
// Make newform 20.8.e.b in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_20_e();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_20_8_e_b();".
// 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_20_e();"
function MakeCharacter_20_e()
    N := 20;
    order := 4;
    char_gens := [11, 17];
    v := [2, 3];
    // 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_20_e_Hecke(Kf)
    return MakeCharacter_20_e();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_20_8_e_b();".
// 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_20_8_e_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_20_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<3,R![165312069975923100792835278874106750812025153126400000000, 0, 0, 0, 642787462226018017116813446144632694302568349696000000, 0, 0, 0, 6471179081930250377601398006676846491766620160000, 0, 0, 0, 12702308689457813009152124947875714402816000, 0, 0, 0, 6431458674338579867288649661998028800, 0, 0, 0, 1336386804590928082970257125120, 0, 0, 0, 126981566234145846279424, 0, 0, 0, 5783684780560368, 0, 0, 0, 123741912, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;