
// Make newform 6468.2.a.be in Magma, downloaded from the LMFDB on 10 June 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_be();".
// 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_be();".
// 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 := [7, 56, 44, -18, -18, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [305, 525, -101, -149, 1, 7], [277, -138, -139, 62, 11, -4], [67, -21, -55, 11, 5, -1], [605, 2595, 121, -1079, -83, 67], [-605, -2271, -121, 1079, 83, -67]];
    Rf_basisdens := [1, 162, 81, 18, 162, 162];
    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]];
    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, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, -1, 0], [-2, 1, 0, -1, 0, 1], [-1, -2, -1, 0, -1, -1], [0, 1, 1, -1, 1, -1], [-1, -2, -1, -1, 1, 0], [0, 1, -1, -1, 1, 1], [1, 2, -1, 0, -1, 1], [-2, 1, 2, 0, 1, -1], [0, 0, 2, -1, 0, 1], [3, 0, 3, -1, 1, 0], [0, 1, -1, 1, -1, -1], [-5, 2, -3, 1, 0, 1], [-2, -4, 1, -2, 1, 0], [1, 0, -1, 0, 0, 0], [-6, -3, -1, 0, 0, 1], [-7, 2, -2, 2, -2, -1], [0, 2, -2, -2, 0, 0], [-6, -1, -1, 1, 1, 1], [-4, -4, -3, 2, 0, -1], [-4, 2, 1, 1, 1, -3], [-6, 1, 2, 1, 0, 1], [4, -3, 5, 0, -1, -2], [-5, -2, 1, 3, -4, -1], [0, -1, -1, 1, -3, -1], [4, 1, 3, -2, 0, 3], [-2, -2, 2, 3, 0, -1], [-6, -1, -1, 2, -2, 1], [-2, 5, 3, -1, 1, -1], [-2, -6, 0, -2, -1, 1], [1, 0, 1, 1, 3, -2], [0, 9, 1, -1, 1, -1], [-4, 2, 1, -4, 1, 0], [7, 0, 1, 0, 2, 2], [-4, 1, 1, 3, 1, -1], [0, 6, -1, 0, 1, 2], [0, 4, -4, 4, 0, 0], [-8, -2, -1, -3, 3, 1], [-8, 2, 2, 3, 1, 0], [2, 3, -1, 1, 1, 3], [4, 0, -4, 0, -3, 1], [0, 4, 4, -5, 2, 1], [2, -8, -2, 0, 0, 0], [2, 0, -4, 0, 0, 2], [-8, -1, 1, 2, 0, -1], [2, -3, -2, -3, 2, 3], [0, 4, 2, -2, 1, -1], [-1, -8, -1, 1, 0, -1], [-7, -2, 2, 0, 5, -2], [-5, 6, 1, 1, -2, -1], [-3, 2, 6, -3, 2, -2], [5, 0, 3, -3, 3, 2], [-6, -6, 1, 0, -1, -4], [9, 6, 1, -2, 4, 0], [6, -9, 1, 2, -3, -2], [1, -2, -1, 3, 2, 1], [-5, 8, -1, 0, 0, -2], [0, -8, -1, -2, 1, 0], [-6, 2, 6, 0, 0, 0], [-2, -6, 4, 4, -1, -1], [-4, 1, 2, 3, 2, -1], [-6, -2, -2, 3, -5, -2], [4, 2, -2, 6, 0, -4], [2, 1, 3, -5, 1, 3], [-4, 0, -2, 2, 1, -1], [-3, -6, 0, 4, -4, 1], [-15, -4, -2, 1, -5, 1], [-6, 2, -2, -2, 3, -1], [-2, 4, -4, 4, 2, 0], [6, 2, -4, -1, -4, 3], [5, -4, 1, 4, -5, -1], [12, 8, 0, 2, -5, 3], [-6, -5, 5, -4, 0, -1], [-16, -7, 2, -3, -2, 1], [-10, -4, 2, -1, 3, 0], [-10, 2, 1, 2, -2, -5], [-7, -4, -5, 7, 1, -2], [5, -6, -3, 4, 1, -1], [-7, 2, -5, -1, 0, -1], [-4, -4, -5, -2, 3, 2], [-7, -2, -3, 6, 0, 0], [2, -1, -3, 0, 2, 1], [2, -7, -7, 2, -6, 1], [2, 9, -3, 7, -1, 1], [-10, 3, -2, -2, -1, -5], [1, 0, -3, 2, 1, 3], [3, -8, -1, -1, 2, -1], [0, 2, 2, -1, 1, 4], [22, 2, 4, -4, -2, 0], [-1, -8, 3, -1, -6, -1], [7, -6, 3, 4, 0, -2], [6, -2, -4, -5, 3, 6], [-10, 2, -5, 6, -4, -1], [-1, -2, 0, -7, 2, 6], [-15, 0, 7, 2, -1, -5], [22, 5, -1, -2, -1, 2], [10, 1, 9, -2, 3, 2], [5, -12, 1, 3, -1, -4], [-2, 14, 6, -5, 7, 4], [-6, 2, 2, 0, -3, -1], [10, 7, -5, 6, -3, 2], [-4, 3, -2, 1, 4, 7], [1, -2, 5, -7, -1, 4], [-22, 1, 0, -8, 3, -1], [-12, 6, -2, 4, -2, -4], [13, -10, 4, 4, 1, -4], [1, 10, 1, -4, -2, 4], [-2, 10, 2, 3, 0, -1], [2, 4, -2, 8, 2, -2], [-2, 2, -6, -1, -6, 1], [-4, 2, -6, 6, 0, 4], [-2, 1, 3, 3, -5, 1], [-8, -12, 2, 1, -4, -5], [9, -16, -7, 3, -5, -2], [-12, 2, -10, 6, 2, -2], [-3, 0, -3, -7, 6, 1], [-10, 19, -8, -5, -2, 3], [-4, 12, -6, -5, -6, 3], [-12, 3, 8, 0, 1, -5], [-4, 4, 4, -7, 1, 4], [22, 0, -6, 8, -6, 2], [6, 4, -2, 2, 2, 8], [13, 6, -1, 0, -1, -3], [-3, -12, -1, -1, -6, -7], [6, 3, -5, -3, -1, 3], [-4, -18, -5, 2, -3, 0], [-2, 3, 1, -5, 7, 7], [1, 8, 3, 1, -1, 2], [17, 0, 3, 4, -8, -4], [9, 2, 5, -4, -2, 0], [-4, 2, -1, -8, 3, 4], [-19, 4, -6, 8, -2, -3], [-3, -6, -2, -1, 1, 5], [-7, 8, -3, 4, 4, 2], [11, 16, -4, 1, -5, 1], [-1, -14, 7, 5, 0, -3], [-1, -12, -1, 6, 0, 0], [11, 4, -5, -3, 1, 4], [3, 16, -9, 0, -3, 5], [-15, 8, 7, -3, 3, -2], [4, 24, 1, -4, -1, 2], [13, 4, -3, 1, -3, 2], [-6, 2, 5, 4, 5, 2], [-6, -12, 7, -12, 1, 0], [-4, 0, 4, -6, 0, -2], [-14, -4, -2, -1, -5, 0], [-8, -4, 4, 0, 8, 0], [5, 4, -4, -9, 3, 1], [-1, 2, -11, -2, -1, 3], [-4, -2, -10, 1, 3, -4], [10, 2, -6, 12, -7, -1], [-4, -13, 7, -6, 2, -1], [10, 9, 3, 3, -3, 3], [15, -2, 4, -3, -7, -5], [0, 4, -5, 2, -5, 6], [0, -4, -5, 0, -3, 2], [-18, -2, 0, 4, 2, 4], [5, 8, 3, -3, 6, -7], [24, 3, 5, -4, -3, -4], [23, 4, -1, 1, -4, 3], [0, 3, 3, -7, 1, 3], [6, 12, 2, 2, -3, -3], [3, -12, 3, 0, -3, -3], [0, 20, 3, -2, 0, 1], [-28, 3, 3, 4, 3, -2], [-29, -8, 0, -1, 3, -5], [-27, 6, -3, 1, -3, -2], [14, 8, 7, -6, 9, 6], [-2, -6, 0, 2, 0, 4], [8, -23, 5, 7, 1, -1], [18, 8, 4, -2, 0, 0], [16, 0, -9, -8, 6, 5], [2, 3, -3, -1, 1, -7], [4, 1, 5, 0, -2, -1], [1, 0, -1, -8, 0, 4], [-23, 0, 0, 2, 3, 0], [1, -6, -13, 8, -9, 1], [-12, -7, -1, 7, -3, -3], [-22, 2, 4, 6, 0, -4], [10, 17, 2, 4, -5, -3], [16, -4, -4, -4, -2, -2], [27, 14, 6, -3, 0, -2], [-8, -20, -6, 8, -6, 6], [14, -8, -4, -7, 0, -1], [-2, 0, 0, 11, 4, -7], [6, -17, 9, 1, -3, -5], [10, -14, -1, 4, -1, -2], [1, 0, -9, -1, 1, -2], [9, -26, -1, 0, -3, -1], [-15, 0, 4, 7, -4, 0], [13, -26, 7, 3, 0, -1], [15, -10, 0, -9, 1, 1], [0, -11, -3, -7, -3, 5], [-6, -20, 2, 2, -2, 0], [-8, -21, -4, 7, -8, -1], [4, -3, 1, -3, -3, 5], [-18, 0, 4, 10, 6, -4], [21, 2, -5, 6, 1, -3], [2, -7, 0, -5, -8, -3], [-23, 6, -8, 12, 2, -1], [4, 14, 8, -2, 1, -7], [-20, -2, -1, 4, 6, -3], [14, -5, 1, 8, -1, -4], [-17, -16, -1, 7, -6, 3], [-24, 18, 6, 0, 3, 1], [22, 7, 1, -3, -5, 5], [0, 11, -11, 0, 7, 2], [1, 6, 8, -7, 1, 5], [3, -6, 3, 6, 2, -6], [18, -8, -6, -4, -2, -2], [20, -3, 5, -6, 2, -3], [-10, -8, 1, 2, 1, 0], [10, 7, 7, -2, 1, 2], [18, 5, -6, -5, -10, 5], [2, 1, -1, -5, 11, -3], [-12, 10, -2, -5, 1, 0], [-10, 0, -4, 3, 2, -1], [13, -10, 11, -2, 1, -3], [0, -1, -7, 3, -13, -1], [15, 0, -7, 8, 2, -2], [4, 8, 2, -6, 4, 4], [-24, 13, -1, 4, -1, 0], [-11, 12, -4, -5, 1, -1], [10, 16, -6, -8, 4, -4], [-11, -22, -3, 10, 0, 0], [-25, -6, -3, 1, -6, -9], [8, -10, 8, 0, 1, 3], [3, -8, -7, 4, -5, -3], [0, -25, 5, -4, -7, -8], [-13, 10, 3, -3, 10, 7], [14, -11, -5, 3, -7, -9], [-12, -3, -15, 8, -4, -3], [14, -2, 9, -17, 5, 1], [35, -8, 9, -1, -2, -1], [14, 9, 3, -5, 7, 9], [-7, 12, -7, 7, -7, -4], [-16, 15, -1, 3, 5, 5], [-12, 12, -6, 0, 2, -4], [34, -2, 4, -7, 2, 1], [0, -6, 6, 6, -1, 7], [-16, 8, -9, 8, -7, 2], [32, 4, -4, 4, 6, -6], [28, -16, 6, 4, 2, 0], [35, 0, 7, -3, 4, 3], [-2, -9, -5, 6, -5, 8], [29, 4, -5, 3, -1, 4], [-8, 13, 1, 8, 3, -12], [-18, 18, -5, -9, 4, 4], [22, -24, -2, -2, 0, 6], [-10, 2, -1, -10, 7, -4], [4, 15, -2, 8, 1, 1], [-26, 0, 4, 4, -3, 5], [-19, 4, -5, 2, -4, 2], [-1, 14, 9, -2, 8, 10], [1, 14, 0, 13, 3, -1], [-11, -10, 0, 0, -4, -3], [-2, -14, -4, 10, -12, -8], [6, 22, -4, 2, -2, 2], [16, 12, -9, 6, -5, 6], [11, 0, 8, -4, 6, 5], [-1, -4, -4, -7, -4, 8], [0, 23, 7, 3, 5, 1], [22, 16, 4, -2, 10, 4], [5, -10, 0, -15, 8, 6], [0, 14, 6, -8, 8, 2], [6, -18, 6, -4, -5, -3], [24, -4, -1, -12, -3, -2], [27, 22, 1, -4, 13, 3], [-22, -11, 2, -5, 0, -9], [2, 26, -6, -5, 4, -1], [15, 10, 6, -4, 6, -9], [-6, 0, -6, 9, -2, -5], [-4, -12, 0, 4, -2, -2], [-34, -20, -4, 6, -14, 0], [-18, 2, -12, 3, -10, 1], [0, -1, 5, 3, -5, -11], [28, 5, 5, -7, -3, 5], [-4, 36, 4, -4, 6, 2], [8, -8, -4, 3, -1, 4], [1, -28, -6, 2, 3, 2], [11, 4, 3, -2, -10, 2], [-26, 4, -10, 7, -5, -4], [-19, -8, 3, 6, -11, -1], [28, -6, 2, 8, -5, 1], [2, -6, -12, 3, -2, -1], [24, -26, 11, 9, -4, -8], [-2, 29, 1, 3, 3, -3], [4, -12, -6, 2, -11, 3], [-1, -4, 9, 9, 1, -12], [2, -6, 2, -9, 5, 10], [-36, -16, -6, 4, -8, -4], [-26, -4, -3, -9, 2, 0], [10, 22, 2, -4, 4, 0], [8, -12, 11, 6, 1, -8], [-14, 6, 18, -7, 7, -6], [-31, 12, -1, -14, 9, 9], [-21, -6, 6, 2, 7, -4], [17, 2, -7, -3, -9, -4], [4, 18, -8, -3, -12, 9], [2, 20, 6, 7, -11, -2], [17, -18, 15, 8, -1, 1], [1, 2, -1, -4, -9, -3], [2, -2, 0, -17, 5, 4], [12, -14, -11, 11, -11, -1], [-10, -2, 8, -13, -1, 2], [-22, 13, 1, 3, -5, -3], [0, 23, -1, 0, -2, 5], [12, -12, -8, 4, -12, -4], [-19, -12, -1, -5, 1, -6], [8, -4, -2, 0, 3, -5], [22, -4, -12, 8, -5, 7], [-50, -10, -3, 2, -2, -1], [-22, -4, -6, 0, 6, 2], [-16, -8, -4, 0, -10, 6], [18, 19, 14, -9, 2, -3], [10, 4, -14, -12, 8, 4], [17, -6, 5, -4, 9, 1], [1, -6, 1, -8, 4, 2], [18, -8, 7, -8, 19, 2], [18, -30, 2, 6, -1, -1], [-41, 8, -19, 7, -2, -1], [20, 4, -3, 12, 1, -12], [-8, 0, 10, -4, -1, 3], [16, -2, 12, 3, -2, -3], [-9, 4, 7, 11, -8, -3], [-11, -10, -17, 11, -12, -5], [-6, 13, 15, -6, -5, 10], [24, -18, 18, -4, 2, -4], [-9, 10, 9, -8, -3, 11], [-42, 19, -13, 3, -5, -7], [0, 16, -6, -15, 4, 3], [-29, 10, -7, -3, 0, 1], [-5, -8, 5, -2, 18, 0], [14, 22, -3, -5, -2, 0], [8, 13, 8, -9, 4, 3], [-38, 6, 8, -8, 8, -6], [-14, 10, -2, -10, 6, -6], [-2, -11, 7, -1, 7, 5], [-16, 10, -1, 5, -4, 0], [11, -6, -5, -16, 10, 8], [18, 11, 1, -9, 11, -1], [-5, 20, -1, -1, 2, -1], [-4, -11, -12, 8, -12, -4], [-2, 2, -10, 2, -7, 9], [20, -12, -8, -2, -8, 2], [-16, 4, 5, 4, 16, 5], [24, 20, 6, -6, 16, 0], [9, -20, 3, 12, -7, -7], [-22, 11, 7, -2, 6, 3], [1, -6, 5, -12, 0, 6], [6, 24, -2, -11, 2, 1], [-28, -6, 5, 5, -10, -8], [-27, 8, -1, -11, 10, -1], [8, 8, 6, -6, 6, 2], [9, 2, 5, -9, 12, -1], [-4, -17, 3, 5, -1, -7], [18, 3, 17, -16, 5, -4], [-40, 14, -3, 10, -7, -6], [18, -1, 5, 14, 2, 3], [15, -4, 13, 1, 7, -6], [-16, -10, 10, -8, -2, 0], [-15, -22, -9, -4, 11, 1], [6, 5, -16, -1, 8, 1], [-12, -7, -9, -4, 1, 2], [12, 3, 11, -10, -1, 16], [-44, 6, -6, 5, 3, 0], [20, -3, 1, 5, -13, -5], [-17, 10, 1, 17, -2, -9], [15, -10, 11, -1, -6, 1], [-40, -24, 0, 0, 4, 4], [-5, 18, 5, 2, 16, 4], [-1, 30, -5, 3, -5, -2], [5, -20, -15, 7, -12, -3], [9, 0, -4, 16, -4, 5], [-12, -43, 2, -4, -3, -3], [-52, 6, -3, 10, -5, -4], [-3, 10, -2, 15, -11, -7], [10, -26, 2, -13, -5, 6], [-4, 9, -1, 4, -5, 8], [-25, -24, -1, 1, 15, -6], [-24, 10, -12, -4, 6, 10], [-31, -14, 5, -5, 12, -3], [-3, 42, 9, 2, -4, 0], [-18, -8, 5, -16, 7, -4], [0, 15, 3, 13, -9, -7], [3, 0, 3, -5, 1, 4], [-3, -32, -13, 4, -8, 0], [44, 12, -1, 6, -6, 3], [-18, -4, -6, 1, -3, -4], [-52, 12, -6, 5, -2, -7], [-36, -1, -1, 6, 8, 1], [18, 2, 18, -3, 8, -3], [-5, -20, -7, -8, 14, 2], [-29, -26, -8, 7, -4, 6], [20, -26, -6, -2, -6, 6], [25, 6, -3, -1, -7, -8], [-45, 16, 6, -12, 7, -2], [12, -14, 8, -3, -3, 2], [0, 23, -9, 9, 5, 3], [6, -22, -3, 0, -7, 10], [-14, -14, 5, 5, -12, 8], [59, -2, 1, 6, 3, -1], [-23, 16, -3, 16, 0, -4], [10, -12, 12, 10, 4, -2], [26, 6, 4, -2, 6, -6], [-18, 10, -6, 6, -13, -5], [-14, -6, -7, -4, 13, 6], [-12, 8, -8, -2, -8, 6], [-14, 5, 3, 0, 0, 13], [-22, -13, -15, 15, -5, 3], [0, -19, 7, 12, 3, 4], [27, -14, -2, 1, -5, 11], [15, -22, 7, 13, -11, -10], [-14, -20, -10, 15, -18, 5], [36, 18, 11, -14, 22, -1], [-12, -16, 8, 11, -11, -2], [-38, -15, -5, -1, -3, -9], [-13, -6, 18, 6, 0, -17], [16, -6, 10, -21, 9, 4], [-26, 21, 5, -19, 7, 1], [2, 14, -13, 2, 5, 2], [11, -10, 12, 3, -4, -10], [20, 4, 16, -3, -7, -4], [-11, -18, -3, 7, -9, -2], [-8, -33, 1, 10, -5, -4], [26, 1, 7, -14, 16, -1]];
    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_be();".
// 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_be(: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_be();".
// 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_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6468_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![98, 224, 79, -64, -18, 4, 1]>,<13,R![142, 72, -203, -128, -4, 8, 1]>,<17,R![-4616, -6720, -2864, -336, 56, 16, 1]>,<23,R![-128, 448, 1724, -72, -94, 0, 1]>],Snew);
    return Vf;
end function;