
// Make newform 680.2.h.c in Magma, downloaded from the LMFDB on 08 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_680_h();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_680_2_h_c();".
// 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_680_h();"
function MakeCharacter_680_h()
    N := 680;
    order := 2;
    char_gens := [511, 341, 137, 241];
    v := [2, 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_680_h_Hecke( : Kf := Kf)
    return MakeCharacter_680_h();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_680_2_h_c();".
// 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_680_2_h_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_680_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2888, 0, 6456, 0, 5100, 0, 1872, 0, 342, 0, 30, 0, 1]>],Snew);
    return Vf;
end function;