
// Make newform 3344.2.a.y 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_y();".
// 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_y();".
// 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 := [2, -18, 11, 12, -7, -2, 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], [-3, 0, 1, 0, 0, 0], [2, -4, -1, 1, 0, 0], [0, 6, -4, -2, 1, 0], [-2, 10, 3, -7, -1, 1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1];
    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], [1, -1, 0, 0, 0, 0], [-1, 0, 0, 0, -1, 0], [1, -1, 1, 0, 0, -1], [-1, 0, 0, 0, 0, 0], [1, -1, 1, 1, 1, -1], [-1, -1, -1, 0, 1, -1], [1, 0, 0, 0, 0, 0], [2, 0, 1, 0, -1, 2], [1, -1, 1, -2, -1, 0], [-1, 1, -2, 1, 0, -3], [-1, 1, -2, 4, 0, -1], [-2, 0, -2, 1, -2, 0], [5, -2, -1, 0, 0, -2], [2, 4, 0, 0, -2, 0], [-2, 0, 2, 0, -2, 0], [1, -1, -2, 2, 0, 1], [-1, 1, 1, -4, 1, -1], [7, -1, 2, 1, -2, -1], [3, -1, 0, 0, 0, -2], [4, -6, 0, 2, 0, -2], [-3, 5, -1, 0, -1, 1], [6, -3, -2, 0, 1, 1], [1, -1, 0, -2, 2, 3], [1, 1, -2, 4, 2, -1], [1, 1, 5, -2, 1, -3], [0, 2, 0, 2, -2, 1], [8, 0, -2, 0, 6, 0], [0, 0, -2, 2, 2, -4], [-5, 3, 2, -2, 0, -1], [-3, -1, -3, 6, 1, -1], [6, 4, -4, 0, 3, -2], [-6, 0, -5, 4, -2, -2], [4, -3, 0, 0, 3, -3], [-4, 0, 2, 2, 0, -2], [-6, 2, 0, 2, -2, -4], [-1, 2, 2, -4, 1, -2], [4, 0, 6, -2, -4, 2], [-5, -1, -1, 0, 1, -3], [-1, 5, 1, 0, -1, -4], [3, 3, -4, 3, 0, 3], [1, 3, 0, 2, 4, 1], [2, -2, 1, -2, 3, 2], [2, 4, -2, 0, 2, 3], [-1, 1, 3, -2, 3, 5], [-2, 0, 4, -2, 2, -2], [3, 1, 7, -6, 1, -1], [5, -1, 2, -2, 0, -3], [-7, -1, 3, 4, 1, -1], [-6, 4, -3, 0, -2, -4], [-9, 3, 1, 2, -3, -1], [-3, -10, -1, 2, 0, 0], [0, 2, -4, 6, 0, -1], [2, 0, -5, 10, -3, 0], [4, 2, -2, 2, -2, 0], [3, 0, 1, -6, -2, 2], [8, 8, -6, 2, 0, -2], [-10, -4, 2, -4, 1, 0], [8, 6, -2, 2, 2, 2], [-16, 2, 2, -4, -6, -1], [5, 3, -1, -4, 4, -3], [-7, 3, -3, 5, 1, 1], [-7, 9, -7, 6, 1, -5], [-4, 10, 2, 0, -2, -2], [-1, -6, 8, -2, -9, -4], [-7, 3, 2, -2, -2, 7], [-5, 9, -8, 2, 0, -2], [5, -3, -5, 1, -3, -3], [1, -1, 3, -4, -3, 3], [7, 5, -5, 4, -7, -3], [2, 0, -1, 6, -1, -4], [-2, 4, 6, 0, -5, 0], [-6, 6, 1, 2, 1, 6], [6, -10, 8, 0, 0, 1], [-7, 1, 0, 1, -4, 7], [1, -1, 6, -1, 8, 1], [-1, -8, 0, -4, 5, 0], [2, -2, -4, -2, -5, 2], [12, 0, 6, -8, -2, 6], [-1, 3, -3, -4, 7, 4], [4, -4, 2, -2, -4, 6], [8, 2, -6, 6, 0, 0], [-6, -2, 4, -4, -4, 12], [3, -3, -8, 0, 8, -5], [-5, -3, -5, 8, -1, -1], [20, -2, -1, 4, 3, -10], [-5, 1, -4, 6, 2, -3], [1, -7, 7, 4, -1, 3], [-6, 0, 6, -2, -2, 6], [10, -10, -1, 10, 3, 0], [-4, 10, -1, -8, -1, 2], [7, -3, 5, 4, 4, -3], [5, 1, -2, 6, 6, 0], [-3, -11, 7, -8, -4, -1], [4, -2, -2, -2, 6, -4], [11, -12, 9, 2, -4, -6], [-1, -1, -8, 2, 8, 1], [-5, 7, -4, 4, 0, 1], [-6, 0, -10, 6, -10, -2], [-11, 9, -1, -4, -9, 1], [3, -9, -1, 0, -3, 7], [12, 4, -10, -2, 6, 0], [-3, -3, 5, -6, -3, -7], [-3, -11, -3, 7, -1, -1], [7, 5, 5, -4, -6, 11], [10, -6, 1, 4, 2, 0], [14, -2, -4, -2, 8, 0], [-16, 4, 0, -2, -6, 10], [6, -4, -8, 4, 2, 1], [7, -5, 5, -7, -5, 1], [-13, 3, 5, -2, 1, -3], [5, 1, -1, 0, -5, 7], [-10, -2, 10, 2, 5, -4], [-2, 4, -5, 6, -5, -2], [10, 10, -1, -6, 3, -2], [-11, -1, 8, 2, -2, -1], [16, -8, 5, 8, -3, -8], [-2, 0, -9, 0, 7, -6], [2, 0, 7, -4, -12, 0], [6, 2, 0, 6, 4, 6], [-11, -3, -10, 6, 0, -5], [-20, 2, -2, -3, 2, -4], [-14, -6, -4, 8, 0, -1], [-10, 6, 2, 0, -2, 8], [12, 2, -5, 2, 1, -2], [-4, 8, 10, -6, 4, 4], [14, -10, 11, -4, -8, -2], [-14, 10, -5, 2, 1, -2], [8, -6, 11, -14, -1, 4], [3, -5, -7, 6, 1, -3], [6, 7, -2, 4, -1, -1], [1, -3, 5, -4, 5, 3], [19, 9, 2, -6, 4, -1], [-21, -4, -1, 2, -14, 0], [-4, -6, 8, 6, -4, 4], [-10, -6, 2, -2, -2, 10], [-16, 12, -6, 0, 0, -2], [6, 0, 6, -4, 6, 0], [-7, 3, 0, -4, 0, -9], [-2, -6, 6, -6, 0, -2], [-11, 1, -1, 8, 7, -7], [3, -9, 3, 0, 5, 7], [-2, 0, 1, 0, -5, 2], [7, -5, -1, -8, 3, -3], [9, -3, -8, 6, -4, 5], [3, -5, 4, -6, 8, -6], [-8, 0, 2, -2, 4, -4], [7, -9, -9, 3, 5, -1], [12, -8, -3, -2, -3, -2], [0, 4, 0, 7, 0, 0], [32, -8, -4, 4, 8, 3], [-2, 0, -3, 12, 0, -2], [16, 8, 2, -10, 2, 4], [-2, 0, 8, 0, -6, 0], [14, 6, -2, 8, -10, -4], [-14, 2, 2, -8, 6, -4], [-20, 11, 0, -10, -1, 9], [-6, -4, 2, -2, -9, 2], [15, -5, 9, 0, 1, 7], [-11, 3, -9, 10, 1, -9], [-2, 2, 1, 12, -1, -2], [7, 1, 1, -2, 11, 1], [11, 13, -7, 4, 7, -11], [15, 5, -12, 2, -8, 8], [-9, 9, 6, -2, 0, 7], [11, -11, 6, -3, 0, 7], [0, -8, -4, 9, 4, -4], [-11, 1, 5, -4, -1, 5], [20, 2, -10, -1, -10, 2], [0, 6, -5, 0, 4, 10], [-4, -14, -6, 6, 12, -10], [-6, -12, 8, -2, 2, 8], [5, 2, 5, 6, -6, -8], [-10, 6, 2, 4, -4, -4], [-11, 7, -10, 2, 4, 5], [-15, 14, -8, -2, -1, 0], [-2, 2, 4, -8, -15, 8], [10, 8, -5, 6, 15, 0], [-17, -5, 3, -8, 7, -1], [9, -5, 7, 12, -9, -7], [8, 16, 8, -4, -10, 4], [-10, -10, 0, 0, 2, -2], [-9, -19, 10, 0, 0, -3], [14, -12, 12, -2, 0, -1], [16, -6, 1, -2, -3, 6], [-22, 4, -6, 0, -10, -4], [15, -11, 3, 10, 1, 1], [-4, 0, 8, -2, 6, -5], [-17, 11, -3, 10, -5, -10], [-5, 6, -3, -2, 8, 12], [-13, 15, -6, 4, 2, -3], [-1, 13, -9, 8, 2, -5], [21, 7, 4, -8, 4, 2], [32, 2, 6, -12, 0, 0], [4, 12, -16, 8, -5, 10], [5, -13, -2, -8, 4, 15], [-16, 2, -10, 2, 10, 6], [-10, 2, -1, 4, 6, -2], [7, -11, 3, 6, 11, -7], [-10, 6, 0, 4, 0, 10], [28, -4, -8, -7, 2, 0], [-12, -12, -2, 4, -8, 0], [10, -8, -10, 8, -4, -2], [10, -8, -2, -6, 6, -17], [0, -4, 7, -12, -1, 6], [12, 6, -7, 10, 13, -14], [-10, 8, -5, -6, -3, -10], [10, -2, -2, 6, 0, -6], [-15, 5, 7, -8, -1, -5], [-4, -8, -6, 2, 0, -5], [29, -11, -5, -6, 3, 3], [-13, 10, -6, 0, 1, -2], [0, 6, -11, 10, 1, -8], [-14, -4, -2, -1, 4, 4], [-17, -1, 13, -2, -1, -7], [-7, 7, 0, -4, -6, 1], [21, -5, 7, 4, -3, 3], [-1, 1, 7, 10, -11, -7], [14, -6, -2, -5, 10, 8], [-42, 6, -1, -8, -2, 14], [-2, 2, -10, 11, -4, 0], [14, -10, 6, 6, 2, 2], [3, 13, -12, 4, 4, -7], [24, -6, 7, 8, -7, -8], [0, 16, -8, 4, 0, 14], [8, 8, -14, 12, -6, 6], [-10, 12, -4, 0, 6, 0], [20, -10, 4, -12, -6, 11], [-16, -10, 8, -10, -6, 10], [14, -7, 20, -4, -5, 9], [-35, 14, -2, 8, 3, 6], [-14, -2, 6, -22, 8, 12], [-10, 10, 0, 6, -10, -8], [-13, 3, -1, -9, -7, -1], [-12, 10, 3, -14, -3, 4], [14, -8, 4, -4, 3, 8], [8, 8, -15, 0, 0, -6], [-26, 16, -6, -2, 2, -4], [32, 4, -3, -2, 15, 0], [-1, 21, 0, -3, -6, -1], [5, -9, 6, 8, -6, 4], [0, -4, 4, -4, -5, -2], [40, -6, -2, -2, 0, -2], [-11, 11, -8, 6, -6, 7], [18, -14, -10, 6, 10, 6], [4, -8, -8, 8, 9, 0], [2, 16, -10, 2, -14, 8], [-16, -4, 8, -2, -14, -3], [-7, -3, -13, 2, 7, 1], [-9, -7, 7, 4, -15, 1], [20, -4, 0, 0, 18, 6], [2, 2, -12, 12, -8, 2], [26, -6, -13, 4, -5, 6], [-10, 6, -4, -6, -12, 8], [8, -6, 10, -12, 0, 12], [16, 3, -2, -4, -13, 11], [19, 18, -9, 10, -8, 2], [-7, 5, -5, 4, -1, 1], [25, -15, 12, -2, -2, 13], [11, -1, 7, 0, 1, -11], [-12, 6, -8, 6, -2, 8], [-7, 11, 5, 10, -3, -5], [17, 5, 17, -14, 1, 13], [18, -14, 8, -14, -4, 3], [31, -18, -5, -4, -2, 8], [-6, 6, -2, -4, -22, 4], [21, -13, 4, 10, 10, -17], [7, 4, -6, 6, -3, -18], [9, 9, 3, -4, 15, 11], [-2, 2, 8, 0, -14, -6], [25, 3, 0, 6, 18, -15], [-5, -35, 18, -2, -2, 4], [-6, -2, -6, 6, 1, -4], [15, -16, 11, -6, -10, -2], [11, 3, 11, -6, 17, 8], [-20, 8, -7, -4, 7, 12], [0, -6, 1, 10, 3, -10], [36, 0, -6, 16, -6, -1], [-24, 10, 4, -8, 0, -5], [14, -17, 4, 12, 13, -9], [-26, 10, 14, -2, 0, 2], [10, 26, -17, 8, -3, -8], [-1, 10, 3, 2, 10, 0], [-10, 6, -12, -2, 8, 3], [14, 1, 16, -20, 5, 5], [12, 14, -7, 2, 5, 0], [30, -2, 0, 6, 1, -16], [-14, 6, -8, 5, 6, -10], [6, -6, -3, -10, -7, 0], [-1, 23, -3, 6, -1, 16], [-20, 16, -11, -4, -8, 4], [4, -24, 2, 0, -4, 4], [-22, -4, 14, -7, -2, 10], [36, -8, -2, -4, 2, 12], [8, -4, -10, 0, 2, 5], [24, 12, 4, -10, 12, -2], [13, 23, -10, 13, 8, -11], [-38, -12, -3, 10, -3, -4], [-17, 9, -15, 8, 1, -1], [11, -18, 13, 0, -2, -6], [-20, -6, -10, 12, 6, -4], [21, -13, 17, 3, -7, 1], [1, -1, -13, -4, -3, 7], [-2, -2, 2, 12, 4, -5], [38, -10, 1, 4, 11, -8], [-5, -11, 12, -12, -6, 19], [-15, -11, 10, -11, -14, -5], [-3, 5, -8, 6, -8, 3], [2, -14, -12, 6, 14, -22], [7, -21, -3, 0, -7, -1], [27, -17, 1, 4, 8, -13], [-3, -6, 5, 2, 10, -2], [18, -4, -8, -7, 4, -12], [27, -15, -6, 14, 2, -16], [46, -4, -14, 0, -1, 8], [28, -12, 14, 6, 6, -20], [-4, -4, 5, -8, 13, 8], [3, 3, 1, -4, 15, 3], [-7, 18, 7, -4, -10, 0], [-30, -10, 2, 2, 8, -8], [7, -17, -5, 14, -11, -7], [-4, -14, -3, 6, -7, 10], [-11, -1, 7, 10, 5, -1], [-21, 17, 10, -6, -16, 11], [-19, 13, -23, 8, 9, 9], [-19, 13, 4, -16, 4, 13], [29, -11, 22, -7, -2, -1], [25, -25, 8, 2, 0, 10], [19, -1, 19, -4, -15, -1], [8, -6, -4, -4, 16, 4], [-3, 5, -11, 2, -15, -7], [10, -4, 6, -6, 12, 8], [19, -6, -7, -6, 14, 8], [-47, 11, -9, 0, -3, -13], [38, -19, 8, 16, 9, -3], [-28, 12, 7, -10, 7, -2], [7, -5, -10, 10, -10, -3], [-16, -18, 0, -1, -4, -8], [-18, 0, -1, -2, -13, 16], [17, -6, 27, -18, 2, -4], [-35, 19, 10, -14, -2, -11], [3, -12, -2, 2, 21, 2], [-40, -4, 12, 1, -10, 0], [29, 9, -18, -8, 18, -8], [-25, 11, -4, -4, -2, -1], [3, -17, 1, 4, 1, 9], [6, -16, 15, 0, 18, -4], [37, 13, -6, 3, 18, -17], [36, -10, -4, 2, 0, 2], [30, -24, 9, 2, -7, -12], [-39, 7, -23, 16, 1, -1], [23, 3, -2, 16, -14, -15], [22, 4, -2, 18, -3, -8], [-4, 4, 0, -18, 10, 0], [33, 7, -5, -1, 27, 3], [-19, 1, -1, -4, -11, -10], [-8, 20, -9, -2, 11, 0], [-25, -18, -11, 16, -14, -2], [-38, 8, 0, 2, -8, -12], [-15, -1, -4, -7, 10, 7], [-34, 12, 0, -16, 4, 14], [3, -7, -5, 0, 9, 11], [3, -5, -14, 4, -10, 6], [-18, 8, -4, -12, -8, -6], [12, -18, 5, 4, -5, -2], [-33, -11, 8, -2, -10, -9], [23, -11, -7, 2, -5, 1], [-33, 21, -3, -6, 5, 11], [-12, -6, 8, 6, 16, -15], [-8, 2, 9, -32, 1, 0], [-16, -10, 12, -9, -14, 2], [35, 1, -5, -12, 5, -7], [16, -8, 8, -8, -4, -11], [13, -21, 9, -12, 3, 15], [-38, 4, 15, 2, 3, 8], [6, -4, 14, -7, -8, 10], [-6, -2, -8, 0, 1, 2], [-15, 11, 1, -16, -1, -11], [11, -11, 3, 6, 1, -1], [-16, 8, 14, 6, -22, 12], [20, 8, 0, 8, 2, 16], [-30, 12, -27, 4, 4, 6], [11, -1, 3, 0, 8, -13], [-15, 7, 1, -20, 9, 9], [4, -2, 4, -16, 2, -8], [7, -7, -18, 10, 16, -21], [-12, -4, -18, 8, 2, -6], [31, -14, 12, -24, -7, 10], [-13, 9, -15, 0, 2, 15], [-27, -19, -8, 14, -4, -6], [-35, 11, -24, 6, 8, -7], [-61, 5, 6, -6, 2, -5], [2, 18, 19, -14, 9, 4], [-36, -6, -5, 6, -3, -12], [39, -15, 10, -5, 12, 21], [16, -26, 8, -16, 0, 13], [12, 4, 6, 2, 10, -22], [20, 18, -2, -8, 4, -4], [-2, 16, 14, -5, 8, 8], [7, -21, 19, -4, -13, 1], [-2, -30, 12, -1, -6, -4], [11, 4, 6, -6, 27, 10], [12, 6, -16, 4, 24, -8], [14, -8, 6, -16, 6, -6], [-1, -3, 11, -20, -11, 10], [-10, -2, -24, 16, -12, -6], [14, -2, -14, 6, 9, -30], [-1, -7, 3, -14, 17, 5], [-2, -6, -12, 12, 12, -8], [4, -28, 13, -14, -7, 8], [-32, -4, -18, 14, 6, -16], [31, -11, 19, -16, -7, 9], [12, -4, 0, 8, 8, 6], [43, 2, 1, -12, 4, 2], [-11, -33, -3, 10, -5, -1], [4, 22, 7, -6, -5, 14], [-3, 9, 5, -6, 1, -11], [15, -7, -16, -5, 18, -9], [0, -2, -16, -2, 1, -4], [-5, -13, 21, 6, -5, 9], [23, 7, 10, -18, 18, 1], [40, -24, 4, 2, 4, 16], [6, 2, -8, 12, 10, 0], [-34, 0, -16, 12, -2, -14], [-36, -12, 8, -8, -8, -10], [-21, 23, -10, 4, 4, 7], [-55, 13, 2, 3, -10, -7], [16, -18, -10, -14, 22, 12], [-12, 12, -9, 6, 13, 16], [-37, 0, -5, 6, -4, 6]];
    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_y();".
// 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_y(: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_y();".
// 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_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3344_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, -8, 0, 16, -2, -4, 1]>,<5,R![81, 88, -20, -46, -8, 4, 1]>,<7,R![178, -206, -65, 116, -23, -4, 1]>],Snew);
    return Vf;
end function;