
// Make newform 3.33.b.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3_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_3_33_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_3_33_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 := [31225030372218346257929044634944667648000, 0, 1938513915962148831841211918581760, 0, 37939920124077893929140224, 0, 302468338607088448, 0, 954745942, 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, 6, 0, 0, 0, 0, 0, 0, 0, 0], [398640447768924348641639738540538024670178168243486720, -35407556469442706594842899659709085678354269470720, 19904574529903728581896468193614499104744800256, -1323557405808714060187144410359102911283200, 248579350147503316782603801211147345920, -14435713677373805300696362864355520, 948801410985586336615193128192, -52709649968373927593486410, 1090600286927067120512, -59633227320101615], [-16555753691274039942295301922045854570789098005463040, 35405888989657021876999795549716004222627500523520, -17903598787082067170171536201916752232622063616, 1323557405808714060187144410359102911283200, -248579350147503316782603801211147345920, 14435713677373805300696362864355520, -948801410985586336615193128192, 52709649968373927593486410, -1090600286927067120512, 59633227320101615], [-18150557188754513393151329580036014513421198478582743040, 441589701534271412296948664504046080788377402081280, -919085840200931200853917954960897273440354959360, -60697171359334841226155598040137246756044800, -11496794944322028401195425806015564748800, -1232724199829376532900019904271148736, -43882065258083368068452682178880, -5463625870662307811665879258, -50440263270376854323680, -6748425254370034631], [-39544588103070603764203024367873599544718686204926959288320, -2415347634433347759661172135718291558994732589944668160, 646916249362852014258641421974558136561932529827840, -126626904659742152729503815327082560036049780736, 25811375565705079312323057384815178271727616, -1624787459152076683950846846367251028416, 127832648772253674808774383620281088, -6271976456514303332525528585090, 165168677402717555021954176, -7253290202712871422403], [-12876918065246757622942862380738644135772214585920480870400, -2446575670542715852211575601616712900371774359983882240, -535006931119558709916410202926366526438892682346496, -127794282291665438530588876697019288803801563136, -6260203394463381014766128471534660574535680, -1637519758615520380226061038413612597056, -24000625452591668769374076605900032, -6318466367786409136662983598710, -28098994266482882343270272, -7305886709209201046833], [20303993052819697404725739919220008228535223977273479659520, 2116511143545111533986804449402519925577837908255047680, 1000722737516932818048196146774351732340008981364736, 82645310993067251716965206094458006834180521984, 12412618675190131867279992446086015811727360, 921225273741721497674511177558107938752, 47404189536089680146141470677799552, 3379233693605605472734703052754, 54616636537615807357500352, 3821640389947315485755], [-887038493243047735583641355934340909001502983758838581166080, 67352436297742361833574316646036448743954629330636636160, -38090526493067233107840296943515728194887302768492544, -975059053403993335335691761049450666486411558912, -443247729599879743787281812862166369216077824, -20310154278569673806896997893124011934016, -1666335945163834138953357575097608960, -83849935149810968170537835209702, -1915552020276173336124268672, -99002483403773996372825], [663965512849069476645870899193805129827594764178669594214400, -141777561351222851802990567506599967599120298496512491520, 28623636740156794758048238299017404042772398697611264, -7399831155135833313723157917017558912255257477120, 334248081429324830335352309135036993546182656, -92386196517360385958043848500554700166976, 1259200438080272932053214825907850496, -352603011978332535838318832543158, 1449470074400225843888100224, -405686191520795612219921]];
    Rf_basisdens := [1, 1, 1000487871410830705862465995848873436061368320, 55582659522823928103470333102715190892298240, 41686994642117946077602749827036393169223680, 1000487871410830705862465995848873436061368320, 1000487871410830705862465995848873436061368320, 250121967852707676465616498962218359015342080, 1000487871410830705862465995848873436061368320, 166747978568471784310410999308145572676894720];
    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_3_b();"
function MakeCharacter_3_b()
    N := 3;
    order := 2;
    char_gens := [2];
    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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3_b_Hecke();"
function MakeCharacter_3_b_Hecke(Kf)
    N := 3;
    order := 2;
    char_gens := [2];
    char_values := [[-1, 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 := 33;
    raw_aps := [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-2138715, 35, -1, 0, 0, 0, 0, 0, 0, 0], [0, 101591, -1092, 1, -1, 0, 0, 0, 0, 0], [-556806241656, 3208, 11574, 594, 0, 1, -1, 0, 0, 0], [-85268, 112444454054, 151954249, -163991, -48806, -212, -8771, 103, -4, -54], [56769755762652894, 445277448, 1654993846, -13172582, -158928, 9434, 87780, -2866, 360, -360], [-39382332, 88193183778324, 118080486780, -123656192, 23729426, 137964, 4123248, -1358886, -48324, 22410], [57100768841239485830, -616153984660, -2266961778411, -27124446349, 117061452, 890590, 30264473, -17603913, 175392, -175392], [-33904266448, 15746726856083428, 61682914026878, -68202285330, -17169921944, 20633408, 234924254, -593029954, -5824448, 3702240], [502051343816, 1842880061075988045, -458263744227316, 594184952323, 670440040929, -362866792, -5589629824, 9135373172, 73858360, -72252108], [233794749974570541022572, 1463522705803840, 5454155030357688, -72426301019760, -466267159224, -3003341225, -35239830661, 53647321746, -330541344, 330541344], [-3530790341198032409474986, -1557927785048872, -6383616996964398, 1215832356274910, 1543075774128, 16431214318, 1659448002428, -474784639014, 7736200920, -7736200920], [-267805319535712, -98073712365116399774, 672700381693062296, -705256494007394, 32209003935906, -37432320928, -4791591600784, -3518090663488, 50312775904, 3220113744], [2013313445733017886418494, 112954635657412988, 419112883017953009, -1936597164703081, -38672174658180, 2267464309746, 23068466852929, -1019839399565, 94804930080, -94804930080], [1959314732793000, -4391042056133807111816, -2362337090488904268, 2844332124888052, 2091146499928988, -1383037898472, -21743455647492, 35558599238640, 76483021752, -326638719180], [-13504913532198080, 6034236783190089499603, 37595322118397905996, -38812799015930299, 4924018521397179, -7762745089856, -477452639831456, -139264012709120, -2451133847104, -2553469734240], [33163535655336312, 111264316019272863445670, 524264923424710113, 9453257740539121, 73420023417003068, 66401596064016, 2861946931502817, 54986207415129, 6033754506960, 18108837642744], [-19666734518434992488300456538, 32311458042023255480, 119867618267471269242, -522595291309267850, -7747211559235536, 449635992383910, -5423755183907124, 1723768066922274, -24093119634696, 24093119634696], [-44255074146718645754251268766, -78045061144700267852, -274144325232794023085, -29032048241091406219, -14829439395991164, 3375926100218920, -12572581775183363, 3237294837278117, -44726440691520, 44726440691520], [31703200374498584, 127005754613447487165788, 34951997184128081618, -46840687929833662, 116915726223794468, 1593456571903256, 51252281637965498, -8634604125904594, -1154769758798408, 109671703276212], [-453028614980332795330100714206, -364489691106527313792, -1348313421061612280448, -1732484891661846720, 85021856071306848, -10332540064161552, 37039029143001168, -13568224588481664, 144873070745760, -144873070745760], [-211511078794440212751213640020, 8881530650025545866560, 32937590067291559056184, -121066782136737064304, -2573531036674935288, -24335980027913545, 383652074275005723, 186669841223631122, 621029687134752, -621029687134752], [21935446298568291340, 13179195104827315316938706, -45024895386292881372917, 48645808663117011563, 6516093145101821562, -4497949098140708, 186705399549839407, 327812044884676225, 8118735567397868, 905196617314290], [8085314880239436548, 58134151299481899722475230, -35553024078733080211948, 35227316529869499414, -14903761348363180148, 4675359031933484, 183174870442846688, 93231188833712426, -18304792691145860, -3407358414803094], [39014896522221877928453838514854, 212841955528185318714008, 787844995770056310351218, 24002497929615160862, -56055268203893900256, -713298456993362298, 2594067100004106916, 5206642300602263098, -11963574307883160, 11963574307883160], [601449301681974030280, 686838404178443391513666095, -976816232197499637953420, 1097660812202496355665, 415038227435654184755, -71439940591009640, 5207355375006714400, 8844251112334678420, -113787516113619400, -46306864176157260], [17525916925474184337536017968144, 367723832397833903200248, 1353845575601523040025210, 14429358741657646454286, -79258803851225264304, 2991781140800458445, 5635882406159359127, 6216824632341265252, 8672082426685440, -8672082426685440], [-823947287462149163152, 549040576127584291187041310, 976914322917202152754229, -1167300447026376275003, -903056120547573180528, -279667840817561320, -18172463938864821811, -10062647624086559551, 620228876409551800, 85140258897997620], [-50771016546344179106493945962890, -599732922745487595334120, -2213399423023004394364542, -12949679662900989574354, 137137248149530149360, -7337331463673694050, 14773540754143623740, -15047302384620126390, 80875941277302360, -80875941277302360], [-1466354036476276902040, 4862379428277827480578278364, 2059197432438450783791240, -2343893562103993449940, -1328290326120079335160, -1859727274184093320, -88131573234400695040, -8956403163807816700, -266357630157736040, -531521226085457340], [-325239887093059949096435628336076, -16225606865909077410133072, -59912224532132350281752004, -292459729304699769322612, 3944987187752255890392, 151480752739189860739, -461661036347671301005, -332311406179840158666, 79608354573232800, -79608354573232800], [-40386714135471091359344, 41884686661569124746923810562, 90035764329213130394430427, -96029597348182236555893, -5521502135245216169168, 1254924610788147304, -473733371841535935293, -572634950656900203761, 8088430663471151048, 2335838818564824588], [7637658857507693320984, 133412178979747083031443773786, 36825096480640564942374304, -31163449472221909710082, 50358311069249620068894, -702534389330396792, 112345142913780620560, 107307654628333815052, 5977095485215803752, 1318640273971351740], [8147644617790544172710835125227410, 39021032014787171489842900, 144090417149632921423198923, 691693640851420994519421, -9884634968888866515420, 155234154236048329840, 2131660171676450740205, 543003311302641261405, 6272989261186538880, -6272989261186538880], [12857454134418754616160, 144521402583216093891937516715, -150908307923546882005771860, 140672074822877578609965, -109052530333165290636285, 113097456882570658080, 3843631721772894668160, -471195467703357576720, -70781153977577866080, 10579075726248198000], [-17328098683900161290424600722003208, -39472737328785073619587800, -151054007690443511545525618, 9720697637890235468977530, 25018689819395597980704, 725632052425082698025, -8133174898423924944529, -1063717492493961397736, -22819515660089499456, 22819515660089499456], [10721934145939050409200221085195294, 163196336444715599916877448, 610279912166086925214288854, -12174433458211185820286086, -60477708508161803202000, 2301145219727490570426, 5711550385817851059364, 4469448169021483685230, 12745494375985100520, -12745494375985100520], [-24853708749161308626794721851718706, 467596002108079892187791708, 1742642637686278670322756921, -23213523022708717003690417, -148730198047209103902948, 508786280975377143754, -13645071403700203828495, 17292311599866867743739, -109455383894214728160, 109455383894214728160], [1360553993857665999432640, 1453705859744006569608104372412, -2854503573177774319387292366, 3056073773247242272176834, 358531161032317881385856, 819741095578091185936, 47202947440995754393666, 14018577914821818680890, -180838472036760226096, 159725655885332739960], [-2384013691523386520037664, -2899923836692536526326160567071, 4515067122218623231807473956, -4904816248181764250041593, -1076422648034906237870519, -1671382097878745842528, -91817692440679406693920, -23069536963354253629552, 179486453758693193248, -372973911030013162320], [-2150459332061211992772184, -4162244690387689859337007032070, 8097418913991727266989882159, -8144912617871251065063457, 2736642113209780356735284, 720080785171664581328, -13154725298822213902129, -33364220768540978617753, -1713717793251866357360, -248409252020050444008], [109877821409711288397315461698405502, -1319833741779095450661859000, -4924416461676612577787989338, 76529529388136744453333930, 468547735887173655249264, 11933111189869948386250, -99620567311494672265564, -29713507195184637169026, -208539384627482326296, 208539384627482326296], [-4532894586269838798999032, -9011824532497192817758070523664, -4001121979464878595511271144, 2221237818072483487306616, -13587779473146567446472084, -4861183289127795405608, -215862202329744637982624, -35475620564068576259468, 3924838357462503732344, -234086232916322918316], [1286568437489319495972306170490323394, -2391897438322489635297506752, -8908057862042738221509927216, 106721894595119333378691184, 741576393742231380866400, -19395687859477695933984, 95875308020487175035504, -88327185954556060792560, 592830237825775323360, -592830237825775323360], [-11622751766676977893229672, 7121876018494753267258461710791, 37631750001707662494215361508, -38608342033049051786950679, 9288932052744055769907043, 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8220888700709555199830653963838921, 3811408841019752769386800173133023, 8449667939721079526797635945120, -8449667939721079526797635945120], [743347763131443821484840274881289560, 747929604328006705499911324641322050794180, -1882648774612277190855201457235606968674, 1973680343346393023612773610823201326, -104748435254498973206104515720011516, -17589803491932900125647944820296, 9369470175536434319161434361399974, 10462536484349054199767756556209610, -62381219656421095714873130184744, -19992755787088498960130268751260], [6389445873051436768852628236887759359617458894, 402090654806129027860251714447590032988, 1475040443021037457848569310004355746169, 26285198668364336121058738545896248079, -76346574117661842622096842993923940, 1731941076723375469205457093707466, 19947463528287645904771723988194609, 3423546742343363262278376602832955, 65668553462833302560067642011040, -65668553462833302560067642011040], [-246981526598282925238660312915288728, 208103681510512573319617082302159331683144, 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13361147115673920719465306864781702, 241118527111912155121243466876100, -24394303314517436176241917155978], [22322769545104172110787548683741058328328933614, 619538470308097068611867592898098361928, 2317026203081576712755030252834157528822, -46722615552086552192775016304498885830, -228189745339993442980807667218454112, -6972585578095200735584532493479662, 26105678762101665889209977165874188, 18993311241410041215136715861970126, 502812743152604095283197159800, -502812743152604095283197159800], [-542237988462163512752643848172407432, -478206140931668843977838689837470621912639, 165688749325311388586865786325875518956, -350873438636828288995442442613415009, -1018900762277614059920678884942602659, 1073976859017246631507747015076392, 27199594668112866070547436892682176, -13694290919296860896872376504475668, -686025787227881721396282518657656, 96987767947341227527866124104684], [679331258928113486885888237455074288, -1864516606090223706450870799792738064327894, 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917292094669801121794442043309319283958, -1226698812302640837106230716723687482, -1502007804270639920874773246669803388, 1221107516637579040545603455350392, 34953212338023051260272143727454142, -23035321611863015896486700392230270, 1218660022655155167648678236314008, 609941884823183552048570422916100], [-19442407718253544280566220836871810683064336428, -1260010931368239982386495592121573095440, -4674795200550435479134027019128860318228, 21224818391617030595951360948827779420, 333675235425442461388680460705688664, 5766322297951234846886307406638195, 80304773059723744290670764651210291, -49654124206982660972819333966147226, 488220214261358417663224482100704, -488220214261358417663224482100704], [275138442787130538090948840055891575383905011654, 3207536889151582501064676096358033332088, 11860945144643674844836144041411711839818, 23909927327409284078387139174428830502, -841644899204877338967071083365853776, -11963654787345608444497579139392298, 142702692506606293461098456915727916, 58510331805542305244348370778783538, 259815488898267553137598946034360, -259815488898267553137598946034360]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3_33_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_3_33_b_a(:prec:=10)
    chi := MakeCharacter_3_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 33));
    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_3_33_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_3_33_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,33,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;