
// Make newform 5766.2.a.bq in Magma, downloaded from the LMFDB on 19 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_5766_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_5766_2_a_bq();".
// 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_5766_2_a_bq();".
// 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 := [2, 0, -16, 0, 20, 0, -8, 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, 1, 0, 0, 0, 0, 0, 0], [-2, 0, 1, 0, 0, 0, 0, 0], [0, -3, 0, 1, 0, 0, 0, 0], [2, 0, -4, 0, 1, 0, 0, 0], [0, 5, 0, -5, 0, 1, 0, 0], [-2, 0, 9, 0, -6, 0, 1, 0], [0, -7, 0, 14, 0, -7, 0, 1]];
    Rf_basisdens := [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_5766_a();"
function MakeCharacter_5766_a()
    N := 5766;
    order := 1;
    char_gens := [3845, 3847];
    v := [1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_5766_a_Hecke( : Kf := Kf)
    return MakeCharacter_5766_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, 1, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, -1, 0, -1, 1], [-3, -1, 0, -1, 1, 0, 0, 0], [-1, 0, 0, 0, -2, -1, 0, -2], [-1, -1, 0, 0, 0, 1, 2, -2], [-1, 0, 0, 1, 0, 1, 1, 2], [-4, -1, -2, 1, 0, -1, 1, 0], [-2, -3, 1, -1, 1, 0, -2, 1], [0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 2, 2, 0, 1, 3, 0], [-2, 0, 2, 0, 1, -3, 0, -2], [-4, 3, 1, 0, -2, 1, 3, 1], [-3, 0, -1, -4, 1, -1, 0, 0], [-2, 1, 2, 2, 0, -2, -1, 0], [-2, 1, -3, 1, 2, 0, -1, 0], [1, 0, 3, 1, 2, 1, -2, 1], [-1, 2, -4, -1, -1, 1, -1, 0], [-1, 3, 0, -3, 1, 4, -1, -1], [0, 4, -2, 1, 2, -1, -4, -1], [-3, 2, -5, -1, -1, -3, 0, 3], [-5, 0, -2, -3, -2, 2, -1, 3], [-2, 1, -5, 2, 2, 4, 0, 1], [-1, -2, 4, 0, -3, -2, -4, 2], [2, -1, 1, 4, -3, 4, -1, -5], [2, -1, 0, 0, 3, -3, 4, 2], [0, 3, -1, 5, 2, -2, -3, 0], [2, 2, -2, -6, -2, -3, 1, -2], [2, -2, 0, -4, -3, 3, -2, -2], [-4, 7, 1, 3, 2, -1, -3, -5], [0, 0, 1, 1, 0, 0, 1, 5], [-4, 4, 9, -3, 3, -3, 0, -2], [-6, -1, 2, 0, 0, -5, -4, 2], [1, 0, 4, -1, -1, 4, 3, -2], [-5, -3, -5, 5, 2, -3, 0, 0], [4, 4, -2, -1, -1, 0, 2, 1], [7, -2, -1, 1, 6, -3, 0, -6], [-4, 2, 5, -5, -1, 0, 2, 2], [1, -3, -3, 1, 2, 4, -4, -3], [-1, 1, -2, -1, -3, -8, 0, 6], [0, -3, 0, -4, -5, 0, 2, -5], [2, 4, 0, -3, 3, -3, 2, 0], [7, -6, -2, -4, 0, -2, 5, 4], [0, -5, -9, 3, 1, 2, -6, -3], [-5, 1, 6, 5, -1, 6, 0, 3], [2, -1, -5, -9, 3, -5, 2, -1], [1, -3, 1, 6, 6, -3, 7, 5], [-2, -3, 4, 3, 1, -5, -6, 4], [-3, 2, 8, 4, 1, 8, -3, -1], [6, -2, 6, 3, 6, -1, -2, -4], [2, 0, -2, 6, -2, 6, 5, 3], [-6, 7, 0, -6, 4, 3, 6, 4], [0, -5, 0, 6, -3, 0, -5, 3], [-4, 2, -4, 0, -4, -2, 0, -2], [-8, -3, -3, -4, 2, 8, -2, 0], [-2, 1, 0, -1, -5, -2, 8, 2], [1, 6, 3, 1, -5, -1, 0, 1], [5, 2, 2, 2, 4, -5, 2, 2], [-2, 4, -1, 1, -5, 5, 3, -3], [1, -5, 1, 0, 0, 0, 1, 7], [-2, 7, 5, 1, -4, 5, -3, 1], [12, -5, 2, 1, -8, -1, 5, 3], [-5, 0, 1, -6, -1, 2, 5, 9], [0, 2, -4, 4, 4, 7, -10, 2], [5, -5, 1, -6, 3, 1, -6, -5], [-10, 3, -1, -2, -6, 3, -7, -5], [4, -5, 0, 7, 12, -3, 4, -1], [1, 4, 0, -5, 2, 6, 7, -5], [-4, 3, -7, 2, 0, 8, 1, -7], [-7, -7, 1, -1, 1, -4, 2, 6], [1, 0, -1, 2, 3, -5, 8, -2], [-7, -7, -6, -3, 5, -8, 2, 7], [12, 0, 4, -3, 9, 6, -2, 3], [1, -3, 3, -4, -3, 0, 7, -1], [8, -4, 9, -1, 6, -1, -4, 7], [-4, 2, -4, 6, -1, -1, -2, -9], [4, -6, -11, 2, -1, 3, 1, -5], [-1, 8, 6, 3, -3, 0, 6, 1], [0, -2, -2, -8, 14, -2, -2, 3], [-6, 1, 2, -2, -4, 4, -12, -3], [-4, -5, 8, 6, -1, 2, 5, 4], [-4, 9, -3, 1, 4, 7, -7, -1], [2, -4, 0, -5, -6, 4, 2, -10], [12, 2, -11, -2, -9, 3, 1, -1], [-10, 6, 1, -4, -13, -1, -3, 4], [-7, 7, -3, -9, -3, 2, 2, -8], [-2, 12, 4, 0, -4, -6, 0, 0], [4, -4, 1, -9, -5, 0, 1, -1], [-1, 9, 8, -7, -7, 4, -2, 3], [-10, -3, -4, 2, -3, 8, 2, -10], [-1, -10, -5, 5, -4, -6, 2, -4], [-3, -3, 1, -8, 6, 7, -7, -3], [6, 0, -5, 5, 4, 5, -6, -2], [-14, -9, 0, 0, -6, 5, 0, -4], [-1, -8, 5, 1, 8, -3, -5, 7], [4, -11, -7, 0, 5, -5, 1, -4], [-14, -1, 3, -11, 3, -7, -5, -4], [-2, -3, -12, -1, -4, -3, 10, -1], [-2, 1, 10, -10, 5, 9, 6, 4], [2, -9, -3, 5, 7, -1, -2, -3], [6, 4, -4, 8, -5, 3, 12, 3], [-14, 4, 6, -1, 5, 7, 2, 1], [0, -1, 5, 1, 3, 15, 1, 9], [-14, -2, -2, 1, -8, -6, 2, 1], [-1, -8, -4, 4, -1, 4, -4, 4], [7, -12, -4, 9, -4, -10, -9, -5], [-2, -2, -5, -1, -1, 11, -3, 13], [-5, -2, 5, -3, -12, -3, 11, 1], [6, 1, 6, 2, -2, 7, -12, -10], [8, 0, 11, 4, 6, -13, 2, -2], [15, 10, -1, -7, -3, 2, -7, 8], [-10, -9, -4, -3, -2, 1, -2, 11], [2, 9, -1, 1, 6, -6, 5, 5], [-4, -4, 4, -1, 0, -16, -3, -5], [-3, -6, -8, -7, 3, 10, 0, 5], [6, 5, -4, -3, 0, -6, 2, 2], [6, -2, 2, 2, 1, 13, 8, -5], [-7, -6, 2, 2, -2, 1, -7, 10], [4, -4, 4, -9, -8, 3, 8, 0], [-6, -3, -8, 7, 14, 1, 4, -3], [6, -13, -2, -9, -4, 12, 4, -1], [14, -6, 7, -10, 0, 0, -8, -5], [-4, 12, 1, 8, -2, -5, 7, 9], [-3, 6, 5, -5, 13, 1, 4, -2], [-1, -2, 2, 0, 8, 5, 11, -18], [-1, 2, -5, 5, -15, -8, -7, 0], [8, -2, 9, -9, -12, 1, 5, 0], [0, -1, 5, -21, 0, 2, -4, -5], [0, -1, 0, 8, 13, 9, -6, 2], [4, 0, 2, -2, -2, -4, -10, -14], [6, 1, -3, -8, 6, -10, 9, 3], [10, -2, -1, 5, -9, 3, 0, -1], [14, 13, -3, -7, 14, -5, 9, 5], [7, -15, 2, 3, 5, -4, -7, -2], [2, 8, -11, -8, -2, -6, 6, -6], [4, -4, -5, -6, -5, -13, -2, 12], [-4, 4, -15, -12, -2, 14, 1, 3], [6, 1, 3, -2, -8, 2, -7, 1], [-7, 9, -1, -3, 3, 2, -8, -14], [-2, -7, -2, -3, -1, -11, -5, 1], [6, -2, 2, 0, -10, -2, 4, 6], [3, 0, 9, 5, 5, 3, -14, 15], [1, 4, -5, -11, -2, -8, -12, 7], [12, 0, -3, 9, -9, -1, 3, 10], [-12, 3, -3, 4, -7, -6, -7, -5], [-4, 3, -8, 7, 4, 6, 1, 5], [2, 3, 16, 0, -9, 7, 6, 11], [8, -1, 6, -4, 0, -11, 16, 2], [0, -2, -14, 12, 4, 4, -1, 1], [2, -10, 17, 5, 6, -3, 3, 10], [1, 8, 6, 17, -7, 0, -8, 5], [6, -9, 14, 10, -4, -8, -4, -7], [6, -12, 4, 8, 0, -21, -3, -1], [9, 2, -12, 7, -2, -3, 19, 8], [2, 4, -7, 11, -16, -7, -6, -9], [2, -2, 12, 6, 3, 4, -11, -1], [-10, 12, 10, 9, -5, -9, -3, -4], [-11, 2, 18, -6, 20, 8, 8, -8], [4, 0, 4, 4, 8, -4, 8, 10], [-9, -2, 5, 5, 16, 6, 6, -1], [-1, 2, 7, -6, 0, 9, 2, 7], [-11, -5, 0, -1, -3, 12, -6, 3], [-2, 3, -9, 1, -7, 4, 1, -7], [-2, -1, 5, 0, -7, -8, 11, 8], [-8, 10, 2, 14, -5, -11, 4, 3], [8, -8, 1, -13, 5, 11, -13, 0], [-6, -3, -15, 7, -1, -3, -4, 10], [-18, 10, -4, -4, -6, -8, -10, 13], [14, -7, 8, -1, 11, 4, -10, 4], [-16, -3, -5, -4, -3, 8, 8, 5], [-7, 0, -23, -7, 3, 2, 13, 2], [-13, 2, 3, -8, -3, -1, -4, 12], [-10, -6, -2, 10, 4, 8, -15, 14], [10, -13, -6, -14, -7, -1, 6, 2], [-7, -7, -5, -8, 2, -1, 15, 0], [3, -10, -5, 5, -3, 7, 2, -2], [-10, -1, 1, -7, -4, 3, 1, -9], [-8, 4, -15, 18, 4, 1, 14, -2], [14, -6, 2, -4, -8, 0, -8, 8], [8, -17, -5, -7, 7, 2, -11, 2], [-11, 4, -5, -5, 8, -12, -2, 7], [-6, -9, -5, -12, -2, 3, 8, 5], [-3, 0, -7, -17, 7, -8, -9, -9], [-25, 12, 7, -4, -9, -13, -4, 2], [10, -6, 0, 9, 13, 0, -1, -8], [4, 3, -10, 21, 4, 7, -6, -9], [5, 8, -1, 11, -6, -11, 8, -2], [2, -1, 0, 13, 18, -6, 4, -2], [16, -9, 5, -1, 0, 5, -17, -19], [-10, 9, 10, 19, 4, -8, -2, 3], [-10, -5, -6, -1, -5, 1, 4, 6], [-8, 1, -2, 1, 8, 13, 4, 3], [22, -2, 4, -18, 3, 11, -8, 1], [8, -2, 3, 6, 12, -3, 5, -13], [-1, 2, -5, 9, 5, 12, -15, -5], [2, 1, -4, 16, -6, 9, -4, 13], [-8, 14, 2, 1, 9, 5, 11, -3], [-6, -2, 10, 8, -1, 7, -10, -10], [-7, -7, 12, -2, 4, 13, 9, -7], [4, 12, -13, 0, -4, -3, -12, 11], [-3, -8, 13, 4, -5, 3, 15, -10], [16, -10, 3, -19, -6, 0, -12, -6], [5, -10, 12, 16, -12, -10, -4, -6], [0, 9, 22, -6, 9, -10, 2, 0], [12, 8, 8, -16, 0, -16, 4, -2], [8, 6, 6, -9, 4, 16, 5, 6], [7, 1, -6, 2, 1, 11, 5, 0], [-8, 16, -2, 8, 5, 5, 6, -8], [21, 5, 3, -10, 0, 0, -7, 1], [-8, 0, 12, 5, -2, -7, 6, -4], [28, 5, 2, -4, -14, 4, 1, -2], [8, -2, -2, 8, 19, 8, 3, 1], [-2, -2, -1, -3, -1, 4, -9, 16], [-12, 11, -8, 3, 11, -4, -7, -17], [-21, 8, -6, -6, -1, 6, 5, -12], [2, -4, -13, 6, 2, 9, 2, 14], [8, 1, 11, 5, 2, -20, 7, -26], [-10, 3, 8, -7, 5, 4, 23, -5], [-13, 1, -4, 12, 5, -18, -6, 2], [2, 8, 6, 2, -6, 10, -14, -12], [16, -10, -16, 4, 2, 8, -2, -8], [-4, -6, -18, 9, 1, 10, 8, -2], [2, -13, 4, 2, -9, -1, 22, 4], [0, -15, -3, 0, 0, -19, 1, -18], [17, -13, -12, 10, 1, 9, -2, -1], [-16, -7, -5, 3, 14, 2, -1, 12], [-14, 5, -8, 1, 1, -4, -2, 10], [3, -9, -22, 0, 7, -2, -3, -8], [0, -7, -7, 13, -14, 8, -3, 14], [33, -5, 13, 1, -10, -4, 13, 7], [-17, 7, -1, -8, 7, -14, -3, 9], [16, 10, 14, 1, -1, -10, -4, 21], [-1, -7, -9, 5, -1, -2, 16, 6], [4, -4, -1, -14, -4, 11, -5, -12], [-4, 6, 7, -5, -2, 9, -22, -15], [33, 10, 3, 0, -7, 6, 8, -12], [-25, -14, 3, -5, 5, -19, 3, 1], [-10, -15, 6, -4, -21, 6, -1, -5], [10, 3, -22, -3, 9, -10, 6, 0], [-20, -5, 6, 14, -1, 12, 10, -12], [2, 5, -2, -7, -18, 0, 2, 0], [10, -6, -9, 3, -1, 5, 3, -8], [-17, 0, 8, -8, -15, -26, 1, -7], [6, 7, -20, 8, -6, 6, 7, -3], [-28, 1, 1, -2, -11, 10, 19, 7], [6, -13, -5, 7, 8, 5, 11, 13], [-8, 18, -16, 9, 10, 3, 7, 16], [-22, 13, -1, 12, 2, -3, 3, -5], [-9, 2, -1, 15, -16, -13, -7, 3], [18, 21, -10, 2, -3, -16, -1, 10], [20, 4, -6, 13, -14, -7, -2, -4], [-8, 3, 9, 16, -10, 6, 12, -2], [-13, -14, 2, 0, 2, -8, 2, -10], [-23, 12, 1, 11, -3, 7, 9, -19], [-2, 7, -5, 22, 3, -5, -11, -3], [-4, -33, -4, 11, 2, -6, -9, 1], [-6, 19, -4, 3, 2, 1, 8, 5], [-8, 7, -12, 29, 10, 8, 7, -12], [17, 0, 4, 4, -11, -4, 8, -2], [-20, -7, 2, -9, -4, -9, 16, -9], [-14, 16, 19, -16, -4, -16, 1, -2], [20, 22, -1, 15, -15, -7, -1, 12], [24, 1, -11, -15, 17, -20, 5, -10], [2, 5, 18, 3, -9, -5, 0, -17], [14, 6, 7, -16, 7, -16, -5, 8], [11, 18, -1, -12, -16, 14, -19, -3], [-16, -8, 12, 4, -5, -15, -2, -14], [20, 1, 11, -5, 3, 13, -10, 11], [-9, -15, 0, -11, -20, 15, 0, 2], [2, -12, 9, -8, -1, -5, 7, 3], [0, 7, 18, 1, -14, 0, -20, 10], [4, 3, -6, -1, -26, -3, 8, -14], [-23, -5, -10, 6, 1, 18, -1, -20], [13, -6, 6, -6, -20, -4, 0, -10], [-4, -3, -11, 7, 14, 0, -4, -15], [-6, 4, 10, 3, 4, 1, -6, -18], [-8, 6, -9, -9, -15, 5, -19, 10], [6, 2, 6, 11, -22, -9, 0, 5], [-9, 11, 1, 5, 1, 8, 17, 20], [-13, 0, -19, -2, 7, 5, -8, 14], [2, 2, -5, -6, -3, -13, 19, 1], [-19, -14, -17, -12, 1, 8, 11, -3], [-14, -8, 3, 0, 1, 10, 10, -21], [32, 11, 1, 7, -3, 1, 6, -5], [-10, -8, 4, 26, -3, -3, -10, 7], [-8, -7, -2, -12, 6, 19, -20, 5], [-4, -9, 3, 4, -12, -14, 13, 1], [20, -4, 3, 15, -2, -1, -1, 4], [5, 0, 13, 1, -5, -4, -1, 17], [3, 8, 17, -2, 2, -20, 17, 1], [0, 23, -7, -15, -19, -6, 9, -19], [-4, 9, -12, -14, 0, -10, -7, -3], [10, 0, -5, -2, 3, 5, 25, -8], [5, 9, 7, 1, -5, -9, 12, -10], [-24, -27, 1, -3, 5, -8, 6, -1], [14, -6, 15, 1, -1, 9, -1, 16], [-15, -3, 17, -17, -24, -6, -4, 3], [-23, -12, -1, 2, 10, -11, -11, 21], [0, -1, 21, 0, 2, 14, 21, -4], [-5, 2, -20, -2, -6, -8, -2, -24], [-14, 11, -6, -1, 31, 4, 4, 6], [3, -5, 4, 0, 3, 8, -7, -8], [-24, -2, 4, 10, -1, 12, -4, -13], [-16, 10, 7, 22, 0, 9, -17, 6], [7, 16, 0, -2, 16, -10, -5, -14], [-3, 10, -13, 4, -2, 7, -3, -23], [-10, 13, -11, 3, -1, 15, 22, -2], [-24, 3, 8, -19, -7, -12, -16, -10], [-40, -12, -6, -9, 3, -1, 5, 1], [-2, 5, -5, -6, 3, 17, 16, 3], [-20, 11, -7, -9, -7, 20, -6, -3], [-4, 1, 4, 23, 17, -18, 10, 13], [-6, -2, 7, -9, -20, -12, -5, 0], [4, 2, -32, -3, 1, -5, 16, -12], [-14, -7, 30, -10, 8, -4, -2, -1], [-29, -1, -7, 5, -9, -6, 3, 14], [-12, -6, 15, 7, -8, -3, -1, -18], [4, 6, 6, -6, -8, 6, 13, -6], [-27, 5, 14, -8, -4, -17, 0, 0], [2, -2, 7, -4, 20, -2, 7, -1], [-20, 14, 0, 1, 22, -7, -8, -9], [34, 5, -1, -3, -8, -3, -7, -3], [14, -6, 5, -5, 3, -3, 8, -23], [-38, -11, 12, 14, 1, -5, 14, -11], [-4, 4, -26, 8, -2, 0, 16, -10], [-11, 17, -11, 4, 17, 6, 1, 3], [6, -4, -17, 2, 10, -12, 3, -22], [-14, -10, -11, 19, -5, -22, 12, -6], [22, -2, 19, 5, -13, 4, -11, -23], [8, 18, 16, 4, 12, -14, -6, -6], [25, 17, 4, 8, -5, 10, -10, -18], [4, -2, -2, 23, -2, -4, -11, 14], [-19, 23, 6, 8, 3, 17, 9, -4], [-1, 6, 2, 15, 22, 9, 11, -8], [-8, 11, 14, -15, -5, 1, 8, 8], [-7, -1, -5, 11, 18, 12, -21, 5], [-14, -11, -5, -1, -10, -10, -17, 20], [-15, -18, 3, 0, 5, 2, -8, -20], [-9, 5, 9, 5, -4, 19, -24, 8], [-17, 4, 20, -8, -4, -9, 2, 4], [7, -14, -24, 3, -9, 22, -2, 7], [-36, -10, 2, -6, -18, 6, -8, -2], [0, -19, 7, 0, -4, 21, 10, -2], [-6, -1, 23, 14, 21, -14, 9, 11], [14, 1, 15, 3, 18, -22, -7, -14], [-16, -8, 16, 11, 17, -20, 4, -3], [-30, 8, -13, 2, 16, 2, -15, 9], [2, 3, -19, 4, 6, -28, -19, -15], [-2, -4, 1, 30, 18, -17, 20, -7], [8, 8, -18, 1, 2, 10, -8, 7], [-8, 18, -8, -2, -2, 31, 12, 1], [25, -4, 7, -4, -8, -4, -13, 3], [8, 9, -4, -6, 23, -10, -14, -5], [2, 5, -6, -5, -2, 0, 11, 9], [15, 1, -5, -19, 23, 2, -2, 10], [-28, -6, 5, -13, 8, -7, -8, -1], [-5, 15, 6, -8, 9, -9, 9, 22], [-12, 3, -19, 7, -7, -17, -1, -9], [-2, -20, -14, 4, 0, 30, -8, -4], [12, -3, 2, -5, -2, -13, 12, 5], [1, 7, -17, -19, -3, 8, -6, 12], [-18, 12, 13, -5, 8, -5, -2, 3], [-34, -7, 6, -6, -2, 8, -24, 1], [9, -33, 1, 0, -9, -8, -5, -9], [-14, 5, 18, 13, -4, -10, 12, -6], [1, 20, -5, 9, -11, -9, -1, 11], [-17, -1, -20, 10, 5, -30, -11, -8], [-35, 18, 0, 12, 9, -16, -8, -2], [-18, -21, 9, -10, 13, -7, 1, -13], [-11, 23, -17, 2, -4, -18, -3, -9], [-20, -7, -11, 3, -8, -3, 11, 15], [23, 2, -1, -3, 23, 7, -3, 11], [14, 18, -3, -15, -25, 9, -18, 1], [-1, 33, 4, 0, 8, 12, -9, -18], [-22, -21, -13, -2, 21, -20, 8, -11], [-7, 5, 6, -14, -10, -14, 14, -12], [-14, -8, 0, -14, -18, 6, 13, 26], [12, 11, 12, -16, 7, 35, -2, 5], [14, 3, 24, 15, 8, -1, -10, 0], [4, -4, -13, 16, -3, -9, 16, 18], [10, 18, 0, 17, -5, -19, -1, -6], [7, -15, -1, -3, 18, 7, 0, 28], [3, -15, -26, 11, 1, -12, -21, -6], [-2, -8, 18, 8, -14, -8, -8, -28], [-12, -17, -21, 10, 7, 2, -7, -7], [-8, 11, 16, -13, 16, 17, -18, -15], [8, -7, -8, -2, 3, 4, 8, 25], [-30, -17, 1, -17, -8, 0, 7, 7], [-6, 14, -1, -9, -40, -15, -1, 14], [-26, -25, -12, 13, 6, -1, 6, -2], [8, 13, 15, -18, 12, -20, 5, 11], [10, -13, 25, 1, -18, -2, 19, -4], [11, 8, 3, 3, -21, -9, 6, 10], [-23, 16, -1, -5, -8, -4, 22, -6], [-3, -12, 6, -12, 16, -2, 0, -22], [24, 29, -15, 2, -4, -9, 10, 2], [4, 8, 10, -31, 2, -1, -10, -6], [-2, 15, -12, -13, -22, -9, 12, -9], [12, 19, 4, 33, 5, -20, 2, -28], [-3, -22, 11, 7, 0, -7, 0, 7], [-4, 4, 8, -20, 3, -9, -22, -4], [-20, -15, 12, 2, -29, -9, -20, -6], [4, -42, 4, -8, -2, 14, -6, -26], [-6, -8, 8, -25, -9, 4, -15, 16], [-6, 4, 16, -18, -1, 24, 25, 13], [-12, 5, 3, 7, -5, 19, -6, -22], [1, 24, -17, -28, -14, 19, -14, 3], [-8, -3, -1, 15, 0, 15, -23, -6], [-19, -23, 8, 8, -17, 5, 5, -20], [-26, -2, 28, 14, 14, 3, -4, 15], [20, 25, -4, 3, 4, 16, -21, 8], [-1, 2, -4, -20, -2, -15, 12, -5], [-24, -14, -4, 9, 14, -9, -10, 32], [-1, 22, -9, 10, 7, 10, 16, 6], [19, -12, 11, -15, 13, 3, -39, -1], [-30, 8, 9, 3, 0, 23, -8, 7], [-6, 18, -10, -13, 11, 0, 8, -1], [10, -2, -41, -3, 3, 17, -9, 17], [2, -36, -14, 8, 12, -17, 19, -1], [18, 1, 17, -8, 2, 22, 7, -9], [15, -11, 17, -9, -5, -31, -16, 24], [2, -50, -7, 7, -7, 4, 5, 5], [-10, -12, -18, -8, 15, -26, -2, 3], [-8, -14, 24, -2, 23, 18, -3, -12], [-12, 13, -23, 16, 16, 8, 7, -7], [-16, -13, -17, 15, -8, 6, -1, -26], [-13, -4, 21, 11, 7, -14, 23, -21], [16, 3, -14, 17, 0, 32, 0, 10], [-16, 11, 7, 0, 7, -3, -25, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5766_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5766_2_a_bq();".
// 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_5766_2_a_bq(:prec:=8)
    chi := MakeCharacter_5766_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_5766_2_a_bq();".
// 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_5766_2_a_bq( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5766_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-1, 40, 68, -40, -110, -40, 12, 8, 1]>,<7,R![382, -176, -408, 112, 160, -16, -24, 0, 1]>,<11,R![-7103, -16024, -10796, -920, 2046, 1048, 228, 24, 1]>],Snew);
    return Vf;
end function;