
// Make newform 3344.2.a.bb in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3344_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_3344_2_a_bb();".
// 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_3344_2_a_bb();".
// 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 := [32, 196, 169, -341, -136, 152, 22, -22, -1, 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, 1, 0, 0, 0, 0, 0, 0, 0], [-5, 0, 1, 0, 0, 0, 0, 0, 0], [-6181, -15117, 12355, 9908, -5225, -1506, 685, 66, -28], [3170, 5868, -6863, -5302, 2986, 954, -404, -48, 17], [-2948, -7644, 6863, 5524, -2986, -954, 404, 48, -17], [-10994, -14538, 34865, 10192, -16300, -1434, 2360, 54, -107], [-13378, -28026, 29365, 19304, -14810, -3048, 2290, 138, -109], [-360, -1005, 1606, 706, -996, -92, 164, 3, -8]];
    Rf_basisdens := [1, 1, 1, 555, 222, 222, 1110, 1110, 74];
    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_3344_a();"
function MakeCharacter_3344_a()
    N := 3344;
    order := 1;
    char_gens := [2927, 837, 2433, 705];
    v := [1, 1, 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_3344_a_Hecke(Kf)
    return MakeCharacter_3344_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+1)]];
    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, -1, 0, 0, 0, 0, 0, 0, 0], [1, 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, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0], [1, -1, 0, 0, -1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 0, 1, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 1, -1, -1, 0], [0, -1, 1, 0, 1, 1, 0, 0, 1], [3, 0, 1, 0, 0, 0, 0, 0, -1], [-1, 1, 0, 0, 1, -1, 2, 0, 0], [2, 1, 0, 1, 0, -1, -1, 0, 0], [-2, 0, 0, 0, 0, 0, 0, 2, 0], [3, -2, -1, 1, -1, -1, -1, -1, -1], [2, 0, 0, 1, 1, -1, 1, -1, 0], [2, -1, 1, -1, 0, 1, 1, -1, 1], [-1, -1, -2, -1, 0, 0, -1, 1, 2], [-1, -1, -1, 1, -1, -1, -1, 1, 1], [2, 0, 0, 0, 0, -1, 1, 1, 0], [-1, 1, 0, 0, 1, 1, 1, -1, 0], [-1, 2, 0, -1, -1, 0, 0, 1, 0], [0, 0, 0, -1, 1, 2, 0, 0, 2], [0, -2, -2, -1, 1, 0, 0, 2, 0], [2, 1, -1, -1, 0, -1, 1, 1, 1], [-3, 1, -1, 0, -2, 1, -1, -1, 1], [0, 0, 0, 0, 0, 1, 1, 1, 0], [4, 0, 0, 0, 0, -1, 1, -1, -2], [4, 0, 2, -1, 1, 0, 2, 0, 2], [-1, 3, 0, 0, -1, 1, -1, -1, 2], [-4, 3, 1, -1, 0, 0, 0, -2, 0], [5, -4, 1, 0, 0, 2, -2, 0, 2], [9, 2, 0, 1, 1, 0, 0, -1, -2], [1, 0, -1, 1, -1, 0, -2, 0, 3], [0, 2, 0, 0, 0, 0, 2, -2, -2], [3, 4, 0, 0, 0, -2, 2, 1, -2], [0, 0, 2, -2, 0, 0, 2, -2, 0], [1, 1, 0, 0, -1, 1, -3, -1, -2], [-1, -2, -2, 1, 0, 0, 0, 0, 2], [-1, 1, 0, 1, 0, 1, -2, 0, 4], [7, -2, -1, 2, 0, 0, -2, 2, -1], [1, 1, 0, 1, 0, 0, -2, 0, 2], [0, -1, 0, -1, -1, 1, -3, -1, -2], [8, 1, 3, 1, 2, -1, 1, -1, -1], [-2, -2, 0, 0, 0, 1, -1, 1, 0], [7, 3, -2, -2, -1, -2, 0, 0, -2], [1, 0, 3, 0, 0, 0, 2, 0, -1], [-3, -1, -2, -2, 1, 2, 2, 2, 2], [-3, 0, 3, 0, 2, -1, 3, 1, -2], [5, 3, 0, 0, -1, -3, -1, 3, -2], [5, 1, -3, 0, 1, 0, -2, 3, 1], [-2, 1, -2, -1, -1, -1, 1, -1, 0], [-3, -1, 0, -1, 0, -1, 1, 1, 0], [2, -2, -4, -2, -2, -2, 0, 0, -4], [-9, 1, 3, 0, 1, 3, 1, -2, -1], [-5, 4, 1, -1, 1, 0, 2, 0, 1], [7, -1, 2, 0, -1, -1, 3, 1, -3], [-5, 6, 3, 1, 3, 0, 2, 2, -1], [-2, 1, 2, 1, 1, -1, -1, -3, 0], [5, 2, -3, 1, -3, 1, -3, -3, 2], [-2, -2, 2, 0, -4, -3, 0, 0, -2], [5, -5, -2, -2, -1, -1, 1, -1, -2], [4, 2, 0, -2, 0, -1, -1, 1, -2], [3, 0, 2, 0, 2, 3, 1, 0, 4], [1, 0, 1, 2, 0, 1, -1, -1, 1], [4, -3, 2, 0, 2, 0, 2, 6, 2], [2, 0, 0, 2, -2, 0, -5, -3, 0], [-3, 1, 0, -2, -1, 3, -1, 3, 4], [4, -7, -1, 1, -2, -1, -5, 1, 1], [-3, 1, -2, -1, 2, 0, 2, 4, 2], [-9, 1, 0, -2, -1, 1, 1, -1, -3], [7, 1, -2, 1, 0, -1, 1, -1, -2], [0, -3, 0, -3, -1, 0, 0, -4, 0], [15, -1, 0, 3, 2, 0, -1, 1, 0], [4, 3, -1, 2, 1, -3, -2, 2, -1], [-7, 2, 2, -2, 0, 0, 4, 1, 0], [0, -1, -3, -1, -2, 2, 0, 2, 0], [4, 2, 0, 0, 4, 0, 0, 2, -2], [-7, -2, 4, -1, 0, 2, 2, -4, -2], [-4, 0, 2, -2, -4, 0, 2, -2, -4], [3, 8, -1, 1, -1, -5, 1, -3, -3], [-6, 2, -2, -2, -2, 0, 0, -4, 2], [-4, -2, 0, -1, 1, -2, -2, -2, 0], [-1, 1, 0, -2, 1, 1, 1, 1, 2], [-5, 3, 2, -1, 0, 1, 1, 1, 4], [2, 0, 2, -1, 1, 2, 0, 0, 4], [-7, -5, 2, 2, -1, 0, 0, 0, 0], [3, 0, 3, 1, 3, 2, 0, 0, 3], [-5, -3, 4, -1, 0, -1, 1, 3, 0], [11, -1, 2, -1, 0, -1, 3, 3, 0], [-12, 2, 2, 2, 0, -3, 1, 1, 1], [-3, -1, -3, -1, 1, 1, -1, 1, -1], [-3, -4, 1, -1, 1, 1, 1, -5, 0], [14, 4, 2, 0, -2, 2, 0, 2, 4], [-13, 5, 3, -2, -1, 2, 0, -3, 1], [9, 0, -3, 0, 0, 4, -2, 0, -1], [-6, 0, -2, 3, -1, 2, -6, -4, 0], [0, 2, 2, 0, 0, -3, 3, 1, -4], [6, 5, -1, -3, -2, -3, 1, 1, -3], [13, 7, -4, -2, -3, -3, -3, 3, -2], [1, 0, 1, 1, 1, 2, 0, -2, -5], [-7, 1, -2, 0, 1, 1, -1, 1, 6], [-16, 6, 2, 0, 0, 2, -1, -3, -2], [3, 0, 5, -3, 3, -1, 3, -1, 2], [9, -6, -3, 0, -2, -4, 0, 0, 2], [-6, 0, 0, 0, 2, 0, -2, -2, -4], [6, -4, -4, -2, 0, 0, 2, 0, 0], [-1, -3, -5, 0, -4, -3, -3, 1, 1], [-8, -4, 0, 0, -2, 0, 3, 1, 4], [3, -1, 0, 2, -1, 1, -5, -3, 4], [-8, 3, 5, 1, 2, 3, 3, -1, -5], [16, -1, -1, 1, -2, -2, -4, 2, -2], [9, -9, 2, -1, -2, 3, 1, -3, 0], [-1, 3, 0, 1, -2, 1, -1, -3, 4], [14, 4, 2, 1, 3, 2, 4, 0, 4], [-1, 3, 0, 1, 2, -5, 3, 5, -4], [-11, -5, 2, -3, 0, -2, 2, 2, 0], [7, -2, 3, -2, 2, 1, 1, -1, -2], [0, 4, 4, 4, 0, -3, -1, 3, 0], [-9, -2, 1, 2, 0, -1, 1, -5, 1], [-9, 3, 0, 0, 1, -1, -2, -4, 4], [2, -1, -2, -3, -1, -2, -2, -2, -2], [-6, -4, 0, -2, 0, 0, -4, -4, 0], [-3, 3, 0, 1, 6, 3, 1, 5, 4], [15, 2, 1, 1, -1, 0, 2, 0, 1], [-7, 0, -7, -2, 0, -1, -1, -1, 6], [-1, -9, -4, 3, 0, 2, -4, 2, 2], [1, -3, -4, 1, 2, 0, -2, 6, -2], [-10, 3, -1, -1, 0, 1, -5, -1, 1], [5, 6, 0, -3, 1, 0, 4, 3, 4], [-13, 5, 2, 2, 1, -3, 1, 3, -6], [-21, 0, -3, 0, -4, 0, -2, 0, 1], [-4, 3, 0, 3, 0, 1, -1, -6, -8], [-16, 6, 2, -4, 0, 3, 1, -7, -2], [20, 4, -4, -2, -2, 1, -1, 1, 2], [11, 4, 1, 1, 3, 3, -1, 5, 1], [-10, -6, 6, -2, 0, 3, 3, -3, 4], [-9, 0, -1, 2, 0, -5, -1, -5, 3], [-16, 2, -6, 0, -2, 1, -1, 1, 6], [7, 7, 2, 0, 3, 0, 4, -2, 6], [-2, 3, -3, -3, 4, -3, 5, 1, -3], [-9, 7, -4, 1, -4, -2, -2, -2, 0], [-5, 1, 6, 0, 3, 4, 4, -2, 2], [5, 4, 3, 4, 2, -1, 1, -1, -7], [1, 1, -1, -3, 1, -5, 5, -1, 1], [13, 2, 3, 3, 1, -2, 4, -2, -1], [0, 10, 0, 0, 2, 0, 1, -1, 4], [5, -5, 0, 3, -2, 3, 1, -5, 4], [-8, 3, 5, 1, 0, -1, 0, -2, 1], [4, 1, 4, -1, 1, 4, 0, -4, 0], [9, 4, -3, -4, 0, 3, -1, 1, 2], [0, 4, -4, 2, -2, -4, -6, 2, 0], [-2, 4, -4, 0, -4, 0, -4, -2, 0], [14, -6, -2, 2, -2, -3, -7, 1, -6], [-15, 2, 1, 1, -3, 0, 0, -4, -3], [6, 0, -1, 0, 0, -3, -1, 0, -1], [8, 1, -1, 1, -4, -3, -1, -9, -2], [5, 3, 0, 2, -3, 4, -4, -4, 6], [15, 7, 4, 0, -1, -4, -2, 0, -2], [-13, 1, -2, 1, 0, 5, -3, -1, 0], [-7, -1, -2, 4, -1, -1, -3, 11, 0], [-10, -3, 1, -3, 2, -3, 5, 5, 1], [2, 1, 2, 0, 0, 1, 3, -5, -6], [-4, 4, -2, -1, 3, -2, 4, 0, 0], [-18, 5, -1, -4, 1, -1, 2, 6, 3], [4, 1, -1, -1, -4, 3, 2, 0, -1], [-8, 5, -3, 1, -2, -7, -3, -3, -3], [-5, -1, 6, -4, -1, -1, 4, 2, 2], [-1, 2, -1, 0, 0, 1, -3, 11, -2], [-4, 6, 0, -2, 4, -1, 1, -1, 4], [9, -4, -1, -5, 3, 3, 5, 5, 5], [-7, -7, -3, 0, 1, 3, -3, 2, 1], [12, 2, 0, 0, 0, 3, -1, 1, 0], [3, 0, -3, 2, 2, 0, -2, 4, 1], [3, -4, -4, 4, 2, 0, -2, -1, -4], [4, -5, -1, 3, -4, 2, -6, -2, 6], [-1, -5, 6, -3, -2, -1, 1, -1, 2], [3, 5, 4, -2, -1, 3, -1, 5, 0], [-2, 1, -1, 1, 0, 7, -1, -3, 3], [-4, 8, 4, 2, 2, -5, -1, -5, 2], [6, 4, 2, -2, 2, -5, -1, -1, -4], [13, 2, -5, 4, 0, -2, -2, 2, -3], [4, 3, 4, -3, -3, 1, 3, 1, 6], [-5, -3, 0, 5, 4, -4, 2, 2, 2], [-26, -2, -6, 0, 2, 4, -4, 0, 6], [-17, 3, -4, 2, 1, -2, 0, 2, 2], [16, -3, 4, -3, 3, 7, 1, 3, 0], [1, -2, -2, 3, -2, -2, -4, 6, 0], [9, 3, 1, -2, 1, 2, 4, 3, -3], [-14, -2, 0, -1, 1, 8, -4, -2, 2], [-7, 4, -1, -3, -5, -1, -1, 7, -2], [4, 3, -2, -2, 0, 3, 3, 3, -2], [-17, 6, 1, -1, 3, 3, -1, -5, -1], [-14, -3, 3, 1, 0, -2, 6, -2, 8], [-16, 4, 0, 1, 3, -2, 6, 2, -6], [20, -2, -8, -2, -4, -3, -1, 3, -4], [19, -2, -3, -2, 2, 1, 1, -3, 4], [3, 1, 6, 0, 5, 2, 0, -2, 2], [27, -8, -1, 1, -3, 3, -5, 3, -1], [5, -9, -2, -4, -1, -3, 4, 0, -6], [-2, -8, -6, 0, 4, -2, -2, 6, 0], [-1, 6, -3, 1, 1, -3, 1, -3, -9], [-22, 1, 0, -1, -3, 1, -1, 1, 6], [1, 7, 0, -3, 4, -5, 7, -3, -4], [3, 5, 4, 3, 2, 3, 1, -1, -4], [25, 3, -6, -3, 0, -4, -2, 6, 2], [-6, -4, -2, 4, 0, -7, -1, -5, -4], [-17, -1, 0, 0, 1, 0, 0, -4, 6], [4, 5, -4, -3, -1, 0, -2, -4, 0], [7, 1, -6, -2, 1, 2, 0, 4, 4], [11, 4, -4, 0, 0, 0, 0, -5, -2], [1, 9, 4, 1, -2, -3, -1, -1, -2], [-11, -3, -6, 0, 1, -2, -1, 3, 6], [-7, -7, -4, -2, -3, 1, -5, -3, 2], [16, 0, -2, 3, 3, 6, -2, -2, 0], [4, 1, -1, -7, 2, -2, 6, -2, -5], [17, -3, 0, 0, 5, 5, -1, 1, -2], [-8, 11, -5, 1, 2, -5, 0, -2, -1], [-1, -2, 1, -2, -2, -5, -1, -1, -4], [9, 9, 2, 2, 1, -1, -4, -8, 2], [-20, 0, 0, 0, 4, 0, 6, -6, 0], [4, 0, 4, 1, -1, 2, 2, -6, -4], [13, -3, 8, 1, 6, 2, 6, 6, 4], [18, -8, 4, 0, 2, 1, 1, -3, -8], [9, -8, -1, -3, -5, 6, -2, 4, 5], [-14, -10, -2, -2, 0, -2, -2, -2, 4], [-7, 9, 1, 0, 4, 1, 1, -7, 7], [6, -2, 2, 0, 2, 0, 6, 0, 6], [-11, -4, -4, 1, 3, 2, -2, -1, 4], [3, 0, -6, 2, 2, -2, -2, -1, 0], [18, -6, -6, 2, 2, -1, -3, 1, 0], [-6, 6, -2, -2, 2, -4, 6, 2, 0], [-22, -2, 2, 2, 4, -2, 3, 1, -6], [-11, -11, -8, -1, 0, 3, -7, 1, 4], [11, -11, 0, 0, 5, -1, 3, 9, -1], [15, -2, 3, -2, 0, -5, 3, -5, -6], [5, -2, 5, 1, 7, 2, 0, -4, 1], [1, -1, 2, -1, 0, 3, 1, -9, 8], [22, 3, 5, 0, -1, -3, 0, -4, -3], [-2, 3, -2, -2, 2, 3, 3, -13, -2], [-4, 5, -3, -1, 2, 6, 2, -10, 2], [-24, 10, 0, -2, 0, -1, 1, -5, -4], [1, 0, 1, -4, 2, 5, 3, 11, 3], [-2, 2, 0, 0, 6, -2, 4, 14, -6], [7, -13, -4, 2, -3, 2, 2, 2, -5], [-8, 2, 8, -2, 4, 4, 2, 4, -6], [0, 3, 8, -1, 5, 1, 3, -7, -4], [27, -9, 2, 6, 1, -5, -3, 3, -10], [-8, -7, -5, -5, 0, 6, 6, 2, 7], [15, 4, 5, -3, 3, 6, 2, 8, -3], [6, 0, 6, 6, 6, 3, 3, -7, 0], [15, -1, 0, 3, 0, 4, 0, -10, -4], [-30, 2, 2, 2, 6, 8, 2, -4, -4], [17, 0, -3, -3, -9, 2, -6, -6, -5], [-7, -8, -4, 1, -1, -2, -6, 3, 2], [16, -1, 2, -1, 0, 3, 1, 6, 2], [17, 1, -2, -2, -1, -7, 5, 3, -14], [1, 4, 3, -6, 4, -6, 4, -2, -1], [-1, -5, 8, 0, 5, -1, 3, 7, 8], [-10, -4, -2, 2, -4, -6, -4, -6, -6], [-9, -1, 12, 0, 3, 1, 3, -1, -2], [-6, -3, -1, 3, 0, -3, -1, -13, -3], [4, -7, -2, 1, -3, -2, -8, 4, -2], [18, 1, -4, -5, -4, 5, -3, 2, 2], [12, -10, 4, 0, 0, -1, -1, -5, -8], [-15, -4, -3, 0, 2, 3, -3, 5, 3], [1, 0, -2, 4, -2, -5, 1, -2, 2], [3, -11, -2, 2, -3, -2, -2, 8, 2], [9, -2, -5, -5, 1, -4, 4, 6, 3], [17, -2, -3, -2, -4, 3, -3, -7, 7], [2, 9, -2, 0, 0, 0, -2, -10, 2], [-8, 1, -3, 1, 0, -3, 7, -7, -4], [-11, -1, -5, 0, -1, -2, 4, -3, 1], [9, 2, 4, -3, -2, -4, 2, 0, -2], [11, 3, -4, -1, -4, 0, 0, 6, -6], [-17, 5, 0, -9, -2, -5, 5, 1, 0], [14, -17, 0, 1, 1, 0, 0, 4, 4], [-6, -5, -8, -3, -1, -5, -1, 9, 10], [7, 0, -8, 3, -3, -6, -2, 1, -4], [0, -10, 6, 2, 4, 2, 4, 8, -4], [-5, -1, -6, -1, -2, -6, 2, 0, 8], [7, -3, -3, -4, 5, 4, 2, 13, 5], [12, -3, -2, -3, 1, 6, -4, 8, 6], [7, 6, 4, -7, 1, 6, 2, 7, 2], [-9, 1, 2, -5, 0, 1, -1, -9, 0], [4, -3, -3, 7, 2, -5, -3, -1, -10], [5, -1, 2, 6, 1, 1, -4, 4, -2], [11, 5, 0, 3, 4, 0, 2, -4, -4], [-5, -4, 0, 3, -2, -2, 0, 2, -10], [1, 0, -3, 2, 0, -5, -5, -5, 8], [6, -8, -2, 0, 4, 0, 0, 16, 0], [-7, 7, 2, 4, 1, 3, -2, 0, -2], [-18, 2, 0, -2, 0, 6, 6, -2, 8], [14, 11, -2, -1, 1, 1, 1, 1, 2], [-19, -8, 3, 7, 1, 6, -4, -6, 1], [12, 9, 1, -4, -3, 1, 4, 0, -3], [-37, 5, -4, -3, -4, -3, 3, 1, 0], [19, -3, -4, -2, -7, 0, -8, 0, 2], [-16, -7, -4, -1, 6, 1, 1, 8, 2], [14, -8, 6, 0, -2, 8, -4, 4, 2], [18, 0, 0, 2, 0, -9, 4, -4, -4], [10, -3, -7, -1, -4, 6, -2, -2, 1], [-22, 5, 12, 1, 1, 8, 8, -14, -4], [23, -9, -6, 3, -6, -1, -9, 9, 0], [8, 4, -6, 5, 1, 0, -4, 4, 0], [3, -7, -2, 3, 0, 6, 1, 3, -2], [5, -6, 3, -2, 0, 1, -1, 1, 5], [6, 4, 6, 0, -8, 2, -4, -8, -4], [-2, 19, 1, -1, -2, -1, -1, 7, -1], [8, -4, -6, -4, -4, 1, -3, 3, -1], [22, -9, -6, 1, 2, 1, -1, 6, 0], [3, 3, 2, 8, 7, -9, 2, -4, -10], [-6, 3, 4, 4, -2, 3, 1, -3, -8], [13, 7, 4, 4, 3, -8, 0, -2, -7], [2, 14, 0, 2, 4, -4, 2, 4, -10], [-9, -9, 4, -3, -2, -3, 1, -9, 0], [31, 1, 0, 4, -3, -3, -3, -5, 2], [10, -3, -6, -5, -4, -2, -2, 5, 10], [-8, 6, 4, -2, -2, 2, 0, -6, 4], [25, 9, 4, -6, -3, 4, 2, 8, 10], [-1, 11, 6, -1, 0, -4, 6, -12, -2], [3, -21, -4, 0, -1, 4, -10, -4, 6], [-9, 0, 5, 4, -4, 0, -2, 0, -11], [19, 11, 2, 2, 5, -6, 6, -6, -8], [30, -10, 2, 3, -1, 0, 4, 6, -4], [7, 9, 2, -1, 0, 10, 1, 7, -2], [-6, -1, 4, -2, -8, -2, 0, -16, 0], [-1, 3, -2, -4, -1, -7, 1, -9, 0], [-7, -8, -7, 3, -7, 4, -8, 8, 1], [24, 9, -3, 1, -2, -3, 3, -1, -5], [13, -18, -3, 3, 1, -1, 1, -7, -5], [-23, 17, 3, -4, 5, -2, 4, 3, 5], [-13, 1, 6, -4, 1, -5, 3, 3, 2], [5, -2, 10, -1, 3, 4, 4, 5, 8], [7, -7, -8, 1, -2, 1, -5, -3, 4], [-9, -4, -1, 2, -2, -1, -3, -5, -11], [25, -5, -6, 0, -3, -1, -4, 2, -10], [17, 1, -6, 3, -6, -6, -4, -6, -2], [11, -9, 1, 0, -1, -2, 0, -7, -5], [17, -6, 5, 0, -6, 3, -3, -11, -3], [-11, 0, 2, 0, -2, 7, 1, -6, 6], [-17, -3, 2, -2, 7, 6, 5, 5, 2], [3, 9, -3, -1, -1, -5, -5, 1, 3], [-17, 12, -3, 4, -2, 3, -5, 9, 5], [-9, 5, 2, -8, 7, 5, 5, -9, -6], [5, -12, 3, -2, 4, 9, -3, -1, 12], [-21, -13, 4, 1, 4, 2, 7, -7, 0], [-20, -8, -2, -4, 0, 6, -4, 6, 0], [17, -7, 2, 3, -6, -3, 3, 1, -2], [-7, 1, -4, 0, 1, -1, -5, -1, -10], [-4, 22, 6, 1, -1, -6, 2, 4, -8], [-20, 7, -9, -3, -4, 4, -2, -2, 12], [-14, -4, -2, -6, -10, -4, -2, 2, -8], [20, 2, -6, 6, 0, -8, 1, -7, 2], [-33, -8, 8, 3, 2, 6, -2, -12, 6], [11, -11, 6, 7, 2, 2, -4, -2, -2], [2, 9, 6, 1, 4, -3, 5, 0, -4], [0, 12, 2, -4, 8, 5, 11, 11, 6], [14, 9, 1, 0, 7, 3, 2, -6, 9], [1, 0, 9, 1, 7, -4, 4, 4, -7], [-1, 5, 4, -6, -3, -1, 3, -7, -2], [35, 9, 1, 1, 1, -3, 1, 3, -5], [-4, -6, 2, -2, -2, -1, 3, 1, 0], [-7, -9, 12, -1, 8, 3, 3, -1, -4], [20, -14, 0, 3, -3, 2, -8, 4, 10], [26, 5, 1, 1, -2, -9, 9, 7, -13], [7, -7, 4, 0, -1, 5, -5, -1, 10], [-14, 7, 6, -3, 1, -9, 9, -1, 4], [37, 7, 2, 3, 2, -5, 7, 3, -8], [21, 19, 0, -2, 1, 0, 1, -3, -8], [-26, -3, 1, 1, 2, -2, -2, -2, -7], [14, -25, -2, 3, 1, -4, -4, 4, -2], [10, 1, 5, 3, 4, -3, 5, -15, -3], [5, 7, 0, -5, 4, -7, 7, 1, 0], [1, 5, 0, 4, 3, -2, -3, 5, 8], [-3, 11, 2, -2, 1, 6, 6, -2, -7], [-11, -1, 8, -2, -1, 1, 5, -1, -6], [-3, 1, -10, -6, -3, 2, 0, 2, 0], [14, -6, -4, 2, -4, 2, 0, 2, -2], [-7, 10, -1, -3, -1, -2, -2, 8, -1], [27, 20, 3, -2, -2, -5, -1, 1, -4], [2, -10, -2, -8, -2, 4, 4, -2, 5], [5, 1, -6, 4, 7, 1, 1, -1, 6], [8, -16, -4, 6, 0, 3, -5, 1, 0], [15, -8, 1, 4, -4, -4, 2, 0, 1], [-8, 6, 0, 2, -2, -5, 3, -13, -8], [13, 16, 6, -4, 0, 2, 10, 1, -6], [17, 8, 3, -1, 5, -1, 7, 3, 4], [17, 3, -5, -3, -7, -9, -3, 3, 1], [32, -8, -6, 3, -5, -6, 4, 4, -2], [7, -10, 3, 4, -2, -1, -3, 5, -9], [-29, 3, 2, 1, 4, 5, -1, 5, 0], [-11, 5, -8, -1, 0, -5, -3, 5, 4], [2, 9, 1, 8, 3, 3, 0, 4, -7], [-26, 9, 4, -5, -3, 1, 5, -1, 10], [-26, 2, -6, 2, -2, -10, -4, 0, -2], [6, 6, 4, -2, -2, -6, 2, 2, 6], [35, 7, 2, 10, 5, 1, -2, 2, 2], [10, -1, 3, -7, 2, 1, 1, 9, 13], [-13, -13, -2, 8, 1, 1, 2, 2, 14], [1, 0, -2, -6, 0, 2, 4, 1, 0], [0, 0, -4, 0, 2, 6, -2, 4, 6], [18, -8, 2, 2, -8, 4, -8, -8, 2], [19, -6, 0, -5, -4, -8, 4, 4, 4], [11, 12, -5, 1, 1, 1, 5, -3, -1], [26, 9, 1, 1, 6, 4, 2, -4, 0], [-23, -3, -4, 4, 5, 1, 3, 13, 2], [-2, -4, 8, 2, -4, -5, -5, 11, -8], [25, -3, -8, -3, 0, -5, 5, 11, 4], [-38, 14, 0, -2, 4, 0, 6, 4, 10], [-14, 13, 5, 3, 4, 1, 9, 3, 3], [14, -4, 4, -4, -2, 1, -3, 1, 16], [6, 15, 0, -1, -8, -7, -3, 4, -4], [-7, 9, 0, -10, 3, -2, 8, 2, -2], [5, -5, -4, -7, -2, 3, 1, -5, 4], [-11, 13, 2, -4, -3, 3, 5, -5, 6], [-24, 9, 1, 0, 1, -1, 12, 0, -3], [0, -5, 5, -1, 4, 1, -5, 3, 12], [21, -3, -4, 2, -7, -7, -1, -3, -6], [3, -2, 5, 6, -4, -6, -8, -2, -13], [-6, 2, 0, -6, 4, -3, -1, 1, 2], [-8, 2, 0, 2, 6, -3, -3, 5, 10], [-20, 4, -4, 4, 2, 1, -3, 3, -12], [2, -8, -8, -6, 0, 2, 6, 4, 2], [-13, -6, -5, 0, 0, -9, -5, 9, -5], [-29, 1, 4, -1, 2, 6, 3, 5, 4], [-16, 8, -4, 0, -2, 1, -5, -9, 4], [19, -15, 6, 1, 4, 7, -3, 7, 8], [-19, -7, -7, -2, -1, -4, -2, -9, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3344_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3344_2_a_bb();".
// 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_3344_2_a_bb(:prec:=9)
    chi := MakeCharacter_3344_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_3344_2_a_bb();".
// 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_3344_2_a_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3344_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-32, 196, -169, -341, 136, 152, -22, -22, 1, 1]>,<5,R![-1336, 316, 2654, 201, -1080, 16, 162, -20, -6, 1]>,<7,R![2592, 1044, -4294, -1614, 1235, 433, -109, -37, 3, 1]>],Snew);
    return Vf;
end function;