
// Make newform 6018.2.a.bc 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_bc();".
// 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_bc();".
// 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 := [-43744, -22032, 138096, 40576, -125592, -28124, 48570, 8315, -9396, -1179, 956, 79, -49, -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], [-121204003017027824, 82447262577467504, 335998399721244300, -114630360428554364, -217396661630863236, 48833287977907775, 50968867163271240, -8545314250127135, -5146738100625936, 664075790819685, 218934418638321, -19196372187770, -2880670567959, 16618726754], [-147416272648448328, -160331873586642872, 386591439015569500, 175862488069791692, -252075259966176062, -77170871949262535, 69366955830930035, 14347689427204105, -9178698567078167, -1128332602443005, 579978221320162, 28753457488965, -14081190935493, 186379188393], [-323879658307596592, -77884611009175368, 730484036257116720, 61232127641237328, -469471921597039298, -28337583971354760, 120335822994201275, 5802375177076970, -14325436667704103, -464256811623320, 798912639958483, 9557085301195, -16961861503452, 202997915147], [-13207959668528272, 7626075980681936, -1669417898377500, -20706135037061272, 17549153084715332, 9313167069656711, -7517982884554071, -1404708310869837, 1220653172685771, 50692387805607, -87670549177272, 3942662986183, 2329023024781, -236276976171], [13827833046029584, 19999582304517216, 2824971170499412, -25513830497582096, -22696271950686584, 9382257151775639, 9814685899795205, -1794366343522957, -1599349359996641, 218559476116059, 114215106567488, -15414700154749, -3008483462819, 443751573941], [-79394039693874016, -151910773512655624, 380724393788637740, 86934350501483764, -295403685028325434, 4465199891261865, 82123225867424865, -7309843756060875, -10443761348377309, 1279836923435835, 621826637080384, -88902889968565, -14073288977621, 2226419771871], [-90032763346833376, -213409310594817704, 48280623164166500, 333908596796940704, -3134468239806414, -174746025060120125, -4934053698709940, 40551053846921025, 1817846619873516, -4646880167421965, -207291278819761, 260235001126690, 7373884997349, -5727717858944], [17065631601733976, -81782923919894696, -60806142225794540, 72685524970824776, 56781193842740004, -41721469365383055, -23909630494110700, 12733747965022935, 4435000064547344, -1863261897938825, -358168627839709, 126833659135550, 10334588586211, -3246512395886], [-25869449120688432, -242528671429568608, -142957183349222400, 389401116803341508, 182706845683949412, -210583446468153260, -76968570525750485, 52688098277426130, 13997174731938057, -6666675407303450, -1121038637019707, 416444703490635, 32343651313568, -10198052289283], [-260037911487082352, 158405026529980472, 676467191148702720, -295590582482895472, -579220686043196398, 152264418850087960, 199267414681539515, -41404355245907430, -31255545769888503, 6115066006045500, 2249746437989783, -443909983325205, -60219666399172, 12229460566967], [166761480409701264, -384292349947742024, -500374230332802640, 966238739011843464, 354134323447064406, -586122971969355440, -103215181842607525, 143065720146530890, 14477216382795401, -16369355249611440, -953369861904061, 879153061774715, 23455557245524, -17924326795869], [182105343465589472, -163056024996551552, -452152041773406660, 460439267349737952, 320335811846240008, -361806925655129755, -116847168938051585, 106085745505489865, 21117514546830813, -14193063964416095, -1730523496563728, 889326633697785, 51088675909607, -21287880786177]];
    Rf_basisdens := [1, 1, 7894197520302920, 7894197520302920, 7894197520302920, 3157679008121168, 3157679008121168, 15788395040605840, 15788395040605840, 7894197520302920, 15788395040605840, 15788395040605840, 15788395040605840, 15788395040605840];
    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, 0, 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, 0, 0, 0, -1, 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, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 1, 0, -1, 1, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 2, 1, 1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 1], [-1, 1, 0, 1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1], [2, 0, 1, -1, 0, 1, 0, 1, 0, 2, 0, 1, 0, -1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 1, -1, 0, -2, 0, 0, -1, 0, 0, 1], [0, 0, 1, 0, -1, 0, -1, 1, 0, 0, 0, 0, -1, 0], [-1, 0, -1, 0, 0, -1, 0, -1, 0, 0, -1, 0, -1, 0], [2, 0, -1, 1, 1, -1, 1, 0, 0, 0, 0, -1, 0, -1], [0, 0, 1, -1, -1, 1, 0, 1, 0, 1, 0, 1, 0, 0], [3, -1, 0, 0, 0, 0, 0, -1, 0, 0, 1, -1, 1, 0], [3, -1, 0, -1, 0, 0, 1, 0, 1, 0, 1, 0, 0, -1], [2, -2, 1, 0, 0, 1, 1, 0, 2, 0, 1, 0, 1, -1], [0, 2, 0, 2, 0, -1, 2, 1, 0, 0, 0, 0, -1, 0], [-2, 2, 2, 1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 1], [2, -1, -1, 1, 0, 0, 2, 0, 2, 0, 0, 0, 1, -1], [3, -1, 0, 0, 0, -1, 0, -1, 1, 0, 0, -1, -1, -1], [2, 1, 1, 0, -2, 0, 1, 0, 0, 0, 1, 0, 0, 1], [3, -1, 0, 1, 0, 0, 1, -1, 0, 0, 0, -1, 0, 0], [-4, -2, 0, -1, -1, -1, -2, 0, -1, 0, -1, 0, -1, 2], [3, -1, 0, -1, 0, 1, 2, 0, 2, 0, 1, 0, 0, -1], [0, 0, 1, -1, 0, 0, -3, 0, -1, 0, -1, 0, 1, 2], [1, 0, 0, 0, 0, 1, 1, -1, -1, 0, -1, 0, 0, 0], [4, -2, 1, 1, -2, 0, 0, 0, 0, 0, 0, 1, -1, -1], [6, 2, 0, 0, 0, 1, -1, 0, 0, 0, -1, 0, 0, 1], [3, -2, -2, -1, 1, 0, 0, -1, -1, 0, 0, -1, -1, 0], [-2, 1, -1, 1, 0, -2, 2, -1, 0, 0, -2, 0, -1, 1], [3, 2, 1, -1, 0, 1, 0, 1, 3, -1, 0, 0, 0, -1], [1, 0, -2, 1, 2, -2, 0, 0, -2, 0, -1, 0, -1, 1], [4, 0, 2, 0, -2, 1, 0, 0, -1, 1, 1, 1, 0, 0], [2, -1, -2, 1, 0, -1, 1, 0, 0, 0, -1, 0, 0, 0], [2, 0, -1, 0, 0, 0, 0, 1, -1, 0, 0, 1, -1, 0], [1, 1, -1, -1, 1, -1, 1, 2, -1, 2, -1, 0, -1, 1], [-2, -3, 0, 0, 0, 0, 0, -1, -2, 0, 0, 1, 0, 0], [3, -2, 3, 1, 0, 2, -2, 1, 0, 2, 1, 1, 2, -1], [0, 2, 3, 2, -2, 2, -2, 3, 1, 2, 0, 1, 1, 0], [0, -1, -1, -2, 0, 0, 0, 0, -1, 0, 1, 1, 0, -1], [4, -1, -1, 1, 0, -2, 0, -1, -1, 0, -2, 1, -1, 1], [0, -2, -1, 1, 0, -2, 0, 0, -1, 0, 0, -1, -2, 1], [0, -2, -1, 0, 0, -1, 1, 0, 1, -3, 0, -1, 0, -1], [2, 0, -2, 1, 2, -2, 0, 1, 0, 0, 0, -2, -1, -1], [-1, 1, 1, 2, -1, -2, -1, 0, -1, -2, -1, -1, 0, 2], [2, 0, -2, 1, 2, -1, 4, -1, -1, 0, 0, 1, -1, -1], [7, -1, 1, 0, -2, -1, -1, -2, 2, 0, 0, 0, 0, 1], [4, -2, 0, -1, 0, 2, -1, -2, 2, 0, 1, -1, 1, -2], [1, 1, 4, 2, -2, 2, -2, 1, 1, 0, 1, 1, 1, 0], [2, -1, -1, -1, 2, 0, 0, -2, 0, 0, 0, -1, 1, -1], [1, 1, 1, 2, 0, 2, 2, 0, 2, 0, 1, 0, 0, -2], [0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 0, 1, 1, 0], [7, -2, 2, 2, -2, 2, 2, -2, 4, -1, 2, 0, 2, -2], [-1, 1, 1, 2, 0, 1, 2, 0, 0, 0, 1, 1, 0, -2], [0, -2, -1, -2, 0, 2, -2, -2, 2, -2, 0, -1, 1, 0], [3, -4, 1, -2, 0, 0, -2, -1, -1, -2, 1, 0, 0, 0], [-4, 3, 1, 1, 0, -2, 1, -1, -1, -2, 0, -1, 1, 1], [4, -2, -1, 2, 0, -1, 0, 1, -1, 0, 0, 0, -1, -2], [-1, 2, -2, 1, 2, -1, 2, 1, 0, 2, -1, 0, 0, -1], [-3, -3, -3, -2, 2, 0, 0, -1, 0, 0, 0, 1, 0, 1], [4, -2, -1, 0, 0, -1, 1, -1, 1, -2, -1, -3, 1, 1], [-1, 1, 2, -2, -1, 2, 3, -1, 1, 2, 1, 2, 1, -2], [-1, 0, 2, -2, -3, 1, -5, -2, 1, 0, 1, -1, 0, 0], [-3, 2, 0, 0, -2, 0, -1, -1, -1, 0, -2, -2, 0, 3], [9, -1, 3, -1, -2, 3, -1, -1, 1, 0, 0, 0, 2, -2], [0, 1, -3, 0, 0, -2, 0, 0, 0, 0, 0, -1, -1, 2], [-12, 6, 0, 0, 0, -1, -2, 2, -3, 0, -3, -1, -2, 3], [1, 1, 1, 1, -2, 0, 2, 0, 1, -1, 2, 0, -1, -1], [6, 2, -1, -1, 0, 0, 2, -2, 1, 0, 0, 2, -1, 1], [-1, 3, 0, 1, 2, 1, -1, -1, 1, 2, -2, 0, 0, 2], [2, 3, 4, 0, -2, 2, 1, 1, -1, 0, 1, 1, 1, -3], [-7, 3, 1, -1, -2, 2, 0, 2, 3, 0, 1, 1, 0, -1], [3, -1, -2, -2, 2, 0, -2, 2, -1, 2, 1, 0, -1, -2], [-4, 1, 0, -1, 0, 2, -1, 3, 0, 0, 0, 1, 0, -1], [6, 0, 1, 2, 0, 0, 4, 2, -2, 0, 1, 0, 1, 1], [-1, -3, 0, 0, 2, 1, 3, -1, -1, 1, 0, 1, 1, 0], [-2, 1, 0, -1, 2, 0, 0, 3, -1, 0, 0, 1, -2, 1], [-4, 1, -1, -2, 2, -3, -2, 1, 1, 0, 0, -1, 0, 2], [6, -1, 2, -1, -2, 1, -1, -2, 2, 0, 3, 0, 0, 0], [-2, 3, 0, 2, 2, -1, -2, -1, 1, 0, 0, -1, 1, 0], [2, 4, 0, 2, 0, -1, 3, 3, -3, 1, -1, 0, 0, 0], [3, 1, 1, 1, -2, 1, -3, 4, -4, 0, 2, 1, -1, -2], [1, 5, 1, -1, 0, 3, 2, 0, 2, 0, -1, 1, 1, -3], [4, -3, 0, -1, 2, 1, 2, -4, 4, 0, -1, -2, 2, 0], [-1, 5, -1, 1, 2, -4, 0, 2, -3, -2, -1, 1, -2, 3], [6, -1, -2, -1, 2, -1, 4, -2, 1, -2, -1, -3, 0, 0], [-7, 2, 4, -1, -2, 0, -2, 0, -2, 0, 1, 1, -1, 1], [-1, -2, 0, 1, -2, -2, -1, -2, -1, 0, 0, 1, -2, 0], [-6, 1, 3, 2, 1, 1, 5, 1, 0, 0, 2, 2, 2, -2], [0, 2, 3, -1, 2, 1, 0, 0, -2, 0, -2, 2, 2, 3], [3, 1, 1, 2, -4, -1, -2, 2, -3, 0, 1, 0, -3, 0], [4, 0, 2, 1, 0, 1, -3, 1, 1, 0, 1, 0, 2, 2], [-6, 1, -2, 0, 2, -3, -1, 2, -1, 3, -2, 0, -3, 1], [-5, 0, 0, 1, -2, -3, -2, -2, -1, 0, -1, -3, -1, 3], [1, 1, -2, 1, 3, -3, 1, 2, -2, 2, -1, -1, -2, -1], [2, -3, -4, -4, 2, 1, 0, -2, 2, 0, 0, 2, 1, 0], [1, 1, -3, 1, 0, -1, 0, 2, 1, 0, -2, 0, -3, -2], [-4, -6, 1, -3, 0, 2, 0, -2, 4, 0, 2, 1, 2, 1], [2, -3, 1, 1, 0, -1, 3, 1, 3, 0, 3, -1, -1, -2], [-12, -2, 0, 0, 1, -2, 1, 3, -5, 0, -2, 0, -2, 2], [-6, 6, 3, 2, 0, -1, 0, 0, -3, 0, -1, 1, -1, 3], [-5, 1, 0, 1, -1, 2, 0, -1, -2, 2, 0, 1, 2, 2], [3, 5, 5, 0, -4, 2, -2, 3, 0, 0, 1, 1, 0, 2], [4, -2, -4, 0, 1, -2, 3, 0, -2, 0, -2, -1, -2, 0], [-7, 0, 0, 2, -2, -2, 0, -2, 0, -4, 1, 1, 1, 2], [-2, 2, 0, 4, 2, -1, 3, 1, 0, 0, 1, -1, 2, -3], [-1, -1, 1, 0, 0, 0, -3, 2, 1, 0, 0, -1, 1, 3], [-4, 1, 1, 4, 1, -2, 2, 0, -1, 0, 1, -1, 0, 1], [-5, -1, -2, 0, 2, 0, 1, 0, -4, 0, -1, 1, 1, 2], [-6, 0, 2, 1, 2, 1, -3, 0, -2, 0, -3, 0, 0, 2], [1, 0, 1, -1, 0, 4, 0, 1, 3, -2, 3, 1, 4, 1], [0, 2, 0, 0, -1, -1, -2, -1, -1, 0, -1, 2, 2, 3], [-1, 2, -4, -2, 4, 1, 5, 3, 1, 2, 0, 0, 1, 1], [3, -3, 1, 4, -2, 1, 2, 2, 2, -2, 2, 0, 0, -1], [1, -1, -4, 1, 2, -2, 2, -2, -3, 0, -1, -1, -2, 3], [-2, 0, -1, 1, 2, -2, 3, -1, 3, 0, -1, -1, 0, -2], [2, -2, -4, 0, 5, -1, 0, 1, -1, -2, -1, -2, 0, -1], [4, -5, -1, 3, 0, 0, 3, -3, 2, -3, 0, -2, 1, 0], [-10, -4, -4, -1, 2, 0, -1, -3, 0, -3, 2, 1, 0, 0], [-8, 7, 0, -1, -2, 0, 2, -1, 2, 0, 0, 1, -2, -1], [-5, 2, -4, -4, -2, 0, -6, 0, 0, 0, 1, 1, -1, 0], [0, 1, 5, -1, 1, 2, 3, 2, 0, 4, 0, 1, 1, -1], [-4, -1, -3, -3, 2, -1, 3, -2, 4, -2, -1, -1, -1, 0], [-5, 1, 0, -1, 0, -1, -2, 1, -1, 2, -3, 0, -2, -1], [-1, -6, -2, -4, 1, 0, 0, 0, -3, 2, 2, 1, 0, 1], [-2, -1, 2, -2, -2, 1, 0, 0, 1, 0, 2, 2, 1, 2], [7, 1, 1, 0, 0, 0, 0, 2, 4, 0, 1, -2, -2, -2], [-6, -1, -1, 1, 2, -1, -2, 1, 3, 0, -2, 0, -2, -1], [-7, 2, -4, -2, 4, -1, 4, -2, 0, 0, 2, 0, -2, -3], [3, -3, -1, 0, 5, 1, 2, 1, -1, 2, 0, 0, 2, 1], [-11, -1, 2, -1, 0, -3, -1, 1, -3, 0, -2, 0, 0, 4], [-14, -2, -3, 0, 4, -1, 3, 0, 0, 2, -1, -2, -1, -3], [1, -2, 1, -3, 0, 1, -3, 0, -3, 0, -2, 1, -2, 0], [-3, 1, 6, 0, 0, 2, 0, 0, -4, -2, 1, 3, 5, 0], [-3, 2, -1, 2, -4, -2, 1, 5, 0, -2, 2, 1, -3, -1], [3, -1, 0, -1, 2, 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[4, 0, -5, 1, 2, -4, 6, -1, -2, 1, -1, -1, 2, 1], [9, 4, -2, 6, 0, -7, 4, -5, 0, -2, -5, -1, 2, 6], [12, -9, -7, -3, 2, 1, -1, 1, -1, 6, 1, 1, -3, -2], [-27, -6, 1, -2, 1, -1, -5, -4, -3, 6, -1, -1, -1, 4], [-13, 16, 1, 5, -4, -5, -1, -1, -7, 0, -2, 2, 0, 10], [-22, -1, 2, 2, -2, 3, 5, 3, 1, 0, 3, 2, 2, -5], [-11, 3, 3, 2, -2, 1, 1, 3, -6, 4, 0, 1, -2, 3], [17, 6, 2, -2, 2, 4, -2, -3, -1, -1, -2, 2, 2, 0], [-12, -5, 2, -1, -6, 2, 2, -1, 1, 2, 0, 1, -2, 1], [-26, 2, 1, 2, 0, -2, -6, 1, -5, 2, -8, 1, -5, 4], [-11, 4, 2, -4, -2, 0, -2, 4, -3, 2, 0, 5, -2, -1], [17, -8, -2, 4, 0, 4, 10, 1, 5, 1, 4, -4, -2, -10], [-18, 6, 7, 0, 0, 2, -8, -1, 0, 2, -4, 4, 5, 2], [0, 3, -4, -5, -2, 4, -2, 7, 0, 2, 6, 1, 0, -5], [-8, 0, 0, -4, -4, -2, -2, -7, 0, 0, -2, -3, -4, 4], [-7, -4, -8, -5, 4, -4, 8, -1, 3, 2, 0, -5, -3, -4], [-2, 10, 5, 4, -2, 2, 6, 3, -1, 0, -2, -1, -1, -4], [21, -4, 3, -3, 3, -1, -4, -6, -2, -2, 0, -1, 0, 0], [11, 7, -4, 4, 0, 2, -1, 5, 2, -4, -1, -2, -1, 0], [-11, 3, -3, -5, 6, 2, 1, -3, 2, 6, -4, -2, -3, -2], [19, -7, -2, -2, 4, 6, 1, -3, 6, -2, 1, -4, 1, -8], [-4, 5, 2, -2, 4, -3, 2, 3, -2, 4, -4, 5, 0, 0], [21, -3, 7, -2, 0, 1, 0, 1, -6, 0, 3, 4, -1, 0], [3, -4, 1, 1, -1, -3, 2, -6, -1, 0, -4, 1, 0, 6], [5, 8, 5, 3, 5, 3, 0, 5, -4, 0, -2, 3, 2, 2], [-8, 0, 4, 0, 2, 2, 0, 2, 2, 0, 6, 0, 4, 2], [13, -1, 1, 4, -4, 4, -2, -1, 0, -2, 1, -1, 2, -2], [8, -3, -1, 4, 0, 3, -2, 0, -3, 0, 0, 1, 4, -2], [10, 1, 0, -1, 0, -5, -2, -3, -2, -2, -3, -3, -4, 0], [-6, 2, 7, -2, -2, 3, -4, 0, -5, 4, 2, 1, 2, 4], [-9, 3, 4, -4, -2, 3, -1, 0, 5, -2, 2, 0, -1, -3], [9, -1, -5, 1, -2, 0, 3, 0, 6, -7, 0, -1, 1, -3], [15, 1, 6, 5, 0, 2, 7, 5, 0, -4, 2, 1, -2, -4], [14, 4, 4, 3, 0, 2, -1, 0, 4, 2, -1, 2, 3, -6], [-13, -3, 0, 6, 0, -5, 5, 3, 2, -2, 2, -4, 1, -1], [-3, -5, 3, -1, 0, -4, 2, -7, 0, -6, 3, 4, 3, 5], [-12, 5, 7, 0, 0, 5, -8, 7, -2, 0, 2, 1, 0, -6], [7, 8, 0, 3, -7, 0, 3, -4, -2, 0, 3, 1, 3, 1], [15, -3, 3, -3, 2, 7, 3, 3, 2, 8, 1, 6, 1, -3], [-14, 3, -2, 2, 4, -7, 4, -5, 1, 2, -2, -3, 3, 4], [-16, 2, -7, -4, 0, -2, 4, -1, 3, -2, -2, 1, -3, -4], [-17, 7, -6, 2, 4, -4, 1, 1, 0, -6, -1, -4, -1, -4], [18, -5, -5, -2, 0, -1, 0, -1, 6, -2, 2, -1, 0, 0], [2, -5, 9, 1, 0, 1, 3, -1, 5, -4, -3, 1, 3, 2], [-6, -3, 6, 1, -8, 0, -2, 1, 0, -4, 2, 3, -2, 1], [14, 4, -2, -10, 2, 7, -7, 2, 0, 7, 3, 6, 2, -2], [0, -2, -6, -2, 4, 1, 1, -1, 1, -2, 5, -2, 4, 1], [-2, 5, 5, 0, 0, 2, 1, 5, -5, 2, 1, -2, 2, -1], [22, -12, 7, -1, 0, 3, 5, 3, 9, -1, 8, 2, 4, -10], [15, 7, 0, 0, 4, -2, 0, -10, 0, -4, -3, -3, 3, 4], [8, -7, -8, -3, 2, 2, 3, 3, 0, 0, 1, -4, -1, -4], [-3, 1, -1, 1, 0, -2, -6, 0, -6, 2, 2, 3, -3, 2], [-2, 5, 0, 4, -4, 0, -1, 3, 2, -6, -1, -3, -1, -5], [18, -7, 1, 1, 0, 5, 2, 3, -1, 0, 2, 4, 2, -1], [-21, 3, 1, -3, -2, 0, -6, -3, -2, -2, -7, 2, 1, 11], [14, 8, 3, 3, -8, 6, -6, 3, 2, 2, -1, -2, 0, 0], [3, 7, 3, 1, 2, 0, -9, 2, 1, 2, -2, -2, 1, 0], [18, 7, 4, 6, 0, 1, -1, 4, -2, -2, 6, 1, -1, -4], [6, 5, 1, 2, -2, -8, 1, -1, 0, 2, -3, 3, -5, 5], [-8, 0, -1, 2, -6, -1, 3, 2, 0, -2, 1, 1, 1, 5], [5, 0, 0, 4, 1, 2, 5, 4, 1, -1, 6, 1, 6, -4], [7, 0, -6, 2, 8, -6, -2, -6, 0, 0, -3, -3, -1, 6], [-12, 6, 12, 6, -6, 3, -1, 5, 0, -6, 1, 3, 4, 3], [0, -5, 1, -1, 4, -4, -1, -3, -7, -2, 0, -1, 1, -1], [-7, 9, -8, -4, 4, -5, -3, -4, -3, -2, -4, 2, -9, 1], [9, 13, 5, 2, -3, 3, 0, 1, 2, 4, -4, 1, -4, 1], [0, -15, 1, 0, 2, -3, -2, -9, -1, 0, 0, -2, 4, 0], [10, -1, 1, 5, 6, 2, 5, 0, 1, 0, -5, 3, 6, 0], [11, -8, 1, -6, 0, -4, 1, -5, 3, -4, 0, -3, -5, -5], [-10, 2, -1, 3, -4, 3, -1, 12, 0, -1, 4, -2, -2, -6], [-14, -7, -7, 1, 8, -5, 9, -2, -6, -4, 3, -3, -1, -4], [-11, 6, -7, -2, 4, -2, 4, -3, 2, -2, -2, -4, -5, -3], [22, -11, 0, -4, -2, 1, 3, -2, 4, 2, -2, -1, -1, -2], [5, 3, -2, 4, 4, -3, 3, 2, -7, 2, 4, -2, 1, 1], [22, 2, -3, 3, 0, 4, 3, -1, 8, -2, 1, -5, 0, -4], [11, -4, -4, 3, 6, -6, 9, -6, 2, 0, -8, -3, -2, -2], [-19, 7, 6, -4, 5, -1, -1, 0, -2, 7, 0, -1, 1, 0], [22, 5, 2, 4, 4, 2, 0, 6, 1, 0, 1, -4, -1, 1], [0, -5, -2, -2, -2, 3, 9, -6, 2, -2, -1, 1, 4, 1], [32, -5, 6, 2, -4, 2, -2, -6, 0, 0, 1, 3, -1, -5], [-28, -3, -3, 3, 5, -4, 3, 0, -8, 0, -2, 1, -1, -3], [-10, -6, 2, 4, 4, 3, 8, 6, 0, -2, 2, 1, 6, -2], [6, 1, 5, -1, 0, 8, 6, 2, 1, 2, 2, -2, 1, -5], [5, 2, 5, 2, -6, 4, 3, 5, -2, 2, 2, 2, -3, -7], [3, -2, -2, -8, 4, 1, -8, -5, -4, 4, -3, 1, 0, 6], [-11, 1, 0, 0, -6, -5, -6, -6, -4, 2, -5, 3, -2, 8], [7, 8, 1, -1, 0, 2, -1, 11, 1, 4, 3, 2, -4, 1], [15, -7, -4, -4, 2, 7, 5, -4, 7, 2, 4, 6, 1, -5], [-3, -3, 0, 2, 4, -4, 3, -3, -2, -4, 0, -5, 2, 1], [-9, -1, -1, -7, -2, -1, 0, -6, 5, -2, 1, -4, -3, -5], [9, 3, 2, -5, 4, 2, 0, 6, 7, 4, 0, 1, -4, 0], [18, -1, 2, -5, -2, 5, -1, 4, -3, 2, 1, -1, 2, 0], [2, -19, -5, -5, 2, -2, 0, -7, 3, -2, 1, 3, 1, 2], [8, 6, -7, -5, -5, 3, -14, 1, -2, -1, 0, -1, 0, 2], [-8, 1, -5, 1, -2, -8, 4, 4, -3, 2, -4, -2, -9, 1], [-1, 2, -1, -7, 2, 4, -13, -4, -1, 2, 4, -1, 0, 2], [9, -1, 2, 1, 0, 3, 10, 1, 3, 0, 7, 1, -1, -7], [-6, -5, 3, -3, -3, 4, -13, 1, -3, 8, -2, 1, -1, 3], [11, 2, -3, -5, 8, -2, -2, -4, -2, 10, -5, 0, -1, 5], [7, 6, 3, 3, -2, 9, 4, 6, -1, -4, 3, 2, 5, -7], [32, 4, 3, -2, 2, 1, 0, 3, 0, 0, -2, 2, -3, 2], [-8, 4, -2, 2, -9, -3, -6, 1, 1, -4, -5, 2, -2, 7], [-23, -3, -11, -1, 4, -4, -3, -9, 3, -5, -4, 1, -3, 5], [-14, -8, -3, 1, 6, -3, 8, -4, 2, -4, -2, -4, 2, -1], [-19, 0, -3, 6, -2, -3, -1, 0, 3, 2, 0, 1, 0, 3], [3, -13, -5, -1, -1, -3, 8, -6, 1, -6, 0, -4, -3, 0], [-8, -2, 1, -7, -8, -1, -8, 4, -2, 6, 4, 0, -2, 1], [-15, -4, -5, 5, -1, -5, 2, 8, -12, 2, 0, 3, -4, 0], [9, -2, 6, 6, -6, 2, 9, -2, 5, -6, 2, -1, 2, -7], [-2, -3, 3, 6, 0, -5, 5, 6, -4, -2, -1, 3, 0, -3], [10, 11, 1, 5, -5, -3, 4, -5, 6, -2, -3, -2, 1, 0], [6, 1, -8, -1, 0, 0, -2, -5, -2, 2, -2, -3, -2, 1], [2, -1, -1, -2, 4, 0, 0, 6, -1, 6, 2, 3, -1, 0], [4, 1, 2, 3, 0, 4, 8, 3, 2, 2, 2, -1, 8, -1], [-16, 0, 4, 2, 0, 0, -6, 4, -4, 2, 0, 4, 2, 2], [-23, -1, -2, -1, -6, 4, -6, 2, 1, 4, -1, 1, 2, 1], [2, -3, 0, 1, -8, -2, 2, 0, -2, 4, 2, 4, -4, -5], [17, 2, 1, 0, 0, 0, -1, -7, -2, 2, 0, 0, 5, 1], [-1, 10, -2, 2, 4, 6, 5, 9, -1, 4, 1, -3, -2, -6], [17, 4, 9, 5, -12, 10, -4, 5, -2, 0, 5, 3, 4, -1], [1, -11, -6, 0, 0, -5, 4, -2, 6, -6, -3, -7, 0, 2], [-26, -2, -5, 3, 0, -1, 0, -1, 2, 0, 0, -3, 0, -3], [1, -4, -5, -3, -2, -8, -6, -10, 1, -2, -7, -5, -8, 5], [-23, 3, 4, 1, 0, -2, 3, 2, -4, -4, 4, 2, 4, 4], [25, 5, 6, 9, -4, -1, -6, 0, -2, 0, 3, 0, 0, 5], [-4, 8, 9, -2, 0, 5, -3, 2, -1, 10, 1, 6, -1, -1], [2, -9, -12, -9, 4, 1, -3, 2, 3, 4, -1, -4, 0, -2], [-25, -9, -6, -4, 4, 1, 0, -1, 0, 8, -3, -2, 3, 2], [11, 3, 4, 2, 4, 2, -1, -3, 3, -5, -1, -1, 3, 3], [9, -1, 6, -2, -4, 6, -10, 3, 0, -2, 5, 1, 3, -2], [-13, -9, -6, -1, 2, -6, -3, -2, -4, 2, 1, 2, 0, 7], [11, -4, -5, 4, -4, -3, -4, -3, -2, -10, 4, -3, 1, 3], [1, 8, 4, 1, -3, -5, 4, 2, 1, 0, -2, -2, 1, 0], [-15, -7, -6, 2, 6, -3, 3, -3, 3, 1, -2, -7, 3, 2], [-9, 7, 5, 6, -4, 3, 5, 6, 5, 4, 4, 1, 0, -1], [11, -5, -1, 2, -2, 0, 8, 4, 0, -4, 5, 0, -4, -4], [-10, 3, 3, -5, 2, 7, -7, 5, 0, 7, -5, 2, -1, -1], [39, -11, -1, 0, 6, -2, 2, 4, -4, -2, 5, 0, 0, 0], [0, 8, -8, 0, 2, -3, -1, 3, -1, 2, -3, -4, -10, -1], [25, 6, -6, 6, -2, -2, 7, -1, -1, 2, -2, 2, -4, -1], [-10, -9, -11, 1, -2, 3, 10, -3, 2, -4, 5, 0, 1, -6], [6, -15, -2, -5, 4, 0, -6, -7, 0, -2, 0, -3, 2, -3], [2, -1, -2, -1, -6, -1, 3, -1, 2, -2, -1, -7, -4, 2], [30, 4, 7, 1, 0, -3, -5, -6, 2, -2, -3, -3, -1, 2], [-9, 2, -5, -5, 10, -5, 6, -6, -1, 6, -3, -1, 3, 3], [7, -2, 4, -4, 0, 9, -4, -1, -2, 6, 1, 3, 4, 4], [-9, -2, -6, -8, 0, 4, 1, -1, 13, 2, -2, -2, -2, -5]];
    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_bc();".
// 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_bc(:prec:=14)
    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_bc();".
// 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_bc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6018_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-43744, -22032, 138096, 40576, -125592, -28124, 48570, 8315, -9396, -1179, 956, 79, -49, -2, 1]>,<7,R![124864, 399160, 242956, -375198, -427029, 43374, 156984, 7863, -25565, -1378, 1998, 66, -73, -1, 1]>],Snew);
    return Vf;
end function;