
// Make newform 6336.2.k.d in Magma, downloaded from the LMFDB on 14 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_6336_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_6336_k_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_6336_2_k_d();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_6336_2_k_d();".
// 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 ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [729, -972, 648, -108, 0, -36, 56, -12, 0, -4, 8, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-80676, -66825, -71064, 2196, -3984, -17111, 1695, 4647, -920, -320, -8, 377], [9963, -1701, 108, 1530, 1032, -211, 96, 177, 104, -46, 20, 7], [-17739, 2673, -1836, -2682, -2010, 23, -336, -381, -166, 62, -52, -11], [508356, -558414, 221670, 18126, -9060, -41546, 21837, 30, -2108, -5018, 3718, -1270], [-36612, 39015, -12846, -1452, 528, 2857, -1317, -9, 136, 352, -254, 89], [-4374, 5265, -2160, -45, 150, 395, -228, 15, 26, 47, -40, 13], [-53460, -110322, 62478, -22743, -22788, -6328, 4614, -3240, -2140, -373, 686, -518], [910764, -1552527, 708642, -10899, -107448, -113141, 65982, -9447, -13952, -12137, 10882, -4789], [-47628, 43740, -15930, -3978, -588, 3248, -1704, -336, 44, 398, -310, 76], [-17982, 15066, -5940, -1665, -366, 1060, -708, -132, -2, 139, -116, 26], [-386856, 1686420, -843156, 197514, 171426, 113704, -68100, 33096, 21346, 9310, -10772, 6668]];
    Rf_basisdens := [1, 53217, 729, 729, 53217, 1971, 243, 5913, 53217, 729, 243, 53217];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_6336_k();"
function MakeCharacter_6336_k()
    N := 6336;
    order := 2;
    char_gens := [4159, 4357, 3521, 1729];
    v := [1, 1, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_6336_k_Hecke();"
function MakeCharacter_6336_k_Hecke( : Kf := false)
    N := 6336;
    order := 2;
    char_gens := [4159, 4357, 3521, 1729];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, -4, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0], [1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, -2, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, -2], [0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [8, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, -2, -1, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 5, 3, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, -2, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -1, 0], [14, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -3, -2, 0], [8, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, -8, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 1, 0, 0, -5, -2, 0], [8, 0, -4, -2, 0, 0, 0, 0, 0, 0, 0, 0], [4, 0, -5, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 3, -1, 0], [0, -2, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, -8, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -5], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, 0], [-2, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -2], [0, 0, 0, 0, 0, 0, 2, 0, 0, -1, 2, 0], [3, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 13, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, -1, 0, 0, 2], [0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, -5], [0, 0, 0, 0, 0, 0, 4, 0, 0, -1, -1, 0], [10, 0, 4, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, -3, -3, 0], [0, 0, 0, 0, 0, 0, 3, 0, 0, -1, 2, 0], [-16, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [7, 0, -5, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, 8, -2, 0], [-2, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, -6, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 6], [0, 0, 0, 0, 0, 0, 0, 1, -6, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 4, 0], [0, -2, 0, 0, -14, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, -3, 0, 0, 6], [0, 0, 0, 0, 0, 0, 0, 0, 0, -4, -2, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, -5, -2, 0], [21, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, -3], [0, 0, 0, 0, 0, 0, 0, -5, 2, 0, 0, 2], [-8, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -6, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, -3], [0, 0, 0, 0, 0, 0, 2, 0, 0, -1, -6, 0], [4, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 9, -3, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, -16, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 4, 0, 0, -2], [0, 0, 6, 4, 0, 0, 0, 0, 0, 0, 0, 0], [14, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, -10, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, -1, 2, 0], [0, 4, 0, 0, 20, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -8, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, -4, -4, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0], [-21, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 2, 0, 0, -2], [0, 0, 0, 0, 0, 0, 6, 0, 0, -3, -3, 0], [0, 0, 0, 0, 0, 0, 0, -2, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, -3, 1, 0], [-14, 0, 4, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, -8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 1], [0, 0, 0, 0, 0, 0, -6, 0, 0, -7, 1, 0], [-8, 0, -1, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, -14, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, -8, 0, 0, 5], [0, 0, 0, 0, 0, 0, 3, 0, 0, 3, -4, 0], [20, 0, -4, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, -27, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, -5, 0, 0, 4], [0, 0, 0, 0, 0, 0, 0, 3, 2, 0, 0, -4], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, -4, 0], [0, -1, 0, 0, -4, 7, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 5, -2, 0, 0, -4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0], [0, 0, 0, 0, 0, 0, 9, 0, 0, -1, -4, 0], [-25, 0, -1, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, -20, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, -2, 0, 0, -8], [-14, 0, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0], [16, 0, -5, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -14, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 4, 0], [2, 0, 2, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 0, 0, 3, -4, 0], [0, -2, 0, 0, 4, -8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -6, 5, 0, 0, -4], [0, 0, 0, 0, 0, 0, -2, 0, 0, 3, -1, 0], [0, 0, 0, 0, 0, 0, -6, 0, 0, -6, 4, 0], [0, 1, 0, 0, 0, -11, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 6, 0, 0, 5], [0, 0, 0, 0, 0, 0, 0, -4, -4, 0, 0, 5], [4, 0, -4, 2, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 0, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, -9, 3, 0], [-25, 0, -1, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 6, -6, 0, 0, 1], [0, 0, 0, 0, 0, 0, -10, 0, 0, -5, -1, 0], [8, 0, 11, 5, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, -10, 2, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 34, 5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, -4, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 4, 0], [0, 0, 0, 0, 0, 0, 0, 2, -5, 0, 0, 4], [0, 0, 0, 0, 0, 0, 0, -2, 6, 0, 0, -1], [0, 0, 0, 0, 0, 0, -4, 0, 0, 6, 2, 0], [0, 0, 0, 0, 0, 0, 3, 0, 0, -1, -6, 0], [0, 0, 0, 0, 0, 0, 0, -4, 3, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, 1, 8, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 4, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, -4, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, -3, 2, 0], [0, 4, 0, 0, 23, 5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5, 4, 0, 0, -4], [-14, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-8, 0, 1, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, -12, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 6, -2, 0, 0, -7], [0, -2, 0, 0, 9, 1, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 8, -12, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, -2], [0, 0, 0, 0, 0, 0, 3, 0, 0, 1, -2, 0], [0, -1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, -3, 0, 0, 0], [8, 0, 14, 4, 0, 0, 0, 0, 0, 0, 0, 0], [5, 0, 3, -8, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, -2, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, -4, 0, 0, -1], [0, 0, 0, 0, 0, 0, -2, 0, 0, -2, 4, 0], [12, 0, -8, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -14, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -7, -2, 0, 0, 4], [0, 0, 0, 0, 0, 0, -6, 0, 0, 1, 5, 0], [-1, 0, 5, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 25, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -6, -3, 0, 0, 2], [0, -1, 0, 0, 4, 7, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, -10, -11, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, -3], [0, 0, 0, 0, 0, 0, 0, 2, -10, 0, 0, 7], [0, 0, 0, 0, 0, 0, -7, 0, 0, -11, 2, 0], [21, 0, -3, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -8, 9, 0, 0, 2], [0, 0, 0, 0, 0, 0, -12, 0, 0, -6, 0, 0], [-2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, 0, 0, -13, 2, 0], [-18, 0, -3, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 5, 0, 0, -10, 15, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -12, 0, 0, -6, 0, 0], [-25, 0, -1, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 16, 8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -9, 4, 0, 0, -8], [-32, 0, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -8, 8, 0, 0, 3], [0, 0, 0, 0, 0, 0, 0, 2, -6, 0, 0, 3], [0, 0, 0, 0, 0, 0, 8, 0, 0, 8, 0, 0], [-18, 0, -3, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, -3, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -8, 4, 0, 0, 9], [2, 0, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, -6, 0, 0, -6, -12, 0, 0, 0, 0, 0, 0], [-7, 0, -11, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, 0, 22, 1, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 14, -7, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -8, 4, 0, 0, -5], [0, 0, 0, 0, 0, 0, 4, 0, 0, 8, -2, 0], [8, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0], [-31, 0, 1, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 5, 0, 0, -32, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 22, 10, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 4, 0], [2, 0, -17, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 5, 11, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 10, -8, 0, 0, -7], [0, 0, 0, 0, 0, 0, -5, 0, 0, 1, 2, 0], [-31, 0, -13, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, -3, 0, 0, 4], [0, 2, 0, 0, -15, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, -8, 0, 0, 2], [0, 4, 0, 0, -6, -12, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 40, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 3, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, -1, -2, 0, 0, 10], [0, 0, 0, 0, 0, 0, -9, 0, 0, -3, 4, 0], [0, -10, 0, 0, -3, -3, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, -12, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4], [0, 0, 0, 0, 0, 0, 0, 0, 0, -5, -9, 0], [0, -6, 0, 0, 31, 11, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 8, -8, 0, 0, -1], [0, 0, 0, 0, 0, 0, -8, 0, 0, -10, -4, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, -2, -12, 0], [-16, 0, -8, -4, 0, 0, 0, 0, 0, 0, 0, 0], [-18, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 26, 12, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, -5, -4, 0], [0, 4, 0, 0, -28, 16, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 13, 0], [0, -9, 0, 0, 8, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 14, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -10, 8, 0, 0, 7], [0, 0, 0, 0, 0, 0, -6, 0, 0, 4, 4, 0], [0, 4, 0, 0, -36, 10, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 8, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 15, -1, 0], [0, 0, 0, 0, 0, 0, 6, 0, 0, -6, -4, 0], [0, 0, 0, 0, 0, 0, 0, -10, -3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 11], [0, 0, 0, 0, 0, 0, -5, 0, 0, 5, 4, 0], [32, 0, -9, 3, 0, 0, 0, 0, 0, 0, 0, 0], [14, 0, -7, -7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 9, 0, 0, 10, 7, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, 0, 0, -9, 3, 0], [-35, 0, 15, -4, 0, 0, 0, 0, 0, 0, 0, 0], [30, 0, 6, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, -9, -13, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 6], [0, 0, 0, 0, 0, 0, 0, -4, -3, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, 11, -8, 0, 0, -4], [0, 0, 0, 0, 0, 0, 4, 0, 0, -4, -2, 0], [-9, 0, 3, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -11, 6, 0, 0, 2], [0, 0, 0, 0, 0, 0, -10, 0, 0, -9, 7, 0], [-20, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 4, 0, 0, 1], [0, 0, 0, 0, 0, 0, 4, 0, 0, 6, 4, 0], [-22, 0, 11, 7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, -9, 3, 0, 0, 0, 0, 0, 0], [28, 0, -12, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 40, 0, 0, 0, 0, 0, 0, 0], [0, -6, 0, 0, 26, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0], [28, 0, -14, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, -40, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, 0, 0, 1, 14, 0], [0, 4, 0, 0, -20, 14, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -10, 0, 0, -4, 4, 0], [0, 0, 0, 0, 0, 0, 0, -6, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 6, 0, 0, 9, 5, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, 13, 6, 0], [20, 0, -1, 3, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, -2, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, -28, -8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 7], [31, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 30, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, -4, 0, 0, 7], [0, 0, 0, 0, -24, -12, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -8, 1, 0, 0, 0], [-2, 0, 9, -7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -52, -6, 0, 0, 0, 0, 0, 0], [55, 0, -3, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 42, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, 0, 0, -3, 5, 0], [-32, 0, 8, -2, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 8, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, -10, 0, 0, 7, -5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 30, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, 0, 0, -10, -2, 0], [-2, 0, -10, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, -26, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 6, 0, 0, -7], [0, 0, 0, 0, 0, 0, 13, 0, 0, 7, 0, 0], [0, 0, 0, 0, 0, 0, 0, 12, 3, 0, 0, 0], [0, 0, 0, 0, 0, 0, -12, 0, 0, -7, -2, 0], [7, 0, -11, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -12, 2, 0, 0, 5], [0, 0, 0, 0, 0, 0, -4, 0, 0, -14, -4, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, -7, 6, 0], [-28, 0, 14, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, -26, 8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 0, 0, -10, -12, 0], [-8, 0, -14, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, -18, 0, 0, 7], [0, 0, 0, 0, 0, 0, 12, 0, 0, 5, -3, 0], [-32, 0, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0], [14, 0, 3, 7, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 24, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 6, 8, 0, 0, 1], [-38, 0, -5, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, -16, -6, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 8, 0], [0, 0, 0, 0, 0, 0, -12, 0, 0, -7, -2, 0], [-20, 0, 8, 12, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, -7, 0, 0, -4], [-11, 0, -5, -8, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 5, 15, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, -56, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 6, 0, 0, -9, -9, 0], [0, 0, 0, 0, 44, 8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, -16, 0, 0, 9], [0, 0, 0, 0, 0, 0, 0, 10, 4, 0, 0, -7], [-44, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, -39, -19, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 9], [0, 0, 0, 0, 0, 0, 0, -5, 6, 0, 0, -14], [-6, 0, 12, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 46, -10, 0, 0, 0, 0, 0, 0], [39, 0, -9, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 7, 0, 0, 14, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 3], [0, 0, 0, 0, 0, 0, -4, 0, 0, 6, -6, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, -8, 12, 0], [-37, 0, -7, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -12, 0, 0, -1, 3, 0], [16, 0, 20, 4, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -11, -12, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 5, 5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 6, -11, 0, 0, 12], [0, 0, 0, 0, 0, 0, 0, 10, -6, 0, 0, -15], [0, 0, 0, 0, 0, 0, 9, 0, 0, 7, -6, 0], [0, -4, 0, 0, -8, 14, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 10, -6, 0, 0, 9], [0, 0, 0, 0, 0, 0, -9, 0, 0, -13, 2, 0], [0, 0, 0, 0, 0, 0, 0, -8, -3, 0, 0, 2], [0, 0, 0, 0, 0, 0, 0, 6, -12, 0, 0, -9], [0, 0, 0, 0, 0, 0, 15, 0, 0, -5, -10, 0], [0, 4, 0, 0, -64, -14, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, 0, 0, -12, -2, 0], [26, 0, 6, -6, 0, 0, 0, 0, 0, 0, 0, 0], [-26, 0, 5, 9, 0, 0, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 13, -7, 0, 0, 0, 0, 0, 0], [-26, 0, 10, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, -40, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 12, 0, 0, 0, -6, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -16, 0], [56, 0, 13, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 21, 15, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 6, 0, 0, -15], [0, -6, 0, 0, -42, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, 0, 0, 1, 6, 0], [38, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -10, -10, 0, 0, 1], [-22, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [17, 0, -23, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -11, 6, 0, 0, -8], [0, -2, 0, 0, 43, 19, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 8, 12, 0, 0, -3], [0, 0, 0, 0, 0, 0, -2, 0, 0, -1, -9, 0], [0, 0, 0, 0, 0, 0, -9, 0, 0, -15, 2, 0], [0, 5, 0, 0, 40, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -12, 0, 0, 4], [0, 0, 0, 0, 0, 0, 4, 0, 0, -3, -3, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 12, 0], [-16, 0, 13, -11, 0, 0, 0, 0, 0, 0, 0, 0], [27, 0, -27, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 8, 0, 0, -20, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, 0, 0, -18, 4, 0], [0, 0, 0, 0, 0, 0, 9, 0, 0, 9, -10, 0], [-16, 0, 10, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 10, 0, 0, -11, -11, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, -6, 0, 0, 6], [0, 0, 0, 0, 0, 0, 12, 0, 0, -12, -14, 0], [-4, 0, -4, 8, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 6, 15, 0, 0, -6], [0, 0, 0, 0, 0, 0, 0, 9, -2, 0, 0, -2], [0, -2, 0, 0, 35, 11, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -7, -2, 0, 0, -8], [22, 0, -14, -8, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 5, 23, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, -4, 0, 0, 12], [0, 0, 0, 0, 0, 0, 0, 8, -6, 0, 0, 3], [0, 0, 0, 0, 0, 0, 8, 0, 0, 10, -10, 0], [0, 10, 0, 0, 26, 4, 0, 0, 0, 0, 0, 0], [-16, 0, 2, 12, 0, 0, 0, 0, 0, 0, 0, 0], [14, 0, 17, 5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 8, 0, 0, -26, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 10, 0, 0, 9, -7, 0], [10, 0, 26, 10, 0, 0, 0, 0, 0, 0, 0, 0], [35, 0, -15, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, 0, 0, -1, 18, 0], [0, 5, 0, 0, -34, 19, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5, -10, 0, 0, 2], [0, 0, 0, 0, 0, 0, -12, 0, 0, 3, 6, 0], [51, 0, 15, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -6, 9, 0, 0, -10], [0, 0, 0, 0, 0, 0, -7, 0, 0, -13, -2, 0], [0, 4, 0, 0, 32, -2, 0, 0, 0, 0, 0, 0], [-48, 0, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-19, 0, 5, 14, 0, 0, 0, 0, 0, 0, 0, 0], [0, 12, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 6, 0, 0, -9], [0, 0, 0, 0, 0, 0, 0, 0, 0, -6, -2, 0], [0, 0, 0, 0, 0, 0, -6, 0, 0, -15, 6, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6336_k_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6336_2_k_d();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_6336_2_k_d(:prec:=12)
    chi := MakeCharacter_6336_k();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_6336_2_k_d();".
// 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_6336_2_k_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6336_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![12, -10, -2, 1]>],Snew);
    return Vf;
end function;