
// Make newform 6027.2.a.bc in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6027_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_6027_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_6027_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 := [1, 8, 8, -27, -4, 23, -6, -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, 1, 0, 0, 0, 0, 0, 0], [0, 1, -10, 3, 14, -7, -2, 1], [3, 16, -10, -15, 8, 2, -1, 0], [-6, -16, 11, 15, -8, -2, 1, 0], [-1, 7, 16, -11, -15, 8, 2, -1], [-6, -26, 14, 29, -15, -4, 2, 0], [0, 8, 42, -24, -44, 23, 6, -3]];
    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_6027_a();"
function MakeCharacter_6027_a()
    N := 6027;
    order := 1;
    char_gens := [4019, 493, 2794];
    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_6027_a_Hecke(Kf)
    return MakeCharacter_6027_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 := [[1, -1, -1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, -1, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, -1, 0, 0, -1, 0], [3, 0, -2, 0, 1, 2, 0, 1], [1, 0, -1, 1, 0, 1, -2, 1], [1, -1, 2, -1, -1, 0, 3, -1], [3, 0, 0, 1, 0, 1, 0, -2], [0, -1, 1, 3, -1, -2, -2, 0], [0, 0, 3, 0, -2, -2, 0, -2], [-3, -1, -1, -1, 1, 1, -3, 1], [1, 0, 0, 0, 0, 0, 0, 0], [1, -1, 1, -1, -4, -1, 2, 0], [4, -2, 0, 2, -1, 1, 0, 1], [-1, 1, 1, -1, 1, -1, 1, 2], [6, 0, -1, -1, 0, 5, 2, 0], [0, -3, -2, -1, 2, 0, -3, 2], [0, -2, 4, 2, 0, -3, 0, -2], [-1, 3, -1, -1, 2, 2, 0, 0], [-2, -1, 3, 2, 1, -4, -1, -3], [-2, -3, 1, 2, -1, -4, 0, -1], [2, -2, 1, 0, -2, -2, 2, 1], [5, 1, -4, 3, 3, 1, -4, 3], [1, 2, 3, 1, -3, 0, 0, 0], [5, -1, 4, 0, -1, -1, 5, 0], [2, 2, -2, 2, -3, 1, -4, -2], [-3, -6, 6, 0, -1, -4, -1, -3], [-4, 0, 5, 0, -3, 1, -4, 0], [-1, 5, 2, -2, 1, 3, 5, -2], [-4, -1, 2, -1, 1, 0, 6, -5], [2, -6, -2, 4, -2, -2, 3, 1], [0, 0, 3, 1, 2, -2, 0, -2], [-3, -1, -2, 2, 1, -8, -3, -1], [-1, 1, 2, 0, -2, -1, 4, -2], [-7, 1, 3, 0, 1, -4, -1, 1], [0, 0, 1, -2, -1, -3, -1, -1], [9, 3, -2, 0, 6, 6, 4, 0], [5, -4, 1, -3, 0, 1, 2, -2], [3, 5, -2, -5, -1, 0, 2, 0], [1, 3, 0, 0, 6, 4, -3, 4], [9, -4, -9, 3, 4, 6, -2, 2], [11, 0, 2, 0, 0, 5, 3, 3], [-1, 0, -3, -3, 3, -5, 0, 1], [-4, 8, 1, -2, -1, 4, 5, 11], [8, -2, -5, 0, 2, 3, -3, 0], [-1, 4, 0, 3, -3, 0, 4, 4], [-1, -2, -2, -2, 0, -9, -1, 3], [-6, 6, -3, -1, 7, 3, -2, 5], [4, 12, -1, -2, 3, 7, 4, 3], [2, -8, 0, -1, 3, -2, -5, -8], [-3, 2, 2, -2, -1, 2, 0, -2], [-9, -1, 5, -2, -2, -5, 2, 3], [1, 8, -3, -3, -1, 8, -1, 1], [5, -7, 6, 2, -4, -3, 0, -8], [10, 4, 1, 0, -6, 3, 5, -1], [0, 3, 6, -1, -10, 2, 2, -3], [-2, 5, 1, -6, -3, 4, 4, 8], [-12, 6, 0, -3, 2, 2, 1, -2], [-5, -3, -2, -1, 1, -7, -5, 2], [-12, 3, 6, -2, 1, -3, 0, 1], [9, 7, -1, 4, 3, 9, 4, 3], [2, 4, 2, -3, -8, 1, -2, -2], [-11, -5, 12, -1, -3, -6, 0, -9], [3, -3, 0, 5, 3, -1, -8, 7], [16, -3, -9, 6, -3, -2, 0, 1], [12, 4, -8, 2, -4, 3, -5, 1], [1, -2, -1, 0, 1, 10, -3, 7], [-21, 3, 7, -9, 0, -4, 1, -2], [4, 8, -1, 4, -5, -2, -5, -1], [6, 3, -2, 0, -3, 4, -5, 4], [6, -1, 3, -1, 2, 3, 11, 4], [2, -8, -6, -1, 10, 1, -9, 0], [2, 3, -4, -1, -3, -6, 1, 0], [-14, 0, 8, 3, -2, -6, -1, -1], [-2, 4, 2, -3, 2, -2, 6, 0], [7, 11, 3, -3, -1, 1, 1, 3], [16, -2, -3, 5, 5, 4, -2, 9], [12, -8, 1, -2, -2, 0, 2, 3], [3, 6, -9, 5, 2, 3, -7, 3], [-17, 2, 9, -3, 9, -8, -5, -7], [7, -1, -8, 6, 8, 12, 2, 5], [-4, 2, 3, -1, -5, 2, 10, 0], [-6, -1, -6, 2, 4, 1, -2, -3], [-2, 1, 0, -2, 0, 0, -8, -8], [1, 5, -4, -1, -11, 5, 0, 2], [-15, -2, 6, -1, -1, -13, -7, 2], [-6, -6, 2, 4, -4, -5, -5, -3], [-7, 2, 7, -2, 4, 7, 0, 6], [-17, 1, 3, -9, 1, -2, 1, -4], [7, -7, -7, 1, 13, 3, -8, 5], [-5, -3, -8, 1, 2, 3, 6, 6], [-16, 0, 3, 4, -5, 3, -2, 0], [7, 5, 1, -2, -2, 1, 12, 0], [-2, -3, 5, 0, -2, -15, -7, -5], [-6, 8, 3, 5, -6, -7, -5, -2], [-4, -2, 6, 2, -3, -4, 5, 2], [-10, 4, -4, 2, 0, -6, 4, 1], [0, 6, 3, -7, -1, -8, 7, -3], [15, -2, -11, -5, 13, 15, -3, 7], [-8, -3, 0, 3, 7, 2, 1, 1], [0, -3, 7, 8, -3, -7, -2, -1], [-11, -3, 12, 5, 9, -5, 5, -5], [11, -5, -5, 5, 0, 1, 0, -1], [-4, -4, -8, 3, -9, -2, -4, 1], [11, 6, 9, -3, 7, 10, 10, 7], [10, 12, -3, 2, 6, 6, 3, 1], [4, -3, 4, 9, -4, -9, -5, -3], [0, -4, -9, -3, 10, 8, -12, 5], [-9, 10, -7, -2, 12, 4, -3, 4], [-3, 6, -6, -3, -5, 1, 1, -4], [8, -16, 7, 2, -8, -4, 8, -8], [-1, 0, 0, -7, 2, -1, 10, 2], [11, -6, 9, 9, -4, -6, 6, -9], [4, -6, -8, 4, 10, 5, -15, -1], [-2, 0, 7, 5, 4, 1, 9, -6], [12, 2, 1, -1, -6, 2, -3, -8], [-2, 14, 8, -4, 6, 4, 6, 9], [-16, -2, 1, -4, 4, 3, 5, 8], [25, -4, 4, 1, 1, 4, 4, 5], [10, 14, 5, -4, 2, 4, -2, 0], [-16, -2, -2, 6, 1, -12, -9, 2], [-5, -1, 4, -11, 6, -8, 2, -2], [-15, 10, -1, -5, 2, -1, -3, -8], [-2, 2, 1, 2, -1, 1, -2, 7], [15, -11, -3, 2, 1, 3, 1, -6], [-12, 9, 6, 0, -4, -8, -9, 1], [-5, 1, 7, 2, -13, 1, -1, -2], [2, -4, 5, -11, -8, 0, 3, 7], [19, -15, 1, 1, -4, -13, 6, -15], [11, -3, -3, 12, 2, -1, 3, -4], [8, -12, 0, -2, 0, -15, 3, 0], [5, 0, -5, 4, 11, 3, 2, 13], [-8, 3, 5, 4, 3, -1, -10, 4], [-2, -2, 0, -6, -5, -10, -1, -5], [0, -2, -2, 3, 1, 10, 5, 3], [8, 8, 1, 4, -12, 0, 2, 8], [-8, 21, 11, 0, 10, 6, -5, 4], [15, 9, 10, 0, -8, -5, -4, 2], [-4, 3, -7, 8, -4, 1, -6, 5], [16, -1, 8, 9, -11, 7, 9, -1], [-11, 5, 0, -3, 3, -11, 4, -7], [-11, 8, 6, 4, 4, 6, 7, -6], [13, -1, -12, -5, 3, 15, -6, 9], [18, 3, -2, -11, 4, 8, 8, -2], [7, 1, 1, 11, 0, -6, -6, -3], [7, -6, -7, 12, 10, 6, -4, 7], [0, -4, 16, 3, -6, -4, 3, -10], [-5, -6, 8, 4, 4, -10, -9, -10], [15, -7, -10, 4, 8, 9, -1, 1], [-2, -10, 4, -2, -13, -7, 13, -7], [0, 7, -3, -7, 5, -6, 1, 14], [10, 8, 9, 6, -4, 5, 15, 6], [14, 16, -18, -1, 14, 15, 4, 15], [-35, 7, 7, 2, 0, -4, -2, -1], [9, 2, -1, 2, -9, -2, -2, -10], [-5, -1, 0, 9, 9, 0, -6, 3], [0, 3, 1, -8, -2, 17, 12, 7], [-8, 3, -2, 5, -5, -5, -1, 3], [-21, -1, 5, -7, 7, -3, -5, -4], [9, -2, 1, 5, -12, 0, -2, -4], [-13, 17, 1, -7, -2, 9, -4, -1], [9, 3, -12, 6, 12, 18, -1, -4], [24, -12, -23, 3, 11, 12, -9, 5], [-16, 3, 11, 4, -8, 2, -5, -3], [9, 7, -11, 5, -3, -1, -5, 10], [-26, 0, 6, -11, -6, 1, -7, 7], [-7, 1, 8, -8, -8, -14, 3, 5], [-3, 8, -14, -15, 5, 13, 8, 4], [-11, -13, 6, 4, -11, -14, -1, 0], [5, -5, -16, -1, 11, 9, -11, 6], [-3, -15, -8, 4, 2, 7, 0, 5], [10, -2, -6, 10, 7, -5, -6, 5], [18, -3, 2, -8, 1, 6, 7, 4], [0, -11, 4, 9, 3, -15, 6, -6], [10, -2, 2, 7, 7, 3, -12, 9], [0, 2, 7, 11, 1, 5, 3, 3], [-15, 2, -10, -5, -3, 5, -7, 6], [13, 2, 5, -6, -3, 3, 17, -2], [-4, -4, 2, -3, 6, -14, -18, -11], [2, 10, 16, -2, -3, -3, 21, 2], [-3, 12, -15, 7, -1, 12, -11, 4], [-4, -4, -1, 6, -7, -1, 8, 12], [16, 6, -3, -3, 2, 6, -5, -10], [-24, -15, 9, 2, 0, -11, -5, 1], [10, -8, 7, 2, 10, -14, 5, -10], [-20, 15, 13, -9, -2, 6, 10, 0], [8, 0, -17, 4, 9, 17, -2, -4], [-11, -8, 10, 7, 11, -2, -5, -5], [-37, 2, 22, -18, 0, -5, 5, -10], [36, 4, 0, 1, 7, 7, 9, 10], [13, 5, 2, 6, 4, 0, -3, 5], [21, -1, -10, -6, 5, 11, 12, -9], [19, 3, -21, -5, 6, 28, 4, 12], [-19, 8, 8, -9, -9, 9, 8, 1], [-15, 6, 8, -9, 0, 5, 12, 3], [-7, -1, -7, 1, 9, 7, -6, 10], [-9, 0, 1, 0, 2, -11, 2, -4], [-9, -1, -9, 2, 3, -13, 8, 9], [1, -3, 8, 2, 2, -11, 9, -2], [21, 9, -11, 7, 10, 19, 0, -13], [13, -12, -15, 0, 12, -4, -12, 7], [23, 12, 0, -4, 4, 5, 13, -4], [10, -14, -12, -8, 5, -8, -5, -5], [-1, 10, -7, 2, 18, 6, 0, -1], [8, 0, 0, -3, -2, -2, 1, -10], [7, -6, -17, 5, -11, 9, 2, 9], [-22, -7, -9, 3, -10, -9, 0, 3], [17, 1, -20, -5, -7, -1, -3, 2], [-33, -3, -2, 7, -3, -9, 2, 0], [22, -4, -22, 5, 0, 14, 4, 6], [3, 2, 24, 4, -7, 2, -1, -3], [20, -9, 4, 2, -17, 3, 14, 5], [-7, 0, 2, 7, -1, -17, 0, -6], [-6, -10, -6, 1, -9, 8, 13, 5], [-15, 8, -9, -15, 7, 1, -13, 11], [-14, 4, 5, 3, -17, -5, 8, 11], [8, 2, -1, -7, 1, 6, -3, 5], [15, 12, -14, 0, -2, -1, -15, 8], [-7, 0, -13, 6, 1, -5, -23, 9], [17, -14, -6, -6, -6, 3, 16, -5], [15, -6, -5, 6, -12, 13, 5, 4], [7, 5, 6, -5, 5, -9, 11, -6], [-12, -9, 16, 11, 3, -8, 3, -8], [11, -2, -12, -11, 7, 14, 8, 1], [-3, -5, -20, -9, 22, 15, -15, 8], [9, -12, -3, -2, -2, -11, -8, -14], [28, -4, 6, 0, -1, -9, 2, -13], [21, -11, -7, -3, -18, -4, 15, -10], [25, -20, -11, 1, -8, 4, -8, 6], [8, 14, -11, -6, -5, 4, 4, 5], [3, -1, 3, -2, 0, 22, 21, -9], [-25, -17, -8, -8, -15, -19, -4, -4], [-21, 3, -2, -8, 10, -4, -9, -2], [-10, 8, -11, 8, 6, -11, -13, 2], [-13, 9, 14, -2, 1, 3, -9, 2], [2, 4, -6, -3, -13, 24, 12, 11], [-12, -6, 5, 7, 10, -9, 1, -8], [3, -4, -7, -5, -4, 7, 3, 6], [9, -3, 7, 5, -6, 12, -2, 4], [3, 9, -6, -14, -6, 9, 3, -9], [-23, 13, 4, -4, -8, -2, -12, -4], [-18, 20, 8, -11, -1, 24, -2, -3], [2, 12, -12, 2, -11, 10, -11, 10], [4, -21, -4, -1, 3, -22, -21, -13], [-24, 13, -12, -12, 8, 8, -6, 3], [2, 20, 5, -8, -13, 7, 11, 18], [-1, -19, -7, 3, 4, 5, 13, -9], [8, 7, -27, -2, 18, 31, -8, 3], [1, -11, 6, 17, -11, -9, 0, -9], [-7, 19, -10, 7, 6, 11, -7, 2], [-18, 3, 12, -21, -5, -12, 6, -9], [27, 0, -15, 8, -10, 4, -2, -5], [-7, 2, 5, -12, 6, 11, 4, 2], [35, -8, -16, 12, 4, 9, 2, -1], [-1, 17, -2, -9, -2, 0, 16, -11], [10, -7, 5, 2, -9, -12, 7, 11], [19, 4, -2, 3, 1, 11, 13, 8], [5, 8, -7, 5, -9, 10, -13, 4], [6, 4, 6, 18, -6, 3, -5, 14], [-10, 11, 9, -1, -9, -9, 5, -2], [-36, 3, 15, -17, 0, -1, 6, 4], [11, -25, -7, 6, -2, -12, -14, 7], [-2, -4, -13, -15, 13, -8, -10, 8], [4, -2, 3, 10, -20, -2, 2, 18], [2, 5, 10, 1, -3, -3, 18, -2], [-20, -11, -1, 4, -7, 4, 6, -11], [6, 3, -3, 13, -8, -16, -10, -13], [29, 0, -22, -3, -2, 6, -9, 14], [13, -9, 10, 4, -17, 4, 21, -5], [37, 16, -5, 12, 14, 27, 14, 0], [5, 18, -2, -2, -2, -2, 0, 12], [3, -23, 2, 2, -17, -14, 1, -11], [-18, 3, -2, -1, 7, -12, -11, 5], [26, -5, 5, -5, -14, 12, 7, 3], [-6, -9, 3, 0, 13, -14, 6, 4], [-30, -11, 24, 8, -6, -20, -11, -2], [-12, 11, -20, 1, 1, 16, -25, 2], [-11, 17, -7, 0, -9, 11, 3, 9], [-22, 29, 2, 0, 10, 2, -7, 1], [17, -14, -23, -9, 17, 18, 7, 6], [4, 5, -15, -12, 11, 26, 8, 21], [10, -4, 2, -3, -7, -16, 8, -8], [-7, 14, -9, -6, 2, -4, 4, -19], [-11, -10, -5, 13, -16, -17, -6, 1], [58, -4, -19, 7, 2, 9, 4, -2], [-46, -2, -3, 1, -3, -17, -14, -3], [44, -17, -6, 6, -6, -6, 12, -5], [-26, 21, -11, 0, 14, -2, 3, -1], [-13, -13, 8, -11, -14, -24, 2, 2], [17, -9, 11, 8, -8, -16, -5, -9], [4, 1, -5, -2, 3, 17, 13, 13], [-16, 18, 2, -13, 5, 20, -7, -10], [-16, -4, 5, 6, -5, -10, 15, -12], [-7, 17, 21, -4, 1, -2, 9, 18], [21, -6, -1, 4, -4, -1, 9, -7], [-27, 1, 14, 14, -1, -14, -11, 12], [16, -5, -11, -22, 13, 13, 7, 4], [38, -4, -11, 3, 5, 8, 12, -8], [-6, 4, -12, -2, 18, 7, 5, -4], [42, -8, -10, 16, -9, -3, -5, 1], [-15, 1, -14, 7, 14, 3, -29, 11], [-4, 4, 14, -1, -8, -5, -8, -4], [-28, 9, 3, -6, -12, 11, 13, 14], [-30, 2, 8, 2, -18, -9, -9, -10], [13, -10, -4, -2, -17, -8, 6, -13], [37, 1, 0, 16, 1, 12, 1, 0], [36, -2, -17, 16, 15, 19, 2, 18], [5, 17, -3, -12, -2, 24, 28, 6], [1, -11, -15, 5, -8, -2, -12, -11], [-31, 14, -1, 2, -6, 2, -10, 16], [7, -15, 9, -7, -16, 2, 4, 12], [4, 0, -10, -10, -9, 12, -6, 17], [1, -9, -8, -9, 7, 18, 13, -1], [-45, 4, 8, -4, 0, -20, -5, 14], [21, -12, -7, 3, -4, 8, 15, -12], [-6, 3, 12, 1, -13, 8, 16, 4], [19, -4, -2, 0, -4, -20, -6, -1], [36, 5, 17, 6, -18, 1, 20, -6], [7, 22, 20, 2, 8, 21, 10, 8], [-10, 17, 4, -14, -22, 16, 9, -9], [9, -2, -23, 21, 13, -9, -16, 2], [-5, -11, 32, -5, 0, -28, 21, -5], [4, -2, -6, -4, -10, -11, 8, -23], [-20, 11, -4, 0, -6, -1, 12, 14], [-17, 13, 3, 3, 16, 12, -12, 13], [-19, 22, 26, 0, -20, -1, 10, 1], [9, -28, -1, 4, -3, -24, -10, 12], [21, -1, 4, -9, 13, -4, 10, 4], [-16, -12, -8, 4, 3, -12, -23, 2], [-30, 10, 0, -10, 9, -2, -7, 25], [41, 1, -3, 2, 0, 5, 13, 3], [15, -17, 1, -15, 20, -5, -2, -18], [3, -19, 3, 11, -15, -17, -9, -2], [-32, 14, -1, -8, 15, -4, -2, -14], [9, -11, -9, -9, 22, 16, -5, -7], [-1, -3, 13, 14, 16, 8, -8, 4], [-33, 3, 18, -15, -11, -3, -2, 1], [-7, 13, 15, 0, -2, 14, 9, -2], [-5, -18, -16, 4, 14, 16, -4, 7], [-17, 26, -1, 0, 20, 0, 3, -9], [22, 11, -1, 8, 10, 0, -6, 6], [-6, -12, -11, -5, -11, -3, -24, 3], [-13, 34, 7, 1, 0, 4, -16, -2], [-4, 19, 5, 1, 4, 13, 34, 7], [19, -31, -27, 5, -11, -3, -4, 0], [33, -9, 17, -6, -15, -10, 13, -9], [-30, -2, 17, 8, 0, -18, -14, 6], [-17, 6, 12, -19, -4, 3, 8, -2], [45, -8, -13, 6, 8, 11, -14, 26], [5, 3, 18, -3, -15, -7, 5, -10], [-4, -22, -12, 0, 8, -10, -5, -1], [30, -20, -15, 19, -6, 2, -9, -13], [-29, -11, 11, 8, -18, -17, -18, 2], [5, -1, 1, 20, -2, 3, -23, 19], [0, 0, -5, 5, 5, -9, 2, -13], [12, -3, 14, 1, -8, -9, 5, -17], [-2, -3, -6, 11, -1, -12, -26, -11], [-13, -11, -9, -1, -1, -31, -16, -3], [7, -12, 15, 14, -4, -24, -5, 9], [-9, -1, 7, -15, 1, -21, 8, -1], [-28, 18, -28, -9, 17, 19, -9, 17], [-24, 7, -6, -22, 4, -6, -7, -9], [-3, 25, -3, -3, -1, 2, -5, 6], [-35, -3, -5, -23, 11, -9, -5, 14], [-2, 16, -1, -2, 0, 22, 6, 15], [2, -7, 2, 5, 10, -18, -9, 3], [22, -21, 15, -1, -9, 1, 8, -17], [8, 10, -21, -21, 15, 13, -2, 18], [-3, -18, -7, 1, -4, -22, -16, -16], [-8, 25, -15, -3, 23, 3, -30, 23], [8, -5, -23, 26, 8, 21, -19, 7], [0, 15, -7, 16, 7, 12, 5, 2], [-13, 0, 21, 8, 4, -2, 11, -19], [-4, 9, -15, -11, 7, 3, -19, 15], [8, -7, -10, 8, -15, -6, -8, -21], [25, -2, -21, 8, 26, 26, -17, -11], [-24, -5, -5, -12, -2, 17, 28, 5], [10, 6, -16, 0, 11, -17, 5, -4], [-8, 18, 19, -3, -10, 9, 9, -4], [9, -22, 16, 5, -10, -34, 10, -11], [-9, 9, 12, 0, 3, -12, 7, -7], [23, 8, -13, 0, 1, 25, -5, -5], [5, -26, -23, 4, 1, -18, -24, -6], [-11, 4, -19, 14, 23, -11, -7, 2], [20, -24, -19, -2, -4, -5, -15, -20], [-44, -14, 20, -5, -4, -20, -4, 7], [19, 26, -10, 2, 0, -3, -15, 9], [23, 8, -10, 2, 7, 10, 11, 11], [10, 7, -2, -1, 27, 12, -17, 10], [-9, 25, -4, -3, -2, -4, -39, 0], [-31, -1, 7, -7, -17, -17, 19, -12], [3, -18, 0, -6, 1, 5, -8, 4], [15, 21, -4, -32, 5, 29, 18, -2], [8, -7, -19, 9, -12, -2, -25, 11], [26, -7, -19, -3, 12, 4, 12, 0], [-22, -7, 1, 5, -8, -19, -7, -13], [-3, 17, 12, -1, 4, -13, 7, -12], [1, -20, -12, 1, -21, -15, 0, -6], [12, -4, -28, 9, -13, 20, -15, 10], [12, -21, -12, 7, -1, 1, -16, -33], [2, 6, 3, 7, -6, 3, -28, 20], [3, 29, -11, -17, 13, 20, 16, -14], [3, 10, 10, 18, 1, -7, 7, -18], [11, -7, -11, -11, 4, 7, -3, 6], [2, 42, 0, -4, 3, 14, -9, 2], [10, 21, -10, -4, 18, 18, -3, 13], [47, 13, -23, 11, 16, 24, -8, 8], [6, 10, -9, -19, -6, -2, -21, 1], [-26, -1, 7, 8, 20, -8, 2, 12], [13, -9, 9, -2, -12, -8, -15, -28], [9, 6, -13, 21, -14, -4, -12, 4], [12, -7, 0, -4, 5, 0, -3, -22], [18, 12, -7, 0, 0, -2, 12, 3], [5, -29, -6, 8, 2, 13, -12, 14], [6, -9, 5, 0, -4, 3, 13, -24], [11, 29, 1, -8, 12, 15, 1, 8], [-27, 10, 28, -8, 0, -9, 22, -17], [-29, 3, -4, -14, 13, 5, 13, 24], [-13, -10, -19, -3, 3, -10, -12, 0], [-12, -11, 9, 35, -14, -20, -10, 1], [12, -1, 14, 25, 12, 0, -3, -10], [-35, 4, 9, 18, -3, -12, -16, 7], [-15, -8, -23, 14, -1, 30, -11, 9], [49, -33, -15, -4, 2, 4, -13, 12], [13, 12, 8, 2, 16, -8, -14, -2], [1, 9, 4, -5, 19, 1, 2, -10], [8, -15, -10, -7, 14, 6, 13, -4], [25, -17, -35, 4, -2, -1, -9, -10], [14, -36, 13, -5, -19, -19, 4, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6027_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6027_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_6027_2_a_bc(:prec:=8)
    chi := MakeCharacter_6027_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_6027_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_6027_2_a_bc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6027_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![113, 44, -148, -39, 70, 11, -14, -1, 1]>,<5,R![-967, 206, 1153, -589, -169, 141, -8, -7, 1]>,<13,R![49, 44, -10, -5, 1]>],Snew);
    return Vf;
end function;