
// Make newform 4338.2.a.w in Magma, downloaded from the LMFDB on 14 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4338_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_4338_2_a_w();".
// 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_4338_2_a_w();".
// 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 := [158, -126, -121, 95, 26, -20, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [-6, 1, 1, 0, 0, 0, 0], [536, -436, -251, 162, 24, -10, -1], [-1276, -224, 1219, 116, -284, -2, 15], [-1397, 102, 1112, -25, -231, 6, 12], [3224, 46, -2597, -72, 534, -4, -27], [-5074, 242, 4067, 26, -842, 12, 43]];
    Rf_basisdens := [1, 1, 38, 38, 19, 38, 38];
    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_4338_a();"
function MakeCharacter_4338_a()
    N := 4338;
    order := 1;
    char_gens := [965, 2899];
    v := [1, 1];
    // 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_4338_a_Hecke(Kf)
    return MakeCharacter_4338_a();
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 := [[1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, -1, 0], [-1, 0, -1, 0, 1, -1, 0], [2, -1, 0, 0, 0, 0, 0], [-1, 0, 0, 1, 0, 0, -1], [1, 1, 0, 0, 0, -1, 0], [0, 0, 0, 1, 0, 1, 1], [0, 0, 1, 0, -1, 0, 0], [1, 0, -1, 0, -1, 1, 0], [3, 0, 0, -1, 0, 0, 1], [0, 0, -1, -2, -1, -2, 0], [2, 0, 2, 1, 0, 3, 1], [3, 0, -1, -1, -2, 1, 1], [2, 0, 1, -2, 0, -1, -1], [0, 0, 0, -2, -1, 1, 1], [4, 0, -1, 0, -1, 0, -2], [-1, 0, -3, 1, 2, -3, -1], [1, 1, 0, -2, 0, -1, 0], [2, 0, 0, 0, 0, 0, -2], [5, 0, 1, 1, 0, 3, 1], [-1, 2, 1, -1, 0, -1, -1], [-4, -1, 0, 0, 2, -2, -2], [3, 0, -2, 1, 0, 2, 0], [-8, 0, -2, 2, -2, 2, 0], [1, 2, 3, 0, -1, 1, 0], [2, 0, 2, -4, -1, -1, -1], [8, 1, -2, -2, 0, -2, 0], [-2, 0, 1, 0, 3, 0, 0], [3, 0, -3, -2, 1, -3, 0], [-1, 0, 2, 0, 0, 1, 2], [-2, 1, 0, -2, 0, -4, -2], [3, -1, 0, 2, 2, 1, 0], [-4, -2, -2, -2, -2, 2, 0], [2, 0, 0, -2, 1, 1, 1], [5, 0, 0, 3, 0, 2, 1], [2, 0, 2, -3, -2, 1, 1], [5, 1, 4, 0, 2, -1, 0], [-2, 0, 0, 0, -2, 0, 0], [-3, 0, -2, 0, 0, 3, 0], [4, 0, 0, 4, -2, 4, 0], [7, 0, 1, 1, 2, -3, 1], [0, 0, -1, 2, 2, -3, -3], [2, -2, -4, -4, 0, -2, 2], [1, -2, -1, -2, 3, -1, 0], [4, -2, -2, -2, -2, -2, 0], [0, 0, 0, 0, -1, -5, -1], [3, 0, 0, -2, 0, 1, 0], [8, -2, 1, 2, -3, 2, 2], [4, 0, 0, 2, -2, 6, 4], [-2, -2, 0, 0, 2, 0, 2], [1, 0, 0, 0, 0, 0, 0], [-2, 0, -4, -2, -1, -5, -3], [4, 0, -4, 2, 0, 0, -2], [6, -2, 4, -1, -2, 3, 1], [6, 0, 4, -4, 2, 0, 2], [3, 0, 5, 1, 4, 1, 1], [0, 0, 0, -6, 0, 0, -2], [-2, 4, -1, 0, -1, -2, -2], [-10, 2, -2, 1, 0, -1, -1], [6, 0, -2, 0, 2, -2, 0], [-5, 1, 1, 2, 3, -1, 0], [-4, 0, 0, 5, -2, 3, 1], [-1, 0, -4, 1, 2, 2, 0], [2, 0, -2, 4, 0, 0, -2], [-10, 2, 2, 4, 2, -2, -4], [-5, 0, 2, 1, 1, 3, -1], [-2, 2, 0, 6, 5, -3, -3], [6, 2, 3, 4, -3, 0, 0], [4, -1, 0, -6, -4, 2, 0], [2, 0, 2, 0, 2, -2, 2], [-9, 0, 4, 6, 2, 5, -2], [-7, -2, 2, 1, -2, 4, 5], [9, -3, 0, -2, -2, 3, 0], [-4, -2, -3, 1, -1, -3, -1], [-6, -2, 2, 4, -2, 6, 0], [-1, 2, 0, -1, -4, -4, -1], [6, 0, 1, 4, 1, 6, 0], [-4, -2, 2, -6, -4, -2, 0], [3, 0, 2, -2, 0, 7, 0], [8, 2, -1, -2, 3, -6, 0], [-1, -1, 0, 0, -2, -3, -4], [-6, 4, 2, 4, 2, -2, -2], [7, -2, 6, 2, 2, 7, 4], [7, -2, 1, 2, -5, 1, -2], [-3, -4, 0, -7, -6, 4, 3], [-12, -2, -1, 2, -3, 4, 0], [-4, 2, -2, -2, 2, -2, 4], [-13, 2, -4, 2, 0, -1, 0], [-3, 2, 5, 1, 2, -3, -3], [-4, 0, 0, 4, 6, -2, -2], [-2, -2, 2, 2, -3, 7, 5], [7, -2, 5, -1, 4, 3, -1], [1, -1, 0, -4, 4, -1, -2], [-5, -1, 1, -4, -1, -1, 4], [-2, -2, 1, 0, 1, 6, 2], [-10, -1, 4, 4, 2, 4, -4], [4, 2, 0, -2, -4, -2, 0], [-10, 0, -2, -2, 0, 0, 0], [0, 0, 3, -1, 3, 3, 3], [3, 0, -2, -1, 3, -1, 1], [-4, -2, 4, 4, 3, 7, -1], [0, -2, -1, -2, 3, -2, 0], [10, 2, 1, -7, 5, -9, -1], [-4, 0, -2, -2, -2, 4, 4], [7, 0, -1, -4, 3, 1, 2], [-5, -2, 4, 5, 4, 8, 1], [-7, 3, -3, 0, -5, 5, -2], [-6, 0, -6, 4, -1, 1, 3], [-2, -4, 0, -4, -3, 7, 5], [10, 3, 0, 2, 2, 0, -4], [6, 2, -5, -4, 3, -6, 0], [4, 2, 9, -3, -1, 3, -1], [-7, 2, -4, 0, -4, -5, 6], [2, -2, 2, 4, -2, 4, 2], [-7, 0, -1, 5, 2, 5, -1], [-7, -3, -4, 0, 6, -1, 2], [8, 2, 4, -6, -4, 6, 4], [0, -2, 2, -2, -3, 1, 3], [8, -1, 4, 8, -4, 6, 0], [8, -2, 5, -2, 4, 1, -1], [10, -2, 6, -4, 2, 6, 0], [-10, 4, -4, 0, 3, -3, 1], [-8, 4, -6, 0, 4, -6, -2], [-18, 0, 2, -8, -2, -2, 0], [11, 0, 2, -5, -6, 4, 1], [1, 4, 5, -5, 2, -3, -3], [0, 0, 0, 4, 0, -4, 4], [0, 4, -4, 0, 0, 2, -2], [1, 0, -5, -5, -2, -5, -3], [0, 0, 0, -4, 0, -4, 0], [-7, -4, 4, 8, -2, 5, 0], [14, 3, 6, -2, -4, 4, 0], [-13, -2, -6, -3, 2, -8, 1], [-18, 4, -8, 4, -2, -6, 0], [-16, 2, 3, -2, 1, 8, 0], [-11, 0, 1, -1, 0, 11, 5], [6, -2, -2, 4, 2, -6, -4], [3, 2, -2, -3, -2, -8, -7], [16, -2, 6, -7, -4, 5, -1], [26, -2, 3, 0, 3, 6, 2], [2, -2, 2, -8, 5, 1, 1], [1, -2, -4, 10, 2, -3, 0], [10, -2, 7, -2, 1, 6, 4], [5, -2, 1, 3, 2, -3, -1], [13, -2, -2, -9, -2, -2, -1], [22, 0, 1, -8, 3, -4, 2], [2, 2, 6, -3, 6, -7, -5], [16, 2, 6, 1, -2, 9, -1], [-10, -2, -1, -6, 3, -4, 6], [-2, 4, 6, 0, 0, 2, 0], [13, 2, 1, 3, 4, -1, -5], [10, -2, 2, 2, 4, 2, 4], [-17, 1, -2, 0, -6, 1, 0], [0, -2, 0, 0, 0, 8, -2], [-6, -4, 0, -2, 5, 5, 1], [8, -1, -6, 4, 0, 2, -2], [-2, 2, -1, 4, 3, 8, 4], [6, -2, 2, 4, 4, 0, -2], [5, 4, 5, -2, -5, 7, 0], [12, -3, -4, -8, 2, 2, 2], [-4, 0, -8, 0, -4, 0, 0], [-11, -6, -2, -4, 2, 1, 0], [11, 6, 2, 7, 4, -4, -1], [-1, 1, 0, 2, -4, -3, 4], [0, 4, 0, 6, 1, -5, -3], [-10, 5, -2, 2, 0, 0, 0], [-3, 0, -2, 3, -4, -2, 0], [-24, -2, -2, 4, 0, -2, 2], [9, -2, -5, -2, -1, -7, 4], [15, 4, 0, -1, 4, -8, -7], [-9, 0, 1, -1, -6, 3, -1], [0, 6, -4, -6, -4, -8, 0], [-24, 6, -2, 6, -1, -1, -5], [-32, -1, -4, 4, 2, -6, -2], [15, 0, -1, 1, -2, 3, 1], [0, -6, -7, -6, -5, 0, 0], [-5, 0, -10, 6, -2, -1, 2], [21, -6, 6, -3, 2, 4, 1], [-7, 0, 3, -1, 0, -1, -1], [-13, 4, 4, -2, 2, 3, 0], [10, -3, 2, -2, 0, -8, 0], [-4, 0, -2, -2, -4, 2, -2], [-1, -3, 7, -6, -5, 7, 4], [3, 4, 3, 5, 4, 1, 3], [13, -2, 0, -1, -4, 2, -1], [-15, -4, -3, 3, 4, 3, 5], [-12, 0, -4, -6, 0, -4, 4], [1, 0, -5, 1, 0, -5, -5], [-12, 0, -4, -2, 10, -10, -4], [11, 2, -10, 2, 4, -17, -2], [-23, -1, -7, 6, -1, -3, -2], [14, 2, 1, 6, -4, -1, -7], [4, -2, 10, 2, -7, 13, -1], [-10, 0, 3, 8, -5, 10, -2], [-6, -2, 5, 0, 1, 4, -4], [12, 0, 7, 8, -3, 10, 4], [14, 2, 5, 4, -3, 2, 0], [-33, 0, -1, 5, 2, 3, 3], [5, 2, 1, 11, 1, 2, 2], [-1, -2, 0, 0, -2, 3, 0], [-2, -2, 1, -12, 3, -12, -2], [0, -2, -6, 10, -2, 0, 0], [7, -2, 1, 9, 2, 7, 1], [-4, -4, -9, -8, -4, -7, 5], [21, 0, 0, -6, 2, 3, -2], [31, 0, -1, 2, 5, -3, 0], [-1, 0, 2, 7, 0, 12, 1], [-14, 0, -2, -17, 2, -9, -1], [20, 2, 0, -10, -6, 6, -2], [-17, 0, 4, 3, 1, 3, 5], [4, 0, -6, 2, 9, -3, -5], [23, 0, -2, 10, 0, 3, 2], [0, -2, -6, 3, -4, 9, -3], [-24, 2, -2, 2, 1, -9, -5], [8, 0, 12, 0, -1, 15, -1], [-1, 4, -4, 1, 4, 0, 3], [21, 0, 1, 1, 4, 1, -3], [0, -2, 8, 6, -8, 6, 4], [-2, 5, 4, 6, 2, 8, 0], [-2, 4, 4, 6, -1, 7, 3], [22, -5, 2, 0, 4, 0, 4], [-1, -2, 5, 4, -1, 5, -2], [9, 0, -4, 6, -4, 3, -4], [25, 0, 8, 3, 0, 14, 5], [7, -4, 2, 7, 0, 4, -3], [-2, 2, 6, -7, 8, -3, 7], [-8, 0, 4, 6, 2, 4, 2], [37, 2, 5, 5, -2, 9, 5], [15, 0, -8, 3, 0, -12, 1], [4, 2, 3, -13, 1, -3, 1], [-28, 0, -4, 10, -2, 0, 2], [0, 5, -6, 2, 6, -2, 0], [11, 1, 4, 2, -2, 9, 2], [-13, 0, -1, 13, 2, 11, 3], [-12, -4, 0, 6, 0, 0, 0], [-8, 4, -2, -12, 1, -7, -1], [13, -2, -1, -11, 0, -11, 1], [-33, 3, 2, -2, -2, -7, 0], [6, -4, 0, -4, 4, -14, -4], [17, -5, -11, 2, -1, 1, -2], [0, -6, -6, -12, -3, -1, 1], [11, 4, 16, -1, 4, 4, 5], [8, -8, 5, -6, 1, 6, 0], [-2, 0, 4, -2, -2, 6, 2], [0, -4, 6, 0, -2, -4, 4], [1, -2, -1, 7, -6, 13, 7], [10, 4, 2, 10, 2, -4, 4], [-20, 4, 4, 9, 2, -1, -7], [-2, 0, -3, 10, 13, -8, 0], [2, 7, 2, -2, -6, 4, -2], [-10, 2, 4, -12, -4, 2, 8], [4, 8, 10, 2, -2, -4, -2], [6, 0, -4, 4, -8, 10, 0], [-3, -4, 4, 8, 4, 1, 0], [17, 0, 10, -1, 6, -2, -3], [18, 0, 6, 6, -2, 8, 0], [0, -2, 13, -8, 2, -3, -5], [7, -8, -8, 9, 2, -2, -4], [5, 4, 9, 9, -4, 3, -3], [-4, 2, -12, 4, -7, -3, 1], [14, 0, -5, 6, -7, -6, 2], [12, 0, -8, 10, 8, -12, -6], [20, -4, 8, -14, -6, 6, 4], [-7, 2, -5, 5, -6, 5, -3], [-16, 4, -4, 2, 0, 6, -6], [19, -2, 0, -9, 2, -4, 3], [8, -4, 2, -18, 0, 8, 4], [-16, 4, 6, 2, 4, 8, -4], [-22, 2, 1, 6, -1, 6, -6], [4, -2, -8, 8, 2, -6, -4], [18, 2, 0, 0, -6, 6, 10], [30, 2, -2, -6, -1, 5, -1], [3, 4, 6, -7, 1, -5, 3], [-6, 4, -2, 8, -7, 9, -1], [37, -2, -1, -5, -2, 5, 7], [24, 3, -4, -8, -4, 6, -8], [-36, 6, 4, -2, 0, -4, -2], [-33, -2, 5, 6, -3, 11, 0], [-9, -1, 4, -4, -10, 9, 4], [-15, 0, 7, 3, 8, -5, -5], [-17, -2, 3, 3, 6, 5, -1], [12, 2, 6, 4, 4, -4, 2], [43, 0, -1, -9, 2, 1, -1], [-18, 0, -2, -15, 0, -7, -3], [-12, -2, -12, -6, 0, -8, -6], [22, 4, 8, 8, 2, -4, 0], [-10, -8, -8, -6, -1, -1, -1], [-18, -1, 6, 12, 8, 8, 4], [16, -2, -3, 8, 4, -9, 1], [7, 6, -11, -6, -1, -11, 2], [-12, -4, -11, -2, -5, 0, -4], [-2, 4, 2, -4, -6, 10, 4], [16, 4, 1, -6, -1, 0, 0], [-14, 0, -6, 12, 8, 0, 4], [5, 0, -8, -4, 4, -9, 4], [-9, 0, 2, -13, -1, -13, 5], [-15, 8, -3, 4, 7, -3, -8], [-1, -4, 1, -13, 0, 5, -1], [12, 6, 6, 10, 2, 0, -4], [32, 4, 4, 10, -6, 12, 4], [-19, 6, -9, 10, 1, -7, 2], [-30, -4, 0, -12, 0, 2, 0], [-3, -4, -1, 1, 8, 3, 5], [-22, 2, -8, 8, -4, -4, -2], [9, 2, 4, 8, 6, 7, -10], [-16, 5, -10, 2, 8, -6, -4], [-3, -1, 2, 10, 6, 3, 0], [10, 2, 6, -8, 0, -10, -4], [2, 0, -2, 8, 10, -6, 2], [15, 0, 4, -17, -7, 3, 5], [38, 3, 0, 0, -2, 0, 0], [18, -2, -2, 17, 0, 3, 3], [1, -8, -5, 5, 0, 3, 3], [-5, 0, 8, 13, 11, 9, 5], [13, -6, -4, 2, 2, -7, -4], [-18, 4, 2, -4, -2, 2, 6], [14, 8, 0, 0, 2, -14, 0], [0, 0, 3, -2, 5, 8, 6], [-4, 4, -14, 3, 2, -19, -5], [-1, 4, 16, -3, 3, 1, -3], [-12, 0, -2, 10, 4, 0, -2], [-23, -6, 4, -6, -2, -1, -4], [0, -2, -3, 2, -5, 8, 0], [-21, 2, -2, 9, 6, 0, 0], [14, 2, -8, -10, -2, -10, 0], [-13, -1, 4, -4, -4, 13, 6], [0, -2, 8, 9, 2, 19, -1], [-14, -8, 8, 4, 2, 8, -4], [15, 2, 10, -1, 6, -4, 3], [-16, 6, 4, -10, 6, -8, 2], [25, 2, 1, -4, -11, 13, 10], [9, -10, 6, 0, -4, 3, 8], [-17, 4, 7, -11, -4, -7, 3], [-10, 0, -10, 12, 5, -11, -7], [-8, -4, -12, -6, -6, -2, 8], [17, 2, 2, 11, 10, -4, -5], [-8, -6, 2, 10, 4, 6, 6], [24, -4, -2, 6, 4, 0, -2], [26, 3, -2, 8, -2, 8, 0], [16, 0, 1, 6, -5, 0, -6], [2, -2, -10, 0, -14, 2, 2], [17, -6, 6, -2, -6, 5, 2], [-8, -2, -10, -8, -4, -6, -2], [11, 4, 6, 12, -4, 17, 2], [0, -1, 8, -12, 0, 14, 4], [-11, -3, 2, 0, -6, 15, -4], [3, 2, -11, 15, 2, -9, 1], [21, -6, -7, 1, 0, -1, 1], [12, -4, -10, 5, 10, -1, 1], [32, 4, -6, -2, -6, -4, 4], [3, 6, -2, 9, -8, 14, 0], [22, 0, 8, -18, -5, 7, -1], [-20, -1, -8, 4, 4, 2, 0], [-21, -2, 0, -1, 6, 0, 3], [26, 0, 6, -5, -2, 13, -1], [-6, 6, 4, 4, -9, -7, -5], [24, 5, -2, -6, 2, -6, 0], [-4, 6, 10, 9, -8, -5, -1], [14, -2, -7, 4, -9, 8, 10], [-8, 9, 2, 2, -6, -14, -8], [56, -6, 13, -15, 3, 5, 3], [-6, -4, -16, 4, 1, -17, -5], [1, -1, -7, 4, -3, 13, 8], [-8, 6, -4, -2, -8, -8, -2], [-7, 2, 6, -1, 17, -7, -9], [13, 4, -1, 2, -1, -13, -4], [-24, 2, -10, 2, -8, -2, -6], [17, -2, 15, -1, 10, 9, 3], [10, -10, -7, 7, -5, 3, -7], [-21, -6, -1, -9, 2, 5, 3], [28, 0, 10, -6, 6, -6, -8], [20, -2, -2, 0, 5, 3, -7], [-28, -6, -5, 16, 1, 6, 6], [-4, -2, 8, -4, -9, 15, -1], [-5, -6, -13, -15, 0, -7, 3], [-10, -2, 1, 12, 9, 0, 0], [10, 1, -2, 14, -2, 12, 10], [8, 2, 0, 10, 8, -8, -6], [9, 2, -6, 11, -3, 1, 3], [0, 2, -5, -2, -15, 4, 0], [14, -4, -4, -6, 3, 15, -1], [46, 2, 3, 0, 3, -6, -2], [0, -4, -10, -6, 0, -8, 0], [-22, 10, 0, 4, -4, -6, 0], [4, -4, -16, 0, 0, -4, 4], [47, 6, 8, 9, -4, 6, 5], [10, -2, 2, -8, -2, 2, -4], [20, 4, 2, 6, -4, -8, -4], [-2, 0, -8, 4, 2, -6, -16], [-16, -2, -6, 6, 2, 2, 4], [6, 4, 4, 2, 1, 1, 3], [26, 0, 7, -5, -3, -9, -5], [29, -4, -11, 1, 4, 5, -3], [-40, 0, -2, 6, -6, 8, 0], [11, 6, 8, -4, 2, 3, 8], [6, 1, 4, -18, -8, -8, -2], [12, 0, 10, 6, -2, 2, -2], [10, 4, 0, -6, -1, -7, -5], [4, 0, -11, -8, -17, -2, 8], [6, 5, -8, 0, 2, -16, -4], [-20, -2, -4, 2, -5, 9, 9], [-2, -5, -14, 12, 12, -20, -8], [-24, -4, 14, 10, 2, 6, 4], [-27, -2, 4, 12, 6, -3, -8], [48, -3, 10, 0, -2, 10, 0], [28, -2, -5, -2, -5, -2, 0], [-7, -1, -2, 16, -2, 19, 2], [1, 2, -4, 2, 8, -13, -16], [22, -2, -9, -12, -1, -20, 4], [42, -4, 6, -12, 8, -8, 4], [-14, 2, -6, 10, -1, -7, 1], [2, 2, 2, 2, 16, -14, -4], [-44, 4, -8, 6, -4, -12, -6], [10, -2, 2, -12, 18, -14, -2], [18, -2, 19, 9, -11, 23, 7], [-8, 4, -10, 2, -4, 4, 2], [26, 6, -3, 0, -1, -16, -6], [-3, -5, 0, -4, 14, -13, 2], [-14, 4, -2, 0, -8, 0, -4], [-12, -8, 1, 6, 11, -4, -6], [15, 5, 2, 2, 6, -3, -6], [-22, 6, 12, -6, 1, 13, 5], [-9, 6, -7, -3, -5, -16, -2], [-30, 10, -12, 12, 8, -2, -6], [-3, 2, -2, 10, 0, -3, 4], [-16, 4, 3, 14, 1, -4, 2], [15, 4, -5, -11, 6, -25, -7], [3, 6, -5, 1, -4, -3, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4338_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4338_2_a_w();".
// 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_4338_2_a_w(:prec:=7)
    chi := MakeCharacter_4338_a();
    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_4338_2_a_w();".
// 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_4338_2_a_w( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4338_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-288, 0, 276, 32, -72, -12, 5, 1]>,<7,R![-832, 1568, -704, -160, 150, -10, -7, 1]>,<11,R![896, -4256, 1440, 816, -176, -54, 4, 1]>],Snew);
    return Vf;
end function;