
// Make newform 6468.2.a.bf in Magma, downloaded from the LMFDB on 09 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6468_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_6468_2_a_bf();".
// 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_6468_2_a_bf();".
// 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 := [-28, -16, 58, 20, -15, -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], [16, -28, -26, -7, 2, 1], [-14, -16, 43, 23, -4, -2], [-34, -26, 26, 13, -2, -1], [40, -52, -47, 23, 5, -2]];
    Rf_basisdens := [1, 1, 18, 9, 6, 9];
    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_6468_a();"
function MakeCharacter_6468_a()
    N := 6468;
    order := 1;
    char_gens := [3235, 4313, 3433, 5293];
    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_6468_a_Hecke(Kf)
    return MakeCharacter_6468_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, 0, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 1], [1, 0, 0, 0, 1, 0], [1, 0, 0, -1, 0, -1], [2, -1, -1, -1, 0, -1], [0, 0, 1, 1, 1, 1], [2, -1, -1, 1, 0, -1], [1, -2, 0, 1, 0, 1], [1, 1, -2, 0, -1, -1], [1, 2, 3, 0, -1, 0], [0, 2, 1, -1, -1, 1], [0, -1, 1, 1, -2, 1], [3, 1, 2, 1, -2, 0], [-1, 1, 3, 1, 1, 1], [0, -1, -3, 1, -1, -2], [2, 0, 3, -1, 0, 0], [2, 3, 1, 2, -1, 2], [-2, 0, 0, 2, -2, 0], [0, 3, -1, -1, 0, 1], [0, 2, 2, 1, 0, 0], [4, -1, -2, -1, 2, -1], [3, 0, 2, 2, -1, 0], [-1, -3, 2, 1, -1, -1], [3, 1, -4, -1, 0, -2], [0, 1, -3, -1, 2, 1], [2, 0, 5, 3, -2, 0], [1, 0, 7, 0, 1, 0], [-4, 0, -3, -3, 2, 0], [-2, 1, 1, 1, 2, -1], [5, -1, 1, 0, -1, -3], [2, -2, -5, -1, 1, -1], [-2, 1, -3, -3, 0, 1], [9, -1, -7, -3, 1, -3], [-6, 3, -3, -1, 3, 0], [-2, 1, 1, 5, 0, 1], [5, 3, 1, -1, 1, -3], [4, 4, -4, 0, 0, 0], [6, -3, 0, 1, -4, 1], [4, 1, 0, 0, -2, 3], [-2, -3, -3, -1, -2, -3], [-1, 1, 7, 0, 3, -1], [3, 2, -7, -2, 1, 0], [-8, 6, 6, -2, 2, 0], [4, 2, 10, 0, 2, 0], [6, 0, 1, -1, 2, -2], [13, -4, 2, 0, -3, 0], [-1, 3, -5, -6, 3, -1], [1, 5, 8, 1, -2, 2], [3, 0, -2, 2, -4, 3], [5, 3, 4, 1, 0, 2], [7, -1, 2, 6, 0, 2], [12, 0, -7, 1, -1, 3], [11, -1, 1, -3, 1, -1], [-4, 5, 1, -3, 1, 2], [-15, 3, -2, 3, -1, 1], [5, -7, 4, 1, 0, 0], [10, -1, -1, 3, -3, 2], [11, -1, -3, 3, -1, 1], [4, -2, 8, 2, -2, 0], [1, 1, -3, 0, 3, 3], [7, 2, 6, 0, 5, 0], [-4, 3, -6, 4, -4, 1], [-14, 0, 8, 2, 2, 0], [-10, 3, -1, -5, -2, -3], [3, 3, -9, -2, 3, -1], [6, -3, 1, 0, 3, 0], [10, -4, 3, -2, -1, -5], [1, 7, 1, 2, -1, 5], [2, 4, -4, -4, 0, 2], [-9, 2, -3, -2, -1, 0], [-3, -2, 6, 3, 0, 1], [7, 3, -9, -4, -1, -1], [2, -4, -3, -3, 0, -4], [1, 2, 8, 4, -1, 0], [-2, 1, -6, 0, -2, -3], [8, 0, -2, 1, 2, 0], [8, 0, -1, -5, 1, -3], [1, 2, 0, -5, 4, -1], [-1, -5, -10, 1, 2, -2], [5, 1, -9, -5, 1, -1], [-16, 1, -7, -3, -3, -2], [4, 2, 7, 1, -2, -2], [-4, 4, -3, -3, 2, -2], [-10, -5, -5, 1, -2, -1], [-3, -1, 8, 0, 3, -3], [-3, -4, 2, -5, 0, 3], [3, 3, -4, 3, -6, 2], [-6, 5, 2, 0, -4, -1], [2, -4, -2, 0, -4, -6], [-7, 7, 4, 1, 2, 0], [-8, -7, -3, -3, 1, -2], [-8, -3, -10, 2, 0, -3], [4, 0, -6, -1, 2, 4], [3, -7, 8, 0, 2, 2], [-5, -2, 2, 3, 0, -1], [-17, 9, 6, 1, 1, 3], [-15, -1, 0, 3, 3, -5], [-12, -2, 1, -5, 3, 3], [-8, -3, 10, 4, 0, 1], [9, -3, -11, 2, 3, 3], [-1, 1, -20, 1, 1, 1], [11, 0, -4, 4, 1, -2], [8, -2, -3, -3, -3, 3], [-5, -1, -12, -6, 1, 1], [10, -6, 4, 2, -6, 2], [-5, 8, -2, 4, 0, 3], [-16, -1, -3, -7, 1, 0], [-7, -8, 7, -4, 5, -4], [-2, 4, 8, 6, 0, 6], [-9, 4, -7, 8, -1, 4], [-2, -8, 8, 2, -2, -4], [24, -1, 3, -3, 0, 1], [3, -4, 1, 0, 5, 2], [6, -2, 5, 1, -1, -1], [0, -2, 6, 2, 0, 2], [11, 5, 0, -5, -2, 4], [-17, 8, -6, 2, 3, 4], [-5, 8, 1, 4, 1, 2], [1, -1, 2, -2, 1, -1], [-2, 1, -12, 2, -4, 3], [6, 4, -8, -6, 4, -2], [-4, -4, -10, 2, -2, 2], [-7, -2, -8, 1, 4, 5], [-5, -3, -2, -1, 8, -2], [-6, 3, 7, -3, -2, -3], [11, -5, -1, -9, 3, -7], [-4, 1, -13, -3, 0, 3], [16, -4, 1, -7, -1, -1], [-16, -1, 5, -3, -1, 2], [6, 5, -11, -5, 5, 4], [-5, 1, -9, -1, -5, -1], [-4, -1, 9, -2, 1, -2], [10, -6, 5, 6, -9, 1], [6, -11, -3, 1, -1, -2], [-2, 2, -3, 0, 1, 7], [3, -5, 4, 5, 0, 2], [4, 7, 1, -5, -1, 2], [4, -2, 7, 1, -3, -3], [-1, -6, 14, 1, 4, -1], [-24, 4, -3, 1, -1, -1], [-9, 1, -15, 1, -5, -5], [-2, -4, 9, 1, -1, 5], [-3, 5, 7, 1, -1, -3], [-7, -3, -3, -1, 3, -7], [-10, -4, 2, 4, 2, 4], [6, -7, 2, 8, -2, 5], [-16, 0, -12, -4, 0, -4], [0, -12, -11, 0, -3, -9], [3, 8, -4, -1, -8, 1], [-6, -1, 18, -4, 0, 5], [-5, -5, -1, -10, 1, -3], [14, 4, -13, -5, 2, -6], [-6, -1, -1, -9, 6, 1], [20, -4, -19, 0, 3, -3], [1, 1, -13, 3, -3, -5], [-19, 3, -5, -1, 1, 5], [6, 4, 10, 0, 4, 2], [1, -1, -10, 1, -4, -8], [-3, -5, 4, 7, 1, 1], [-1, -3, 8, 3, -2, -4], [0, -1, 31, -3, 2, -1], [-7, 7, -5, -10, 7, 1], [-5, 2, -10, 5, 0, -5], [-6, 12, -4, -1, -2, -2], [1, 7, -12, 5, -3, 7], [-4, -2, -5, 4, -5, -1], [-2, 6, 1, -1, -7, -3], [3, -3, 11, -1, 1, 1], [18, -2, -14, -8, 4, -8], [-2, 9, -7, -3, -4, 1], [2, -2, -4, -4, -4, 8], [0, -4, -2, 3, 0, 6], [-6, -5, 5, 9, 2, 1], [-24, 8, 7, -7, 4, 2], [-26, 5, -3, -1, 7, 2], [-9, 6, 16, 0, 4, 1], [-19, 6, -8, 1, 4, -3], [-18, -3, 5, 3, 0, -7], [6, 6, -14, -4, 0, -4], [31, -9, 12, -4, -3, -3], [10, 2, -2, -4, 10, -2], [9, -1, -2, 6, 4, 0], [-8, 4, 8, -2, -8, 6], [-33, 10, -13, 2, -1, -2], [-7, 16, -3, -2, 5, 0], [-16, -9, -1, 1, -4, -3], [31, 3, 13, -1, 1, 3], [20, 2, -15, 1, -1, 5], [7, 4, 10, 1, -10, 3], [-9, 9, 4, 8, -8, 8], [-5, -1, -18, 1, 2, -6], [6, 4, 19, 0, -7, 1], [-12, -3, -5, 5, 2, -3], [0, -4, -10, -2, 2, -2], [5, -10, -8, 4, 3, 0], [-32, -3, -1, -3, 2, 5], [2, 2, 28, -4, 4, -2], [-15, -8, 14, 7, 2, 5], [33, -6, 18, -2, -3, -2], [-20, -5, -5, 4, -3, 2], [9, -3, -5, 0, 1, -7], [-8, 0, 8, -5, 4, 6], [-1, -5, -14, -3, 1, -5], [-3, 7, 0, -3, -2, 0], [-15, 15, -1, -2, 9, 5], [-14, -13, -5, -1, 2, -3], [-1, -11, -2, -1, 3, -1], [-4, -6, -1, -4, 9, -3], [12, -1, -3, 5, 5, 4], [42, -4, -8, 2, 0, -2], [-32, -10, -11, -5, 4, 4], [-33, 9, 3, -7, 5, 1], [-29, 5, 20, 9, 1, 9], [7, -10, -10, -2, -9, -6], [16, 3, -1, 1, -4, 5], [-4, 1, -16, 8, -2, 3], [-21, 6, 15, -6, -1, 6], [-23, 0, 6, 11, -6, 5], [14, -9, -3, -1, 0, 1], [-18, 3, -7, -1, -1, -4], [-2, 4, -18, -6, -6, -4], [11, 5, 12, 5, -1, -5], [10, -14, -17, 0, -1, -3], [16, -2, 26, -2, -2, 0], [-20, -3, -15, 1, 5, 6], [11, -5, 32, -1, 0, 0], [3, 5, 9, 8, 3, 5], [-23, 20, 2, 1, 4, 3], [-31, -1, -8, -5, 1, 1], [9, 5, -6, 5, -6, -4], [14, -7, 5, -3, -6, -5], [-24, 4, 9, 3, -8, 6], [2, 1, 6, -5, 6, 3], [3, -7, -2, 3, 0, -4], [20, -1, -21, 1, 0, -7], [-24, 4, -5, 9, 1, 5], [22, -1, 13, 5, 4, 3], [-30, 6, -6, -2, -2, -2], [-7, -4, -5, -2, 1, 0], [17, -7, -21, -2, -3, -9], [5, 7, 3, -9, 5, 5], [6, 10, 14, 0, 2, 6], [-6, 6, 2, 6, 6, -6], [7, 9, 0, 3, -6, -4], [-7, 3, -18, 5, 3, -5], [-10, 4, 15, 9, -3, 3], [17, 7, -14, -9, -7, -1], [-15, 0, 5, -3, -3, 2], [-18, 2, -4, -6, 0, 0], [-23, 3, 15, 11, -5, 11], [-11, -5, -2, -4, 5, -1], [-15, 5, 15, -8, -1, 7], [-2, 3, -13, -3, -3, -10], [-20, 3, 11, -7, 7, -6], [6, 0, -11, -8, 3, 3], [18, -7, 5, 4, -9, 4], [6, -10, -14, -8, -4, -8], [24, 0, -12, -4, 2, -6], [37, -7, -5, 11, 1, 3], [-4, 1, 15, 4, -1, 6], [5, -9, -4, 0, 12, 0], [30, -1, -3, -3, -8, 3], [-38, 10, -12, 4, 4, -2], [-3, 1, 24, -8, -2, 0], [-28, 4, -10, -2, 8, 8], [7, -9, -9, 2, 5, 1], [-3, -9, -5, 7, -3, 9], [-17, 6, 32, 3, -4, 7], [1, 10, -2, -4, 9, 6], [5, -8, 15, -4, 5, 0], [-4, -19, -11, -2, 11, -6], [-39, 4, -7, -4, 1, -4], [-10, -2, 2, -12, 6, 2], [6, -2, 20, 4, 0, -2], [-11, 0, 7, 2, 1, -2], [-24, -5, 17, 5, -2, 7], [-4, 13, 3, 9, -2, 5], [10, -14, 6, 8, -2, 2], [4, -9, -6, -2, 6, -3], [-1, 2, -8, -14, 12, 1], [42, -3, -21, -9, 9, -12], [-6, 5, -2, -12, -6, -7], [19, -2, 4, -7, 10, -1], [1, -5, -5, 2, 13, 5], [19, 0, 9, -2, -5, -4], [11, -6, -11, 5, 1, 2], [2, -7, 21, -1, 2, 5], [-7, -7, 17, 6, -7, 1], [-2, 0, 13, 7, -13, 1], [6, -1, -16, 0, 2, -9], [14, -10, -14, 10, -2, -4], [11, -2, 1, -13, 7, -12], [-24, -14, -28, -6, 2, 0], [-1, 3, 17, -9, -5, -7], [-18, 5, -4, 0, -6, -3], [-3, 0, -4, -9, 2, 1], [7, -12, 24, 6, -2, 1], [-20, -4, -1, 3, -5, 3], [-9, -2, 23, -6, 1, 0], [-20, 11, 8, 8, 4, 7], [-23, 2, 12, 9, -8, 13], [-11, 10, -4, 9, -4, 13], [12, -5, 6, 2, -12, -1], [14, -5, 0, -13, -4, -13], [-14, -3, -16, 2, -2, -3], [18, -13, 7, 9, -2, 3], [-26, 10, -3, -3, -6, -4], [0, 8, -8, -8, -4, -12], [-4, -22, 9, 1, 3, -3], [1, -9, 9, 14, -15, -1], [7, 15, -3, -4, -3, -11], [-16, -8, 2, 5, -6, 0], [22, 8, 20, -2, 12, 2], [6, 2, -34, 4, -2, 2], [-13, 12, -14, -4, -9, -4], [16, -10, -14, -6, 2, -12], [19, -8, 24, 1, 4, 1], [28, -5, -25, 3, 3, 2], [19, 7, -13, -1, -3, 3], [-1, -19, -9, 2, 1, -9], [31, -15, 18, 3, 0, 0], [1, -9, 11, 9, -7, -3], [-11, -5, 13, -2, -3, -9], [-1, 14, -5, 2, 1, 8], [-1, -3, -34, -3, 0, 2], [-37, -13, 6, 1, 2, -10], [-9, 17, 6, -3, 9, 7], [-34, 10, 12, 6, -6, 10], [7, -10, 22, 9, -10, 1], [0, -3, -9, 3, 4, 15], [-9, -2, -11, -4, 9, 4], [-1, 1, -18, 5, 2, 16], [-4, 3, -1, -1, -3, 8], [-5, -2, 3, 11, 3, 8], [9, 0, -8, 2, 7, 10], [-12, 6, -24, -4, 14, 4], [14, -2, -6, -2, 8, -6], [-4, -5, -1, 5, -8, 9], [-17, -14, 5, 1, -3, -6], [-18, 11, 15, -11, -5, 0], [-8, 23, 13, 3, 4, 9], [17, -13, 12, 13, -10, -8], [48, -12, -3, 0, 0, -4], [1, 15, 5, -2, 7, 13], [-14, 12, 2, 0, 6, 16], [-2, 18, 28, 1, 6, 8], [-18, -8, -26, -10, -6, -8], [-11, 2, -10, 9, -12, -1], [-56, 16, 11, -1, 2, 6], [-22, 21, -1, 5, 3, 6], [27, -12, 7, 4, -13, -6], [-3, 12, -1, -5, 11, -8], [-19, -3, 16, 3, -10, 10], [-16, -10, 4, -18, 8, -6], [5, 5, -12, -5, -8, 6], [30, 5, 9, 1, 8, 1], [-5, 3, 24, 11, -3, 1], [-57, 3, -13, 1, 7, 5], [-26, -10, 17, 1, 8, -2], [12, 10, 11, -5, 1, -5], [18, 6, 16, -2, -2, -6], [17, 14, -24, -7, 0, -7], [21, -8, 20, -4, -7, -4], [-7, 19, -2, 9, -3, 9], [23, -13, 18, -11, 11, -1], [-16, 7, -40, 4, 2, 5], [58, 5, -13, 3, -4, 5], [-21, -1, -8, 3, 0, 4], [43, -7, 0, 1, -4, 4], [-4, 8, 20, 0, -12, 0], [8, 11, -7, -1, -7, -2], [26, -12, -7, -13, 5, -9], [-13, 15, 28, 7, 2, 10], [2, 5, -1, -12, 11, 4], [5, 3, -4, 8, -7, -5], [-7, -11, -35, -3, -7, -1], [-10, 14, -15, -2, -5, 5], [-20, 15, 6, -12, 0, -3], [-9, 5, 8, -11, -1, -1], [26, -8, -7, 1, -1, -13], [10, -2, 0, 0, 10, -2], [5, -5, -14, -7, 14, -4], [-14, 7, 13, -1, -7, 6], [15, 11, 11, 5, -3, 11], [14, -9, 19, -5, 0, 3], [16, 6, -19, -5, -1, -11], [40, 3, -7, -3, 3, -14], [22, 0, -12, 3, -2, -6], [-6, 7, -18, -8, 10, -1], [17, -2, -13, -8, -9, -8], [-12, 6, -23, 7, -4, -4], [31, 0, -15, 4, -1, 0], [8, 5, -15, -15, 11, -4], [35, 3, 20, -8, 10, 4], [-32, -18, -10, -6, 8, -2], [-16, -2, -53, -5, 3, 1], [-11, -20, 16, 10, 0, -3], [4, -1, -20, -2, 6, -9], [-20, 13, -15, -1, -10, -7], [45, 5, -1, 9, -1, 9], [11, 8, -21, -5, -1, -2], [-17, 8, -26, 9, -2, 3], [-6, -3, 19, 3, 1, -10], [40, 4, 10, 4, -2, 0], [32, 0, 4, 0, -2, -2], [35, 1, 23, -6, 1, -5], [-27, 9, 3, -15, -1, 1], [2, 0, -2, 8, 10, -4], [-34, 0, -5, -1, 8, 4], [24, -3, 19, 13, 8, 3], [-29, -3, 0, 9, 11, -9], [-42, -6, 15, -2, 1, -5], [18, 16, -17, 9, -9, 9], [11, -4, 19, 4, -9, 8], [10, -12, -10, -1, 10, -10], [-14, -11, 28, 2, 0, -1], [2, 1, -3, -7, 2, -13], [4, -21, 13, 6, 11, 8], [0, -3, 20, 8, 2, -5], [38, -23, 11, 5, 6, 3], [7, -11, -27, 7, -3, 1], [23, 3, 0, 4, -10, -4], [-2, 13, -12, -10, -4, -1], [-2, -16, 25, 7, -3, 7], [-15, 21, 2, -3, 5, 3], [14, 2, -39, 5, 0, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6468_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6468_2_a_bf();".
// 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_6468_2_a_bf(:prec:=6)
    chi := MakeCharacter_6468_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_6468_2_a_bf();".
// 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_6468_2_a_bf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6468_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![18, -96, 23, 40, -10, -4, 1]>,<13,R![14, -136, 13, 144, -12, -8, 1]>,<17,R![56, -64, -240, 160, -8, -8, 1]>,<23,R![6272, -3584, -132, 344, -30, -8, 1]>],Snew);
    return Vf;
end function;