
// Make newform 6422.2.a.bl 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_bl();".
// 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_bl();".
// 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 := [-1, 4, 10, -28, -35, 35, 14, -12, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 14, -5, -50, 3, 21, -1, -2, 0], [3, -4, -23, 15, 32, -7, -11, 1, 1], [-4, -3, 28, 44, -13, -20, 2, 2, 0], [3, -9, -12, 38, 26, -17, -10, 2, 1], [5, -14, -21, 38, -14, -15, 10, 1, -1], [1, -14, 15, 72, -9, -31, 2, 3, 0], [-1, 8, -18, -44, 53, 18, -22, -1, 2]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 1, 1];
    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], [0, 0, 0, 0, 0, 0, 1, 0, 0], [1, -1, 0, 0, -1, 0, 0, -1, 0], [-2, 0, 0, -2, 0, 0, -1, -1, -1], [0, 0, -1, 2, 1, -1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 2, 0, 0, 0, 0, -1, 1, 1], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [-4, -2, 0, -3, 0, 2, -1, 0, -4], [-3, 1, 2, 1, 2, 0, -1, 3, 1], [-2, 2, 1, 2, 2, 1, -1, 3, -1], [-2, -1, 0, 0, 2, 0, -1, 0, 1], [-1, -1, -2, 0, -1, -1, 1, -2, 0], [-3, -2, 0, -2, 0, 0, -2, 1, -2], [-2, 2, 1, 2, 1, 0, -1, 1, 0], [3, 3, 1, 3, -1, 0, -1, 3, 2], [-1, 1, 1, -1, 0, -3, -2, 1, 2], [2, 0, 0, 1, 0, 0, 3, -1, 3], [-4, -1, 0, -2, -1, 3, 0, -3, -1], [-1, 0, -2, 1, -2, -1, 3, -4, -2], [-3, 1, 1, 0, -1, 0, -3, -1, 0], [1, 2, 0, 6, 1, -4, 2, 3, 4], [-5, -3, 2, -3, 2, 3, 0, -2, -3], [3, -4, 1, 0, -1, 2, 0, -3, 3], [-4, -1, 1, -1, -2, 1, 0, -1, -5], [7, 0, 0, 8, 1, -4, 4, 0, 1], [2, 2, -2, 4, -3, -2, 0, 3, 2], [0, 2, 2, 1, 0, 0, 3, 2, 1], [5, -1, 0, 0, -2, -5, 1, -4, 4], [4, -4, 5, 1, 0, 4, -2, 0, 3], [0, 2, -5, -4, 2, 1, 2, -2, -5], [-7, -4, -2, -2, 2, -3, 5, 0, -4], [-3, 0, -6, -6, -2, 1, -1, -5, 1], [7, -1, -2, 6, -1, -4, 5, 1, 2], [-2, -1, 1, -7, 2, 1, -1, -1, -10], [-4, 0, -1, -4, -2, -3, 2, -5, 4], [0, 0, 3, -4, 0, 1, -5, -1, 3], [-5, -1, 2, 0, -5, 1, -1, -1, -8], [10, 1, -3, 6, -4, 6, -2, -1, 10], [0, 2, 0, -9, -1, 0, -2, -4, -3], [-8, 0, 2, -1, 3, -3, 0, 1, -9], [-1, 1, -4, 0, -2, 3, -4, 3, -1], [3, 3, 0, 4, -6, -3, -1, 0, 3], [-4, 1, -1, -4, 3, -1, 2, 0, 2], [-8, 3, -2, -3, 1, -1, 1, 3, 2], [-1, 5, -1, -1, 6, -3, 4, 2, -1], [2, 2, -4, -1, -6, 1, 1, -1, 8], [-10, -8, 2, -4, 4, 1, 1, 2, -5], [-1, 7, 3, 3, 4, -2, -4, 2, -1], [-1, 0, 1, 7, 2, 7, 2, 4, -1], [3, 11, -1, -1, -3, -1, -5, -1, 7], [-1, -5, 0, -8, 0, 2, 1, -2, -6], [-6, 1, 1, -2, -1, 6, 0, 4, -3], [6, 0, 6, 4, 4, -6, 6, -3, 3], [-3, -2, 2, 1, 0, 1, 3, -6, -8], [-2, 4, 0, -4, -7, 4, -2, -4, 8], [3, -1, -3, 5, -3, 2, 1, 1, 7], [-7, -6, 1, 7, 4, -2, 2, 5, 1], [-4, -7, 4, -9, 2, 5, -2, -8, -13], [-8, -2, -6, -6, 4, 0, 6, -7, -5], [4, -1, -5, 6, -5, -2, 0, 5, 5], [-15, 3, -7, -1, -3, -2, 2, 7, -2], [-3, 1, 0, 9, 8, -6, 2, 5, 4], [-5, -4, 6, 4, 4, 1, 3, 0, -1], [-10, -6, -3, -11, -1, 4, 1, -6, -20], [-7, -1, -2, -17, 0, 1, 1, 2, -9], [-13, -2, 2, -10, -2, 2, 1, -1, -9], [-4, 0, 0, -3, 3, -3, -8, 4, 0], [0, -2, 7, 1, -3, 1, -8, -1, -1], [9, 6, 2, 5, 5, -6, 4, 4, 11], [-1, 5, 0, -4, -2, -2, -2, 2, 11], [-12, 6, -12, 0, -5, 0, 0, 4, 1], [4, 1, -3, 8, 4, -8, 11, 0, 7], [4, 0, 5, -6, 0, -4, 7, -4, 2], [9, 5, -3, 3, -6, 2, 3, -7, 2], [-10, 9, -5, -9, 1, -2, 9, 0, -8], [7, -7, 7, -2, -2, 4, -3, -7, -2], [5, 0, -4, -7, 0, -4, 2, -3, 1], [1, 3, 3, 1, 8, -5, 1, 7, -1], [-3, -8, -2, -3, -4, 4, 1, -3, -8], [-10, -3, 2, -6, 1, 4, -6, 6, 0], [-14, 6, -7, -7, -1, 4, -3, 2, 1], [-4, 6, -4, 2, 2, -5, 5, -3, 6], [18, 0, 7, 6, -1, 0, -4, 2, 10], [1, -6, -4, 10, -4, 3, 3, 0, 1], [-10, -5, 1, -10, 2, 2, -3, 3, -11], [-3, -6, 7, 10, 11, -1, 2, 2, 4], [-9, 5, -5, 3, 3, -1, 7, 9, 5], [1, 8, -6, -3, 0, -5, -1, 5, -1], [-4, -8, 4, -1, -1, 8, 2, -7, -8], [7, 7, -1, -3, 8, -8, 6, 5, 0], [6, 4, 5, 1, -2, 5, -2, -3, 9], [4, 2, -1, 13, -6, -3, 1, 12, 12], [-6, -5, 8, 5, -1, 3, 4, -7, -8], [-14, 0, 4, 2, 3, 2, 9, 3, -2], [-1, -1, 5, -3, 3, -6, -5, 4, -4], [3, -13, 1, 3, -5, 6, 1, -2, -3], [-3, 0, -6, -3, -1, 3, -10, -3, -3], [9, 2, 3, 5, -1, 4, -3, 0, 8], [5, -3, 2, 16, 2, -5, 3, -2, 11], [-4, -2, -12, -11, -3, 2, 1, -7, 0], [2, 4, -4, 5, 4, -2, 13, 4, 2], [-14, -5, -4, -10, 4, 12, 3, -7, -12], [4, 0, -1, -6, 12, -5, -1, -3, -2], [9, -1, -4, 10, 1, -2, -3, 0, 0], [-4, -3, 0, 3, -2, -1, 0, 9, -1], [-8, -12, 6, -4, -4, 11, -13, 2, -9], [5, 5, -1, 5, 4, -5, -5, 5, 6], [-2, 8, -9, 3, -4, -2, 2, 7, 12], [6, -8, 2, -8, 4, 3, 0, -1, -2], [10, 4, -6, -9, -4, -5, 2, -6, 5], [0, -11, -2, 11, -1, -1, 9, -2, 7], [14, 4, -5, 10, -7, 1, -3, -6, 2], [-6, 6, -14, -6, -3, -5, 4, 0, 2], [-5, -1, 6, 12, 12, 4, -2, 7, 3], [4, -7, 2, -2, -4, 10, -3, -3, 3], [1, 4, 2, -21, 5, -2, -5, -2, 3], [-11, -2, 1, 0, 10, 6, -6, 12, -3], [16, -2, 4, 9, -3, 0, -8, 5, 11], [-4, 5, -8, 0, -3, 3, -4, 2, 9], [-5, -7, -11, -14, -10, 7, 0, -9, 1], [-12, -4, 8, -21, 9, 2, 1, -13, -15], [3, -7, -3, -16, -2, 4, -4, 2, -11], [-14, 2, -6, -14, -13, 10, 0, -3, 0], [5, 10, -2, 7, 2, 4, -5, -2, 14], [7, -6, 10, 4, 4, -3, -4, 2, 2], [7, -9, 6, 9, 12, 3, 5, 6, -2], [-8, -8, 6, -9, 3, 4, 1, 3, 2], [-1, -5, 2, -19, 1, 6, -6, -17, -12], [-10, 8, -11, -3, -2, -8, 7, -1, -2], [-6, -9, 11, 2, 0, 3, 2, 1, -9], [-14, -4, -1, 1, -6, 7, 4, -5, -15], [0, 5, 5, -13, -5, 4, -4, 0, -6], [-6, 5, -2, 9, -5, 3, -7, 12, 9], [10, 6, -7, -2, -9, -3, 4, 3, 8], [12, -2, 8, 13, 4, -9, 2, -1, -6], [-17, 1, 6, 1, 12, 6, -9, 6, -4], [-15, -6, -2, 8, -9, 5, -5, -9, 3], [18, -1, 7, 21, 2, -3, -8, -3, -5], [1, 13, -5, -4, -2, -2, 9, 2, -1], [-12, -1, 10, -4, 2, 4, -5, 2, -9], [-3, 8, -5, 5, 3, -1, -6, -9, 2], [-7, 8, -1, 5, -5, 5, 1, 9, 12], [12, 14, 0, -9, 0, -8, -10, 2, 10], [11, -7, 0, -10, -8, 3, 3, -3, 1], [-7, 11, -5, -2, 8, 0, -1, 3, 10], [-5, -2, 0, 4, -20, 7, -5, -4, 12], [-12, -8, 10, -13, 11, 4, 4, -5, -16], [0, -8, 1, 3, -6, 9, -11, -2, 13], [-12, -1, 11, -5, 2, 6, -6, -10, -13], [-9, -14, 10, 2, 5, -5, -4, 1, -2], [8, 3, 3, -2, -4, -11, 2, 3, -2], [2, -2, -17, -11, -7, 9, -5, -8, -6], [-13, -10, 9, -9, 5, -3, 4, -1, -10], [11, -9, 14, 2, 5, 1, 6, 3, 2], [-5, -5, -6, -28, -6, -3, -5, -9, -17], [1, -4, -11, 6, -5, -4, 18, -9, -10], [6, 3, -6, 2, 3, -2, 11, -14, -7], [2, 10, -3, 5, -11, -14, 1, -6, 8], [-2, -14, 0, -14, 0, 5, -12, -6, -9], [5, 5, -3, -9, -10, -14, 3, -10, 9], [3, -16, -3, 2, -2, -4, 1, -14, 10], [-21, -12, -2, -17, 1, 4, 0, -12, -25], [-14, -5, -11, -16, -3, 0, 5, -5, -16], [9, 7, -9, 13, 2, -10, 2, 3, 25], [16, 7, -4, 9, -8, 1, -11, 6, 16], [5, -1, 2, 11, 14, -3, 8, 7, 2], [16, 15, 7, -10, 5, -9, -4, 3, 9], [1, -11, 12, 10, -7, 0, -4, -1, 0], [14, 10, 8, 10, 1, -7, -3, 11, 9], [-7, 2, 8, -9, -1, 0, -11, -8, -13], [-7, -12, 0, -14, 5, 2, 4, -11, -33], [-6, -5, -5, 6, -7, 10, -3, 1, 7], [-1, -7, -1, -15, 5, 3, 7, -4, -3], [14, -3, -1, 3, -2, -5, 12, -15, 9], [-8, -15, -8, 0, -1, -4, 16, -4, -7], [-4, -10, 1, -13, -3, -3, -9, -5, 7], [23, 5, -5, -1, -5, 9, -7, 1, -2], [18, -11, 2, 1, 5, -1, 2, -10, 9], [-3, -1, -6, -26, 0, -1, 2, -5, -7], [-9, -1, 7, -4, -8, 5, -21, 9, 1], [0, 3, -6, 6, -3, 3, -19, 7, 7], [11, 4, 1, 11, -5, 4, 0, -12, 6], [14, -23, 10, -9, -5, 18, -11, -4, -12], [23, -5, -4, 1, -4, -10, 6, -4, 6], [9, 18, -6, 13, -4, -6, 10, 3, 25], [1, 7, -10, -3, -2, -4, -14, 5, 13], [19, -6, 0, 19, 1, 2, 2, 3, -5], [15, 12, -7, 2, -11, -2, -11, 2, -12], [4, 1, -7, 2, -2, -3, -3, -6, 21], [20, -7, 11, 22, 4, -10, -5, 6, 16], [9, -5, 8, 6, -1, -7, 7, -5, -14], [-2, 1, 5, 12, -13, -2, 4, -6, 13], [-6, -8, 0, -10, 8, -3, 9, -5, -15], [21, 12, -4, 14, -2, -4, -3, -11, 19], [-9, -2, 4, 17, 13, 6, 2, 19, -12], [-21, 7, -3, 8, 0, -2, -1, 6, -18], [-20, 15, -2, 0, 0, 8, -9, 11, 10], [5, -9, 10, 13, 2, -1, 14, -2, -2], [-22, 7, 0, 13, 6, -6, -7, 4, 5], [-9, 2, -1, 0, 17, -5, 0, -5, 4], [-16, 3, -1, -19, -9, 6, -14, 9, -25], [-14, 2, 3, -10, 6, -10, -9, 3, -14], [14, -2, -8, 10, 0, -5, 4, 0, 9], [11, -1, -2, -19, 3, 5, 3, 1, -14], [21, -14, 7, 3, 6, 10, 5, 2, -9], [10, -14, -7, -3, -14, 8, -8, -9, -9], [-19, -13, 9, 0, 7, 0, 10, 10, -7], [19, -8, -10, -4, 1, -6, 2, -9, 2], [-3, -11, 2, -10, 7, 5, -18, 3, -11], [-14, -3, 4, -10, -7, 1, 9, 1, -16], [-9, 12, -10, -7, 4, -9, -1, 16, -13], [-8, -2, 15, -9, 4, 0, -10, 3, -3], [4, -2, 6, 1, 0, -1, -19, 14, 6], [-13, -7, 3, -3, -1, 3, -3, -7, -9], [-1, -14, 2, 4, -9, 4, -9, -21, -6], [-6, -11, 2, 22, -8, 5, -3, -7, 19], [-2, 6, -17, -4, -8, 4, 0, -12, 9], [0, -2, -6, -1, 0, 4, 5, -7, -26], [14, -8, 6, -6, 0, 0, 7, -10, 19], [16, 5, -7, -3, -5, 8, -5, -7, -1], [-3, 0, 13, 2, -8, 13, -10, 9, 3], [32, 8, -13, 22, -14, -1, 1, 7, 30], [6, 2, 9, 31, 4, -3, 23, 9, 26], [10, -5, -12, 13, -9, 3, -3, -12, 14], [29, 2, 2, 8, -3, -3, 4, -10, 22], [-4, 11, -4, -19, -14, -1, -11, 3, -20], [6, -16, 3, 10, -8, -3, -1, -13, 5], [-21, 11, 6, 3, 6, -5, 9, 1, 7], [21, 2, -6, 8, -5, 6, -2, -4, 8], [1, 16, -8, 1, 0, -8, -7, 10, 27], [-1, 6, -9, -10, -13, 4, 6, -8, 4], [24, -12, -1, 10, -6, -1, -5, 2, 14], [10, -4, -21, -3, -3, -8, -4, -2, 3], [-19, -12, 1, 16, 14, -6, -4, 0, 7], [-5, -15, 0, 0, 0, -2, -1, -2, -10], [-14, -5, -12, -7, -2, 6, 10, -16, -32], [12, 20, 6, 4, 7, -4, 8, 12, 0], [1, -16, 8, 5, -1, 11, 0, -5, 3], [-16, -8, 10, -30, 9, 4, 6, -7, -20], [31, 13, -7, 6, -12, -1, 2, 2, 18], [3, 6, 9, -5, 4, 2, -20, 7, 10], [-25, -8, 13, -12, 0, 20, 7, 5, -26], [-5, 11, -5, -13, 1, -17, 6, -1, 13], [9, 3, -9, -4, -4, 2, 16, -11, -9], [6, 1, 15, 47, 7, -1, -3, 11, 19], [1, -6, -4, -19, -4, 12, -12, -3, -18], [8, 5, -7, 30, 2, -5, 12, -9, 8], [-10, -2, -11, -9, -1, -11, 20, -5, -1], [-12, -10, -10, 3, -16, 19, -5, 2, -3], [5, 13, 3, -1, 4, 6, -20, 6, 6], [15, -8, 7, 20, 0, 0, 1, 14, 17], [-24, -2, 9, -25, 9, 1, 6, -6, -2], [4, 19, -9, -8, -1, -7, -4, 11, -7], [7, 21, 5, -4, 9, -3, -2, 4, -10], [22, -14, 6, 15, 2, 2, -2, -9, 22], [41, 11, -19, 16, -17, -11, 3, -1, 18], [-20, 9, -9, -14, 7, -4, 11, -21, -2], [1, 10, -3, -7, 1, -15, 9, 2, 4], [-9, 20, -17, -1, -5, -12, -8, 2, 25], [-5, -10, 19, 4, -5, 2, -11, -4, -9], [11, 28, -16, 5, -13, -15, 3, 0, 19], [-9, -9, 2, 6, 4, 16, 4, -1, 6], [18, -3, -7, 8, -5, -10, 4, -17, -2], [22, -15, 20, 10, 11, 9, 3, 5, -4], [-10, 11, 12, 33, 3, 2, -15, 27, 8], [-26, -3, 8, 11, 17, 6, -1, 1, 8], [4, 3, -12, -12, -6, 11, -7, -2, -6], [9, 8, -10, 31, -7, -3, 2, 12, 30], [18, -25, 17, 28, 6, -3, 3, 10, -9], [-6, -13, 8, -13, 18, -1, 3, -9, -32], [-11, -12, 13, -21, 20, 3, 7, -19, -33], [-8, 12, -10, -28, -3, 0, -2, 7, -7], [-42, 9, -12, -20, -8, -4, 9, -4, -31], [10, -5, -9, -18, -24, 3, -5, -37, 6], [14, 2, 6, 17, 0, -10, 18, 2, 13], [-14, -10, 4, -19, -6, 11, -26, -1, -14], [-12, -4, -3, 5, 6, 9, -9, -6, -7], [-29, 3, 4, -24, 7, 2, -13, -12, -32], [-14, 3, -9, -22, -5, -6, 30, -15, -14], [1, 16, -18, 5, -10, -2, -16, 3, -6], [4, 0, -3, 9, 4, -8, 13, 20, 9], [-13, -2, -9, -22, -3, -7, -4, 9, 12], [-8, -3, 12, -9, -8, -5, 16, -10, -24], [11, 16, -6, -2, 2, -9, 7, -10, 7], [-5, -3, -13, -11, -12, 5, -2, -26, -12], [-25, 3, 9, 7, 22, -1, -9, 13, -1], [9, 6, 24, 30, 7, -11, 4, 24, 3], [-10, -24, 4, 15, 11, 4, 9, 6, -12], [-9, -10, 10, -3, 10, 14, -18, -4, 1], [11, 6, -4, 22, -8, 11, -15, 12, 7], [-9, 15, 14, 24, 1, 2, -8, 11, 1], [-11, -21, 21, 5, 0, 1, 9, 12, -15], [9, 4, 7, 8, 11, 12, -6, 5, 3], [-17, -8, -19, -22, -3, -4, 9, -16, -18], [-11, -2, -9, -30, -3, 13, 0, 4, -1], [-12, -5, -5, -32, -5, 19, -8, -12, -1], [2, 1, -27, -14, 2, -10, 4, -5, 5], [-25, 15, -6, -3, -3, 9, -9, 18, 2], [-36, 3, -1, -4, 1, 8, 3, -9, 1], [-23, 9, -5, 12, 0, -8, 13, 11, 10], [-14, -10, -5, 2, 2, -12, 16, 5, -7], [-31, 19, -14, -3, 3, -6, 10, 13, 3], [-6, -22, -5, -9, -5, 3, 6, -6, 15], [-25, 10, 8, -15, -5, -6, 4, -11, -22], [-9, 1, 20, 8, 13, 15, -12, 12, -1], [28, -6, 3, 4, -7, -3, -17, -8, 8], [-1, 8, -2, -6, 0, 11, -15, 8, 7], [-9, -6, 12, -7, 14, 5, 8, 1, -20], [2, -21, -13, -19, -9, 5, 3, -18, -9], [9, 12, 16, 16, 14, -13, 23, 11, 19], [-6, -4, 16, 0, 9, -5, 25, 2, 3], [-15, -21, 12, -13, 15, -5, 4, -14, -11], [-18, 5, -5, -12, 4, 13, -19, -11, 7], [-16, 2, -14, 13, 6, -4, 3, 8, 2], [14, -4, 18, -2, -9, 5, -15, 7, 4], [23, 1, 6, 26, -7, 11, -20, 15, -6], [39, 5, -9, 18, -9, -8, 5, 14, 20], [-5, 6, 17, 15, 3, 3, -7, 8, -10], [7, 18, 18, -5, 5, -6, -4, 5, -10], [-4, 0, 18, 24, -3, 13, -7, 7, 8], [10, -6, 11, -12, -7, 13, 6, -3, 1], [16, 15, 0, 30, -11, 4, -9, 17, 33], [-44, 0, 1, 0, 0, -4, 7, 8, -20], [18, 18, -5, 4, -11, -5, -7, -1, 18], [-13, -8, -9, 6, 3, 5, 9, -3, 0], [-12, 10, 8, 5, -3, -1, 13, 2, -8], [-18, -24, 7, -9, 18, 8, 7, -8, -13], [15, 16, -2, 33, -3, -13, 5, 20, 15], [4, -19, -4, -17, -4, 17, -16, -4, -24], [-3, 17, 4, 9, -7, 11, -1, 0, 16], [5, -4, 2, -31, -13, 0, 9, -17, -24], [-22, 15, -4, -16, -3, 6, -13, 9, 19], [-22, 8, -1, -20, -20, 8, 3, -15, -1], [-30, -7, -16, -15, -3, 7, 5, -9, -34], [2, 16, -8, 7, -6, -5, -6, -6, -2], [-8, 22, 3, 28, 1, -3, 9, 5, 11], [-29, -12, 8, 0, 5, 0, -10, 3, -18], [21, 19, 5, -5, 7, -19, -14, 11, 11], [-8, 11, -4, -3, 16, -18, 18, 6, 15], [-5, 4, -6, -20, 3, -11, 14, 0, -21], [12, -1, 13, 18, 19, -2, 7, 20, 2], [21, 6, 1, 24, 8, -17, 13, 5, 1], [-16, -22, 17, 15, 9, 2, 6, 5, -10], [2, -3, -2, 8, 4, -8, 0, -3, -12], [9, 19, -17, 0, -11, 0, 16, -6, 16], [5, -7, 16, 10, 7, -6, 9, 26, -9], [-16, -4, 2, -10, 5, -2, 2, -6, 4], [28, -1, -2, 9, -9, -17, -9, -8, 27], [-6, 6, 5, -7, 15, 3, -8, 13, -6], [-47, 0, 12, -19, -7, 14, -23, -1, -3], [-1, -19, 9, 13, -10, 8, 1, -4, -9], [0, 0, 12, 12, -10, 16, -15, 14, -14], [9, -15, -3, 2, -22, 15, 6, 3, 2], [26, -1, 7, -23, 2, -8, -5, -12, 5], [9, 16, 17, -25, 8, -7, -9, -7, -7], [-8, -18, 3, -11, 1, 15, 9, -13, -24], [-49, -7, 5, -13, 1, 11, 6, 12, -22], [-1, 5, -3, -11, 0, -9, 13, 4, -19], [-3, 16, -11, 11, -16, 8, 14, 14, 22], [-28, 1, -6, 17, 1, 3, -4, 13, -1], [2, -20, 14, -1, 24, 0, 8, -4, 7], [-7, -22, 5, -4, -8, 15, -14, -7, -24], [-10, 1, -15, 8, 8, 4, 20, 10, 15], [-15, -10, 11, -23, 22, -7, -1, 6, -25], [30, 2, 10, 33, 2, -12, 19, 12, 21], [10, 24, -1, 5, 3, 4, -21, 4, 18], [-9, -7, 6, -23, 15, 5, -21, 2, -6], [11, -17, 3, -6, -8, 29, -10, -1, -4], [10, 7, -16, 15, -5, -12, -14, -6, 27], [28, 17, 14, 22, 15, -17, 11, 7, -3], [34, -5, -13, 21, -10, -5, 0, -5, -13], [0, 5, 0, 13, -7, 3, 8, 8, -8], [9, 9, -3, 21, 1, -4, 9, 4, 19], [21, 3, 1, 24, 0, -7, 11, 15, 45], [34, -26, 4, 4, -10, 13, -6, -5, 3], [-12, -13, 6, 4, -1, -1, 4, 12, 21], [-7, -25, 8, 22, 14, -1, 13, -1, -1], [-10, 11, -11, 14, 2, 16, 9, 0, 16], [1, -3, 12, 3, 6, -7, 12, 13, -21], [7, -7, -4, -5, -14, 7, 11, 8, 6], [-33, 6, -26, -13, -14, 18, -3, 0, 2], [-9, 4, 20, 4, -1, 4, -20, 10, -4], [-8, -7, 0, 18, 3, -7, 1, 1, 34], [25, 2, -12, 16, -11, -8, 2, 6, 25], [-39, -3, 20, -30, 15, 7, 3, 3, 0], [8, -6, 7, 22, -3, -3, 3, -9, 30], [-4, -10, -9, -18, -4, -12, 15, -29, -4], [-26, -2, -12, -2, 4, -5, 16, 9, -3], [15, -2, -8, -24, -3, 11, -11, -26, 19], [27, -29, -1, -19, -13, 6, 7, -17, -20], [-14, 20, 0, -7, 0, -14, -12, -5, -10], [10, 10, 9, 12, -3, -8, -10, -6, -17], [-5, -4, -17, 8, 9, 5, -1, 2, 29], [19, -27, 12, 16, 6, 0, 15, -8, 9], [-26, 8, 0, -13, -14, 13, -19, 1, -3], [8, 20, 0, 10, 9, -24, -1, 27, -6], [21, 5, -4, 1, 1, -19, -4, 11, -15], [10, -23, 9, -10, -17, 9, -10, 2, -34], [6, 26, -10, -10, -10, -7, -4, -6, 5], [-25, 14, 1, 1, 12, 11, -13, 25, 11], [-28, 0, -11, -14, 2, -1, -10, -19, -20], [-33, -15, 13, 7, 5, 14, 4, -17, -7], [16, 8, 4, 7, 6, -2, -32, 3, -8], [-12, 19, -16, -5, -11, -9, 12, 1, 16], [-21, -25, 6, -27, 14, 12, 14, 0, -18], [-46, -15, 7, -15, 3, 18, -2, 16, -27], [2, 20, 16, 9, 8, -2, 6, 4, -8], [-5, 21, 8, -4, 14, 8, -12, 15, 4], [8, 16, -3, 9, 1, -3, -10, 21, 4], [3, 3, -26, 3, -14, -19, 15, -6, -13], [52, -5, 2, 19, 2, 3, -2, 2, 43], [27, -1, 12, 7, -1, 6, -9, -3, 15], [-2, 2, -9, -27, -8, 5, -6, -13, -4], [20, 2, 0, -19, -20, 12, 0, 2, 5], [36, 11, -7, -6, -16, 10, -3, -27, 0], [0, 1, -1, 2, -2, -10, 10, 21, 4], [30, 19, -10, -18, -10, -5, -2, 4, -1], [-19, 13, -15, 15, -1, 6, 15, 0, 8], [11, 4, 6, -3, -10, 3, -15, 9, 24], [34, -20, -8, -5, -15, -8, 0, -20, 7], [34, -1, 5, 6, -8, -15, -14, 4, 13], [10, -2, 15, -16, 4, 2, -22, -19, -19], [51, -7, 1, 3, 1, -12, -16, -4, 21], [-34, -1, -5, -17, -4, 11, -8, -1, 25], [13, -5, 0, -2, -6, -1, -4, -17, 12], [-12, 0, -15, -29, -22, -1, 7, -21, -21], [-37, 27, 6, 10, 11, -1, 0, 10, 20], [-37, -6, 7, -14, -14, 21, -3, 5, -27], [5, -8, 1, 10, -2, 11, 8, -25, -28]];
    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_bl();".
// 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_bl(:prec:=9)
    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_bl();".
// 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_bl( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6422_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, 39, 10, -77, -14, 49, 7, -12, -1, 1]>,<5,R![-169, 130, 402, -371, -168, 182, 21, -26, -1, 1]>,<7,R![-533, -1375, 1291, 1631, -357, -574, -70, 44, 13, 1]>],Snew);
    return Vf;
end function;