
// Make newform 6042.2.a.bh in Magma, downloaded from the LMFDB on 09 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6042_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_6042_2_a_bh();".
// 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_6042_2_a_bh();".
// 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, 9304, 16812, -9466, -22200, 6938, 9639, -2914, -1707, 537, 123, -40, -3, 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], [-15131253915640, -69505164946224, -30721425263802, 66746578762844, 30037232426544, -24476327772653, -8515787410070, 3980299416517, 1022243116203, -276650415923, -55180929014, 6679376015, 1104465971], [-33713612784760, -150515869605488, -30222105954378, 165465205024724, 20272416631968, -65484111774973, -443211211426, 11062731147833, -786204074125, -780424603231, 89540587722, 18882316439, -2632284209], [-25057438289328, -106411914676176, -60768123000, 123875804125670, -2416172669298, -50218247672710, 4585058152073, 8558511877653, -1209915311616, -605287792477, 102820445846, 14646261136, -2727750661], [-955171318320, -1427999257888, -25651497176616, -20469284564146, 30424461859070, 13755460366038, -12585440057303, -2859488827735, 2181905208796, 219870360311, -156363041234, -5504342876, 3825967735], [-20324471216384, -102911563743592, -69773601238538, 94219150878618, 68240550554438, -32224399272003, -21545221318909, 4949970794130, 2999393383979, -333593208456, -186403032696, 7962412967, 4180312254], [-15375912065432, -8967190274096, 99166364858438, 64874222641024, -89720193617028, -39050414261665, 33013088468736, 7818704648743, -5407907514181, -592303202345, 376726997738, 14744154019, -9076089647], [-57489181360808, -261526193845840, -182388318974730, 219914719422692, 174198902752088, -69934638097797, -53794158630402, 10188017922425, 7352334071027, -664510529295, -450595476454, 15569453759, 10002084743], [-1802645531584, -9936848006134, -23030561263496, -909737621208, 26156616453175, 3874256295364, -10090594369310, -1017129760174, 1657562870917, 84482827509, -115007316091, -2173491833, 2761861453], [10900451890120, 23466620266904, -114796805478126, -104200446673436, 127116995436076, 63229291380089, -52033807512746, -12861301367465, 8996584023733, 983188381227, -643851470174, -24588080919, 15738407933], [7501450230024, 22658914077896, -31658973603658, -43741985100670, 34875401841134, 22580897844015, -14800691726179, -4339680041921, 2627378123161, 324090176126, -190884176433, -8016452396, 4706616569], [19380400213704, 34044755921264, -170364844943950, -143491522737224, 177219409007748, 84837751822269, -69906482897300, -17070643833519, 11863059795713, 1299036592121, -841326116794, -32421181759, 20467926511]];
    Rf_basisdens := [1, 1, 887834031664, 887834031664, 443917015832, 443917015832, 443917015832, 887834031664, 887834031664, 221958507916, 887834031664, 221958507916, 887834031664];
    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_6042_a();"
function MakeCharacter_6042_a()
    N := 6042;
    order := 1;
    char_gens := [2015, 4771, 2281];
    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_6042_a_Hecke(Kf)
    return MakeCharacter_6042_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], [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], [1, 0, -1, 0, 0, 0, 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, 1, 0, 0], [0, 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], [1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [2, -1, 1, 0, 0, 0, -1, 1, 0, 0, -1, 0, 0], [0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [2, 1, 0, -1, 0, 0, 0, -1, 0, 0, 0, -1, 0], [1, 1, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -1, 1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0], [2, 0, 0, -1, 1, 0, 0, 0, 1, -1, 0, 0, -1], [-1, 0, -1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0], [2, 0, 1, 1, 0, 0, 0, 0, 0, -1, -1, 1, 0], [1, -1, 0, 2, 0, 1, 0, 1, -1, 0, 0, 1, 0], [2, -1, 1, -1, 0, 0, 0, 0, -1, 1, -1, 0, 0], [1, 0, 2, 0, 0, 1, 0, 0, -1, 0, -1, -1, -1], [0, 1, -1, 0, 0, 1, -1, -1, 1, 0, 1, -1, 0], [4, -3, 2, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0], [1, -2, 0, 0, -1, 0, -1, 0, 1, 0, 0, 0, 0], [1, -1, -1, 1, 0, -1, 1, 0, 0, -1, 1, -1, 1], [0, 0, 0, 1, -1, 1, 0, 0, -1, 1, 0, -1, -1], [2, 0, 0, 0, -1, 2, 0, 0, 0, 0, 0, 0, -2], [1, -2, 1, 0, 1, -1, -1, 0, 0, 0, -1, 2, 0], [4, 0, 0, -1, 1, 1, -1, 0, 2, 0, 0, 1, 0], [1, 0, 0, -1, 0, -2, 0, 0, 1, 0, 0, 0, -2], [-1, 0, -1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0], [3, -2, 1, 0, -1, 1, 0, 0, 0, 0, 0, 2, 0], [-1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1], [1, 0, -1, 1, 0, -1, 1, 0, -1, 0, 0, -2, 2], [2, -2, -1, -1, 1, -1, -1, 0, 2, 0, 1, -1, 0], [-5, -1, -1, 1, 1, 0, 0, 1, 1, -1, 1, 0, 1], [0, 1, 2, -1, 1, -1, 0, -1, 2, -1, -1, 1, 0], [2, -3, 0, 0, -1, 0, -1, -1, 1, 0, 0, 1, 0], [1, 1, 0, 1, -2, 0, 1, -1, -2, 2, 0, -1, 0], [3, 1, -1, -2, 0, 0, 1, -1, 1, 1, 1, -1, 0], [-3, -5, 1, 0, 0, -1, -1, 1, -1, 1, 1, 1, 0], [0, 0, -2, -1, 0, 0, 1, 0, 0, 0, 0, 0, 2], [-2, 0, 4, 2, 0, 1, 0, 0, -1, 0, 0, -1, 1], [4, 0, 1, -1, 0, -1, 0, 0, -1, 1, -3, -2, 1], [0, 1, 1, -1, 0, -2, 1, -1, -1, 0, 2, -2, 1], [3, 1, 0, -2, 0, -1, -1, -1, 0, 1, -2, 0, -1], [0, -1, -3, 1, 1, -1, 1, 0, 1, 0, 1, 0, 0], [3, -1, 0, 0, 0, 2, 0, -1, 0, 0, 0, 2, 1], [-1, 1, 0, -1, -1, 0, 0, -1, -2, 3, 0, -1, 0], [-8, 1, 0, 1, 0, -1, 0, -1, 1, 0, 0, 0, -1], [2, 1, 0, -2, -1, 0, 1, -1, 1, 0, 2, 1, 0], [-3, -2, 1, 3, 0, -1, 0, 2, -2, 0, 1, -2, 0], [-4, 1, -2, 1, 0, 0, 0, 1, 0, 0, 2, -1, -1], [-3, -2, -2, 1, 1, 0, 0, 0, -1, 0, 0, 0, 2], [-1, 3, 1, 0, -1, 2, 0, 0, 0, -1, 1, -2, 0], [-2, -5, 0, 3, -1, 0, -1, 2, -2, 0, 0, 3, 1], [-1, 1, 4, -1, 0, -1, 0, 1, -1, 1, -2, 0, 1], [2, 0, 4, 1, 1, 2, 0, 0, -1, -1, 0, 2, 0], [0, -1, -2, -1, 0, -3, -2, 1, 2, 0, 0, 0, -1], [2, 1, 2, 0, 0, -2, -1, 1, -1, 0, -4, 0, 1], [0, -1, -4, 2, 1, 0, 1, -1, 1, -2, 2, 0, 1], [-2, 1, 1, -2, 0, -2, 0, -1, 2, 1, 1, 1, -2], [6, 0, 2, 2, -1, 2, 0, 0, -1, 0, -2, 1, -1], [1, 0, 2, 4, -1, 3, 0, 1, -1, 0, 2, 1, 0], [-3, 0, -1, -1, 1, -3, 2, -1, -2, 1, 3, -1, 0], [-4, 1, -2, -2, 0, -3, -1, -1, 2, 1, 1, 0, -1], [-5, 4, -1, -4, 2, -1, 0, 0, 0, -1, -1, 0, -2], [4, -1, 0, -1, 1, -1, 1, 1, 1, 1, 0, -1, 2], [-10, 2, 0, -2, 1, -1, 1, 0, 1, 1, 0, 1, -3], [3, -1, 3, 1, -1, 1, 0, -1, -2, 1, -1, 1, 0], [4, -9, 2, 4, 1, 0, -1, 1, -1, 0, 0, 1, 1], [0, -1, -1, 0, 1, -1, -2, 1, 1, -1, -1, 3, 0], [5, 4, -1, 0, -2, 4, 0, -2, 0, 1, -1, 0, -2], [2, -3, -1, 1, 0, -2, 0, 0, 1, -1, -1, 0, 0], [-4, 2, -2, -1, 2, 0, 1, 2, 1, -1, 1, 0, 2], [2, 0, 1, 1, -1, -2, -1, 0, 0, 0, 1, 0, -2], [-1, -3, -2, -1, 1, 1, -1, 1, 2, -1, -1, -1, 0], [-7, 0, 2, 0, 2, -1, 1, 2, 1, -2, -2, 3, -1], [2, -4, 0, 2, -1, 2, -2, -1, 0, 1, 0, 0, 1], [0, 6, -2, -2, 0, -2, 4, -2, -1, 0, 2, -1, 1], [-4, -1, -1, 1, 0, -1, 1, 1, -1, 1, 3, 0, -1], [4, 5, 1, 1, -3, 3, 3, -1, -4, 3, -1, -1, 0], [-1, -4, -3, 0, -1, -2, 1, 0, 0, 0, 1, 0, 0], [-9, 3, -3, 0, 0, 0, 0, -1, 0, 0, 1, -2, 3], [2, 1, -2, 0, -1, 1, 0, -1, 0, 1, -2, -2, 3], [-1, 1, -2, 0, 1, 3, 0, -1, 0, 0, 2, -2, -3], [-2, 1, -3, 1, 0, 0, 1, 1, 0, -1, -1, 1, -4], [3, 7, -2, -5, -2, 1, 0, -3, 1, 1, -2, 0, -3], [-8, 1, 2, 1, 0, 1, 0, -1, 0, 0, 0, 1, 0], [1, 5, -2, -4, -1, 0, -2, -3, 3, 0, 0, 1, -2], [4, 4, -2, -2, -2, 4, 1, -2, 1, 1, 0, 1, -1], [-2, -2, -2, -4, 1, -2, -2, 1, 2, 1, -2, 0, -1], [-7, -2, -4, 3, 1, 1, 0, 2, 1, -4, 4, -1, 3], [2, 2, 0, 0, -2, 1, 0, -2, 1, 0, 2, -1, 1], [-3, -4, 1, 3, 1, 2, -3, 0, -1, 1, -3, 0, 2], [4, -2, 4, 3, -1, 5, -2, 2, 0, 1, 0, 1, -1], [-10, 4, -1, 4, -1, 2, -1, 0, -2, 1, -1, -4, 3], [-1, 0, -1, 0, 1, 0, 0, 2, 1, -1, 1, -1, 1], [-6, 0, 6, 1, 0, -1, 1, 2, 0, 0, 2, -1, 0], [-9, 5, 1, -1, -2, 2, 2, -1, -1, 1, -1, 2, -2], [-1, -6, 1, 4, -1, 1, -2, 2, -2, 1, -1, 2, 0], [-3, 2, -4, -4, -2, 0, 1, -2, 1, 0, 0, 0, 0], [7, -5, -1, 1, -2, 3, -1, 0, 2, 1, -1, 3, 1], [6, 5, 2, -1, -3, 0, 0, 1, -1, 1, -5, 0, -1], [1, 4, 3, -1, -1, 5, 1, 0, -3, 3, -1, 0, -2], [2, -4, -1, 1, 2, -1, -2, 0, 3, -1, 1, 0, 3], [-1, 1, -4, 0, -2, 1, 2, -1, 1, 0, 2, -1, -2], [9, 5, -1, -2, -1, 3, -1, -1, 5, 0, 0, 3, -2], [-7, 4, -1, 2, 0, -2, 3, 0, -7, 2, 1, -5, 3], [4, 4, 2, -1, -2, 2, -2, -2, 0, 0, -1, 1, 1], [3, 1, 0, 4, -1, 4, -1, 1, 2, -1, 2, 0, -1], [-11, 3, -3, 2, -1, 2, 3, -1, -3, -1, 3, 1, 2], [-3, 0, 2, 0, 1, 2, 0, 2, 1, -3, 4, 1, 1], [-3, 3, -5, 4, -1, 0, 0, 0, 0, -1, 1, -2, 2], [0, 1, -6, 1, -1, 0, 4, -1, 1, 0, 4, 0, -1], [-4, -7, -1, 1, 3, -1, 0, 2, 2, -3, 3, 1, -1], [-14, 5, -1, -2, -1, -2, 0, -1, 1, -1, 1, 1, 2], [-7, 4, -3, -2, 3, -1, 2, 0, 0, -1, 3, 0, 2], [0, -3, 6, -2, 1, -5, 0, 1, -2, 1, -4, 0, 3], [3, 3, 3, 2, 0, 0, -1, 1, 0, -1, 1, 0, -1], [3, -3, 2, -2, 0, 1, 0, -1, 1, -2, -2, 1, -4], [-4, 2, -1, -2, 2, -2, 2, 0, -1, -2, -1, -2, 2], [3, 4, 2, 0, 0, 1, -3, 0, 1, 2, 2, -3, 0], [1, -5, 3, 4, -2, 2, 3, 1, -3, 2, 4, -1, -2], [3, -2, 0, 0, 0, -1, 0, 0, 3, 1, 6, 1, -2], [5, -6, 7, -1, 2, 1, -1, 0, -1, -1, -3, 0, 2], [-10, 1, -2, 3, 0, 2, 3, 0, 1, -2, 4, 2, 0], [-1, 0, -1, 3, -1, 3, -1, 2, -2, 1, -1, 1, -3], [0, 2, -3, 0, 0, -1, 2, 3, 2, -1, 1, 2, 1], [-8, 1, 1, 2, -1, -1, 0, 1, 0, 1, -1, 0, -1], [-3, 5, -2, 1, -2, 3, 2, -1, -2, -1, 2, -3, 0], [4, -3, 3, 3, -1, 3, -3, -1, 0, -1, 1, 3, -4], [-8, 2, 0, -1, -3, 1, 0, 0, 1, 1, -2, -2, -4], [5, 2, -1, 1, -1, 2, 3, 0, -1, -1, -3, -2, -2], [8, 2, 1, -2, -2, -3, -1, 0, 1, 3, -5, -1, -1], [3, -3, 4, 0, -2, -3, -2, 1, -1, 2, 0, 1, 0], [-1, 1, -1, 0, 2, -6, -1, 1, 2, -3, -1, -2, 3], [-9, 1, 4, -4, 3, -4, -1, 1, 2, -1, -2, 0, 5], [0, 0, 3, 2, 0, 2, 1, 2, -3, 1, 1, -3, 2], [-2, 0, -1, 2, 0, -5, -1, 2, 2, -1, 3, 4, 1], [-5, -3, 1, 1, -4, 3, 0, -1, 0, 1, -1, 0, -3], [-11, -8, 1, -1, 2, -3, -5, 2, 5, -3, -1, 2, -2], [2, 3, 1, 4, 1, 1, 0, 1, -5, 3, -3, -3, -2], [0, 0, 2, 1, -2, 6, 1, 0, -4, 4, 0, -1, -3], [1, 5, 3, -1, 0, -2, 3, -1, -2, -1, 2, -5, 2], [2, -2, 0, 3, -1, 5, -2, 0, -2, 1, -4, 3, 1], [4, 8, 1, -6, -1, -1, 1, 0, 0, -1, -7, 0, 2], [0, 1, -3, 3, 2, 1, 5, -1, -3, 1, 1, -2, 3], [7, -2, 3, -3, -2, -1, -2, 0, -1, 4, -5, 0, -1], [-4, 1, 5, -2, 1, 0, -3, -1, 2, 0, -3, 2, -5], [11, 4, 4, -3, -2, -2, -1, -1, -1, 2, -8, 0, -3], [13, 0, 8, 1, 1, 4, -2, -2, -1, 1, -5, 0, 0], [-9, 10, 0, -4, 0, -2, 2, -3, 0, 0, 0, -6, 1], [3, -2, 5, -1, 3, 0, -1, 2, 1, -3, -3, 2, 2], [1, 4, 1, 1, -2, 4, -2, -2, -2, 2, 1, -1, 3], [-12, -1, -8, -1, 0, 0, 0, -1, -2, 1, 1, -1, 4], [5, 6, 3, 0, 1, -1, 2, -2, 3, -3, 1, 2, 0], [-2, 3, -2, 1, -4, -1, 1, -2, -2, 0, 0, 1, -3], [2, -7, 4, -3, 0, -1, -4, -1, 2, 0, 0, -1, -2], [8, 1, 0, 3, 0, 2, 1, -1, 2, 1, 2, -5, -2], [3, 5, 1, 2, -2, 1, 3, 1, -4, -1, -5, -3, 0], [-12, 2, 4, 1, -2, 6, 4, -2, -6, 0, -1, 1, 3], [0, 1, 5, -3, 2, -1, -3, 3, -1, 0, -6, 0, 1], [2, -3, -1, -2, 1, -1, 0, -1, 0, 1, -3, -2, 3], [-2, -5, -5, 0, 4, 3, -3, 1, 6, -2, 5, 4, -1], [2, -3, -1, 0, 1, 2, 2, -1, -1, 1, 1, -1, 0], [12, 5, -2, -2, -3, 3, 1, -3, -1, 4, -2, -1, -2], [-4, -6, 2, 2, -4, -1, -2, 2, -1, 2, -2, -1, -3], [-3, 0, -1, 1, -2, 4, 1, -2, 0, -3, 3, 1, -3], [-2, -2, 3, 2, 2, -1, 1, 2, -2, -1, 3, 2, 2], [-6, 3, -1, 0, 1, -4, 2, -1, 1, -3, 1, -2, 1], [12, -2, -2, 1, 0, 4, -1, 0, 2, -2, 2, 1, 1], [5, -2, -8, -3, -1, -2, -2, -2, 3, 2, 0, -2, 2], [5, 3, -4, -2, -3, 3, 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1, 1, 0, 2, -8, -1, -1, -1, 0, 1, 1, 6], [20, -8, 0, 7, -1, 10, -7, 4, 3, 0, 2, 5, -5], [-11, -2, -2, -2, 0, 0, -3, 2, 1, -5, 1, 5, -3], [-12, 10, 2, 0, 2, -2, 2, 0, -2, -2, 6, 2, -4], [11, 6, 2, -2, 2, 3, -4, 3, 2, 2, 0, 0, -1], [32, -10, 0, 7, 0, 3, 1, 2, -1, 0, -4, 0, -4], [20, 10, 2, -4, 4, 8, 1, -2, 2, 0, -1, -6, -2], [4, -2, 0, -4, 6, 0, 0, 2, 0, 1, 3, 0, -2], [8, 4, -6, -3, -1, -2, 3, -4, -3, 4, -4, -8, 2], [-25, -1, -2, -5, 3, -4, -3, 5, 0, 0, -2, 1, 0], [4, -2, 1, -2, 4, 1, -1, 0, 1, 1, -1, 3, 1], [-11, 2, -3, 0, -3, -2, 5, 0, -1, 2, 1, -5, 8], [19, 6, -5, 3, 2, 5, 0, 0, 5, -4, -1, 0, -3], [29, -6, -4, 3, -3, 1, -3, 0, -2, 5, 4, -1, 3], [19, -1, 0, -3, 1, 1, 3, -3, -6, 2, 0, -3, -2], [12, 4, -4, 4, -1, -3, 1, 4, -6, -1, -2, -2, 2], [27, 0, 2, 5, 0, 5, -4, 2, 2, -4, 0, 3, -1], [8, 2, 0, -1, -5, 3, 0, 0, -3, -1, -4, -2, -2], [-7, 0, -2, -4, 1, -7, 5, -4, 4, -4, 2, -2, 2], [19, 1, 0, -4, -1, -8, 0, -3, 3, -2, -2, -1, -1], [-19, -5, 3, 8, 2, 3, 0, 0, -2, 1, 3, 0, -2], [9, 0, -1, 4, 0, 2, -3, -4, 6, 0, 1, -2, -2], [5, -16, -3, 7, -2, -2, -1, 2, 2, 1, 3, 3, 1], [-13, 7, -3, -5, 5, -2, -2, -3, 3, -1, 1, -1, 2], [5, -5, -8, -5, 2, 3, 1, -1, 7, -2, 2, 4, 2], [23, -7, -2, -7, -2, 4, -3, -1, 4, -2, 2, 3, -4], [-4, -3, 7, -6, -3, -7, -8, 3, 2, 3, -9, 2, -1], [5, -1, 2, 2, 5, -4, 3, 6, 5, -6, 8, 3, 1], [9, 9, -2, -4, -1, 2, 2, -7, 1, -2, 0, -1, -6], [9, -7, 3, -1, 1, -4, 2, 1, -4, 1, 1, 0, -3], [-8, -1, 5, -1, 3, -2, -2, 1, -3, 4, -3, 2, 2], [-4, -5, 1, 0, 2, 3, 4, 3, -1, -4, 0, 1, -2], [0, -11, 7, -1, -3, -6, -1, 1, -1, 3, -7, 4, 1], [15, 10, -3, 7, -2, 6, 4, 0, -4, 2, -1, -7, -4], [13, -6, -1, -7, -2, -1, 0, 0, 2, 0, -5, 3, 1], [8, -13, -1, 2, 0, 5, 0, 3, -1, 1, 1, 3, 0], [3, -1, 10, -1, 3, -2, -3, -1, -1, 0, -8, 2, 1], [-15, -2, 1, -1, 0, -5, -2, 0, -2, -4, 5, -3, 7], [-3, -5, -2, 2, -2, -7, -2, 3, -3, 4, -4, -1, -2], [-2, 13, -3, -3, -1, -5, 3, -1, -4, 3, -5, -1, 2], [12, -4, -5, 3, 0, -1, 8, -2, -1, -1, 7, 0, 4], [-2, 5, 3, -4, -1, -4, 3, -2, 1, 3, 1, -3, 5], [-1, -4, -9, 2, -2, -6, 4, -4, 1, 3, 7, -5, -2], [-4, -1, 0, -7, 2, -6, -2, -3, -3, 4, -2, -1, 2], [30, 1, 1, -1, 1, 3, -5, -1, -2, 1, -5, 4, 1], [-4, 7, 5, 4, 1, 0, 0, 1, -4, -1, 3, 6, 1], [-5, -1, 0, -1, 1, 1, -1, 5, -6, 2, 2, -3, 2], [21, -7, 11, 2, 2, -6, -5, -1, -3, 1, -5, 1, 4], [9, 4, 6, -3, 3, -5, 6, 0, 0, 2, 2, -3, 5], [-24, 3, 3, -4, -1, -5, 2, -1, -6, 2, -6, -4, 9], [-1, -2, -13, -3, 0, -4, -1, -2, 4, -1, 7, -1, -1], [1, -3, 0, 7, -2, 4, 1, -3, -6, 0, -2, -5, 8], [-11, -9, -4, 4, -2, -3, 4, 5, -6, 0, 4, 2, 2], [29, 7, -11, -7, -4, -1, 6, -9, 3, 0, 0, -8, 1], [-9, -15, -4, -1, -2, -2, -4, 3, 9, 1, 4, 4, -3], [6, -9, 3, 1, 4, 1, 1, -1, -5, -3, -5, 1, 4], [8, 2, 0, -1, 0, -1, -3, 2, 7, -3, -3, 4, 0], [-18, 0, -5, 0, 1, 2, -3, 0, -2, -3, 3, -3, -1], [1, 10, -4, 0, 2, 3, 10, -2, -3, -1, 2, -5, 9], [2, 5, -2, -2, 1, -7, 5, -1, -3, 0, 6, 0, -7], [-10, 1, -3, -6, 2, -6, 1, -6, 4, -1, 5, 0, 2], [11, 7, 2, -5, 6, 1, 3, -3, 3, 0, -2, 2, -3], [10, 2, -8, 0, 3, -1, -3, 4, -2, 1, -2, -2, 4], [-6, 3, 5, 3, -3, -1, -1, 0, -3, 0, 9, 2, 2], [-14, -7, 12, 2, 3, -1, -3, -1, 0, 2, -6, 0, 0], [14, 5, 9, -1, 3, -5, 5, 1, -6, 1, -1, 1, 6], [-1, -8, 8, 7, 2, 4, 4, 0, -7, 2, -2, 0, -6], [24, -3, 7, -7, -6, -2, -7, -1, 0, 5, -7, 3, -4], [-15, -8, 4, 5, -1, -4, 0, 2, -2, 4, -2, 1, -1], [-18, 0, 4, 1, -1, 2, 1, 0, 5, 2, 8, -2, 0], [4, 3, -1, -4, -1, 1, 9, -6, 3, 3, -3, -3, -7], [12, 2, 5, 7, 0, 1, -8, 4, 3, -3, 5, 0, 4], [3, -1, -2, -3, 6, -7, 4, -1, 0, -1, -2, 1, 6], [10, 0, 9, 3, 3, -1, -2, 4, 4, -5, -1, -3, 6], [-12, -11, 8, 11, -2, 13, -2, 5, -2, -3, -5, 7, 2], [14, 5, -5, 9, -5, 4, 1, 1, -5, 3, 1, -2, -5], [0, 12, -13, -6, 3, 0, 2, 2, 5, -2, -3, -1, -1], [12, 1, -5, 4, 0, 0, 3, -1, -3, -2, 7, 3, 0], [-1, 6, -5, 3, -4, 6, 2, 0, -3, 2, -1, 4, 0], [30, -5, 5, 1, 3, -1, -7, -1, 2, -2, -4, -1, 10], [-3, 13, 4, -8, -3, 5, -2, -3, 2, -2, -10, 0, -3], [26, -2, -5, 4, 2, 2, -2, 0, 2, -2, 11, 8, 0], [-8, 3, -3, -7, -2, 1, -3, -6, 2, 0, -5, 1, -7], [5, 5, -3, -6, 0, -5, -5, -1, 2, -5, -7, 5, -2], [-19, -1, 6, 5, 0, 10, -3, 3, -3, 0, -4, 6, -2], [-8, -9, 0, 9, 1, -11, 2, 5, -7, 2, 2, 0, 5], [37, -3, 1, -2, 2, 2, -5, 3, 6, 1, -7, -2, -3], [-12, -7, 11, 0, 1, 2, -4, 7, -5, 3, -7, -5, 2], [27, 16, -1, -6, -1, 3, 0, -8, 6, -1, -3, 2, -8], [-3, 0, -4, -5, 0, -3, 6, -2, 3, 2, 0, -8, -4], [18, -1, 7, -3, -1, -9, 3, 3, -4, -1, -9, -3, -4], [15, 5, -12, -3, -5, 2, 1, -5, 1, 4, 6, 0, -3], [-12, 3, 13, -3, 4, -7, 1, -3, -5, 3, -5, -2, 7], [-1, 9, -8, -1, -2, 1, 6, 3, 0, 1, -2, -5, -2], [-6, -8, 0, 7, -5, -1, 2, 2, -7, 5, 4, 0, 0], [30, 8, 2, -2, -1, 1, 4, 0, -2, -2, -2, -6, 3], [-2, 12, 0, -7, -2, 3, 1, -4, 3, 2, 2, -2, 4], [-7, 9, 2, -7, 3, 1, 1, -3, 1, -2, -4, -4, 1], [-8, -3, 1, 2, 2, 2, -2, -1, 0, -5, 5, 0, -1], [0, 0, 1, 5, -3, 1, -1, -2, 0, 0, 1, -1, 9], [36, -2, -3, 2, -3, -3, 3, 0, -4, 3, -1, -4, 2], [15, 3, 7, 7, -1, 1, 1, 2, -5, 5, 5, -2, 8], [11, -9, -10, -5, -4, -6, 1, 1, -2, 2, -4, -5, 0], [-19, -6, -2, 6, -4, 1, 3, 2, -7, 3, 13, -3, 9], [-11, 11, -6, -2, -5, -7, 6, -3, -4, -1, 5, -4, 1], [3, -10, 6, 12, 4, 0, -1, 2, -4, -2, 2, 4, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6042_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6042_2_a_bh();".
// 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_6042_2_a_bh(:prec:=13)
    chi := MakeCharacter_6042_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_6042_2_a_bh();".
// 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_6042_2_a_bh( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6042_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1200, 9304, 16812, -9466, -22200, 6938, 9639, -2914, -1707, 537, 123, -40, -3, 1]>,<7,R![3968, -15168, -62896, 25904, 82128, -42164, -28197, 23604, -2746, -1634, 446, 6, -12, 1]>,<11,R![53760, -401536, 182464, 504464, -342376, -107416, 117034, -7136, -10792, 1681, 365, -75, -4, 1]>],Snew);
    return Vf;
end function;