
// Make newform 8464.2.a.cf in Magma, downloaded from the LMFDB on 26 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_8464_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_8464_2_a_cf();".
// 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_8464_2_a_cf();".
// 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 := [484, 0, -1272, 0, 1264, 0, -616, 0, 157, 0, -20, 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, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -30, 0, -64, 0, 231, 0, -138, 0, 29, 0, -2], [0, 1282, 0, -2152, 0, 1419, 0, -452, 0, 69, 0, -4], [60, 0, -114, 0, 65, 0, -14, 0, 1, 0, 0, 0], [0, 1234, 0, -2426, 0, 1771, 0, -587, 0, 89, 0, -5], [78, 0, -128, 0, 67, 0, -14, 0, 1, 0, 0, 0], [0, 2370, 0, -4470, 0, 2959, 0, -901, 0, 129, 0, -7], [0, -2252, 0, 4534, 0, -3190, 0, 1039, 0, -158, 0, 9], [0, -2898, 0, 6736, 0, -5379, 0, 1902, 0, -305, 0, 18], [134, 0, -336, 0, 287, 0, -105, 0, 17, 0, -1, 0], [-188, 0, 404, 0, -309, 0, 107, 0, -17, 0, 1, 0]];
    Rf_basisdens := [1, 1, 44, 44, 2, 44, 2, 44, 44, 44, 2, 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_8464_a();"
function MakeCharacter_8464_a()
    N := 8464;
    order := 1;
    char_gens := [7407, 2117, 6353];
    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_8464_a_Hecke( : Kf := Kf)
    return MakeCharacter_8464_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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, -1, -1, 0, 1, 0, 0, 0, -1, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, -1, -1, 0, 0], [2, -2, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, -2, 0, 0, 0, 1, 0, 1, 0, 0], [0, 0, 1, 1, 0, -2, 0, 0, 1, 0, 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, 2], [-1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, -2], [0, 0, -2, 2, 0, 0, 0, 0, 1, 0, 0, 0], [-3, -1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 2], [0, 0, 1, 2, 0, -1, 0, -3, 0, -1, 0, 0], [-3, 1, 0, 0, -1, 0, -2, 0, 0, 0, 0, -1], [0, 0, 2, 2, 0, -2, 0, -2, -1, 1, 0, 0], [-5, 0, 0, 0, 4, 0, -1, 0, 0, 0, 0, 1], [0, 0, 4, 0, 0, -6, 0, 0, 1, -1, 0, 0], [0, 0, -3, 2, 0, 0, 0, 2, 0, 3, 0, 0], [2, 2, 0, 0, -1, 0, 1, 0, 0, 0, 1, -1], [3, -2, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 5, 0, 0, -2, 0, 1, 1, 0, 0, 0], [0, 0, 2, 2, 0, 2, 0, -3, -1, 1, 0, 0], [0, 0, -2, -4, 0, 5, 0, -1, 1, -1, 0, 0], [0, 0, -2, 0, 0, 5, 0, -1, -1, -1, 0, 0], [-2, 2, 0, 0, 3, 0, 0, 0, 0, 0, -1, 0], [0, 0, 3, -2, 0, 0, 0, 3, 3, 0, 0, 0], [0, 0, -1, 1, 0, 2, 0, 3, 2, -1, 0, 0], [0, 0, 4, 0, 0, -4, 0, 4, -2, -1, 0, 0], [0, 0, 4, 0, 0, 3, 0, 2, -1, 0, 0, 0], [-6, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0], [-8, -5, 0, 0, -2, 0, 2, 0, 0, 0, -1, 1], [0, 0, 3, -6, 0, 0, 0, 2, 1, 0, 0, 0], [-1, -2, 0, 0, -1, 0, 0, 0, 0, 0, 1, -2], [0, 0, -4, -4, 0, 2, 0, 4, 3, -1, 0, 0], [3, 1, 0, 0, -7, 0, 0, 0, 0, 0, 0, -1], [0, 0, -8, 4, 0, 8, 0, -2, -2, -3, 0, 0], [-1, -3, 0, 0, 3, 0, 2, 0, 0, 0, 0, -3], [-7, -1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2], [-6, 4, 0, 0, -5, 0, 0, 0, 0, 0, 0, -2], [-2, 2, 0, 0, -3, 0, -2, 0, 0, 0, 0, -5], [0, 0, 2, 2, 0, 0, 0, -4, -1, 0, 0, 0], [0, 0, -8, 2, 0, 10, 0, -2, -2, -2, 0, 0], [4, 0, 0, 0, -4, 0, 0, 0, 0, 0, 2, 3], [2, 4, 0, 0, -1, 0, -4, 0, 0, 0, 0, 2], [0, 0, -3, -1, 0, 1, 0, -2, -2, 3, 0, 0], [-4, 1, 0, 0, 6, 0, -2, 0, 0, 0, 1, 7], [-12, -4, 0, 0, 1, 0, 1, 0, 0, 0, -3, -3], [0, 0, 7, -4, 0, -5, 0, 1, 2, -1, 0, 0], [0, 0, 0, 2, 0, -2, 0, -2, -3, 2, 0, 0], [-5, 4, 0, 0, -2, 0, 2, 0, 0, 0, -2, -6], [-7, 3, 0, 0, 0, 0, 1, 0, 0, 0, 5, -2], [0, 0, -4, 0, 0, -5, 0, -2, 1, 0, 0, 0], [0, 0, -8, -1, 0, 0, 0, -3, 0, 0, 0, 0], [-4, -2, 0, 0, -2, 0, 2, 0, 0, 0, 1, 7], [0, 0, 7, -1, 0, -7, 0, -4, 0, -5, 0, 0], [-6, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -4], [2, -10, 0, 0, 1, 0, -1, 0, 0, 0, -1, 9], [-8, 0, 0, 0, -2, 0, 2, 0, 0, 0, 0, 8], [0, 0, -1, 0, 0, 7, 0, 3, 1, 1, 0, 0], [0, 0, -13, -2, 0, 12, 0, -4, 2, -1, 0, 0], [0, 0, 2, 4, 0, 2, 0, 0, 1, 5, 0, 0], [-10, 6, 0, 0, 0, 0, -5, 0, 0, 0, -1, -6], [-1, -3, 0, 0, -3, 0, 2, 0, 0, 0, 2, -1], [0, 0, -1, -2, 0, -4, 0, 4, 3, 6, 0, 0], [-8, 6, 0, 0, -3, 0, 4, 0, 0, 0, 2, -4], [-10, 1, 0, 0, -5, 0, 3, 0, 0, 0, -2, -6], [0, 0, 5, 4, 0, -4, 0, -2, -2, 2, 0, 0], [0, -4, 0, 0, -2, 0, -1, 0, 0, 0, 3, -4], [-4, 8, 0, 0, 4, 0, -2, 0, 0, 0, -2, 2], [-2, 6, 0, 0, -6, 0, 4, 0, 0, 0, 4, -3], [0, 0, -4, -1, 0, 13, 0, -1, 1, 3, 0, 0], [0, 0, 1, -1, 0, 3, 0, -6, -4, 1, 0, 0], [0, 0, 10, -6, 0, -2, 0, 2, 1, 4, 0, 0], [0, 0, 4, -1, 0, 7, 0, 0, 0, 0, 0, 0], [0, 0, 7, 4, 0, -8, 0, 3, -3, 2, 0, 0], [0, 0, 2, 2, 0, -4, 0, 0, -3, 4, 0, 0], [4, -2, 0, 0, 3, 0, 4, 0, 0, 0, 1, 2], [0, 0, 4, 2, 0, -7, 0, -6, -4, 4, 0, 0], [-9, -1, 0, 0, -3, 0, -1, 0, 0, 0, -1, -3], [0, 0, -1, -4, 0, 3, 0, -3, 6, -1, 0, 0], [0, 0, -8, -4, 0, 4, 0, -4, -1, -2, 0, 0], [0, 0, -2, 9, 0, 1, 0, -5, 3, -5, 0, 0], [0, 0, -2, 2, 0, -1, 0, 5, 0, 7, 0, 0], [-8, 2, 0, 0, 8, 0, -4, 0, 0, 0, -6, 2], [7, 1, 0, 0, -5, 0, 5, 0, 0, 0, -1, -1], [-12, -10, 0, 0, 2, 0, 2, 0, 0, 0, -1, -1], [0, 0, 6, 0, 0, -5, 0, 1, 0, -5, 0, 0], [-10, 10, 0, 0, 3, 0, -4, 0, 0, 0, 0, -2], [-5, -1, 0, 0, -6, 0, -3, 0, 0, 0, -1, 2], [0, 0, -10, -2, 0, -3, 0, -1, 0, 2, 0, 0], [0, 0, -7, -10, 0, 12, 0, 7, 7, -2, 0, 0], [0, 4, 0, 0, 5, 0, 3, 0, 0, 0, 3, 1], [-10, 6, 0, 0, -2, 0, 2, 0, 0, 0, -2, 2], [-2, 6, 0, 0, -1, 0, 0, 0, 0, 0, -2, 5], [0, 0, -3, 5, 0, -7, 0, -4, -6, -1, 0, 0], [-12, -2, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0], [0, 0, -3, 6, 0, 5, 0, 2, 2, 0, 0, 0], [0, 0, 4, 4, 0, -7, 0, -7, -4, -4, 0, 0], [0, -6, 0, 0, 3, 0, 2, 0, 0, 0, -4, 4], [-12, 0, 0, 0, -8, 0, 4, 0, 0, 0, 0, 4], [0, 0, 0, 0, 0, -4, 0, -4, -3, 1, 0, 0], [0, 0, -7, 0, 0, 2, 0, 6, 0, 7, 0, 0], [0, 0, -6, 2, 0, -7, 0, -2, -7, -2, 0, 0], [0, 0, -1, -7, 0, -9, 0, 3, -1, 0, 0, 0], [0, -3, 0, 0, -3, 0, 1, 0, 0, 0, -2, 6], [-7, -10, 0, 0, 1, 0, 4, 0, 0, 0, -5, 2], [1, 1, 0, 0, 3, 0, -7, 0, 0, 0, 3, 5], [-7, 5, 0, 0, 9, 0, -6, 0, 0, 0, 0, 1], [3, 5, 0, 0, 5, 0, -1, 0, 0, 0, 3, 10], [0, -10, 0, 0, -1, 0, 1, 0, 0, 0, -9, -3], [0, 0, -6, 4, 0, 14, 0, -10, -5, -3, 0, 0], [0, 0, 1, -2, 0, -3, 0, -1, -3, -7, 0, 0], [0, 0, 5, -4, 0, -5, 0, -6, 1, 4, 0, 0], [0, 0, 16, 0, 0, -12, 0, 2, -2, -6, 0, 0], [0, 0, 1, 0, 0, 0, 0, 4, 0, -2, 0, 0], [0, 0, 16, 3, 0, -16, 0, 4, 1, 5, 0, 0], [1, 17, 0, 0, -1, 0, -4, 0, 0, 0, 2, -1], [-12, -4, 0, 0, -5, 0, -4, 0, 0, 0, 3, 4], [0, 0, 1, 0, 0, -6, 0, -4, -2, -3, 0, 0], [0, 0, 8, -8, 0, -8, 0, 6, -3, -7, 0, 0], [1, 9, 0, 0, -13, 0, 1, 0, 0, 0, -1, -3], [0, 0, 2, -2, 0, 2, 0, 4, 5, 7, 0, 0], [11, -10, 0, 0, 7, 0, -4, 0, 0, 0, 1, 2], [1, -1, 0, 0, 2, 0, -1, 0, 0, 0, -1, 16], [0, 0, 6, 2, 0, 2, 0, 4, -5, -6, 0, 0], [0, 0, -6, 2, 0, 12, 0, 6, 3, -5, 0, 0], [12, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, -12], [0, 0, -12, 5, 0, -5, 0, 3, -3, 3, 0, 0], [0, 0, 4, -4, 0, -10, 0, 8, 0, -3, 0, 0], [-13, 5, 0, 0, 3, 0, 2, 0, 0, 0, 4, -15], [0, 0, 12, -3, 0, 3, 0, 3, 5, 7, 0, 0], [0, 0, 7, 1, 0, -7, 0, 6, 8, 3, 0, 0], [0, 0, -20, -2, 0, 4, 0, -6, -3, -5, 0, 0], [-2, 6, 0, 0, -2, 0, -4, 0, 0, 0, 6, 7], [0, 0, 7, 6, 0, 1, 0, -4, -2, 0, 0, 0], [0, 0, 22, 8, 0, -6, 0, 8, -3, 2, 0, 0], [0, 0, -23, 3, 0, 17, 0, 1, -3, 8, 0, 0], [0, 0, -10, -2, 0, -6, 0, -6, 7, -8, 0, 0], [-5, 7, 0, 0, 9, 0, -3, 0, 0, 0, -3, -1], [-10, 13, 0, 0, 4, 0, 2, 0, 0, 0, -1, 3], [-14, -8, 0, 0, 1, 0, 2, 0, 0, 0, -1, 6], [-5, 9, 0, 0, 3, 0, 0, 0, 0, 0, -4, 9], [0, 0, 8, -1, 0, -12, 0, -7, 0, -4, 0, 0], [0, 2, 0, 0, -1, 0, 4, 0, 0, 0, 4, -8], [0, 0, -16, -9, 0, -1, 0, -3, 1, 3, 0, 0], [16, 16, 0, 0, 0, 0, -6, 0, 0, 0, 6, 4], [-13, -4, 0, 0, 0, 0, 0, 0, 0, 0, -10, -8], [-10, -6, 0, 0, -10, 0, 5, 0, 0, 0, -3, -4], [0, 8, 0, 0, -1, 0, -3, 0, 0, 0, 1, -1], [14, -14, 0, 0, 0, 0, 8, 0, 0, 0, 2, 6], [0, 0, 0, 6, 0, -7, 0, -7, -7, 1, 0, 0], [-1, 8, 0, 0, -2, 0, 1, 0, 0, 0, 6, -9], [17, 7, 0, 0, 0, 0, -1, 0, 0, 0, -1, -2], [0, 0, -2, -3, 0, 1, 0, 10, 6, -4, 0, 0], [0, 0, -12, -4, 0, 26, 0, 12, 6, 2, 0, 0], [0, 0, -3, -7, 0, 5, 0, 2, -4, -3, 0, 0], [-32, 8, 0, 0, -2, 0, 2, 0, 0, 0, 2, -6], [0, 0, 23, 0, 0, -19, 0, 3, 5, -1, 0, 0], [0, 0, 8, 4, 0, 8, 0, 6, 3, 8, 0, 0], [-12, 4, 0, 0, -5, 0, 6, 0, 0, 0, 0, -7], [0, 0, -12, 6, 0, 1, 0, 1, 4, -7, 0, 0], [14, 4, 0, 0, 1, 0, 5, 0, 0, 0, 3, -9], [0, 0, -12, -5, 0, 8, 0, 5, 4, 0, 0, 0], [0, 0, 8, -4, 0, 15, 0, 3, 2, 1, 0, 0], [0, 0, -5, -16, 0, 2, 0, 9, 5, 6, 0, 0], [-10, -4, 0, 0, -2, 0, 0, 0, 0, 0, 6, 8], [10, 12, 0, 0, -7, 0, -4, 0, 0, 0, 10, -2], [0, 0, 7, 0, 0, -6, 0, -2, 5, 2, 0, 0], [-8, 2, 0, 0, -6, 0, 2, 0, 0, 0, -6, -10], [0, 0, 0, 1, 0, -20, 0, -5, 0, -4, 0, 0], [10, 8, 0, 0, -13, 0, 4, 0, 0, 0, 0, -10], [0, 0, -13, -3, 0, 7, 0, -4, 4, 3, 0, 0], [0, 0, -11, -8, 0, -6, 0, 4, -1, 0, 0, 0], [3, -7, 0, 0, 3, 0, -2, 0, 0, 0, 2, 1], [0, 0, -1, -8, 0, 16, 0, -4, 6, 2, 0, 0], [17, 7, 0, 0, -6, 0, 1, 0, 0, 0, -3, 4], [12, -2, 0, 0, 7, 0, 2, 0, 0, 0, 1, -2], [0, 0, 0, -2, 0, -16, 0, 2, 4, 4, 0, 0], [0, 0, -4, -4, 0, -8, 0, 2, -5, 1, 0, 0], [-15, -1, 0, 0, -1, 0, 10, 0, 0, 0, -2, 1], [0, 0, -14, 1, 0, 9, 0, -6, 2, 8, 0, 0], [26, 10, 0, 0, 3, 0, -8, 0, 0, 0, 4, 6], [-17, 3, 0, 0, -7, 0, -5, 0, 0, 0, -5, 4], [0, 0, -2, 1, 0, 13, 0, -5, -1, 3, 0, 0], [0, 0, 2, -12, 0, 10, 0, -2, 6, -3, 0, 0], [-10, 2, 0, 0, 11, 0, -4, 0, 0, 0, 3, 4], [0, 0, 7, 4, 0, 1, 0, 5, 12, -3, 0, 0], [5, 0, 0, 0, 4, 0, 10, 0, 0, 0, -3, -4], [-5, -1, 0, 0, -9, 0, -6, 0, 0, 0, 8, -1], [-8, 6, 0, 0, 6, 0, -2, 0, 0, 0, 3, 9], [10, -17, 0, 0, 3, 0, -1, 0, 0, 0, 0, 8], [0, 0, 4, -6, 0, 5, 0, 9, 6, -10, 0, 0], [-6, -4, 0, 0, -5, 0, 14, 0, 0, 0, -7, -8], [0, 0, 9, 4, 0, -23, 0, -5, -2, -7, 0, 0], [0, 0, -3, 2, 0, -13, 0, -9, -1, -5, 0, 0], [0, 0, -4, -4, 0, 7, 0, 11, -4, 13, 0, 0], [0, 0, -10, -4, 0, 12, 0, -18, -3, -10, 0, 0], [0, 0, 10, -14, 0, 9, 0, 3, 8, -5, 0, 0], [11, -9, 0, 0, 14, 0, -3, 0, 0, 0, 3, 6], [0, 0, 0, -4, 0, -4, 0, 6, 9, -4, 0, 0], [9, 3, 0, 0, 1, 0, -6, 0, 0, 0, -2, -1], [8, 2, 0, 0, -11, 0, 8, 0, 0, 0, 8, -10], [0, 0, -3, -6, 0, 11, 0, 12, 6, 12, 0, 0], [0, 0, -8, -2, 0, 18, 0, 3, 1, -3, 0, 0], [-22, -8, 0, 0, 6, 0, 2, 0, 0, 0, 2, 6], [0, 0, 3, 2, 0, 8, 0, -7, -7, -2, 0, 0], [-3, -13, 0, 0, 6, 0, 3, 0, 0, 0, -5, -16], [-24, 6, 0, 0, 12, 0, -4, 0, 0, 0, 5, -1], [-16, 8, 0, 0, 8, 0, -6, 0, 0, 0, -6, -12], [12, -6, 0, 0, 2, 0, 2, 0, 0, 0, -7, 1], [-4, 0, 0, 0, -5, 0, -4, 0, 0, 0, -3, 2], [0, 0, 7, 8, 0, 0, 0, -1, -11, 2, 0, 0], [0, 0, 7, -4, 0, 19, 0, 8, -1, -8, 0, 0], [11, -19, 0, 0, 0, 0, -1, 0, 0, 0, -5, 26], [0, 0, -9, -4, 0, 16, 0, 0, 2, 12, 0, 0], [6, -20, 0, 0, -1, 0, 1, 0, 0, 0, 3, 5], [-20, 0, 0, 0, 4, 0, -10, 0, 0, 0, 0, 11], [0, 0, -10, 11, 0, 11, 0, -9, -5, 11, 0, 0], [10, -4, 0, 0, -5, 0, -4, 0, 0, 0, -3, 8], [0, -16, 0, 0, -1, 0, -6, 0, 0, 0, -1, 8], [0, 0, 18, 1, 0, 9, 0, 1, -3, 1, 0, 0], [-8, -16, 0, 0, -6, 0, 10, 0, 0, 0, -2, 5], [0, 0, -24, -5, 0, -9, 0, -9, 3, 1, 0, 0], [-20, 16, 0, 0, 3, 0, 0, 0, 0, 0, 2, -11], [12, -4, 0, 0, 0, 0, 2, 0, 0, 0, -10, 4], [0, 0, 13, -6, 0, -15, 0, 10, 6, 0, 0, 0], [18, -4, 0, 0, -11, 0, -1, 0, 0, 0, -3, 3], [0, 0, 6, 7, 0, 17, 0, 3, 5, 5, 0, 0], [9, -13, 0, 0, 6, 0, -8, 0, 0, 0, -8, 10], [16, 2, 0, 0, 7, 0, 4, 0, 0, 0, -2, -18], [0, 0, -21, -3, 0, 6, 0, -6, -5, -8, 0, 0], [0, 0, 8, 4, 0, -10, 0, 2, 8, 8, 0, 0], [7, 18, 0, 0, -2, 0, 4, 0, 0, 0, 5, -12], [0, 0, 19, -2, 0, 5, 0, 10, -7, 10, 0, 0], [0, 0, -9, -4, 0, 2, 0, -15, -3, 2, 0, 0], [0, 0, 4, -8, 0, -15, 0, 2, 3, 4, 0, 0], [0, 0, 18, 16, 0, -6, 0, -2, -2, 1, 0, 0], [-21, -3, 0, 0, -5, 0, 0, 0, 0, 0, -2, -15], [-17, 13, 0, 0, 4, 0, -1, 0, 0, 0, -1, -10], [0, 0, -3, -19, 0, 4, 0, 7, 0, -13, 0, 0], [-23, -5, 0, 0, -2, 0, 4, 0, 0, 0, 0, -6], [-6, 18, 0, 0, -1, 0, 3, 0, 0, 0, 11, -1], [20, -8, 0, 0, -7, 0, -2, 0, 0, 0, 1, 2], [-4, -9, 0, 0, 5, 0, 1, 0, 0, 0, 4, 8], [23, 3, 0, 0, 13, 0, 0, 0, 0, 0, -2, -3], [-15, 7, 0, 0, 4, 0, -11, 0, 0, 0, -3, -2], [0, 0, -14, 0, 0, 1, 0, -4, 5, 7, 0, 0], [0, 0, -20, 0, 0, 35, 0, 3, 4, 4, 0, 0], [0, 0, 2, 4, 0, -20, 0, 2, 2, 4, 0, 0], [0, 0, 0, 6, 0, -4, 0, -8, -5, -14, 0, 0], [0, 0, 13, 24, 0, -15, 0, -15, -7, 7, 0, 0], [0, 0, 6, 3, 0, -35, 0, -3, 7, 7, 0, 0], [0, 0, -4, 14, 0, -9, 0, -1, 2, 3, 0, 0], [-8, 12, 0, 0, -7, 0, -8, 0, 0, 0, -1, 0], [-6, 10, 0, 0, 8, 0, -7, 0, 0, 0, -11, 10], [0, 0, -10, 14, 0, -8, 0, 0, -3, 3, 0, 0], [0, 0, 6, 12, 0, -4, 0, -5, 1, 19, 0, 0], [4, -10, 0, 0, 15, 0, 0, 0, 0, 0, -2, -16], [-7, 9, 0, 0, 7, 0, -3, 0, 0, 0, -1, 4], [0, 0, 7, 9, 0, 17, 0, -8, 2, 1, 0, 0], [0, 0, -16, -13, 0, 0, 0, -3, 0, -8, 0, 0], [14, -4, 0, 0, 4, 0, -2, 0, 0, 0, -8, -8], [0, 0, 14, -8, 0, -12, 0, 14, -10, 5, 0, 0], [17, -4, 0, 0, 12, 0, 6, 0, 0, 0, -8, -6], [0, 0, 3, 4, 0, 13, 0, 1, 4, 9, 0, 0], [0, 0, 22, 12, 0, -20, 0, -12, -4, -9, 0, 0], [0, 0, 0, -4, 0, 5, 0, 5, -10, 1, 0, 0], [0, 0, -9, 16, 0, 1, 0, 2, -9, 0, 0, 0], [2, 22, 0, 0, 11, 0, -8, 0, 0, 0, 0, -2], [14, -14, 0, 0, -3, 0, -2, 0, 0, 0, -8, 20], [0, 0, -5, -12, 0, -2, 0, 0, 2, -17, 0, 0], [0, 0, 8, 4, 0, -33, 0, -9, -1, -3, 0, 0], [0, 0, 24, -5, 0, 13, 0, 5, 7, 5, 0, 0], [11, -9, 0, 0, 13, 0, -1, 0, 0, 0, -9, -3], [-9, -11, 0, 0, 1, 0, 8, 0, 0, 0, 0, 3], [-13, -3, 0, 0, 12, 0, -1, 0, 0, 0, 3, 6], [4, -20, 0, 0, 13, 0, -4, 0, 0, 0, 7, 12], [0, 0, 19, 16, 0, -32, 0, -10, 2, 2, 0, 0], [0, 0, -7, -5, 0, 21, 0, 7, -1, -6, 0, 0], [-7, 5, 0, 0, -4, 0, -9, 0, 0, 0, 13, 0], [0, 0, -5, -8, 0, -8, 0, -9, -15, 2, 0, 0], [0, 0, -1, 7, 0, 9, 0, 2, -10, 1, 0, 0], [0, 0, 20, 10, 0, -14, 0, -12, 4, -13, 0, 0], [-11, 15, 0, 0, 5, 0, -7, 0, 0, 0, -5, 5], [2, -2, 0, 0, -2, 0, 16, 0, 0, 0, 0, 6], [0, 0, -16, 0, 0, 9, 0, 13, 4, 9, 0, 0], [0, 0, 2, -8, 0, -8, 0, 12, 4, 9, 0, 0], [-5, 5, 0, 0, 6, 0, 1, 0, 0, 0, 1, 12], [-18, -5, 0, 0, 1, 0, -7, 0, 0, 0, 0, 0], [0, 0, -8, 6, 0, -6, 0, 6, -7, -6, 0, 0], [0, 0, -18, -13, 0, 11, 0, 2, -6, -4, 0, 0], [-21, 20, 0, 0, -6, 0, 2, 0, 0, 0, -1, -16], [0, 0, -14, -2, 0, -2, 0, -9, -3, 7, 0, 0], [18, -8, 0, 0, -1, 0, -2, 0, 0, 0, 2, 14], [0, 0, -26, 2, 0, 24, 0, -2, -6, -1, 0, 0], [0, 0, 1, -10, 0, -16, 0, 5, 1, 12, 0, 0], [16, -14, 0, 0, 0, 0, 0, 0, 0, 0, -8, 10], [-13, 5, 0, 0, -2, 0, -10, 0, 0, 0, 6, -10], [-24, 8, 0, 0, -14, 0, -10, 0, 0, 0, 10, 6], [0, 0, 15, 4, 0, 29, 0, -1, -1, -1, 0, 0], [0, 0, 18, -6, 0, 14, 0, 8, 2, -3, 0, 0], [0, 0, -16, -1, 0, 7, 0, -1, -1, -15, 0, 0], [-1, 12, 0, 0, 5, 0, 6, 0, 0, 0, 1, -14], [0, 0, 0, -14, 0, 17, 0, -4, 3, -5, 0, 0], [15, 12, 0, 0, 2, 0, 4, 0, 0, 0, 10, 0], [-7, 11, 0, 0, 9, 0, -11, 0, 0, 0, 7, -3], [0, 0, 6, 2, 0, 10, 0, -6, 8, 3, 0, 0], [0, 0, -43, -8, 0, 38, 0, -3, 5, 0, 0, 0], [48, 2, 0, 0, 5, 0, -10, 0, 0, 0, -1, 0], [-8, -2, 0, 0, 1, 0, -11, 0, 0, 0, 3, -13], [0, 0, 0, -2, 0, 12, 0, 8, 1, 10, 0, 0], [0, 0, 12, 6, 0, 16, 0, 3, -2, -1, 0, 0], [-9, 4, 0, 0, 12, 0, -5, 0, 0, 0, 0, 5], [0, 0, -4, 2, 0, 12, 0, -2, -8, 4, 0, 0], [0, 0, -27, -12, 0, 26, 0, 4, 7, -2, 0, 0], [-7, -1, 0, 0, 0, 0, -5, 0, 0, 0, -9, -10], [-8, -10, 0, 0, 0, 0, 2, 0, 0, 0, -8, -10], [0, 0, -19, -8, 0, 9, 0, 5, -3, 7, 0, 0], [21, -11, 0, 0, 9, 0, 1, 0, 0, 0, -11, -1], [0, 0, -22, 0, 0, -7, 0, -12, -7, 7, 0, 0], [0, 0, 3, 18, 0, -5, 0, -11, -1, 11, 0, 0], [10, 2, 0, 0, -4, 0, 12, 0, 0, 0, 2, -4], [0, 6, 0, 0, -1, 0, -1, 0, 0, 0, 1, 21], [0, 0, 22, 12, 0, -29, 0, -7, 0, -5, 0, 0], [0, 0, -11, 4, 0, 1, 0, -19, 1, -1, 0, 0], [0, 0, 1, -3, 0, 12, 0, -1, -8, -5, 0, 0], [12, -9, 0, 0, -6, 0, 6, 0, 0, 0, -1, 17], [0, 0, 6, -19, 0, -3, 0, 15, -1, 11, 0, 0], [0, 0, -14, 8, 0, -18, 0, -6, -14, -1, 0, 0], [0, -10, 0, 0, 1, 0, 12, 0, 0, 0, -4, -14], [-8, -8, 0, 0, 2, 0, 6, 0, 0, 0, 4, -18], [-10, 22, 0, 0, 8, 0, -4, 0, 0, 0, 4, -12], [-34, 21, 0, 0, -8, 0, -10, 0, 0, 0, -1, -13], [0, 0, 1, -1, 0, 22, 0, 13, -2, -7, 0, 0], [-38, 2, 0, 0, -5, 0, 12, 0, 0, 0, 2, 5], [0, 0, 10, 14, 0, -18, 0, -2, -14, 1, 0, 0], [0, 0, 18, -14, 0, -11, 0, 4, -10, -2, 0, 0], [1, -6, 0, 0, -8, 0, 0, 0, 0, 0, -14, -14], [0, 0, 12, 6, 0, 18, 0, 14, -2, -6, 0, 0], [58, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, -4], [0, 0, 7, -2, 0, -18, 0, -6, -10, -10, 0, 0], [-2, -18, 0, 0, -9, 0, 2, 0, 0, 0, -7, 6], [0, 0, 10, 15, 0, -27, 0, -23, -5, -25, 0, 0], [0, 0, -22, 2, 0, -6, 0, 12, 5, 5, 0, 0], [7, 5, 0, 0, 8, 0, -3, 0, 0, 0, 5, 14], [-4, 16, 0, 0, 0, 0, -20, 0, 0, 0, 2, -6], [-1, 9, 0, 0, -13, 0, 5, 0, 0, 0, 3, 11], [0, 0, 1, -11, 0, -7, 0, -6, 0, 9, 0, 0], [0, 0, -28, 2, 0, 2, 0, -8, -3, -8, 0, 0], [-43, 15, 0, 0, -3, 0, -6, 0, 0, 0, -4, 3], [16, -5, 0, 0, -5, 0, 5, 0, 0, 0, 4, 16], [-32, 14, 0, 0, 10, 0, 0, 0, 0, 0, 6, 6], [0, 0, -7, -17, 0, 1, 0, 6, 6, -1, 0, 0], [0, 0, 6, 20, 0, 6, 0, -2, -6, -9, 0, 0], [-8, 8, 0, 0, 4, 0, -2, 0, 0, 0, -6, 26], [0, 0, 28, 12, 0, -24, 0, -4, 4, 8, 0, 0], [0, 0, 21, -6, 0, -5, 0, 10, 7, -2, 0, 0], [11, -9, 0, 0, 11, 0, -5, 0, 0, 0, 7, -4], [-22, 12, 0, 0, 1, 0, -5, 0, 0, 0, 13, 3], [0, 0, -26, -8, 0, 2, 0, 8, 8, -15, 0, 0], [-8, -11, 0, 0, 9, 0, 5, 0, 0, 0, -6, 6], [-11, -15, 0, 0, 7, 0, -10, 0, 0, 0, 0, 15], [0, 0, 14, 3, 0, -22, 0, 11, -4, 6, 0, 0], [-17, 2, 0, 0, 14, 0, -15, 0, 0, 0, -12, 1], [55, -16, 0, 0, 12, 0, -2, 0, 0, 0, 1, 4], [14, -2, 0, 0, 12, 0, -12, 0, 0, 0, 2, 0], [0, 0, 8, 23, 0, -3, 0, -15, -1, -3, 0, 0], [0, 0, 11, 8, 0, -2, 0, 0, 3, 18, 0, 0], [-47, 11, 0, 0, 1, 0, 2, 0, 0, 0, 12, -13], [8, -12, 0, 0, 18, 0, -4, 0, 0, 0, -4, -10], [39, -5, 0, 0, -4, 0, 1, 0, 0, 0, 3, 16], [0, 0, -8, -4, 0, 28, 0, 14, 7, 0, 0, 0], [0, 0, -3, 6, 0, -2, 0, 5, -1, -2, 0, 0], [14, 4, 0, 0, -24, 0, 2, 0, 0, 0, -4, -6], [-3, -5, 0, 0, -10, 0, 11, 0, 0, 0, -1, -8], [0, 0, -3, -4, 0, -8, 0, 7, -7, 4, 0, 0], [0, 0, -2, 4, 0, 9, 0, -13, 1, 5, 0, 0], [0, 0, 7, -10, 0, -20, 0, 14, -3, -10, 0, 0], [-21, 16, 0, 0, 26, 0, -2, 0, 0, 0, 8, 2], [0, 0, 34, -2, 0, -30, 0, 4, -7, 1, 0, 0], [0, 0, -23, 10, 0, 9, 0, 2, 2, -8, 0, 0], [-21, -21, 0, 0, 9, 0, -2, 0, 0, 0, -8, 25], [-32, 10, 0, 0, -16, 0, -10, 0, 0, 0, -3, 3], [0, 0, 14, 3, 0, -24, 0, 2, -5, 1, 0, 0], [-4, 18, 0, 0, -10, 0, 2, 0, 0, 0, -4, -16], [6, 12, 0, 0, -14, 0, -2, 0, 0, 0, 8, -20], [12, 6, 0, 0, -1, 0, -8, 0, 0, 0, 8, 6], [0, 0, -13, -2, 0, 1, 0, 14, 1, 2, 0, 0], [0, 0, -16, -9, 0, 27, 0, 21, -3, 15, 0, 0], [0, 0, -5, -4, 0, 7, 0, 3, 15, -7, 0, 0], [-4, -2, 0, 0, 2, 0, -12, 0, 0, 0, -10, 14], [23, 4, 0, 0, 12, 0, -13, 0, 0, 0, 0, 5], [22, -2, 0, 0, 9, 0, 9, 0, 0, 0, -5, -7], [0, 0, 23, -9, 0, -27, 0, 14, -14, -7, 0, 0], [0, 0, -18, 14, 0, -5, 0, -10, -3, 8, 0, 0], [0, 0, -19, 8, 0, -6, 0, -13, 3, -10, 0, 0], [0, 0, 0, -2, 0, -19, 0, 6, 8, -2, 0, 0], [0, 0, 18, -13, 0, 7, 0, -2, 14, 0, 0, 0], [22, 10, 0, 0, -20, 0, -8, 0, 0, 0, 10, -16], [6, -12, 0, 0, 15, 0, -12, 0, 0, 0, 3, 12], [-35, 6, 0, 0, 0, 0, 8, 0, 0, 0, 3, 4], [0, 0, -8, -26, 0, 8, 0, 12, 2, -2, 0, 0], [0, 0, -28, -4, 0, 13, 0, -19, -21, -11, 0, 0], [-28, 6, 0, 0, 27, 0, -8, 0, 0, 0, -5, 12], [7, -19, 0, 0, 24, 0, -3, 0, 0, 0, 5, 2], [0, 0, -8, 2, 0, -10, 0, 2, 10, 15, 0, 0], [-16, -9, 0, 0, -1, 0, 13, 0, 0, 0, -8, 0], [0, 0, -17, -6, 0, 24, 0, 0, 2, -13, 0, 0], [-28, -8, 0, 0, -6, 0, 1, 0, 0, 0, 5, 12], [-22, 10, 0, 0, 2, 0, 0, 0, 0, 0, -6, 10], [0, 0, 0, 0, 19, 0, -12, 0, 0, 0, 1, 12], [0, 0, -21, 11, 0, 5, 0, -11, -19, -6, 0, 0], [0, 0, 5, 26, 0, -8, 0, -4, -18, 5, 0, 0], [0, 0, 30, 30, 0, -16, 0, -5, -6, -3, 0, 0], [-14, -4, 0, 0, -8, 0, 6, 0, 0, 0, -12, -24], [48, 2, 0, 0, 5, 0, 3, 0, 0, 0, 1, -13], [32, -26, 0, 0, 8, 0, 2, 0, 0, 0, 4, -2], [0, 0, 32, 14, 0, -24, 0, 7, -12, 11, 0, 0], [0, 0, 27, -15, 0, 7, 0, 4, -8, -5, 0, 0], [0, 0, 22, 12, 0, 8, 0, -4, -1, -1, 0, 0], [0, 0, -6, 10, 0, 12, 0, -4, -1, 3, 0, 0], [-13, -19, 0, 0, -1, 0, 12, 0, 0, 0, 4, -19], [-28, -13, 0, 0, -7, 0, 7, 0, 0, 0, 2, -8], [17, 21, 0, 0, -3, 0, -11, 0, 0, 0, -7, 8], [-14, 22, 0, 0, 2, 0, -4, 0, 0, 0, -8, -18], [0, 0, -6, -9, 0, 2, 0, 3, 4, 6, 0, 0], [-7, 2, 0, 0, 8, 0, 8, 0, 0, 0, 6, -34], [7, -18, 0, 0, 14, 0, -3, 0, 0, 0, 0, 9], [33, -17, 0, 0, 4, 0, -3, 0, 0, 0, 5, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8464_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8464_2_a_cf();".
// 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_8464_2_a_cf(:prec:=12)
    chi := MakeCharacter_8464_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_8464_2_a_cf();".
// 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_8464_2_a_cf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8464_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, 8, -16, -24, -3, 4, 1]>,<5,R![16, 0, -1648, 0, 2537, 0, -1392, 0, 330, 0, -32, 0, 1]>,<7,R![256, 0, -5376, 0, 14608, 0, -7456, 0, 1048, 0, -56, 0, 1]>,<13,R![-368, -448, 197, 240, -30, -8, 1]>],Snew);
    return Vf;
end function;