
// Make newform 420.2.t.d in Magma, downloaded from the LMFDB on 25 June 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_420_t();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_420_2_t_d();".
// 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_420_t();"
function MakeCharacter_420_t()
    N := 420;
    order := 4;
    char_gens := [211, 281, 337, 241];
    v := [2, 4, 3, 4];
    // 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_420_t_Hecke( : Kf := Kf)
    return MakeCharacter_420_t();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_420_2_t_d();".
// 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_420_2_t_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_420_t();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![1478656, 0, 90385408, 0, 63369280, 0, 16368064, 0, 2125296, 0, 152544, 0, 6076, 0, 124, 0, 1]>,<13,R![1478656, -4825088, 7872512, -225280, 1152256, -2728448, 2785792, -678144, 96320, -28032, 28416, -7072, 896, -32, 32, -8, 1]>],Snew);
    return Vf;
end function;