
// Make newform 91.8.a.e in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_91_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_91_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_91_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 := [59402280000, -835334324470, -344602049215, 491049966, 10752389517, 186018392, -117081910, -1739560, 567554, 5598, -1243, -6, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-209, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1679148611865916382256857279040, -23436379137177392576984407981510, -3683784470633856804373932985433, 951549587174881606995743281219, 66349192792533587939476333054, -11698074756976265164966939470, -432682308525274351930308880, 60895066450672328807569768, 1178692702718681078134506, -139797096343619060307636, -1138016347474817517247, 116429657912251546397], [3446270623281713659494357549120, -28939668772391818786596548000150, -4571284511998824301781850392977, 1187070934316815045629774870251, 79911352078036297609303854446, -14512440979280355872025645630, -510859156602059802197533040, 74888085287797374072248072, 1376189494391125315953594, -170525503250158274999604, -1321368208330912848743, 141031799835156954133], [-9168059273764298644104549965760, 145824718509625566957099538645930, 24048047849053299449847971927951, -5644056424542822771207113820373, -433045831613775202229351305138, 68087284560539008509126878370, 2799096589087306991508056560, -351936065708782709495083096, -7542437021495047512502182, 806513670413135042019372, 7201001136259246898329, -672297715825811331659], [-11995431178475375449376816873280, 114045612110390167858989308805310, 20745653678817745884840227142221, -3991627290446877087145103977183, -370139735993344499256041019478, 46003512553343583784819821990, 2343747323291820081413066320, -233456620972939522657384456, -6171321615378891390862002, 532198440675710379396612, 5757076556160489338539, -444215779571142725249], [1048418219356569337167844866240, -9314994513411804651098692684610, -1540810224746301919403569695571, 375001404612388190581224707153, 27744496546709916819675850058, -4597163497409732536978558170, -180343773913668276713264720, 23928732437573602807670936, 489020560147228722315822, -54989797161806797472892, -469732263709633455509, 45868994553761432239], [-3479949461714552153204912293440, 47453870473811818471895339682310, 7477561322500770210009917648729, -1932224301429983448516094511707, -134393655495755014587681106282, 23747945291740541668316217210, 875136723060146885144020780, -123539260255985288273224324, -2382072504396417685996398, 283435243550555522451768, 2298951677556646950931, -235934583564866268761], [3937168746418691106708027855168, -53820802387928806951216943909278, -8666771897379350497491877262117, 2133923567171396736435749146939, 155842610229818604650945057350, -25989152392685012223846362958, -1010754805819621938004777792, 134793767492426742679126360, 2736443413617789664855170, -309088744126651132111836, -2625872673146483638435, 257443220169661359413], [-1404670124409058383656692565568, 22703000562199630555553247973894, 3688010565269392815003437362961, -887222437218943234060039931767, -65979650366847306993437667382, 10714468327424970704488511406, 424956174241481052496767688, -55287477018474159628548928, -1143559512197128739100690, 126396278362775742892884, 1091928376244149378735, -105097028809302622289], [47423945977332090939100555448640, -689711203535751280755640835858990, -111315596224730355848768113995877, 27197561210274725383179521872991, 1992415007509288983913547114486, -329657296467709068917868022230, -12861517571267114511289085120, 1703570694832613150614662392, 34694386975894305181994514, -3896559705367117031102244, -33202403758684091513123, 3239997280258850960593]];
    Rf_basisdens := [1, 1, 1, 2520330500171562437167994880, 2520330500171562437167994880, 2520330500171562437167994880, 1260165250085781218583997440, 93345574080428238413629440, 315041312521445304645999360, 126016525008578121858399744, 42005508336192707286133248, 1260165250085781218583997440];
    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_91_a();"
function MakeCharacter_91_a()
    N := 91;
    order := 1;
    char_gens := [66, 15];
    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_91_a_Hecke(Kf)
    return MakeCharacter_91_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 := [[1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [7, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [87, -3, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0], [343, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [992, 46, 8, 16, 3, 0, 1, 0, 1, 0, 0, 0], [2197, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [7062, -302, -2, 52, -6, 12, 1, 4, 1, 0, -4, 3], [-293, -1093, 9, 52, 20, 22, 4, -6, -9, 2, 3, 9], [7833, 719, -62, 99, 42, -8, -15, -10, 11, -16, 12, 10], [29316, -131, 213, 401, -101, 50, -18, -12, 39, 10, 7, -34], [5013, -1259, -146, -1333, 64, -24, 30, 4, -2, -16, 20, 15], [30816, 1514, -831, 135, -368, -26, 25, -40, -16, -10, -59, -11], [114183, 2580, 766, -525, 392, 188, 51, 20, -81, 64, 32, -55], [47309, -167, 544, -719, 147, 272, -1, -52, -139, -44, -2, -143], [108490, 10363, -785, 1419, 506, 258, 89, -24, 24, 78, -51, -240], [169261, 7163, -4083, 2234, 50, -622, 15, 198, -56, 214, 1, 80], [19019, 27100, -1753, 848, 260, -1134, -235, 92, 46, -366, -65, -237], [17199, 42358, -1510, 3389, -2780, 796, -167, -298, -103, -372, 238, 495], [-266779, 21612, -5342, 4077, 642, 1052, 107, 578, 287, -604, -2, 189], [481431, -15448, -239, -11872, 270, 1342, 502, -320, -865, 26, -101, 496], [536053, 95355, 7401, -5200, 1649, -1658, -254, -420, 1047, -62, -21, -662], [362150, -10071, -226, -1080, -537, -1744, -850, -208, 392, -724, -794, 258], [208126, 29793, 1348, 11664, 15028, -3896, -81, 770, 471, -532, -126, 520], [-983622, 129535, 17781, 11401, -3399, -2458, -951, -156, 2136, -866, 1033, 83], [-1958825, -145141, -4413, 19600, -12721, 894, 1080, -1132, -1249, 1102, -167, -1266], [1991337, 282380, 31427, -103604, 2021, 5742, 500, 1172, -1363, 90, -113, 2937], [-528010, 22166, -12947, -9595, 6735, 4810, -380, -2114, 199, -910, -889, -829], [-2086531, 250974, -5074, -13161, -699, 6404, -1637, 2478, 297, -1640, 664, -526], [-1616777, 318358, -21118, 36729, -4645, 120, -276, 3386, 5804, 1556, 1786, -373], [557738, 307167, 18761, -95215, -10932, 2222, 398, -3136, -6747, 2422, 749, -1747], [8528770, 657814, 32612, -19962, 11375, 9068, -2542, 4162, -3296, 2224, -412, 5297], [14728450, -303846, 70167, -198605, 24071, -13658, -1594, 5482, 727, 662, -1163, -503], [530923, 168984, 17202, -91107, 7999, -6360, 3068, -13958, -10336, -1140, 2534, -3543], [4574312, -413136, -41453, -59325, -34533, -24606, -4371, -1242, 6324, -2310, 4251, -3024], [277200, 520086, -149856, -200696, 1509, -6432, -1393, -10874, 8391, -4212, -3778, 5310], [7638141, -909618, -26060, -85913, -1427, -36944, 10066, 2570, -820, -920, -5576, -6661], [14447547, -1086032, -115816, -337243, 35212, -6424, 1796, 894, -8628, 1276, 942, -3558], [13442040, -999922, 61741, 353955, 8503, 2470, -4364, 15158, 22295, 4082, 2959, 2599], [8562699, -493461, -138713, -376390, 27492, -15738, -7216, 870, 15449, -11706, 11865, 5301], [25426568, -1810794, -24772, -528324, 24876, -6720, -14045, 4368, 12641, -10692, -6222, -13749], [18827565, 1193717, -135366, 401547, 47027, -86968, 7147, 7052, -14053, 14032, 2852, -25819], [-1580219, -2238716, 190651, -25200, 58949, -40562, 4900, -1668, -10083, 11754, 4719, -9991], [614095, 883518, 52110, 959745, 121851, 8944, 961, -14758, -17009, -25144, -15156, 22282], [29551958, -2395392, 26742, -997842, 40728, -10684, -11842, -11356, 11816, 5564, 202, -11778], [25935913, 885114, -182717, -538580, -65277, 40294, 12721, 16498, 11172, 170, -6797, 7414], [44187008, 2814556, 156386, 326896, 37692, -41504, -15953, 6640, 23719, -6408, 6704, 21433], [-48276178, -1560069, -20908, 1053846, -71244, -38472, -4932, -21366, -12846, -4296, -7236, -35151], [-10561697, -6262101, 215296, 845051, 4106, -86736, -40854, 24572, 37388, -7668, 27058, 8503], [16844566, 5153848, -237356, 1149366, 240570, 60556, -3900, 24380, -7684, -10168, -14316, -11042], [52642185, -1566708, 123259, -1009100, 57736, -6990, -37881, 2380, 66160, -49590, 599, 33725], [-55732663, -248645, -299241, 368658, -170527, -46822, 16781, 42274, -30386, 49174, 6453, 23813], [8996140, -81242, 377001, -1837981, -200079, 74686, 24012, -5178, 12231, 9530, 34703, 23713], [35037585, -4728357, 17644, 801649, 285163, 20500, 14440, 13556, -10048, 14968, -32156, -25870], [59419435, 450206, 138418, -644289, -126938, 1668, -20230, 3576, -13696, -30612, -12798, 72282], [57308814, 2882990, -479089, 392863, -180095, -64154, 9133, -15342, 8056, 24098, -13285, -61340], [105659131, -8960953, -680496, 1016339, -9907, 27300, -20751, 29136, 20235, -25932, 12342, 56439], [78198615, -1561082, -89977, 608828, 495655, -156930, -11827, -34156, -66660, 52134, 26977, -43416], [-60596769, -8035522, -113022, 308295, -133010, -202744, 27023, 9426, -6465, -1812, -6894, -72509], [18772054, -6382969, -406301, -220511, -543631, 54186, 35652, -25108, 40295, 56470, 38149, -5316], [14481685, -8493610, -234996, -1452107, -143411, -159964, -38052, 582, 61738, 54040, 39592, -86239], [144364251, -1770174, -591045, -2052872, -429473, 586, 15367, 7706, -139186, 55234, 29023, 65336], [37979616, 10245681, -193825, -986091, 113953, -31622, 97677, 109432, 55842, 96698, -44729, 20029], [-148716294, -3399119, -1097188, -1126564, -23180, -226680, -52377, -156494, -89993, 9212, 68234, -30384], [74956764, 17112164, 676899, 1670623, 474553, 155658, -56095, -95450, -49940, 72378, 52703, -28962], [-83286892, 13620808, -2618356, -3930260, 534720, -179856, -97359, -40992, 42399, 15380, 53898, -70085], [-36130431, 10874472, 2025101, -97198, 747537, -217654, -187191, -9950, 15326, -122546, 10377, -24214], [55113753, 28396276, 1680898, -3486091, -372686, 10124, 28301, 152466, -95619, 132028, 17706, -21429], [-246483664, -7408009, -312191, -3477265, 46542, 122986, 90004, 143824, -75797, 35726, -83611, -15739], [147674460, -9891320, -365018, 8778930, -345890, -397052, 30156, -128356, 13158, 70964, 12570, -208658], [-231338752, -7621137, 1511143, -1240791, -1277535, 237170, 13631, 68972, -144988, -112558, -92069, 21861], [207744648, -10181488, -1994981, -3234477, -119114, 151086, 60646, 41596, 146625, 1302, -27451, -40632], [-17460581, -18127700, -255963, 969612, -24162, -589738, 69812, 132688, 119189, 51994, -73725, 187454], [-8678002, -14190160, 715351, 819905, -70167, -811062, -148137, 16818, 218658, -129566, 4659, -72856], [-327032265, 11079226, -1499360, 2023385, -588887, -396300, -86817, -16274, -259433, -115876, 10454, 179202], [-302251761, -24875164, 1712166, -6262341, -567419, 625412, 94154, -86838, 162210, 73188, -9990, -69071], [180956669, -4666250, -3146758, -3979063, -656204, 526864, 285055, -152754, -261669, 64268, -196462, 58309], [-231327529, -16205482, 2725358, -2407273, 1700207, -2936, 79881, 12298, -263569, -133120, 209644, 82222], [-31240490, 12825467, 1945583, -7184273, 934481, 391738, 1243, 95328, 243096, 114698, -92841, 359577], [88438748, 4074108, 1631963, 6465203, -1707527, -134838, 47124, -25358, 392189, 83214, -29539, 104955], [-210905584, 13435211, -103739, -6848305, -177321, 686106, -101361, -434628, -385554, -82202, 93297, -280357], [103731321, -12632170, -962742, -15257711, -829648, 185700, 9054, 188100, 153256, 153036, 81426, 278052], [-676700968, 13678734, 3888553, 467475, -485094, 56110, 141451, -171812, -379958, 324382, 19529, -284859], [72652462, -28185024, 4326329, 5081565, -359233, 1143262, 85664, 42554, -266509, -233470, -112101, 135913], [211073418, 16456160, -1242387, 10282923, -106805, 783298, 302818, 296702, -490555, 468194, -80441, 87307], [-109603544, 51533660, 2255203, -14816075, 523319, -1281642, -427560, -6910, 632669, -116402, 240625, -168687], [44190960, -58172649, -761976, -2133160, -73372, -980388, -380298, -329166, -340164, 43824, -120660, -216009], [-821433898, -5767428, -1033953, 5139445, -1532619, 179626, -149048, -396910, -326739, -334906, -50007, -97589], [-408449047, 34251446, 2387417, 10548572, -3029418, -1810950, -206544, 100032, 416937, -414614, 243243, -25000], [863034914, 8573384, 758700, -5149056, 289762, 1209020, -344866, 57628, -35382, -277880, 247296, -15268], [-409753331, 14373588, 1534354, -2077435, 54659, -830724, 293256, 363122, -429388, 236316, 78214, 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-2322641887, 610253987, -258030400, -13086436, -109010378, 82945348, -29265124, 48174422, -45132983], [71756066070, 13502935484, -399730836, 642455518, 149968073, 266074816, -68189598, -17433240, -18592296, 70312244, 29340530, -32814353], [113571312213, 27052512068, 1540656489, 579907206, 690243612, -465296658, -18553861, -66535880, -20942990, -147869970, -92085907, 155861937], [161945010682, -5650284518, -753931142, -63976536, 356537495, -386842112, 27806062, -84579486, -100150302, -6535620, 24627750, 108501329], [-19947704750, 19932913742, 2360453327, -2475384505, 379185206, -232278302, -173105359, -9343468, -33043398, -101990686, -29574609, 27378283], [-4745220455, -33090400830, 850496954, -285711389, -160448491, 69814096, -8210531, 4645218, 70571651, 30013160, 62837900, -167298188], [171285190530, 6200106592, 196043440, 767212058, -471244046, 11773776, -70654827, -112019192, -92255933, -87936340, 30139490, 31041753], [-179885355903, -994587596, -737522187, 1156132394, -210837056, 185929526, 22890200, 72394836, 163887069, -43180886, -74375037, -76786512], [-42623293910, 3260806265, -107476901, -2679895757, -5014488, -919085894, -104970639, 82924220, 117538324, 21333086, 16717389, -197419928], [175635845205, -15164711468, 1414910201, 1076557906, 640148653, -110579254, -18892218, -68474648, 164351977, 13053286, 102755025, 80229395], [-411365059831, 2783703235, -1433823825, -2309474936, -205768308, -6548414, -71339150, 25987342, -90101177, -110460882, 13539773, 89569447], [227106840459, -4128389628, -8486909, 1531865464, 790764724, 190327286, -29530132, 27440400, -39868065, -131074978, 52319097, 162168752], [63152230242, -8813805592, -325543348, 1825551626, 798399473, 38593360, 202391983, 25971210, 13421455, 141778820, 25870322, 83953408], [154788884500, 6765055684, 2551582857, 577794507, -459269695, 219850770, -31912110, -212037910, -242230195, -2570230, -8756213, -16387783], [-160928554169, -5122956014, 139652136, 3716849853, -259834813, 109059000, 118210191, -89897734, 51902555, 41115620, 45167878, 65955164], [137577607936, 25984059300, 1575703220, -2306182624, -715269530, -168447144, 48006107, 90089000, -193473831, 126488436, -47761814, -18679101], [-147097037945, -20178070225, 844953566, -5971629523, -873213232, 29862520, 106693958, 136802804, -48166590, 99934952, -187422408, -37790255], [47225995300, -18853729427, 1463495840, 1262813206, 1003001752, 482796580, 81206653, 173002818, 52724053, 87407876, 34579778, 21499774], [-13798989396, -14193004288, 2090397430, 6330316974, -424962381, -96088848, -103894860, 51017972, 188523324, -63187176, 30205708, 69688155], [-93336693091, 33715233030, 765466133, 2089098898, -168372488, -70461746, 10682674, -41846484, 99980383, 61853642, -13371013, 188089360], [396614425702, -3698854340, -103224540, -2348485380, -313307476, 492085664, 85620914, -13517708, 22214338, 106469584, 66367460, 92057162], [45764807078, -10836516782, -766952579, 4188426247, -731343669, 130615722, -59756376, -5691538, -29573509, -1032846, 52001647, -70050729], [107936942602, -38412546771, -856425732, -2655639762, -349502849, -57060352, -64269394, 22238740, 143855886, 19617304, -16455564, -72154690], [117735305292, 1133020934, 2162190917, -2600235429, 97338158, -14376850, -570807, 20430610, 66238080, 93197098, -74742009, 154796741], [23815282895, -16767103834, 463440750, -5801313949, 248571738, -309588316, -78938716, 36765728, -124382662, -110559780, -49807694, -124575052], [-81031600529, 9461429783, -2181645, -4080563564, -316859221, 684623574, 1330995, -57480694, -91310944, -51237338, -1457639, -68778423], [138418216635, 10527989485, 609877266, -808288931, -110850943, 284887888, -65812270, 21748480, -3590852, 155378972, 170488942, 33982540], [23928919377, 14161018122, 55679071, -946969172, 1223359960, 19112242, -71710610, -115642060, 217387733, 40452718, 113927761, 171108632], [334749728430, -13362436251, 1873745029, 7914711553, -200286644, 171608978, -87412138, 51654528, -47480595, -124848082, -67032747, -27861601], [-507032093865, -6828695138, -573457408, 811755507, 97429359, -672469732, -10487235, 74477650, -433907, 75605212, 5733882, 92406218], [-201322832157, 17970422636, -1159680611, 638578056, -868855943, -343161798, 150051282, 76468330, 146896177, 168899362, -16359577, -163316423], [-302864805793, -19014768832, 112376367, 2906969602, -1066611662, -175002542, -49099202, 79034384, -181287779, -177115690, 81194013, 24765076], [186413808466, 13380031269, 391608825, -308407061, -838272443, -60300610, -40699466, -192211936, -8858865, -256953254, 87389435, -6388000], [-129269689462, 11463576318, -555391485, -229660131, 726427497, 204556490, -26530939, 41031590, -114318940, -2010814, 181699435, 232929108], [-63638270748, -24015195839, -1533095521, -1815818509, -395198807, 285186794, 95337368, -22042308, -80169621, 233581318, 16539765, -193938270], [-142522942496, -11346160108, -2951942714, -8638452118, -211825466, 121475748, -141489150, -130276660, 11014592, -71592228, -42986810, 88713396], [63389688774, 16832272994, -46754754, 2337636620, -661748969, 1061194932, -65593124, 52998926, -60977216, -62476500, 72006922, 255769227], [-369176100580, -10980267428, -1276398307, -3557338595, -820578384, -238547014, 54701198, 121609736, 201356207, -158171606, -108450941, -3352478], [-328403494916, -12706913324, -967048945, -9181518373, -178322239, -93185714, -6383126, 361905630, 239900779, -94520530, -139371103, 45842101], [-69573544076, 46689783332, -2697897372, -1033909118, -27237616, -386826440, -12423950, 255560988, -51995142, 112293272, -58254372, -17717202], [129682002147, 11242413345, -191621236, 3625613195, 568805071, 1021311340, 110086162, -235346940, -116598078, -39370000, 44214596, 57750946], [-304785720092, 7410574696, -3409565090, 3367561136, -5552566, 683853268, -571488, -85974412, -141848638, 1723292, 74193810, -53751914], [132044304463, 20606061182, 187597342, 6436485355, -760037983, 36464700, -139259454, -187225054, 79772222, -81120404, -72254906, -99744585], [41509648375, -18823918968, -3014744346, 273556329, 1535134249, -57984416, 233581412, 165329818, -35466968, 145096460, -127017242, -88159181], [-333434384977, -8475330303, 2178029489, -6068204340, 146940162, 854177178, 59589864, -150347046, 111960747, -47824710, 15205683, -22799619], [-126714161250, 4079439033, -1239878858, 67741678, 1092712662, 284754716, 209896983, -3659158, -417738555, 161737600, -33940300, 10809746], [-184117165683, -5802992510, 460337188, 12976769123, -686807189, 811573912, 128687806, -44973334, -127972924, 80075568, -13067036, 76238241], [322580388587, 44749920598, 395828683, 8581107292, 924535569, -437068666, -32441521, 65575196, 54407978, 45029854, 36495445, -61075736], [4179551035, 8613506476, 639293815, -248261754, 1015228040, -497410694, -95434388, 148860396, 164004579, 48525918, -44265687, 105124604], [-291204089952, 3019470230, -4765595113, 2283895419, -292310009, 137114410, 95209991, -11738794, -254983046, -56410102, 55384287, 121945648], [29430059142, 37475994394, 1326083893, 2358546851, 2232840871, 422567510, 93779012, 103996950, -103484769, 44710714, 90504375, -129663093], [-101700178111, 3647809187, 5250657558, 2068100151, 1196691701, 452187120, 70630458, 83429664, -87930028, 68447388, -107593554, -170318202], [-120657117425, -12503277358, 1487473002, -3672528957, -746028554, 6839020, 64307137, 1536194, -526258387, -26379532, -77307906, -16150439], [-96715882524, 50818559056, 2726658272, -4076640436, -1550392724, -404070816, 63336378, 173509624, 120088150, 43611176, -94349956, -13352382], [73160106430, -7466902288, -190733770, 2119637456, 284806730, 1062209260, 46343222, -332169772, -399822636, 172010564, 87756510, 63486136], [-97640965142, 42274683195, 3401417197, -2270434111, 571670630, -287509950, -9410970, -170623780, -93207787, -43518762, 10529161, 153213355], [-282944592229, 28193396430, 2404692170, 1108389483, -367514034, -78299332, 91148399, 77891822, -295928209, 75624236, 29249518, -62765421], [144302303558, -38569558318, -2972673047, 7854132463, -1061516263, -192817782, -83079618, 7594754, -46942355, 42435618, 170764039, -215136909], [126769189821, -11624882589, 21449222, 2005531783, 843872215, -325840192, -62823949, 110655072, 90731259, -50033416, -101704608, 20376059], [553255499188, 1771926700, 310426871, 11463968863, -980148953, -305684742, -67785050, -127051818, -63068845, 77297334, 71168469, -191206845], [312728196098, 8839643113, -1313630831, -7953703055, 330072183, -105872334, -122348517, -137736492, -100354422, -129160258, 41291553, 173095765], [-2568091803, 21604963459, -1580461779, -1351833418, 1311820424, -233276510, -66057665, -177356206, 209590444, 205166, -46799175, 59565154], [230793419491, 25474056832, -2406119302, 1450576263, 551130473, 155417460, 212518914, -86742550, -190537474, 142449548, -131959802, -100822135], [-327075380572, -105048057, -309828704, 745142268, 44279473, -786547440, -42480029, 11278540, 89685009, -128835852, 90567686, 276628137], [580662756976, -5520682003, 746336409, 9030960159, 198471985, 761623026, -37337873, 149097428, -116114958, -29306154, -79056563, -46427049], [-337296136552, 3032588892, 545930582, 713749032, -163634124, -701934208, -116764717, -227274268, -37470937, 95588936, -15817400, -193163631], [385493983271, 51337133330, -460968536, -8128619263, 602812464, -426237012, -105644277, 81036696, 281146941, -70224644, -7962906, -314385073], [26809929925, -2712217040, -390504426, -3215740125, 2146465763, 471737816, 5466640, -324725146, -174959656, 102735340, -55526386, -47296195], [-360875348113, -2972962875, -2994315139, 65629404, -739079310, 174664354, -121291520, 102283294, -57901577, 32028778, -61107373, 15168885], [-371217416069, 2354653969, 4138668257, -3760041000, -779602706, -204557698, -24871330, -140596078, 126202845, 83081434, 97128239, -121666129], [647456718730, 22569474438, -3067927660, -337441684, -575720728, 227796816, 29108556, 214149668, 93156524, 141757224, -97111100, -156420170], [495470382213, 15759938465, -830198109, -3521821556, -634557408, 85663130, 96590149, -44690458, 872312, 163874386, 148579031, -242446870], [-61047257880, 23293697645, -2107590326, 5396791502, -3256360752, -221573936, -162937142, -139421186, 256430390, -266697100, 78106, -118836719], [266129448665, -1302971238, 1116167338, 10496183283, 790030713, 69952292, 139518131, 227831138, 51718445, 50517128, 119472640, 120056334], [79190221644, 21766625545, 808984177, -1765489213, 341818540, -901751774, -185027522, -2971900, 381783785, -15194514, 88367221, -165537249], [382922863526, -8155355722, 578075170, 1486043336, 1099282772, -179922188, -145537285, -31113744, 89134791, 318056, -28476076, 125750155], [-158802835826, -1414385638, 4624181785, 3538031741, -1374502221, -790968090, -149923606, 426405450, 347759933, -110858286, 47732115, 46017213], [-22758280280, 39909334158, -1351682651, 5833983503, 1886950139, -205505806, -77390308, -18139142, 158789383, -2656870, -58933501, -215554037], [268732695337, 17092913248, -425415958, -1084285723, 30159910, 637901032, 83575071, -439012436, -102575021, 62600360, 44132992, 74703187], [-587391122588, 4973434720, 2930032028, 21799141822, 1214009880, -470734852, -198791757, 29767228, 251224653, -66958316, -47197470, -8212855], [297512557615, 4427327123, 1662840389, -3228300968, -2628190058, -442681146, -103971385, -138060310, -132374096, -51390186, -16865651, -3523572], [221264446481, 14905417392, -444727651, 7388898486, 892800648, -381225242, -895577, 1793296, 29818858, 158221414, -93287479, -52365675], [122272219755, -8137465772, -2758911013, 1331580914, -3011961688, -575849126, 57876882, 189307708, -119495935, -185588530, -82132727, -95175482], [274009922672, 8076142006, -40442321, -3189296825, -1379394001, 1157684154, 258711785, -330882582, -417077128, 115621842, -102529185, 215653818], [-786748521680, 16475629764, -477179016, 12922651970, 1943077682, 521131768, 4824346, 269787144, 192572082, 207620392, 108779040, 305342980], [330961009194, -5342211271, 3076264843, -6344036073, -134819987, 172454206, 70638931, -56106296, -103486768, 23819474, -117257105, 431711959], [-422111909809, -3761031511, -2777161474, -1194123259, 1776081721, -56027332, 164308517, -250271608, -155810663, -227034680, -97355116, -183867811], [227365712480, -35217264252, -880898729, -8056541329, -1025567099, -1209926278, -111440590, -10682166, 93876523, 38298094, -24408875, -416450935], [54751573963, -14924559743, 1749542477, 190266072, 556581716, -24325318, 116373896, 319327162, -51169805, 280365898, 101614475, -229107457], [-249343928328, -25435849766, -1197045600, -3111625032, -1783016146, 210830880, -42794520, -55285248, 27507296, -154944048, 43488528, 205679640]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_91_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_91_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_91_8_a_e(:prec:=12)
    chi := MakeCharacter_91_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_91_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_91_8_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_91_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<2,R![-1109222885376, 1479835713024, -278531100864, -42990372480, 9905539840, 520830672, -113159964, -2852052, 561506, 6942, -1243, -6, 1]>],Snew);
    return Vf;
end function;