
// Make newform 1344.2.bl.i in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1344_bl();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1344_bl_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_1344_2_bl_i();".
// 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_1344_2_bl_i();".
// 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, -16, 4, 4, -6, 2, 1, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [-8, 4, 0, -6, 4, 1, 0, 1], [-24, 4, 16, -6, 8, 1, -4, 1], [40, -12, 0, 18, 0, -3, 4, -3], [56, -20, -16, 14, -8, -5, 4, -5], [12, -8, -2, 4, -3, 1, 1, -1], [20, -4, -2, 6, -3, 0, 1, -2], [104, -52, 8, 30, -20, -1, 8, -9]];
    Rf_basisdens := [1, 16, 16, 16, 16, 4, 4, 16];
    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_1344_bl();"
function MakeCharacter_1344_bl()
    N := 1344;
    order := 6;
    char_gens := [127, 1093, 449, 577];
    v := [3, 6, 6, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1344_bl_Hecke();"
function MakeCharacter_1344_bl_Hecke(Kf)
    N := 1344;
    order := 6;
    char_gens := [127, 1093, 449, 577];
    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], [0, 1, 0, 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+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], [-1, 1, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, -1, 0, 0], [0, -1, 0, 1, 0, 0, 0, -1], [0, 0, -1, 1, -1, 0, 0, 0], [1, -2, 0, 1, 0, 1, -1, -1], [0, 0, -1, 0, 0, 0, -1, 1], [0, -1, 0, -1, 0, 0, 1, 0], [0, 0, 1, -1, -1, 1, 0, 1], [2, 0, 0, -1, -1, 0, -1, 2], [-3, 3, -2, -1, -1, -4, 2, 2], [0, -1, 1, 0, 1, -1, -1, 1], [2, -4, 0, 2, 0, 0, -2, -2], [1, -2, 0, 0, 1, -2, 0, 1], [0, -2, -1, 1, -1, 1, 0, -1], [2, -2, 1, 2, 2, 2, -2, -2], [-2, 2, 0, -1, -1, 0, -1, -1], [0, 0, 2, -2, 0, 2, -2, 0], [3, 3, -2, 1, -1, 0, 1, -1], [2, -4, 0, 1, -1, -1, -1, -2], [-1, -1, 1, -1, 1, 0, -2, 2], [2, -1, 4, 2, 3, 4, -1, -3], [0, 0, -4, 0, -1, -2, 0, 1], [0, 0, 0, -2, -2, 0, 0, 2], [2, -4, 0, -1, 0, 1, 1, 1], [-2, -2, 2, -2, 2, 0, 0, 0], [0, -1, -2, -1, 2, 2, -1, 2], [4, -2, 2, -1, -2, 2, 1, 2], [-1, 1, -1, -1, -1, -2, 0, 0], [6, 0, 2, -1, 3, 1, -1, -2], [3, -6, 0, -1, 0, 0, 1, 1], [0, -8, -1, 4, -1, 1, -3, -1], [4, -4, 2, 0, 0, 4, -2, -2], [3, 0, 4, 0, 3, 2, 0, -3], [0, 8, -2, 0, 0, 2, 0, 0], [8, 8, -2, 0, 0, 0, -1, 1], [0, 0, -4, 4, -4, 0, 0, 0], [4, -2, 0, -2, -2, 0, 0, 2], [-2, 0, 2, 1, -3, 1, 1, 2], [0, 0, -1, 1, -1, -1, 2, 1], [6, 6, 5, -2, 2, 0, -1, 1], [-1, 2, 0, -1, 0, 5, 1, 1], [12, -6, -3, -3, -3, -3, 0, 3], [-9, 9, 0, 2, 2, 0, 2, 2], [0, 0, 6, 1, 3, 3, 1, -4], [-8, 8, 2, 0, 0, 4, -2, -2], [2, -4, 0, 0, 0, 6, 0, 0], [-2, 0, -4, 1, -1, -2, 1, 0], [-8, 8, -3, -3, -3, -6, 2, 2], [-14, 7, 3, 4, 5, 3, -1, -5], [0, 10, -1, -1, 1, 1, 0, 1], [10, -20, 0, 0, 0, 2, 0, 0], [6, 6, 7, -3, 3, 0, 2, -2], [-2, 0, 4, 0, -5, 2, 0, 5], [4, -2, 5, 3, 3, 5, 0, -3], [4, 4, -2, 4, -4, 0, 4, -4], [-4, -4, -2, 2, -2, 0, -1, 1], [0, 2, 2, 3, 1, -2, -4, 1], [-3, 3, 1, 1, 1, 2, 0, 0], [-12, 0, -2, 3, -7, -1, 3, 4], [-13, 13, 0, 1, 1, 0, 3, 3], [-6, 12, 0, -1, 3, -4, 1, 4], [3, 0, -4, 0, 1, -2, 0, -1], [-14, 14, 2, -4, -4, 4, 2, 2], [-14, 7, 0, 0, -4, 0, 4, 4], [0, 6, 0, 1, -3, 0, 2, -3], [14, -7, -2, 1, 4, -2, -3, -4], [1, 0, -8, 0, 2, -4, 0, -2], [0, 0, -3, 4, -4, 0, 1, -1], [0, 0, 0, -2, -2, 2, 2, 0], [-2, -2, -5, 0, 0, 0, 3, -3], [-8, 4, 0, 0, 0, 0, 0, 0], [-15, 15, 0, -1, -1, 0, 2, 2], [0, 5, 1, -6, 3, -1, 3, 3], [9, -18, 0, 2, 1, -2, -2, -1], [0, 8, 5, -1, 3, -5, -2, 3], [6, -6, -3, -2, -2, -6, 5, 5], [-2, 1, -5, -4, -7, -5, 3, 7], [0, 8, -3, 1, -1, 3, 0, -1], [-5, -5, -4, 4, -4, 0, 4, -4], [14, 0, 8, -4, 6, 4, -4, -2], [-1, 0, -10, 1, -4, -5, 1, 3], [6, 6, 1, -2, 2, 0, -5, 5], [-5, 10, 0, 5, 4, 7, -5, -1], [0, -14, 2, -5, 1, -2, 4, 1], [12, -6, -4, -1, -4, -4, 3, 4], [-2, 0, 6, 1, -1, 3, 1, 0], [0, -5, 4, 0, 4, -4, -4, 4], [8, -16, 0, 8, 0, 8, -8, -8], [9, -18, 0, 0, -1, 4, 0, -1], [0, -6, 3, -3, 5, -3, -2, 5], [-4, 4, 1, 8, 8, 2, -5, -5], [9, 9, 6, -5, 5, 0, -4, 4], [2, -4, 0, -9, -2, -9, 9, 7], [-18, 9, -10, 3, -6, -10, 9, 6], [-2, 0, 4, 4, -2, 2, 4, -2], [24, -12, 7, -2, 2, 7, -4, -2], [-4, -4, 6, -6, 6, 0, -8, 8], [0, 3, -6, 11, 0, 6, -11, 0], [0, -7, -3, 10, -3, 3, -7, -3], [2, -4, 0, -2, -2, -2, 2, 0], [-4, 4, 6, 4, 4, 12, -5, -5], [-8, 8, 6, 5, 5, 12, -5, -5], [0, 14, 4, 0, -4, -4, 4, -4], [3, 3, -4, -1, 1, 0, 3, -3], [7, 7, 0, 4, -4, 0, -4, 4], [-10, 0, 2, 7, 2, 1, 7, -9], [28, -14, 2, 0, 4, 2, -4, -4], [12, 12, 0, 0, 0, 0, 0, 0], [-1, 2, 0, 0, -4, 0, 0, -4], [0, 9, -4, 2, -9, 4, 7, -9], [14, -14, 2, 0, 0, 4, 0, 0], [-14, 0, -4, -4, 4, -2, -4, 0], [-1, 1, 4, -3, -3, 8, -3, -3], [4, -8, 0, 9, 1, 4, -9, -8], [4, -4, 5, -2, -2, 10, -7, -7], [1, 0, 4, 2, -5, 2, 2, 3], [-14, 14, -6, -8, -8, -12, 12, 12], [0, 16, -2, -1, 5, 2, -4, 5], [0, 0, 0, 3, 7, -1, -3, 4], [-5, -5, 9, 3, -3, 0, -4, 4], [-7, 0, -4, 2, -2, -2, 2, 0], [28, -14, 2, -5, 3, 2, -8, -3], [-2, -2, 8, 1, -1, 0, 9, -9], [0, -21, -2, -3, 0, 2, 3, 0], [-2, 0, -10, 0, -6, -5, 0, 6], [0, -18, -8, 8, -4, 8, -4, -4], [0, 22, 2, -10, 10, -2, 0, 10], [1, 0, -4, 1, -6, -2, 1, 5], [6, -3, -1, 6, 7, -1, -1, -7], [3, 3, -4, 1, -1, 0, -3, 3], [-2, 4, 0, -4, -2, -4, 4, 2], [-2, 1, 4, -4, -9, 4, 5, 9], [-6, 0, 8, -2, 6, 4, -2, -4], [24, -12, 3, -3, -7, 3, 4, 7], [-3, 6, 0, -4, -6, -4, 4, -2], [-2, -2, -14, 6, -6, 0, 0, 0], [-16, 16, 4, -6, -6, 8, 4, 4], [14, -28, 0, -2, 0, -1, 2, 2], [22, -22, -3, 6, 6, -6, -1, -1], [8, 0, -4, -10, -2, -2, -10, 12], [0, 24, 7, 2, -2, -7, 0, -2], [10, 10, 0, -2, 2, 0, 4, -4], [4, -8, 0, 10, 3, 10, -10, -7], [3, 3, -7, -5, 5, 0, -4, 4], [12, 0, -4, 0, 0, -2, 0, 0], [3, -6, 0, 5, 4, -5, -5, -1], [-8, -8, -4, 10, -10, 0, 4, -4], [0, 28, -4, -2, 0, 4, 2, 0], [36, -18, 12, 4, 4, 12, 0, -4], [0, -10, 0, -6, -4, 0, 10, -4], [16, -32, 0, 3, -3, 3, -3, -6], [1, -2, 0, 2, -3, -6, -2, -5], [0, -10, -10, 8, -4, 10, -4, -4], [-11, -11, -6, 5, -5, 0, -5, 5], [4, -8, 0, 5, 9, -9, -5, 4], [2, -1, -4, -9, -4, -4, -5, 4], [4, -2, -19, -5, -15, -19, 10, 15], [-5, 10, 0, 2, 4, -6, -2, 2], [-10, -10, 1, -2, 2, 0, 4, -4], [-4, 2, 12, 0, 8, 12, -8, -8], [-24, 0, 2, 1, 9, 1, 1, -10], [-5, 10, 0, -7, 2, -4, 7, 9], [0, 18, -4, -5, -2, 4, 7, -2], [-6, 6, -8, 0, 0, -16, 0, 0], [-12, 12, 3, 6, 6, 6, -3, -3], [-7, -7, -10, 7, -7, 0, -4, 4], [13, 13, 3, 1, -1, 0, -6, 6], [-9, 0, -4, 2, -8, -2, 2, 6], [28, -14, 12, 7, 3, 12, 4, -3], [-10, -10, 1, -7, 7, 0, -10, 10], [-8, 16, 0, -6, -2, -16, 6, 4], [-40, 20, 2, 4, 0, 2, 4, 0], [-15, 15, 4, 14, 14, 8, -6, -6], [-5, 5, -2, -7, -7, -4, 8, 8], [-8, 16, 0, 0, 0, 8, 0, 0], [-1, 2, 0, 0, -1, 2, 0, -1], [8, -8, 1, -8, -8, 2, -2, -2], [16, 0, 4, -10, 10, 2, -10, 0], [-14, 7, 5, 12, 5, 5, 7, -5], [34, -17, -4, 4, 1, -4, 3, -1], [14, 0, 22, 1, 8, 11, 1, -9], [15, 0, 6, 3, 4, 3, 3, -7], [24, -12, 5, 7, 5, 5, 2, -5], [-14, -14, -16, 4, -4, 0, 0, 0], [-18, -18, 10, -4, 4, 0, -1, 1], [-2, 2, -4, -2, -2, -8, 8, 8], [3, -3, -10, -5, -5, -20, 11, 11], [0, 5, 4, -6, 8, -4, -2, 8], [18, -18, 2, -6, -6, 4, 2, 2], [-22, 11, 0, 2, 0, 0, 2, 0], [-8, 16, 0, -10, -5, 2, 10, 5], [-20, 10, 4, -8, -4, 4, -4, 4], [0, 0, -7, 3, 1, -7, 2, -1], [-2, -2, -6, -3, 3, 0, 7, -7], [4, 4, 6, -12, 12, 0, 4, -4], [5, -5, 7, -5, -5, 14, 4, 4], [0, 6, -4, -6, 6, 4, 0, 6], [22, -44, 0, -2, -2, 6, 2, 0], [0, -20, 9, -15, 1, -9, 14, 1], [8, -8, -4, 0, 0, -8, -1, -1], [9, 0, 12, 2, 7, 6, 2, -9], [0, 0, -4, -8, -4, -4, -4, 4], [4, 4, 10, 4, -4, 0, 6, -6], [-10, 0, -16, 4, -10, -8, 4, 6], [-8, -8, -4, 6, -6, 0, 7, -7], [0, 16, 2, 2, 2, -2, -4, 2], [-32, 16, 1, 2, -3, 1, 5, 3], [-4, 0, -4, -2, 4, -2, -2, -2], [16, -16, 0, 4, 4, 0, 2, 2], [0, -9, -5, 2, 1, 5, -3, 1], [10, -20, 0, 7, -5, 2, -7, -12], [-12, 24, 0, 5, 1, 4, -5, -4], [0, 22, 4, 7, -2, -4, -5, -2], [16, -16, 7, 8, 8, 14, -15, -15], [-10, 5, -21, 0, -7, -21, 7, 7], [-24, -24, 0, 4, -4, 0, 1, -1], [-20, -20, -14, 14, -14, 0, 2, -2], [-16, 8, -13, 1, -5, -13, 6, 5], [22, 0, -10, 1, 9, -5, 1, -10], [0, -37, 1, -2, -1, -1, 3, -1], [-3, 0, -16, 8, -7, -8, 8, -1], [0, 2, 1, -5, -7, -1, 12, -7], [8, -4, 0, -3, 5, 0, -8, -5], [0, 0, -2, -3, -8, -1, -3, 11], [14, 0, -8, 0, -10, -4, 0, 10], [16, -8, -12, 12, 8, -12, 4, -8], [-10, -10, -8, -6, 6, 0, -6, 6], [0, -1, -6, 8, -5, 6, -3, -5], [-20, 10, 3, 9, 3, 3, 6, -3], [-21, 21, -3, 1, 1, -6, 10, 10], [32, -32, 8, 0, 0, 16, -2, -2], [-23, 46, 0, -7, -4, -14, 7, 3], [6, -6, -4, 10, 10, -8, -12, -12], [-7, 0, 4, -4, 5, 2, -4, -1], [18, -18, 9, 8, 8, 18, -5, -5], [-10, 5, -8, 0, -4, -8, 4, 4], [0, 40, 2, -3, 3, -2, 0, 3], [12, -24, 0, 4, 2, -8, -4, -2], [6, 0, -18, 9, -13, -9, 9, 4], [-6, -6, 5, 6, -6, 0, -9, 9], [0, 9, 12, -11, 2, -12, 9, 2], [3, -3, 0, -3, -3, 0, 10, 10], [0, -10, 2, 0, 10, -2, -10, 10], [8, -16, 0, -2, 0, -4, 2, 2], [0, -4, 13, -1, 7, -13, -6, 7], [-23, 0, 16, -13, 4, 8, -13, 9], [2, -2, 3, 2, 2, 6, -7, -7], [-14, -14, 14, -2, 2, 0, 4, -4], [12, -24, 0, 2, -8, 10, -2, -10], [-11, 0, 14, -1, 2, 7, -1, -1], [-16, 8, 7, -7, 1, 7, -8, -1], [-10, -10, 14, -6, 6, 0, -4, 4], [17, -34, 0, 7, 8, 1, -7, 1], [-14, -14, 10, -8, 8, 0, -13, 13], [-12, 6, -19, -2, -3, -19, 1, 3], [21, -21, 13, 9, 9, 26, -18, -18], [9, -9, 10, 9, 9, 20, -9, -9], [8, -16, 0, 8, 8, -11, -8, 0], [-20, 10, 13, -2, -5, 13, 3, 5], [-3, -3, 10, -9, 9, 0, -3, 3], [-14, 28, 0, -5, -2, -21, 5, 3], [-3, -3, 3, 9, -9, 0, 10, -10], [-15, 30, 0, -1, -2, 1, 1, -1], [-6, -6, -14, 0, 0, 0, -12, 12], [0, 5, 0, -1, -4, 0, 5, -4], [24, 0, 0, -3, 11, 0, -3, -8], [6, -12, 0, -17, -7, -19, 17, 10], [-22, 44, 0, -8, 10, -6, 8, 18], [-16, 16, -4, 12, 12, -8, -7, -7], [6, -3, 7, -12, 1, 7, -13, -1], [-2, -2, -14, 10, -10, 0, -6, 6], [17, 17, 8, -10, 10, 0, -6, 6], [22, -11, 18, 6, 5, 18, 1, -5], [2, -4, 0, 3, 4, -11, -3, 1], [0, -4, -6, 10, 2, 6, -12, 2], [32, -16, 2, -9, 6, 2, -15, -6], [23, -23, -1, 1, 1, -2, -8, -8], [0, -11, 3, 12, -7, -3, -5, -7], [0, -52, -5, 11, -7, 5, -4, -7], [28, -28, -10, -10, -10, -20, 14, 14], [11, 11, 0, -11, 11, 0, 4, -4], [-26, 0, 4, -8, 8, 2, -8, 0], [13, -26, 0, -11, -4, -9, 11, 7], [0, 44, 10, -18, 10, -10, 8, 10], [-36, 18, -6, -6, -4, -6, -2, 4], [39, -39, 4, 8, 8, 8, -8, -8], [-12, 0, -10, -8, 0, -5, -8, 8], [25, -25, 2, -13, -13, 4, 10, 10], [10, -20, 0, 7, -1, 11, -7, -8], [20, -20, 0, -4, -4, 0, 10, 10], [34, -34, 3, -7, -7, 6, 4, 4], [0, 10, -7, 11, -1, 7, -10, -1], [-32, 0, -4, 10, -2, -2, 10, -8], [3, 0, -2, -7, 4, -1, -7, 3], [-16, -16, -12, 6, -6, 0, -3, 3], [0, 25, -2, -7, 14, 2, -7, 14], [-12, 24, 0, -5, -3, 7, 5, 2], [0, 22, -1, 13, 5, 1, -18, 5], [17, 0, 20, 2, 5, 10, 2, -7], [-24, 12, -22, -2, -2, -22, 0, 2], [0, -28, 2, 17, -9, -2, -8, -9], [-28, -28, -4, -4, 4, 0, -1, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1344_bl_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1344_2_bl_i();".
// 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_1344_2_bl_i(:prec:=8)
    chi := MakeCharacter_1344_bl();
    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_1344_2_bl_i();".
// 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_1344_2_bl_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1344_bl();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![16, 96, 236, 264, 125, 0, -11, 0, 1]>,<11,R![400, 240, -212, -156, 125, 78, -1, -6, 1]>],Snew);
    return Vf;
end function;