
// Make newform 531.8.a.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_531_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_531_8_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_531_8_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 := [-2385018853548032, 3193975642099712, -493048066650624, -494039551757568, 99470801612800, 16796366777600, -3353576629440, -226132003872, 42885295136, 1488374176, -269092298, -5107725, 890490, 8791, -1493, -6, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-189, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [67, -298, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [981418811144741929782073660320589253156602382145569800318976, -458439489854631649894932217132810128320856011874955172921344, -273826719218794098795105945699170867977587556583610030011136, 39509797905556573829487630657431872425875390392819956037632, 11903941638133989263223526039579799183877050542348257214464, -861622039114945884709271565291691044878397476902249992832, -197167027722867637252127690869794240912613596277931671264, 6710350307985913550475184379701605694401384275836423168, 1506135362097229385117263352956068410258360693850061168, -20975240937210409010507157367787813269173962012898952, -5779256502908570911746064510823955116603011851309715, 14133951682856921189514293489249705078263951354320, 10860688524025305928237333170647058491350754564615, 49831136758234138131066253200973502096551178517, -7962150382781891011397658124568438749328032672, -72532190443349592980224939674071553018102081], [184996531606687738222038534852154703419431521219970373668864, -77833070212640702189393610608743609220984733375973669687296, -52784154425248985885394184863489376167672763194715065235712, 5398722475854531176920768918034456252455960648537228170496, 3004551462055620476870599740404993478993171113576501110912, -90077674646785673625522084108084463216167414729811164480, -63543399284378562965049545360056020544073209308373084416, -523575353344287141484631109918557009551858125668295120, 556412928635306708004152408020564923663748256471230600, 13694499294455100137074890969024589093643507313773460, -2316239452750700238525353274759230508062118576043923, -76484329367728243419596195851656265301359303695924, 4584014396312103136735463834296998267750741018299, 174022955207776787762159962381351013989893180069, -3476973948520470591306808127935482280835724924, -143084456061778931953264783332387790253876649], [12669157124210206254916104575138134065518553521622436170600448, -4788065094624843533109525716195101936500151877577610741030912, -2396778653732913508555656815573992776542849090256948365540608, 516127986618844850548451605259064698769615185853658705555456, 87426214102112326573634279870603345992648438041962869102592, -15013063325993943163912957045386528692749298612533810514816, -1282272500618610905487395540500426414800002084873435624992, 179472852554905808833311258847191576266563950627354072704, 9126838396363212736619969083038761268450844974083578384, -1076295971191828282998388919256656235967159330160544056, -33480852647688469734240927450232419372335478923894765, 3432069071408240177374052458387813886585212596303600, 60923586326310417208009714006205425943686816175545, -5566674640100689023560517668592241643942169753109, -43543020698230311319755068171561208621278625056, 3616121502233286625521346328773206995809195457], [1075991525628574573615579299084273230049130473921271008407552, -3133987977982723818332004883272099509357950341987827182419968, -291082926216027078447415632859103243541055286831460167436032, 298586802801850950718038979980084469686711370096476048656384, 4839454171932183415788698132712366595669612066730871257088, -9811944355255416205009798412677490344054091750463654521984, 102472335478047614367715538702577595312091877072468506272, 143383097695372725835763143634154071162281636492248238336, -1766452665341606733723078074121933248849685682004280784, -1029545308999717533591749425004573793408706169969400104, 9505548914392367783050953407635493812851899032778105, 3810723129974781034079825560966715791148566608544720, -21774883775648316687236068663165790176760959848725, -6988840001933344679249220012687410223430172586111, 18232204043597009404159432662201717835500872096, 5029308229786134423822080760577024516915255683], [-4850870898845563804612381492759145903796854142158740836184064, 3019106731084659328177607199472512106860417121431124374716416, 1275392797756054778512248437346592512000859371849548100981504, -535088783890545988430495837694876436682135546442527225289728, -46814510728043007985624710421317717502775777430458095264256, 17959523949551025204240958611916121925509587074209511753088, 636726581680504061981623508251018098013915444919794977376, -228619111789622796709001765970217250758398359537560508352, -4083441974961144558247667019201062308006231339726726992, 1423471692065388636464406029106663265579679833183816088, 13263086030746368822028484970645362438629675145586235, -4661204207833195548157554058387513601282247630750720, -21089576120021789450488795891116107358074747331615, 7717222754010701476149168860667618660348464755347, 13023354139016638127686455665935287879734527568, -5097556660509098324170951844173289722768229991], [104738593787527582742738686150058255491551158515508487115264, -61855582688144915190352645899915970415468758404440871433856, -16402577110600015143098063195170067128939355979208381461504, 8576044747225616965078048304433220759943042041509177517408, 412202756831696869119091430745869258312036743224538007536, -285434468815061960660645688216403019640739255508590344808, -2759536780525354125002522681578433871912446101442550816, 3862979113398256784904124459422113154171815499157322142, 1603806471654124931895937286679773725279213192972357, -25798251702160975641338723655139897019080796880822053, 43539063377121979870806311638211093716210597602185, 90274894861098507498741950868967147088535840212400, -147708842161061122043854474001610563721875350945, -158751709158205363035950955685349956696199999077, 143589250188114864200720779426545341571396267, 110707548373849166695650093906463396167609116], [7914210379526965885672046138041349573219765473753451742294016, -2739911856209735175453994928559368246532987390134251776425984, -1738809691967154315934011278870012581489900342375091463529216, 529945096417440316594018512898914864143201218970662663473152, 53986139305448553599996028539918963740852961653864584387584, -18929163403431766350327139268463320127716920064904820975232, -510489618567448943584955938353663214832152002914967671264, 259037539413690239345824373241976579463697461286959235328, 1907869994797554781687228902864791551403061151449773488, -1724588615840972736754202725659451907670679357152925512, -2062255471682212547386215675564877899079983602558195, 5997632262993012924617548348664429193972675736041680, -3198438617019950309105333383259285339578399104025, -10479695913339127087991596161516139960809859016523, 6165905073089299997499407053744814010325478688, 7264343152238016351592021961435333469383883359], [-8749324980845770454483460106026856574703546499165882508587008, 6325868404993517303897198222247265551280102921169220049559552, 1102836150714671168882779128746170425482175565739398846521088, -629029290482653906625959264252503354777199831251480760918016, -22871629789232060172663285980936425332384650077057519982592, 19662588757688186847902032647564610918898921749373698242176, -7029458210644396785048929922670987965598369008558737568, -264355207250913822750826423070177042326018164198092407424, 1972379111104090261951707390352210508859764530712399696, 1772586935830522239461235080870583844487298132253004776, -12451209122618242348947033982406456551942288961173265, -6237494955511832246512676005280869646452455613802960, 29715708075235071592614382919215862020318342933965, 11024855532250427699678721898750385541535518115239, -25048504423772526377815128292062856981089962784, -7720395604270371699545471661268529169360338187], [-2005113015136766569294444998766188499344904187909282635575296, 2009519915119217955907195105491671714479318879718161054552064, -87078407433119372196202629365083033274016328774697104918784, -228851496154490993543033521659934950308187702044633472077312, 21842930263770836411166030351334762934128892308388704914176, 7777168361450421904579763545390871319335443270298869139712, -611919564226605248081397489604119226416901515360502575456, -115996465385011149269118287031179717560133092576698599968, 5895145205242689802691615816129395008402953919782533792, 843442637682818575644099345588678340988318845641356752, -25736366085314979647165031991913373628157571434362045, -3151030291694766733510764828730900349462881995680840, 52441555644669072951352813308267623946220735302305, 5823974900460273220966326579076415128171391603643, -40582522729316779182619948735788479536497035368, -4219628579760384715204025522570781822614345439], [-412805785631783147830221474715453817843787912830649837819904, 310035365765500475003389087315906477481765555859030232508416, 63490952785384886676254812338828786852796797943543545267584, -43444713160043949412980908432205290239168968755005489888448, -959597702940176412827621986050366499926282765083844242336, 1499813632839424888664234939264635453804070682048349236688, -10667093085961781305849353276904118702997450485524768184, -20964512993649012432781477997054300162746510576243729332, 211017276135746071155312376717867737989402347722274818, 143281336221099428699774156819768230670386153651331883, -1140227890490342627348153901031781423863522950705780, -509532638828593553019609424608139342339647963878185, 2576122868003195131483631286279023506480165354585, 906125846533274771404059728434031727755923138992, -2116169818880076837590409484876250128988100027, -636767690535804699119920905345721163447977826], [-2280152709900375738887416703392741614778640061121521970253824, 1287702794060428814060700045204165254740446036450030904359936, 329083342362221969707223140944966032884627197617544586599424, -150640949164222162973816585840214108096435396333408010864768, -10299441514129589216282749119081465378297503099875892744256, 4617138683404036191064466886818299721375064850999047136608, 132000573407991569236143063229799880056175103859269269776, -57641835408592340954490810792272690227655692241772889752, -826393974617403291791559479013216792154294933768250492, 359692672528747875217291033550126481345448405084312158, 2666655221130280358227502302827585203210875008384875, -1190901957034973452227980938765009311696201136907490, -4264800095056088712832255309658144187926289920245, 2001435847177820259750029044625790953949031060667, 2678709944021728782760435005055643844972986938, -1344142178610721980848805253868410578201601411], [1120366420090373758294263766486851206096660417937305369454592, -895933597789943890755353829907303294987991088859491347652608, -202555307892274450491286933050132532282519419842524203612032, 105005118441357814604111275511823546903284063681821591057984, 8096062422665483397399509882081584292923829860104743091168, -3128685635602867939664727530054990789580143168927087330864, -131462658072166510415394936932938211251678788504899946808, 36908891714035762725970176138474484009120841840705345436, 996126759226518628164692504367726924950210209717929706, -216829080334734475206192035758190992557590116607931509, -3796716029070210167586366633102097444284953769423435, 678570921643493323830760234902545760804772290347815, 7088424018096430409312812891662014548494755124820, -1084722369197905760218255164195938510040166401711, -5164578266747195797354374637783935345268730259, 697485077755115788728697507560219358225087433]];
    Rf_basisdens := [1, 1, 1, 1, 1603980578468012296786861463843480337158629514386095800320, 80199028923400614839343073192174016857931475719304790016, 1603980578468012296786861463843480337158629514386095800320, 1603980578468012296786861463843480337158629514386095800320, 1603980578468012296786861463843480337158629514386095800320, 25062196538562692137294710372554380268103586162282746880, 1603980578468012296786861463843480337158629514386095800320, 1603980578468012296786861463843480337158629514386095800320, 801990289234006148393430731921740168579314757193047900160, 100248786154250768549178841490217521072414344649130987520, 200497572308501537098357682980435042144828689298261975040, 100248786154250768549178841490217521072414344649130987520];
    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_531_a();"
function MakeCharacter_531_a()
    N := 531;
    order := 1;
    char_gens := [119, 415];
    v := [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_531_a_Hecke(Kf)
    return MakeCharacter_531_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 := 8;
    raw_aps := [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [5, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-148, 5, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [-73, 37, -3, 0, 3, 0, 1, -1, -1, -1, 0, 0, -2, 0, 2, 0], [-527, 42, -15, 3, -2, 2, 0, 0, -2, 0, 2, -1, -3, 1, 0, -1], [2857, -110, 44, -1, 9, 1, -2, 3, 2, 2, 0, 4, -3, 7, 0, 1], [-2481, -118, -109, 2, -14, -6, 4, 5, -2, 1, -4, 7, 2, 6, 3, 8], [90, 277, 70, 8, -6, -9, -8, 24, -5, 4, 23, 4, 12, 0, -15, 6], [-8188, -2323, -91, -44, 17, 21, -7, 18, -29, -19, -38, -6, 10, -14, -19, 20], [-10334, 3825, -214, -4, -28, -53, 42, 2, -6, -6, -14, -29, 22, 14, 10, -10], [-20829, 5539, -21, -14, 397, 68, 9, -9, 15, 146, -2, -65, -96, 72, -7, -30], [-38117, -8031, 186, -118, -166, -123, 9, -84, -38, -379, -123, -54, -22, -96, -29, -52], [-93992, 11917, -807, -6, -194, -13, 67, -173, -70, 30, -73, 47, 180, -142, -72, 10], [105509, -20309, -2207, 149, 1235, 199, -42, -71, 215, 254, 223, 222, -1, 43, -36, -133], [-30712, -18819, -2750, 440, 1836, 50, 139, 65, 252, 207, 145, 182, -2, -30, -132, 118], [-205379, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-386219, 9867, 56, -500, -667, -512, -202, 100, -67, -221, -291, -372, 440, -848, 180, -198], [-1018619, -34597, 118, -655, -766, 721, -163, -47, 207, 318, -298, -371, -43, 577, -47, 137], [680947, -12116, -5457, 1339, 4414, -1343, 1399, -281, -753, -1766, -226, -226, -753, -585, -124, 167], [-1328867, 25474, -6451, 557, 1803, 158, -712, 181, 429, 257, -135, 358, 271, 119, -739, -565], [-209482, -66750, 3869, -1154, -2484, -1050, -303, -119, 952, -163, -487, -491, 1176, 188, -763, 904], [108681, 30161, -2422, -3368, 3210, 41, -1807, 234, -854, -613, -2327, -2878, 2292, -2366, 341, -502], [-642950, -31348, 1116, -1367, 1029, 1090, -830, 424, 3038, -239, -924, -501, 3135, -2365, -1418, -785], [-1504208, -198778, -18212, 1005, 11283, 438, 264, -1150, -1885, -438, 427, -1835, 2793, -2651, -206, -1015], [548148, 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3562401231, 463017172, -53681361, -828195737, -289764438, 142489261, 66067051, 138105464, 256491277, 133594535, 43455530, 302973045, 205870981, -158531183, 30545571], [-233174656546, 28699460293, 67410835, -262447691, 406116110, 90795400, 375822, 34206831, 135441713, -870291560, 35832935, -41579625, -191428935, 236550705, -19477126, -68057223], [305209268755, -6049736655, -45436462, 243316357, -1177304743, -272060491, 378609429, -130405526, -137029999, 199945403, 248732296, 86951225, -251597751, 255134089, 66734017, 168225229], [-15996867024, 12681320377, 804877905, -560916473, -679263067, 347063975, -322322849, -216283789, 65211130, 155784955, -320352109, 87334746, -42506133, -244881127, -188253852, -198811159], [161233024190, 813883370, 2729752823, -336619117, -1557639613, -55147978, 175837063, -212219949, -72388239, 37350441, 185532728, -203526318, 350082705, 71565963, -98516209, 33784791], [-28439990668, -14628634830, 3517701602, -253929841, 220300981, 402684141, -106618571, 20035261, -134094465, 191407999, -24375022, -315818234, 166561857, 113476369, 229406210, 25329265], [-332479481480, 7192072975, -1237574115, 345787634, -534218123, 74434930, 91575572, 170883051, -51458572, 393563208, 185806292, 168981218, -381662622, 26154662, 115345210, 147245706], [69664726075, -21933013108, -1726234153, 122088539, -2201389611, 184050192, 109465922, -92468741, -3633, -232645773, 193540601, 298383164, -203136387, 232210821, 203248415, 101673969], [-89736735621, -3042105256, 774882896, -308632378, 1197369131, -211992656, 166057955, -79123671, 56019608, -359236027, -248513189, -243952390, 207520638, 78238538, -178235406, 14985802], [375720028647, -13703998316, 675282405, 172018728, -1144319362, 216078286, 128390368, -9459521, 1606950, 791474539, 305973768, 111813957, 99925652, 286059148, 176867815, 114633038], [-335044251222, 21098432290, -1218740573, -638873148, 767827667, -12012832, -45688059, 205480484, 80155296, -381714149, -194940963, -196350550, -121415320, 129557044, 86298588, -241742160], [227689080154, -9993476608, -1817998279, -283531509, -593034292, -8394409, -211714756, 283689089, 194808685, 447218089, 348580731, -4102669, -153145461, 117201891, 199926241, 66300287], [389262088142, -8532614144, -1940338856, 102733434, -533630604, -136579334, 91244598, 317649328, -140562942, -268296840, -56409646, 254314190, -251802300, -32455244, 103735996, 282657238], [-57144445024, -41383205766, -1329213169, 273369511, -1655204271, -3064678, 148779019, 348784815, 183268863, 3708987, 172037046, -91697902, 42182877, -30298447, -234748339, -103271759], [567904922188, -32347373509, -3454053514, 155218498, 1200066457, -71257389, 126677140, -365834469, -215446520, 151519684, -152699938, -209940928, 123056154, -611741840, 258774517, -151694788], [354390569404, -7077328884, -52110621, -102051569, -529528958, 278023817, -43469194, -299861311, -158649173, 296129960, -41883425, -362156292, 237595597, -402661667, 68305150, 50304821], [174452417026, -30515013486, -1301805839, -393028668, 585033565, -7304457, -285737652, 262019409, 169476045, 78624667, 183206619, -80819065, 171704716, 102701204, -438985940, -69705550], [101714130440, -12364628625, 1162788393, -128251645, 5712476, 13474102, 210586348, -212677133, -304520936, -967889714, -274245108, -162430462, -195496417, -174438809, 147978989, 18278069], [-188593897961, 27293934297, 1886631544, -12724885, 619310757, -219721204, 37947042, 34357109, 165740049, -346589864, 6719747, 111916694, -425083523, -106771945, 71824091, -57031697], [-59244007824, -12501370791, 884072273, 140929878, 1489931634, 265298471, 12863731, -112435229, 228881034, 160498412, 85165911, -72300279, 242486180, -143253814, -300799138, 17414502], [-749341810921, 14019324400, 588419368, -78686583, -2253653540, 51968693, -23079532, -27000080, 128941550, 468201861, 65953258, -29101292, 14268123, 374668411, -19253092, 142237617], [235881013894, -23983392689, 2894387944, 435868995, 195492754, 188235662, -118745771, -330144676, 129657868, 285642520, 474236901, 209259352, -93827441, -50711935, -314401593, -288797499], [-292308903104, -6664325904, -395941170, 57308774, -109976284, 372018219, 32354467, -638101491, 55207575, 65521361, 179298806, -105040982, 46594082, -4915770, -55739585, -115583532], [176422990974, 12198841001, -1310837612, 229567244, -610128206, 324657731, 142146410, -392425306, 89772507, -64923968, 34528557, -119576924, 33964020, -226474576, 248379217, -2994450], [52093129476, 27650406200, 2557011112, 144702935, -2438127819, -95580128, -191496298, 57060050, 535534311, -42205662, 116790083, 480848817, -68475359, -139223599, -324972556, -131189753], [-201792164351, -38334487411, 1270990965, 635645654, 98707755, 5445833, 49317942, 173432858, -341027180, -633823305, 186682798, 143195217, -157082698, -161203352, -16312336, 20717378], [150770276994, 13332289839, -1687500127, -257043543, 416010930, 279202814, -603288706, -67919801, -80852906, 771969830, 37268804, 25635636, 116689477, -226863571, -56265107, -270254193], [24422333619, 10158772316, 4206082214, -676999008, -1521862511, 132543338, -190272959, 144488341, 13796928, 53988953, 111447325, -262550458, 134413760, 408116824, 186914362, 119131032], [257702631843, -22229560354, 3189899525, 224170612, -620016968, -177629789, 250919537, -68802490, 19245938, 546346895, 265699903, 115134677, -125532640, 220905600, 160815652, 356181120], [105910343922, -22520452466, 1628037954, 662195231, 1242213713, -204553287, 179636687, 235802991, 106576284, -149032152, 22483931, 110162466, -136264219, 226172505, -208543094, 171337607], [168192381401, 25954932485, 1339292398, -42426273, 1266677233, -32708614, -225697256, -81365173, 233485863, -345855983, -253379279, 484102613, 956425, -699589723, 143146254, -158144587], [-678308655851, 53708630583, -4818723119, -376468264, 1469111534, 269978061, -177176388, 92431735, 39862842, 215199868, 325505220, -17315193, 121053858, 91357822, -330336384, 70622094], [-24651290188, 24133884539, -2687465896, -630838971, 1254359623, 161666675, -384274912, -128943194, -189538875, -403267685, -345666473, -230090095, 73493125, -202371799, 170861037, 137566029], [-226869683229, -9241180758, 3977955616, 124140297, -471265136, 125092013, -12499878, -167092538, 19126776, -297501954, -6980976, -41788877, 172259899, -274362361, 84318919, -89164911], [-406647341978, -21291777087, -502612089, -478605992, -69062343, -60887930, -359333524, -197128299, 392465090, -45045396, -261935554, -96294704, 817122356, -790843840, -233031630, -147467344], [-265337356903, 51797008294, 1920067325, -89195703, 510023838, -261937218, -165592240, -21305170, 377240191, 194044642, 117606587, 213295142, 692572611, 23346689, -327948517, -49345975], [250485723100, -31711957069, -1506994289, 277531845, -580692234, 70641388, 12329732, -8808629, 136184066, -337932258, -47526206, 68341448, 224310529, -557439691, -66126353, 299412735], [-333285194000, 27187844060, -1337507079, -506348034, -515383809, -247053093, -77345296, 3428647, 326138941, -169840054, 369431093, -77577082, 391877498, 221831238, -154781455, -235512348], [-56464156450, -47404772656, -1782694695, 713430914, 1226625554, -257198253, 145908426, -122378522, -496532622, -162678486, -139142498, -198807334, -35096466, 520703684, 367246759, -112185752], [48635823415, -29369356736, 5786343849, -45276037, 1262953877, -190342374, 236359422, 194908547, -435909277, -880984315, -235737091, -108579386, -417076527, 590882909, 215870519, 266802129], [-704258577759, -4137615068, -7933987208, 344073382, 455023361, 383662819, 94487672, -127455416, -159340580, -762936829, -38051368, -351977924, -200879380, 100467636, 14253133, 33480380]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_531_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_531_8_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_531_8_a_b(:prec:=16)
    chi := MakeCharacter_531_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 8));
    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_531_8_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_531_8_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_531_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<2,R![-2385018853548032, 3193975642099712, -493048066650624, -494039551757568, 99470801612800, 16796366777600, -3353576629440, -226132003872, 42885295136, 1488374176, -269092298, -5107725, 890490, 8791, -1493, -6, 1]>],Snew);
    return Vf;
end function;