
// Make newform 6422.2.a.bi in Magma, downloaded from the LMFDB on 12 May 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_bi();".
// 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_bi();".
// 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, -48, -84, -11, 54, 14, -12, -2, 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], [-2, -7, -41, -42, 31, 23, -9, -3, 1], [3, -2, -87, -87, 63, 46, -18, -6, 2], [3, -16, -118, -81, 86, 42, -22, -5, 2], [5, -26, -171, -119, 120, 64, -31, -8, 3], [-5, 27, 212, 162, -151, -87, 40, 11, -4], [-5, 31, 298, 249, -214, -133, 58, 17, -6], [13, -78, -527, -388, 366, 215, -97, -28, 10]];
    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], [-1, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0], [2, -1, 0, -1, 0, 0, 0, 0, 0], [2, 0, 0, -2, 1, -2, 1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 2, -1, 2, -2, 0, 2, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -2, 0, 0, -1, 0, -1, -2], [2, 0, -2, -1, 1, -2, 2, -1, 1], [-1, -2, 0, 2, 1, -1, -1, 3, 1], [2, 1, 0, 1, -1, 1, 0, 0, -1], [-2, 3, 1, 0, -2, 0, 0, 0, -1], [-2, -1, -2, 2, -2, 0, 0, 1, 0], [2, 0, 1, -3, 2, -2, 3, -1, 1], [2, 1, 3, -3, 3, -2, 1, 0, 0], [-2, -3, 0, 1, 0, 1, -1, 3, 0], [4, -1, 2, -1, 2, 3, -2, 0, 0], [7, -2, 1, 1, 0, 0, 0, 2, 1], [8, 4, -5, -4, -1, -1, 2, -3, -3], [4, 3, -1, -5, -1, -3, -1, 0, 0], [-3, -1, 0, -2, 0, 3, 0, 1, -3], [0, -3, 1, 7, 2, 0, 0, 1, 2], [-2, -2, -1, 2, 0, 0, -1, 4, 1], [-1, 1, -2, 0, 2, 1, 3, -1, 0], [-4, 0, 4, 1, 4, 1, 2, 2, 5], [-2, -3, 0, 0, 0, -3, -2, 3, 3], [1, 6, -6, -1, 0, 5, 4, -5, -2], [-1, -1, -1, 4, -2, 3, 0, -2, -4], [-5, 5, -3, 0, 0, 4, 3, -5, -2], [1, 3, -4, 0, -5, 3, -1, -3, -3], [2, 0, 1, -6, 1, -8, -2, 3, 1], [3, -1, -3, 5, 1, 2, -2, -4, -3], [3, -2, -4, 4, -1, 0, 0, -3, -6], [2, 3, -2, -5, 6, -3, 5, -4, -2], [0, -2, 0, 1, -1, 1, 1, 2, -1], [-1, -8, -2, 4, 1, 7, -1, 1, -1], [3, 6, 1, -10, -2, -2, 6, -5, -3], [-3, 5, 3, -1, -3, 0, -1, 1, 1], [1, -2, 0, -7, 2, -8, 0, 2, -1], [-8, 5, 1, -3, 1, 4, 4, -3, 0], [1, 4, -1, -9, 0, -1, 0, -2, -4], [-11, 3, -1, 9, -2, 5, 2, 2, 0], [3, 0, -4, 5, 2, 5, 1, -2, 1], [4, -4, -1, 8, 6, 3, 4, 1, 5], [-3, 2, -4, -4, -2, 1, 3, -2, 0], [-4, 5, -3, 2, -3, 5, 0, 0, -5], [11, 4, -1, 1, -5, -2, -2, -1, 1], [-1, -6, 3, 2, -1, 3, -1, 6, 1], [-2, -1, 4, 2, -1, 6, 1, -5, 1], [8, 1, 8, 2, 0, -4, -7, 6, 3], [3, 1, -2, 4, 1, -7, 0, 0, -5], [3, -4, -1, -2, 1, -3, -1, 0, -6], [-3, -3, -2, -1, 0, -12, -2, 5, -2], [2, 8, -5, 4, -3, 5, -4, -5, -5], [5, -4, -4, 0, 0, -5, -2, 6, -1], [2, 3, 1, 2, 1, 6, 5, -4, 2], [11, -4, -5, -6, 0, 0, -1, 2, -6], [5, 1, 3, 1, -2, -2, 2, -3, -4], [-7, 5, 0, 1, -8, 2, -4, -3, -2], [8, -5, 1, 0, -1, -5, -1, -1, 2], [-3, 5, -3, -9, 1, -6, 1, -7, 2], [-3, 2, -2, -2, 1, 8, 10, 0, 3], [-4, -4, -1, 1, 1, -4, 6, 7, 1], [5, -9, 5, -3, 8, -1, 1, 4, 3], [-1, 5, -11, -1, -4, 4, 8, -8, -2], [11, 1, -2, 3, -6, -6, -8, 0, 0], [11, 2, -5, -12, 1, -8, 6, -2, 0], [-1, 2, 0, 4, 1, 11, 1, 0, 0], [4, 4, 2, -9, 4, -6, 2, -2, 5], [-4, 1, 2, -7, 3, -1, 4, 5, 3], [9, -8, 6, -3, 0, -9, -4, 6, 3], [-5, -10, 11, 2, 7, 8, 1, 7, 3], [6, -1, 5, -3, 4, -17, -1, 2, 0], [9, -7, 7, -7, 5, -3, -3, 2, -3], [-2, -8, 13, 1, 5, 4, -3, 7, 0], [9, -3, 1, -3, -7, 0, -3, -8, -5], [12, 5, 2, -5, 2, -1, 0, -1, -6], [5, 15, -8, 0, -4, -3, 7, -10, 0], [1, 3, -2, 0, 4, 3, -2, -6, 2], [-4, 7, 2, 4, -3, 6, 0, 1, 6], [-9, 3, -7, 12, -6, -1, 3, -2, -1], [10, -5, 7, -2, 1, 2, -10, 0, -1], [0, -3, -1, -3, -4, 6, -5, 1, -9], [2, 10, -5, -5, -3, -2, 0, -3, -3], [2, -9, 9, -2, 9, -4, 3, 8, 1], [-17, -5, 6, 9, 3, 8, 3, -7, 4], [6, 1, -12, -4, -2, -4, 1, -6, -1], [7, 3, 3, -5, -5, -13, -8, 0, -1], [14, -3, 2, 0, 2, -2, 4, 3, 3], [-2, 3, -1, 5, -3, -3, 5, -3, -1], [-6, 4, 0, 10, 3, 4, -1, 0, 5], [8, -13, 2, -3, 7, -2, -7, 8, -1], [1, 2, -7, 2, -6, -3, -2, -2, -7], [17, 9, -4, 0, -2, -3, 0, -2, 1], [1, -2, -3, 7, 3, 0, -1, -1, 6], [7, 0, -3, -2, -7, 2, 1, -5, -6], [1, 1, -1, 1, -3, 0, -2, -3, -2], [-4, -5, 1, 9, -4, 9, -9, -2, 1], [9, -3, 1, 7, 0, 9, -4, -4, -8], [7, 3, -4, 6, -6, 2, -6, 0, -7], [11, 6, -4, 0, -2, 9, 6, -9, -2], [1, -4, 2, 0, 3, -7, -4, -5, 7], [-9, 10, -14, 9, -1, 1, 9, -9, -7], [4, -11, 4, -2, 3, -8, 8, 8, 2], [-2, 0, 5, 11, 4, 6, -6, 6, 4], [-3, -8, -1, 3, 1, -6, -2, 3, 3], [5, 7, 6, -11, 2, -11, 9, -6, -2], [2, 8, -13, -3, -10, -4, 3, -10, -4], [-9, 15, -7, 8, -3, 14, 4, -7, -3], [-2, -2, -11, 1, 5, 5, 10, 0, -1], [-1, -5, 7, 3, 2, 3, 0, 2, 13], [5, 13, -4, 9, -5, 0, -5, -4, -4], [-5, -12, 3, 18, -3, 5, -10, 14, 2], [-15, 4, 2, -5, -1, -3, 2, 2, -3], [-5, 8, -6, 5, -7, 7, -8, -9, 1], [17, 2, 0, -11, 2, -14, 0, -1, -1], [8, 3, 5, -10, -2, -11, 1, 5, -2], [22, 5, -2, -21, -2, -12, 6, -5, -3], [-4, 7, -3, 3, 0, -6, 2, -5, 3], [-8, -9, 0, 6, -4, 4, -1, 11, 2], [-5, -7, 0, 5, 2, -2, 10, 4, -1], [-5, 10, 3, -4, 5, 2, 5, -8, -5], [-2, -1, 0, 10, -2, -1, -12, -1, 1], [10, 2, -10, -6, 0, -15, 4, -9, -6], [-3, 2, -1, 10, 13, 2, 6, -6, 3], [-15, 17, -9, 14, -12, 12, 2, -6, 0], [-5, -5, 12, 4, 6, 8, 0, 8, 3], [10, -2, -6, 8, -1, -3, -8, 2, 4], [-5, -6, 1, 1, 0, 12, 1, 9, -6], [1, 3, -4, 4, -4, -18, -1, -1, 0], [-6, 0, 0, 12, -3, 6, -7, 4, 1], [3, -6, 1, -13, 7, -6, 9, 6, -2], [15, -7, 11, 1, 2, -7, -4, 8, 5], [25, -14, 0, -7, 5, -6, 5, 4, 2], [11, -15, 13, -8, 9, -9, -6, 11, 9], [-3, 7, -4, 16, -1, 4, -6, -4, 1], [4, 1, -1, -9, -7, 1, -6, -4, -2], [5, 1, -4, -16, -4, -5, 5, -1, -8], [-10, 12, -13, 2, -7, -2, 3, -7, -1], [-18, 7, -2, -5, 7, 7, 10, -4, 1], [-18, 6, 6, -7, -5, 3, -1, 0, 4], [7, 6, 2, -5, 3, 9, -1, -3, 4], [2, 10, -10, -8, -10, 9, 4, -8, -4], [-20, 6, -9, 11, -8, 9, -4, -3, -4], [5, 5, 3, -19, -1, -4, 3, -6, -5], [2, 8, -13, -6, -5, 8, 8, -5, -3], [-8, 5, -8, 2, -2, 0, 6, -5, -7], [-16, 3, 0, -2, -5, 6, 3, 10, 3], [-2, 2, 1, 10, 1, 9, 5, -8, -1], [13, -7, 3, 4, 1, -13, -12, 3, 0], [15, 9, -12, 3, -10, -4, -5, -15, 0], [-13, -13, -2, 7, -5, -1, 1, 0, 2], [2, 10, -4, -11, -7, -3, -3, 0, 2], [-2, -10, -1, 4, -2, -10, -2, -2, -7], [-10, -12, 11, -5, 12, -9, -6, 13, 10], [-2, -2, 3, 3, 2, 2, -3, 10, 9], [5, -19, 10, -3, 13, 10, 0, 6, 6], [3, 7, -13, 13, -6, 5, 5, -12, -1], [18, 0, -5, -13, -1, -8, 6, 0, -1], [-18, 0, 5, 0, -5, 1, -1, 7, 3], [7, -9, 7, 5, 10, 3, 9, 0, 2], [8, 0, 2, -3, 4, -4, 4, 4, 13], [-11, -3, 11, 1, 3, -1, 3, 0, 12], [-20, -9, 9, 14, 5, 1, -7, 7, 5], [26, -1, -15, -8, 0, -8, 0, -6, -6], [11, 2, 5, 10, -4, -12, -16, 0, -8], [0, -3, 0, 8, -8, 11, -3, 7, -3], [4, 10, 4, -6, 3, -24, 6, -4, 2], [1, 7, 5, -9, -3, -1, -2, -12, 8], [-7, -16, 4, 11, 6, 8, -4, 3, 7], [11, 1, -10, -9, -2, 11, -4, -1, -3], [9, -1, -15, 0, -7, -9, 5, -6, -10], [21, -2, 2, -10, 10, -2, 11, -3, 9], [-29, -1, 6, -6, 0, -7, 1, 13, 0], [22, -3, 10, -11, -3, 6, -16, -3, -8], [24, 6, -14, -12, -7, -11, -1, -11, -14], [2, -21, 12, -9, 10, -10, 1, 16, 5], [2, 5, 1, -7, 14, -13, 10, -5, -1], [-3, 2, -7, -7, 6, -17, 12, -2, 8], [1, 1, -2, 26, -2, 8, -5, -4, -2], [12, 6, -1, 7, 0, 12, -2, -1, -1], [-3, 9, 7, 7, -12, 11, -3, -1, -9], [15, -3, 8, -26, 1, 1, -8, 6, 0], [8, 5, -6, -20, 1, -6, 0, 1, 1], [-6, -9, -6, 5, 2, 7, -2, -3, -2], [-3, 4, 22, 1, 7, 6, -10, 12, 13], [-5, 9, -6, 7, -14, 14, -14, -7, -7], [-6, -9, 1, -5, -2, -4, -1, 0, 3], [-3, -6, 0, -4, 0, -18, 5, 9, 2], [11, 2, -3, -25, 1, -20, 1, -15, -7], [-1, 0, -9, 14, -2, 2, -8, -7, -3], [23, 18, -17, -24, -9, -13, -1, -22, -10], [-8, -13, 11, -19, 17, -12, 10, 16, 1], [-21, -9, -8, -7, 0, 16, 2, 4, -2], [37, 1, -2, 14, -10, -2, -8, 1, -1], [8, -4, 5, -3, 11, 4, -3, -5, -5], [13, 2, 2, 10, -1, -3, -4, 11, -3], [8, 3, -7, -12, -6, -10, 14, -3, -6], [-16, 5, -14, 19, -15, 16, 4, -4, 3], [-22, 10, -8, -5, -3, -12, 7, -6, -2], [18, 27, -11, -10, -11, -16, 9, -17, -6], [-1, 2, 9, -7, 4, -3, -7, 15, 4], [-2, -8, 21, 5, 9, -20, -4, 12, 11], [3, -16, 11, -26, 6, -7, 6, 9, -1], [-7, -3, 1, -4, 2, 8, 5, -10, -12], [-16, -6, 7, -12, 8, -12, -1, 16, 6], [4, -18, 5, 4, -3, 15, -11, 4, -12], [-11, -7, -4, 28, -6, 19, -4, -5, 1], [-6, -6, 9, -11, 2, 14, -8, 8, 7], [-8, 12, 9, -2, -2, 2, 8, -9, 9], [2, -18, 3, 15, 3, 7, 8, 1, 3], [-16, 8, -3, 4, -12, 6, 1, -12, 8], [9, 4, 1, 6, -11, -5, -12, -9, 5], [-30, 13, -8, -6, 4, -2, 27, -18, 3], [-14, 9, -14, 22, 4, 5, 8, -10, -1], [-5, 0, -3, 23, 0, 35, 8, -5, 4], [-7, 7, -9, -24, -2, -5, 9, -13, 0], [-20, -13, 16, 8, 12, 1, 10, 14, 12], [5, 8, -2, -29, -6, -22, -2, -5, 4], [-27, -13, 9, 26, 13, 16, 3, 16, 14], [-1, -14, 12, 6, 9, 7, 1, 10, 3], [21, -8, 10, -7, 15, -1, 1, 3, 7], [-7, -10, -10, 19, -5, 6, -13, 1, -15], [17, -1, -11, -4, 4, -1, 8, -10, 3], [-12, -2, -5, 6, 3, 25, 6, 2, -2], [-5, 2, 3, 16, -8, 12, -6, 9, 16], [5, 6, -14, 6, -1, -7, 1, -13, 1], [-20, -4, -7, 23, -6, 16, 8, -5, 6], [-22, 4, 4, 10, 4, 21, 6, -4, 7], [-12, 12, -6, -7, -14, -1, 0, -9, -10], [13, -23, 3, 7, 16, 2, -5, 1, 11], [0, 9, 8, -13, -1, 3, 3, 8, -3], [-20, -19, 7, -6, 6, -11, -9, 17, 7], [-12, 7, 1, -2, 8, 3, 7, 11, 4], [-9, -5, 10, 28, 7, 3, -4, 8, -3], [-5, -15, -6, 12, -7, 16, -4, 1, -3], [9, -2, -14, 6, -2, -13, -2, -2, -3], [3, -1, 7, -1, -13, 2, -18, -1, -6], [-22, -19, 18, 10, 4, 5, -6, 13, 23], [-1, 10, -20, 3, 8, 4, 21, -20, -6], [-11, -6, -7, 21, -8, 22, -5, -4, -6], [-14, -2, 3, -1, -2, -1, 5, -13, 4], [1, -5, -6, 8, -6, 20, -3, -13, -1], [0, 19, -15, 4, -15, -6, 13, -15, -9], [-17, -19, 18, 20, 0, -2, -21, 20, -3], [-18, -13, 4, 26, 10, 17, -6, 9, 4], [-21, 7, 0, 13, 0, 8, -7, -9, 4], [23, 14, -8, 4, 3, 17, -3, -16, -2], [11, -4, 7, 11, -5, 13, -8, 9, -5], [15, -4, 5, -5, 7, -9, -5, 4, 7], [-25, 21, -11, 12, -6, 11, 15, -4, 4], [-22, 9, 4, -21, 9, 7, 11, 7, 4], [-20, 9, -12, 16, 0, -17, 11, -1, 1], [-4, 6, -12, -5, -6, -1, -3, 2, 7], [-1, 1, -6, 8, -6, -1, 4, -9, -8], [-6, -19, 8, -9, 8, 3, -3, 0, 3], [7, 5, 9, -15, 13, -1, 9, -6, 16], [20, 9, -6, 23, -5, -25, -1, -2, -4], [17, 9, 3, -20, 2, -25, 5, -13, 7], [2, -9, -3, 6, 6, 2, 5, -16, 1], [-21, 10, -1, -13, -7, 6, 6, -3, 4], [21, 18, -10, -24, -13, -13, 10, -10, -5], [-12, 3, -7, 17, 3, 19, -3, -13, -4], [8, -16, 1, 2, 10, -4, 4, 5, -13], [-24, 4, 6, -6, 5, 13, 4, 5, 6], [-8, 0, 2, 14, -9, 21, -4, 11, -6], [-20, 3, 7, -4, 6, -5, 14, -2, 12], [16, -14, 11, -22, 3, -12, -2, 6, 8], [-29, 12, 0, 4, 0, -6, 19, -1, 14], [-5, 10, -1, -24, 2, -14, 8, -17, 0], [10, -24, 10, -16, 17, -8, 3, 17, 5], [6, 5, -10, -11, 0, -25, 4, -9, -9], [0, -9, 8, 3, -7, 23, -14, 10, -5], [0, -9, 6, -11, 10, -22, -1, 0, -2], [-30, 20, -14, 15, -2, 13, 14, -8, -6], [17, 1, 13, -18, 17, -29, 4, 11, -1], [14, -1, -6, -10, -3, 10, -1, -10, -1], [-2, 7, -12, 6, -5, 20, 14, -10, 2], [-18, 3, 11, 13, -1, -3, 9, -9, 10], [-18, -17, -10, 4, 2, 19, 4, 1, 8], [-10, -21, 13, -26, 12, -10, 1, 18, -6], [30, -10, 10, -13, 5, -2, 3, -4, 2], [-1, -5, -1, -4, -16, 16, -12, 1, -8], [-26, -26, -3, 21, 3, 26, -12, 17, 3], [-23, -12, 6, -7, 8, 1, 5, 12, -2], [1, -1, 8, 14, -4, -11, -9, -2, -2], [30, 10, -21, -27, -9, -5, 6, -10, -8], [-2, 0, -16, 12, -8, 16, 6, -23, -9], [23, 12, 6, -31, 6, -17, 2, 2, -4], [-2, -18, -1, 3, -3, 8, 4, 3, 9], [8, -2, 0, -11, -7, 16, -2, -4, -4], [-26, -4, 10, 16, -6, 33, -17, -3, 0], [12, -4, 13, -8, 10, -26, -9, 14, -3], [-20, 11, 5, -5, 12, -4, 3, -4, 7], [19, 11, -14, 14, -7, 2, -3, -19, -14], [13, 8, 10, -20, 2, -17, -17, 4, 3], [9, 7, -7, -10, -3, 12, 17, -5, -17], [-8, -7, 13, -2, 4, -11, 4, 10, 5], [-17, -5, 7, 14, 5, 5, -5, 8, 12], [-21, 11, -1, -19, -9, 6, -7, -10, -1], [-12, -12, 7, -2, -6, -5, -3, 20, 6], [-16, -18, 4, 7, 1, -1, -10, 4, 6], [2, 5, 8, 2, -13, 15, -7, -15, 2], [15, 1, -12, -2, -6, -10, -2, 7, -7], [15, 3, -25, -11, -2, 12, -8, -9, -19], [-19, -9, -2, 15, 7, -18, -9, 10, 6], [2, -8, -7, 15, -21, -8, -10, 1, -9], [10, 13, 1, 0, 11, 5, 10, -21, 3], [-4, 2, -4, -12, 6, 12, -1, -7, -3], [-7, -15, 5, 7, 5, -10, -12, 4, -7], [0, 12, -1, -15, -19, -1, 2, -13, -4], [21, -3, 15, -17, 12, 15, -12, 1, 13], [-24, 13, -6, -8, -4, -10, 13, -8, 2], [15, 22, -14, 0, -22, 11, -2, -15, -26], [-19, -5, 7, 22, 5, -3, 0, 22, 6], [4, 2, -21, 6, 2, 18, 2, -6, 1], [13, 28, -9, -33, 1, -23, 19, -27, 2], [10, 7, -4, 3, -9, -3, 1, -18, -6], [-5, -11, 0, -12, 0, 6, 8, 13, 3], [-7, 15, -23, 10, 7, 19, 10, -18, -10], [17, -18, 6, -6, -7, 8, -1, -2, -8], [-22, -4, 6, 9, 3, 0, 2, 3, -12], [14, 1, 1, 15, 2, -11, 5, 6, 0], [34, -2, 12, -1, 3, -22, -3, 7, 0], [1, -18, 6, 13, 7, -4, -1, 1, 10], [8, 18, -5, 6, -7, -12, 2, 1, -2], [8, 17, -21, -10, -14, 16, 3, -24, -22], [9, -10, 3, -3, -15, -10, -8, 21, 6], [-1, 11, -5, -8, 0, -2, -2, 9, 10], [-28, -5, 5, 30, -4, 23, 8, -10, 13], [20, -25, -16, 22, 0, 14, -4, -16, 1], [10, -6, -12, 1, -13, -10, -8, 15, 2], [-23, -16, 7, -14, 14, -19, 5, 21, 12], [16, -16, 1, -6, -4, 7, -6, 3, -5], [22, 0, -3, -20, -3, -13, -18, 6, 1], [36, -10, 12, 0, 7, -4, -23, 19, -2], [-11, 17, -4, 24, 2, 27, 0, -3, -5], [-11, -7, -4, 10, -10, 34, -11, -1, -11], [25, 1, -14, -11, -19, 6, -10, -11, -19], [34, -6, -3, -9, -7, 1, -12, -4, 2], [-6, -20, 31, -17, 3, -9, -19, 37, 4], [-14, 20, -12, -10, -27, -3, -5, -15, -7], [-41, -12, 11, 27, 10, 3, -7, 0, 11], [-14, -6, 6, -6, 9, 5, 6, -2, -7], [-24, 2, 9, 7, 11, 7, 13, -4, 4], [-4, -15, 2, 21, -1, -2, -12, 8, 4], [19, 0, 0, -10, -2, -1, 6, 10, -1], [-33, -7, -7, 21, -4, 9, -8, 10, 1], [-23, 22, -20, 1, -11, 35, 19, -12, -4], [8, -9, 10, 25, -4, -14, 6, -2, -1], [22, -22, -5, 10, 5, 7, -1, 11, -9], [3, 16, -16, -6, 12, 6, 8, 1, -2], [-2, 1, -6, 1, 14, 0, 11, 5, -12], [11, 8, 13, -18, -1, -19, -21, 21, 1], [-2, -12, 8, -4, -13, -16, -5, 9, 1], [-10, -14, -6, 11, 1, -3, -9, 7, 10], [10, 28, 2, -11, 4, 2, 5, -12, 9], [-8, -2, 14, 14, 10, 5, -3, 13, -2], [-14, -3, 5, -3, 0, -2, 5, 13, 8], [-45, -12, 11, -5, 6, -1, 0, 28, 7], [-8, 14, -6, -5, -4, 3, 14, 2, 8], [17, -4, 0, -27, -3, -19, -5, 10, 1], [-12, 2, -7, -6, -7, -10, -17, 14, -3], [47, 5, 8, -40, -5, -40, -9, 2, 11], [-33, 16, 2, -1, -4, -3, 16, -9, 2], [17, 41, -9, -41, 2, -15, 27, -22, 5], [34, 21, -11, -20, 2, -1, 2, -10, -1], [-20, 3, -14, 5, -8, 36, 5, 3, 4], [-10, -1, 8, -25, -1, -15, 0, 3, -4], [-2, 4, 11, -19, -2, -12, -6, 3, 7], [13, 13, -6, 4, -10, 27, 3, -13, 12], [-6, 7, -11, 10, -8, 21, -2, 15, -7], [-7, 2, 1, 18, -15, -16, 1, 3, 0], [5, 15, 8, -8, -4, -16, 1, 14, 6], [1, 13, 5, -31, -9, 6, -2, -1, -21], [38, 11, 5, -7, -2, -2, 0, -15, -11], [36, -6, 1, 21, 6, 19, -8, -1, -8], [-15, 6, 3, 15, -3, -20, 5, 7, -1], [7, -4, 15, -19, 6, -8, 16, -6, 10], [21, 11, -7, -8, -8, -26, -2, -12, -10], [6, 6, -4, -2, -12, -19, -4, 1, 4], [34, 2, -2, 12, -26, 9, -6, 0, -7], [-4, 28, -3, 20, -6, 36, 4, -25, -13], [-8, -8, -18, 11, 10, 3, 4, 0, -1], [41, 2, -13, -6, -6, -2, -2, -25, 1], [0, 6, 8, 1, -13, -10, -11, 5, 10], [-3, -2, -7, 7, 2, 11, 9, 17, -2], [-1, 3, 29, -1, -7, 10, -6, 1, -9], [-39, -2, -19, 15, -5, 3, 16, 5, 14], [-22, 21, -23, 9, -11, 42, 13, -22, -16], [-17, -3, -4, 30, 10, 27, 18, -5, 12], [18, 7, 3, -6, -4, -18, 3, 3, -1], [8, -3, -20, -2, -1, -32, 1, -4, 0], [-17, -23, 12, -19, 17, -15, 8, 16, 17], [-38, -13, -15, 37, -11, 19, -16, 10, -5], [31, -7, 8, -16, -4, -27, -18, 8, 1], [4, 6, 11, -1, 4, 19, 10, -8, 17], [7, -4, -14, 3, -12, -5, 3, -10, -7], [-14, 2, 7, 17, -1, 8, 20, -12, 3], [-6, -26, -2, -6, 5, -28, 1, 29, 5], [-28, -8, -8, 39, -11, 9, -19, 2, 7], [7, 13, -5, -28, 1, -14, 15, 1, -2], [-3, 23, -8, 0, -6, 7, 2, -19, -10], [-13, -2, 5, 24, 2, -7, 6, 16, 13], [20, 3, -6, -6, 7, -26, 10, -5, -3], [-5, -6, 25, -10, 19, 4, 5, 6, 6], [6, 2, -2, -16, -6, 9, -8, 4, -10], [37, -14, 10, 0, 7, 9, -10, 10, 1], [-15, -8, 0, 3, 11, -3, -7, -6, 2], [21, -9, 7, -17, 4, -22, -22, 6, 6], [15, 7, 15, 3, -2, -2, -16, 11, 4], [-39, -30, 4, 51, -11, 21, -20, 15, 6], [-4, -24, 16, 3, 11, -6, 13, 20, 9], [-8, 20, 0, -17, 2, -15, -8, -4, 2], [26, 11, -3, -25, -7, -14, 13, 4, 7], [17, -3, 21, -21, 5, 4, 5, 13, 11], [-7, -3, 3, 9, -2, -5, -16, 1, -16], [-3, 10, -1, 5, 1, 28, 21, -1, 14], [-21, -3, -11, 12, 5, -3, 2, 16, -13], [-11, 0, -4, 19, -20, 32, -8, -14, -5], [16, 0, -12, 12, -8, 22, -3, 6, -12], [-4, 10, -5, 16, -6, 27, 15, -14, -16], [-10, -8, -1, -26, -1, -10, -3, 0, 6], [5, -9, -10, -16, 4, 5, 5, -15, 8], [-21, -2, -9, -16, -6, -13, 16, 4, 12], [31, -8, 10, -12, -1, 4, 5, -11, 9], [-11, -3, -17, -8, 6, -24, -6, 7, 1], [4, 2, 18, -6, -15, -9, -19, 5, -7], [11, 36, -10, -23, -7, -3, 11, -14, -7]];
    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_bi();".
// 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_bi(: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_bi();".
// 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_bi( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6422_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-7, 63, -126, -1, 136, -11, -53, -6, 5, 1]>,<5,R![-1, 20, -32, -175, 64, 118, -21, -24, 1, 1]>,<7,R![-7, 77, 7, -587, 863, -380, -14, 52, -13, 1]>],Snew);
    return Vf;
end function;