
// Make newform 6018.2.a.ba in Magma, downloaded from the LMFDB on 10 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6018_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_6018_2_a_ba();".
// 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_6018_2_a_ba();".
// 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 := [5653, 5149, -10082, -8325, 6915, 4690, -2301, -1101, 381, 111, -31, -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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-435582988, -84801718, 631646387, 53318216, -319841161, 4825579, 69490919, -5503599, -6728760, 833951, 239615, -37967], [-541927834, 94398553, 993409765, -192373155, -617251320, 126683562, 151503348, -30426407, -15664289, 3132406, 575314, -118047], [546225598, -139596031, -1020839309, 237183129, 634988602, -143333204, -156149770, 33453103, 16221775, -3402596, -600100, 127547], [870706107, 64214944, -1492728506, 66506323, 886236402, -113127340, -211704075, 34189684, 21598935, -3900430, -789735, 154847], [-622275386, -73257917, 1085960100, -1932955, -652711197, 63583367, 156316299, -21914010, -15934575, 2636977, 581451, -107852], [1047731024, -295295825, -1967030013, 503450773, 1220063292, -289297782, -298396854, 65032295, 30758583, -6439990, -1126760, 237037], [-625188159, -3011304, 1141470325, -93300582, -707942363, 102692323, 172914262, -29046033, -17843661, 3243284, 656217, -128019], [1621505661, 65188905, -2893219847, 171457996, 1762942612, -231916382, -426411439, 68395961, 43783498, -7780026, -1606687, 310088], [-1533113265, 22537731, 2814323071, -266922102, -1749467342, 267410740, 428531541, -74242225, -44361950, 8235372, 1636811, -324090], [5703167183, -32595192, -10269561860, 915294055, 6293405452, -937729774, -1528293261, 261108042, 157286279, -28952712, -5777705, 1137721]];
    Rf_basisdens := [1, 1, 1323382, 2646764, 2646764, 2646764, 1323382, 2646764, 661691, 1323382, 1323382, 2646764];
    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_6018_a();"
function MakeCharacter_6018_a()
    N := 6018;
    order := 1;
    char_gens := [4013, 1771, 1123];
    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_6018_a_Hecke(Kf)
    return MakeCharacter_6018_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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 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, 1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, -1, 1, -1, 0, 0, 0, 0], [2, 1, 0, 0, -1, 0, 0, 0, -1, 0, -1, -1], [2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [1, -1, 0, 0, 1, 0, 1, -1, 0, 1, 1, 2], [0, 1, 1, 0, 0, -1, -1, 0, 1, -1, 0, 0], [1, 0, -1, 0, 0, -1, 0, 1, 1, -1, 0, 0], [2, 1, -1, 0, 0, 1, 0, 0, -1, 0, -1, -1], [0, 0, 0, -1, 1, -1, 1, 0, 0, 1, 0, 0], [0, 0, 1, 0, 1, -1, 0, 0, -1, 0, 1, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, -1, -1, 0, 1, 0, 1, -1, -1], [2, 0, 0, 1, -1, 1, 1, 0, 0, 0, -1, 0], [2, 0, 1, 0, 0, -2, 1, -2, 0, 0, 1, 2], [2, 0, -1, 1, 0, 1, -1, 2, 0, 0, -2, -1], [1, 1, 0, 1, 0, 1, -2, 1, 0, -1, 0, 0], [-1, 0, 0, -1, 1, 0, -1, -1, 2, 0, 1, 1], [0, 2, -1, -1, 0, -1, 1, 0, 1, 0, 0, -1], [3, -3, 0, 2, 2, 0, 0, -1, 0, 1, 0, 1], [1, -1, 0, 1, 1, 1, -1, 0, -2, 1, -1, 0], [2, 1, 1, 0, 0, 0, 0, -1, -1, 0, 0, 1], [2, 1, -1, 0, 1, 1, 1, 0, 0, 1, 1, 2], [-1, 1, 0, -1, -2, 1, -1, -1, -1, 0, 0, -1], [1, 2, 0, 0, -1, -1, -2, 1, 2, -2, -1, -1], [1, 1, 0, 1, -2, 1, 0, 1, 0, 0, -1, 0], [2, 0, 1, 3, -1, -1, -1, 2, 2, -3, 1, -1], [2, 2, 0, 1, -1, 1, -1, 0, 0, 1, -2, -2], [2, -1, 0, 1, 2, 0, 1, -1, -2, 1, -1, 1], [1, 0, 1, 0, 1, -1, -1, -1, -1, 1, 0, 1], [1, 1, 1, -1, 0, 2, 1, 0, -1, 1, -2, -2], [-3, -3, 0, 2, 2, -1, 0, 1, -1, -1, 2, 0], [-2, 3, 0, 0, -2, -1, 0, 0, 0, 1, -1, -2], [-4, 2, 1, -3, -1, -3, 2, -4, 0, 1, 1, 0], [3, 1, -1, 2, -1, 0, 0, 3, 0, -1, 0, -2], [2, 1, -2, -3, 0, 0, 3, 0, -2, 2, -1, 0], [-3, -2, -1, -1, 2, 0, 2, 1, 0, 1, 0, 1], [-2, 0, 0, -2, 0, -2, 1, -2, 0, -1, 1, 3], [1, 0, 0, 1, 0, -2, 1, -3, -2, 1, 0, 1], [-2, 3, 1, 2, 0, 1, -1, 2, 0, -3, 2, -2], [-1, -1, 0, 2, 3, 3, -3, 2, 1, -2, 1, -2], [-2, 1, 2, 3, 1, 2, -2, 2, 2, -4, 2, -1], [1, 2, 0, 1, 0, 0, 1, 1, 2, 1, 0, -1], [-1, 1, -1, -1, 0, 1, 0, 1, 3, -2, -1, 1], [-2, -1, -2, 1, -2, -1, 0, 0, 3, -1, 1, 2], [5, -1, 1, 4, 1, 0, -1, -1, 0, -2, 3, 1], [-2, 5, 2, -1, 0, 0, 0, -1, -2, 0, 1, -2], [-3, 3, 0, -1, -2, -1, -2, -1, 0, 0, -1, -2], [-7, 2, 1, -1, 0, 0, 0, 1, 2, -2, 1, 1], [-3, 2, -1, -3, 0, 0, -2, 1, 4, -1, 2, 3], [-1, -1, 1, 1, 1, 4, -2, 1, -1, 0, 2, 0], [5, 0, -1, 1, -1, 1, -1, -1, -1, 3, 1, 3], [-3, -2, 1, -2, -1, -1, -1, -1, -2, 2, 0, 1], [-4, -2, 1, -3, 0, -2, 1, -2, -1, 3, 2, 4], [0, 0, 1, -1, 1, 1, -2, 0, 0, 1, -2, 0], [-4, 4, 0, -2, -1, -2, -1, 2, 2, 0, -1, -1], [2, -4, 0, 0, 2, -2, 4, -2, -4, 2, -1, 2], [2, -1, -1, 0, -1, 3, 2, 0, 0, 0, 0, 3], [-7, 5, 1, -1, 0, -1, 0, 1, 2, -1, -1, -3], [6, -2, -1, 1, 0, 1, 3, -1, -2, 2, 1, 2], [0, 1, 0, -2, -2, 5, -2, 2, -3, 1, -1, -2], [-1, 1, -2, 2, 2, 2, -1, 1, -2, -2, 0, -2], [-8, 2, 1, 0, 0, -4, 0, 1, 6, -2, 0, -2], [8, -9, 0, 4, 4, 3, -1, 2, 1, 2, -2, 3], [0, -1, 0, -3, -2, -1, 1, 0, 1, -2, 0, -1], [0, 2, -2, 1, -2, -1, -1, 2, 4, -3, 0, 0], [0, 3, -2, -1, -4, 1, -2, 2, 1, 0, -2, 0], [-1, 1, -2, -1, -2, 1, -1, -1, 0, 1, 2, 3], [3, -2, 0, 3, -3, -2, 0, -2, -2, 2, -2, -1], [-8, 0, 3, -1, 0, 1, -4, 1, -2, -1, -1, -3], [3, 2, -1, -5, -1, -1, 2, -3, 0, 1, -2, -2], [-2, 1, 2, -1, 1, 0, -3, 4, 4, -5, 1, 0], [-2, -2, -1, 1, 4, -1, -1, 1, 2, 0, 0, -2], [-4, 4, 2, -1, -1, -3, 0, -2, -4, -4, 0, -3], [1, -2, 3, 1, 0, -2, 1, 1, 0, 0, 1, 0], [-3, 3, 1, -3, -3, -1, 1, -4, -4, 2, -1, -1], [-7, 0, 1, -2, 0, 2, -4, 1, -1, -2, -1, -1], [2, -1, 1, -3, 0, -3, 2, -4, -2, 3, 0, 3], [-4, -1, 3, 3, 0, -2, -2, -2, -1, -1, 1, 0], [-4, -1, 1, 4, 0, 1, -5, 2, 3, -3, 2, 0], [3, 3, -3, 1, -3, 5, 0, 3, 0, 2, -3, -1], [9, 3, 0, 5, -1, 3, 1, -1, -4, 0, -3, -5], [8, -4, 0, -2, 2, -2, 3, 0, -2, 1, -3, 3], [2, 1, 1, 1, -2, -3, -3, 0, 3, -1, -3, 0], [-3, -6, -1, -2, 3, -1, 3, 1, -1, 1, -1, -1], [-2, 0, -1, -4, 0, 2, -3, -2, 2, 0, 0, 0], [9, -3, -1, 6, 1, 6, -1, 1, 2, 0, 1, 1], [-10, 4, 0, -1, 2, 4, -1, 2, 2, -3, 3, -1], [2, -1, 0, 1, 2, 0, 1, 1, -2, 1, -3, -1], [-2, 3, -1, 1, -1, -2, -1, 2, 5, -1, -1, -1], [-5, 2, 1, 2, 2, -6, -1, 1, 7, -5, 2, 2], [-2, -5, -1, 1, 0, 0, 1, -1, 0, 2, 2, 0], [-2, -1, 0, 3, 0, 0, -1, 0, 0, -3, 2, -2], [-4, -6, -1, -2, 2, -2, 1, -2, -2, 2, 4, 4], [-8, -2, -1, -2, -1, -4, 0, 3, 4, -1, -2, 0], [1, 2, -1, 0, -4, 1, -2, 0, 0, 1, -4, -1], [8, -3, 1, 6, 2, 3, -2, 0, 2, 1, 2, 3], [-10, 0, 3, -1, 0, -3, 3, 0, -2, -1, 4, 1], [3, -8, 1, 4, 4, -1, 1, 1, 2, 0, 2, 1], [-9, 3, 1, 0, 0, 4, 0, 1, 0, 2, -1, -2], [-8, -3, 0, -1, 1, -6, -1, -2, 5, 0, 2, 4], [4, -3, 1, -3, 4, -4, 1, -2, -2, -1, 2, 3], [3, 0, -2, 5, 0, 0, 2, 1, 0, -2, 3, 0], [1, -6, 0, 2, 2, -4, -1, 1, 3, -1, 2, 5], [1, 0, -1, -5, -1, 0, 6, -3, -5, 3, -2, 1], [4, -2, 3, 1, -2, -2, 4, -4, -3, 1, 2, 1], [2, -4, 1, -1, 0, -2, -3, -2, 1, -3, -5, -4], [3, 0, -1, 3, -3, 6, -2, 1, 0, 1, 2, 3], [-4, 2, 0, -2, -1, 0, 2, 1, -2, 2, 1, -1], [-15, -1, -2, -2, -1, 0, 1, -1, 6, -1, 5, 2], [8, -2, -2, -3, -3, -3, -2, 1, 2, 3, -1, -2], [-1, 0, -1, 4, -3, -3, -2, 1, 2, 0, -2, 2], [7, 3, 0, 3, -1, 6, 2, 3, -2, 4, -3, -3], [5, -5, 2, 2, 4, 0, 3, -4, 0, 3, 1, 1], [1, 1, 2, 4, 2, 0, 2, -3, 0, -1, 1, 1], [-6, -3, -2, 2, 2, 7, 1, 0, -2, -2, 4, 5], [4, -1, -2, 1, 1, 4, -5, 3, 2, -4, -3, 1], [3, 1, -1, 0, -1, 6, -2, 5, -1, -2, -2, -4], [3, 5, -2, -5, 2, -1, 3, -5, 2, 3, 1, 1], [-10, 5, 1, -3, 2, -4, 2, 0, 0, 2, 2, 2], [-4, -1, 2, -4, -4, -3, -1, -4, -2, 2, -2, -3], [2, 4, 1, 0, -2, 0, 0, 0, 4, -3, 5, 1], [-5, -4, -1, 7, 5, 4, -4, 6, -2, -7, 4, -2], [-1, -1, 2, 0, -1, 2, -2, -1, 0, -2, 0, -1], [-2, 0, -2, 0, -3, 0, 2, 1, 4, 2, -3, -1], [-5, 7, 0, 3, -4, 7, -5, 3, 2, -2, 1, -1], [5, -1, -3, -3, 0, -3, 5, -5, -6, 0, -2, 0], [-7, 3, 0, -3, 0, -1, 2, -1, 4, 0, 3, 6], [3, -10, -2, 3, 2, 4, 0, 3, -2, -1, 0, 2], [4, -1, -1, 3, -3, -2, -2, -2, 2, -1, -2, 2], [4, 0, -1, -3, 3, -1, 0, 0, 0, -3, -2, -2], [5, 2, -4, 2, -2, 5, -4, 6, 0, -2, -7, -4], [-12, 8, -2, 4, 0, 2, -2, 2, 3, -3, -2, -4], [-2, 1, -1, -1, -5, -4, 3, -4, 4, 1, 0, -1], [0, -8, 0, 3, 3, 3, 0, -2, 4, 6, 6, 9], [-3, 3, 2, -4, -2, -6, -4, 5, 6, -5, 0, -3], [3, 3, 1, -3, -3, -6, 1, -1, 1, -1, -5, -1], [-4, 5, -3, -4, 0, -1, 0, -4, 0, 3, 3, 3], [3, 4, -1, -3, -1, -2, 5, -3, 0, 3, -2, 0], [-9, 2, -3, 2, -1, -1, 0, 3, -3, 0, -1, 2], [0, -2, 2, 3, -1, -1, -2, -2, 0, -2, 0, -1], [-1, 1, 3, 2, 3, 0, 3, -3, -8, 2, 0, 1], [7, 0, -3, 5, 0, 4, -5, 7, 2, 0, -4, 2], [1, -2, 3, 5, 0, -4, 2, -5, -4, 0, 3, 3], [-6, 3, 0, 3, -5, 4, -2, 4, 0, 2, -1, -3], [5, 4, -4, 1, 1, -1, 1, 3, 1, -4, -3, -4], [1, 0, 1, -3, 2, -2, 0, -1, -4, 3, -3, -1], [-9, 7, 2, 3, -1, -5, 3, 3, 2, -4, -1, -1], [3, -3, 1, -3, 3, -4, 0, -1, 0, -3, 6, 4], [3, 0, -1, 1, 0, 4, 1, 1, 0, 1, -2, 4], [-4, 4, -2, 0, -2, 2, 2, 4, 0, -2, -3, -2], [-3, 0, 3, -5, 2, -2, 1, -3, -6, 5, 1, 2], [2, -4, -1, -9, -3, -3, 6, -6, 0, 5, 1, 4], [-4, 1, -2, 5, -1, 0, 1, -2, -2, -3, -2, -6], [18, -3, -1, 4, 3, 6, 2, 2, 1, 0, 1, 4], [10, 5, 0, -5, -2, -2, 4, -2, -4, 5, -4, -1], [5, 1, -1, 2, 1, 2, 0, -1, 2, -1, 3, 6], [15, -3, 0, 4, -4, 1, 3, -5, -5, 2, -5, -1], [-2, -3, -1, 5, 7, 6, 0, -2, 1, -2, 8, 2], [10, -10, 1, 0, 4, -2, 1, 0, 1, 3, 0, 8], [1, -11, 1, 4, 2, -3, -4, -1, -3, -3, 2, 3], [-12, 2, -6, 2, -4, 4, -1, 2, 0, -1, -5, -1], [3, 7, 2, 4, -2, 9, -4, 5, -3, -1, -3, -4], [18, -3, 0, 2, -4, 5, 0, 2, -4, 6, -5, -4], [0, -6, 1, -1, 1, 7, -3, -2, -6, 6, -2, 1], [1, -4, -4, 7, 3, 2, 0, 2, 6, 0, 0, 9], [6, -1, -5, 3, 0, -1, -5, 6, 3, -5, -1, -2], [3, 3, 0, -5, -2, 1, 6, -3, -6, 6, -2, -2], [-14, 4, 2, -2, 2, 4, -2, 2, 4, -2, 2, -2], [2, 0, 4, 4, 0, -2, 1, 0, 2, 3, 1, 5], [8, -3, -2, 3, 0, 6, -7, 4, 3, -5, 3, 0], [-2, -4, -2, 9, 7, 9, -4, 6, 0, 0, 1, -1], [6, -1, 0, 1, 2, -5, 4, 2, -3, -3, -1, 0], [3, -3, 3, 1, -2, -5, 2, -3, -2, 5, 0, -1], [2, -2, 1, 0, 4, -6, -4, 2, 2, -2, 2, 1], [-3, -3, -1, -3, 0, 1, 3, -1, -6, 2, -1, 2], [16, 2, 1, 0, -1, 0, -2, 2, -4, 0, -4, -1], [-6, 1, -1, 3, 0, 0, 2, 1, -6, 1, -1, -1], [14, -3, -1, -4, -2, -3, 7, -6, -4, 2, 2, 6], [-15, -3, 0, -3, 1, 2, -1, 5, -1, -3, 5, -4], [-6, 2, -5, -4, 0, 4, -1, 0, 2, 3, -1, -2], [1, 0, -2, -3, 0, 4, -4, 1, 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1, 2, 3, -8], [-13, 3, 0, 1, -4, 3, -5, 5, -6, -3, 4, -3], [-3, -7, 1, 8, 11, 6, 0, 0, -2, -4, 4, 3], [-13, 4, 0, 0, -5, 3, -8, 7, 5, -5, 4, -1], [9, -9, -2, -6, 3, -10, 8, -3, -6, 2, 3, 7], [-14, 7, 1, 7, 1, 5, -1, -1, -9, 5, -8, -7], [9, -10, 2, 4, 4, 8, 5, -5, -9, 4, -3, 1], [25, -6, 1, 10, 4, -2, 1, -3, 10, 2, 4, 4], [-28, 9, 1, -8, 1, -10, 7, -4, 7, 5, 9, 5], [14, 6, -1, 0, -6, -6, 8, -2, -3, 0, -2, -1], [-6, -1, -3, 3, -5, 6, 4, 2, -4, 0, -2, 2], [0, 7, -2, -13, -1, 4, -1, -6, -2, 5, 2, 2], [34, -11, -2, -1, -3, -4, 3, -4, -2, 5, 0, 4], [-3, 0, 1, 8, 4, -1, 0, 6, 6, -5, 10, -1], [-1, 0, 1, 5, 0, -8, 0, 9, 4, -4, 6, -5], [-10, -9, 5, 9, 7, 6, 3, -3, -14, 9, -3, -2], [10, -5, 4, -4, 0, -6, 15, -9, -9, 4, -1, -7], [24, 1, 4, 1, 1, 0, -1, 0, 2, -3, 9, 6], [-4, -1, 0, -6, 2, -1, -7, 2, 2, -6, 0, -5], [-9, 7, 1, -6, 5, -10, -4, 3, 3, -6, 6, 6], [-6, 0, -3, 6, 2, 4, -3, 10, -6, -2, 2, -4], [4, -3, -2, 13, 8, 2, 0, 3, 2, -2, 0, -2], [13, 6, -1, -4, -3, -5, 3, -1, -6, 3, -7, -5], [-6, 8, -1, 6, 4, -4, -1, -8, 6, -4, 4, 2], [14, 6, 1, -6, 1, 4, -5, -2, 2, 0, 3, 6], [3, -15, 8, -3, -1, -6, 0, -3, -5, 5, 3, 7], [2, 3, 5, -3, -2, -8, 2, -6, 4, 0, -1, 0], [0, 4, -2, 9, -1, -9, 2, -2, 2, -1, 2, 1], [7, 3, 0, -6, -3, -10, 6, 1, 2, 4, -2, -1], [-1, 12, -5, -2, -3, -1, 6, 4, 2, 4, 0, -5], [-33, 3, 2, -4, -1, 0, 2, -3, 2, -2, -6, -5], [8, 4, 0, 6, -4, 6, -2, -10, -4, -8, 0, -4], [12, -11, -1, 3, 1, -10, 1, 2, 4, 0, 5, 5], [-3, 7, 2, 2, -4, -7, -1, 3, -3, 2, -9, -13], [33, -17, -3, 8, 8, 11, 3, 1, -1, 7, -5, 8], [13, 3, -3, -7, -1, -5, -1, -1, -3, 2, -4, 8], [5, -7, 5, 7, 8, 3, -6, 5, 2, 7, 0, 7], [12, -7, 2, -6, 4, -5, 4, -3, -4, 1, -2, -9], [-14, 12, -3, 0, 3, 1, -2, 6, 5, -8, 5, -4], [16, -6, 1, -3, -3, -1, 5, -8, -2, 0, -1, -1], [27, -1, 6, -9, 5, -6, -1, -5, 3, 0, 4, 10], [2, -6, 6, -1, 3, -5, -7, 2, 8, 1, 2, 2], [-12, 1, 4, -9, -3, -2, -8, 4, -6, 0, -1, -3], [-10, 0, -1, -15, -4, -5, -1, 0, -4, 1, -12, -9], [29, 1, 4, 10, 7, 2, -9, 7, 0, -3, 3, 4], [13, -10, -1, 15, 4, 6, -9, 15, 8, -8, -4, -2], [-35, 10, 2, -5, -5, 6, -3, -5, -10, 4, -3, -9], [30, -6, 0, -4, 4, -10, 7, -4, 5, 2, 5, 5], [19, -12, 1, 6, 6, 1, -5, 3, 3, 2, 1, -1], [32, -7, -4, 0, -2, 5, -1, 6, -10, -2, -5, -9], [-3, -3, 2, 2, 7, 2, 1, 3, 7, -10, 10, 6], [4, 2, 1, 1, 1, -5, 8, -4, -2, -3, 6, 6], [1, -6, 2, -11, 2, -8, 12, -9, -4, 6, 4, 6], [-13, -5, -1, -1, -2, -9, 0, -3, 1, -2, -1, 5], [22, 8, -1, -5, -13, -3, 8, -2, -8, -3, -10, -8], [-1, 2, -7, 10, 4, 0, -3, 11, 1, -1, -7, -4], [-4, 3, 1, -11, -3, 0, 1, 2, -8, 4, 0, -1], [3, -11, -1, 6, 9, 0, 4, 1, 3, 8, 9, 14], [9, 1, 4, 6, -5, 0, -12, 3, 6, -2, 0, -9], [12, 6, -8, -4, -5, -7, 2, 2, -1, -7, -6, -7], [7, 0, 4, -3, 1, 0, 9, -10, -12, 13, -4, -4], [12, 4, 2, -11, 3, -1, 3, -3, 0, 2, 12, 7], [-11, 3, 2, 5, -8, 1, -3, 1, 8, -5, -6, -3], [-4, 5, 1, 15, 5, 10, -9, 6, -4, -4, -2, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6018_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6018_2_a_ba();".
// 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_6018_2_a_ba(:prec:=12)
    chi := MakeCharacter_6018_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_6018_2_a_ba();".
// 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_6018_2_a_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6018_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1056, 6688, 8812, -8180, -9364, 5783, 2550, -1695, -180, 199, -9, -8, 1]>,<7,R![-9932, 16812, 11101, -23458, -2758, 11381, -655, -2324, 314, 196, -35, -5, 1]>],Snew);
    return Vf;
end function;