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

NN := ideal<ZF | {5, 5, w - 1}>;

primesArray := [
[2, 2, w],
[5, 5, w - 1],
[8, 2, -w^3 + w^2 + 6*w + 1],
[9, 3, -w^3 + w^2 + 5*w + 1],
[9, 3, -w^3 + 2*w^2 + 3*w - 1],
[19, 19, -w^3 + 2*w^2 + 4*w - 1],
[23, 23, w^2 - 2*w - 1],
[29, 29, w^2 - 3*w - 1],
[29, 29, -w^2 + w + 3],
[37, 37, w^3 - w^2 - 5*w + 1],
[43, 43, -w^3 + 2*w^2 + 3*w - 3],
[47, 47, -w^2 + 2*w + 5],
[47, 47, 2*w^3 - 2*w^2 - 11*w - 5],
[53, 53, -w^3 + 2*w^2 + 2*w + 1],
[59, 59, -w - 3],
[73, 73, w^2 - w + 1],
[73, 73, 2*w^3 - 2*w^2 - 12*w - 5],
[79, 79, 4*w^3 - 4*w^2 - 22*w - 7],
[97, 97, 2*w^3 - 3*w^2 - 9*w + 1],
[101, 101, 2*w^2 - 2*w - 9],
[103, 103, -w^3 + 2*w^2 + 3*w - 5],
[107, 107, 2*w^3 - 4*w^2 - 6*w + 1],
[113, 113, 2*w^2 - 4*w - 7],
[113, 113, w^3 - w^2 - 7*w + 1],
[125, 5, -2*w^3 + 5*w^2 + 7*w - 11],
[137, 137, -w^2 + 4*w + 3],
[137, 137, -w^3 + 3*w^2 + 3*w - 7],
[139, 139, -3*w - 1],
[151, 151, 2*w^3 - 2*w^2 - 11*w - 1],
[157, 157, -2*w^3 + 3*w^2 + 11*w - 3],
[163, 163, 2*w^3 - 3*w^2 - 8*w + 3],
[163, 163, -w^3 + w^2 + 7*w + 1],
[167, 167, 2*w^3 - 5*w^2 - 5*w + 1],
[167, 167, -w^3 + 7*w + 3],
[173, 173, -3*w^3 + 4*w^2 + 16*w - 3],
[181, 181, 2*w - 3],
[197, 197, -2*w^3 + 6*w^2 + 6*w - 17],
[199, 199, 2*w^3 - 4*w^2 - 5*w + 1],
[211, 211, w^3 - 2*w^2 - 6*w + 5],
[227, 227, -w^3 + 4*w^2 + 2*w - 13],
[227, 227, w^3 - 2*w^2 - 5*w - 3],
[229, 229, 2*w^2 - w - 7],
[233, 233, 2*w - 5],
[233, 233, 3*w^3 - 4*w^2 - 16*w - 1],
[239, 239, -2*w^3 + w^2 + 12*w + 11],
[239, 239, -3*w^3 + 2*w^2 + 19*w + 9],
[251, 251, 3*w^3 - 2*w^2 - 17*w - 7],
[257, 257, -w^3 - w^2 + 7*w + 9],
[257, 257, 3*w^3 - 4*w^2 - 15*w + 3],
[269, 269, -3*w^3 + 4*w^2 + 14*w + 3],
[271, 271, 2*w^3 - 14*w - 15],
[277, 277, w^3 - 2*w^2 - w + 3],
[277, 277, 2*w^3 - 6*w^2 + w + 5],
[281, 281, w^3 - 2*w^2 - 3*w - 3],
[281, 281, -2*w^3 + 3*w^2 + 10*w + 1],
[289, 17, w^3 + 2*w^2 - 10*w - 17],
[289, 17, 3*w^3 - 6*w^2 - 13*w + 13],
[293, 293, 2*w^3 - w^2 - 13*w - 5],
[311, 311, w^3 - w^2 - 3*w - 3],
[311, 311, 2*w^3 - 3*w^2 - 11*w - 1],
[313, 313, -2*w^3 + 4*w^2 + 9*w - 3],
[317, 317, -3*w^3 + 3*w^2 + 17*w + 7],
[331, 331, -2*w^3 + 3*w^2 + 8*w - 1],
[337, 337, -4*w^3 + 6*w^2 + 19*w - 3],
[347, 347, w^3 - 3*w^2 - 5*w + 1],
[347, 347, -2*w^3 + 2*w^2 + 9*w - 3],
[347, 347, 3*w^3 - 6*w^2 - 12*w + 11],
[347, 347, w - 5],
[349, 349, -w^3 + 5*w^2 - 3*w - 5],
[349, 349, 2*w^3 - w^2 - 13*w - 11],
[353, 353, w^3 - 2*w^2 - 4*w - 3],
[353, 353, 2*w^2 - 6*w - 5],
[359, 359, -2*w^3 + 3*w^2 + 6*w + 1],
[367, 367, 2*w^3 - 2*w^2 - 8*w + 1],
[367, 367, w^2 - 7],
[373, 373, w^3 + w^2 - 7*w - 11],
[373, 373, -2*w^3 + 2*w^2 + 12*w - 1],
[373, 373, -2*w^3 + 15*w + 13],
[373, 373, -3*w^3 + 3*w^2 + 15*w + 7],
[379, 379, -2*w^3 + w^2 + 11*w + 9],
[397, 397, -w^3 + 3*w^2 + 3*w - 11],
[397, 397, -w^3 + 9*w - 1],
[401, 401, -2*w^3 + 4*w^2 + 7*w - 7],
[419, 419, -w^3 + 2*w^2 + 2*w - 5],
[419, 419, -2*w^3 + 4*w^2 + 6*w + 3],
[421, 421, 2*w^2 - 2*w - 7],
[421, 421, 4*w^3 - 5*w^2 - 22*w - 1],
[421, 421, w^2 - 3*w - 9],
[421, 421, 3*w^2 - 5*w - 5],
[431, 431, w^3 - 4*w^2 + 5],
[431, 431, -w^3 + 6*w + 1],
[433, 433, -2*w^3 + 4*w^2 + 10*w - 9],
[443, 443, -w^3 + 2*w^2 + 4*w - 7],
[443, 443, -3*w^3 + 6*w^2 + 10*w - 7],
[479, 479, 2*w^2 - w - 9],
[479, 479, 2*w^3 - 4*w^2 - 7*w - 3],
[487, 487, 4*w^3 - 3*w^2 - 24*w - 9],
[487, 487, -w^3 + 4*w^2 - 3*w - 3],
[491, 491, -w^3 + 9*w + 3],
[491, 491, -2*w^3 + 4*w^2 + 8*w - 1],
[503, 503, 3*w^3 - 5*w^2 - 15*w + 3],
[503, 503, w^3 - 2*w^2 - 8*w + 5],
[509, 509, 2*w^3 - 2*w^2 - 8*w - 5],
[509, 509, 2*w^2 - 4*w - 1],
[521, 521, 3*w^3 - 4*w^2 - 15*w - 3],
[541, 541, 3*w^3 - 2*w^2 - 18*w - 7],
[541, 541, -2*w^3 + 5*w^2 + 6*w - 1],
[547, 547, 2*w^3 - 2*w^2 - 9*w - 7],
[557, 557, -2*w^2 + 4*w + 9],
[563, 563, -2*w^3 + 5*w^2 + 7*w - 9],
[571, 571, 3*w^3 - 7*w^2 - 7*w - 1],
[571, 571, -2*w^3 + 5*w^2 + 7*w - 3],
[577, 577, -3*w^3 + 4*w^2 + 16*w - 9],
[587, 587, w^2 + w - 5],
[593, 593, 4*w^3 - 5*w^2 - 20*w - 3],
[593, 593, 3*w^2 - 5*w - 11],
[601, 601, -2*w^2 + 6*w + 3],
[601, 601, 3*w^3 - 6*w^2 - 11*w + 3],
[607, 607, -w - 5],
[613, 613, w^3 - 6*w + 1],
[619, 619, -2*w^3 + 3*w^2 + 6*w - 1],
[647, 647, -3*w^2 + 6*w + 5],
[653, 653, 2*w^3 - w^2 - 11*w - 11],
[661, 661, -2*w^3 + 3*w^2 + 6*w - 5],
[661, 661, -2*w^3 + 3*w^2 + 12*w + 3],
[673, 673, w^2 - 2*w + 3],
[683, 683, 2*w^3 - 2*w^2 - 8*w - 1],
[691, 691, 2*w^3 - 3*w^2 - 12*w + 1],
[691, 691, 3*w^3 - 4*w^2 - 14*w - 1],
[691, 691, -3*w^3 + 6*w^2 + 10*w + 1],
[691, 691, -2*w^3 + 2*w^2 + 10*w - 3],
[727, 727, -w^3 - 2*w^2 + 9*w + 15],
[733, 733, 2*w^3 - 5*w^2 - 3*w + 5],
[733, 733, 2*w^3 - 2*w^2 - 5*w + 1],
[761, 761, -w^3 + 3*w^2 + w - 7],
[769, 769, -2*w^3 + 4*w^2 + 5*w - 5],
[773, 773, -2*w^3 + 2*w^2 + 13*w + 3],
[787, 787, -w^3 + 4*w^2 + w - 5],
[809, 809, 6*w^3 - 9*w^2 - 30*w + 5],
[809, 809, 2*w^2 - 4*w - 11],
[809, 809, -3*w^2 + 4*w + 13],
[809, 809, w^3 - 8*w + 1],
[811, 811, -5*w^3 + 11*w^2 + 11*w - 1],
[823, 823, 3*w^3 - 6*w^2 - 7*w + 1],
[827, 827, 3*w^3 - 20*w - 17],
[829, 829, 4*w^3 - 4*w^2 - 23*w - 3],
[839, 839, -w^3 + 6*w + 11],
[841, 29, 4*w^3 - 4*w^2 - 22*w - 11],
[853, 853, -w^2 - 3],
[853, 853, 2*w^3 - w^2 - 11*w - 3],
[857, 857, 2*w^3 - 2*w^2 - 12*w + 3],
[863, 863, w^2 - w - 9],
[877, 877, 4*w^3 - 5*w^2 - 20*w - 1],
[877, 877, w^3 - w^2 - 3*w - 5],
[881, 881, -w^3 + 2*w^2 + w - 5],
[887, 887, -3*w^2 + 6*w + 11],
[919, 919, -3*w^3 + 22*w + 19],
[919, 919, 5*w^3 - 5*w^2 - 27*w - 9],
[937, 937, 4*w^3 - 4*w^2 - 23*w - 9],
[947, 947, w^3 + w^2 - 7*w - 13],
[967, 967, 3*w^3 - 3*w^2 - 13*w + 1],
[971, 971, -w^3 + 6*w^2 - 5*w - 7],
[971, 971, 4*w^3 - 4*w^2 - 21*w - 5],
[977, 977, -2*w^3 - 2*w^2 + 17*w + 21],
[983, 983, 3*w^3 - 3*w^2 - 17*w - 1],
[991, 991, -w^3 + 4*w^2 + 4*w - 5],
[991, 991, -3*w^3 + 6*w^2 + 13*w - 5]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - 12*x^6 + 38*x^4 - 35*x^2 + 4;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1, 1/2*e^7 - 6*e^5 + 18*e^3 - 23/2*e, -1/2*e^7 + 6*e^5 - 19*e^3 + 35/2*e, -e^7 + 11*e^5 - 28*e^3 + 16*e, e^5 - 10*e^3 + 17*e, -1/2*e^7 + 6*e^5 - 19*e^3 + 39/2*e, -e^7 + 12*e^5 - 38*e^3 + 33*e, e^6 - 11*e^4 + 27*e^2 - 11, e^7 - 12*e^5 + 38*e^3 - 33*e, -e^6 + 10*e^4 - 18*e^2 - 3, -e^4 + 9*e^2 - 8, 4*e, e^4 - 7*e^2 + 8, e^4 - 7*e^2 + 6, 2*e^6 - 20*e^4 + 40*e^2 - 12, 3/2*e^7 - 18*e^5 + 57*e^3 - 97/2*e, e^7 - 10*e^5 + 18*e^3 - e, -2*e^7 + 23*e^5 - 64*e^3 + 37*e, e^6 - 10*e^4 + 16*e^2 + 7, e^4 - 9*e^2 + 12, -e^7 + 10*e^5 - 20*e^3 + 7*e, -e^6 + 10*e^4 - 18*e^2 + 7, -1/2*e^7 + 6*e^5 - 19*e^3 + 35/2*e, 2*e^6 - 21*e^4 + 45*e^2 - 20, 2*e^7 - 22*e^5 + 56*e^3 - 30*e, -e^6 + 10*e^4 - 20*e^2 + 5, -2*e^7 + 22*e^5 - 54*e^3 + 18*e, 1/2*e^7 - 7*e^5 + 29*e^3 - 65/2*e, -3/2*e^7 + 18*e^5 - 57*e^3 + 89/2*e, -e^7 + 10*e^5 - 20*e^3 + 7*e, 2*e^7 - 23*e^5 + 66*e^3 - 45*e, -e^7 + 12*e^5 - 38*e^3 + 35*e, -3/2*e^7 + 19*e^5 - 65*e^3 + 115/2*e, 1/2*e^7 - 9*e^5 + 47*e^3 - 117/2*e, -e^4 + 9*e^2 - 22, -e^6 + 12*e^4 - 38*e^2 + 29, -3/2*e^7 + 18*e^5 - 59*e^3 + 137/2*e, -2*e^6 + 21*e^4 - 45*e^2 + 10, e^6 - 10*e^4 + 22*e^2 - 9, -2*e^6 + 20*e^4 - 36*e^2 + 6, 2*e^6 - 22*e^4 + 50*e^2 - 8, 2*e^2, -2*e^7 + 20*e^5 - 38*e^3 + 2*e, e^6 - 12*e^4 + 30*e^2 + 5, -1/2*e^7 + 5*e^5 - 9*e^3 - 19/2*e, e^7 - 9*e^5 + 10*e^3 + 10*e, 2*e^6 - 21*e^4 + 49*e^2 - 16, -4*e^7 + 46*e^5 - 130*e^3 + 82*e, e^7 - 11*e^5 + 24*e^3 + 4*e, -e^4 + 11*e^2 - 2, 4*e^6 - 42*e^4 + 88*e^2 - 20, 1/2*e^7 - 6*e^5 + 19*e^3 - 35/2*e, -4*e^6 + 43*e^4 - 99*e^2 + 42, -2*e^7 + 24*e^5 - 74*e^3 + 54*e, 4*e^6 - 40*e^4 + 80*e^2 - 22, -3*e^6 + 30*e^4 - 58*e^2 + 17, -e^7 + 14*e^5 - 54*e^3 + 47*e, 2*e^6 - 22*e^4 + 58*e^2 - 30, -3*e^7 + 34*e^5 - 92*e^3 + 53*e, e^7 - 10*e^5 + 18*e^3 - 3*e, 2*e^7 - 23*e^5 + 60*e^3 - 17*e, -1/2*e^7 + 3*e^5 + 11*e^3 - 87/2*e, -1/2*e^7 + 2*e^5 + 21*e^3 - 101/2*e, 5/2*e^7 - 30*e^5 + 97*e^3 - 199/2*e, -3/2*e^7 + 16*e^5 - 37*e^3 + 9/2*e, 3*e^6 - 33*e^4 + 87*e^2 - 37, 3*e^4 - 27*e^2 + 36, e^7 - 16*e^5 + 76*e^3 - 95*e, -4*e^2 + 2, e^4 - 9*e^2 + 22, 7/2*e^7 - 38*e^5 + 95*e^3 - 113/2*e, 3*e^7 - 31*e^5 + 66*e^3 - 22*e, e^7 - 14*e^5 + 54*e^3 - 49*e, 4*e^4 - 28*e^2 + 16, -e^6 + 11*e^4 - 23*e^2 - 13, -2*e^7 + 22*e^5 - 54*e^3 + 12*e, 2*e^6 - 20*e^4 + 32*e^2 + 16, 5*e^7 - 55*e^5 + 140*e^3 - 80*e, 2*e^6 - 24*e^4 + 74*e^2 - 36, -2*e^6 + 20*e^4 - 40*e^2 + 20, 3/2*e^7 - 20*e^5 + 75*e^3 - 145/2*e, e^6 - 10*e^4 + 22*e^2 - 27, 4*e^6 - 41*e^4 + 83*e^2 - 10, -e^6 + 14*e^4 - 56*e^2 + 39, 2*e^6 - 19*e^4 + 29*e^2 + 6, 2*e^5 - 20*e^3 + 32*e, -e^6 + 13*e^4 - 41*e^2 + 19, 2*e^7 - 23*e^5 + 64*e^3 - 37*e, e^6 - 9*e^4 + 11*e^2 - 3, -e^7 + 10*e^5 - 20*e^3 + 11*e, -3*e^6 + 30*e^4 - 58*e^2 + 5, 4*e^6 - 44*e^4 + 102*e^2 - 26, e^6 - 14*e^4 + 48*e^2 - 11, -2*e^4 + 10*e^2 + 12, -e^6 + 8*e^4 + 2*e^2 - 37, -2*e^7 + 21*e^5 - 44*e^3 - 11*e, -5*e^7 + 60*e^5 - 186*e^3 + 139*e, -3/2*e^7 + 20*e^5 - 81*e^3 + 217/2*e, -1/2*e^7 + 6*e^5 - 15*e^3 - 9/2*e, 9/2*e^7 - 50*e^5 + 133*e^3 - 171/2*e, e^6 - 15*e^4 + 57*e^2 - 35, -e^6 + 11*e^4 - 23*e^2 - 5, 5*e^7 - 55*e^5 + 138*e^3 - 78*e, 13/2*e^7 - 72*e^5 + 185*e^3 - 207/2*e, -2*e^7 + 22*e^5 - 56*e^3 + 34*e, 2*e^5 - 20*e^3 + 40*e, 2*e^6 - 18*e^4 + 12*e^2 + 32, -e^6 + 12*e^4 - 40*e^2 + 27, -2*e^6 + 21*e^4 - 39*e^2 - 20, -2*e^6 + 20*e^4 - 32*e^2 - 6, e^7 - 13*e^5 + 44*e^3 - 20*e, -4*e^4 + 42*e^2 - 44, 5*e^6 - 52*e^4 + 114*e^2 - 27, 6*e^7 - 67*e^5 + 176*e^3 - 113*e, -2*e^6 + 24*e^4 - 70*e^2 + 42, -2*e^5 + 20*e^3 - 32*e, 1/2*e^7 - 7*e^5 + 27*e^3 - 57/2*e, 3*e^6 - 35*e^4 + 97*e^2 - 49, 4*e^7 - 48*e^5 + 150*e^3 - 140*e, 4*e^7 - 41*e^5 + 84*e^3 - 15*e, -1/2*e^7 + 3*e^5 + 9*e^3 - 75/2*e, -12*e^2 + 42, -2*e^6 + 21*e^4 - 47*e^2 - 10, 1/2*e^7 - 3*e^5 - 11*e^3 + 67/2*e, 4*e^6 - 43*e^4 + 103*e^2 - 22, 2*e^7 - 25*e^5 + 88*e^3 - 93*e, 6*e^7 - 69*e^5 + 192*e^3 - 117*e, 2*e^7 - 23*e^5 + 64*e^3 - 43*e, 1/2*e^7 - 6*e^5 + 15*e^3 + 41/2*e, -4*e^7 + 46*e^5 - 134*e^3 + 116*e, -2*e^6 + 20*e^4 - 42*e^2 + 16, 2*e^4 - 6*e^2 - 22, -7/2*e^7 + 37*e^5 - 83*e^3 + 83/2*e, -2*e^4 + 14*e^2 - 14, -6*e^6 + 66*e^4 - 164*e^2 + 66, e^7 - 11*e^5 + 32*e^3 - 52*e, 3*e^6 - 36*e^4 + 106*e^2 - 41, 7/2*e^7 - 42*e^5 + 133*e^3 - 245/2*e, 4*e^6 - 40*e^4 + 72*e^2 - 10, -2*e^6 + 22*e^4 - 58*e^2 + 8, 4*e^7 - 46*e^5 + 136*e^3 - 136*e, 3*e^7 - 32*e^5 + 72*e^3 - 11*e, -9/2*e^7 + 54*e^5 - 169*e^3 + 307/2*e, -2*e^6 + 18*e^4 - 12*e^2 - 40, -5*e^7 + 56*e^5 - 148*e^3 + 91*e, -2*e^6 + 19*e^4 - 31*e^2 + 14, -2*e^7 + 24*e^5 - 74*e^3 + 46*e, 6*e^6 - 57*e^4 + 89*e^2 + 24, 13/2*e^7 - 73*e^5 + 201*e^3 - 309/2*e, -2*e^7 + 24*e^5 - 76*e^3 + 56*e, 5*e^6 - 50*e^4 + 88*e^2 - 3, 1/2*e^7 - 5*e^5 + 7*e^3 - 5/2*e, 6*e^6 - 60*e^4 + 110*e^2 + 2, 2*e^6 - 24*e^4 + 70*e^2 - 46, -2*e^6 + 19*e^4 - 29*e^2 - 8, 2*e^6 - 16*e^4 + 6*e^2 + 48, 9/2*e^7 - 46*e^5 + 91*e^3 - 15/2*e, -8*e^7 + 89*e^5 - 228*e^3 + 117*e, -2*e^6 + 16*e^4 - 12*e^2 + 6, 3*e^7 - 38*e^5 + 140*e^3 - 169*e, 7*e^7 - 81*e^5 + 236*e^3 - 166*e, 5/2*e^7 - 33*e^5 + 121*e^3 - 233/2*e, 2*e^6 - 27*e^4 + 99*e^2 - 60, -6*e^7 + 65*e^5 - 154*e^3 + 63*e, -5*e^6 + 57*e^4 - 149*e^2 + 73, 9/2*e^7 - 48*e^5 + 115*e^3 - 107/2*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {5, 5, 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;