
// Make newform 98.10.a.l in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_98_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_98_10_a_l();".
// 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_98_10_a_l();".
// 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 := [-5372164, -32180328, 12807408, 16168, -7165, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [10243942375796, 22368923366, -8636476956, -100542547, 1210854, 16557], [114577250368, -91498140874, -46012148, 64030099, 2982, -10717], [2902579423454, -2355511160234, 13263914945, 1169730299, -3910461, -142991], [-10581294778128, -21107028046, 8820525548, -155577849, -1222782, 26311], [8276645371622, -6442323297722, -58197203515, 3145018667, 15601227, -374123]];
    Rf_basisdens := [1, 3101451260, 1860870756, 18608707560, 1240580504, 4652176890];
    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_98_a();"
function MakeCharacter_98_a()
    N := 98;
    order := 1;
    char_gens := [3];
    v := [1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_98_a_Hecke(Kf)
    return MakeCharacter_98_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 := [[16, 0, 0, 0, 0, 0], [0, 0, 1, 1, 0, 0], [0, 0, -2, -3, 0, -1], [0, 0, 0, 0, 0, 0], [18448, 11, 0, 0, -1, 0], [0, 0, 140, 237, 0, -25], [0, 0, 3271, 984, 0, -100], [0, 0, -995, 3009, 0, 0], [40586, 178, 0, 0, 112, 0], [1744037, 498, 0, 0, 207, 0], [0, 0, -51446, 11430, 0, 2260], [4056625, 2938, 0, 0, -653, 0], [0, 0, 151337, -26244, 0, -900], [12456752, -7883, 0, 0, 1153, 0], [0, 0, -16242, -258162, 0, -6920], [22972348, 10700, 0, 0, 1310, 0], [0, 0, -277999, -168267, 0, 38940], [0, 0, 1618678, -260397, 0, -17575], [-33828738, -21068, 0, 0, -6542, 0], [102397488, 56350, 0, 0, 10270, 0], [0, 0, -2404929, 113934, 0, -70010], [-30390664, 27998, 0, 0, 14942, 0], [0, 0, -6327087, -119799, 0, -43180], [0, 0, 8660755, -1840248, 0, -113580], [0, 0, 3117303, 1014690, 0, 182930], [0, 0, -8586326, -1925469, 0, -109295], [0, 0, -1471362, -2777190, 0, 253420], [229429226, -517908, 0, 0, -44082, 0], [-160630877, -125050, 0, 0, 182705, 0], [807039333, 177694, 0, 0, -79619, 0], [-1695717674, 277228, 0, 0, 95542, 0], [0, 0, -3642449, 6807783, 0, -1249820], [-2405426298, 1614268, 0, 0, -193598, 0], [0, 0, 74950113, 2182077, 0, 1439600], [-5597259350, -841600, 0, 0, -248320, 0], [506306354, 139608, 0, 0, -744498, 0], [0, 0, -13031332, -31758909, 0, 1963825], [-2952871580, 1840681, 0, 0, -425471, 0], [0, 0, -60463266, -17140698, 0, 85760], [0, 0, -44084168, -94633761, 0, -1181315], [-3966821358, 1452256, 0, 0, -424046, 0], [0, 0, 73061700, -46207041, 0, 2270685], [-6650324884, 398774, 0, 0, 411266, 0], [4105469999, 5170054, 0, 0, 839071, 0], [17684873612, 1471852, 0, 0, -633182, 0], [0, 0, -261611766, 32568102, 0, -6365580], [6047178498, -7660672, 0, 0, 2911562, 0], [0, 0, 130119040, 69433572, 0, -6578300], [0, 0, 86023495, 226985523, 0, 4462840], [0, 0, 153383032, 46956603, 0, 8880985], [56609672784, 4525364, 0, 0, -1435804, 0], [21558988458, -18218422, 0, 0, 3258032, 0], [0, 0, 24137905, -179401266, 0, 18992710], [0, 0, -1114528047, -194214831, 0, 4019180], [0, 0, 783136521, -268606920, 0, 17996160], [-14414750028, -12479414, 0, 0, 466054, 0], [0, 0, -330292412, -467730267, 0, 8707455], [0, 0, -529056576, -63915684, 0, 16326020], [-56191810982, 7856728, 0, 0, 4022332, 0], [-31093056284, -17724500, 0, 0, 10401760, 0], [0, 0, 1240093041, -165166479, 0, 43122760], [0, 0, -327535146, 98726247, 0, 51601685], [0, 0, 306544767, -247262721, 0, -14935700], [0, 0, 282770648, -479121960, 0, -22363640], [0, 0, -1507022759, 979939812, 0, 45624820], [27225817180, -14063196, 0, 0, 14164386, 0], [73047134200, -46932991, 0, 0, -6427579, 0], [-41921660203, -83427150, 0, 0, -4553025, 0], [-17969419980, 42719301, 0, 0, 22483869, 0], [0, 0, 2306788420, 1504342563, 0, -8001695], [0, 0, -3176873689, 332941146, 0, -107254050], [-20778743942, -8503424, 0, 0, -23438066, 0], [0, 0, 3185496060, 95946192, 0, 131812540], [95146780330, 40838784, 0, 0, -958464, 0], [-256839842156, -11023953, 0, 0, -18709857, 0], [0, 0, 5326345706, -2109249414, 0, -44681720], [-109881874759, 73605378, 0, 0, -311373, 0], [0, 0, 5455039884, 17581413, 0, 44213255], [331904631747, -41603250, 0, 0, -3520455, 0], [0, 0, 5943906803, 2629324026, 0, -76850890], [0, 0, -637434793, 1777713567, 0, -272527920], [366241867170, -187078392, 0, 0, 37972932, 0], [-121849142298, 36160318, 0, 0, -4638248, 0], [0, 0, -948287389, 5246515452, 0, -6907000], [0, 0, -4941874948, 2250753504, 0, -324019780], [-137008942174, -306126976, 0, 0, -32631214, 0], [145145227350, -6319584, 0, 0, 72775944, 0], [212555212419, -243187846, 0, 0, 48096671, 0], [0, 0, -4669474928, -2194941153, 0, -218362875], [328513123336, 260671100, 0, 0, -19799620, 0], [0, 0, -2777446783, -3026501811, 0, 329924520], [0, 0, -3983175818, 2330189730, 0, 9408140], [751661990142, -296490726, 0, 0, 29952276, 0], [-776620039482, -481257924, 0, 0, -35276886, 0], [268809211646, 308267916, 0, 0, -107032086, 0], [0, 0, 21654876852, 6185634936, 0, -256308340], [0, 0, 4743712044, 5765520573, 0, 765322895], [0, 0, 7640323031, 4010305950, 0, -75028230], [0, 0, -18635761129, 11185917627, 0, 324906500], [2067208679590, -144749200, 0, 0, 75594500, 0], [-1223803365476, -278490297, 0, 0, -31594113, 0], [-533359927680, -31895084, 0, 0, -138989546, 0], [0, 0, 6684567041, 6484083237, 0, 801747300], [-404481508055, 251697114, 0, 0, -239471589, 0], [994062956788, 78483827, 0, 0, -97435357, 0], [0, 0, 27660351065, -2129641290, 0, 460129610], [0, 0, -2481596179, -21168113247, 0, 660403040], [0, 0, 25217468795, -8335239156, 0, -497603420], [-2109544909516, -750512016, 0, 0, 5556636, 0], [0, 0, -35316333251, 10696024248, 0, -1429758260], [0, 0, -47354450628, -5233059780, 0, -703646760], [2906664257565, 667352162, 0, 0, 188069543, 0], [1821784315999, -1417659914, 0, 0, 110359069, 0], [0, 0, 32867445027, -23253940845, 0, -1629169960], [2801900358824, 13514050, 0, 0, 91045930, 0], [566579133041, -112184102, 0, 0, 421663747, 0], [0, 0, -12614186407, 6735128373, 0, 1021449580], [0, 0, -15516926086, -29745811818, 0, -558490420], [1357825695747, 1547593110, 0, 0, -146802255, 0], [-3683019753900, 64375955, 0, 0, 414490115, 0], [0, 0, 50392039314, -23727256503, 0, -517465725], [3864996069558, -991721400, 0, 0, 380361630, 0], [0, 0, -9562971528, -47176746813, 0, 847099425], [-695377438278, -357053960, 0, 0, -226028270, 0], [0, 0, -48153254049, 322166607, 0, -396789900], [680808181875, -500475498, 0, 0, -118336527, 0], [4070898456625, -743918598, 0, 0, 276806643, 0], [0, 0, 4051875118, 38119477494, 0, -3897491440], [0, 0, -53929722706, 5430782526, 0, 3530234320], [0, 0, 47383470410, -15863962419, 0, -1511858185], [2491953114060, 1272124685, 0, 0, -34649155, 0], [-8018628890054, 837500808, 0, 0, -88931178, 0], [5054061412406, 1635769158, 0, 0, -184631088, 0], [2056993324997, -764581558, 0, 0, -654411337, 0], [0, 0, -9827194885, 8004905592, 0, -4148090720], [0, 0, -54371314011, 23446967154, 0, -3624311550], [0, 0, -50169954354, 12146636733, 0, 2079030215], [0, 0, 131099282761, -39358163403, 0, 1853942720], [0, 0, -11676145620, -79511435067, 0, 673517295], [2220998996849, -2069997650, 0, 0, -191303075, 0], [0, 0, 28176372581, -12504860427, 0, 565691240], [-7451329459758, -756579504, 0, 0, 921424764, 0], [-7084630768580, -1397815188, 0, 0, 758505888, 0], [-1432151312802, 683466072, 0, 0, -278433882, 0], [0, 0, 72482789102, 80114575323, 0, 1033026105], [0, 0, 142619293106, 39400548102, 0, -2225237340], [0, 0, 155048667220, -67988459817, 0, -5000789555], [0, 0, -81815187483, 49168004046, 0, 898940570], [0, 0, -5162150501, 68420439723, 0, -3202112620], [-26625673972264, -432702314, 0, 0, 166751974, 0], [-1486635623011, -2060092494, 0, 0, 2404293639, 0], [0, 0, -126795172851, 10534687626, 0, 6594195990], [-2414294996506, -7583581232, 0, 0, -79226498, 0], [0, 0, 137690856426, -60329674686, 0, 21282800], [20738109336346, -2320717816, 0, 0, 447301706, 0], [6339961782530, -1752885160, 0, 0, -460100410, 0], [19278653480428, -4749383000, 0, 0, 431040220, 0], [0, 0, 425314066607, 17551105656, 0, 1544796180], [0, 0, 175883193943, 176838453030, 0, -1487407050], [0, 0, 199319693022, -38182951413, 0, 5120054545], [21344945324772, 3988817827, 0, 0, -593213837, 0], [-13331233182056, -4556162196, 0, 0, -623554374, 0], [21784896762978, 112237446, 0, 0, -1941538836, 0], [0, 0, 129092506617, -176128079091, 0, -3825648980], [-17114687939718, -2386346064, 0, 0, 3009367194, 0], [0, 0, -303199693954, -46067926062, 0, 7148165620], [-963167583856, 11400567148, 0, 0, 245353732, 0], [0, 0, 160391346662, 178127985525, 0, -9825890505], [14692930719481, -5282891670, 0, 0, -1304823255, 0], [0, 0, 141085137004, -110201689767, 0, 2415751275], [29150633723622, -173394496, 0, 0, -3300667474, 0], [0, 0, 39123575808, -33304101063, 0, -7205907605], [31312482576540, 526270622, 0, 0, 1366905458, 0], [-1681810248440, -1553924852, 0, 0, 1907003812, 0], [0, 0, -49979021418, 108820876782, 0, 8181505000], [0, 0, -114678891023, 62564899716, 0, 6746785040], [-12479946429462, 4791284592, 0, 0, 1402545618, 0], [-6856281196155, 2367321594, 0, 0, -50908569, 0], [0, 0, 190921633920, -149969502024, 0, 20504228280], [0, 0, -683127650894, 1952072613, 0, 5529745215], [32706088130782, 8341867578, 0, 0, -3072127908, 0], [0, 0, -725487779725, -74955018969, 0, 5863299100], [-37426076151285, 15654589342, 0, 0, -2346309167, 0], [0, 0, -777346891685, 200782671522, 0, 12621080010], [9032819478670, 14212734744, 0, 0, -314292294, 0], [0, 0, 326242373788, -339124994529, 0, -15010550835], [6571367189724, 15015921940, 0, 0, 3091882030, 0], [0, 0, -649906528947, -101699519391, 0, -9053733480], [-43066170210634, -440515164, 0, 0, 1033438074, 0], [0, 0, 1019577688526, -83272845750, 0, -4267580], [0, 0, -252889972479, -50581105074, 0, 10314027450], [-15011837195202, -6777076412, 0, 0, 1462304362, 0], [-26850211666774, 2951077944, 0, 0, 1233290706, 0], [0, 0, -165067590664, -105979116039, 0, 28355107635], [-16220822605278, 9327121488, 0, 0, 281015442, 0], [0, 0, 343569367591, 290056406364, 0, 22536605600], [-54634091455995, -19914619934, 0, 0, 924282559, 0], [-35511043434455, 3094850890, 0, 0, -1916828405, 0], [0, 0, 420888784829, 536010305010, 0, 334978790], [0, 0, 333345582200, -390768634500, 0, -29553092820], [55790901091524, -5430863700, 0, 0, -6764362770, 0], [0, 0, -939306299238, 326827552542, 0, 13967330220], [0, 0, 404726324026, 335058029955, 0, -38236257735], [0, 0, -824157821563, 346877812554, 0, 12332000950], [0, 0, -222508988953, -531780897141, 0, 5257336740], [0, 0, -72429534950, -188317834011, 0, 29119282815], [0, 0, 931972607838, -209432430798, 0, 43049998900], [63552197358096, -17171050571, 0, 0, 673666081, 0], [-39753803647951, -24305254694, 0, 0, 847749709, 0], [0, 0, -1278485360787, 318832132773, 0, -13317905220], [-57142850423015, -37555692534, 0, 0, -2965278021, 0], [0, 0, -1031938717362, 256452627711, 0, -26788728435], [91438454204146, -15559612208, 0, 0, -2597350442, 0], [0, 0, 61928451417, -344301454719, 0, -8765936720], [0, 0, -288774420152, -126235299228, 0, -20887637220], [0, 0, 754487405515, -791989764480, 0, -25096655040], [18194847619964, -29318208612, 0, 0, 4434239832, 0], [0, 0, 1463345068579, -9128445282, 0, -24782093850], [38896067605584, 6236955900, 0, 0, -18025916100, 0], [14871996854595, 31072501278, 0, 0, 821514297, 0], [-34684433964337, -45216794778, 0, 0, -7592521827, 0], [0, 0, 1136747349680, 123359517132, 0, 29013482420], [-5401273617648, 16982915452, 0, 0, 11831545978, 0], [-88525352553994, -44928797306, 0, 0, -464320304, 0], [0, 0, 601741346005, 578452382697, 0, 18245013920], [-89107788600281, 4003050758, 0, 0, 7855492037, 0], [0, 0, -2031503070845, -369876609816, 0, -8407455780], [-84310049350994, -73139730238, 0, 0, -6402890812, 0], [0, 0, 525485068784, 642987003204, 0, -7432417100], [-24603979733988, 40297133771, 0, 0, -4305799021, 0], [11095996683287, -3974144850, 0, 0, -1809478035, 0], [0, 0, -490316015467, 233388177597, 0, 12988203200], [-76416230992424, -1029479252, 0, 0, 1406693692, 0], [-182734616097180, -5662492900, 0, 0, -7731148870, 0], [0, 0, 2164064156445, -366889297779, 0, 53389371040], [0, 0, 1767710464192, 80028299124, 0, -27753871700], [0, 0, -1208702297337, 551318766888, 0, -138836963460], [-42200304183826, 57607867824, 0, 0, -4169749884, 0], [-133356338318206, 2256791376, 0, 0, -86739246, 0], [0, 0, -2616706997972, 728626610052, 0, 13092627680], [26507159895264, -45857892017, 0, 0, 20505012787, 0], [0, 0, -293024724969, -154487697477, 0, 116470435620], [0, 0, -991602149324, -413825547408, 0, -29336031260], [81667562791152, 1380485940, 0, 0, -15030125790, 0], [0, 0, -2152677331285, -853054292610, 0, 80192370], [0, 0, -1431258538802, 208552820202, 0, 1113636020], [0, 0, 2702396399868, -612829786596, 0, -18726429200], [0, 0, 671447966149, -709191341427, 0, 70465073100], [292630032432236, -3697657863, 0, 0, -2954029287, 0], [-176114892369580, 88284141968, 0, 0, -9367476508, 0], [304193546013122, 9510780192, 0, 0, -13347939072, 0], [0, 0, -131379369497, -1416506681430, 0, -120250254530], [194430807392108, -64451536210, 0, 0, 343265810, 0], [0, 0, 2200731205435, 404868716232, 0, 4302622580], [0, 0, 3619024284886, 1469172580587, 0, -51966083055], [-154190985402698, -65648420996, 0, 0, 15171016186, 0], [-93745484476997, 84873306038, 0, 0, 7669769417, 0], [0, 0, -2377798932731, 477292143177, 0, 92445091320], [0, 0, 781157466866, 526018179699, 0, -104828739135], [0, 0, 1489341073429, 66116416398, 0, 113409523730], [-189633044047934, -31036589934, 0, 0, -17266520976, 0], [442094495545468, -32721301823, 0, 0, 6095658433, 0], [0, 0, 1626673344794, 1307373126171, 0, -19454508175], [0, 0, -4580574356146, 1518053072265, 0, 134759172955], [0, 0, -3954574237135, 334931169414, 0, 31076818910], [0, 0, 1008344287511, 2011309219755, 0, -5945794300], [86247945641286, 128141399008, 0, 0, 3585627892, 0], [0, 0, -7148685037949, -231663945336, 0, 4518693460], [-28544403350026, 20318890628, 0, 0, 27539675882, 0], [69813027406363, 28869404742, 0, 0, 10306913433, 0], [0, 0, 5221860088392, -74502198657, 0, 119976886965], [434214017174788, 31574952387, 0, 0, -20138735997, 0], [0, 0, 2297553473407, 376548299532, 0, 117961459720], [-180680881541342, 107637507876, 0, 0, -19543263726, 0], [0, 0, 1784456411039, -1608944101416, 0, 108204560040], [0, 0, -5366436763004, 642576509028, 0, 27731707640], [-223671881355444, 27471150933, 0, 0, 20654127477, 0], [-44836677224644, -119492975316, 0, 0, 26231112426, 0], [189393769682561, 205932208002, 0, 0, 2513233413, 0], [0, 0, 1902590588299, 781387999059, 0, -178474679500], [0, 0, 5679339884940, 708833340456, 0, 100782729580], [-271290226338560, 93563591924, 0, 0, -13157398204, 0], [0, 0, 1029850566654, -2703500594034, 0, -163434414760], [0, 0, -112569499282, 1341510222159, 0, -30349162675], [0, 0, -11577731967675, -449446254963, 0, -42960635080], [-438768853997646, 90645743104, 0, 0, -24040328114, 0], [274735748909728, 59243062012, 0, 0, 19219757818, 0], [401530738801052, 108474122932, 0, 0, -25750023062, 0], [0, 0, 2137963687302, -1007691446706, 0, -12677916080], [0, 0, 7415190338337, 1682165584638, 0, -22101319310], [-211779191417792, 102240230556, 0, 0, -11652907686, 0], [0, 0, -5152553785277, 3784437064347, 0, 162414038420], [-146140481213377, 64968383302, 0, 0, 935382253, 0], [0, 0, -590050154209, -2671523802225, 0, 146982940840], [-496349901513562, -58928785456, 0, 0, -6555597064, 0], [0, 0, -8747691633852, 611941453941, 0, 139067456975], [0, 0, 1715462043984, -1239589022940, 0, -51171060700], [0, 0, -3183178249272, 432124701267, 0, -428764270655], [0, 0, 1497690876939, 326525753883, 0, 224504356780], [0, 0, 5350800815795, -5693271514629, 0, -331829675600], [0, 0, -2315039729701, 1568486728074, 0, 148105807070], [31101095616507, 109038169710, 0, 0, 44244887265, 0], [65673821382768, 44666842432, 0, 0, 22772331688, 0], [-11797848788424, -216284522599, 0, 0, 11045760269, 0], [-107523812892616, -170731020635, 0, 0, -33684598535, 0], [679365998341878, -83788553552, 0, 0, -8971114538, 0], [640945210433880, -82220464171, 0, 0, 28288231481, 0], [0, 0, 991715174294, 1364896713291, 0, 95147874265], [259777428608490, 27157339550, 0, 0, -64460701540, 0], [626344033731046, -64878107352, 0, 0, -17133983448, 0], [0, 0, -51424109216, -2916446167044, 0, 188903582100], [91958178669405, 68156717714, 0, 0, 26707650551, 0], [-282843281349094, -33557416448, 0, 0, 1425151258, 0], [1158248056229332, -28696858971, 0, 0, 24997013901, 0], [315462720432948, 144933383222, 0, 0, -37364906062, 0], [0, 0, 9452519369957, 1768866672312, 0, -75207086540], [0, 0, 8594647576929, -3859850616555, 0, 188623479340], [-1115994467557520, 115030977704, 0, 0, -24665827504, 0], [0, 0, 3158985707835, 1565310507222, 0, 207560542210], [-11173897632468, -23844633976, 0, 0, 66045087236, 0], [0, 0, -2131596713411, 1933934022789, 0, 293503161720], [565267588368442, -58931304116, 0, 0, 72527268886, 0], [0, 0, -6183291796946, 221727313617, 0, 167091535155], [-42696912068292, 276059151762, 0, 0, 87105999678, 0], [-4481840484147, 188477710846, 0, 0, -81552389111, 0], [0, 0, 235751503349, 3656204497500, 0, -526984940400], [1018663342571048, 65880035439, 0, 0, -11184395109, 0], [0, 0, -1697125842649, 8228767848855, 0, -67304883400], [219909205845290, 107020760484, 0, 0, -45946230054, 0], [538454108986151, 259134443158, 0, 0, -13576761563, 0], [-325881787099104, 180811482692, 0, 0, 88865531558, 0], [376165537431883, 118940611014, 0, 0, -21159087639, 0], [0, 0, -8621105716814, -2811834919302, 0, -385008771880], [0, 0, -213607967401, 882345974691, 0, -147196707540], [-612242648745022, -214060452456, 0, 0, 37985159586, 0], [0, 0, 20389760416407, 103220196339, 0, -240039478220], [130119067234155, 3877566150, 0, 0, -143197530135, 0], [0, 0, -16138087862903, 2542762748262, 0, -133954563150], [0, 0, -14364760692171, 6064700339550, 0, 481503217590], [0, 0, -4083312756106, -6815828433102, 0, -342920948980], [740650028514896, 439415722428, 0, 0, -36026799978, 0], [-539463684405277, 153586824846, 0, 0, 17866576089, 0], [0, 0, -33578645924, -4442888219457, 0, 950779605], [151503292840908, -410349734834, 0, 0, -113410150526, 0], [-1165809967881817, -18417662914, 0, 0, 102228808109, 0], [132796808663168, 340215271291, 0, 0, -57178920881, 0], [0, 0, 16390035631336, -7708315044147, 0, 124303700935], [-129733894425644, -128759325209, 0, 0, -50010387881, 0], [0, 0, -11966898406536, 5545329964788, 0, 374087336020], [0, 0, -14416652923750, 3736565114697, 0, -308173919125], [0, 0, 2021968644833, 1559827265025, 0, 737026620800], [948359915573342, -571815848112, 0, 0, 58794279672, 0], [-492905724646436, 588499718164, 0, 0, -136855639394, 0], [0, 0, -6127055276382, 665713079694, 0, -1040191305620], [425259916798244, 83918554356, 0, 0, 1176937554, 0], [0, 0, 15718961109457, -7245532834716, 0, 131167396160], [0, 0, -1796535530054, 6072689965230, 0, -43294148460], [0, 0, 57521278467, 5032035086943, 0, -85145095660], [-172829451479064, -503662627564, 0, 0, -101547695986, 0], [-790854361495898, -504243644430, 0, 0, -73402329660, 0], [-149774092288502, -65150560088, 0, 0, 86383171528, 0], [0, 0, -1828010916713, 10267525064406, 0, -35483028370], [-135860362160082, -383023830488, 0, 0, -13740356582, 0], [175307172440310, 322946299128, 0, 0, -115312786338, 0], [0, 0, -6867534055667, 6633172334481, 0, -653101863320], [-1374489471753095, 381324806394, 0, 0, -91403132949, 0], [0, 0, 23723376516032, -4954879890873, 0, -61195487435], [-627786445994, -244104858690, 0, 0, -170416846320, 0], [-851752134166285, 43211717938, 0, 0, 31234012567, 0], [2885436204939762, -37659724208, 0, 0, -31094662142, 0], [0, 0, -7495627373927, 5521309701, 0, -573996872600], [-1649626197785012, 249940577132, 0, 0, 86121871208, 0], [772766458453146, -282231718752, 0, 0, -244319823888, 0], [0, 0, 13956584526916, 2107460228964, 0, 17563592080], [312996796807026, 344866518664, 0, 0, -24861316664, 0], [0, 0, -3515313060627, 15516380749425, 0, 63351281400], [125933637760698, 315710156810, 0, 0, 214214531360, 0], [0, 0, -668731179517, 3373523257614, 0, 318832266290], [0, 0, -21724798468279, -11644787277978, 0, 528824009630], [0, 0, 3573872589411, 13888508164602, 0, -952565010330], [0, 0, 7938680798108, 8196120543243, 0, -277333092455], [51334499251201, -55459345730, 0, 0, -137091830435, 0], [509649348309912, 87487231508, 0, 0, 57348343052, 0], [770503939294316, 875428445900, 0, 0, 52794594170, 0], [0, 0, 20019889098835, 11595412848903, 0, 464245916940], [0, 0, -10881287444524, 14312556785424, 0, 781454440820], [-127471239493428, -202873254858, 0, 0, -279108873102, 0], [0, 0, 5841727130914, 18276209856225, 0, -338662017485], [1039971613315702, 219798171108, 0, 0, 141195772722, 0], [0, 0, -18254649100278, 7372580142846, 0, 662786755380], [-1944746126606203, 527464810306, 0, 0, -188109673601, 0], [0, 0, -3179015572510, 9872164967619, 0, 397204849505], [-326767338924490, -442724116092, 0, 0, -31142272038, 0], [0, 0, 36012589792109, 5909367888021, 0, -74064621160], [50890089589558, 643813129744, 0, 0, 236040449266, 0], [-204946034406738, 942799075604, 0, 0, -211152389374, 0], [0, 0, 17512758602262, -12979107562542, 0, -1508723945300], [0, 0, 25218925657343, -1487076014250, 0, 228620175010], [-161878859853342, 244127224440, 0, 0, 18588110610, 0], [1762216778433649, 40479213714, 0, 0, -274889007909, 0], [0, 0, -16793400497110, 2801204506053, 0, -221788669825], [-1777026323949978, -317803834192, 0, 0, 163451507192, 0], [1495254610074980, -24147859004, 0, 0, -208168718936, 0], [0, 0, -18504507643485, 5895392094666, 0, 671265883070], [0, 0, 13427388967172, -10364924088033, 0, 2166053310485], [0, 0, -22240561395282, 2466242144826, 0, -1425571782380], [-2901157808109362, 51208069456, 0, 0, 42557154304, 0], [1525504394965231, 223331988350, 0, 0, 277559975975, 0], [0, 0, -24110644174865, 5916784722927, 0, 197439553360], [0, 0, -35531195939967, 14611770826989, 0, 1085221851900], [0, 0, 16043233025725, 5617368872970, 0, -1064868881810], [3222148663676547, 139327031310, 0, 0, -129640084455, 0], [1751658268656900, 1239789691547, 0, 0, 45848897963, 0], [-4463433722599418, 12364640016, 0, 0, 42329456454, 0], [-1633813280098124, -228128868812, 0, 0, -95425763288, 0], [0, 0, -29470655205968, -3167403953487, 0, 2358984595], [3206097613895820, -13056256110, 0, 0, -165531471810, 0], [0, 0, -32460445951342, -43095510702, 0, -1928833311720], [0, 0, -33449547898399, 2982919651134, 0, 789140050270], [0, 0, 14556071071316, -3386403592992, 0, -606184445940], [4548214751013999, 121664265862, 0, 0, 65635860973, 0], [0, 0, -18729828542242, -7281891570099, 0, -1539004183905], [625923864766932, -181924940800, 0, 0, 154739802980, 0], [0, 0, 20597191965167, -19265775984201, 0, -1372063636740], [0, 0, 20955856458767, -8452019274228, 0, -596065020340], [0, 0, 47086009354566, -10210041490629, 0, 1648313551625], [-2124176663609328, -788985732941, 0, 0, 201202101031, 0], [5690489258487440, 566920474284, 0, 0, -6511543284, 0], [0, 0, -15567959781681, -15321413190537, 0, 1210917108100], [0, 0, 30244293930714, 9162252537450, 0, 166099607760]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_98_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_98_10_a_l();".
// 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_98_10_a_l(:prec:=6)
    chi := MakeCharacter_98_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_98_10_a_l();".
// 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_98_10_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_98_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,10,sign)));
    Vf := Kernel([<3,R![-18134891776800, 0, 2418290244, 0, -95324, 0, 1]>],Snew);
    return Vf;
end function;