
// Make newform 169.4.e.h in Magma, downloaded from the LMFDB on 16 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_169_e();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_169_4_e_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_169_e();"
function MakeCharacter_169_e()
    N := 169;
    order := 6;
    char_gens := [2];
    v := [5];
    // 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_169_e_Hecke(Kf)
    return MakeCharacter_169_e();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_169_4_e_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_169_4_e_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_169_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![14003408896, 0, -733815736320, 0, 33713801137920, 0, -241200040669696, 0, 1414950649352560, 0, -1104690208428372, 0, 550763625040545, 0, -169109004288036, 0, 37515808927932, 0, -5894967845306, 0, 700620402282, 0, -62734700289, 0, 4367945071, 0, -233733762, 0, 9732537, 0, -304748, 0, 7095, 0, -109, 0, 1]>],Snew);
    return Vf;
end function;