
// Make newform 384.10.a.c in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_384_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_384_10_a_c();".
// 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_384_10_a_c();".
// 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 := [561570, -13768, -2070, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [1012188, 76928, -100, -88], [-1362252, -15488, 1396, -8], [-3249600, 246656, 4736, -160]];
    Rf_basisdens := [1, 483, 483, 483];
    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_384_a();"
function MakeCharacter_384_a()
    N := 384;
    order := 1;
    char_gens := [127, 133, 257];
    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_384_a_Hecke(Kf)
    return MakeCharacter_384_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 := 10;
    raw_aps := [[0, 0, 0, 0], [-81, 0, 0, 0], [60, 0, 1, 0], [1210, 1, 2, 1], [24916, 5, -13, 4], [15210, 5, -23, 19], [-108738, -41, -83, 10], [-157944, -55, -29, 30], [187348, -146, 216, 174], [1977136, 556, -41, -202], [2837810, 663, -1072, -171], [3398230, 1571, -1123, 2949], [-4709722, -1171, 9263, 1018], [-3544480, -1509, 8681, 30], [-9444780, -6662, -2440, -3450], [-28834128, -1480, -31753, 2334], [28757020, 4356, 19436, -1924], [-43307162, -5041, -38587, -4043], [57946276, 17388, 34324, 42716], [49369052, 51562, 20804, 39230], [11157350, -8672, -38004, 33290], [-88943646, -7939, 98534, -25099], [151903332, -20983, 79319, -60312], [-289286606, -6142, -68962, -114996], [-399884118, 49562, 41750, 36136], [27112240, 79510, -25479, 37042], [170255422, 284405, 232832, 195243], [535364772, -58578, -746966, 134184], [26037506, 30853, 524501, -25945], [-642590110, -133734, 642662, 28400], [-657660314, -244631, -703948, 137519], [1554670476, 108768, 874832, 59912], [-808941666, 77655, -818507, 499238], [321069404, 703368, 76016, 518764], [3010667404, 589584, 775449, 1056968], [609215234, 77863, 1171484, 432613], [1838027366, -656057, -2421775, -255007], [4321488872, -236133, -995871, 23006], [-4526867408, -1527020, -1288380, -1829100], [3352423436, -1458190, -1244069, -943692], [5740706004, -876364, 2927180, -2652828], [5263001362, -968527, -6673527, -1637581], [12796187208, -519880, 907584, 1970672], [-11773066226, -1469112, -4300116, 1887198], [7822634760, -1026144, 12046995, -2152982], [1541315766, -1491839, 7357264, -815105], [14863339956, 628892, 3677884, 919356], [17215626422, -3364325, -18158210, -6008273], [9308273476, -4510461, 12360605, 3383224], [18181098298, -1091619, 18333745, 5161759], [-14895570406, 4978754, 17342838, 3311552], [32736926176, -4117776, -13897884, -7634140], [-7693345618, 3680318, -34924842, -5522898], [32040297668, 2520605, 869259, -7415644], [-48202795054, -4258160, -20226504, -2008632], [18782101576, 8269136, -25615728, 11547756], [20205809792, -8036534, -27907247, -6062110], [30349120982, -12490653, -47149762, 2698351], [82524342242, -2180711, 28516801, -6375269], [-82877951566, 19302232, 16817640, 15582324], [66182437364, -12844904, -11774960, 893704], [-31967513984, -36782036, -5947581, -9935218], [36439162644, -34611140, 20977348, -10543380], [10324291056, -32146212, 41808944, -11521176], [-160279668362, 14466960, 50767908, -10064538], [48234532024, 11138798, -78669575, -5219306], [177830071988, 39497214, -27418126, -26072792], [-2919707090, 46048, 64652300, 24268138], [118158048252, -656239, 51808095, 27268072], [51178665494, -32020373, 41976937, 18192725], [94501698050, 48023466, 81720910, 35820008], [153065080396, -9030642, 36501312, -41549226], [18020703250, -2024557, -144131188, -19016915], [162717234806, -56740265, -51768275, -7616215], [269564962144, 46243001, 27408155, 119981566], [-210757154840, -88479908, 46091348, -11300592], [348248629548, 45765606, 5897299, -24100252], [74707052702, 30109343, 234866053, -49121883], [38682668742, -28479827, -21154033, -115813010], [-57251988334, 39854502, -61655222, 122602824], [-17803910876, 7409351, -8243783, -44000968], [438285689090, -41724043, 34727653, 51835531], [267765041484, 67540122, -26383836, -52978366], [590641522162, 20638468, -122750076, 1570780], [239415770382, 51332737, 200117860, 61960155], [173606269836, 8504441, 61372503, -72739520], [287594166158, -58518485, -332051807, 174411786], [67910413682, 31333218, -58854050, -79303672], [-983022049956, -67258586, -144626041, 28942772], [-264865861330, -43117011, -200542620, 16345443], [-46774195180, 64325487, -408663807, 147184968], [-309928954220, -13658998, -332570684, 167309986], [735124610730, 240614553, 188136634, 149116353], [-655946912964, -161709034, 201138050, 47543380], [-305158990596, 151561336, -208908576, -52788600], [-1417430369796, -133944946, 4147608, -64023274], [-1315983538424, 188657780, -93508685, 25374194], [-513642431090, 81267625, 950688955, 252021286], [-83149092368, 216150361, -480933213, 248449646], [-855357539654, 81044489, -480041583, 257535007], [-625099554640, -77984163, 189784815, 160499058], [-2214506395132, 278501684, 450458797, 146739860], [-58971649812, -158258223, -384476025, -158290476], [-1276019231610, -96991683, 181762639, -593543598], [-1710078249476, 128599190, 1139325930, -111284804], [2923508939830, -29994440, -248864436, -271492914], [-854661933316, 472345618, 436407414, 10510372], [1021989280194, -121609894, 703277654, -145816992], [-1387573843636, 420203166, -1430727588, 288677266], [590190570286, -44568892, -1890836896, -52444642], [399092993462, 294891205, -111286544, 661576127], [703377017638, 21936907, 323636609, 181209265], [430343765778, 321702756, -1341704556, 703247452], [-2890734740284, -392008732, -845596084, 262176504], [-332562205478, 404242697, -1077023998, -320685407], [2162569142838, -685379721, 670437381, -114314442], [-1502709366128, 429117273, -601595157, 94537250], [-4715151538044, -594440878, -126023196, 197024830], [-3532026295340, 648622358, 691262379, 524769480], [901008996196, -103110318, -1295150970, -325223236], [5446545952238, -310807733, 686118237, 22405597], [4613608104258, -466964696, 1537177856, 35524892], [-1036008063756, 445628104, -1022583867, -664354652], [909974890140, 803181567, -1920390095, 677882584], [-1142415225080, 1305381323, -52838223, 961035030], [-1596269610072, 107953350, -2440463815, 996910966], [4862393482890, -201337079, -3082392907, -777105681], [2705860062380, 305944062, -469989708, 189840166], [3783645081186, -525156285, -955849464, 1215393477], [1158254214226, -57657915, 4143484921, -698601693], [-6022490720908, -202625554, 1550497346, -218360420], [-996701280168, -286673656, 3422733292, -1929470136], [5654448492610, 280872589, 3183127190, -326462747], [10358596784458, -815356363, -1407876231, -574322161], [-5309808964906, -179163369, 1137347925, 753320594], [6155037219870, 332622372, -1052916848, 451553766], [-4883735309976, -866412934, 334299097, 988428686], [-8611681927904, -605530893, 3653642825, 108558746], [-2340711569004, 560883464, -2999458551, -1479708596], [-2790002375346, 1386298511, -3176883403, 363130494], [4206013138520, 788027987, 1296974105, -933141950], [5114035203464, 383211806, -1470191299, -1525918306], [7312790389378, -1162033487, 1139674358, -2409546563], [6187739347636, -1695940894, -9542476890, -465728584], [11492889307810, -739203491, -2042912979, 1103728655], [-1458742047724, 1230453886, 3533428004, 2716961930], [12005881165822, -728226301, 5781571757, 152020189], [-5254081343370, 1882396365, 6064152439, 1159316650], [-3323541069032, -2069602747, -7138694337, 1497758690], [-4856187622728, 65202480, -142702136, -343043424], [1917868481078, -2152775605, -54671487, 1028507197], [-6742213963374, 1934360972, -4996032260, 2702817664], [4322382699528, 1593929967, 5775035733, 1404045782], [-4117928429816, -1798451624, -6805534936, -22885212], [2827372970088, 1065049817, -7247893365, -1391633998], [11965308271264, 1486869636, -4875415976, 2754984612], [19159410003602, -1455477351, -1480822338, -485622179], [-7829771275434, 3779970801, 1548962707, 3732915550], [4534882661402, -2598883400, -260176216, 937755792], [-4805621525220, -1185227758, 9654605875, -716547228], [6437378507796, -2967857900, 6386739420, 446509828], [-7441183333762, 4460252841, 8790787163, 3702676822], [18616115994502, -1141881805, 6050997626, -2655395249], [-5396930797300, -321930533, 17616131333, 5758475376], [-14816430995706, -1728497975, 8019778123, -2266706226], [-1252413615976, 4607573648, -8530376072, -210158564], [17026166600866, -1132556439, -1446832342, 1043838261], [-5903825706258, -2319091893, -14248202703, 4962626865], [-6739785607478, -5194479778, 6841789010, -5547913016], [5481528136052, 489206778, -3276035661, 522954824], [-2417610482036, -3034769622, 2043644814, -7567771516], [23732046741094, -1801720677, 5114249125, -2311098279], [-11070090533256, -4188885548, -10944894848, -487816048], [28417163233114, 1196552592, 4250173696, -3867897184], [17656135084826, 441237197, -2320166374, 3056274153], [-7955907916478, -11561329832, -3292023704, -8044934596], [-25711724486924, 2061268264, -20081830000, -1176532728], [-14873105622132, 3546405264, 164516809, -3292574592], [12801218783118, -2884263999, 16088040132, 4924913491], [21387124691374, -9374347109, -15304976731, -7340379079], [-9050136331438, -1284554187, -3709642786, -9147875779], [24415310568276, 1236170789, -12904793573, 8618461976], [-1699300175598, -387126323, -3343537631, -8073126201], [25035403096258, -3424987248, -27469899232, -3598160236], [23126291482072, -4165771244, -28911003684, -6683484296], [-45645536196140, -7195760906, -8802751937, -2375481688], [10000291508546, -10206978687, 6796572789, -3209711781], [-16981606696324, 7560441244, -7495416420, 8947278064], [36594454800534, -4043837098, 6299779470, -6673227706], [-6478333468320, -3171821300, -23552551740, 10387163608], [26039760149682, 7332863524, -8954974644, 12862031424], [-11151669709092, 17691666232, 1796948504, 15294035620], [11311506407324, 15308549200, -2712390664, 13801616968], [-31299282427596, -7415418894, -591364137, -2865373144], [2992562263388, -1109727184, 4800255760, -13896004720], [26766455896562, 737522588, -23322480340, -8672117748], [-195291638010, 1510448304, 10153484068, 5218786638], [-23190773210574, -752257203, -3666155879, 5299396307], [-29164824050102, 1017714764, 1402353204, -5452336516], [-54642357270608, 510090584, -27586792452, -4845435168], [-8651330542212, 4741232796, 3954804593, 12200248864], [-10856803655554, 6403642837, 18447017860, -8704746469], [-30523021028790, 14657020941, 18221407861, 13431928451], [-15725385825938, -3494529512, -48917634732, -5107898110], [-8122429979484, 15942774527, -371556903, 390319108], [-6596609620440, 5882767956, -20478899621, -19636545902], [-87781677215874, 13610192887, -47244797850, 13536085675], [-19667129878980, -14456522963, 13559743563, -6345288524], [10296937967610, -6945435570, -23498282878, 9070594480], [46923749024536, 13543416585, 5563811059, 11739681146], [7757459295214, 259393060, 34277772224, -16558905130], [-25468908878556, 22211605400, -21103394827, 9855365452], [15391052991838, 1606566717, -11487838272, 23845916299], [-36264382572020, 311566224, -20097723776, 19920629060], [-12783506192796, 7177463030, 41223818300, 15183359322], [58054185209050, -8122663172, 637939276, 9488159148], [-49821068261510, -18163522709, -19672874624, 9732407897], [11431561915958, -4015054553, 18234975085, 2385294830], [-1401634274592, 19674112256, -14143019912, 9104390516], [14502024490140, -924457780, -1553030703, 22378050040], [-12732315382870, -13294487843, 32338942653, 7553768963], [99305484727302, 29446014603, -11919874558, -1461969865], [-1745857944898, -18619813655, -52141476565, -5888182390], [-1051514882138, 6296366331, -87578922466, -15564699401], [39602474537924, 3640854993, -37819193889, 1545850920], [-101075327643378, 7829116599, -3324520959, -13700408707], [-56765833724606, -31035107762, 19937372522, -3261486796], [-16160411097516, -5391359914, -15121732760, -23197492662], [402177144398, -13237313637, 15366429598, 10056929035], [-55947145388052, 25561698198, -14800725758, 4791500316], [63156384675490, 15089316129, 31254316617, 8196826015], [-27554858397872, 11003869365, -92215308121, 2917499498], [-57620713718770, 18045366461, -82863875108, 6226040275], [21303827880610, 18230099644, -64236249276, -10879991884], [23803792216, -34143970383, -64387440917, -18310906262], [156169272917412, -18689768850, -30720188968, -3565311926], [22029231768714, 30752690022, 62313779034, 42910207808], [-948383291012, 7954624556, 57047438525, 34000980256], [-108685491720548, -5001134829, -25158715075, -26956254128], [-38932217969452, -14372114582, -27891709044, 8734345230], [65064355533804, 3710646681, -27049031953, 4814695268], [-6282991100044, -35453108114, 57261903146, 1861524720], [49957825008274, 7325724979, -73176445744, -2340649099], [-4570443823514, -11503290481, 114651980129, 7745201673], [127769185403354, -4050266680, 1223502192, -12576308532], [-95685994541628, 4989381514, -133491818480, 12544732898], [-149294543679222, 24299954341, 150243138370, 12380995633], [108663717741980, -38352271474, 66036718634, -48374390192], [215886225315172, 15025715446, -9613166190, 41975666592], [338496443190280, -13497167680, -71623870468, -10171060348], [-107564017212462, -10965433787, 7100839109, 20171764519], [11441831956434, -19150975094, 93148815190, -6764567312], [126712741814868, 41119246530, 2646433740, -4857319114], [-222495091469302, 18255716444, 80474586708, -6125228400], [54003461214880, 30432531568, 35743214683, -20391830010], [-10670220707700, -79972240378, 49933721858, -30567650048], [24910159082474, -16457859911, 46960111065, 23230273427], [384872049479500, 20468015198, 19416685098, 2551693380], [114784586558984, -38451437716, 34002320519, -9877321334], [-368563083035734, 20951266796, -25155057076, 16513106452], [-321637591865466, -7891510551, 30871146236, 19767852039], [281832453629380, -32909113844, -42849588220, -33885872300], [75365171529530, -30127717891, -62006222963, -11966563045], [-338547011468666, -27174777977, 32116013721, -86906956863], [-45731528169842, -14135105883, 9555706183, 45796316562], [43734579256508, -44216746678, -136848328834, -10934862008], [25028661702580, -15297313276, -46890316387, 47432020524], [-28410772150502, 38959880726, 123922391082, -62367404568], [158247144179368, -65681471657, -28146112555, -47564121070], [321880095090060, -34488798144, -92935866567, -30224094688], [-67345681495258, 88897368031, -13661414883, 48613685165], [148370598149668, -68227845358, 118271764998, -3313282148], [-425073524497654, 5659594688, -128823698336, 13192844920], [-379532158012342, 5164504711, -63374693348, 14120585009], [-119896687011134, 6315179552, -16008490560, 48471676416], [-167484180225898, 63004376579, 14567177042, 74968618511], [-128175773522076, 85385966915, -127571347147, 72701109244], [-125043667723922, 59342474823, -65848457291, 41592217557], [-166798280112914, 25218738540, 215482953016, 33881784294], [199403083207404, 47569043622, -140935863710, -78562462160], [124897526696532, -22288332426, 122270603016, 7422424330], [263972798151310, -19780807011, 104827727080, -47850410117], [-268157051358532, 30066845738, -27295439928, 65646007490], [129298670738870, -16412160853, -126750328567, 48417636137], [222192695665448, -44170283325, -149192487895, -73239740950], [796609034648328, 11055426380, -29807455400, 45783748564], [-44995962959778, -145312483750, -22493667558, -101672406490], [228907558937732, -68496078000, -118783100659, -7873440396], [-323861389313110, -7342142355, 37569696526, -104000428727], [324776499360834, 21530438794, 13357837838, 20191463144], [295719260727268, 48356438814, 12892425987, 29128044840], [142393986597108, -68387608480, -125217606624, 74343544468], [-325868808616174, -126195009922, -80018071526, 66761476092], [-529105115006580, 8835715680, -4321530272, -46854295796], [-423387000479778, -98108718149, 21529675925, 36804296801], [-481445416495908, -55954362682, 52751490727, -110006314180], [-391834116835250, 120595756231, 210735127366, 81163029883], [-4362986677168, -58042732222, -334884070019, -134604480206], [-373388622429868, 96490630279, 75144451057, 72803932900], [-283156144660384, -28951841649, -57940522659, -84142841742], [96547602023478, 57426629164, -323663234936, -80856095794], [-179060981624012, 46620315010, 28640925431, -36878401584], [-279755633874790, -95063212383, -292308102114, 42502204489], [-2340053873380, 50180161196, 206363425108, 96518074872], [-295902248494864, -124354340837, -97219185607, -39121716434], [259971972805774, -33747514850, 179961132038, 87193025862], [-457685883069468, 4617632877, 243576371259, -143747770948], [46795609827506, 213048403729, 137620097429, 100092754635], [-668614315714140, -2001095158, 112500404472, 1789794058], [-620733473742666, -51682335789, 133864973177, -60775369287], [260615146342184, -120525149588, 156250204508, -147789000848], [-303629802287040, -29525900898, 11373917057, 104595159626], [236872567944022, -189144045173, -5110364399, -107692785514], [-203248702716072, 79700518513, 250004405795, 42009159438], [-524154042230784, -50873149348, -117183390976, -97488943280], [-51648032292358, 1882975244, 192193967380, 183348415344], [-23969818171164, 22190187246, 156856513066, -84982669492], [-454690349481876, 122190478970, -229741120320, 109181039742], [-259548054108418, -41560422160, 239765753548, -48336491190], [497968537783210, 152108722446, -73419115726, 88184174364], [-225712533280180, 49294294700, -224842133140, -41351152816], [33355251575430, -66276853956, -337330666984, -161342787122], [13899743998304, -63395588784, -161570787597, 106056278398], [-1010848970463614, 109551509259, 282118691292, 57913006173], [-38049305585642, -113059084531, 102629326487, -130202652046], [273419678354330, 89514319930, -252620145674, -150375096144], [-243001057822680, 45934814137, -337930233493, 46765490038], [113640040993824, -24981994361, -722012318675, -217654629818], [-1062591167476656, -57058411092, 3031475636, 119309449080], [-243857799657252, 156175632406, 163560326155, 12297117316], [-28389161145454, -47250931243, 366046338317, -237736915609], [109705329007548, 17112110572, -267434164727, -171425732928], [-1127231466840154, 23608033291, -134028423810, 102297706879], [593959147447404, -65475249374, -504014023690, 113288496232], [343207313954508, 127752441356, -149561589236, 252119921620], [86316607531988, 32792873759, 371532082761, -51943884508], [609434115895558, -71418744169, -16559114103, 1699178897], [-646864936063254, -178913060162, 183686798922, -322362971708], [1578809338855430, 27393555316, -122570296056, 95674302954], [528491919835662, 1075621519, -166668409114, -69265298773], [-554463248404434, -231721279521, -289130940031, -36967551219], [-428882164374370, -252785394699, -185569956233, -324480175082], [-788811932218384, 65619183644, 280772846315, 126405989230], [938764021581950, 23696477239, -284249430094, 50288239843], [690354181104488, 78719889784, -258429090377, 97635134338], [808068524078316, 147903356173, 8516855531, -224299032052], [316168912458834, 123321720869, -325855245567, 295941227871], [-703328397841132, -254703531370, -114187088558, 36668684232], [1552318215960876, -109372935890, 684715388852, -93092698826], [-358358013344904, -35407823068, 230782592599, -81153631638], [-1465061136773408, -248750997089, -136865059651, 35101191554], [71790357705426, 130756246250, 171886419846, 30948812400], [668630074497272, -263545695024, -548259858057, -257017497590], [1718971889172702, -55467530101, 312654165906, 118747099819], [213329534868286, -57144952833, -615052223839, -226994986947], [183758774455482, -59831269652, 210365346524, -126112651968], [-771462153170800, -25083890312, 249448973748, -120157442604], [41032195397996, 134672172322, -124785604778, 243132035020], [-974963165861082, 278472155803, -77592953199, 202761357542], [-272583940438032, 187531507904, 336641399652, 21300185896], [642377376026390, 521935655783, -81088447919, 184342881925], [-778333608511470, -103714573214, 209267458430, 146502143236], [1330252916250600, -94412177380, 546167651212, -97705776016], [1118020701008148, 231547643367, 361555989329, 289911542052], [1274907849317700, 206963777678, -121454508630, -71792068380], [1126575605454238, 20184019220, -74294653632, -150303637282], [361718687866992, -13033610718, -984423792279, 310695189146], [733643436430962, -238357651831, 164776287222, -72371577171], [823118666066490, -137566502096, 331203962968, -70858416316], [46974769027676, -429999222697, -25044088703, -40265801572], [140858940979856, 145881803681, 352087407811, -301269897490], [1031108609125584, -261704190516, -423338794340, 141872822968], [899485208836336, 335947659428, -220841811021, 219845711550], [482209273776450, -313404365717, -1155544773968, 199099197269], [1390700374890882, -247639403411, 521447702705, -88899344749], [379474197239420, -430450851359, 60356500695, -121777114004], [478524012125992, 336067869928, 105003484928, 514749325776], [-279263426137182, -64330558144, 410648811472, -548310298896], [-1820581531818898, 24167772625, 140451731611, -103071310406], [297228043350278, -210211012246, 1285737412314, -108378292410], [773411796401564, 76968224014, 214809222487, -78134361796], [-187698491599434, 168551428607, -1394569820435, -292398829606], [-780661062022046, -77392840665, -634057585268, -626719488491], [-1230870556153390, 16869005832, -177759315800, 244692619352], [-187902879725152, -206798955995, 1073621320559, -286170751094], [908047853606976, 214617282844, 464301433268, 414043352668], [3259769361308514, 118973338925, 38604301302, 209174755365], [122872446656450, 109596835557, -197836164955, 366373669891], [-918963721672452, -421772743272, -315387391952, 325957142276], [-879119270542708, 500571572026, 613239909732, 122718870554], [-1261098063456878, -13129396956, 850375936332, 128660266800], [127026847987428, 345480671414, -554634738977, 531837171824], [76469701782532, -564217009440, 196467815232, -455716237544], [818801330561768, -279839330331, -761977023641, -369365639186], [-90421231084680, -88065145216, -882852922444, -331736163392], [1933402489421218, 60881923482, 886672713926, 341808921752], [-828701827645370, 425233939629, 404419138160, 524018704983], [-821874332070546, -382471181955, -828524176033, -101747245242], [-625884205415008, 317746084593, -28477418181, -2560523978], [-121544418540868, 115754600958, 779166958259, -373738761060], [280108153887566, -474799129097, 776334160381, -283707363035], [-2136460475974258, 290840131989, -1118425963929, 156638271922], [1076140557174910, -471783861021, -486978111974, 87900947259], [-2096272811960538, -120745869327, -498565283029, 43118023154], [-63997477702160, 409269952486, -203077968967, -148802068518], [778004046955258, -631746699853, 147968484364, -243170911727], [-941524203091566, -407001196663, 737047758641, 310141603847], [-537195883551034, 454526880701, 324606390143, -31706643026], [289666097565944, 436471678385, 1364528575435, 496088803318], [-63133607857636, -383218964659, -581999067285, 257447192436], [275321325056086, -269065397800, 1150977765820, 322259409142], [-1958524106862156, -438156955660, 102593135337, 149721207580], [775796359597396, -359044882871, -166662481073, -610529659172], [-1342719735284536, 276340946415, 1468396661221, -53495737170], [420527819849270, -378193331644, -1714885878216, -325977060734], [2185092887230256, 941577383754, 230310421657, 464561754994], [-4414709764141080, -63735167296, 929668092312, 41724571920], [-399691356810174, -152824706583, 1340801196814, -898884058171], [437888695622886, -52386792031, 1884050002163, -123210790098], [1767900698677380, 545749840270, 655964098036, -42150937902], [-2077081494060436, 172165464076, -2162959884671, 133408777040], [441461999723038, 43520758735, 179891509401, -194887161683], [204245936172472, 138306530432, 1398049588452, 735147822460], [-450236014793508, -774282067625, -669955397639, -335504619648], [-343961492844122, -307344093924, -346335662808, 324547089398], [-1459634212618048, -78440628990, -1188738125375, -258951735742], [-3815286160194796, -386571310377, -458549501815, -976531808104], [802421783082882, 241313377758, 305353693474, 210311026996], [-3101381781666456, -415950181199, -457749822069, -637106064886], [-648790882512876, 360542525570, 68085033796, 249632074358]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_384_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_384_10_a_c();".
// 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_384_10_a_c(:prec:=4)
    chi := MakeCharacter_384_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_384_10_a_c();".
// 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_384_10_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_384_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,10,sign)));
    Vf := Kernel([<5,R![-217486924800, 2588500480, -6873216, -240, 1]>,<7,R![107350205672080, -39372619552, -73699080, -4840, 1]>],Snew);
    return Vf;
end function;