
// Make newform 6027.2.a.bj in Magma, downloaded from the LMFDB on 19 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_bj();".
// 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_bj();".
// 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 := [16, -112, 44, 579, -384, -1007, 662, 716, -438, -237, 134, 36, -19, -2, 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, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-272668, 965410, 1001107, -3817394, -797891, 4814290, -530532, -2228088, 523923, 409674, -121798, -24595, 8468, -115], [145640, -575125, -89430, 1847591, -509978, -1814842, 636556, 741431, -270514, -131524, 47189, 8640, -2845, -66], [-25044, -234844, 678535, 323980, -1345959, 274678, 730612, -324642, -152093, 97154, 11712, -11191, -186, 433], [61772, 143024, -696899, -305616, 1345959, -274678, -730612, 324642, 152093, -97154, -11712, 11191, 186, -433], [-166252, 922034, 193241, -3322458, 481015, 3821874, -1196788, -1635072, 661925, 265034, -132302, -9569, 8616, -657], [-12872, 56360, -628747, -330636, 1748099, 148858, -1457088, 103978, 514897, -80530, -80292, 16835, 4502, -1101], [-9076, -620446, 1777283, 1321658, -4549267, 78546, 3272368, -633748, -1000485, 266526, 138006, -42039, -7032, 2301], [13459, -161239, 236449, 299871, -735873, 93287, 579567, -216860, -191269, 85888, 28201, -13286, -1518, 720], [57572, 66047, -821800, 343411, 2012748, -1192380, -1600106, 953861, 554328, -315048, -85969, 46216, 4825, -2470], [41296, -151836, 465907, -203184, -2298535, 878202, 2811688, -1042982, -1272953, 475158, 233648, -87939, -14586, 5521], [-261192, 577848, 2807311, -2959952, -7406035, 5292938, 6893844, -3990986, -2725189, 1391798, 464720, -219659, -27814, 12521]];
    Rf_basisdens := [1, 1, 1, 18364, 9182, 18364, 18364, 18364, 18364, 18364, 4591, 9182, 18364, 18364];
    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 := [[0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 1, 0, 1, 1, 1, 0, 0, -1, -1, 0], [2, -1, 1, -1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1], [2, -1, 0, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 1], [-1, 0, -1, 0, 0, 0, -2, 0, 0, -1, 0, 0, 1, -1], [-1, 1, 0, 1, 0, 0, -1, 0, 0, 0, 1, 0, 1, 0], [-1, 0, -2, 1, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0], [1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, -1, 0, -1], [0, 1, 0, 0, -1, -1, 1, 0, 0, 0, 1, -1, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -1, 0, 0, 1, 1, 0, 2, -1, 0, 0, -1, -1, 0], [2, 1, 0, 0, -1, 2, 0, 0, 0, -1, -1, 1, -1, 0], [-1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 1, 1, 0], [2, -2, 0, 0, -1, -1, 0, 1, -1, 1, 1, -1, 0, 0], [2, -3, 2, -1, 2, 0, 1, 2, -2, 0, 2, -1, 0, -1], [-1, -1, 1, 0, 1, -1, 1, -1, 2, 0, 0, -1, -2, 1], [-1, 1, -1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1], [2, -2, -3, 1, 0, -3, 0, 2, 0, 0, 2, -1, 1, -2], [2, 2, 1, -1, -2, 0, 2, -1, 2, 1, -3, 0, -1, 1], [1, 1, 1, 0, -1, 1, -1, -3, 2, -1, -2, 0, -1, 0], [3, -1, -2, 0, -1, 0, 0, 1, 0, -2, 0, -1, 1, 0], [1, 3, 0, 1, -1, 1, 0, -1, 0, -2, 0, 1, 1, 0], [4, 1, 0, 0, -2, 0, 0, -2, 2, 0, -2, 0, -1, 1], [1, -4, -1, 2, 2, -3, 1, 2, 0, 1, 0, -3, 1, -2], [-1, 3, 0, -1, 1, 3, -1, 2, -1, -1, -1, 2, 0, 0], [-2, 1, 1, 3, 1, -1, -1, -1, 0, -1, 1, 1, 2, 0], [1, 0, 2, 1, 3, 1, 0, -1, 1, 0, 0, 0, -1, 2], [0, 1, 0, 1, -2, 0, -3, -4, 0, -1, 0, 0, 4, -1], [0, -1, -2, 2, 0, -1, 1, 0, 1, 2, 0, -1, 2, 0], [0, -4, 3, -1, 1, 2, 3, 2, -3, 0, 3, 2, 2, 1], [0, -3, 4, -2, 1, 1, 3, -2, 1, 4, -1, -1, -2, 3], [0, -1, -2, 3, -1, -3, 3, 1, 1, 1, 1, -1, 0, -2], [-1, 2, 3, -1, -1, 4, 0, -1, -2, -1, 1, 4, -1, 1], [10, 3, 2, -4, 0, 5, 2, 0, 1, -1, -2, 3, -2, 3], [-1, -3, 0, -2, 1, -2, -1, 4, -3, -1, 3, -2, 1, -3], [1, 2, -2, 2, 2, 2, 1, 2, 0, -1, 1, -1, 1, -1], [9, -4, 2, 1, 2, 1, 3, 0, 1, 2, -2, -1, -3, -1], [5, 0, -2, 1, -1, 2, 3, 3, 2, -1, -3, 0, -2, 0], [3, 0, 4, 1, 1, 1, 0, -2, 1, 1, -1, 0, 0, -1], [-8, 2, -2, 0, 0, 0, -1, -2, 1, -4, -1, 1, 0, 2], [5, -4, 6, -2, 0, -2, 4, -2, 2, 5, 1, -2, -4, 1], [5, 0, 2, -2, -1, 2, 0, 0, -2, -1, 0, 2, 2, -1], [-1, -1, 1, -4, 1, 1, -2, 0, 0, 0, 1, -1, -2, 1], [-2, 4, -3, 0, -2, 2, -3, -2, -2, -1, -1, 1, 4, 2], [5, 2, 1, 0, 2, 4, 2, 2, -1, -1, -1, 1, 1, 2], [-2, 2, -3, 2, -2, 0, 0, -3, 2, 2, -6, -1, -1, 0], [1, 5, 0, 0, -3, 1, -2, -5, 0, 0, 2, 3, 3, 0], [-7, -2, -3, 2, 0, -1, -4, 1, -3, -1, 3, -1, 3, -3], [-6, -3, -1, 0, 0, 0, -3, -3, -1, 0, -3, -1, 2, -1], [3, -5, -2, 0, 3, -3, 1, 1, 0, 1, -1, -3, 0, -1], [-3, 2, -2, 1, -2, -1, -1, -4, 1, 1, -1, 1, 3, 3], [7, -2, 1, 2, 2, 1, -2, 0, 0, -4, 1, 1, 0, -3], [-2, -5, -3, -1, 3, -1, 1, 5, -2, 0, 3, -2, 2, -2], [-2, 3, -2, 0, -2, -1, -1, 1, -4, 1, 6, 0, 4, -2], [1, 4, -3, 4, -1, 0, -3, 0, -3, -3, 2, 2, 1, -2], [-1, 0, 0, -1, -3, 0, -3, -5, -1, 1, -2, 0, 3, 1], [4, -5, 0, -1, 4, 0, 5, 4, 0, 0, 2, 0, -1, 3], [-5, 8, -3, 2, -3, 1, -2, -5, -1, 0, 1, 4, 3, 2], [6, 7, -3, 3, 1, 2, -5, -2, -3, -3, -2, 2, 4, -2], [-3, 4, 2, 0, -3, 1, -1, -3, -1, -2, -1, 0, 3, -1], [-2, -4, 0, -2, 7, 0, 2, 6, -1, -3, 3, -1, -1, 3], [3, 1, -4, 0, -2, -3, -1, 0, 3, 0, 0, -1, 2, -1], [-6, 6, 3, 1, 2, 5, 0, -2, -1, 0, -1, 0, 1, 2], [-6, 5, 1, -3, -3, 3, 1, 0, -4, -1, 2, 4, 5, 0], [6, 5, 1, 1, -3, -2, -1, -2, 1, 1, 1, 1, -1, -3], [-2, -3, 0, -1, 2, 1, -1, -1, 0, -1, 1, -1, 2, -1], [-2, 4, -2, 4, -3, 2, -3, 1, -4, -1, 3, 3, 0, 0], [1, 3, -1, 3, -3, 0, -4, 0, 1, -1, -4, 1, 2, -5], [-9, 2, 5, -3, 1, 4, 1, 1, -4, 1, 2, -1, 2, -1], [3, 9, -6, 3, -1, -1, 1, -4, 0, 0, 1, 2, 4, 2], [6, 4, -5, 1, 2, 2, -3, 2, 3, -4, -4, 1, -4, 1], [0, -2, -5, 2, 1, -6, -2, 3, -1, 2, 3, -4, 0, -3], [8, -5, 0, 4, -1, -1, -2, 0, 3, 2, -1, -3, -2, -2], [2, 4, -2, -1, 0, 3, 0, 3, 2, -5, 0, 0, 0, -1], [7, 8, -3, 3, 2, 0, -2, -3, 1, -1, -2, 4, 0, 2], [1, -5, 4, -4, 1, 2, 4, 1, 2, 2, -6, -4, -7, 4], [2, -1, -3, 2, -1, -3, -3, -2, 2, 2, -2, -5, 0, -3], [7, -6, -3, 1, 2, -6, 0, 6, -1, 2, 3, -3, -3, -1], [2, -2, 1, 2, -2, -2, -3, -5, -1, -1, -1, 3, 2, -4], [-4, 2, -4, 4, -2, -5, -1, -2, 2, 3, -1, 0, -1, -1], [6, 4, -1, 1, 1, 2, 2, 5, 0, -3, 3, -2, 0, -2], [5, -5, 6, -2, 5, -3, 4, 2, 0, 2, 0, 1, -3, 1], [-7, 10, -1, 2, -5, 3, -2, -6, 1, 2, -5, 3, 1, -1], [2, 1, -3, 4, 2, 1, 0, 5, -1, -1, 8, 3, 4, 1], [8, -3, -3, 1, 1, -3, 3, 6, 4, 3, 4, -6, -3, -2], [5, 4, -4, 5, -3, -4, -5, -2, 2, 2, -3, 1, -2, -1], [-6, 3, -9, 0, -3, -3, -7, -4, 0, 1, -5, -1, 3, -2], [8, 1, 0, -2, -5, -1, 1, 2, -3, 0, 4, 2, 2, 0], [1, -1, -1, -2, -4, -1, 2, 1, 3, 1, 1, -3, 0, -2], [1, 1, -2, 1, -3, 3, 4, -5, 0, 4, -5, -1, -1, 4], [4, -1, -1, -3, -2, 3, 7, 1, 0, -1, -2, 3, -2, 5], [-4, 5, -6, 4, -3, -4, -3, -4, 0, 2, 0, 2, 2, -4], [-2, 8, -6, 6, 0, -3, -6, -5, -1, -5, -1, 5, 6, -2], [14, -5, 6, -6, 2, 6, 4, 9, -6, -1, 6, 4, 3, 3], [-2, 4, 2, 1, -3, 0, 1, -6, 0, -1, -1, 0, -1, 2], [9, 1, 4, -4, 4, 0, 6, 3, 0, 0, 5, 1, -2, 4], [9, -5, -1, -1, 4, 2, 0, 4, 3, 2, -2, -5, -6, 3], [10, -1, 5, -3, 0, 3, 4, 6, 1, 0, -1, -1, -6, -1], [2, 1, 3, -3, 0, 2, 2, 1, 4, 1, -6, -2, -4, -3], [-5, 7, 1, 0, -4, 4, -2, 0, -2, -1, -5, 0, 0, -3], [3, -1, 0, -1, 0, 2, -2, -2, 0, -4, 0, -2, 2, 2], [2, 1, 2, 0, 5, 1, -3, -2, -1, 0, 1, 2, 1, 4], [7, -4, 3, -1, 0, -2, 4, -4, 4, 2, 2, -4, -3, 3], [-2, 6, 3, -1, 2, -1, 1, 3, 3, -2, 1, -1, -5, 0], [0, -4, 4, -2, 5, 4, 3, 0, -1, -2, -1, 2, -1, 1], [-2, 3, 0, -1, 0, 1, -2, -1, 0, -1, 0, 2, 1, 2], [-2, 1, 3, -5, 1, 0, -1, -2, 0, -2, 0, 5, -1, 5], [7, -6, -1, -5, 0, -7, 1, 7, 3, 1, 2, -3, -5, 1], [13, 2, -4, 6, 5, 3, 0, -1, 7, -3, -6, 1, -4, 4], [-3, -5, -3, 3, 0, 1, 2, 2, 4, 1, 1, -4, 1, -2], [3, 0, 6, -2, 2, 5, 5, -4, 3, 2, -8, 0, -7, 7], [7, -2, 1, -4, -1, 3, 2, 0, 0, -3, -6, -2, 2, 2], [-6, -2, -2, -2, 0, -1, -4, 3, -5, -3, 7, 1, 2, 0], [4, -5, 8, -10, 1, 1, 9, 1, 1, 6, -2, -1, -2, 8], [-2, 5, 0, 0, 1, 4, -2, -1, -3, -2, 1, 4, 7, 3], [6, -4, -3, 5, -1, -6, 3, 9, -1, 1, 1, -7, 1, -6], [0, -8, 2, -1, -3, 2, 2, -1, -2, -1, -1, 2, 0, -5], [14, -1, -1, -1, 1, 1, -1, 6, -1, -2, 1, 0, -4, -1], [-9, -4, -4, 5, 2, -1, -6, 2, -4, -3, 5, -1, 5, -4], [-2, 8, -5, 0, -7, 5, -3, -2, 0, -1, -4, 1, 1, -1], [4, -6, 2, -6, 2, 0, 1, 11, -8, 0, 2, -4, -2, -2], [-1, 4, -1, 3, 5, -2, 2, 8, -3, -4, 12, -1, 3, -3], [9, -2, 8, -3, 7, 6, 10, 7, -2, 2, 3, 0, 0, 5], [-6, -6, 5, 0, -1, 3, -3, -1, -9, -1, 7, 5, 5, -2], [6, 6, 12, -7, -2, 5, 7, -7, 6, 7, -9, 0, -8, 7], [-5, -1, 3, -7, -1, -3, 8, 1, 1, 1, -1, -3, -5, 2], [2, 6, 0, 4, 1, 3, -5, 1, -6, 1, -3, 1, 1, -1], [12, -3, 2, -1, 3, 0, 1, 5, 1, 1, 5, -4, 1, 1], [-7, -2, 0, -1, -4, 5, -1, -7, -4, 3, -4, 1, 4, 5], [-15, 4, 1, 3, -2, 0, -5, -7, 4, -2, -2, 3, 1, 4], [-6, -2, -6, 9, 1, -7, -9, 0, -3, -2, 5, 0, 7, -12], [-18, 6, -4, -2, -1, -6, -8, 0, 1, 1, -1, -2, 2, -4], [10, -11, 2, -2, 3, -1, 4, 6, 7, 1, -3, -10, -10, -2], [1, -1, 0, -3, 3, -5, 1, 8, -6, 2, 10, -1, 8, -3], [4, -1, 1, 0, -2, -1, 4, 1, -7, 5, 0, -2, 6, 1], [16, 1, -2, -1, -5, 0, 4, -2, 3, 5, -4, -2, -1, -1], [-7, -5, -1, 2, 2, -10, -3, 4, -1, 4, 7, -5, 6, -6], [2, 2, -5, 1, 3, 4, -3, 1, -2, -8, 2, 7, 4, 4], [-5, 3, 5, -2, 0, 3, -2, -1, 0, 0, 0, 5, 2, -2], [6, 4, -9, 0, -4, 0, -3, -2, 1, -5, -3, -1, -5, -2], [-4, -10, -2, -9, -3, -9, 7, 10, 1, 3, 4, -5, -2, -3], [5, -9, 1, -3, 3, -2, 3, 4, 3, 0, -1, -6, -6, -5], [10, -6, 10, -11, 1, 2, 3, 5, -1, 0, 1, 2, -5, 5], [-1, 0, -1, 5, -4, -3, -8, -7, -3, -1, 3, 3, 8, -4], [0, 17, 1, 4, 0, 1, -6, -9, 5, -1, -4, 0, 1, -1], [-6, -11, 0, -7, -2, -9, 1, 3, -4, 5, 9, -5, 7, -1], [-18, 1, 2, -4, -5, -6, 1, -2, 1, 2, 3, 2, 0, 1], [2, 3, -4, 0, 0, -5, -3, -3, -2, -5, 0, 2, -2, -1], [-8, 1, 6, -4, 0, -1, 3, -6, 4, 5, -7, -5, -5, 4], [4, 4, -3, 3, -2, -4, 3, 1, 6, -1, 0, -2, -5, 0], [-2, -9, 4, -2, 0, -6, 4, 4, -3, 3, 3, -7, -4, -2], [14, 2, 2, -6, 4, 6, 2, 1, -2, -2, 4, 3, -3, 5], [-18, -1, -5, -2, 1, -4, -3, 4, 2, -3, -3, -2, -3, -5], [2, -2, 2, -2, 0, 4, 4, 8, 0, -3, 2, -2, -1, -9], [-15, 0, 3, -3, 4, -2, 6, -3, 2, 3, 3, -1, 2, 3], [-8, -8, 2, 3, 7, -10, 4, 0, 5, 8, 4, -7, -4, -1], [-1, 6, 2, -6, 2, 2, 0, -3, -2, 1, 7, 6, 3, 3], [-8, 8, 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5, 12, -4, 2, -2], [14, -4, 8, -6, -4, -3, 14, -12, 13, 6, -9, -3, -11, 8], [2, 19, -7, 8, -9, 6, -14, -6, 0, -4, -4, 13, 3, 3], [-11, 5, 2, -2, 3, -6, 4, -3, -1, 7, -1, 6, -4, 6], [-2, 10, 6, -7, 10, -1, -7, 0, -7, 1, 10, -1, 4, -2], [2, -5, 2, -15, 3, -14, 0, 2, 0, 3, 7, 0, -5, 2], [-17, 5, -15, 4, 0, 1, -5, 0, 4, 1, -9, -10, 3, 2], [-8, 5, -14, 5, -10, -9, -4, 7, -3, 1, 11, -5, 8, -10], [4, 2, 4, -7, -2, -5, 2, 6, -2, 5, 2, -8, -9, -5], [0, 4, -5, 9, 1, -3, -4, 2, 2, -2, 9, 3, 4, 1], [1, 3, -4, 3, -4, 3, -8, 0, 2, -4, -4, 6, 0, 2], [-2, 2, -4, 2, 3, -2, -3, 2, 9, -3, 2, -2, -8, -3], [11, 16, -10, -3, -13, 8, 1, -12, 2, 5, -17, 8, 0, 6], [18, -23, -2, -3, -4, 3, -1, 4, -3, 3, -3, -1, 7, 0], [26, 11, 6, 0, 1, -1, 5, -14, 9, 7, -2, 3, -4, 9], [1, -9, -11, 0, -2, -5, -14, 4, -2, -2, -5, -8, 6, -17], [-8, -18, 1, -5, 11, -17, 1, -4, 3, 2, 8, -6, 2, 6], [-12, 10, -5, 4, -10, 4, -8, 1, -4, -6, 7, 4, 13, -7], [8, -10, -1, 1, 11, 3, 10, 10, -1, 0, 1, -5, -8, -4], [-5, 3, 8, -8, -7, 5, -7, -3, -5, -4, -7, 10, 7, -5], [-4, 4, -1, -4, -6, 9, 8, 0, -2, -2, -12, -3, 1, 4], [10, 11, 1, -4, -13, -9, -5, -8, 4, -1, -3, 0, 2, -5], [0, 1, -9, 8, -13, -9, -4, -2, 6, 1, -3, 0, 4, -8], [-18, -2, -7, -7, -4, -15, -2, -3, 9, 6, -2, -6, -2, -1], [5, -11, 3, -4, 1, -2, 10, 0, 6, 11, -7, 0, -16, 11], [14, -27, 8, -6, 14, -7, 19, 15, 1, 6, 4, -12, -11, 5], [4, 0, 3, -1, -3, -7, 5, 5, -6, 8, 8, -5, 5, -1], [0, -1, 8, -2, -4, 8, 6, -9, 8, -6, -10, 0, -11, 8], [18, 3, 22, -12, -2, 3, 11, -10, 3, 15, -5, -3, -10, 6], [-13, -18, -10, 0, 2, -13, 0, 16, 2, 1, 11, -10, 5, -15], [16, 9, -11, 2, 2, -5, -13, -3, 4, -8, -7, 2, -2, -4], [4, -1, -3, 1, 7, -6, -4, 9, -7, -1, 6, -2, -1, -11], [10, 0, 3, -2, 1, 17, 0, 4, -4, -8, -4, 15, -1, 2], [-11, -2, 1, -9, -3, -1, 0, -10, -5, 6, -7, -4, 7, 0], [-3, 11, -7, -3, -6, -5, -10, -15, 2, -8, -9, 2, 2, 0], [14, 22, -4, 6, 2, 6, -5, -8, 11, -5, -6, 5, 1, 4], [3, 28, -5, -3, -8, 16, -4, -17, 2, -8, -6, 22, 6, 8], [14, -15, -7, 3, -1, 1, -1, 8, 2, 0, -2, -2, 0, 3], [-25, 19, 3, 6, -9, -2, -8, -21, 1, 0, -5, 8, 11, 1], [20, -8, 9, -8, 5, 14, 8, 0, 9, 1, -16, -4, -13, 11], [-15, -18, 4, 0, 5, 1, 2, 10, -1, -4, 10, 7, 4, -2], [-5, -3, -6, -4, 6, -4, -5, 4, -5, 6, 3, -1, 8, -10], [-3, 12, -5, 4, -8, 5, -8, -11, 2, -6, -2, 11, 2, -8], [-4, 6, 5, 2, 0, 1, 9, -8, 5, 8, -5, -6, 2, 3], [-1, -3, -21, 9, -1, -11, -3, 1, 9, 4, 2, -6, 8, -6], [17, 5, -4, -2, -5, 0, -1, -2, -5, 0, 12, 12, 9, 2], [1, -11, 4, -10, 5, 5, 2, 9, -9, -4, 11, 5, 8, 10], [16, 0, 22, -17, -1, 19, 6, 5, -4, -7, -4, 12, 0, 7], [-9, 18, 4, -2, 1, 10, -8, -4, -3, -11, 10, 20, 8, 6], [5, -20, -12, 10, 2, -5, 1, 9, 1, -5, -3, -3, -3, -4], [-11, -12, 4, 1, -1, 7, 0, -5, -14, -2, 2, 6, 13, 9], [-19, 1, -8, -3, 5, -10, -13, 2, 0, -2, 10, 3, 9, -8], [0, -23, 2, 6, 3, 0, 3, 6, 4, 2, -3, -3, -9, 3], [1, 11, 10, -8, -3, 6, 7, -1, 4, -5, -2, 2, 1, 2], [8, 2, 4, -9, 6, 4, 7, -7, 14, 10, -14, -6, -14, 9], [-3, 17, -5, 8, -9, -3, -4, 0, 7, 2, 5, -1, -1, -7], [-9, 18, 10, -1, 1, 10, -4, -9, -2, 0, -15, 4, -4, 6], [15, 6, -5, -1, -7, -2, -5, -3, 2, -4, -10, -6, -8, -9], [-15, 12, -13, 5, -13, 3, -15, -2, -2, -9, -6, 0, 9, -6], [3, -9, 8, -7, 3, -5, 12, 21, 2, 4, 6, -10, -10, -11], [-8, -3, -3, -8, -1, 2, -7, 10, -10, -10, 3, -1, 3, -3], [-1, -9, 3, -12, 12, -1, 9, 7, 3, 1, -1, -9, -5, 7], [-7, 5, 2, 6, 13, 7, -8, 12, -13, -11, 9, 5, 13, -3], [-18, -6, 0, 5, -7, -5, 0, -19, -6, 9, 0, -2, 10, 0], [11, -14, -11, 9, 1, -6, -5, 7, -12, 0, 5, -5, 13, -7], [-8, -4, -6, -6, 3, 0, 4, 11, 1, -5, 0, -3, 4, -3], [10, 8, 0, 6, -8, 2, -2, -4, -2, 4, 1, 3, 4, 3], [-8, -10, -5, -4, 13, -9, -15, 9, -1, -3, 5, -4, -4, -7], [-17, 0, 8, -5, 1, -7, -8, -4, -3, -6, -5, -1, 1, -3], [20, -13, 6, 6, -2, -3, 6, 3, -2, 6, 0, -3, 5, 0], [-4, -5, 4, -10, 9, 5, 0, 0, 8, -10, -5, -4, -5, 9], [-26, 2, 6, -8, 5, 5, -6, 2, 7, -6, -7, 1, -2, 1], [-2, -19, -6, 0, -1, -17, -3, 6, -4, -3, 15, -3, 7, -9], [27, 11, 0, -1, -15, -1, -12, -7, -6, -1, 0, 9, 7, -7], [4, 11, 0, 9, -7, 1, -2, -16, 0, -4, -4, 2, 12, 8], [-15, 0, 8, -18, -3, 11, 5, 1, 1, 5, -2, 3, 1, 9], [-5, -3, 9, -6, -3, -6, 3, 0, -3, 7, 2, -7, -2, -7], [17, 6, 4, -4, -5, 1, 10, 4, 7, 0, -3, -4, -10, 3], [12, 21, -4, -5, -4, 7, -11, -6, -7, -4, -3, 14, 10, 4], [-2, -17, -15, 7, 4, -17, 6, 13, 1, 2, 0, -15, -3, -11], [16, -6, 0, -5, 6, 5, 12, 12, -4, 3, 8, -8, -4, 1], [-27, 11, -8, 0, 0, 5, -4, 1, 5, -6, -7, 2, -7, -3], [-33, 3, -9, 1, 5, -9, -8, -4, 3, -3, 5, 1, 2, 4], [2, -20, 1, -2, 4, 2, -3, 4, -2, -4, -8, -14, -7, -10], [-21, 13, -11, 2, -5, -9, -12, 5, -7, -3, 5, -3, 14, -20], [-12, -6, -4, -7, -7, -13, 6, 5, 3, 12, 10, -7, -4, 1], [-4, -14, -2, -10, 14, -2, 4, 22, -7, -6, 8, -12, -7, -2], [30, -7, -1, 8, 7, -13, 16, 5, 13, 4, 9, -10, -8, 0], [15, 3, -12, 7, 2, -5, -1, -3, 2, -3, -2, -4, -3, 0], [-3, 5, 7, 8, -4, -7, -1, -5, 5, 6, -9, 3, -6, 1], [5, 2, -18, 2, 3, -10, -7, 7, -2, -6, 7, -6, 5, -9], [26, 12, 0, -1, -2, 10, -7, -2, 7, -6, -6, 7, -5, 2], [-18, 14, -16, 11, 3, -4, -5, -6, 0, 4, -1, -1, 3, 4], [11, -2, 7, 10, 3, 1, 8, 10, -1, 3, 4, -4, -13, 1], [32, 2, 1, 0, 0, -2, 10, -4, 15, 2, -17, -3, -9, 4], [-24, 10, -2, -4, -6, -7, -11, 2, -10, 4, 1, 5, 7, 1], [-17, 9, -1, 8, 0, 7, -10, -2, -7, -2, -5, 7, 9, -6], [23, -4, 16, -7, 8, 5, 14, 13, -5, 4, 6, -2, -6, 7], [27, 7, 1, 2, -11, 11, 6, -12, -1, 6, -7, 8, -6, 7], [2, -4, 3, -2, 4, -8, -3, 8, 0, 5, 13, -2, 9, 3], [-27, 12, 3, 11, -3, 6, -17, -27, 3, -10, -7, 5, 6, 0], [2, -10, 11, -10, 6, 5, 6, 1, -3, 1, -5, -5, -9, 8], [15, -6, 1, -6, 0, -10, 9, 9, 8, 6, -2, -11, -19, -1], [-36, 23, -8, 3, -10, -2, -15, -19, -6, -7, 0, 17, 15, -3], [-5, 11, 2, 19, 4, 15, -10, -11, -1, -9, -8, 12, 7, 3], [20, 5, 5, -4, 6, 16, 4, 7, 4, -3, -7, -1, -2, -2], [10, -18, -3, -6, -4, -5, -9, -6, 9, -1, -4, -5, 1, -1], [-21, 1, -3, -2, 2, 0, -8, -3, 3, 0, -3, 0, -4, 9], [15, -1, 9, 5, 2, 2, 13, 6, -1, -4, 9, 4, -2, -5], [14, -10, 7, -10, -1, 7, 5, -4, 5, 2, -13, -7, -14, 3], [3, 23, 9, -3, 0, 6, 0, -3, 8, -3, 1, 9, -11, 4], [-7, 5, -9, 10, 1, 0, -1, 4, -6, -3, -4, 0, 7, 0], [12, -21, 9, -25, 5, -2, 16, 13, 8, 2, -1, -7, -9, 7], [5, 0, 9, 2, 0, 10, 6, 11, -5, -1, 2, -2, 1, -8], [1, -12, 7, -8, 4, 4, -3, -1, 7, -6, -8, -3, -3, 0], [3, -6, 13, -2, 13, -8, 16, 22, 7, 4, 16, -11, -5, -4], [-3, -2, 12, -8, 5, -8, 8, -3, -4, 2, 10, 3, 5, -4]];
    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_bj();".
// 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_bj(:prec:=14)
    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_bj();".
// 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_bj( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6027_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![16, 112, 44, -579, -384, 1007, 662, -716, -438, 237, 134, -36, -19, 2, 1]>,<5,R![-166, -1288, 8528, -8523, -18545, 41760, -23539, -4389, 8102, -1475, -705, 253, 4, -10, 1]>,<13,R![50897351, -88891192, 25101906, 45196705, -41479876, 11188967, 1531858, -1540976, 272365, 19825, -12416, 1136, 89, -21, 1]>],Snew);
    return Vf;
end function;