e^2 + 4, -1/2*e^6 + 7/2*e^4 - 4*e^2 + 16, -1/2*e^6 + 5*e^4 - 9/2*e^2 - 20, -3/2*e^7 + 19*e^5 - 143/2*e^3 + 92*e, 1/2*e^7 - 6*e^5 + 41/2*e^3 - 27*e, 3/2*e^7 - 20*e^5 + 155/2*e^3 - 75*e, -1/2*e^7 + 6*e^5 - 37/2*e^3 + 5*e, 3/2*e^6 - 29/2*e^4 + 31*e^2 + 16, 13/4*e^6 - 33*e^4 + 315/4*e^2 - 18, -5/4*e^6 + 27/2*e^4 - 145/4*e^2 + 24, -3/2*e^6 + 31/2*e^4 - 28*e^2 - 38, -e^4 + 15*e^2 - 38, -3/2*e^6 + 19*e^4 - 123/2*e^2 + 34, 13/4*e^6 - 69/2*e^4 + 385/4*e^2 - 56, -1/2*e^6 + 6*e^4 - 47/2*e^2 + 34, -1/2*e^4 - 9/2*e^2 + 26, 7/4*e^7 - 23*e^5 + 377/4*e^3 - 138*e, 5/2*e^7 - 57/2*e^5 + 89*e^3 - 69*e, -3/2*e^7 + 33/2*e^5 - 46*e^3 + 33*e, -1/4*e^7 + 13/2*e^5 - 177/4*e^3 + 80*e, -e^7 + 29/2*e^5 - 143/2*e^3 + 124*e, -e^6 + 21/2*e^4 - 55/2*e^2 + 36, -1/4*e^6 + 7/2*e^4 - 101/4*e^2 + 46, -5/2*e^7 + 28*e^5 - 173/2*e^3 + 77*e, 13/4*e^7 - 38*e^5 + 487/4*e^3 - 98*e, -5/2*e^7 + 57/2*e^5 - 85*e^3 + 59*e, 5/4*e^7 - 19*e^5 + 359/4*e^3 - 123*e, 3/2*e^7 - 20*e^5 + 171/2*e^3 - 137*e, -3/2*e^6 + 12*e^4 - 13/2*e^2 - 34, -3/2*e^5 + 37/2*e^3 - 43*e, e^7 - 10*e^5 + 29*e^3 - 40*e, -5/4*e^7 + 15*e^5 - 187/4*e^3 + 12*e, -e^7 + 9*e^5 - 5*e^3 - 51*e, -15/4*e^7 + 42*e^5 - 497/4*e^3 + 83*e, 9/4*e^7 - 55/2*e^5 + 357/4*e^3 - 61*e, -3*e^6 + 30*e^4 - 59*e^2 - 8, 21/4*e^7 - 61*e^5 + 775/4*e^3 - 157*e, -3/4*e^7 + 25/2*e^5 - 271/4*e^3 + 114*e, -5/4*e^6 + 15/2*e^4 + 39/4*e^2 - 34, 11/4*e^7 - 30*e^5 + 313/4*e^3 - 30*e, 1/2*e^6 - 11/2*e^4 + 13*e^2 + 8, -1/2*e^6 + 6*e^4 - 23/2*e^2 - 12, -17/4*e^7 + 49*e^5 - 615/4*e^3 + 128*e, 7*e^4 - 51*e^2 + 48, -3*e^6 + 57/2*e^4 - 127/2*e^2 + 28, -4*e^6 + 44*e^4 - 116*e^2 + 56, -5*e^6 + 54*e^4 - 139*e^2 + 56, 3/4*e^6 - 7*e^4 + 5/4*e^2 + 52, -3/4*e^6 + 4*e^4 + 23/4*e^2 - 24, 7/2*e^7 - 42*e^5 + 291/2*e^3 - 151*e, 4*e^7 - 44*e^5 + 122*e^3 - 70*e, 3*e^7 - 35*e^5 + 114*e^3 - 98*e, -5/2*e^7 + 31*e^5 - 223/2*e^3 + 115*e, -5/2*e^6 + 51/2*e^4 - 45*e^2 - 24, -7/4*e^7 + 25*e^5 - 453/4*e^3 + 155*e, e^7 - 31/2*e^5 + 141/2*e^3 - 94*e, 2*e^6 - 24*e^4 + 70*e^2 - 8, -15/4*e^6 + 36*e^4 - 285/4*e^2 - 20, -5/2*e^6 + 22*e^4 - 79/2*e^2 + 10, 3/4*e^6 - 17/2*e^4 + 55/4*e^2 + 28, -5/2*e^6 + 45/2*e^4 - 32*e^2 - 18, -9/4*e^7 + 53/2*e^5 - 349/4*e^3 + 79*e, -5/2*e^7 + 31*e^5 - 229/2*e^3 + 132*e, 2*e^4 - 8*e^2 - 6, -2*e^6 + 24*e^4 - 80*e^2 + 34, -5/4*e^6 + 27/2*e^4 - 97/4*e^2 - 26, -3/4*e^7 + 21/2*e^5 - 187/4*e^3 + 69*e, 3/2*e^7 - 27/2*e^5 + 19*e^3 + 25*e, 13/4*e^6 - 33*e^4 + 307/4*e^2 - 10, -5*e^6 + 53*e^4 - 138*e^2 + 64, -1/4*e^6 - 5/2*e^4 + 143/4*e^2 - 42, -e^7 + 13/2*e^5 + 15/2*e^3 - 41*e, 15/4*e^7 - 83/2*e^5 + 499/4*e^3 - 109*e, e^7 - 9*e^5 + 15*e^3 + 5*e, -3/2*e^7 + 18*e^5 - 129/2*e^3 + 66*e, 7/4*e^7 - 20*e^5 + 245/4*e^3 - 28*e, -e^7 + 11*e^5 - 34*e^3 + 30*e, -17/4*e^6 + 43*e^4 - 423/4*e^2 + 36, 11/2*e^6 - 107/2*e^4 + 116*e^2 - 18, -e^6 + 19/2*e^4 - 9/2*e^2 - 38, -9/2*e^6 + 46*e^4 - 215/2*e^2 + 14, 7/2*e^6 - 37*e^4 + 171/2*e^2 - 18, e^6 - 13/2*e^4 - 3/2*e^2 + 2, 4*e^6 - 75/2*e^4 + 143/2*e^2 - 4, -21/4*e^6 + 115/2*e^4 - 633/4*e^2 + 70, 1/2*e^6 - 17/2*e^4 + 36*e^2 - 16, 1/2*e^7 - 7*e^5 + 57/2*e^3 - 48*e, 1/2*e^7 - 1/2*e^5 - 30*e^3 + 74*e, -1/4*e^6 + 5/2*e^4 + 15/4*e^2 - 26, 3/2*e^5 - 37/2*e^3 + 42*e, -3/2*e^7 + 11*e^5 + 11/2*e^3 - 81*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {23,23,w^3 - w^2 - 3*w + 4}>] := 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;