
// Make newform 354.8.a.e in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_354_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_354_8_a_e();".
// 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_354_8_a_e();".
// 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 := [15300107670464479800, 237398587398357570, -12153827093891319, -191825159847522, 808091684078, 16696953355, -2627755, -260588, -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], [-2027834810736579130218015207089154601444816851262259400, 110074072158696633046291681610817452915364371270148690, 1077202071723610663643339322640899163240208989312515, -14198952877994518486481308166268388028461601892661, -75920706305805909138164910976987871519071561109, 212610580222705259305153124089274632924641212, 1024232305562256697349930174921364047203213, -808927757691660685532640411115107139705, -3668939840838464900208001479756826739], [-82923706136157192792692017368131668590520608308631352200, -382066949961677418104366215782574455171318015023080230, 28085418254449554328902761310538957319792572619315871, 103699244460244789558284990033847253771063309891891, -1922596629885236661623187863842062927199207654049, -3024192307844304357116007988924504830312840232, 30481371877807375743089022476056177451694229, 13595699219220410996763709815180968169899, -118684004847550059995542261743618877787], [-5866505261182097230307101835202254223855567679656393000, -154186562029489947330374028704137958411651456534163870, 4091864013234158861313598244750788959450083475778731, 28796906905906049775833620338336523148969058584911, -444607696400972519555722297814460838206382362229, -825141565071695543897330770845570883887518312, 7719432430908895424475638453797638120292889, 3711933721783212859865161893471312142279, -30918435317443156228807597915841337847], [18341088375876563304019673790659987067685186088026138200, -356910380139749914086961920172649359385781254132019870, 40449564277301131082425014715196369287298989265108139, 167544239394045192027393523996789645313102088060319, -4663310076628940346921864556576657455273674847541, -6306084734750921318874707963766803202932692088, 79954026153943303988433609082440025706807561, 29524247885641074757696978878953649672391, -318346690634034368473388623817452193383], [170536040550175901532968458378464126389486546422809054600, -519087784007291069767282093447308206760109342682999610, -156688934826991019487174603181202478108491480983533903, -372984438916463045266074516946704333801386818326083, 15251630441157334077543879746241556671884163610577, 16887482994280777627315395389364439379555021416, -252880741903334146927288918493442503118261637, -80487732975776562781914311641126142686907, 994989197480974901132417176878043862411], [77006632551331802382072715087977635057390902132337836200, -73023582675783362477543204436452132177915836802025570, -52291904204423494126970708285245358169326839638416971, -176376640121537010890823039863823065662417633842191, 5157702614710903890188443308645317525834042363349, 6850882490986822323610434493952680305370034632, -86828181697893048469423336655249951235850329, -32199428189865214136469283558127322037799, 343883527816357216173114211852959877687], [38391130682115345448181474326295742567362258757557483400, -58998475006516095505055677529803116478719268419862890, -31193007741511147854424399553870945417110675241829687, -79542398652486879592363498787696402203982378745887, 3121813203000535098371355426778026920870559330113, 3612837746963555301204000567795550672694760724, -52240207679110604791748286552129880570658233, -17291292699056025928059149170821621093803, 206267896829254349511224388063445300199], [-9484897774275819470615195419733617188435922954531066200, 3082733564840701057422315887334955185869278943556870, 5785830220549247645774640772969558173804580724359817, 10719879155884022086517718381188220339174378456997, -522183859029337187069718019940855438842372426543, -531815687794417709774989855680608938127892944, 8534473203765996872973613103837858546008403, 2564671763761851894228729142951429296133, -33420522753055636948940132034854277909]];
    Rf_basisdens := [1, 2531334875083578265657622457138246449156087356442400, 50626697501671565313152449142764928983121747128848000, 10125339500334313062630489828552985796624349425769600, 50626697501671565313152449142764928983121747128848000, 50626697501671565313152449142764928983121747128848000, 16875565833890521771050816380921642994373915709616000, 6328337187708945664144056142845616122890218391106000, 843778291694526088552540819046082149718695785480800];
    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_354_a();"
function MakeCharacter_354_a()
    N := 354;
    order := 1;
    char_gens := [119, 61];
    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_354_a_Hecke(Kf)
    return MakeCharacter_354_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 := [[-8, 0, 0, 0, 0, 0, 0, 0, 0], [-27, 0, 0, 0, 0, 0, 0, 0, 0], [-26, 0, 0, 1, 0, 0, 0, 0, 0], [-37, 0, 1, -1, 0, 0, 0, 0, 0], [-608, 2, 0, 1, -1, 0, 0, -2, 0], [346, -1, 4, 9, 2, 0, 0, 2, -2], [-416, -3, -10, 15, 1, -2, -6, 4, 7], [62, -14, 4, 33, -1, 6, 14, 1, 1], [-10265, 4, -37, 40, -41, -8, -8, 17, 4], [-4521, -36, -55, -91, 15, -8, -14, -17, -40], [-35981, 107, -14, -223, 8, 37, -29, 47, -2], [37877, -189, 32, -458, -15, -71, -7, -40, -10], [-11555, 42, -24, -337, 66, -93, 36, 113, 119], [111169, -28, -161, -394, -293, -58, 158, 106, -67], [-40638, 111, -121, 54, 32, 181, 54, 30, 116], [224690, -278, 288, -577, 834, 80, 132, -318, 4], [205379, 0, 0, 0, 0, 0, 0, 0, 0], [566576, 377, 210, -410, -34, -560, 293, 455, -4], [1215582, 203, -226, -3575, 561, 158, -984, -537, -74], [397941, -1143, 344, 6206, -860, -363, 264, 476, -524], [1439021, 2004, -1966, -1679, 387, 719, -1071, 370, 240], [831143, -1276, 788, 2594, 217, -1107, 1610, -1509, -692], [303351, 1424, -650, 3687, -1584, -59, 892, 1763, 239], [-1110551, 983, -828, 9685, -2198, -775, -2540, 2852, 1773], [4242613, -1476, -3475, -6046, -714, 1636, 3625, -4689, -1714], [203612, -652, -3085, -1648, 167, 1407, -7213, 5438, 1424], [1958625, 2487, -6, -12857, -4359, -131, 5938, -3121, -426], [-69999, -1169, -2984, -21382, -1656, 813, -967, 7403, 2326], [-2369613, -155, 1157, -1446, 826, 810, -5844, 7924, 4262], [-1158358, 3758, -2385, -3985, 6348, -1045, 2393, -1927, -1874], [-6032043, -10718, 3879, 516, -3584, 1212, -53, -8127, -3932], [-17092940, 1664, 10611, -15168, -1810, -2547, 2720, -304, 3659], [-10132708, -4014, 15619, -13973, 18239, 1679, 5684, -22364, -8377], [-8255423, -5574, 16030, -40190, 19874, 385, -1, -19465, 2055], [-13965743, -3845, -5809, -40534, -15357, 6544, 1956, 4687, 5488], [-15168927, 17843, -4834, 1975, -6088, -3165, -605, -10345, -7828], [-5928726, -18023, -25967, 75459, -5675, 1047, -315, 20186, 6950], [-11990681, 20238, -14471, -25921, 32865, -3228, -7289, -15646, 843], [-33306092, 14318, -3367, 3192, 5875, 7159, -7356, -25333, 3872], [-23541123, 16537, -6382, 51535, -5121, 5363, -13831, -11316, -4464], [-49552138, 20434, 27067, 9727, 23459, -1193, 34666, -23348, 13665], [-24913692, 19636, -10606, 118580, 29805, -3878, 8527, 21862, -7698], [-52007717, -62779, 48268, 51734, -325, -7927, -8250, 19245, -24814], [-20799075, -15953, -25063, 61823, 8080, 13412, -5100, -47583, -1659], [-18142025, -21607, -36274, -96389, 31749, 5449, -3634, -32303, -18458], [-45339541, 46664, -36142, 107875, 15933, -4633, -28165, 20366, -6658], [-21440333, 70090, -31605, 84560, -116274, 7142, -12653, 75516, 8593], [-21166776, 51336, 70333, -90922, -48771, -43025, 22303, 19086, -2192], [-41775070, -29693, 101276, -343727, 52509, 4568, 76206, -10534, -25333], [25204518, -37839, -9965, -215892, -72955, 4809, -14744, -35151, -14432], [-75140012, 30770, -42721, -136124, -138602, -32045, 43080, 55978, 12075], [-12348445, -74541, -38203, -582522, 8447, 40226, -14981, -14234, 10760], [-14076415, 138091, 62786, -197775, 184542, 60047, -39936, -86873, 37252], [-58549494, -1968, -44337, -503743, -14778, 76813, -12361, 44925, 70500], [-42199271, 28235, -31891, -191857, -56778, -32644, -85930, 197573, 105443], [-50922708, 7926, 37190, -138868, -71968, 42190, 37105, -88475, -63406], [-101952746, 54952, -101666, -88427, -39623, -42284, -22924, -28659, -35912], [-43954077, 31346, 192630, 113898, 29941, 78027, 7714, 39959, -59764], [61006228, 17537, -60366, 59873, -113285, -87134, -110126, 188919, 149532], [-79297407, -144935, -62463, 179843, -31968, 31696, 82592, -206449, -59233], [-21433487, 121432, -70703, -426174, 92544, 123962, -78077, -54668, 29005], [15935982, -108763, -14973, -369871, 194458, -14889, -13606, -206122, -17996], [50544694, -100916, -42915, 751000, -250798, 108449, 102798, -89048, -38147], [-194990604, 89494, 86663, -706205, 176228, -19881, 71638, 49442, 41156], [122750398, 158852, -105159, 319209, -154993, 77451, -167476, -132327, -43204], [-21816692, 195140, -390866, 245885, -293264, 33660, -197438, -1450, -73236], [-45628830, -158960, -147763, 74874, -158152, -5927, 166202, -16116, -214793], [-3595108, -218530, 158027, -423897, 167145, 47575, -79516, 45121, 279382], [-15169582, -299881, -66050, -708200, 54716, -247526, -122260, -217464, -20781], [171209982, 26081, 223938, 617177, -50414, -100990, 463988, -309422, -104078], [-18869139, 178959, 288322, 614026, -309494, -40883, 221947, 60441, 60608], [28502827, -218949, 153355, 276579, 196236, 158158, -55165, 302507, -138442], [178278741, 526092, 12494, 190084, -154512, 118283, -105374, 180806, 146878], [484878748, -278273, 402808, -1105163, 93356, -37246, 232417, -51831, -183298], [505050131, -268382, -21089, -1808142, 36418, -217010, 181371, -264243, 47340], [278144063, -718655, 316154, 969348, 288036, -56843, 83322, 317782, 119194], [99805053, 141301, 1155486, -1367795, 445105, 134599, 389844, -634893, -7158], [450498774, 664669, 335093, -942871, 654769, 107775, -124815, 114662, 220442], [-78264229, 465713, 153587, -800548, 625036, 147766, -45251, -398903, -389351], [231169168, 39882, 133000, -2377282, -320886, 98694, -346940, -282880, -91340], [-596496333, -181533, -305469, 1631519, -1378086, 47458, -96126, -42189, -167571], [663528826, -631181, -186326, -2285811, -170068, -94070, -224996, -316, -49598], [-505555180, 353211, -1172975, 1478488, -426324, -148943, 31374, 143030, -21864], [-2527701, 792348, -245446, 2597824, 47670, -100179, 469987, -129854, 311506], [-179256534, -246958, 714704, 362601, 497144, 92258, 282319, -234263, 419626], [-671989092, 847899, -761792, -1003905, 181919, -177678, -313692, -328492, -362371], [-548063744, 47188, -1117519, -1629685, -464439, 368997, 210002, 505886, 362241], [-77452133, 130044, 1042548, 580857, -522197, 295847, -86499, 650412, -308806], [19819820, 637719, -405475, -1126661, 533432, 199051, 231906, -115370, -35498], [-121100025, 311350, -827906, 3074476, -406727, -409397, 187431, -654198, -643878], [-398610047, 331251, -1156122, -1430691, 599390, 502793, -664132, -435685, 235312], [-889704122, 54638, -7664, -3658723, 917379, -234668, -740859, -380768, -29412], [-24799106, 541462, -1391138, -105793, -253578, 620472, 202391, 334075, -99026], [-1928732801, 178521, 1450489, -2601489, 367742, 320816, 81924, 912250, 245622], [-418045881, 353790, 320439, -5347584, 905628, -12874, 1307859, -592573, 333764], [-985068454, -589171, 1429717, 2345314, 1452126, 356459, 727268, -161794, -251474], [-1346220005, -859149, 1318729, -3343451, 622360, -574314, 503178, -584512, -404710], [-1357939400, -931299, -591309, 5542780, -564002, -576307, 1478434, -377438, 846276], [-468689757, 496564, -2835799, 2497478, -367182, 552576, -526633, 1185707, 349326], [-435924888, -1508879, -561744, 3180341, 322380, 104246, -370456, -811564, 789746], [76450316, 65754, -412083, 2528309, 864703, -803245, 1260590, -1913071, -629262], [868125975, 439363, -1362194, -4637981, 2224429, -206279, -1104684, 1111607, -308458], [-673011469, -718699, -1797841, -1226494, 2207745, -282794, -829948, -262133, 1089203], [-1260755375, 1379369, -2926397, 5604671, -1645072, 719022, -1292604, 1335098, -37584], [295685768, -1237947, -1845986, 544119, 638345, -373936, 449638, 361341, -419822], [253417729, -1469056, 259664, 2468088, 50431, -788145, 283162, -1179914, 71212], [501280396, 674666, -143074, 6017255, 625051, 289890, 544614, -992450, -1749842], [-470773304, 1395119, -1206081, 2821712, 440148, 27117, -201574, -1245060, 1364516], [-2445512375, 443743, 3760721, -948339, 659466, 863662, -1082381, 445905, 955966], [52178694, 1374330, -1933899, -20255, 3193441, 418055, -617832, -120761, -1017098], [-244740627, -1989582, -1236096, 2272644, -2456067, -1674433, -1810200, 1687409, 602636], [-290972691, 2584345, -2369491, -3433651, 55000, -910348, 763937, 1303479, -82452], [315859256, -266919, -2346332, -8240331, -1413423, 177656, -692870, 2549942, 1415511], [-1867818293, -1349868, 55553, 1333270, -679920, 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-1157116, -17750378, -130611997, 17085048], [-159589381280, 7899530, 27831953, 158538179, -140112889, -13398897, 16023472, -41558222, -76309155], [86754248102, -96776977, 15845026, 689391875, -128011405, 29249732, -42810, -49975045, 18309360], [-191167149042, 80993458, -141947648, 1037852773, -18030607, -54977670, -46262834, 42750555, 7815090], [55664300697, 40651970, -190938825, -113595691, 26757814, 4185984, -125225406, -13777512, 73878584], [-152782203670, 197966148, -162208210, 148512843, 161765400, -23293662, -27842068, -57525660, 9852152], [-47655946363, -78446688, -83434531, -347414857, -118796610, -18930440, -14047910, -78945429, -84313070], [-27200765078, -80176657, -98902352, 202349513, 6044777, 10852776, 73060512, -19459305, 48875872], [101175284832, -217934942, -26573350, 304514476, -93948395, 26284546, 83066479, 38019284, 18986528], [56353030230, -111490990, 240450586, -147975899, -42885909, 5563404, 125146042, 17941817, -914184], [31415578167, -124937020, 156209950, 161665660, 51643954, -43824035, 153168254, 18119996, 18751100], [-29104308297, 32660237, 116516205, -31304461, 27938766, -19448416, 56452102, -247206327, -118075449], [205368691112, 51298480, 1379979, 491347106, -273611828, 630949, -23491180, 148674536, -53296503], [19540128661, -13537536, -226297097, 121131196, -62042086, -76594750, 28109039, -131551157, -18825842], [-88988098600, -73196568, 237120101, -361930825, 50955598, 20945405, -47530915, -46057573, -24962814], [105140110746, 83343701, 72993585, -39784000, -163565693, -55627273, -117115666, 199532197, 53736960], [-110688784564, -218623703, -109469141, -420473520, -85648778, -42411173, -45887590, 22223900, -41649526], [72941147843, 288267265, -229557638, 330264319, -76406761, -50606719, -49459520, -127981691, 13075860], [-45323687223, 80833899, -57125936, -240475960, -67742802, 64752609, 70695426, -82040832, -62248594], [-97390939544, -77926383, -6997709, 817095113, 106086272, 80879781, 40320250, -170771126, -91618642], [-124820931640, -295945832, -4007009, -72104303, -149552559, -38175909, 163511960, 56386100, -81729555], [170177380241, -142697589, 41111721, -81037705, 174416423, -69082104, 58647455, -282510695, -65271685], [-140353306980, 201532638, 117169657, -663241657, -2766750, -7430091, -80575532, 78395790, -24650520], [-89358787443, -33944752, 41087055, -380965066, 123050550, -16382414, -101345865, -104542441, 86161006], [-113261943622, 106122609, 40026353, 218063876, 141078348, 22102055, 91593446, -5370928, 9387832], [-174897668569, -141390319, 258668452, 97755846, 85809674, 32432031, 10971550, 24205292, -26272656], [-88782610112, 106365174, 20996389, 464895814, -28010680, 1948445, -133348592, 191627746, 119047021], [-56598582306, 77939288, -38793875, -33972335, 203571761, 45107915, 71432900, -288836274, -16809191], [-524621936, -55288220, 213368636, 620682738, 81480194, -16727124, -61683700, -139386514, 8801464], [213039497738, -69315574, -129921218, -674524776, -27234620, -33090170, -174246966, 19002992, 36158498], [-97291858891, 81692284, -57442903, 764192979, 232049623, -52645450, 19240902, 14061381, -22153670], [31104434770, -55437870, -185332060, 386894764, -168727498, -3708370, -104825262, 245647800, -4399018], [93062733707, -233595567, -106025774, -146947200, -1459931, -39473235, -57823800, 170944465, -80669936], [-173342327516, 56043724, 17239134, -1241946106, -65389302, -51628280, 69563460, 51555946, -6356170], [168575805445, -15003362, -122729875, -1147971398, -81253533, 67685144, -130425138, 77000780, 42057875], [42664880146, 64762202, 189947634, 718466740, 156144522, 69832052, 5692712, -184133922, 10922084], [-58317041448, 104275644, 114574228, -458954088, 260567572, 16972960, -142523270, -82744922, -116364394], [93862628949, -198666701, 152301339, -1121263843, 171257134, -6560014, 30771748, -208564424, -9798216], [83556538089, -198541887, 118227786, 724358294, -177177427, -60360201, 76532861, 68640332, -75373118], [-328663951534, -60288480, 4263880, 55660090, 123410318, 50606484, -186767620, 79534236, 135168610], [-121986545653, 6289867, 254075790, 311492525, -100411935, 73125583, -24682870, 32310623, 91704420], [130518381328, 129850898, 241260087, -500313027, -36805254, 45173147, -17057699, 181756743, 105789244], [290057779362, 216304904, -277762741, 972272937, -275467169, 64457191, -45126796, 189260195, 91876526], [-108067067069, 5678859, 209161379, 287365291, -178219596, -130126370, -157330666, 236302134, 16186292], [20586617802, -29221030, 133761674, -1540087726, -110594075, 117654488, -138060515, 90541682, 86411212], [-16686101494, 398033692, -254061617, -509063354, -82186122, -7721379, 11790628, 56967590, -4640387], [-167455188585, 977040, -7561860, 182631545, 204884943, 72216703, -74376777, -32844422, -54316960], [-84546434283, -301624230, 240284281, 981025212, -43024842, -166024218, 296276465, -257928831, -98918562], [129122852760, -43819508, 76381860, -2333287726, 112530452, 183714538, -93909124, 39256044, 66418118], [-175790289968, 35064304, 171371930, -696330165, -61284070, -103050926, 214342193, 20818653, -132295749], [-102602803026, -185140048, -95676649, 927407342, 196841971, -52234203, -51703421, -121988430, -34105700], [23610920807, -116590021, 265370832, -56551559, 364824695, 69534945, 12543408, -247473453, -71480732], [260476277554, 123692312, -473682627, -390881510, 231600651, 22714147, -230269701, 242411382, 247259450], [93133937740, 245667578, -316648453, 996636513, 51182217, -39327223, -80327696, -192484904, 21142561], [-261411840706, -368115, 201505854, -204042559, 120041692, -76590438, 55619156, -167706344, 9241730], [-89667536064, 163454296, -434987410, 1394094856, -54097358, -48584882, -145657318, 137372996, 167350060], [-23312006919, -292840633, 216221838, -377024780, 201767661, 71610065, -161974428, -99173765, -30547278], [-118797412836, 67507636, 243861753, 949137055, -24552182, 11392123, 36772139, -219846045, -132655590], [225887340940, 161187381, -496784170, 1405330187, -172430982, -70337222, 48276271, 190679691, 150819572], [-570498062761, -256932587, -171973267, 495549165, -65398777, 6566880, -64149109, 225514581, -63108117], [277095130481, 392486512, -114346749, 148192253, 26326669, 103162120, -22320920, 144557761, 59719638], [360732863149, -463663033, 476392174, 666822639, -69605224, 30207659, 117842401, -374631369, -84690400], [84037903068, 72328066, 3565142, 575894654, -203191337, -10788718, 176095317, -182376678, 67630454], [-130588453830, 395248310, -606033272, 506336290, -107528332, 130592220, -240234154, 224500100, 177385274], [-349892597476, 33242694, 561641164, -1082601318, -128353012, 59517348, 91118676, 23919682, -5621640], [271536423809, 75627073, -237761465, -382873445, -105140202, -159122360, 287297386, -255444590, -33261924], [2414617123, -248640606, 345533057, -506211338, 245751991, -13977764, -82030836, -43759035, 81649054], [52271099199, -78946959, 111937136, 392199168, 320455202, -44985441, 308264008, -125183918, -56245578], [-206962288720, 110396117, -255421542, -1343910819, -66963280, 80821768, 176074610, 97589922, 6861354], [-163350026352, -210898869, -1890092, -234199413, 65821003, 72651004, 23268658, 212290138, 10443517], [-97379403839, -113588127, -173794087, 503080257, -320069992, -9901162, 54899347, 61945479, -11822692], [20352407850, 252642216, -344996449, 706372913, -2606669, 137439887, -199318360, 125219862, 197768013], [-144545757511, 167900998, -465912713, 1469281399, -5941011, -18538416, -92721947, 17293224, -150023523], [-96917891793, 150503920, -86850014, 845518679, 405206331, -904079, 97500673, -27086668, -109213454], [-152564126262, -279157992, 265594131, 1691963416, -72291901, -117482879, 260752091, -44330210, -107083950], [187390844266, -165539934, 29984283, 961474000, -303247037, 61401531, 117134299, 146445892, 117467500], [-257061979690, -291799590, 357984100, 92354236, 20864918, -185781084, 335959820, 77306278, -170463096], [259460474763, -35444305, 8823319, 632326123, 183645174, -99623178, -56181012, 151488758, -33108956], [-386780878165, -51895698, 456767345, 899063354, 457115405, 113278282, 84061478, -380910734, -155006111], [522061913065, 55310321, 203068298, 575912462, 210554217, 5125537, -65274296, 204420231, 148624756], [22082090526, 49625661, -233598877, 627836037, -256826698, 138809807, -55776578, -149289178, 118015972], [518532091107, 46887027, -99186460, -202286439, 128041969, 118590837, -123995570, -15866005, 63855956], [11296904802, -298978840, -98673020, 653491398, -257076383, -45862954, 208078513, 16520192, 33511580], [304054962329, 401411090, 292782888, 1354457002, -335647809, -30551119, -69320780, 162551163, 82322603], [-112138271215, 157269320, 146501979, 987311292, 17255899, -47271838, 160208604, 331092709, 66831040], [242938386549, 580444834, -114118034, -547029649, 293357119, 168591435, -32505089, 51132330, -544446], [424994934334, 33718302, 112016187, 1005690290, 303410584, 89064985, -326416030, 252326860, 99054079], [-316779456280, -121729595, -62557647, -217926637, -327522159, 17758743, -5723572, 92925297, 76961497], [248006397967, 245507201, 676511660, -159517839, 346846931, 71014315, 277874970, -769146451, -61752116], [-101806740445, -55139065, 158967028, -1292215257, -459494304, -159822241, 297856838, 6632652, -167245583], [-283790095502, -18222786, -340692574, -2287816099, -49797482, -89055988, -116344129, -115930909, -88286488], [-604973831818, -435195338, 577717342, 145089661, 495608780, -109464988, 124936531, -108467745, 72711979], [178661755663, 215577424, 152624981, 1127921576, -496439422, -83518880, 133098737, 281309309, 161891666], [112167527133, 253132997, -707412394, 901956994, -18048453, -142930715, -296513756, -247849431, 144384600], [68537278103, -7003582, 124334300, -1196143448, 92097531, -102759343, 24488752, -566163445, 63226036], [-358352976016, -311155850, -181764066, 1410942908, -265424073, 23433386, 381344925, -107544500, 104686290], [489768131328, 551761940, -14063314, -600948407, 315404057, 184773386, -17207494, -199225581, -11381061], [291366370203, -264281407, 206243348, -968221918, 117186621, 101754337, 322418622, -279908677, -133111580], [-67014732791, 24651564, -187794363, -123926536, 290603034, -28054152, -373422795, 73096773, -204331952], [464539803413, 3632287, -345701817, 1739768910, -228577981, 176234412, 107744438, -239605211, -13492032], [323095959568, 71836789, -150613327, 799960253, -106017761, -159933503, 9428664, 142739482, -96793774], [290354575095, -231204495, -374218397, 8028851, 371462506, -23602002, -121520196, 211824247, 110306201], [406410958990, 142598511, 166429031, 1124690276, -215024634, 133090341, 395638744, -76732110, -145820552], [512929245394, -62829792, -510813946, 1128628874, -29898026, -82386134, 254126570, -455518554, -52902862], [481578877301, 291152570, -118407840, 1160528643, 158441109, -54095525, 177012661, -320828940, -283728536], [246361252506, -164288748, -509944838, -1540281228, -321081840, -175156438, 152966278, -394580586, -105536524], [-95076561830, -5859934, 266579768, 1266860013, 150612445, 107411942, -191609312, 242916479, 180339769], [-178636466150, -96481348, 639012096, -849117085, -47612748, -77665844, -43036504, -141176728, -27270048], [-46306089810, -298657527, 182748490, -596374605, -332280231, 24184052, 53775828, -272443745, 65837954], [440224260466, -20387416, 465918329, -1774235658, -35244806, 47653015, -115721724, -205084288, 194413871], [342065850334, 397470484, -99914505, 3519455386, -399086969, 64095195, -47618669, 334051020, 162151472], [109238128977, -331736482, 198250754, -266645727, -62088468, 271830845, -15549924, 480107361, -154044739], [264751884044, -185255339, 137618787, -826810799, 602154080, 203857329, -72885794, 33586320, -59557892], [-224180256157, 88662846, 294252878, -1490692636, -149970954, -254544475, -426652420, 580742466, 97794178], [-731591490569, -244985853, -693515178, -13097557, -826463343, -48447307, -251556972, 57431947, 19976220], [-61392440743, -381951873, 56816583, -234834931, 489067440, -8241146, -373073376, 26535917, -340959699], [479643932257, -173719904, -115366033, -554122894, 105727921, 77957518, 229081342, -150548908, -38978185], [-619509503736, 35580762, 16195553, 233688215, -400885142, -163294819, 440246301, 124401045, -21588570], [-50401018463, 683403940, -643649508, -1078169794, 382990408, 145166757, 91988533, 85581068, 338100310], [-82701019305, -125261874, 210040809, -18977465, 242502885, -133888660, -224289356, 951391737, 165856610], [761248904265, -690053552, 618118813, -677951460, 241115111, 80072560, 349736284, -218507787, 8918560], [-89154142091, 288277300, -490153064, 4056576998, -448586700, 124170167, -265632395, -87589771, -53899689], [85587671123, -257267516, -3444354, 1446007896, 95443631, -121149145, -508963052, 845012094, 330744858], [212886542785, -167463715, -294417592, 127104467, 43219275, -16019319, -373032962, 371428087, -163489938], [169510854394, 250806850, -789990278, 439041438, 244351454, -18979700, 179226598, 341950342, 195503204], [-395993887159, 201822367, -360169872, 490738707, 226108028, -133082923, 89005673, -23416389, -117249990], [252382886674, 152883216, 380232677, 142906541, -233979514, -48277443, 218962625, 213876981, -443453228], [158879931690, 347398874, -723113040, 246885324, -67273372, -323834690, -423562411, -75076923, 70869978], [-679934886997, 134991607, 923900276, -698843548, -473339754, -72991379, 342307459, -309075433, -230051212], [-698495488012, 284324787, 261480552, -30725633, -209236031, 344241460, -31318288, 631689901, 231319846], [-561084596312, -160092482, -532534620, 1146972631, -334771569, 830654, -675260809, 647466354, 187518386], [-117430278254, 64881668, -719726654, -582770342, 871720994, 216565648, -145498544, -498898936, 107251660], [-301520780753, -517942421, -253527407, 1104801321, -796068032, -20129882, 455129532, -66705499, -370548431], [-470725830019, 395147549, 192248202, -92109605, -222647037, 393362071, -76635446, 198956521, 208283756], [888117640049, -757770702, 592305109, 1721856594, -491925259, 52965860, -56300400, -129885601, -341799252], [-246181239105, -233917702, -220134472, -1775546095, -224858120, 38569197, -28482920, 462011661, 277772987], [151815412895, 185156162, -349125614, -1409961431, -635722853, 65355175, 447229085, -635981576, 118091922], [-983483083845, 386593581, -232083231, 1740761062, 62128147, -12943314, -315280555, 655186436, 90008424]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_354_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_354_8_a_e();".
// 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_354_8_a_e(:prec:=9)
    chi := MakeCharacter_354_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_354_8_a_e();".
// 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_354_8_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_354_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<5,R![-747536929115760000000, 24050654230926750000, 18450513376662500, -1946294901908750, 1672119703500, 43153434275, -40574510, -362478, 230, 1]>],Snew);
    return Vf;
end function;