
// Make newform 4028.2.a.d in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4028_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_4028_2_a_d();".
// 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_4028_2_a_d();".
// 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 := [-49, -826, -2375, 5796, 26515, 219, -64854, -10941, 71354, 3416, -38897, 2544, 11179, -1598, -1714, 332, 132, -30, -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, 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], [-4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-36307996555354849920, -4164221605907178213362, -6989008533635252643771, 39595263533206333032434, 69513017742215683190015, -103779970138095754748039, -128499761875924242856430, 135462042252905651618449, 77255857497718219255495, -82627527926263228597648, -18076037847766854541528, 25399920257422537055929, 855756579969074611934, -4057421456854802221336, 289170464821686701843, 320664495983429416571, -43375197057086977459, -9882149061894242460, 1746095294249905525], [483921811060713145533, 5689248964462977037139, 874953622100007023511, -55943767001779800289086, -46789161173125655425506, 148143090411297334733782, 91287776562331314493241, -186042123550325361577403, -45746760297992265654926, 110332407390778069031044, 3431892679435488472801, -33332936791207512527838, 2963377924552730128550, 5268285624886140485261, -839866884795173919115, -413624666729155238549, 84076917574960256843, 12694984241530833769, -2943780406006641765], [-1163592902846952508077, -10716774429709574464823, 1557379993306038135353, 111813032790893018762455, 64803520076310100110799, -291230576907575657777941, -122717780039396478438628, 346388332779359323049071, 51617732662797384306580, -198272510073702647686792, 4317230792299849092388, 58657910015823290199629, -7164617461856096041454, -9154097347335590647765, 1674137724859097326569, 713001287691853722889, -157775075461770482860, -21770149302925042489, 5367366175361990662], [-767982636210474333887, -10555286174427386953037, -7902964447253402551451, 112359898898568469548753, 121318731949279369464324, -304946572937529267890946, -224004286752909750898195, 387071868444202644915812, 116850638684083221365791, -230632984523399313959543, -14283739980853695730737, 69849775976451597633276, -4865977009331272204910, -11055794589174623184396, 1616710294741622158139, 868798848549651489109, -168178430493991923146, -26680747861818192613, 5986540821363048083], [-425409066286894832191, -11787538564722251354047, -13988656566371121836817, 124479998890880575687817, 143986447003899364514933, -349870200047557082743712, -251510394749040653365510, 455968562412609108317925, 122805265553987816991443, -275051090647484797769281, -9376119487851360027860, 83812009843926202606782, -7685249755764291736826, -13310513813794211859868, 2182009360343441981983, 1048160645118184606647, -218246461204391413265, -32234704197001569255, 7634075689117315443], [-1571164249019956057369, -20001792233774411403989, -11348298894438304802815, 203416584074287209867702, 199384122108507089701670, -545424751792344385783695, -366628352549431740785521, 683279953541843260445629, 185842642247020067427678, -404093377305337244125151, -18448630739731778650887, 121870727971300884271619, -9727120821862602810878, -19241747310288014109107, 2948870386732726476653, 1509736642100354109756, -300153156956505182157, -46317271424766431009, 10583488570732583080], [1423735123658409096612, 22881633776118358856299, 17288575867973303529433, -236977996666831952475416, -225126564890101466278021, 629862347828509404003707, 395419269034118877063649, -772172666021715621775968, -194246883387390224925121, 450451640075255791543425, 16183010811149986450309, -134764644450956180041717, 11471880999951571487763, 21174240238043555130948, -3314682900513242189374, -1656236244441767709207, 333006147432104940562, 50707552311051149897, -11670781974338432578], [-2227351681199340420998, -26260223104617142553418, -11570439594905860465419, 261986926110365744568328, 235680140090945263367131, -691072682416700603098456, -433707429116295140456111, 850195825097354486442429, 217340664863567212748095, -497659654230141998572598, -19508279410526756576822, 149226602100555301111447, -12335770759364466632990, -23480695927427830562029, 3630523757720748227307, 1838393784492939905825, -366733316124510233831, -56320378776851402420, 12887337244309102310], [422929458039533903359, 5506401536397342947630, 3325967357404050000805, -56048267335871887838477, -53721696104914722775357, 148560170897994524266440, 97207718232382746830615, -183275383715474459093238, -49008248157728226951319, 107424995269575245141776, 4852906604258270437399, -32235027301369914602315, 2554374730253101640076, 5074222031303082478672, -773719932918447123382, -397380567530873041216, 78708560151114998678, 12176056338712085544, -2774197342220715149], [-2899363062180072381215, -38505505587758768950801, -27127912924828904331696, 393788704637746081556702, 401836188374829621113285, -1051964580499689009862854, -725952232729224716384505, 1309212673190232652360595, 371603559829635867753852, -771230824813105073147062, -41705165044881695029030, 232074251927807865451673, -17060810333536869116116, -36591913083644724216675, 5426248967727490103921, 2868546645954305222135, -558589882809057011560, -87950799774851853575, 19791322111485126204], [2872552231573007321058, 42072549651667420730118, 29895819272271390186238, -434094089528405914136182, -428362716991615958333398, 1163341240616084616448015, 767925470706985433199518, -1447524651396993508147989, -383718486771389100286389, 852117377936432109055784, 35640582461401394679617, -256270775630394694065260, 21079302661832812973500, 40391910531833465482195, -6261465310879068713631, -3165720527951712937811, 633802788787607721493, 97050618546137680209, -22289867140185408278], [3061469864996299447894, 47174575583883340369582, 35926639666028662100641, -480279123762301493691939, -495044373763106739194155, 1291257565269313627230584, 892123991041871910598122, -1619413654701392804562061, -452810635090310303890993, 958002424674232127312982, 47451143554738106380962, -288938484730705145961233, 22393926429258917147637, 45618730798133385926743, -6921202134071797919604, -3579202409104159368513, 707647289178142272786, 109803036871893909119, -24998369798017516548], [4320986960026640338519, 58078743700198732636821, 37102300925453892343376, -588597017763378213232673, -581346941904714610628111, 1566178638821494910920206, 1055431957751715212449711, -1946576197255065457127915, -536095155374584599520342, 1146171567520701030543707, 55934569543284961625347, -344834623061650905277647, 26650501384434821329586, 54366704678379851026705, -8222505495437665605209, -4261845844975398059130, 840355296619823602257, 130670739174724825698, -29681376895996197944], [-4269370263534605129033, -62045186303231683408650, -41733388729028138072672, 636235023035425463017489, 620498805560384532184000, -1698478842811708781813908, -1116865682365930558280660, 2105826776065409602738968, 560778203360560864323400, -1237272726567812173162473, -53976877019970239794073, 371721318395513424412158, -29944337197098411648676, -58553492280121244504793, 8989055740957247679851, 4587401677629310734476, -912383701329069111656, -140598400564189312527, 32126750325000417763], [-4043661676142879980322, -62369066267108048404051, -49686465088446824450421, 645175837406879853982374, 668892019683462644248792, -1745435831325722933254093, -1196825848752823800956711, 2193802864638411868169019, 601711962492870212285946, -1298635512620329536916196, -59144568971774230229251, 391759239964210643251937, -31532631822481732262996, -61857820411663542311015, 9529404779309219804362, 4853538593332586374159, -968844809569255216252, -148902808973472428057, 34139576512779733606], [-5256536700393855125760, -72413223583601264960432, -41271629821614059947905, 733853893038264699002650, 678401614172280322061840, -1947900714257381645846663, -1224609347834873327463388, 2401546521589485076123624, 606407207630109272285654, -1406601328489586758783901, -50686847457296408089964, 421858313389172862318606, -36017717369322207106991, -66383634490649735076473, 10418007751645399158371, 5197633440043365121890, -1047361719567607781222, -159238822068501653493, 36723603444103622524]];
    Rf_basisdens := [1, 1, 1, 57367504414897716263, 57367504414897716263, 57367504414897716263, 57367504414897716263, 57367504414897716263, 57367504414897716263, 57367504414897716263, 57367504414897716263, 8195357773556816609, 57367504414897716263, 57367504414897716263, 57367504414897716263, 57367504414897716263, 57367504414897716263, 57367504414897716263, 57367504414897716263];
    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_4028_a();"
function MakeCharacter_4028_a()
    N := 4028;
    order := 1;
    char_gens := [2015, 2757, 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_4028_a_Hecke(Kf)
    return MakeCharacter_4028_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 := [[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, 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, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, -1, 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, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 1, -1, 0, 0, 1, 1, 0, 0, -1, 0, -1, 0, 1, 0, 0, 0, 0, 0], [-1, 0, 0, 1, -1, 0, 0, -1, 0, -1, 0, -1, 0, 0, 0, 0, -1, 1, -1], [-1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, -1, 0, 0, 0], [-1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, -1, -1, 1, -1, 0, 1], [2, 1, 0, -1, 0, -1, 0, 0, 1, 1, 1, 0, 0, -2, 1, 1, 0, 0, 0], [0, 1, 0, -1, 0, -1, 0, 0, 0, 2, 1, 0, 0, -1, 0, 1, 0, 0, -1], [-1, 0, -1, 0, 0, 0, 1, -1, -1, 0, 0, -1, 1, -2, -1, 2, -1, 0, 1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 1, 0, 1, 1, 1, 0, 1, 1, 0, -1, 0, 0, -1, -1, 1, 0, -1, 1], [-1, 1, 1, 1, 0, 1, 0, 0, -2, 0, -1, -2, 1, 1, -2, 0, -1, 0, 2], [-2, 1, -1, 0, -1, 0, 0, -1, 1, -1, 1, -1, 0, -2, -1, 1, -1, -1, 0], [0, 0, 0, -1, 0, 0, 1, 0, -1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0], [0, 1, 1, 0, -1, -1, 0, 0, 0, 0, 2, 1, 0, -3, 0, 1, -2, 1, 0], [-2, 1, 1, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [-1, 0, 2, 1, 1, 0, 0, 1, -2, 2, -1, 2, 0, -1, -1, 0, 0, -1, 1], [-3, 1, 0, -1, -3, -1, -1, -2, 1, -1, 1, 1, -2, 0, 1, 0, 0, 1, -2], [-3, 0, 2, -1, -2, 0, -1, -1, 0, -1, 2, 1, -1, 2, -1, 0, -1, 2, -1], [-1, -1, 0, -1, -1, -1, -2, -2, 1, 0, 0, 0, -2, -1, 0, 1, -1, 1, 0], [2, 0, -2, -2, -1, -3, 1, -2, 0, 1, 0, 0, 0, -4, 1, 2, -1, -1, -1], [2, 0, 0, -2, 3, 0, 1, 2, 1, 4, 0, 3, 1, 0, 2, -2, 4, -2, -2], [-3, -1, 0, 1, -1, 0, 0, 0, 1, -3, 0, 0, -2, 0, 1, -2, -1, 1, 1], [3, 0, 1, 0, 3, 1, 0, 1, 0, 1, 0, 0, 0, -1, 0, 0, 1, -2, 2], [-4, -1, 0, 1, 1, 0, -1, -2, -2, 0, -2, 0, 1, 1, -2, 0, 0, -1, 1], [1, 0, -3, 1, -1, -2, 3, -2, 1, -1, 1, -1, 0, -4, 2, 2, -1, -2, 1], [2, 0, 0, 0, 2, 0, 1, 1, 2, -1, -1, 2, -1, 0, 2, -3, 2, 0, -1], [1, -2, 0, -1, 1, 0, 0, -1, 2, 0, 1, -2, 0, 1, 1, 1, 0, 1, -1], [-1, -1, 1, 1, -3, -2, 0, -2, 0, -2, 2, 1, -1, -3, 0, 2, -2, 1, -1], [-4, 0, 2, 1, -1, 2, -1, 1, -1, 0, -1, 0, -1, 5, 1, -2, 0, 0, -1], [-1, 0, -4, -2, -2, -1, -2, -2, 1, 1, 0, -2, 1, -5, -1, 4, 1, -3, 1], [-4, 0, -1, 2, 2, 1, 0, 3, 0, -1, -1, 0, 0, 1, 1, -1, 0, -1, 0], [-1, -1, -2, 0, 2, 1, 2, 1, -2, 0, -3, -2, 2, 3, 0, 1, 3, -1, -1], [-4, -2, -1, 0, 0, 2, -3, -1, -1, -2, -3, 0, 0, 4, -2, -1, -1, 1, -2], [3, -3, 3, -1, 2, -3, -1, -1, 0, 3, 0, 1, -1, 0, 0, 1, -1, 1, 0], [2, -1, 0, -1, -1, -2, -1, -2, -1, 2, -1, -2, 0, 0, -1, 1, 1, 0, -1], [-4, -1, -1, 2, 1, 1, 1, 1, -1, 1, -1, 0, 1, 1, -1, 0, 2, 0, -3], [3, 1, 0, 2, 1, -1, -1, 0, -1, 2, 0, -1, 1, -3, 1, 1, -1, -1, 0], [-2, -1, 0, 0, 1, -2, -1, -1, 0, -2, 0, 0, -1, 0, 1, -1, 1, 0, 2], [0, -1, 1, 0, 3, 1, 0, 0, -1, 3, -3, -1, 1, -1, 0, 0, 1, -3, 0], [-3, -1, 0, -3, 4, 1, -2, 2, 0, 3, -2, 1, 0, 3, -1, -3, 2, -3, 2], [0, 0, 0, 0, 3, -2, 4, 4, 0, 0, 1, 3, 0, 3, 3, -3, 3, -1, 0], [0, -2, 0, 1, 2, 1, 2, 3, -2, -2, -1, -1, 1, 2, 1, -3, 1, -2, 2], [-5, 0, -1, 0, 0, 2, -4, -1, -1, 0, -1, -3, -1, 2, -2, -1, -1, 1, 0], [-7, -1, 1, 2, -1, 2, -1, 1, -2, -1, -2, 1, -2, 3, -3, -1, -3, 1, 2], [2, -1, 0, 0, 0, -1, 0, -2, 1, -1, 2, -2, 1, -4, -1, 3, -1, 0, 2], [-4, 1, -3, -2, 0, 3, 0, 0, 3, -1, -2, -2, 0, 3, -1, -1, 2, -1, -2], [3, 1, 3, 0, 1, 2, 0, 1, 2, 2, 1, -2, 0, 0, 0, -1, 1, 1, -2], [1, -1, -2, -3, -2, 0, 1, -2, 2, 0, 0, -1, 1, -1, -1, 1, 1, 1, -1], [-2, -1, 3, 0, 2, 3, -2, 5, 0, 2, 1, 3, 1, 3, 0, -2, 1, 2, 1], [1, -2, 2, 1, -2, 0, -2, -1, 0, 1, 0, -1, 1, 4, 1, 0, -1, 2, -1], [-6, -3, -1, 3, -1, 1, 0, -1, -2, -1, 0, -1, 1, 0, -1, 1, -2, 1, 1], [-5, -3, -3, 1, 0, -1, 0, -1, 0, -4, 0, 0, -1, 1, -1, -1, 1, 0, 1], [4, 6, 0, 2, 0, -2, 1, 0, -1, 0, 2, -1, 0, -4, -1, 3, -4, -1, 4], [-5, -3, 0, 3, -1, 3, 3, 4, 0, -4, -1, 2, -1, 6, 3, -3, 3, 2, -2], [-7, -3, 2, 4, -3, 4, -2, 0, -2, -4, -1, 1, 0, 6, -3, -3, -3, 2, -1], [-1, 0, 0, -3, -1, 0, -3, 0, 0, -1, 2, -4, 1, -1, -1, 2, -3, 1, 4], [3, -1, 0, 0, 0, 0, -1, -3, -2, 1, -1, -3, 2, -2, 0, 2, 0, -2, -1], [-1, 1, -4, -2, -2, -4, 2, -2, 1, 2, -1, 0, 1, -4, 1, 2, 1, -2, -3], [0, -3, -2, -2, -2, -3, 1, -2, -1, 0, -1, 1, -1, -3, 2, 0, -1, 0, 0], [1, -3, 0, 0, -1, 0, 3, -1, 1, -2, 4, 0, 0, -2, 3, -1, -1, 1, -2], [5, -3, -2, 0, 0, 3, -1, 1, -1, 1, -2, -1, 4, -2, 0, 1, 2, -1, -2], [-3, -2, 0, 1, 5, 4, 1, 2, 2, 1, -2, -4, 0, 6, -1, -3, 4, -2, 0], [-3, 1, -2, -3, 0, 1, 0, -1, 1, -1, -4, 1, -1, 4, -1, -3, 5, 0, -5], [4, 3, 2, 0, 4, 3, 1, 6, -1, 4, 0, 2, 2, 4, 0, -2, 3, -2, 0], [3, -1, -2, -4, -1, 0, 1, -3, 0, 2, 2, 1, 2, -2, 0, 2, 1, 0, -2], [-3, -4, 1, -1, -1, -1, -1, -2, -1, -1, 0, 2, -1, -1, 0, -1, 0, 0, -1], [0, -1, 0, -2, 4, 1, -1, 3, 0, 3, -2, -1, 3, 2, -1, 1, 0, 0, 2], [-7, 2, 4, 2, -5, 3, -2, 0, 0, -3, 3, -2, 0, -3, -1, 2, -4, 2, 1], [-4, 3, 0, -2, -1, -1, -1, 0, 0, -2, 0, 5, -3, 2, 2, -4, 3, 0, -2], [-2, -1, 0, -2, 2, 1, 2, 0, -2, 0, -1, 0, 1, 5, -1, -4, -1, -1, 2], [-4, 2, 0, -2, 0, 2, -2, -1, 2, 4, 0, -1, -1, -1, -3, 1, 1, 0, 1], [1, 0, 5, 4, 4, 0, 2, 4, -1, -1, -1, 4, -1, 5, 0, -5, -1, 1, 1], [-2, 2, 3, -2, -2, 4, -1, 4, 2, -1, 1, 1, -1, 4, -1, -4, 2, 3, 0], [-8, -1, 1, -1, 0, 2, -2, 2, 0, -1, -3, -1, -2, 8, -2, -6, 1, 0, 1], [0, 2, -1, 3, 0, 2, 0, 5, 1, -1, 1, 1, 1, -5, 2, 1, 0, -1, 4], [-6, 0, -1, 0, -3, -2, 0, 0, -2, -2, -1, 1, -1, 0, -1, 0, -3, 2, 2], [10, 1, 2, 1, 1, -4, 1, -1, -1, 0, 1, 3, -2, -4, 2, -1, -2, -3, 2], [6, -2, 1, -3, 6, -3, -1, 4, -2, 5, -2, 5, 1, -1, 3, -2, 2, -1, 1], [3, -2, 1, 0, 3, -1, -1, 1, 0, 2, -6, 0, -1, 3, -1, -2, 3, -4, 0], [-5, -2, 0, -4, 3, -1, 0, 2, 0, 2, -1, 6, -2, 6, 0, -3, 4, -1, -1], [-2, 2, -1, 7, -4, -1, 6, -1, 0, -6, 3, -3, 0, -5, 1, 4, -6, 1, 4], [5, -6, -2, -2, -1, -1, 0, -3, 2, 0, -1, 1, 0, 0, -1, 1, 2, 0, 0], [-7, 0, 3, -1, 2, 0, -2, 1, 1, 0, 0, 4, -1, 5, 3, -5, 4, 2, -3], [4, 1, -4, -1, 1, 3, 0, 1, -1, 1, -3, -4, 5, 0, -4, 0, -1, 0, -2], [6, 6, 3, 1, -5, -3, 4, -4, -1, 0, 5, -1, 0, -6, 2, 4, -3, 1, -3], [-7, -3, 3, 2, -1, -2, -2, 0, -2, 3, -1, 3, 2, -3, -2, 4, -2, -1, 3], [2, -1, 0, 0, 1, 7, -5, 0, 0, 0, -2, -3, 3, 0, -4, 2, -2, 0, 0], [-4, 1, -4, 2, 3, -1, -1, -1, 0, -3, -2, -2, -1, -1, -2, -2, -1, 0, 1], [-1, 0, 1, -3, 4, 2, 1, 4, 1, 2, -4, 3, 2, 8, -2, -5, 5, -2, -2], [2, 0, 5, 3, -1, -2, -1, 0, -1, -2, 3, 7, -4, 3, 1, -2, -2, 5, 2], [1, -2, 2, 5, -1, 0, 3, -1, -4, -3, -1, 4, 1, -5, 0, -2, -3, 1, 1], [-3, 1, 1, -1, -5, -1, -4, -3, 0, -1, 3, -1, -2, -2, -1, 4, -2, 3, -3], [-1, 1, -5, -3, 5, 3, 3, 1, 2, 2, -4, -6, 3, -2, -4, 1, 4, -6, 3], [-6, -2, -1, 0, -1, 6, -4, -1, 1, -2, -2, -2, -1, 4, -3, -1, 3, 1, -1], [7, -5, 1, 0, 5, -4, 1, 1, -2, 1, 0, 3, 2, -1, 1, -2, 1, -1, 4], [-3, 2, 0, 3, 4, 1, 4, 2, -1, -1, -1, 1, 0, -1, -1, -5, -2, -2, 5], [14, 1, -2, -7, 4, 1, 1, 1, 4, 4, 2, -3, 2, -3, -3, 3, 2, -1, 1], [2, -6, -4, -4, -3, -2, -3, -6, 2, -1, -1, -3, 1, -1, 0, 1, 2, -3, -4], [-5, 1, 1, 2, 1, 2, 2, -1, -1, -1, -2, -1, -1, 2, -2, 1, 1, 1, -1], [1, -4, -2, -3, 6, 1, 0, 5, 4, 0, -4, 3, -2, 7, 1, -4, 5, 0, 2], [-2, 3, -4, -1, 1, 1, 1, -1, 2, -5, 0, -2, 0, 5, 3, -2, -1, 1, 0], [4, 6, 0, 2, -1, 4, 0, 1, -4, 0, 0, -5, 2, -1, -2, 2, -4, -3, 5], [-2, -1, -1, 5, -3, -3, 1, -5, -1, -7, -1, -1, -6, -4, 0, 0, -6, 0, 3], [-3, -1, 0, 3, 3, -3, -2, 0, -2, 1, -3, 0, 0, -2, 2, 1, -2, 0, 0], [-7, -6, 1, -1, -3, 2, -1, 3, -1, -1, 1, 7, -2, 6, 1, -5, 3, 3, -5], [-3, 0, 5, 4, -2, -3, 2, 3, 1, -1, 4, 3, -1, -1, 2, 3, 0, 4, 2], [-3, 3, -1, -1, 0, -1, -4, -4, 0, 0, 0, -4, -3, 1, -2, 0, -1, -1, 1], [-11, 1, -1, 0, -9, -4, -3, -3, 0, -4, 2, -1, 1, -3, -2, 4, -2, 5, 2], [3, 2, -1, 1, -4, -1, -2, -1, -6, 0, 0, -1, 1, -6, -5, 3, -5, 1, 2], [-2, 1, 0, -7, 2, -2, -3, 2, 3, 3, 0, 0, -3, 9, 3, -2, 2, 1, 0], [7, -4, 5, 0, 1, -4, 2, 1, -3, 6, 4, 1, 1, -3, 3, 0, -3, 1, 1], [2, 3, -4, 0, -3, 1, -3, -2, 6, -1, 4, -3, 1, -9, -1, 6, 0, -1, 0], [1, 2, -2, -2, -3, -6, -1, 0, 3, 1, 2, 0, -5, -2, 6, 2, 2, 0, -4], [-3, 2, -4, -2, 1, 3, -2, 2, 1, 0, 0, 4, 1, 5, -1, 1, 6, 0, -1], [12, 1, -2, -3, 3, -2, 3, -2, 1, 4, 1, -4, -1, 1, 1, 3, 0, 1, -5], [-7, -8, 5, 3, -2, 5, -5, -2, -2, -5, -1, -2, -4, 7, -3, -2, -2, 2, 2], [5, 1, -1, -2, 3, 0, -3, 2, 6, 4, 3, 1, -1, -3, 1, 3, 4, 1, 1], [0, -2, 4, 1, 1, -1, 3, 4, -6, -1, 0, 3, 1, 4, 4, -2, 0, 1, 1], [-7, -1, -3, -3, -3, 3, -3, -2, 3, -3, -3, 5, -5, 7, -1, -4, 5, 4, -6], [-7, 4, 8, 4, 8, 7, -3, 9, 2, -1, 0, 4, -2, 12, -3, -7, 3, 2, 2], [-5, 1, -3, 0, -2, 2, 2, 0, -1, -4, -3, 2, 2, 5, 1, -2, 4, -3, 3], [-8, -1, -1, 0, 3, 2, -4, 6, 0, -2, -4, 2, 1, 6, 2, -5, 3, -2, 5], [-5, 1, 1, 2, -8, -1, -3, -5, 1, -1, 1, -2, -3, -9, -3, 9, -5, 0, 3], [-1, 5, -5, -1, -2, 1, 0, 1, 4, -1, 1, 1, -3, -2, 2, 0, 3, -2, -1], [0, 3, -2, 4, -5, -5, -1, -4, -1, -3, 1, -3, 1, -9, 4, 2, -5, -3, 2], [7, -5, 4, 3, -1, 1, 0, 3, -6, 0, -1, 1, 6, 3, -4, 2, 0, 2, 3], [1, -7, -1, -2, 5, 3, 1, 5, 0, 6, -5, 2, 5, 7, 2, -3, 7, -3, -3], [4, 0, -1, -3, -1, -3, -1, -3, -1, 1, -2, -1, -3, 2, 0, -1, 4, -3, 4], [4, 6, 4, 2, 6, 4, 4, 8, 0, 6, 0, -2, 4, 0, -2, 1, 3, -4, 0], [3, 4, 4, -1, 0, -3, 0, -1, -2, 3, 4, 1, 3, 0, -2, 4, -3, 4, 2], [-8, 2, -1, -1, -1, 0, 2, 0, 4, -2, 5, 6, -4, -1, 2, -1, 1, 2, 0], [8, 3, 1, -1, -1, -5, -2, -2, -3, 2, 1, -1, 3, -3, -1, 0, -5, 1, 0], [8, -9, 2, -4, -1, -1, -8, -1, 1, 3, -3, 4, -4, 2, 1, -1, 3, 1, -1], [-2, 1, -2, 0, -3, -3, -3, -7, -5, 2, 0, -3, 0, 3, 0, 1, -4, 3, -5], [9, -9, -2, 2, -4, -6, -1, -7, -2, -1, 1, 2, 1, -9, 4, 1, -2, 3, -6], [-10, -6, 2, -2, -4, -2, -5, 3, 0, 0, -1, 8, -5, 7, 5, -7, 5, 3, -3], [5, 0, 1, -2, 0, -1, 0, 2, 2, 3, 6, 0, 3, -5, 0, 3, -3, 0, 4], [5, 0, -2, 0, -2, -1, 2, -5, -2, 3, 2, -8, 5, -11, -3, 10, -3, -1, 0], [1, 9, 4, 2, 0, -2, 4, -2, -3, 0, 2, -3, -2, -1, 1, 4, -6, 2, 1], [11, 5, -1, 0, 6, -6, 5, 2, 4, 1, 4, -2, 0, -10, 5, 1, -1, 0, 3], [9, -6, 8, 3, 1, 4, -1, 5, -2, 3, 5, 4, 1, 5, 4, -4, 0, 5, -5], [-1, 1, 5, 8, 2, 4, 5, 2, -9, -4, -3, -6, 1, -2, -7, 3, -10, 0, 11], [-3, -2, 0, 6, -2, -1, 0, 0, 1, -5, 4, 4, -3, -3, 3, -2, -2, 3, 3], [-16, -2, 2, 3, -2, 3, -3, -2, -9, -5, -5, 1, -1, 7, -6, -5, -6, 3, -1], [18, 7, 3, -4, 2, -3, 3, 1, 4, 9, 3, 3, -4, -5, 3, 0, 5, -4, -5], [-11, 0, -4, -2, -2, 1, 1, 0, -2, 0, -1, 2, 5, -4, 1, 2, 2, -2, -1], [2, 4, 2, 1, 5, 0, 2, 5, 3, 2, -1, -4, 2, 3, 0, -2, 6, -2, -3], [-10, 3, -5, 3, 3, 0, 5, 2, 3, -4, -3, -2, -4, 0, 1, 2, 0, -4, 3], [-2, -1, 0, -3, -7, -4, -5, -2, -2, -5, 0, 2, -3, 6, 1, -4, -2, 2, -3], [-8, 4, -4, -5, -3, -3, -1, -3, 8, -3, 3, 5, -5, 0, 3, -1, 6, -1, -6], [10, 4, -4, 6, -4, -3, 9, -1, -2, -3, 3, -1, 2, -13, 0, 2, -5, -5, 4], [2, -2, -4, 0, -6, -4, -2, -2, 7, 1, 3, 3, -3, -4, 8, -1, 3, 2, -9], [-2, -2, 1, 0, 1, 6, -3, -4, -4, -1, -5, -5, -2, 8, -6, -4, -1, -2, 4], [-1, -1, -1, 1, -6, 0, -8, -3, 0, -6, 3, -5, -2, -3, -4, 8, -8, 4, 2], [4, -3, -1, -2, -3, -1, 0, -2, 0, -1, 1, -4, -2, 2, 3, 0, 1, -1, 0], [-8, 3, 6, 0, 3, 4, -7, 5, 4, 1, 4, 5, -1, 5, 1, -6, 0, 5, -3], [-6, 1, 2, 8, -1, 7, 0, 2, -2, -5, 2, -5, 2, -4, -5, 3, -7, 2, 6], [-5, -8, -4, -3, -1, -5, -6, -4, -3, -2, -5, -2, -5, 3, -4, -2, 0, -2, 3], [3, 3, -3, 0, 0, 2, 4, 0, -3, 1, -9, -2, 1, -3, -4, 0, 3, -9, 3], [-6, 3, 0, 3, 2, 2, 2, 0, -1, -2, 2, 1, 3, 2, -2, 0, 1, 2, -1], [-8, -3, 4, 4, 2, 5, 6, 2, -2, -3, -2, -6, 0, 1, -4, 3, -3, 0, 1], [1, 2, 0, 0, -2, 2, 1, 1, 7, -2, 2, 1, -1, -4, 4, 5, 3, -1, 3], [-2, -2, -2, 4, 2, 1, 2, 6, 2, -4, 3, 3, -1, -6, 3, 0, -1, -2, 1], [1, -1, 2, -1, -2, -9, 6, -1, -5, 0, 4, 5, 2, 0, 2, 2, 0, 2, -1], [16, -1, 0, -1, 6, -3, -1, 1, -1, 1, 1, 3, 0, 0, 3, -5, 2, -4, -1], [1, 2, 7, 3, 3, 1, 0, 2, 1, 3, 9, 2, -3, -3, -1, 3, -4, 5, 3], [-8, -2, -3, 2, -1, 6, 8, 0, -6, -2, 1, -7, 7, 0, -3, 2, -5, 0, 6], [-11, -4, -1, -4, 10, 6, 3, 10, 6, 1, -8, 7, -1, 15, 6, -10, 13, 1, -2], [3, 2, -1, -3, 3, -7, -1, -4, 2, -2, 5, 2, -4, -3, 1, -5, -2, 1, 4], [2, 8, 1, -6, 6, -3, 10, 7, 9, 4, 4, 4, 1, -3, 8, 0, 6, -2, 0], [-7, -1, -2, 8, -2, 3, 2, 0, -2, -9, 1, -3, 3, -5, -1, 2, -9, 1, 4], [-12, 1, 3, 3, -5, 1, 2, -1, -1, -6, 1, -2, 0, 2, 0, 1, -8, -1, 7], [4, 5, 2, -1, 4, 5, -2, 6, 5, 2, 4, 1, 4, -3, -4, 1, 3, -2, 2], [-2, 7, -1, -3, -1, 2, 4, 0, 6, 5, 7, 1, -3, -10, 1, 3, 0, -1, -1], [3, 3, -5, 3, -4, 6, -2, -2, 2, -5, 0, -2, 2, -1, -7, 5, 1, -1, -1], [1, 3, 0, 0, 0, -1, -1, -5, 1, 4, 1, -3, 2, -6, -7, 8, -5, 4, -3], [3, 0, -3, -2, -5, -3, 1, -4, 7, 3, 7, 5, -2, -15, 2, 9, 0, 1, 1], [0, 4, 3, 1, -2, -1, 4, 1, -1, 0, 2, -2, -4, -1, 6, -2, -2, 3, -3], [-13, 1, 5, 0, -7, -5, -6, -2, 5, -2, 4, 6, -3, -1, 6, -1, 0, -1, 0], [3, -7, 2, 1, -6, -7, -4, -7, -8, 0, 3, 1, -1, -7, -1, 7, -10, 5, 1], [-6, -5, 5, -2, 4, -1, -9, 2, -2, 11, -1, 10, 1, -2, -2, 1, 6, 0, -2], [17, -1, -3, 2, 9, -2, 2, 3, 5, 3, -2, -4, 0, -7, 4, -3, 0, 0, 4], [2, 1, -2, -1, -5, -2, -7, -3, 2, 0, 0, -1, 3, -7, -1, 6, -2, -2, 0], [-13, 4, -2, 4, -5, -1, 1, 1, 0, -5, -6, 4, 1, -4, 0, 0, 0, -5, 2], [-5, 6, 3, -2, 0, 1, -1, 4, 2, 2, -4, 5, -1, 8, -1, 1, 5, 3, -3], [-7, 2, -2, -3, 0, 2, -2, -4, 8, -1, 2, 1, -5, -2, 2, 1, 1, 1, -5], [4, -3, -6, 0, 6, 7, -6, 4, 8, 2, -6, -3, 0, 8, 1, -5, 11, -5, -4], 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4, -4, -3, -4, 0, 2], [-1, 0, -3, -2, 1, -4, 3, 0, 6, -3, 4, 11, -3, -10, 4, -2, 2, -2, -4], [7, -11, -1, 4, -1, -3, 5, -2, -6, -2, -5, 1, -3, -4, 0, -4, -4, -3, 4], [-4, -2, 5, -1, -4, -2, -4, -3, 0, 1, 11, -3, 1, -6, -3, 6, -5, 5, -1], [-4, 10, 5, 2, -6, -4, 2, -3, -7, 2, 6, 2, 4, 0, 2, 4, -5, 2, -1], [-8, -3, 0, 4, -10, -2, -1, -12, -7, -11, 0, -4, -5, -5, -6, 1, -12, 4, 5], [-2, 4, -2, 6, -5, 0, 3, 5, 2, 0, 0, 1, 0, -7, 4, 4, 4, -1, -1], [2, 4, -8, -3, 4, 3, 0, 4, 5, 0, -3, -1, 2, 7, 1, -1, 7, -5, -5], [-1, 1, 4, 0, 6, 2, 0, 3, -1, 5, 1, 1, 1, 5, 1, -7, 1, 0, -2], [3, -9, 1, 0, 5, 0, -1, 3, 6, 3, 2, -1, 2, 1, 3, 0, -1, 5, 0], [4, 4, 0, -3, 9, 1, -2, 5, 2, 6, -3, -1, 0, 10, 0, -3, 9, -9, -2], [-5, -3, -3, 1, -1, -3, -1, -3, -4, 0, -2, 0, 0, 1, 6, 2, 0, -1, -1], [4, -4, -1, -6, 5, -5, 3, 0, 4, 7, -2, 1, -3, -3, 5, -1, 7, -3, -6], [-8, 0, -3, -2, -2, 9, -6, 0, 4, -3, -1, 2, 1, 2, -2, -2, 1, 6, -6], [7, 0, -8, -3, -7, -4, -4, -5, -2, -6, -2, -7, 0, -3, -1, 3, 1, 0, 1], [7, 6, 0, -3, -2, -6, 6, 0, -2, 2, 1, -1, -2, -8, 1, 6, -2, 0, 4], [1, -4, -2, 1, -5, -2, 1, -2, 0, -5, 0, -4, 1, -1, 1, 1, -1, -1, 2], [-1, -2, 2, 2, -6, -3, 4, -1, 3, 0, 5, 2, -2, -3, 7, -2, 0, -1, 0], [4, -5, 0, -5, -5, -4, -5, -4, -2, -1, 0, 4, -4, 11, 1, -6, 7, 1, -5], [11, -1, 6, -6, 4, -3, 4, 0, 4, 8, 1, 8, -5, -2, 7, -1, 8, 0, -3], [-12, 4, 2, 1, -10, -5, -2, -2, 1, -3, -1, 5, -1, 1, 4, 0, -2, 5, -3], [13, 1, -2, 3, 2, -5, 7, -1, -1, 0, 3, 0, 3, -13, 4, 2, -6, -6, 7], [-13, 5, 0, 4, -4, 4, 2, -1, -3, -5, 2, -6, 4, 4, -1, -3, -3, 0, 0], [19, -4, -3, -12, 8, -1, -3, 0, 1, 9, -8, 2, 2, 8, -1, -7, 11, -8, -1], [-11, -1, 6, -1, -2, -1, -3, -2, -3, 4, 0, 2, -3, 9, -5, -2, -2, 2, -2], [3, 8, -6, 3, 9, 0, 5, 6, 5, 0, 1, 3, -6, 1, 5, -6, 2, -1, 1], [-19, -5, 6, 4, -3, 1, 2, -2, 5, -4, 5, 6, 0, 0, 1, 2, -1, 5, -2], [16, -10, 5, 2, 0, -6, 6, -2, 0, 3, 4, 6, 0, -3, 5, -1, 0, -3, -4], [-6, -6, -3, 3, 5, 3, -1, 0, -3, -5, -7, -5, -1, 10, -3, -11, 2, -1, -1], [-12, -7, -2, 5, -8, 6, 0, -4, 6, -11, -3, -3, -6, 8, 2, -3, -6, 2, -1], [6, -1, 6, 3, -8, -3, -1, -4, -5, -1, 3, -5, 3, 5, -2, 6, -4, 5, -3], [12, -7, -5, 1, -2, 0, 5, -4, -5, -3, -1, 0, 3, 1, 1, -2, -1, -2, -7], [0, 14, -4, -4, 6, 5, -3, 4, 9, -1, 2, -5, 3, 2, 1, 0, 7, -3, 3], [-5, 5, 5, 0, -4, 1, 5, -1, -5, 0, -2, 3, -1, 14, -2, -4, 1, 0, -5], [-15, 2, -7, -2, -4, 5, 1, -4, -2, 0, -2, -6, 6, 7, 1, 1, 6, 0, -10], [-5, 6, -3, -1, -2, -3, -6, -2, 1, 1, 0, -1, 1, -5, -4, 1, -5, 0, 3], [16, -6, 2, -4, -2, 0, -4, -3, 1, -4, 6, 0, -3, -5, -2, -4, -7, 9, -1], [-11, 9, -4, 0, -5, 2, 1, -9, 6, -6, 7, -8, -1, -9, -2, 6, -4, 4, 0], [0, -5, -2, -6, 5, -3, 9, 3, 5, 4, 4, 7, 2, -7, 10, -2, 8, -1, -1], [-5, -2, -6, -11, 3, 3, -6, -1, 6, -3, -2, -1, -2, 11, 1, -9, 6, 3, -2], [-5, -4, 6, 8, -10, -1, -8, -5, -4, -5, 6, -4, -2, -8, -1, 5, -12, 6, 4], [-8, -9, 3, 7, -2, 4, -6, -3, -4, -14, -3, -2, -2, 5, -6, -2, 0, 4, 6], [-27, 1, -2, -3, -6, -4, -4, -5, 0, -9, 3, 3, -4, 4, 1, -7, -5, 5, -8], [-10, 3, -4, 4, -9, 13, 1, 0, 0, -9, -1, -9, 4, 12, -3, -3, 3, 3, -3], [7, -4, -1, 3, 1, -8, 2, -1, -8, -2, -4, -9, -1, -8, -1, 1, -5, -7, 7], [-1, 0, 4, 0, -4, -2, 2, -8, 2, -6, 5, 2, -4, -1, 5, -2, -5, 10, -2], [-12, 6, 6, 7, -7, 5, -2, -1, -5, -3, 5, -8, 3, -6, -6, 12, -11, 0, 8], [3, 3, -3, -3, -3, -9, 5, -4, -2, 2, 4, 2, -2, -5, 5, 1, -3, 0, 3], [-16, 0, 4, 2, -8, 6, -4, -2, 0, -5, 1, 1, -2, 7, -3, 0, -1, 6, 0], [-9, 0, -6, -7, -9, -2, 3, -7, -1, -2, 1, -11, 6, -3, -2, 10, -1, 0, -7], [-3, 2, 9, 5, 3, -2, 1, 4, 3, -1, 4, 1, -6, -4, 4, -1, -6, -1, 8], [2, -2, 2, -2, 0, -10, -4, 3, -1, 5, -3, 9, -1, 0, 7, 0, 5, -5, -3], [12, 0, -3, -8, 2, 1, -5, -6, 4, 4, 3, -4, -4, -6, -1, 5, 7, 0, -4], [-15, 4, 3, 1, -13, -2, -3, -1, -6, -6, 8, -1, 4, -1, -5, 7, -10, 10, 1], [-8, -9, -1, -1, -2, 4, 0, -5, -2, -1, -1, -2, 1, -7, -5, 4, 2, -4, -2], [-7, -1, 1, 1, 1, 3, -4, 3, 1, -5, -6, -1, 3, 5, -10, 0, 4, 0, 7], [-21, 4, -2, -6, -5, 2, -2, 0, 3, -1, 3, 12, -6, 7, 2, -4, 8, 1, -10], [-11, -1, -6, -4, 0, -3, 1, 1, 6, 0, -6, 1, -5, 1, 2, -3, 3, -1, -3], [-14, 5, 3, 7, 3, 0, -4, 1, -8, 3, -6, -5, 5, -2, -6, 3, -1, -1, 8], [7, 6, 2, 13, -4, 1, 7, 5, -1, -2, 3, -1, 4, -6, -1, 9, -7, 1, 6], [-8, 3, 0, 1, 1, 0, -1, -3, -6, 2, -5, -5, 7, 1, -3, 2, 0, -1, -3], [-18, 0, -1, 4, 4, 9, -1, 5, -1, 1, -7, 4, 3, 4, -3, 3, 2, -1, 2], [-14, 12, 1, 7, 1, 10, 3, 9, 1, -2, -5, -7, 2, 10, -1, -1, 2, -2, -1], [-29, 3, -5, 9, -9, 4, 2, 1, -1, -13, 1, -2, -2, 1, -1, 3, -9, 2, 6], [-12, -6, 3, 4, -4, -3, -5, 1, 5, -3, 3, 3, -4, -3, 4, 0, 3, 4, -4], [-7, -6, 4, 0, 2, -2, -5, -4, -1, -2, -3, 7, -4, -1, -2, -1, 2, -1, 4], [13, -8, 2, -2, 4, 1, -8, 4, -2, 2, 0, 7, 2, -1, -5, -7, 5, 0, 1], [1, 5, 9, 5, -7, -7, 0, 1, 0, -3, 10, -6, -7, -5, 3, 2, -6, 7, 1], [-9, 3, 3, -2, 1, 3, 1, 1, -4, -1, 0, -5, 2, 3, -4, 1, -1, -1, -1], [-12, 1, 1, -4, -5, -5, 4, -1, 9, 4, 8, 7, 1, -8, 6, 5, 6, 4, -9], [19, 0, 4, 0, 6, 1, 3, -3, -9, 5, 1, -6, 7, -1, -8, 3, -11, 1, 12], [31, -3, 0, -8, 11, -5, -1, -1, 5, 8, 2, 6, -2, -5, 8, -4, -1, -1, -5], [13, -2, -4, -2, -2, 1, 7, -1, -3, -3, 1, -5, 2, -2, 2, 3, 1, 2, 1], [-11, 0, 2, 7, 4, -5, 6, 2, -6, 0, -2, 2, 2, -4, 5, -5, 0, -4, 3], [17, 1, 2, -4, 5, -2, -4, -2, -4, 2, -1, 3, 0, -2, -5, -3, -1, 0, 3], [3, -3, 1, 0, 2, -4, 3, -5, -3, 13, 2, 6, 5, -7, 3, 1, 1, -5, -6], [-10, 3, 9, 5, -7, 4, -2, 1, 3, -8, 8, 4, -6, 6, 6, -3, -8, 10, -2], [22, 3, -6, 1, 7, 0, -1, 3, 3, 6, -6, -5, 2, -10, 4, 5, 3, -8, 3], [8, 2, 0, -7, -2, -3, -3, 1, -3, 5, 0, 7, -1, -3, -1, 3, 7, -1, 0], [2, 3, 4, 0, -1, 2, -3, 5, -5, 3, 2, 6, 1, -5, 2, 0, 6, 0, -3], [-2, -2, 2, -5, -10, -12, -9, -6, 6, -1, 10, -2, -7, -7, 5, 5, -4, 5, -2], [4, 5, -12, -6, 2, -5, 3, -1, 6, 4, 4, 1, 0, -2, 12, -5, 4, -4, -8], [12, -1, 5, -3, -1, -9, 1, 0, 4, 2, 5, 8, -2, 2, 11, -2, 0, 8, -5], [-21, -8, 6, -2, 2, -1, 0, 9, -4, 1, 0, 8, 0, 8, 1, -9, 5, 2, 0], [12, 7, 2, 2, -2, -7, -1, -9, -9, 2, 4, -9, 2, -18, -5, 9, -15, -7, 15], [8, 3, 5, -6, 7, -8, 1, 1, -4, 4, 0, -4, -3, -2, 4, -3, -3, -1, 6], [6, -2, 1, -2, 14, 8, -2, 7, 0, -3, -9, -2, 0, 17, -4, -11, 7, -3, 3], [-9, -2, 3, 2, -7, -3, 5, 4, -1, -2, 7, 5, -1, -1, 1, -4, 1, 1, 1], [-12, 14, -9, -10, -3, 6, 1, 0, 7, 2, 6, -3, 4, -3, -2, 5, 3, 3, 0], [17, -10, -3, -8, 3, -11, 1, -5, 4, 7, 0, 12, -4, -7, 16, -4, 11, -2, -8], [-5, 8, -1, -2, 8, 6, 1, 13, 3, 1, 0, 11, 3, 11, 5, -13, 10, 0, -3], [-19, -4, 8, 11, 8, 4, 9, 11, -7, -3, -3, 3, 0, 13, 2, -10, -3, 5, 5], [7, 4, 3, 3, -5, -2, 8, 2, -4, -2, 5, -3, 0, 3, 4, -1, -7, 2, 2], [-14, 0, 6, -1, 6, 1, 2, 5, 5, -1, 4, 5, -5, 3, 5, -10, 2, 5, 4], [-16, 1, -7, -2, -2, 6, -5, -1, 6, -6, -6, 2, -5, 9, 2, -7, 4, 2, 2], [7, 1, -1, -4, -4, 0, -7, 0, 0, 2, 5, 3, -2, -5, 0, -3, -4, 2, 3], [2, 4, -2, 1, 4, -1, 0, -5, -3, -6, -4, -11, 0, 5, -6, -2, -8, -5, 4], [-10, -9, -2, 0, 3, 1, -2, -1, -5, -3, 0, -3, 8, 0, -5, 5, -5, 0, 5], [-7, 7, -7, -4, 10, 4, 9, 5, 3, 6, -5, -1, 7, 4, 0, -3, 12, -11, 2], [1, 2, -9, -3, -6, -4, 4, -6, 2, -7, 0, -4, -3, -8, -2, 6, 0, 3, -4], [10, -4, 3, -2, -3, -5, -7, -2, 2, 4, 8, 4, -4, 1, 4, -1, 1, 6, -9], [-3, 4, -1, 5, -1, -4, 7, -5, -1, 1, 3, -5, 0, -20, 1, 9, -6, -9, 5], [20, 15, 0, -7, 13, -7, -2, 0, 8, 12, 2, -7, -2, -6, 5, 5, 2, -3, -1], [-11, 3, -5, 3, -1, -3, 0, -1, -6, -2, -4, 4, 10, -15, -4, 6, -4, -5, 7], [16, -1, 2, 3, 4, -2, -4, 7, -1, 3, 2, 5, 2, 2, 2, -2, -5, 1, 9], [-23, 0, 2, 6, -15, 7, -8, -6, -1, -12, 6, -10, -1, -3, -5, 3, -9, 10, 1], [-4, -4, 1, -1, 13, 5, -2, 12, 1, 0, -8, 8, 1, 13, 2, -14, 12, -1, 7], [-11, 4, 2, 9, 3, 10, 3, 6, -10, -8, -9, -4, 6, 11, -6, -4, -5, -4, 3], [-12, -5, 3, 8, -6, -3, 4, -2, -3, -3, -5, 3, -3, 12, 3, -4, 3, -3, -2], [-8, 1, -5, -4, 10, 7, 6, 7, 2, 6, -9, -5, 5, 15, -4, -8, 9, 0, -9], [-17, 4, 5, 5, -13, -3, -6, -4, -4, -10, 2, 5, -9, -2, -3, -2, -11, 11, -2], [-11, 2, -6, 1, -2, 1, 0, -5, 5, -1, -4, -8, -2, -5, 2, 4, -5, 3, -6], [-1, -7, -2, -4, 0, -4, 4, 0, -5, 2, -3, 6, 1, 4, 3, 0, 3, -4, 1], [3, -7, 0, -6, -3, -9, -10, -5, 2, 6, -1, 5, 0, -4, 4, -2, 1, -1, -3], [-15, 15, 0, 3, -5, 6, 2, 2, 1, -7, 10, -2, 0, -2, -5, 5, 1, 3, 1], [-10, 10, 1, 13, 3, 2, 6, 7, 0, 3, -1, 0, 2, -1, 4, -1, -2, -5, 4], [29, -1, 6, 1, 1, -7, 4, 0, 0, 5, -1, 2, -2, -4, 4, 1, -3, 2, 0], [18, -3, 7, 0, -2, 2, -11, 5, 3, 8, 5, 5, 1, -3, 6, 1, 1, 4, -7], [-14, -3, -12, -12, 7, 7, -2, 3, 10, 4, -2, -2, 2, 7, 5, -5, 18, -2, -9], [-16, 4, 4, -2, -3, 4, 4, 2, 8, -2, 1, 4, -3, 0, 1, 0, 8, 2, -3], [13, -1, 12, -2, 11, 4, 3, 11, 3, 11, -2, 5, 2, 9, 0, -4, 5, 0, 1], [2, -5, -1, -10, 12, -4, 1, 2, 7, 9, 5, 3, 1, -6, 12, -1, 7, 1, -3], [8, -3, 3, -1, -4, -2, 3, -6, -7, -1, 3, -10, 6, -2, 1, 11, -6, 3, 1], [-16, 5, 3, 5, -2, 8, -1, -2, 0, -4, -1, -5, 1, 4, -7, -3, -2, -2, 1], [-6, 3, -7, -3, -9, -3, -2, -2, -5, -2, -2, -6, 3, -7, -4, 9, -5, -3, 7], [-8, -13, 1, 5, -7, 3, 8, -5, -5, -8, -1, 1, 0, 4, 0, -3, -6, 2, -3], [-7, 9, 0, 7, -7, 4, 4, 4, -9, 1, 3, -4, 5, -11, 1, 10, -8, -4, 9], [-6, -5, -5, -3, 3, 10, 2, 9, -4, 7, -7, 0, 8, 7, -8, 2, 11, -5, -2], [-10, -2, -7, -8, 4, 6, -2, -3, 5, -2, -8, 4, 0, 6, 0, -3, 10, 0, 3], [1, -4, 8, 1, -4, -8, -4, -2, 4, 5, -1, 5, -3, -10, -2, 6, 0, 0, -2], [-5, 8, 1, 6, 0, -9, 4, -2, -1, -3, 0, -2, -6, -6, 5, -2, -12, -2, 3], [-15, -8, 6, 0, -6, -1, -2, -2, 2, 2, 5, -2, 3, -4, -1, 4, -7, 8, -7], [-18, -8, 11, 12, -2, 5, 2, 5, -11, -4, -4, -2, 2, 16, -3, -6, -2, 4, -4], [3, 7, -1, -1, 0, 3, -2, 3, 9, 0, 0, 3, -3, 3, 4, -1, 6, -3, -5], [-23, 4, 0, 3, -12, 6, -3, -1, -7, -4, 0, -6, 7, 3, -11, 8, -4, 2, 3], [-2, 0, -2, -4, 1, 3, 10, -2, -4, 9, 0, -6, 10, 2, -5, 7, 1, -1, 1], [9, 6, 1, -5, -1, -2, 4, 0, 1, 2, 1, 2, 2, 9, 3, -4, 4, 1, -6], [0, -1, 2, 5, -4, -6, 7, -1, -6, 9, 2, 1, 3, -7, -1, 8, -4, -3, 0], [-11, -4, -6, 1, -1, 1, 2, -3, -11, -3, -4, -9, 5, 1, -9, 3, -5, 1, 8], [-8, -6, -3, 1, -8, -1, -2, -7, 1, -4, 1, 2, -6, -9, -2, 4, -8, 2, 8], [20, 3, 0, -4, -3, -6, -1, -1, -9, 1, 0, 2, -1, -1, -6, -3, -1, -4, 3], [-19, 3, 4, 2, 6, -9, 8, 0, 0, 1, 0, 7, -3, -3, 0, 0, -5, 2, 3], [10, 5, -1, -2, 2, 7, 1, 9, 10, 7, 6, -1, 4, 1, 7, 3, 7, 3, -4], [17, 15, 6, -3, 10, -5, 11, 2, 0, 10, 7, 10, 0, -6, 2, -5, -1, -3, 0], [9, -2, 5, -2, -6, -14, -3, -12, -1, 0, 5, 3, -6, -15, 1, 6, -6, -2, 0], [6, 1, 3, -3, 8, 6, -1, 9, -2, 9, -3, -4, 7, 7, -5, 5, 3, 0, 2], [-18, -3, 1, 4, 8, 15, -2, 18, 8, -3, -10, 3, 2, 25, 0, -14, 13, 0, -2], [-20, 1, -8, 5, -9, -5, 12, -6, -1, -13, -1, 0, -2, 5, 7, 2, 1, -1, -4], [5, 0, -6, 0, -6, 1, 4, 1, 6, -9, 4, -2, -4, -10, 3, 3, -1, 0, 4], [20, -18, -2, -3, 2, 1, 1, 2, -3, 9, -6, 6, 6, 12, 6, -7, 9, -3, -5], [6, -17, -5, -4, -1, -1, -4, -3, 6, 0, 0, 6, 1, -4, 3, 6, 8, 1, -7], [-11, 7, -4, 0, -9, -4, -12, -10, 2, -11, 3, -5, -5, -8, -8, 4, -15, 8, 3], [-20, 0, -3, -5, 1, 1, 9, -5, 9, -5, 0, 4, -6, 9, 6, -3, 9, 0, -11], [5, -5, 8, 10, -2, -10, 2, -5, -7, -2, 3, 7, 0, -9, 2, 7, -8, -2, 6], [18, -4, -2, 1, 1, -5, 1, -5, 3, -1, -1, -2, -7, -2, 2, -3, -5, -2, 0], [10, -2, 4, 2, 3, -4, -1, 1, -4, 6, -1, 6, -4, 3, -2, -2, -1, -2, 3], [-19, 12, -8, 2, -9, 6, -1, -1, 10, -11, 1, -10, 1, 2, 2, -4, 2, 2, -7], [-6, 2, 4, 9, 3, 5, 0, -1, -2, -6, -6, -5, -6, 12, 0, -8, -8, -4, 7], [-26, 12, -4, -1, -10, 5, 1, -1, -2, -1, -3, -7, 2, 0, -6, 6, -4, 6, -4], [10, 4, 2, 0, -1, -7, -3, -5, -5, -3, -8, -3, -3, 5, -1, -2, 1, -1, 1], [-3, 5, 0, -1, -4, -4, 8, -2, 0, 4, -4, 1, -5, -6, 3, 4, 2, -3, -5], [-14, -5, 5, 0, -3, 0, -10, 1, -2, -10, 1, 7, -6, 12, -2, -1, 0, 7, -4], [-22, 5, 7, 5, 10, 5, 1, 11, 0, -7, 0, 5, -1, 15, 1, -11, 4, -2, 7], [-1, 6, -3, -2, 5, 4, -10, -2, 5, 0, 2, -7, -4, -8, -4, 5, -1, -2, 5], [9, -12, 5, 5, 2, -5, 0, -2, -13, 6, -3, -6, 3, 0, -5, 3, -7, -4, 1], [-1, 8, -3, -4, 1, -4, -4, -2, 7, -2, 7, -2, -5, -3, 0, -1, 0, 0, -1], [-7, -3, -1, -2, -3, 5, 3, -4, -7, 4, -3, 4, 5, -4, 1, -2, 4, -2, -4], [0, -7, -4, -11, -2, -1, -7, -6, 11, 5, -2, 7, -6, 5, 0, -1, 13, 2, -10], [-7, 13, 2, -5, -7, -2, -8, -2, -5, 3, 8, -2, 3, -8, -6, 4, -11, 5, 5], [-18, 1, 8, 5, 15, 14, -2, 17, 4, -4, 0, 5, 0, 10, -1, -13, 0, 1, 10], [5, -9, -7, 6, -3, 2, 8, -4, -6, -8, -2, -4, 0, -2, 5, 1, -5, 1, 1], [22, 0, 3, -2, 7, -1, -3, 5, 5, 5, 1, -3, -2, -4, 4, 3, 2, 3, 2], [2, -2, 4, -8, 7, 4, -1, 3, 11, 9, 5, 5, -2, 0, 4, -2, 9, 10, -7], [-4, 15, -1, 0, 4, 0, 10, 8, -1, -2, 1, 3, 2, 7, 3, -8, 3, -1, -7], [-25, 1, -13, -3, -2, 5, -3, -5, -3, -4, -9, -4, 4, 4, -10, 6, 6, -1, -5], [-9, 6, 0, 3, 2, 11, 4, 9, 12, -7, -5, -9, -1, 12, 1, -4, 8, -1, 2], [20, -7, -3, -10, 0, -7, -8, -2, 13, 6, 2, 8, -5, 7, 8, -4, 17, 3, -16], [-13, -1, 12, 8, -6, 6, 2, 2, -3, -4, 2, -3, 4, 5, -3, 4, -6, 4, -1], [9, -13, 5, 0, -3, -8, -7, -10, -2, -5, 4, 4, -1, -14, -2, 8, -10, 7, 5], [-27, -1, -5, -3, 5, 7, 4, 11, 12, -2, -4, 6, -6, 13, 2, -14, 10, 3, -9], [4, 10, 5, 10, -2, 6, 10, 7, -2, 0, 9, -12, 7, -1, 1, 5, -7, 1, 1], [32, 0, 2, 1, 12, 5, 0, 8, 3, 13, 0, -4, 8, -8, 6, 5, 1, -9, 3], [4, 0, 9, -3, -9, -4, -4, -6, 1, 0, 11, -2, -3, -11, -2, 12, -13, 8, 8], [13, 6, 3, 3, -2, 4, 11, 15, -3, 4, 2, 2, 3, 11, 9, -5, 4, 2, -3], [-29, 4, -3, 9, 2, -1, 12, 4, -7, -1, 2, 1, 6, -1, 1, -2, -5, -6, 5], [-6, 1, 4, 6, -2, 4, -1, -6, 3, -5, 1, -9, -5, -2, -1, -1, -5, -2, 3], [-18, -5, 12, 6, 4, 6, 1, 8, -4, -1, -3, 12, -5, 6, 0, -7, 3, 7, -1], [21, 5, 7, 0, 3, 4, -2, 7, 4, -1, 4, -5, 2, 15, 1, -8, 6, 1, -2], [7, -5, -4, 2, 6, 0, 10, 12, -1, -5, -1, 8, 4, 10, 8, -10, 10, 0, -1], [-15, -5, 6, 4, 1, 4, 3, 3, 0, 0, -1, -4, -6, 16, 0, -9, 0, -1, 2], [19, 3, 1, 2, 13, 2, -4, 9, 5, 9, -4, 0, 0, -1, 2, 4, -1, -2, 6], [-2, 15, -6, 5, -10, 2, 6, -3, -1, -3, 1, -8, 1, -5, -2, 7, -7, 5, -5], [4, 12, 10, -7, 12, 3, -2, 10, 11, 7, 1, 1, -3, 13, -2, -9, 8, -3, -3], [18, 0, -3, -7, 2, 3, 2, -2, 9, 5, -1, 5, -4, -2, 4, -7, 9, 4, -9], [5, -7, -7, -6, 0, -6, -8, -5, 9, 8, 0, 2, -2, -11, 0, 8, 5, -7, -8], [-6, 2, 3, 3, -7, 12, 1, 12, -6, -2, -2, -4, 6, 6, -7, 2, 4, 1, 4], [1, 1, -7, 1, -6, -5, 0, -16, 6, 0, 1, -10, -1, -17, -4, 14, -4, -2, 6], [-2, -16, -3, -4, -1, 6, -12, -6, -3, -7, -11, -2, 0, 16, -13, -7, 0, -1, 4], [-13, -5, -3, 3, -11, -7, -1, -6, 1, -9, 6, -4, 0, -16, -3, 13, -6, -4, 8], [19, 3, -1, -5, -7, 0, 3, -4, -4, 0, 9, -3, 8, -1, 2, -1, 1, 6, -12], [15, -3, 1, 4, 6, -6, -5, 1, -9, 1, -13, -5, -2, 0, -3, -7, -4, -7, 10], [-11, 0, -9, 3, -9, -4, -1, -16, -4, -12, -1, -9, 1, -12, -6, 7, -11, 5, 4], [22, -10, 7, 0, 2, -2, 3, 10, 6, 6, 0, 14, 1, 3, 6, -2, 8, -5, 0], [18, -3, -3, -1, -2, 3, 0, 0, 5, -1, -2, -6, 5, 1, 4, 1, -1, -6, -9], [-18, 9, 16, 3, 6, 3, 6, 12, -1, 4, 3, 16, -4, 16, 3, -12, 8, 8, -2], [9, -18, 2, -7, 4, -4, -11, -5, 2, 0, -1, 2, -4, 0, -2, -5, -1, 1, 7], [0, 13, 4, -4, 6, -1, -4, -2, 7, 5, 1, -3, 0, -5, 7, 4, -2, 0, -2], [12, -12, -2, 2, -6, -3, 1, -14, 0, 0, 1, -1, 2, -4, 0, 3, 6, 2, -11], [14, -11, -1, -2, -9, -3, 1, -6, -2, -1, -1, 4, -2, -2, 1, -2, -2, -3, -2], [18, 0, -1, -3, -6, -11, 7, -7, 4, 11, 8, 2, 0, -16, 12, 7, -8, 1, -8], [-16, 2, 4, 1, 7, -3, 3, 6, -5, -1, 0, 4, -3, 6, 7, -10, 2, -2, 3], [-4, -1, -3, 1, 6, 6, -7, 6, 4, -3, -4, -2, -4, 8, -2, -10, 5, 5, -5], [2, 9, 4, -4, 8, 3, 2, 3, -5, 9, -3, -4, 1, 8, -2, -3, -2, -2, -1], [8, -1, 7, -3, -9, -7, -10, -4, 6, 4, 4, -1, -9, 5, 5, -3, 1, 10, -15], [19, 6, -4, -15, 5, -2, -1, -2, 0, 16, -5, -1, 5, 0, 0, 2, 13, -13, -8], [-15, -7, -6, 7, 1, 1, 9, -2, -10, -10, -6, -7, 0, -1, -3, -5, -4, -10, 12], [31, 7, -6, -17, -4, -9, -5, -19, 3, 5, 4, -5, -3, -12, -1, 7, -1, 0, -11], [-13, -6, -7, -3, 0, 2, -1, 1, 12, -5, 3, 4, -3, -2, 15, -2, 8, 1, -9], [3, -1, -11, 0, -2, 4, -2, -6, 7, -4, 2, -7, 0, -17, 3, 7, 0, -4, 3], [-29, 1, 5, 4, -2, 12, -5, 2, 6, -1, -1, 9, -4, 0, -9, -3, 6, 5, 5], [-21, 2, 0, 8, -3, 7, 8, -4, -5, -14, -1, -17, -2, 6, -9, 4, -10, -2, 13], [15, 4, 6, 3, 8, 4, 0, 5, 3, 1, -1, 3, -8, 11, 4, -8, 6, 1, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4028_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4028_2_a_d();".
// 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_4028_2_a_d(:prec:=19)
    chi := MakeCharacter_4028_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_4028_2_a_d();".
// 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_4028_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4028_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![49, -826, 2375, 5796, -26515, 219, 64854, -10941, -71354, 3416, 38897, 2544, -11179, -1598, 1714, 332, -132, -30, 4, 1]>,<5,R![-1664, -17344, 29920, 144384, -143456, -387496, 261548, 458865, -207748, -268671, 81751, 82537, -16828, -13735, 1831, 1234, -98, -56, 2, 1]>],Snew);
    return Vf;
end function;