
// Make newform 3344.2.a.w in Magma, downloaded from the LMFDB on 24 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_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_3344_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 := [-30, -21, 69, 13, -17, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [137, -48, -253, 1, 24, 1], [-354, -487, 42, 93, 1, -4], [29, 101, 13, -89, -2, 8], [-199, 588, 68, -182, -3, 12]];
    Rf_basisdens := [1, 1, 97, 97, 97, 97];
    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, -1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 1], [1, -1, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, -1, 0, -1], [0, 0, 0, 0, 2, 0], [1, 0, 0, 0, 0, 0], [-1, 1, -1, 1, 1, 1], [3, -2, 1, 1, 1, 1], [0, 1, 1, 0, 1, 0], [1, -1, -1, -1, 1, 1], [1, 0, 0, 1, 0, -1], [-2, -1, 1, -1, -1, 2], [0, 0, 0, 0, 2, -2], [4, -2, 0, 0, 0, 2], [-1, -1, 3, -1, 1, -1], [4, -2, 0, 0, 0, -2], [-4, 1, 1, -2, 1, 0], [0, 1, -2, 0, 0, -2], [-4, -2, 0, -2, 2, 0], [2, 2, 0, 0, -2, 0], [0, 1, -1, 1, 1, 0], [1, 1, -1, 1, 3, -1], [5, 1, -1, -1, 1, -3], [2, 2, 0, 0, 2, -2], [-3, -2, -1, 1, 3, -3], [0, 2, -4, 0, 2, 2], [8, 0, 2, 2, 0, -2], [5, 1, 3, -1, -1, -1], [6, 2, 2, -2, -2, -2], [7, -3, 3, -2, 2, 2], [-4, 2, 0, 1, -1, -3], [0, 3, -5, -1, 1, 0], [0, -4, 0, 2, 0, 2], [-4, -2, 0, 0, -4, 0], [1, -2, 0, -2, 0, 1], [2, -2, 0, 0, 2, 4], [8, 0, 0, 2, 2, -2], [1, 0, 1, -1, -1, 1], [-2, -3, 1, 0, 1, 2], [-1, -1, 1, 1, -3, -3], [1, 1, -1, -3, -5, 1], [9, -4, 3, 3, -1, -1], [-4, 4, -2, 0, -4, 0], [6, 2, 2, 0, -4, 0], [-10, 0, -2, 0, 0, 0], [1, -3, 5, 1, 3, -3], [6, 2, 4, 0, 0, -2], [4, -4, 0, -3, -1, 1], [8, 2, -4, 4, 0, 2], [10, 1, -3, 1, -1, 2], [5, 0, 1, -1, -5, 5], [-5, 1, -1, -1, -3, 5], [-8, 2, -2, 0, 2, 2], [2, 1, 1, -3, 3, 2], [-8, -2, 2, -4, 0, -4], [9, 5, 1, -2, -2, 2], [4, -2, -2, 0, 0, 4], [1, 2, 3, -1, -1, -1], [-11, 1, 1, 0, 0, -4], [-9, 2, 2, -1, -2, -3], [0, 4, -4, 2, -2, 2], [-2, -6, 2, 0, -2, 2], [-5, 2, 2, -2, -2, 3], [5, -1, 5, 1, 1, -5], [-10, 3, 0, 2, -2, -4], [5, 4, 0, -3, 0, -5], [6, 2, -2, 2, -2, 2], [8, 2, -4, 2, 0, 2], [3, 5, 1, 1, -1, 3], [7, 3, -1, -2, 2, -2], [-11, 3, -3, -3, 1, 1], [7, 2, -1, 1, -3, 7], [-16, 1, 5, 0, 1, 0], [-4, -5, 3, -2, 3, -4], [13, 0, 2, 0, -2, -3], [3, 5, 3, -2, -2, -4], [0, -4, 0, 2, 0, 6], [15, -8, 5, -3, 5, 1], [-2, 0, 2, -2, 2, 2], [8, 6, 0, 0, 6, 0], [6, 0, 0, 4, -4, -2], [-13, 7, -3, -1, -1, -5], [-12, -4, -4, 2, 4, 4], [-9, 3, -3, 5, -11, 3], [7, 1, -3, -1, -5, -5], [4, -4, 0, 2, -2, -4], [4, 2, 4, -2, 0, 6], [-7, -1, 7, 3, 1, -1], [7, 1, -1, -5, 9, -3], [-1, 1, 3, 2, 6, -6], [4, -1, -4, 0, -4, 0], [3, 1, -1, 2, 6, 6], [-8, -4, 4, 4, -8, 4], [-10, 1, -9, 5, 1, -2], [-3, -7, 3, 5, -1, 3], [7, 5, 1, 1, 1, -7], [8, -6, 2, 2, 4, 2], [6, 6, 4, 2, -4, -4], [-6, 2, 4, -2, 4, -6], [-6, 2, -2, 2, 2, 0], [4, -8, 4, 2, 0, -4], [-1, -4, 4, -3, 0, -3], [5, -7, 5, -4, -4, -4], [0, -8, 0, 3, -3, 7], [-6, -8, 0, 2, 2, -4], [-4, 2, 0, 2, 6, 0], [9, -2, 5, 1, 3, -5], [-5, 2, -2, 1, 2, 1], [0, -2, 0, 2, 0, 4], [-4, -2, 2, -8, 2, -6], [-9, -1, 7, 4, 4, -2], [1, 9, -3, 3, -3, -3], [3, 1, -3, 5, -7, 3], [5, -7, -1, 5, -1, -1], [3, -1, 7, -3, -1, -5], [-7, -1, -9, 3, -5, 1], [-4, -8, -4, 1, 3, 1], [-8, -10, 2, 0, -2, 0], [-11, 7, -7, 5, -3, -5], [-3, 0, -8, -1, -8, 7], [-11, -2, 1, -9, 7, -3], [-16, 4, 0, 0, -4, 6], [1, -7, 1, 5, 5, -3], [-4, 6, -2, 4, -6, -2], [18, 0, 2, -1, 1, -1], [-1, 1, -1, -3, -3, 1], [-1, 1, -5, 1, 1, 1], [-10, -4, 4, 0, -6, -2], [8, 3, 1, 7, -1, 0], [-22, 0, 0, 0, -2, -4], [3, 3, 3, -1, 5, -9], [-2, -3, 1, -7, -9, 4], [-34, 8, -4, -2, -4, 0], [-26, 4, 2, 2, -4, -2], [-4, -2, 4, -4, 0, -2], [2, -6, 2, 0, -2, -6], [-21, -3, 3, 1, -7, 3], [-10, -4, -8, 2, -4, 8], [-16, 8, -4, -4, 2, 2], [18, -4, 6, -2, 6, 0], [15, 7, 1, -1, -3, -5], [-4, -4, 6, -10, 10, -8], [1, -3, -7, 3, -5, 9], [-12, -5, 4, -4, 4, -8], [8, 2, -6, 0, -10, 6], [21, -6, 2, 1, 6, 7], [-11, 7, -5, 3, -7, -5], [-13, 8, 0, 3, 2, 5], [1, 0, 5, -5, -5, 1], [4, 2, -4, -3, 7, -7], [-8, 0, 2, 0, 4, 6], [4, 2, 4, -4, 12, 0], [-4, 0, -8, 4, -12, 2], [-16, 4, 0, 4, 4, -2], [-8, 11, -13, -5, 1, 4], [-5, 9, -1, -6, 2, -8], [8, -12, 0, 2, 0, 6], [10, 6, 8, -4, 6, -4], [7, 5, -5, -1, -1, 11], [14, 6, -4, 4, 2, 0], [-10, 6, -6, 2, -10, 2], [-4, -1, 2, 0, 0, -6], [5, -13, 7, 5, 3, -5], [-26, -9, -1, 4, -1, 6], [-3, 0, -12, -3, 2, 3], [-18, 8, -6, 2, -10, 0], [13, -2, -2, 5, 6, 3], [-16, 0, 0, 1, -1, -3], [-6, -12, -2, 4, 0, 4], [-6, 4, -4, 0, 12, 4], [8, -5, 3, -1, -3, 8], [-6, 6, -4, -8, -2, -2], [3, 7, -9, 7, -3, 7], [9, -2, -4, 2, -8, 1], [-1, 11, -5, -4, -4, 8], [-1, -7, -3, -3, -5, -1], [-10, -2, 6, 2, -4, 4], [4, -8, -2, 2, -4, 8], [30, 4, -4, 2, 4, -2], [-8, 10, 2, 0, 2, -4], [1, 15, 1, -3, 1, 3], [1, -2, 9, -9, 3, -7], [-15, 9, -9, 9, -3, 3], [22, -8, 4, -4, 0, 0], [-8, 0, -6, 6, -4, 6], [21, 6, 1, 1, -5, 7], [-1, -4, -5, 5, -3, 7], [-26, 11, -5, -1, -7, -2], [-19, 11, -3, 1, -3, -9], [-7, 5, 1, -2, -2, -2], [-24, -3, 4, -8, 4, -8], [0, 8, -4, -2, 2, 4], [-9, -9, -5, 2, -2, 10], [-3, -5, -1, -5, 7, 3], [14, -6, 4, -6, -8, 0], [4, -2, -4, -1, 1, 7], [-8, 0, -14, 4, -8, 8], [14, -10, 2, 2, 6, -6], [-21, 14, 2, 5, -6, -7], [-8, 6, -2, -8, -2, 0], [6, -6, -8, 6, -6, 2], [15, -8, 7, -7, 1, -1], [-21, -7, -1, 1, -1, -1], [5, -1, -1, 7, 13, 5], [17, 7, -1, 3, 3, 1], [4, 8, 0, 0, 0, 0], [-2, -2, -8, 6, -2, -6], [-11, -10, 5, -3, 7, -9], [4, -2, 4, -8, -6, -8], [7, -4, 2, 0, -6, 7], [-3, 3, -1, -7, -3, -1], [9, 6, 6, -5, 0, -1], [-10, -8, 2, 2, -4, -12], [19, 5, 7, -1, 13, -9], [18, 2, 6, 2, -2, -6], [-12, 0, 6, 2, 8, -14], [-11, 10, -2, -5, -4, -5], [28, -2, 12, -1, 1, -1], [-5, 0, -4, -5, 0, -7], [-4, 2, -12, 6, 4, 0], [-9, 5, -5, 5, -7, -9], [23, 1, -3, -5, 7, -3], [16, 6, -2, 6, -2, -4], [-2, 6, 8, -2, 0, 4], [-4, 6, -2, 4, 8, -8], [-7, -6, 13, -11, 3, -5], [18, 4, -8, -2, -10, 12], [14, -5, -3, -9, -9, -2], [-15, 0, -2, 0, 6, -3], [-20, 6, -2, 10, 10, 4], [-4, -8, 2, 10, 8, 2], [5, 6, -6, -7, -2, -1], [27, -15, 3, 3, 7, -5], [-17, 11, -1, 4, -8, -4], [-14, -6, -2, 3, 17, -9], [-30, 8, -8, -2, -2, -6], [-9, -9, -3, 3, 9, 11], [8, -11, 1, 6, 5, 4], [-32, -5, -4, 0, -12, 8], [27, -5, 11, -2, -6, -6], [2, 0, -8, 2, 6, 4], [-7, -7, 1, 3, 9, -9], [-22, -2, -4, 8, -10, 4], [17, -13, 13, 4, -4, 0], [12, 4, 0, 4, 6, -6], [15, -10, -1, -5, 5, 13], [24, -4, 6, 0, -6, 8], [-4, 8, 0, 0, 12, 8], [-2, 0, 2, 6, 10, 8], [-14, 2, -4, -2, -10, 6], [19, 9, -3, 11, -1, -7], [6, 2, -2, 6, -12, -2], [6, -6, -4, 6, 6, -10], [30, 7, -1, -1, 1, -2], [14, 3, 3, -1, -19, -4], [-6, 2, -4, 8, -6, 0], [-9, -5, -9, -1, 3, 9], [4, -8, -4, -4, 12, 2], [38, -6, 14, -4, -4, 0], [22, -10, 2, -2, 10, 2], [16, -8, 0, 0, 2, 8], [-17, -14, 1, 1, 5, -7], [-6, -7, 13, 1, -5, -4], [-4, -4, 4, 4, 0, 14], [-13, -1, -1, -3, -5, 1], [5, -2, 4, -2, 4, -7], [8, 8, 0, 4, -8, 0], [4, -10, -4, 0, 2, 4], [15, -5, 5, -9, -13, -5], [12, 3, 6, 0, 4, -6], [-41, 3, 7, 0, 0, 2], [16, -5, 5, -1, 7, -8], [-13, 6, -5, -3, 1, -1], [13, -9, 9, 1, 1, -3], [-3, -11, -5, 3, -1, 5], [3, 6, 3, 1, -7, 11], [-9, 12, -3, -3, 1, -3], [-26, 1, -5, -11, 9, 6], [10, 2, -4, 8, -4, -6], [11, -15, 7, 3, 1, -7], [8, -15, 5, 5, -1, 4], [3, -10, 1, -3, 13, 5], [34, -5, 15, -1, -7, -6], [13, 5, -9, 1, -1, 5], [-19, 1, 13, 4, 0, 2], [17, 0, 0, 3, 12, -13], [23, 3, -7, 1, 9, 11], [-23, 0, 5, 3, 3, -3], [-2, 16, -6, 9, -5, -3], [-2, 4, 0, -10, -4, 2], [-13, -2, 6, -11, -2, -3], [12, 12, -2, -6, -6, -4], [-15, 6, 1, 3, 11, 13], [26, 2, 8, -4, 22, -2], [4, 13, -15, -6, 1, 4], [11, 5, 1, 1, 7, -1], [24, 2, 12, -6, 6, 2], [18, -11, 9, 1, -5, 14], [16, 0, 6, 2, -14, 10], [21, -2, -10, 9, -2, -1], [26, -14, 6, -10, 2, -2], [5, -14, -1, 1, 11, -7], [-9, 3, -9, -5, 15, -5], [-7, -13, 3, -1, 5, -15], [-24, -3, 1, -4, 13, -8], [15, 7, 9, 3, -15, -3], [8, 6, -18, -2, -4, 10], [-2, 4, 2, 0, -8, 14], [-23, 9, -11, -4, 0, 8], [-26, -3, 3, -3, 13, -4], [-9, -6, -2, -1, -10, -3], [2, -3, 0, 2, 10, -4], [-5, -9, -5, 2, 6, 10], [-4, 0, 2, 6, -16, -6], [7, -19, 11, 5, -3, -7], [0, -14, 4, 6, -2, 2], [-2, -13, -3, -1, -1, 4], [-20, -2, -8, -6, 18, -8], [-26, 0, -4, -4, -10, 8], [-23, 15, -5, -1, -11, 5], [30, 10, -4, -8, 2, -8], [-21, 15, -9, -1, 1, -13], [4, 12, -6, 4, -2, 6], [-11, 15, -5, -1, 5, -11], [-28, 9, -3, 0, -7, 4], [-4, 19, -2, -4, -4, -6], [-16, 8, -16, 10, -8, 8], [8, 14, -6, -6, -4, -8], [6, 0, -2, -6, 22, 0], [20, 14, 4, -4, 0, 2], [26, -1, -3, 3, -1, -10], [-46, 2, -2, 6, 6, -2], [14, 3, 3, 7, 13, -2], [39, -3, -1, -5, -7, -7], [-1, 1, 3, -3, 1, 7], [-3, -2, 2, 3, -14, -1], [-17, -3, 1, -7, 3, -5], [-14, -15, -5, 1, 9, 6], [-13, 3, -13, 7, -1, -11], [27, 4, -6, -8, 10, -1], [-21, -2, 10, -7, 10, -7], [-10, -13, 12, 2, -10, 4], [-25, 15, -5, -3, -15, -1], [22, -4, 10, -2, -4, -8], [-10, -2, -2, 5, 7, 5], [-30, 5, -3, -6, -3, -2], [-2, 12, -6, -2, 10, -10], [15, -15, 9, 1, 7, -1], [2, -6, -4, -2, 2, 8], [-23, -7, -3, 1, -17, 5], [-15, -15, 5, 0, -4, 6], [26, -6, -4, 0, -12, -2], [13, 2, 2, 7, -14, 11], [-13, 0, 5, 1, 5, 13], [-11, 7, -5, -5, 7, 1], [6, 13, 13, -3, 3, -10], [-8, 6, 2, -4, -4, -12], [-62, 3, 3, 2, 3, -6], [22, -10, 10, 2, 2, 8], [34, -2, 14, 0, 2, -4], [-14, 7, 8, 2, 2, 0], [-42, 2, 4, 6, 10, 0], [-13, 7, -3, -1, -13, -11], [9, 3, 3, -11, 15, -9], [-34, 18, -10, -6, -14, 4], [38, -6, 0, 0, 10, 12], [19, 8, 3, -7, 9, 7], [-7, 1, 5, 5, -13, 11], [-19, 6, -2, -9, -4, -9], [-20, 12, -16, 4, -8, 4], [11, 4, 11, -3, 21, -9], [24, 8, 4, 0, 12, -8], [17, -17, 11, 3, -3, 1], [-11, -4, -8, -11, -2, 7], [47, -3, 11, -2, -10, -6], [6, -16, 12, 2, -6, 2], [-24, -12, 4, 4, -6, 8], [4, 4, -2, -2, 0, -10], [-26, 0, -4, 4, 2, 6], [22, 8, 14, -5, -7, 3], [-15, 1, 1, -2, -14, 14], [-36, 14, -14, 2, -2, -10], [-42, 10, -4, -12, -8, 2], [25, 1, 1, 1, 15, -3], [18, -8, -2, 6, 14, 2], [-7, 10, -2, -10, -6, 1], [37, 1, 9, 0, 8, -4], [6, -21, -2, 6, -6, -2], [9, -5, -3, -7, -17, 5], [-7, -9, 3, -5, 1, -3], [7, -1, -5, -7, -3, -7], [-7, -1, -15, 1, -7, -3], [-10, -15, 13, -10, -3, 6], [5, -16, -5, 3, -1, 3], [16, -6, -6, 0, -4, 14], [-20, -18, -2, 8, 2, -2], [-19, 2, -10, 11, 0, 3], [4, 0, 10, 8, 16, -8], [-47, 6, -2, 5, -6, -1], [-7, -2, 10, -6, -10, 13], [-10, 6, 18, 0, 0, -16], [28, -10, -8, -4, 6, -8], [-3, 10, 1, -5, -17, 13], [10, 14, -2, 14, -2, -6], [-13, 5, -5, -10, 2, 8], [-12, 4, 2, 6, -8, -6], [-8, 0, -12, 4, -20, -6], [17, 7, -11, 21, -11, 17], [-12, 8, 6, -10, -4, 6], [12, -4, 14, -12, 12, -8], [8, 8, 8, 4, -4, -14], [-18, -19, -9, 9, 1, 10], [18, 8, -2, 6, 0, -10], [37, 11, -1, 3, -7, 9], [24, -18, -10, -8, 0, 14], [-8, 3, -5, -2, 3, -8], [27, -15, 7, -6, 10, -8], [-4, 14, -6, -6, 8, 18], [-9, 5, -11, -5, 3, -5], [38, -6, -8, -8, -6, 0], [22, -4, 20, -10, -2, 4], [6, 12, -12, 0, -2, 0], [-28, -2, 6, -4, 8, -4], [-35, -7, -1, -11, -7, 9], [12, 3, 3, 14, -1, 4], [-8, 6, -6, 12, 18, 14], [-21, 5, 7, 1, 1, 1], [-28, 3, -1, -1, -3, 0]];
    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_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_3344_2_a_w(:prec:=6)
    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_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_3344_2_a_w( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3344_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-30, 21, 69, -13, -17, 1, 1]>,<5,R![18, 9, -99, 77, -9, -5, 1]>,<7,R![-96, -156, 81, 58, -25, -2, 1]>],Snew);
    return Vf;
end function;