
// Make newform 4334.2.a.c in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4334_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_4334_2_a_c();".
// 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_4334_2_a_c();".
// 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 := [16, -2368, 4764, 10764, -17609, -15232, 23882, 9174, -16035, -2233, 5845, -25, -1172, 112, 121, -19, -5, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4964148712500, 583468762702830, -1094164592554507, -2148171715451517, 3823696790702626, 1809259031989422, -3958689755510945, -243727627646008, 1778312632644080, -241566735409474, -380428946367245, 104049504854782, 34534806713985, -15372280076054, -276196522876, 786165390578, -91723570911], [-67533919204240, -190539934184444, 1250907254720430, 1880353988143381, -4188162954706204, -3642185079524894, 5014433712810048, 2957304042810433, -2877690011328757, -1208965498831175, 890270534285541, 260132191219108, -152957243097834, -28005362835511, 13788245789985, 1184304247799, -509076733869], [162671423964068, -653207391574582, -259140355245027, 2974914526161209, -637811258428806, -4377893946856266, 1399869917022719, 3049473086762270, -1020614188326326, -1137379446949248, 371564728816921, 232297282536018, -73288570687139, -24371480448752, 7484292506002, 1021132038096, -308784775837], [7745938632420, -738484297008254, 1433570651843203, 2763784513173272, -4532670283647386, -3219633454980436, 4770019421347531, 1676629791575657, -2352361040909301, -448074396137981, 612993286559696, 62761530204714, -86308107482081, -4233286317113, 6139477315997, 101241374385, -170336316612], [-71489484909776, 699171181058340, 553722445375278, -4844986973009815, -643781242374876, 9331178508472058, -385514686185256, -7673676783924067, 966410756702031, 3154673661644085, -554074901045703, -681673033236444, 142237379376294, 73891352351813, -17190975335075, -3155392683109, 792005346607], [55249696318436, 284793480825738, -383591168856383, -2625578094942696, 827860531961810, 5638018424007148, -1050848965066127, -4899898478366905, 875140346691645, 2086739538050589, -405685489914624, -462294466169742, 99007575554409, 51037527914021, -11895288323205, -2209375896877, 551710330124], [-98451203132072, -1612354330066280, 1857398741300468, 9234594545289095, -3807166580939632, -15402786527022570, 3869815539651202, 11666951225319209, -2506215018753661, -4593848250975307, 986191176025127, 971208140521624, -219253669780124, -104178816010035, 24984965598945, 4428902062779, -1123737079895], [178861298752392, -1825362262669216, 1135223370299772, 9270217038809193, -3273334092249312, -15626846103463182, 3930628145790070, 12044462867808527, -2752846504349003, -4777756363650341, 1100935009107897, 1008802697283824, -241932951972556, -107498069830549, 26996550686535, 4528746272453, -1187737792121], [23820479731352, 3475406967932200, -4144145790814952, -15485113384874845, 12827251314474584, 22395644634392166, -14368967508564538, -15087835246362591, 7962004552994011, 5386138146173405, -2436418421323981, -1048002194817432, 420742355803808, 104709554236013, -38465640240887, -4190428658205, 1445357887021], [-12473615943512, 346142064758124, 44988271684781, -2113472564481506, 108790546795678, 4117830218063230, -527197725468509, -3462404729991225, 606574573474745, 1449122506411195, -299039518477814, -316690240094120, 72444713374627, 34515483633765, -8493223265449, -1475838823975, 383736935734], [16731542670136, -2790162624296960, 3951919641881596, 15234945854121633, -12462398010825744, -23408078923966270, 14235089730434214, 16248943297829479, -8080186123962899, -5907893263690797, 2535124096333457, 1163669685606672, -447824660101724, -117236889965981, 41714738148287, 4717981222477, -1590141174737], [-125227015713032, -1837200248163568, 3715669632931636, 12484017019792205, -9767678056660192, -22572212212191654, 10864406052087158, 17767005988662707, -6562435204181839, -7109172898477521, 2282204656442653, 1507572914869376, -451643483317444, -160966593761713, 46859464594843, 6784781527537, -1962307634045], [119091380419960, 2653170565039744, -5766309642938088, -15818173966589971, 14659368392183800, 27088420199961090, -15314923376756766, -20501251571511737, 8657402276946813, 7970289941119843, -2837599922010899, -1655987550453792, 535269693491576, 174333688782267, -53559488228625, -7277989925923, 2185052574659], [3463324569132, 990980801586774, -1870428444124007, -6480869452916690, 5160975111594602, 12201022833911164, -5822963705597639, -9977361588247115, 3598239953725903, 4099973184415807, -1283089987028538, -885116348957066, 259525375274885, 95663028125819, -27394453721783, -4066576751287, 1162170575998]];
    Rf_basisdens := [1, 1, 1, 20274514939304, 40549029878608, 20274514939304, 10137257469652, 40549029878608, 20274514939304, 40549029878608, 40549029878608, 40549029878608, 10137257469652, 40549029878608, 40549029878608, 40549029878608, 20274514939304];
    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_4334_a();"
function MakeCharacter_4334_a()
    N := 4334;
    order := 1;
    char_gens := [1971, 199];
    v := [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_4334_a_Hecke(Kf)
    return MakeCharacter_4334_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, 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, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, -1, 0, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1], [-1, 0, 0, 1, 0, -1, 1, -1, 0, 0, 0, 0, 1, 0, 1, 1, 0], [1, 0, 0, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 1, 0, 0], [1, -1, 0, 0, 0, 0, -1, 0, -1, -1, 0, -1, 1, 0, -1, 0, 0], [1, -2, 0, -1, 1, 1, -1, 0, 0, 0, -1, 0, 0, -1, 0, -1, -1], [0, -1, 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, 1, -1, 0, 0, 0], [-4, 1, 0, -1, 1, -1, 2, -1, 1, 0, 2, 3, -1, 0, 2, 0, 1], [-1, 0, 1, 0, -1, 0, 1, 0, 1, -1, 0, 1, -1, 1, 0, -1, 0], [-1, 1, -2, 1, 0, 0, 1, -2, -1, 1, 0, 0, 0, -1, 2, 2, 2], [1, 0, 0, 0, 0, -1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0], [1, -1, 0, 0, 1, -1, -2, 1, 0, 0, 1, -1, 0, -1, 0, 1, 0], [-1, -1, -1, 0, 0, -1, -1, -1, -1, 0, -1, -2, 1, 2, -1, 0, -1], [0, -1, 1, 1, 0, -1, 2, -2, 0, 0, 0, 1, 0, 1, 1, 1, 0], [-1, 0, 0, -2, 1, 0, 0, 2, 1, 0, 0, 2, -1, 1, -1, -1, 1], [-3, -1, 1, 0, 1, 0, -1, 3, -1, 0, 0, 0, 0, 1, -2, 0, 0], [-2, 0, 0, -2, -1, 1, 0, 0, 1, 1, 0, 2, -4, -1, -1, -1, 1], [-1, -2, 0, -2, 1, 2, 0, 1, -1, 0, 1, 1, 0, -1, -1, -1, 0], [-1, 1, -1, -1, -1, 1, 1, -1, 0, 2, -1, -1, -1, -1, 1, 0, 0], [2, -1, -1, 1, 0, -1, 1, 0, 1, 2, -3, -2, 1, 1, -3, 1, -2], [-6, 3, 1, 0, -2, 1, 2, 1, -1, 0, 4, 1, -1, 2, 1, 0, 1], [0, -1, 1, 2, -1, -2, 2, -4, 0, 0, -1, 0, 0, -1, 1, 0, -1], [4, -2, 1, -4, 2, -1, -2, 0, 3, 2, -1, 0, -1, -1, -1, 0, 0], [-4, 0, 1, 1, 1, 2, -1, 1, -2, -1, 1, 0, 2, -1, 0, 1, 1], [1, 0, 1, -4, 1, 1, -3, 2, 2, 1, -1, 0, 1, -1, 0, -2, 0], [-3, -2, 1, -1, -1, 2, -1, 2, 0, 1, 1, 0, -3, -1, 0, -1, 0], [0, 0, -1, -1, -2, 1, -2, 5, -1, 1, -1, -3, -1, 0, -2, -1, 0], [-6, 0, 1, 1, 2, -1, 0, 1, -2, -1, 4, 2, 2, 2, 2, 0, 3], [-2, -1, 0, 2, 2, -2, 1, -3, 1, 0, 0, 1, 1, 1, 0, 2, 1], [3, -3, 4, -1, 2, 0, 1, -3, 4, 0, 3, 1, 0, -1, 2, 3, -1], [-5, 1, -1, 1, 0, -3, 2, 3, -2, 1, 2, -1, 0, 4, -2, 0, 1], [2, 2, 0, -1, 2, -3, -2, 1, 2, -1, -1, -1, 1, 1, -1, 0, 1], [-1, -2, -1, 0, 2, 2, 1, -2, 0, 1, 0, 2, 0, -2, 5, 0, 2], [0, 0, 0, 7, -1, -3, 1, 0, -3, -2, 1, -3, 3, 3, 0, 1, 0], [-5, 2, 0, -1, 0, 1, 3, -1, 3, 2, 2, 3, -6, -2, 2, 1, 2], [7, 1, -2, -3, 0, 3, -2, 3, 1, 1, -4, -3, 2, 1, -6, -2, -4], [-1, 2, 0, 0, -3, 2, 1, -3, 4, 1, -3, 0, -1, -1, 3, 0, 0], [2, 3, 1, 3, -3, 0, 2, -1, 2, 2, -3, -2, 1, 1, -1, 1, -1], [-3, -2, 1, 1, 2, 1, -2, 4, 2, -1, 3, -2, -1, -1, 1, 1, 1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-6, 1, 2, -1, 0, 2, 0, 6, -3, -2, 2, 1, -1, 0, -1, -3, -1], [-3, 2, 0, -1, -2, 2, -2, 3, 2, 1, -1, 1, -1, -2, 0, -3, 0], [-2, 3, 0, 1, 3, -1, 1, 3, 3, -1, 4, 2, 0, 1, -1, 1, 1], [-4, 1, 0, 0, 1, 4, 0, 3, -3, 2, 0, 0, 0, 1, -1, 1, 1], [-4, 0, 0, 0, 0, 0, -3, 5, -3, 0, 1, 1, 1, 1, 0, -1, 0], [4, -3, 3, 1, 1, 1, -4, 2, 2, -3, -1, -1, 2, 1, -2, 0, -3], [0, -1, -1, -2, 0, 2, -2, 2, -5, -2, -3, -2, 1, 2, -5, -3, -2], [-7, 1, -1, -1, 1, 2, 0, 1, -3, 0, 0, 0, 0, 1, -1, -1, 0], [5, 0, -2, -1, 1, -2, -1, -2, 2, -1, -1, 2, -2, 2, 0, -1, 0], [1, 2, 0, 1, -2, -2, 3, -2, 1, 1, -3, 0, 3, 0, 0, -2, -2], [3, 1, -3, 3, 2, -3, -1, -2, 0, -2, -1, -3, 2, -1, 1, 1, 3], [2, 1, 3, 0, -2, -2, 1, -1, 3, 1, 3, -1, -3, 1, 0, 0, -2], [-2, 1, 0, -1, 2, 1, -6, 1, 1, -1, -3, -2, 3, -1, -1, 0, -2], [-2, 4, -5, -1, -1, -5, 6, -7, 3, 3, -4, 2, -3, -1, 1, -2, 0], [-7, 0, -2, 1, -2, 1, -3, 4, -7, -2, -1, -1, 4, 1, 0, -2, 3], [3, 2, -2, -3, 1, 1, 4, -8, 4, 4, -4, 1, -2, -4, 4, 2, 1], [-2, 2, 1, -2, -3, 3, 1, 3, -1, -3, 0, 0, -5, 3, -6, -5, -3], [4, -2, 3, -1, 4, 4, -1, 2, 1, 0, -1, 0, 3, 1, -3, 0, -2], [1, 2, 1, -4, 2, 6, -1, 4, 1, 0, -3, 0, 1, -2, 1, 0, -1], [-10, 4, -4, 1, -4, 0, 2, -3, -1, 3, -1, 2, -3, -3, 5, 1, 5], [5, 3, -6, 5, 0, -6, 5, -7, -3, -2, -2, -1, 3, 1, 0, 1, 0], [-4, 4, -2, 4, -1, -4, 1, -5, -3, -2, 1, 0, 3, 2, 1, 2, 0], [-4, 3, -1, 1, 1, 1, -3, -2, 0, -3, -2, -1, 0, 0, 0, 0, 0], [5, -3, 4, -1, -1, 0, -1, 6, -2, -1, -1, -1, -2, 3, -4, -2, -1], [-1, 0, 2, -2, -1, -2, -3, 1, 1, -3, -1, -4, -1, 0, -3, -2, -1], [4, 0, 1, -2, 2, 1, 3, -1, 1, 2, 0, -2, 0, -2, 0, 4, -1], [-1, 0, 5, 3, -1, -1, -5, 3, 2, -2, 1, -3, 0, 0, 1, 1, 0], [-3, 4, 3, -2, 0, 1, -2, 6, -3, -1, 6, 0, 2, 3, -2, 1, 1], [-5, 2, -6, 3, 0, -7, 4, -6, -2, 0, -1, -1, -1, -2, 4, 3, 2], [0, 0, -2, 5, -2, -5, 0, 0, -1, -1, -2, -5, 5, 2, -4, 4, 0], [6, 1, -3, -3, -1, 1, 3, -2, 4, 5, -3, 2, 0, -3, 3, 2, 3], [3, 2, 0, 1, -4, -2, -1, -2, 2, 1, -1, -2, 0, -1, -1, 0, -1], [-4, 3, 1, 0, 1, 2, -2, 1, -2, 0, -1, -2, 1, 0, 0, 0, -1], [1, -1, -3, -1, 0, 0, 2, -4, 0, -1, -7, -2, 3, -2, -1, -1, -3], [-21, 8, 1, 4, -5, 2, 4, -2, -1, 0, 4, 4, -2, 3, 7, 0, 3], [4, -1, -4, -2, 1, 0, -1, 3, -6, 0, 1, -1, 5, 1, -2, 2, -3], [-2, 4, -2, -2, -3, -3, 6, -8, 4, 3, -2, 2, -4, -5, 5, 2, 2], [-3, -1, -2, 6, 0, 0, -1, 0, -3, 0, -5, -1, 2, 0, 2, 1, 1], [-5, 0, -2, -3, 1, 2, -2, -2, 3, 0, 0, 1, -4, -4, 2, -2, 0], [5, -4, 3, -3, 2, -4, -3, 3, 6, -2, -1, 0, 0, 0, -5, -1, -2], [3, -1, 2, -3, 1, -2, -1, 0, -2, -1, 2, 0, -1, 1, -2, -2, -3], [2, -2, 0, -1, -5, 3, -3, 1, 0, -3, -4, -3, -2, 1, -2, -5, -2], [-3, -4, 6, 3, -3, 4, -5, 5, -1, -2, 2, -5, 2, 3, -1, 0, -2], [-7, -3, -1, -1, 3, 1, -2, -2, -3, 2, 0, 2, 0, -1, 1, 0, 0], [7, -1, -1, -5, -3, 3, -2, -2, 5, 3, -6, -3, 0, -4, -3, 0, -1], [2, 0, 2, 7, 0, 3, 3, -1, 2, 1, 1, 0, -1, -3, 0, 2, 1], [8, -10, 5, -1, 2, 0, -7, 2, 2, -5, -6, -3, 4, 0, -5, -4, -6], [-11, 7, -2, 3, 0, -2, 0, 1, -2, -1, 7, 2, 1, -1, 5, 1, 6], [-18, 5, -8, 5, -1, -6, 8, -8, -4, 2, 0, 2, -1, 0, 6, 2, 3], [-12, 4, 1, 0, 0, -2, 7, 2, 0, 0, 4, 7, -4, 1, 1, -1, 0], [-1, 1, -4, 6, -3, -3, 1, -2, -4, -2, 1, -2, -1, -1, 1, 3, 0], [3, -5, 0, -1, 2, 2, 0, 0, -2, 0, -3, -1, 5, -1, -3, -2, -7], [-10, 5, -5, 2, -1, -2, 3, -6, 0, 0, 1, 5, 0, -3, 7, -1, 4], [4, -8, 5, 4, 1, -1, -8, 6, -3, -4, 4, -7, 7, 4, -7, 2, -3], [-7, 4, -3, -1, 1, 1, 0, 0, -5, 2, -2, -3, 3, -1, 0, 0, -1], [-1, 0, -6, 2, -2, -6, 7, -4, -2, 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1, -1, -1, 1, 2, 1, 7, 0, -1], [-7, 7, 4, 2, 1, 12, 1, 8, 5, -1, 0, 7, 9, -7, 8, 1, 6], [3, -2, -1, -7, -4, 1, 3, 1, 15, -1, -8, 5, -3, -7, -6, -3, -1], [7, -11, 3, -8, 7, 13, -15, 10, -2, 0, 2, -3, 0, -4, 0, -13, -3], [-1, -7, -4, 7, 6, -16, 5, -8, -6, 2, 0, -8, 11, 12, -7, 13, -1], [0, 1, 14, 1, 10, -6, -5, -4, 4, 2, 9, -4, 2, 3, -2, 8, -3], [6, 1, 8, -5, 16, -4, -4, 13, 3, -10, 2, 1, 2, 7, -7, -1, 1], [-11, 10, 0, 2, -13, -1, -9, 6, -2, -1, -7, -8, -9, 3, -9, -10, -4], [-12, -7, -9, 4, -6, -8, -2, 1, 1, -2, -7, -7, -6, -2, -4, -4, 2], [4, 13, 5, -7, 7, 2, -18, 16, -5, -4, 7, -7, 13, 3, -9, -2, -4], [-8, -6, -7, 6, 3, -6, -4, -6, -14, -4, 1, -10, 11, 13, -1, 2, -5], [-3, -2, -9, 19, -13, -6, 8, -8, -17, -2, -13, -16, 2, -5, -1, 0, -5], [1, 1, 1, 12, 1, 9, 4, -3, 2, -5, 2, 6, 0, -5, 4, 0, -1], [5, 0, -8, -9, 14, -13, 5, -7, -4, -1, 8, 7, 5, 2, -4, -2, 2], [-25, -3, 6, 11, -3, 0, -5, 24, -17, -13, 8, 2, 2, 14, -3, -12, 0], [-7, -1, -9, 10, -4, -16, 5, -11, -13, -4, 12, -3, 0, 6, 6, 2, 11], [-7, 7, 12, 9, -9, 1, 0, 0, 8, 0, 3, 2, 0, 6, 9, 5, 3], [-1, -3, 14, -3, 8, 9, -16, 19, 4, -7, -2, -1, 3, -1, -10, 0, -9], [-5, -1, -9, 2, -2, 3, 17, -23, 3, 6, -5, 5, -9, -11, 9, 0, 1], [9, 10, -12, 13, -2, -15, 4, -28, -8, 1, 0, -4, 11, 0, 11, 12, 4], [17, -9, 3, -2, 1, -11, -12, -5, 13, -3, -10, -10, 0, 1, 1, 5, 0], [-12, -2, -2, 1, -4, 12, -4, 7, -3, -1, 2, -3, -3, -4, 3, 0, 7], [6, -11, 0, -7, 8, 8, -2, -6, 9, 8, -3, 3, -6, -11, 10, 3, 3], [11, -9, 3, -13, -3, 7, -2, 3, 18, 6, -9, 8, -7, -10, -7, -2, -9], [-10, 6, 5, -9, 4, 5, -2, 6, 0, 0, -2, 0, 1, 4, 0, 2, -6], [16, -2, 1, 11, -6, -6, -4, 6, -7, 0, 5, -8, 3, 9, -5, 10, 6], [-29, 5, -1, -6, 3, 9, -11, 10, -3, 0, 11, -1, 1, 2, -2, 2, 7], [17, 9, 5, -12, -1, -3, 8, -9, 13, 11, -3, 12, -11, -1, 13, 4, 5], [32, -17, 10, -11, -4, 8, -12, 2, 2, 1, -5, -10, -1, 6, -10, -8, -17], [4, -4, -9, 6, 3, -6, 8, -15, -6, 5, -2, -2, -4, -5, 7, 9, 10], [14, 11, 6, -8, -1, 4, -2, -1, 16, -1, 0, 5, 2, 3, 5, 1, 2], [-2, -2, 10, 8, 0, -4, -2, -5, -1, -2, -5, 1, 13, 7, -1, 3, 0], [20, 2, -12, 10, 0, -15, 11, -21, 0, 8, -6, -4, -2, 5, 7, 8, 0], [12, -3, -6, 2, 7, -12, 3, -6, 2, 9, -6, -6, 2, -2, -6, 6, 0], [-12, 9, -2, 19, -3, -1, 13, -20, -8, 0, 15, 7, 0, -8, 29, 12, 13], [-22, 6, 0, 0, 6, -6, 10, -9, -6, -7, 13, 5, 3, 5, 3, 7, 5], [25, -9, 5, -9, 1, 5, -6, 13, 4, -6, -3, 3, -10, -1, -14, -13, -4], [-8, -3, 8, -2, 0, -4, 6, -2, 2, -6, 14, 3, 4, -2, -3, 2, 0], [14, -2, -1, -11, 1, 4, -7, 5, -2, 0, 3, 9, -9, 1, -8, -11, 1], [28, -9, 9, -1, 2, -9, -3, 1, 14, 3, -10, -1, -6, -2, -9, 2, 1], [-6, -9, 6, -15, 14, 6, -3, 13, -4, 2, 4, 7, 5, 1, -1, -6, 2], [7, 6, -5, 5, -5, -7, -1, 7, -12, 0, -3, -7, 0, 4, -13, -6, -4], [10, -9, 12, 0, 6, -1, -12, 12, 0, -12, -2, -4, 11, 8, -9, -7, -9], [-3, 6, -13, -21, -5, 9, -3, -3, 7, 0, -7, 1, -4, -4, -4, -11, -5], [0, 9, -5, -4, 1, 0, 15, -13, 12, 10, -4, 13, -7, -13, 8, 7, 10], [-17, -5, 1, 0, 6, -1, -10, 11, -8, -9, 12, -2, 3, 6, -2, -4, 7], [0, -20, 5, -13, -1, -8, -14, -4, 0, 1, -9, -3, -7, -1, -12, -4, -3], [-5, -4, 9, 14, 2, 1, -3, 9, -4, 0, -8, -7, 12, 10, -5, 3, -2], [-17, 2, 7, 13, 1, 0, 13, -12, 2, 8, 1, 6, 3, 3, 9, 16, 0], [-4, 0, -1, -1, -6, -5, 1, -9, 10, 10, -5, 7, -8, -1, 11, 12, 4], [8, -22, 19, -1, -1, 16, -4, 8, -7, -13, 13, 8, -2, 6, 0, -9, -4], [-11, 7, -10, 9, -1, 2, -6, 12, 3, 3, -11, -16, 9, 3, -12, 1, -5], [5, -6, 7, 8, -4, -4, 8, -14, -1, -2, 1, -2, 7, -5, 4, 4, -1], [-2, -1, 9, -5, 9, 5, 1, 7, -14, -7, 10, 13, 4, 5, 0, 4, 1], [10, -8, -10, 5, -5, -3, 6, -10, -13, -5, 1, -4, 6, -1, -4, 8, 2], [5, -6, -5, 6, -8, -9, 3, -27, 5, 2, -3, -4, 2, -2, 15, 13, -2], [-9, 7, -9, 3, 5, -10, 13, -15, -6, -2, 2, 2, 0, 8, 4, 2, 3], [12, 14, -5, 1, 1, -10, 0, -12, 4, -5, -4, -9, 13, 3, 5, 5, -4], [-21, -3, -11, -1, 2, -2, -6, 10, -3, 6, -10, -8, 8, 0, -7, 0, 0], [-18, 1, -22, 15, -1, -10, 14, -12, -10, 3, -6, -6, 9, 0, 5, -4, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4334_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4334_2_a_c();".
// 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_4334_2_a_c(:prec:=17)
    chi := MakeCharacter_4334_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_4334_2_a_c();".
// 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_4334_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4334_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, -2368, 4764, 10764, -17609, -15232, 23882, 9174, -16035, -2233, 5845, -25, -1172, 112, 121, -19, -5, 1]>],Snew);
    return Vf;
end function;