
// Make newform 6018.2.a.z in Magma, downloaded from the LMFDB on 24 April 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_z();".
// 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_z();".
// 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 := [1200, -692, -4372, 562, 3403, -484, -1023, 200, 117, -27, -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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-6530016, -30704680, -16641720, -1530071, 41682174, 12213703, -16451050, -2438247, 1972995, 100902, -68377], [-277474058, 17946790, 966550127, 301247970, -599066657, -155628260, 147883587, 21958815, -14837488, -806963, 478220], [437533206, -115397832, -1555875710, -362823799, 944194472, 200728579, -224589340, -28769331, 21837033, 1061578, -690399], [1824506128, -230680552, -6482502258, -1934315723, 3999281264, 1024667347, -981973340, -145767653, 98175115, 5380560, -3156869], [-599619469, 65727972, 2258150935, 728576639, -1432388663, -387860873, 359948269, 55439219, -36518360, -2051153, 1183567], [-2734225624, 558006780, 9978785870, 2853612557, -6227037308, -1563255225, 1539701372, 224826723, -154431069, -8328764, 4973375], [-894863368, 108639160, 3317249635, 1067360641, -2085306979, -567120079, 521178581, 80817374, -52746178, -2986333, 1707617], [-5650580696, 615967628, 20866262830, 6833246697, -13063793500, -3608727329, 3262430232, 513011107, -330397033, -18943456, 10702739], [10921660816, -1489912936, -40564710178, -12726418695, 25332890104, 6746297163, -6282554076, -959470865, 632398343, 35419276, -20411677]];
    Rf_basisdens := [1, 1, 9881348, 4940674, 4940674, 9881348, 2470337, 9881348, 2470337, 9881348, 9881348];
    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], [-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, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, -1, 1, -1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0], [2, 0, 0, 0, 1, 0, -1, 1, 0, 0, -1], [1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 1], [2, 0, -2, 2, -2, -1, 0, -2, 0, -1, 0], [1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0], [1, 0, -1, -1, 0, 0, 0, 1, -1, 2, 0], [0, 0, 1, -1, 1, 1, 0, 1, -2, 1, 0], [1, 0, 0, -1, 0, 0, 0, 0, -1, 1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, -1, -1, 0, 1, -1, 0, 0, 0], [2, 1, 1, -1, 2, -1, -1, 1, 1, 0, 1], [2, -1, 0, 0, 0, -1, 0, 0, 0, 0, -1], [3, 1, 2, 0, 2, 1, -3, 2, 2, -1, -1], [1, -1, -1, -1, -1, 0, 1, -1, 1, 0, 0], [1, 0, 0, 2, -1, 1, 0, 0, -1, 2, 0], [0, 2, 0, 0, 0, -1, 0, 0, 1, 1, -1], [2, 2, 1, -1, 1, -2, 0, -2, 2, -2, 0], [0, 0, -1, 3, -1, -1, 0, 0, 1, -2, 0], [2, -1, -1, 0, 2, 0, 0, 0, 1, 1, -1], [3, -2, 0, 0, -1, -1, 0, 1, 1, 1, 0], [5, -1, 4, -2, 1, 1, 0, 1, 0, 0, 0], [3, -2, -1, -1, 0, 0, -2, 0, 0, 0, 1], [3, 0, 1, -1, 0, 0, 0, -1, 0, -1, -1], [1, 2, 0, 0, 2, -1, -3, 0, 2, -1, 0], [4, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0], [3, -1, 2, 0, -1, 1, 0, 1, 2, -1, -1], [2, 0, 1, -3, 1, 1, 0, -1, 1, 2, 1], [-3, 0, -1, 2, 1, -1, -2, 0, 0, -3, 0], [0, 0, 2, -1, 3, 0, 0, 1, 2, 0, 0], [3, 0, 2, -2, 4, 0, -1, 2, 3, 1, -1], [2, 1, 3, -1, 1, 2, 0, -1, 2, -1, 1], [1, -5, 0, 0, -1, 2, 0, 2, -4, 2, 0], [1, -2, -1, 1, 2, -1, -2, -1, 0, -1, 1], [1, -1, -4, 2, -1, -2, 2, -3, 0, -1, 0], [-3, 0, -3, 3, -2, 1, 0, -3, -1, 1, 0], [7, 1, 2, 0, 1, 1, 0, 2, 1, 0, -1], [3, -1, 2, 0, 0, 3, 1, 0, -1, 1, 0], [5, 1, 4, -1, 0, 3, 0, 1, -2, 2, 1], [4, 0, 0, 2, 2, 0, 0, 1, -2, 3, 0], [-3, -1, 0, 0, -5, 1, 2, -2, -1, -2, -1], [1, -2, 0, 0, -1, 2, 0, -1, -2, 4, 1], [0, -1, 1, 0, -2, 2, 0, 2, -4, 1, 1], [4, 3, -1, 1, -3, 1, 0, 1, -1, 1, 0], [4, 0, -5, 3, -1, -2, 0, -1, -2, 1, -1], [1, -2, 1, -1, -2, -2, 2, -3, 0, -1, 3], [1, -2, 0, -2, 2, 0, -1, 5, 1, 2, -1], [1, -1, 0, 0, 1, 2, -2, -1, 0, 1, 0], [-6, 0, -1, -2, -2, 0, 2, -3, -1, 0, 1], [-4, 3, 0, -1, 1, -1, 0, -4, 2, -1, 1], [1, -1, 1, -1, 4, -1, -2, 4, 0, 0, 0], [9, -5, 1, 2, 1, 1, -2, 4, 0, 1, 0], [2, 2, -1, -1, -3, 0, 4, 0, -3, 0, -1], [1, 0, -3, 1, 0, -1, 0, 0, 2, -2, 1], [-2, 1, 0, 0, 0, 2, -2, 1, 1, -1, 0], [6, -2, 1, -3, 1, -2, 0, 0, 2, 0, 0], [0, -1, -4, 2, 0, 0, -4, 0, -1, -2, 0], [5, 1, 1, -3, 6, -3, 0, -2, 1, 0, 0], [10, -3, -4, 6, -2, 0, 0, -2, -2, -1, -1], [-7, -1, -6, 0, -3, -1, 0, -4, 0, -2, 2], [9, 0, 0, 0, 1, 0, 2, -1, 1, 3, 1], [7, 1, 3, -1, 0, -1, 2, -1, -1, 0, -1], [3, 1, 2, 1, 0, -1, 0, 3, 3, -6, -1], [2, 4, -1, 1, 4, -1, -3, 3, 3, -4, -2], [-2, 2, -1, 4, 3, 0, -3, -2, 5, 0, -2], [0, 0, -2, 2, 2, 1, -2, 2, -1, 4, 1], [6, -5, -4, 2, -4, 0, 2, -1, -6, 3, 2], [6, -6, 0, 0, -2, 2, 0, 0, -3, 2, -2], [4, 2, 1, 0, 2, -4, 0, 0, 6, -2, 0], [1, -2, -1, -1, -4, 1, 2, 2, -1, 4, 0], [7, -2, 3, -1, 4, 1, 0, 5, -2, -1, 0], [1, 3, 0, 2, -1, 0, 2, -1, 2, -3, -2], [0, -1, 3, -3, 3, -1, 2, -2, 3, 0, 2], [3, -3, -2, 0, -3, -1, 2, -1, -2, 1, 3], [12, -1, 5, 0, -4, 4, 2, 1, -2, 2, -1], [-4, -2, -4, -3, -1, 0, -4, 1, -3, 7, 1], [0, 1, -1, 5, 3, 0, -4, 1, 2, -4, -1], [-2, 6, 4, 0, 2, -1, -2, 0, 4, -5, -2], [4, -3, 2, 2, 0, 2, -2, 3, -4, 1, 1], [2, 1, -4, 0, 2, -5, -2, 0, 1, 2, -2], [3, -2, 1, -1, -4, 3, 2, -3, -5, 1, 2], [-6, -1, 3, -6, 0, -3, 2, -1, 1, -2, 2], [-6, -1, 1, 1, -3, 1, 4, 0, -2, -3, 1], [-8, 0, 6, 0, 2, 1, -2, 5, 5, -3, -1], [-16, -1, -1, 3, -5, 1, 4, -3, -1, -1, 2], [5, -2, 2, -5, 4, 3, -2, 3, -1, 5, 0], [-5, -1, 2, -4, 5, -1, -2, 1, 4, -3, 1], [-7, -2, -2, 2, -1, -1, 2, 2, -3, -1, -2], [-5, 1, 2, -1, -4, 1, 2, 0, 1, -5, 1], [-6, -1, -1, 1, -3, 3, -2, -1, 0, 1, 0], [-1, -5, -9, 5, 1, -1, -3, 1, 1, -2, -2], [2, -6, -2, 2, 2, -1, -2, 1, -3, 1, 1], [7, -4, 0, 4, -5, 2, 4, -2, -2, 2, 0], [-2, 1, -3, 1, 1, -2, 0, -3, -1, 1, -1], [-6, 0, -3, -1, 1, 4, 2, 3, -3, 3, 1], [-5, -5, -4, 4, -3, -1, -2, -1, -2, -1, 0], [-5, -1, 1, 1, 1, 1, 1, -2, -1, -3, 0], [-1, -2, -8, 2, -5, 0, 2, -2, -2, 0, 1], [0, -2, 3, 1, 3, -3, 0, 0, 1, -3, -3], [0, 6, 4, -2, 2, -1, 0, -1, 2, 5, 0], [-2, 1, -3, 3, 5, -2, -4, 3, 0, -1, -1], [-10, -2, -5, 4, -1, -3, 1, -2, 2, -6, 1], [4, -4, 0, 0, -4, 0, 4, -3, -2, 4, 0], [0, 8, 2, -1, 0, 0, -3, 3, 0, -3, -1], [7, -4, 1, 0, 5, 1, -4, 1, 1, 0, 2], [-3, 3, 0, -2, 2, 0, -3, 3, 3, 0, 0], [-2, 0, 5, -5, 1, 0, -2, 2, 2, 2, 1], [-7, 0, 0, 0, -1, 1, -2, -2, 6, -2, 1], [-3, 3, -1, -1, -4, 0, 4, -4, 1, 2, 2], [-2, 6, 2, -6, 3, 0, 3, -1, 0, 0, -1], [-2, 3, 6, -3, 9, 0, -2, 6, 4, 0, -1], [1, -3, -1, 5, -2, -1, 2, -7, 1, -9, 0], [-8, 1, 3, -5, 1, -6, 4, -3, 3, -3, 0], [0, 8, -1, 1, 6, -5, -3, -3, 8, -4, -1], [3, 1, 4, 2, -5, 3, 0, 0, -1, -6, 2], [-6, 4, 2, 0, -2, -4, 2, -1, 3, -4, -3], [-8, -3, 4, -4, 2, 5, -2, 3, -4, 7, 1], [-1, 5, 3, 3, 2, 2, -2, 3, 2, -3, -2], [-2, 4, -7, 3, 0, -5, -1, -1, -2, 0, -1], [-7, 2, -1, 1, 2, 0, -4, -2, 0, -2, 1], [-15, -2, 3, 1, -2, 0, 2, 2, -1, -2, 1], [-2, -6, 1, -5, -7, 2, 6, -1, -7, 5, 1], [2, 2, 3, -3, 1, -4, -2, -2, 4, -4, 1], [-8, 7, 0, 0, 1, -3, 5, -4, 4, -1, 2], [-7, 1, -5, -1, 2, -3, -2, -5, 0, -1, 0], [-5, 4, 7, -5, 6, 0, 0, -1, 4, -1, 1], [10, 4, -3, -3, 2, -4, 3, -3, -1, 2, 0], [-3, -1, -3, 3, 2, 2, 2, -2, 1, 2, -3], [-12, -2, 3, -1, -1, 2, 0, -2, -7, 6, 1], [-5, -5, -2, 0, -5, 0, 6, -5, -7, 1, 2], [-1, -1, -8, 2, -5, -6, 4, -5, 1, -4, 1], [0, -4, -6, 4, -3, -3, 3, -1, -1, -2, -2], [1, 1, 0, 0, 7, 0, 0, 5, 4, -3, 0], [10, 0, 7, -5, -1, 2, 0, -2, 2, 0, 2], [-11, 0, -4, 1, -4, -1, -4, 0, 0, 3, 1], [-10, 3, 3, 5, -5, 2, 0, 0, -6, -4, 1], [0, -2, 3, -1, 1, -1, 0, 2, -5, 2, 3], [9, 0, 8, -2, 5, -2, 2, 2, 0, 5, -1], [-3, 3, 6, 2, 3, -2, 0, 3, 6, -9, 0], [-8, 2, -5, 7, -3, -1, -2, 2, -6, 0, -2], [-2, 0, -3, 1, -1, -1, 0, -7, 0, -5, 2], [-3, 4, -2, 2, -4, 0, 1, -1, -3, -4, 1], [-6, -1, -6, 8, -4, 0, 2, -8, 1, -2, 2], [11, 2, 7, -4, 9, -1, -2, 7, 1, 0, -2], [-9, -2, -2, -2, 1, -4, 2, 0, -5, 6, 0], [-6, 7, -2, 0, -2, -5, 4, -8, 3, -6, 2], [-7, -9, 3, -2, 1, 6, -2, 4, -4, 2, 2], [16, 0, 7, -5, 2, -3, 1, 0, 4, -3, -1], [-6, -1, -4, -4, 0, 2, 0, -1, -7, 3, 4], [-4, 5, 3, -1, -6, -3, 3, -3, 1, -2, 3], [18, -5, 5, -5, -1, 0, -2, -1, 2, -3, 1], [7, -5, 1, 3, -4, -1, 2, 1, -2, 5, 0], [4, 3, 4, -1, 5, 0, -4, 6, 1, 3, -2], [12, 2, 4, -6, 0, 2, 2, 2, -1, 8, 1], [7, 0, 0, 0, 1, 0, -4, 3, 3, 1, -2], [-7, 0, -7, 6, 1, 1, -2, 4, 1, 1, -2], [-12, 0, -1, 1, 3, 0, -2, -3, 8, -2, -2], [-7, -5, 6, 0, 5, -1, -4, 2, 2, 3, 0], [-7, 1, 11, -2, 0, 0, -1, 3, -1, 5, 2], [7, -2, -3, -1, -6, 3, 4, -1, -9, 3, 2], [4, 5, -2, -5, 1, 2, 0, 0, -5, 3, 2], [15, 0, 0, 2, 7, 4, -8, 1, 5, -3, 0], [-1, -1, -8, 0, -1, -3, -2, 0, -1, 4, 1], [-12, 7, 8, 0, 4, 2, -4, -1, 4, -3, 0], [-6, -7, -11, 4, -2, -5, 2, 1, -1, -2, 0], [-9, -1, 7, -11, 0, 4, 0, 2, 3, 0, 1], [2, -8, -8, 2, -12, -2, 6, -8, -6, 2, 2], [-19, 4, 3, -1, -2, 1, 4, -5, 5, -3, 0], [-8, -1, 7, -3, 11, 1, -4, 10, 1, 1, -2], [1, 4, -2, 2, 5, -2, -2, 4, -5, 2, 0], [4, -1, -3, -4, -2, 6, 2, 0, -2, 5, 3], [4, 0, 4, 0, -8, 2, 8, 1, -4, 3, 2], [-19, -3, -5, -1, -5, 2, -1, -2, 1, -1, 0], [-9, -2, -3, 1, -6, -1, 0, -2, -5, 4, -2], [1, -9, -9, 7, -2, -4, 0, 0, -2, 0, -2], [9, -3, -2, 2, 3, 0, 0, 2, 5, -4, -6], [1, -3, -12, 10, -1, -3, -2, -3, -1, -1, -3], [-14, -3, -5, 4, -1, -5, -3, 4, 3, -2, -3], [-6, 5, -2, 0, 3, -7, 3, -1, 4, 0, -2], [10, -9, -7, -1, -7, -4, 2, -7, 0, -3, 1], [-5, -7, 8, -6, 2, 3, 1, 5, -3, 0, -2], [5, 0, -1, 3, 2, -1, -2, 0, -1, -2, 1], [-10, 6, 0, 1, 1, 1, 2, -6, -2, -3, 1], [7, -10, -1, -1, 0, 1, 0, 8, -2, 7, 1], [4, 2, 0, -6, 0, 4, 6, 1, -4, 11, 0], [-4, -9, -1, -5, -3, 2, -2, 3, 1, 6, 4], [-7, 3, 6, -7, 0, -3, -2, 3, 3, -2, 1], [-17, -8, -6, 2, -9, -1, 6, -3, -6, 5, -1], [1, -6, 0, -2, -5, -3, 2, -5, 6, -1, 1], [-1, -2, -3, -5, -2, -1, 0, 2, -5, 2, 6], [-8, -3, -3, -5, 3, 0, -2, -5, 8, -5, -3], [-5, -1, 6, -6, -3, 0, 6, 5, -6, 3, 2], [-13, -4, 8, -2, -5, 5, 6, -1, -2, 7, 3], [-24, 2, -5, 3, 1, -5, 4, -9, 5, -9, 2], [-2, 0, 1, -3, 3, -5, -2, 4, 3, -2, 4], [6, -3, -9, 7, -16, 5, 3, -3, -7, -2, 1], [-8, 2, -1, 1, 7, -3, -2, 0, 6, 0, -2], [-3, -4, -3, -1, 4, -2, -2, 4, -1, 4, 1], [-4, 3, 2, 0, 6, 0, 2, -3, 10, -8, -1], [-8, 3, -2, -4, 6, -3, 0, 3, 2, 9, 3], [-1, 0, 10, 2, 3, -3, 2, 3, 9, -5, -2], [-8, -8, -10, 10, -8, -1, -4, -3, -5, -3, 1], [3, -5, -6, 4, -7, 2, -2, 4, -3, 2, 0], [17, -5, 6, -2, 0, 4, 3, -1, -2, -4, 3], [6, -4, -9, 0, 4, -4, -4, 6, -3, 1, -1], [-1, -2, -5, 2, 3, 1, 0, -2, -1, 1, -2], [-2, -2, 0, 6, -10, 3, 2, -8, -4, -4, -1], [0, -9, -6, 0, 0, 4, -2, 7, -9, 9, -2], [9, 4, -2, 10, -5, -3, 6, -5, 0, -3, -2], [0, -3, -3, 1, -1, -7, 0, -2, 5, -6, 0], [-4, 3, 11, -9, 3, -1, 4, -4, 3, 2, 4], [-3, 2, 0, -4, 11, -2, -4, 6, 6, -1, -1], [-8, 1, -6, 3, 1, -2, -2, -8, 7, -1, 0], [9, -3, 5, -2, -5, 4, 6, 6, -5, 5, 3], [-4, 5, 8, -2, 4, 2, -4, 0, 3, 0, -2], [-8, 5, 0, -8, 8, -3, 0, 0, 2, 1, -1], [-7, -3, -2, 0, -5, -2, 6, 2, -3, 0, -4], [-5, 2, 1, 9, -4, -1, 2, 5, -6, -1, 1], [2, -4, 5, -4, 0, -3, 4, -3, 0, -8, -2], [-5, 3, 10, -12, 1, 4, 2, 12, -5, 11, 1], [-9, -15, -5, 2, -7, 7, 2, 5, -12, 10, 0], [-1, -3, -12, 5, 2, -1, -4, 2, -4, 3, -1], [6, -6, 5, 3, -5, 5, 0, 3, -1, -1, -1], [0, 3, -10, 1, -5, -3, 2, 0, -8, 1, 1], [4, 5, -2, 4, -2, -2, -6, -2, 4, -6, -1], [26, -4, 8, 1, 1, 7, 4, 3, -9, 6, 1], [-4, -3, 7, -5, 7, 2, 0, 11, -4, 1, -2], [-7, 5, 6, 0, 5, 3, -2, -1, 4, 4, 2], [3, -3, 5, 4, 0, 1, -3, 1, -6, -5, 3], [-18, 7, -1, -1, 1, -5, 0, -7, 8, -3, 2], [-24, -4, -2, 3, 1, -2, 0, 5, -2, -6, 0], [5, -3, -2, -8, 0, 4, -1, 8, -12, 11, 1], [-25, -4, -5, 2, -7, 3, -4, 3, -3, 2, -2], [5, -3, -2, 4, -3, 0, -8, 4, 1, 0, -1], [2, -5, -3, 3, 3, 3, -2, 3, -4, 8, -1], [-12, 6, -1, 7, -1, 0, 2, -4, 5, 0, -1], [13, -2, 2, -2, -11, 4, 4, -3, -6, -1, 4], [9, -2, -1, 5, -5, 1, 1, -1, 0, -2, 2], [-14, 11, -1, 1, -1, 1, -4, -6, 3, -2, 2], [17, 5, 5, 1, 0, 3, -2, 3, 0, 1, -2], [-10, 9, 3, -5, 6, -2, -5, -1, 8, -4, -4], [-17, 11, -1, -3, 4, -5, -6, -3, 11, -3, 1], [-5, -2, -2, -5, 4, 7, -4, 5, -9, 8, -1], [4, 4, -6, 10, 7, 2, -9, 2, 6, -1, -3], [0, 5, -5, 3, 1, -3, -6, -4, 7, 2, -2], [13, 4, 8, -6, 3, -7, -4, -2, 4, -8, 3], [-6, -2, -4, 6, -8, 0, 6, -6, -4, -2, -2], [-17, 2, -15, 8, -7, -3, 0, -6, 3, -5, 0], [-8, 2, -2, -6, 2, 3, -2, 0, -4, 5, 2], [0, 0, 3, -8, -4, 5, 8, 4, -8, 5, 4], [15, 1, 7, -1, 2, -3, -2, 12, 0, 2, -2], [5, 5, 3, 0, 1, 4, 0, -1, -5, 4, -3], [18, -8, 1, 1, -10, 5, 7, 2, -5, 9, -2], [-5, 5, 0, 2, -7, -2, 10, -5, -2, -1, 2], [23, 5, 7, -5, 4, -7, 2, 2, 7, -7, -3], [5, 6, 2, 3, 4, 5, -6, -2, 6, -5, -1], [4, 5, 3, -6, 4, -4, -2, -3, 4, 2, 3], [-3, 4, 9, -7, 2, -2, 4, -6, 0, 2, 4], [7, 3, 1, 1, 6, -1, -4, 4, -3, -2, -2], [3, 15, 0, -4, 5, 4, -2, 5, 0, 1, 2], [-1, 1, 1, 1, -8, 0, 0, -6, -3, -6, 0], [7, -3, -2, 0, -3, 7, 0, -1, -2, 2, 2], [-5, -6, -11, 10, -7, 2, -2, 3, -4, -6, -3], [4, -1, -11, 11, -8, -1, 3, -2, 1, -5, -3], [-1, 7, 11, -9, 12, -2, -4, 5, 11, -3, 1], [-6, -8, -11, 1, -9, -5, 4, -2, -8, 4, 6], [10, 1, 2, 8, -12, 10, 6, -3, -3, -1, -2], [15, 4, 1, -3, 6, -3, -8, 6, 9, 5, 0], [-16, -5, 11, -5, -3, 9, 8, 6, -8, 9, 5], [22, -5, 7, -3, 3, 2, 0, 8, -7, 5, -2], [-3, 1, -1, 11, -4, 5, -4, 10, -9, 0, -6], [19, -13, -6, -2, -7, -2, 2, -5, 2, -7, 0], [12, -10, 2, -4, 0, 10, 8, 5, -15, 11, 0], [-1, -3, -12, 3, -10, -8, 6, -8, -3, -3, 2], [0, 4, -5, -5, 5, -8, 2, -8, -1, 4, -1], [-10, -6, -9, 13, -15, -1, 0, -2, -7, -2, 4], [18, -7, 6, -6, 2, 6, -2, 14, -15, 12, -4], [20, 0, 0, 4, 4, -3, 0, -8, 7, -8, 3], [6, 11, -3, 5, 7, -6, -2, -3, 2, -5, -5], [24, 4, 2, 0, 7, -4, 1, 1, -3, 0, -2], [18, 6, -2, 0, 2, 0, -4, -3, 8, -1, -2], [3, -5, -8, 3, -10, -7, -2, -13, 2, -6, -3], [-7, 4, 1, -1, 6, 3, -6, 4, 5, 2, 0], [3, -3, 10, -4, 7, -1, 2, 3, -3, 5, -3], [10, -7, 2, -2, 2, 0, -2, 3, 6, -1, 1], [7, -7, 14, -4, -3, 13, 0, 6, -5, 8, 3], [7, 6, -1, 9, 2, -4, -6, 1, 3, -6, -2], [9, 5, -4, 2, -3, -1, 0, -7, -1, -1, 3], [-7, -5, -10, -1, 2, 1, 4, -1, -3, 4, 1], [-10, -4, -5, -5, 5, -6, -6, 3, 3, 3, -1], [5, 10, -4, 3, 0, -5, 4, -3, 0, -8, -5], [0, 4, -2, 4, -6, -2, -4, 2, 3, -3, 1], [3, 7, 12, -12, 5, -5, 6, 2, 5, 0, 5], [-6, 5, 10, 4, 1, -9, 1, -2, 11, -15, 1], [9, -7, 0, -8, -3, -5, 0, -5, 2, 2, 4], [9, -1, 3, -1, -2, -6, 6, -5, 3, -3, 7], [-19, 11, 0, -2, -3, -7, 4, -1, 4, 0, 2], [11, 0, -5, 5, -6, -3, 4, 0, -10, 0, 0], [22, 4, 6, -4, 0, -3, 4, -6, 8, -10, 1], [24, -16, -1, 3, 0, 3, -1, 0, -4, -3, -5], [17, -6, -1, -5, -9, -2, 5, -6, -8, -1, 6], [-16, 4, -2, 2, 0, 6, -6, 2, -8, 8, 2], [13, 0, 11, -8, 3, 5, -4, 5, 5, -2, 4], [7, 6, 5, -9, 2, -2, 4, 0, 0, 0, 3], [-6, 4, 4, -10, -2, 8, 4, 0, 1, 3, 1], [17, -13, 11, -5, 2, 6, -2, 6, 0, 2, 3], [14, -1, 11, -3, 11, 0, -6, 12, -2, 8, -3], [11, 6, 2, -8, 9, 3, -10, 11, 1, 10, -2], [1, -14, 1, -1, -8, 8, 4, 3, -8, 5, 3], [-14, 12, 7, -1, -1, 3, 0, -7, 3, -7, -1], [1, 15, 5, -3, 10, -3, 0, -2, 6, -2, 0], [3, -6, -6, -2, 3, 2, -4, 7, -5, 7, 4], [6, 2, 5, -5, -7, -7, 12, -6, 0, 0, 4], [4, -11, -13, 7, -8, -7, -1, -4, -3, -7, -1], [13, -7, -2, 6, 1, 6, -6, 4, 1, 1, 1], [6, -13, 1, -8, 0, 9, 0, -1, -5, 12, 2], [-5, 9, -2, 4, 3, 3, -12, 2, 7, -4, -3], [31, -7, 10, -4, 3, 3, -4, -1, 4, 2, 0], [20, -2, -5, -1, -1, -3, 0, 4, -10, 4, -4], [26, -5, -5, 1, 5, 0, 0, 1, 4, -3, -3], [-6, 0, 5, -1, 7, -2, 0, -3, 5, -9, -1], [15, 1, -3, 5, 2, 2, -6, -4, 10, -9, 2], [25, 14, 2, 3, 2, -1, -4, -3, 4, -12, -3], [20, -2, -7, -3, 7, -3, -4, -2, 4, 0, 2], [4, -1, 6, 6, 8, 0, -4, 2, 5, -2, -6], [-16, 7, 1, -5, 11, -4, -4, 6, 4, 2, 3], [-1, 2, -2, 0, -3, 0, 6, -5, -8, 7, 6], [4, -1, -6, 6, -12, 1, 0, -8, 5, -9, 6], [4, 10, 0, 4, 0, -9, 0, -10, 4, -8, -4], [-3, 1, -12, 8, -9, 2, 0, -1, 1, -1, 4], [-17, 14, 0, -2, -3, -2, 2, -5, 11, -7, 4], [-7, -3, 6, 0, 1, 8, 0, 6, 0, 1, -2], [34, -3, 7, 1, -1, 3, 2, 1, -9, 3, -2], [-19, 5, 14, -4, 13, 0, 0, 1, 7, 0, -3], [3, -8, -14, 0, -21, 3, 6, -10, -12, 4, 4], [-14, -1, 2, -5, 1, 8, 6, 3, -9, 8, -2], [5, 9, 13, -3, 15, -2, -3, 0, 13, -5, -2], [2, 0, -7, 15, -15, 8, 4, -4, -6, -6, 1], [1, -2, 6, 2, 9, 3, -10, 10, 6, -10, -1], [18, -7, -2, 2, 2, 2, -8, 2, 2, -5, -7], [14, -3, -5, 1, 1, 0, 0, -6, -4, 8, -3], [-6, -6, 2, -1, 16, -7, -1, 10, -5, 6, 0], [20, -6, -1, 4, -2, -3, -6, -3, 0, -8, 0], [2, 15, 4, -4, 8, 5, 2, -1, -3, -1, -1], [6, -1, -16, 14, -6, 1, -8, -2, 0, -6, -3], [-1, 1, 8, -8, 11, 1, -4, 12, -1, 2, 1], [-3, 11, -13, -1, 8, -7, -4, 1, -4, 3, -2], [26, -1, 7, -3, -3, 1, 4, -3, -2, 12, -1], [-19, 1, 8, -10, -11, 0, 10, -7, -3, -8, -3], [-2, 5, 3, -1, -5, 0, 0, -2, -4, 4, 3], [-7, -2, 4, 2, -9, 2, 4, -1, -1, -7, 0], [8, 0, -10, 2, -14, -5, 6, -1, -7, 3, 4], [12, -4, 2, 2, -7, 2, 9, 2, -18, 9, -3], [-13, 5, 8, -8, 5, 4, -4, 5, -7, 4, 7], [4, -8, -5, -1, -3, -4, 6, 6, -9, 12, 0], [7, 7, 0, -1, -10, -4, 12, -6, 3, 4, 3], [10, -13, -2, 4, -19, 5, 11, -10, -4, -1, 4], [4, -4, -12, 0, -4, -2, -4, -8, -3, -6, 1], [13, 3, 1, -3, 10, 1, -8, 3, 2, 7, -2], [4, 9, 4, -9, 7, 5, -2, 7, 12, 2, 1], [-18, 2, 12, -4, -2, 3, 4, 12, -3, 10, -1], [7, -2, -3, -3, 6, -6, -2, -4, 2, 4, -6], [-22, 18, -1, 5, 7, -8, -4, -4, 18, -17, 2], [4, 7, 9, -1, 7, -4, -2, 4, 4, 4, 1], [-4, 1, 2, -5, -1, -5, 4, -1, -9, 8, 3], [26, -5, 1, -2, 0, -3, 2, 0, 9, -5, -2], [-28, -12, 10, -3, 1, 1, 0, 7, -2, -5, 6], [15, -2, -13, 6, -9, 1, 2, -10, 1, -7, 0], [-10, 4, 3, 3, 3, 8, -2, 14, -6, 7, -4], [5, 3, -7, 13, 9, -6, -11, 1, 7, -2, -4], [-11, -11, -1, 5, -12, 9, 6, -3, -4, -7, 2], [-21, -7, 5, 3, -4, -1, 2, 3, -10, -5, 2], [-8, -3, -1, -3, -3, -8, 4, -5, 5, -7, -1], [-5, -13, -2, -2, -2, -4, 9, 0, -4, 1, -1], [-16, 6, -5, -5, -1, -5, -4, -5, 4, -3, 3], [22, -16, -3, 4, -2, 7, 6, 2, -7, 7, -4], [-11, 8, -5, 1, -8, -3, 2, -12, 3, 0, 8], [-6, -6, 1, 1, -6, 9, -3, 1, -6, 8, 1], [8, 8, -4, 10, -4, -2, -4, -5, 3, -4, 3], [-4, 14, -4, -6, 8, 1, 2, 5, -7, 5, 1], [18, -5, -3, 3, -9, 0, 6, 6, -3, -6, 3], [-15, 2, -1, 1, -3, -3, 1, -10, 7, 1, 3], [6, 4, 4, -2, 8, 7, -8, 1, 5, -1, 1], [2, 3, 2, 6, -7, -1, 3, 0, -3, 1, 3], [9, -8, 12, -3, 6, 12, -6, 11, 3, 2, 4], [1, -2, 1, 3, 2, -4, 0, 4, -11, 8, 2], [13, 11, 4, -2, 10, -4, -17, 4, 14, -11, -5], [4, -8, 7, 3, 5, 2, -4, 13, 1, 7, 1], [-16, 1, -3, 3, -1, -4, 4, 2, -10, 8, 0], [2, -6, -7, 1, -7, -6, 4, 1, -9, 7, -5], [14, -8, -14, 8, 0, 2, 0, 0, -12, 8, -4], [16, -11, -2, -4, 2, 9, 2, 1, -4, 11, -3], [-7, -2, 9, -2, 1, -9, 0, 6, 7, -9, -2], [4, -5, 16, -5, -3, 3, 4, 3, -4, 0, 7], [-8, -2, 3, 5, 7, -11, 0, 1, 6, -10, -8], [-14, 0, -2, 6, -12, 10, 0, 1, -6, -5, 0], [25, -14, 9, -7, 10, 6, 0, 16, -10, 12, -6], [-9, -10, 8, -5, 10, 6, 0, 0, -1, -1, 4], [-33, 4, -4, 2, -3, 1, 2, -3, -10, 5, 9], [26, -12, 5, -4, 0, -5, 12, -2, -4, -5, 2], [-22, 0, -5, -1, 17, -5, -8, 11, -8, 15, -4], [-26, -5, -8, -4, -2, -4, -4, -4, 9, 2, 2], [5, 3, 0, 0, 9, -5, -12, 5, 16, 1, -2], [5, 10, -8, 6, 1, 3, -4, 2, 0, 2, -5], [22, -10, 7, 1, 1, 9, -4, 13, -16, 5, 2], [3, 2, 18, -18, 1, 9, 4, 2, -3, 8, 4], [16, 1, -15, 3, -7, -9, 6, -6, -10, 7, -1], [14, -7, 5, 3, -5, -3, -4, 7, -1, 6, 4], [2, -3, 7, -3, -6, -7, 9, -12, 5, -5, 9], [18, 0, -10, 10, -8, -3, -2, 2, 0, -3, -2], [2, 0, -1, -3, 3, 0, 0, 4, -1, 10, -1], [16, -6, -5, 9, -1, -2, -6, 0, -3, 6, -1], [-13, 10, -8, 6, -1, -2, -2, 3, 4, 4, -5], [-18, 6, 3, -3, -15, 2, 6, -2, 3, 0, -1], [17, 12, 8, -1, 0, 1, 2, 5, 4, -8, -3], [4, 2, 0, -2, -2, 3, 10, -2, -13, 8, -1], [-14, -11, -12, 10, -16, -2, 18, -10, -14, 1, 3], [-20, 6, -3, -3, -11, 4, 8, -7, -1, -9, 5], [0, -16, -5, -6, 2, 2, -6, 4, 3, 5, -1], [-23, 2, -7, 7, 6, -1, 0, 1, -6, 1, -6], [22, 17, 8, 0, 14, -7, -10, 3, 12, -11, -6], [-12, -8, 4, 4, -12, 7, 8, -6, -11, 2, 5], [-20, -4, 4, 0, 2, 9, -4, 10, -3, 10, -3]];
    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_z();".
// 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_z(:prec:=11)
    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_z();".
// 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_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6018_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1200, -692, -4372, 562, 3403, -484, -1023, 200, 117, -27, -4, 1]>,<7,R![-5472, -38747, -17272, 22766, 9257, -5627, -1628, 704, 118, -43, -3, 1]>],Snew);
    return Vf;
end function;