
// Make newform 208.10.a.i in Magma, downloaded from the LMFDB on 04 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_208_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_208_10_a_i();".
// 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_208_10_a_i();".
// 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 := [48215700, 2867800, -87027, -5049, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [-1, 3, 0, 0, 0], [113831770, 2053669, -405502, -3558, 113], [-337166700, -20196261, 307443, 7497, -87], [-363924080, 4578036, 785247, 2118, -168]];
    Rf_basisdens := [1, 1, 59735, 119470, 59735];
    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_208_a();"
function MakeCharacter_208_a()
    N := 208;
    order := 1;
    char_gens := [79, 53, 145];
    v := [1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_208_a_Hecke( : Kf := Kf)
    return MakeCharacter_208_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]];
    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 := 10;
    raw_aps := [[0, 0, 0, 0, 0], [29, -1, 0, 0, 0], [345, 1, 1, 0, 0], [-674, -25, 0, 1, 3], [-1408, -235, -11, 9, -14], [-28561, 0, 0, 0, 0], [-29279, 69, -51, -32, -62], [-22151, -222, 48, 283, 157], [-333473, 4301, -49, 112, 217], [1395379, -22685, 805, -1592, -941], [-2828484, -16703, 1575, 101, -132], [5652498, -15008, 3894, -4148, 935], [11109075, 16099, -927, 7096, -2539], [-12307708, -97874, -8265, -110, -10285], [-15833435, -158880, 847, 12069, 7364], [21113707, -193617, -3147, -6800, -4487], [11920666, -60441, 38987, 26285, 40392], [-8508649, -542821, 10185, -144, 38753], [43017835, 385048, -117798, 9915, 16267], [95268626, -588603, -55712, 21259, 38025], [-40225643, 193437, 42027, -37756, -98971], [116525058, 98116, -202632, -42210, -97142], [227491105, 1582174, -32070, 66215, -111225], [-96672256, -3277538, -53550, 7812, -159674], [295408537, 1392617, 573759, 31876, 11097], [55576356, 1430002, -36286, -342712, -105092], [214574932, 3071340, 2508, 497624, 161540], [-387616543, 6209909, -319453, -69490, -426913], [675466274, 2961728, -303474, 653076, 53399], [906594366, 10175500, -1032500, 401988, 192444], [347068385, -12670321, 1246089, -1014364, -897533], [-915696000, 2252860, -1767163, -58972, 755731], [1967873395, -23145581, 174585, 515792, 1166361], [1963004705, 10424331, 125370, 1357200, 1169550], [-263118710, 29663008, -97512, -1309576, -452340], [2976902835, 17443038, 2311983, -1042455, -1147392], [-237770075, 24389921, -2735937, 990976, -142059], [6491696388, -1715571, -493473, -363307, -1933804], [6633672662, 20324377, 1755171, 2160135, 1969006], [3118589796, 16152546, 2671114, 834704, 50716], [5443983573, -86362415, -3071388, -965438, -2086702], [-3278795482, 86268940, 2624952, -445392, 855886], [678672787, -70542977, -2626577, -4661210, -4755677], [2936672316, -177045430, 1896450, 113436, -2470672], [6864442616, -12768038, -11777760, -1342260, 934915], [1638347498, 88004270, -339780, -918192, 4507004], [-1500740376, 87729912, -4617687, 10374488, 192467], [-13052700377, 51446048, -2351487, -251241, 221960], [1210924648, -155425505, 8304177, 15874331, 11401620], [3055761163, 211491855, 1691679, 4955824, -1151084], [20155742420, 104312902, -3281152, 1223140, 9657003], [-31024782266, 105452205, -11827134, 7314439, -9109877], [-3160692176, 214277854, 5873634, -13441372, -14662902], [-26836125989, -79484263, 16285203, -5415948, -13113991], [27044791951, 129106359, -32777613, 23717428, -15742304], [-40206364931, 127477197, -9859377, -4840682, -9562373], [-24041882869, -85609869, 16891065, 11890568, 6679109], [51795352927, -243032066, 10705305, 8337669, -2255918], [-15581984353, 365736187, 7769073, -24565512, -8178683], [-12900174932, 407030906, 18678950, 12336964, 23618362], [7142484037, 606537291, -18525891, 10592744, -32033509], [-39743456862, 29258928, -59747818, 26903668, -9807709], [68094164086, -433693229, 45382335, 38376109, 7106372], [-131192634256, -134080032, -30279978, 14646416, 21900410], [-4567504046, -613404824, 28817994, 3472176, 4783027], [147450881144, 463230542, -6810870, -28986764, -30616362], [29912203135, 1436265596, -55046976, 6518867, 56403085], [-37817487082, -590336044, -39690042, 74355056, -24222809], [-64134938737, 889288313, -37289556, -47847212, -105078792], [-120390992437, 130383583, 44102403, 14743568, -15955938], [75253307012, -10116482, 40132166, -45841232, 33628020], [-262291640654, -585392879, 43598931, -76261669, -32176086], [-99740383184, -737945646, -27056016, -37100934, 16298022], [-204456350928, -2294278470, 89065146, -20815088, -55647728], [126994301536, -1443073353, -97420071, 81979955, 48152084], [-54509392979, 2569591694, 47362299, -63990621, -28544614], [106043591515, 406037611, -149490315, -12828880, -60400485], [-256736716206, 222536052, 351671340, 8732620, 78962920], [6728841463, 985569395, 320219625, 52126596, 144439433], [-590902946491, -925073783, 161354043, 5503712, 109459959], [207087913881, 3099543295, -106192224, 38677036, 101965244], [352819225987, -84839737, -60239517, 221460704, 84705490], [-84525076382, -734228543, 279301086, -137117065, 107559831], [555937018560, 805616818, 253210008, 37834092, -45135491], [-261445137379, -4968071617, -208850457, 126661056, 114198521], [-4209383234, -952636718, 69988317, -101346928, -178816497], [-184850686731, 4159424205, 146534747, -152878100, -269579803], [-306921158078, -1245141712, -270103632, -273711472, -208801928], [-447306740684, 5129601470, -98062656, 261495972, 256204325], [-145375996426, 620290693, 18998367, -275845061, -360551266], [235090077459, -2609599103, -322674049, -10849876, 94559893], [383833031163, -568377574, 225701853, -47201011, 259774390], [793507744570, -3046830069, 344099529, 200345133, 123188418], [-571518260922, -6817274512, -145690547, 169303438, 29346377], [-1063039863796, 4079706867, 123581061, 174780243, 283832280], [638950159488, -3197650626, -177423608, 479792270, 236487202], [375492609738, -2100510212, 98068004, 417958460, 466907704], [1436546671352, -2716049670, 422189040, -512808016, -365748163], [-39973596823, -3652682935, 449003295, -222680966, -27149913], [452238763122, 6711302400, 462164394, -432348572, 272335753], [240909508355, -2784569615, -262970418, 274149904, 465231194], [-802806041996, -6063358506, 617624796, -161877452, -654834305], [-229928598747, 4675409313, 822017192, 490451026, 30401030], [-290404139271, 2191342509, -1515237151, 283173724, -237921424], [736356849277, 8865880333, 194216526, -526912186, -350535012], [1922290136921, 3551105973, -37215453, 378811568, -82885339], [2151029484494, 8412955513, 979095779, -52325093, 743971650], [-974077329173, 10172985907, 1186813693, 496072024, 670887715], [-493488186679, 979634429, -1516302899, -1429606710, -1328601239], [-237905717135, -11015449779, -1303600419, 49400616, -800256366], [146692430439, -13165660533, 159900183, -1095402042, 71199627], [488363977670, 10782081628, -60231324, 915168520, -347577244], [-3466032685882, -5930598648, 808597316, -387365556, -255858962], [666902933654, 13806958163, -1181652711, 319863725, 42438754], [-866891993069, 3009780574, -469223997, 813728409, 912259540], [1259293733022, -27924257716, -837643948, 310766028, 1268089296], [166364759809, 7225853702, -1346444772, -1175283721, -941936435], [-3487100978989, -13654443999, 544465333, 583001440, 871687083], [-49960753216, 15688367834, 263462730, 859111288, 1270142484], [5264552470111, 1516210923, 332816013, 234802210, -699128207], [-619592160332, -19330763214, -2476147962, 1154821748, -429280798], [4451872388351, 2152420335, -1338613269, 823712704, -1256532332], [-130808519820, 45700738, -1574126094, -816218000, -85719484], [410436232547, -20520550878, -1475116358, -546252131, 866394969], [-2497386964095, -15010117822, -1969523922, -188693013, -1645481521], [1518166306503, 1014515471, 1266007821, 2001171632, -806698643], [-1172595457876, -25691955590, 2052814398, -869832548, -371473846], [824811586872, 19746586280, -2793628282, -114230424, -606241734], [-4074528423075, 38997888639, 1076461311, 298367632, -1870033343], [1783208607589, 49950504965, 206068833, -2446622000, -1414546962], [5135749260968, -6378282175, 1677079035, 538986301, 788434608], [411608491699, -48881790264, 3279824347, -2366953077, -1954121494], [-3375571358923, 38683580359, 5109780585, 1095103632, 1675661431], [-865127756191, 52346018801, 949081995, -1281933200, 1683770355], [-3018752602489, 46745461111, -360675231, -2800736688, -2099933045], [-3718750199150, 38604214876, 2886836616, 2272836068, 1025048370], [-848734357449, -5771238317, -598211585, -751608744, 3718180398], [734423399358, 31327741987, -2126154939, -3676089843, -3324244298], [-3527869777300, -7212416570, 1080295122, 706067400, 2781694282], [-5047319875110, -17781995760, 2079087834, -933864720, -3145333101], [4551409034017, 6112523074, -3103791582, 344299747, 488567099], [1821117235943, -36052302077, -1116131369, -1281496552, 1718527010], [-2411774767359, -13506731033, -3416314989, 4140290236, 179396849], [2406225973067, -20223672854, -4425844464, 2114425989, -3555167357], [-8027234187062, -76594697852, -3548279316, 2301461416, -1271768968], [8268178932018, -33475291587, 2369491799, -408240281, -894143494], [7991200292083, -16018954585, 1641300159, 2426475608, -849414692], [8649247624472, 113101018810, -1564548226, -1733282336, -1498425594], [10777509457625, -136724446143, -3452025201, 135041218, -943275857], [1523698362874, 59569272793, -6450410558, -1158687833, -1641827357], [9631886011103, 43516543291, 3160263747, -3463640832, -2214359864], [3109097424867, 111305740115, 1630430151, -5392584988, -6676113608], [-6370212203322, 10007388104, 777501231, 476955174, -4348922137], [588521408369, -29598314351, -5583679337, -3505265986, -4287088045], [8831673493578, -100488837784, 6319371153, -2622308230, -1317302927], [1823100434657, -117240110251, -1345713735, -3590930942, 1704843069], [-599386032800, -121341040290, -3600289086, -1601676074, -8173678208], [9092954957082, 30814334180, 3400419840, 807160004, 1935244462], [-852111649930, 79065888628, 6892021668, 4692400128, 9474121432], [-9941699223409, -86379297277, -7346197381, -1761960064, -1192792408], [-10804932538656, 64187863053, -9887811315, 9122048601, 1673892018], [4894567913067, -174379091789, 2194808563, 8937221868, 7490997582], [13684695557998, -13541798139, -10249939116, -406244313, -661365735], [-8225814289440, -112004231020, 10688587131, -9297960916, -5847356387], [9308784687194, -22310065900, -10184466108, 10225422676, -6037042200], [-6364477786222, 61884457339, -6494322244, -4515436487, -1409629665], [13037655155370, 152380500480, -4280835690, -2826663462, -6494663376], [11447086855335, 61857007995, 5185150149, -1077648704, -5501207213], [-17357231599453, 38746830695, -4960296447, 1087484276, 8691426211], [20875634843720, -43450752546, 11958808570, 9044964640, 7711570290], [32058387953821, -131249479918, 5348958096, -491896425, -6832907239], [6931265815542, 197905742012, -7521003114, 8305328396, -1223374751], [22765034986029, -84326021833, -611239079, 7978734832, 3724503767], [9528190334950, -52733096824, -6726468228, 843079484, -1501422026], [16445629147898, -6062540170, 18022247208, -1595172416, 6653328840], [8507368168050, -86017838776, 10136943698, -18277059948, -4683434301], [-29473830464279, 75671685784, 11640617916, 1807439809, -2415115117], [22681133599180, 42615858106, -2956397410, -10299245172, -5195677966], [6321230448236, 74808782784, 11086268436, -3961426860, -394243344], [-27966270600091, -160952438775, -23721243021, 13501351680, -2491680921], [-1176831344314, 7997058337, 6034351179, -8881514937, -20367372122], [-32415068548069, -53256673247, 6139837447, -209288572, 10775397461], [1374062899913, -57072284911, -21879956793, -5590844880, -6498311863], [-24402519234169, -59738601261, -4255768283, -5434887548, -19065212217], [-46050353263212, 53778179599, 12164019937, 11237765211, 7357710060], [10412022295535, 34419370359, 24280406897, 5858639616, -3252400565], [-5634758105251, 86863799353, -30891904641, -5068211728, -8498873835], [-21677334977244, -155236315373, -16259933109, 1202292931, 1240134074], [-31033181348344, -102090420926, -11920036782, -1557565640, -4889063992], [-15738860759212, -108247756649, -10669308651, -8901463869, -13741242608], [32768126365194, 6820407280, 26531981814, 4526730288, 13720769773], [-62082416580911, 13717571514, 20937454050, -1892215829, 8728882843], [-13894708345185, 16658295577, -36375287058, -4702014460, -10981548498], [4186581808750, 118268627428, 27590204116, -4161810728, -6081920648], [6956187755180, 82983175510, 3991339321, 23955460754, 25235242433], [11242188055216, -258422828710, 7334454054, -27148321128, -19908547510], [-37244641857785, 212460754663, -12399227547, 14877249048, 9435121297], [-27749596249620, -109864836154, -2903274306, -7801198064, -21703856688], [3758147137660, -232158567154, 17143693294, 9457975328, 6207000108], [49425907431197, 99136760445, 6834076849, 12404951230, -12987795831], [8915080978905, 234995587769, -30752399151, 13810913288, 25696097608], [11350152258244, 457358089006, -4165167162, 8227723330, 13530652808], [-21509799453484, -141618170674, 22324387332, 27503440676, 15632166055], [-30751555279008, 116647789506, -23546129052, -2493622720, -38360253277], [27607600865675, 171589834954, 30341607454, -18510232963, -12714625195], [23946203330153, 578500066577, -9330580473, -18377243384, -22539670035], [58152199600883, -76276746760, -10550552955, 12994084571, 19511046808], [1686466873347, -40056811577, 21233402913, 18149417554, 2307538925], [6411255699245, 148302070525, 24982047147, -22896996660, -9392712263], [-33053766600987, 138136722861, -12909453102, -5454451794, -5670827424], [-42945036846364, 464540965774, -22624497606, -9380304824, -3965608318], [38788852419207, 300886958651, 28419797733, -30239815744, -25100103329], [-14734165548768, -253276359630, 8168895258, -32742073750, 16901215508], [94866752258850, 332793875039, 32991060655, 10207091965, 32654552540], [48047328057068, -142374261777, -35421072030, 4514976647, 25251594371], [6688200519796, 224428917006, 36340847406, -11616193776, -115314672], [-5763156032675, 463176773361, -38134020117, -36735407038, -54775818781], [46796266247632, 41515139834, 37227859708, -8110375540, 21264031023], [21846508466218, 707979282999, -20268028295, -33394141527, -5156294026], [82210785396768, -50062727302, 44760427320, 13478336396, -1574033237], [88295862134030, -480883461832, -13113570540, 3440063352, 48244205786], [43880448964553, -49416143776, 18561391485, -3470017239, -50070651142], [58742421930820, 660389781898, 6495589626, -9680028796, 2567312148], [108770453434507, -962153063644, -13778849871, -39666515105, -8414110536], [-1997920589236, 372875723818, -49149369975, 18826544502, -31899252659], [106364881824161, -835958558183, 21194622015, -28383669752, -16160663131], [48797608752428, 72908779046, 2999267740, -12414112476, -29539020743], [118390536459083, 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-119442440073, -130535124735, -343480562248, -142651577209], [-1131332959318776, 4419572231986, 27316512930, -20168012616, 71763076132], [-177311383400991, -3544218657911, 295188544131, 126811465504, 316138649783], [-546606558493788, -3537349310399, 58551187908, -208481295159, -162369384209], [-477034071382436, 796066163286, -34710497618, 61159542288, -282610982704], [-183840806165886, -800564435888, -205560145392, 214729224624, 218489546080], [311953811210771, 2187733781950, 167420209092, 42479061337, 78114611603], [98915008250276, -5386489845101, -120226369299, 100490579427, -155550212288], [-247097926110538, -3273484382534, -311691712706, 65141345240, -243052626526], [761567110179502, 3351098358452, -399215869376, 132566093840, -152155842246], [-525556109774624, -7001498603226, -44480354310, 170004726516, 231665649138], [168082075951156, -4730406619650, 131665981750, -384146163264, -253535487112], [-180061058332237, 1578961996939, -660962055663, -16725708790, -122325482283], [366487183374144, -2744003328803, 223041555699, -376671132863, 48954706580], [-595671394852795, 2915945148786, 49591833438, -187203352073, -99703686701], [160914795305983, -6656913312844, -7231206628, -168063291701, 66507704569], [803962871349340, -3652754647158, -624901011834, -171301461364, -178031855266], [-237158275277454, -1999716066008, 167618469532, -518095799256, -482606770146], [327378014188357, 655475545437, 392599045767, 312151271044, 170943605899], [900901527859545, 4456851023959, 245392576059, 113409768156, 482482953937], [-205801632301861, -3132805355537, -695182463277, 231970436208, -371171213378], [683274165814156, -824601402058, -163802883468, 232330846992, 449007285773], [-145910804655554, -3764235914884, 389498376506, -173977542836, 508598981143], [-726779064386436, -2229599461268, 426386333538, -335796089224, -57769196154], [76335675332375, 6518675789315, 504217882675, -161506002984, 127900516828], [-1025214749895631, 1210577095065, -328373885612, 238480173238, -90354139578], [192754504788539, -4526429964989, 155296995525, -31006083736, -568497623793], [405054567089074, -3744904621509, 388125463797, 296948242929, -278288906046], [-560567474489086, -3306326600249, 398990003042, 23159773621, -267811048395], [-118925536175548, -4552936219842, 496297855386, 395853151088, 526192880690], [-572354656153066, 5794385424081, -19067242095, -100819974933, -271631656164], [-293467562942741, -2516018159973, 354622935573, 433885340520, 28756415103], [-1201250348105420, -1541900066026, 229338053400, 249817632348, 250568290289], [388182820804132, -3105019047104, -124672937160, -337926171884, -384333145732], [-66399677506928, 2184718075358, 261773255346, -431798316648, 179824323602], [-802359845923664, 6650150970970, 170738451006, 289303864916, 368851490874], [201983330243028, 7679656679185, 137144670300, -503054365991, -212719981993], [-1202814549849273, 8623129398680, -330833905998, 120360461663, 161790435391], [-84762181774102, 1066505296672, 321688429410, 377708912892, 475613687071], [-248820766829192, 2092911047091, -72535744194, 557538667207, 201588519967], [1050573881593270, 746588139168, 516411851688, -196159234044, 211185383976], [-2170726980333736, -2542269979926, -262667258730, -74313499948, -333272134234], [1259989136705790, -3015043922672, -355147724050, 47993307230, -20014565900], [426401372138935, -3672467966128, 613547406236, 78171250639, 831986187013], [221103099255769, -640597808509, 673746689685, 89080962668, 136410741439], [-127095612219893, 9508834230475, -541544940849, 568651627288, 160146129500], [-949346258344870, -4487348559504, -685263684030, 98963938436, -511261553587], [1417089771130276, -10602611105113, 151864793901, 146599076731, -130436835216], [-947344361920166, -813414608528, -608524844850, 166864859888, 347572678689], [-860943745057807, -2630079739565, -639111258132, 344193574440, -622851642932], [811297525493399, 11462473808039, -338777117448, -263898474238, -29936027882], [752435972899119, -2001130443358, 315930072305, 65995853241, 873668326278], [317322063376806, -6245560911224, -76709843952, -219416110472, 344155584392], [42671608917361, -6137718858007, -591523978737, -446159843274, -398549456989], [-760052598437150, 14244891177564, 489499197276, -7409855552, 283327264540], [-721425161138198, -4317961132087, 335467811109, -182483445945, -395327655556], [-667633554309348, -413445271316, 89837170074, 294479110192, 825370816550], [242120759096387, -13782183058821, -446006445399, 27904663968, -85195454055], [-135062136927568, -12725714840038, -625272310154, -5242919664, -591523197566], [848707497263434, 6667235947424, -6772764192, -159604682764, -261408239204], [-1197968693085066, 1156285690764, -356122091100, 632580837864, 293224920252], [228002538027751, -307017051457, -326520954287, -399372052192, 206058798987], [1093656213513014, -8094292411155, -230118239253, 19954793051, -368040918910], [1079270193067951, 276515526047, 296293490609, -205483279688, 691439175175], [-54358053343065, 7445487995890, -60769591074, -1066357950535, -814670692599], [-84319987526274, 15939125619395, -720178997856, 401049032237, -367146997025], [909746421726046, 10788847261077, 324341453910, 130089234439, 188809026247], [1940813005795197, -7416726648227, -919971164601, -582519450976, -494858504949], [-1467205907919761, 2350064957516, -387685717926, -101557304005, 180925381099], [1025699022194828, 1851415130914, 964263833618, 685122317422, 1409521710384], [556102485772602, 6304886889056, -550595796864, -599348483200, 21939865456], [-162502664665032, 3218682757546, -434218712896, 729023007524, 1091823871951], [850838078090879, 1382392798746, -810941250648, 44496795785, -307947845113], [-227627229867176, 16075013589132, 127891150173, 293166876156, 814303172091], [973002119697588, 6810566187477, -401916285558, -480725089923, -586291681107], [1184815264408785, -1077502021915, 1146363408981, 1024067110036, 465958895454], [2321526572099565, 4040326178402, 491934801351, -61536819769, 569703432514], [-49428129735216, -16936030617774, 657568001596, -19401883600, -431727983129], [-2085253057851481, -3036220250837, 1334604837045, 402332431122, 588321366841], [-1453734044980524, 8387121857070, 681707770008, 319485070284, 846409966005], [-2156144977179190, 8735174237920, 304865587998, -350987258812, 538953555087], [2418896476772634, 15831959132960, -129024223928, -77544149924, 210735243920], [423524416475460, -4972022100047, 1435293786852, 144147136697, 503062350423], [-1017519813229358, -12105905105616, -92592421348, 235497997712, 401908424058], [-1051025123313217, -6823763031477, -736173939045, -413806306864, 963833373668], [2804757350030006, 9519895926052, -1388280407886, 659450491298, -511254507252], [909004451472287, 5408069067163, 1966723723179, -571492657912, -508878450552], [537749096148010, 4380252182820, 1871916621072, 315986651548, 882587563906], [-1586253639544828, 9754147619345, 1475672773449, 960788126513, 454990943998], [953804000825189, 10187138263040, -335560865904, -1183254539171, -873976278597], [438222868509287, 1756097685775, -1822320937485, -92306075884, -903835423392], [-508032785243753, 15729184909699, 859975617285, 105069271160, -612841297365], [-406509145074718, 12882044047822, 506641777985, -613948762640, 868125934875], [-1494463693234405, -4332939098433, -42821829990, 115592215326, -87296182632], [292695205686070, 13319263639020, -761815662222, 1269578781524, 362836136645], [3894853741638, 12791399748696, 33811028748, -1274669681592, -805003735638], [1157450977873903, 2813524583392, -1035524638563, 253585918031, 528477665340], [-624097289539292, -4531681969990, -1246124206026, -952828520522, -306677950932], [-1980369261872546, -13512973293584, -476859865872, -972153865420, -675152123412], [-1959080955038808, 4923506911956, -1486155428946, -503288247780, 155410601334], [1202246653603962, 14614206820596, 2178226838608, -853768692592, 619650288142], [1682974011169136, -13490120007590, -1690887876642, -1164566104192, -661546044258], [220356090830194, -2941209309317, -1259834391447, -1469238025219, -1453481644974], [-730481877183224, 8341820346772, 812161130103, 235344159668, -391190713143], [692726025310292, -3182529713562, 2410792147062, 420841998696, 649459025220], [-428387364116778, 19572627558172, 130874008076, 96339376832, -1144050762128], [-1903253911221327, -3664656952159, 2201415627404, -785461792770, 82063176294], [-1309850774895178, 15866405638872, -779738280516, 708599270972, 1089186988550], [1247732714389740, -17619842364019, 108435358131, -1733197184619, -902849366296], [260808951452543, -19640870857457, 804706160059, -249537836950, -91509357061]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_208_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_208_10_a_i();".
// 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_208_10_a_i(:prec:=5)
    chi := MakeCharacter_208_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 10));
    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_208_10_a_i();".
// 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_208_10_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_208_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,10,sign)));
    Vf := Kernel([<3,R![-15365376000, -27656640, 6185619, -36801, -147, 1]>],Snew);
    return Vf;
end function;