
// Make newform 387.8.a.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_387_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_387_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_387_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 := [238240894976, -60918607872, -14180732672, 3410917248, 209758592, -53409536, -1160956, 344686, 2592, -977, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-178, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-152547096099166884427779072, 20251203606917738939518208, 5816603974274884482789248, -86962422456018334411072, -52006805018854165868720, -3472321812967918657468, 67209956153990217778, 24515803722851066612, 664832435218620337, -39183746302943970, -1500599134629001], [68998149783161691615417344, 6432997678553769789887744, -2974328985066183090142336, 29643268429800372462784, 10430567452590387855440, -453344375740036975004, 242092152331931307554, -804903210724535564, -1628799915307686999, 3943244548083470, 2537638357898367], [-1859002308800284282174464, 548442611133774445781248, 91159995442397982641536, -20534728206647225735744, -1573313378247863242480, 203453192555128423636, 11771785592905208858, -735871367912450396, -38489149606439611, 868207465743318, 44082674675011], [-251762078824321604521729024, 2670286972075641915863296, 3478856203926691302541696, 658304812827375950986176, 89208792813294927465360, -9761906883295197975276, -1298843231045805189414, 43770032756855928324, 5406184685771909989, -58848026278326730, -6970668805319197], [416612947090106170508833792, -67138267300992344115170048, -24842951078069978264709248, 2350790628670385956897472, 404662938221299312555600, -25434543944531325289372, -2790356918408797448798, 103828813201334493428, 8717368078423901673, -135598692615217010, -9798468099969729], [-43714760928708310030678016, 16812743664771525034465024, 3237789793856720396293504, -688305088482367009769536, -79521382930120278299360, 7872425702454351826796, 721865558146059121354, -33154890484697645824, -2655475128449888139, 44072850657173290, 3266643518058507], [36527953719388860818871296, -4200110409497461229513472, -3035229804891562332200064, 203856762786007024154816, 63692959965362967409744, -2404703196261121554844, -500526086076007585886, 10005406636137431924, 1634408559024771305, -13025999644003570, -1847875868774081], [-516049764394654432562250752, 70841626666134288569022208, 37619968029519414788702848, -3553201609758346553394112, -754689293610768064087760, 42670881479478950332172, 6052126047627667082278, -182317709313163411828, -20678531372967524613, 242974029137269690, 24395717495229789]];
    Rf_basisdens := [1, 1, 1, 278996920407780897300480, 278996920407780897300480, 2426060177458964324352, 278996920407780897300480, 278996920407780897300480, 34874615050972612162560, 18599794693852059820032, 139498460203890448650240];
    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_387_a();"
function MakeCharacter_387_a()
    N := 387;
    order := 1;
    char_gens := [173, 46];
    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_387_a_Hecke(Kf)
    return MakeCharacter_387_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 := [[2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [68, -1, 1, 0, -1, 0, 0, 0, 0, 0, 0], [-4, 12, -2, 1, -2, -1, -1, 0, 1, -1, -2], [-94, -145, -12, -6, -5, 2, -4, 6, 6, -1, -1], [-1641, 93, 20, -12, 2, 16, -3, 5, -3, -6, 1], [5691, 333, 71, -33, 47, -21, -3, 2, -13, -15, 6], [-5047, 279, 33, -7, -46, -21, -4, 39, -4, 54, 2], [12158, 1748, 4, 87, -135, -81, 57, 34, -53, -99, 19], [27822, 1742, -154, 18, 66, 194, -211, -133, 69, 127, 51], [-19050, -33, 295, -199, -112, -31, 180, 283, -98, -84, -26], [-26532, 96, 698, -663, 576, -579, 102, -1065, -660, -486, -111], [120903, 9334, 2539, -1377, 395, 69, -594, -489, 132, 633, -141], [79507, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-48052, 11048, 611, 1209, -819, -81, 30, -259, 10, -839, -203], [195623, 30275, 77, -1194, 791, 1130, 1361, 1413, 955, 817, 405], [532654, -5798, -1593, -2214, -159, 1992, 252, -3180, 1142, 67, -2308], [-410657, 29014, 1770, -3693, -1919, 755, -724, 3455, 288, 592, 2462], [-623972, 4139, -428, -3506, 655, -2496, 3896, -5066, 1658, -3551, 1133], [977207, -20634, 1368, -5124, -1847, -3720, 1401, 5401, 2967, -3125, -450], [-431280, 130632, 3046, -6159, -5554, 365, -3109, 476, 4245, 1885, 448], [-1422246, 76915, -1928, -4229, 10952, 2359, 3949, 4546, 5513, -6520, 184], [1018726, -33503, -3843, 4578, -12288, -1760, 406, -3494, -7478, -5702, -2383], [1267095, -130482, -21278, -15147, 5923, -4871, -302, -18869, 3378, -5494, -8368], [-992439, -22346, -39602, 5411, 1776, 11903, 4606, -10783, 7790, 4774, 10525], [263761, -166165, -6537, 5874, 20003, 8290, 763, -14733, -1271, -16899, -9559], [-208260, -65942, -40555, -2987, -19914, -2327, -12237, -9392, 1347, -13522, -25847], [1864972, 123312, -43891, -18408, -35009, 9332, -3412, 21886, 11766, 31557, -1952], [4518429, 97813, 27022, -2514, 25764, 8188, -12773, -49807, -4129, 1680, -14821], [2587034, -119178, 27944, -2832, -17710, -3788, -21530, -19698, 12006, -22472, -18004], [3836732, 351903, -25472, -14171, -65897, -16123, 45674, 47883, 4426, 251, -5240], [-107835, -129972, 22398, -11271, 2305, -65285, 38455, 16036, -4863, -55441, 7719], [7065428, 252750, -93790, 53520, 26636, 39600, 1002, 16198, -26726, 13686, -17612], [-2114136, -392693, 121977, -8926, 95854, -47470, 15624, -70308, -11668, -62308, 32535], [11340107, 1216242, -145172, 65295, -122531, 34727, -60793, 70582, -18017, 62483, -369], [15085508, 153348, 125760, 44765, -69728, 10651, -11113, -10312, -50947, 54657, -40050], [-14472406, -744087, -301193, 54949, -145305, -6039, -15217, 38012, -21971, 44411, -2194], [9053090, -1496307, 74883, 22680, -5171, 83368, -3584, 77466, 37312, -8672, 52092], [10463794, -276845, 22079, -23088, 71670, -6286, 97568, 13210, -18036, 40128, 12809], [28627858, 74310, 146084, -20742, 15118, 93452, 32996, 35186, 89748, 17572, 13070], [-10047543, -1004301, -75095, -112176, -222840, -182704, 51551, 64745, 5509, 40419, 6884], [10651866, 102, -199583, 61878, 99883, 10706, 56298, 111354, 106720, -92263, 53808], [-10884060, -1659354, -246844, -5412, -205592, 652, -47108, -104320, 69864, 158406, -5860], [42346367, 52244, 195524, -257199, 38682, -69997, -46042, 59687, 68982, 125248, 4413], [13176668, 742214, 232043, -121422, 139399, 96374, 54842, 197306, -41592, 463, 56042], [21358531, 74242, -399984, 197968, 24013, -13940, -305077, -225673, -99927, 129597, -5448], [21804384, -6635040, -341984, -257717, -159876, -29711, 35590, -304835, 28706, -147472, 51741], [31686453, -7554838, -321054, 325629, -150629, 265471, -220768, 248825, 151372, 201642, -85416], [42543912, -3239949, 561179, 221493, 366313, 63999, -201681, 157878, -15959, 191255, 8098], [-24192897, -4103747, -56766, -38806, 409896, 98176, -162757, -301647, -100841, 129316, -274905], [51344984, 2511100, -503280, 180972, 136930, 216168, -59844, 416280, 85048, 265376, 341190], [5268386, -7547285, -62056, -202329, 512548, -172089, 28617, -330182, -295931, -126648, -118336], [-43332269, 188672, -261526, -434776, -394651, 87664, 276767, -148605, -76307, 77125, 217918], [13408088, -12427817, 101020, 722160, -190039, -75534, 42762, 3662, 12680, -283281, -241193], [76491930, -3892096, -449256, -431031, 299296, -339891, 15903, -608388, -391059, 23163, 107336], [-14836237, -3536350, -1062486, -865440, -486877, 184628, 12845, 329529, 383827, 169945, 292094], [182212577, -307545, 1045722, -93282, 199980, 349030, 85, 574495, -91523, 373102, -10117], [46898418, 5617580, -281633, -318031, -175658, -108273, -148379, -170510, 173489, -377570, 15507], [71021499, -5386027, 1671447, 364250, 822968, 368334, -723519, -340669, 326243, 384065, -440374], [-34511189, 6997561, -964830, -918075, 16638, -281795, 82825, -508194, -39887, -244332, 408190], [-23248564, -21226623, -382813, 13830, 83366, 131430, -407390, 664866, 97346, 459774, 328095], [73686524, 6264118, 377131, 294822, 236889, 6652, -762122, -609634, -591844, -64979, -841326], [51777188, 226513, 265767, -737964, 229292, -76058, -789130, -717044, -332794, 433376, -31395], [30316292, 451505, 3488306, -53136, 2356521, 435366, -762528, -240764, 670906, 48249, 75879], [-85298116, -8611006, -1113822, -650957, 1106586, 313295, -168419, -1884620, -224165, 633, -624688], [179174285, -13079609, 1841240, -1161624, -28128, 285848, 462305, -933299, 108805, 214184, 754539], [212904457, 24667454, -2437010, 257461, 1349761, 497723, 400374, 459939, 528138, -143272, -406872], [-225164915, 21166673, 2218898, 1644085, 290828, -527879, -319293, 799088, -460081, -462702, -625742], [-165252512, -1306217, 247347, -900780, 933313, 781908, -290166, -1128184, -409566, -636364, 306370], [-275386531, 32555568, 2898116, 476804, 2114991, 2328, 967744, 1406128, 343370, -604862, 764035], [120425871, -6646491, 583898, 1045572, 470592, 1482208, -636743, 307729, -187603, -494700, -6695], [73608146, 32757867, -622387, -1020861, 810814, -908003, 957784, -948895, 591242, -1428536, -155150], [78310866, -27789778, 4660143, -1500945, 2926093, -1428257, -25846, -1533133, -107678, 41649, -1337727], [-246747954, 14164885, -3631811, 354305, -1203737, 2738213, -655653, 1117780, 917653, 2102601, 489584], [-124330408, 6193999, 1654810, -171014, 386473, -1739106, 383298, -1680612, -1450264, -1589301, 1559549], [-91571951, 53668962, -305240, 1340160, 174181, -529884, -564411, -1037911, 199415, -345489, -1375834], [117908037, -38682934, -2369776, -582, -2494649, 552970, 45985, 2065507, 518531, 2552103, -1105494], [-203079740, 9021530, -505133, 1062912, -4104021, 1706948, -1671142, 364380, 1537160, 1891163, 335768], [-48636935, 21418397, 1374890, -705513, 1203504, 2201797, -300875, -889816, -126595, 2073610, 1057388], [-153676669, 25332418, 2554538, 284249, 2308475, 65521, 338202, -720049, 63478, 272382, 2299828], [-726534377, -34648548, -3849490, 209007, 2797829, -568239, 1225830, 1883837, -783690, 947522, 821086], [55906587, 32734951, -751785, -1004313, -4673164, -2487881, -401902, 2530733, 1090330, 364920, 397088], [-805673152, 7085321, 3932632, -4117227, -1943191, -2287829, 965968, -1076875, -200140, -947, 20098], [-51988639, -26121594, -9115948, 1445223, -2903493, -1149, -1528116, 913199, 182836, 1086662, 253834], [253265670, -19289743, 7932615, -818151, 3158754, 1643489, 895706, 3603481, -1166624, -1436664, 699826], [-358626032, -75219934, 10145559, -2358432, 241705, -51596, -1361984, 794714, 1224426, -280995, -347000], [-209783875, 29316148, 1577360, -74013, -443183, -1691541, 1589100, -4546961, -2159108, -4767264, -3751012], [52458267, -21807952, 8011330, -1521937, -1449951, 3030911, 102714, -1024247, -992990, 2429100, 1075032], [-201578762, 35399728, 4633175, 1337574, -5686289, -759268, -121486, 1672822, 1778128, 1652675, -887656], [-271204464, -44006938, -7649746, -3366249, 2549010, -2851843, -351307, 44928, 218855, 206927, 54518], [212087297, 81052366, -1341182, 1983825, 3658895, 3842343, -2434917, -3431456, 2530529, -1681915, 759669], [-571847470, 20292582, -8406584, -1045449, -1952589, -3616289, -884399, -4704566, 730735, -1483949, -2365043], [-677752408, -54048985, 480065, 3484565, 3998667, 2768955, -5322325, 374828, 870945, 2105753, -2909986], [-76207115, 77825071, 9092849, -2067162, -704730, -2181190, -4632967, -3252959, -2884725, -1890507, -3281302], [-244209699, -66361294, 19502638, 540984, 488765, -2850416, 1069276, -600064, 2192756, -438240, 1542201], [-509732076, -198875106, 4089598, -1501686, -4609560, -817966, 1961210, -1371580, -2418950, 3324378, 1024706], [585421109, 29208769, -5566817, -952674, 3418293, -2877514, 460259, -2112049, -2735171, 1014627, 2536595], [84175309, -49833848, -3277350, 98697, 3544039, 3025689, -1166148, 4245521, 5078568, 4591238, 5543518], [190934712, 53192840, -8161084, -8753468, -1378930, -4134512, -1244613, -4673237, 1859337, -1454209, 1911083], [624432557, -160539897, 16482999, 5719696, -7834495, -2479014, -1108095, 8152571, -1619205, -742351, 1740761], [825849498, -48307157, 4724669, -239532, -2717574, -460238, -4881710, 5021904, 937126, 6360556, 733405], [1717586107, 74327735, 25977696, -6193062, 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2771354995, -131186853, 250039680, 248183025, 391894410, 409843239, -482416024, 831553030, -562800836], [119327390525, -47468555488, 709015046, 559523643, -1928981109, 158954079, -162971957, 1651874154, 204258911, 443457951, 710455943], [185435143221, 14335299844, -625833592, 499568295, -2464725713, -871693785, -747065580, -320085045, -506039092, 248879008, -640783846], [-141972349042, -2866455964, 669055642, 435396789, 177621124, -532283369, 151753243, 746662828, -859062815, -1247221721, -70446060], [-232634821111, 2217177157, -2184174681, 1961797329, -217037668, -286525545, -1104773316, 135443707, 638580212, 1188935486, 121761046], [-277380695810, -48815028387, 445439174, -629595160, -133623813, -998342712, 188487676, -54021396, -222054258, -929630241, -143049193], [-178940341616, 11972962972, 1451756526, -302112742, 1233962818, -862019386, 1107478050, -300486344, -516561482, 145959660, 381476564], [-61495441442, 2890140355, -1051880149, -1026120015, 780867457, 569589241, 205917979, 1721410644, 806545877, 683210777, 570763460], [126982870204, -13945492034, 698542871, 115839085, -1593257973, 210745773, -1531358720, -545419265, 65315256, 84795187, -450343397], [-15367706852, -26623013824, -31416100, -249489345, -1757119158, -14651077, 333129017, -167164220, -108430973, -820787247, -551574758], [47994978949, -23002953967, 2814504733, -831528543, 184756298, 593992315, 1072687960, 1618779651, 214028812, -204227952, 912310786], [-43273357812, 3641848300, -2100922456, 2016722722, -1053556902, -898725338, -690074121, -562351859, 742902597, -667239205, -1275810455], [192383951824, -27829702195, 2302525650, -75200025, 1084868922, 176442085, 385309291, 260339412, -427151029, -295503508, -575840668], [-94345959768, 6221241121, 3330782865, -272282580, 1337648271, 1414864136, -695703268, 1255993484, 1364951072, 1115271194, 724516516], [39041644828, -11202670807, 1141488079, -2041720209, 270772444, -326277477, 504448878, -27075813, 849670356, 783259128, -491222566], [39579942699, -22830093148, -800397110, 614068898, 1958355805, 591444390, -853139765, -458375591, 475288649, -193695791, 916573956], [-160756617536, -11196037237, 1863320547, 1178775531, 828970539, -752584131, -1349261823, -1686619386, -154771329, 279257025, -830439888], [152912352826, 5661095580, -1288472891, 1684601898, -1826846273, 382041972, 117631074, 735985950, 1615219076, 1074805873, -105235278], [-74514820725, 25210694755, 87498682, 1943316705, -874309824, -160746425, 516271717, -730635334, -1341746167, 1131488112, 133262402], [-40672857277, -2951564840, 692370732, 170886147, 1586053187, -490789249, 172480478, -282862879, -438616878, -544130138, -1159918608], [-162930227171, -17503247596, -611727398, -386885975, -2420741991, 291258841, 258541172, 1275167133, -110970640, 578896714, -593444508], [244789422669, -8269743185, 923324891, 102142440, 468645355, -1742096396, -157574537, -1078843447, 44431669, -797255107, -256849137], [465679483011, -10198516608, 603188823, 2299797069, 833553669, -1343495431, -1577503648, -1494127357, -69971450, -1223236089, -976579629], [239997433596, 17520069753, -756651448, 375723235, 700507017, -2229807965, 543540256, 399351119, -1332214928, 56347131, 499689122], [-230601552470, -30731848508, -3631513538, 37083465, 2298602758, -747687923, -689820347, -2120744742, -839229469, -1882525549, -139956330], [-63380901936, -11166107007, -183985530, -1236688382, -367071259, 538642280, -26556266, 1418653256, 949276068, -448681661, -38436505], [32964272396, 22484998827, -3264565652, -94141227, 1259736539, 110078203, -340309310, 1874582303, 664882362, -249714015, 994126934], [-202080207390, -557126232, 2875731232, 1944121981, -2004535614, 311871, -112968643, -277604550, -987586005, 566371323, -22703426], [-15664725569, -67000585805, 596343914, 1615545206, 1614969172, -804218596, 141340683, -2116612815, -389793205, -1667821078, -953368809], [-476820828858, 18258260968, -1659164891, -27293074, 203394325, 1493928310, 824440690, 1692201138, 848456084, 1052504131, 408368190], [-343790188696, -51216191, -3060488094, -359592030, -1768803463, 727730590, -595646642, 143081464, 267480748, 1086949549, -463846875], [91546616421, 18980935253, -2630826930, 1606985411, -2192832236, 265125243, 967795033, 250093596, -21570447, -719237460, 1651294438], [-120092124274, -50728521436, -891122084, -948311868, -331560678, -159992816, 1561588831, 679570747, 729313203, -820777903, 1676081709], [-227804936200, 16196747196, -5948568374, 628404390, -735194750, 2161384598, -919440514, 628379328, 511564822, 1394519812, -148840000], [56729095924, -21461567289, 2860664035, -1358026238, 471493374, -935195856, 986283206, -80981246, -84948226, -686240438, -343890975], [-46554267344, -28031833004, 590902772, -690059487, -1230334034, -75824369, 1004033353, 456849422, -179430037, -684756185, 575545366], [76452947054, -10823464973, 3557013900, -1365661199, 1161667486, 2104635345, -391927515, 1772318162, 1061042061, -165175372, 254462124], [-418751627070, -35550862587, 336743054, 206765221, -2065447495, -939376065, -1720329960, 344015129, 1606567668, 374157071, -775976932], [500987831733, 3031080512, 1921204088, 3647620167, 458890929, 770837635, -654491786, 688376097, -49804166, 324220050, -1344056854], [-184360347527, 37815219683, -612172215, 1571810338, -1666141204, 999973674, -458770955, 647787905, 216474407, 1451139329, -1875414152], [153952346746, -3323334449, 405179683, 152574318, 612779929, 673358390, 487375436, 2466804634, 212851840, -685209442, -288510846], [105001155517, -8275874395, -1957527717, 233588172, 684801483, -457223272, 340135301, -101433433, -80771289, -2548385075, -921180759], [520651582858, -24581516211, -3172579390, -1781676651, 1950510818, -10832177, 1847858089, 752850922, 997470689, 537603018, 878143100], [-364854466188, -8898329576, -2493161267, 318512725, 22551458, 54622437, 343554035, 524023044, -923508009, 702927240, 1334212673], [119742978620, 23198112119, 774195646, -1767033171, -922241824, -577454725, 606260051, 400461076, -8119565, -564021322, 1555458524], [-285583749721, 9865432812, 5242160384, -330863307, 2775393251, -132672087, -156144159, -1100605274, -1718486975, -1892381337, -631157039], [-212880725758, -7489527704, -6724537314, 2634359223, -853300138, 247521645, 841660851, -1249605726, 371777493, -1447659489, -1728793460], [351725124225, 6152505172, -1642282782, -1456185471, 178071463, 1286056637, 642957155, -2537139416, -400455869, -410244405, -1128366603], [249456249886, -5215834652, 4306318207, 2237935503, -220802443, 193099519, -545857790, 2960289535, -254944332, 1772779433, 221019523], [93558283929, 27925789923, -2749398337, -2179072461, -1557027358, 1037560705, -540689430, 723627755, 1401739506, 54390786, -356032880], [-466181170490, -28836627865, 5120458109, -1914169764, 1064728828, -966162762, 606216462, -1030631104, -1544997026, -948994676, -440882193], [561994033626, -8490655411, 4582983076, -1386499969, 809899092, -1679459731, 2529535095, 369115486, -535489961, -3089154762, 74087148], [-332365497517, -55969732586, -2659867692, 99300732, -114503911, 496255760, -650161927, -2945715379, -1447066029, -84795557, -327914418], [-430016257032, -23760366560, -3360211676, -597573777, 2284822992, 1226338345, -1087006592, -2505832109, -140919552, 1429946090, -723731469], [-588039856158, 9811855501, 1086063117, -664727430, 4114550192, 1483828044, 1476051528, -407907548, -224291140, -2054685728, 1670285415], [146722167123, -18864768954, -4263386370, 3108100469, 1397652887, 1537639709, -1248135036, -529650675, 511786196, -194904592, 319533470], [192937658081, 22704956921, 1354826525, 434819826, 1761780201, -569156784, 1355112561, -1511612455, -42087269, -825437699, -1305325283], [-474090508410, 14714732823, -61173330, 2462675484, -3449259611, 788856680, 157634006, 1634936162, -432251056, 775567559, -476098101], [318600628982, -29963313335, 71142956, 2096859501, 254712518, 1239893595, -1035043101, 2332719302, 453050071, 2137731554, 1263986840], [416496863142, -4402146116, 2069354223, -1302556788, -2139829345, -126805306, -1192196986, -499677880, 1254555748, 551078871, -920524334], [361432339120, 71039966684, -4254561688, 1555085415, 422677214, 1448439773, -154262335, -958636728, -32689365, 1777488179, 459338876], [-361251255841, 26212140071, 2729987689, 477093438, 1014921192, 767129982, -737219067, -773549585, -339042929, 2505327069, -1093655300], [33240499604, 3543380054, -5000888259, -158689842, 1409391175, 209874652, 42470234, 1450843426, -405672796, 2866968179, 2126261014], [-500122393944, -8746461726, -1113765294, 2056899627, -1643601576, -601568095, 348661157, -1326884808, 446907571, -1860659093, -2841909788], [217500271190, 43632423643, -2657107606, -2332729992, -437344971, -1446057638, 2222207368, -904524456, -676738090, -2028637681, -938651649], [-254105878921, -25485500468, 403447406, -286500156, 1944985183, 650926172, 353307893, 435051713, 76323591, -1222188245, 1944945278], [150048205688, 23360766987, 8965905419, -708726045, 2711879227, 1813323119, -1453817509, 323775806, 293949253, 2018631333, 11626404], [27065752498, 33735721492, -2213904430, -1398130617, -1640132444, -150839815, -592694458, -190456979, -659511746, 560117546, -1933210021], [177636425957, 143372816, -2430354634, -3903159639, 4758075567, -737178331, 637642028, -5005550571, 512291504, -886002354, -983221218], [-440235934414, -17641769123, -1944432767, 3661727846, -1557028958, 537996626, 1118504712, 3707253060, 1461138872, 1155854042, -1366596573], [-697849578003, -159722428, -2591465482, 3203021265, -1919902617, 188562551, -654236110, 4118317463, 753775998, 856646542, 887293638]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_387_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_387_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_387_8_a_b(:prec:=11)
    chi := MakeCharacter_387_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_387_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_387_8_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_387_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<2,R![281015823360, 26122731136, -25840429824, -34580064, 584457696, -13609592, -5061216, 169726, 18498, -717, -24, 1]>],Snew);
    return Vf;
end function;