
// Make newform 9800.2.a.da 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_da();".
// 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_da();".
// 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 := [4, 0, -38, 0, 85, 0, -18, 0, 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], [-5, 0, 1, 0, 0, 0, 0, 0], [2, 0, -9, 0, 1, 0, 0, 0], [0, -20, 0, 83, 0, -18, 0, 1], [0, -38, 0, 85, 0, -18, 0, 1], [-20, 0, 83, 0, -18, 0, 1, 0], [0, -96, 0, 397, 0, -88, 0, 5]];
    Rf_basisdens := [1, 1, 1, 1, 2, 2, 1, 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_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, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 1, 0, 0, -1, -1, 0, 0], [0, 1, 0, 0, 1, 1, 0, -1], [0, 0, -1, 1, 0, 0, 0, 0], [2, 0, 1, 1, 0, 0, 1, 0], [0, 2, 0, 0, 2, 1, 0, 0], [0, 0, 1, 1, 0, 0, -1, 0], [0, -1, 0, 0, 0, -2, 0, 0], [-2, 0, -1, -1, 0, 0, 1, 0], [0, 0, 0, 0, 4, -2, 0, -1], [4, 0, 0, 2, 0, 0, 1, 0], [0, -2, 0, 0, -1, 1, 0, 1], [0, 2, 0, 0, -2, 1, 0, -1], [-5, 0, 0, 2, 0, 0, 0, 0], [2, 0, 1, -1, 0, 0, 1, 0], [0, 1, 0, 0, 5, 0, 0, -1], [2, 0, -1, -3, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, -1], [0, -5, 0, 0, 1, -2, 0, 0], [0, 0, 0, 0, 5, -1, 0, 1], [0, 4, 0, 0, 4, -1, 0, 1], [0, -2, 0, 0, 0, 0, 0, -1], [-9, 0, -1, 1, 0, 0, 2, 0], [2, 0, 1, -3, 0, 0, 0, 0], [3, 0, 0, -1, 0, 0, -2, 0], [-2, 0, 1, -3, 0, 0, 1, 0], [0, 0, 0, 0, -5, -2, 0, 1], [3, 0, 1, 2, 0, 0, -1, 0], [0, 1, 0, 0, 4, 0, 0, 0], [4, 0, -1, 5, 0, 0, 1, 0], [4, 0, 1, 1, 0, 0, -3, 0], [0, 4, 0, 0, -8, 1, 0, -1], [3, 0, -1, 4, 0, 0, 0, 0], [0, 6, 0, 0, -4, -1, 0, 0], [0, 4, 0, 0, -4, 2, 0, -1], [3, 0, -1, -1, 0, 0, 1, 0], [0, 6, 0, 0, 0, 5, 0, -1], [10, 0, 0, -2, 0, 0, 1, 0], [7, 0, 2, -3, 0, 0, -4, 0], [2, 0, -3, -1, 0, 0, 2, 0], [0, -2, 0, 0, 4, -3, 0, -1], [-5, 0, -3, -1, 0, 0, -2, 0], [0, -2, 0, 0, -12, 1, 0, 1], [0, 6, 0, 0, -1, 2, 0, -2], [0, 0, 0, 0, -6, 2, 0, 1], [12, 0, -1, -1, 0, 0, 0, 0], [-4, 0, -4, 4, 0, 0, 3, 0], [0, -3, 0, 0, 10, 1, 0, 0], [0, 7, 0, 0, -11, 1, 0, -2], [0, -4, 0, 0, -1, -3, 0, 1], [6, 0, -1, -1, 0, 0, -1, 0], [0, 0, 0, 0, 8, 3, 0, 0], [0, 4, 0, 0, -2, 3, 0, -2], [6, 0, -2, 0, 0, 0, 2, 0], [8, 0, -1, 5, 0, 0, 1, 0], [0, -1, 0, 0, 6, -5, 0, 0], [0, 0, 0, 0, 2, -5, 0, 0], [0, 1, 0, 0, 3, -1, 0, -2], [0, 2, 0, 0, 10, 0, 0, 2], [0, -4, 0, 0, 1, -4, 0, -2], [-8, 0, 1, 3, 0, 0, 2, 0], [11, 0, -4, -5, 0, 0, -1, 0], [9, 0, 1, -5, 0, 0, -1, 0], [-25, 0, 0, -1, 0, 0, -1, 0], [0, -10, 0, 0, 0, -6, 0, -1], [0, -8, 0, 0, 1, -5, 0, 3], [14, 0, -1, 1, 0, 0, 1, 0], [0, -8, 0, 0, 0, -9, 0, 1], [4, 0, -3, 1, 0, 0, -1, 0], [17, 0, 2, 1, 0, 0, 2, 0], [0, 0, 0, 0, -10, 5, 0, 0], [-8, 0, -1, 3, 0, 0, 3, 0], [0, -8, 0, 0, 0, -6, 0, -1], [17, 0, 0, -2, 0, 0, 3, 0], [0, -1, 0, 0, 2, -1, 0, 2], [0, 5, 0, 0, 0, 2, 0, 3], [16, 0, 1, -1, 0, 0, 1, 0], [-4, 0, -2, 6, 0, 0, 3, 0], [0, -3, 0, 0, 2, 1, 0, -3], [0, 2, 0, 0, -12, 3, 0, 0], [-5, 0, 1, 1, 0, 0, -3, 0], [3, 0, 4, -2, 0, 0, -1, 0], [-5, 0, 0, 3, 0, 0, 0, 0], [0, 4, 0, 0, -2, -3, 0, 2], [0, 0, -2, 2, 0, 0, 2, 0], [0, 2, 0, 0, 5, -2, 0, 2], [0, 8, 0, 0, 8, 5, 0, 0], [-6, 0, 5, -7, 0, 0, -2, 0], [0, 0, 1, 3, 0, 0, 1, 0], [12, 0, 2, 6, 0, 0, 4, 0], [0, -12, 0, 0, -4, 0, 0, -1], [0, -4, 0, 0, 12, -1, 0, -2], [0, -5, 0, 0, 6, 3, 0, 0], [0, 1, 0, 0, -1, 3, 0, 2], [18, 0, -1, 1, 0, 0, 3, 0], [-11, 0, -2, 5, 0, 0, -2, 0], [8, 0, 1, -5, 0, 0, 1, 0], [0, -2, 0, 0, 5, 6, 0, -4], [3, 0, 0, 4, 0, 0, 4, 0], [6, 0, -3, 1, 0, 0, 1, 0], [0, -3, 0, 0, -15, -1, 0, -1], [0, -5, 0, 0, -9, 0, 0, -3], [0, -3, 0, 0, -9, -1, 0, 2], [8, 0, 1, -3, 0, 0, 2, 0], [0, -5, 0, 0, 6, -3, 0, -1], [0, -2, 0, 0, 6, 5, 0, -2], [-10, 0, -7, -1, 0, 0, 0, 0], [6, 0, -3, 5, 0, 0, 4, 0], [0, -2, 0, 0, -15, -1, 0, 1], [-4, 0, 1, 5, 0, 0, 0, 0], [14, 0, -1, -5, 0, 0, -4, 0], [0, -8, 0, 0, -3, 0, 0, -3], [0, 2, 0, 0, 6, 10, 0, -1], [8, 0, -2, 8, 0, 0, 4, 0], [5, 0, 3, -4, 0, 0, 1, 0], [0, 8, 0, 0, -4, -4, 0, 1], [-16, 0, 0, 2, 0, 0, -3, 0], [0, -6, 0, 0, 6, 6, 0, -1], [-7, 0, 2, -4, 0, 0, -2, 0], [0, -11, 0, 0, -7, -4, 0, 1], [-8, 0, -4, -4, 0, 0, -2, 0], [0, 0, 2, 2, 0, 0, -8, 0], [0, -12, 0, 0, -6, -3, 0, -1], [0, -6, 0, 0, -10, 2, 0, 0], [0, -8, 0, 0, -2, 0, 0, 2], [22, 0, -3, 3, 0, 0, 2, 0], [-4, 0, 4, -4, 0, 0, 1, 0], [-6, 0, 2, -8, 0, 0, 2, 0], [14, 0, 1, 7, 0, 0, 0, 0], [0, 13, 0, 0, -6, -1, 0, -3], [0, 3, 0, 0, 7, 0, 0, -3], [0, 6, 0, 0, 2, -5, 0, -4], [0, 4, 0, 0, -13, 9, 0, 0], [0, -18, 0, 0, 6, -8, 0, 4], [-20, 0, -1, -3, 0, 0, -3, 0], [0, 6, 0, 0, -7, 8, 0, -2], [-8, 0, -6, 2, 0, 0, -2, 0], [-10, 0, -7, 5, 0, 0, 0, 0], [-23, 0, 0, 4, 0, 0, 5, 0], [0, 12, 0, 0, -2, -2, 0, 0], [0, -6, 0, 0, 2, 5, 0, -3], [0, -2, 0, 0, -12, 0, 0, 4], [0, 3, 0, 0, -4, -2, 0, -2], [0, -11, 0, 0, -6, 5, 0, 2], [16, 0, -1, -1, 0, 0, -5, 0], [-8, 0, -4, -6, 0, 0, 1, 0], [0, 0, 0, 0, 7, 7, 0, 1], [3, 0, 6, -4, 0, 0, 0, 0], [0, -6, 0, 0, 6, -7, 0, 3], [4, 0, 4, 0, 0, 0, 4, 0], [0, 0, 1, 3, 0, 0, -2, 0], [6, 0, -5, 9, 0, 0, 5, 0], [0, 6, 0, 0, 5, 1, 0, 6], [0, 1, 0, 0, 20, -10, 0, -2], [0, 6, 0, 0, 2, -3, 0, 2], [-14, 0, -2, 4, 0, 0, -6, 0], [19, 0, 0, 2, 0, 0, 3, 0], [-6, 0, -2, 4, 0, 0, 4, 0], [0, 1, 0, 0, 5, 5, 0, -1], [9, 0, -2, 7, 0, 0, 6, 0], [0, 6, 0, 0, 4, -1, 0, 4], [-8, 0, 3, 13, 0, 0, 3, 0], [0, -8, 0, 0, -6, 2, 0, -5], [7, 0, 0, 9, 0, 0, 5, 0], [0, 2, 0, 0, 18, 0, 0, -4], [-21, 0, -2, 2, 0, 0, -9, 0], [0, 8, 0, 0, 16, -2, 0, 3], [-22, 0, -3, -1, 0, 0, -1, 0], [-9, 0, 1, -10, 0, 0, 5, 0], [0, -16, 0, 0, -2, 1, 0, 1], [0, -5, 0, 0, -18, -11, 0, 3], [1, 0, 4, -4, 0, 0, 1, 0], [-8, 0, -2, 0, 0, 0, 7, 0], [0, 4, 0, 0, -30, -1, 0, 0], [0, -4, 0, 0, -12, -1, 0, 6], [20, 0, -5, 11, 0, 0, 4, 0], [0, -1, 0, 0, -14, -7, 0, -1], [-14, 0, -1, -9, 0, 0, -3, 0], [0, -1, 0, 0, 10, 1, 0, 0], [10, 0, -1, 1, 0, 0, 6, 0], [0, 2, 0, 0, 12, 0, 0, -5], [26, 0, 5, -5, 0, 0, -2, 0], [0, -7, 0, 0, -15, 7, 0, 1], [-7, 0, -4, -1, 0, 0, 5, 0], [0, -2, 0, 0, -14, -3, 0, -3], [0, 1, 0, 0, -7, 9, 0, 1], [-13, 0, -4, 4, 0, 0, 1, 0], [17, 0, 8, 0, 0, 0, -3, 0], [0, 6, 0, 0, -2, 4, 0, -1], [-25, 0, 2, -12, 0, 0, -2, 0], [0, 2, 0, 0, -11, 7, 0, -2], [5, 0, -8, 0, 0, 0, 2, 0], [-10, 0, 3, 7, 0, 0, -7, 0], [0, 13, 0, 0, 10, 9, 0, 0], [0, -16, 0, 0, -22, -3, 0, -2], [-2, 0, 8, -12, 0, 0, -6, 0], [0, 0, 0, 0, 14, -14, 0, -1], [0, -12, 0, 0, 4, -9, 0, 2], [0, 3, 0, 0, -13, 14, 0, 1], [0, -20, 0, 0, 9, -13, 0, 0], [0, -4, 0, 0, 14, -5, 0, -3], [0, -8, 0, 0, -2, -2, 0, -5], [-42, 0, 0, 0, 0, 0, 3, 0], [-4, 0, 3, 3, 0, 0, -8, 0], [0, -4, 0, 0, -5, -1, 0, 0], [6, 0, 1, -7, 0, 0, -6, 0], [0, 4, 0, 0, 20, -14, 0, -2], [0, 0, 1, -3, 0, 0, -8, 0], [0, -1, 0, 0, -3, 7, 0, -2], [0, -18, 0, 0, 12, -3, 0, 3], [0, 17, 0, 0, 6, -1, 0, 3], [16, 0, -3, 3, 0, 0, -3, 0], [0, 11, 0, 0, -13, 9, 0, -3], [-8, 0, -5, 9, 0, 0, -7, 0], [0, 0, 3, -11, 0, 0, 4, 0], [8, 0, 5, 7, 0, 0, 3, 0], [0, -2, 0, 0, 18, 6, 0, 0], [-9, 0, 8, -12, 0, 0, -6, 0], [-8, 0, -7, 5, 0, 0, 6, 0], [0, -3, 0, 0, -25, -6, 0, 4], [-8, 0, 2, 2, 0, 0, -8, 0], [0, -3, 0, 0, 24, -2, 0, 4], [-6, 0, 1, 11, 0, 0, 3, 0], [0, 8, 0, 0, -4, 0, 0, -2], [17, 0, 10, -1, 0, 0, 3, 0], [-18, 0, -1, 3, 0, 0, -4, 0], [0, 1, 0, 0, -29, 0, 0, -2], [10, 0, 6, -4, 0, 0, 2, 0], [-5, 0, 10, -2, 0, 0, -2, 0], [0, -6, 0, 0, -5, 1, 0, -1], [0, -2, 0, 0, 18, -3, 0, 4], [0, -15, 0, 0, 9, 0, 0, -3], [-10, 0, -3, 11, 0, 0, 0, 0], [-29, 0, -4, -2, 0, 0, 4, 0], [0, -8, 0, 0, 10, 0, 0, 0], [-31, 0, 4, 9, 0, 0, -3, 0], [0, -15, 0, 0, 0, 12, 0, -3], [0, 4, 0, 0, 22, 9, 0, -2], [22, 0, 11, 1, 0, 0, 4, 0], [0, -1, 0, 0, 20, -8, 0, -5], [0, 10, 0, 0, 2, 8, 0, -8], [0, 8, 0, 0, 6, 2, 0, -3], [0, 2, 0, 0, 1, 0, 0, 4], [19, 0, 5, -6, 0, 0, 2, 0], [-6, 0, -6, -4, 0, 0, 10, 0], [2, 0, -4, 6, 0, 0, 9, 0], [0, -3, 0, 0, -3, 0, 0, 1], [6, 0, 6, -12, 0, 0, -12, 0], [0, 9, 0, 0, 26, 5, 0, 1], [0, 6, 0, 0, -24, -2, 0, 1], [-31, 0, 3, 5, 0, 0, 0, 0], [16, 0, 7, 1, 0, 0, 1, 0], [0, -1, 0, 0, -7, -11, 0, 0], [0, 2, 0, 0, -20, 8, 0, -3], [0, -3, 0, 0, -7, 1, 0, -3], [-14, 0, 0, -12, 0, 0, -7, 0], [-25, 0, 6, -17, 0, 0, -2, 0], [0, -4, 0, 0, -14, 1, 0, 0], [0, 6, 0, 0, 12, 11, 0, -8], [0, 11, 0, 0, 7, 8, 0, 3], [0, 7, 0, 0, 2, -7, 0, 5], [-2, 0, 3, -3, 0, 0, -8, 0], [0, 22, 0, 0, 1, 9, 0, -2], [-33, 0, 0, -8, 0, 0, 1, 0], [8, 0, -3, -9, 0, 0, -7, 0], [0, 4, 0, 0, 10, -8, 0, -6], [25, 0, 1, -6, 0, 0, 1, 0], [0, 21, 0, 0, -6, -3, 0, -3], [34, 0, 7, -1, 0, 0, -4, 0], [0, -6, 0, 0, 15, -6, 0, 3], [0, -8, 0, 0, 10, 6, 0, 4], [27, 0, 10, -1, 0, 0, -2, 0], [26, 0, 4, -2, 0, 0, -10, 0], [-16, 0, -5, 1, 0, 0, 7, 0], [0, 13, 0, 0, 19, 1, 0, 2], [0, -6, 0, 0, 32, -7, 0, -3], [-20, 0, 6, 4, 0, 0, -2, 0], [0, -6, 0, 0, -6, -15, 0, 2], [0, 10, 0, 0, 10, 0, 0, 2], [0, 5, 0, 0, -16, 0, 0, -4], [-16, 0, 5, -3, 0, 0, -4, 0], [-26, 0, 7, -11, 0, 0, 1, 0], [26, 0, -5, -5, 0, 0, -4, 0], [0, 20, 0, 0, -12, -4, 0, -2], [0, -8, 0, 0, -11, -8, 0, -2], [-38, 0, -7, -1, 0, 0, -2, 0], [0, 2, 0, 0, 13, 5, 0, 5], [-1, 0, -6, 10, 0, 0, 0, 0], [0, -5, 0, 0, -12, -4, 0, 4], [-2, 0, 3, 3, 0, 0, -6, 0], [0, 0, 0, 0, -14, -8, 0, 10], [0, -14, 0, 0, 18, -5, 0, -1], [0, 6, 0, 0, -10, 9, 0, 4], [0, -4, 0, 0, -3, 1, 0, 6], [0, -9, 0, 0, 19, -3, 0, -1], [0, 9, 0, 0, 4, 12, 0, -2], [-10, 0, -8, 2, 0, 0, -4, 0], [18, 0, 3, -5, 0, 0, -10, 0], [-13, 0, 4, 1, 0, 0, -2, 0], [-9, 0, -12, 9, 0, 0, 2, 0], [-29, 0, 3, -6, 0, 0, -2, 0], [-22, 0, -3, 9, 0, 0, 5, 0], [0, -10, 0, 0, 34, -3, 0, 0], [0, 0, 1, 1, 0, 0, 2, 0], [-10, 0, 4, 10, 0, 0, 8, 0], [0, -2, 0, 0, -16, -13, 0, 3], [-32, 0, 2, 0, 0, 0, -3, 0], [-41, 0, -2, -3, 0, 0, -4, 0], [-22, 0, 6, 16, 0, 0, 0, 0], [6, 0, -9, 11, 0, 0, -2, 0], [0, -14, 0, 0, -15, 6, 0, -3], [0, 11, 0, 0, 10, 7, 0, -2], [-10, 0, 6, -6, 0, 0, -7, 0], [0, -19, 0, 0, -3, -2, 0, -1], [-11, 0, -10, 5, 0, 0, 1, 0], [0, 12, 0, 0, -15, 12, 0, -1], [9, 0, -15, 0, 0, 0, 1, 0], [0, -6, 0, 0, -2, 6, 0, 3], [-8, 0, 3, -9, 0, 0, -11, 0], [-4, 0, -7, 1, 0, 0, 4, 0], [0, 20, 0, 0, 7, 9, 0, 1], [-19, 0, 2, 11, 0, 0, 3, 0], [0, -14, 0, 0, 35, -4, 0, 2], [-4, 0, -5, -5, 0, 0, 4, 0], [68, 0, -3, 3, 0, 0, 0, 0], [-30, 0, 0, 2, 0, 0, -7, 0], [14, 0, 7, 7, 0, 0, 1, 0], [0, 6, 0, 0, 18, -8, 0, -5], [0, 3, 0, 0, -34, -3, 0, 1], [19, 0, 5, 11, 0, 0, 8, 0], [0, 6, 0, 0, -1, 8, 0, 0], [18, 0, -9, -15, 0, 0, -1, 0], [0, -16, 0, 0, -1, -11, 0, -1], [0, -11, 0, 0, 12, -4, 0, -1], [0, -14, 0, 0, -12, -11, 0, 9], [-48, 0, -2, -4, 0, 0, 6, 0], [5, 0, 0, 2, 0, 0, -9, 0], [0, -12, 0, 0, 2, 15, 0, 1], [-24, 0, -9, 5, 0, 0, 0, 0], [26, 0, 1, -3, 0, 0, -2, 0], [-30, 0, -9, 1, 0, 0, -2, 0], [0, -12, 0, 0, -10, -1, 0, -5], [5, 0, -4, -3, 0, 0, 1, 0], [0, 0, 0, 0, 8, -6, 0, 3], [0, 14, 0, 0, 6, 3, 0, -3], [0, 27, 0, 0, -13, 0, 0, -7], [10, 0, -3, -9, 0, 0, -5, 0], [-54, 0, 4, -4, 0, 0, -2, 0], [0, 12, 0, 0, 12, 14, 0, -1], [-56, 0, -3, 5, 0, 0, 8, 0], [0, 1, 0, 0, -6, 3, 0, 4], [0, -8, 0, 0, -18, 15, 0, 6], [0, 1, 0, 0, 1, -4, 0, 1], [-14, 0, -4, -12, 0, 0, 6, 0], [-14, 0, -2, 2, 0, 0, 4, 0], [56, 0, 1, -3, 0, 0, 7, 0], [0, -1, 0, 0, 12, 6, 0, 1], [30, 0, -4, 8, 0, 0, 8, 0], [-12, 0, -1, -7, 0, 0, 5, 0], [0, 15, 0, 0, -18, 13, 0, 2], [-31, 0, -5, 3, 0, 0, -4, 0], [0, -6, 0, 0, -8, -2, 0, -7], [0, 0, 7, 3, 0, 0, 3, 0], [23, 0, 4, 5, 0, 0, 1, 0], [12, 0, -7, 17, 0, 0, 8, 0], [0, -17, 0, 0, -12, 6, 0, 5], [14, 0, 1, 17, 0, 0, 14, 0], [30, 0, 2, 0, 0, 0, 9, 0], [0, -8, 0, 0, -36, -1, 0, -6], [4, 0, 15, -3, 0, 0, -3, 0], [0, 15, 0, 0, -3, -9, 0, 0], [-6, 0, -11, 1, 0, 0, 9, 0], [0, -19, 0, 0, 18, -9, 0, -1], [0, 13, 0, 0, 26, -7, 0, 2], [0, -18, 0, 0, 5, 5, 0, 0], [0, -4, 0, 0, 2, -5, 0, -5], [31, 0, 2, 3, 0, 0, -7, 0], [36, 0, -3, -3, 0, 0, -2, 0], [-37, 0, 2, -6, 0, 0, -8, 0], [0, 0, 0, 0, 21, -13, 0, -2], [0, -14, 0, 0, 0, -6, 0, 6], [24, 0, 7, -3, 0, 0, -3, 0], [0, -26, 0, 0, 0, -3, 0, 4], [0, 0, -5, 13, 0, 0, -3, 0], [0, 22, 0, 0, -6, 9, 0, -3], [25, 0, 4, -8, 0, 0, 8, 0], [0, 2, 0, 0, 14, 9, 0, 0], [-31, 0, -2, 12, 0, 0, -6, 0], [0, 13, 0, 0, -15, -1, 0, -3], [-2, 0, 5, -11, 0, 0, 8, 0], [-15, 0, 2, -1, 0, 0, -8, 0], [0, -4, 0, 0, -30, -8, 0, 8], [0, 15, 0, 0, -16, 6, 0, 5], [11, 0, -7, 9, 0, 0, 5, 0], [-4, 0, -3, 7, 0, 0, 0, 0], [0, 4, 0, 0, -10, 11, 0, 0], [11, 0, 10, -4, 0, 0, -11, 0], [18, 0, -1, 1, 0, 0, -4, 0], [0, 3, 0, 0, 40, -4, 0, 1], [0, -6, 0, 0, -16, 4, 0, 1], [0, 0, 0, 0, -6, 19, 0, 2], [4, 0, -9, -3, 0, 0, -2, 0], [-48, 0, -2, 2, 0, 0, -2, 0], [0, -16, 0, 0, -39, 7, 0, -2], [0, 1, 0, 0, 27, 2, 0, -8], [0, 9, 0, 0, 4, 0, 0, 3], [12, 0, -3, -3, 0, 0, 1, 0], [-55, 0, 5, 2, 0, 0, -3, 0], [20, 0, 3, -7, 0, 0, -9, 0], [3, 0, 0, -7, 0, 0, -7, 0], [0, 20, 0, 0, 0, 10, 0, -5], [-18, 0, -8, 2, 0, 0, 4, 0], [0, -6, 0, 0, -24, -5, 0, 1], [0, -9, 0, 0, -30, 1, 0, -3], [0, -4, 0, 0, -50, 3, 0, -2], [-24, 0, -2, -2, 0, 0, 5, 0], [0, -30, 0, 0, -2, -5, 0, 1], [24, 0, 2, 18, 0, 0, 12, 0], [0, -3, 0, 0, 7, -9, 0, 2], [0, 15, 0, 0, -31, 6, 0, 0], [0, 4, 0, 0, 14, 3, 0, 7], [-42, 0, -1, -5, 0, 0, 9, 0], [5, 0, 9, -10, 0, 0, 2, 0], [0, -17, 0, 0, 26, -13, 0, -1], [0, 14, 0, 0, 26, 0, 0, 1], [0, 13, 0, 0, 1, 12, 0, -1], [-7, 0, -4, 13, 0, 0, -7, 0], [-25, 0, 5, -16, 0, 0, -5, 0], [0, 18, 0, 0, 38, 10, 0, -2], [0, -12, 0, 0, -16, -2, 0, 6], [0, -13, 0, 0, 26, -1, 0, 3], [41, 0, -2, -4, 0, 0, -3, 0], [-18, 0, -11, 13, 0, 0, 15, 0], [6, 0, 2, 2, 0, 0, -3, 0], [0, 10, 0, 0, 0, -12, 0, 8], [0, -8, 0, 0, -19, 1, 0, -6], [39, 0, 5, -3, 0, 0, 0, 0], [2, 0, 5, -15, 0, 0, -8, 0], [-32, 0, 7, 25, 0, 0, 1, 0], [0, -13, 0, 0, 15, -10, 0, -3], [27, 0, -1, 14, 0, 0, 0, 0], [0, 3, 0, 0, 3, 13, 0, -7], [0, 6, 0, 0, 8, 6, 0, -2], [0, 8, 0, 0, -23, 0, 0, 2], [0, 2, 0, 0, -4, -1, 0, 5], [6, 0, -6, -12, 0, 0, 0, 0], [0, 24, 0, 0, 14, 0, 0, 7], [-3, 0, 0, 3, 0, 0, 4, 0], [0, 0, 0, 0, -1, 0, 0, 2], [24, 0, 5, 3, 0, 0, 1, 0], [2, 0, -5, 15, 0, 0, -7, 0], [-4, 0, -1, 7, 0, 0, -7, 0], [0, 9, 0, 0, -1, -12, 0, 0], [0, -18, 0, 0, 0, -5, 0, 8], [11, 0, 6, -1, 0, 0, -5, 0], [30, 0, -1, 17, 0, 0, -2, 0], [0, 0, 1, 11, 0, 0, 1, 0], [-26, 0, 7, 3, 0, 0, -5, 0], [8, 0, 8, -20, 0, 0, -12, 0], [0, -21, 0, 0, -38, 1, 0, -4], [28, 0, 13, 5, 0, 0, 6, 0], [66, 0, 3, 5, 0, 0, 5, 0], [0, 9, 0, 0, -34, 9, 0, 1], [7, 0, 3, -13, 0, 0, 4, 0], [0, 5, 0, 0, 27, 14, 0, -2], [8, 0, -13, 1, 0, 0, -2, 0], [21, 0, 7, 9, 0, 0, 6, 0], [0, 12, 0, 0, -36, 10, 0, -1], [15, 0, 14, -3, 0, 0, -6, 0]];
    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_da();".
// 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_da(:prec:=8)
    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_da();".
// 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_da( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, 0, -38, 0, 85, 0, -18, 0, 1]>,<11,R![257, 62, -38, -2, 1]>,<13,R![64, 0, -288, 0, 340, 0, -38, 0, 1]>,<19,R![84100, 0, -119588, 0, 7997, 0, -160, 0, 1]>,<23,R![-8, 156, -47, -2, 1]>],Snew);
    return Vf;
end function;