/*
  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![-5, -7, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {25, 5, w^2 - 7}>;

primesArray := [
[5, 5, w],
[8, 2, 2],
[11, 11, -w^2 + 2*w + 4],
[11, 11, -w + 2],
[11, 11, w - 1],
[13, 13, -w^2 + w + 4],
[17, 17, -w^2 + w + 8],
[17, 17, -w^2 + w + 3],
[19, 19, -w^2 + 6],
[23, 23, -w^2 + 3],
[25, 5, w^2 - 7],
[27, 3, -3],
[31, 31, w^2 - 2*w - 8],
[37, 37, w^2 - 2*w - 6],
[41, 41, -w - 4],
[41, 41, w^2 - 2*w - 7],
[47, 47, 2*w^2 - 3*w - 7],
[53, 53, -w^2 + w + 9],
[61, 61, 3*w^2 - 2*w - 17],
[67, 67, 2*w - 1],
[73, 73, 2*w^2 - 2*w - 9],
[89, 89, 2*w^2 - 3*w - 11],
[97, 97, 2*w^2 - w - 9],
[97, 97, w^2 - 3*w - 6],
[97, 97, 2*w - 3],
[101, 101, 3*w^2 - 4*w - 13],
[107, 107, 2*w^2 - 3*w - 14],
[113, 113, 2*w^2 - 3*w - 12],
[127, 127, -w^2 + w - 1],
[127, 127, 2*w^2 - 2*w - 7],
[127, 127, w^2 + 3*w - 1],
[131, 131, w^2 - 3*w - 8],
[137, 137, 2*w^2 - w - 8],
[139, 139, 4*w^2 - 5*w - 19],
[149, 149, w^2 + w - 7],
[151, 151, w^2 - 4*w - 6],
[157, 157, w^2 + w - 9],
[163, 163, w^2 + w - 8],
[169, 13, 2*w^2 - w - 7],
[173, 173, w^2 - 5*w - 3],
[179, 179, -w - 6],
[191, 191, w^2 + 2*w - 4],
[197, 197, 3*w - 1],
[229, 229, 3*w^2 - 4*w - 18],
[233, 233, -w^2 + w - 2],
[233, 233, -3*w^2 + 4*w + 11],
[233, 233, 3*w^2 - w - 14],
[239, 239, 3*w^2 - 5*w - 14],
[241, 241, w^2 - 4*w - 11],
[257, 257, -2*w^2 + 2*w + 17],
[263, 263, 3*w^2 - 3*w - 23],
[269, 269, w^2 - 2*w - 12],
[277, 277, w^2 - 4*w - 9],
[281, 281, 4*w^2 - 2*w - 21],
[283, 283, 3*w^2 - 3*w - 13],
[293, 293, 3*w^2 - 4*w - 9],
[331, 331, w^2 + 3*w - 3],
[337, 337, w^2 + 2*w - 6],
[343, 7, -7],
[361, 19, w^2 - 5*w - 1],
[367, 367, w^2 - 5*w - 8],
[373, 373, w^2 + 2*w - 16],
[373, 373, 2*w^2 + w - 11],
[373, 373, 3*w^2 - 2*w - 13],
[379, 379, -w^2 + w - 3],
[383, 383, 2*w^2 - 6*w - 9],
[401, 401, w^2 + 2*w - 7],
[419, 419, 3*w^2 - 6*w - 13],
[421, 421, 4*w^2 - 5*w - 26],
[431, 431, 4*w - 1],
[439, 439, w^2 - 5*w - 9],
[439, 439, w^2 - 2*w - 13],
[439, 439, 4*w^2 - 7*w - 17],
[443, 443, 3*w^2 - 3*w - 8],
[443, 443, w^2 - 13],
[443, 443, 3*w^2 - w - 12],
[449, 449, 4*w^2 - 6*w - 13],
[457, 457, w^2 + 2*w - 11],
[461, 461, -w - 8],
[461, 461, -w^2 + 5*w - 1],
[461, 461, 2*w^2 + w - 12],
[463, 463, 3*w^2 + w - 9],
[487, 487, 5*w^2 - 3*w - 27],
[499, 499, 4*w^2 - w - 24],
[499, 499, w^2 - 5*w - 11],
[499, 499, 5*w^2 - 4*w - 27],
[503, 503, 3*w^2 - 5*w - 22],
[503, 503, 3*w^2 - 5*w - 17],
[503, 503, 5*w^2 - 8*w - 22],
[509, 509, -w^2 - 4],
[529, 23, 4*w^2 - 5*w - 16],
[541, 541, 2*w^2 + w - 17],
[547, 547, -w^2 + 2*w - 4],
[547, 547, 3*w^2 - 6*w - 14],
[547, 547, 2*w^2 - 5*w - 17],
[557, 557, 5*w^2 - 2*w - 31],
[563, 563, 3*w^2 - 5*w - 18],
[571, 571, -w^2 + w - 4],
[571, 571, w^2 - 6*w - 2],
[571, 571, 3*w^2 + w - 8],
[577, 577, 2*w^2 - w - 19],
[587, 587, 2*w^2 - 5*w - 13],
[593, 593, 2*w^2 + w - 16],
[599, 599, 3*w^2 - 19],
[599, 599, 3*w^2 - 5*w - 19],
[599, 599, 4*w^2 - w - 18],
[607, 607, 4*w^2 - 2*w - 19],
[617, 617, 4*w^2 - 3*w - 19],
[619, 619, 4*w^2 - 7*w - 18],
[643, 643, 3*w^2 + w - 7],
[643, 643, 2*w^2 - 2*w - 19],
[643, 643, 5*w^2 - 8*w - 16],
[653, 653, 3*w^2 - w - 9],
[653, 653, 2*w^2 - 6*w - 11],
[653, 653, 3*w^2 - w - 7],
[659, 659, w^2 - 2*w - 14],
[661, 661, 5*w^2 - 6*w - 31],
[661, 661, w^2 - 14],
[661, 661, w - 9],
[673, 673, -2*w^2 + 4*w - 1],
[673, 673, 4*w^2 - w - 26],
[673, 673, 6*w^2 - 5*w - 33],
[677, 677, 3*w^2 - 8],
[719, 719, 6*w^2 - 7*w - 29],
[719, 719, 4*w^2 - 6*w - 29],
[719, 719, 4*w^2 - 5*w - 14],
[727, 727, 4*w^2 - 4*w - 31],
[743, 743, w^2 - w - 14],
[757, 757, 5*w^2 - 8*w - 23],
[761, 761, w^2 - 6*w - 11],
[761, 761, 4*w - 7],
[761, 761, 4*w^2 - 3*w - 18],
[773, 773, -5*w^2 + 10*w + 18],
[787, 787, 5*w^2 - 7*w - 19],
[797, 797, 4*w^2 - 5*w - 12],
[797, 797, 3*w^2 + w - 16],
[797, 797, 3*w^2 - 6*w - 16],
[811, 811, w^2 + 3*w - 7],
[821, 821, 4*w^2 - 6*w - 23],
[829, 829, 2*w^2 + 2*w - 11],
[829, 829, 2*w^2 - 7*w - 11],
[829, 829, 5*w^2 - 5*w - 24],
[853, 853, 2*w^2 - 6*w - 19],
[857, 857, w^2 + 4*w - 4],
[877, 877, -w^2 - w - 6],
[881, 881, -4*w^2 + 4*w + 9],
[883, 883, 2*w^2 - 6*w - 13],
[907, 907, 3*w^2 - 6*w - 17],
[911, 911, w^2 + 3*w - 8],
[919, 919, 4*w^2 - 8*w - 17],
[937, 937, 4*w^2 - 6*w - 27],
[941, 941, 4*w^2 - 2*w - 17],
[947, 947, 3*w^2 - 8*w - 13],
[961, 31, 6*w^2 - 7*w - 39],
[967, 967, 5*w - 2],
[971, 971, 5*w^2 - 9*w - 21],
[977, 977, 3*w^2 + w - 17],
[983, 983, 2*w - 11],
[991, 991, 6*w^2 - 7*w - 38],
[991, 991, w^2 + 3*w - 9],
[991, 991, w^2 + 3*w - 13],
[997, 997, 4*w^2 - w - 16],
[997, 997, 5*w - 12],
[997, 997, 3*w^2 - 2*w - 26]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - 19*x^2 + 9*x + 34;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1/3*e^2 + 11/3, 1/3*e^2 + e - 11/3, -2/15*e^3 - 2/15*e^2 + 16/15*e + 13/15, -1/15*e^3 + 4/15*e^2 + 23/15*e - 26/15, 1/15*e^3 - 4/15*e^2 - 23/15*e + 86/15, -e + 1, -1/15*e^3 - 11/15*e^2 + 8/15*e + 94/15, 1/5*e^3 + 8/15*e^2 - 18/5*e - 22/15, 1/5*e^3 + 1/5*e^2 - 18/5*e + 26/5, 1, e - 3, 1/5*e^3 - 2/15*e^2 - 18/5*e - 32/15, -1/5*e^3 - 13/15*e^2 + 13/5*e + 62/15, -1/3*e^3 - 1/3*e^2 + 17/3*e - 4/3, 1/3*e^2 + 7/3, -4/15*e^3 + 2/5*e^2 + 62/15*e - 13/5, -4/15*e^3 - 4/15*e^2 + 47/15*e + 86/15, -1/3*e^2 - e - 4/3, 7/15*e^3 + 7/15*e^2 - 116/15*e + 52/15, 1/3*e^3 + 1/3*e^2 - 8/3*e + 10/3, 8/15*e^3 + 8/15*e^2 - 79/15*e - 22/15, -2/3*e^2 - 2*e + 10/3, 2/3*e^3 + 1/3*e^2 - 34/3*e + 4/3, -1/5*e^3 + 17/15*e^2 + 33/5*e - 193/15, -7/15*e^3 - 7/15*e^2 + 146/15*e + 83/15, -8/15*e^3 + 2/15*e^2 + 109/15*e - 148/15, 14/15*e^3 + 4/15*e^2 - 187/15*e + 64/15, 7/15*e^3 + 17/15*e^2 - 101/15*e - 13/15, -1/3*e^3 - 4/3*e^2 + 8/3*e + 23/3, -4/15*e^3 - 3/5*e^2 + 77/15*e + 77/5, -2/5*e^3 - 2/5*e^2 + 21/5*e + 43/5, -3/5*e^3 + 1/15*e^2 + 44/5*e + 91/15, -1/15*e^3 + 8/5*e^2 - 7/15*e - 102/5, 1/5*e^3 - 4/5*e^2 - 8/5*e + 61/5, -1/5*e^3 + 32/15*e^2 + 13/5*e - 358/15, 1/3*e^3 - 14/3*e + 8, 4/15*e^3 - 11/15*e^2 - 92/15*e + 94/15, 1/3*e^3 + 2*e^2 - 11/3*e - 12, 1/3*e^3 + 1/3*e^2 - 17/3*e - 11/3, -8/15*e^3 - 28/15*e^2 + 124/15*e + 227/15, -1/5*e^3 + 2/15*e^2 + 43/5*e - 13/15, 6/5*e^3 + 28/15*e^2 - 78/5*e - 152/15, -11/15*e^3 + 4/15*e^2 + 163/15*e - 161/15, -1/3*e^2 - 6*e + 8/3, -1/5*e^3 + 22/15*e^2 + 33/5*e - 248/15, -4/5*e^3 + 1/5*e^2 + 57/5*e - 59/5, -1/15*e^3 - 11/15*e^2 - 67/15*e + 34/15, -1/15*e^3 + 4/15*e^2 + 8/15*e - 281/15, 4/15*e^3 + 19/15*e^2 - 2/15*e - 266/15, 2/3*e^3 + e^2 - 28/3*e + 11, -4/5*e^3 - 32/15*e^2 + 57/5*e + 223/15, -1/3*e^2 + 4*e - 1/3, -8/5*e^3 - 29/15*e^2 + 109/5*e + 106/15, -6/5*e^3 - 11/5*e^2 + 63/5*e + 94/5, -6/5*e^3 - 38/15*e^2 + 78/5*e + 247/15, -1/5*e^3 - 1/5*e^2 + 13/5*e + 44/5, -7/5*e^3 - 26/15*e^2 + 86/5*e + 94/15, 11/15*e^3 + 11/15*e^2 - 283/15*e - 139/15, 4/5*e^3 - 1/5*e^2 - 67/5*e + 14/5, 8/15*e^3 + 43/15*e^2 - 109/15*e - 212/15, -1/3*e^3 + 1/3*e^2 + 23/3*e + 4/3, 11/15*e^3 + 31/15*e^2 - 178/15*e + 106/15, 2/3*e^3 - 5/3*e^2 - 40/3*e + 43/3, 16/15*e^3 + 2/5*e^2 - 308/15*e - 38/5, 14/15*e^3 - 26/15*e^2 - 232/15*e + 319/15, -2/15*e^3 - 2/15*e^2 - 89/15*e + 28/15, 1/15*e^3 - 14/15*e^2 + 7/15*e + 121/15, -2/15*e^3 + 1/5*e^2 + 31/15*e - 64/5, 1/15*e^3 - 34/15*e^2 - 68/15*e + 416/15, -3/5*e^3 - 3/5*e^2 + 54/5*e - 58/5, -1/15*e^3 + 8/5*e^2 + 83/15*e - 142/5, -28/15*e^3 - 8/15*e^2 + 419/15*e + 52/15, -1/5*e^3 - 23/15*e^2 + 23/5*e + 277/15, 13/15*e^3 + 43/15*e^2 - 149/15*e - 197/15, -1/5*e^3 - 8/15*e^2 + 8/5*e + 247/15, -1/5*e^3 - 16/5*e^2 - 17/5*e + 189/5, 1/3*e^3 + 4/3*e^2 - 5/3*e + 4/3, 2/15*e^3 - 13/15*e^2 - 106/15*e + 47/15, 8/15*e^3 + 16/5*e^2 - 169/15*e - 169/5, -7/15*e^3 + 8/15*e^2 + 86/15*e + 173/15, 4/3*e^3 - 2/3*e^2 - 65/3*e + 37/3, 9/5*e^3 + 22/15*e^2 - 117/5*e + 127/15, 22/15*e^3 - 13/15*e^2 - 311/15*e + 167/15, -22/15*e^3 - 17/15*e^2 + 296/15*e + 43/15, -4/15*e^3 + 16/15*e^2 + 122/15*e - 224/15, 1/5*e^3 + 1/5*e^2 - 23/5*e - 14/5, -4/3*e^3 - 10/3*e^2 + 65/3*e + 62/3, 1/15*e^3 + 11/15*e^2 - 128/15*e - 154/15, -14/15*e^3 + 1/15*e^2 + 202/15*e - 209/15, -13/15*e^3 - 1/5*e^2 + 209/15*e - 56/5, -2/3*e^3 + 22/3*e - 1, 1/15*e^3 - 3/5*e^2 - 98/15*e + 37/5, -2/3*e^3 - 4/3*e^2 + 16/3*e + 50/3, -4/15*e^3 + 11/15*e^2 + 92/15*e - 64/15, -4/3*e^2 + 26/3, 7/15*e^3 - 43/15*e^2 - 26/15*e + 737/15, -5/3*e^3 - 2/3*e^2 + 94/3*e - 8/3, -8/5*e^3 - 3/5*e^2 + 124/5*e - 8/5, 2/15*e^3 + 9/5*e^2 + 44/15*e - 46/5, 1/3*e^3 - 4/3*e^2 + 1/3*e + 68/3, 3/5*e^3 + 3/5*e^2 - 79/5*e + 28/5, 1/3*e^2 - 2*e - 26/3, 13/15*e^3 - 4/5*e^2 - 239/15*e - 4/5, 22/15*e^3 + 17/15*e^2 - 356/15*e - 73/15, -4/5*e^3 - 4/5*e^2 + 62/5*e - 54/5, -7/15*e^3 - 4/5*e^2 + 161/15*e + 91/5, -1/15*e^3 + 29/15*e^2 - 37/15*e - 541/15, -2/3*e^2 - 9*e + 31/3, 4/3*e^2 - 5*e - 119/3, 8/15*e^3 - 2/15*e^2 - 139/15*e - 92/15, 4/3*e^3 + 1/3*e^2 - 59/3*e + 28/3, 4/3*e^3 - 1/3*e^2 - 56/3*e - 34/3, 8/15*e^3 - 22/15*e^2 - 139/15*e - 37/15, 19/15*e^3 - 7/5*e^2 - 272/15*e + 48/5, 3/5*e^3 - 7/5*e^2 - 54/5*e + 168/5, -16/15*e^3 - 17/5*e^2 + 263/15*e + 98/5, 2/15*e^3 - 38/15*e^2 - 46/15*e + 262/15, -13/15*e^3 - 23/15*e^2 + 164/15*e + 382/15, 1/3*e^3 - 20/3*e - 38, 2/5*e^3 + 7/5*e^2 - 46/5*e - 53/5, 4/3*e^3 + 5/3*e^2 - 68/3*e - 22/3, 13/15*e^3 + 23/15*e^2 - 194/15*e - 22/15, -2/15*e^3 - 17/15*e^2 - 29/15*e - 167/15, 6/5*e^3 + 1/5*e^2 - 88/5*e + 41/5, 17/15*e^3 + 17/15*e^2 - 226/15*e - 163/15, -28/15*e^3 - 6/5*e^2 + 434/15*e + 44/5, -13/5*e^3 - 44/15*e^2 + 184/5*e + 16/15, -2/5*e^3 - 17/5*e^2 + 26/5*e + 203/5, -19/15*e^3 - 3/5*e^2 + 167/15*e - 68/5, 4/5*e^3 + 32/15*e^2 - 47/5*e - 223/15, 4/5*e^3 + 2/15*e^2 - 52/5*e + 212/15, 4/3*e^3 + 10/3*e^2 - 47/3*e - 35/3, -32/15*e^3 - 2/15*e^2 + 526/15*e - 212/15, 9/5*e^3 + 62/15*e^2 - 97/5*e - 628/15, -19/15*e^3 + 1/15*e^2 + 212/15*e - 179/15, 1/15*e^3 - 3/5*e^2 - 113/15*e + 32/5, -2/5*e^3 - 16/15*e^2 + 11/5*e - 136/15, -16/15*e^3 - 22/5*e^2 + 218/15*e + 98/5, -8/15*e^3 + 2/15*e^2 + 154/15*e + 92/15, -14/15*e^3 - 18/5*e^2 + 52/15*e + 222/5, 29/15*e^3 + 18/5*e^2 - 307/15*e - 112/5, 1/15*e^3 - 4/15*e^2 + 82/15*e - 214/15, -8/15*e^3 - 8/15*e^2 - 11/15*e + 52/15, 11/15*e^3 - 4/15*e^2 - 103/15*e + 71/15, e^3 + 3*e^2 - 7*e - 32, -19/15*e^3 + 16/15*e^2 + 167/15*e - 374/15, 1/5*e^3 + 23/15*e^2 + 17/5*e - 232/15, -6/5*e^3 + 9/5*e^2 + 98/5*e - 71/5, 1/15*e^3 + 31/15*e^2 - 68/15*e - 584/15, -19/15*e^3 - 14/15*e^2 + 332/15*e - 14/15, 2/5*e^3 - 8/5*e^2 - 31/5*e + 107/5, 22/15*e^3 - 8/15*e^2 - 401/15*e + 202/15, -3/5*e^3 + 7/5*e^2 + 39/5*e - 78/5, -19/15*e^3 + 1/15*e^2 + 272/15*e - 344/15, 23/15*e^3 - 7/15*e^2 - 274/15*e + 188/15, 8/15*e^3 + 28/15*e^2 - 154/15*e - 377/15, -2*e^3 - 8/3*e^2 + 21*e + 70/3, -7/5*e^3 - 12/5*e^2 + 106/5*e + 148/5, -9/5*e^3 - 17/15*e^2 + 147/5*e - 182/15, 2/15*e^3 - 13/15*e^2 - 31/15*e + 242/15, -1/3*e^3 + 5*e^2 + 29/3*e - 40, 4/15*e^3 - 16/15*e^2 - 107/15*e - 91/15, -34/15*e^3 + 26/15*e^2 + 647/15*e - 184/15];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {25, 5, w^2 - 7}>] := -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;