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

NN := ideal<ZF | {7, 7, w + 1}>;

primesArray := [
[2, 2, -w],
[4, 2, w^2 - w - 7],
[5, 5, w - 1],
[7, 7, w + 1],
[11, 11, -w^2 + 5],
[13, 13, w + 3],
[13, 13, -2*w + 1],
[19, 19, 2*w^2 - w - 15],
[25, 5, w^2 - 7],
[27, 3, 3],
[31, 31, w^2 - 2*w - 1],
[37, 37, -3*w^2 + 2*w + 23],
[41, 41, -2*w - 1],
[47, 47, w^2 - 3],
[49, 7, w^2 - 2*w - 5],
[59, 59, 2*w - 3],
[67, 67, w - 5],
[71, 71, w^2 - 2*w - 11],
[73, 73, w^2 + 1],
[83, 83, -4*w^2 + 3*w + 27],
[89, 89, w^2 - 2*w - 7],
[97, 97, w^2 - 6*w + 1],
[109, 109, -w^2 + 4*w + 1],
[113, 113, -w - 5],
[113, 113, -3*w - 1],
[113, 113, 2*w^2 - 3*w - 11],
[137, 137, -2*w^2 + 3*w + 17],
[149, 149, 2*w^2 - 13],
[151, 151, -2*w^2 + 1],
[163, 163, 2*w^2 + w - 7],
[167, 167, 2*w^2 - 5*w - 5],
[191, 191, -6*w^2 + 4*w + 41],
[191, 191, 4*w^2 - 12*w + 1],
[191, 191, 3*w - 7],
[193, 193, w^2 + 2*w - 5],
[197, 197, -4*w^2 + 4*w + 27],
[211, 211, -10*w^2 + 8*w + 69],
[211, 211, 3*w - 5],
[211, 211, 2*w^2 - 4*w - 1],
[223, 223, 2*w^2 - 3*w - 7],
[227, 227, 2*w^2 - 15],
[229, 229, -2*w - 7],
[233, 233, 2*w^2 - 3*w - 13],
[233, 233, -w^2 - 4*w + 3],
[233, 233, 2*w^2 - 2*w - 9],
[239, 239, -5*w^2 + 4*w + 33],
[241, 241, -2*w^2 + 2*w + 1],
[257, 257, 2*w^2 - 6*w - 3],
[257, 257, 2*w^2 - 3*w - 15],
[257, 257, -w^2 - 2*w - 3],
[269, 269, 2*w^2 - w - 9],
[269, 269, 2*w^2 + w - 11],
[269, 269, -2*w^2 + 7*w + 1],
[271, 271, -7*w^2 + 6*w + 49],
[277, 277, -3*w^2 + 2*w + 25],
[277, 277, 3*w^2 - 8*w - 1],
[277, 277, 2*w^2 - 4*w - 9],
[281, 281, w^2 - 4*w - 3],
[283, 283, 4*w^2 - 2*w - 25],
[283, 283, -3*w^2 + 10*w - 5],
[283, 283, 2*w^2 + w - 5],
[293, 293, -5*w + 3],
[307, 307, -4*w^2 + 2*w + 33],
[317, 317, 2*w^2 + w + 1],
[331, 331, 2*w - 9],
[337, 337, -10*w^2 + 6*w + 75],
[337, 337, w^2 + 2*w - 7],
[337, 337, -8*w^2 + 6*w + 55],
[347, 347, w^2 - 13],
[347, 347, 2*w^2 - 4*w - 19],
[347, 347, -5*w^2 + 4*w + 39],
[349, 349, 2*w^2 - w - 19],
[353, 353, -8*w^2 + 7*w + 55],
[359, 359, -4*w^2 + 3*w + 25],
[361, 19, -w^2 - 6*w + 3],
[367, 367, 2*w^2 - 3*w - 3],
[367, 367, 2*w^2 - 2*w - 7],
[367, 367, w^2 - 2*w - 13],
[373, 373, -4*w - 7],
[373, 373, 3*w^2 - 2*w - 17],
[373, 373, -11*w^2 + 8*w + 77],
[379, 379, -7*w + 3],
[383, 383, w^2 + 2*w - 17],
[397, 397, 2*w^2 - w - 7],
[401, 401, -4*w - 3],
[409, 409, 12*w^2 - 7*w - 91],
[419, 419, 5*w + 1],
[419, 419, 2*w^2 - 2*w - 3],
[419, 419, -7*w^2 + 4*w + 51],
[421, 421, w^2 - 4*w - 13],
[431, 431, 3*w^2 - 19],
[449, 449, w^2 + 2*w - 9],
[457, 457, -8*w^2 + 7*w + 59],
[461, 461, 2*w^2 - w - 5],
[463, 463, -4*w - 5],
[479, 479, -4*w^2 + 4*w + 23],
[487, 487, -3*w^2 + 4*w + 23],
[499, 499, 2*w^2 - 4*w - 11],
[509, 509, 4*w - 7],
[541, 541, -w^2 + 4*w - 7],
[557, 557, -2*w^2 - w + 19],
[569, 569, 8*w - 1],
[569, 569, -w^2 + 2*w - 5],
[569, 569, 2*w^2 - 6*w + 5],
[577, 577, -w^2 + 6*w - 11],
[587, 587, w - 9],
[593, 593, 2*w^2 - 8*w - 1],
[599, 599, 4*w^2 - 10*w - 5],
[599, 599, 2*w^2 - 6*w - 5],
[599, 599, -4*w^2 + 3*w + 33],
[613, 613, -3*w^2 - 2*w + 3],
[617, 617, 2*w^2 - 3*w + 3],
[619, 619, -10*w^2 + 7*w + 75],
[631, 631, 3*w^2 + 1],
[643, 643, -10*w^2 + 7*w + 69],
[661, 661, 2*w^2 - 7*w - 3],
[673, 673, 5*w - 13],
[673, 673, -w^2 - 2*w - 5],
[673, 673, 2*w^2 + 2*w - 11],
[677, 677, 6*w + 1],
[677, 677, 2*w^2 + w - 17],
[677, 677, 3*w^2 - 2*w - 15],
[683, 683, 3*w^2 - 25],
[691, 691, -5*w - 9],
[701, 701, -5*w^2 + 2*w + 37],
[701, 701, -3*w^2 + 4*w + 1],
[701, 701, w^2 - 4*w - 9],
[709, 709, -6*w^2 + 4*w + 39],
[727, 727, 2*w^2 - 3*w - 21],
[727, 727, -8*w^2 + 6*w + 61],
[727, 727, 2*w^2 - 4*w - 15],
[733, 733, 2*w^2 + 2*w - 23],
[739, 739, -4*w^2 + w + 31],
[743, 743, 2*w^2 + 4*w - 7],
[751, 751, 4*w^2 + 2*w - 19],
[751, 751, -4*w^2 + w + 29],
[751, 751, -6*w^2 + 6*w + 41],
[761, 761, -10*w^2 + 6*w + 77],
[769, 769, -w^2 - 5],
[773, 773, -3*w^2 + 2*w + 27],
[787, 787, -10*w^2 + 6*w + 73],
[797, 797, -5*w^2 + 6*w + 31],
[797, 797, 3*w^2 - 6*w - 5],
[797, 797, 2*w - 11],
[811, 811, 4*w^2 - 16*w + 9],
[821, 821, 4*w^2 - 8*w - 15],
[827, 827, -2*w^2 + 9*w - 11],
[839, 839, -3*w^2 + 8*w + 7],
[853, 853, -6*w + 5],
[863, 863, -w^2 + 8*w - 13],
[877, 877, 5*w - 7],
[881, 881, -7*w^2 + 6*w + 45],
[881, 881, -2*w^2 + 10*w - 7],
[881, 881, 3*w^2 - 4*w - 11],
[883, 883, 9*w - 1],
[883, 883, 2*w^2 - w + 3],
[883, 883, -2*w^2 + 8*w - 9],
[887, 887, 4*w^2 - 5*w - 27],
[887, 887, -2*w^2 + 5*w - 5],
[887, 887, -5*w - 7],
[929, 929, w^2 + 4*w - 7],
[941, 941, -2*w^2 + 2*w + 21],
[947, 947, -3*w - 11],
[961, 31, -4*w^2 + w + 21],
[967, 967, -2*w^2 + 11*w - 13],
[977, 977, 5*w^2 - 16*w - 1],
[983, 983, 2*w^2 + 2*w - 13],
[997, 997, 3*w^2 - 2*w - 13],
[997, 997, -2*w^2 - w - 3],
[997, 997, -4*w^2 + 4*w + 21]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 + 3*x^7 - 7*x^6 - 28*x^5 - 5*x^4 + 49*x^3 + 44*x^2 + 12*x + 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -e^7 - 3*e^6 + 7*e^5 + 28*e^4 + 5*e^3 - 50*e^2 - 44*e - 9, -7*e^7 - 19*e^6 + 54*e^5 + 180*e^4 - 13*e^3 - 334*e^2 - 217*e - 31, -1, 3*e^7 + 10*e^6 - 21*e^5 - 94*e^4 - 16*e^3 + 172*e^2 + 134*e + 22, -e^6 - 2*e^5 + 9*e^4 + 19*e^3 - 13*e^2 - 37*e - 12, -2*e^7 - 8*e^6 + 12*e^5 + 75*e^4 + 30*e^3 - 137*e^2 - 126*e - 21, e^7 + 3*e^6 - 6*e^5 - 28*e^4 - 13*e^3 + 49*e^2 + 56*e + 8, -3*e^7 - 8*e^6 + 23*e^5 + 76*e^4 - 5*e^3 - 141*e^2 - 94*e - 17, 4*e^7 + 11*e^6 - 32*e^5 - 104*e^4 + 18*e^3 + 195*e^2 + 102*e + 12, -5*e^7 - 13*e^6 + 37*e^5 + 123*e^4 + 4*e^3 - 222*e^2 - 178*e - 34, 3*e^7 + 8*e^6 - 23*e^5 - 77*e^4 + 5*e^3 + 147*e^2 + 94*e + 12, -11*e^7 - 29*e^6 + 86*e^5 + 275*e^4 - 30*e^3 - 512*e^2 - 328*e - 48, 7*e^7 + 20*e^6 - 52*e^5 - 190*e^4 - 7*e^3 + 354*e^2 + 258*e + 38, 7*e^7 + 17*e^6 - 55*e^5 - 161*e^4 + 24*e^3 + 294*e^2 + 194*e + 32, -5*e^7 - 11*e^6 + 42*e^5 + 104*e^4 - 42*e^3 - 191*e^2 - 95*e - 16, -5*e^7 - 14*e^6 + 39*e^5 + 131*e^4 - 11*e^3 - 237*e^2 - 157*e - 30, 5*e^7 + 17*e^6 - 36*e^5 - 159*e^4 - 18*e^3 + 290*e^2 + 205*e + 34, 3*e^7 + 7*e^6 - 24*e^5 - 67*e^4 + 14*e^3 + 124*e^2 + 77*e + 16, 13*e^7 + 35*e^6 - 98*e^5 - 331*e^4 + e^3 + 608*e^2 + 458*e + 72, e^7 + 2*e^6 - 8*e^5 - 18*e^4 + 4*e^3 + 26*e^2 + 31*e + 15, -14*e^7 - 38*e^6 + 108*e^5 + 360*e^4 - 22*e^3 - 668*e^2 - 452*e - 65, 4*e^7 + 10*e^6 - 30*e^5 - 96*e^4 - e^3 + 182*e^2 + 148*e + 14, 2*e^7 + 5*e^6 - 14*e^5 - 49*e^4 - 6*e^3 + 93*e^2 + 69*e + 11, 5*e^7 + 15*e^6 - 37*e^5 - 140*e^4 - 7*e^3 + 251*e^2 + 189*e + 33, 16*e^7 + 42*e^6 - 125*e^5 - 400*e^4 + 45*e^3 + 750*e^2 + 467*e + 62, 4*e^7 + 7*e^6 - 36*e^5 - 67*e^4 + 57*e^3 + 126*e^2 + 37*e + 2, 13*e^7 + 40*e^6 - 92*e^5 - 377*e^4 - 56*e^3 + 690*e^2 + 555*e + 92, -20*e^7 - 51*e^6 + 162*e^5 + 486*e^4 - 107*e^3 - 919*e^2 - 497*e - 50, 24*e^7 + 67*e^6 - 183*e^5 - 636*e^4 + 24*e^3 + 1186*e^2 + 787*e + 111, -e^7 - 3*e^6 + 3*e^5 + 27*e^4 + 42*e^3 - 37*e^2 - 124*e - 35, 16*e^7 + 42*e^6 - 127*e^5 - 397*e^4 + 62*e^3 + 733*e^2 + 439*e + 56, -15*e^7 - 46*e^6 + 113*e^5 + 434*e^4 + 3*e^3 - 808*e^2 - 527*e - 80, -e^7 - e^6 + 9*e^5 + 11*e^4 - 18*e^3 - 26*e^2 + 8*e + 2, 27*e^7 + 64*e^6 - 219*e^5 - 610*e^4 + 155*e^3 + 1139*e^2 + 643*e + 82, -e^7 + 3*e^6 + 16*e^5 - 27*e^4 - 85*e^3 + 46*e^2 + 134*e + 29, e^7 + 7*e^6 - 6*e^5 - 67*e^4 - 19*e^3 + 132*e^2 + 76*e + 16, e^7 - 4*e^6 - 17*e^5 + 33*e^4 + 93*e^3 - 42*e^2 - 139*e - 41, -e^7 + 10*e^5 - e^4 - 23*e^3 + 9*e^2 + 6*e, 12*e^7 + 31*e^6 - 90*e^5 - 293*e^4 - 4*e^3 + 532*e^2 + 438*e + 78, 5*e^7 + 17*e^6 - 37*e^5 - 160*e^4 - 12*e^3 + 300*e^2 + 206*e + 19, -8*e^7 - 27*e^6 + 55*e^5 + 254*e^4 + 55*e^3 - 468*e^2 - 395*e - 67, -e^7 + 2*e^6 + 16*e^5 - 18*e^4 - 80*e^3 + 28*e^2 + 117*e + 35, 16*e^7 + 48*e^6 - 121*e^5 - 454*e^4 + 3*e^3 + 850*e^2 + 549*e + 60, 2*e^7 + e^6 - 18*e^5 - 10*e^4 + 31*e^3 + 12*e^2 + 14*e + 9, -31*e^7 - 82*e^6 + 240*e^5 + 776*e^4 - 62*e^3 - 1433*e^2 - 969*e - 143, -14*e^7 - 41*e^6 + 109*e^5 + 389*e^4 - 30*e^3 - 735*e^2 - 433*e - 41, 16*e^7 + 39*e^6 - 129*e^5 - 371*e^4 + 82*e^3 + 691*e^2 + 407*e + 56, 20*e^7 + 52*e^6 - 154*e^5 - 492*e^4 + 34*e^3 + 903*e^2 + 631*e + 89, -30*e^7 - 74*e^6 + 242*e^5 + 705*e^4 - 156*e^3 - 1323*e^2 - 757*e - 90, 37*e^7 + 104*e^6 - 283*e^5 - 984*e^4 + 37*e^3 + 1825*e^2 + 1227*e + 170, 23*e^7 + 55*e^6 - 190*e^5 - 524*e^4 + 158*e^3 + 986*e^2 + 511*e + 52, -4*e^7 - 11*e^6 + 35*e^5 + 109*e^4 - 49*e^3 - 233*e^2 - 40*e + 28, 7*e^7 + 18*e^6 - 52*e^5 - 172*e^4 - 2*e^3 + 318*e^2 + 243*e + 44, 6*e^7 + 19*e^6 - 46*e^5 - 179*e^4 + 8*e^3 + 334*e^2 + 190*e + 27, 38*e^7 + 103*e^6 - 293*e^5 - 976*e^4 + 66*e^3 + 1813*e^2 + 1202*e + 168, 5*e^7 + 15*e^6 - 41*e^5 - 139*e^4 + 29*e^3 + 253*e^2 + 116*e + 17, -16*e^7 - 41*e^6 + 125*e^5 + 387*e^4 - 46*e^3 - 705*e^2 - 463*e - 87, 15*e^7 + 39*e^6 - 119*e^5 - 370*e^4 + 57*e^3 + 692*e^2 + 417*e + 43, -9*e^7 - 16*e^6 + 83*e^5 + 156*e^4 - 144*e^3 - 310*e^2 - 54*e + 18, 16*e^7 + 36*e^6 - 131*e^5 - 343*e^4 + 108*e^3 + 639*e^2 + 339*e + 40, 19*e^7 + 52*e^6 - 151*e^5 - 494*e^4 + 71*e^3 + 931*e^2 + 529*e + 64, 4*e^7 + 10*e^6 - 35*e^5 - 96*e^4 + 40*e^3 + 187*e^2 + 79*e + 5, -20*e^7 - 51*e^6 + 152*e^5 + 484*e^4 - 21*e^3 - 885*e^2 - 645*e - 112, -36*e^7 - 100*e^6 + 274*e^5 + 945*e^4 - 29*e^3 - 1746*e^2 - 1189*e - 169, 30*e^7 + 83*e^6 - 227*e^5 - 785*e^4 + 13*e^3 + 1449*e^2 + 1021*e + 145, -e^7 - 8*e^6 + 76*e^4 + 67*e^3 - 144*e^2 - 143*e - 8, 31*e^7 + 89*e^6 - 235*e^5 - 841*e^4 + 14*e^3 + 1557*e^2 + 1044*e + 140, -7*e^7 - 16*e^6 + 53*e^5 + 155*e^4 - 6*e^3 - 293*e^2 - 242*e - 36, 12*e^7 + 33*e^6 - 90*e^5 - 310*e^4 - 7*e^3 + 559*e^2 + 439*e + 80, -31*e^7 - 88*e^6 + 236*e^5 + 829*e^4 - 22*e^3 - 1522*e^2 - 1033*e - 161, 29*e^7 + 85*e^6 - 216*e^5 - 803*e^4 - 20*e^3 + 1485*e^2 + 1033*e + 152, -17*e^7 - 51*e^6 + 125*e^5 + 484*e^4 + 28*e^3 - 904*e^2 - 645*e - 99, 5*e^7 + 7*e^6 - 45*e^5 - 68*e^4 + 76*e^3 + 125*e^2 + 24*e + 8, 17*e^7 + 47*e^6 - 127*e^5 - 447*e^4 - 5*e^3 + 834*e^2 + 595*e + 64, -5*e^7 - 16*e^6 + 38*e^5 + 153*e^4 - 6*e^3 - 297*e^2 - 157*e + 2, 20*e^7 + 50*e^6 - 159*e^5 - 473*e^4 + 80*e^3 + 872*e^2 + 543*e + 75, 38*e^7 + 104*e^6 - 290*e^5 - 986*e^4 + 41*e^3 + 1829*e^2 + 1231*e + 183, 40*e^7 + 105*e^6 - 315*e^5 - 999*e^4 + 131*e^3 + 1872*e^2 + 1154*e + 146, 6*e^7 + 23*e^6 - 33*e^5 - 214*e^4 - 116*e^3 + 374*e^2 + 435*e + 95, -19*e^7 - 47*e^6 + 152*e^5 + 448*e^4 - 92*e^3 - 840*e^2 - 467*e - 59, 7*e^7 + 12*e^6 - 58*e^5 - 115*e^4 + 50*e^3 + 207*e^2 + 169*e + 36, -37*e^7 - 100*e^6 + 289*e^5 + 951*e^4 - 99*e^3 - 1788*e^2 - 1097*e - 141, 15*e^7 + 39*e^6 - 121*e^5 - 372*e^4 + 77*e^3 + 703*e^2 + 380*e + 36, 24*e^7 + 74*e^6 - 175*e^5 - 696*e^4 - 60*e^3 + 1274*e^2 + 947*e + 153, 24*e^7 + 71*e^6 - 178*e^5 - 672*e^4 - 31*e^3 + 1249*e^2 + 903*e + 133, 10*e^7 + 25*e^6 - 84*e^5 - 239*e^4 + 87*e^3 + 462*e^2 + 167*e - 12, -42*e^7 - 115*e^6 + 326*e^5 + 1091*e^4 - 93*e^3 - 2035*e^2 - 1283*e - 175, 2*e^7 - e^6 - 17*e^5 + 10*e^4 + 21*e^3 - 36*e^2 + 38*e + 25, -36*e^7 - 90*e^6 + 285*e^5 + 855*e^4 - 137*e^3 - 1585*e^2 - 991*e - 146, 6*e^7 + 18*e^6 - 47*e^5 - 171*e^4 + 20*e^3 + 325*e^2 + 155*e + 20, 28*e^7 + 85*e^6 - 207*e^5 - 804*e^4 - 39*e^3 + 1499*e^2 + 1046*e + 133, -4*e^7 - 9*e^6 + 35*e^5 + 85*e^4 - 49*e^3 - 156*e^2 - 39*e - 20, -25*e^7 - 62*e^6 + 199*e^5 + 592*e^4 - 109*e^3 - 1114*e^2 - 665*e - 81, 9*e^7 + 24*e^6 - 70*e^5 - 228*e^4 + 22*e^3 + 431*e^2 + 267*e + 27, -6*e^7 - 13*e^6 + 53*e^5 + 127*e^4 - 73*e^3 - 256*e^2 - 78*e + 20, 19*e^7 + 58*e^6 - 140*e^5 - 548*e^4 - 30*e^3 + 1017*e^2 + 701*e + 97, -37*e^7 - 96*e^6 + 296*e^5 + 911*e^4 - 165*e^3 - 1702*e^2 - 970*e - 119, -10*e^7 - 28*e^6 + 77*e^5 + 267*e^4 - 19*e^3 - 498*e^2 - 298*e - 57, 43*e^7 + 124*e^6 - 322*e^5 - 1175*e^4 - 19*e^3 + 2184*e^2 + 1546*e + 219, -47*e^7 - 130*e^6 + 361*e^5 + 1236*e^4 - 69*e^3 - 2318*e^2 - 1497*e - 198, -25*e^7 - 78*e^6 + 174*e^5 + 736*e^4 + 134*e^3 - 1350*e^2 - 1121*e - 190, -48*e^7 - 136*e^6 + 367*e^5 + 1287*e^4 - 53*e^3 - 2388*e^2 - 1556*e - 209, -10*e^7 - 31*e^6 + 78*e^5 + 295*e^4 - 26*e^3 - 559*e^2 - 286*e - 39, -39*e^7 - 110*e^6 + 299*e^5 + 1040*e^4 - 50*e^3 - 1929*e^2 - 1255*e - 179, 57*e^7 + 164*e^6 - 432*e^5 - 1552*e^4 + 29*e^3 + 2883*e^2 + 1908*e + 261, 5*e^7 + 22*e^6 - 29*e^5 - 208*e^4 - 84*e^3 + 384*e^2 + 333*e + 68, -e^7 - 7*e^6 + 8*e^5 + 66*e^4 + 3*e^3 - 131*e^2 - 46*e - 16, -2*e^7 - 11*e^6 + 17*e^5 + 105*e^4 - 13*e^3 - 215*e^2 - 34*e + 17, -11*e^7 - 34*e^6 + 78*e^5 + 321*e^4 + 50*e^3 - 592*e^2 - 487*e - 79, -12*e^7 - 33*e^6 + 93*e^5 + 312*e^4 - 28*e^3 - 572*e^2 - 355*e - 56, 16*e^7 + 40*e^6 - 131*e^5 - 380*e^4 + 96*e^3 + 717*e^2 + 387*e + 32, -26*e^7 - 83*e^6 + 185*e^5 + 780*e^4 + 109*e^3 - 1425*e^2 - 1112*e - 182, 16*e^7 + 30*e^6 - 136*e^5 - 288*e^4 + 155*e^3 + 534*e^2 + 261*e + 28, -2*e^7 + e^6 + 20*e^5 - 9*e^4 - 46*e^3 + 25*e^2 + 6*e - 23, -40*e^7 - 111*e^6 + 304*e^5 + 1049*e^4 - 21*e^3 - 1939*e^2 - 1364*e - 191, 25*e^7 + 78*e^6 - 177*e^5 - 735*e^4 - 109*e^3 + 1346*e^2 + 1077*e + 155, 49*e^7 + 133*e^6 - 377*e^5 - 1261*e^4 + 75*e^3 + 2347*e^2 + 1573*e + 215, -23*e^7 - 72*e^6 + 169*e^5 + 680*e^4 + 45*e^3 - 1262*e^2 - 887*e - 142, -11*e^7 - 19*e^6 + 96*e^5 + 186*e^4 - 131*e^3 - 356*e^2 - 141*e - 27, -6*e^5 - e^4 + 60*e^3 + 13*e^2 - 127*e - 26, -28*e^7 - 72*e^6 + 221*e^5 + 681*e^4 - 99*e^3 - 1251*e^2 - 788*e - 129, -8*e^7 - 28*e^6 + 50*e^5 + 264*e^4 + 99*e^3 - 481*e^2 - 457*e - 95, -5*e^7 - 18*e^6 + 25*e^5 + 169*e^4 + 113*e^3 - 288*e^2 - 369*e - 97, -24*e^7 - 64*e^6 + 187*e^5 + 608*e^4 - 67*e^3 - 1141*e^2 - 686*e - 72, -18*e^7 - 42*e^6 + 145*e^5 + 403*e^4 - 96*e^3 - 762*e^2 - 436*e - 43, -e^7 - 17*e^6 - 10*e^5 + 155*e^4 + 171*e^3 - 273*e^2 - 351*e - 57, 25*e^7 + 71*e^6 - 187*e^5 - 673*e^4 - 11*e^3 + 1250*e^2 + 893*e + 140, 24*e^7 + 62*e^6 - 185*e^5 - 590*e^4 + 46*e^3 + 1097*e^2 + 759*e + 129, -7*e^7 - 12*e^6 + 62*e^5 + 117*e^4 - 95*e^3 - 230*e^2 - 52*e + 19, 15*e^7 + 47*e^6 - 106*e^5 - 444*e^4 - 65*e^3 + 819*e^2 + 634*e + 104, -30*e^7 - 90*e^6 + 217*e^5 + 848*e^4 + 93*e^3 - 1551*e^2 - 1241*e - 211, 59*e^7 + 161*e^6 - 453*e^5 - 1524*e^4 + 87*e^3 + 2818*e^2 + 1875*e + 263, -35*e^7 - 94*e^6 + 264*e^5 + 891*e^4 - 15*e^3 - 1644*e^2 - 1168*e - 167, -57*e^7 - 161*e^6 + 430*e^5 + 1527*e^4 - 15*e^3 - 2837*e^2 - 1928*e - 281, 50*e^7 + 132*e^6 - 389*e^5 - 1253*e^4 + 125*e^3 + 2328*e^2 + 1498*e + 209, -5*e^7 - 16*e^6 + 36*e^5 + 150*e^4 + 15*e^3 - 265*e^2 - 194*e - 43, -5*e^7 - 18*e^6 + 33*e^5 + 167*e^4 + 43*e^3 - 297*e^2 - 247*e - 55, -11*e^7 - 46*e^6 + 65*e^5 + 433*e^4 + 179*e^3 - 795*e^2 - 737*e - 135, -17*e^7 - 45*e^6 + 136*e^5 + 428*e^4 - 71*e^3 - 805*e^2 - 472*e - 46, -48*e^7 - 133*e^6 + 364*e^5 + 1259*e^4 - 24*e^3 - 2337*e^2 - 1632*e - 221, 7*e^7 + 20*e^6 - 48*e^5 - 190*e^4 - 43*e^3 + 349*e^2 + 332*e + 65, -38*e^7 - 99*e^6 + 300*e^5 + 938*e^4 - 127*e^3 - 1746*e^2 - 1109*e - 147, -28*e^7 - 76*e^6 + 224*e^5 + 721*e^4 - 129*e^3 - 1357*e^2 - 705*e - 70, -17*e^7 - 50*e^6 + 121*e^5 + 468*e^4 + 67*e^3 - 835*e^2 - 713*e - 133, e^7 + 10*e^6 + 5*e^5 - 93*e^4 - 122*e^3 + 157*e^2 + 270*e + 62, 3*e^7 + 16*e^6 - 6*e^5 - 141*e^4 - 162*e^3 + 210*e^2 + 423*e + 119, -47*e^7 - 125*e^6 + 360*e^5 + 1183*e^4 - 57*e^3 - 2184*e^2 - 1544*e - 242, -3*e^7 - 7*e^6 + 24*e^5 + 70*e^4 - 26*e^3 - 142*e^2 - 17*e + 8, -31*e^7 - 90*e^6 + 234*e^5 + 854*e^4 - 3*e^3 - 1606*e^2 - 1082*e - 120, -37*e^7 - 98*e^6 + 284*e^5 + 931*e^4 - 59*e^3 - 1729*e^2 - 1166*e - 163, -7*e^7 - 16*e^6 + 56*e^5 + 150*e^4 - 36*e^3 - 275*e^2 - 162*e - 23, 6*e^7 + 21*e^6 - 43*e^5 - 193*e^4 - 22*e^3 + 337*e^2 + 246*e + 71, 17*e^7 + 34*e^6 - 151*e^5 - 329*e^4 + 221*e^3 + 644*e^2 + 181*e - 20, -10*e^7 - 30*e^6 + 75*e^5 + 279*e^4 + 4*e^3 - 486*e^2 - 349*e - 97, 32*e^7 + 91*e^6 - 242*e^5 - 858*e^4 + 9*e^3 + 1579*e^2 + 1097*e + 164, -e^7 - 2*e^6 + 6*e^5 + 19*e^4 + 14*e^3 - 29*e^2 - 59*e + 8, e^7 + 4*e^6 - 8*e^5 - 35*e^4 + e^3 + 63*e^2 + 44*e - 4, -71*e^7 - 202*e^6 + 541*e^5 + 1913*e^4 - 60*e^3 - 3563*e^2 - 2348*e - 332, -61*e^7 - 163*e^6 + 471*e^5 + 1545*e^4 - 115*e^3 - 2864*e^2 - 1919*e - 280, 34*e^7 + 109*e^6 - 239*e^5 - 1028*e^4 - 160*e^3 + 1897*e^2 + 1462*e + 206, 37*e^7 + 100*e^6 - 289*e^5 - 954*e^4 + 98*e^3 + 1804*e^2 + 1112*e + 129, -15*e^7 - 36*e^6 + 118*e^5 + 339*e^4 - 49*e^3 - 612*e^2 - 418*e - 63, -46*e^7 - 129*e^6 + 355*e^5 + 1223*e^4 - 86*e^3 - 2284*e^2 - 1423*e - 174, -e^7 - 3*e^6 + 22*e^5 + 30*e^4 - 127*e^3 - 87*e^2 + 202*e + 72, -45*e^7 - 123*e^6 + 343*e^5 + 1165*e^4 - 36*e^3 - 2161*e^2 - 1507*e - 209, -45*e^7 - 116*e^6 + 361*e^5 + 1099*e^4 - 203*e^3 - 2047*e^2 - 1213*e - 168, -e^7 + e^6 + 8*e^5 - 12*e^4 - 7*e^3 + 38*e^2 - 17*e - 7, -e^7 - 5*e^6 - 2*e^5 + 49*e^4 + 90*e^3 - 89*e^2 - 221*e - 28, -18*e^7 - 45*e^6 + 142*e^5 + 432*e^4 - 65*e^3 - 820*e^2 - 496*e - 73];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {7, 7, w + 1}>] := 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;