
// Make newform 6027.2.a.bi in Magma, downloaded from the LMFDB on 10 May 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_bi();".
// 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_bi();".
// 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 := [-4, -12, 69, 84, -311, -174, 478, 50, -263, 26, 56, -11, -4, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4612, -1463, 35530, 56201, -35248, -71846, 19248, 32643, -7424, -6122, 1471, 402, -105], [11577, 44430, -91598, -298916, 119516, 428762, -110536, -219999, 58087, 44868, -12772, -3120, 944], [44488, 182903, -321158, -1003841, 499382, 1279188, -418518, -614931, 187114, 120936, -37363, -8228, 2617], [9559, 80294, -63961, -495112, 43093, 723044, -93135, -378472, 75852, 78197, -19453, -5483, 1537], [7851, 50433, 5531, -398713, -146249, 714974, 21737, -412026, 55005, 89304, -19212, -6432, 1680], [-31592, -171323, 153280, 1126967, 9716, -1793936, 151906, 981221, -177062, -207380, 49447, 14730, -4041], [31592, 160513, -153280, -1124805, -9716, 1793936, -151906, -981221, 177062, 207380, -49447, -14730, 4041], [-23055, -113389, 104974, 751304, 23336, -1209623, 90681, 666901, -117946, -141614, 33585, 10088, -2764], [-47218, -275121, 294962, 1799105, -204722, -2709136, 375044, 1438983, -297124, -299544, 75787, 21092, -5977], [78110, 347655, -502024, -2142659, 548460, 3091452, -611328, -1604297, 376628, 329744, -88523, -23044, 6745]];
    Rf_basisdens := [1, 1, 1, 2162, 1081, 2162, 1081, 1081, 2162, 2162, 1081, 2162, 2162];
    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], [1, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, -1, 0, 0, 0, 1, 0, 0, 0, -1, 0, -1], [-1, 0, 0, 0, 0, 0, -1, -1, -1, 0, 1, 0, 0], [-2, 0, 0, -1, 0, 0, 1, 0, 1, -1, 0, 1, 2], [1, 0, 0, -1, 2, 0, 0, 1, 0, -1, -1, 0, -1], [-2, 2, -1, 1, -1, 0, 0, 1, 0, 1, 1, 0, -1], [1, -1, 2, 0, -1, 1, -1, 2, -1, 1, 0, 0, -1], [0, 0, -2, 0, 2, -1, 0, 0, 0, 0, 0, -1, 0], [2, -1, 0, -1, 0, 0, -1, 0, -2, 0, -1, -1, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, -1, -1, 0, -1, -1, 0, 0, -1, 1, 2], [-2, -1, 1, 1, -1, 0, -2, 0, 0, 1, 2, -2, 1], [-2, 3, -1, 2, -1, -2, 1, -1, -1, 1, 0, 1, 1], [0, -2, 0, 0, 0, 1, 0, -1, 0, 1, -1, 0, 0], [-5, 3, 1, 0, -1, 0, 0, 0, 1, 2, 2, 2, 1], [-2, 2, 1, 1, -1, 1, 0, -1, 0, 0, 2, 0, 1], [-2, -1, 1, 0, 0, -2, 1, 0, -1, -3, 0, 0, 2], [-1, 1, -2, 3, 1, 0, 1, 1, 1, 0, -1, -1, -1], [0, -1, -1, 2, 0, -1, 0, -1, 0, 2, 0, 0, 0], [-2, 2, 1, -2, -2, 2, 1, 1, 0, -1, 0, 2, 0], [-6, 1, 2, 0, -3, 3, 1, 0, 2, 0, 1, -1, 1], [-3, 1, 0, -1, 2, 0, -1, 0, -1, -2, -2, 0, 0], [-5, 0, -1, 3, -2, -2, -1, -3, 1, 3, 1, 1, 0], [-3, 0, 1, 0, 0, 1, 0, -1, -2, -1, 0, 1, 1], [-2, 0, -1, 2, -1, -1, 1, 1, 1, 1, -1, 0, -2], [4, -1, 0, -2, 1, 0, -2, -1, -1, -1, -1, -1, -2], [1, -4, -1, 0, 1, -5, 0, -3, -4, -1, 0, 0, 1], [0, 0, 2, 0, -2, 1, 2, 1, 0, -2, 1, -1, 2], [3, -5, 2, -1, 1, -1, -3, -1, -2, 0, 0, 0, 1], [0, 0, -1, -2, 1, 0, -1, 0, -1, 0, 0, 1, 2], [-1, 1, 3, -1, -1, 2, 2, 1, 2, 0, 0, 0, 1], [-4, 1, -5, 3, 1, -4, 2, -3, 2, 2, 2, -1, 1], [3, 1, -1, -1, 0, -5, 0, -1, -3, 1, 1, 0, 0], [-3, -2, -2, 1, 1, 0, -1, -2, 1, 2, -1, 1, -1], [2, 1, 0, -2, 0, 2, 2, 2, 1, 0, 1, 2, -3], [-4, 4, -3, 2, -1, -1, 3, -1, -1, 2, 2, -2, 2], [-3, -3, -1, 2, 1, -1, 0, 0, -2, -1, -1, -3, 1], [-6, -2, 0, -1, -3, 0, -1, -4, -1, 3, 3, 4, 3], [-1, -3, 3, -2, 1, 0, 2, 2, 1, -3, 0, 0, -1], [-5, -6, -2, 0, -1, -1, -3, -6, 0, 2, 2, 2, 4], [6, -1, -1, -1, 2, -1, 0, 2, -2, -5, -5, -1, -1], [-3, 3, -1, 2, 1, 2, 2, 3, 3, 2, -1, 0, -5], [0, 1, 5, 1, -4, 3, 0, 5, 0, -2, -2, 1, -1], [10, -5, 1, -5, 5, 1, -2, 0, -4, -5, -3, -3, -2], [1, 2, 2, 5, 1, -2, 0, 6, 0, -3, -2, -2, -2], [1, -2, 1, 1, 3, 1, -1, 4, 3, -1, -4, -3, -5], [-3, 3, -2, -1, 0, 2, 5, 2, 0, 0, -2, -2, -4], [2, 1, -2, 6, 2, 1, -3, 0, 0, 5, 1, -7, -6], [-11, -4, -3, 0, 1, 1, 0, -2, 0, 2, -2, -2, -1], [2, -5, 0, -5, 2, 2, -1, -4, -1, -4, 1, -2, 2], [-7, 2, -3, 4, 2, -2, 1, 1, 1, 1, 0, -3, 0], [-8, 2, -1, -2, -2, 1, 0, -1, -1, 5, 3, 3, -3], [-1, -4, 5, -2, -4, 1, -4, 0, -2, 1, 6, 4, 3], [-7, 3, 4, 3, -3, 6, -5, 1, 3, 6, 4, 2, -4], [0, -4, 6, -3, 2, 3, -3, 0, 0, -3, 2, -2, 1], [-4, 5, -1, 3, -2, -1, 4, -2, 1, -1, -4, 2, 0], [3, 4, 2, -2, 1, 3, 1, 7, 3, -2, -2, -2, -3], [-2, -4, 4, 0, 0, 0, -2, -2, -1, 1, 5, 3, 5], [-13, 4, 0, -1, -3, 1, -1, -2, -1, 3, 3, 6, 1], [-7, -4, 3, 2, -4, 1, -3, -5, 3, 3, 4, -3, 2], [-11, 2, 0, 1, -4, -4, 5, -5, 2, 3, 3, 1, 8], [-1, -1, -1, 2, 4, 5, 4, 4, 5, -1, -6, -5, -5], [6, -5, 1, -6, 3, 0, 2, 1, 0, -5, 0, 4, 0], [4, 1, -2, 3, -3, -4, 1, -4, -2, 0, 2, -1, -1], [-1, 3, -4, 2, -3, -1, -3, -5, -2, 6, 4, 2, -1], [-3, -1, 2, 0, 4, 2, -1, 4, 7, -3, 0, -2, 0], [1, 0, 3, -7, 4, 3, 2, 7, 0, -3, -4, 2, -1], [-7, 2, 1, 1, 0, 1, 1, 2, 1, -3, 1, -2, -4], [4, -10, 7, -2, 2, 4, -9, 1, -2, 0, -1, -4, -3], [3, -5, -3, -1, 1, -1, -2, -1, 1, -2, -2, -5, 2], [-7, 8, -4, -1, -1, -2, 4, -4, 0, -1, 3, 2, 6], [-3, 1, -3, 0, 1, 1, 1, -3, 2, 6, 1, 3, -4], [0, -7, 3, -7, 3, 3, -3, 2, 0, -2, 3, 2, 0], [4, -4, 0, 1, 1, 1, -6, -2, -2, 2, 1, -4, -1], [-1, 0, 0, -2, 1, 0, 1, 3, 0, 0, -1, -4, -3], [6, 5, 1, -2, -2, -2, 2, 3, -6, -2, 0, 3, -1], [0, -1, 0, 8, -2, 5, -2, 2, 3, 3, 0, -5, -8], [5, -5, 3, -8, 2, 0, -4, -3, -1, -6, -2, 0, 3], [15, -4, 8, -1, 0, 3, -4, 3, -2, -3, 2, 0, 0], [0, 2, 0, -1, 1, 1, -4, 4, -1, 4, -4, -3, -5], [-13, -2, -2, 0, 0, -3, 3, -2, 0, 0, -4, 1, -2], [3, 0, -4, -1, 6, -2, 2, 2, -2, -2, 0, 2, -5], [2, 1, 3, 0, -3, -1, 4, 1, -1, -5, 2, 5, 4], [-2, -5, 2, -2, 0, 4, 1, -3, 2, -1, -3, 0, -2], [6, -9, 2, -5, 5, -4, 5, -4, 0, -6, -2, 5, 9], [-3, 2, -3, 3, -2, -3, 2, -5, -6, 1, -2, 1, 1], [5, 9, 1, 3, 1, 4, -4, 4, 1, 0, 2, -2, -3], [-7, -5, -3, 0, -2, -4, -2, -4, 3, 1, 2, 6, 5], [9, -1, 5, -9, 0, 3, 0, 1, -3, -2, -2, 4, 0], [-12, 11, -6, 9, -3, -3, 1, -7, 3, 8, 1, -3, 0], [-6, 3, 1, 8, -1, -1, -1, -3, 3, 1, 1, 1, -2], [8, 11, 1, -4, 1, 5, 2, 10, 2, 0, -6, 1, -5], [-8, 2, 0, 7, -4, -1, 2, 1, 7, 6, 4, 2, 2], [-9, -2, -2, 4, -1, -2, 4, 2, 5, 2, -5, -1, 0], [-2, -4, 1, 2, 1, 2, 1, -3, 0, 2, -4, -3, 0], [-3, 8, 4, -1, -7, 4, 5, -1, 1, 1, 0, 4, 3], [2, 1, -3, 0, 4, -1, -2, 4, -1, -2, 2, -5, -3], [-3, 1, -1, 3, -2, 1, -6, -1, -3, 2, 2, 3, -2], [3, -1, -1, 0, 0, 0, 8, -2, -2, -4, -4, -7, 0], [-8, 10, -6, -3, 6, -1, 10, 0, 2, -5, -3, -1, 5], [-1, 7, -4, 4, 4, -2, 4, 2, 1, -2, -8, -4, -2], [-10, -2, 1, 0, -6, 5, -2, -1, 3, 5, 4, 3, 1], [-4, 5, -6, 3, -2, -4, -1, -1, -4, 5, 5, 3, 0], [1, 2, -4, 0, 4, -6, -1, -3, 0, 1, -2, -2, 2], [-2, -7, 4, 0, 4, -4, 3, -2, 0, -6, 5, 3, 8], [1, -2, -3, -8, 3, -3, 3, -1, -1, -3, -2, -1, 1], [-1, 2, -2, 5, 0, -4, -2, 4, -2, 1, 0, -4, -1], [-3, -1, -1, 9, -4, 1, -3, -3, 0, 12, 5, -3, -5], [-4, 4, 2, 0, 1, 2, -4, 2, 2, -1, -4, 1, -2], [-6, 7, -5, 1, -1, -2, -3, -5, 2, 0, 3, 5, 4], [9, -7, -4, -4, 0, -2, -1, -9, -9, 3, 2, 3, 1], [1, 5, -7, 2, 1, -1, 7, 4, -2, -3, -4, -4, -4], [-10, 2, -4, -7, -6, -1, 2, -5, -3, 2, 0, 6, 4], [0, -12, -1, -1, 2, 1, -3, 0, 6, 0, -1, -3, -2], [7, -7, 4, -2, 0, 3, 1, 5, 2, -4, 2, 3, 0], [9, 3, -4, -2, 3, -9, 4, 1, 0, -6, -2, 4, 6], [10, 2, 2, -7, 7, 5, 8, 13, 3, -11, -10, -6, -8], [3, -1, 3, -7, 2, 1, -1, 6, 5, -6, -5, 4, 1], [20, -2, -5, -3, 4, 1, 0, 4, 4, -4, -4, -5, 0], [-9, -6, -4, 2, 1, 1, -5, -4, -6, 3, -1, 2, -1], [3, -5, 6, 4, 0, -3, 0, -3, 2, 4, 5, -4, 2], [6, -3, 7, 1, -5, 1, 2, -2, -2, -3, 0, -1, 1], [-4, 5, 0, 5, 3, -3, 1, -1, 0, -5, -6, -3, 5], [-8, 5, -8, 7, -6, 0, -4, -3, -1, 10, 5, -3, -1], [3, 0, 5, -4, -1, 9, 0, 9, 7, -3, -3, 2, -2], [-11, -2, 1, -4, -5, 1, 4, 1, 3, -3, 2, 8, 10], [-1, -7, 4, -3, 1, 0, -1, 0, -1, -2, 6, -4, 2], [0, -4, -9, 3, 5, 0, -4, -3, 5, 7, -3, -3, -2], [-9, 4, 4, 0, -1, 3, 5, 3, 5, -3, 3, -1, 0], [10, 4, 2, 6, 0, 0, -1, 8, 5, 0, 3, 2, 2], [16, -5, 3, -8, 4, -3, 2, -1, 0, -11, -6, 5, 5], [-2, -3, 2, -2, -3, 3, -3, -3, -1, 2, 3, -2, 0], [-7, -6, 4, 1, 0, 6, -1, 1, 1, 2, 6, -1, -1], [-15, 1, -3, -4, 4, 1, -2, 2, 2, -1, -4, 5, 0], [-14, 15, -7, 4, -3, 3, 8, 2, 6, 1, 2, -2, 0], [2, -1, -5, -1, -2, -2, -2, -5, -7, 3, 3, 2, 0], [-2, -1, -1, 2, 1, -3, 7, 2, 4, 4, -2, -2, -4], [-15, -3, -4, -3, -3, 0, -1, -10, 6, 9, 3, 4, 8], [1, 6, -3, 0, -3, 6, -3, 2, -1, 9, 0, -2, -12], [2, -2, 7, -2, -2, -1, 0, 6, -1, -8, -1, 7, 5], [7, -3, 9, 5, -4, 1, -7, 6, -6, 0, 1, -6, 0], [9, -13, 7, -6, 3, 4, 1, 4, 7, -4, -2, 3, 8], [-5, 2, -5, -5, -1, 2, 5, 1, 0, 5, 3, -2, -5], [-3, -10, 5, -1, 0, -1, 0, 1, -6, -6, 4, 2, 10], [-2, -4, 9, 0, -2, -3, -10, -2, 2, 3, 5, 8, 10], [21, -6, 4, -6, 7, 2, -3, 8, -7, -9, -3, -6, 0], [6, 1, -2, -2, 3, -8, -3, -6, -4, -2, 0, -5, 0], [-20, -1, -5, 6, -2, 2, -4, -10, 6, 7, 4, 1, 4], [-5, 11, -2, 6, -3, 6, 0, 2, 2, 7, 5, -4, -11], [-2, -1, -5, -2, -1, 1, -5, -7, 4, 2, 2, 3, 3], [-8, 4, -5, 1, -4, -6, 4, 0, 4, 1, -1, 6, 6], [10, -8, 3, -6, 6, 1, -2, 3, -8, -8, -6, -2, -11], [-5, -5, 1, 1, 2, 8, 0, 6, 6, 9, 0, 0, -8], [6, 0, -1, 3, 2, -4, -2, -3, 0, -5, -5, 2, 7], [5, -6, -2, -2, 6, 6, -5, 2, 2, 0, 0, -4, -8], [-13, 2, -6, -1, -4, 1, 0, -2, -2, 9, 1, -1, -1], [-9, 5, -2, 0, -10, 1, 6, -1, 1, 2, 4, 2, 4], [-15, 2, -5, 5, 1, -4, -2, -8, -2, 5, -1, 4, -1], [0, 3, 11, -1, -4, 0, 0, 0, 2, -5, 7, 2, 6], [-2, 6, -10, 6, -7, 2, -6, -6, -4, 14, 0, -3, -8], [3, 7, -4, 12, -1, -8, 3, -5, 4, 1, -1, -4, -3], [0, 7, 10, -1, -2, -4, 2, 3, -4, -4, 0, 3, 4], [4, -8, 3, 0, 8, -2, -6, -3, 2, -3, -3, -1, 7], [-15, 5, -2, -1, -5, 7, 9, -4, 2, 7, 5, 1, -1], [-11, 7, -4, 10, -8, 1, 7, 3, 13, 13, 5, 1, -3], [14, -7, 0, -5, 9, -3, 1, -5, 1, -5, 4, 8, 8], [-1, 4, -2, -2, 3, -2, 3, 4, -8, -3, 2, -4, 2], [-9, 5, 8, 0, -5, -3, -8, -1, 3, 3, 10, 11, 12], [-26, 4, -6, -2, -1, 1, 7, 4, 5, 3, -1, -2, -3], [-8, 5, 3, 7, -6, 6, 2, 2, -1, -2, -3, -3, -4], [8, -15, 4, 0, 3, -4, -8, -4, 2, 2, 8, -5, 7], [1, 4, -3, 14, -5, -1, -1, -3, 7, 10, 2, -6, -1], [5, -8, 7, 1, -4, 3, -9, -8, -3, 4, 8, 10, 9], [17, -6, -3, -2, 5, -2, 1, -2, 1, -8, -2, -8, 0], [1, 7, 9, -3, -1, 2, 1, 5, -8, -9, -3, -7, -3], [3, 1, 2, -7, 1, 4, 0, 7, 0, -6, 0, -3, -6], [3, 3, -7, 5, -5, 3, 3, 3, 1, 1, -4, -9, -6], [12, -7, 0, 3, 2, 2, 0, 1, 10, 3, -2, -8, -6], [-7, 5, -4, 6, -6, -10, -3, -10, -2, 1, 3, -2, 0], [12, -3, -4, 6, 1, -4, -8, -3, 1, 7, -1, -9, -2], [19, 2, 0, -3, 3, -1, 2, 6, 1, -7, -12, -1, -10], [-34, 3, -1, 3, -9, -1, -3, -4, 3, 3, 3, 8, 2], [-1, 3, -3, -2, 4, -2, 3, 2, -7, -11, -1, -3, -2], [-4, 0, 4, 9, -2, -3, -4, 2, -1, 7, 4, -10, -2], [7, 6, -1, -4, 5, 3, -2, 3, 0, -4, -4, 2, -5], [-7, 2, 5, -4, 1, 7, 3, 13, 2, -6, -5, 1, -11], [17, -2, 1, 3, 0, 0, -5, -3, 0, 3, 4, 1, -6], [2, -9, 3, -1, 2, 0, 1, -9, 3, 4, 4, 4, 4], [6, -9, 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-7, 9, 6, 4, -2, -1, -2], [2, 9, 0, 0, 7, 1, 9, -2, -4, -16, -8, -2, -4], [-11, 3, 1, -16, 17, -10, 3, 2, -5, -11, -4, 12, 6], [-13, 2, -3, 12, -6, -9, -7, -6, -3, 13, 6, -10, -3], [19, 2, -5, -5, -1, 4, 0, 1, 1, -4, 0, -9, -2], [23, -28, 11, -11, 4, 11, -16, 2, 7, -3, 4, -1, -6], [-24, 28, 6, -6, -8, 1, 7, -1, 3, -1, 1, 19, 1], [-20, 6, -22, 2, 1, -7, 8, -8, -1, 7, 4, -4, -7], [-20, 9, -4, 14, -10, 0, 3, 0, 1, 14, 10, -1, -8], [-15, 21, 1, 10, -4, 0, -2, -1, 0, 9, 6, -1, -3], [10, 0, -6, -4, -4, 9, -9, 2, -4, 5, -7, -5, -18], [-35, -4, -16, 14, 1, -8, -1, -21, -1, 20, 12, -7, 5], [-8, -12, 9, -9, -5, 23, -8, 3, 7, -1, 3, 5, 3], [-5, -16, -8, 13, 5, -11, -3, -7, 2, 13, 5, -1, -3], [6, 15, -19, 5, 5, -9, 19, 4, -2, -7, -17, 5, -10], [10, -4, 9, 8, -3, -1, -2, 5, -1, 3, -2, -11, -4], [7, 5, -3, 6, -6, 5, 12, 0, 9, 2, -6, 4, -5], [21, -7, 2, -14, 15, -1, 5, 8, 1, -20, 0, -7, 2], [21, -10, 2, -7, 5, 12, -5, 12, 10, 4, -6, 7, -7], [8, -16, 8, -10, 3, 1, 3, -6, -4, -11, -4, -5, -2], [12, 2, -6, 9, 15, -17, 4, 7, 7, -5, -3, -9, -5], [-24, -10, 11, 3, 7, 0, 3, 10, -3, -10, -4, 6, 5], [3, 5, -2, 12, -11, 3, -9, -7, -6, 15, 4, -6, -13], [-13, -2, 1, -1, 6, 9, -5, -1, 4, -8, -6, 7, -1], [-24, 4, -3, 2, -14, 9, 4, 4, 7, 18, 5, 4, -4], [43, -4, 3, -2, 11, 1, -5, 11, -3, -2, -7, 5, -11], [2, -4, 16, -19, -1, 6, -2, 11, -9, -19, -12, -4, 6], [1, -28, 18, 2, -1, 7, -2, -1, 10, -1, 5, -2, 11], [-2, -7, -8, -2, -7, -11, 0, 3, -6, 1, 0, 9, 4], [-2, 5, 12, -8, 0, 8, 1, 14, -1, -19, -5, -4, -10], [7, 2, -11, -15, 6, 6, 3, 10, 6, -8, -2, -3, -3], [11, 6, -12, 0, 6, 1, 4, -3, 3, -3, -3, -1, -4], [4, -3, -9, -12, 12, -14, 5, -7, 8, -3, -2, 9, 7], [-11, -26, 15, -8, 13, 8, -11, 5, 3, -7, -8, 4, -5], [-7, -7, -6, 12, 7, -3, -6, -4, -8, 11, 2, -11, -15], [25, 11, -6, -10, -2, -9, 7, 5, -4, 1, -4, 11, 9], [5, -16, 6, -6, 7, -6, -7, -16, -7, -2, -2, -1, 9], [4, 0, 3, 6, -1, -2, 1, 2, 3, 9, -6, 7, 2], [-30, 1, 13, 1, -9, -3, 0, -1, 5, 0, 12, -1, 11], [3, 23, 0, -2, 9, -10, 10, 13, 0, -15, -14, -11, -1], [-11, 3, -11, -6, 0, 1, 6, -1, 3, -1, -5, 3, 8], [-11, 5, 4, 2, 1, -18, 1, 1, -11, -11, 4, 2, 7], [6, -4, -4, 1, 13, -7, 8, 0, 2, -8, -11, -11, -7], [-12, 1, -14, -7, 12, -3, 12, -6, -5, -7, -16, 3, -3], [-11, 3, 3, -2, -12, -3, -2, -1, -2, 9, 5, 2, 2], [-5, 9, 3, 1, -6, 6, 4, -1, 1, 6, -4, -3, -7], [20, 5, -4, -7, 3, 0, -6, 8, -7, -9, -6, -4, -9], [-1, -10, -4, 4, 5, -6, 2, -2, 1, -1, 2, 5, 21], [-14, -6, 6, 13, 0, 2, -5, -7, 2, 2, -1, -12, -4], [29, -2, -9, 1, 0, -10, -8, -18, -12, 4, 3, 5, 2], [-16, -20, -7, -10, 5, 6, 5, 1, 0, -6, -6, 2, 2], [22, 9, 4, 0, 3, -10, 9, 15, -19, -2, -2, -11, -8], [-13, -9, -2, 8, -8, -5, -9, -12, -3, 8, 5, 4, 13], [-26, -2, -6, 18, -5, -5, 6, -7, 16, 15, 12, 1, 5], [11, -23, 12, 7, -3, 2, -19, -4, -7, -1, 4, -11, 1], [-21, -11, 19, -9, -10, 6, 2, 1, 0, -2, 17, 4, 10], [5, 14, -4, -6, 14, -15, -9, -4, -5, -3, -1, 6, 8], [8, -17, 18, -8, -5, 15, -11, 8, 8, -5, 6, -3, 1], [12, -17, 14, -14, 9, 12, -3, 10, 11, -9, -6, 9, 4], [19, -18, 9, -16, 10, -4, 4, -4, -7, -12, -13, 14, 6], [13, -23, 20, -8, 0, 1, -5, -7, 1, -8, -3, -1, 5], [12, 31, 7, 7, -3, 3, 6, 5, -17, -10, -3, -4, -4], [7, -22, 15, -15, 9, 8, 14, 13, 7, -15, -15, 5, 10], [-9, 20, -2, 1, 9, -4, -7, 1, -5, 1, -7, -12, -13], [1, -22, 0, -10, 4, -13, -2, -10, -3, -1, 5, 2, 24], [-17, -1, 1, -1, -5, 2, 4, -9, 12, 5, -2, -1, 16], [-23, 14, -2, 6, -3, -3, 3, 4, 2, 5, -7, 4, -6], [24, -2, 1, -9, 9, -5, 5, 7, 1, -3, -16, -1, 0], [9, -5, 7, -7, -13, 0, 1, -8, -8, 1, 7, 7, 0], [-17, -2, -6, -3, 11, 2, -6, 5, -8, -4, -6, 4, 5], [10, 20, -11, 3, -6, 8, 10, 11, 7, 1, -1, -1, -10], [-14, 13, -18, -4, -6, 5, 5, 0, 3, 10, -5, -4, -12], [-33, 1, 6, 3, -1, -4, 4, 2, 8, -7, -8, 9, 3], [25, 9, 2, -10, 14, 0, 1, 5, -7, -10, 4, -2, 5], [-6, 0, -6, -3, 2, -2, 5, -6, -9, -7, -2, -4, -11], [-13, -1, 12, 6, -14, -1, 7, -5, 0, 6, -6, 2, 5], [42, -6, -14, 0, 3, -3, -9, -8, -10, 0, 1, -3, -15]];
    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_bi();".
// 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_bi(:prec:=13)
    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_bi();".
// 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_bi( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6027_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, -12, -69, 84, 311, -174, -478, 50, 263, 26, -56, -11, 4, 1]>,<5,R![438, 82, -5001, 1717, 9154, -2511, -5759, 980, 1489, -85, -177, -8, 8, 1]>,<13,R![2025, 31617, 70137, -384480, 412788, -78601, -86349, 26659, 9174, -2063, -615, 29, 16, 1]>],Snew);
    return Vf;
end function;