
// Make newform 8967.2.a.bb in Magma, downloaded from the LMFDB on 23 February 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_8967_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_8967_2_a_bb();".
// 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_8967_2_a_bb();".
// 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, 12, 30, -26, -81, 14, 54, -2, -13, 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, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0], [11, 30, -26, -81, 14, 54, -2, -13, 0, 1], [-3, -29, 20, 80, -45, -36, 19, 4, -2, 0], [-6, -31, 53, 62, -94, -9, 38, -6, -4, 1], [6, 33, 8, -100, -35, 81, 17, -23, -2, 2], [6, 51, 7, -141, -17, 99, 8, -25, -1, 2], [-9, -61, 11, 148, 8, -100, -7, 25, 1, -2], [7, 48, 8, -141, -49, 117, 25, -34, -3, 3]];
    Rf_basisdens := [1, 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_8967_a();"
function MakeCharacter_8967_a()
    N := 8967;
    order := 1;
    char_gens := [2990, 5491, 3235];
    v := [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_8967_a_Hecke(Kf)
    return MakeCharacter_8967_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 := [[0, -1, 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, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, -1, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, -1, -1], [1, 0, 1, 1, 0, 0, 0, 0, 0, -1], [-2, 1, 0, 0, -1, 0, 0, 0, 1, 0], [-1, 0, 0, 0, -1, 2, -1, 0, 0, 0], [-1, 1, 0, 0, -1, 0, -1, 0, 0, 0], [-2, 0, -1, -1, 0, -1, -1, 0, 1, -1], [1, -1, 1, 0, 0, 1, 1, 0, 0, 1], [-2, 0, -1, 0, 0, 1, -2, -1, 0, 1], [0, 2, 2, 0, 0, 1, 1, -2, 1, 0], [1, 1, 1, -1, 1, 0, 1, -1, 0, 0], [-2, 1, -1, -1, 0, -1, 2, 0, 0, -2], [-2, 0, 0, 1, 1, 0, 0, 0, 1, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, -1, -1, 1, 1, 1, -1, 1, 0, 1], [0, 0, 3, 0, 0, 1, 1, 1, -1, -2], [-2, -2, -1, -1, 1, -2, -1, 2, 1, -1], [-4, 0, -2, 2, 0, -1, -3, 0, 0, 2], [4, -2, 0, -2, 1, 1, 1, 2, -1, 1], [0, 0, 2, 0, 2, 0, 0, 2, 0, 0], [-1, 1, -2, -2, -2, 1, 1, 0, -1, -1], [0, 2, 0, 1, 0, 0, -2, 1, 3, -1], [-2, -3, 0, 2, -1, 2, -3, 2, -4, -1], [2, -5, 2, 1, -1, 1, 1, 3, -2, -2], [0, 1, 2, 3, 2, 0, 1, 0, 0, -1], [-3, -2, 2, 1, 0, 2, -2, 2, -3, -1], [-2, 1, 1, -2, 2, -3, 3, -1, 4, -1], [-4, 0, 0, 1, 1, 0, 2, -2, -2, 0], [0, 4, 3, 4, -1, 0, 0, 1, -1, -3], [-5, 2, -2, 1, 1, -1, -3, 0, -1, 0], [-4, 0, 0, 2, 1, 4, -5, 1, -1, 2], [0, 2, -2, 0, -2, -1, -1, 2, 0, 0], [2, -3, -1, -3, 2, -2, 2, -2, 0, 0], [0, -3, 2, -1, -1, -3, 4, -1, 2, -2], [4, -4, 5, 4, -2, 3, -2, 1, -2, 1], [2, -1, 4, -1, 0, 1, 4, -2, 1, -1], [-2, 7, -3, 0, 0, -1, -1, -1, 1, -3], [-3, -3, 0, -1, 4, 2, -2, 1, 0, 2], [-2, -3, 2, 1, 1, -2, 5, -2, -2, -2], [-1, 3, -3, -3, 1, -4, 6, -3, 2, 3], [-11, 5, -5, -1, 2, -3, 1, -1, 4, 2], [-5, 3, 1, 5, 0, 1, 0, -2, 1, 1], [-4, -5, -1, -2, 3, 0, -1, 1, -4, 3], [-5, -2, -7, -2, 1, 0, -2, 0, -4, 5], [6, 4, -2, -4, 0, 2, -2, -2, -2, 0], [-6, 1, 0, 2, -2, 0, -4, 0, 0, -3], [0, -3, 2, -1, 0, 4, 1, 1, -3, -2], [2, -1, -1, -5, 3, 1, 2, -3, -2, 3], [-1, -1, -2, -2, -2, -1, 0, -2, 2, 4], [3, -6, -3, -1, 0, -3, 2, -1, -1, 4], [1, -1, 3, -4, -2, -2, 7, -3, 2, -3], [-6, -3, -4, -2, 5, -1, -1, 0, 4, 6], [-2, 0, -8, -1, 2, -5, -3, 2, 1, 2], [-4, -4, -2, -4, 1, 2, 2, -4, 1, 1], [-1, -1, 3, 3, -3, 0, 0, -3, -4, -1], [-7, 4, 3, 1, -2, -3, -3, 1, 3, 0], [1, -3, 4, 4, -1, -3, 1, 2, -4, -6], [7, 9, 3, 0, -6, 0, 1, -3, 4, -2], [4, -2, 10, 8, -1, 4, -1, 1, -2, 0], [0, -4, -2, -5, 1, 1, 6, -1, 4, 4], [2, -1, -1, 3, -4, 2, -3, 2, 0, -1], [-11, -4, -3, -3, 1, 0, -4, -3, 2, 2], [6, -3, 1, -3, 0, 2, -1, 0, -5, -1], [-3, 4, -2, -3, 0, 4, 1, -2, -4, -2], [6, -4, -3, -9, 0, 0, 5, -3, 0, 3], [-4, 1, -4, -2, 0, 3, -4, 4, -1, 4], [1, 3, -6, -4, -1, 0, 6, -4, 1, 5], [-3, 3, -2, 4, -2, 0, -5, 3, -5, -4], [-4, -6, -1, -1, 2, -6, 6, 0, 1, 3], [-5, -2, 0, -5, -2, -2, 3, 0, 5, 4], [-10, -1, -7, 1, 5, 1, -6, 1, -2, 10], [6, 4, -2, 0, 3, -6, 6, -3, 2, 2], [3, 0, -2, -1, -4, 2, -4, 0, -2, 2], [-1, -4, 1, 4, 1, 0, -2, 2, 2, -3], [0, 0, 5, -2, -1, -1, 1, -4, 2, -4], [-2, 1, -1, 3, -5, 4, -7, -1, 2, 8], [7, 1, 5, -1, -3, 7, 4, -4, 0, 3], [-2, 3, 5, 0, -5, 2, 7, -3, -1, -3], [2, -8, 3, 1, 3, -2, 0, 0, -3, -1], [6, -2, 5, 3, 0, 5, 1, 4, -3, -3], [2, 0, -3, -1, 1, 4, -4, 4, 2, 4], [7, -4, 3, 2, -1, 6, -2, 0, -6, 0], [-10, 8, 1, 2, -1, -2, -4, -1, 5, -1], [-6, -3, 1, 3, -4, 3, -2, 0, -3, 1], [-6, -2, -8, -1, 3, -6, -4, 0, -2, 1], [-4, 6, 4, 7, -1, 4, -3, 4, 0, 3], [-3, -1, -9, -8, 2, -8, -2, 1, 0, 1], [5, 3, 0, 1, -3, -1, 7, 0, 2, -7], [6, 0, 8, 5, -3, 2, 7, 0, 0, -1], [-7, 2, 3, 0, 0, -6, -3, 5, 7, 2], [2, 5, 5, 4, -1, -7, 0, 1, 1, -3], [2, -8, 5, 3, 2, 4, -3, 5, -5, -4], [-8, 12, 0, 6, -2, -4, -6, 0, 6, -2], [11, 2, -3, 0, 0, -1, -4, 0, 0, 3], [11, -8, 2, -1, 3, 5, 4, 0, -1, -1], [-3, -4, -2, 2, 1, 3, -6, 5, -1, 10], [-12, 0, 4, 4, -4, 7, -5, 2, 1, 2], [-12, -7, -2, 0, 2, -6, 0, 0, 2, -2], [1, 7, -8, 1, -1, -5, 1, -2, 2, -1], [-1, 1, -2, -1, -4, -7, 2, 3, 4, 1], [-1, 2, 0, -3, 0, -1, -2, -1, 2, -4], [3, -11, 1, 0, 2, 5, -2, 3, 1, 6], [-3, 0, -9, -1, 7, -6, -2, -1, 2, 3], [-7, 8, -1, -2, 2, -3, 4, -4, 6, -3], [6, -1, 6, 6, 1, 4, 0, 3, -5, 2], [-5, 2, -6, -4, 0, -5, 9, 0, -1, 2], [11, -2, 8, 2, -3, 5, 1, 0, -4, 3], [-12, -4, -12, -8, 5, -7, -7, 3, 2, 1], [-3, -5, 3, 1, -5, 7, 5, -2, -2, 6], [-3, -5, 3, -1, 1, 1, 1, -6, -1, -5], [-14, 8, -6, -3, -1, -5, -5, -7, 9, 6], [-8, 0, -10, -6, 2, -10, 2, -4, 6, 6], [-6, 10, 3, 5, 5, 1, -3, 3, -3, 0], [-1, 2, -3, 5, -2, -2, -8, 6, -6, -5], [2, -12, 0, -1, 0, -4, 5, 1, -5, -4], [-16, 0, -8, 3, -1, -6, 1, 2, -3, 0], [0, -1, 0, -2, 0, -1, -3, -4, -1, 5], [-7, -5, 1, -3, 3, 2, -2, 5, -1, -1], [-6, -2, 2, 8, 4, 2, -8, 4, -6, 0], [9, -15, 6, 1, -1, 7, -1, 4, -3, -1], [3, 11, 9, 5, -10, -1, 4, 0, 3, -7], [-2, 10, 6, -1, -4, 2, 1, -5, 7, -4], [-2, -2, -2, -5, 7, -3, 5, -3, 2, 2], [12, -2, 9, 9, -2, 10, 1, 3, -11, 0], [-6, 10, 2, 2, -4, 4, -6, 0, 2, -2], [4, -11, -7, -9, 7, 0, -1, 0, 3, 8], [5, -12, 4, 7, 2, 4, 1, 0, -5, 0], [-3, 2, 6, 5, 1, -2, 4, -5, 1, -4], [1, 4, -5, 0, 5, -9, -1, 1, 4, -5], [-9, 9, -7, 6, 0, -3, -7, 1, 2, -5], [6, -11, 2, 2, 0, 1, -2, 5, -4, 4], [3, 8, 6, 1, -3, 5, -5, -4, -5, -6], [13, -5, -1, 1, 5, -2, 5, 1, -4, -4], [6, -10, 1, -7, 2, 5, 8, -2, -1, 4], [-6, -4, -7, -4, -7, 4, -5, -3, 1, 1], [-17, 10, -6, -5, -7, -4, 1, -3, 14, 1], [0, 10, 1, -1, 0, -9, -1, 0, -1, -3], [4, 1, -6, -3, 7, 0, -5, 6, 0, 2], [-3, -10, -6, -8, 3, -6, 1, -3, 1, 8], [-3, 7, 0, -1, -7, 3, -5, 4, -1, -11], [2, 10, 2, 6, -4, 6, 3, -2, 1, -1], [-2, 0, 4, 1, -3, 0, 2, 4, 8, -2], [4, 5, -2, -4, 0, -2, 0, -10, 4, 1], [12, -4, 7, 2, 3, 4, 1, 3, 1, 2], [-13, -2, -5, -11, 1, -9, 2, -6, 6, 7], [7, 6, -8, -4, -2, -6, 6, -7, 5, 3], [-1, 4, -4, -2, 1, -5, -2, 3, -1, -2], [14, -12, -2, -12, -4, 6, 6, -2, -6, -2], [-4, 2, -2, -9, -7, -1, 7, -3, 8, 6], [8, 10, 6, 5, -3, -1, 4, 1, 2, 2], [-4, 3, 3, -1, -6, 7, 0, -8, 3, -5], [6, -9, 7, 0, 2, 1, 3, -5, -1, -3], [-3, 6, 0, 5, -6, 3, -6, 7, -4, -12], [11, 3, 6, -1, -3, 13, -1, 2, -2, -4], [-2, 19, -6, 4, 2, -2, -3, 0, 3, 2], [20, 7, 2, 4, 0, 4, -4, 0, -7, -5], [-6, -8, -1, 6, 2, -3, -3, 3, 5, -4], [-7, -3, -4, 3, 4, 4, -8, 2, -8, 4], [4, -1, -1, -6, 2, 5, 3, 1, -6, -1], [-15, 3, 0, 3, 8, -6, 0, 6, 3, 0], [-11, -2, 4, 7, -4, 0, -2, -2, -1, 3], [6, 8, 4, 11, -5, 10, -8, 2, -5, -3], [-6, 6, -5, 4, -2, -8, 10, -4, 6, -2], [0, 4, 3, -1, -8, 3, 7, -4, 3, -5], [-12, -5, -9, -2, 3, -14, 1, 3, 12, 3], [-1, -1, 7, 6, -4, 1, -4, 4, -3, -8], [-1, 2, 2, -1, -1, -4, 0, -1, 9, 4], [0, 1, 10, 2, 4, 10, -4, 2, -2, -5], [21, 1, 0, -8, 0, -2, 13, -1, 5, 2], [4, -18, 5, -1, 2, 9, 4, 5, -12, -4], [13, -6, 9, -2, -7, 2, 0, 4, -2, -7], [-1, 11, 9, 16, -8, 6, -7, 3, 6, -11], [-3, 5, 2, 6, -5, -2, -2, -2, -4, 6], [-4, 0, 6, 14, -2, 4, -6, 6, -2, -12], [-16, 0, -4, -4, -3, 0, -10, 5, 2, 1], [-7, 7, -6, 1, 4, -6, -4, -3, 10, 4], [8, 2, 2, 1, -3, 2, 8, -3, 2, -10], [2, -7, 2, -3, 1, 0, 2, 2, -4, -10], [3, -11, 3, 5, 5, 3, -2, 5, -14, 4], [-15, 5, -6, 1, -3, -5, 7, -6, 0, 0], [-7, -4, -2, -3, 3, -16, 9, 1, 0, -3], [14, 9, -2, 7, -4, -4, 0, 1, 6, -2], [-13, -3, 3, -1, -7, -1, -4, -1, 6, -4], [-19, -16, -2, -4, 11, -5, 3, 6, -6, -1], [-20, 6, -8, -1, 1, -5, 3, 1, 6, 0], [0, 9, -8, -10, -4, -4, 2, -8, 11, 7], [-2, -3, -5, 1, 0, -10, 3, -2, 0, 3], [-9, -12, -6, -10, 8, 2, -2, -1, 3, 3], [3, -4, -5, -3, -4, 1, -6, 1, 8, 4], [2, -6, 13, -1, -3, -1, 6, -3, 2, -12], [-3, -3, 2, 5, 2, 10, 0, 0, -4, -2], [6, 20, 0, -3, -1, -4, 0, -6, 10, 5], [-12, 5, 5, 14, -5, 0, -10, 2, 0, -6], [0, 3, -1, -3, 2, 3, 10, -4, 3, 3], [-4, -2, -4, -2, 8, -6, 6, 4, 0, -1], [1, -11, 4, -2, 5, -2, 4, -2, 5, 8], [-13, -2, 3, 1, 4, 2, -1, 4, 9, 4], [-4, 10, 5, -3, -2, 3, 6, -2, -1, -8], [11, 3, 2, -1, -3, -6, 5, 0, -4, 2], [-15, -5, 0, 1, 5, -8, 5, 4, 4, 2], [6, -12, 6, 11, -5, 14, -10, 6, -8, 0], [18, -2, -1, -1, -1, -13, 10, -1, 6, 1], [5, 8, 5, 10, -12, 6, 0, 1, -4, -4], [13, 3, -9, -1, 6, 11, -5, 1, -7, 8], [6, -10, -5, -9, 1, -3, 4, -7, -7, 2], [-3, 6, 5, 6, -1, -2, -1, -2, 4, 6], [3, -5, 3, -1, -6, 10, -11, -4, 0, -3], [3, -7, -1, 3, 2, 7, -2, 1, 1, 5], [-29, 1, -15, -10, 3, -10, 4, 4, 12, 6], [-1, -7, -1, -7, -14, 5, 4, -5, 0, 3], [-2, -2, -16, 1, 5, 2, -10, 4, -11, 15], [-7, 7, -5, 0, 0, -9, 7, -3, -5, -9], [-5, -5, 7, 7, -2, -6, -7, 6, -2, -7], [8, 10, 9, -1, -6, 2, 5, -1, 7, 9], [-4, -3, -9, -5, 8, 1, -9, 0, 3, 1], [3, 1, -6, -1, -5, -8, 0, 1, 11, 2], [11, 9, 13, 7, -6, 0, 7, -6, 10, -5], [-27, 14, -10, 8, 2, -17, -6, 2, 4, 7], [0, -1, 0, -3, 6, 0, -3, 1, 3, -10], [-3, 7, 2, 2, -5, -4, -6, 0, 10, -8], [9, 3, -3, -3, -5, -1, -6, -2, -2, 13], [19, -6, -4, -3, -6, -8, 9, -2, -2, 0], [-11, -15, -7, -11, 10, -2, -4, 8, -3, 2], [0, 11, -18, -15, 0, -7, -6, -6, 7, 2], [13, -2, 3, -6, 3, -2, 5, 4, -4, -6], [19, -1, -3, 2, -8, 7, -11, 4, -10, 4], [-24, 4, -10, -5, 9, -13, 7, -3, 6, 0], [-31, 5, -14, -6, 3, -1, -9, -4, 10, 18], [13, -17, 3, -1, 1, 9, -9, 7, -11, 5], [-23, 4, 5, 9, -3, 7, -7, 0, 6, 4], [24, 0, -2, -10, 5, 4, 8, 2, -11, 5], [-1, 4, -7, -8, 7, -19, 16, -5, 6, 3], [-14, -2, 12, -4, 2, 0, 16, -4, -4, 0], [17, -5, 3, 1, -7, 7, 3, 4, -14, -8], [-2, -7, 2, 0, -3, 10, 6, -9, -5, 6], [-11, 7, -7, 3, -3, -2, 7, -7, 8, 4], [2, -7, 13, 0, 7, 5, -1, 6, -4, 0], [-6, 2, 13, 13, -5, 3, -3, 9, -1, -4], [10, -4, 17, -5, 5, 1, 8, -1, -1, 13], [-3, 17, -3, 0, 2, -5, 1, -2, -5, -1], [17, 9, 3, 6, 8, -2, 10, 3, 2, -9], [2, -2, -2, -6, -2, -8, 14, -2, 0, -6], [8, -2, 12, 8, 1, 0, 3, 2, -4, -14], [-9, 5, -4, -8, 0, -17, 3, -4, 12, 1], [7, 4, 2, -6, -5, 9, -2, 9, -3, 0], [-15, 10, -10, -2, -1, -6, 5, -6, 0, -9], [13, 3, 1, 1, -7, 12, -10, 3, 2, -1], [13, -13, -3, -6, 4, 4, 11, 3, -14, 3], [-4, 2, 3, 8, 4, -7, 1, -3, 5, 0], [10, -4, 11, 9, 4, -1, -2, 2, -9, -2], [19, 1, 0, -1, 3, 1, -3, 0, -8, 1], [19, -4, 14, 8, -11, 10, 13, -2, -6, 4], [-26, 7, -9, -10, 7, -10, 13, -3, 9, 5], [-10, 1, -1, -9, 7, 2, -1, -2, 9, 4], [20, 0, 2, 3, -2, -6, 0, 7, -3, -9], [-4, 15, -12, -6, 1, 3, 3, -1, -1, 2], [-11, 5, -5, 1, 0, -12, -5, -2, 6, 7], [23, -1, 8, 4, 1, 7, 5, 5, -7, -15], [-18, -3, -7, -5, 1, 2, -6, 10, 0, -7], [37, -5, 8, -4, -2, 5, -1, -2, -7, -1], [14, 4, 16, 9, -8, -3, 0, -4, -6, 0], [11, -1, 8, 8, -7, 11, -11, 0, -4, 4], [0, 8, -1, 10, -1, 0, -8, 3, -1, -19], [0, -10, -11, -11, 3, -9, -2, -3, -6, 6], [-10, 23, 6, 3, -1, 3, -2, -9, 10, 4], [-28, 11, -12, -12, 0, -17, 8, -7, 11, 6], [-5, 15, 4, -1, -2, 12, -8, -1, -2, 8], [-3, -7, 3, -8, 3, -2, 14, -6, 0, -4], [-1, 0, 9, -5, -1, -13, 13, -2, 7, 2], [14, 6, 2, -6, -6, -16, 18, -2, 6, 0], [-4, -17, -5, -7, 3, 4, -9, 1, -8, 6], [-4, 18, 6, 0, -8, 2, -3, -2, 13, -3], [-1, -15, -3, 3, -6, 7, -13, 1, -1, 4], [-9, -8, -6, -1, 1, 8, -1, -6, -20, 1], [-1, -7, 3, -5, -7, 2, 2, -7, 9, -7], [5, -15, 3, 11, 6, 13, -8, 1, -11, 1], [-6, 3, 4, 11, -11, 5, -6, 3, 3, 5], [-7, 4, 2, 3, 4, -1, -4, 5, -6, 6], [-13, 13, -13, -14, 8, -11, 12, -10, 11, 6], [32, 4, 14, 4, 0, 4, 0, 6, 2, -12], [-11, 4, 19, 2, 5, 10, -3, -6, 4, 6], [10, -12, 5, 5, -7, 4, -8, 6, -7, -5], [-10, 16, 6, 7, 3, -9, -7, 5, 6, -2], [10, -6, 11, 8, 3, 2, -4, 1, 3, 7], [-6, -11, 2, -4, 0, 8, -10, 0, 6, 11], [-15, 2, -3, -1, -2, -2, 0, -8, 10, 7], [-22, 5, 9, -1, -4, 1, -2, 2, -2, -2], [7, 9, 10, 14, -8, 13, -7, 10, -2, -1], [3, 12, 4, 15, 0, 8, 0, 0, -5, -7], [-24, 0, -12, 4, -5, -5, -2, 2, 11, 11], [4, 14, -10, -3, 10, -13, 4, -1, 0, -10], [16, -8, 8, 8, -12, 18, -18, 4, -10, 1], [-4, 11, -15, -7, 1, 0, -9, 2, -5, 4], [8, -5, 4, 4, 8, 3, 0, 1, -14, 2], [-6, -24, -1, -3, 5, -1, -2, 9, -8, -3], [0, -4, 8, 8, 6, 2, -4, 8, 10, 6], [-2, -19, 5, -7, 9, -2, 11, -4, -9, -8], [-7, -10, 6, 9, -7, -8, -1, 9, -2, -17], [7, 8, 3, 6, 3, 5, 1, 1, -20, 3], [-7, -8, -4, -6, 13, -8, 11, -2, 2, 20], [24, 13, 7, 9, -2, 8, -1, 0, 0, -9], [-7, 8, -1, 5, -5, 8, 1, -4, -2, -15], [-8, 2, 6, 3, -1, -4, -9, 8, 15, -6], [-20, -3, -3, 9, 5, 6, -10, 4, 0, -9], [-2, 11, -7, -12, 3, -15, 9, -4, 12, 0], [-10, -16, 0, 2, 12, -6, 12, 2, -15, 6], [13, 7, 1, 7, -3, 4, -8, 3, 11, 16], [0, -6, -5, -8, -6, 8, 1, -6, -16, -5], [-3, -8, 8, -4, 6, 4, -9, -1, 11, 18], [-17, 9, 7, 8, -1, 6, -10, -2, -8, 2], [-6, -14, -12, -10, 2, 2, 4, 2, -6, 8], [13, -9, -3, 1, 3, -7, 12, -4, 6, 8], [1, 7, -2, 6, 4, 3, -3, 8, -10, -7], [-27, 23, -3, 4, 8, 1, -8, 4, 6, 2], [11, -6, -8, 6, -4, -5, -1, 0, 1, 0], [8, -14, 2, 5, 5, 2, 1, 18, -4, 0], [-17, 4, -12, 4, 9, -7, -1, 0, -8, 9], [11, -13, -7, 1, -3, 4, -4, 5, -15, 5], [-6, -9, 2, -4, -6, 1, -8, 9, 6, -12], [17, 15, 5, 11, 5, -1, -1, -1, 1, 5], [-12, -12, 3, -7, 6, 3, 6, -8, 6, 4], [-14, 22, 1, 9, -1, -3, 2, -7, 9, -7], [24, 5, 5, -9, -2, 4, 7, 0, -1, -7], [15, 16, -17, -14, -6, 0, 5, -7, 6, 7], [-3, 13, -12, -11, 10, 0, 6, -4, 8, 4], [6, 5, 23, 2, 4, 1, 9, -3, 3, -18], [-4, 9, -11, -4, -3, -6, -6, -2, 6, 2], [-3, 11, 8, 5, 6, -7, 0, 1, 10, 7], [-4, 6, 8, 14, 2, 0, 7, 10, 0, -9], [-11, 13, -20, 5, -1, 3, -11, 12, 4, -1], [-5, 12, -8, -7, 4, 1, -10, 2, -3, 6], [22, -6, 2, -7, 4, 2, 8, -5, -9, 5], [14, 0, -15, -3, -2, -3, -2, -1, -4, 2], [12, -2, 9, 9, -11, -9, -4, 11, 2, -6], [6, 26, -5, -11, 2, -1, 5, -6, 5, 3], [-43, -4, -14, 5, 6, -15, 0, 5, -5, 0], [-17, 3, 17, 16, 8, -3, 1, 7, -4, -13], [1, 16, -6, 2, 0, -10, 13, -5, 7, 6], [-8, -10, -13, 1, -4, 2, -11, 5, -13, -5], [-19, -3, -13, -5, -1, 5, -11, 1, -17, 1], [8, -2, -11, -12, 9, 0, -16, -1, -1, 15], [10, 29, -5, 8, -5, -11, -9, 2, 14, 0], [-8, 5, -18, -4, -1, -5, -6, 3, -7, 5], [-4, -19, 11, 10, 3, -2, -5, 5, -5, 1], [6, 11, 10, 3, 11, -10, 16, -2, -6, -8], [-17, -9, 3, -10, 4, -10, 1, -5, 18, 5], [-14, -12, -5, -15, 16, -3, 9, -6, -1, 21], [-21, -4, -4, -8, -11, 11, -2, 1, -5, -2], [-5, -12, -14, -14, 1, -14, 5, -2, 2, 18], [-21, 6, -13, -5, 15, -7, -3, 0, 3, 8], [-9, 7, -7, -12, 2, 5, 17, -10, -1, -3], [-25, 9, -6, 6, 4, 1, -3, 0, -6, -5], [27, 7, 25, 12, -5, -1, 2, -13, 5, -11], [-1, -26, -11, 7, 9, -2, -8, 3, -4, -5], [3, 4, 14, 17, -8, 4, 6, 0, 2, -10], [-30, -5, -14, -11, -3, 4, -7, 0, -10, -14], [22, -13, 8, -1, 1, 11, 14, 5, -8, -20], [12, -12, -6, -4, 0, 12, -12, 2, -4, 5], [5, -4, -12, -15, -5, 10, -12, -3, -9, 2], [14, 20, 1, -12, -4, -6, 3, -6, 14, 5], [-14, -5, 1, -7, 0, -4, 6, 6, 5, 4], [-18, -4, -16, -5, 9, -7, 1, 11, -3, -6], [0, 18, -4, -4, -6, 1, 2, -1, 3, 11], [13, 3, 13, -7, -5, 3, 9, -1, -4, 3], [-2, -5, 0, -7, 6, -6, 9, -2, -14, -15], [14, 14, 7, 12, 10, -5, -3, 5, 3, -4], [-18, 20, 11, 5, -4, 3, -5, 4, 5, -1], [27, 13, 21, 10, 4, -3, 11, 0, 2, -16], [-7, 4, -14, -5, -1, 6, -11, 5, 10, 19], [16, 2, 12, -4, -10, -2, 1, -2, 4, -17], [-2, -1, -1, 3, -8, 0, 3, -6, 4, 3], [22, 1, 4, 5, -11, -12, 14, -6, 8, -6], [4, -3, 4, 11, -7, -10, 5, 10, 0, -4], [7, -11, -2, -9, 5, -6, 6, 0, -12, -9], [12, 2, -11, -7, 1, 1, -14, -3, 4, 6], [33, -21, 14, 15, -9, 24, -9, 12, -22, -2], [39, -11, 9, -4, -6, 12, 10, 3, -18, -1], [-40, 7, -18, -10, 2, -12, -2, -4, 6, -4], [6, -9, -9, 1, 10, -2, 7, 4, -13, -5], [-10, -8, 17, 20, -4, 16, -7, 8, -10, 11], [-7, -29, 6, -2, -7, -1, 4, 2, 5, -3], [-15, -13, -13, -2, 3, -8, -4, 4, 10, 3], [-2, 18, 3, 10, -8, -12, 6, 0, -2, -4], [18, -4, 1, -13, -10, 3, 10, -6, -5, -6], [23, 3, -5, -3, 14, -16, 21, -6, 6, -11], [-5, 9, -6, -18, 10, -5, -1, 0, 6, 9], [7, -6, -3, 14, -8, 12, -20, 2, -6, -10], [15, -23, -18, -7, 7, -1, 3, 12, -14, 10], [-21, 13, -11, 8, -1, -1, -32, 5, -5, -14], [10, -4, -6, -14, 1, -6, 21, 1, -2, -6], [4, -16, -12, -10, -8, -8, -2, 8, 0, -4], [-32, -9, 1, 1, 8, -5, 6, -10, 5, -11], [9, -2, 1, 2, -4, -16, 18, -1, 2, -12], [-10, -2, 1, -6, 12, -17, 18, -5, 6, 4], [-15, -19, -18, -19, 10, -18, 3, -10, 6, 21], [6, 3, -2, 0, 4, 15, 6, -1, -15, -4], [-2, 3, -9, -9, -1, -18, 11, -2, 5, -4], [-28, 2, -8, 0, 4, -6, -8, 12, -6, 2], [20, -9, 8, 15, -11, 15, -24, 7, -12, 6], [26, -20, 10, -9, 4, 22, 12, 1, -19, -3], [-6, -20, 2, -4, 0, -7, 8, -1, 9, -11], [2, -5, 0, 0, 16, 4, 6, 0, -10, -3], [-15, -2, 10, 4, -2, 10, -4, -8, -8, 14], [8, -4, 8, 0, -9, 6, 7, -7, -5, 14], [24, 6, 13, 3, -6, 4, 8, -10, 3, 3], [23, -3, 7, 2, -2, -4, 1, 1, 16, -6], [-38, 12, -23, -11, 5, -18, -5, -2, 1, -3], [-13, -4, -15, 0, 6, 7, -12, 10, -4, 10], [-5, -9, 18, 5, -4, 0, 4, 2, 0, 0], [44, 1, 18, 5, -3, 5, 10, 1, -2, -6], [-7, -4, -2, -13, 2, -14, 11, -10, 17, 8], [1, -14, -1, -3, 2, -6, 0, -2, 6, 17], [-23, 13, -9, 11, -1, -3, -22, 0, 5, 2], [-10, 19, 2, -2, -7, 10, 13, -10, 10, -7], [-5, 14, -7, -9, -4, 3, -6, -9, 19, 22], [71, 5, 7, 2, -7, 1, 6, -3, 1, 7], [10, -24, 3, 2, 6, 6, 13, 4, -4, 21], [-2, -5, 12, 12, -8, -1, 24, 4, -3, -12], [0, 7, -24, -15, -1, -11, -3, 3, 8, -8], [-16, 5, 6, 2, 5, -5, 0, -2, -6, -8], [30, 24, -2, 8, -12, 0, -1, -4, -7, -7], [12, -25, 6, 5, 12, 5, 2, 4, -7, -1], [9, 6, 13, 16, -1, 9, -4, 3, -11, -2], [36, -7, 12, 8, 4, 5, 2, 5, -5, -16], [-38, 6, -17, -3, -1, 9, -6, 7, 11, -7], [5, 5, -3, -14, 11, -11, 0, -7, 13, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8967_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8967_2_a_bb();".
// 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_8967_2_a_bb(:prec:=10)
    chi := MakeCharacter_8967_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_8967_2_a_bb();".
// 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_8967_2_a_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8967_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -12, 30, 26, -81, -14, 54, 2, -13, 0, 1]>,<5,R![1, 30, 202, -4, -328, 0, 150, 0, -22, 0, 1]>,<11,R![-1860, 2868, 2091, -3202, -1223, 966, 295, -108, -29, 4, 1]>],Snew);
    return Vf;
end function;