
// Make newform 2340.2.dj.d in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2340_dj();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2340_dj_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_2340_2_dj_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_2340_2_dj_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 := [16, -32, 20, 4, -11, 2, 5, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [8, -8, 2, 3, -4, -1, 2, -1], [12, -12, 3, 5, -3, -1, 3, -1], [-4, 11, -6, -2, 4, 0, -2, 1], [32, -40, 11, 15, -11, -3, 7, -3], [22, -27, 7, 11, -7, -3, 5, -2], [-88, 116, -28, -45, 30, 11, -20, 7], [-104, 132, -30, -47, 32, 13, -22, 9]];
    Rf_basisdens := [1, 8, 4, 2, 4, 2, 8, 8];
    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_2340_dj();"
function MakeCharacter_2340_dj()
    N := 2340;
    order := 6;
    char_gens := [1171, 2081, 937, 1081];
    v := [6, 6, 6, 5];
    // 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_2340_dj_Hecke();"
function MakeCharacter_2340_dj_Hecke(Kf)
    N := 2340;
    order := 6;
    char_gens := [1171, 2081, 937, 1081];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, -1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, 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+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, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 0, 1, -2, 0], [-2, 0, 0, 0, 1, 0, 0, 0], [-1, 0, -2, 1, 1, 0, 2, 1], [-1, 2, 0, -2, 2, 1, -2, 0], [0, 1, 2, 0, -1, 1, -2, -1], [-1, 1, 2, 1, 1, -2, 0, -2], [1, -3, -1, -1, 0, 2, 4, 1], [2, -2, 0, -4, 0, 4, -2, 0], [1, 3, 0, -3, 1, 0, 0, 0], [-2, 6, 0, 0, 1, 0, 0, 0], [2, -5, 0, -4, 0, 2, 2, 3], [2, 0, 0, 0, -4, 0, 0, 0], [-2, 0, 1, 3, -3, 3, -4, 0], [3, -2, -4, 0, 3, 0, 6, 2], [-2, 5, 0, -2, 3, 1, -2, 3], [-9, -5, 0, -1, 5, 0, 0, 0], [0, 3, 6, 0, 3, -3, -6, -3], [2, -8, -1, -3, -1, 3, 6, 2], [-4, 6, 3, -1, 1, -1, 4, 0], [0, 8, -2, 0, 0, 0, -6, 4], [2, -2, 2, 0, -1, 0, 0, 2], [1, -2, -4, 0, -4, 5, 2, 2], [1, -1, 1, -1, 2, 2, 0, -1], [4, 6, 3, 1, -1, 1, 2, 0], [2, -8, -1, -2, -2, 4, 12, 1], [-2, 2, -3, 1, 3, -1, 2, 6], [-3, -8, 0, -2, 4, 1, 2, -2], [1, 3, 0, -1, 9, 2, -8, 0], [-4, 0, 5, -3, 3, -3, -2, 0], [0, 1, 2, 0, -6, 6, -6, -1], [6, 3, 0, -2, -5, 1, -4, -3], [-3, -2, -4, 0, -3, 0, -6, 2], [0, -2, 1, 1, -1, -1, 2, -2], [4, 4, -3, 1, -1, 1, 6, 0], [-4, 1, 2, 0, -2, -2, -8, -1], [10, -2, -1, -6, -2, 0, 0, -1], [4, 1, 0, -4, -2, 2, -4, -3], [4, 2, -2, -4, 5, 8, -12, 2], [-2, 4, 9, 1, -1, 1, -6, 0], [-12, -1, 0, 2, 11, -1, 4, 5], [-7, 15, 2, 1, 3, 0, 0, 2], [15, 9, 0, 3, -9, 0, 0, 0], [3, -20, 0, -4, -1, 2, 8, 0], [-1, 3, 3, 1, -2, -2, -4, -3], [-2, -4, -3, -2, 2, 0, 0, -3], [8, -5, -10, 0, 2, 6, 6, 5], [4, -12, -4, -8, 0, 8, 8, 8], [16, -4, -7, 1, -1, 1, 2, 0], [6, 4, 8, 0, -12, 0, -12, -16], [2, 5, 10, 0, 7, -5, -8, -5], [-11, 4, 0, 8, 7, -4, 2, 0], [0, -2, 1, 0, -20, 0, 4, -1], [2, -10, -9, -2, 2, 4, 16, 9], [-15, 2, 0, 8, 11, -4, 4, 2], [-7, 11, 1, 3, 2, 0, 0, 1], [4, 3, 0, 0, -4, 0, 0, 3], [-4, 8, -2, 6, 2, -6, 0, 4], [-6, 4, 3, 6, -4, -12, 4, -3], [2, -1, -2, 0, 2, 0, 0, 1], [6, 2, -1, -1, -11, 1, -2, 2], [0, -4, 7, -5, 5, -5, -6, 0], [-4, 4, 0, 2, -2, 2, 2, 0], [2, -2, -4, -6, 2, 6, 0, 8], [-2, -1, -2, 0, 1, -3, -12, 1], [14, -8, -9, 5, -5, 5, -4, 0], [11, -8, 0, 6, -14, -3, 2, -10], [19, -3, -3, 9, -14, 0, 0, -3], [-20, 4, -1, 0, 10, 0, 0, -1], [6, -4, 4, 0, -12, 0, 0, -8], [-4, 2, 3, 4, -12, -8, 4, -3], [17, -6, 0, -6, -14, 3, 0, 0], [25, 3, -1, 7, -16, 0, 0, -1], [-11, 2, 4, 0, 4, -15, 12, -2], [10, 14, 5, 1, -1, 1, 8, 0], [16, -5, -10, 0, 8, 8, -4, 5], [-9, 3, 3, -3, 6, 0, 0, 3], [5, 0, 0, 0, 3, 2, -4, 0], [-4, -2, 2, 4, -11, -8, 12, -2], [-4, 20, -2, 6, 2, -6, -12, 4], [-2, -22, 4, 0, 1, 0, 0, 4], [6, 13, 0, 8, -10, -4, -4, -3], [2, 4, 0, -2, -15, 4, -12, 0], [6, 6, 3, 3, -3, 3, 0, 0], [5, -2, -4, 0, 5, 0, -6, 2], [-24, -8, -1, 0, 12, 0, 0, -1], [-1, 4, 8, 0, -1, 0, -6, -4], [0, 10, 5, 3, -3, -3, -12, -10], [-10, -6, -10, 0, 0, 0, 4, 0], [6, 8, -2, 12, -9, 0, 0, -2], [-13, 2, 4, 0, -8, -5, -2, -2], [-10, 10, 2, 10, 1, -20, 0, -2], [-2, 14, 2, 2, 2, -2, -14, -4], [-2, 17, 0, 0, 2, 0, -8, 1], [-10, -16, -2, -12, 11, 0, 0, -2], [10, 12, 10, 6, -6, 6, -4, 0], [0, -16, -9, 0, 16, 0, 32, 9], [2, -16, 3, 7, -11, -7, 20, -6], [26, 6, -3, -3, 3, -3, 12, 0], [2, 4, -7, 6, -4, 0, 0, -7], [23, -12, 0, 2, -24, -1, 6, -2], [7, -1, -11, -7, 26, 14, -12, 11], [-8, -2, 0, 0, 0, 0, -2, 0], [14, 24, 5, -5, -23, 5, -34, -10], [35, -13, -2, 3, -19, 0, 0, -2], [2, 4, 5, 3, -7, -3, -6, -10], [0, 10, -7, 5, -5, 5, 12, 0], [-3, -15, -3, 3, 20, -6, 36, 3], [-2, -33, 0, 0, 2, 0, 18, 3], [9, -1, -2, 9, -9, 0, 0, -2], [4, -7, -14, 0, 10, -6, 18, 7], [-2, 20, 8, 8, -4, -8, -20, -16], [-8, -3, -6, 0, 5, -13, 8, 3], [-15, 2, 0, -4, 17, 2, 4, 14], [10, -1, -2, 0, 4, 6, -30, 1], [-2, 10, 9, 2, -20, -4, -16, -9], [-3, 15, 12, 3, 3, -6, -24, -12], [-1, -26, 0, 0, 1, 0, 14, 2], [6, 6, 0, -12, 3, 0, 0, 0], [0, -6, -15, 0, -2, 0, 12, 15], [-4, 0, -13, 3, -3, 3, 10, 0], [0, -1, -2, 0, -6, 6, -12, 1], [17, 13, 5, 3, -10, 0, 0, 5], [-26, 8, 2, -2, 2, -2, 8, 0], [-13, -2, -4, 0, -17, 4, -8, 2], [15, 2, 0, -16, -7, 8, -14, -10], [-6, 14, -3, 1, -1, 1, 16, 0], [-6, 26, -2, 0, 12, 0, -24, 4], [2, 2, -2, 12, -7, 0, 0, -2], [-10, -26, 5, -6, 8, 0, 0, 5], [-7, 3, 9, 7, 0, -14, 8, -9], [6, -14, 3, -6, -4, 12, 16, -3], [-10, 3, 6, 0, 5, -15, -18, -3], [23, -13, -7, 5, -14, 0, 0, -7], [-4, 1, 2, 0, -16, 12, -18, -1], [9, 0, 0, 0, 10, -1, 20, 0], [-19, -20, 0, -6, 22, 3, 2, -10], [4, 4, -6, -4, -11, 8, -16, 6], [-14, 16, -2, 10, 18, -10, -4, 4], [15, -3, 3, -3, -6, 0, 0, 3], [11, -26, 0, -10, -6, 5, 8, 0], [6, -22, -5, -9, -3, 9, 18, 10], [-8, 12, -6, 8, 3, -16, -8, 6], [9, -4, -8, 0, -9, 18, -6, 4], [-8, -12, -7, -11, 27, 11, 8, 14], [-8, 12, -2, 0, 0, 0, 14, 0], [-8, 12, -9, 3, -3, 3, 18, 0], [6, -22, -5, -21, 9, 21, 6, 10], [-12, 7, 14, 0, -4, -8, -8, -7], [4, 8, 4, -4, 11, 8, -24, -4], [12, -10, -12, 4, -4, 4, -2, 0], [-3, -3, 12, 3, 3, -6, 12, -12], [7, -3, -18, -7, 15, 14, -8, 18], [12, 10, 2, 2, -2, 2, 6, 0], [16, -27, 0, -14, -9, 7, 14, 15], [-2, 7, 14, 0, -5, 3, -12, -7], [-4, 16, 9, 5, -5, 5, 2, 0], [2, 8, 13, 9, -13, -9, -12, -26], [-6, 32, 1, 6, 0, 0, 0, 1], [-5, -2, 0, 26, -8, -13, 14, 0], [4, 24, 0, 6, -14, -6, -18, 0], [-17, -2, 0, -4, 19, 2, 2, 6], [41, -7, -8, 9, -25, 0, 0, -8], [-20, -6, -9, -3, 43, 3, 12, 18], [5, 5, 9, -5, -24, 10, -20, -9], [-3, -18, 0, -26, 16, 13, -4, 0], [0, -22, -6, -2, 2, 2, 26, 12], [36, 8, 1, 1, -1, 1, 6, 0], [8, 30, -3, -3, -13, 3, -30, 6], [16, -1, -2, 0, -1, 17, -22, 1], [4, -25, 0, -6, -1, 3, 16, 13], [-4, 11, 22, 0, 4, -8, -14, -11], [-12, 16, 12, -4, 4, -4, 8, 0], [3, -4, 0, -4, -1, 2, 4, 8], [-17, 9, 11, 1, 8, 0, 0, 11], [-10, 6, -9, -9, 29, 9, -6, 18], [3, 15, 18, -3, -29, 6, -36, -18], [13, -7, 3, -13, 6, 26, -12, -3], [2, -16, 6, -14, 10, 14, -4, -12], [4, 4, -6, 14, -14, 14, -4, 0], [-8, 16, 6, 0, 0, 0, 10, 0], [0, 16, -1, 3, -3, -3, -12, 2], [-2, 8, 7, -18, 10, 0, 0, 7], [-2, 15, 0, -2, 3, 1, -12, -7], [8, 0, 6, 0, 0, 0, -6, 0], [-8, 4, -14, -8, 24, 8, 2, 28], [-2, 4, 10, -8, 5, 0, 0, 10], [4, -1, -2, 0, 7, -3, 24, 1], [8, 3, 0, -4, -6, 2, -8, -9], [8, -13, -26, 0, -4, 12, 18, 13], [24, 18, 9, 5, -5, 5, 4, 0], [-32, 20, 10, -6, 19, 0, 0, 10], [-20, 7, 0, -4, 22, 2, -10, -9], [5, -13, 0, -5, -25, 10, 16, 0], [14, 32, 1, -3, -25, 3, -36, -2], [6, 21, 0, 8, -10, -4, -8, -3], [-12, -20, 5, -3, 27, 3, 12, -10], [8, 14, 9, -5, 5, -5, 10, 0], [-8, 9, 18, 0, 7, -15, 0, -9], [-17, -24, 0, 0, 17, 0, 12, 0], [-14, -14, 5, -6, 10, 0, 0, 5], [-2, 12, -3, 1, -1, 1, 14, 0], [-13, 5, 1, -3, 8, 0, 0, 1], [4, -30, -19, -4, 0, 8, 52, 19], [-8, -8, 1, 1, 15, -1, 8, -2], [17, -38, 0, -6, -14, 3, 20, 8], [-21, 29, 13, -3, 12, 0, 0, 13], [-9, -28, 0, 8, 5, -4, 18, 0], [-1, -9, -12, 1, 25, -2, 20, 12], [-2, 10, -3, -7, 7, -7, 20, 0], [-3, 7, -6, 3, 23, -6, -8, 6], [-5, -4, -8, 0, -2, -3, -12, 4], [-14, 1, 2, 0, -23, 9, -30, -1], [20, -20, 2, -6, -34, 6, 12, -4], [8, 22, 6, 2, -2, 2, 14, 0], [4, -35, 0, -10, 1, 5, 8, -9], [1, 15, 6, -3, 1, 0, 0, 6], [14, 8, 13, -9, -19, 9, -30, -26], [0, -6, 0, 0, -11, 0, 12, 0], [-6, -28, -4, -4, 16, 4, 28, 8], [-14, 16, -7, 9, 19, -9, 0, 14], [-1, -2, -4, 0, -10, 9, 30, 2], [-4, 0, -5, 4, 18, -8, 8, 5], [-14, 8, -6, -2, 2, -2, 16, 0], [11, -15, -2, -11, 9, 22, 8, 2], [5, 20, 0, 4, -7, -2, -20, -24], [-2, -45, 0, -8, 6, 4, 26, 15], [20, 0, 3, -9, -31, 9, -12, -6], [1, 7, 6, -1, -1, 2, -16, -6], [-1, 3, 9, 1, 12, -2, -4, -9], [17, -9, 5, -5, -6, 0, 0, 5], [-10, -10, 3, 7, -7, 7, -20, 0], [-16, 30, 15, -3, 3, -3, 18, 0], [-10, 22, -1, 3, 17, -3, -18, 2], [-5, 10, 20, 0, 19, -24, -18, -10], [11, 7, -4, 15, -13, 0, 0, -4], [0, -9, 0, 6, -3, -3, 6, -3], [1, -21, -7, -1, -6, 2, 40, 7], [-3, 39, 0, 21, -9, 0, 0, 0], [8, 8, 0, -8, 1, 16, -32, 0], [-3, -20, 0, -22, 14, 11, -4, -6], [-19, -15, 11, -5, 12, 0, 0, 11], [-29, 2, 4, 0, -8, -21, 0, -2], [-8, 0, -8, 0, 0, 0, 8, 0], [-12, 13, 26, 0, -14, 2, -34, -13], [-1, -3, -9, 21, -10, 0, 0, -9], [6, 3, 6, 0, -5, 11, -40, -3], [-5, -3, -10, 5, 21, -10, 16, 10], [-3, -13, -8, 9, -3, 0, 0, -8], [51, -19, -5, 9, -30, 0, 0, -5], [14, 22, 17, -3, -25, 3, -42, -34], [-2, 16, 24, 2, -15, -4, -28, -24], [-32, -24, -11, -3, 3, -3, -10, 0], [10, 1, 2, 0, 13, -3, 18, -1], [-21, 22, 0, 20, 11, -10, -10, -18], [-11, -17, -4, 9, 1, 0, 0, -4], [-10, 4, 19, 5, -5, 5, -20, 0], [-17, 4, 8, 0, 10, -27, 24, -4], [20, -2, 9, -13, 13, -13, 2, 0], [-6, -2, -11, 6, 16, -12, 16, 11], [12, -4, 4, 0, -24, 0, 0, -8], [-12, 0, 3, 12, -20, -24, 24, -3], [-5, 14, 28, 0, 10, -15, -12, -14], [-10, 27, 0, 30, -5, -15, -6, -15], [14, -9, -18, 0, 11, 3, -18, 9], [16, 28, 8, -12, -20, 12, -48, -16], [31, -2, -4, 0, 16, 15, -12, 2], [16, 21, 0, -8, -12, 4, -18, -7], [-2, 18, 0, 0, 0, 0, 18, 0], [-6, -14, -7, -3, 15, 3, 18, 14], [42, -6, 3, -18, -12, 0, 0, 3], [-1, 4, 0, -4, 3, 2, -4, 0], [-12, 7, 0, 20, 2, -10, 2, -9], [-28, 0, 1, 8, 10, 0, 0, 1], [-4, -1, -2, 0, -10, 6, -18, 1], [-16, 12, 10, 26, 6, -26, 4, -20], [2, -7, -14, 0, -5, 7, -32, 7], [-31, 24, 0, 8, 27, -4, -6, 4], [-4, -12, 1, 4, 0, -8, 32, -1], [-11, 43, -7, 15, -2, 0, 0, -7], [2, 14, 12, -2, 2, -2, 4, 0], [12, -34, 2, 2, -7, 0, 0, 2], [-26, 10, -3, 29, 23, -29, 22, 6], [-28, -24, -10, -6, 6, -6, -8, 0], [40, -12, -33, 3, -3, 3, 18, 0], [14, 12, -3, 3, -31, -3, -6, 6], [0, 5, 10, 0, -21, 21, -12, -5], [-14, -1, 0, -18, 23, 9, -2, 13], [-5, -11, 3, 5, -32, -10, 32, -3], [-24, -9, 0, -12, 30, 6, 6, 15], [-45, -11, -10, 3, 21, 0, 0, -10], [6, 1, 2, 0, -9, 15, 0, -1], [9, -10, 0, 10, -14, -5, 16, 12], [10, 4, 24, -4, 4, -4, -16, 0], [32, 8, -15, -3, 3, -3, 26, 0], [-8, 22, 15, 8, -26, -16, -28, -15], [13, -11, -9, -13, -8, 26, -4, 9], [-7, 1, 2, 9, -1, 0, 0, 2], [6, -3, 0, 4, -8, -2, 2, -3], [-18, 26, 4, 30, 6, -30, 0, -8], [-6, 18, 0, 6, 13, -12, -24, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2340_dj_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2340_2_dj_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_2340_2_dj_d(:prec:=8)
    chi := MakeCharacter_2340_dj();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_2340_2_dj_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_2340_2_dj_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2340_dj();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![9, -54, 126, -108, 3, 36, 6, -6, 1]>],Snew);
    return Vf;
end function;