
// Make newform 23.16.a.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_23_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_23_16_a_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_23_16_a_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 := [3479819676773674797376, -1120833250638227525562, 3701665188552036536, 3084648153192808082, 50805039593451715, -702635376772966, -15614385123802, 50352376602, 1644150380, -977032, -68942, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [361186256051082415877485114918184632939950128401354840891520, -280883382998417448139711653832482685895079743573156066360356, 3242658212205642157133437169963257970305689333740547258246, 484575452468805796984074764874256970697243657027101359924, 1548590110219564806963074201008864101213828842792446938, -139073990163087963534613231631177158790779115543311133, -474121710128263579194085874204132827276060684475194, 15191616390897367959807743109063836499343380390395, 34708925214666121632146842071146422080027869094, -674874541164462136947823054608474602966841251, -736992050280179635407655608388323821085900, 10418555600359609147819306969220622149941], [-4747517081454103102319754977738616476141896141255780099411840, -292673297299772734654895461061187066761219729739006065351716, 3687560638671879384121505367272857248273235981508471748486, 484575452468805796984074764874256970697243657027101359924, 1548590110219564806963074201008864101213828842792446938, -139073990163087963534613231631177158790779115543311133, -474121710128263579194085874204132827276060684475194, 15191616390897367959807743109063836499343380390395, 34708925214666121632146842071146422080027869094, -674874541164462136947823054608474602966841251, -736992050280179635407655608388323821085900, 10418555600359609147819306969220622149941], [-90143157627556186395814630300305567776487383522134293739303616, 2168355374245582099080881178247826711265289766950030921557478, -78600420749643847072990591623190862286392495340982119259758, -52302201240215297543606937145194898664019964296892241510560, -740294623149893560752149471463876773517637616420231695149, 15439475477702605823543596323899876213986227900691181447, 160066712365186283412677346727750452636085943995377823, -1712292291185360891828341879223854884907145111919941, -9839413845268313473103054796186027412842534803439, 79510980607304287751348765578276666789345792967, 185497412513766231667687126299973264558387371, -1299599033383903911996240125027464880137807], [14415559785413012588865540496025639807767263582263549257052736, 10631494099887234057373307275022060118483209952530629838896894, -612393676554090302930591136939075519945776173294346367432734, -24539923346174378957506021806140038216045121262089526516688, 81491376358412928031371147198096821767994758439969602063, 7343887603033519872988685139826314965426333683387045879, 5925929025588498855017406244858078055014055980748635, -810071069723111428173780528191975845192673807379909, -981988644328401475853331737331780883801184320059, 35846252169547554385195609709103614359408028895, 26117659416497604737305746230910421679859119, -548336516275392787099551814257198966197055], [486439180479985846761690061206643752542466090438782646086757056, -102420278895048930444314383370880459871382636391167080099964326, 1135767296747783978079939839226561370151219515812141896862366, 221503550322400940801419697948532661910503731475783146879232, 806805442136559327828491685351658672505834673939489713133, -67278390691693721679136476579355758490200955612433867471, -233649073977534356566004472962312313446428635597956095, 7564846117360692891250143751940281411892367417837821, 17140058263230816085730160867131166761708379657711, -340650182365039709137998095664191684356831152095, -366142124671731193030978153322795713698890811, 5292782413451219636744520121257055885090135], [-270224909537835494972419757480849672094114976904822586900963520, -27526103255959403099557007379145020274521668269721798447583658, 4339046232319263205033753761452139232268387767856112994686078, 164262244954877068817171013942892749774214292027127180193912, -600212681509366752642912519452432670937648882291112230221, -50279916185758462551934031277911642015627391824512987259, -27801987661638525587569093182755812015162139969465737, 5486197219394339109784328889492992748224979428720825, 5729374955171363799767419326191150003484500600097, -239502419464750170129864716182563836110591159343, -156851605771141632683181742659253098287440905, 3616907306514768710617313270262495246649123], [-242949821525907530351870754960372549510971212884086774676035136, -21840857233686754776271264228711282416654321653848940309651038, 1366589376801003891733182701466677900546239430019415828671098, 72283602249805258214537863149741211042743716901575242468264, -53238079407066792608003080138403080653113968997697349591, -22555826216478551839825884778812460909169496414438679441, -39531947506043281970852494430052886105881304185591691, 2529502332465310316475950167143032784193896587140299, 3920818353199610194195939940177482330988321453715, -112740542518057494019666776038078464607427762189, -93468252498502047768818936103726484446934379, 1730272771249178494450500316538252408833929], [-13550214004252660862261036255159914128482467915423208111800512, 54070505160710408592468540495882914360900798599721212392679126, -1800626263873592619794417297104435812925903858192806864128926, -164605106971952050029208974768075547393558525755400815046560, -346282382615902737758004262041551047419785424025003903413, 50790546274157923795492161549363813683634647046574250399, 147645206706809392702134992441214406764311926863434231, -5687312500344314698093217839748310910253753531209997, -11629483589561621390249284305858068938970785182663, 254618483212452150572399089101322309463805003999, 255534207578864464164493715717098541149824227, -3935961961541647731952440539754678970241959], [-1129894438841038961074327053145406977692852344354406874149944896, 189589517747186310851112643111935526351707578391877143871634786, -552586972050181461732416168818097008286606399675180301501066, -307587598516233507737451442165517427521245160551780838062592, -1202017744900586329650233300149384311203589002685849775503, 88708203012476073777657256314591464004990461953071200581, 302653960023728937871868995385927948841089638949214485, -9684546336510082718675925511075727600294036304931711, -21438532125312267879243506817534526501476531758181, 428763201636659156894268089271282786830167345525, 451471054670961720265027455576762686359381161, -6590426094387205027046720394919592262938525], [-52994006982028903594019991977572010362744941755031964764926656, 62958772767811138692270586987488749249968747456150449458700646, -1610400429290952258051515923406913604838210257715348926007706, -147195096691263534478527412532381869001391397092640439500632, -282026063150462803719775545171869246863761208185850071413, 43148540469401021620550307794726136129371540371735184761, 122364600264441632036549635540547584404074344192188575, -4693627489111180019840710767482143191029112404233011, -9521821415514608575988018517866265311464000962231, 206913519101459769409906773412547489708033846365, 207140384179342981141544742946762868016828391, -3170004073269862474167082176364991599008945]];
    Rf_basisdens := [1, 1, 222451213233118613494034098654799638983773323883962245120, 111225606616559306747017049327399819491886661941981122560, 889804852932474453976136394619198555935093295535848980480, 296601617644158151325378798206399518645031098511949660160, 889804852932474453976136394619198555935093295535848980480, 444902426466237226988068197309599277967546647767924490240, 148300808822079075662689399103199759322515549255974830080, 296601617644158151325378798206399518645031098511949660160, 444902426466237226988068197309599277967546647767924490240, 148300808822079075662689399103199759322515549255974830080];
    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_23_a();"
function MakeCharacter_23_a()
    N := 23;
    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_23_a_Hecke(Kf)
    return MakeCharacter_23_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 := 16;
    raw_aps := [[-21, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-145, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-17125, -257, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0], [-476268, 533, 1, -9, 1, -2, -5, 0, -1, 0, 1, 0], [-7317109, -42519, 628, -508, -1, 37, -19, -30, -47, 10, 44, -12], [-24477585, -272462, -28552, -494, 113, 18, 64, -15, -2, 95, -45, 24], [-205595840, -1155016, -7015, 5582, -142, -727, -2152, -281, -235, -1049, -925, 858], [11243043, -10040243, -73565, 16920, -218, 667, -3793, -341, 2341, 1017, 649, 1496], [3404825447, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-13762699091, -116790602, -3839798, 316840, -50542, 49941, -114282, 112285, 63163, 55131, -48081, 72692], [1715923163, 90744664, 993136, 12168, -110342, 157399, -194694, 71332, 9210, -120883, -1777, -47262], [-72097759911, 1198908326, -63329864, 3339223, -58146, -110598, 139512, 197245, 182117, 161797, 134690, 146526], [-626176679493, -1028101675, 32085355, -3648881, 209485, 1043, -23731, 347880, 106939, 75161, 386236, -159578], [-472826501435, 2102472806, 34558255, 14208187, 293758, -214433, -1614028, -495654, -1485618, 353618, -664661, -622918], [-485444475341, 857172032, -179601300, 25824872, 2405092, -1288791, -3857246, -2690492, -1012248, -1182391, 1802511, -1857822], [119495911135, -4139160545, -547998819, 46326258, 905622, -235785, 9730133, -2214163, -6696337, -3744181, 6331293, -6700200], [-962498960260, -10237184374, -912261118, -214196490, -2913396, 11297206, 19538046, 2364312, 15071284, -3980200, 2023418, -1862484], [-1957991861337, 22873280072, 313721445, -245194303, 3220144, 14611199, 12374818, -12462768, -2299346, -6339062, -4720919, -5165282], [5148341974379, 66056527332, -1424197082, -356561107, 29443607, -5514131, 2248956, 1729987, 17485976, -13027475, 5466710, -10061058], [16517696934927, 79622094997, -1142970046, 439022649, 24623165, -951892, -86265759, -42875343, -261824, 34611190, -4382427, -12718776], [-1868217844273, 116983236157, 562676281, -153391265, 47945946, -37931374, 10854311, -34057980, -64485152, 11839159, 35474096, 52567966], [19177271647446, 47242136365, 2883763462, -2288767945, -74043694, 82756028, -6886415, 67146279, -31310181, -57164129, -77314184, -43357518], [30831864176065, 162618818946, -14038527778, -456520913, -53126311, -53162695, -715618770, 175801003, 143438478, 70833317, -107072192, 103982898], [-7100640120950, -47957094097, 9153978840, 3334515655, -171722942, -6725234, -577619645, 165174907, 456833679, 297899061, -272381066, 95457722], [-420483878028578, 213692698971, 43517849509, -2736835007, 57835938, -46998561, -257681577, 515807470, 629475654, 289348198, -662777365, 383660396], [-507794538401504, 1638563638070, -9709105480, 3957508546, 1309632761, -846015937, -833621888, -1105582326, 9266451, 109180480, 680358708, 151089596], [-642863911417474, 1236516154275, 81166257221, -6802795293, 212808817, -46880902, 1062069565, -949174576, -166206385, 400878960, 732630477, 267154720], [250754548393181, 2791943398864, -18914539993, 13965592815, -1658114688, 697697983, 4082928974, 571971428, -120077230, 607424360, -770342731, 3133534], [-184810447592331, 2880937351480, 87094973441, -15501886447, -2040783124, -387834055, -522199358, 1873435170, 406169172, -722057460, -1012347553, -117990614], [-243247245395610, -3761599196248, 1167055687, 2459916244, 1442382942, 854013075, -1878094780, 321576147, -402093679, -1363313205, 455382725, -1022156958], [426697150554525, -4109726838244, 19192285066, -21952716940, -3127988986, -577241757, -1019535410, 1164246126, 5312865068, 4642665281, -1277818073, 2841464562], [-1087466421633963, -13476760791375, -73263324297, -66953791157, 4616433717, -1058950005, 9248474719, 160675507, -802336626, -2015214469, -1958776882, -1353109434], [559277087327594, 7995122757642, 28274195748, -14226997344, 476916172, -6033301562, 5513633554, -10540555448, -1652418852, 110825830, 607049966, -3640251680], [-860997090523701, -1541947962673, -663387783917, -40579397003, 5212267443, 6537492899, 14776398237, -2015443371, 2432126466, -7403021549, -181646508, 609460438], [-212936572438171, -15912831526090, -370915031413, 177634648453, -3585636986, 3817654471, -58352421896, 6580671822, 4517880266, 5994411976, -14702347553, 2976193222], [-3530016537262389, -33037109954401, 558892582030, 142481746639, -4221421677, 704185210, 56266994227, 8198414875, 6261730648, -11110043116, 1601482689, -5068821044], [-6152112247533715, -24144204314045, 4183432547229, 19608145054, -16721573708, 19498347007, 11151543613, 10826483271, -5716716377, -3907930459, -7105443349, -7150380164], [-11055860625845827, -7943164763110, -142095572307, 243433422150, 2389338818, -26973675228, -33548086190, 9639687374, -1819534308, 15902995852, 3188175726, -7183040208], [-26088089704500060, -90106672711738, 2370563012756, -521298208010, -7624026894, -23203821146, -73093114966, 3178230268, 6651893734, 1204666126, -20351636572, 2727364844], [-14724806513041186, -16914317582302, 2131083924242, -151721447904, 12773599139, -11234878589, 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23293819286259107889, 16787722970787490972], [26024760436792335280430107, 121538149036073127337342, 6028823511759032500152, 453961426530655104906, -1756596634838946444, -70752025669692839907, -76563034502497973332, -37911437182028015918, 11767669620338787390, 1174895270800326703, -16175611074441227597, 17507224631904965214], [18983756227017369381877233, 23820556315711719801293, -3260363758046184646989, -92470824073122151466, 19179575724222501088, 12279753502448801899, 281865740470593986107, -3605793174420428301, -6031593683273831161, 9810700716773795333, 5835849797552418659, -35080374949408067804], [785398661681104958814289, -102862048215210937535573, 7680691371578008214047, -51531281145714483115, 30960242959081996853, 3596277368296982589, 53825488709349572283, -44423950193672104980, -58199211206782426101, -41646244242675663091, 40576778950401591906, -27083893943254661214], [10183576274512823277225195, 75122760952228643744780, 3556740300497245842464, 86094219521706312507, 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7498858914266148248], [3670693253011823703695837, -39160993237253636599319, -4824888307373666075570, 182773930784815643970, 35994766588602802555, -7732819367525268125, -195662438296725666255, -12037001453303382086, 11501738547727283497, -15611487861729181026, 18945901360881670082, 40323813772407164832], [-9777276977698657552501637, -5951464292321179684217, -1240805024540291991129, 189831280519731171538, 12206335549689881076, 23327666092116199649, 96936583414691058741, 17363219002474146181, 2051483264558337601, -13156793698834232901, 51698579098998760777, 12270265511547757464], [-41104025433976096739907576, 2944465826984412874447, 3388814018710191775962, 26240878702579427163, 54242240673683741174, -33652836479654674770, -8637420916323624609, -64564940133965914139, -35992077530900693939, -793083292984344503, 69371957973929658486, 16701934596912430882], [22546527782908580969127883, -33537028176042212032044, -1654608858134233773442, -70761316503832760574, 17967690512873275195, 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-14729559769319168378], [-20751405456050138349264060, 11434131601163907907641, -9650027878438837915122, -169440425414455105129, -44007579048011732248, -43336901421425556188, -134380011829158777795, 53501422668610507031, 12096680551383132257, -11753170187237765459, -67718784333754818246, 30271283535053536274], [33421488839679358939749131, -75112317187711065960849, -3832569897223318443132, -116704315987038446042, -45864692603542930158, 46958904358292414258, -422702175932042210791, 69406003623305517258, 39159270630869316560, 13156776446500796002, -43166502958994643416, 24559871612166962188], [12711523343215458779774479, -88418755729583392528530, -3317464534541560339409, -38797313245742785211, -41853158720935250264, 48150599039336441663, -63777146060151792460, 46910345543042432444, 56955054500285104346, 8977891242311024294, -34168839042344480359, 7838627929028693918], [-9606456073459040448425752, -81954707211084328717837, -11877834810725481802462, 680215469470718031327, 22144223209865630957, -15944168190090140609, 126723195941894986441, 28176879164568682409, -25468239445481095148, 49810536183906542683, -27899870613977282722, -2400038652428384246], [-17302978066668567308960396, -30958253517054921543890, 1152394284446927166762, -763969324794273339914, 12921761831683228418, -3145931585214275196, 114250853111402462034, 62569112237688932626, 17995384407436353044, -47937865067506741384, -7017384345567104230, 29863938857959784664], [-924452770976951875496793, -43447702159110500501547, -4590964267638138480201, -483714939728400748211, -84493869477539005411, 73536904354991126553, 62637315337573739317, 97021732205800491276, -7606773739015437121, -36931905075560446871, -54927786698998939190, 7582013543353384614], [45472154715864811463068341, -21004868745276934436044, 4772777899011759224438, -24822660864708626368, -73815218661047861354, 15214802047361805703, -136288520439876964212, 29841850786522437709, 40895475899451326215, 7924000431654365909, -66108947896233755879, -31751949995704717484], [7224306735408001363622481, 135923889275445018048755, -2276527449956030541635, -22673295800919398620, -66730266535123021082, 15190646770771538065, 81769048591768125929, 33066688792481980673, 20803255308242922099, -11773711304366109645, -31203300045664522337, 38806037398216470208], [-64208405493874888554295028, -288756637664236753066444, -4933546459181310670232, 157271965813599135996, -7465305311171854283, -16876362741371885845, -242353802661725479146, -28100877918560335924, 50855313714044758781, 22186145988235663610, 23070445987457541736, 27621767739743187132], [-20810359986571630596145367, 101559690728069629698475, 10076101001736399594459, 1332090354900242323661, -26593313027609105347, -64440237365443937849, -271546291181074048467, 34541156642867061539, 5256772304848782730, -21679810954850957987, -25784140374657910374, 22010952200527359750], [59789474148441886091290671, 21194106515163121152334, -4767203244674610974238, -1028119722870480991854, -19669565902244600572, 35962861877219390667, -77799900229811190792, 46347747975755142188, -39512533741297375508, -19158242477666327011, -50159027970313353043, -60991992018048475798]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_23_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_23_16_a_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_23_16_a_a(:prec:=12)
    chi := MakeCharacter_23_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 16));
    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_23_16_a_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_23_16_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_23_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,16,sign)));
    Vf := Kernel([<2,R![-15120112467951573696249856, 584026570212381298786304, 132965155784742993395712, 2096409906654081712128, -3121844114192269312, -190371743943798784, -415151179919360, 5756110997824, 18854461712, -66355616, -245760, 256, 1]>],Snew);
    return Vf;
end function;