
// Make newform 2268.2.i.l in Magma, downloaded from the LMFDB on 15 November 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2268_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2268_i_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_2268_2_i_l();".
// 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_2268_2_i_l();".
// 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 := [1, 1, 5, -4, 15, -2, 4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [92, -1586, 437, -391, -14, -92, 23, 14], [-240, 305, -1140, 1020, -353, 240, -60, 64], [326, 1693, -1197, 2227, -414, 541, -63, 125], [-773, -854, -854, -3434, -52, -961, -121, -237], [-229, 1730, -1160, 4080, -562, 1096, 15, 273], [-444, -86, -86, -4471, 344, -1290, -111, -344], [-189, -970, 1920, -5916, 990, -1545, 25, -412]];
    Rf_basisdens := [1, 289, 289, 289, 289, 289, 289, 289];
    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_2268_i();"
function MakeCharacter_2268_i()
    N := 2268;
    order := 3;
    char_gens := [1135, 1541, 325];
    v := [3, 2, 2];
    // 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_2268_i_Hecke();"
function MakeCharacter_2268_i_Hecke(Kf)
    N := 2268;
    order := 3;
    char_gens := [1135, 1541, 325];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 0, 0, 0, 0, 0], [-1, 0, -1, 0, 0, 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]];
    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, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0], [1, -1, 1, 1, 1, -1, -1, -1], [-1, -1, -1, 1, 1, -1, -1, -1], [0, 0, -1, 1, 0, 1, 0, 1], [-2, 1, -2, -2, -2, 2, 1, 2], [0, -1, -1, 1, 2, 1, 0, -1], [0, 2, 1, 0, -2, 0, 0, 2], [1, 0, 0, -1, 1, 2, 2, 1], [1, -1, 1, 0, 1, -1, -1, 0], [0, 0, 0, 1, 2, -2, 0, -1], [0, 2, 2, 0, -1, 0, 0, 1], [-3, 0, 0, -1, 0, 1, 2, 0], [0, 1, 6, 2, 2, 2, 0, 0], [-2, 0, 0, 2, 2, 0, -1, 2], [4, 0, 0, 1, -1, -2, 2, -1], [0, 0, 0, 2, 1, -1, -5, 1], [-1, 0, 0, 2, 1, -1, 2, 1], [0, -1, 3, -2, 0, -2, 0, -2], [1, 0, 0, 2, 0, -2, 1, 0], [0, -2, -10, 1, 1, 1, 0, 0], [-6, -3, -6, 4, 2, -2, -3, -4], [0, -3, 3, 0, -1, 0, 0, 1], [2, -1, 2, 2, 1, -1, -1, -2], [0, -4, -4, 1, 1, 1, 0, 0], [3, 0, 3, 4, 2, -2, 0, -4], [0, 3, 6, -2, -2, -2, 0, 0], [2, -2, 2, 1, 3, -3, -2, -1], [-2, 0, 0, -1, -2, -1, -1, -2], [0, 3, 1, 3, 2, 3, 0, 1], [5, -1, 5, -4, -5, 5, -1, 4], [-9, 0, -9, -3, -1, 1, 0, 3], [0, 4, 3, -1, -1, -1, 0, 0], [-9, -6, -9, 1, 1, -1, -6, -1], [2, 0, 0, 3, 0, -3, 0, 0], [4, 2, 4, -2, -3, 3, 2, 2], [-1, 2, -1, 2, 1, -1, 2, -2], [-5, 0, 0, 3, -1, -4, 0, -1], [0, 3, 6, 2, -2, 2, 0, 4], [13, 0, 0, -1, 1, 2, 2, 1], [9, 0, 0, 4, 3, -1, -2, 3], [-1, 0, 0, -3, 2, 5, -1, 2], [-1, 0, 0, 0, 1, 1, 3, 1], [0, 1, -4, 2, -1, 2, 0, 3], [7, 5, 7, -2, -3, 3, 5, 2], [0, 3, 8, 3, 0, 3, 0, 3], [9, 4, 9, -1, 0, 0, 4, 1], [0, -2, 6, 1, -1, 1, 0, 2], [7, -3, 7, -1, -1, 1, -3, 1], [-6, -1, -6, 3, 1, -1, -1, -3], [-8, 2, -8, -5, -6, 6, 2, 5], [18, 0, 0, -3, 0, 3, 0, 0], [0, -3, 6, -4, 1, -4, 0, -5], [15, -3, 15, 4, 5, -5, -3, -4], [0, -4, -8, 0, 1, 0, 0, -1], [-1, -5, -1, 4, 2, -2, -5, -4], [-8, 1, -8, 1, 1, -1, 1, -1], [0, -6, 5, -2, 3, -2, 0, -5], [3, 0, 0, 1, -1, -2, -2, -1], [-9, 1, -9, -7, -6, 6, 1, 7], [6, 0, 0, 5, 2, -3, -4, 2], [1, 0, 0, 0, -1, -1, 6, -1], [18, 0, 0, 0, -1, -1, 0, -1], [9, 0, 0, -2, -3, -1, 4, -3], [-4, 0, 0, 5, 3, -2, 2, 3], [-1, 4, -1, -7, -8, 8, 4, 7], [0, 0, 0, 0, -3, -3, -3, -3], [0, -6, 12, -3, 2, -3, 0, -5], [0, 5, 0, -9, -8, 8, 5, 9], [5, 5, 5, -7, -5, 5, 5, 7], [7, 3, 7, -3, 3, -3, 3, 3], [0, 4, 3, 1, -7, 1, 0, 8], [-6, 0, 0, 4, -2, -6, 0, -2], [0, -1, 5, 1, -1, 1, 0, 2], [26, 1, 26, 1, 0, 0, 1, -1], [-13, 1, -13, 2, 1, -1, 1, -2], [0, 3, 0, -4, -8, -4, 0, 4], [11, 0, 0, -4, 2, 6, 1, 2], [9, 0, 9, 4, 2, -2, 0, -4], [0, -1, -2, -7, -5, -7, 0, -2], [0, -9, 9, -4, 4, -4, 0, -8], [14, 0, 0, -4, -3, 1, 5, -3], [-2, 0, 0, 7, 4, -3, -8, 4], [-1, 0, 0, -3, 1, 4, -6, 1], [1, 0, 0, -6, -4, 2, 3, -4], [-3, 0, 0, 7, 2, -5, -11, 2], [0, -3, -20, 3, 8, 3, 0, -5], [-3, -7, -3, -2, 2, -2, -7, 2], [-19, 3, -19, -6, -7, 7, 3, 6], [-2, -12, -2, 5, 5, -5, -12, -5], [0, -4, 17, 3, -1, 3, 0, 4], [12, 8, 12, 0, -2, 2, 8, 0], [0, -4, -15, -2, 0, -2, 0, -2], [24, 0, 0, -4, -3, 1, -4, -3], [0, -3, 0, -4, -5, -4, 0, 1], [0, 2, -3, -7, -3, -7, 0, -4], [-14, 1, -14, 5, 3, -3, 1, -5], [-10, -5, -10, -1, 1, -1, -5, 1], [0, -4, -11, 7, 5, 7, 0, 2], [0, -4, 16, 0, 1, 0, 0, -1], [-19, 0, 0, 1, -2, -3, -2, -2], [24, 0, 0, -2, -3, -1, 1, -3], [19, 0, 0, -1, 1, 2, 5, 1], [0, 1, 6, 5, 1, 5, 0, 4], [0, -1, -31, -2, -1, -2, 0, -1], [-9, 0, -9, -5, -4, 4, 0, 5], [-6, 0, 0, -1, -3, -2, -7, -3], [0, -5, 17, 3, 4, 3, 0, -1], [0, 8, 7, -1, -5, -1, 0, 4], [0, 8, 4, 1, -7, 1, 0, 8], [25, 10, 25, -6, -3, 3, 10, 6], [-3, -4, -3, 2, 4, -4, -4, -2], [-4, 0, 0, 3, 7, 4, -5, 7], [19, 4, 19, -6, -3, 3, 4, 6], [-2, 3, -2, 0, -3, 3, 3, 0], [0, 8, 12, -1, -12, -1, 0, 11], [0, -10, 15, -4, 9, -4, 0, -13], [0, -10, 2, 1, -1, 1, 0, 2], [6, 0, 0, -1, -4, -3, -1, -4], [0, -4, -19, 0, -1, 0, 0, 1], [-21, 0, 0, 5, -3, -8, -7, -3], [0, -5, 3, -1, -4, -1, 0, 3], [11, 0, 0, -7, -3, 4, -7, -3], [-11, 0, 0, 0, 5, 5, -9, 5], [-4, 0, 0, -3, 1, 4, -2, 1], [-9, -4, -9, 12, 10, -10, -4, -12], [0, 8, 6, 1, 0, 1, 0, 1], [0, 8, 19, 2, 1, 2, 0, 1], [0, -1, -2, 2, 1, 2, 0, 1], [-18, -8, -18, 2, 0, 0, -8, -2], [0, 5, -2, 5, -5, 5, 0, 10], [-17, 0, 0, -2, -4, -2, -1, -4], [0, 3, 2, 1, -3, 1, 0, 4], [-18, 2, -18, 1, 2, -2, 2, -1], [0, 3, -2, 3, 8, 3, 0, -5], [-5, 0, 0, -8, 2, 10, -1, 2], [-18, -1, -18, 0, 1, -1, -1, 0], [0, 1, 16, 0, -9, 0, 0, 9], [3, 0, 0, 2, -1, -3, -7, -1], [17, 0, 0, 10, 7, -3, -5, 7], [15, 0, 0, -10, 0, 10, 12, 0], [-21, 0, 0, -7, -3, 4, 8, -3], [0, -8, 21, 1, 5, 1, 0, -4], [0, 2, -19, -2, -4, -2, 0, 2], [0, 3, 20, -3, -9, -3, 0, 6], [6, -12, 6, -2, -1, 1, -12, 2], [0, 13, -1, -1, -10, -1, 0, 9], [5, 10, 5, 1, -3, 3, 10, -1], [0, -3, 3, -3, 2, -3, 0, -5], [3, 0, 0, -5, 0, 5, 4, 0], [26, 0, 0, -4, 0, 4, 8, 0], [0, 2, 15, -1, 3, -1, 0, -4], [5, -4, 5, 4, 10, -10, -4, -4], [0, 5, 12, 2, 9, 2, 0, -7], [6, 4, 6, 0, -5, 5, 4, 0], [-6, 0, 0, -13, -3, 10, 11, -3], [26, 0, 0, -1, -7, -6, 4, -7], [22, 0, 0, 10, 5, -5, -5, 5], [-11, 0, 0, 3, -7, -10, -12, -7], [-4, 0, 0, 8, -2, -10, -4, -2], [10, -5, 10, -1, -3, 3, -5, 1], [-12, -6, -12, 7, 11, -11, -6, -7], [-6, 0, 0, 1, 6, 5, 1, 6], [15, 7, 15, -7, -6, 6, 7, 7], [0, 2, 14, -8, -5, -8, 0, -3], [7, -6, 7, 3, 6, -6, -6, -3], [-10, 0, 0, 9, -1, -10, -7, -1], [0, -1, -3, 5, 3, 5, 0, 2], [23, 5, 23, -7, -8, 8, 5, 7], [-24, 4, -24, 3, -5, 5, 4, -3], [0, 5, -21, 2, -6, 2, 0, 8], [-1, -2, -1, 14, 7, -7, -2, -14], [7, 0, 0, 2, 9, 7, 10, 9], [-7, -8, -7, 10, 12, -12, -8, -10], [0, 2, 4, -4, -8, -4, 0, 4], [0, -1, 14, 7, 14, 7, 0, -7], [-4, 0, 0, -6, 4, 10, 4, 4], [38, 0, 0, 2, -3, -5, 2, -3], [-11, 0, 0, 11, -2, -13, -1, -2], [0, -2, -10, 4, 10, 4, 0, -6], [12, 7, 12, -3, -5, 5, 7, 3], [-4, 9, -4, 0, -10, 10, 9, 0], [22, 0, 0, -2, 8, 10, 10, 8], [-9, 9, -9, -2, -7, 7, 9, 2], [0, -4, 13, 4, 14, 4, 0, -10], [-11, 5, -11, -10, -10, 10, 5, 10], [0, -2, 13, 0, 2, 0, 0, -2], [0, 9, 33, 5, 7, 5, 0, -2], [-24, -13, -24, 7, 2, -2, -13, -7], [0, -8, 0, 2, 2, 2, 0, 0], [3, 0, 3, -14, -5, 5, 0, 14], [-7, 0, 0, 6, 10, 4, 3, 10], [0, -6, 9, 8, 4, 8, 0, 4], [2, 0, 0, -3, 4, 7, -3, 4], [-4, 0, 0, 11, 3, -8, -16, 3], [3, 0, 0, -10, 0, 10, 6, 0], [0, -8, -9, -10, -1, -10, 0, -9], [-18, 12, -18, -14, -16, 16, 12, 14], [-12, 0, 0, 4, 0, -4, 4, 0], [0, -11, -2, -6, -1, -6, 0, -5], [0, -17, -13, 2, 11, 2, 0, -9], [-10, 10, -10, 4, 2, -2, 10, -4], [15, 0, 0, 1, 0, -1, -5, 0], [0, 1, -15, 8, 5, 8, 0, 3], [10, -10, 10, 1, 9, -9, -10, -1], [0, 4, -9, -1, 5, -1, 0, -6], [0, 8, 12, -7, -3, -7, 0, -4], [-34, 0, 0, 3, -5, -8, -5, -5], [-8, -4, -8, -6, -8, 8, -4, 6], [5, 4, 5, 10, 12, -12, 4, -10], [0, 8, -8, 4, -1, 4, 0, 5], [16, 0, 0, -4, -10, -6, 8, -10], [-24, 0, 0, -4, 0, 4, -4, 0], [-10, 0, 0, 0, -7, -7, -1, -7], [0, 0, 0, -4, 2, 6, -7, 2], [-2, 3, -2, 5, 8, -8, 3, -5], [-14, -11, -14, 6, 3, -3, -11, -6], [7, 13, 7, -3, -3, 3, 13, 3], [0, -7, -9, 8, 8, 8, 0, 0], [21, -2, 21, 6, 4, -4, -2, -6], [0, -3, -6, 5, 1, 5, 0, 4], [15, -2, 15, 0, 4, -4, -2, 0], [-3, 5, -3, -9, -8, 8, 5, 9], [0, 1, -31, 2, 5, 2, 0, -3], [11, 0, 0, -5, 1, 6, 7, 1], [0, 13, -8, 9, -6, 9, 0, 15], [10, 0, 0, -13, 1, 14, 8, 1], [6, 0, 0, 4, 9, 5, 7, 9], [1, 0, 0, -7, -2, 5, -4, -2], [0, 14, -18, 1, -9, 1, 0, 10], [-3, -7, -3, 16, 17, -17, -7, -16], [-18, 0, 0, -5, 3, 8, 4, 3], [0, -14, -13, -5, 0, -5, 0, -5], [24, -4, 24, -3, -8, 8, -4, 3], [0, -14, 9, -8, 5, -8, 0, -13], [12, 6, 12, 1, 2, -2, 6, -1], [6, 1, 6, -13, -6, 6, 1, 13], [13, 0, 0, -10, 5, 15, 2, 5], [7, -7, 7, 1, 1, -1, -7, -1], [-14, 9, -14, -3, 6, -6, 9, 3], [8, 0, 0, 9, -3, -12, 3, -3], [-34, -2, -34, 10, 5, -5, -2, -10], [10, 13, 10, 3, 3, -3, 13, -3], [-15, 0, 0, 8, -6, -14, -13, -6], [1, 0, 0, 20, 0, -20, -2, 0], [-43, 0, 0, -1, 1, 2, -10, 1], [-34, 0, 0, -6, 3, 9, -6, 3], [0, 0, 0, 2, 0, -2, 17, 0], [-21, 4, -21, -6, -5, 5, 4, 6], [24, -7, 24, -3, 7, -7, -7, 3], [3, 0, 0, -6, -9, -3, 3, -9], [0, -2, 36, -1, 7, -1, 0, -8], [-23, -8, -23, 9, 9, -9, -8, -9], [-17, 5, -17, 7, 4, -4, 5, -7], [0, 2, -4, 1, 4, 1, 0, -3], [29, 1, 29, 8, 7, -7, 1, -8], [-17, 0, 0, 13, 2, -11, -14, 2], [-5, 6, -5, 6, -6, 6, 6, -6], [18, -4, 18, -2, 5, -5, -4, 2], [0, 8, 37, -9, -11, -9, 0, 2], [18, 0, 0, -10, -5, 5, 13, -5], [-9, 0, 0, 10, 1, -9, -12, 1], [36, 0, 0, 6, 0, -6, -9, 0], [6, 0, 0, 12, 2, -10, 0, 2], [3, 0, 0, -18, -3, 15, 15, -3], [0, -2, 4, 3, 12, 3, 0, -9], [27, 13, 27, 6, -2, 2, 13, -6], [3, 0, 0, 3, 0, -3, 6, 0], [0, 2, 7, -1, -11, -1, 0, 10], [0, 1, 11, -1, 8, -1, 0, -9], [5, 16, 5, -1, 4, -4, 16, 1], [0, -4, 14, -5, 7, -5, 0, -12], [9, 4, 9, -15, -8, 8, 4, 15], [44, -2, 44, 7, 5, -5, -2, -7], [0, -8, -7, -11, -8, -11, 0, -3], [11, -5, 11, -1, 13, -13, -5, 1], [3, -1, 3, 4, -1, 1, -1, -4], [30, 0, 0, -7, 3, 10, 11, 3], [-3, 0, 0, 5, -6, -11, 5, -6], [4, 0, 0, 8, -5, -13, -10, -5], [0, -14, -6, -7, -1, -7, 0, -6], [0, -4, -1, -2, 4, -2, 0, -6], [0, 11, 25, -5, -16, -5, 0, 11], [-38, -12, -38, -4, 5, -5, -12, 4], [0, 16, -8, 3, -19, 3, 0, 22], [22, -19, 22, 10, 10, -10, -19, -10], [-29, -14, -29, -1, 6, -6, -14, 1], [-39, 0, 0, 8, -12, -20, -19, -12], [-15, 15, -15, 4, 8, -8, 15, -4], [0, -7, 0, 5, -6, 5, 0, 11], [0, -11, -6, 2, -1, 2, 0, 3], [23, -3, 23, -6, -10, 10, -3, 6], [0, 6, 35, -3, -3, -3, 0, 0], [-6, 0, 0, -1, -9, -8, 5, -9], [-25, 0, 0, -11, -7, 4, 18, -7], [0, -7, 6, 5, 3, 5, 0, 2], [9, -7, 9, 0, 7, -7, -7, 0], [0, -2, -47, 3, 8, 3, 0, -5], [0, 9, -30, -6, -7, -6, 0, 1], [12, 16, 12, -10, -9, 9, 16, 10], [0, 11, 9, 4, -6, 4, 0, 10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2268_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2268_2_i_l();".
// 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_2268_2_i_l(:prec:=8)
    chi := MakeCharacter_2268_i();
    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_2268_2_i_l();".
// 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_2268_2_i_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2268_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![729, -243, 405, 0, 135, -6, 16, 2, 1]>,<13,R![9, 48, 220, 210, 195, -4, 21, 3, 1]>,<19,R![97969, -66043, 53911, 1322, 2275, 182, 94, 8, 1]>],Snew);
    return Vf;
end function;