
// Make newform 8040.2.a.bc in Magma, downloaded from the LMFDB on 01 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8040_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_8040_2_a_bc();".
// 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_8040_2_a_bc();".
// 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 := [3524, -4752, -3864, 5969, -92, -1708, 358, 132, -37, -3, 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], [-6029040447, 470820744, 5781898709, -1218757803, -1158625792, 547588947, 40103260, -58839604, 648324, 1820505], [24517354298, -34441127948, -32331453390, 24641558311, 8604438743, -5438935403, -622469771, 405600855, 13049510, -9643245], [28946015068, -42982345244, -36860976572, 44010196469, 9509001303, -11857995913, -442082665, 996434659, -252840, -26086723], [28921908461, -29564834748, -42939304662, 31178512266, 12215412133, -8641020986, -798514962, 723832368, 12691238, -18725497], [50050926048, -56651824652, -81281718972, 62299880455, 23732873701, -17859830925, -1501477001, 1524928207, 21312224, -39985935], [97540858780, -85015093680, -140121203272, 90980987213, 40011083555, -25227887457, -2752544585, 2092831555, 49948754, -53420509], [44873983953, -42694551972, -67762156460, 45461921568, 19656566495, -12902838205, -1328495831, 1085047513, 22939251, -27977725], [60525448260, -46084337074, -89483442459, 50178947424, 25929421256, -14093727028, -1880005231, 1166227892, 37943916, -29521266], [-75217331604, 63721166168, 112947152263, -72543332013, -32209590738, 20720606549, 2162038384, -1746534425, -37028569, 45139551]];
    Rf_basisdens := [1, 712963777, 1425927554, 1425927554, 712963777, 1425927554, 1425927554, 712963777, 712963777, 712963777];
    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_8040_a();"
function MakeCharacter_8040_a()
    N := 8040;
    order := 1;
    char_gens := [6031, 4021, 2681, 3217, 5161];
    v := [1, 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_8040_a_Hecke(Kf)
    return MakeCharacter_8040_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, 0], [-1, 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, -1], [1, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [-1, -1, 0, 0, 0, -1, 0, 0, 0, 0], [1, 1, 1, -1, -1, 1, 0, 0, 0, -1], [0, 0, 1, 0, 0, 0, -1, -1, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [3, 0, 0, 0, 1, -1, -1, 0, 1, 1], [0, -1, -1, 1, 0, 0, 2, 0, 0, 0], [-2, 0, -1, 0, -1, 1, 1, 0, 1, -1], [2, 0, -1, 1, 0, 0, 0, -1, 0, -1], [3, 0, 0, 0, 2, 0, -1, 1, 0, 0], [2, -1, 1, 0, -1, 0, 1, -1, -1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 1, -1, 0, 1, -1, 0, 0, -1], [2, 1, 1, 0, -1, 1, 1, -1, -1, -1], [0, 0, 1, 0, -2, -1, 0, -1, -1, -1], [0, 2, 1, 0, 0, 0, -1, -1, -1, -2], [2, -1, -1, 0, 0, 0, 0, 0, 0, 0], [1, -2, -1, 0, -1, 0, 1, 1, 0, 1], [2, -1, 0, 2, 0, -1, -1, -1, -1, 0], [1, 2, 0, 0, 1, 0, -2, 0, 0, 0], [0, 1, 2, 0, 1, -1, -2, 2, 0, -1], [3, 0, 2, 0, 0, 0, -2, 1, 0, -1], [5, 0, 1, 0, 2, 0, -1, 0, 1, 0], [-4, 0, 0, 0, 1, 0, -1, -1, -1, 1], [0, -1, 0, 0, -2, 0, 0, 1, -1, -2], [-1, 2, -1, 0, -2, 1, 2, 0, 0, -1], [-1, 0, 1, 0, 1, -1, 0, 1, -3, 0], [2, 0, -1, 0, -1, -2, 0, 0, 0, 0], [2, 0, 1, 0, 0, 1, -1, 1, 3, 2], [-1, 2, 1, -2, 2, -1, -3, 2, 1, 0], [-2, -3, 1, 0, 1, 0, 0, 1, 0, 2], [-5, 0, 3, 0, -1, 1, 0, -1, 0, 0], [4, 2, 1, 0, -2, 1, 1, -1, -2, -4], [0, -1, -1, 0, 0, 1, 2, 2, 0, 0], [1, 1, 0, 0, 2, 1, 0, 0, 0, 1], [-2, 1, 0, 0, -1, 1, 1, 0, -3, -2], [5, 2, 2, -2, 0, 1, -2, -1, -2, 0], [8, -1, 3, -1, 1, 0, -4, 1, 2, 2], [-3, -2, 0, 2, -2, 0, 1, -1, 1, 1], [0, 1, -2, 0, -2, 2, 2, -1, 2, 1], [-1, 2, 1, 0, 1, 1, -1, 1, -1, -2], [-5, 3, 1, -2, 0, 2, 2, -1, 0, 0], [8, -1, -1, 2, 0, -3, -2, 0, 0, 1], [5, -3, -1, 0, 3, -2, -1, 0, 1, 3], [-1, 0, -1, 2, 3, -2, -1, 1, -1, 2], [7, 1, -2, 0, -1, -1, 2, -1, 0, 1], [8, -3, -1, 2, 1, -1, 2, -1, 0, 0], [0, 2, 0, 0, -1, 0, 1, -1, -2, 0], [0, -3, -2, 0, 1, 1, 1, 2, 3, 1], [6, -1, 0, 0, -1, -1, 1, 0, -1, 1], [1, -2, -2, 2, 1, -2, 0, 0, 1, 2], [-6, 3, -1, -2, -1, 1, 4, -1, -2, -2], [2, 3, 1, 0, -2, -1, 0, -2, -2, 0], [4, -2, -1, 0, -1, -1, 3, 0, 1, 4], [10, 1, -2, 0, 2, 0, 0, 1, -1, -1], [-3, 1, 2, 2, -1, 2, 1, -3, -1, 0], [-1, -1, 1, 0, -1, -2, -1, 4, -1, -1], [0, 2, 1, -1, -3, 1, -1, 0, -2, -3], [6, -1, 0, 2, 0, 1, -1, 1, -2, -2], [4, 1, 6, -2, 3, -2, -4, 0, -3, -2], [3, 1, 1, 2, -1, 0, -1, -2, 0, 0], [-4, 1, 4, -2, -2, 1, -1, -1, 1, -2], [4, -3, -2, 2, 4, -2, 0, 3, 0, -1], [2, -2, 2, 2, -1, -1, -3, -1, 1, -1], [-1, 0, 0, 2, 3, 0, 1, 0, 0, 3], [-3, -1, -3, 0, 0, 0, 3, 1, -1, 2], [1, 0, 3, 0, -2, 0, -3, 0, 2, -2], [4, 3, -2, 0, 0, 0, -2, 1, -1, -2], [2, -3, -1, 1, 0, -3, -1, -2, -1, -1], [12, 3, 4, 0, 2, 2, -2, -3, 0, 0], [-3, -2, 1, 0, -3, 0, 2, -1, -2, -3], [7, -2, 2, 0, 1, 0, 0, -2, 0, 2], [6, -1, 0, 2, 0, -3, -1, -1, -1, -2], [4, 1, 1, 2, -1, -1, -1, 1, -4, 0], [8, -1, -1, 4, -1, -2, -2, -1, 0, -2], [8, -2, -1, 0, -1, -2, 0, 4, -1, -1], [5, 4, 0, 1, -2, 2, 1, -3, 0, 0], [1, 2, 0, 0, -2, 4, -1, -1, -1, -5], [6, 0, -4, -2, -2, 3, 5, -2, 1, -1], [8, 1, -1, -2, 3, -2, -4, 1, 2, 4], [-8, -1, 3, -2, 3, 0, -2, 3, -2, -2], [7, -1, 1, -1, -5, 0, 3, -2, -3, 0], [2, 0, -2, 0, 1, -2, 0, -1, 2, 1], [-3, -3, -4, 2, 2, 0, 2, 0, -2, 3], [3, 3, 5, -3, -2, 1, -4, 1, -3, -4], [-1, 0, -3, 2, 4, 1, 0, 0, 2, 2], [14, 0, 1, 2, 0, -2, 2, -3, 1, 0], [-3, 1, 3, -2, -3, 4, 1, 2, -2, -2], [-1, 2, 6, -3, -4, 0, -3, -3, -3, -3], [13, 0, 0, 0, 2, -1, 0, 1, 4, -1], [10, -1, 1, 0, 3, 2, -5, 3, 3, 1], [6, -2, 1, -2, -3, 0, 2, 0, 2, 1], [-3, 0, 2, 2, -1, -3, -1, 2, 1, -4], [-7, 5, 3, -4, -4, 2, -1, -3, -3, 1], [1, 1, -1, -2, -5, 2, 5, 2, -2, -3], [8, 0, 2, 0, 2, 1, -5, 2, 1, 5], [6, -4, 1, -4, -3, -1, 3, 2, 1, 2], [5, -2, -3, 1, -2, -2, -1, 4, 1, 0], [-8, -1, 3, 1, 2, -1, -1, 0, 0, -2], [-1, 1, -3, 2, -4, 2, 3, -3, -3, -3], [3, 4, 1, -2, 1, -1, -4, -3, 0, 1], [1, -2, -1, 0, -1, -1, 0, 3, 0, 3], [3, 4, 3, -1, 0, 0, 3, -2, -3, 0], [-7, -1, 4, 0, 1, -3, -1, 1, -5, 0], [2, 1, -2, 0, 3, 0, -1, 6, -1, -2], [3, -4, -3, 4, 3, -5, 3, 1, -4, 3], [7, -6, 0, 3, 2, -2, -1, -3, 1, 1], [-3, 0, -2, 2, 0, 1, 3, -3, -2, -3], [-4, -3, -3, 0, 1, -2, 3, 5, 1, 5], [4, -1, -2, 2, 2, -2, 4, 5, -2, 1], [0, 4, -2, -2, 0, 2, 2, -2, 0, 0], [4, -1, -3, 1, -2, -1, 7, 0, 1, -1], [4, 2, 3, -4, 0, 6, -2, 3, 3, 0], [2, -1, -3, 4, 4, -2, -3, -2, 3, 4], [-5, -9, -3, 4, 3, -4, 3, 0, 0, 4], [10, -1, -3, -5, 1, 0, 2, 1, 0, 0], [7, -1, 1, 0, 2, 1, -4, -1, 4, 2], [3, 2, 2, 0, 2, 1, 0, 1, -2, 2], [17, 4, 1, 0, 2, 0, -3, 0, 3, 2], [-6, -1, -2, 2, -3, 2, 4, -2, 0, -3], [8, -4, -3, 4, 1, -2, 4, -2, 1, 1], [-12, -1, -2, 2, -4, 4, 2, -1, 0, -2], [7, 4, -3, 0, 1, 4, 1, -3, 0, 0], [7, 1, -1, 0, 7, -1, -4, 4, 1, 3], [6, 3, 4, 2, 0, 2, 0, -1, -2, -2], [-5, 1, 4, -2, 1, -1, -4, 1, 0, -1], [-6, -4, 0, 2, -1, 0, 0, -1, 1, 0], [0, -3, 3, 0, -4, 1, 3, 6, -3, -1], [-13, -1, 3, 0, -4, -1, 4, 1, -4, -6], [-4, -2, 1, -2, -7, 0, 2, -2, 3, -3], [-7, 2, 0, 1, 1, -3, 0, -2, -2, 0], [-3, 1, 5, -2, 0, -1, -6, 1, -1, -2], [4, 3, 3, 0, -4, 3, 2, 0, -4, -8], [4, -2, 0, -2, -3, 3, 3, -3, 4, 0], [9, 0, -2, 4, -2, 2, 5, 1, -3, 2], [9, 5, 1, -2, -2, 6, 2, -3, -2, -2], [-7, -4, -1, 2, -1, 0, 2, 3, 0, -3], [-7, -1, -4, 0, 0, 4, -1, 2, -1, -2], [-2, 1, 2, -1, -3, -1, -1, 2, 0, 2], [2, 1, 2, 0, 2, 2, -5, -1, 1, 4], [3, -4, -3, 4, 3, 0, -3, 5, 7, 0], [5, 4, -2, -3, 5, 0, -3, 4, 5, 3], [8, -1, -3, 4, 2, 0, 0, 0, -2, 1], [1, 2, -1, 0, -2, 0, 4, -8, 0, 1], [3, 0, 2, -2, -5, 1, 2, -4, 0, 0], [-2, -7, -5, 7, 4, -5, 5, -2, 1, 3], [-15, -1, 2, -2, -1, -3, 2, -3, 0, 6], [0, 4, -5, 2, 0, 2, -2, -1, 4, -1], [-1, 1, -2, 0, -3, 0, 5, -1, 0, 0], [-4, 1, 1, -2, 3, -4, -1, -1, 4, 1], [3, 0, 0, 2, 3, -1, 1, -6, 3, 3], [-4, 0, -9, 1, 3, -1, 1, 2, 4, 5], [12, -1, 4, 0, -1, -3, -1, 2, -1, -2], [-10, -3, -4, 4, 2, -5, 0, -3, 2, 3], [-13, -2, 1, 0, -3, -1, 1, 1, 2, 1], [-3, 5, 0, 0, 2, 0, 0, -8, 2, 1], [12, -1, 1, 2, -2, -2, -2, -2, -3, -7], [8, 3, 4, -2, 5, -2, -5, 0, 2, 3], [-4, 0, -1, -3, 0, -1, 3, 1, 1, 4], [-10, -2, 5, 0, 2, -5, -2, 3, -3, -1], [0, 2, 3, 0, 3, -1, -2, -2, -6, -3], [-18, 4, 0, -1, 3, 2, 2, 1, 2, 0], [14, 0, 2, -2, -1, -2, -4, -1, -2, -2], [-11, -3, 2, 2, -4, 1, 2, -4, -1, -2], [-4, 1, -3, 2, 8, 1, -7, 0, -1, 3], [-6, 0, 4, -3, -3, 0, -2, 3, -1, -2], [8, -3, 4, 2, 2, -2, 0, -3, -5, 4], [-10, 0, -1, 2, -1, 1, 0, 2, 1, -3], [-1, 2, 6, -4, -1, 2, -6, 0, 0, -6], [10, -3, 4, -2, -3, -3, -3, 0, 0, -5], [13, 1, 2, 4, 3, 1, -4, 1, -2, 2], [-2, -4, 0, 2, 3, -3, -1, 1, 1, 10], [-1, 2, 2, 0, 3, -4, -7, 0, 1, 1], [-8, 2, 5, -4, 1, -2, -7, 2, -5, -4], [9, 0, 5, -2, -1, 0, 1, -5, -2, -1], [-5, 2, 7, 0, -3, 0, -2, -5, -2, 3], [12, 0, -3, -2, -3, 5, 6, -2, -1, 0], [4, 7, 2, -2, -2, 4, -4, -3, 0, 3], [2, -4, -3, 3, -3, 1, 3, -2, -5, 3], [-14, 3, 2, -4, 0, 3, -3, -1, 4, -4], [4, 0, -1, -1, -4, -2, 0, -5, -4, 1], [-17, 0, -1, -4, 2, 0, 1, 0, -5, 4], [13, 0, -5, -4, 4, 2, -3, 2, 9, 2], [-11, 5, 1, -2, -6, 4, 1, -1, -1, -1], [-9, -6, -5, 2, 0, -3, 7, 0, 3, 6], [-11, 2, -1, -2, -3, 5, 2, -1, 3, 4], [8, -5, -2, 2, -2, -3, -3, 5, 5, 4], [-6, 1, -4, -2, 5, 2, -2, 4, 6, -1], [-12, 2, 5, 0, -2, 4, 1, 5, 4, 3], [9, -1, 2, 0, -6, -2, 3, -6, 1, 0], [-16, -3, -4, -2, 0, -1, 5, -3, 6, 4], [-2, -1, 0, 2, -1, 1, -3, -8, -1, -2], [-1, 3, 5, 0, -4, 5, 4, -3, -4, 1], [0, -7, -1, 1, 1, -3, 5, 5, 1, 4], [-6, 5, -5, 2, 1, -2, 2, 3, -2, -4], [5, 1, -3, 0, -1, 5, 2, -2, 1, 6], [-17, 3, 1, 0, -2, -2, -3, -1, -7, -3], [4, 4, -3, 2, 0, 4, 1, -7, -1, -4], [-1, -4, -2, 4, -1, -3, 4, -2, 1, 8], [-6, 2, -2, -2, 2, 2, 2, -4, -2, -6], [3, 2, 5, 0, -5, 4, 4, 1, -4, -3], [-15, 5, 2, 0, -4, 2, 3, -2, -11, -10], [17, 1, -1, 6, 1, 1, 0, 0, -1, -3], [-29, 1, -1, 0, -3, 2, 7, 0, -3, -7], [12, 1, -2, 0, 1, 2, -6, -2, 2, 3], [0, 3, 4, -2, -1, 5, -7, 4, 3, -1], [-5, 0, 5, -2, 1, -3, 2, -1, -2, -3], [0, -5, 0, 2, -2, -3, 1, -7, 1, 6], [-7, 3, 3, -2, -2, 2, 3, -1, -1, -5], [-14, 1, 6, -4, 2, -1, -6, -1, 0, 3], [0, -4, 0, 0, 3, 2, 2, 3, 2, 6], [-22, 3, -3, 1, -5, 3, 3, 3, -5, 1], [-1, -2, 4, 0, 5, 0, -1, 2, 2, 1], [-9, -1, -6, -2, 5, -4, -6, 5, 6, 4], [-9, -6, -2, 0, 0, -4, 3, 5, 3, 9], [-16, -1, -3, 0, -4, 0, -1, 0, 7, -5], [-13, -2, -1, -2, -12, -2, 6, -2, -2, -7], [-25, 4, 1, 0, -4, -1, 2, 0, -8, -7], [-23, 1, 2, 0, -3, 0, -5, -3, 3, 0], [-9, 0, 0, -2, 2, 0, -2, -1, 9, 1], [-4, 3, -2, 0, -2, -1, 1, -1, -7, 0], [-2, -5, 0, 0, 5, -2, -3, 6, 5, 2], [13, -1, 0, 0, -2, 0, 0, 2, 6, -3], [5, -1, -11, 4, 0, 5, 10, -1, -1, 2], [-16, 0, 2, 2, -3, 4, 4, -1, 1, -1], [-21, 1, 0, 4, -6, 6, 3, -2, 1, -2], [10, 4, -3, -2, 1, 0, 1, -4, -3, -4], [13, -1, 0, 0, 4, -2, -3, -4, 3, 6], [-29, 2, -2, -1, 1, 3, 4, 2, 6, 2], [-16, -2, 2, -9, 2, -2, -2, 4, 2, 6], [4, -4, -2, 2, 4, -3, 3, 4, -2, -7], [-6, -2, -3, -4, -3, -6, 0, 6, 3, 2], [11, 3, 5, -4, -1, -4, -5, -6, 2, -5], [-8, 4, 6, -4, 4, 0, -5, 2, -1, 1], [5, 9, 1, -2, 5, 2, -1, 0, 2, 2], [20, 9, 4, -4, 1, 2, -4, -2, -6, -4], [-30, -5, -1, 2, 0, -1, 2, 6, 0, 3], [0, -4, -2, 2, 1, 0, 2, 3, 4, 0], [-24, -1, 0, 0, -4, -1, 3, 3, -1, -8], [-26, 1, -4, 2, -4, 2, 1, 1, 5, -4], [23, -6, -2, 2, 8, -6, -4, 1, -4, 4], [-12, -1, -2, -2, -5, -1, -1, 0, 0, -5], [-18, -6, 0, 2, 4, -1, 3, -2, 3, 5], [-15, -7, -7, 6, 2, 0, 8, 3, 4, 8], [14, -2, -4, 2, 8, -5, -5, 4, -5, -1], [-11, -1, -2, 5, -1, 5, 4, 5, 0, 3], [1, -4, -5, 5, 1, -1, 12, -3, -8, -1], [0, 1, -8, 0, 5, -3, -1, 2, -3, -6], [-1, 7, -4, 0, 6, -1, -1, 10, -1, -4], [-15, 1, 2, 2, 4, 4, -6, 2, 0, -6], [-4, -5, -2, 4, 5, -2, -10, 10, 8, 3], [5, 2, -3, 2, 0, 0, -5, -6, 0, -2], [5, 6, 0, 0, -3, 1, 0, -4, 0, 4], [-6, 3, 0, -2, 3, 1, 0, 6, -2, 1], [-22, 2, -3, -4, -2, 0, 7, 3, 3, -6], [-9, 0, 2, 4, -3, 0, 4, 0, -8, -1], [-19, 1, 1, -2, 2, -3, -6, -5, 5, 2], [-14, 11, 5, -6, -8, 6, -1, -4, 2, -4], [-5, 7, -1, 1, -6, 4, 3, -5, 1, -1], [-9, 2, -1, 4, -7, 2, 1, -1, -1, -8], [3, 0, 0, -2, -10, 7, 7, -1, -1, -11], [16, -4, -1, 3, 6, -1, 3, -5, -3, 2], [4, 2, -10, 4, -1, -4, 0, -1, 2, -4], [-6, -7, -3, 0, 4, -8, -3, 4, -3, 0], [1, 2, 8, 5, 2, -3, -10, -3, -2, -2], [-10, 1, 2, 0, -2, -1, 1, 7, 2, -5], [1, 0, -7, 4, 5, 1, -4, 1, 0, -1], [3, -8, 0, 2, 4, -7, 0, 1, 4, 4], [8, 3, 2, -2, 3, 1, -5, 2, 3, -5], [-8, -2, -8, 0, -3, 0, 2, 5, 0, -6], [-12, 0, -8, -2, -3, -1, 3, 3, 3, 7], [-2, -4, -3, -3, 0, -5, -7, 7, 5, 2], [8, -2, -1, 4, -1, -4, 7, -2, -1, 10], [-8, 5, 3, 2, -4, 4, 4, -2, -2, -1], [-8, 2, 4, -6, -13, 1, 1, -7, -3, -5], [-27, 1, -5, 2, -1, 3, 4, 0, -6, 0], [-13, 2, 6, -4, -2, 0, -4, -3, 2, -3], [-15, 7, 4, -3, -8, 5, 2, 2, 2, -3], [12, 2, 4, 4, -3, -2, -2, -5, -1, -6], [-14, 0, -1, -2, 3, -8, -4, 0, -4, -3], [5, 4, 2, -2, 7, 1, -4, -4, 1, 4], [14, -2, 1, 2, 4, -2, 1, -1, 8, -2], [-13, -4, 6, 0, 5, -4, -6, 0, -4, 0], [24, -5, 0, 1, 4, -6, -4, -5, 2, 4], [17, -7, 3, -2, 8, -5, -8, 3, 0, 6], [-3, 3, 2, -4, 7, 2, -6, 7, 1, 0], [5, -3, -9, -2, 4, 3, 2, -3, 5, 3], [-16, -3, -2, 6, -7, 4, 4, -8, -3, -1], [4, 1, -2, 4, -1, 3, -1, -2, 1, -1], [-4, -2, -1, 6, 3, 1, 2, 2, 3, 13], [-17, 6, 5, -2, -6, 1, 0, -6, -10, -5], [10, -6, 7, -2, 3, -5, -9, -4, 6, -1], [12, -1, 0, 4, 1, -1, 11, -2, -7, -2], [4, 0, 0, -6, -2, 5, 1, 0, 6, 2], [19, -3, -8, 6, -2, 3, 10, 0, 7, 15], [-9, 11, 1, -4, -1, 3, 0, -6, -2, 2], [-17, 2, -2, -4, -1, -4, 4, 2, -6, 6], [36, 3, 7, 0, 8, -1, -13, 2, -3, 11], [-12, 2, -3, 1, 5, 1, -5, -4, 0, -9], [-3, 2, 1, -2, 9, -2, -12, 5, 2, 7], [-2, 0, 9, -4, 0, -2, -6, -3, -8, -3], [14, -1, 2, 3, 10, 1, -1, -1, 0, 4], [-3, 5, -1, 2, 5, 3, 0, 6, 1, 2], [-15, 0, -5, -2, 2, -2, -1, -4, 9, 2], [10, 5, 5, 2, 2, 3, 1, -8, 3, 4], [-16, 1, 5, -2, 0, 0, 2, -6, 4, 2], [23, -3, 0, 2, -2, -4, -5, -2, 11, 6], [-9, 1, 0, -8, -2, -3, -2, 0, -4, 0], [6, 5, -2, -6, 9, 6, -6, 4, 8, -3], [0, -6, -9, 6, 3, -2, 8, 0, 3, 10], [11, -3, 7, 0, 1, -1, -6, -4, 2, 4], [-13, -6, 2, 6, -3, -2, 5, 2, -1, 1], [-11, -2, 0, 2, 0, -1, -1, 1, 1, -3], [21, -3, -2, -2, -1, -8, -7, 5, 1, 0], [6, 3, 3, -2, 2, 6, -6, 2, 0, 8], [1, 10, 3, -9, -6, 5, 2, -2, -2, -4], [21, -3, 1, -2, -6, -1, 0, -5, -2, -7], [-3, -1, -3, 2, 0, 3, 7, 7, -8, 2], [-2, 1, -6, 6, 5, -1, 3, 2, -5, 2], [12, 0, 3, -2, 1, 4, 4, 6, -4, 3], [16, 5, 8, -4, -4, 2, -4, -5, -8, -9], [-32, 7, 0, 2, -3, 4, 2, -4, 2, -5], [10, 3, -4, -4, -3, 9, 5, -2, 5, 0], [-14, 3, 5, -6, -6, 0, 2, 4, -4, -4], [16, -7, -1, 2, 5, -5, -7, 3, 4, 9], [-1, 4, 8, 2, -3, 4, 4, -6, -9, -6], [-11, -4, -3, -6, -7, 7, 4, -3, 6, 1], [16, -5, -1, -1, 2, 2, 8, 6, -2, 2], [25, -1, 1, 0, 7, -5, -5, 4, -5, 7], [-14, -6, -1, 4, -2, -1, 0, -7, 0, 8], [10, -6, -6, 4, 6, -2, 1, -4, 0, -2], [19, 0, -5, -4, 3, 0, 5, -1, 1, -2], [-12, -8, 0, 0, 4, -8, 1, 0, -3, 3], [-17, -5, -4, 8, -2, -1, 5, 0, 1, 4], [26, -3, -1, 2, 0, -2, -3, 4, 0, -6], [6, -1, 10, -6, -1, -7, -5, 2, 3, 2], [0, -4, 1, -4, -1, 3, -3, 4, 7, 6], [-12, 3, -5, 8, 8, 4, 4, -4, -3, 0], [0, 5, 5, -4, -6, 6, 5, 2, -11, -5], [-14, -1, 3, -2, -3, 3, 2, 5, 0, -7], [10, -6, -7, 9, -2, -4, 8, -1, 0, -1], [-6, -8, -8, 6, 1, -5, -1, 5, 4, 11], [12, 0, 8, -2, 3, 0, 0, 3, -8, 4], [-4, 2, -8, 2, -7, 2, 8, -7, 0, 4], [-9, -2, 4, -3, -8, -6, 3, -5, -2, 0], [0, -1, 4, 3, 5, -4, 0, 2, -6, -8], [1, 1, 3, 0, 2, 7, -4, 5, 2, -5], [-17, 2, -6, 6, -5, -1, 3, -6, -1, -2], [4, -7, 0, 0, 6, -2, -6, 3, 2, 1], [0, -4, -2, 5, -3, 0, 0, -5, 1, 6], [40, 4, 3, 2, -2, -2, -3, 1, 3, -2], [27, -6, -5, 8, 7, -5, 0, -5, -2, -3], [-22, -4, 3, -5, -4, 0, -4, -3, 4, 1], [-5, 0, 6, 0, -10, 2, 6, -1, -6, -14], [17, 1, 3, -8, 5, 0, -7, 6, -1, 1], [7, -4, 0, 6, 0, 3, -8, -1, -4, 3], [1, -3, 5, -4, -7, 3, 2, 2, -1, -3], [12, -8, -3, 4, 0, -2, 8, -7, 0, 15], [-6, -7, -9, 3, -1, -1, 11, 3, 2, 0], [6, 3, -2, 4, -2, 11, 5, -3, -2, -2], [0, -4, 8, -6, -5, -1, 4, 7, -2, 0], [-9, 2, 1, 4, -6, 4, 3, -4, 3, 4], [-1, -6, -12, 8, -4, 2, 6, 3, 8, 10], [14, -6, -6, 6, -5, -1, 9, -3, -5, -1], [-8, -3, -9, 4, 8, -3, -3, 4, -5, 8], [3, -2, -9, -4, -2, 3, 8, -4, 6, 13], [3, 2, -2, -2, 5, 3, 3, 4, 4, 11], [-11, -5, 3, 4, 6, -2, 1, -7, -6, 3], [19, 12, 2, -2, 3, 1, -6, -4, 1, -3], [-9, -2, -3, 2, -6, 2, 3, -6, -7, 0], [12, -1, 2, -4, 5, -1, -7, 2, -3, 4], [-15, -8, 0, 2, 0, -4, 2, -7, -2, 2], [-3, 2, 4, 0, 1, -6, -7, -10, 3, 7], [-8, -2, -2, 2, 5, -2, 2, 1, 2, 16], [-5, -5, -5, 8, 2, -5, 2, -1, 0, -6], [4, 2, -2, 2, 4, -6, -8, 0, 4, 3], [11, -10, -1, 2, -4, -4, 1, 4, 7, 8], [28, 0, -12, 4, 1, -4, 4, 5, 3, 3], [-2, 6, 7, -3, -2, 4, -4, 1, 4, -5], [-18, 1, 1, 2, -4, 5, 2, 2, 9, 4], [-15, -7, -5, 2, -9, 4, 10, -4, 2, -2], [14, 9, 2, -2, -8, 6, 0, -13, -2, -5], [-13, 5, 3, 4, -3, 6, 3, -6, 5, 1], [-22, -3, 10, -5, -3, -7, -1, -2, -3, 0], [-14, 1, 1, -3, -6, 5, 7, 12, -7, -9], [36, 1, 6, 0, -3, 1, 1, -4, -4, 3], [0, -10, 0, 0, -3, 0, 6, 5, 2, 6], [9, -3, -9, 4, 7, 1, 10, 4, 2, 12], [39, 1, -4, -2, 1, -1, -2, 3, 4, 3], [-9, 3, 7, -6, -4, 6, -7, -3, 9, 1], [-2, -6, -7, 6, 3, -6, -2, 4, 6, -1], [19, 5, -1, 6, 3, 4, -5, -2, -4, -2], [-15, -3, 3, -8, -7, 0, 1, -2, 3, -3], [7, 1, 4, 2, 2, 2, -14, -4, 0, 4], [-13, -5, -6, 8, -3, 2, 10, -5, 5, 6], [4, 8, -2, -6, 0, 5, 5, -4, -1, -6], [11, -4, -3, -2, 8, -5, -4, 2, -6, 5], [-6, -3, 4, 2, -2, -1, -1, 3, 7, 6], [-22, 3, 3, -4, -10, 1, -2, 0, -9, -10], [9, 6, -5, 0, 4, 6, -3, 8, 6, -2], [-5, -6, 0, 2, 3, 3, 6, 0, 6, 12], [15, -7, 0, 4, -5, -5, 2, -5, -2, 5], [16, 5, 7, -6, -5, -1, -6, 1, -2, 6], [34, -1, -1, 6, 1, -2, -2, -5, 4, -2], [3, 6, 4, -4, -1, 2, 1, 4, -2, -7], [-9, -2, -1, -6, -4, 2, 6, 6, 0, 1], [26, -6, -1, 2, 4, -5, 1, 1, 5, 2], [4, -5, 2, -6, 1, 4, 4, -2, 9, 0], [1, -1, 7, -1, 0, 4, -5, 1, 5, 8], [5, 5, 2, -4, 2, -2, -5, -4, -6, -12], [8, 2, 0, -8, 6, 4, -3, 8, 5, -1], [-7, -1, -11, 0, 2, 2, 8, 3, 5, 9], [-7, -3, 2, 2, 1, 2, -5, 7, 3, 7], [7, 2, 1, -2, 0, 8, 1, 2, -1, 0], [-4, 5, -4, 0, 5, -4, 1, 4, -3, 0], [-6, 3, -2, -2, -4, 2, 0, -3, -8, 7], [-7, -10, 0, -2, 9, -4, -2, 12, 8, 12], [10, -4, -3, 5, -3, 0, 6, 2, 5, -4], [22, 0, -1, 2, -5, 0, 0, 2, -5, 9], [-20, 4, 0, 8, -4, 3, 6, 2, -2, -2], [-19, 3, -2, 0, -5, 0, -3, 5, -7, -2], [29, 8, 1, 0, 2, 7, -10, -4, -2, 1], [-21, 2, 1, 2, 4, -6, -7, -4, 3, -2], [12, -4, 10, -2, -7, -9, -3, 1, 1, 1], [9, -12, -2, 4, 5, -7, 1, 6, -1, 13], [14, -9, -9, 6, 3, -1, 7, 1, -7, 9], [7, 9, 6, -9, -6, 11, 6, -2, -6, -5], [8, 3, 5, -2, -2, 5, 2, 0, -2, -2], [-15, 9, -4, 6, 4, -2, 1, -4, -5, -8], [-17, -8, 1, 8, 4, 0, 7, -4, -5, 0], [-30, 4, 3, 0, 5, 3, -6, 4, 0, -2], [-5, -5, -5, 6, 0, -7, 2, -1, -2, 12], [-9, 1, 0, -4, -1, 4, 7, 9, -5, -4], [-1, 4, 1, -6, -10, 8, 10, -8, -2, -9], [-39, -3, -6, -2, 2, 1, 3, -2, -1, 8], [7, -9, -2, -2, 5, -4, -6, 5, 8, 0], [25, 2, 0, 1, -3, -3, 2, -8, 5, 0], [11, 7, 11, -5, -2, -1, -4, -9, -2, -6], [-6, -2, -4, -2, 2, -1, -10, 0, 11, 13], [6, -1, 5, 2, 1, -3, -5, 9, 5, -7], [-9, -2, 0, -3, -1, -3, 2, 2, 6, -2], [-5, 0, 5, -4, -1, 2, -1, -7, 7, 0], [9, -1, -4, 0, 0, 3, -5, 0, 11, 1], [-17, -6, 3, -4, -6, 5, -6, -2, 2, -5], [-1, -6, -1, -6, -2, 1, 2, -2, 4, 3], [2, 2, -3, 8, -8, -1, 8, -9, -6, -5], [-3, 9, 6, -4, 5, 1, -8, 3, 8, -5], [23, 5, 0, 6, -5, 3, 4, -3, -6, -9], [10, -7, -2, 6, -8, -6, 6, 1, -2, -3], [-37, 1, -2, -4, 2, -7, -7, 4, -1, -5], [35, 3, 6, 2, 2, 4, -7, 0, 2, 9], [29, 3, 2, 6, -1, -2, -2, -3, -2, -3], [2, 4, 2, 0, 3, 1, 3, 5, 1, 1], [6, -3, -8, 2, 1, -4, 0, 6, 7, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8040_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8040_2_a_bc();".
// 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_8040_2_a_bc(:prec:=10)
    chi := MakeCharacter_8040_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(3259) - 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_8040_2_a_bc();".
// 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_8040_2_a_bc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8040_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-39168, -7680, 32864, 4416, -9504, -766, 1117, 48, -56, -1, 1]>,<11,R![-32768, -63488, 2304, 32620, -322, -5740, 399, 384, -46, -7, 1]>,<13,R![73856, -151552, 54112, 38920, -20940, -2590, 2135, 19, -79, 1, 1]>,<17,R![-3264, -62464, 28760, 38594, -15138, -5747, 2100, 247, -86, -3, 1]>],Snew);
    return Vf;
end function;