
// Make newform 8.18.b.a in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_8_b_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_8_18_b_a();".
// 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_8_18_b_a();".
// 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 := [1286374137827816254118965326485913600, -5242030267748791654842336509165568, 13720647095471028734661620662272, -78950720850572326734309359616, -217909345031306501735579648, 1127435088957285706235904, 5699397839986467274752, 163121042717484777472, -1108241988138827776, 7317489121656832, -3763915816960, 2100004864, -373411968, -1102776, 4022, -7, 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], [-5011931281375373950898113452441600, 14337747111082586657860026892288, -77137501951444740399800778752, -226652362390948291813572608, 1092877169955223759749120, 4319739875257743310848, 172457245591090495488, -1167025359200714752, 7347921382735872, -3804032634880, 5014602240, -364324672, -1135912, 4142, -15, 1], [-1307918326365733351197882895792625243033117392896, 7185287161653279738519647172874520152177115136, -8388768239043845953173344711830221929578496, 316935254505413433227207893255938658795520, -1603238282147148494127706912682382721024, 5888327403336175402603233454457880576, -27672768791325398176464031063212032, 81854179062030283121283913220096, 280741080760090838567177879552, -2879891060237120944769241088, 16587271450622714462478848, -73336152594651528733504, 1751547800883521228632, -1990638038811968562, 3579309129031585, -202665796662607], [18845741216881567653212805560342540108427165696, -1350933380440929000176333980599806887496515584, -11075180427959602430732688041681686202679296, -35181402519514491276159315828300165677056, 361463220342815868770366645375810404352, 3789155521825886529756236065988935680, -9285428078136832336174350401011712, 55131719807716946056351234654208, 522182811371385852168883994624, 204604437956683790291259392, -9707113364261484455623168, -49449917647326078950848, -220289109152326083704, 89379930216345162, 20981730382621171, 113144783158307], [-42842242313433409157198097861383892281595179040768, 2075806875838358259300866372084555064669697474560, -326948898370078637694841793787316059452211200, 10973650936597761955187703075305555455115264, -54004544780781897263939623376473978568704, 233310146532465547264623475010178121728, -1166374515623648011467010342438043648, 4141978656388196889091461967511552, 8739527854070694564618943791104, -96434234990842499939510554624, 499305254572666965941182976, -2701371333148351121886016, 60355599933393004743256, -71924520144668149362, 303875298361647553, -6904520336896303], [-162155573135491747530070259641225519352391615381504, 899252032187111568334586168320385540416969113600, -4437296663921489070336972831413605674252238848, 54291785038192160883119877680631421660561408, -322640640016353060856259701913380081106944, 325374793801576608358648619010076180480, -5467130684642266757417518517157101568, 18159091488291785697025022761631744, 32227772239726044875414375825408, -463318105746970631653198196736, 2824217736569580899843395072, -34754118447671531705815744, 219229276392453852169192, -435367860605531963582, 420160533246763159, -23764046739574521], [-212443539810488962984196497540684037439650663497728, 178736169347404892368177768942825281716993654784, -16923636404437021961466804923357199661371752448, 142278047432660488442110883577556849420926976, -569119037815830353697816284670009704185856, 1910949747772507075274658287658939187200, -18481221721368640562218758921605087232, 63643655274265293287480993897775104, -281753824313032317479569124425728, -1046000012667212196888999260160, 6373070160238365102187412992, -95583249796152773009666240, 313220997282730427849256, -737936359670639398382, 14505856015322668527, -79475357111706657], [-185505802241528092001646840062476114531302468747264, 11476051827805283267140964024566040046546286804992, 14408588659297117394966785788323710307514974208, 14656898996034777893422374283628069370462208, -4564102114671747672792696816077617582047232, 10843777390362171761104709106924771082240, -62355199183567530836001057888981221376, -351390499559859802037740660073168896, 74394255645491163039954628444160, 4820922237134929736732351115264, 52692484229758254649334405632, 50732439176719779586927040, 1375415759975075434553848, -17693109570428398788458, -88115910645405087971, -139983251471800851], [331946290865197439861686822309551297062156504137728, -2102192050691586533652990130030102196069282611200, -4255390062751649230471162667754652451796418560, 39205383190523538084812673473512606357520384, -219931962883335819398058191014151707426816, -1106895031135065237798415622567880556544, 4092461963329089856867872568986566656, 96967607656334634431739171845963776, -339499814763947505167181307379712, 638555207063349773900630675456, 186650728376932474571326976, -85984181553503126367440320, -427812697316982600911480, 657846057797970546378, -3604090098844291149, -90518228670632477], [-176086927171074639335724822640507316605077977825280, 537829294102933816125243913276909033631200051200, 123594757277887966953595262819735960011935842304, 177140163256641667371663042089684284477538304, -681794279929093603505250138470822111608832, 3096927582130604018972432126246169608192, -37220936146863178452860080491396595712, 80478227778838442252152514991882240, 58101688193789209621228531417088, -1748849473503961998524538032128, 3654038634456230028312678912, -255592500160806746470668608, 77828540666637231364760, -1661406249058542285602, 40453859256237604057, 63696133524337513], [-446857289916422880756135548656061687822758361186697216, 3857047130380332794886889840997897896952994434383872, -15813083982427836541159518258047853080572512436224, 36803522566620037670305689212618852531351584768, -403920180774726801327742756366723320243027968, 2228284684745780277936512201220566672211968, -9888217948094796216552292194205623648256, -11357093663122621295404663240836775936, 278307737343036769692121693640785920, -2762808019810760602276337509511168, 5354214846714736507609408013824, -23072525327384513398590451136, 257293577497698492156445064, -2279072692149939495203382, 1594408635764797793331, 639837761265273443], [-12457638142722266620463918081398695939515689973317632, 73063726408568484940622133076307540880439369531392, -107434315118504475965017458719426522634849878016, 823346967437778828348323076877508672076382208, -14244916012747063543009727707369015540187136, 38850529002016158683401494610575611658240, 194897461450219154663438397896740831232, -1333541570291061349432256331500748800, 6785174686098384494970410338942976, -76442953692418415550333524996096, 143660568804565270993412251136, 934392191454225580282622784, 17208567183505367921423016, 73910045733907667271346, 84577590717130483999, 1403534495675861519], [-131476165426658160879723990389618499871368040402649088, -535520245609857682789875188619586047388431805317120, 5860053499256353413376159982407075174193691623424, -10862297532646645381361143130010816270934474752, 49400018421813078085665870465132908215533568, -316584023354429134414533678869198676164608, 4645979012483082114565276225903708340224, -28538844713705839309936868289805811712, -14560372787432912758932499738722304, -82357863234112472660066142392320, -5610932669816068884115138899456, 1243961011910623144525399744, 54345578054040030035805848, 663441832793891708576734, -4513334916052844702375, 4416227129565238249], [-14371339549409824527938625463321567727006567343587328, 167723589065161578292934608133552318914552764825600, -2882626426938584657413114843753098701201662279680, 21161041115278965463095134648902976986289274880, -22990456373827086939273437450140188298706944, 321484992072320973415693419507954894241792, -2442550614328793702853673461174931292160, 5919262543080642382305655323620802560, -25540948847039136017814423664132096, -163540930893504629339610855395328, 2266442335524504741752548874752, -9772499022515832362066492992, 36024113928126723727693048, -68404303676981589363626, 1615368927941353293373, -6728427736961304819], [3787539807794520433833808653907703995020190490296320, -38068889786964383755617586598909590428132619321344, 114254197459737699406252288890724189238087122944, 840877143882642954035132407467114901857632256, 5000757058880931594479619079525709241122816, -19144569611500297305207552231399156088832, 127146173838695861549379458949120524288, -34487418142976115230945298492162048, -2216917920529689478334208845742080, 18727112226378226946032194113536, -58332026979339556575491701248, 164472886995999387208780352, -3674772470073441078887288, 28334617387145520434570, 38657826300279070547, 335557145064303491], [18682904434239177348428442721016529498953081784631296, -621935794034622382697864818891716030673188814848000, 5307615328242954476198030259209039938775483940864, -27544087622214577694098551156413277901269499904, 103640121663088523422524554309577278912200704, -801110964490051851333431693321536756776960, 3065394645944282097999341842411118657536, -14827120702670160869576434304345964544, 3312367996946440558223578957938688, 407815836691381071688843785457664, -3402665664519586461636712016384, 12265870697776638720995115072, -70097611116970604350249656, 511571073214509969501050, -3000941267738359526245, 4596605843708830027]];
    Rf_basisdens := [1, 20282409603651670423947251286016, 56317154117405836067971416567429555214614528, 7039644264675729508496427070928694401826816, 56317154117405836067971416567429555214614528, 168951462352217508203914249702288665643843584, 56317154117405836067971416567429555214614528, 84475731176108754101957124851144332821921792, 42237865588054377050978562425572166410960896, 21118932794027188525489281212786083205480448, 168951462352217508203914249702288665643843584, 168951462352217508203914249702288665643843584, 168951462352217508203914249702288665643843584, 168951462352217508203914249702288665643843584, 5279733198506797131372320303196520801370112, 42237865588054377050978562425572166410960896];
    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_8_b();"
function MakeCharacter_8_b()
    N := 8;
    order := 2;
    char_gens := [7, 5];
    v := [2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_8_b_Hecke();"
function MakeCharacter_8_b_Hecke(Kf)
    N := 8;
    order := 2;
    char_gens := [7, 5];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 18;
    raw_aps := [[17, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 3, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-6, 63, -4, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [721574, -7724, -16, -11, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [-14371, 106244, 2365, 718, 7, -149, -10, 2, 1, 0, 0, 0, -1, 0, 0, 0], [-145744, 1180147, -4135, -1048, 75, -208, 1, 12, -1, 0, 0, 2, 0, 4, -1, -2], [-468326394, 2068552, 6232, -21710, -370, -165, -1766, -263, -8, 6, 4, -9, 8, -48, -12, -8], [-1431619, 11519612, -45827, 24426, 439, -6201, -4578, 762, 45, -8, -32, 48, 19, -32, -24, 16], [46665420895, 99184513, 246935, -245992, -2403, -8430, 42937, -1700, -614, 55, -210, -148, 92, -552, -82, -92], [32427454, -299052279, 8479012, 7397561, -1328, -28561, 199384, -18656, 4168, 4688, 1216, -1328, -928, 672, 664, -336], [-19890520757, -336321781, 867737, -15845509, 37507, -248722, 221383, 32420, 4205, 7497, -686, 2708, -860, 5160, 1490, 860], [-781872320, 5250720265, 282565939, 75199440, 228857, -444968, -1785621, 39684, 29077, 39744, -7424, 4694, 12160, -1876, -2347, 938], [468610503174, -7424991414, 180184, -161410156, 695288, -2821314, 4452274, -576314, -39302, 78062, -23308, 41578, -11800, 70800, -10460, 11800], [924933786, -6716603231, -359193417, 200311636, 214606, -10342098, 5924348, 472500, 157626, 134256, 19904, 49760, 2630, 68544, -24880, -34272], [-23554803794147, 88805215129, 213457323, -209803853, -5454295, -3169110, -2435995, -444244, -839887, 121803, 8342, 492156, -1748, 10488, 1686, 1748], [-27065118220, 234725009873, -5338620715, -988901894, 15644503, 101464854, 38946941, -2815780, -774813, -219088, 155968, -129830, 458400, 58548, 64915, -29274], [134444010324, -1084476769513, 5002784559, -3132543020, -71308084, -40662520, -72774912, -986880, -2481616, -1954312, -270368, 88624, -8432, 398304, -44312, -199152], [-226598464000, 1871797658247, -17418398943, -3426912092, 116121443, -146251940, 57734705, 4822860, -420113, -2209456, 186560, 83842, -1381792, -95228, -41921, 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-111982068643659236, 27580032907263632, -148833332030271349, -848844161794524, 752728043729224, 1566538324870736, -150058137929072, 900348827574432, 924745966804808, 150058137929072], [355689302280246758203, -3981104083098633696474, 372250854107806720053, 9101525095798722494, -141845107879148591, -1177081378174004967, 178521677301647746, 59071957837610086, 8586093825329123, 5376377667950136, -288153260579616, 4617407119256496, 2325697297635805, 1102543963290336, -2308703559628248, -551271981645168], [202656425342417438228, -867639261084452154397, -241412395801670594989, -13115520433227456174, -133946923282863599, 3648339967624645406, -230899897634828037, -18043619075697468, -7122182626749403, -6618986971993008, -930598425779008, -2343501904010474, 491262405738080, -4433918032776404, 1171750952005237, 2216959016388202], [-1121781430316472611964026, 2151171453213477507204, 2421843169499385344, 22671601416819938541, -183428125888192656, 347029183060314528, -13314461880459632, -20734313030179904, 117567236051857369, -11536105301467536, -94429485634976, -13188987958018368, 841380837030080, -5048285022180480, -1396012278072480, -841380837030080], [544959489818473605308, -5252831047154874462783, 312149672737059226121, -20595701628525904994, -282376994487918301, 1969180651071363650, -84154233935533999, -38309878100689716, -15325006725283249, -10060258570455792, 318408249176512, -1536342520403646, -459065995077920, 3737708618476420, 768171260201823, -1868854309238210], [-781753083852823217432146, 3203920765224432429294, 8145505545376354664, -11294482880860306276, -96572786282286616, -211331066498179590, 76529792477445894, -6880496598167694, -211549671666124866, 6859139496272858, -362316874713668, 15484470659036830, -254479494330504, 1526876965983024, 394901288643532, 254479494330504], [-417879417065734430806, 2079426125794746370429, 435991473017879813611, -23523807912073664152, 235019731233091374, -3195031603771353870, 251489742752975188, 14958412518635580, -10344635204087154, -15313925994584168, 346520651526752, -328709194442384, -1530097422253774, -3585736508380064, 164354597221192, 1792868254190032], [2650139705930531080459783, -2282999444390784997129, -7139268200976436291, 22808696390515974751, 184054277757046287, 292067719481912550, 415297178298153091, 10803668345814196, 46744777301119705, -8831018954246707, -1089580380922918, 26792569922783332, -1233219796025932, 7399318776155592, 1743976761836890, 1233219796025932], [1284887095628767941690, -9383687298260477533735, -310271745061000708742, 18305827559314454879, -547445128869421126, -2107329343787923087, 892333480974871918, -48696443743096536, 8515852254987410, 12494195215308032, 4023016102857728, -3577373506466596, 3510788505536000, -875489702057544, 1788686753233298, 437744851028772], [1069335569931343977532346, -101318259084206049924, 930883016027276688, -14450959032331507736, -313030331035382432, -40025937373635588, -792451678048567268, 3208047139472380, 476585690942577820, 2962694739690148, 3057461102042136, -43102399780553308, -801772258029520, 4810633548177120, -1112069859086920, 801772258029520], [-1037283717721056650597, 9352542772988486347114, -337165082460798786039, -20569555238179907058, 499870473138479921, 2163407955204294473, 2970110616519330, 25663802987685702, -7907059118231533, -10331150594303912, -718099803442848, 1364588869749232, 680913464528685, -1776221439420064, -682294434874616, 888110719710032], [-3704285442131961612850092, -8190924052252443350762, -12774869439164208034, -52174006929875715987, 401052564955783354, -730666914587057820, 9133447985467346, -75979067799090376, 102776136573113021, 20608180461017102, -1523721941320932, -33939343786576040, 1027122274876984, -6162733649261904, -3756986164888292, -1027122274876984], [-472803035875175022440, 5541784333881779742143, -620168487534189216973, 49415348580540360712, 250564674939086184, -2745666041575157664, 232546701179976608, -25046892247681824, 33258689872339152, 23435273954706800, 1959820180473280, -5179555794975904, -8648551246917200, -6984378182705728, 2589777897487952, 3492189091352864], [7061800450224624895679675, -3297450646822128654417, -4106751776990248915, -31149777118017897523, 261486637568950367, -505986366047602842, 89101955099492067, -125424246450588108, -194005437378072705, 14479888654958189, -1848155118965414, 21324885869266660, 2261530808501940, -13569184851011640, -4056342214184870, -2261530808501940], [-7687364167315211821525879, -4388107573120287278597, -11654994198970973643, 20583363280824072362, 60233061652938119, 371915269333702678, 201192313909594267, 53717165122896468, 540505910584122824, -11599587124221899, -58193575668054, -7349268468933756, -1206546909314220, 7239281455885320, 1942793550824490, 1206546909314220], [709544987149081646429286, -1068391710798886970640, -1791058013563924456, -5357038219894278110, -133918734510986546, -124894201838016045, 1292936011457387810, -116327757542156143, 631676513629746848, -569503082457986, -6352552794806060, 11649753507273503, 284380971888040, -1706285831328240, -2989682846244220, -284380971888040], [-3123573942159320936258110, -2019199863754028557610, -596891512728422816, -37880321549135590026, 282141360361539022, -645885414195691855, 114143688803700008, -60177809218584417, -341635134930477162, 21725636347668824, -827242816863984, 94278938598144801, 1277770851661344, -7666625109968064, 598268560677840, -1277770851661344], [-28318202890423982146, -1319424553602450052411, 494078131518358868940, 31996031782678771933, -118814956825085528, 3304256460544166771, -1134858151359781240, -119835938379648928, 2617656760310456, 17646719543180320, -2299456804816000, -7992588782906800, 3194023836082624, -2453301032388960, 3996294391453400, 1226650516194480], [520200047903621268905, -4753879041627581852560, 127837428378326668557, 97746074020182430858, -366582612076010789, -10071247368769781101, -1279015579500174058, -166173079069325086, 32126598427149953, 44530595400816040, -1213448163990880, -12655342974693616, -3592853422633025, -4199612229732192, 6327671487346808, 2099806114866096], [5915461294938919990651822, -3805673685340345857250, -7609939796321597848, -3347257679818381988, 458456151423563976, -177412513233435270, 1417196592048682006, 121747460545570834, -208764893463402770, -4095344400738358, -6138718986226852, -134945119495567490, 1006210680793784, -6037264084762704, -437529994620692, -1006210680793784], [-13909414963164567054174734, 30236503449315423838256, 66660452512526443172, -6296485867654509415, -1603690993891541828, -285255672896572840, 773193841903356076, -80425420259381680, -479657527739576179, 2653027004045460, -4813356147396440, -8916984529928944, 5637594012144976, -33825564072869856, -4781387957887576, -5637594012144976], [-265487567448548556064, 4687259919467400709031, -853306865590117855261, -1801883942286020168, 114991853587498048, 2466810934049200728, -661541708401402864, 235594049087867312, 15421449498971064, -2474849522309008, -4857916980286016, 20046513808834144, -8987131152595896, 16302685864281024, -10023256904417072, -8151342932140512], [-2806786100444292668298154, -10782626289439946181504, -16126934698809952904, -75097109829209893182, 656347112068067598, -1041415568354653965, -540841070969047342, 175266368824899665, -63032828133230992, 38282713746145230, 3969291538386740, 81886288027086303, -2683354095694232, 16100124574165392, 8878052406851172, 2683354095694232], [-16707557821836676689432942, -5857544991415761227444, -11757964730617506048, -17000721121567410553, -413658215321722416, 160907031914594400, -2155218943300922640, 31548718348431936, 1196152532723488779, 915982011436048, 8952447310220192, -2904680964846784, 2717673504328512, -16306041025971072, -768456649785184, -2717673504328512], [15365775943353452655157384, 31811764600380545936912, 60682801728643448096, 86790044107173192908, -2172877997984737760, 1161427397819976128, 1370821433716288096, -263859971105901952, 734985166176249116, -39578337543828576, -6811859566465472, 65243583687306368, 1583516274640000, -9501097647840000, -6424324370499520, -1583516274640000], [-4046358588994090654912, 35191722382771599981225, -955264971101381188037, 25377927356107194136, 1901930314215669713, 8627024025342380064, -309719777419868221, -17482650590870748, 11262351740260605, 22346887338916576, -996629784479616, 815205202174022, 9272296889578560, 2038416727858828, -407602601087011, -1019208363929414]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8_18_b_a();".
// 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_8_18_b_a(:prec:=16)
    chi := MakeCharacter_8_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 18));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_8_18_b_a();".
// 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_8_18_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,18,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;