
// Make newform 4680.2.l.j in Magma, downloaded from the LMFDB on 01 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4680_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4680_l_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_4680_2_l_j();".
// 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_4680_2_l_j();".
// 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 := [128, -1792, 12544, -43872, 91808, -111264, 79076, -25304, 3808, -4696, 6361, -976, 32, -40, 150, -8, 0, 0, 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], [-68565952232414162187528016, 692776020724577646191190592, -2524144530329422204749450696, 4540598497335753486687335804, -3850899916201963107870542040, 1096370273722153492906699780, 316241963949501348147598822, 128903738329370493388078471, -390961529601349289925988345, 6237738507450128557650641, 16075405080062638225266940, -937990942159963409708790, -10507555813301979209878898, -1107235851566488504042706, 139877778373472263571070, -21380934606708396430641, -74816531663221108681441, -11907746234648643444495], [32231579070800384671282064, -341116536045083242826180864, 1323567921505648772591404288, -3195631428726102279996865900, 4203857065504314738627621504, -3243174258044189122267234964, 1083794399834343241734204558, -143335065738050578214842091, 147278129804198506370300447, -270575135338550260831570549, 42548065411152503631888604, -1560370300669182093177690, 305067522437638617466966, -6525230512258776752650238, 350454672019112697432766, -3738932717577904769547, -9899430031205324744569, -44022066037115180404173], [272346251338929167109470848, -2809243799886769563470686720, 10704146218295810019270696272, -21982558054543139856898849024, 24191596558989135359463839256, -14472159480695918830246155712, 3515562915719299184767853228, -784411673576613073446060746, 1410381490764708797190333118, -1027153912427544042520190607, 129818843959422951591365256, -2325033799553998837177700, 24236526643165404252070460, -22112141189536697518370702, 1052069950003334912776156, 44962097382019178485350, 127395174065218568210982, -139793463203358946176127], [-6238344009455475703877024, 778968983201466234097909632, -4988023519684510296743548208, 14354307251336355623356202784, -21138133493816256266812685584, 16178230344096991966204984356, -5135520209884017275565387916, 113613613246564575603057688, -1041786447272687620293414130, 1265506546997985047208745545, -203115344584232834015369064, -14099142185869825825128464, -13431911892369800090513852, 29216438345419395820581094, -1675553830889703963807868, -155991623906635775633800, -47193255106133817460714, 192458429560395857394305], [-363125508114690949204767712, -2186166136020952345047681024, 13999913574515664858854857776, -40301461586179775858726715360, 57115876539773675396835996416, -45849928916553296139398055332, 15368999240904465833578903116, -1495562784400427784779791516, 2011688409311220856638634184, -3825942009839365498534658563, 609690943878472312083784840, -7400567679497258236938648, 4442960371252285357896072, -92182260400544099887324570, 5035544586312857588324476, 52640661688198920998372, -132599507573809711755280, -621534373008443244782523], [-21205642838434425382097584, -959613573847328533115542224, 6408781201571083190652051864, -18454675086146835626039820672, 25727386034846190737149191260, -21183363640313916895452801698, 7455806939960242576766231334, -1202717973713771977359700537, 555482230301327987998807838, -1872182161620194392310438197, 296528450488721823937764580, -24024397273308534463461606, -11873803091703282804712680, -46678278102714350548864288, 2451563715208084176196726, -144934913101817581261117, -172926241165279621102906, -320294570363696444423037], [536402629722202052159600288, -5731782438062354707547580256, 23734730761840717106613397392, -51296013463441707167413443288, 65654788294673565367660725200, -48926716021622053790578099560, 17156728512784474693526598484, -1995834615846999468757727608, 2469751452142329796288573114, -4074494639810805982878941513, 668947866234094511486492616, -12940661662727224743946112, 11521206302688294004220652, -98143369094393459318481826, 5497038887404604880332916, 38934587558230772215584, -99280627860275351455694, -661661416643658239613505], [4301169480182196106895152, 1140661129615319118686480304, -7042069982544551931143347608, 20277979565170840892341241376, -28265225264874295344797537912, 23266005910461722977039651202, -8169550847680435226885678886, 1293137851220229826998667639, -628385115495180074291553917, 2050530638538477534733526080, -324876130659340029095886292, 25276810396524271791214338, 12328795654147771765445142, 51045723459855256344964150, -2685801895133585100123862, 150100481935243818662971, 184143039318013335411547, 349992860457312418656744], [-570211576438706510710005152, 5929558885738893759761655520, -25001308323787654587595988880, 54942622421489717700016284696, -70730466754729774582957418504, 53092000561917665953751798568, -18584216328224859993765493588, 2176674370859915168035661812, -2615557222530033968874065272, 4431191593647372267725117279, -725643226575330921802736040, 15445487909158699399451576, -10611221177799316082755728, 106878259808675270910681550, -5965515247255606728187188, -28603449891378297411876, 121714224165742780072976, 721057996830890188080919], [436379939497335316223444416, -2399703834860226045708910912, 4034857499834607920673656352, 3262941448017351167867712468, -18404714814936776031935681072, 22281874317798384482649697840, -9612743032723679612044437122, -406210707574478823946578065, -52518308245011700365831273, 2074093150363163102763361296, -397373457118326698351132660, -19634664760925954553069686, 22561727388088353067940166, 53068653822017565195806080, -3314072842792951466298050, -166245092674182130498465, 240393553987329286719183, 368686100698569195172976], [71817640120381736345581984, -483391356948271052752384448, 1008684702011637795181092480, 11881978985281054984020369960, -33518470651973229753038227760, 41708224850608778704990700568, -19073496812752331172825145468, 1270387507627616897536650100, 961247301505981946586320138, 4168740930516673966729612481, -779890581067417095204415752, 36607909852650558951487664, 82865395947729014941278444, 110701439862204394088563978, -6488663041087207400262876, 273068581022650103134980, 778452773236316086876034, 782730547332396401831233], [-13356599924293353683887040, -518180172114703510006434976, 3318207687451036713656657816, -9550316229597148644241462624, 13947259850136665702685796660, -10807035349651722785190344770, 3512848887822043738075832966, -194488519820057824296087891, 607551592320942227959486923, -869550018268763761210013518, 139121939725658972021848844, 4631655241856188842872130, 5700732770644577697351026, -20464623912531577672608230, 1148264984327112977142766, 64840321797813902696549, 5334803654711449347275, -136252564316050277752970], [293650224279838884461235488, -3798150827330801596059702336, 24321624228145336105594117632, -69999788996152799509460041056, 102377056274222495845602547376, -79158683340152562895740676256, 25631495414129324459049107712, -1281749405319307407945855202, 4555616610956705939719215104, -6340128048163321441337319301, 1014885210302480249308131376, 39704054079944069209315116, 45667697390495687929153512, -148755551743663122506492410, 8375798459154419592807232, 522520662795998389547822, 70410482179582194371720, -988737642650038889597589], [455434398992851441451377920, -4754200680580481952278485312, 19099363241853645610345272720, -40629635904345043167056393504, 49165275553718140501000710832, -34083794664931968934121163296, 10787028066988318852850650808, -1536483888728946274940601354, 2210806644112177096965770908, -2708516281727000796036252487, 415474995656522218205044080, -8195864628048600827982452, 23065227713606262150966544, -63322288645511778874903998, 3403938424211350999143736, 49040349468209538181590, 44079595598100124833140, -420132740648168731317175], [408995598322255652232907104, -6270484387170042834871814272, 28821616650242321476165039488, -68076116752162750903772124108, 85778813735067100005960369408, -58409831660250144712072636408, 16550311815995586240840201046, -1763443362511118155672982019, 4548440725051757799839362139, -4399155467971695395991441991, 638636445918569690312043404, 18793687397516143268292790, 67625289098143988692545462, -98972746254023813170847814, 5236321930164448804421558, 349579288474962519164317, 300472898500059006287195, -642485138256937123041895], [15735957753899561530320448, 865561768778931666020776736, -5543054271706851481170932760, 15958126713184371830315465248, -22717048480694755468212325896, 18200053921645832855919564514, -6185588723951491418517528742, 716033616997689684909817575, -722196490223567617598287541, 1543720498970225913021565624, -245563910352211381776322348, 7857844296076185150237334, 1309154290924256396619502, 37569027785913026518269786, -2028748833241111887716430, 19481338497761959214463, 77805761335065285184203, 254633159574342096906900], [667959225929295445316815952, -7024393264733327852576936512, 27641317409342515442263894248, -63146317265055723678950653980, 80167192380695754859183259648, -59268636060432824339958047652, 19089830048262305744851611928, -2697261816684084599376042757, 3151771948643165380592223379, -4837161326904968603679443450, 743833269170549091643358592, -25229684706064080840906062, 19524200173846320882571350, -115066509459593004476251744, 6114668930940015354578544, -26915223240145250538309, -76380127710498629579933, -770685309376171817332618]];
    Rf_basisdens := [1, 13693221988286843243983408, 13693221988286843243983408, 27386443976573686487966816, 27386443976573686487966816, 82159331929721059463900448, 41079665964860529731950224, 82159331929721059463900448, 41079665964860529731950224, 82159331929721059463900448, 41079665964860529731950224, 82159331929721059463900448, 13693221988286843243983408, 82159331929721059463900448, 27386443976573686487966816, 41079665964860529731950224, 13693221988286843243983408, 41079665964860529731950224];
    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_4680_l();"
function MakeCharacter_4680_l()
    N := 4680;
    order := 2;
    char_gens := [3511, 2341, 2081, 937, 1081];
    v := [2, 2, 2, 1, 2];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4680_l_Hecke();"
function MakeCharacter_4680_l_Hecke(Kf)
    N := 4680;
    order := 2;
    char_gens := [3511, 2341, 2081, 937, 1081];
    char_values := [[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], [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], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 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, -1, 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, -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, 0, 0, 0, -1, 0, 1, -1, 0, 0, 0, 1, -1, 0, 1], [-1, 0, 0, 0, 0, -1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 0, 1, 0, 0, 0], [1, 0, 0, 0, -1, 1, -1, -1, 1, -1, 0, 0, 0, -1, 0, 0, -1, 0], [0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, -1, 0], [0, 1, 2, -1, 0, 0, 0, 1, 0, -1, 1, -1, 0, 0, 0, 1, 0, 0], [-1, 0, 0, 0, 0, -1, 1, -1, -1, -1, -1, 0, 1, 0, 0, 1, 2, 0], [0, 0, -2, 1, 0, 0, 1, 1, 1, -1, -1, 1, 0, 0, 0, -1, 0, 1], [0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, -1], [0, -2, -1, 0, 0, 0, -1, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 1, 1, 0, -1, 0, 0, 0, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, -1, 2, 1, 2, 0, 0, -1, 1, 0, 0, 0, 0], [0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, -1, 0, 0, -1], [-3, 0, 0, 0, -2, 1, -1, 1, 1, 1, 1, 0, -1, 0, 0, -1, -2, 0], [0, -3, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 1, -1, 0, 1], [0, 0, 0, 0, -1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, -4, 2, 0, 0, 2, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 2, 0, 0, 0, 0, 0, -1, 0, -2, -1, 0, 1, 2, 0], [0, 0, 1, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 1, 0, 0, -1], [-7, 0, 0, 0, 1, -1, -2, 2, 2, 2, -1, 0, -2, -1, 0, 1, -1, 0], [0, 0, 2, -1, 0, 0, -1, 1, -1, -1, 0, 1, 0, 0, 0, 0, 0, 1], [0, 4, 3, -1, 0, 0, -1, 2, -1, -2, 1, 2, 0, 0, -2, 1, 0, -1], [3, 0, 0, 0, -3, 3, -2, 2, 2, 2, 2, 0, 0, -1, 0, -2, -3, 0], [0, -3, -5, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, 0, 0], [0, -2, -4, 1, 0, 0, 1, 1, 1, -1, -1, -1, 0, 0, 2, -1, 0, 1], [-3, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, -1, -1, 0], [0, -2, 0, 1, 0, 0, -1, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, -3, 0, -3, 0, 3, 0, 1, 0, 1, -1, 0, -1, -2, 0], [3, 0, 0, 0, -2, -1, -3, 1, 3, 1, 2, 0, 1, 2, 0, -2, -2, 0], [-2, 0, 0, 0, 3, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 3, 0], [0, -2, 0, 1, 0, 0, 1, 1, 1, -1, -1, 1, 0, 0, 0, -1, 0, 3], [0, 0, 0, -2, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 3, -1, 0, 1], [0, -2, 0, -1, 0, 0, 3, -2, 3, 2, 0, -1, 0, 0, 2, 0, 0, 3], [0, 2, -2, -1, 0, 0, -3, 1, -3, -1, 0, 3, 0, 0, 0, 0, 0, -1], [5, 0, 0, 0, -3, 1, -2, 0, 2, 0, 3, 0, 0, 1, 0, -3, -1, 0], [0, 0, 0, 0, -1, 0, -1, 2, 1, 2, -1, 0, -1, -1, 0, 1, 0, 0], [-4, 0, 0, 0, -2, 0, 3, -1, -3, -1, 1, 0, 2, 2, 0, -1, 0, 0], [0, 2, -3, 3, 0, 0, 2, 5, 2, -5, 2, 0, 0, 0, -5, 2, 0, -3], [0, 4, 0, 1, 0, 0, -1, 2, -1, -2, 0, -1, 0, 0, -2, 0, 0, -1], [-4, 0, 0, 0, 2, 1, 1, 3, -1, 3, -1, 0, -3, -3, 0, 1, 0, 0], [-2, 0, 0, 0, 0, 0, -4, 4, 4, 4, 1, 0, -2, 0, 0, -1, -4, 0], [0, -2, 3, 1, 0, 0, 2, 1, 2, -1, 1, -3, 0, 0, 0, 1, 0, 0], [0, -2, -2, -1, 0, 0, -1, -2, -1, 2, -2, 1, 0, 0, 2, -2, 0, -1], [-1, 0, 0, 0, 3, -3, 2, 0, -2, 0, -4, 0, 0, -1, 0, 4, 5, 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-2, 0, 0], [0, 8, -6, -7, 0, 0, -11, 11, -11, -11, 2, 3, 0, 0, 0, 2, 0, -5], [0, -5, 4, -5, 0, 0, -4, 1, -4, -1, 5, -7, 0, 0, -2, 5, 0, -2], [0, -7, 20, -2, 0, 0, -12, -1, -12, 1, -3, -10, 0, 0, 3, -3, 0, 1], [-9, 0, 0, 0, -22, 7, -7, 7, 7, 7, 8, 0, 1, 2, 0, -8, -10, 0], [8, 0, 0, 0, -8, 1, -1, 1, 1, 1, 0, 0, 9, 5, 0, 0, 6, 0], [0, 8, 32, 11, 0, 0, 1, 0, 1, 0, -2, 1, 0, 0, -2, -2, 0, -3], [-3, 0, 0, 0, -1, -5, -16, 4, 16, 4, -1, 0, -4, -3, 0, 1, -7, 0], [-28, 0, 0, 0, 11, -2, 5, 4, -5, 4, -2, 0, 1, -1, 0, 2, 4, 0], [0, 2, -10, 2, 0, 0, 16, 8, 16, -8, -3, 7, 0, 0, 3, -3, 0, 9], [-14, 0, 0, 0, -8, 0, -2, 1, 2, 1, 3, 0, -4, -4, 0, -3, 0, 0], [0, -4, 21, 4, 0, 0, 4, 5, 4, -5, 5, -7, 0, 0, 0, 5, 0, -4], [31, 0, 0, 0, -10, 1, 2, -3, -2, -3, 8, 0, 6, 4, 0, -8, 2, 0], [0, 0, 11, 7, 0, 0, 0, 11, 0, -11, -4, 8, 0, 0, -1, -4, 0, 5], [9, 0, 0, 0, 4, -3, 13, 7, -13, 7, -4, 0, 1, 0, 0, 4, 0, 0], [0, 4, 36, -1, 0, 0, 9, -5, 9, 5, -1, -3, 0, 0, 10, -1, 0, 5], [6, 0, 0, 0, -11, 4, -3, 8, 3, 8, 8, 0, 1, 1, 0, -8, -2, 0], [0, -9, 29, -5, 0, 0, -6, -2, -6, 2, -3, 4, 0, 0, 5, -3, 0, 0], [-5, 0, 0, 0, -8, 2, 0, -2, 0, -2, 5, 0, 8, 13, 0, -5, -2, 0], [20, 0, 0, 0, -2, -1, -5, 0, 5, 0, -6, 0, 1, -1, 0, 6, 6, 0], [0, -5, -7, -1, 0, 0, 9, 5, 9, -5, -2, 6, 0, 0, -1, -2, 0, 10], [0, -4, -12, 5, 0, 0, 5, 2, 5, -2, -6, -7, 0, 0, 8, -6, 0, 3], [0, 8, 12, -4, 0, 0, 0, 2, 0, -2, -4, 0, 0, 0, 8, -4, 0, 2], [-6, 0, 0, 0, -2, 6, -3, -5, 3, -5, -3, 0, -2, -6, 0, 3, 4, 0], [0, 5, 21, -8, 0, 0, -5, 7, -5, -7, 1, -3, 0, 0, 5, 1, 0, 3], [-2, 0, 0, 0, 12, 1, 1, 6, -1, 6, 2, 0, -3, 5, 0, -2, -4, 0], [0, 10, 2, -5, 0, 0, -11, 3, -11, -3, 1, 3, 0, 0, -4, 1, 0, -3], [0, -4, -4, -2, 0, 0, -2, -10, -2, 10, -6, -6, 0, 0, 12, -6, 0, 10], [0, 4, 6, 5, 0, 0, 5, -2, 5, 2, -4, 13, 0, 0, 6, -4, 0, -1], [0, 3, 19, 2, 0, 0, 2, 6, 2, -6, 0, 8, 0, 0, 2, 0, 0, -4], [18, 0, 0, 0, 1, 12, -7, 4, 7, 4, 5, 0, -3, -5, 0, -5, -6, 0], [17, 0, 0, 0, 13, -5, 10, -6, -10, -6, -3, 0, 2, 3, 0, 3, 7, 0], [10, 0, 0, 0, -12, 4, 2, 8, -2, 8, 8, 0, 2, 2, 0, -8, -4, 0], [0, 2, 18, 3, 0, 0, -1, -12, -1, 12, -4, 5, 0, 0, 6, -4, 0, -5], [17, 0, 0, 0, -7, -7, -8, 10, 8, 10, -3, 0, 0, -7, 0, 3, -1, 0], [-4, 0, 0, 0, 3, -5, -2, 3, 2, 3, -4, 0, 2, -6, 0, 4, 12, 0], [0, -6, 18, -6, 0, 0, -10, -4, -10, 4, -1, -6, 0, 0, 10, -1, 0, 2], [-17, 0, 0, 0, -8, -4, -2, 8, 2, 8, -2, 0, 0, -1, 0, 2, 0, 0], [-2, 0, 0, 0, 24, -4, 14, 0, -14, 0, -10, 0, 0, 6, 0, 10, 12, 0], [0, -8, 1, 1, 0, 0, 2, -1, 2, 1, -8, 6, 0, 0, 3, -8, 0, 5], [-8, 0, 0, 0, 0, 4, 14, -6, -14, -6, 2, 0, 8, -4, 0, -2, 4, 0], [0, 0, 38, -2, 0, 0, 12, -2, 12, 2, 0, -8, 0, 0, 6, 0, 0, 2], [-16, 0, 0, 0, 0, 2, -6, 13, 6, 13, -2, 0, -6, -6, 0, 2, -8, 0], [0, -4, -40, 3, 0, 0, -1, -2, -1, 2, -4, 9, 0, 0, 2, -4, 0, 5], [0, -4, -23, 5, 0, 0, -14, 9, -14, -9, -5, 7, 0, 0, 2, -5, 0, 6], [0, -4, -8, 5, 0, 0, 11, -2, 11, 2, 2, -5, 0, 0, -14, 2, 0, -1], [-7, 0, 0, 0, -6, 5, 0, -7, 0, -7, 4, 0, 8, 6, 0, -4, -6, 0], [0, 6, 8, -5, 0, 0, 1, 2, 1, -2, 4, -5, 0, 0, -2, 4, 0, 3], [10, 0, 0, 0, -13, 10, -12, 1, 12, 1, 14, 0, -4, 3, 0, -14, -15, 0], [0, -6, 10, 0, 0, 0, 2, 0, 2, 0, 6, -12, 0, 0, -12, 6, 0, -10], [0, 5, -22, 1, 0, 0, -4, 11, -4, -11, 0, 6, 0, 0, -9, 0, 0, -5], [0, -17, -4, 9, 0, 0, 1, 0, 1, 0, -5, -3, 0, 0, 7, -5, 0, 2], [2, 0, 0, 0, -12, 2, -6, -2, 6, -2, 2, 0, 4, -6, 0, -2, -2, 0], [0, -4, 20, 0, 0, 0, 0, -3, 0, 3, -2, -8, 0, 0, 4, -2, 0, 0], [-3, 0, 0, 0, 4, -4, 4, -10, -4, -10, -6, 0, 0, 1, 0, 6, 12, 0], [0, 4, -16, 12, 0, 0, 0, 8, 0, -8, -6, 8, 0, 0, -4, -6, 0, -4], [-11, 0, 0, 0, -2, 5, -5, -5, 5, -5, -3, 0, -9, -16, 0, 3, -14, 0], [0, -6, -22, 2, 0, 0, -8, -4, -8, 4, -3, 0, 0, 0, 0, -3, 0, -2], [18, 0, 0, 0, -15, 6, -2, 9, 2, 9, 2, 0, -6, 5, 0, -2, -7, 0], [10, 0, 0, 0, 2, 6, -15, 3, 15, 3, 5, 0, -6, -2, 0, -5, -8, 0], [6, 0, 0, 0, 23, -4, 3, 0, -3, 0, -2, 0, 1, 9, 0, 2, 6, 0], [4, 0, 0, 0, 4, -4, -12, 7, 12, 7, 0, 0, -8, 4, 0, 0, -12, 0], [17, 0, 0, 0, -11, -1, 10, 0, -10, 0, 2, 0, 8, 5, 0, -2, 7, 0], [0, -1, -32, -4, 0, 0, 0, -1, 0, 1, -5, 4, 0, 0, 9, -5, 0, 5], [0, 5, -15, -10, 0, 0, -2, 6, -2, -6, 6, -6, 0, 0, 4, 6, 0, 2], [0, -4, -14, 3, 0, 0, 11, 4, 11, -4, -4, -1, 0, 0, 2, -4, 0, 9], [3, 0, 0, 0, -12, 1, -3, -1, 3, -1, -2, 0, -3, -6, 0, 2, -8, 0], [16, 0, 0, 0, -7, 2, -13, 8, 13, 8, 6, 0, -3, 11, 0, -6, -8, 0], [0, -16, 14, -4, 0, 0, 0, -16, 0, 16, -6, -8, 0, 0, 16, -6, 0, 4], [4, 0, 0, 0, 4, 5, 9, -10, -9, -10, -1, 0, 5, -7, 0, 1, 0, 0], [0, 12, 16, -6, 0, 0, -16, -4, -16, 4, 1, 1, 0, 0, -3, 1, 0, -5], [-7, 0, 0, 0, -6, -10, -4, 8, 4, 8, 0, 0, 0, -3, 0, 0, 2, 0], [0, -2, 6, -4, 0, 0, -4, 8, -4, -8, 10, -10, 0, 0, -2, 10, 0, -2], [0, -2, -13, 4, 0, 0, -11, -2, -11, 2, 2, 5, 0, 0, -4, 2, 0, 0], [0, 6, -23, 1, 0, 0, -14, -1, -14, 1, 2, 2, 0, 0, -6, 2, 0, -5], [-18, 0, 0, 0, -1, 0, 1, 8, -1, 8, 7, 0, -7, 7, 0, -7, -6, 0], [0, 12, 22, -4, 0, 0, -2, 8, -2, -8, 0, 10, 0, 0, -6, 0, 0, 0], [33, 0, 0, 0, 3, 3, 6, 6, -6, 6, 7, 0, 4, 5, 0, -7, -11, 0], [9, 0, 0, 0, 12, -2, 8, 16, -8, 16, 1, 0, -4, -1, 0, -1, 6, 0], [0, 8, -22, 5, 0, 0, 9, 9, 9, -9, 5, -1, 0, 0, -6, 5, 0, -3], [0, -10, 20, -7, 0, 0, -1, -14, -1, 14, -2, -9, 0, 0, 12, -2, 0, -3], [0, -3, -24, -3, 0, 0, -12, 1, -12, -1, 0, 5, 0, 0, 3, 0, 0, 0], [-9, 0, 0, 0, 5, -1, 0, 2, 0, 2, 3, 0, -2, -3, 0, -3, 3, 0], [0, 7, 0, 5, 0, 0, 6, 7, 6, -7, -3, -5, 0, 0, 2, -3, 0, 4], [0, -10, -10, 5, 0, 0, 11, -8, 11, 8, -8, -5, 0, 0, 6, -8, 0, 9], [-11, 0, 0, 0, 5, -3, 0, -8, 0, -8, -5, 0, -4, -7, 0, 5, -5, 0], [0, 11, 19, -2, 0, 0, 6, 4, 6, -4, 9, -5, 0, 0, -13, 9, 0, -1], [0, -8, -16, -8, 0, 0, -4, -15, -4, 15, -4, 0, 0, 0, 16, -4, 0, 8], [0, -4, -1, -3, 0, 0, 2, 1, 2, -1, -2, 8, 0, 0, 0, -2, 0, 9], [-23, 0, 0, 0, 2, -14, 4, 2, -4, 2, -9, 0, -8, -9, 0, 9, 10, 0], [-1, 0, 0, 0, -2, 7, -2, 9, 2, 9, 2, 0, -6, -6, 0, -2, -12, 0], [-4, 0, 0, 0, 8, -10, 13, 7, -13, 7, -7, 0, 2, 2, 0, 7, 10, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4680_l_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4680_2_l_j();".
// 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_4680_2_l_j(:prec:=18)
    chi := MakeCharacter_4680_l();
    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_4680_2_l_j();".
// 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_4680_2_l_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4680_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4096, 0, 258048, 0, 829952, 0, 982528, 0, 529040, 0, 145120, 0, 21344, 0, 1681, 0, 66, 0, 1]>,<11,R![992, 1296, -2072, -1332, 1384, 276, -214, -37, 6, 1]>],Snew);
    return Vf;
end function;