
// Make newform 336.4.h.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_336_h();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_336_4_h_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_336_h();"
function MakeCharacter_336_h()
    N := 336;
    order := 2;
    char_gens := [127, 85, 113, 241];
    v := [1, 2, 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_336_h_Hecke(Kf)
    return MakeCharacter_336_h();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_336_4_h_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_336_4_h_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_336_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![7274042944, 0, 6129603648, 0, 693441840, 0, 18224416, 0, 176364, 0, 708, 0, 1]>],Snew);
    return Vf;
end function;