
// Make newform 9800.2.a.cw in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9800_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_9800_2_a_cw();".
// 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_9800_2_a_cw();".
// 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 := [-8, -15, 35, 8, -12, -1, 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], [-4, 0, 1, 0, 0, 0], [8, 17, -4, -10, 0, 1], [-16, 11, 14, -8, -2, 1], [4, -45, 0, 14, 0, -1]];
    Rf_basisdens := [1, 1, 1, 4, 4, 4];
    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_9800_a();"
function MakeCharacter_9800_a()
    N := 9800;
    order := 1;
    char_gens := [7351, 4901, 1177, 5001];
    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_9800_a_Hecke(Kf)
    return MakeCharacter_9800_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 := [[0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 1], [-1, 0, -1, 0, 0, 0], [-1, 0, 1, 1, -1, -1], [1, -1, 1, 1, 0, 0], [-1, 1, -1, 0, 0, 0], [0, 1, -1, -1, 2, 0], [1, -1, -1, -2, 2, -1], [-2, 0, 0, -1, -1, 1], [-2, 1, 1, 0, -2, 1], [0, 3, 0, 2, -1, 0], [-5, 0, 1, 0, -1, 0], [1, 0, 1, 0, 1, 0], [3, 1, 1, 0, -2, -1], [-1, 0, -2, 1, 0, 1], [-3, -1, 1, -1, 0, -1], [1, 0, -3, -1, 0, -1], [-3, 0, -1, -3, 1, -1], [-3, 1, -1, 0, 2, -1], [-1, 1, -1, 1, 1, -1], [-1, -4, -2, -1, 2, 1], [-2, 0, 2, 2, 0, 2], [-2, 0, -1, 0, 2, -2], [-4, 3, 2, 2, -2, 0], [-1, -1, -1, -1, 4, -1], [-3, 4, 0, -1, 0, -1], [5, -4, -1, 1, 2, -1], [7, 2, 1, 0, -2, 2], [6, -3, 2, 0, 0, 1], [-4, 0, 0, 2, -1, 2], [6, 0, 2, -2, 0, -2], [3, 0, -2, -3, 0, 1], [1, 1, 3, 0, -2, -1], [-10, 4, 0, -1, -1, 1], [1, 4, 1, 1, -3, -1], [0, -1, 0, -3, -1, -1], [-2, -4, 4, 1, -1, -1], [1, 7, -3, 1, 0, 0], [0, -3, -1, -3, 2, -2], [-2, -1, 0, 1, 2, 2], [-6, 0, -2, 2, 1, 2], [12, 0, 2, 1, -2, -1], [8, 5, -2, 3, 2, 2], [-3, 0, 3, 0, -2, 2], [1, 2, 1, -3, 4, 1], [-7, 0, 5, 3, -3, 1], [-4, -3, 4, 1, -4, 0], [-5, -8, 1, -5, 1, -3], [5, 0, -3, -3, 0, 1], [-6, 7, 0, 2, -2, 5], [7, 0, -3, -2, 2, 0], [-12, 4, 4, 4, -1, 0], [-2, 1, -4, -4, 2, 2], [-11, 0, 6, 3, -4, -1], [-11, -3, -1, -2, -2, -3], [-9, 4, -3, -3, 1, -1], [-4, 0, 0, 1, 2, 3], [-9, -4, -1, 1, 3, -1], [4, -4, -2, -1, 2, -3], [-3, 3, 1, 3, -1, 1], [-3, 3, -1, -2, -2, 3], [-6, 0, -2, -2, 5, 2], [-5, 4, 1, 5, -1, -1], [1, -5, -3, -3, 4, 0], [9, -4, -5, 1, 2, -1], [-3, 3, 5, 5, 0, 1], [-6, -5, -1, -1, -2, -6], [1, 0, -5, -3, 5, -5], [-13, 1, 1, 0, -6, 3], [-13, -7, 3, -2, 0, -2], [4, 0, -4, 0, 3, 4], [3, 4, 5, 6, -6, 4], [-7, -5, -3, -4, 4, -4], [0, -5, 4, 1, -4, 2], [-20, -4, 4, 0, -5, 0], [7, -4, 4, 2, 4, -2], [-10, 4, 3, 4, 4, 4], [14, 4, 0, 3, 2, 5], [-5, -5, 2, 2, 0, -1], [-6, -3, 0, 1, 2, 0], [-15, -4, -1, -1, -2, 1], [-8, 7, 2, 3, -6, 4], [-4, -3, 4, -2, -4, 2], [-4, -7, -1, -2, -2, -1], [2, 0, -2, -2, 5, 2], [-10, -4, 0, 2, -4, -2], [2, -1, -2, -3, 7, -1], [-11, -3, -3, -1, 4, -5], [9, 1, -1, 2, -2, -3], [-5, 0, 3, 3, -5, 1], [-3, -12, 1, -7, 4, -3], [-6, -3, -4, -5, 6, -2], [1, 1, -1, 7, -1, -1], [0, 4, -5, -4, 2, -2], [-2, 3, 8, 0, -2, -1], [-5, 0, 3, 3, -5, 1], [10, -8, 5, 0, -2, 2], [9, -3, 1, -1, 1, -7], [-8, 12, 2, -1, 1, -3], [3, -1, 7, 1, 0, 1], [6, -1, 4, 6, -10, 1], [7, 1, 5, 8, -6, -1], [-8, -4, 4, 2, 1, -2], [-13, 2, -5, -3, 4, 1], [-7, 8, -1, 3, -5, 5], [0, -5, -2, -1, -2, 4], [-2, 0, 8, 0, -4, 0], [-7, -3, -3, -2, 4, -2], [3, 0, -1, -6, 1, -6], [16, 0, 4, 6, 0, 6], [18, -7, -2, -4, 0, 1], [16, 0, -4, 6, 4, 2], [2, -4, 1, 5, 2, 3], [-11, -2, -3, -1, -4, 3], [-3, 1, 1, -4, 4, 0], [-1, -4, -1, -4, -1, -4], [7, -8, -1, 3, 0, -5], [1, 0, 4, -3, 0, -3], [0, 0, -8, -4, 4, 0], [-9, -8, 3, 2, 3, 2], [6, -3, -6, -6, 4, 2], [18, -7, 0, -1, -2, -2], [1, -9, -4, -2, 4, -1], [12, 0, -7, 2, 6, 0], [0, -5, -6, -7, 2, -6], [-10, 3, -2, -5, -1, 1], [-12, -4, -2, -5, 3, -3], [2, -13, 2, -4, -4, 3], [2, -5, -2, 1, 3, -5], [-12, -5, 2, 3, -6, -8], [-1, 12, 1, -1, -2, 5], [-6, 3, -4, 0, 2, -1], [0, -4, -4, -7, 6, 3], [13, -4, 1, -6, 3, 2], [-17, 5, -1, 1, -8, 5], [8, -4, 4, -6, 0, 2], [3, -4, -2, 4, 0, 0], [5, -12, 3, -2, 2, 0], [-8, -3, -4, -5, 0, -2], [-23, 5, -1, 3, 2, -3], [0, -3, -8, -6, 9, 4], [-16, 9, -4, 3, 0, 2], [6, 4, 6, 4, -4, -4], [-7, 4, 1, 0, 4, 0], [2, 4, -6, 4, 5, 4], [6, 3, 4, 5, -2, -6], [-16, -7, 0, 1, 4, 5], [7, 0, 7, 0, 1, -4], [-16, 5, -1, 4, -6, 1], [-5, 2, -1, 3, 0, 3], [-5, 1, 1, -3, -6, 3], [-12, -5, 4, 4, -12, 4], [26, -4, 2, 8, -4, 0], [-11, -11, 7, 2, 2, 1], [-2, 12, 5, 9, -6, -1], [-16, 0, 4, 6, -4, 2], [-20, 7, -4, -1, 0, 4], [-7, 3, -3, 5, 0, 5], [10, 12, -2, 2, 4, 2], [1, -3, -5, -5, -1, 7], [24, -3, -4, -6, 0, -5], [-4, -4, 4, 4, -9, 0], [4, 7, 0, -3, -4, 1], [5, -7, -5, 2, 6, -3], [1, 8, 3, -1, 3, 5], [-15, 1, -6, -5, 4, 2], [-1, 0, -1, -4, -3, 4], [14, -5, 0, 1, 2, -2], [8, -11, -5, -3, 2, 2], [-3, -5, -1, 2, 6, -1], [5, -4, -1, 5, -9, -1], [15, -3, -3, 2, 6, -3], [-25, -5, 0, -2, 2, -3], [-23, 0, -1, 4, -6, 6], [16, 5, 0, 1, 4, -4], [-13, -11, 11, 0, -3, -2], [7, -12, 0, -9, 8, -1], [18, -9, 6, -1, 0, 7], [23, 8, 1, 0, -6, -2], [-10, 8, 4, 9, -6, -5], [4, 20, 0, -2, -1, 2], [11, -20, -3, -2, 3, -10], [-12, 8, 3, -4, -6, 6], [-10, 8, -4, 1, 9, -5], [-17, 8, -1, -5, 7, 1], [10, 12, -1, 3, 6, 1], [-4, 7, 10, 5, -2, 2], [4, 4, 0, 6, 1, -6], [7, -10, -1, -1, 0, 3], [0, 7, -4, 4, 0, 3], [-15, -4, 0, 5, -8, 5], [19, -8, -9, -7, 11, -5], [14, -8, -6, 2, -1, 2], [-22, -4, -3, -1, -6, -7], [4, 16, -4, 4, -1, 4], [13, -16, -1, -7, -2, -1], [19, -8, 3, -5, 3, 1], [-12, 12, -3, -2, -2, -4], [-3, 4, 5, 3, 0, -9], [9, -8, -5, 3, 5, 1], [-15, 8, -2, 2, 0, -2], [13, 0, -11, 1, 8, 1], [11, -4, -1, -6, 1, -10], [-9, -5, 1, 0, 6, -3], [-7, 8, 5, 3, -7, 9], [-13, 5, 0, 7, 0, 2], [14, 7, 10, 16, -16, 3], [-1, 8, 1, 5, -9, 3], [-7, -3, 6, 0, 4, -1], [-16, -1, 6, 6, 1, 6], [10, 5, -6, 6, 0, 6], [-6, 13, -4, -1, -2, 10], [10, 3, 0, 2, 6, -3], [18, -17, -2, -11, 4, -7], [-7, 4, -4, 10, 4, -2], [13, -4, -1, -6, 5, -6], [24, 4, -2, -3, 6, -1], [7, -16, -4, 3, 0, -9], [12, -4, 8, -8, -4, 4], [9, 4, 4, 6, -8, 2], [27, -16, -7, 0, -1, 0], [20, -5, -4, 0, 7, -2], [-1, 5, 4, 0, -4, -1], [2, 12, 6, 2, -3, 2], [-20, 3, -2, 5, -6, -2], [10, 5, 2, 5, 4, 13], [11, -13, -11, -14, 10, -9], [-5, 20, 3, 8, -11, 8], [-5, 5, -1, -3, -8, -2], [14, 4, -6, -2, 0, 2], [18, -4, 9, 3, -2, 1], [-36, 5, -4, -2, 1, -8], [5, -3, -1, 3, -2, 9], [6, -8, -1, 0, -4, 4], [-7, 0, -7, -12, 7, 0], [-3, 16, 7, 6, -6, 4], [22, -12, 6, 4, 0, 4], [17, -4, 5, -5, -3, -11], [18, -15, -2, 2, 0, 3], [-17, 12, -5, 3, 3, -3], [10, 0, -1, 0, -2, -2], [8, 4, -4, -2, 0, 2], [-10, -5, 0, -5, 2, 4], [9, 11, -7, 6, -5, 0], [21, 3, 5, 1, -8, 4], [10, 11, -4, -7, 6, -2], [-11, -10, -1, -8, -6, -2], [-18, 4, -2, -4, 12, 0], [-9, -20, 10, 2, 0, -2], [-17, -5, 3, 8, -4, 0], [-6, 11, 3, 10, 2, 5], [21, 0, -1, -1, 1, -3], [-13, 1, -9, 9, -4, 6], [6, 12, -3, 4, 6, 6], [10, -7, -14, -14, 12, 6], [-26, 12, 0, 7, -10, 9], [29, -8, -5, -9, -7, -7], [12, -7, 8, 3, 0, 11], [-7, 5, 9, 1, -3, -1], [-12, -7, 8, -1, 0, -2], [-13, -4, -3, -5, -10, 1], [28, 0, 8, 6, -7, 6], [18, 3, -2, 8, -4, 7], [-18, 8, -12, -8, -4, 0], [-16, -8, 8, -7, 2, -5], [-22, -9, 6, 4, -1, -2], [12, 0, 6, 7, 7, 5], [8, 0, 1, 6, 6, -8], [7, 8, -9, -1, -1, -3], [-33, -4, -3, -1, -3, 1], [-14, 17, 8, 15, -10, 6], [-2, -8, 6, 4, 4, 8], [-15, -4, -1, 6, -11, -2], [3, -12, -1, 5, 7, -5], [-16, 16, 1, 14, -2, 12], [-8, -4, -13, -11, 14, -1], [33, 17, -7, 5, 8, 6], [23, 11, 7, 12, -8, 8], [-26, 5, 0, 3, -10, 6], [-7, 6, 3, -2, -2, 0], [-6, -11, 1, 1, -2, 2], [26, -3, 6, 10, -8, 2], [-6, 7, 12, 5, -6, 6], [-19, -8, -1, -11, 11, -1], [-10, 4, -16, -5, 6, -3], [-3, 9, 3, -2, 6, 1], [4, 4, -8, -7, -2, -5], [-7, 12, 10, 7, -12, 11], [11, 1, -5, -7, 0, 1], [-3, 4, -9, 1, 13, 3], [16, 8, -2, -3, -2, -5], [-21, -12, 3, -4, -4, -4], [-3, -4, -6, 1, 0, -7], [-33, -3, -7, -8, 2, 7], [23, 4, 9, 3, -10, 9], [12, 3, -8, 2, 0, 1], [-17, 6, -1, 13, -4, 5], [9, -8, 3, -15, 11, -5], [-19, 4, -7, -2, -7, -2], [-14, 15, -4, 9, -14, 2], [-12, -1, 20, 6, -13, 4], [-5, 7, 9, 6, -10, -9], [-27, 4, 7, 5, -6, 3], [23, -8, 11, 3, -1, 1], [22, -1, 7, 13, -2, 4], [25, -15, 11, 4, 6, -5], [17, -12, 5, -6, -1, -14], [10, 3, 6, 5, 4, -3], [24, -17, -8, -17, 8, -4], [3, -24, -2, -13, 6, -15], [16, 17, -8, 8, 8, 0], [-20, -3, 4, 3, 9, 5], [13, -8, -10, -14, 4, -6], [12, -20, -2, -11, 2, -1], [-43, 1, 3, 2, -6, -3], [-16, -4, -8, -8, 16, -4], [16, -7, 9, -1, -4, 2], [19, -13, -7, -6, 6, -9], [9, 8, 7, -5, -5, -3], [-19, 5, -6, 6, 4, 13], [-1, -5, -5, -8, 3, 2], [1, 8, -13, -11, 3, -1], [18, 0, -5, -1, -6, 1], [5, 13, 9, -3, -12, 2], [15, -5, 7, -9, 4, 3], [-9, -12, 3, 3, -5, 5], [-17, -4, 1, 3, 6, -7], [12, 9, 8, -1, 0, 2], [12, 12, -12, 0, 7, 8], [-13, 0, 11, -1, -4, 3], [14, -1, 6, -6, -16, -6], [22, 9, 2, 8, 4, 5], [15, 15, -1, -5, -9, 5], [14, -4, 0, -2, -4, -14], [16, 16, 0, 0, -11, 4], [10, -3, 9, -6, 2, 7], [7, -12, -15, -10, 19, -10], [-42, 0, -6, 2, 1, -2], [-34, 4, 10, 10, -4, 10], [-19, 3, 4, -2, -8, 3], [25, -12, -3, -11, 4, -11], [-28, -4, 4, -4, -1, 0], [7, -12, 3, 7, 0, 3], [-9, -16, 2, 1, 8, 1], [9, -13, 1, -13, -8, -5], [3, -12, 3, -7, -4, -11], [28, -12, -6, -15, 13, -1], [12, 17, -8, 0, 4, 1], [9, 8, 3, 13, 5, -5], [-19, 3, -8, -2, -4, 7], [-38, -17, 2, -7, -4, -3], [-26, -20, 7, 0, -10, -6], [38, -8, -10, -8, 8, -8], [-1, 11, 5, -2, -6, -1], [14, 15, 12, 1, -18, 2], [27, 0, -19, -7, 7, -1], [1, 19, -9, -1, 1, 3], [-25, 0, 11, 10, -12, -2], [-28, 3, 12, 13, -16, 6], [24, -1, -14, -6, 10, 0], [4, -3, -20, -12, 20, 1], [-2, -15, 6, -2, -8, -10], [41, 4, -1, 3, -3, -3], [-23, 8, 1, 8, 0, 0], [7, -8, -7, -14, 3, -2], [-8, -9, -7, -18, 10, 1], [-13, -11, 11, 7, -12, 4], [-3, -1, -7, -5, 20, -2], [-6, -7, 16, 12, -10, -2], [6, 4, -8, 2, 8, 14], [-8, -9, 14, 9, -2, 6], [37, 8, -9, -5, 15, 1], [7, 13, -1, 15, -8, 4], [8, -12, 0, -10, -12, 6], [-37, 0, 5, 1, -11, 7], [16, 9, -16, -5, 8, 2], [27, -20, -3, -11, 10, -5], [8, 0, -19, -8, 14, 2], [1, 4, 7, 15, -10, 9], [8, 17, 0, -7, 13, 3], [12, -4, 12, -12, -5, -12], [-17, -4, -7, 2, 22, 0], [10, -17, -18, -13, 20, -9], [8, 5, -6, -11, 10, 8], [-10, 4, -20, -7, 11, -9], [-8, -7, -12, -8, 16, 4], [4, 15, 6, 6, 5, 2], [14, -8, 0, 0, 8, 0], [18, -4, 6, 10, -15, -2], [-1, -5, -3, -2, 10, 3], [25, 20, -15, -3, 5, 3], [24, 7, -2, 1, -2, -10], [-5, 4, 1, -1, -5, -11], [-27, -11, 3, -8, -14, 3], [-13, -25, 6, -3, 0, 2], [-38, 4, 0, 5, -10, -5], [33, -16, -14, -7, 8, -3], [-11, 4, -10, -1, -4, -9], [10, 24, 2, 12, -5, 8], [9, -1, -15, -8, 12, 12], [4, 8, 8, 6, -21, 2], [17, 12, 10, 17, -12, 5], [-29, 9, 9, -8, -2, -5], [14, -16, 0, -13, 9, -3], [19, 3, -8, 2, 6, 13], [-42, 7, -2, 8, -4, 12], [19, 7, 3, -1, 4, -2], [-27, 12, 3, 3, -5, 1], [-24, -12, 16, -4, -7, 0], [43, 6, -1, 1, 12, -7], [24, 15, -12, 12, 8, 11], [-20, 4, -12, -12, 8, 0], [-12, 19, -16, -11, 4, -14], [28, 21, -14, -1, 10, 6], [-5, 16, 5, 10, -13, -2], [-12, 0, -12, 0, 4, -12], [-4, 23, -8, -3, 4, 1], [1, -4, 4, 11, -4, -5], [8, -8, 6, -11, -7, -5], [-13, -7, 5, -11, -5, -7], [25, -4, 5, -7, 0, -3], [-22, 8, -10, 6, 0, 6], [4, 16, 2, -11, 7, 3], [-19, 20, -3, 11, 9, -11], [-16, -3, -5, -4, 2, 5], [-8, 15, -4, -14, 16, 5], [-22, 1, 4, 1, 10, -4], [-17, -12, -4, -1, 2, -7], [-12, 16, 10, 9, -14, 7], [18, -23, 2, -12, -4, 1], [31, 14, -11, -6, -6, -4], [-47, -8, 7, -7, -6, -1], [-20, -12, 7, 6, -16, 14], [23, 0, 4, 15, 2, -7], [-11, 20, 6, 9, -8, 1], [5, -2, -3, -5, -4, -9], [-18, -24, 2, 4, 12, 0], [18, -13, -14, -14, 12, 2], [7, -32, 2, -6, 16, -10], [5, -19, 14, -6, 8, -3], [-20, -25, -2, -9, 10, 0], [32, -20, 0, 5, -10, -1], [13, 12, -1, -3, 2, 11], [-49, 13, -1, 1, -3, 7], [7, 12, -3, 4, -9, 8], [35, -8, -8, 2, 8, -2], [-4, -5, -12, -13, 8, -8], [-8, 11, -4, 14, -16, -2], [19, 20, -1, -9, 12, 7], [1, 5, 1, -7, 12, 1], [18, 1, -4, 14, -6, 3], [-32, 4, -8, 0, -5, 8], [-27, 3, 16, 8, -12, -7], [-8, 15, -4, 5, 4, 2], [30, -9, -26, -14, 12, 1], [5, -8, -3, 4, 1, 16], [6, -4, -14, -2, 9, -6], [6, -9, 14, 2, -16, 1], [-13, 15, 9, -8, -10, 1], [29, -3, 3, -2, 6, -7], [0, -12, -13, -2, 14, -4], [17, -16, 9, 5, 1, 7], [17, 0, 7, 3, 5, 9], [-11, 4, -7, 5, 0, -3], [-3, 3, -19, 7, 8, -1], [-3, 4, -14, -3, 14, 7], [-13, 0, 11, -1, 4, -1], [-28, -3, 8, 5, -4, -3], [5, 18, 1, 7, -4, 7], [-15, -8, 5, 3, -20, -5], [-4, 16, 16, 13, -14, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9800_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9800_2_a_cw();".
// 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_9800_2_a_cw(:prec:=6)
    chi := MakeCharacter_9800_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(3361) - 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_9800_2_a_cw();".
// 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_9800_2_a_cw( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-8, 15, 35, -8, -12, 1, 1]>,<11,R![512, -832, 296, 69, -37, -1, 1]>,<13,R![256, 320, -8, -100, -10, 7, 1]>,<19,R![568, -1300, -94, 301, -47, -5, 1]>,<23,R![-337, 478, 123, -108, -19, 6, 1]>],Snew);
    return Vf;
end function;