
// Make newform 4028.2.a.f in Magma, downloaded from the LMFDB on 24 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_f();".
// 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_f();".
// 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 := [139, -454, -4299, 5074, 30973, -17877, -73816, 23819, 72868, -18074, -36567, 8368, 10113, -2310, -1554, 364, 124, -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], [9549212368394962571709, 90955650844496841415443, -95863405384260650578292, -486508306277487080639637, 199437779162941782881844, 628050699125868748022486, -173273403885715628219601, -386387527001845234846065, 78774350849770433685435, 134639173642389041239377, -21632651398845249336587, -27975363851656900935375, 3904702873986854643991, 3427742929074332447811, -464929689331149538134, -228419041336493582073, 32803688712224653227, 6383747831994437549, -1013664970932510359], [-705048640521550753042, -2596169087667389090749, 2463678410242470840173, -17339171104576617579725, 1526271337661236731546, 75554189902004790177990, 1974045361748312551583, -80261934709718916524839, -2126285279123905811034, 39188082322529543612125, -101997297017966586210, -10269789484355555618923, 425689341513369750520, 1491567589135895220569, -122100450800907992989, -113206383041965395179, 13579116821905178591, 3504924600062958270, -539666137422440460], [-27761816022030137605193, 30696628439091594640205, 409776759302680392011964, -227160207345741293466219, -1251133748221990524239808, 498719127530284526140679, 1313330090608220013976793, -560547808941881549889402, -642414053222074731899579, 328269697028020825287738, 156111636410667783119248, -103019545288071940947301, -16346936089987529154161, 17414925175315486603474, -106264229225035104222, -1492920352180888609775, 149810450509960698612, 50880960665791710281, -8082393497238969918], [-10377632966827837230239, -31727649377226224441820, 208367847416842091060431, -192745004320722586135416, -596066650789253892258349, 823567850258607734691640, 706422386390206413846371, -929821432914397749569504, -377008214805426624184168, 492197278713807875634399, 91128082993182265376372, -140062845135953860708398, -6799678529958853409030, 21925206662121998889480, -955354606015150023233, -1774315165620840355805, 192444453247398587102, 57926039052261232522, -9012255479615042506], [15374816921928772387641, -57209576941038616681447, -234975891591382458211335, 519748650162003555440563, 625724025506759966537372, -1321405476326058704896156, -713224542559588122446075, 1334621888720261551464630, 369120574768379920994051, -677647246369639428662873, -81682667251389768988723, 188792773780458340703538, 2617817530125671121032, -29172991237797969942832, 1830443128106006764405, 2340433431752486897347, -279042796602546905350, -75947048383905243843, 12279512726675507013], [-15853588812280295893235, 23840706565177952489302, 267273604049339077042987, -268415003301992744582801, -785844487179013523942315, 636223365416096458777228, 688097937337630306295553, -686917593354695955136902, -230459265695755928256657, 380259478726129276646478, 9828945992025776641311, -113935358339789835760057, 12819124770709822621272, 18593376947389501907842, -3348151150958097388752, -1550580703664022927300, 335701280791985868818, 51660710244299223716, -12314233074030226871], [-28540922552046344408613, 93150844803515409831409, 350995415744919634018826, -622937693458533872437694, -946705048039643406597082, 1056358072940628201207339, 772620603092657335620073, -917172825624740629980752, -230905172735736471947896, 453114600578601704464099, -3016754497444860404492, -128638473769920492132139, 17408020904773011271906, 20488657664749485257454, -4076081956140805156372, -1691009143344334751529, 392201561392315144161, 56121672911821323774, -14065420372976239962], [262481099435760130156, -139956340876172755996760, -39479411448802007040965, 968395259204709981854911, 265372142123995710209070, -2016364357746995762333272, -448004696282821758586931, 1836166095699660343346776, 250297060329871709304802, -880863656053346109871933, -42092416564172393413605, 237678327165159094674093, -6868373979917272840906, -36042418775253329863245, 3252798257142975761126, 2857419119741421916394, -394095446186709103493, -91966583363222535462, 16072902599680908164], [11724294891590491667200, 49020417227303367772001, -179606709602492481236308, -760900293840186507497893, 791993991768000869119075, 2180901680454689586443078, -1058318120853383259078135, -2239724891565664824511173, 744046916765897486514004, 1128173850045330519126837, -313689368711917707970084, -310036125983548622083787, 80170999974298067452935, 47245043857622210665744, -11963509787421702685666, -3740690915509558295648, 950561701022030194182, 119883384573197873174, -30920828473910836670], [1702381593629708054002, -3661243503266635980568, -9804675742813541314583, 59232482816746354496069, -2496555985410708691581, -172099709947074829946152, 26888038421775232406730, 182933524201420133860161, -33475877988139306145394, -95597437847383147171360, 19341402264776327604609, 27142175615340616021689, -5923377110657869476954, -4248080377837881910399, 981957540622806796858, 343539561760627625000, -83094520879379915030, -11193644850226234750, 2810991684136139728], [-677863154453489493139, -7840357659589812655154, 19433210783559378052114, 80504940697230352291768, -71019286251562866791270, -211203652985939143543248, 87958646698080945085998, 212283232873006321803260, -60921824645513997515995, -106323157980190747086538, 26237875672467871584328, 29205864935468996547239, -6918026893091333773760, -4456501003206093014746, 1061232596093721452725, 353490448330280482628, -86108385019162773642, -11349244696320364505, 2843677740234168613], [18658611605802252907286, 23092845362882775090641, -236425795844853268020665, -762538587432516170978728, 820001318022588237116666, 2523077391398104754291646, -972178791369964142248127, -2735691555041965902330833, 670278700785601694237276, 1421761132329473018744018, -299779358367535526932361, -398916321013084576590828, 83153170959761190732914, 61679278046621658447683, -13340827181414440749721, -4935523460323896609933, 1120729210703677838193, 159449436967766400861, -37978614188213183709], [556041491076014525149, -113067419744114310145875, -134077632441870097460973, 1146406281163259773760714, 291400103820138825132342, -2983358528152073364113123, -118537219926767515035213, 3047221868344953263091880, -170599429571660391077600, -1557247265878266347682706, 181268757821956946544273, 435638704099058343964752, -69043082073825151131539, -67477650887082219075842, 12745454425647953208988, 5416654855583962870974, -1146351490718162460997, -175567919116157259598, 40260786094638775650], [15780177101141905520027, -183519035076096841260150, -243669449289305650873372, 1557597897636070494502675, 487122365658243491064730, -3747246844666897741476244, -283478766103769053494878, 3727403505749017453402842, -116988033924023668491712, -1890216667950960941706585, 187312962802747769677365, 528414033628651687262510, -77276600504063208667028, -81978738033841550109093, 14770922372314263261637, 6594657794653130538136, -1354654963745146272946, -214183933471934477069, 48161166165251651829], [-1760874733024935089596, -41897785519851591635960, -31233698366619744425441, 859923878877118650833277, -6987864269487478484626, -2646330299676823810790046, 240849246121361947629705, 2962668592805649058520512, -408904352130513796813489, -1614284798142203414385807, 279312317316166220169993, 473026411609922601696842, -94140475891623742215449, -75769169127617525350201, 16540275904376039157620, 6232225832453837317484, -1452874990210943747302, -205635435554699590005, 50398135494972326796], [-25629137576845305236733, -140612089369578959957683, 327642677232636835042171, 1757835191063551020382195, -1282054769126796767569368, -5009650161716357552256666, 1732825690188432852621703, 5253982706384872413598083, -1319563541418099291012535, -2703288083749504306601356, 614094686929332164171070, 755874140813028078170070, -170512673658002814491048, -116715759953417738577203, 27005489836129121170361, 9332831270331460449619, -2234608040469850149678, -301318492504557167969, 74710486437446437192]];
    Rf_basisdens := [1, 1, 1, 3478013382692242784767, 267539490976326368059, 3478013382692242784767, 3478013382692242784767, 3478013382692242784767, 3478013382692242784767, 3478013382692242784767, 3478013382692242784767, 3478013382692242784767, 267539490976326368059, 267539490976326368059, 3478013382692242784767, 3478013382692242784767, 3478013382692242784767, 3478013382692242784767, 3478013382692242784767];
    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, 0, 1, 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, 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, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 1, 1, 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, -1, 1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0], [1, -1, 1, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0], [1, 0, 0, 1, -1, 0, 1, -1, 2, 0, 0, -1, -1, 0, 2, 1, 0, 1, 0], [-1, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, -1, 0, 2, -1, -1, -1], [2, -2, 1, -1, 1, -1, 0, -1, 0, 1, -1, 1, 1, 1, 0, 0, 0, 0, -1], [1, 0, 0, 0, 1, 0, 0, 1, -1, 0, -1, 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, 0, 0], [1, 1, -1, 0, 1, 0, -1, 2, -1, -1, 0, 0, 0, 0, 0, -1, 0, -1, 1], [1, 1, 0, -1, -1, 0, 0, -1, 1, 0, 1, 0, 1, -1, -1, 1, -1, 0, -1], [5, 1, 1, 1, 1, 1, 0, 1, -1, 0, 0, 0, 1, 2, 1, -2, 0, 0, 1], [0, 1, 0, 1, 1, -1, 1, 1, -1, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0], [-1, -1, 0, 0, -2, 0, -1, 1, 0, -1, 0, 1, -1, 1, 2, 0, 0, -1, 1], [1, -1, 1, -1, 0, 1, -1, 0, -2, -1, -2, 2, 1, 2, -1, -2, 2, 0, 1], [2, -2, 0, 0, -2, -1, -1, -1, 1, 1, 0, -1, -1, 1, 2, 0, 0, 2, -1], [0, 1, 0, 1, -1, 0, -1, 0, 0, -1, 0, 1, 0, 0, 1, 0, 0, 0, 1], [0, -2, 0, -1, 0, -1, 0, -2, -1, 0, 0, 0, 1, 1, -1, 0, 1, -1, 1], [1, -2, -1, 0, 2, -1, 0, -2, 1, 3, -1, 0, 0, 1, 0, -1, 1, 2, -1], [2, 2, -2, 1, 1, 0, -1, 1, -2, 0, 2, -2, 0, 0, -1, -2, 0, 0, 1], [1, 0, -1, 2, -1, 0, 1, 1, 0, 1, 0, -2, -3, 0, 4, 0, 0, 0, 1], [3, 1, 0, 1, 2, 1, -2, 0, 0, 0, -2, 1, 0, 2, 0, -2, 0, -2, 1], [-1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, -1, -1, 0, -1, 1, -2, 0, 0], [0, 4, -3, 1, 0, 0, 0, 0, 0, -1, 2, -2, -2, -5, 0, 3, -3, -1, 0], [-1, -3, 2, -1, -1, 0, 1, 0, 0, -1, -2, 0, 1, 1, 0, 1, 1, 0, -1], [0, 2, -2, 1, 1, -1, 1, 1, 1, -1, 1, -1, -2, -1, -1, 2, -1, -1, 0], [3, -3, -1, 1, -1, 0, -1, -1, 1, 0, -1, 1, 0, 0, -1, 0, 3, 0, 0], [1, 0, 0, -1, 0, 2, -2, 1, -3, -1, -2, 2, 0, 1, -2, -1, -1, -1, 0], [1, 3, -1, 0, 1, 2, 1, 1, -2, -2, 1, -1, 1, -1, -4, -2, 0, -1, -1], [2, -5, 1, 0, -2, -2, 0, -3, 2, 1, -1, -1, -1, 0, 3, 1, 1, 3, -1], [4, -2, 1, -1, -3, 1, 0, -1, 1, -1, 0, 0, -1, 0, 1, 2, -1, 0, 1], [-1, -3, 2, 0, -3, -2, 2, -5, 3, 1, 0, 0, -1, 0, 1, 3, 1, 2, -1], [0, 2, -1, 0, 0, 0, 1, 0, 0, 0, 1, -2, -2, -1, 1, 2, -1, 0, 0], [0, -5, 1, 2, -1, -2, 2, -2, 3, 1, -1, -1, -1, 1, 3, 1, 2, 3, -2], [3, 1, 2, -1, 3, 2, 0, 2, -3, -2, 0, 2, 5, -1, -4, -3, 1, -2, 1], [6, -1, 1, 2, 2, -1, 1, -1, 2, 1, 1, -2, 1, 2, 2, -1, 0, 1, 1], [-2, 0, -1, 0, -1, -1, 0, -2, 1, 0, 1, 0, -2, -1, 1, 1, 1, -3, 0], [1, 0, 0, 2, -2, -1, 1, -1, 0, -2, 1, -3, -2, 0, 1, -1, 0, 2, 1], [0, -2, 1, 0, -2, 0, 1, 1, -2, -2, -1, 2, 0, 2, -2, 1, 2, -2, 1], [-1, 0, -1, 0, -2, 0, 1, -2, 4, -2, 0, -1, -4, -2, 2, 1, 0, 2, -1], [2, 1, 0, 0, 0, -2, 1, -4, 3, 2, 2, -2, 0, -2, 0, 4, -2, 1, -2], [3, -2, 0, -1, 3, 1, 0, 0, -2, 2, -1, 3, 5, 1, -2, -3, 0, -1, -1], [6, -2, 2, -1, 0, -2, 0, -3, 0, 0, -1, 1, 0, 3, 0, -2, 2, 0, 2], [1, -7, 5, -1, 1, 0, 0, -1, -3, 0, -3, 4, 4, 3, -2, -3, 3, 0, 2], [4, -3, 2, 0, 0, 0, 0, -3, 1, 0, -3, 2, 0, 2, 0, -1, 1, 1, 1], [-3, -1, -2, 2, -4, -3, 0, -1, 3, 2, 1, -4, -5, 0, 8, 0, 0, 3, -1], [-1, 1, -1, 1, -1, 0, 3, -1, 2, 1, -1, -3, -1, -1, 4, 0, 0, 4, -1], [1, -6, 1, 0, -1, -2, 0, -2, 5, 2, 0, -1, -1, 2, 4, 0, -1, 3, -1], [2, -3, 1, 2, 0, 0, -2, 2, -1, -2, -2, 1, -1, 3, 0, -1, 1, -1, 1], [4, 2, -2, 1, 0, 1, 0, -1, 1, 2, 0, 0, 1, 1, 3, 0, 1, -1, 1], [6, -6, 2, 0, 0, -1, -3, -1, 3, 1, -1, 2, 0, 2, 3, 1, 1, -1, -1], [4, -5, 3, 0, -1, 2, -1, 1, -1, -1, -1, 2, 2, 3, -2, -1, 0, -3, 0], [-4, 1, 0, 0, 0, 1, -1, 2, -3, 0, -1, 0, -1, 3, 0, -3, 0, -1, 1], [5, -4, 1, -1, 0, 0, -2, 1, -2, 1, 0, 2, 2, 2, -1, -3, 1, -1, 1], [-1, 0, 0, 1, 1, -1, -1, 0, -2, -2, 1, 1, 2, 1, -2, 2, -1, -3, 1], [5, 3, -1, 0, 4, 1, 0, 2, -2, -3, 0, 2, 2, -1, -6, -2, 1, -1, 0], [1, -2, -4, -1, 2, -2, -2, 3, 1, 2, 0, 0, -1, 0, 1, 2, -1, -3, -1], [-5, -1, 0, 0, -1, 2, 2, 4, 0, -2, -2, 3, 2, -2, -3, 0, 2, -1, -1], [1, 2, -1, -1, 3, 1, -3, 1, -2, -2, 2, 1, 2, -1, -5, 0, -3, -1, 1], [0, -3, -3, 0, -3, 0, -1, 2, 3, -1, 0, -3, -2, -4, 3, 3, 0, 1, 1], [2, -8, 3, -3, -3, 1, -2, -3, 2, 1, -4, 5, 2, 1, -2, 0, 2, 0, 0], [-1, -4, 1, 1, -1, 1, 0, 0, 2, -2, -3, 0, -2, 0, 3, -1, -1, 1, 1], [3, 1, -1, -1, -1, -2, 2, -4, 3, 0, 0, -1, 0, -4, 1, 3, -1, 2, -1], [2, -4, 2, -3, 6, 0, 2, 0, -2, 3, -1, 1, 6, 0, -5, -1, 2, -1, -1], [-4, 2, -3, -2, -1, -1, 2, 1, 0, 3, 3, -3, -3, -3, 1, 2, 0, 4, -2], [6, -1, 2, 4, -2, 2, -2, 2, 0, -1, 1, 2, -1, 3, 0, -3, 0, -3, 2], [1, -2, 3, -1, 0, 1, 2, 2, -3, -2, 0, 0, 0, -1, -2, -1, -2, 1, 0], [5, 7, -4, 4, 4, 4, -2, 4, -2, -1, 0, 1, -3, 3, -1, -3, -1, -1, 2], [-3, 0, -1, -1, -1, 2, -1, 2, 2, -2, -3, 3, -1, -2, 1, 1, 2, 1, -2], [4, 2, 0, 2, 0, 3, 0, 5, -1, -3, 2, 3, 2, -3, -4, 1, 0, -4, -1], [-2, 0, 4, 1, -1, 1, -2, 5, -4, -2, -3, 4, -1, 5, 1, -7, 3, 1, 3], [-6, 1, -2, 1, 1, -1, 2, -2, 3, -1, 0, -2, -1, -2, -2, 5, -3, -1, -1], [-3, -2, -4, 4, 0, -3, 1, 2, 5, 3, 4, -5, -4, 0, 6, 2, 1, 2, -1], [0, 2, -4, 0, 0, 0, 3, 1, 2, 0, 0, -1, 0, -1, -1, 3, 0, -2, 0], [5, 1, 1, 0, 0, -1, 2, -3, -1, 1, 1, -1, 1, 2, -1, 1, -2, 2, -1], [0, 4, 3, -3, -1, 2, 1, 0, -3, -3, 1, 2, 2, -1, -3, 3, -3, -1, 1], [-7, 2, 2, 1, -1, -3, 4, -5, 1, 1, 0, -2, 0, -1, 0, 3, 1, 3, -3], [2, -2, -2, 2, -2, -5, 1, -5, 1, 3, 0, -4, -2, 4, 7, -4, 4, 4, 1], [-2, 2, -2, 2, 2, 0, 1, 3, -1, -1, 2, -2, 0, -1, -3, 1, -2, -2, 1], [0, -2, 1, 1, 0, 3, -4, 6, -5, -4, 0, 2, 2, 3, -2, -2, 0, -3, 0], [1, 7, -1, -2, 4, 3, 0, 2, -4, -1, -2, 1, 1, -2, -7, -2, -4, -3, -2], [-6, 1, 1, 3, -2, 2, 2, -1, 3, -1, 0, -2, -3, -3, 4, 2, 1, 2, -2], [4, 7, 1, -1, 1, 2, 1, -2, -5, -2, 0, 1, 2, -3, -5, 0, -2, 0, -1], [0, 0, -3, 0, 0, -1, -3, 0, 3, 0, 5, -2, -4, -5, -1, 3, -5, -1, -2], [5, -4, 3, 0, -1, 1, -3, 3, 0, -1, -4, 1, 0, 6, 5, -7, 4, 2, 5], [10, -3, 0, 0, 3, 0, -1, 2, -1, 3, -1, 0, 1, 3, -1, -5, 1, 3, -1], [-3, 2, 1, -1, -1, 2, 1, 4, -2, -2, -1, 3, 3, -2, -3, 1, -1, -3, -2], [-1, -5, 1, 1, 0, -1, 3, -3, 3, -3, 0, -2, -4, -1, -1, 4, -1, 2, -1], [-7, 4, -2, 1, 3, -2, 1, 0, -1, -1, -2, -1, -2, -1, -1, 3, 0, 1, -1], [-8, 9, -4, 3, -1, -2, -2, -2, 3, 1, 2, -7, -6, 1, 7, -1, -4, 4, -1], [-7, -4, 1, -2, 1, 0, -1, -2, 0, 3, 1, 0, 2, 0, 0, 0, -1, 0, -2], [1, -2, 4, 0, -6, 1, 0, 3, 5, -3, -2, 1, -3, 0, 8, -1, 2, 4, 1], [5, -3, 0, -3, 1, -1, -3, 3, -2, -1, -1, 2, 2, 1, -1, -3, 0, -2, 3], [-2, -4, 1, -3, -2, -4, 1, -5, -2, 5, 3, 2, 1, -1, -1, 6, 1, -1, -2], [4, -6, 4, -2, -2, -1, 0, -7, -1, 5, 0, 2, -1, 6, 3, 1, 2, 0, -1], [8, 7, -1, 5, 7, 2, -1, 5, -3, -2, 0, -2, -1, 4, 1, -6, -2, 0, 3], [-6, -2, 5, -1, -3, -3, 1, -5, 1, 1, 2, 1, -1, 1, 2, 1, 0, 4, -1], [7, 3, 3, -2, 1, 3, 1, -3, 2, -4, -1, 3, 1, -1, -4, 0, -1, -1, 1], [3, -3, 3, 0, -1, 3, -2, 1, -3, -1, 0, 4, 2, 2, -2, -4, 2, -1, 3], [0, 5, 0, 2, 3, 1, -5, 7, 1, -2, 3, 1, 0, -1, 1, -2, -4, -5, 2], [0, -6, 1, 1, -2, -2, -1, -3, 2, 6, 0, 0, 0, 4, 4, 3, 1, 2, -2], [-7, 5, -5, 1, 4, 0, -2, 4, -3, -3, 0, 2, -1, -4, -5, 0, -4, -4, 1], [1, -6, -3, -3, 1, -1, 1, -1, -1, 4, 0, -1, 1, 0, 1, -2, 3, 1, 1], [-4, 6, 0, -1, 3, 3, 0, 4, -5, -5, -1, 3, 3, -1, -9, -1, 0, -7, 0], [4, 1, 3, 2, -1, 2, 2, -5, -1, 1, 2, 0, 0, -3, 0, 0, 1, 3, -1], [3, -3, 1, 3, -3, 0, 2, 0, 2, 3, 0, 1, -5, 5, 8, 1, -2, 4, 0], [-3, -5, 0, 1, -1, -4, 2, -2, 0, 3, -1, 0, 1, 3, 2, 2, 0, -2, -2], [1, -2, 1, -4, -3, 0, 0, -1, 0, 1, -3, 5, 4, 1, -1, -1, 3, 0, 0], [-2, 0, 3, 1, -2, 1, -3, 5, 1, -1, 2, -1, 0, 2, 2, -1, -1, 1, -1], [-4, 2, -5, 1, -7, -3, 2, -6, 4, 2, 2, -5, -7, -5, 7, 6, 1, 1, 1], [0, 2, -4, 1, -5, 1, -1, 2, 0, 0, 3, -1, -2, -2, 2, 1, 1, -2, 0], [3, 12, -5, 5, 4, 0, 1, 0, 1, 2, 2, -4, -3, -3, 1, 0, -3, 1, -1], [2, 5, 0, 1, 0, 0, 0, -1, 1, 3, 0, -1, -2, 3, 5, -3, 1, 4, -2], [-2, 10, -1, 2, 4, 2, 2, 2, -4, -2, 1, 0, 0, 4, -3, -6, 0, 0, 2], [-2, 7, 3, 3, 3, 3, 0, 6, -3, -8, 2, 2, 2, 3, -8, -2, -2, -4, 3], [-4, -10, 1, 1, -4, -7, 2, -10, 8, 2, -4, -5, -4, 0, 7, 5, 2, 6, -4], [-6, 0, 0, -1, -1, -2, 1, -5, 2, -4, -1, 3, 1, -3, -2, 7, -2, -5, -1], [3, 0, 0, -1, -2, 0, -1, -2, 2, 1, 2, -4, -6, -4, 4, 4, -3, 1, -2], [-9, -5, 0, -5, -3, -1, 3, -2, 2, 1, -4, 3, 3, -2, -1, 5, 3, -2, -1], [1, 11, -2, 2, 0, 3, -1, 6, 1, -4, 6, -3, -4, -5, 1, 3, -10, -3, 0], [0, -3, 0, -4, -3, -5, 2, -12, 6, 6, -2, -3, -1, -2, 5, 1, 3, 9, -1], [-3, -6, 1, -2, 2, 1, 0, 4, -5, -1, -2, 1, 0, 2, 0, -3, 1, -2, 2], [-3, 0, 4, 1, -1, -2, 1, -5, 8, -6, 4, -2, -2, -3, -2, 7, -4, 1, -2], [3, 0, 4, -1, -2, 3, -4, 0, -4, -5, 0, 1, 2, 0, -5, -4, -2, -3, 2], [4, -4, 2, 0, -2, -2, -3, -4, 6, 0, -3, 0, -4, 0, 6, 5, -2, -2, -3], [-1, -4, 3, 0, -1, -1, 2, 1, 2, -3, -5, 0, -1, 1, 4, 4, 2, -2, 2], [-3, 5, -6, 5, 3, 2, -1, 0, 0, 1, 0, -3, -3, -2, 1, 1, -2, 2, -3], [5, 4, 3, 3, 4, 2, -2, 4, 3, 2, 1, -2, -2, -1, 4, -1, -4, 1, -1], [5, 5, 1, -1, 3, 3, 0, 0, -2, -3, 1, 2, 5, -3, -7, -4, 2, -2, -1], [1, 2, 1, -1, 0, 4, 3, 2, -5, -2, -3, 1, 3, -2, -1, -5, 7, 0, 0], [2, -8, 2, -3, -5, -1, 5, -8, 8, -1, -1, -1, -3, -4, 1, 6, 1, 7, -2], [2, -1, 3, 2, 4, 0, 1, -2, 0, -3, 1, 2, 7, 5, -5, 0, -2, -7, 3], [4, 2, 6, 3, 1, 3, -1, 6, 0, -4, -2, 2, 1, 7, 3, -4, 0, -3, 4], [4, -10, 0, 1, -1, -3, -4, -4, 1, 3, -2, 1, 0, 1, 3, 3, 2, 1, -1], [-9, 7, -4, 0, 2, 2, -1, 1, -2, -1, 0, 0, -4, -2, -5, 2, 0, -3, 1], [6, 14, -6, 1, 4, 5, 1, 1, -2, -1, 4, -2, -2, -6, -5, 7, -10, -5, -1], [0, 6, 0, -4, 5, 3, 0, 0, -6, -3, 1, 3, 7, 0, -10, -7, 1, -2, 1], [3, -6, -2, -1, -5, -5, 1, -3, 3, -1, 3, -3, -2, 0, 2, 4, -1, 1, 1], [-2, 9, -1, -2, 0, 1, 3, -7, 4, -3, -1, -3, 0, -5, -3, 3, 0, 1, -1], [-2, -2, -2, -4, 5, -2, -2, -3, 0, 7, -1, 4, 7, -3, -5, -1, 1, 0, -3], [3, 4, -3, 6, 3, 4, -5, 12, -4, -1, 3, -2, -4, 4, 2, -4, -4, -4, 2], [4, 4, 0, -1, 4, -1, -4, 1, 4, -3, -3, 3, 3, -1, -2, -2, 0, 1, 0], [-12, 1, 1, -4, -2, -2, 0, 1, 0, -1, -1, 3, 1, -4, 1, -3, 0, 3, -1], [-4, 1, 1, 0, 5, 6, -3, 5, -8, -2, -4, 2, 6, 2, -5, -7, -1, -3, 0], [5, -3, 0, 2, -5, -3, 0, -7, 4, 2, -3, -2, -6, 6, 10, -1, 3, 3, 2], [9, -7, 2, -1, -5, -4, 4, -4, 8, 5, -1, -2, -2, 0, 8, 3, 6, 6, 0], [0, 4, -1, -3, 1, -1, -2, 0, -5, 0, 6, 3, 5, 1, -11, 1, -3, -5, -1], [0, 2, 2, -6, 2, -1, 0, -2, -1, 1, 1, 2, 8, 2, -7, 1, -1, 0, -4], [-8, 10, -5, 1, -5, 0, 6, -2, 9, -1, 5, -7, -10, -6, 4, 6, -5, 5, -1], [6, 0, -3, 2, -2, -2, -2, -4, 2, 0, -2, 4, -5, 2, 3, 6, 0, -4, -1], [-7, -4, -4, 0, 2, -5, -2, 0, 4, 6, -2, 1, 2, 1, 6, 1, 3, 2, -2], [-5, -1, -3, 0, 1, 2, -7, 2, -3, -1, 6, 1, 0, -2, -3, 0, -4, 1, 1], [-5, -2, 4, -3, 0, 1, 1, -3, 0, 1, 0, -3, 3, 0, -3, -1, -1, 3, -2], [7, 1, 2, 4, 3, 1, 3, 2, 2, 1, 3, -2, 1, 2, -1, -2, 2, 3, -1], [-14, 14, -7, 5, -1, -1, 2, -4, -1, -5, 3, -4, -5, -4, -1, 1, -3, -3, 1], [-4, 4, 0, -3, 7, 3, 0, 5, -8, -1, 1, 3, 8, -3, -13, 2, -6, -10, -2], [-1, 9, -1, 0, 3, 4, 1, -3, -1, -1, -1, -2, 3, -6, -6, 1, -1, -1, -2], [-15, 2, -5, 0, 1, 0, 7, -3, 3, 3, 0, -5, -3, -4, 1, 2, -1, 4, -3], [-15, 3, -1, 4, -2, 0, 0, 0, 1, -2, -3, 0, -4, 1, 7, 1, 2, 1, 0], [-2, 2, -5, -1, -5, -5, -1, -3, 4, 6, 4, -2, -10, -3, 10, 8, -6, 1, -3], [10, -11, -4, 3, -6, 1, 4, 6, 3, 3, 2, -2, -6, -1, 8, 4, 2, 2, -2], [-6, -7, -1, 3, 0, -1, -1, 0, 1, 3, -6, -3, -5, 1, 8, -3, 1, 5, 0], [-10, 5, 6, 3, -1, 2, 4, -5, 1, -1, -5, 2, 0, 0, 4, 1, 1, 0, -2], [3, -6, -1, -2, -1, -2, -2, 2, -2, -5, 0, -2, 2, -2, 2, -1, -1, 0, 3], [-5, 2, 0, 0, -2, -3, -3, 3, 5, -5, -2, 0, -6, 1, 5, -1, -2, 1, 2], [-10, 0, -1, 6, -3, 1, 5, 0, 3, 2, 5, -3, -2, -2, 2, 2, 1, 3, 0], [-10, 0, -3, -1, -5, -2, 5, -4, 1, 3, 3, -4, 0, -3, -1, 9, -1, -3, -3], [-2, 5, -5, -1, -7, -3, 1, -9, 7, 2, 0, -3, -6, -7, 5, 12, -2, 0, -1], [2, 5, 1, -1, 4, -1, 0, -1, -10, 3, 0, 0, 5, 2, -1, -4, 2, -1, 3], [7, -1, -3, 3, 1, 3, 1, 0, -1, 2, 5, 2, 1, 1, -6, -2, 2, -4, 2], [-1, -11, 1, 3, -1, -1, 1, -3, 0, 5, -2, 0, -5, 6, 5, -6, 5, 6, 2], [0, 1, -3, -1, -3, -3, 1, -5, 4, 5, 0, -6, -6, 0, 6, 1, 1, 8, -4], [-1, 7, -3, -4, 6, 2, 1, 3, -8, 1, 2, 2, 7, -1, -12, -3, -1, -5, 0], [13, -1, 1, 0, -3, -5, -3, -6, 7, 5, 3, -5, -4, 0, 5, 4, -3, 5, -4], [0, -7, -4, 6, -4, -2, -4, 1, 5, 2, 4, -7, -8, -5, 12, 7, -5, -1, 2], [5, -1, -4, -1, -1, -3, 5, -10, 5, 5, 0, -7, -4, -5, 1, 7, -4, 0, 1], [-11, 2, 3, 0, -4, 5, -1, 6, -5, -9, 1, 3, 0, 2, -5, -4, -1, -4, 3], [2, -7, 0, -3, -5, -1, 5, -4, 0, 3, -2, 3, -1, -6, -3, 4, 4, 4, -3], [4, -9, 3, -4, -5, -2, -2, 1, 2, 0, -2, 3, 0, -3, 5, 2, 4, 5, -2], [-6, -7, 0, -6, 2, -4, 0, -2, 3, 4, -4, 0, 1, 0, 6, -3, 6, 5, -2], [15, -11, 10, -3, -2, 4, -2, 2, -2, 0, -2, 8, 7, 5, -3, -6, 4, 4, -1], [8, -9, 8, 2, 2, -2, -1, -1, 2, -1, -1, 4, 4, 5, -3, -8, 7, 3, 0], [-10, -1, -7, 1, -6, -1, 0, -1, 8, 0, 4, -4, -9, -9, 8, 7, -2, 10, -2], [-2, -12, 7, 2, -5, -4, 3, -5, 2, 5, -3, 1, 0, 12, 9, -4, 11, 4, 2], [-2, 2, -4, -4, 1, -3, 3, -1, 2, -1, 1, -2, 2, -6, -7, 7, -1, -2, -5], [3, 0, 3, 4, 0, 3, 3, 8, -2, -6, 2, 2, 2, -5, -7, 5, -1, -4, -1], [2, 11, 0, 1, 6, 1, -1, 0, -1, 1, 2, -5, 1, -1, -1, -2, -6, 4, -2], [5, 3, -4, -1, 2, 5, -1, 1, -6, 2, 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4, 1, 1, 9, -3, -6, 0], [13, 7, 5, -1, -3, 4, -4, 7, -2, -7, -2, 6, 4, 5, -7, -8, 5, -2, 4], [-8, 7, -7, 4, 6, 1, -1, 4, 3, -2, -1, 0, -2, 1, -7, 5, -3, -7, 4], [-14, 8, 1, 1, -6, 1, 2, -8, 3, -4, 2, -2, -5, -4, 2, 7, -2, 5, 0], [4, -2, 0, 0, -2, 4, -1, 1, -3, -5, -6, 3, -5, 0, 2, -1, -2, -2, 1], [15, -2, -1, -1, 1, 5, -4, 13, -6, -4, 0, 5, 3, 5, -3, -9, 2, 1, -1], [6, -10, 3, -6, -2, -1, -3, -5, -7, -3, -5, 8, 2, 2, -2, -6, 6, 2, 3], [16, 5, 6, -3, 7, 2, -6, 7, -12, 1, -2, 9, 11, 8, -7, -11, 4, -6, 3], [14, -6, 7, -3, -1, 5, -2, 4, -5, 2, -3, 5, 6, 6, 1, -14, 9, 4, 6], [-3, 3, 4, -1, 0, 6, -8, 5, -7, -3, 0, 7, 5, 9, -3, -6, 2, -4, 4], [7, 5, 5, -2, 6, 8, -3, 8, -10, -3, 0, 9, 14, 1, -15, -2, -6, -9, 0], [7, 4, -6, 11, 2, 0, 0, 7, 0, -1, 3, -7, -7, 1, 9, -3, -4, -1, 1], [-3, -10, -3, 3, -1, -8, 2, -6, 4, 10, 2, -1, -3, 2, 10, 0, 6, 3, -1], [2, 8, 3, -5, -2, 3, -3, 12, -4, -4, 8, -3, 1, -3, -3, -3, -3, -1, 4], [1, -8, -1, -4, 2, -2, -4, 2, -5, 3, -6, 9, 2, 7, 1, -2, 0, -6, 4], [-12, 9, 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0, -8, 3, -5, -5, 0, 1, 0, 0, -2, 3], [5, 0, 4, 7, 3, 5, 3, -6, 2, -2, -1, 1, 0, -1, -4, 2, 1, -1, -1], [-8, 7, 4, 1, 2, -6, 5, -9, 7, 7, -3, -4, -1, 1, 4, 1, 5, 8, -3], [15, 0, -1, 7, -2, -3, 5, -7, 3, 2, -3, -8, -1, 0, 12, 3, 2, 3, 1], [-9, 4, -2, 3, 7, -1, 4, 4, -8, -4, 4, 0, -2, 3, -9, 0, -5, -5, 0], [0, -6, 1, 8, 0, 1, 0, 7, -2, 0, 1, 2, -4, 3, 7, -9, 8, 6, 0], [3, -12, 1, -2, -6, -7, -1, -12, 3, 10, 1, -7, 0, 1, 10, 3, -3, 9, -4], [2, -8, 6, -9, -5, 0, 0, -10, 4, 4, 3, 4, 9, -6, -8, 6, 1, 1, -5], [8, 14, -4, 7, -3, 0, -2, -2, 10, -6, 5, -5, -11, 0, 8, 5, -14, 3, -1], [4, -1, 2, 3, 0, -1, -6, -3, -5, 4, 2, 5, 0, 0, -2, -2, 0, -1, 1], [-1, 9, -8, 0, 0, 1, -3, 7, 11, -8, 0, -7, -12, -9, 5, 5, -4, -1, 0], [-3, -6, -5, 6, -7, -2, -4, -1, 6, 3, -4, 3, -4, 5, 8, 3, 4, -2, 0], [1, 6, -2, 2, 6, -1, 4, 6, -4, 6, 5, 0, 3, 2, 0, 2, 0, -5, 2], [9, 0, -4, -1, -5, 0, 5, -6, 5, 1, 5, 1, 1, -5, -9, 9, -2, -1, -6], [3, -15, 3, 3, -6, -13, 1, -4, 2, 8, -9, -2, -2, 10, 18, -8, 15, 6, 3], [17, -1, 4, 1, 2, 0, -1, 3, 6, -8, 1, 2, -1, -3, -1, 3, -2, -2, 6], [5, 0, -1, 2, 1, 7, -7, 5, 4, -5, -3, 11, -1, 4, -7, -5, 0, -6, 4], [6, 9, -4, -3, 3, 1, 7, -6, 0, 5, -5, -2, 1, -2, -4, 2, -2, 1, -5], [-6, 10, -2, 4, 0, 2, 2, 2, -4, -6, 1, -6, -11, 0, 0, -2, -1, 2, 1], [1, 2, -4, -9, -3, -3, -3, -10, 9, 9, -1, -1, -3, -2, 3, 6, -3, 7, -6], [9, 5, 0, 3, 1, 6, -2, 8, -4, -7, 4, 4, 5, 2, -10, -11, 4, -4, 5], [-4, -2, 2, -1, 4, 0, 3, 4, 1, 0, -1, -1, 6, 0, 0, 1, -2, -1, 2], [0, -14, 6, -6, -9, -10, -3, -10, 1, 6, -1, -1, -2, 7, 9, -2, 7, 6, 6], [12, 5, -5, -5, 6, 7, 0, 0, -7, 7, 1, 3, 8, -1, -10, -5, 2, -8, 0], [-6, 12, 0, 1, 6, 5, -2, 13, -20, -5, 3, 8, 4, 2, -13, -8, 1, -7, 4], [4, -3, 4, 3, -10, -4, 0, -2, 6, -6, -1, 1, -8, 1, 14, 2, 1, 4, 6], [-20, 27, -15, 3, 0, 0, 4, 2, 1, 3, 8, -12, -12, -11, 5, 12, -11, -4, -5], [13, -15, 6, -2, -5, -7, -1, -11, 9, 9, 0, -1, -1, -4, 8, 11, 1, -2, 1], [4, 7, -1, -2, 1, 8, -4, 5, -7, -5, -2, 12, 4, -9, -8, 10, -4, -15, 5], [-7, 8, 4, 0, -4, 4, 3, -1, -3, 0, 2, 5, 4, 4, -3, 0, -2, 5, 0], [-3, -6, 6, 0, -8, 3, -1, 6, 0, -6, -8, 4, -4, 0, 8, -4, 7, 8, -2], [-3, 9, -4, -1, 3, -5, 0, -6, 12, 3, 8, -11, -5, -9, 3, 11, -7, 4, -3], [8, 8, -3, 4, -2, 0, 8, -2, 2, -6, -1, -6, -6, -2, 1, 0, -3, 2, 0], [18, -2, 2, 0, -2, -2, 2, -4, -2, 6, 3, -1, 2, 3, 1, 10, -6, -10, -1], [-8, -9, -4, -2, -1, -12, 4, -5, 10, 6, 1, -7, -11, -6, 7, 3, 2, 11, -2], [-1, 10, 0, 3, 2, 3, 1, 0, -5, 2, 1, 2, -4, -4, 3, 5, -1, -3, -2], [8, -19, 9, -4, -6, 8, -2, -6, -1, -2, -6, 11, 7, -3, -6, -3, 1, 1, -1], [-10, -2, -3, -6, -6, -5, 5, -9, 6, -1, -4, 0, -4, -8, 2, 17, -3, 3, -4], [7, -4, 0, -8, -3, -7, -3, 2, 0, 1, -5, -2, -3, 4, 13, -2, 0, 0, 1], [11, 3, 2, -6, -1, 2, -4, 6, -4, 4, -4, 5, 2, -4, 1, -4, 2, -1, -1], [13, -11, -1, -8, 0, -7, -4, -2, 1, 0, -6, 7, 2, -1, -5, 2, 3, 1, -3], [4, 2, -11, 0, -3, -6, 8, -4, 16, 7, 5, -13, -7, -13, 9, 18, -5, 8, -4], [0, 11, 6, -2, -1, -1, 1, 3, -3, 0, 1, -1, 2, 6, 3, -11, -2, 2, 4], [2, 6, 2, -1, 1, 3, 2, -2, -11, 1, 3, 3, 5, 1, -9, -3, 3, -3, 4], [-19, 21, -6, -2, 3, 3, 12, 4, -7, 0, 2, -10, -2, -6, -2, 4, -5, -4, 1], [7, 12, -3, -2, 4, 1, -3, 2, 8, 0, 2, -1, 0, -7, 2, 10, -11, -6, -3], [6, 18, -6, -5, 10, 0, -2, 4, -4, -3, 2, -2, 1, -6, -6, 6, -12, -13, 0], [-3, 4, 0, -3, 10, 5, -8, 16, -17, -2, -5, 6, 8, 3, -9, -9, 7, -9, 2], [23, -9, 6, -4, -5, -1, -4, -6, 9, -5, 3, -3, 1, 2, 4, 0, 0, 6, 0], [8, -17, 2, 2, -4, -1, -4, 3, 3, 10, -5, 3, -5, 8, 17, -6, 7, 7, 3], [2, 21, 0, 3, 4, 8, 5, 0, -3, -5, 2, -2, 0, -7, -12, 1, -6, 2, 0], [-3, -6, -3, -5, -1, 1, -8, -4, 8, 11, 0, 3, 1, -6, 4, 6, 0, -6, -4], [12, 10, -1, 3, 3, -2, 4, -4, -5, 3, 5, -2, 3, -2, 1, 1, -1, -3, 3], [16, 10, -1, 13, 5, 3, 2, 4, 1, -7, 0, -1, -3, -1, 1, -1, 0, -5, 2], [5, 11, -1, -1, 6, 10, -9, 13, -13, -9, -6, 7, 7, 3, -8, -11, 8, -8, 5], [22, -2, -1, 0, 0, -6, -1, 2, 9, 1, 3, -6, -6, -3, 10, 7, -1, 4, -5], [12, -3, 4, -7, -5, 6, -1, -7, -1, -4, 3, -2, 3, -6, -9, 6, -6, -2, -5], [-1, 5, -1, 4, -4, -1, -8, 7, -2, -7, 3, -3, -2, 8, 7, -10, 2, 5, 5], [-5, 1, -4, 3, 7, -4, 8, -3, 8, 5, 3, -4, -4, -6, -2, 15, -5, 0, -4], [16, -1, -3, -1, 7, -6, -2, 9, 3, 4, -3, 3, 1, 7, 1, -10, 6, 3, -1], [5, 13, -6, 1, 8, 0, 8, 6, -6, -2, 5, -2, 4, -3, -7, -4, -4, 4, -2], [3, 0, 5, 0, -4, -1, 3, -5, -1, 1, -1, 1, -1, 8, 8, -4, 1, 4, -2], [3, 15, -5, 0, 0, 3, -5, 4, 6, -7, 1, -2, -9, -5, 0, 0, -1, 4, 4], [-13, 1, 2, 3, -1, 0, -4, 9, -2, -9, -4, -5, 2, 5, 1, -9, 2, -4, 7], [9, 9, 2, 2, 5, 6, 4, 4, -8, -6, 1, 5, 3, 2, -11, 4, -1, -5, 2], [8, 4, -1, 2, -12, 1, -4, 2, 4, -4, 1, -2, -10, -5, 6, 3, -2, 4, 0], [14, -9, 8, -3, -3, 5, 1, -3, -3, 4, -6, 3, 5, 4, 6, -1, 3, 2, -2], [-28, 3, -2, 1, -1, 0, -1, 6, -5, 1, -8, -5, -3, 4, 5, -7, 1, 3, 1], [5, 4, -12, -7, 1, -4, -1, 10, -11, -2, -3, -3, 3, -2, -2, -4, 0, -2, 3], [-8, -6, 3, 5, 4, -12, 0, -10, 7, 2, 2, -4, -1, 7, 8, 4, -7, 3, -2], [-7, -6, -1, -6, -4, -2, 4, -6, 1, 1, -8, -4, -5, 0, 8, 8, 0, 4, 0], [-18, 17, -8, 4, 4, -6, 4, 2, 5, 4, 12, -5, -5, -7, -7, 8, -3, 0, -5], [4, -2, -4, 1, 3, 7, -9, 13, -6, -4, -3, 11, 3, 6, -2, -6, 2, -13, 3], [3, -16, -8, 2, -4, -6, -1, -2, 6, 10, 4, -4, 0, 3, 6, 3, 7, 0, 0], [-15, 2, -4, 2, 3, 0, 11, -2, 10, 3, -1, -12, -6, -10, 9, 9, -8, 11, -6], [-7, 9, 0, 6, 2, -3, 3, 7, 3, 0, 0, -3, -4, -1, 3, 3, 0, -3, 0], [-9, -3, 4, -6, 8, 1, -8, 4, -18, 3, -3, 8, 12, 9, -13, -12, 3, -7, 0], [13, -1, 2, -5, 3, -7, 1, 0, -2, -1, -3, 1, -2, 8, -5, -7, -4, 6, 0], [-13, -1, -2, 6, -7, -3, 5, -6, 5, 3, 3, -14, -8, 3, 17, 2, -4, 3, 1], [4, -2, 2, 3, -5, -5, -1, -8, 10, -2, 0, 0, 0, -4, 1, 5, 1, 5, -6], [-5, 9, -6, 7, 0, -1, 0, -3, -6, 2, 1, -1, 3, 7, 0, -9, 7, -12, 6], [9, -2, 6, -1, 3, 4, -3, 5, 7, -3, -2, 8, -1, 2, -10, 1, -4, 1, -5], [5, -3, 1, -1, 1, -4, -5, -4, -1, 0, -1, 6, -4, 5, 1, -12, 4, 0, 5], [9, -5, 5, 0, -2, -4, 3, -13, -7, 14, 4, -1, 3, 12, 9, -8, 4, 3, 1], [-12, 3, -1, -1, 6, 6, 4, 10, -4, -4, 4, 9, 5, -10, -19, 5, -2, -8, 4], [1, 19, -8, -2, -1, 1, 0, 1, -7, 3, 3, -2, -3, -5, 3, 3, -2, -9, -1], [4, 10, -4, 0, 7, 1, 2, -2, 6, 0, 8, 0, -4, -3, -10, 3, -12, 3, -1], [2, 8, -4, -2, 0, 1, 14, -14, 4, 13, 0, -12, 2, -9, 3, 7, 0, 12, -9], [-20, 7, -3, -1, 9, 0, 3, 3, -1, -1, -1, 1, 5, -7, -16, 4, -7, -4, -7], [9, -16, 8, -1, -2, -6, -6, 0, 2, 3, -3, 3, 5, 12, 9, -10, 10, 5, 5], [-8, 7, 4, -4, -5, 1, 5, -14, 9, 0, -2, 1, 7, -8, -7, 9, 5, 0, -7], [-3, -11, -2, -2, 0, -4, -9, 2, -2, 2, -1, 9, -2, 3, -1, -6, 6, 3, -1], [-13, 17, -7, 4, -2, 12, 4, 15, -3, -11, 1, -1, -11, -7, -3, 0, -3, -4, -1], [-7, -4, 3, 2, 1, 6, -9, 20, -16, -4, 2, 15, 4, 11, -8, -12, 5, -8, 8], [-21, 8, -6, -7, 5, 4, -2, 8, -12, -1, 0, 5, 4, -10, -13, -2, 2, -5, -5], [4, -2, -6, -2, -2, -2, 5, -8, -11, 0, 3, -5, 2, -6, -5, 3, 5, -4, 3], [12, 4, -7, 4, 1, -9, -4, 2, 2, 1, 5, -4, 5, -2, -1, -6, 5, -2, 1], [7, 14, -1, 9, 12, 3, 6, 3, 0, -5, 6, 1, 1, -2, -16, 6, -8, -5, -3], [-6, 4, 0, 5, 6, 4, 0, 3, -5, -4, -8, 9, 2, 2, 4, -8, 5, -9, 9], [-8, 11, -8, 0, 1, 0, -6, 13, 1, 5, 2, -5, 0, -6, 4, -2, 1, 3, -2], [-11, -3, -8, -4, -1, -3, -2, -2, 3, 6, 2, 1, 5, -6, -3, 1, 8, 1, -3], [21, -4, 4, 12, -6, 4, 1, -2, 4, -4, -9, -4, -11, 1, 14, -3, 0, 6, 3], [-1, -11, -8, 1, -7, -3, 4, -7, 10, 11, 3, -10, -6, -13, 11, 16, 3, 4, -4], [11, 6, -2, -1, 2, -1, 0, -3, 0, 5, 3, -1, 1, -4, -9, 2, 0, -1, 1], [8, 13, 3, 4, 4, 11, 1, 13, -4, -3, 3, 3, 0, 3, -5, -1, -6, -3, 5], [-5, 13, -5, 8, 2, 3, 2, 8, -9, -8, 4, -6, -3, 0, 1, -2, -8, -3, 3], [7, -2, -8, 0, 0, 3, -5, 2, -14, 4, 2, -2, 8, 0, -4, 0, 1, -16, 4], [4, -12, -3, -1, -10, -7, -5, -6, 8, 1, 0, -6, -6, 4, 17, 3, -4, -1, -1], [9, -7, 6, 2, 4, 1, -4, -9, 6, 9, 1, 4, 4, 7, 5, -2, -3, 1, 0], [13, -3, -2, 4, -7, -2, 2, -14, 13, 9, 6, -5, -5, 1, 7, 2, 10, 9, -2], [-2, -7, 1, -1, 3, 0, 9, -11, -3, 10, 4, 3, 10, -3, -13, 6, 2, -6, -5], [-1, -11, 1, 0, -3, -2, -3, 1, 6, -6, 1, 4, -2, -6, 3, 2, 1, 5, -1], [7, 17, 2, -4, 3, -1, 9, -3, -1, 1, 5, -8, 2, -10, -7, 3, -8, 1, -3], [-18, -15, 3, 1, -12, 1, -4, 0, 12, -4, -7, 0, -8, -2, 16, 4, -3, 0, 1], [-16, 12, -6, 2, -2, 7, 6, 6, 1, -7, 6, -2, -7, -8, -1, 9, -3, 0, 1], [0, 5, 1, 6, 6, 3, 6, 8, -9, -2, 5, 2, 5, -9, -11, -6, 5, -3, 2], [1, 21, -6, -5, 7, 6, -7, 18, -4, -5, 9, 1, 1, -3, -12, -2, -11, -8, -1], [1, 22, -3, 1, 9, 3, -1, 5, 2, 2, 6, -4, -5, -4, -6, 2, -11, -1, -7], [1, -2, 4, 8, 1, 1, -3, 13, -8, -3, 0, -3, -6, 5, 12, -6, 1, 2, 10], [16, -7, 5, -6, 5, 2, -1, 2, 0, -2, -8, 8, 10, 2, -9, -7, 1, 7, 0], [3, -1, -5, 2, 3, 3, -1, 4, -8, 8, -2, 6, 1, 4, 3, -4, 4, -7, 2], [18, 2, 4, -6, 2, 2, 11, -12, 0, 8, -3, 3, 5, -7, -7, 9, 6, 4, -3], [3, -17, 8, 7, -7, 1, 4, -3, -4, 3, -9, 5, -1, 16, 12, -4, 9, 0, 2], [11, -5, -2, -1, -1, 2, 2, -1, -1, 2, -1, 2, -5, -3, 7, 9, 0, -4, -1], [-19, -5, 2, 7, -6, 4, 10, 3, 5, -1, -1, 1, -4, 9, 3, -4, 4, 2, -1], [9, 8, -8, 4, -5, -1, -4, 9, 4, -5, 0, -1, -6, 0, 12, -4, -2, -4, 10], [-11, 11, -6, -1, 9, 9, 3, -5, -5, -11, 5, -3, 5, -8, -21, 0, -5, -3, -4], [7, -3, 5, 3, -3, -3, 6, -14, 17, -4, 4, -4, -3, 2, -2, 4, -9, 8, -5], [-4, -6, 7, 1, -9, 10, 3, 3, -5, -1, -1, 7, 2, 0, -1, 9, 3, -5, 2], [-9, 14, 0, 2, 12, 7, -2, 17, -14, -4, 4, 12, 7, 6, -18, -9, -2, -10, 2], [-4, 8, 0, 0, -10, 2, 4, -1, -2, -1, 1, -6, -7, -9, 5, 3, -5, 3, -2], [20, 13, 3, 2, 0, -1, -1, 8, 7, -11, 5, -2, 0, -7, -2, 3, -7, 5, -1], [14, -5, 6, 10, -13, -3, 11, -11, 10, 2, 1, -15, -8, 2, 20, 2, 1, 11, 1], [6, 0, 4, 3, 2, -2, -1, -6, 9, 0, -5, -6, -3, 2, 11, -4, -4, 7, 0], [-12, 11, -5, -3, 2, 2, 3, 0, 4, 4, -1, -8, -3, 3, 4, 2, -6, 0, 1], [-14, 4, 1, 5, -5, -5, 2, -7, 5, 1, -5, -10, -13, 4, 21, -5, -2, 11, 1], [-4, -12, -6, -5, -9, -8, 3, -4, -3, 1, -11, -2, -5, -3, 10, 7, 4, 3, 0], [10, -2, -1, 6, -3, 0, -2, 7, 1, 0, 8, 0, -6, -4, 7, 3, -6, -4, 0], [-2, 10, -4, 2, -1, 2, 0, 5, 10, -3, 3, -7, -11, -6, 3, 5, 1, 4, 0], [10, 4, 0, 8, 1, -7, -1, 7, -1, -2, 0, -10, -6, 13, 12, -15, 4, 6, 7], [-14, -2, -10, 2, -6, -5, 1, -5, 11, 6, 4, -3, -11, -4, 13, 5, 1, 13, 0], [9, -21, 9, 0, -2, 3, -1, -1, 7, 0, -4, -3, 5, 3, 2, -3, 0, 9, -5], [-8, -18, 4, -1, -7, -3, -1, -12, 8, 4, -3, 0, 2, 1, 5, 4, 9, 1, -1], [-7, -13, 7, -7, -1, -6, 4, -19, 9, 10, -4, 1, 4, 0, -1, 1, 1, 10, -9], [12, 2, -3, -5, -5, 1, 1, 9, -1, -5, -3, 6, 0, -2, -11, -4, 5, -1, -1], [-13, 10, -3, 7, 7, -2, -3, 1, 2, -4, 8, -7, -5, 0, 6, -3, -5, 3, 4], [-2, -10, 6, -14, -8, -6, 11, -12, 0, 6, 0, 2, 10, 0, -3, 3, 9, 6, -7], [-8, 9, -9, 1, 0, -6, 1, -3, 2, 2, 1, -8, -9, -2, 4, 3, -4, 8, -3], [12, 18, -6, 4, 2, 4, 4, 9, -4, -10, 11, -8, -5, -6, -6, -2, -10, 0, 5]];
    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_f();".
// 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_f(: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_f();".
// 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_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4028_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![139, -454, -4299, 5074, 30973, -17877, -73816, 23819, 72868, -18074, -36567, 8368, 10113, -2310, -1554, 364, 124, -30, -4, 1]>,<5,R![25088, -268352, 1038048, -1652304, 375400, 1892192, -1649600, -537325, 1030638, -36807, -298423, 46799, 47508, -9947, -4277, 1000, 204, -50, -4, 1]>],Snew);
    return Vf;
end function;