
// Make newform 6027.2.a.bm in Magma, downloaded from the LMFDB on 17 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_bm();".
// 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_bm();".
// 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 := [-49, 112, 500, -960, -1276, 2424, 1218, -2708, -400, 1532, -54, -456, 64, 68, -14, -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, 0, 0], [0, 1, 0, 0, 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, 0, 0], [-16891, -26662, 120024, 89961, -281488, -129437, 285731, 97946, -146235, -38854, 39832, 7439, -5554, -525, 316, -2], [13062, -27262, -129487, 189327, 360236, -399882, -426432, 380344, 251383, -186478, -77407, 48798, 11870, -6475, -711, 344], [17073, -24545, -202558, 198515, 623019, -467859, -790497, 494579, 489196, -266506, -155889, 75753, 24523, -10822, -1501, 615], [18179, -26757, -203111, 199068, 623019, -467859, -790497, 494579, 489196, -266506, -155889, 75753, 24523, -10822, -1501, 615], [-4621, 7243, 48620, -55090, -141097, 123884, 173926, -124989, -105797, 64712, 33328, -17793, -5199, 2471, 316, -137], [-7202, 12184, 80223, -86909, -240319, 193949, 300321, -197609, -183812, 103836, 58088, -28999, -9080, 4089, 553, -230], [-153286, 248333, 1695969, -1874925, -5090446, 4268299, 6390473, -4373500, -3928944, 2297459, 1246429, -639991, -195436, 89943, 11929, -5042], [-144991, 253310, 1686015, -1876584, -5088787, 4268299, 6390473, -4373500, -3928944, 2297459, 1246429, -639991, -195436, 89943, 11929, -5042], [78449, -122629, -872178, 960374, 2591188, -2197811, -3224355, 2243872, 1970354, -1172474, -622481, 324855, 97309, -45416, -5925, 2533], [-72814, 120697, 828099, -949454, -2504444, 2185400, 3152381, -2241205, -1941640, 1174231, 617063, -325870, -96924, 45612, 5925, -2547], [-35294, 58774, 395619, -449193, -1186931, 1018766, 1487065, -1037609, -912900, 542200, 289313, -150389, -45329, 21060, 2765, -1177], [286916, -517798, -3202236, 3790041, 9619586, -8416079, -12056686, 8477201, 7401420, -4404277, -2345744, 1218296, 367622, -170457, -22436, 9529], [170730, -295615, -1934572, 2249010, 5857389, -5072879, -7373853, 5150163, 4536144, -2687259, -1438573, 745048, 225401, -104349, -13746, 5834]];
    Rf_basisdens := [1, 1, 1, 1659, 553, 553, 553, 79, 79, 1659, 1659, 553, 553, 237, 1659, 553];
    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, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 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, 0], [-1, 0, 1, -1, 1, 0, 1, 0, 0, 0, 0, -1, 0, -1, -1, 1], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [-1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, -1, 0, 0, -1, 1], [1, 1, -1, 1, -1, 0, 0, 0, 1, 0, 0, 2, 0, 1, 1, 0], [-1, 1, -1, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 1, 1, 0], [-1, 0, 0, 1, -1, -1, 1, -1, 1, 0, 1, -1, -1, 1, 1, 0], [1, 0, 1, -1, 1, 0, -2, 2, -1, 1, 0, 0, 0, -2, 0, -1], [-4, 0, 0, 0, -1, -1, 1, -1, 1, -1, -1, 0, 1, 2, 0, -1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, -2, 0, -2, -2, -2, 0, 1, 0, -1, 2, 1, 0, 1, -1], [-4, -1, 1, -1, 1, -1, 2, -2, 2, -1, 0, -2, 1, -1, -1, 1], [-5, 0, -1, 0, 1, 3, -1, 0, -4, 1, 1, 0, -1, -1, -2, 0], [-1, -1, -1, 0, -1, 0, -1, 1, -1, 2, 1, 0, -2, 0, 1, 0], [2, 0, -2, 2, -4, 0, -2, -1, 0, -1, 1, 2, -1, 0, 2, -1], [-2, -2, -1, 0, -2, -2, -1, 0, -1, 2, 1, 0, -4, 1, 0, 0], [1, 1, -1, 2, -1, 1, -1, 2, -1, 1, 2, 0, -1, 0, 1, -1], [-2, 1, 0, 0, 2, -4, 2, 0, 2, 1, -1, -2, 0, 1, -1, 1], [-3, 2, -1, 1, 1, 1, 0, 1, -1, -1, 1, 1, 0, 2, -1, 0], [-3, 1, 0, 0, 0, -1, 0, 0, -2, -1, 0, -1, -1, 1, -3, 1], [0, 0, 1, -2, 1, -1, 1, 1, 0, -1, -1, 1, 1, -2, 2, 0], [4, 0, 1, -1, -1, -2, -1, 1, -2, 1, 1, 0, -2, 1, -1, 1], [2, -3, 1, 2, -4, -1, 1, -2, 5, -1, 2, 1, 1, 1, 5, -1], [1, -2, 4, 0, -1, -1, 3, -4, 2, -3, 0, -2, 1, 0, 1, 0], [-4, -1, 3, 0, 2, 3, 0, -2, 0, -2, 0, 0, 0, 1, -1, 0], [-5, -1, -2, -1, -1, 3, -1, 0, 1, 1, 2, 1, 2, -1, 0, 1], [-3, 3, 0, -2, 3, 1, -2, 3, -3, 1, -2, -1, 1, -2, -4, 1], [1, 3, -3, 3, -3, 0, -3, -1, 0, 0, 1, 3, 1, -1, 1, -3], [-2, 1, 0, 1, -1, 2, 0, 1, -2, 1, 1, -1, -1, -3, -1, 0], [-5, 0, 0, -1, 0, -1, 0, 0, 3, -1, -3, 0, 3, 0, -2, 0], [-3, 1, -1, -2, 3, -1, 3, 1, -1, -1, -4, -2, 3, 0, -1, 1], [-6, -2, 5, -3, 1, 1, 3, -2, 2, -1, 2, -2, -2, 0, 2, 3], [1, -2, 1, 1, 0, -2, 1, 3, 3, 3, -1, -1, -1, 1, 1, -3], [2, -2, 3, 0, 1, 1, -2, 0, -1, 2, -2, 0, 2, 0, 0, -1], [-4, -4, 1, 2, -1, -1, 2, -4, 4, -2, 1, -2, 1, 2, 2, -2], [-4, -3, 2, -4, 4, 1, 0, 2, -3, 2, -1, -3, -1, -1, -3, 2], [-3, 1, 0, -3, 3, 3, 0, 1, -2, -1, -1, 3, 3, -1, -1, 2], [2, 0, 1, 2, 0, 2, -1, 0, 2, 2, 3, 1, -1, -3, 2, -1], [6, 0, -2, 2, -6, 2, -1, -2, 5, -3, 1, 3, 1, 4, 4, -2], [1, 5, -1, 2, -2, 1, -2, 3, 1, 0, -4, 3, 1, 1, 0, -1], [0, 1, 3, -2, 3, 1, -4, 3, -4, 2, -1, 0, -1, -2, 1, 0], [-3, 3, -1, -3, 1, 2, -5, 4, -4, 1, -3, 1, 1, 0, -3, 0], [3, 4, -1, -1, 0, -1, -3, 3, 0, 2, -5, 1, 2, -2, 0, -2], [0, -3, 0, 2, 2, 3, -2, 0, -4, 0, 3, -3, -3, -3, 1, -1], [3, -1, -1, 3, -1, -1, 0, -2, 0, -7, 5, -2, -2, -1, 0, 0], [-7, -2, 0, -1, 2, -1, 2, -4, -3, -1, 1, -1, -1, 0, 0, 1], [2, -1, -2, 0, -5, 0, -2, -3, 2, -3, 0, 2, -2, 1, 3, 1], [-1, 5, -2, 0, -3, 2, -4, 1, 0, 2, -1, 3, 2, -4, 0, -2], [3, 0, -3, 3, -4, -3, -2, 1, 5, -4, 0, 2, 4, 1, 4, -4], [5, -3, 4, 1, 1, 6, 0, -1, 0, 2, 4, 0, 0, -2, 0, 2], [0, -1, 0, 3, -3, 2, -1, 0, 2, -2, 1, 4, -2, 5, 3, 0], [-1, 5, -1, 1, -1, -6, 6, -1, 1, -3, -4, 0, 0, 3, 0, 3], [-1, -5, 5, -1, 3, 3, 2, -4, -2, -2, -1, 0, 0, 3, 3, 3], [-1, 1, -2, 0, 2, 2, 0, 2, -2, 5, 0, 2, 1, -3, 1, -1], [-2, -3, 3, -2, 6, -1, 4, 0, 0, 4, -3, -4, 1, 0, -1, 2], [-3, -3, 4, -1, 7, 2, 2, 2, -1, 7, -1, -1, 5, -3, -1, 0], [-5, -2, 3, 2, 3, 2, 7, -4, 3, -4, 1, -2, 3, 4, 0, 3], [7, 3, -6, 4, -5, 0, -3, 1, 3, 0, 2, 2, 4, -1, -1, -2], [-2, -2, 3, -2, 3, 0, 4, 1, 5, -4, -4, 0, 2, 2, 1, 0], [-3, -1, 1, -1, 3, 3, 0, 1, -4, -1, 0, 0, 1, 3, -4, -1], [-5, -1, 1, 1, 0, 0, 3, -2, 4, -2, -1, -2, 0, 3, -1, 2], [1, 3, -3, -1, 3, 4, -1, 2, -1, 2, -5, 3, 4, -1, -3, 0], [-10, 2, -5, 2, 1, 0, -1, 3, -2, 4, -1, 1, -1, -2, 0, -3], [-2, -2, 3, 0, 2, -5, 2, -2, 0, -1, 1, -3, 0, -2, -4, 0], [1, -7, 4, 4, 0, -1, 4, -1, 2, 2, 3, -2, -3, 1, 0, 0], [-2, 0, -1, -1, -5, -1, -1, 0, -4, 3, -2, 2, -2, 0, 0, 1], [10, -5, -3, 2, -2, 4, -4, 5, 1, 3, 0, 4, 0, -2, 0, -3], [2, 0, -5, 1, 0, -2, -2, 5, -1, 4, -2, 1, 0, -1, -2, 0], [-4, 0, 6, -4, 4, 1, 3, 2, -6, 0, 1, -5, -5, 1, -3, 4], [0, 1, 0, 2, 3, 1, 2, -4, -4, 2, 0, -2, -4, 2, 0, 2], [-8, -1, 3, 2, 0, 1, 3, 0, -1, -3, 1, 0, -3, 1, -4, 1], [0, -5, 2, -1, -3, 0, 4, -2, 4, -5, -1, 1, 0, 0, 1, 1], [-3, 0, -5, 2, 0, 3, -7, -2, -3, 4, 0, 1, 2, -1, -1, -6], [-8, -1, 5, 0, -2, 2, 5, -4, 2, -4, 3, -1, -3, 0, -1, 2], [1, 2, 0, 3, -1, -7, 2, -1, -5, 6, 4, -4, -9, 2, -4, 0], [-3, -3, 2, -3, -4, -2, -2, -1, -3, 2, -1, -2, -4, -2, 0, 3], [-3, 4, -3, -4, 1, 4, -3, 0, -6, 1, 2, -2, -4, -1, -3, 5], [-2, -4, 0, 0, -1, -5, 5, -1, 2, -1, -1, 1, -3, 3, 2, 4], [-8, 0, -3, -2, -1, 2, -3, 0, -4, 1, -1, 3, -2, 0, 0, 1], [3, -1, -2, 0, 7, 1, -2, 3, -7, 4, 0, -3, -2, -8, -3, 1], [-3, 4, -1, 0, 2, 2, 5, -6, -1, -5, 5, -4, 0, 2, -1, 4], [-5, -5, 1, 0, 1, 4, -1, -1, 0, -5, 5, 0, 2, 2, 2, -1], [1, 1, -2, 3, 0, 2, 0, -1, 3, -1, 1, 0, 3, -2, 3, 5], [0, -6, 6, -4, 5, -3, 5, -1, -1, 1, 2, -5, -5, -1, 2, 5], [-5, 5, -8, 1, 0, 3, -3, 3, -3, -9, -1, 3, 1, 2, -2, 0], [-6, -3, 6, -4, 3, -1, 3, -1, -4, -4, 1, -4, -4, -2, -7, 1], [-3, -5, 4, -1, 1, -2, 2, -3, 5, -3, -1, -5, 3, 0, 4, 1], [-7, 2, 2, 3, 2, 2, -2, 3, -4, 6, 2, 0, -4, -1, -2, -3], [-10, -3, 5, -6, 2, -5, 7, -2, 3, 1, -8, -3, 1, 1, -1, 3], [-1, -1, 3, 2, 3, -6, 5, -4, 0, 3, 4, -6, -5, 0, -4, 0], [2, -2, 4, -5, 3, -1, 2, 4, -2, 0, -3, -1, 0, -2, -3, 2], [4, -7, 1, -2, -2, 3, -3, 2, 1, 2, 0, -1, 2, -5, -1, -1], [0, 3, -1, 1, -2, 3, -6, 1, 0, -4, 0, 4, 5, -1, 0, -6], [6, 0, 2, -1, 3, 2, -1, 1, 1, -3, 3, 0, 1, -1, -2, 1], [0, 2, -3, 2, -1, -3, -1, 3, 4, -1, -1, 3, 3, -4, 2, -2], [-2, -3, -2, 1, -6, -4, 5, -3, 5, 6, 0, 0, -3, 6, 2, 0], [0, -7, 11, 0, 0, 5, 5, -4, 3, -7, 4, -2, -3, -2, 5, 3], [4, -2, -4, 2, -2, -5, 1, 1, -2, 2, 3, 0, -1, -5, -2, -2], [-1, -5, 3, 2, -6, -7, 2, -3, 3, -4, -2, 1, 3, 1, 4, -1], [11, -3, -2, 1, -2, 9, -5, -3, -1, -3, 4, 2, 2, -6, 6, -2], [-3, -5, 5, -4, 5, -4, 6, -3, -1, -1, 2, -5, 1, -3, -5, 3], [6, 0, -1, -2, -2, 8, -9, 1, -4, 3, -2, 7, -2, 0, 2, -4], [0, -5, 4, 0, -1, 4, 2, -3, 3, -4, 0, -2, 2, 3, -5, 1], [-4, 2, -1, -3, -2, -2, 6, -3, -1, -4, 1, 1, 2, 4, 3, 2], [0, 1, 10, -5, 1, -6, 2, -2, -1, 3, -4, -5, -4, 1, -2, 4], [-4, -6, 0, -7, 8, 5, -4, 5, -7, 5, -2, -3, 2, -8, -5, -3], [13, -7, -2, 1, -6, -2, -1, 0, -1, 1, 1, 1, -5, -1, 1, -4], [1, -2, -3, -1, 3, 2, -2, 2, 3, 6, -2, -2, 0, -2, 4, -1], [-12, 0, -7, 1, -4, -6, 3, 2, 3, 6, -3, -1, 0, 5, 1, -2], [-1, -3, 1, 0, -5, -8, 0, 1, 7, 2, -2, 5, 2, 5, 4, -3], [-4, -1, 1, 0, 2, 1, 6, 1, 6, -2, 0, 2, 2, 4, 1, -1], [-4, 0, 0, -3, -2, 4, -2, -1, -2, -1, -5, 3, 4, 1, -3, 1], [2, -4, -5, 1, 0, 4, 0, 1, 3, -1, 2, 3, 0, -2, 3, 2], [-2, 11, -2, 0, -5, -5, 2, -5, 4, -7, -2, 2, 1, 9, -3, 2], [-9, 0, 1, -1, 2, 4, 2, 3, -1, -2, -2, 0, 0, 1, 0, 2], [-12, -3, 0, 3, 0, -2, 0, 0, 2, 4, 4, -5, -7, 0, 0, -3], [-4, 0, 2, -1, 1, -6, 4, 0, -5, 2, 3, -5, -3, 4, -6, 1], [3, 3, -1, -2, 2, 2, -7, 5, -3, -3, -3, 2, 1, -1, 0, -2], [3, -9, 0, 1, 6, 2, -4, 9, -6, 8, 3, -2, -3, -7, -2, 0], [-5, -1, 3, -3, 8, -3, 10, 1, -1, 3, -2, -6, 7, -1, -5, 1], [-4, -1, -2, 0, -1, -5, 4, -3, 4, 1, -3, -4, 2, -3, -5, 0], [6, 2, -11, 4, -7, -2, -3, 0, -1, 0, 1, 7, 5, 6, 2, -2], [-2, -3, -3, 2, -3, 3, 0, 1, -1, 2, -5, 2, 1, 5, -3, -1], [-6, 9, -10, 0, 3, -8, 1, 0, 2, -6, -5, -1, 1, 5, -1, 2], [-13, 2, 0, -2, 2, -6, 1, 8, 0, 10, -2, -4, -3, -3, -2, 1], [-3, -7, 6, -4, 8, -3, 7, 1, 3, 5, -1, -10, -1, -2, -6, 4], [-1, 10, -8, 2, -4, 4, -13, 5, 0, 3, 2, 4, 2, -5, -4, -8], [-3, 0, 3, -5, 4, 2, -1, 2, -4, -2, 2, -9, 1, -7, -8, 1], [2, -2, -7, 5, -3, 0, -6, 6, 0, 4, 1, 5, -8, 5, 3, -2], [0, -4, 3, -2, 0, 6, 0, 0, -2, 0, 1, 5, -2, 1, 0, 2], [-1, 1, -6, 11, -11, -6, -2, 4, 10, 0, 6, 4, -3, -3, 8, -5], [-5, 10, -2, -3, -1, 4, -12, 7, -2, 6, 1, 1, 2, -6, -2, -1], [7, -6, 5, 3, 2, 2, 12, -9, 4, 3, 2, -6, -3, 5, 2, 7], [1, 0, 4, 2, 6, 2, 3, -6, -4, -4, 2, -2, -5, -1, 0, 1], [-4, -5, -4, 3, -10, 1, -7, -4, 6, 0, -3, 6, 4, 2, 3, -9], [0, 6, -10, 6, -7, 7, -8, 2, 0, -4, 3, 8, -2, 5, 4, -4], [-2, 3, -4, 1, -8, 0, -3, -6, 0, -7, 2, 6, -3, 4, 7, -1], [-1, 7, -3, -2, 7, 5, 1, 2, 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15, 0, 4, 4, -7, 3, 0, -12, -4, -8, -10, 9], [1, -8, 2, -2, 5, -6, 3, 5, -5, 11, 0, -10, 6, -7, 0, -1], [1, -14, 0, 9, 3, -5, 8, -3, 7, 1, 4, -2, -4, 5, 7, -2], [-5, -8, 0, -1, 6, 7, -4, 8, -1, 3, -5, -4, 5, 6, -1, -5], [0, 8, 1, -1, 8, -16, -1, 7, -12, 18, -4, -9, -10, -4, -13, -4], [4, 5, 2, 0, 3, 6, 0, -3, 3, -2, -9, 0, 10, 3, 9, -5], [13, 3, 6, -3, 0, 2, -2, -4, -9, -9, 4, -3, -1, 0, 0, 0], [-10, 15, -6, 1, 15, 10, -4, 11, -2, 5, -1, -3, 1, -4, -7, 0], [-1, -4, -13, 5, -2, 0, -8, -8, -2, 3, 5, 8, 0, -5, 6, -6], [2, 3, 10, -4, 1, 1, 5, -1, 13, -4, -1, 5, 4, -1, 1, -2], [-6, -9, 10, -3, 10, 6, 4, -4, 2, -8, 4, -11, 3, -2, -10, 5], [0, 6, 3, -4, 10, 1, 1, 3, -12, -4, -2, -3, -2, 3, -14, 4], [-11, 7, 0, -1, -6, 2, 2, -3, -3, 5, -1, 4, 5, 4, -1, 1], [13, 1, -3, 8, -11, -4, -14, 7, 1, 14, 2, 5, -3, 2, 0, -5], [3, 8, 2, 2, 0, -5, 8, -6, 3, 17, -9, -1, -6, 1, -1, 1], [-19, -4, 1, -2, 6, 4, -6, 4, -3, 17, 3, 1, -3, -7, -2, 2], [4, 4, -3, -3, 2, -6, 8, -7, -3, -2, -6, 3, 2, 10, -2, 6], [-19, 14, 2, -7, 11, -2, 2, 8, -8, 10, -9, -10, 5, 2, -15, 8], [-13, 12, -13, 2, 1, -9, 2, 0, -1, -10, 1, 1, 0, 3, -8, -1], [-5, 10, -2, -3, 1, 11, -14, 11, -15, -11, 2, 8, 0, -7, -9, -4], [-5, 8, 3, -7, 4, -3, 10, 3, -2, -11, 2, -6, 2, -3, -10, 9], [30, -5, 2, 5, -7, 3, -10, 4, 6, 7, 0, 13, 2, -4, 6, -13], [5, 5, 12, -1, -7, -11, 6, -9, 4, 7, 1, -13, -11, -3, -5, 4], [-4, -13, 22, -3, 7, -4, 13, -8, 6, -13, 7, -11, -2, 3, 0, 10], [5, -11, 1, 2, -1, -7, -1, -3, 5, -6, -5, 4, 14, -4, 0, -6], [4, -7, 8, -5, 8, 8, -7, 6, -13, 11, 11, -8, -10, -10, -10, 7], [-17, -9, 7, -8, 8, -8, 9, -2, 8, -8, -6, -10, 1, 6, 8, 5], [13, 6, -4, 4, -5, -1, -1, -5, 6, 17, 6, 6, -2, -4, 11, -2], [-1, 9, 3, -5, 7, -4, -8, 11, 1, -3, -5, 6, 2, -10, 8, -3], [9, -8, 10, 5, -3, -5, -3, 2, 2, 8, 2, 7, -6, -12, 3, -5], [5, -6, 2, 4, -4, 0, 3, 1, 4, -9, -4, 6, 3, 7, 12, -10], [9, -8, -10, -3, -1, 9, -7, 4, -8, -2, 1, 11, 0, 1, -3, 2], [7, -9, 16, -4, 8, 2, 1, -2, 10, -2, -6, 0, 7, -6, 5, -2], [0, -10, 5, -8, 3, -7, 6, -5, 7, 8, -5, -8, 7, -9, 1, 0], [2, -15, -3, 4, -12, 6, -3, -4, 11, 1, 8, 11, 7, -5, 17, -11], [22, 4, -5, 0, -11, -13, 6, -9, 11, 17, -4, 3, -1, 2, 6, 0], [-16, 16, -16, -4, -1, -6, -6, 3, -2, 12, -6, 8, 2, -3, 2, 0], [4, 11, -8, 3, -7, -20, 7, -8, 8, -16, 4, -6, 1, 13, -3, -2], [1, 4, -1, 1, -2, -10, 1, 4, 10, -14, 1, -1, 1, 2, -5, -1], [-5, -3, 10, 2, 1, 2, 7, -7, 5, 2, 3, 0, 3, -7, 2, -1], [17, -16, 14, 3, -5, -12, 6, -5, 10, 1, 3, -12, 2, 1, 1, 1], [-22, -6, 7, -9, 8, 10, 2, -7, -7, -6, 6, -11, -4, -11, -6, 4], [2, 3, 7, -7, 5, -4, 10, -6, 6, -25, -3, 1, 5, 1, 3, 4], [-16, -8, -4, 0, -2, 8, 4, 2, -11, 11, 0, 2, -6, 0, -6, 2], [2, 6, 5, 3, 11, 13, 3, -4, -4, 1, 4, 0, 2, -6, 2, 5], [23, -10, 8, 3, 2, 1, -2, 6, 2, 8, 1, -4, -9, 8, -6, -3], [-9, -8, 5, -2, 2, 8, 8, -4, -4, -10, -1, -2, 4, -11, -9, 6], [8, -4, 6, -1, 3, 7, 4, 1, -4, 6, 3, -1, -8, -15, -2, -1], [5, 9, -9, 3, -1, -4, -3, 11, 2, 7, -4, 7, 1, -3, 6, -3], [-26, 14, -9, 3, -1, -10, 4, -2, -6, -9, -1, -5, -5, 1, -7, -2], [2, 2, -11, 9, -15, -8, -2, 1, 3, 0, -8, 17, 0, 19, 7, -9], [-1, 5, -2, -5, -14, -6, -4, 0, 3, -1, 0, -1, 1, -2, -4, -4], [6, -8, -4, -1, 8, -5, -1, -3, -8, 6, -1, -14, -2, 11, -7, 6], [-3, 11, -16, 12, -5, -20, -2, 6, -4, -8, 5, 1, -12, 8, 1, -1], [5, -21, 8, -2, -5, -17, -4, 1, 3, 10, 6, -3, -6, -12, 11, -3], [-6, -1, -1, -6, -3, -1, -5, 3, -3, 6, -7, 11, -2, 9, 3, 2], [-21, -3, -1, -4, 13, 8, 4, 3, -5, -3, -12, -9, 9, 3, -3, 5], [10, -6, 23, -9, 13, -7, 9, 5, -2, 1, 0, -8, -8, -9, -9, 9], [-16, 5, -19, 11, -13, 13, -15, -5, -9, 2, 16, 7, -10, -4, -1, -4], [4, 8, -3, -8, -2, -2, 0, 4, -2, -4, -10, 2, 2, 10, 2, 6], [9, -5, -7, -5, 1, 5, -18, 15, -25, 20, -2, 1, -7, -9, -13, -3], [-10, -4, 0, -6, 11, 2, 6, -3, -8, -8, -1, -6, 7, -1, -8, 3], [14, -1, -2, -6, 8, 4, 12, 0, 10, -3, -6, 5, 13, 12, 5, 7], [-6, 15, 2, -5, -5, 0, 5, -8, 5, -6, 1, -2, 2, 2, 0, 0], [11, 8, 5, 2, 9, 8, -7, 7, -6, 1, 5, -4, 1, -5, -5, 4], [-22, 4, -7, 10, -4, 3, 3, -15, 8, -16, 4, 3, 4, 5, 8, 0], [-9, -5, -10, 0, -5, -3, 4, -1, -1, 5, 2, 9, 5, 2, 5, -1], [-12, -15, 7, 6, -6, 9, 14, -7, 4, 2, 6, 0, -6, 4, 3, -2], [-13, -3, 3, 5, -7, -4, 8, -11, -5, 1, 6, 4, -1, -1, 0, -5], [-17, -7, 12, 2, -3, -15, 10, 1, 7, 4, 8, -13, -14, 4, -4, -1], [7, 6, -8, 0, 6, -12, 9, 4, 10, 2, -8, -2, 10, 1, 1, 2], [-8, -2, 0, 2, 7, 10, -12, 9, -17, 5, 2, -4, -3, -1, -6, -6], [-11, 4, 8, -5, 12, -2, 11, -2, -18, 12, 2, -15, -7, 1, -22, 13], [-23, -1, 3, -1, 4, -4, 16, -13, 3, -23, 11, -3, -5, 0, 1, 3], [13, -4, -4, -1, -8, 2, -7, 6, 2, -13, 2, 9, 1, -10, 0, 3], [-13, -12, 5, 2, -4, 3, -2, -10, -1, 5, 0, -1, 6, 5, 9, -5], [-12, 3, 17, -13, 4, -3, 13, -7, 3, -10, -7, -3, -1, 3, -2, 6], [-2, -11, 3, 3, -10, 2, -6, -7, 8, -7, 11, 3, 6, 5, 10, -7], [20, -7, 10, 2, -8, 4, -2, 4, 0, 7, -1, 2, -2, 9, -1, -3], [11, 22, -8, 9, -3, 2, -3, 5, 4, 2, -8, 19, 5, -1, 13, -4], [9, -18, 6, 3, -1, -8, 0, 5, 7, 21, 3, -2, -2, -8, 14, -8], [13, 0, -15, 0, -20, -14, -5, 7, 3, -5, 2, 8, -7, 7, 6, -5], [10, 19, -8, 8, -7, -2, -7, 3, 18, -4, -3, 11, 12, -3, 15, -15], [14, -1, 3, -2, -15, -15, 2, -13, 1, -13, -1, -2, 4, 7, 5, 1], [9, -5, 12, -8, 8, -5, 6, -8, -2, -2, -1, -11, 1, 2, -1, -4], [-8, 9, -8, -1, 0, -19, 7, -1, -3, -4, -9, 1, 7, 13, -6, -2], [-8, 4, -8, 0, -1, 9, -7, -3, -1, 4, 7, -5, -3, -4, 2, 5], [9, 16, -9, -3, -1, 1, -9, 13, -9, 17, -6, -5, -10, 2, -6, -3], [12, 9, -13, 4, 2, 7, -8, 8, -8, -1, -2, 18, 7, 3, 3, -4], [12, 9, -7, 12, -10, -9, 3, 7, 1, 23, 1, 5, -7, 3, 5, -8], [0, 12, 5, -6, 19, -13, 19, 8, 2, 14, -3, -13, 3, 0, -12, 7], [6, 17, -9, -5, 6, 10, -5, 8, -5, 8, -8, 8, 11, -8, -7, -3], [7, 2, 9, 8, -13, -13, 11, -9, 16, -19, -3, 4, 8, 21, 12, 2], [2, 13, -1, 3, -7, -8, 1, 3, 12, 8, -3, 3, -3, -13, 1, -5], [1, 9, -11, 9, -14, -4, 3, -2, 13, -12, -7, 13, 6, 7, 1, -11], [8, -4, 0, -2, 6, -2, 2, 0, -4, -4, 2, -2, -4, 4, -10, -2], [8, -8, 13, 3, -5, -4, -14, 2, 0, 15, 9, -9, -5, -7, -1, 2], [24, -10, -3, 1, -6, -8, -2, -1, 13, -1, -2, -1, 12, -10, 1, -8], [-8, 4, 12, -3, 5, -5, 5, -3, -11, 9, -4, -10, -11, -6, -10, 10], [-6, 5, -1, 2, 6, 1, -10, 16, -12, 4, -8, 4, -6, 15, -9, -7], [-2, -3, 5, -10, 11, -8, -6, 14, -13, 7, -5, -8, 1, -11, -9, 2], [-20, 16, -6, -2, 6, 7, -7, 2, -10, -3, 9, 7, 0, 3, -10, 6], [2, -14, 7, -3, -4, -4, -5, -2, 10, 8, 0, 0, -5, -10, 6, -11], [-12, 6, 3, -8, 15, 8, 9, 2, -10, -13, -4, -5, -3, 15, -11, 15], [15, -12, 21, -1, 2, -17, 4, -1, -6, 0, 2, -12, -11, 0, -4, -4], [0, 10, -12, 2, -16, 1, -1, -8, 11, 3, 1, 19, 1, 10, 9, -2], [-12, -24, 21, -7, 6, -15, 31, -15, 21, 10, 1, -20, 2, 2, 3, 9], [11, -4, -9, -1, 12, 4, -3, 10, -6, 4, 2, -7, 3, -5, -20, -1], [16, -8, -6, 9, -4, -6, 9, -1, 9, -6, 9, 6, -12, 17, 17, 1], [4, 20, 2, 2, -2, 0, -11, 4, -9, 15, 7, -1, -15, -14, -4, -4], [-12, 1, 12, -5, 11, 14, 0, -1, -12, 5, -1, -1, 7, -7, -7, 10], [-1, -13, 2, -1, -2, 13, -7, 6, -10, 14, 4, 6, -1, -9, -2, -2], [-8, -2, 0, -2, 0, 0, 0, -10, 3, 5, -4, 4, 0, 4, -10, 8], [-3, 10, 10, 2, -6, -22, 7, -5, 18, 19, -4, -4, 3, -5, 6, 0], [2, 13, -20, 9, -5, -6, -11, 10, 6, -11, -5, 15, 14, -1, 4, -10], [-6, -12, 4, -2, -4, 15, -2, 0, 6, -15, 4, 0, -1, -4, 6, 0], [23, 6, 0, 4, -9, -6, 9, -5, 10, -11, 0, -12, -5, 5, 7, 0], [4, -1, 0, 4, 6, 2, -1, -5, -5, 13, 13, -3, 6, -7, 1, -4], [4, -1, -3, 5, 1, -3, 3, 2, 4, -17, 4, -10, 1, 2, -4, -5], [-6, -16, 7, -8, 10, 20, 1, 0, -6, -3, 1, 9, 8, 0, -3, 2], [-5, -3, -10, 3, 7, 0, -4, 6, 4, 4, -2, -2, 8, -8, -6, -13], [18, 1, -2, -5, 9, 18, -16, 4, -5, 8, -2, 7, 7, -14, -2, -10], [-25, 11, -10, -5, 3, 7, -3, -4, -4, -2, -5, 4, 2, 0, 1, 10], [-21, -6, 4, 3, 2, 8, 9, -8, 3, -16, 6, -3, 0, 8, -2, 0], [-27, 12, -7, 10, -7, 4, -2, -8, 3, -23, 1, 9, -1, 14, 1, -5], [1, 17, -3, -4, 4, -6, 1, 3, -1, -11, -5, 4, -5, 5, 6, -6], [4, 6, 14, 6, -2, -9, 5, -6, -9, -8, 2, -7, -10, 11, -1, 1], [-13, -4, 8, -3, -8, 0, 7, 2, 12, 4, 6, -3, 1, -3, 2, -3], [25, -1, -4, -1, 2, -11, 2, -1, 0, -1, -1, -4, -6, 4, 0, -4], [-7, 6, 10, -8, 12, -3, 8, 8, 8, 6, -2, -8, -2, -14, 2, 10], [2, 7, 2, 0, 13, 23, -5, 13, -5, -8, -1, 1, 3, -5, -1, 6], [-22, -19, -1, -4, -5, 4, 4, -9, 4, 4, 1, 2, 10, -8, 5, 4], [-6, 4, 5, 2, 15, -8, -2, -3, -6, 3, -9, -10, 5, -6, -9, 5], [22, 4, 9, -3, -7, -7, 4, -7, 5, -12, -2, 13, 5, 1, 11, 6], [-17, -9, 16, 3, 11, -8, 13, -4, 14, -10, -5, -1, 8, 11, 9, -6], [-31, 10, -2, -4, 0, -12, 9, 2, 2, 9, -9, -8, -5, 15, -2, 7], [-30, -5, 14, -12, 9, -4, 0, -7, -7, -8, -1, -8, -3, -20, -7, 1], [17, 11, -14, 5, -4, 15, -14, 1, -14, 5, -2, 14, 3, -1, 6, -10], [11, 6, -12, 9, -9, 17, -14, 2, 8, -2, 7, 13, -2, 10, 17, -4], [-5, -15, 17, -9, 18, 3, 5, 6, -12, 4, -5, -13, 5, 0, -2, 4], [-22, 25, -5, -3, 7, 11, -14, 7, -9, 1, -7, 2, -1, -10, -9, -3], [-1, -9, 6, -5, 16, -3, 7, 5, -5, -1, -6, -11, -4, 7, -8, 5], [15, -14, -3, 1, -10, -20, -5, -3, 7, 13, 3, -1, -3, -4, 8, -7], [7, -19, 9, -2, 6, -10, 11, -10, 6, -2, 1, -4, -5, 8, 4, 12], [20, -13, 9, -4, 16, 1, -6, 14, -9, 19, 2, -18, -15, -12, -13, -4], [1, -7, -3, 6, -2, 7, -21, 0, -16, -3, 2, 4, -7, 0, -1, -12], [-12, 16, 1, 13, -3, 6, -3, 5, 7, -8, 3, 12, -1, -3, 11, -1], [22, -3, -13, 11, -10, -9, 9, -7, 1, 7, 3, 4, -7, 12, -2, 1], [12, 24, 0, -5, 2, 5, -17, 14, -1, 9, -11, 3, -4, -11, 4, -6], [-8, 5, 1, 0, 1, -8, 14, -9, 18, -14, 7, 10, 0, 1, 16, 2], [-14, -21, 20, -10, 15, 1, 1, -1, 4, -4, -7, -12, 4, -5, -11, 3], [-11, -6, 2, -6, -1, -1, 15, -15, 12, -16, 8, -6, -8, 4, 6, 10], [10, 6, -6, 1, -10, 8, -22, 13, -7, 20, 3, 7, -9, -3, 10, -7], [-4, 8, 6, -4, -6, -3, -4, -8, -5, 15, 4, 4, -5, 4, -6, 5], [-11, 0, -4, 6, 7, 4, -9, 14, 6, 14, -1, 1, 8, -8, -1, -3]];
    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_bm();".
// 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_bm(:prec:=16)
    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_bm();".
// 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_bm( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6027_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-49, -112, 500, 960, -1276, -2424, 1218, 2708, -400, -1532, -54, 456, 64, -68, -14, 4, 1]>,<5,R![7, -120, -756, 404, 6906, 7724, -6132, -12676, -1464, 6192, 2800, -732, -730, -100, 36, 12, 1]>,<13,R![6489, -415572, 1297286, -699296, -1691066, 2155768, -208090, -706696, 212080, 87176, -36046, -4612, 2588, 88, -84, 0, 1]>],Snew);
    return Vf;
end function;