
// Make newform 8450.2.a.ct in Magma, downloaded from the LMFDB on 26 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_8450_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_8450_2_a_ct();".
// 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_8450_2_a_ct();".
// 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, 8, 26, -33, -69, 53, 17, -14, -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, -8, -7, 33, -4, -11, 1, 1, 0], [-5, 3, 9, -34, 35, 6, -12, 0, 1], [3, -9, -8, 35, -35, -6, 12, 0, -1], [-3, -10, 24, 30, -46, -5, 13, 0, -1], [2, -14, -23, 42, 35, -18, -11, 2, 1], [0, -19, 0, 73, -42, -18, 13, 1, -1], [0, 19, 24, -74, -30, 29, 10, -3, -1]];
    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_8450_a();"
function MakeCharacter_8450_a()
    N := 8450;
    order := 1;
    char_gens := [677, 3551];
    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_8450_a_Hecke( : Kf := Kf)
    return MakeCharacter_8450_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]];
    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, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, -1, 0, 1], [0, 0, 0, -1, -1, -1, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, 0, 0, -1, -2, 1, 2], [0, 0, 0, 0, -1, -1, 1, -1, -2], [-2, -1, 1, 1, 0, -1, 1, -3, -1], [2, 2, -1, 2, 2, 0, -1, 1, 1], [1, -1, 1, -2, -1, 2, 1, 2, -1], [0, -2, -1, 0, 1, -1, 0, -1, 1], [4, 2, 0, 1, 3, 1, 0, 1, 1], [-5, 2, -1, 1, 1, -1, 1, 1, -1], [-1, 0, -2, -1, 1, 3, 4, -1, -2], [-4, -2, -3, -2, -3, 3, 2, -3, -1], [1, -4, -2, 0, -2, 3, 5, -2, -2], [2, 3, 3, 0, 3, 1, -3, 4, 1], [-1, 5, 1, 1, 2, -2, 0, -1, 1], [2, 2, 3, 4, 1, -3, -6, 5, 3], [2, 0, 2, -1, 0, 2, 0, -2, 1], [-2, -2, 2, 0, 0, -2, 2, 0, 0], [-3, -1, 2, 0, -2, -1, 7, -4, -2], [4, 3, 1, 0, -2, -1, -3, -3, 0], [-3, 0, -3, 0, 2, -1, 2, -1, 0], [2, 0, 3, 2, 4, 3, 3, 0, 0], [-4, 0, 1, 1, 1, -1, -7, 3, 0], [2, -1, -1, -5, 0, 2, -1, 3, -2], [3, 2, -1, 0, -1, 2, 0, 5, -1], [-4, -2, 0, 1, 0, -2, 6, 0, -3], [-7, -4, -2, -4, -2, 3, 4, -6, 0], [4, 4, 0, -2, -1, -1, 3, 1, 0], [6, 0, 2, -1, 6, -2, -2, 6, 1], [0, -4, 0, -1, -3, -1, -9, 2, 0], [4, -4, -2, 2, 1, -1, 3, 0, 0], [-5, 5, 1, -2, 3, 2, 7, -2, -1], [-5, -3, -2, -2, 0, -1, -1, 1, -4], [6, 1, 0, 2, 0, -4, -4, 7, 3], [0, 1, 0, 0, 0, 4, 1, 1, -2], [-6, -5, -2, -3, -2, 6, 4, -5, -5], [-2, 2, -2, -1, 1, -1, 1, -4, -4], [-4, -3, 1, 3, 0, -3, -1, -1, 3], [2, -3, -5, -1, 1, 3, 0, 6, -4], [1, 5, 0, 0, 3, -2, -12, 10, 4], [7, -5, 1, 0, -1, 2, -5, -4, 1], [-3, 1, 2, -2, -4, -1, 3, 3, -4], [4, 8, 2, -1, 3, -3, -3, 2, 2], [3, -1, 1, -4, 0, 3, -2, 0, -3], [1, 0, 6, 6, 3, -2, -4, 0, 0], [4, -3, 3, 3, 2, 2, 1, -4, 0], [-4, -2, -4, -3, 0, -4, -8, -2, -3], [1, 3, 4, 4, 2, -3, -9, 7, 8], [-1, -5, 0, -6, -1, 1, 6, -4, -5], [4, 0, 4, 1, 1, -1, -3, 0, 2], [-1, 5, 2, -4, 5, 2, 10, 2, -4], [1, 6, 0, 2, 2, -1, -8, 16, 4], [2, 7, -1, 3, 4, 2, -7, 10, 8], [-7, -1, -1, 0, -3, 0, -5, -4, 3], [-7, 4, -2, 5, 2, 1, 1, -4, -1], [-1, -5, -5, -1, -3, 0, -5, -8, -1], [-13, -1, -4, 0, 4, 7, 9, -4, 0], [11, 1, 0, -2, 4, -3, 1, 11, 2], [4, -7, -2, -4, 0, 2, 0, -3, -3], [-8, 6, -1, 0, 1, 1, 6, -7, -1], [-3, 4, 5, 2, 4, 3, -6, 5, 0], [-4, -5, 3, 3, 3, -1, 6, -8, -2], [-10, -6, -2, -8, -4, -2, 4, -2, 0], [-5, -1, 0, 0, -1, 2, -4, -4, 4], [7, -8, -7, -1, -6, 5, 2, -1, -2], [0, -3, -5, -1, -1, 3, 0, -4, -4], [1, 10, -1, 4, 10, 3, 4, -1, 4], [3, -2, -2, 3, 2, 3, -9, 4, -1], [-12, -3, 0, 1, -3, -1, 7, -1, 3], [-13, 2, -2, 1, 2, -5, 3, 4, 3], [-5, -2, 4, -2, -4, -5, -1, -8, 2], [-12, 3, -7, 1, 4, -4, 7, -4, -4], [-10, -1, -2, -5, 1, 2, 11, 0, 2], [18, 9, 4, 3, 6, 2, -2, 9, -1], [9, -5, -1, 0, 4, 7, -1, 0, -4], [8, -2, 4, -2, 2, -6, -7, 2, 0], [0, -2, 0, -4, -2, -2, -12, 0, 8], [-8, 0, 1, 6, 2, -6, 3, -1, 3], [-16, -4, -6, -2, -8, -2, 12, -18, -4], [-24, -6, 0, -3, -2, -2, 6, -8, 3], [-2, -9, 4, 3, -4, -6, -6, -1, -3], [0, 5, 2, 6, 2, 0, -6, 1, -1], [0, 6, 6, -2, 4, 0, -2, -1, 4], [-12, 4, -2, -4, 0, 2, 4, 6, 2], [5, 6, -2, -1, -1, 3, -4, 9, -2], [1, 8, 0, 3, 0, 5, 1, -2, -1], [-1, 5, 5, 5, 0, -2, -10, 5, 3], [-2, 13, 3, 1, 11, -7, -6, 16, 8], [5, -9, -4, -2, -1, 1, 14, -8, -5], [-1, -8, -2, -4, -10, 3, 3, 0, 4], [-14, -6, -6, -7, -5, 1, 5, -2, -6], [-3, -5, 2, 3, 0, 1, 8, 5, -1], [6, -1, -2, -9, 1, 5, -3, 3, 5], [10, 1, 7, 2, -6, -5, -19, 17, 4], [-6, -6, 3, -3, -5, 4, 2, 6, -2], [-3, -6, -5, -1, 2, -1, 2, 7, -8], [5, 4, -6, 5, 4, 3, -4, 8, -1], [5, -5, -2, -4, -2, 1, -7, -5, -4], [1, 10, 1, 2, 1, -6, -10, -1, 1], [-8, -6, -4, 6, -8, -8, -1, -6, 2], [-13, -4, -12, -4, -10, 9, 11, -2, -2], [-1, -5, -4, -6, -7, 16, 10, -8, -6], [11, -1, 2, 1, 3, 6, -9, 10, 1], [-13, -6, -3, -8, -12, 1, 6, -17, 2], [2, -7, 5, -7, -9, 3, 6, -4, -10], [-2, 0, -4, -5, -3, 5, 12, -5, 1], [-15, -8, -5, 1, -4, 4, 6, -14, -5], [2, -4, -5, 4, -1, 7, -6, -1, 1], [0, 8, 8, 1, 0, 0, -6, 6, -3], [-10, 2, -8, -10, 4, 8, 8, 4, -2], [-11, -2, -3, 5, 3, -4, 5, -5, -2], [0, 14, 2, 3, -1, -7, -6, 1, -3], [-4, -9, 3, 3, -10, -10, -1, 3, -8], [8, 5, -6, 1, 4, 8, 14, -9, -1], [8, 4, 2, -2, 16, 0, -2, 12, 2], [-19, 0, 4, 4, -4, -9, 11, 2, -4], [-11, 7, -1, 0, 10, 2, -8, 11, 6], [-8, 6, 8, 3, -2, -10, 0, -4, -3], [-9, -7, 8, 0, -2, -3, 9, -9, -2], [3, 14, -1, 5, 3, -3, -9, 21, 5], [4, 6, 4, -6, 3, 9, -3, 3, -2], [-25, -6, 2, -2, -4, -3, 14, -24, -2], [-7, -5, -10, 3, -4, 3, 8, -13, -5], [11, 2, 6, -3, -4, 3, 7, -4, 3], [-24, 1, -6, 1, -7, 4, 13, -14, -2], [6, 3, 10, 3, -2, -10, -16, 1, 1], [1, -10, 4, 2, 2, -1, -3, 4, 0], [-3, 3, -4, -3, -8, -7, -16, 7, 5], [-13, 2, -8, 5, 4, -3, 13, -6, 3], [-17, 4, -7, -3, 3, 6, 13, 3, 2], [-13, 7, -1, 11, 3, -6, -3, -14, 5], [-17, 1, -11, -6, -6, -1, 11, -6, -6], [-4, -11, -5, -11, -5, 7, 14, -20, -2], [-7, -5, -1, 5, 4, -5, 2, -18, 5], [4, -6, 0, -4, -4, -4, -8, 0, 4], [-14, 10, 8, 7, 3, -5, -6, 3, 1], [-14, 2, 4, 1, 1, -1, 1, 4, 2], [-1, -1, 4, 2, -3, 5, -2, 4, 7], [9, 9, 8, 2, 8, 5, -3, 13, 2], [-5, 4, 6, -6, -7, -2, 4, 8, -2], [21, -1, 0, 10, -1, -2, 4, -1, 0], [-21, -13, 1, 2, -11, 0, 1, -14, 3], [-6, -11, 2, -5, -10, -8, 6, -9, -9], [5, -14, -1, -16, -6, 7, 8, -1, -16], [1, -2, 4, -2, -6, 11, 15, -6, 2], [-2, 3, 7, -7, 0, 2, 7, 10, -4], [-20, 5, -8, 1, -4, 4, 22, -13, 1], [6, -1, 2, 4, -1, -9, -1, 3, 3], [-9, -11, 1, -1, 6, -9, 0, -4, -1], [0, -11, -7, -4, -9, 9, 1, -8, -17], [-13, 2, -4, -3, 10, -1, -2, 6, 3], [14, 2, 1, 6, 5, -5, -18, 7, 11], [2, -14, -3, -2, 1, 13, 14, -11, -13], [4, -7, -1, 1, -15, -2, -14, 10, 5], [-5, -9, 4, 0, -9, -4, -22, 10, -4], [5, -13, -5, -1, -7, 2, 12, -6, -10], [-23, -9, -10, -1, -17, 0, 13, -18, -5], [1, -1, -4, -10, -3, 2, 20, -6, -16], [5, 4, -8, 1, 4, -3, -3, 6, -7], [-8, 0, 4, 0, 1, 1, 5, 19, 4], [11, -5, 0, 6, -9, -6, -8, 0, -8], [6, -4, -3, -2, 10, -4, -5, 17, -1], [-6, -11, -1, 5, 1, -5, 4, -16, 2], [10, 2, 3, 8, 17, 5, -4, -1, 1], [8, 13, 5, 5, 9, -4, -12, 26, 17], [16, -4, -2, -10, 0, 2, 14, 0, 0], [-7, 8, -6, -2, 0, 15, -3, 6, 6], [-1, 3, 3, 3, -7, 8, -6, -2, 6], [-4, 6, -3, -8, 3, 3, 10, -3, 7], [21, 4, 7, -8, 4, -1, -10, 13, 8], [-3, 0, 0, -1, 2, -7, -17, 6, 13], [4, 4, 0, 11, -5, -3, -13, 6, 11], [-4, 6, -8, 4, 0, 0, -2, 0, -4], [0, 4, 0, 6, 7, 10, 5, -9, 5], [4, 10, 3, -2, 1, 11, 4, 15, 1], [6, -1, -1, 1, 6, 1, 5, 9, -7], [-19, 6, 0, 4, 4, -3, -2, 0, 8], [-14, -2, 8, 8, 1, -7, -11, -3, 4], [-4, -11, -2, 5, -10, 4, -16, -5, 7], [11, -2, 3, -10, -6, 5, 0, 23, 6], [-18, -7, -2, -1, -9, 3, 17, -21, -1], [-21, 3, -2, -3, -2, 3, 8, -17, 7], [-17, 9, 0, 4, 8, -3, 19, -19, 2], [5, 4, 0, 9, 0, -5, -20, 0, 5], [-1, 3, 3, 5, 1, -4, 7, -14, 1], [-12, 1, -1, -3, -5, 1, 4, -10, -12], [-3, 7, -4, 6, 1, 2, -12, 6, -6], [6, -5, -6, 5, 3, 7, -20, 7, 4], [4, -8, -4, -8, -9, 1, 13, -15, -10], [7, -10, 3, 7, 0, -13, -2, -13, 0], [0, -2, 1, -7, 1, -1, 20, 3, 0], [1, -7, 4, -10, -13, 0, 6, 6, -6], [-13, 9, 5, 2, 8, 5, 21, 8, 2], [17, 5, 3, -10, -3, 6, -15, 14, 13], [-9, -7, -8, 6, -15, -2, 10, -10, -4], [3, 1, 3, -2, -4, -5, 16, -4, -9], [11, 8, 7, 17, 12, -6, -20, 2, 5], [-10, 0, -1, -10, 5, 13, 10, -7, 1], [-8, 5, -5, -3, 17, 3, -2, 10, 6], [-23, 3, -5, -8, -8, -5, 1, 2, 2], [0, -8, -6, 1, -13, -9, -13, 4, 2], [13, 4, 12, 5, -8, -9, -23, 6, 3], [3, -8, -3, -17, -5, 10, 3, -1, 6], [16, 6, -1, -5, 9, 10, -2, -4, 2], [26, -2, 2, 1, 3, 3, -17, 24, 11], [16, 4, 14, -11, -3, 1, -27, 20, 4], [-21, -14, 5, -12, -12, -3, 16, -27, -8], [12, -7, -3, -8, -1, -1, -11, 12, -3], [-10, 9, -8, 5, 1, -3, 9, -13, -9], [-10, 4, 1, 3, -9, 6, 4, -4, 0], [-15, 7, -4, 0, 12, 13, 1, 17, 0], [9, 8, -1, 8, 1, 8, 0, 7, 1], [3, 20, 3, 9, 6, 1, -6, 25, 16], [11, -10, 3, -12, -7, 2, 0, -7, -15], [-8, -6, -4, -10, -5, 8, 17, -5, 3], [4, 23, 0, 1, 18, -8, 0, 11, 7], [5, 10, 6, 7, 17, -4, -4, -10, 17], [-6, -19, 3, -3, -5, 3, 18, -28, -12], [-6, -7, 5, 2, 0, -5, 9, -9, 0], [-3, 23, 5, 22, 2, -6, 2, -7, 10], [27, 7, 16, 0, 3, 9, -2, 18, 1], [-8, 6, 8, 8, -3, -5, 5, -2, 4], [-4, -2, -4, -4, 4, 6, 10, -20, -8], [17, -2, -1, 3, 11, 12, -5, 13, 0], [8, 11, -5, -5, -6, -2, -29, 16, 4], [21, -2, -2, -12, 4, 7, -9, -2, -10], [1, 10, 1, 5, -5, -2, -25, 17, 20], [-23, 12, -2, 6, -2, -7, 11, -28, 6], [-8, 0, -12, 6, 10, 6, 6, 8, 2], [-2, 2, -4, -2, 2, 4, 4, -7, 0], [-7, 5, 3, 13, -2, -11, 0, 6, -3], [12, 11, 12, -3, 3, -4, -15, 0, 4], [14, 13, -11, 2, 2, 11, -1, 1, 2], [-1, -2, 2, -5, -10, -9, -3, -10, 3], [2, -10, -12, -14, -7, 3, 25, 1, -18], [28, 8, 5, 14, 7, -15, -22, 19, 5], [-9, -7, -8, -18, -13, 7, 38, -14, -13], [22, 8, 8, 14, 7, -13, -23, -3, 0], [-22, 6, 7, 3, -3, 1, -11, -21, 10], [31, 15, 9, 2, 14, 4, -20, 39, 2], [14, 4, -2, -4, 0, -8, -12, 12, -10], [-18, -5, -8, -9, -12, 12, 24, -29, -5], [2, 7, 4, -6, 6, 0, -3, -1, -14], [0, 10, -4, 0, -8, 2, 16, 0, 2], [5, -14, 2, -2, -8, -5, -1, -6, -16], [29, 7, 6, -2, 9, -4, -30, 31, 6], [19, -9, 1, -6, -7, 0, 21, 10, -15], [21, -10, -3, -8, -3, 2, 15, 7, -11], [4, -4, 5, -1, -9, -15, 1, -13, -12], [10, -19, -7, -8, -10, 15, 7, -3, -20], [6, -5, 0, -3, -4, -4, 16, 1, 3], [-14, 2, -14, -6, 4, 0, 4, 14, -8], [-11, 0, -5, 1, 0, -3, -4, -9, 2], [-29, -8, 1, -3, -2, -11, 14, -11, 2], [22, 0, 9, 14, 15, -3, -20, 7, 11], [-8, -12, 4, -5, -16, 0, 6, -4, 7], [17, -7, 0, 8, -6, 11, 23, -7, -10], [6, 10, -1, 17, 17, -4, -16, -4, 16], [2, -13, 1, -5, -6, -13, 7, -3, -1], [-3, 10, 12, 11, 10, 5, 15, -12, 13], [-19, -4, -9, -10, -18, 1, 28, -27, -10], [-14, 22, -6, 9, 15, -5, -3, 12, 6], [19, 13, 7, 4, 12, -9, 10, 26, -1], [21, 6, 3, 2, -13, -10, -16, 17, 13], [-14, 10, -3, 13, 7, -5, -4, -13, -2], [12, 5, 6, -3, -6, -2, 4, 7, 7], [4, -1, -10, 0, 14, -8, -1, 19, -4], [-7, -23, 4, -11, -13, -4, -7, -18, -19], [17, 3, 0, 16, -13, -6, -18, 8, 12], [4, 5, 13, 13, -11, -15, -30, 26, 14], [-3, 3, -12, 12, -9, -6, -30, 2, 12], [-17, 1, 2, 2, -5, 8, 18, 5, 2], [23, 13, 2, 7, 5, -12, -13, 14, 11], [7, -4, 0, 13, 6, -5, 3, 4, 13], [4, -1, 16, 18, -1, -3, -21, -1, 7], [21, -14, -2, -8, 2, 5, 15, -2, -14], [15, 10, 2, 5, -5, 8, 2, 6, 3], [18, -6, 1, 2, 11, -7, -32, 9, 15], [26, -21, 1, -12, -1, 7, 19, -18, -13], [-12, -15, -3, 1, -14, -9, -3, -13, 11], [-10, 12, 5, 5, 9, -6, -18, 38, 8], [-29, 2, 0, 9, -4, 3, -1, -18, 13], [19, 5, 12, -12, -1, -2, 16, 12, 0], [3, 15, -3, 6, 5, -4, -11, 24, -3], [-3, 2, -1, 13, 7, -6, 3, 25, 14], [-40, 1, -11, 4, -6, 9, 21, -13, -2], [22, -2, -3, 2, 8, 6, -21, 29, 5], [9, 11, 11, -3, 4, -5, -6, 18, 1], [4, -8, 2, 0, -2, -8, -2, 12, -8], [7, 10, 6, 14, 8, -7, 19, 0, 4], [17, 3, -7, -2, 23, 6, -11, 2, 15], [0, 5, 0, 3, -7, 4, 3, 8, 6], [-13, -8, -3, -21, -11, -2, -3, -3, -6], [6, 14, -11, 10, -5, 5, -6, 13, 11], [1, 8, 8, 6, 22, -13, -39, 18, 14], [12, 10, -7, -3, -9, 12, -2, 20, 8], [13, 0, 11, 12, 4, -5, -20, 29, 2], [-3, -10, -10, -15, -8, 1, -3, -4, 3], [-33, -24, -15, -13, -21, 2, 3, -41, 2], [13, -19, 2, 5, -1, 0, -9, -16, 9], [7, -6, 4, -12, 6, -1, -13, 12, 10], [-21, -4, 19, 10, -14, -5, 14, -17, 4], [29, -3, -8, -5, -5, 12, -1, 4, 1], [9, -6, 10, 9, 5, 0, -20, 26, 3], [-16, -1, -8, -9, 0, 2, 8, -11, -9], [-9, -24, -1, -7, 1, -6, 5, -23, 0], [36, 1, -6, -1, 5, 3, -7, 1, -3], [9, -12, 3, -14, -15, -6, 14, 9, -11], [17, -2, -3, 5, 6, 19, -6, -1, 0], [-22, -6, -5, -7, -13, 7, 26, -29, -12], [0, 2, -7, -7, 9, -1, -7, 17, 4], [16, -7, 3, -2, -18, 3, -13, -1, -8], [-24, -14, 4, 10, 2, -10, 4, -28, 6], [16, 5, 2, -9, 14, 4, -26, 37, 7], [3, 19, 4, 0, -5, 0, -20, 2, 12], [1, 21, 7, 22, 8, -9, -5, -20, 8], [19, -12, 4, -14, -16, -5, 15, -18, -14], [-3, 3, -8, 0, -1, 14, -28, -6, 14], [11, 9, 8, 16, 6, -9, -13, 15, -4], [-3, 0, -25, 2, 15, 2, 14, 1, -7], [-9, -5, 0, 6, -17, 0, 12, 8, 6], [-9, -14, 0, -2, 4, 6, 17, 11, -2], [-7, 14, -8, 4, 8, 1, -5, 10, 10], [20, 2, 15, -13, -7, 8, -6, 10, -2], [-2, -12, -9, -12, -16, -6, -9, 15, -3], [40, 3, 7, -3, -3, 3, -14, 0, -8], [3, -3, 3, -13, 7, 6, 22, -6, -6], [-1, 17, -4, 8, -12, -9, -11, -21, 6], [14, 19, -1, -7, -1, 3, -2, 2, -6], [-21, -6, 1, -12, 1, 12, 24, -15, -15], [18, 4, -8, 1, -9, 11, 7, -16, -6], [-8, -5, -6, 6, 6, 0, 22, -7, -1], [42, 2, 10, 4, 3, 7, -33, 10, 0], [-1, -9, 2, 0, -3, 6, 28, 0, -26], [17, 8, 6, 4, -4, 12, -11, 9, 18], [-40, 10, -7, -6, 6, -2, 37, -11, -19], [8, 8, 17, 16, -3, -1, -24, -3, 9], [-1, 5, 16, 16, -17, -14, -10, 0, 2], [29, -1, -17, 2, -8, 2, -6, -9, -4], [-27, -7, -1, -10, -9, 8, 3, -4, -19], [8, 1, 10, -7, -10, -2, -2, 1, -7], [12, 22, -6, 14, 7, 7, 25, -11, 2], [-27, -6, -1, 7, -18, -2, 4, -28, -3], [21, 7, 6, 10, 5, -5, -8, -14, 13], [-8, 5, 12, 3, 10, -4, -20, 29, 25], [12, -7, 2, 1, 4, 10, 10, 31, -9], [3, 10, 24, 18, 4, -19, -35, 10, 16], [18, -4, 6, 19, 2, -18, -44, 38, 9], [25, 19, 2, -2, -7, -1, -10, 22, 11], [-15, -8, -8, -19, -17, -1, 10, -13, -20], [8, 18, 3, -13, -11, 1, 9, -3, -6], [5, 13, -2, -8, 1, 12, -14, 16, 12], [-43, -9, 6, -8, -4, -13, 13, -17, -10], [10, -14, -4, -14, -7, 21, 13, -9, -12], [13, 17, 0, 12, 5, -14, -22, 20, 0], [-5, 2, 15, 15, 6, -10, -22, 34, 11], [14, 8, 13, 0, 5, -17, 14, 1, 1], [-11, 6, -11, 2, -21, 6, -6, -7, -9], [52, 1, -4, 10, 12, 4, -15, 9, 12], [-12, 6, -18, 2, -3, 9, 1, -5, 4], [-28, 26, -1, 13, 11, -4, -28, 22, 24], [11, -5, 4, -8, 7, 0, -2, 20, 16], [-3, 23, 6, 16, 20, 3, -13, 25, 14], [-7, -17, -22, -26, -3, 8, 22, -11, -22], [2, -8, -4, 5, 1, 5, 7, -4, 5], [-5, -2, 10, 6, 6, -9, 19, -2, 2], [11, 0, -12, -4, -2, -3, -13, 16, 10], [-30, -10, 12, -12, -17, 4, 31, -39, -11], [-24, 5, -14, 3, -15, 3, -9, 3, -7], [17, -7, 2, 2, -24, 5, -13, 21, 14], [-45, -3, -18, 2, -8, -9, 11, -39, 2], [2, 16, -4, 8, 14, -8, -12, -16, 2], [-17, -11, -14, 0, -14, -3, 25, -13, -24], [-7, 5, 2, 12, 3, -8, 16, 14, 0], [25, -8, -13, 3, 0, 3, -4, 24, -2], [-11, -11, 4, 2, 1, -10, -30, -4, 0], [10, -4, 5, 11, 9, -21, -25, -5, 16], [-7, 6, 15, 6, 2, -9, 16, -5, -10], [12, -6, -1, -2, -12, 5, -27, 0, -24], [-33, 8, -1, 2, -6, 7, 18, -7, -10], [8, -4, 8, 10, -21, 3, 33, -9, -8], [10, 0, -13, 2, -8, 5, -1, 12, -8], [-14, 18, 3, 12, 11, -5, 14, -35, 13], [-21, -6, 8, 15, -6, -5, -1, -18, 1], [5, 38, 3, 3, 23, 9, -3, 23, 15], [-22, 11, 0, 18, 8, -2, 13, -29, 4], [7, -15, 3, 18, -14, -5, -25, -4, 6], [5, 0, 9, -1, -5, -2, -25, 13, 18], [-31, 2, -6, 20, 6, -7, -13, -4, 8], [-17, 15, 8, 13, 13, -12, -15, 22, 9], [5, -1, 9, 2, -1, 2, -49, 28, 1], [31, 8, 7, 6, 2, 5, -16, 19, 14], [-16, -23, -5, -1, -19, 3, 18, -12, 0], [-29, -16, -2, 2, -18, -12, 17, -37, -4], [28, 20, 26, 12, 25, 9, 5, 17, 20], [25, 8, -14, -7, -9, 17, 4, 11, -2], [-10, -5, -2, 11, -12, -8, 10, -35, -3], [-4, -12, -4, -17, -8, 6, 0, 8, -15], [44, -13, -10, -3, -3, 11, -11, 31, -3], [-22, -18, -12, -7, -6, 24, 30, -16, -13], [7, 2, 15, -11, -6, 6, -2, -8, -11], [4, 3, 10, 17, -8, -12, -16, 1, 3], [-16, 6, 8, -6, 2, -4, 20, -4, 4], [7, -6, 11, -5, -4, 13, 18, -4, 4], [4, 14, -7, 1, -7, -9, -20, 7, -4], [-11, -20, -8, 14, 8, 3, 19, -12, -2], [-20, 0, -4, -11, -8, 14, 16, 0, -1], [-15, -10, -11, -11, -1, 28, 25, -11, -8], [-49, -14, -29, 1, -7, -3, 25, -33, -1], [8, 6, 6, 7, -3, -7, -9, 30, -2], [-32, -12, -12, -9, -22, 16, 6, -18, -13], [-18, 1, -14, -5, 7, 4, -7, -4, 2], [-40, -12, -1, -2, -21, -1, -8, -17, 13], [-3, 16, 6, -5, 23, 5, -12, 37, 4], [-27, 12, -5, 14, 2, -19, 6, -5, 10], [3, -1, 3, 8, 14, 13, -10, 28, 21], [16, 1, 17, 1, -2, -1, -7, -21, 19], [27, -4, 11, 1, -5, -28, -45, 21, 2], [16, -14, 8, -14, -1, 13, 13, -24, -6], [-22, -6, -2, 14, -4, -16, 2, -38, 12], [-16, 4, -20, -17, -18, 6, 22, -20, -13], [30, 1, -4, 9, -2, 8, 0, 1, -3], [-13, -2, 8, -1, -6, -21, -10, 4, -9], [46, 8, 2, -7, 7, -9, -10, 19, -17], [14, -14, -8, -19, -21, 3, 21, -20, 2], [-23, -13, 7, 10, -11, -22, 15, -24, -17]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8450_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8450_2_a_ct();".
// 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_8450_2_a_ct(:prec:=9)
    chi := MakeCharacter_8450_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_8450_2_a_ct();".
// 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_8450_2_a_ct( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8450_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, -14, 21, 111, 49, -70, -40, 7, 7, 1]>,<7,R![104, -80, -380, -24, 310, 108, -55, -26, 1, 1]>,<11,R![-1688, 3900, 2084, -3185, -938, 672, 126, -45, -4, 1]>,<17,R![-28664, -72684, -26144, 17737, 7138, -994, -552, -11, 12, 1]>,<31,R![3017216, -831872, -1037568, -64200, 58096, 6852, -1052, -152, 6, 1]>],Snew);
    return Vf;
end function;