
// Make newform 4029.2.a.j in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4029_a();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_4029_2_a_j();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
// To make the character of type GrpDrchElt, type "MakeCharacter_4029_a();"
function MakeCharacter_4029_a()
    N := 4029;
    order := 1;
    char_gens := [2687, 3556, 3163];
    v := [1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_4029_a_Hecke(Kf)
    return MakeCharacter_4029_a();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_4029_2_a_j();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_4029_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4029_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1820, -7204, -14348, 75684, 32100, -310190, 17430, 661566, -185786, -824589, 342451, 634369, -327389, -307229, 188611, 92400, -68652, -15984, 15860, 1129, -2249, 83, 178, -20, -6, 1]>,<5,R![9459, 75834, -225647, -3158938, -6082653, 9053688, 28866608, -7481197, -46429599, 2006543, 37851819, -154084, -17953707, 271606, 5284646, -220795, -991666, 71378, 118280, -11938, -8636, 1094, 350, -52, -6, 1]>],Snew);
    return Vf;
end function;