
// Make newform 8670.2.a.cl in Magma, downloaded from the LMFDB on 27 June 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_8670_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_8670_2_a_cl();".
// 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_8670_2_a_cl();".
// 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 := [17, -72, -104, 48, 72, -8, -16, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [-340, 1338, 404, -917, 24, 174, -12, -7], [-833, 2780, 2649, -1491, -1053, 241, 93, -18], [102, 1676, -43, -1464, -14, 332, 7, -20], [697, -725, -971, 1352, 403, -474, -57, 39], [272, 38, 3835, 1587, -1675, -656, 115, 43], [119, 7163, 1409, -6468, -872, 1770, 147, -131], [238, -5972, -1976, 6903, 1299, -2070, -216, 163]];
    Rf_basisdens := [1, 289, 289, 289, 289, 289, 289, 289];
    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_8670_a();"
function MakeCharacter_8670_a()
    N := 8670;
    order := 1;
    char_gens := [2891, 6937, 6361];
    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_8670_a_Hecke( : Kf := Kf)
    return MakeCharacter_8670_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], [-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, -1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, -1, 0], [1, 0, 0, 0, -1, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0], [2, -1, 0, 0, 1, 0, 1, 1], [0, 0, 0, 1, -1, -1, 0, 1], [-1, 1, -1, -1, -1, 1, 0, 0], [1, 1, 1, -1, 0, 0, 0, -1], [0, -2, 0, 2, 2, 0, 0, 0], [0, -2, 0, -1, -1, 0, 1, 1], [2, 1, -1, -1, -4, 0, 0, 0], [4, 0, -2, -1, 2, 1, 0, -1], [4, 1, 2, 0, 3, 0, -1, -1], [3, -1, 1, 2, -1, -2, -1, 0], [-1, -1, -1, -2, 3, -1, 0, 0], [2, 2, 0, 1, 0, 2, -1, -1], [0, 0, 0, 0, 1, 1, 1, 0], [-4, 2, 2, -2, 1, 1, -1, 0], [-3, 1, -4, 0, 1, 1, 0, 0], [4, 1, 4, 2, 3, 0, 1, -1], [0, 0, 0, 0, -2, 2, 0, 0], [0, 0, 0, 2, 3, -1, -1, -2], [0, -1, -5, 0, 0, -1, 0, -1], [6, 4, 0, -3, -2, -1, -2, -1], [3, -1, 1, -2, 5, -1, 2, 0], [-6, 2, 4, -4, -4, -2, 0, -2], [3, -1, 0, 3, -3, -2, 2, -1], [3, -1, -3, 3, 3, 0, 0, -1], [2, 0, -6, -2, 0, 0, 0, 0], [4, 1, -2, -6, -3, 1, 1, 0], [-1, 3, -1, 5, -1, 0, 0, -1], [-2, -4, -2, -2, 0, -1, 3, 2], [6, -1, 0, 6, -1, 1, 3, 2], [3, 1, 7, -1, -1, 1, 1, -1], [-1, 1, 1, -3, 5, 1, 2, 0], [0, 4, 5, -1, -4, 1, 0, -1], [1, 3, 5, -4, 1, -1, -2, 0], [3, 2, -2, -7, 5, 1, -2, 0], [-2, 0, -4, 3, -2, -1, -2, 1], [3, -3, -1, 4, 1, -3, 2, 2], [2, 3, -6, -2, -2, -1, -2, -3], [-1, 3, -3, 2, 9, 3, 0, 0], [-3, 3, -3, 3, 0, 0, 0, -1], [-2, 4, 8, -3, 2, -1, -2, -1], [10, 4, 4, 5, 2, 3, -2, -1], [7, -1, -5, 4, -7, -1, 2, 0], [4, 6, -6, 4, 2, 2, -4, -4], [3, -1, -5, -6, 2, -1, 0, 2], [3, -5, 7, -7, -3, -4, 2, 5], [-2, 0, 9, 2, 2, 0, 0, 2], [-1, 3, -3, 0, 1, 2, -3, -2], [10, -1, -4, 3, -1, 2, 1, 3], [-2, 0, 10, 3, -10, 0, 2, 2], [-2, -2, 4, 1, -6, -1, 0, 1], [8, 3, 0, -8, -3, 1, 1, -2], [1, -2, 9, -2, -2, 0, 3, 2], [-2, 0, 4, -5, 10, 1, 2, -1], [3, -1, -7, 5, -3, 1, 2, 0], [4, 0, -10, -2, -2, 0, 0, -2], [10, -4, -12, -1, 4, -2, -1, -1], [-4, 3, 2, 4, -6, 1, -2, -1], [2, -6, -4, 6, 5, 1, 1, 4], [3, 3, -5, 7, 1, 0, -4, -3], [-2, 1, -6, -4, 1, -2, -1, 1], [4, -3, 0, -2, -4, -1, -2, -1], [5, 1, 3, 8, 1, -1, 2, -2], [12, 0, 2, 2, -6, -4, 0, -2], [-6, 2, 2, 0, -4, -4, -2, 0], [0, -4, -10, 3, -4, 1, 2, -1], [2, 0, -2, -1, 5, -2, -3, -1], [-5, -3, 3, -1, -1, -3, 3, 5], [7, 1, 3, -5, -1, 2, 2, -1], [6, -4, -2, -1, 10, -2, -2, 0], [-4, -1, 7, 4, -12, 1, 0, 1], [3, 0, -1, 0, -2, 2, -3, 0], [-6, -1, -6, 1, 10, 2, 2, 0], [-10, 4, 4, -3, -4, 5, 0, 1], [5, 6, -7, -4, 0, 2, -1, -2], [-2, -1, 2, 8, 7, 0, 1, -1], [2, 1, 0, -4, -14, 1, 0, -1], [-7, 3, -1, 3, -13, -2, 0, 1], [7, 1, 4, 10, 3, 1, 0, 2], [-4, -3, -10, 2, 7, 6, 1, 1], [6, 2, 4, 9, -1, 2, -1, -1], [1, 5, -5, 9, 1, 2, 2, -1], [-8, -1, -7, -8, 2, 1, 2, -1], [0, 2, -4, -3, -2, -1, -6, 1], [2, -1, 10, 2, 5, 0, -3, 1], [4, 3, 0, 0, 4, 5, -2, 3], [0, -6, -4, 3, 3, -2, -1, 3], [3, 2, 4, 1, -9, -1, -4, -4], [0, 0, 6, -4, -12, -2, 4, 6], [12, 0, -7, 1, 4, 1, 0, -1], [-2, -4, 2, -8, 14, 3, 3, -2], [-2, -5, -2, 7, 2, 2, 6, 4], [-4, 0, 10, 3, 0, 0, 3, -1], [-7, 7, 11, 0, -7, 1, 2, 2], [-1, -3, 3, -9, -3, -3, 4, 2], [6, 4, 2, -8, 4, 2, -2, 0], [4, 2, 14, -2, -8, -6, -2, 0], [-5, -3, -7, 11, 11, 2, 0, -1], [-7, 3, -1, 6, 5, 1, -2, -4], [-5, 3, -1, 2, -9, 1, -6, -4], [6, 7, 12, 6, 5, 0, 1, 1], [-8, 4, 6, 6, 0, 0, 2, 0], [0, 0, -4, -5, 2, 1, -2, -1], [-4, 0, 5, -8, -4, -2, 0, -2], [0, 1, -6, 7, 2, 4, 2, 2], [-13, -2, -12, -10, 3, 0, 6, 1], [5, 5, 1, 8, 2, 3, -2, -6], [-2, -8, -6, 2, 4, 2, 6, 6], [-2, -3, 4, 1, 7, 2, 3, 3], [10, -2, -10, -1, 1, 2, 3, -1], [-1, -1, 9, 7, -9, -5, 0, 4], [8, 6, -4, 1, -4, 2, -2, -2], [14, -4, 2, 5, -8, -7, 0, -3], [-7, -2, 0, 1, 3, -3, -2, -2], [0, -1, -6, -4, -5, -4, -1, -5], [-2, 8, 0, -8, 5, 3, -3, 2], [3, -5, 11, -1, 5, -2, 4, 7], [22, -6, -2, -13, -4, 1, 4, 3], [-10, 0, -14, 0, -6, -2, 2, -2], [-4, 2, -4, 6, 12, 1, -5, -2], [1, -1, -3, 15, -13, 2, 2, -1], [24, 0, 14, 8, -5, -5, 1, 2], [-7, -3, 5, -2, 9, -1, 4, 4], [-7, -5, 5, 13, 3, 1, 5, 1], [-2, -8, -6, 4, -2, 2, 8, 4], [0, -2, -3, -13, 8, 1, 4, -1], [-19, 3, 2, 2, 3, -1, -6, -4], [1, 1, -3, -5, -13, 1, -3, -3], [-5, -5, 1, -2, -1, -3, 6, 4], [-4, -2, 14, 7, -2, 1, 0, 3], [-6, -3, 12, 10, -9, -2, -1, -5], [-2, -6, 0, -1, 6, 3, 0, 3], [8, -7, -8, -8, -7, -2, 1, -1], [12, 9, -4, -11, -6, 0, -6, -4], [-17, -3, -5, 2, 17, -1, 2, -4], [-17, 1, -1, 23, -3, 1, 2, -2], [0, 3, -2, 5, 4, 0, 6, 2], [20, -6, -14, 5, 0, -3, 2, 3], [4, 5, 8, -16, -1, 2, -5, -3], [6, 5, 4, -10, -2, 5, 2, -3], [11, 8, -7, 8, 2, 10, -1, -2], [-5, 3, -10, -7, -3, 2, 2, 3], [0, -8, -6, -2, -2, -2, -2, 0], [-8, 6, 6, 0, 8, 7, 0, -3], [-4, -8, -12, -5, 0, -1, 8, 1], [-8, 4, 0, -9, 9, -2, -5, -3], [-6, -3, -7, -9, 2, 0, -6, -6], [4, 8, 1, -3, -6, 1, -2, 1], [6, -4, 0, 8, -4, -4, 0, 6], [-14, -9, 2, 14, -2, -5, 2, 3], [0, 1, -4, 2, -5, 1, -1, -4], [12, -7, -8, 3, 8, 2, 4, 4], [-20, 0, -4, -4, 6, 2, 0, 0], [-13, 1, 1, -11, 3, -1, -2, 4], [15, -1, 9, 13, -1, 6, 6, 3], [-16, 6, -4, 5, -4, -5, -4, 1], [-10, -4, -8, 2, 0, 2, -4, 4], [-7, -4, -6, 11, -3, -3, 0, 0], [8, 2, 8, 4, -8, -2, 4, -2], [10, -4, -1, -7, 0, -7, -2, 3], [-21, 1, -4, 2, 5, 7, 2, -2], [-2, 6, 8, -6, 4, -2, 2, -2], [-2, 4, -1, 12, -6, 2, -8, -4], [8, -8, -12, -3, -6, -7, -2, 1], [-13, 8, -2, 5, -13, -1, 0, 0], [-12, -7, 14, 6, -11, -4, -1, 3], [-2, 3, -6, 14, 2, -1, 0, -3], [-24, 4, 8, -13, -4, 1, 0, -1], [-10, -8, -6, 10, -4, 0, 2, 4], [12, 6, 8, 9, -3, 4, 1, 1], [-4, -8, -2, -2, -2, 0, -4, 0], [-6, 4, -2, 7, -12, 3, 0, -3], [-4, -6, -4, -6, -12, -4, -4, -4], [8, 9, -6, -2, 11, 8, -1, -1], [-6, -2, 6, 4, -12, 0, 8, 6], [-16, -2, -10, -8, 0, 0, 0, -2], [-5, 2, 1, -14, 6, -2, 3, 2], [2, -1, 14, -4, 13, -3, -3, 2], [2, -6, 12, -7, 12, 1, 0, 5], [-14, 1, 11, -6, -6, -3, 0, 3], [-6, -14, 10, 1, 0, -7, -2, 3], [-2, -11, -25, 9, -2, 2, 6, 4], [-12, 4, -4, 2, -7, -4, 0, 4], [6, 1, 14, -4, 0, -5, -2, 3], [-4, 5, -2, 16, 0, 5, 2, -7], [11, 7, 11, 5, 3, 6, -2, -1], [4, -6, -2, -4, -4, 0, -2, 2], [6, 4, 6, 2, 14, -3, -1, -4], [2, 2, -4, -6, 0, 6, 0, -4], [11, 5, -5, -16, -4, 3, 2, -4], [-6, 4, 2, 2, -11, 2, -6, 2], [6, -2, -8, 18, 4, -6, 6, 2], [-3, -7, 1, 0, -6, -7, 2, 4], [18, -14, -16, -8, 6, -6, 2, 0], [-7, -1, -3, 9, 11, 5, 2, 0], [-21, 7, 19, -5, 8, -2, -6, 1], [7, -12, -4, -6, 7, 0, 6, 1], [0, 4, 4, 8, -8, -7, 0, -3], [7, -1, 1, 18, 1, 2, 1, -6], [12, -4, -8, 10, 6, 0, -2, -4], [0, 2, 4, 17, -4, 5, 0, -1], [6, 4, 14, -14, -16, 4, -2, 0], [4, 8, 2, -13, -13, -4, -1, -1], [6, -10, -10, 0, 20, 2, -4, 0], [-20, 3, 4, -2, 6, -5, 2, -1], [12, 5, 15, 8, 18, 1, -2, -5], [-4, 6, -4, 5, -11, 8, 1, -1], [12, -1, -8, -6, -15, 4, 1, 3], [0, 7, 10, 2, -14, 9, 4, 1], [-6, 10, 15, 0, 2, 0, -4, -2], [-6, 6, 6, 16, -12, 0, -8, 2], [0, -10, 4, -1, -6, -3, 6, 5], [-12, -4, 8, -9, 17, -3, -2, 1], [-8, -3, 0, 0, -13, 8, 1, 5], [-12, 6, 8, 2, 8, 0, -4, -2], [19, 7, 2, -4, -5, -7, -4, -6], [-18, 6, 14, 1, -6, -5, -6, -7], [-14, -1, 10, 5, -6, 2, 2, 2], [20, 0, -6, -6, 0, 6, 0, -6], [-6, 11, -10, 8, -19, 2, -5, -5], [5, 7, -7, -6, -8, 3, -2, -2], [16, 0, -12, 10, 9, -5, -1, -4], [5, -7, -15, 5, 7, -2, 8, 3], [-4, -4, -4, -13, 0, -1, 4, 7], [1, 7, 13, 1, -5, 1, -7, -7], [-4, -6, 2, -5, 22, -7, 2, -1], [-18, 13, 28, -13, -5, 0, -1, -3], [-1, -5, -25, -3, 3, 0, -2, 3], [4, 3, 1, -5, -12, 0, 2, -4], [20, 8, -6, -16, -2, -3, -6, -7], [14, 2, -11, -2, -4, 2, -2, 2], [-1, 5, -3, 12, 19, -1, -2, -6], [-4, -4, -20, -7, -6, -1, -2, -3], [0, 4, -2, -12, -22, -4, -2, -6], [7, -3, 7, -12, -5, 3, 0, 6], [14, 1, -6, 8, 1, -4, -5, -5], [17, -9, -17, -2, -11, -5, -4, 0], [-4, -1, -6, 8, 17, 2, 3, 5], [11, -15, 11, -4, 16, -3, 6, 6], [18, -14, 0, 16, 19, -5, -1, 2], [-18, -5, -12, 1, 2, 4, -4, -6], [-14, -10, -4, 2, 6, -6, 0, -2], [-6, 0, 10, -32, -4, -2, 4, 0], [-8, -8, 8, -4, 0, 5, -2, 9], [-17, -1, 3, 9, 5, -3, 5, 7], [-10, 2, 2, -15, -17, 4, 1, 3], [-2, -2, 4, -5, 10, 0, -8, -4], [4, 6, 24, -4, 11, 2, 2, -4], [0, -1, 6, -24, 13, -2, -1, 1], [-23, -7, 3, -16, -3, -4, 3, 4], [-26, 2, -4, 6, 10, -2, 4, 6], [-30, 6, 0, 3, 0, 5, -2, -5], [6, -6, -6, 11, 0, 1, -2, 3], [-14, -16, -16, -11, -10, -1, 6, 5], [4, -9, -6, 7, 6, -4, 6, 14], [-2, 11, 4, 6, -19, -4, 1, 1], [-13, 7, 11, 18, -5, 1, 0, -2], [17, -20, -19, -2, 12, -6, 5, -2], [-13, -1, -8, 11, 21, -6, -6, -5], [18, 2, 14, 2, -6, -4, 6, 8], [-14, -4, 2, 6, -10, -5, -1, 6], [-23, 3, 3, 1, 9, 4, 4, -1], [-15, -5, -7, 27, 13, -1, -2, 0], [8, 8, 6, 2, -2, -4, -10, -4], [-17, -13, 1, 0, -5, -7, 4, 10], [-10, 0, 4, 1, 16, 0, -5, 5], [15, 7, 21, 10, 5, 5, 2, -4], [-20, 0, -4, 1, 22, 7, 2, 1], [-24, -12, -20, 4, 6, 4, 4, -2], [-4, 0, 14, -6, 12, 6, 2, 0], [10, 9, -20, 12, 7, 0, -3, -7], [6, 6, 14, 28, -2, 4, 0, 4], [-8, 2, -6, 0, -12, -5, -2, -3], [-15, 6, 16, -11, 5, 5, -8, 0], [-2, -2, -2, 7, 0, 6, -4, 10], [-11, 3, 12, -4, 13, -3, -8, -8], [-16, 8, 3, 15, -2, -1, 2, -1], [6, -1, -12, 0, 9, 0, 9, 7], [-10, -6, -16, -2, -2, -2, -2, 2], [-10, -4, 10, 10, -16, -4, 4, -2], [14, -7, 22, -6, -8, -13, 0, 7], [-16, -6, 6, -11, 6, 7, 4, 9], [26, 14, 6, 2, -2, 2, -8, -6], [-13, 3, -5, -7, -5, -12, -2, -5], [-11, 1, -25, -3, 5, 7, 2, -2], [15, 1, 7, -3, -31, -2, 4, 5], [-4, -10, -12, -3, 16, -5, 0, -1], [-12, 6, 2, 5, 2, 3, -8, -9], [-15, 0, 5, 2, 10, 6, 7, 0], [-1, 3, 19, -1, 1, -5, -6, 0], [-4, 4, -16, 9, -2, -3, 10, -7], [-10, 8, -12, 24, 20, 6, -6, -4], [40, 6, 16, 9, 14, 1, 2, -1], [-20, 1, -3, 3, -2, -6, 8, 0], [-18, 0, -30, -15, 8, 3, -4, -5], [-6, -9, -18, 0, -15, -4, 3, 5], [-11, 5, -3, 11, -8, -6, -2, -1], [0, 6, -22, -4, -8, 2, -6, -12], [-26, 0, -2, -5, 18, -5, 6, 3], [-16, -6, 14, 14, -19, -3, -3, -2], [2, 4, 16, 12, -2, 0, -6, 0], [-36, -6, 10, 14, -2, 2, 2, 6], [-12, -6, 12, 24, -10, -4, -2, -2], [-9, 3, 15, -9, -25, -11, 1, -1], [8, -6, 16, -7, -1, 1, -2, 1], [2, 14, -12, 5, 14, 11, -4, -7], [-10, 10, 0, -17, -17, 0, 1, 3], [-24, -10, -14, 5, -2, 2, 5, -1], [-12, -2, 10, 8, 12, 3, 8, 5], [-16, -4, 12, -10, -4, -2, 0, 6], [1, -3, -17, -22, 23, -2, 1, -4], [-4, 1, -22, 10, -4, 9, -2, -5], [30, 1, 0, -12, 12, 5, 0, -7], [-27, -7, 1, -9, 1, -4, 14, 9], [-19, 7, -9, 8, -11, 3, -14, -6], [-2, 12, 8, 0, -7, 3, 7, -4], [10, -8, 6, 4, 4, 1, 3, 0], [6, 8, -8, -14, 12, 2, -10, -14], [15, -15, 10, 3, -1, -4, 16, 7], [6, -2, -8, -1, 8, 6, 12, 4], [-3, -7, -11, -18, -9, 5, 0, 2], [26, 2, 34, -9, -16, -1, 0, -1], [16, -10, 0, 13, 9, -5, 6, 3], [46, 8, 6, -11, 6, 3, -8, -5], [0, -12, 16, -10, -8, -8, 4, 2], [10, 10, -8, -10, -8, 0, -10, -2], [3, -11, -5, 5, 4, -4, 6, -1], [26, -8, -6, 18, 12, 4, 8, 14], [-17, -1, -1, -1, 21, 12, 4, -1], [22, -14, -10, -13, -6, 1, 0, -1], [6, 7, 8, 2, 1, -8, 1, -5], [19, -7, -12, 1, 11, 6, -4, 5], [-12, -8, -4, 32, 9, 4, 4, -4], [-8, 12, 26, -16, -4, -8, -2, 2], [-29, 1, -1, 7, 25, -3, 1, -5], [12, -8, -4, 4, 6, -2, 10, 4], [-2, -6, -18, -8, -6, -2, 12, -2], [12, -4, -2, 25, -8, -3, -2, 5], [-10, -6, -2, 28, -6, 2, 0, -2], [16, -14, 26, 3, 12, -1, 6, 9], [-6, 0, -2, -16, -2, -2, 10, 4], [-8, 8, -22, 3, 8, 6, -3, -9], [0, -7, 10, 28, 14, 1, 2, -5], [-7, 1, 13, -5, -17, 6, 4, 1], [8, -7, 18, -18, 1, -2, 3, 7], [2, 1, -20, 8, -2, -1, 8, 3], [26, 1, -1, -2, 26, 3, 0, 1], [12, 5, 8, 5, -8, -10, -8, -10], [26, 1, 4, 16, -1, 4, 3, 1], [-14, 2, 16, -14, -22, -10, 2, -8], [-20, -2, -2, -7, 6, 1, -8, -7], [18, -10, -2, -10, 0, 6, 4, 2], [3, 15, 15, 6, -8, 17, 2, -2], [-10, -4, 4, -15, -20, -6, 2, 4], [19, -2, -3, -8, 26, 0, -5, -4], [4, 7, 16, 29, -4, -2, -6, -2], [16, -6, 8, -5, -12, -3, 0, 9], [-8, 2, -4, 5, -30, 5, 2, 3], [12, -10, -2, -9, 10, 10, 3, 1], [12, -2, -10, -4, 2, 2, 2, -2], [6, -2, -4, 11, -20, 1, -2, 7], [-1, -11, -9, -17, -5, -2, -2, 3], [-16, 0, 0, 24, 11, -10, -4, -2], [3, 5, -3, -4, -27, 2, 3, 0], [12, -2, 8, -4, -4, 4, 10, 2], [10, 2, 6, 3, -21, 3, 8, 11], [-14, 2, -10, 18, -18, 3, -5, -2], [18, 3, 26, 2, -7, 3, 3, 4], [-4, 12, -10, -13, -30, -1, 0, 1], [10, 2, -2, -3, -2, 7, 8, 1], [18, -16, -10, -24, 11, -1, 3, 2], [-34, -8, -16, 13, -2, 7, 0, -1], [-4, 22, 12, -7, -6, 11, -8, -7], [-6, -16, 0, 15, -6, -7, 2, 3], [-34, 10, -12, 14, 2, -6, -2, 0], [-28, -5, -10, 3, 23, 2, 7, -5], [-10, 0, -20, 13, 19, 10, -1, 5], [3, 13, 7, -8, -8, -15, -12, -12], [12, 16, 22, 5, -4, 1, -8, -7], [40, 0, 16, -1, -9, -5, 0, 1], [0, 1, -14, -4, 21, -7, -1, -8], [-13, 4, 4, 12, -9, 10, 6, -1], [-16, 8, 12, -11, -20, 5, -6, -3], [8, -4, -2, 20, 18, -4, 0, -4], [6, -4, 6, -12, 15, 0, -4, 2], [27, 11, 11, 12, -1, 11, -4, 0], [-23, -2, 8, 13, -25, -3, 6, 2], [-4, 6, 10, 1, -18, 6, -5, 9], [11, -1, -13, 3, 17, -8, 6, -3], [18, 7, 4, 16, -2, 13, 6, 3], [34, -10, 6, -26, -18, -4, -4, 10], [-8, 2, 24, 11, -10, -1, -8, -3], [-17, 3, -9, 7, -9, 0, -4, 7], [28, -6, 0, -2, -2, -9, 3, 0], [-13, 7, -11, -27, -25, -2, 0, -1], [-24, 9, 14, 14, 13, 3, 1, -2], [-16, 12, -26, 3, 2, 5, -8, -7], [21, -13, 0, -5, 3, -8, 6, 11], [-10, -5, -19, -12, -4, 7, 6, 3], [-21, 3, -9, 5, -20, -10, 2, 5], [9, 14, 14, 4, -9, 4, -12, -7], [-1, 3, 1, 8, -9, -1, -2, -10], [40, 5, 25, 13, 10, 0, -4, -2], [13, 2, -3, 18, -12, 12, 3, 4], [44, -2, -10, 10, -23, -3, -11, -6], [10, 1, -8, 2, -27, 4, 3, 9], [6, 14, -20, 8, -4, 0, -4, -10], [3, -9, 15, 0, -11, -1, 0, 2], [-13, -1, 3, -25, -7, 2, -4, -5], [21, 1, 3, -11, 11, 3, 0, -4], [40, 8, 0, -10, -7, 1, -7, 0], [-20, 10, -10, 5, -9, 2, 5, -7], [23, -5, -10, 11, -13, -2, 0, -7], [20, 0, -6, 35, -18, 1, 4, 3], [8, 4, 8, -18, 6, 5, -7, 8], [2, -2, 4, -5, 8, 9, 4, 1], [0, -20, -5, 1, 0, -7, 8, -5], [17, -8, -10, 5, 1, 9, 8, 12], [8, 13, 26, -6, -20, -3, -4, 3], [14, 7, 34, -12, -15, -4, 5, -1], [10, -8, -20, -4, -16, 4, 6, 0], [4, 10, -18, 19, -25, -4, -3, 1], [18, -5, -18, 4, -3, 8, 13, 5], [-20, 8, -12, -18, -11, 3, 9, 2], [-10, -2, -10, 1, 22, -8, 8, 6], [13, -4, 4, 5, 33, 3, 4, 2], [0, -8, -18, 19, 8, -3, 0, -3], [28, 8, 5, 15, 2, -7, -2, -3], [20, -6, 20, -7, 18, -3, 14, 5], [29, -3, -9, 4, 1, -1, -4, 2], [22, -12, 18, 8, 9, 2, 12, 4], [14, 14, 6, 4, -18, -6, -12, -2], [-33, -9, 17, -3, -3, -7, 2, 12], [10, -3, 0, -12, -23, -5, 1, 2], [-6, 4, 0, 13, 0, 7, 2, -13], [-18, 13, 26, -3, -16, -8, 0, 0], [30, 11, 14, -18, -3, 0, -15, -5], [12, -14, -8, 2, -10, -4, 4, -4], [-8, -2, -12, 0, -7, 2, 6, -8], [4, 7, 0, 30, 1, 5, 7, -2], [24, 0, 22, -3, -20, -8, 9, 5], [-12, 8, -9, -9, -6, 11, 10, 3], [0, -6, 21, -6, -10, -8, -6, 2], [26, 3, 6, 10, -11, 10, -3, 13], [30, 9, 11, 3, 4, -4, 0, -12], [-8, -3, -12, 6, 12, 9, 4, 1], [22, -6, 4, 33, -8, -11, 0, 3], [-32, 10, -8, -6, 4, 10, -8, 4], [12, -12, -8, -15, 8, 3, 0, 5], [20, 7, 13, 26, -2, -1, -6, -7], [26, -18, -20, -14, -16, -6, 0, -6], [37, 19, -5, -6, 19, 12, -7, -6], [16, -4, -18, 13, -2, -7, -8, -5], [29, -21, -11, -14, 26, 3, 8, 4], [-21, 13, -19, -24, -15, 3, -10, -4], [-13, -17, -16, 18, 7, -9, 8, 12], [17, -2, -12, -23, 41, -1, -4, -2], [2, -2, 14, -11, 18, -1, -16, -1], [18, -5, 3, -1, 4, -10, 4, -2], [-8, -8, -4, 35, 0, -1, 4, -3], [26, 5, -4, 3, -5, 18, 7, -1], [-10, 2, -8, -24, 8, 2, 8, 4], [-8, -1, -2, 7, -18, -10, -10, 2], [18, -1, 4, -14, -11, -4, 5, 9], [-10, 5, -38, -3, -4, 6, 0, -12], [14, -11, -4, -4, 23, -4, 13, 5], [-28, 10, 2, 14, -17, 3, -11, 2], [-6, 4, 21, -12, -8, 6, -10, -6], [-3, -2, 25, -12, 16, -4, -5, 0], [17, -8, -17, -20, 4, 6, 15, 4], [-18, 14, 10, 22, 6, 0, -8, -4], [20, -1, 4, 4, 29, -3, 5, 2], [-14, 10, -14, -3, 27, 11, 6, 5], [24, -6, -16, 2, 24, 0, -6, -12], [-3, -1, -13, -19, 31, -6, -6, -5], [-23, -7, -1, 25, 7, -4, 4, -5], [16, 8, 34, 10, 17, -3, -13, -4], [-4, 12, -18, 13, -10, 7, 2, -3], [18, 10, 12, 13, 31, 2, -5, -11], [14, 10, 20, -22, 2, 0, -4, 2], [8, -7, -10, 6, -11, -8, 3, 3], [-17, 0, 15, 6, 0, 0, -3, 8], [-10, -14, -14, 25, 16, 3, 0, 13], [-8, -8, 16, -1, 14, -1, 6, -5], [17, -11, 1, 3, 17, -1, 3, 13], [24, 18, 4, 5, -12, 3, -4, 3], [-18, -4, 2, -8, -3, 0, 4, 10], [6, -12, 0, 6, 2, -10, 16, 2], [10, 12, 8, 11, 6, 1, 14, 5], [-14, 2, 16, 19, -22, 1, -8, -3], [7, 11, 1, 19, -31, -7, -14, 0], [28, 7, 14, 4, 19, 6, -5, -7], [-43, -17, 8, 4, 5, -3, 4, 4], [0, -10, 0, -14, -4, 4, -2, 2], [-3, -11, 13, -15, 29, 1, 20, 6], [-22, -6, -48, 5, 24, 7, 6, -1], [-29, 9, -4, -23, -13, 14, -4, 5], [0, 13, -4, 13, 10, 6, -8, -2], [-1, 0, -2, -12, 13, 0, 8, -9], [16, 5, -20, 19, 15, 10, -7, 1], [16, 4, 8, -32, 12, -3, 6, -7], [-9, -5, 10, 20, 15, -1, 4, 0], [-15, -5, -23, 17, -9, 5, -1, 3], [7, 11, 11, -39, -1, -7, -2, -8], [17, -1, 3, -6, -11, 4, -1, 4], [-8, 0, -32, -28, 4, 6, 10, 0], [-27, 13, 1, -35, -5, 8, -4, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8670_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8670_2_a_cl();".
// 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_8670_2_a_cl(:prec:=8)
    chi := MakeCharacter_8670_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(3673) - 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_8670_2_a_cl();".
// 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_8670_2_a_cl( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8670_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-992, -2432, -1504, 384, 472, -16, -40, 0, 1]>,<11,R![1552, -10048, 13184, -4576, -360, 400, -32, -8, 1]>,<13,R![-18428, 8064, 8320, -4624, -188, 384, -32, -8, 1]>,<23,R![-29308, -66208, -23408, 10688, 3924, -208, -120, 0, 1]>],Snew);
    return Vf;
end function;