
// Make newform 6422.2.a.bd in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6422_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_6422_2_a_bd();".
// 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_6422_2_a_bd();".
// 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 := [5, -24, 13, 24, -12, -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], [-31, 10, 36, -12, -3, 1], [-14, 29, 33, -24, -3, 2], [17, -56, -33, 27, 3, -2], [-8, 25, 12, -13, -1, 1]];
    Rf_basisdens := [1, 1, 6, 6, 6, 2];
    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_6422_a();"
function MakeCharacter_6422_a()
    N := 6422;
    order := 1;
    char_gens := [4903, 4733];
    v := [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_6422_a_Hecke(Kf)
    return MakeCharacter_6422_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 := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, -1, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0], [-1, 0, 1, 1, 1, 2], [-1, 0, 0, 0, 0, 0], [1, -1, 1, 0, 0, 1], [3, 0, 0, 0, 0, 1], [2, -2, -1, 0, 1, -1], [-2, 1, 1, 1, -2, -1], [0, 0, 0, -2, 1, 1], [3, -1, -1, -1, 0, -1], [3, 0, 1, 0, 0, 0], [3, 0, 0, -2, -1, 2], [-1, 0, 2, 2, 2, 3], [1, 3, 0, 1, 1, -2], [-7, 2, 1, 0, -2, 1], [7, -1, -1, -1, -2, -1], [1, 1, 1, 0, -1, -2], [3, 3, -1, 1, -1, -2], [0, -4, -1, 0, -1, 1], [2, 0, 1, 2, 1, -1], [0, 2, 0, 4, 4, 0], [-1, -1, -3, 1, 0, -1], [4, 2, -1, 2, -1, -3], [9, -2, -1, 1, -2, -1], [-6, -1, 0, 1, -2, 2], [-1, 3, -1, 1, 2, -1], [0, 0, -1, -2, -3, -1], [3, 3, -2, -1, 0, 1], [6, -3, -2, 1, -1, -1], [2, -2, -1, 4, -1, 1], [3, 3, 3, 1, -3, 0], [1, 1, -1, -1, -1, -2], [8, -4, -3, 0, -4, -2], [1, -3, 0, 3, 4, 1], [3, -3, 4, 3, 4, 7], [3, -5, -5, 3, 1, -4], [2, 1, 2, 7, 5, 4], [3, 3, 0, 5, 0, 1], [-1, 2, 1, 1, -4, -1], [1, -7, -2, -1, 1, 2], [-6, -2, 0, 2, -1, -3], [-2, -1, 5, -2, 5, 5], [2, -3, 1, 1, -2, 2], [1, -3, -3, -1, 4, 1], [-6, 3, 3, 0, 1, 1], [-6, 0, 0, 0, -1, -1], [-6, 5, 4, -3, -1, 0], [6, -6, 0, 1, -2, 0], [-7, 3, 0, -1, -4, -1], [-14, 2, 2, 2, 1, 5], [-5, 1, 0, -5, -1, 2], [3, -2, 1, 4, 3, -1], [-4, -5, -1, -7, -3, -2], [3, 2, -4, -1, -1, -4], [4, 0, 0, -2, 1, -3], [2, -6, -5, 2, -2, 0], [-5, 7, 5, 5, 1, -4], [4, 2, 3, 1, 1, 1], [-3, 2, 2, 0, 4, 4], [-8, 3, 2, 5, -1, -7], [2, 3, 2, 6, 0, 2], [2, 3, 0, 0, 4, -2], [-3, 2, 0, -5, 1, 4], [3, 3, -3, -3, 3, 4], [2, -6, -5, 2, 3, -3], [-12, 0, -7, -2, 0, 0], [-6, 0, -4, -1, 4, 2], [-6, -5, -3, -1, 0, -2], [10, -1, -1, -3, 4, 1], [10, -1, 5, 2, 3, -1], [5, 3, -6, -4, 1, -2], [17, -9, -4, 1, 5, -2], [0, 4, 1, 0, -1, -1], [-3, 1, -4, -7, -2, -1], [7, -1, 5, -3, -6, 1], [8, 4, -4, -2, -1, -7], [-7, -9, 3, 7, 2, 1], [-6, 4, -2, -5, -4, -8], [-3, 3, 2, -3, -1, -4], [10, -8, -7, 0, 0, -2], [1, 1, 8, 1, 6, 7], [6, -2, -1, 2, -5, -5], [6, -4, -2, -8, -1, 1], [-3, 4, 2, 1, -4, -7], [4, 4, 1, 6, 5, 3], [-4, 0, 4, 2, -3, 3], [-15, -1, 4, -3, -4, -5], [14, 2, -4, -2, 2, 0], [-17, -3, -3, -1, -2, 3], [19, -3, -1, 3, -5, -6], [15, -3, 3, 1, 4, 7], [-9, 0, 3, 2, 2, -3], [-16, 2, -3, 0, -1, 3], [17, -9, -6, -1, 1, 0], [4, -8, 2, -1, -2, 4], [-1, -3, 7, 3, 1, -2], [9, -5, -5, 7, 5, 0], [11, 1, 1, -1, 1, -2], [4, 0, -9, 0, -4, -4], [-14, -4, 5, 6, 1, 1], [-3, 5, 0, -5, 2, 1], [-8, -1, -5, -5, 3, 1], [3, -7, -9, -7, 1, 2], [-14, 5, 5, 3, 4, 7], [9, 7, -1, -3, -2, 3], [10, -2, -3, 4, 1, -5], [13, -1, 4, 1, -9, -2], [17, -5, -5, -11, -9, -2], [4, 5, 5, -11, 1, 6], [10, -2, 3, 0, 2, -2], [-15, 2, 2, 6, -6, 0], [11, -3, -3, -1, 1, 4], [-4, 14, 3, -4, -3, 1], [-5, 4, 2, -4, -7, -2], [-4, 2, -2, -4, -11, -5], [6, -6, 5, 1, -6, -2], [-3, 9, 0, 0, -8, -5], [23, -7, -5, 5, -2, -3], [-4, 5, 4, 0, 5, -1], [18, -3, -3, -1, -3, -8], [10, 0, 10, 6, 5, 9], [-2, 2, -3, 4, -1, -3], [-11, -11, -2, -3, 4, 1], [-2, 6, -2, -3, -4, -2], [-13, 1, -2, -6, 2, 1], [-3, -7, 5, 7, 2, 1], [-3, 1, 0, 1, -1, 0], [-16, 4, -2, 0, -8, 2], [5, 1, -3, 3, -7, -8], [-14, 4, 3, -4, -9, -5], [-1, 4, 1, -9, 0, 3], [5, 4, 4, 0, 4, 1], [-24, 13, 4, -1, -2, -4], [-9, 1, 6, 14, -5, 2], [2, 0, -10, 3, -2, -6], [8, 1, -1, -2, 1, -5], [-1, 4, -4, -1, 4, 7], [-9, 1, -2, 7, 0, -7], [0, 11, 4, 0, 3, 5], [2, -2, 6, 3, 0, 4], [-5, -1, 2, 0, -1, 6], [8, -2, -6, 2, -3, -5], [6, 0, 0, -2, 6, 6], [-4, 0, 2, 0, 2, 0], [-19, 15, 5, -3, 0, 7], [14, -2, -3, -10, -8, -4], [-11, -16, 4, 0, -2, 9], [7, 2, -1, 14, 2, 0], [15, 9, 2, 1, 1, 0], [16, -6, 5, -2, -9, -1], [-8, 5, 2, -7, 8, 2], [-16, -4, 4, -2, -5, -5], [-1, 1, 5, -10, -3, -6], [-18, 0, 7, -1, -8, 6], [-5, -5, -1, -4, 0, 7], [-19, 5, 6, -6, 0, 7], [17, -3, 5, 1, 4, 3], [16, 4, 10, 6, -1, 3], [-3, -6, -1, 4, 7, -1], [0, -1, 9, -3, 1, 10], [12, 2, -10, -2, 1, 3], [6, 2, -6, 0, -9, -1], [-4, -8, -9, -2, 4, 0], [-3, 11, 7, -3, -4, 1], [13, 9, -5, -3, -2, 1], [-6, -8, 5, 4, -2, 0], [11, 4, -2, 1, -1, 0], [-20, -1, 7, -9, 1, 1], [5, 10, -6, -6, 2, -4], [16, 7, 1, 7, -5, -3], [-19, -5, 7, -1, -2, -3], [8, -5, -5, 7, 5, 0], [5, 9, -5, 5, -3, -14], [21, -3, 6, 1, 1, 6], [-9, 5, -2, 7, 8, -5], [17, -3, -6, -11, -5, -6], [27, -4, 3, -3, 0, 3], [-6, 11, 3, -1, -6, 6], [1, 6, 3, 4, 6, 6], [-3, -5, -8, -3, 4, 3], [3, 2, -2, 1, 0, 3], [2, -8, 5, -2, -3, 7], [14, 16, -6, 1, 0, -8], [-7, -6, -5, -10, -5, -3], [-31, 3, -1, -5, 10, 5], [26, -1, 2, 2, 7, -1], [8, 0, -2, 4, 5, 3], [-21, 9, 4, -5, -4, -3], [11, -8, -5, -8, -5, -5], [17, -8, -6, -2, 11, 8], [3, 3, 10, 3, -2, -1], [5, -5, 0, -3, 0, 5], [3, 6, -1, -1, -6, 1], [4, -10, -5, 2, 8, -4], [9, 2, 8, -7, 0, -3], [5, 10, 10, 7, 6, 3], [-2, -3, -2, -3, -2, 3], [16, -2, -7, -12, -4, 6], [17, 2, -5, 7, -2, -13], [1, 1, -4, -11, -8, -5], [2, 0, -6, -4, -4, 0], [15, -3, 0, 1, 8, 11], [12, -1, -7, 3, -2, -6], [-17, 4, 2, 0, 3, 14], [1, 1, -8, -8, 1, 4], [-8, -1, -8, 3, -10, -8], [-1, 11, 4, -2, 3, 0], [-17, 1, 5, -7, 10, 7], [-4, 17, 8, 9, -7, 2], [18, 9, -3, 2, 0, 6], [10, 2, 2, -4, 0, 0], [-5, -1, -3, 7, 5, 10], [-4, 3, -4, -1, 10, 6], [-14, 4, -7, 2, -6, -4], [-2, 4, 5, -10, -8, 6], [9, -1, 0, 11, 11, 8], [26, -4, -9, 4, -8, -12], [4, 4, 11, 10, -1, 1], [12, 6, -3, 10, 8, 2], [1, 0, -7, -11, 9, 0], [-11, 3, -2, -2, 1, 2], [-3, 1, -10, -3, -10, -15], [6, -2, -6, 12, 1, -9], [0, -14, -10, -4, 6, -6], [-11, 13, 4, -3, -7, 0], [35, -7, -8, 1, 5, -14], [-25, 3, 0, -9, -4, -5], [-10, -9, -17, 4, -3, -11], [12, 14, -4, 4, 2, 4], [13, 17, 1, 3, -4, -3], [20, -10, -4, -10, -1, -5], [9, 2, -10, 3, -14, -17], [1, 2, 3, 6, -6, 4], [-10, -6, 4, -6, 16, 10], [-23, -3, -4, -5, -1, -6], [-14, 0, -5, 6, 3, -9], [8, -4, -1, 6, 1, 3], [-18, 12, -2, -6, -14, -10], [10, -2, 12, 9, -7, 1], [15, -9, 5, 2, -3, 4], [8, 2, -6, -8, -4, 0], [2, 2, -1, -6, 2, 8], [-20, 3, 4, -5, 9, 16], [-10, 15, 1, 3, -10, 1], [-1, 0, 2, -4, -9, -1], [-3, 6, -1, 10, 3, -10], [-6, 4, -11, -8, -11, -3], [16, -13, -10, 1, -4, -4], [13, 3, 2, -6, -7, 0], [-24, -2, 0, -8, -2, -6], [-8, -2, 11, -6, -1, 13], [23, -11, 7, 9, 9, 16], [18, -8, -9, 0, 10, -8], [7, -10, 5, 4, 0, -4], [12, -11, -6, -3, -4, 1], [-25, -1, 3, 3, 7, -2], [5, -11, -11, -13, -8, -11], [7, -14, -14, -4, 0, -8], [29, 7, 1, 9, -9, -12], [17, -2, 6, 12, 7, 8], [17, -7, 2, 7, 18, 5], [36, -1, -4, -11, 3, -1], [-7, 4, -13, 7, 6, -5], [-13, -1, 10, -8, -11, -2], [-1, 4, -11, 3, 8, -1], [6, -8, 2, -14, -2, 0], [-9, 19, 2, 2, -3, 0], [-24, -5, 1, -6, 1, 15], [-10, -5, 1, 10, 8, -6], [8, -12, -2, -3, 6, 10], [7, 1, 1, -5, 5, 6], [8, 20, 6, 9, 2, 6], [21, 3, -11, -11, 7, -2], [5, -2, 13, 11, -2, 7], [-22, 10, 12, 10, 6, 0], [-19, 7, -11, -7, -9, -12], [-3, -19, -7, -13, -5, 10], [-6, 5, 4, -11, 6, 4], [19, 1, -2, -8, 5, -2], [11, 12, -6, -5, 1, -2], [18, -8, 5, 8, 14, 4], [32, -5, -8, 5, 0, -3], [7, -6, -8, 2, 0, -9], [-21, -9, 3, -7, -2, 3], [22, 3, -9, 3, -2, -24], [8, -4, 1, -8, 2, 0], [-16, -10, -5, -4, -4, -8], [-30, 12, 9, 6, 12, 0], [8, -8, -1, -6, -5, -5], [31, 20, -9, 4, -5, -9], [5, -8, 9, 16, 3, 7], [-21, -3, 11, 3, 6, 1], [-11, -3, 4, -17, -2, 5], [6, 10, 2, 4, -4, 2], [-10, -5, 15, -11, -9, 9], [4, 0, -6, 4, -3, -19], [25, -18, -13, 8, 1, -7], [-14, -1, 2, 3, -3, -1], [-2, 1, 0, -3, -4, -2], [37, -6, 1, 2, 3, -4], [1, 19, 8, 9, 10, -7], [14, -4, -16, 10, -10, -20], [17, -5, -13, -4, 5, 10], [9, -4, -11, 1, 1, -4], [24, -5, -1, 3, -1, 6], [13, 5, 14, 5, 3, 8], [4, 3, 1, -3, -10, -5], [-8, -8, 7, 12, 11, 1], [2, -10, 2, -6, 10, 24], [16, 0, 5, 2, -7, -5], [4, 1, -5, 1, 5, 10], [14, 8, -8, 6, -1, -17], [5, 3, -11, -13, -15, -10], [6, 0, -15, -8, 13, -5], [23, -4, -3, 1, 7, -4], [37, 13, 0, 7, -2, -11], [-16, -4, 11, -5, 6, 12], [-27, -2, 7, -3, -6, -1], [28, 11, 16, 8, 3, 5], [-5, -5, 4, 15, -4, 3], [8, 2, -13, 11, 4, -4], [-1, -9, -4, 5, 16, 3], [-9, -11, -3, 0, -10, -1], [-16, 13, -1, -7, -14, -4], [14, -4, -2, 14, 11, 5], [20, -4, -4, 0, -2, -20], [-24, -2, -8, 0, -10, -12], [-41, -2, 4, 5, 0, 7], [-29, 2, 11, -9, 17, 14], [3, 3, 2, 7, 9, 18], [-1, -1, 12, 5, -1, 12], [-23, 13, 8, -7, -8, 1], [-45, 14, 7, -2, 0, -3], [-26, -4, 4, 2, -3, -1], [17, 14, 4, 13, 4, -3], [3, 1, 17, -10, -3, 10], [-1, 2, -9, 1, 2, 1], [20, -21, -5, -3, 0, -4], [-29, -8, -11, -3, 4, -5], [13, 5, -2, -5, -3, -2], [3, 12, 13, 3, -1, 0], [-25, 0, -3, 0, 9, 5], [-21, 19, 14, 11, -10, -3], [-21, -14, -9, -14, 3, -2], [32, 4, -3, -8, 5, 1], [-9, 9, -1, 1, -5, -10], [1, 10, 5, 8, 2, -6], [-5, -9, -9, -5, 10, 13], [-31, -13, -5, 1, 0, -3], [-19, 15, 10, 13, 5, -4], [-21, -12, 14, 4, 21, 13], [23, 1, -3, 5, -8, -21], [32, 5, 15, -2, -2, 14], [-14, 10, 0, 14, 11, 3], [42, -7, -4, 1, 9, -7], [12, -24, -13, -2, 2, -4], [7, 5, 3, -1, -9, -10], [1, 20, -4, -6, 7, -7], [16, -8, -7, 12, 12, 8], [-21, 16, -3, -8, -9, -6], [23, -3, -1, 5, -9, -4], [-27, 3, 8, -7, 4, 19], [-5, 11, -6, 5, -1, -14], [-22, 4, -7, 6, -6, -8], [7, 9, 11, -1, -5, 2], [4, 10, -11, 0, -11, -11], [31, 0, -4, -3, 7, -8], [24, 13, -8, 2, 4, 4], [33, 6, 9, -15, 7, -2], [3, -14, -13, 14, 0, -10], [38, -4, 10, 2, 2, 6], [-13, 13, 2, -2, -8, 9], [16, -15, -9, -5, 16, 7], [20, 10, 18, 10, 14, 4], [-2, -8, 14, -9, 4, 2], [23, -1, -2, -19, -18, 1], [-17, 0, 1, 14, 16, 6], [16, -2, 1, 8, -5, -9], [-22, -1, 5, 13, -3, -3], [-16, 7, 11, 11, 1, 16], [-24, -4, -13, -1, 1, 1], [-6, 0, -14, -8, -11, -9], [17, -2, -5, -19, -9, -14], [40, -14, 8, 8, -10, 0], [1, 5, -8, 7, -2, -21], [26, -4, -4, 14, 10, 6], [-14, -4, 6, 4, -10, -6], [1, 8, -4, -8, 8, 11], [-10, 6, 2, 2, 7, -3], [7, -9, -19, 7, -3, -4], [-10, -2, -10, -12, -1, 1], [14, 2, -15, -4, 3, -7], [12, -4, -10, -20, 16, 2], [-2, 4, -3, 28, 4, 8], [-20, 9, -1, -1, 1, -4], [-31, 15, 3, 7, 12, 13], [-14, -5, 4, -9, -12, 3], [48, -15, 0, 3, 6, 1], [11, 13, -7, -9, -1, -14], [-32, -6, -20, -10, -1, -11], [-3, -6, -9, 18, -1, -3], [25, -18, -10, -6, 9, 0], [-11, 17, 12, -1, 4, 17], [45, -21, -12, 1, 1, 2], [30, 5, -12, -4, 7, 3], [6, -12, 4, 22, -5, 3], [3, -9, 0, -3, 3, 4], [-22, 0, -3, 10, 6, 10], [17, 10, -9, 4, -18, -15], [6, 2, 12, -4, 7, -5], [46, 4, 4, 9, -1, -1], [-12, 12, 8, 4, -3, -1], [16, 0, -2, 12, -3, -3], [-9, -3, 10, -13, 6, 15], [15, -8, -6, 4, 4, 7], [-9, -1, 15, -9, -5, -6], [-18, -7, -22, -5, 5, -8], [30, -5, -9, 11, 5, -9], [2, -5, -5, -9, 0, -10], [-1, 26, 2, 4, -4, -3], [24, 26, -5, 0, -9, -5], [-44, -2, 9, -13, -6, 0], [30, 3, -7, -6, -12, -2], [24, -9, -9, 3, 9, 7], [1, 1, 2, 11, 16, -5], [-16, -17, 3, -7, 3, 13]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6422_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6422_2_a_bd();".
// 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_6422_2_a_bd(:prec:=6)
    chi := MakeCharacter_6422_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_6422_2_a_bd();".
// 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_6422_2_a_bd( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6422_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![5, -24, 13, 24, -12, -2, 1]>,<5,R![-64, -144, 64, 51, -21, -2, 1]>,<7,R![51, -117, 57, 25, -17, -1, 1]>],Snew);
    return Vf;
end function;