
// Make newform 6422.2.a.bh in Magma, downloaded from the LMFDB on 25 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_bh();".
// 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_bh();".
// 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 := [-2, 6, 7, -28, 3, 22, -4, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [1, -19, -7, 36, 12, -11, -2, 1], [-4, 4, -20, 18, 33, -10, -7, 2], [-19, 1, 91, -45, -78, 23, 14, -4], [-34, 19, 169, -66, -141, 35, 26, -7], [-29, 26, 170, -99, -153, 46, 28, -8], [70, -58, -349, 201, 306, -92, -56, 16]];
    Rf_basisdens := [1, 1, 3, 3, 3, 3, 3, 3];
    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, 0], [0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0, 0, 0], [-2, 0, 0, -1, -1, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 1, 1], [-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, 1, 0, -1, 0, 1], [1, 2, 0, 1, 1, 1, 1, -1], [-1, -2, 0, -1, 1, 1, 0, 2], [0, 1, -1, -1, 1, 2, 1, -1], [1, 0, 1, 0, 3, 0, 0, 0], [-4, 2, -1, -1, -1, 0, -2, 0], [1, 1, -2, 1, -1, -1, 0, 0], [-4, -1, -2, -2, -4, -1, -3, -3], [-1, 1, 5, 0, 0, -3, -2, -2], [-4, 3, -1, 3, 0, 1, 2, 2], [0, 1, 1, 0, -1, -4, -2, 0], [-5, 1, 0, 2, 1, 1, 0, 3], [0, 2, -1, -4, -2, 0, -4, -4], [-1, -2, 1, 1, 1, -1, 1, 0], [3, 0, -2, 1, 0, 1, -1, -3], [-2, -2, -2, 0, 2, 2, -2, 3], [-4, 6, 0, 0, -2, -2, -2, -4], [1, 0, 2, -2, 3, 2, -1, -1], [-6, 4, 2, 0, -4, -4, -4, -3], [-2, -2, -3, 0, 0, 3, 1, 6], [0, 0, 2, 0, -1, -5, -1, 2], [-1, -1, 0, 3, 1, -3, 3, -2], [-4, 1, 3, 0, -1, 2, 0, -3], [2, -2, -3, 2, 5, 2, 0, -1], [-3, -3, 2, 2, 1, 3, 1, 4], [-8, 6, 2, 0, -5, -6, 1, -2], [-2, -1, 2, -1, 3, -2, -2, -2], [5, -4, -4, -2, -1, 2, 4, 2], [-4, -4, -2, -4, 3, 2, 0, 3], [1, -2, 2, 2, 2, 2, -1, -2], [2, -1, 1, -1, -1, 3, -2, -2], [-8, 8, 2, -1, -5, -3, -1, -6], [-9, 2, 1, 0, 0, 2, 4, 3], [2, -1, 2, 5, 9, 0, 2, 3], [-7, -1, -5, -4, 3, 5, -3, -2], [-6, 5, 5, -1, -5, -5, -3, -7], [0, -3, -5, 1, 2, 2, 3, 3], [-4, 5, 1, -1, 0, 1, -1, -3], [1, -1, -5, 6, 6, 6, 2, 3], [3, -3, -4, 2, 1, 5, 0, 5], [4, -7, 0, 0, 1, -2, 3, -2], [6, -1, 3, 1, 6, -2, 3, 1], [-2, -4, -3, -1, 4, 9, 1, 3], [2, -3, -1, 1, -1, -4, -2, -6], [7, -3, 4, 0, -2, 1, 8, 4], [-4, 2, 0, 3, -1, 6, -1, -1], [-6, -1, -2, -6, -6, -2, -5, -2], [11, -9, 5, -1, 0, -2, 0, 1], [-3, -1, 0, 3, -1, -6, 0, 0], [10, -8, -1, 4, 4, 7, 11, 3], [-9, -7, -1, -3, 0, 1, 7, 4], [-3, 6, -5, 6, -1, -2, 6, -3], [-2, -4, 3, 4, 2, -2, 3, 1], [11, 4, 0, 3, 4, 5, 0, 0], [-1, 3, 0, 2, -8, -7, 1, -2], [-7, 6, 7, 1, -4, -7, -8, -5], [3, -6, -8, -6, 0, 6, 1, 1], [6, -3, -2, -5, -2, -4, -3, -1], [3, -7, 1, 2, 8, -3, 6, 2], [-5, -8, 0, -7, -3, 0, 1, 3], [2, 1, 3, -2, -6, -8, 1, -4], [-6, 2, 6, -3, 2, 0, 0, 5], [-3, 0, -4, 5, 4, 5, 4, 7], [1, -3, 2, -1, 3, 1, 12, -2], [-4, 2, -4, -10, -10, -2, -10, -8], [-10, -3, 2, -1, 3, 6, 0, 5], [8, 0, -1, -2, 2, 2, 5, 4], [-6, -8, -3, -5, -1, 2, -2, 3], [9, -8, 0, 0, 4, 5, 0, 6], [4, -10, -3, -8, -1, 4, -4, 3], [4, 3, 3, 5, 3, 3, 7, 6], [-1, -4, -1, 2, 3, -4, 4, 6], [12, -7, 4, -1, 4, 0, -2, 0], [10, -9, -6, -7, 1, 9, -2, 3], [-2, 8, -3, -9, -9, -2, -10, -9], [-7, 0, -1, 2, 2, 0, 9, -4], [-5, -8, 5, -7, -3, -1, -7, -3], [0, 2, -6, 6, -1, 0, -3, 6], [5, -2, 5, -1, -3, -2, -3, 1], [-1, -1, -4, -3, -4, 10, -2, 1], [-6, 5, -7, 3, -4, 8, 8, 5], [-4, 0, -2, 1, 4, -2, 6, 2], [6, -6, 1, 6, 5, 0, 4, 1], [0, -4, -2, -2, -4, -2, -10, -2], [-9, 9, 0, 0, -3, 3, -4, 1], [14, 1, 6, 6, 9, -2, 11, -1], [15, -2, -2, -7, -1, -2, 4, -2], [-6, 9, -3, -2, -7, -4, -8, -2], [-7, 4, -5, -7, -10, -6, -15, -6], [-16, -1, 0, -4, 2, 4, -17, -4], [-10, 3, 2, 6, -1, 5, 1, -2], [-2, 7, -8, 7, 1, -2, 0, 8], [-2, -2, -7, -14, -7, 0, -16, -6], [10, -1, -6, 5, 1, 4, 0, -4], [-3, -5, 11, -6, -1, -1, 1, 5], [-1, -4, -4, 1, 9, 3, 2, 6], [-8, 2, -1, 0, -7, -10, -8, -5], [11, -4, -2, 0, -1, 0, -5, 0], [-2, 2, 4, -3, -8, 1, 2, 1], [-6, -5, 5, 1, -1, -2, 7, 7], [-6, 7, 3, 0, -7, -3, -2, -5], [-17, 7, 1, -8, -6, -3, -17, -9], [-4, -4, 0, -6, -2, 3, -8, -6], [10, 7, 4, 7, 3, -6, 2, 0], [9, -6, 6, 1, -6, -3, 2, 1], [6, 4, 10, -5, 1, -2, -13, -2], [-15, 5, -3, -3, -5, 0, -7, -7], [11, 0, 2, -5, -1, -8, 7, -1], [-5, -1, -1, -1, 5, -1, -7, -5], [-7, 9, 6, 14, 0, -10, 5, 2], [-7, -6, 2, 0, 3, 11, -12, 3], [11, 1, -2, -2, -5, -2, 0, -7], [7, 12, 3, 1, -9, -3, 2, -6], [-9, -1, -8, 3, 9, 1, 5, 0], [-15, 11, -2, -4, -2, -3, 3, -2], [5, 10, -7, 5, 2, 8, 6, 4], [-13, 7, 1, -6, -6, -3, -10, -11], [-3, 13, 7, 1, -5, -1, -1, -3], [10, -10, -7, 7, 5, 2, 17, 6], [9, 6, 2, -2, -4, 1, -11, 0], [-5, -16, -4, -5, 3, -2, 2, 5], [4, 4, -3, 2, -1, 2, 1, -5], [-4, 0, -2, -8, -13, -1, -5, -9], [-6, -5, -1, 6, 7, 8, 4, 10], [5, 5, 0, 5, 2, -1, 1, -2], [-9, -1, -1, 3, 3, 9, 7, 7], [3, 11, -3, 12, 9, 7, 3, 4], [8, 3, -2, 0, 2, 7, -2, -2], [8, 4, 6, -4, -7, -3, -1, -2], [-10, 8, 1, 0, -8, -2, -1, 2], [-3, 5, 16, 5, -3, -6, -2, -6], [13, -4, -1, 0, 6, 3, 1, -1], [2, -1, -12, 7, 4, 15, 2, 8], [4, -4, 5, 0, -3, -8, -6, -7], [-2, -10, 3, 3, 2, 12, 9, 11], [-5, 11, 12, 9, -5, -4, 2, -6], [-6, 5, -6, -5, 1, 6, -4, -5], [-6, 2, 7, 8, -1, -13, 4, 5], [16, -6, 4, 1, -2, -8, 7, 5], [10, -11, 3, 5, 1, -4, 13, 9], [-10, 2, -3, 13, 1, -10, 4, 8], [7, 7, 6, -1, -3, -2, 2, 11], [13, 0, -4, 11, 6, 5, 14, 6], [3, -4, -9, -4, -4, -1, -17, -1], [4, 2, -3, 13, 10, 10, 11, 4], [0, 1, -15, 0, -5, -4, -10, -5], [-2, 10, 2, 2, -3, 5, 15, 10], [11, -11, -4, -3, -2, -10, -7, 1], [2, 0, 9, -7, -6, -12, -1, -1], [-1, 9, -10, -3, 2, 14, 2, 1], [15, 0, 4, 13, 8, 2, 12, 0], [7, -7, 2, 2, 12, 13, -7, 3], [13, -12, -6, 5, 7, 6, -2, 6], [8, 1, -14, 8, 4, 5, 8, 3], [-3, -16, 11, 4, 8, -7, 4, 2], [6, -4, -5, 6, 2, 11, -7, 0], [3, -2, -15, -4, 9, 12, 6, 4], [0, 4, -7, -2, -3, -1, 2, -9], [-21, 12, 13, -7, -7, -4, -17, -14], [-5, 4, 8, -2, 3, 6, -1, 3], [-4, 5, 1, 2, -5, 5, 2, -5], [16, -2, 14, 2, 1, -10, -4, 3], [-3, -15, 5, 6, 12, 1, 8, 6], [5, 18, -9, 7, -1, -5, 4, -2], [10, -11, 4, -9, 1, -4, 1, -7], [-3, -4, 4, 0, 7, 10, 7, 0], [-4, -7, -9, 4, -3, 7, 12, 11], [18, 10, -6, 12, 2, -2, 7, 3], [22, -7, 15, -1, -7, -13, -2, -1], [-20, 17, 0, -12, -15, -12, -6, -11], [11, 0, 13, 5, -3, -11, 5, -9], [-2, -12, 4, 2, 7, 15, 17, 11], [0, 5, 11, 9, 7, 2, -8, 0], [3, 6, 2, 8, 1, 6, 10, 1], [2, 7, 0, 9, -1, -6, -8, -7], [1, -9, 2, -11, -4, 1, 5, 3], [-14, 15, -6, -9, -1, 1, -1, -1], [1, 2, -12, 4, -8, -3, 4, 4], [1, 1, 2, -1, 1, -12, 4, -2], [-15, 10, 10, 3, -10, -18, -1, -4], [-16, 5, -7, -7, -7, 7, -15, -4], [5, -2, -6, -1, -2, 2, 11, 7], [22, -4, 9, -7, -3, -1, -5, -2], [7, -1, 11, 4, 9, -6, 5, 0], [-1, 7, -2, -4, -8, -9, 3, -7], [-15, -11, 2, -6, -1, -2, 0, 11], [12, -4, 13, 4, 1, -2, -10, -1], [12, -5, -9, 3, 5, 17, 22, 12], [1, 4, 0, 1, -4, -7, 2, -7], [2, 2, -4, 2, 3, -6, 2, -5], [9, -7, 0, -1, 4, -4, 6, 5], [-3, -12, 5, 4, 7, 6, 19, 20], [19, 6, 10, -1, 0, 1, -4, -6], [9, 7, 0, 1, 10, 0, 7, 0], [16, -11, -4, -3, -9, 5, 8, 3], [24, 0, 4, 6, 8, 3, 9, 17], [-6, -1, -9, 2, -1, 0, 4, 0], [-1, 1, -19, 3, 1, 4, -6, -3], [-13, -19, 10, -9, 1, 3, 0, -4], [5, 14, -9, 0, -6, 0, 2, -3], [-40, 13, 9, -2, -6, -9, -4, -12], [2, -6, 6, 4, 7, 15, 9, 10], [-5, 4, -9, 6, 4, 9, 17, 1], [9, -11, -9, -7, -12, 8, -6, 4], [-37, 5, -2, -6, -6, -4, 1, 2], [-29, 7, -1, -7, -7, -4, 3, 8], [-16, 14, 6, -6, -2, -4, 12, -6], [17, -13, 2, 8, 6, 7, 0, -3], [10, -13, -8, -5, -13, -2, 7, -1], [7, 16, 1, 0, -4, -8, -13, 2], [19, 9, 0, 15, -1, -2, -2, -7], [-10, -7, 4, -3, 11, 2, 1, 0], [-27, 8, -8, -1, -1, -1, -7, 7], [13, -8, 12, 11, 13, 9, 18, 8], [3, -8, -1, 0, -7, -6, 0, 1], [-18, 3, -1, -2, -7, 4, -4, 10], [19, -9, 3, 14, 10, 0, 24, 9], [-7, -3, -1, -5, 1, 2, -4, 9], [-4, 0, 9, 9, 7, 7, 23, 6], [20, -2, -6, 2, 0, 6, 14, 8], [-4, 5, -1, -12, 1, -3, 0, -6], [-8, 1, -19, 0, 8, 11, -5, -1], [21, -15, 5, 9, 9, -2, 10, 11], [-17, 3, 1, 6, -2, -2, 10, -5], [-24, -3, -13, -10, 3, 6, -14, 2], [-5, 5, 6, 11, 5, -3, -3, 8], [10, -12, -14, -23, -9, 12, -13, -3], [-14, 3, 2, 5, -9, -7, -13, -9], [-10, -6, -6, 5, 7, -1, 19, 3], [0, 0, -16, -3, -4, -4, -15, -5], [0, -2, -7, 1, 12, 20, 5, 4], [8, -11, -1, -10, 7, 0, 4, -3], [-18, 15, 12, -3, -4, -1, 6, -14], [-4, -15, 5, -12, 3, 4, -2, 6], [10, -6, 2, -2, -8, -16, 0, -16], [8, 9, -7, 9, 12, 5, 10, -3], [14, 8, 2, -7, -6, -10, 3, 1], [46, -5, 2, 4, 10, 9, 2, -2], [28, -17, -16, -1, 11, 7, 11, 5], [-20, 13, -5, -8, -15, -12, 2, -6], [4, -18, -14, -3, 17, 15, 13, 4], [15, -4, 12, -14, 6, -1, -17, 0], [-4, 11, 15, 8, 2, -10, 7, 12], [-36, 7, -2, -3, -3, 10, -21, -3], [1, -14, 9, -11, -4, 11, -6, 4], [-4, -3, 15, 5, 5, -14, 11, 1], [-17, -3, 13, 1, 5, -13, -7, -7], [-18, 5, 7, 9, -3, -5, 2, -11], [0, 3, 17, 9, 6, -2, 7, 4], [18, 8, -5, -2, -12, -10, -23, -6], [9, -8, -4, -10, 3, 17, -14, -4], [-31, 9, -14, -2, -11, 5, 1, 2], [-10, 8, -4, -16, -4, 1, -4, -13], [2, 21, -13, 10, -2, -9, 5, -8], [-2, 5, 20, -1, -7, -6, 2, 10], [4, -9, -10, -2, -6, -13, -15, -5], [12, 3, 7, 8, 10, -1, -4, 6], [40, -10, -5, 0, 1, 6, 8, 14], [24, -14, 3, 2, 10, -1, 1, 10], [-2, -7, 13, -16, -8, -11, -8, 8], [-7, -12, -7, 1, -8, -6, -16, 2], [-2, 3, 15, -7, 7, -8, -11, -9], [-8, -3, 0, 19, -1, -4, 2, -1], [-30, 17, 2, -6, 9, 6, -13, -4], [3, 2, 16, 8, 9, -14, 1, -2], [-25, -16, 6, -4, -8, -7, -9, 4], [-5, 9, -5, 14, 0, 0, 18, 2], [-44, 9, 5, 3, 1, -8, -8, -6], [22, -19, -9, -3, 15, 17, 12, -1], [-4, -3, 10, -11, 7, 3, -28, -9], [-18, 10, -12, -1, -4, 2, 1, 7], [0, 4, 1, -6, -16, 0, -7, 3], [-15, 4, -27, 3, -7, -1, 3, -2], [-21, -5, 5, 17, 11, -6, -1, -3], [-8, 23, -7, 15, 0, -3, 15, -9], [41, 8, 10, 3, -2, -1, -8, 0], [-18, 26, 12, 15, -7, 0, 17, -2], [-8, -3, 6, -12, -20, -17, -10, -10], [-22, -12, 13, -1, -8, -1, -8, 0], [6, 5, -7, 19, 16, 11, -2, 9], [-13, 15, -6, -15, -15, -9, -24, -28], [-21, 26, -9, -18, -20, -12, -15, -16], [10, -13, -7, 9, 8, 12, 15, 14], [-8, 17, -5, 7, -3, -12, -6, -15], [0, 2, -10, 1, -6, -8, 11, 4], [1, 4, 10, 17, 10, -4, 15, 8], [-19, 14, 14, -1, -16, -10, -7, -22], [1, -2, 18, -14, -17, -10, -9, -11], [-7, 9, 6, -13, -8, -17, -3, 3], [18, -17, -3, -1, 9, -2, 17, 5], [23, -9, 12, 2, 4, 7, -3, 17], [-7, 20, 4, 14, 5, 9, 2, -1], [-15, 4, -1, 13, -4, -17, 7, -3], [30, -25, 6, 5, 8, 4, 13, -7], [-26, -8, -11, -7, 0, 4, -11, 3], [-28, -18, 3, -12, 6, 11, -1, 2], [16, -21, -14, -4, 14, 2, -2, 3], [-26, 8, 6, -7, -8, -14, -19, -11], [-16, 0, 2, -6, -15, -3, -7, -8], [44, -22, 2, -1, 10, 12, -3, 3], [2, -3, 25, -5, -9, -14, -8, -12], [-10, -7, -9, 13, 10, 17, 10, 24], [2, -11, 7, 6, 3, -6, 16, 10], [22, -11, 9, 6, 8, -12, 13, -6], [12, -13, 9, 7, 9, 14, 11, 17], [-25, -4, -1, 2, -6, -12, 4, -3], [-16, 21, 1, 7, -1, -14, -2, -24], [-14, -14, -5, -5, -9, 7, -9, -8], [2, 5, 7, -11, -13, -9, -8, -15], [-31, 8, -2, -1, -13, -1, 6, -8], [6, -2, 8, -6, 0, -10, -12, 4], [-1, -6, -3, -2, 4, 1, 5, 13], [-7, 7, 2, 8, 5, -10, 12, 5], [-23, 5, -15, 2, 1, 17, -5, 4], [-5, -15, 2, 5, 6, 9, -10, 11], [-24, 3, 9, -3, -15, -1, 2, 8], [-17, 20, -3, -6, -9, -18, -9, -2], [18, -8, -8, 6, 4, 15, 13, 5], [-8, 2, 24, 3, -9, -1, -3, 6], [-10, -9, -4, -6, -11, 16, -7, -8], [-3, 6, -5, -15, -9, 2, -19, -9], [6, -7, -17, -5, -7, -8, -9, -3], [-10, 2, 24, 1, -2, -8, -9, -3], [-2, 3, 4, -12, -10, -1, -17, 7], [-12, 0, 32, -8, 4, 2, -12, -6], [-18, 19, 3, -14, -8, -5, -21, -17], [46, -11, -9, -4, 2, 13, 1, 7], [-27, -2, 2, -10, -8, -12, -11, -14], [2, 7, 5, 9, 9, 8, -5, 5], [-17, -2, -17, 3, -13, 6, -1, 5], [5, -3, -4, 7, 2, -4, 10, 13], [-12, 12, -7, 3, 2, -12, -11, 1], [-14, 6, 5, -6, -4, -5, -5, -4], [-1, -16, 15, -10, 12, 3, 1, -5], [-12, -9, 8, -9, -7, 9, -6, -1], [-28, -8, -6, -6, 8, 13, -8, 16], [14, -22, 4, 2, 10, 12, 10, 2], [-7, 4, 1, -20, 3, 1, -17, -17], [-17, -7, 3, -9, 7, 10, 0, 11], [7, -6, -3, 11, 8, 20, 26, 16], [3, -17, 7, 3, 2, -5, 24, 30], [18, 11, -1, 27, 11, 1, 24, 17], [-8, 14, -3, -2, -8, -6, 21, -2], [-8, -15, 7, 0, -1, -5, -8, -13], [38, -24, -7, -6, 1, 2, 5, -1], [-24, 5, -1, -10, 5, 9, -4, -4], [-26, 14, -6, 14, 17, -9, -8, -7], [-11, -9, -4, -29, 3, 9, -13, -4], [-45, -8, 5, -6, 10, 11, 6, 14], [17, 2, 22, -14, -5, -9, -10, -5], [3, 14, -4, 7, 14, -2, 6, -2], [-9, -6, 9, 3, 0, -12, 12, 15], [-2, -15, -4, -5, 15, 8, -8, -6], [13, 14, -10, 5, -20, -10, -12, -8], [3, -5, 12, -7, -6, -12, 13, 8], [-23, 12, 5, -3, -14, -24, 6, 0], [-2, -28, 5, -10, -12, -20, -21, -3], [2, -23, 9, 11, 17, 17, 12, 12], [-14, 7, -16, 9, -7, 16, 17, 2], [-5, -5, 4, 3, -9, -8, -2, 5], [1, 20, 12, 11, 7, -12, 1, -13], [-6, -13, -3, 12, 16, 12, 21, 14], [-3, 11, 15, -6, -5, -7, 1, 3], [9, -24, -14, -6, 22, 14, 15, 5], [4, 20, 16, 13, 10, 4, 5, 11], [34, -23, 15, 7, 20, 15, -5, 11], [-10, -5, 11, 13, 3, 2, 15, 25], [21, -20, -12, 3, 22, 30, 21, 24], [-18, 4, -2, -20, -6, 0, -10, -8], [14, 2, 6, 4, 10, 2, -12, 0], [-2, 12, -4, 8, 10, 8, -3, 15], [-16, 8, -2, 22, -2, 3, 3, 9], [15, -5, -19, 4, 8, -2, 0, 17], [13, -11, 2, -15, -8, 2, 2, -11], [-19, -15, -8, -22, -13, 3, -14, -14], [25, -3, 5, -12, -3, 5, 3, -8], [13, 8, -12, 10, 13, 7, 18, 14], [-26, 7, 0, -8, -2, -7, -32, -21], [-2, 21, -7, 12, -1, -4, 4, 14], [-48, 20, 10, -14, -12, -6, -40, -24], [3, -8, -12, -16, 8, 4, -15, -10], [16, 10, -22, 0, -4, 4, -6, -2], [-12, 0, 10, 11, -4, -8, 3, -1], [5, 10, 6, -5, -2, -1, -8, 0], [-19, 26, -16, 4, 3, 23, 4, 1], [6, 18, -13, 6, 6, -6, 11, -3], [20, -6, 3, 10, 27, 13, 22, 15], [-3, 0, 4, 21, -3, -5, 12, 8], [6, 29, -5, 24, 11, 0, 12, -6], [11, -5, 4, 5, 15, 10, 16, -5], [20, 21, 2, 16, 9, 5, -3, -10], [-12, 8, -11, 4, 5, 6, 11, -3], [15, 11, -25, -2, 1, -13, 3, -5], [12, -14, -13, -18, -22, -11, -15, -13], [45, 3, 17, 0, -16, -14, 10, 2], [-11, 8, -7, 20, 7, -14, 10, 8], [31, 2, 29, 13, -9, -19, 2, -4], [-17, -1, 8, 8, 0, 10, -7, 6], [24, 8, -5, -10, -21, -8, -6, -15], [16, 19, 11, 23, -9, -13, 3, -7], [-13, -19, 8, 6, 16, -1, 25, 14], [-21, 3, 3, 6, 2, 9, 12, 11], [-18, -7, -12, -2, -2, 12, -7, 26], [22, -10, -12, -8, -5, -12, -16, 1], [-12, 14, -16, 2, -6, 12, -2, -2], [-32, 11, 15, -11, -20, -18, -23, -23], [0, 4, -2, -3, 4, -10, 23, 1], [-31, 3, 7, 9, 6, -9, 5, -4], [-4, 16, -27, 8, -9, -3, 4, -3], [-21, 10, -17, 1, 5, -3, -9, -9], [16, 13, 0, -4, -16, 1, 6, -6], [-35, 11, 0, 1, 6, 13, 5, 10], [10, -9, -2, 0, 8, 1, 1, 5], [11, -36, 5, 1, 7, -3, 18, 20], [-21, -12, -1, -9, 12, 15, 10, 12], [-29, 10, -6, -12, -26, 3, -27, -19], [-46, 10, -2, -8, -16, -18, -14, -13], [23, -4, 19, -5, -10, 0, 7, 6], [-29, 23, 16, 0, -2, -7, 7, 5], [37, 10, 6, -6, -3, 6, -3, -12], [-49, 11, 13, -14, -19, -18, -9, -8], [-9, -3, -19, -1, -18, -26, -2, -9]];
    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_bh();".
// 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_bh(:prec:=8)
    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_bh();".
// 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_bh( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6422_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, -6, 7, 28, 3, -22, -4, 4, 1]>,<5,R![-3, -12, -2, 36, 25, -20, -11, 2, 1]>,<7,R![-128, -240, 5, 220, 86, -38, -19, 2, 1]>],Snew);
    return Vf;
end function;