
// Make newform 1045.4.a.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1045_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_1045_4_a_b();".
// 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_1045_4_a_b();".
// 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 := [17756160, 74703872, -36068096, -240447168, 43097920, 280461480, -45064360, -143049638, 19879768, 37893109, -4539454, -5673747, 623613, 498499, -53768, -25467, 2826, 700, -82, -8, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-11, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1017159496214166664214486016191601664, -2329726121906236985086313708643948032, 2196886235169306852566914883296940608, 4252188966502451270525469251950234080, -3271777394076768207204843345982328264, -1790791735729018198746659363302980112, 1578521028488970797364761785995709262, 198085568359096724337640351591040988, -344301464212946926615494487263102241, 27434935899659633171073838280947898, 39413166042518481211660739504734195, -8171520444160025742823813980127157, -2448536271935896202403179600966319, 746589678647592933774296964151868, 77339322142504523651326528848543, -31977884292236989195472482000258, -923107893843079783060498828404, 625066262857449266832597516650, -2068542207323588132054066868, -3994758687856231406973990197], [4664437484124174047741111094178184192, 142563194527083089747765934440353280, -23378004946055384041090275439684645696, -5759415910098472092671399035237446624, 39621902476614788525001950457908784872, -1203309044444085910162194680450206192, -23070425776809799643676640135954563990, 1939029677254957333468449895515352148, 6430863539589623308426801741067616173, -599594536111287517524935040312557906, -979149914399412277680283066635040151, 94233607657203418323730381949943441, 86143269676362073279113111487566563, -8659930089323499943783232641532940, -4376178938835631186279749556826035, 467725626189972904355446044544218, 119270484596985938766333089252900, -13687071315594442380125675315010, -1350495829557780830759355380060, 166814599350794488141432427409], [3142791990008086600077831119235435520, 2632528548535226185924042257073942016, -23101342128943368141515133626058691392, -5897747318654480042458969942050423776, 39621902476614788525001950457908784872, -1203309044444085910162194680450206192, -23070425776809799643676640135954563990, 1939029677254957333468449895515352148, 6430863539589623308426801741067616173, -599594536111287517524935040312557906, -979149914399412277680283066635040151, 94233607657203418323730381949943441, 86143269676362073279113111487566563, -8659930089323499943783232641532940, -4376178938835631186279749556826035, 467725626189972904355446044544218, 119270484596985938766333089252900, -13687071315594442380125675315010, -1350495829557780830759355380060, 166814599350794488141432427409], [-1329244071393430309523537521953195008, -10144907672843579159114893303871769600, -10034170681597074030736142591149411776, 26739538941256078877668668087072672032, 19854755213702739095329327187805049176, -27988242192264774218813060363520818496, -8047507698795493216902414469786217658, 11132734379886535172119459065236117876, 1164867671037673598327653015324640235, -2149997278392931104706938314303505974, -18025503401113937512314347452176345, 223459282065681754221609041313385807, -11811100299285459166347631362155691, -12658017693772790992567986909535916, 1244743144907356472200649976769643, 356586203117742551888711227503318, -50417106465790611531663074990932, -3218159341213545867236191384110, 744644081528315025676207590020, -26152855439014240173295064393], [1521267749819177792137684795876447232, 2683680119996150974228672074304516096, -7138104342338140829232890637267942208, -5948126867410425429233267277639761184, 11993682506515188099848864466897739304, 5368341915983778265270891041088581632, -9099812817933363722157414892016663094, -1930037663951864385956752721731255668, 3158320267917432831310031782110492109, 307584326086485916512104733722100038, -569797159612921949834815581264606639, -16310541738680620271054746129520823, 57395254796601216420525272419658419, -1086723356047383297337294684510676, -3257211061780901147762200403967507, 177055479878767121877216584110554, 97335502882183415209384026183988, -7966235422166388747313871075746, -1190773904247417592817979932036, 123525159950971870551890158465], [3724432510963413575686242205536678912, 27124024119619926581672445451504713984, 20573387140105406671025972809331013312, -78328469315135693884048098191217004992, -36358297946051292134739217134654305400, 83897021834069761829070172734935327576, 7677927161047544448763676929001684762, -33553728922783085584400759347850092680, 1782540497724091382340341526010228413, 6435928273463147005514946681670478466, -843629500783099673286229847671112171, -649033748390459326151768888199421003, 122151233154196149396081148356995739, 33881456868452938910175506114755672, -8509922019006161965735985242328835, -736580371944334180771692721878950, 290883149420540625425402243790620, -2942805648054103984962522981362, -3914936949461812342108126806600, 260360902712202528391180952921], [9159915562842141515496671134866762752, 20623346724848320991524703380622993920, 12018941177854988117823521471997361088, 10398608918078701231943840352877346336, -70787462678325229420290484609570954168, -64229392528317324467544228882665564560, 90050947670258750205026205821265800194, 31315665438121986549723857270260197332, -37672282502285085103167767439726811431, -5845884607147256909761407330672796154, 7437940801693670996697949514596624213, 424819230910228461772661565935728765, -789270710872714718685700904947670249, 3143932966026059038087650389450228, 46317554304015108272222429045970617, -2145333590005402777557331432325326, -1416458647639749555548518336823052, 109817102890800573327380021906566, 17620625610680817554726180833956, -1788373734203393270729163483443], [-4040234531058161861380181304955294720, -22250699430089036423923158006959744000, -22261615337626343840943998258305178560, 52345939195684171199645047395635759648, 54425421573622519590226523969991369144, -49878906261970738325712844078405770400, -33384763213679149364602165595163349714, 19441714532741249898070168076960390132, 9316857742809739607682145589208458383, -3843190529847116593055288243543609102, -1383850153329613567319711367995462069, 435621330276460378770608046219302739, 116830919776915455522710931276841681, -29775333237570180853781011942974124, -5632642022576766085663414823154385, 1220387755625786830143037930548574, 144567089182818611607456436064732, -27762295847733549523072724546806, -1532994348055985889852408388988, 270631979056161931538377927979], [-4585554694708791960516092041655262208, -6488722222104081019675877650164081152, 38590126843121565972847276832453133888, 27174505133151770345542711129734800864, -72669954115903765261340127309533153624, -24974661312420451876773029926241065312, 51159680011017506364198762472799039842, 8617872799206939005076759533053393548, -16749965191545894588045659758802987951, -1275396675139807482309801321556223362, 2915855658058193305228109859042936213, 49152812977964960852651264525094461, -287233282239453651371099585493855121, 7433220755266536899691579940661676, 16062621008592056323542871974760129, -940276825354377113746643638748638, -475105107838455376976991782443996, 40068708103979170900708632185606, 5768879043064883284018041288732, -606735301470424531560367103803], [3028833422376758239915819722763836416, -37744287474427449805916379618904297984, 19085173842419991668351144677263064000, 164578565496505962916066614172484308896, -86480460564257256804979291227186852856, -206228610661151742702250237747011448368, 118842757478767702707146612522384627538, 84917062152136417536129689123716303012, -51742504702700706730303205824261627743, -15745772516642533368318557269282927962, 10514206951398952337561196792996274893, 1369116854255621981106733954404625109, -1141277687554219765450319462867812945, -39935024725118885708686490013354172, 68211031844784043497652849153653761, -1686174618301662166841336836378814, -2117402209616253534347091463202380, 138516335443058192084806522373238, 26667155422505892509686888565940, -2526990315991479089613411173931], [1686516275898058668146233210685871872, 12616556194272009725631699511159088832, 18506734721934176294206901861211219328, -26420895231700353389990207881691066232, -43414785106996972475387319139399627056, 22790290853551431471783371834774037402, 29588314024106391902157517427781421090, -8548891525163274223746464224468704511, -9196654933759586094171566144269490493, 1723117424565402392873800594726283875, 1517024326433960104853370853714683586, -215985527950874961354236415166376600, -142020285167817803035246324972266521, 17638617012993194849076977133867889, 7584536471876917095136911563918787, -906342286093101609257379218832354, -215378220151877471120335459560238, 26226833805703484327121799515942, 2523430745321182509741525497485, -322110730455045326351781591139], [32130619130711692779488453151696002048, 83245588353134266419335019750371993088, -177894154458912902726902379542509481536, -220048506363736353752904209181598042720, 333106525975707064139691062251794023240, 156547691847078294963028803307203841040, -232673117757990695207356487502292907870, -47687572364152184089631805094650619484, 75828841172607309470084358969969321225, 6688304822622065981151389474842671734, -13166806452301108599035940705207235403, -276596555524681255603009881706843715, 1295125238580121506290799808583585159, -31805510278174905588040903723339964, -72360823327143083125584087136181527, 4215271811969510885322664893037842, 2139027652052384986936135482936852, -180534291368196176802780946225306, -25960975634296432951773789470284, 2734152877994688274310085418013], [41404525471547362382781934010256050176, 167168282198202823223642312275894786816, -15460043563048985235383592413689688128, -303057496372158469040224203351886861184, -65332774450185673166824998924649224760, 159832150835194479332006249082043704488, 91605745515222116795029612773332628330, -41085242382295321735141271248561303296, -38191876369926948550330412419396272315, 6497714981597840600274478844794621018, 7404080636894780700052916364411170133, -774249552990611950560226063352144251, -769486358710604031469242161845589277, 70649464444231661736192823107424608, 44269001827933218141611786689127397, -4252050758864421185836519934535958, -1329892414657902353036207406768468, 141106858120813943537320812242814, 16287040824113915720418958816672, -1919313306468480675248340463471], [25985517162189879743030209058957093888, 74301629338761126769418014138538850304, -63268263686124659224791415378002972096, -329605989292145423043887524251492424288, 159143692118104562221003527005901843704, 447945430523716351548517140775104032576, -250978141154510968053896759010267872514, -188507364007700683283051246246075921276, 114458037507572873325837443002465580215, 35215687580361853970131096562530006866, -23731187734056638811482300307705084061, -3058531810236390439132199567749105141, 2602274091016702038009725681506144777, 86576312434407992012429187608312420, -156401216132856434257004328875922793, 4078178482968792209399747248640558, 4870548366671082701532424860106908, -323184664283558102681155459577862, -61455534139715025644526163315244, 5860862776188473328983662662227], [46217886103058991717909125565034193920, 118488278756469740761564753031855086080, -194784778257691164791602763440105589952, -414280763967850314213711456002677685024, 363886863416399985029125223169593658840, 468391033428631407807866899845139030704, -352058982302904859873612266450803173818, -185679589634918657738628976580453594388, 138532519311185431874831280983538443691, 33427325911953030677855447873940510530, -26930123403804907456498800174815477841, -2753470503145951901670297015451511161, 2852400493983402729047485902351608325, 59753165782527919679906368208597708, -167852297007722567124040900078831989, 5401809518929685296087502879429974, 5154825361696526204378795230840060, -358056776142994516698966362061102, -64412829750065273239199976483172, 6243587762078502040184754164967], [-35139485451163901040223031902672918528, -126806643885529669770104961027518524160, 128546586646400237964351273269060650048, 352344334799787582275312942835609298048, -233353683049101476351614860541327197064, -335930920639816473412237031550580438600, 211786758996958649335043575841388152310, 125356958142191471677724075521773470224, -81283771947994434360723045893569299557, -22100195742101185678405281587529904234, 15644403771251129141597974718346723003, 1846965064480131146812643157999681803, -1650129410363802786557437880292585059, -48834303749444178998814691508173840, 96937215687017315911723109259004571, -2646310538798880004313874813034218, -2975221374372404180002181960083660, 198340779699890366472307985246882, 37175121062452472296264218597616, -3543949853526097963471903435761], [-48947594072994128246929152723342875648, -136506169652057135709109574486510304768, 288413892034579136974596559321102377792, 459946410071502261473173009555896858592, -599506804310975338461275442890470571688, -497178628316881260665452957843906220752, 534736019777413251578960456775440894790, 191868205835992371865177381821778730220, -200354969365438833189659805632203941109, -33152112578567773088545747911580532926, 37780024308437588173137340613069419407, 2403094063973111432614286252550222631, -3919110691319537738302265125159284443, 153456878855132463785513764075340, 227156930967178655028896385127746987, -9979254176322390402926725014205034, -6896571287350775712905175358391684, 524727998844334322855700449465042, 85410402679933920810414122333468, -8594010954294800399321506317305]];
    Rf_basisdens := [1, 1, 1, 34582852139001987446892726703244288, 138331408556007949787570906812977152, 138331408556007949787570906812977152, 9880814896857710699112207629498368, 19761629793715421398224415258996736, 34582852139001987446892726703244288, 138331408556007949787570906812977152, 19761629793715421398224415258996736, 34582852139001987446892726703244288, 138331408556007949787570906812977152, 17291426069500993723446363351622144, 138331408556007949787570906812977152, 69165704278003974893785453406488576, 138331408556007949787570906812977152, 138331408556007949787570906812977152, 69165704278003974893785453406488576, 138331408556007949787570906812977152];
    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_1045_a();"
function MakeCharacter_1045_a()
    N := 1045;
    order := 1;
    char_gens := [837, 761, 496];
    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_1045_a_Hecke(Kf)
    return MakeCharacter_1045_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 := 4;
    raw_aps := [[-1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-7, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-11, -1, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [-23, 0, 0, -1, -1, -1, 1, 2, 0, -1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1], [19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-30, -3, -2, 1, 0, 0, -2, 2, -3, -1, -1, -1, -2, 0, 0, -1, -1, 0, 1, -1], [8, -4, 3, -4, -3, -4, 1, 3, 3, 0, 1, -2, -2, 2, 2, 2, -1, 2, 0, 1], [-5, -3, 3, -1, 0, 0, -2, -5, -3, 1, -1, 0, 3, -3, -4, 1, 5, -4, 0, -1], [-8, -6, 1, 2, 3, -5, -1, -6, 0, 5, 2, -1, 5, -4, -5, -1, 3, -1, 0, -1], [-54, -18, -6, -5, -1, -5, 1, 3, 0, 0, -2, -2, -1, -1, -1, 1, -2, 1, -2, 1], [-85, 7, -7, 1, 2, -6, -3, -7, 0, 0, -2, -4, 0, 1, -4, 1, 2, -1, -2, 0], [-102, 4, -10, 1, 1, 8, -4, 6, 3, -7, -4, 0, -4, -1, 2, 0, -1, -4, 1, -4], [-35, -11, -9, 3, -3, -17, -1, 10, -2, 6, 7, 2, -3, 7, 8, -2, -8, 2, 7, 1], [-16, -41, -4, 8, 8, -23, -2, -2, -8, 0, -4, 1, 1, -4, -5, 0, 2, 1, 5, -1], [15, 0, -15, 6, 5, -17, 7, 4, 4, -5, -4, 9, 3, 1, 6, -1, -5, -7, -4, -4], [-80, -13, 11, 1, -2, -15, 8, 4, 13, -5, 1, -10, 3, 5, 3, 1, 1, 5, 2, 10], [-9, -17, -4, 11, -1, -16, 0, -16, -6, 3, -2, -11, -5, -9, -11, 2, 9, -2, -4, -1], [-183, -23, 12, -10, -3, -21, -4, 0, 9, -7, -3, -2, -3, -3, 6, 5, -8, 5, -1, -5], [6, 0, 20, -5, 4, 10, 7, 1, -1, -8, -6, 10, -6, -6, 7, -5, 6, -5, -4, 1], [-140, -61, 2, 5, -7, 16, -11, 3, -11, -8, -6, 9, -2, -10, 4, -3, 8, -10, 3, 8], [120, -77, -7, 1, 17, -20, -11, 7, -16, -4, 5, -12, 5, -13, 1, 0, 6, -7, -7, -2], [-51, -78, 23, -2, -17, -17, 5, 30, 17, -22, 3, 0, -14, 8, 21, -2, -3, 7, -10, 10], [-75, -58, 1, -7, -14, -41, 7, 45, 25, -9, 9, 4, -2, 27, 16, -4, -12, 21, 13, 11], [-154, -51, 21, 21, 2, 35, -19, -6, -6, 2, 1, -2, 6, 8, -10, 4, -12, -3, 1, -29], [-325, -33, -30, 25, 4, 37, 8, -24, -5, 18, 1, 8, 9, 4, -24, 1, -9, -2, 6, -17], [-24, -35, 8, 34, -5, -26, 18, 0, 10, 18, 17, 15, 20, 23, -14, 7, 4, 2, 8, 7], [9, -22, -37, 13, 34, 3, -15, -16, -15, 1, -10, -2, 2, -13, 0, -7, -12, -20, -6, -16], [-110, -67, -23, 28, 19, 51, -6, 4, -3, 17, -8, -7, 10, 2, -10, -4, -14, 6, -7, 8], [-510, 147, 15, 27, 24, 14, -28, -18, -19, 7, 0, 2, -13, -10, 3, -8, -11, -5, -1, -17], [-470, 66, -118, 64, 13, -8, -8, 12, -16, 12, 11, 24, 5, -6, -18, 3, 4, -29, 1, -15], [-215, 59, 61, -10, -1, 7, -23, -5, 2, 13, 18, 0, -2, -2, 23, -5, -23, -3, 1, -21], [-241, 102, -41, 11, -2, -58, -18, -9, -10, -8, 3, 0, -29, -39, 18, 11, 1, 3, -14, -1], [-341, 48, -6, 46, 20, 96, -50, -78, -45, 3, -33, -15, -5, -46, -18, -9, 22, -28, -16, -1], [-117, -74, -47, 33, 8, -45, -31, -85, -11, 34, -6, -13, 21, -30, -17, -1, 10, -22, -17, -17], [-261, 2, 118, 1, -53, 21, 61, 15, 14, 39, 28, 17, 10, 49, 0, 15, 22, 42, 35, 22], [-361, 209, -15, -40, 18, 13, 35, 1, -18, -16, 2, 45, 2, -6, 46, -21, 1, -28, -28, 0], [-268, 55, 62, -14, -50, 80, 69, 63, 72, -19, -17, 27, 6, 51, 17, 8, -13, 16, -13, 4], [187, -38, 20, -11, -63, 74, 46, 28, 67, 4, 30, 15, 7, 61, 19, 16, -25, 15, 42, 17], [-218, 76, 69, 7, 26, -98, -14, -5, 4, -6, -6, 8, 57, -2, -16, -2, 41, -19, -9, 26], [-311, 266, -22, 1, -25, 83, -22, 14, 22, -11, -14, -34, -28, -17, 20, 8, -17, 13, 9, -14], [-417, 152, 95, 7, 27, -168, -8, -10, -28, 56, 2, -52, 16, -6, -19, -6, -9, 16, -7, -3], [-1410, 249, 14, -81, -26, -69, 75, 130, 88, -47, 5, -6, -10, 66, 26, 2, 6, 28, -13, 67], [-16, -220, 56, -2, 14, 176, 7, -13, -35, 23, 8, -8, -7, -37, -18, 3, 27, -13, 27, -8], [-44, -146, 34, 7, 21, -117, -12, -46, 23, -25, -16, 20, 16, -16, 3, -14, -35, -8, -5, -9], [-164, -23, 49, 55, -3, -62, 1, -88, -44, 56, -4, -32, 15, -45, -53, 9, 98, 16, -18, 49], [-963, 270, -13, -107, 11, -148, 29, 69, 55, -35, -8, -32, -15, 13, 27, -3, -10, 7, -32, 71], [491, -7, 95, -2, 56, 22, -21, -75, -29, -2, 33, 23, 36, 24, -10, -22, -23, -22, 9, -35], [-977, 50, -6, 24, 29, -265, -45, 23, -69, -35, -25, -20, 10, -80, -14, -12, 27, -15, -10, -19], [336, 279, 135, -93, -38, 175, 38, 66, 36, -20, -8, -11, -66, 44, 50, -10, 15, 31, -14, 56], [-96, -287, -111, -1, 19, -20, -23, 15, -33, -16, -56, -14, -49, -95, -28, 5, 18, -46, -24, -32], [-1015, 237, -113, 55, 108, -169, -5, -17, -34, 5, -58, -37, 29, -64, -81, 7, 6, -39, -39, -25], [129, 378, -150, -6, 8, 39, 55, -9, 21, -9, -32, 122, -23, -25, 60, -17, -1, -75, -53, -46], [-1495, 433, 2, 55, -3, 1, 50, 1, 68, 17, 39, -1, 71, 71, -88, 29, -9, 6, 7, 41], [1357, 87, -144, 20, -46, -290, 62, 106, 49, 45, 9, 42, -16, 67, 24, -23, -18, 57, -1, 13], [-670, 450, -16, -105, 24, 31, 41, -19, -69, 18, -60, -12, 28, -67, -76, -23, 42, -19, -30, 36], [-174, -72, -131, 0, -88, -185, 19, 70, 55, 60, 31, -32, -11, 52, 19, 44, -77, 57, 52, 20], [520, 577, 4, -96, -73, -188, 95, 99, 64, 4, -11, 26, 20, 21, -20, 0, 79, 31, 46, 33], [-1032, -570, 117, 68, 14, 98, -74, -198, -31, 10, -31, -51, 41, -87, -67, 3, 0, -43, -62, -54], [-1457, 403, -149, 15, 25, -1, -36, -4, -49, -77, -2, 6, 22, -61, 46, -2, -71, -31, 3, -34], [-1086, -483, -81, 110, 34, -124, -32, -160, -7, 8, 14, 20, 16, -94, 20, 38, -48, -40, -32, -72], [-785, 113, -135, -33, 7, -49, -81, -125, -46, -8, -32, -50, -7, -57, -6, -15, -78, -65, -36, -92], [-133, -58, 68, -193, -34, 155, 47, 53, 101, -16, 23, -24, 80, 32, -46, 40, 8, 3, -85, 43], [1754, 67, 129, -84, -151, -62, 127, 226, 135, 16, 50, 3, 9, 141, 34, -13, 15, 86, -8, 80], [1607, -24, 99, -115, -1, 80, 48, 244, 11, -147, 8, 143, -31, 104, 81, 0, -67, -55, -38, -11], [-461, 348, -257, 111, 102, 37, -20, -24, 7, 31, 49, 65, 66, 64, 34, 23, -43, -15, -5, -21], [-757, -117, -198, 207, 87, -386, -43, -81, -36, 9, -11, 40, 127, -48, -62, -8, 32, -96, -49, -51], [45, -371, 86, -52, 67, -312, 137, 108, 36, 32, 36, -48, -8, 100, 27, 1, -67, 82, -33, 85], [-1248, 464, -140, 92, 73, 135, -149, -27, -165, 13, -11, -70, -42, -78, -27, -12, -75, -52, 38, -96], [-2113, 432, -106, -122, -61, 397, 79, 22, -41, 11, 20, 134, -36, 37, 68, 31, -7, 13, 35, 74], [33, 211, 167, 190, 125, 163, 49, -185, 78, -30, -15, 119, 139, 31, -48, -12, 5, -54, -19, 7], [-653, -200, -54, -106, -133, 229, 88, 178, 124, -23, 63, -9, -86, 94, 142, 19, -24, 65, -22, -37], [-892, 235, -207, -3, -9, 42, 40, 34, 105, 17, -27, 20, -54, 116, 119, 1, 15, 110, 16, -50], [-2149, 764, 443, -132, 47, 534, -92, -195, -84, -5, -19, 100, 53, -34, -38, -7, 28, -30, 71, 5], [1785, -539, 3, 134, 12, 306, -81, 35, -102, 91, 141, -68, 43, 9, 14, 20, -45, 10, 96, -105], [898, -533, 326, -84, 94, 536, -82, -268, -74, 40, -52, -131, -33, -135, -58, 58, 40, 57, -5, -26], [932, 1223, 1, 64, 21, 306, -16, -64, 63, 102, 5, -100, -9, -13, -66, 3, 66, 116, 21, 49], [442, 225, 90, -70, -99, -331, -82, 50, -23, 6, 76, -41, -100, 152, 42, -24, -58, 115, 129, 66], [-924, -210, -73, -110, 60, 319, -60, -26, -48, -94, -1, 69, 29, 1, 154, -31, -16, 97, 60, 99], [3936, -618, 42, -176, 52, -259, -56, 25, 65, -5, -4, 55, -71, 78, 88, 14, -108, 85, 21, 18], [564, 541, 63, -271, -96, 219, -25, 171, -102, -14, -20, -60, -26, 8, -46, -43, 120, 2, 12, 136], [-906, 680, 228, -53, 72, -155, 144, 76, 15, 52, 15, -27, 12, 127, 95, 19, -89, 140, -5, 18], [-525, -1394, -3, -16, -120, 165, -10, 20, -51, -124, 9, 139, -27, 172, 39, -41, 38, -46, 60, 48], [-740, 232, 411, 92, -15, 13, 77, -215, 53, 12, -55, -34, 16, -41, -83, -12, 121, -74, -24, 22], [-908, 351, -128, 28, 146, 180, -60, -110, -134, 46, 28, -39, -13, -9, -132, 42, 40, -1, -27, -28], [1646, -1132, -6, -78, 237, -599, 40, 158, 68, -112, -98, -60, -123, -26, 104, -12, -110, 10, -120, -4], [-1029, 949, -602, 51, 25, 344, -2, 137, 10, -115, 64, 141, -63, 29, 130, -8, -43, -37, -38, -112], [-1659, -443, -189, -33, 246, 126, -32, -74, 11, 85, -37, -88, 113, -129, -115, 11, 115, -87, -116, -42], [-4147, 687, -358, -82, -2, 94, -18, 178, -36, -224, 45, 31, -39, -6, 20, -21, 6, 35, -72, 33], [-1048, -827, -156, -61, -154, 166, 197, -112, -23, 192, -32, -47, 47, -49, -160, -13, 92, 27, -101, 40], [-450, -69, -388, 21, 209, -107, -69, 22, -59, 68, 52, 53, -54, -10, 60, -21, -100, -54, -9, -68], [-2032, 543, -174, -33, -329, 195, 201, 279, 233, 50, 195, 79, 74, 244, 40, 28, -96, 51, 62, 33], [-1373, -83, -238, 38, 229, 301, 32, 136, 42, -135, -185, -156, -131, -42, 64, -13, -80, -21, -101, 19], [-2801, 307, 322, 125, 11, 1213, -9, -268, -27, 22, -143, 15, 26, -200, -100, 48, 276, -205, -76, 47], [1513, 978, -434, 137, 34, 57, 12, -355, -23, 96, 111, 157, 52, 94, -55, -63, -203, -131, 57, -194], [1284, -311, -333, 330, -19, 414, -90, -210, -64, 61, 1, -49, -88, -56, -164, 71, -94, -84, -96, -95], [-3203, -461, -106, 31, 48, 475, -240, -196, 14, -35, 9, -41, 4, -53, 178, -77, -93, 196, 43, -67], [2509, 835, -356, 108, -49, 256, 45, 226, -155, 83, 148, 175, -105, 43, 271, -1, -47, 140, 28, -49], [-916, 582, 411, 137, -8, -74, -303, -281, 87, 7, 26, -204, 2, 126, 59, -29, -86, 34, 86, -89], [-2669, 313, -286, 68, 55, 230, -24, -143, -111, -147, -103, 0, 123, -258, -12, -15, 72, -140, 33, -57], [-2696, 1011, 29, -320, -351, 343, 276, 184, 69, 120, 189, 336, 77, 269, -27, 85, 24, 125, 151, 68], [1198, 699, -273, 178, 54, -123, -142, -190, -247, 124, -110, -2, 18, -162, -49, -88, 232, -73, -54, -49], [1231, -866, 240, -471, -402, 504, 249, 899, 191, -120, 65, 44, -162, 353, 51, 58, -114, 291, 175, 122], [-424, -1560, -69, -57, -83, -1027, 7, 117, 136, -198, 27, -107, -104, 105, 28, 126, 118, 14, -150, 113], [-2963, 398, 333, 391, 43, 1143, -4, -590, -125, 129, 46, -121, 237, -53, -322, 124, 185, 5, 60, 76], [-1257, -116, 148, -338, -80, 1316, -155, 397, -37, 7, -4, -112, -129, -105, 129, -77, -31, 66, 17, -23], [1857, 1035, 88, -443, -189, 921, -45, 130, -70, 351, -42, -202, -201, 46, 47, -45, -10, 202, 184, 173], [400, 953, 899, -88, -104, 416, -53, -313, -82, 120, 5, 31, 84, -237, -101, -90, 229, -137, -129, 34], [490, -245, -457, 129, -53, -248, -112, -278, -45, 40, -126, -25, 57, -196, -80, -106, -11, -138, 36, -140], [-2325, 700, -573, -25, -47, -306, 232, -99, 117, 45, 85, 115, 41, 133, 7, -29, -81, -63, 5, 54], [-1032, -776, 376, 532, 227, -345, -244, -250, -189, 351, 167, 1, 174, 106, -45, -81, -170, -29, 176, -98], [2844, -218, -441, -640, -150, 441, 130, 162, 156, -18, -111, 125, -85, 45, 44, -39, 16, 147, 138, -109], [-3834, 1096, -309, 683, 190, 537, -405, -557, -436, -68, -26, -139, 133, -325, -262, -39, 246, -273, 15, -260], [3355, -607, 365, -329, -176, 911, 381, 11, 238, -232, -99, -46, -56, 259, 142, 68, 86, 30, -95, 207], [-4600, 813, -101, -484, 45, 588, 94, 164, -259, -14, -33, 8, -11, -335, 154, -55, 34, -154, -81, 95], [-2003, -1927, 203, 110, -182, -806, -214, 72, -56, 21, -1, -81, -281, 11, 232, 101, -66, -17, 32, -213], [-2386, -498, 859, -704, -149, -502, -110, 558, 378, -454, 80, -19, -54, 114, 465, -64, -140, 138, -117, 349], [1407, -1293, -152, -473, -112, 511, 53, -391, -120, 102, 60, 273, 181, 31, -45, 75, 92, -103, 53, -79], [3234, 185, 408, -325, -345, 930, 71, 306, 118, -117, -15, 4, -97, -60, -76, 30, 130, -30, 9, 251], [-849, 220, 559, -326, 215, -16, -163, 71, -169, -168, -88, -318, -300, -150, -152, 93, -11, -7, 77, 18], [863, -1702, 741, -125, -433, -727, 184, 378, 5, 114, 117, -96, -50, 179, -3, -129, -119, -1, 30, 176], [-5660, 354, 243, -254, 154, 420, 79, 67, 164, 63, -83, -139, 302, 65, -259, 74, -31, -113, -75, -20], [792, 84, 201, 164, 336, 513, 128, 416, 113, -31, 33, 68, 18, 208, 220, -3, -192, -163, 29, -77], [1477, 700, 94, 505, 322, 1145, -346, -81, -201, 166, -121, -203, 8, 140, -91, -61, -135, 31, 315, -64], [3010, -795, 108, -347, -216, -492, -27, -47, 118, -71, -10, 116, 246, 14, 187, 57, -254, -16, -96, -13], [-1233, -448, 888, -233, -426, 224, -256, 100, 246, -175, 17, -208, -5, 132, 35, -45, 41, 52, -56, 122], [-1840, -1138, -380, 162, 226, -669, -42, -608, -202, 165, -167, 115, 43, -136, -58, -55, 254, 29, -198, 137], [-3929, -776, -275, 100, 69, -659, 44, 135, -109, -248, -47, 61, 70, 139, -69, 53, 186, -181, -202, 42], [-2200, 288, 118, 547, 40, 231, -250, -712, -7, 18, 4, 77, 266, 185, -18, 96, -35, 14, 74, -274], [-7935, -689, 1031, 491, 55, -317, 29, -25, -175, 216, 149, 6, 109, 90, -65, 108, 22, 77, 263, 0], [2404, -666, -970, 468, 25, 371, -460, 280, -140, 422, 317, -38, 35, 285, 155, 5, -264, 49, 239, -268], [-139, -1077, 692, 235, -276, -436, 221, -107, 186, 534, 258, -69, 281, 366, -208, 147, -162, 163, 93, -22], [-729, -363, 755, -247, 143, -1210, 78, 208, -57, -139, -36, -34, -32, -113, -5, -88, -101, 320, -75, -7], [-3814, 786, 237, -407, -258, 502, 277, 187, 171, -39, 106, -280, -444, 229, 91, 93, 276, 297, -174, 363], [1362, -1456, -369, 183, 66, -1228, -21, -750, -47, 95, -11, -80, 214, -147, -119, 88, 110, -214, 103, -182], [-3096, -1189, 358, 278, -246, -901, -229, -293, 71, 135, 30, 5, -304, 102, 281, -59, 133, 277, -14, -73], [3659, -1264, 683, -288, -254, 191, 337, 600, 333, 60, -176, 118, 27, 176, -121, 33, -57, 2, 43, -169], [-5731, -1239, 632, -274, -745, 617, 213, 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3451, 3309, -5308, -2016, 1314, -1651, -764, -641, 186, -1738, -1443, 989, -417, -1199, -180, -806, -2082], [5720, -20285, -4643, 3558, 2338, -10923, -1208, -1110, -187, -228, -424, 226, -542, -634, 979, -1038, 640, -383, -1150, -1151], [23755, 22740, -1260, -2133, 1033, 16524, 744, 1306, 139, -1756, 512, -865, -67, 447, -183, 1020, -613, 541, -428, 71], [1356, 5548, -1753, 1903, -1386, 3554, 1224, -2006, 273, 963, -674, 1757, 2082, -325, -1851, -358, 1465, -616, 242, -97], [10031, 5823, -5916, -3346, -971, -7274, -1044, 34, 2082, 337, 223, -1904, 1667, 88, -79, -147, -526, 974, 305, 10], [8235, -7423, -2354, -1487, 1026, -1783, 1998, -2832, 2316, -1205, -336, 1435, 958, -286, 39, 920, 635, -1957, -548, 487], [-8325, 9068, -2001, -164, 1033, -4605, 645, 1699, 8, 105, -144, 1947, -383, 617, -1041, 753, 927, 1326, 1794, -757], [-11257, -11861, 2100, 4437, -16, 767, -1117, -1678, 618, -1749, -677, 1684, 1296, 982, -1327, -66, -38, -1067, 309, -1114], [-19413, 878, -2231, 2383, -370, -4985, 2079, 2795, 1582, -750, -979, 221, -687, 142, -546, -1432, -396, -973, -318, -431], [46787, -11797, -4360, -2483, 1813, 3269, -235, 671, -1586, -1562, -1955, -163, -862, -965, 499, -748, 662, -1496, -648, 50], [28800, 3989, 2166, 1974, 347, -2590, 894, -3152, 119, 313, 378, -2216, 349, 960, -3525, 892, 806, 750, -1208, -39], [-45639, -74, -546, 2538, 902, -2420, 958, -660, 2454, 1120, -225, 192, -190, 861, 170, -203, -2058, 572, -2875, 557], [26176, -1974, 6524, -932, 366, 3128, -2455, -1819, -971, -1331, 8, -248, 545, 70, 282, -1391, -501, -264, 1181, -1040], [-9635, 3491, 1654, 3990, 739, 7696, -2220, -2205, -84, -788, 711, -1709, 1280, -861, -1084, 429, 2558, -66, -758, 39], [21500, 6318, -2807, -4508, -526, -3489, 803, -79, -1172, -15, -2345, -1467, -564, -2206, -2176, -190, 3605, -1037, -2627, 1359], [-23370, -7354, 1906, -722, -2757, -8867, 926, -678, 555, 715, 995, 1711, 2904, 992, 697, -37, 1356, 1427, 1318, 24], [-2079, -12280, 1233, -538, -121, -4316, -1488, 173, -390, -193, 2075, 260, 1413, 1547, 740, -503, 356, 199, 1697, -1221], [35555, 10937, -531, -828, 2032, 7504, 231, -1779, -1018, 1465, -11, 338, 1023, 103, -798, 180, 873, -2762, -216, -989], [-3300, 6247, 2393, -2267, 1845, -4166, 181, -2129, -1500, -2066, -1193, 208, -851, -2145, -345, -50, 100, -1533, -553, 328], [-43227, -5842, -2752, 1696, 367, -3495, -1992, -725, -2297, 673, 1693, 1597, 136, -1453, 503, -487, -814, 212, 2756, -480], [-4011, -9885, 1173, -3539, -3595, -5876, 3812, 4232, 1613, 1221, 1274, -928, -1444, 615, 2361, -122, -645, 2834, 397, 1191], [26934, -8163, 6605, -1242, -3207, 3753, 3772, 1120, 4480, 330, 2198, 456, 481, 5023, 2236, 1442, -1440, 2470, 1758, 32], [15019, 7504, 2409, 2710, 513, 8824, 2550, 660, 4557, -676, 1408, 631, 2162, 1149, 726, -96, 254, 767, -11, -306], [-33801, -4307, -8145, 552, 1785, -7, 873, -3139, -1354, 148, -1705, 1133, 285, -704, -1428, 310, 747, -849, 627, -143], [16869, -7708, 2840, -306, -474, -18047, 1644, 264, -230, 124, -343, -1208, -416, -426, -1835, 1081, 2296, -1088, -1887, 462], [-12403, 17715, -4957, 1100, 2829, 6267, -3228, -6276, -4714, 2460, -747, -1460, -17, -3323, -1838, 777, -1298, -881, 827, -1785], [55579, -13101, -5474, -1264, -1071, -14185, 2432, 6396, 1939, -3043, 331, 896, 185, 2492, 112, -1429, 872, 1350, -99, 1745], [-35247, 16811, -1925, 489, 479, -432, -220, 863, -1543, 1515, 607, -866, 1542, 943, -2269, -211, -669, -13, -794, -448], [-21272, 4810, 1094, -3887, -339, 2045, -420, 1910, -182, -2287, -2352, 172, -808, -1422, -329, -24, 1233, -9, -2059, 1925], [74465, -11317, -4445, 3849, 1503, 5720, -2156, 236, -2359, -753, -1474, -2784, -512, -1541, -1529, -660, 1493, -1362, -597, 379], [-31612, 10175, 2964, -5839, -67, -455, -321, 5471, 3254, -4986, 528, -1611, -1976, 1584, 3863, 39, -2656, 2300, -485, 1951], [4125, 845, 1006, 1457, 2887, 6800, -905, -29, -1951, -666, 62, 977, -430, -198, 999, -1303, -3274, -733, -703, -3547]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1045_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1045_4_a_b();".
// 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_1045_4_a_b(:prec:=20)
    chi := MakeCharacter_1045_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_1045_4_a_b();".
// 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_1045_4_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1045_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![-3840, -10653408, 103133888, -105488216, -313740532, 95862386, 290994701, 6991187, -105293959, -16253206, 18581116, 4319428, -1700735, -521840, 74163, 32533, -727, -1004, -44, 12, 1]>],Snew);
    return Vf;
end function;