
// Make newform 1680.2.cz.f in Magma, downloaded from the LMFDB on 13 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1680_cz();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1680_2_cz_f();".
// 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_1680_cz();"
function MakeCharacter_1680_cz()
    N := 1680;
    order := 4;
    char_gens := [1471, 421, 1121, 337, 241];
    v := [4, 4, 4, 1, 2];
    // 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_1680_cz_Hecke( : Kf := Kf)
    return MakeCharacter_1680_cz();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1680_2_cz_f();".
// 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_1680_2_cz_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1680_cz();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![38848, 40320, -84352, -39744, 49072, 15552, -11456, -2720, 1172, 184, -56, -4, 1]>,<13,R![2316304384, 4336910336, 4060086272, -5730467840, 2883190784, 677642240, 3303538688, -4519297024, 2677641216, -864534528, 855539712, -953081856, 634910720, -244082688, 60030976, -11300864, 3334784, -1112320, 260608, -34688, 3456, -544, 128, -16, 1]>],Snew);
    return Vf;
end function;