
// Make newform 441.2.f.h in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_441_f();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_441_2_f_h();".
// 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_441_f();"
function MakeCharacter_441_f()
    N := 441;
    order := 3;
    char_gens := [344, 199];
    v := [1, 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_441_f_Hecke(Kf)
    return MakeCharacter_441_f();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_441_2_f_h();".
// 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_441_2_f_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_441_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 14, 206, -116, 261, -74, 184, -58, 63, -10, 11, -2, 1]>,<5,R![2401, 0, 219030, 0, 19694250, 0, 25896660, 0, 22667115, 0, 10937664, 0, 3775328, 0, 782970, 0, 117495, 0, 11580, 0, 831, 0, 36, 0, 1]>],Snew);
    return Vf;
end function;