
// Make newform 125.8.a.b in Magma, downloaded from the LMFDB on 30 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_125_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_125_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_125_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 := [5997063924716, -33211379316578, 16618021055705, 311267728793, -870619507266, 7842422605, 17205581695, -7402790, -149769280, -554520, 635719, 2493, -1290, -3, 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], [-76040241383306613217388075649036092, 3538616274208278089877927754654784574, -2340193685446253667393191397792374071, 238907566490490354754475599649758628, 76905434645374899337079854135002250, -9450446391494292356991682966120635, -986044323285087203164854610234450, 108424484855761026354409861292780, 5634974188538612489326514443940, -546837729312314247485693061420, -14242141218626558267120650623, 1252676526740954660182682456, 13035704311013695199809046, -1062667873129158437656383], [31544159039136312767482034127500092, -3316135862487426587628397547047104574, 2340193685446253667393191397792374071, -238907566490490354754475599649758628, -76905434645374899337079854135002250, 9450446391494292356991682966120635, 986044323285087203164854610234450, -108424484855761026354409861292780, -5634974188538612489326514443940, 546837729312314247485693061420, 14242141218626558267120650623, -1252676526740954660182682456, -13035704311013695199809046, 1062667873129158437656383], [-22060009591685459936633564354818918488, 6188214310198034776780083067556024936, -3377507768397141770727624296536609344, 323174192661145395574252494168440617, 113888877416125266950982846306253375, -14221996505181851620551926624594515, -1468809886106519204802303811971300, 167159393955581481723080830466170, 8456398704077135597832991172410, -851297602898522202505054439630, -21498501167400387635355782772, 1958720157673175575373810109, 19764479165654659698656119, -1664924268059686143467687], [-332554000581594597206035212529804690497540344, -1680283730285637486460639157235061103488874992, 651224974645212852143266214249930299932615668, -1251232318180448919280673599571055935992429, -19900301173095910975174891346156456687108675, 941618373822079465393038883457436887287455, 232622098531857050376917882605609868219600, -12980370024264039182896742652804192354490, -1242095298988036049553263387119124754270, 70803110017918308131924942678362109910, 3001618147897450469155010320482336164, -169874739325736916073675945032806073, -2663478590289138246652118512712143, 148746950509611296677718604534619], [-8151979326508409350128713473354096808, 45095536647823286243412126356399262976, -26804514613072852278834712592871743004, 2571501229267688396889164869441237597, 877066125662125932103233648071275875, -108911768091886427530726364987455115, -11288226723672944836670937500794300, 1269494909200951514033043597501970, 64821890274540127946471013637810, -6443838931741868002468044443830, -164461070711508171245261516352, 14804306895329696517599780369, 150965213072328079292516779, -12575292832815064467963967], [4732629297002164698427491250621515852754440188, -19185902810356434476195348100710285079282894786, 9988931681961522894337134085896528209969029869, -747907919297865646932372909024910418382237267, -326928643002540823208327490331767753993396875, 34933659885336327179004059800785749860356390, 4129129663460911655736047643934303437348550, -414899168428076394598053212356272970167670, -23326423461219698375207367891525511523910, 2124796189403872160768597937239061257130, 58519766511671174227989943899521255197, -4908645326823957709236838873923545759, -53302743371439100547866476648353019, 4186034350943235317388608636591962], [-49932990827346333509625784197560723262879990116, 22776073219186805576017503169155450888550628202, -12616251426083941107233014901714325391013488533, 2115619118748946346374775710030381853301956694, 464958876192499140213664025837157782595885500, -78715438344800808516243068918011263453444855, -6402284770506972362569920021379912342712350, 877810819937962606659888975076911572963440, 38095444804237985668073809850015404557120, -4338842274268793926734924720298380707160, -98565016544837534346077793872865567829, 9798072356328149212832605117227518738, 91570110972343382479170199638116208, -8227337268410724784257139075715059], [-91583101084515219337645492437233610262997277972, 249077768816786791510971460206739516659783879434, -98224234583166352153281583678294337373048802861, 344928301619525888678317869603181517732054948, 3306302851624000239558430230650402108688557750, -152224393257439317021992680182866801935462785, -39965026316304600450636324771943195780870950, 2186423330911479412020848067328008195800980, 216492462011525632658511710942705971176540, -12325107554231983805210621944614669599220, -526994397872865639031209736222571073093, 30234190967823402699975688741897631296, 469709620949978477532024182892370586, -26861282656688757785277146580464653], [-1149228235708430191474483070919526027410545956, -6093367484119529781691162442490059796396741618, 6375191741360940920361738182650241361178295397, -1086615175850156688244825177343951711455620531, -218202819466244605048908125991277074999926675, 40042767962112249979724188566722841675053070, 3099367973330743769459895411074527221566350, -454572256777766733957956900018099676368110, -19036899768783937112613433733028803745630, 2281691244194269737776834837601435233490, 50321051582226829542700084065369297661, -5210732027671505350064650621015334567, -47428624331424358086695051719965147, 4410467963469195826452788996402266], [-25518905864170030949596877586727820386630252412, -52260156476426219755140276235248611627741938686, 54298636720851315749478574727482689214220175219, -6661527296795507237985600445364960562294413517, -1779114931392696293348485845658922985041831125, 237361844584818139651779752719137083718988890, 22844962686909550364881994405695803982365050, -2653371044216820723033819187193900466156170, -130753646653342582323549593818913362732410, 13219247801910928401440626864453779708630, 330964333713251690502907941433718548547, -30071672318383202246130546105598828609, -303380459817881751167946321081952869, 25399755992242968132021193843919462], [13413996337538352406102111138704482897350125348, -60162478907668431448061382302785167459565588466, 36823702819509056992187727539166226005811092769, -3302450555523354797108138122906221035006479292, -1200212731515155592936473523164853377304011750, 141412413468368187689579534551307535251616765, 15277345589943596745845020935024053118285150, -1647537256760485354393008938534718660080020, -86915973496460866881454757665762672192860, 8348576368852098926455558228706337103380, 219116852460835319077439594976736961337, -19153559842710834993135030359582842504, -200258012291143170311168178733219994, 16255271943341439147941217212798137], [18607766205457099115361203273356043003524830964, -95662075785182424466662111023795956411119190358, 58853913447639503093247683614813333629376643407, -5750802351435462069369698600464453407880692701, -1926900890715689880598570196606479142512832625, 238636803506043338110162929059643110998558170, 24698367968726846978235484431953057535780650, -2762937819753586791149507866208031179289510, -141245306413614276398182484079785454608730, 13965035325796677532089434007393771775890, 357304121330749821673651163011217091191, -31993271896383503658623028784208302277, -327324837475166894490845336565144857, 27125506153438537717479416904660786], [8556193511560780663125969084798793199745403348, -81876138219257667766553951117968571019489051846, 62039451094608375784735104241973765717460497559, -8147973395360193986274775209576215052389559477, -1997667664028315451017153101135103757590232325, 306235233223449098562862759118455938091811790, 26022445210965720393843262803226862460055850, -3447345248653332668315619391005548994115370, -151266646594773942635061397821285599929410, 17138163760479750880351221065364569329830, 387175935565718290405148592159706136687, -38826319921971866554335663723765220249, -357727970697777085906145622809675709, 32655716782807488766523243794597322]];
    Rf_basisdens := [1, 222480411720851502249530207607680000, 88992164688340600899812083043072000, 111240205860425751124765103803840000, 2881736467875393936449985914590470474432000, 111240205860425751124765103803840000, 28817364678753939364499859145904704744320000, 28817364678753939364499859145904704744320000, 57634729357507878728999718291809409488640000, 5763472935750787872899971829180940948864000, 28817364678753939364499859145904704744320000, 11526945871501575745799943658361881897728000, 14408682339376969682249929572952352372160000, 5763472935750787872899971829180940948864000];
    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_125_a();"
function MakeCharacter_125_a()
    N := 125;
    order := 1;
    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;

function MakeCharacter_125_a_Hecke(Kf)
    return MakeCharacter_125_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], [-7, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-201, 7, -21, -1, 0, 1, 1, 0, -1, 0, 0, 0, -1, 0], [-471, 30, -2, -10, -10, 15, 0, 3, 2, 2, 1, -1, -1, 0], [-1588, -63, -72, -27, 0, 28, -35, -3, 3, 2, -3, -3, 0, -2], [-2280, -9, 59, -41, 16, -17, -38, -7, -1, -7, 9, 7, 8, 2], [-3159, -25, 506, 12, 25, 175, -190, 27, -15, -9, -10, 14, -23, 3], [-13946, 734, -325, 28, 113, 88, -147, -17, -50, 7, -35, 17, 43, 4], [17129, 4361, -1709, 21, -58, 748, -587, 5, 111, -26, 65, -31, 0, -6], [10150, 6166, 5659, -227, 17, -902, -117, 19, 64, 13, 4, 44, 118, 37], [-58243, 7785, 20086, 269, 101, 355, 1183, -156, -43, 100, -32, -136, -13, 64], [25706, 4962, -173, -4, 509, -2703, 294, -173, 140, -2, -135, -67, -169, -64], [-212653, -6876, -26109, 519, 363, 86, 506, 137, 258, 125, -98, 114, 306, -3], [-234182, 9584, 17610, 667, -348, -78, 3720, 498, -150, -276, 471, -81, 243, -45], [-328679, -8895, -36272, 870, -2, -1773, -4474, -570, 297, -216, -745, -79, 1330, 27], [304287, -8571, -2295, 3966, -828, -4003, -1306, 121, -567, 531, -94, -534, -183, 265], [194252, -44175, -26691, 2028, 1049, 2941, -13847, -231, 1089, 591, 580, 100, 1505, -141], [-854366, -566, 150559, 973, 113, 9924, -17469, 675, -224, -489, 448, -484, -1094, 113], [408650, -70614, -217183, 2608, -3023, -5076, -24654, 418, -404, -1530, 3646, -386, -2102, -256], [-667208, 50300, -230978, 10923, -2185, -15202, 7763, -1120, 1200, -1116, 1055, 221, 183, -323], [49570, -84242, -118181, 7239, -2163, -4835, -21883, 2453, -1282, 1052, 2224, 770, -1736, -277], [-1507490, -32272, -83352, 11931, -5114, 16639, -16701, -339, -654, 2826, -852, 674, -56, 399], [949573, -33038, 170973, 7187, 765, 16728, -37455, -2676, -1078, -1112, -2607, -2253, 1295, 71], [-585449, -220640, 191580, -11986, -2613, 9218, 8693, -8171, 3668, -2063, -1645, 885, 8669, 120], [-427584, -61175, 156185, 14313, 3994, -63003, 28716, 6133, 3321, 1213, -4763, 715, -630, -1694], [-1684173, -117876, 409106, -44543, -30722, -20902, 6767, -1040, -1658, 790, 1495, -65, 607, -395], [-847317, 232708, -627152, -33587, -24526, -47703, -74798, 6135, -3846, 4302, 3868, 910, -6124, 501], [3194, -122105, -22421, 32714, -7271, 76755, 10777, -4921, 8183, 4085, -2502, 3862, 9317, -1033], [1018613, 459525, 858517, -17342, 20948, -13461, -40754, 7714, 1753, 1204, -5281, 1769, -4358, -1409], [-2276347, -376096, -1237642, -94423, 41716, -51936, 32040, 471, -8176, 8775, -20392, -2660, -1870, 2101], [-2071252, 12136, 1271117, -26832, -24113, -497, 68053, -10247, -7470, 13704, -3743, -629, -14025, 3482], [10143021, 789593, -1814433, 62989, 32897, 62305, 121763, -18920, 6457, -6494, -7452, -2224, 28027, 2128], [-3818258, -517234, 216628, -22328, 56172, 116306, 51095, 2521, -8540, -2951, 12813, -1811, 7253, 2578], [-1241972, -1435773, 625687, 53552, 27932, -122484, 78831, 10844, 18101, 13683, -14839, 3281, -21556, 6103], [-6957340, -1263116, -672455, 59611, -36015, -177230, 115561, -4797, -4182, -9465, 27316, 4536, 11952, -4035], [-7646397, 125273, -1816209, -171719, -30969, -179671, -1645, 14202, 3313, 9576, -25134, -2526, -2723, 3120], [-10831027, 2658304, 2986831, -110354, 7421, -209522, -35801, -11216, -12026, -6380, 8112, 16520, -2642, -1936], [-12772097, -10089, -4067254, 22187, 9741, 542325, 39231, -19714, -27163, -17224, 17984, -1192, -20999, -1726], [-7950299, 1131966, 3776059, 63725, 20791, 120821, -162638, -25428, -11376, -21645, -34769, 7235, -29609, -4803], [-10190075, 1010483, 3082075, -97773, 14576, -211343, 853993, -6906, -14169, 21282, -4598, -7366, -14127, -21856], [8429635, -1112162, 3785088, 77922, -45620, -53921, 949459, 48306, -45960, -24591, -4682, -5776, 11266, 3228], [5666768, 3083558, -6642126, 5026, 35772, 90036, 1248750, -26154, -12842, -8332, 22230, 90, -16988, -1172], [-51813203, -2587164, 114980, -184664, -127601, 80562, -363543, -23137, -11912, 33647, 19227, -35195, -22787, -2546], [-58338769, -594177, -511181, 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-1183730011, 5333603720, -189391726, -41642758, -7028373, 431500675, -110497455, 310499663, 970067], [-90543182521, -3883792500, -3492140520, 1973738341, 791226984, -2973969524, 1643386018, -233302517, 245030836, -46646597, -114148276, 3226192, -432790, 44922997], [13972222617, -25658779513, -20378214946, 1390772656, -23802775, -826608571, 617028392, 284544897, -329962383, 3949479, -255141700, 227265112, -223476529, -2874561], [-310063128568, 22502275916, 75500420596, 122144904, 106726745, 867463011, 3275857974, 169932087, -82458258, 2976658, -92788265, 164551703, -271964661, 109008286], [-90468984565, -6122185568, -43029016567, -576699174, -946868261, 1364725196, -308391949, 100005015, -68375964, -160479591, -123527803, 52368799, 81635427, 63624004], [170216081543, 10107769003, 22293325709, -1041880045, -758411400, 1439690746, -3808987907, -88578112, -276322999, 52704095, 176565684, 110935130, -189891971, -7674712], [-192550134824, 7702408042, 33192145565, 1544135754, -592926035, 1311147786, 2054424798, 77663958, -540714264, -50319808, -192588988, 98756788, 40667136, 38314022], [-37435838446, 9024374896, -14279052175, 820138002, -76502393, -690100502, -3833762516, 47337844, 179563122, 41289384, 665779900, -77585252, -224072058, -152864464], [-291651035851, 16138679783, -24714262557, 1131772681, -207249654, -1896473286, -2381075715, -378519363, 258368389, -83766082, -25625529, -101321013, 427450912, -89628884], [-213313558736, 10297546016, -16836227202, 61294211, 141673396, -2545482149, 707097111, 411485753, 105338150, -110640754, 438263636, -131103998, -1181668, 9418811], [-17198829832, -11800537199, -60252926447, -2928696182, -1036466284, -1665722172, -3913921575, 111526115, -35548361, 121405522, 189778760, 103425458, -450190487, 99033141], [62058179591, 14253448240, 21568656166, -129750983, -866363854, -973020464, -4172651452, 212595581, 37376288, -23545999, 20822758, -86606154, -700382964, 59296505], [-123708888891, -5964697949, -106743043466, -1177022804, 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-786541705, -338928274, 3375103732, -1747899149, 17300046, 94034597, -67513601, 83122006, 373838336, 378515963, 23210464], [-547749866432, 16139436408, -54826882478, -2394918478, -492685090, -2704168303, -7060014112, 269487876, 169917578, 132585207, 287316430, 127807868, 447842120, -95078566], [59162742467, -18732297078, -68562455626, 1389726988, -491279234, 1907270217, 186823167, -171460094, -320887832, -10670049, -255404660, -117941258, -473367032, -29871706], [-10774671719, 4957097843, 60320712444, -366556853, 698770812, 2285186038, -650150087, 146752971, 312925285, -115027040, 442375935, -404433189, -509518892, -56504458], [79224693866, -37420327843, 11062078839, 1226218966, 1256904, 2326769262, 3774388407, -253826514, 823153783, -320783755, 253910847, -324553389, 152122228, -4780649], [359056142293, 15841604463, 4851756833, -2162864219, 408721582, -4096181282, -2432346899, -368205935, -259777259, 92019446, -304899089, -181290197, -178142120, 41035632], [127374634561, -4794095848, 24388076588, 2830832111, 1519354733, -1145434971, 2118876382, -193560778, -245347582, 16295411, 611654229, -20747679, -173204985, -123024955], [155887744443, 4857460204, 12204455630, 1109113668, -1443715124, -93827037, -70266750, 877364251, -69888264, 74918452, 216045805, 116983711, -465263727, -26998028], [-183469612178, 8732606815, 49704679063, -67224417, 130452051, -1696139262, 3589940310, -363185268, -77998323, 198551055, -1058980956, 313253802, -203525313, -89830968], [281605937449, -1171095679, 25340587480, -2620586543, -664689739, 1110040951, -6501210901, 202479208, -383338805, -44638514, 37407234, -183081670, 315007129, -98223770], [341941761350, 29323168495, 47577141212, -1903548640, -566183297, 997106558, 984021169, -31537207, -37504561, 262720346, -374378952, 60589330, -67510867, 292705151], [139633086252, -31455354828, 53442437387, 1253840396, 747263669, 1390859684, 1172111530, 5134664, -211796710, -36270982, -119033898, -6243270, -193146972, -200983150], [191606862892, -8928171461, 33409207830, 295840778, 1140795339, 1398633940, -5570489372, 51137837, -741991243, 288441970, 49263388, 198434418, -166751779, 63026107], [-400816013733, -7135540038, 35660219048, -1323765230, 312941170, 274140299, -2541428719, 147704038, -206064236, -104860499, 464653678, -230882300, -873065762, -136092464], [258019451279, 3961226811, -53026981682, -1028197434, -1886844605, 4430657466, -2860834156, 178840003, 250344007, 62063440, -398302632, -275183094, -78703131, -1957787], [-59975081986, -47572325580, -19574401661, -2577438095, 78395158, 245608965, 2932816823, -144438651, -131664090, -152871336, -143511718, -139290448, 56750602, 77901609], [-154026753988, -26080495430, -39477389190, -2428122362, -491679852, -626764112, -3168475714, 547061226, 25345470, 208961340, 525777758, 70225350, 420894636, -260353224], [491865010494, 5146432210, 30307281694, -2094403682, -1036147486, 6933170380, -753509525, -488427615, 5424292, -92057881, 35639743, -503721325, -114232445, 66845352], [80326998791, -25717153128, 2637372631, -695279875, -311649529, -4720824925, 4143495416, -110166397, -102823312, 72381334, 26724794, -32133244, -149278516, 103662915], [-338575239031, 16506025046, 42878760348, -638408975, 1969987128, -3395339526, -2481527064, -198520629, 35508106, -187409437, 857508474, -362681682, -288778358, -96116873], [-50770589657, 14038082282, -47254756630, -1968814456, -1414927430, 1256608201, 9453476311, -85054914, -232937200, -365485357, 67671308, 156499222, -619591288, -120291190], [-219844193348, -311921306, -1100365483, 4205164726, 282228549, -5597327539, -4394270256, -203877054, 565921830, -146245335, 452248620, 275891466, 203967810, 31106622], [138267511970, -6171105422, -11892495395, -1657120727, 158408864, 1545187327, 3380697733, 488611555, 47186644, 34187854, 474065370, 226159352, 991736416, 109070363], [-71035193979, -20683892304, -52435032245, -622098181, -483811375, -1466192542, 3249221453, 514878588, 53280204, 458697192, 39518363, 339146851, -1000292695, -12312339], [-114091511606, -17698298842, -72049643326, 1629823097, 1857056425, -1270152526, -496330321, 457648652, -254540626, 15818378, 346413135, 17568097, 15665009, 219287233], [-307082616762, -971633430, -46410705794, 78005882, 1367832636, -1037988034, -976470772, -342065110, -562618486, 408568982, -209963774, -63814762, -365172692, 81194540], [-281330967709, -14682532684, 121005878288, -930156119, 526662696, 199636824, -10235881998, 347114031, -234315372, -730318005, 43615596, 38662152, -449454798, 23420721], [-86763698448, -35158305254, -30044579931, 1097204582, -613527345, 1802390488, 1444482115, -9461985, 477666266, 302633103, 530659947, 28144215, 189859649, -54587586], [257506445839, 4541301580, -55710235164, -2215132586, 1454499514, 447557477, -70816367, -82871028, -4340706, -111105993, -684830978, 35144312, 877238056, 68388138], [-354873652745, -24631082056, -103020560154, 640848949, -2087766543, 742405953, -3570550482, -811949744, -33297806, -121353143, 120232781, -224997831, -612583685, 21679467], [-188552996161, -26392307578, 21441857563, 1281626045, 521274801, -769026392, 7083557292, 52284939, -484945260, -637579281, 186580452, 184826832, -194983590, 99706149], [76734923638, 30310788840, 59714885577, -3422618295, 1327584753, -494991463, 4622411018, -686542638, 683250702, -193437389, 232763559, 149469831, 830857029, 30114935], [-142944029415, -7036320775, 51663900328, 26876043, -962840731, -8851258860, 15297591510, -237868618, 21747615, 137283677, 281754992, -279485554, -51212859, -53908558], [62236234065, -22287870806, 2725958692, -946999385, 1749858128, -7505024983, 836787020, -640449853, 91064368, 108434288, -696647632, -412643770, 885614878, -67588339], [-156341199191, 299138319, -100778144463, 542617080, 771831996, 3309376192, -7710887588, 235221385, 228533073, 209914234, -309761794, -4314684, 599597565, -195542295], [204366081896, -39890511289, -76080415961, -839061390, 1141946510, 2493249970, -3422859173, 632359652, 238298229, -102997835, -214398265, -71635729, 602599650, -27277743], [-460632374540, -8090784417, -71943509598, 4218488388, -2780346078, -4035872327, 9980490094, 474727708, 385569079, -23168930, 17040461, 148798123, -3821266, 40478983], [-557321465442, 13246556258, 111752460592, -4001993427, -1413082524, 2863380343, -1400166945, 114099261, -157311564, -222146748, 160579176, 18294570, 516459174, 25571307], [640419457044, -42984963930, -52332281855, 916367422, -1664603469, 6121601641, -3046482982, -841340278, 98947562, -113324455, 106493560, -377114446, -544708106, -121781230], [-28723905406, -49870180816, -14263019328, 2865229999, 1057792458, -4360096123, 3059274371, 430249759, -654943910, -266073394, 533440550, 68161116, 152002174, 53458427], [-294692120113, 6678095819, 139728183637, 49871151, -1425789850, 6819619178, -828271795, 323472823, -417522039, 668991028, 479402043, -8995225, -876170484, -65729434], [220628999293, -11827722854, -96955041, -718454962, 801084063, -1181794834, 2457977657, -1033755734, 132436136, 616982992, -349542364, -366625824, -258733168, 105040578], [595701452952, -819829626, 92364846973, 790064672, -1920187471, -6654738657, 3328351269, 266933097, 310009004, -26487394, 662462347, -100919887, 52022279, -197897520], [-349997236017, 13038573845, -101899741515, 2336082516, 2301416592, -937895823, -600751284, 244238408, -890765011, -86407876, 17657279, 2416181, -181524470, 220375865], [552765853590, 29001593529, -45680763828, 1425007671, -411262023, -1037478762, -6947461700, -190761420, 658212309, -6662829, 715191472, 75851562, -321650967, -131547552], [144223246015, 21471679056, 125044829892, 2989680660, -198666270, -4117352217, 4960253655, -552153418, 139219110, 124402363, 240554458, -370530912, -72889608, -216422684], [79017975171, 10259259030, 39511541833, -757645117, 897797954, 4329158404, 8482479754, -28925725, -1179598462, 7832345, -89174656, -238340224, 431389832, -96177415], [-144998803007, 8582290113, 43727759759, -340444780, 599887114, 1562082699, -6666005440, -466264707, 371147529, 117913641, -194880040, -1068265012, 168083635, -31646289], [-494436621122, -1698123020, 50007909297, 3390522689, 283818945, -2585103602, -913199438, 632631392, -85846384, 400398496, -829203671, 757061625, -560663965, 246504571], [-219128103309, 36563231696, -16150419306, 422844613, -821867826, -7058110356, 7115146968, 1030536713, 113376952, 299932641, -90049874, 117248862, -37939196, -111741019], [83157585131, -2546620836, -53742857545, 554692789, 470901325, 11722161053, -567437118, 849271870, -522188630, -137535833, -341708093, -127938789, -84266259, -68764769], [-1026694488146, 28052180138, 39065099198, 4430142245, 308758254, -7915520697, 4849427709, -365202101, 336352324, 216437662, 575833078, -139471808, -356278612, -272269985], [-258014300394, -3444741421, 31746032844, -1497296594, 1349122389, 6982217027, 2618994527, 305784807, 763071835, -191308815, -284018214, -278065218, 92911441, 94151319], [359881407698, 4613150007, 80048228792, -2033872804, 553103109, -5830664473, -9133316321, 659666019, -320789787, 327201771, 416147554, 254514354, 164012877, 59158551], [387609123284, -3218299842, 20397343795, -1002495202, 789742703, 8834546359, -2910391676, 38129982, 93856958, -777689009, 215472436, 128257382, -9234886, -165964718]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_125_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_125_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_125_8_a_b(:prec:=14)
    chi := MakeCharacter_125_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_125_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_125_8_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_125_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<2,R![1359046823936, 19534487615488, 13482409130240, 523785924352, -885948475136, -17244543360, 17882255200, 65206560, -156309200, 444240, 659029, -2533, -1320, 3, 1]>],Snew);
    return Vf;
end function;