
// Make newform 1058.4.a.v in Magma, downloaded from the LMFDB on 26 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1058_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_1058_4_a_v();".
// 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_1058_4_a_v();".
// 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 := [-774554923, -2187721107, -248273080, 2280616242, 73251838, -487850935, 1945215, 38601508, -201553, -1439134, 3762, 27588, -21, -264, 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], [2925199718867511743278121183943047928404262229817, 4388583299505687824233796890524479348603432323120, -1966186012359524553104560258116880317073311068390, -3080304895450211974223364509131837200773155014873, 148245784215230688043291665370306977381811176947, 549345172287176699946367773439832455326229109473, 19732248293082892232283537620194513849067059967, -37818642540431983262619961569206834777717321448, -2862700300398161885797551433467328136977905301, 1106922913051077949751212145941349366918143918, 104880164102456076815308467528029520101522167, -14361463857378540077280230529959696759126072, -1511984148152831955914397971974483948425710, 68416346911880486676399178602877915014371, 7643816924386050878776716774430886010791], [5045891656471174773452873854582197758217795040480, 6562426734535646196200511651747645231641144697072, 1189191449105016642573191521844472585703964986433, -7302401031950562031621764952660119715577536994029, -875938706162223397852896215121108385355960757621, 1451608156234931687686034140992427285201359999416, 108323032457712996338333485243174345680794620568, -98685001147452857611624177209246725557846706831, -7257105738668395036710625116235711761160623967, 2831543381612382145107011753016477264068008938, 218801144473507813961252235410286474958424812, -36146665820720581647235940227508217114326402, -2916781674580804292153058182114542937659996, 169973450989140904440903835828508784972601, 14169249986338902082199364077992284942538], [6436854790875600368371029592925765896345847379330, 3229819864673444322260619026709264255042813144589, -18235642105038327310937023065610748182598457495931, 6888474242103929950703468848849138144996440206744, 5433488873618175722602171898096369373235568044094, -1817738463792533422007067327475353266927238791937, -545738404522080196309487535829940111036045071816, 133591641567623957683612940467449374979034465496, 27523140061234780040214778781535090982070834966, -3960916696741340018333403311727998217773699549, -695005262696612436167173124638708288990408937, 51534344269934438074892693776197115538968797, 8356828745578694421535125662844758232453006, -245702181275708154172703485633134695937621, -38119723881790537152739771937657444446886], [1183985957511810964332656964276508419413520355505, -7547450534088621371320004658702205440198327205485, -11910863689728855162088973678216799064524286094454, 11542470317179299037327006152802139057649157749901, 4179528094033474630613057920399346809737879808309, -2485960028544056790088507572140407318713167658119, -498887898207320390292465561705074557957869849585, 179572693833396327826274245304958300310297388605, 29561715002491071569941268138455010200147425543, -5332591500185900781669783275161281099581952504, -822192713037749197538446040678593966527198290, 69566404572714921830136624746231644521027423, 10455195585704572558583234029491163822235344, -332270140553031428932801619003846929075465, -49332943903067802380575498275797753085830], [-31603612627572629435186228278773658133411462134785, -27822391451794620332317638126830259390032139609835, 57577281688717469886240897009283892852084954768698, -325642407731068324962720088047863304923207439754, -14079526949334529547074177405841718721505118273596, 1313469397264986552403006900065238521872036874210, 1223800906240545825976479784170299959550836712259, -111094659161699615374955979262681233117862036191, -52181386840717033636182012441863610794952209716, 3397491141278721607962431452215080746051849437, 1151514744089012455328706302177336706899832082, -44550778450777743012269822855777477037092291, -12546344579264793289017240261668968292311704, 212692928991005644266002765885959655416757, 53262193231675498432792041540567621591899], [3897759376881934858752373957044584284478137100229, -11056831279235571346293262680492236637194183817841, -20434369008857106747647851412350574139606344312636, 14335372370731918354529232785371838264562772767384, 6277879213559978769337899322011360467141460018021, -2977333602227573045459354140507987775984327989598, -680743166428282444668192709095157624149718158336, 209275225563066988907962621436547207653281606384, 37294951794730317937552984598927363774531397573, -6128330489564582630981767144608774560769029875, -994051637827696940382923761617581835169890081, 79359172097041033598962851466812717324203110, 12356927870257625652449949555335246614102403, -377530804768412237353181247149217507528270, -57570675394817872964416555037569702747748], [28002566843064015580121708391651971338628919475254, 22870254811181856215389761902948202063550794514179, -49907629458840428518287834446624238655206999355311, 3165762851875203543189517503207526654282023173939, 12882541277872163938085747012931511601060670017974, -1927459804604793634727822188937739929540450426906, -1191130630975804378034751670517203533094491009419, 161064631009852144787292330869774471649547042297, 55253249981227870480913338779254797260356405120, -4992138103441849031736238078253753182630103952, -1312787853518829939352996685980793473829395146, 66310501648581314951865865806114925984684942, 15144418836866309573446386852276834466183993, -319853701856707057760965102340492166953952, -67122693719540951018057560041375354928360], [49944411259943746429453931069196447195842885467179, 78815294682323397465318424313691773011519471092633, -52784726404553762237538047446813492785463760899936, -41853463414151505042253012484690984040509868247024, 8985580866944341481954643673174467156835520050834, 6894980427384167610732410450117103805579743445336, -373603171820331653228033684400710505120276234821, -467258169475067136461950160056625797060602006871, -10901269146305078253961195549238375559025967636, 13654045314720906896214115723969807936262026353, 804153173353148009292307980151014319530837552, -177361019079744489456392450062730867232758351, -13796492523997334259494398066400001842521548, 846244876725383009375293975612594429890829, 75375550497012980527693620625397514260475], [-12646703890021044585084754348974097283023718168090, 4670781344253883108820556523177741897910626150553, 42060366778121160762668546877726184707540178657459, -20143010282274614243754082650764851112873272488698, -12241657620720403749725165961127794607751931273574, 4793585951724393353193690792828430177816386059101, 1264910220288905639637418435475898008268837108390, -352932311322999928868313786106742367623673501582, -67210672502593952468800836373857372834815680298, 10539435795567496040028572931852958464642853087, 1763314552276078098842133697865788895366854053, -137852253550262863342431584223301024685150793, -21726010322123897918669789209252341489696600, 659540749071225120119901969057628405539937, 100657082714440290714645256775594944202530], [-6585267989326371675306811459892377299686486748664, -32722610733438980992871232663742071114188662328936, -12360662751944135879331146078819651479790919419807, 31060963373027021892089869363895386952847043895357, 6140954510188820138283048999404988317719828486990, -6123625200184499144907358704842056965688104450336, -856992569099533005363439786174572576764858402427, 429560455080298039366992018374509632868420088372, 56339927550379713741141773692838010456084835038, -12611790530733903043309387752986641445137790722, -1650015793135888886625492918112887385111721337, 163644333570211716010648244311467956101915511, 21572805067215459827041725974916076306959521, -779295633290636679797152883699507085897892, -103502710110455333894298126203818776511335], [-32462109345377477605122103911330503024157291055682, -18044866210653980091744134555152101667103982841939, 65104512252581526141580149675466921750290036613558, -13695154511425667445673535139509195198666233502148, -17305844892711306074609941991737920422659417703728, 4159998951153610799233546074593056296655521980072, 1651701928763144565488534988321009641163669155990, -316794094917694631829845006859688974025933450595, -80019933431315817747279527655139001820227127862, 9515991060171026323138752533245202347592082660, 1967127904102838277242254197952495866507627617, -124494509532724925673624318276414504217156157, -23233177513753816016848326220886291911945376, 595020159954376308358290868399940338302418, 104648317416903560910112760619161136237689], [-46152520628049726047378770473472440033701491079537, -77846495308696976900265551113102939040781303854057, 36146867199389758737009011992717037008208541542761, 54306261419466326691567214939224797047295296426931, -3083725703571027045157325972456063738634665107768, -9871493052756247175499103018594250619560308196318, -344199143631289903310360058798603391526990978102, 688747916947360743362880409360504504664749825803, 53047051793326036227347667166601030089940264004, -20314624555233762416288670867722635532231846401, -1964856341888544600809531979149499493084866523, 264881764401855887796979435490124079324029284, 28458492789583321519545570199534601692683421, -1266389058547391971880775807346658198635135, -144263558037924218636156134188668059220104], [19504741165766201652915848463304354746796349781040, 54602899328347112295657849356031210109812186261815, 16560070007978823884200201730622744443079177844673, -54127441133476037720850522245218179161552148798913, -10077550940143799273929202633986262786115246089403, 10890048397709571209547361096684903166221233036126, 1499515413064604389094495894130616750195782820690, -775556605732997251034871754101837345788012337056, -102221559889296256387606542469511169613088487273, 22953205412416995956021134822768474191604541314, 3043103877854067731441194278894690551525115405, -299198029071736260563482812944698185449166015, -40119816260184872668211633979857616045559102, 1428938488582483995708411870638509693118607, 193415023155925714441764033939834921019567], [-17055089727990475162454962064415741642210625882343, -60159514179659436328473772486115806502294113676824, -27271337470335339622197810274177636864870743091549, 62146768395035878731040715369099975774347211095486, 13073398215097490185162426377442758525693298998567, -12536791403497153074510632720297915230367571991957, -1807352426381784674267383506579583416900750948720, 889025311649993585261507955607916761700080636036, 118844761706751026394899540773355326435249305515, -26235744526246598124826307285316929332830895266, -3485545402942827891504458682848039442336447360, 341382036876806582168986009719483376493715969, 45625508734864513301090991633943034403675847, -1628553983198356479083091471145206434685239, -219097273124356667386929755507571428659220]];
    Rf_basisdens := [1, 212072618512813742903498053024609321858032851078, 212072618512813742903498053024609321858032851078, 212072618512813742903498053024609321858032851078, 212072618512813742903498053024609321858032851078, 212072618512813742903498053024609321858032851078, 212072618512813742903498053024609321858032851078, 212072618512813742903498053024609321858032851078, 212072618512813742903498053024609321858032851078, 212072618512813742903498053024609321858032851078, 212072618512813742903498053024609321858032851078, 212072618512813742903498053024609321858032851078, 212072618512813742903498053024609321858032851078, 212072618512813742903498053024609321858032851078, 212072618512813742903498053024609321858032851078];
    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_1058_a();"
function MakeCharacter_1058_a()
    N := 1058;
    order := 1;
    char_gens := [5];
    v := [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_1058_a_Hecke( : Kf := Kf)
    return MakeCharacter_1058_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]];
    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 := 4;
    raw_aps := [[2, 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], [-3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [-6, -1, 0, 0, 1, 1, 0, 1, 0, 0, -1, 0, 0, 0, 1], [-5, 0, 0, 0, -3, 0, -1, -2, 0, -1, 1, 1, 0, 0, -2], [2, -2, 0, 0, -3, -1, 1, -1, 0, 2, -1, -3, 0, -1, -2], [-20, -5, -1, 1, 2, 0, 3, 1, 1, 1, -3, 1, 3, 0, 1], [-25, -8, -2, -3, 5, -3, 3, 1, 0, 2, -1, -4, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-23, -6, -7, -6, -1, 0, -2, 5, -4, 0, -2, 0, -4, 7, -2], [-6, -5, 6, 9, 7, 1, -9, -7, 1, -4, 4, 3, 0, 2, -1], [-83, 3, -3, -12, -20, -3, 6, 5, 4, 2, 2, 0, 0, -9, 0], [8, -15, 8, 3, 9, 5, -16, -11, -6, -8, -2, -2, 3, 6, 15], [-100, -38, 6, 9, 4, -7, -5, -1, -7, -2, -13, 6, -3, 10, 17], [-55, 15, 3, -19, -12, -9, 24, 14, 1, 10, -3, -12, 15, -13, -7], [-128, 15, 6, 21, 24, -5, -11, -16, 10, -12, 2, 17, -5, -2, -10], [100, 0, 6, 19, -4, 30, -28, 6, 8, -10, 17, 14, -24, 3, 26], [-164, 30, 23, 25, -8, -3, -38, -16, -7, -14, 10, 25, -12, 15, 16], [-202, 11, -35, -27, -7, 9, 15, 20, 5, 5, -17, -6, -1, -3, 5], [-9, -5, 19, 28, 41, 9, -14, -4, -9, 2, 18, 19, -43, -10, 11], [54, -13, 6, -8, -36, 7, 29, 22, 17, 0, 16, -11, -4, -4, -10], [-143, 50, -21, 0, -10, -3, -2, 15, -6, -15, 5, 39, -4, -23, -4], [-334, 21, 31, 23, -8, 0, -9, -5, 17, -5, -2, 0, 9, 7, 17], [-357, 40, 9, 21, -3, 38, -49, -40, -19, -16, 15, 7, -9, 13, 16], [-380, 68, -3, -6, 14, 8, 27, 29, -7, 10, 39, -5, -24, -22, -18], [155, -49, -31, -14, -18, -36, -9, 10, -21, 17, -32, -39, 18, 20, -12], [-355, 50, 34, 13, 32, -7, -11, -28, 12, 14, -14, -18, 13, 16, 1], [-128, -18, -12, -50, -69, -90, 107, 15, -11, 67, -45, -54, 30, -24, -78], [-305, 81, 21, 9, 22, 20, -1, -7, -30, -21, -2, 12, -4, 20, 9], [-381, 131, -13, -39, -15, -9, 1, 28, -4, 29, 40, -52, 18, -10, -46], [92, 18, -35, -81, -41, -73, 75, 39, 4, -20, -13, 14, 58, -34, -38], [102, 139, 44, 27, 20, 37, -51, -38, 7, -20, 77, 8, -73, 2, -3], [-78, 22, -78, -95, 33, 42, 41, 5, -4, 56, -80, -58, 69, 16, 33], [-258, 32, -57, -5, 56, -43, 39, -18, 30, 48, 50, -9, 22, 25, -73], [-382, 12, -30, -92, 21, -99, 46, 57, -17, 6, -20, -33, -9, 51, -53], [273, 131, -11, 121, -66, 37, -52, -139, 12, -66, 56, 50, 3, 10, -43], [-148, 130, 130, 236, 20, 33, -164, -72, 29, -152, 25, 141, -174, 37, 71], [-282, -75, -1, -59, -91, -12, -9, -68, -42, 17, 5, -44, -6, -58, -58], [387, -66, -77, -138, -14, -110, 129, 25, -39, 144, -48, -184, 167, -15, -15], [-417, 49, 16, 120, 90, 71, 1, -6, 24, 5, -40, 85, -62, -102, 37], [116, -85, 52, 25, 50, 26, 40, 94, -19, -85, -71, 62, -84, 67, 59], [-278, 140, -143, -199, -2, 41, 222, 132, 65, 18, -76, -80, 60, -52, -127], [340, -44, -32, -133, -92, 46, 43, 171, 37, 3, -73, -44, -109, -14, 37], [844, -85, -13, 48, -59, 1, -94, -10, -31, 11, -83, 52, 11, 25, 55], [130, 7, -41, 62, 213, 78, -75, 36, -17, -16, -169, 48, 48, 104, 205], [-398, -104, -20, 36, 157, 11, -31, -29, -55, -25, -5, 64, 85, 153, 166], [900, -51, 107, 18, -96, 11, -45, -84, -4, -15, 0, 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-7212, -1976, -2152, 1492, 6810, -6097, 3634, 5675], [-22602, -966, -3349, 566, 7328, 4234, 3202, 1804, 1031, -1988, 4381, 449, -2668, 1082, 1149], [34921, -5954, 2256, -4456, 3882, -5039, 264, 595, -2382, -869, -4815, -1971, 605, 2456, 2005], [-39228, 5670, 3837, 5007, 4641, 4657, -10927, -4123, -2447, -3098, 4403, 4363, -4370, -907, 4976], [-9508, 1547, -1563, -1707, 3195, 501, 3804, 2777, 748, 4127, -6920, -3382, 87, -1550, 52], [-10498, 11420, -494, -2599, -831, 2630, 1585, -1492, 399, -109, 3176, -1677, -1398, -4098, -5205], [-45652, -2830, -322, 318, 4592, 3134, 3274, 2274, 1260, 618, 2951, -2511, 396, 1068, 2688], [20952, 2138, 2097, 234, 5622, 4615, -1560, 3637, 1364, 3761, 3857, -673, -4159, -1235, 789], [-19628, 10152, -3079, -1105, 4728, 6695, 4078, -277, 2134, -1357, -4112, -1980, -378, -3817, -1872], [-37153, -6313, -1679, -4759, -7985, -6026, 5052, 675, -322, 4892, -605, -141, 3122, -4258, -4880], [12616, -39, 989, 1682, -5164, -5490, -1782, -564, -1950, 787, 378, 5270, 3178, 1854, 3599], [-3276, 7363, 5164, 1349, -724, -4095, -4258, -7292, -2176, -5043, -1652, 1753, 3304, 2829, -2194], [-30570, 6548, -3003, -5612, -5193, -5873, 7742, 5860, 3486, 1157, 2408, 344, 2002, -4841, -6275], [32127, 974, 4785, 8870, 4666, 688, -4436, -4594, 1017, -5771, 3716, 7131, -628, 3938, 960], [20610, -7763, 1048, 5246, 1665, -3981, 213, -1821, -2609, 4068, -4567, 152, 2783, 223, 397], [30082, 4035, -2823, -4783, -786, 889, 5318, -834, 2601, -702, -5391, -6544, 3539, -4697, -1098], [11848, 642, -2971, 4357, 5932, 1428, 4447, 5810, 1411, -4135, 1386, 4127, -1233, -34, 4491], [-10292, -918, 3024, 5515, 4629, 2360, -2026, -3819, 2584, -6453, 247, 1302, 480, 1290, 3738], [21721, 5433, 975, -959, -3293, -1625, 211, -554, -869, 3388, 3066, 1911, 2953, -1505, -437], [35124, 15091, -7187, -9987, -4530, 760, 3903, -179, 223, -1335, -623, -2803, 5447, 1127, -3617], [21969, -6155, 5323, 5497, 4226, 1443, 669, -2803, 1311, -2567, -1896, -1551, 1676, 4282, -1299], [-6019, 2883, 5509, 7896, 8324, 2204, -7964, -2059, 229, -4616, 1272, 6088, -7483, -1428, 3641], [-5009, 1826, 937, 9189, 3676, 5001, -5986, -1984, -159, -4767, 9638, 6960, -2862, 4503, 2532], [413, 1757, -3086, 1303, -4843, -988, -566, 1103, 2348, -2612, -6908, -646, 4619, 4538, 2286], [7974, 14818, 2427, -6665, -9203, -5618, 5682, 1488, 1117, 1137, 2469, 939, 3334, -3215, -5322], [-2537, -7309, 2182, 657, 2605, 195, 1407, 3090, -948, 4383, -246, 2856, -1751, -3480, 1504], [20474, -4994, -5676, -1458, 7648, 4649, -2946, -1633, -1072, -309, -3819, -401, -2024, 2612, 908], [36825, 13327, 876, -1734, -2939, -1679, -293, -2896, 1197, -200, 4542, -1210, 4539, 2489, -1790], [27866, 8679, 6940, 7537, -4569, 4278, -8498, -4269, -157, -48, 4761, 3707, -9080, -2808, -1334], [-31712, 4410, 930, -9081, -9545, -6357, 10352, 1467, 1413, 4403, -4640, -7498, 5723, -6288, -7690], [12123, -4046, -6301, -8117, -5158, -2197, 7533, 4538, -477, 8996, -2993, -5381, 2434, -4171, -3860], [25212, -5416, 4321, 6613, 2027, 1723, 1491, -1865, -269, 3311, -890, -2496, -3370, -1341, 2329], [61985, 1206, 2385, 7466, 3660, -600, -1939, -2113, -1070, -609, -2774, 2161, -1784, 1682, 4586], [-43996, 6616, 2487, 916, -836, -1143, -379, 4921, 839, 789, -724, 4782, -2936, 1496, 649], [67040, 2619, -530, -4387, -3471, -847, 5768, 1485, 943, 3117, 3448, -3812, 2598, -5757, -6597], [-25383, 1488, -3427, -3982, -2032, -3828, -431, -3319, -1465, 974, 3885, 809, -125, -2308, -3559], [-3946, 2822, 2250, 6420, -6126, 1169, -8305, -4619, -650, -1847, 1151, -714, -689, 524, -925], [-6939, 6908, -1129, -5348, -7628, 526, 3875, 6824, -772, 981, 2496, -2659, -3873, -2669, -2413], [-61630, -4011, -342, -307, 4100, 2688, -89, -3547, -98, 4462, 3780, -8010, 281, -787, -3139], [-1452, 10795, 1381, 12570, 4703, 5677, -6946, -5573, -1310, -3916, 4515, 4511, -1033, 1682, 6333], [10905, -437, -208, -165, 2192, -4843, -1366, -2872, -1637, -627, -2546, -123, 5482, 7401, -4502], [16072, 9230, 2500, 2246, 1719, 3811, -3274, -993, 1807, -692, 4669, -322, -2195, 895, 3235], [-38253, -4186, -1903, 695, 9135, 5374, 1356, 2886, 4926, -4113, -1204, 5389, 1395, -1252, 2895], [40471, -193, 4020, 3216, 2531, -3668, -2585, -1022, -2544, -1166, -552, 6900, 434, 1494, 3287], [21447, 3633, 2564, 1826, 3267, -4598, -580, -2315, 1061, -1668, -179, 5080, 940, -1181, 753], [47453, 11664, -2211, -693, 1661, 4814, 1285, 2282, 4633, -1660, -2203, 2262, 2201, -690, -275], [7504, 3775, 3317, 6348, -1394, 639, -9410, 17, -1059, -6856, 2983, 3009, -12242, 4722, 1258], [24158, 7756, -727, -945, 2869, 2881, -4032, -6666, -2469, -5316, 2994, -5161, -1832, 2716, -1750], [-11552, 863, 3989, 6584, -7574, 2330, -3575, -2563, 937, -508, -6112, 4662, -7821, -5917, 2370], [25449, 11335, -5304, -9496, -24, 1004, 7207, 7950, 96, 2730, -2089, -7246, 6878, 1769, -4834], [26343, -8938, -1714, -4962, 6662, 870, 2696, 8588, 605, 204, -2605, 1465, -4740, 2334, 1328], [24797, 1882, 4491, -3274, -6909, -10525, 6443, -4942, -1428, 122, 5048, 83, 3313, -1790, -9120], [-17172, -5167, 1178, -7275, 2435, -1221, -1323, 419, -1758, 6233, -1930, -3870, 6325, 648, 4975], [3497, -2341, 4572, 9098, -2743, 754, -1998, -4637, -55, -1826, 8057, 5838, -4134, 3067, 2381], [-15569, -6208, -6342, -6470, -9254, -2105, 1913, 1027, -1236, 3045, -2642, -1756, 3468, -5698, -3687], [-15557, 2847, 4204, 1706, -6989, 3263, -6862, -3568, 1911, -2554, 1799, 2868, -7138, -3208, -1695], [-60809, 1034, 4018, 3314, 7194, 2788, -6982, -3557, -104, -2103, 4488, 4311, -1272, -324, 6810], [2276, -5471, 2698, -5082, 4777, -5806, -80, 5118, -2761, 1900, 1911, -2519, 2792, 2149, 73], [24149, -5675, 2847, 3519, 1245, -794, 688, 4297, -131, -425, 4857, 4822, -8287, 340, 1395], [-30277, 6506, 2184, -9370, -3798, 1193, 643, 4948, -74, -858, 2417, -5398, -474, -266, -5066], [6394, 360, -454, -6469, -8154, 692, 7914, 1160, -1089, 1390, -8577, -7653, 2008, -1340, -958], [-1376, -3450, -2067, -77, 3977, 2701, 960, 2658, 2238, 2632, 18, 2294, 1854, -1598, 6068], [30024, -12014, 3790, 3094, 3862, -9065, 7186, -1615, -2627, 1903, -1070, 605, 12, -761, -4146], [8962, 14871, -1645, -2963, -6432, 3695, 4454, 2570, 2910, 3504, 1721, -4186, -1799, -4646, -4623], [28836, 9563, 1181, 622, 2864, 3092, -4347, -3882, -835, -3209, 2476, 1359, -634, 1373, -1267], [-1225, 9399, -2315, -686, 2055, 10235, 3361, 7671, 6321, -2946, -5898, 4306, -961, -3863, 6791], [27165, -226, 1101, -1024, -987, -3570, -105, -1940, 108, -537, -3167, -373, -1971, -1817, -5858], [-79900, 5879, -2846, -2520, -3020, -2607, 2845, 702, 2347, 312, -896, 5186, -610, -3457, 23], [-4369, -5625, 3383, 1202, 9680, -4110, -4667, -5172, -2047, -1632, -4366, -4117, 1415, 8381, 2463], [-33238, 3136, -5109, -8761, -7253, -10752, 15407, 5815, 154, 3709, -6889, -999, 4459, -2888, -8784], [25595, 5627, -6475, -1074, 1099, 4198, 5698, 5186, 1917, -7003, 2205, 2274, -1523, 802, 94], [6991, -1412, 3362, 9030, -2528, 1421, -8132, -12876, -668, -977, 6770, 2303, -1294, 2346, 21], [17398, -4624, -4409, -1532, 7519, 2143, -5387, 1586, -2566, -127, -3494, 4001, 5857, 9296, 6824], [26561, 2564, 6637, 12411, -287, 2019, -12435, -7401, -2396, -3297, 8137, 5501, -8540, 275, 2523], [-1748, -4217, 1546, 3885, 2747, -1192, 2583, -1480, 1944, 913, -2340, -1420, 2741, -9444, -1621], [-11684, -7569, -839, -7630, 4742, 2054, 5837, 8592, -17, 7187, -67, -3878, -639, -2392, 6781], [16977, -6623, 1786, -6058, -9053, -9314, 3573, -1778, -61, 155, 5579, -1069, 4831, 1255, -2748], [10435, 8357, -1615, -4809, 5358, -3442, 1204, -310, -2532, 8778, 223, -2446, 6983, -1620, -310], [14789, 6459, 1128, 141, -3871, 3638, -49, 724, 3544, -3938, 1186, -1675, -2787, 7082, -1464], [33999, -790, -5401, -10618, -8041, 423, 8721, 4026, 730, 903, -2033, -11013, 8388, -2513, -7172], [19897, -7952, 2538, 1060, 8060, 3640, 3380, 1488, 660, -802, -1393, -894, -6094, 1742, 6294]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1058_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1058_4_a_v();".
// 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_1058_4_a_v(:prec:=15)
    chi := MakeCharacter_1058_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_1058_4_a_v();".
// 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_1058_4_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1058_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![430200319, 3740741834, -7997190269, 3129320386, 1192015345, -624045948, -82716052, 43545641, 3139460, -1501427, -65560, 27690, 702, -262, -3, 1]>,<5,R![1858594139159, 5347176749168, -1622413897824, -2734234188125, 272173328857, 235804288394, -1463098259, -6108292505, -182441449, 65769138, 3452990, -292443, -22510, 255, 50, 1]>],Snew);
    return Vf;
end function;