
// Make newform 208.10.a.m in Magma, downloaded from the LMFDB on 04 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_208_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_208_10_a_m();".
// 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_208_10_a_m();".
// 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 := [1558753465556664, 60107523849756, -44131402568826, 598076469, 5213582205, -24398, -135356, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-33839, -4, 1, 0, 0, 0, 0, 0], [25171873882561838510431704, 16984478171516176633383180, -38633733783377335154142, -1978975850824891444347, 3075474092645384840, 43606952374955798, -39044211581394, -253915866823], [164157322886542727743525524, -8421975206264412446491260, -16412918673655690975647, 970558562520558500148, -936624278573222210, -21645710670467672, 18240898422141, 128047192432], [301796847905241257012257164, 920829405167636301889782336, -261345093685128067302693, -102527567025158164565919, -3827971026661375066, 2677358148043976414, 193965141915267, -20167518840415], [2183292927255433174954289946, -351483117751082695028206500, -576124374229498514487753, 40919206889539567545237, 25704280413317710870, -1071531612430390298, -294583135275621, 8045120983423], [-3003945246374967062553055788, 2086913499159241514096780112, 2417792523035358240292437, -273215632144996444689945, -232223784629747428342, 7899539280420562226, 3338720387277165, -64477563979993]];
    Rf_basisdens := [1, 1, 1, 65409994973554708417200, 25157690374444118622000, 109016658289257847362000, 40881246858471692760750, 327049974867773542086000];
    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_208_a();"
function MakeCharacter_208_a()
    N := 208;
    order := 1;
    char_gens := [79, 53, 145];
    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_208_a_Hecke( : Kf := Kf)
    return MakeCharacter_208_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]];
    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, 0, 0, 0, 0], [-18, 1, 0, 0, 0, 0, 0, 0], [256, 2, 0, 1, 0, 0, 0, 0], [301, 4, 0, 0, 1, 0, 0, 0], [6639, -3, 0, -7, 2, 0, 0, -1], [28561, 0, 0, 0, 0, 0, 0, 0], [54440, -685, 0, 118, 2, 3, 4, -1], [54047, 964, 12, 103, -7, -5, 0, -11], [139980, 1917, 0, 432, -7, 17, -14, -24], [-237311, -4436, 27, 669, -144, -42, 64, -18], [-657032, 14087, 0, 1939, 37, 65, 95, 77], [-54843, 4111, -279, -1635, -530, 165, 0, 65], [1502037, 43508, -153, -589, 220, 170, 0, 358], [-4024524, 21229, -120, 4759, -855, -174, -322, -583], [-7223355, -43444, 360, -3803, -2914, -6, 322, -229], [34112719, 63284, 585, -7003, 264, 1490, 220, 306], [-6297401, -135325, 84, -18451, 520, 1430, 0, -353], [17343727, 227014, -1287, -20229, -7904, -288, -324, -1652], [-66083649, 206664, -1200, 25689, 5403, 1443, 3024, 1127], [-42026153, 425874, 3156, -43826, 2547, -1738, 1216, -378], [78998029, -378830, -4137, 2963, -4640, 308, 224, -5676], [-108001692, -349080, -3180, 3834, -11840, 6436, -2482, -4518], [-190132879, 300260, -3156, 40703, -14207, -593, -1134, -4451], [111792152, 371710, -17646, 257084, 17084, -4406, -8648, -458], [122778081, -633070, -13509, 106639, -63452, -1940, 8120, 6192], [526282098, 1661302, 8604, 283248, -58000, -14670, -14140, -9106], [-66267098, -308342, -1740, 160206, 57334, -19388, -1134, 9638], [-243225161, -2267609, -26796, 311702, -55668, 2998, -6561, 19566], [435302433, 469285, -7779, -260405, -126146, 15407, -224, 11775], [877722630, -3092984, 7476, -294776, 59080, 22024, 27412, -1948], [387791194, -2009567, -51024, 462578, 174313, -8677, -19936, -59786], [506847086, 4247003, -27948, -1879383, -134547, -54842, 14, 17091], [646190585, 4490888, 4755, 983047, 498832, -7410, 31640, 51886], [1624292046, 8092187, -39504, 1431582, 62602, -6112, -4496, -62174], [1709562222, 1523572, 92184, 1288940, 139380, 28580, -36360, -6520], [1952317841, 9509974, -118764, -1161759, -289266, -11558, 11326, -129717], [850681959, 20983616, 25185, 976431, 203572, 1630, 63040, 19258], [4159289927, 7581479, -156, 1301859, -780484, -77772, -40176, 27897], [6027313462, -7562897, 137520, 186505, 367591, -75217, -4481, -70265], [3184439250, 19222588, 103644, -4957458, 953824, 40476, -55932, -95444], [4968405880, -8428659, -121284, 974660, 824746, 45668, 10864, 42840], [-997935484, 73335762, -22782, -3456798, 161768, 80814, 78352, -46654], [4669518444, -8513377, -256764, -4727838, 2301261, 57969, 67952, 179226], [7429048490, 9316872, 301080, 3034202, -1170568, -55000, -103320, -240364], [-494139147, 71401409, 6597, -2630671, -596722, -176953, -67984, -194593], [1862776172, 7486292, -531912, 5135046, -208848, 95870, -295120, -133422], [-4086655446, -22673107, -328404, -1104261, -637567, 373248, 101934, 545737], [16155631271, -37346942, 1074000, 17179173, -275186, 224564, 126882, 685459], [-7815688317, 58511119, 553272, -11785169, -1156882, -180444, -138222, 336113], [11788603008, 5009568, 116340, 45437, 334032, 104174, 401072, 267426], [-1421499731, -62055946, -260043, -17086796, -2545740, -114562, 66624, -892778], [-19121995417, -73030206, -578628, 2655818, -773823, -80504, -429952, -212126], [25531624476, 20572868, 1238718, 2302274, 149408, -29044, 395408, 560232], [-22093353753, 59664127, 592560, 12048976, 2797744, -113310, 936685, -564352], [-5650475138, -16524795, -687282, -34618, 4894610, -621283, 170016, -63359], [-14749428012, 72940143, -589164, 38180604, 4524661, 649627, 124916, -43088], [-20412507185, 89671202, -1233555, -9493397, 4457164, -208600, 874640, -133856], [-23112145195, -419104260, 149844, 16853535, -983504, -514702, -279646, -236355], [-5446867121, -89047806, 1383945, 15023571, -2445304, -33396, 1093012, 739648], [-16205307056, 228916706, -1241310, -15225112, 7661204, 442014, -1438528, -900966], [-20587044269, 169260737, 775692, 10149540, 6695674, -1752096, -88403, 981880], [802030293, 574857573, 2767089, -27748377, -200646, 1709079, -363048, 952499], [-4193748847, -7523005, 2077092, -1434579, 246242, 1768784, -377298, 1408543], [-10773598104, -386352468, -2217948, 32069996, 3870450, -1702742, -1451596, 791128], [-73666779627, 218379398, -248031, 6572622, -21801876, -2754038, -343444, 1424526], [-70876056140, -216003276, 507906, -31710946, -3399896, 872668, -1962416, -5116568], [37755658787, 386313416, 3258372, 4377621, 6507121, 3554981, -1503572, -244257], [-20144925379, 413589420, 3107841, 71677152, -9585444, 2881420, -1188300, -3746964], [-13258721310, 184394605, -721932, -96306750, 2905592, 2910490, 4337360, 3457490], [-147831251558, -234429934, -7251366, -33445845, 6889836, -99236, -1429888, -4128760], [-9728000622, -332544180, 3777348, 39556050, -10347924, 2793168, 345744, 2616540], [57510966874, 39806267, 1187484, -101477691, -35600445, -3064405, 993215, -1935113], [-16327983506, 541550646, -5208408, -26268526, 3327890, 2305122, 1679076, -1748938], [90718977406, -1475166640, 7057476, 44691058, 31464408, -4633476, -2045328, 3396924], [-175779622667, -654294637, -9929124, 105522999, 7591160, 3340482, -297684, 5089997], [150042286171, 796466138, 9997560, -53276313, -4972872, -3283796, 4531142, -1219639], [4305705569, 1908655698, -1961493, 79357419, 9591400, 2816960, 5382080, 11550532], [68312636806, -720705776, 607548, -119682928, 29409884, 4288332, 3370096, -7330644], [-90181903663, 890293646, -1277817, -236336279, -32980976, -1623468, 845456, 472228], [78902007659, -631602850, -10842027, -98175733, -6609668, -1055820, 100920, -5260324], [49826479298, 45790139, 7339596, 45951034, -56285068, 8052950, 783580, -10329838], [386227715354, 620060842, 5988114, -282383769, 35061892, -2694828, 881736, 3882728], [-44186095069, 1161988432, -7605672, -64796340, 8140153, -13853118, -8658204, -6847620], [279096863579, -2168294814, 282783, -109605716, 654432, -1582558, 8249536, -3608498], [131359630306, 1395304585, 1482168, 20339666, -3993093, -604199, 4044348, 7955758], [-78473251274, -5556579, 5355924, 55199505, -21613595, 2037290, -4270210, 11982395], [-59189308495, 1036867198, -13704117, 74478451, 9164860, -18960, -4245920, -17093652], [330091377902, 854173388, -16839204, 30458552, 18175956, 169660, 6952120, 23286696], [7270476955, -707157187, -5615121, 65669705, -38445274, -8265061, -560728, -3799245], [137409501492, 1813866063, -1118928, -2195391, -52902627, 16602973, 7286649, -8170113], [-274546890767, 2339441719, 13555308, -373721840, -36972142, 2500028, 11699979, 9459500], [58374615861, -486753164, -22723416, 42939611, 44686584, -2695186, -1632198, -14070987], [14288488642, 3598522517, -14205348, -118934849, 56968437, 3073505, -12840195, 766813], [727165176404, -3318137523, 10681980, 124140241, 83561119, 3821186, -3139542, -1890397], [-317062781657, 2421461775, 19147368, 121062511, 54896666, 6264224, -1025938, -2909887], [680478175016, -3482550584, 9779244, -182504168, -102357292, -2394450, 13601070, 22987316], [-427920812666, -497337480, -26770428, 28380868, -19772504, -1741484, -7201632, -2810336], [602646082955, -2540825888, 15909495, 589832458, 206130368, 12574740, 14828700, -3277696], [-188352991889, -2254548233, 30061620, -55921974, -13611764, -9246706, -12098133, 30334938], [402555386663, 3548781065, 23022231, -703526933, 52691002, 12581739, -13568968, -14127849], [-762125986006, -6072773475, 13224480, -278758686, 60523738, -3230236, 14710472, -17860818], [-451382832135, -13438449597, -4917711, -173652929, 42674018, -14453459, -1631512, 15565333], [192920386068, -659183773, -18000228, 73507182, -37982366, -14545866, -41351828, -38493874], [-695595004090, -13051917097, 30699330, 292646114, -43993946, 14574271, 1214208, -893173], [-1618251764884, 2144540599, -21945408, -191328282, 50425896, 14982204, -6625268, 5148874], [611670431837, -9402717448, -38454645, -337242267, -12814128, 4406818, -8367016, 28425834], [498522514897, 3276919525, 15764748, 993349275, -68141866, -28785252, 13693004, -38645815], [-1409278327309, -1163079050, 34676217, -527260715, 247468584, -15513240, 14648840, -12954660], [799693281166, -6785463789, -4954176, 233947296, 343409113, -14813473, 11865806, 12251848], [271180674604, 4399693405, 36476988, 204274860, -150340106, -21226767, -6205716, -3766355], [119097981184, -5605728441, -4024704, -607148714, -54099451, 18048777, -50715424, -6334222], [958917152354, -7563891720, 99488340, -525453804, 191617648, 21369544, 26775152, -57602112], [-639674604852, 2318572424, 22175370, 267694096, 261270372, -7573500, -16202160, -55650612], [-1216315875279, 4544570533, 19426992, -1198846777, -43788672, 2466244, 23566122, -43566511], [691999999389, -6267767520, 77747820, -1260044525, 154773770, -3917744, 33557258, -2436015], [-1311358358042, 14751036812, -70136244, -601633932, -33131500, 17193780, -2933340, -92308116], [-3556232824061, 1268584802, 18573864, -437053059, 84457141, -30432927, 15277024, 9095403], [2840387208362, -6581388121, -826776, -778118990, -129297367, -43976993, -16721984, -47262290], [-1048878348810, 15262323426, 32959092, -1453586120, -162984676, 13274970, -45962560, 1160170], [-3679408356799, -5825837851, -61152156, 340542762, 186839768, -39972802, -406671, 16955146], [1193368529752, -3352728504, -91179450, 244320822, 297735800, 22149576, 11060248, -2410244], [-1240299972038, 6641863293, 21395082, 197957986, -242288838, -65609787, -27591836, -72214947], [1020950796662, -8656881894, -7435248, 1366845944, -12092124, -3849490, 10116380, -53620658], [-965665855005, 2765649990, 32571324, -1164265417, -235383765, 25753535, -16630090, 45695029], [-3633560635711, -2188253288, 37704504, 2250280463, -80016471, 2899027, -40468554, -33285751], [1760201945411, 8331078912, -25289343, -259975807, -162500916, 38901578, 46406864, -142257146], [1874744888004, -11637478148, 193570290, -201866930, 37857848, -56371060, 65321280, -44257520], [1337271899682, 9540278446, -37926648, -91400410, -132395808, 41685930, -9089070, 188826850], [-2603141832614, 21251037345, 199909596, -3113151026, 281280635, 66724569, -29879008, 39998342], [3358367613446, 20052808876, -120586854, -305872633, -102239832, -151865186, -78328848, -119645054], [-3920536805825, -7640675989, 146814360, 4140184235, -442299544, 88384240, 87299120, -29385835], [2839140914075, -2537431538, -72787476, 964882513, 49000420, -109744902, -66194366, -205017581], [3382915835190, -3191155175, 88476444, -1842053710, 230978541, -90237449, -46232172, 64672370], [1494070523533, -11073929374, 3433623, -329365587, 340189388, 19211464, 33584172, -57760856], [-870742113949, -11096234798, 96850017, -242260979, 666221340, 53105188, -114443416, -3158852], [2081867973144, -10843135534, 103399638, 672274078, -171524512, 76780698, -35537696, -1064006], [-1862895952754, 16661975200, -59698410, 2041460217, -552595416, 13419430, -71111600, 2201826], [-572437910989, 24992861401, -86186880, 1378250311, -346441138, 153408006, 69617988, -83559247], [1003184702384, 14159447408, 71093874, 3910755866, -637040668, -63582876, 84001232, 235531328], [-1260426935515, 41208765616, 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-2970435725299, 2455881210, -26229340840, 17112233690, -6868743031, 1594553072, 6717529445], [447669327300768, -257364962432, -12567749232, -166419357145, 32831553376, -1401158862, -833372416, 2882256310], [-102084440126802, -193743292872, -10091285904, -118412003146, -18752336078, -1388587832, 7724503894, -7550273078], [140557342855057, -1112556059125, -12741746784, 212047461474, -3949377458, 318065784, -5301817103, -9198722978], [-267026323934618, -1410326090083, -12300902388, -67826135085, 9426160243, -4671817662, -4010018006, -3193659455], [197801111313001, 1032208313060, -750161121, -119442464673, 43582833388, -13185440834, 5805904768, 8049599926], [168712351006472, 149622124927, -12244854576, 52752929559, -16541144833, 6873848376, 10789388918, -3527254223], [-183677279627168, 1097867595426, 5050558578, 90892887062, -3038861296, -13281749942, 217388604, 13079830486], [606879009236694, -1132988159361, -94626888, 184356846425, 40123774727, -1496612765, -326222325, 2431858335], [-366381711857823, -224399600142, -2281237713, 177052999461, 15793018420, 5896765628, -11501310736, -11794851812], [652553788289310, -658520717033, -10856807616, 58451598608, -49596470137, 804023617, 1941251746, 9783698824], [161890951428555, -1010759849417, -1496292309, -410142687259, -77187851910, -291359947, -5400784296, 903605253], [-406211976800084, -2542869118489, -3773687568, -75564505112, 38023187054, 14632634443, -2721306836, 1969233659], [-292383584637351, 730455318747, -15756394140, -34715018794, -2998547906, 6613018156, -10286174327, 7262266318], [-73870966764375, -867694499530, 24990070356, -13095647045, -10648352392, 3467892862, 1398169546, -2749255935], [242260328011622, 501251262857, -1792259082, -120884248778, 14084272790, 11011111217, 3405125116, -11971022859], [139350875643003, 3143966475526, 13265580972, -178332254405, 25304799775, 9370321961, 8060968228, 22782374065], [318467511187980, 940801403475, -18061564584, 245289123727, -49583878071, -3027664731, -1462867193, 1862700841], [254004471817, 2475415632340, 28927620531, -68206527441, -34663519844, 8988874050, -3848637840, -5283927538], [269891525997442, 1698860536788, -17497352160, 91354055456, 95008234628, -16443475316, -1693688148, 7436156068], [-127080883710343, -1078201713267, 14967739488, 4980944768, -93255796020, -709069882, 603090719, -14516351296], [397340026608610, 2307523289718, -18655210668, 121362538800, 13528454860, -6039636570, -8092718200, -8609218850], [278438772203593, -1604132434200, 8231975091, 110879347563, -86867995056, -2600126734, 2245674588, 7116440594], [277847680340925, 154196133108, -11887366236, 24431057130, 8019545989, -23736267906, -6512416708, -15188741670], [103925229892678, 2972180133682, -9480219120, -21742050176, -19398389900, 68179850, -2973629000, -13034869718], [-235170637453490, 3924091935572, -12970140564, 282795537956, 39348826700, 9539403388, -10587738856, -4281468752], [-681287885175777, -1951499525194, 1875062520, -239229103773, 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4193211144, 247799438529, -53703265239, 4475271221, -8309257712, 9464389287], [80372737996072, -4883937047612, -11169208866, -362958833738, -46607840256, -16845174132, 5571879944, -12188210104], [-449248714325300, 713919533420, 10276163886, 222538505224, 59083068184, -23719470108, -4281308584, 9631548452], [514887482204139, 4834476949548, -116121183, -51729015691, -21188916540, 3728950730, -1940748520, -20793048438], [45255750114808, 4548806837291, -12612928980, 176110557248, 78775696273, 7010009881, -14429999022, -10729861996], [-404942040821722, 575722876278, 6824959470, -524471588353, 55082893288, -5268456024, 16108046968, -2356409384], [44299542676131, 3982029142978, 2367948735, -33794550564, -44319380352, -4451282822, -3039679316, 3282311578], [10724551014941, -2089622197099, -16551255399, 254864147147, -10519727874, 16879044767, 2094603736, -2688089], [-493197106528988, -4325804517220, 24423597108, 235775960480, -67460439070, 7148754150, 9883935800, 5581584820], [125311994603088, -3693943569256, 24222772188, -361621075407, 26039045584, -22689600266, -5050052288, 8443728594], [-415659991807272, 4148460935391, -20251139568, -333170860770, -4724664630, -5084770898, -7626417144, -14738900110], [-505638518275783, 2291673477122, 22075279875, 190731615087, -40177459064, 4917389280, 17218364320, 24085635116], [-1077217739753163, 2559067375765, 6298907220, 363101287487, 1066513860, 1081662028, -9732786126, -10272830419], [-148598320277227, 4664489219462, 8343477792, -696002913586, 30992479263, -4249640128, -14899530404, -26116134558], [-783944524211076, 1836795280056, 14039536302, -743647847658, 65428425184, 6041458372, 12475535936, -497147084], [-202324633751979, -615515354309, -9700096452, 162053368013, -65053327180, -10710679484, 19590189208, -4076181481], [950343424757187, -2821619903324, 17532773337, -369691197473, -232957755860, -3746707134, -10250398232, -1182605754], [717923629567151, -2856551185717, 2291495499, 281098415283, -112234557226, -1425150731, -8954339448, 3470646289], [-32008259002268, -1381345570298, 1709164776, -92452771602, 97979469596, 10666940838, 3983471744, 22029436954], [473263332303884, 103432533082, -108006966, 389245808136, -165208940504, -9573825682, 10309768884, -16436799502], [-672454742864236, -1373717580184, 4858914666, -447296025866, 66490560140, 13901870628, 25519368844, 15849859252], [-1139689610068259, 1504918582368, 12647931972, 137098124266, -121989012955, -15733906702, 1448836784, 14258956698], [-495005261001507, -5282342254896, -18971529912, -862933705271, 20145846705, 3642153801, -15689407622, 3965335167], [-91287852341467, -1992612880362, 35004108369, -346005895662, 20357947992, -10327616254, -13936633632, 7688137694], [719966078412563, -2578514318980, 1298804100, 204798027862, 94404840133, 2197483216, -8439669712, 16954462686], [-302482389660462, -3910741486244, -12000529896, -512209963976, 3123049392, 831246904, -3968480128, 23933026332], [450044519622276, -389138802944, -15684519462, -204730633670, 181800332604, 12469903212, -13364377804, 18756269100], [-795647966880138, 866765282600, -18012413712, -111255244310, -36154176244, 4184194440, -2434973180, 22663584310], [535906332228087, -8889443533244, 16299935052, 388626307589, 65926330233, -5127172475, -9636340680, 11606511767], [-61620285799123, -5175940771463, -64363127520, 59317277892, 59792054804, -7228347066, 2466832727, 1871051516], [65245154780934, 2971368592383, 56637958542, -278158196800, 37999025578, -11065628829, 14917498828, 11186163095], [-429106943055285, -3124653162211, -28708439925, 959181307095, 67584070978, 1998140447, 3303892816, -22107728085], [-83738059816382, 1356892358157, 190212840, 854084219113, 70387470717, -1363879199, 1951341403, -42264862641], [-90535681572837, -9492496763048, -8190437853, 448991120856, 26100914232, -21163120424, 18746985508, -14248129612], [-448300129643270, 1681275178769, -38540575764, -372563453504, -7689184692, 487949536, -8045799672, -21646874292], [909114417808868, -3405406373359, 17289539388, -207008514206, 58019582502, -17689963442, -15907102056, -18063933318], [301154766869397, 906123282980, 39048912936, 116597685259, 220640553264, 13721426802, 7211303166, 21834811077], [-236515121990606, -2225070463098, 40702063008, -377570187814, 51540154828, -5030897890, -38932035140, -24869465842], [-533872043743156, -5702274668883, 23341698876, -387635356478, -50779310963, -4588772407, 11113633884, 43920720786], [341625858790766, 4237101516690, -65461180836, 273658665882, 33955150048, -20531308390, -19557477940, -8990228994], [802777448094277, 4387761806211, -9864056616, -784650908923, 66613910440, -4729136976, -1136204528, -33601291189], [-674013049536002, 4751054443764, 55735060116, -341679004488, 6419524456, 4657428288, 31418078234, 10268870636], [-145300359377447, 2773496924642, 17847922155, -312262195311, 148522241780, 27876251124, -24815798568, 15699149692], [-504329996678100, 4244566700484, -15258237534, -924252109478, 148179529040, -8829487952, 2747248904, -6425675544], [613407792365630, -2862898557080, -11316747084, -287364911724, -154078218456, -9108591320, 3613470700, 15066638788], [-37045520225810, 1308126019560, 1851658704, -336397353216, -141914306000, 12102766344, 6663980552, 24176624432], [-39982274540653, -6227542245896, -1770971175, 184178050427, 101106392716, 29782309598, 29992158944, 30272963666], [-96672677718754, 441335923527, 5535158520, -148430406423, -295406217721, -523390557, -11634284861, -19526124529], [160986570120747, -10714397597698, -26512552431, -115166042779, -159788480608, -5731113680, -17165273800, -14803820532], [460903557254425, 12301359102, 25366566564, -1497006463545, 4373779701, 3701799417, -16472862744, -7889503715], [-214853693403095, 5150801315974, -21037584, -10930902102, 129805560705, -30251986370, -1799800640, -10906870746], [357795953218011, 1347643543502, 21023011164, 114446617026, -159179003479, 19352141604, -18223834248, -9989039062], [290591205563697, 5788427675718, -7949826597, 352237907033, 14739792324, -12767070612, -21073102636, -42789766356], [795045791134293, -8937522872156, 28402311420, 80494863017, 73273544921, -40925288535, -11780362310, 5566554851], [-1115006842898012, 2084154576194, 5626617612, 85481634358, -313707493858, 16386265062, 2141294266, -10300290106], [533834533807202, -478041551308, -13219605240, -128161802140, -214739537780, -32596818660, -6498864840, -26976847040], [-1429503522738187, 2927491710891, -12521058171, -438988453357, -346355912786, -22348583259, 15535576168, 19171639449], [44403625518053, 6895223218804, -52339009764, 504410324145, -137541683393, 1357863181, 5212163268, 42510095035], [931946972928338, 347936693141, 39016596108, -133233343561, -119762056247, -17000210056, -20947493478, -1307173523], [-2074251380842783, -2711025963338, -12404843868, 854769848844, 74497145227, -13059211158, 6174901956, -18359010488], [1245634382391296, 8807305117001, 26978987184, 1410056841436, -19757749794, -15659374747, 15247174944, -22411737667], [136735449796987, -1396290394476, -22768755432, 226546473097, 67880233204, -34373265878, -2892327354, 41093680231], [1339118169240701, -3021622873510, -37638765483, -127612034544, 82247064344, 9158402058, -15135256476, -30988525462], [-474501965304521, -2106661909195, 687230496, 300917078754, -88914875614, -32695859448, -27772943229, -55862037058], [-578736156755513, -1488175807497, -61846169181, 659736496539, 118155600934, -1414409823, -16461927544, 13839777397], [1963008472483765, -6932400403287, 23473930209, -21312366217, 231197247378, -12224353973, 2507852456, -17200421641], [245513695199166, 550179126340, 4098112956, -711193769392, 289371001900, 31075907636, 22612662308, 20620468404], [-406710314244287, 5104452535290, -49952602308, 47586998912, 210183914465, 7530154268, 22107383004, 19661911196], [-139914315331576, -1521611768304, -69032274966, 484113410332, -66876875252, 19672674920, 3270141120, 46863197196], [-260953357009856, -6150025542008, 70677268716, 308888918993, -31146575664, 34125062494, 25909717632, -4809580846], [-1645296605551816, 1243512766264, -58802576148, 165229396686, 687289718, 9491589306, 24557067878, -55665894002], [546924587641548, 11997992898810, -65028237144, 321461238967, 182322277832, 34667749636, 2508999968, -15620901684], [-395173886502912, -8540337887874, 19674723870, -606986400710, -12581714472, -42583457378, -8432155304, 32136313910], [2389287344464725, -930248678257, -53187621600, -677225620533, 43982442300, 15111077756, -6829226302, 19897803421], [-9993981674491, -2950044835062, -32205274584, 614064888665, -36594563421, -26931024019, 12019533548, 2014933615], [-277765554617450, -408252866097, 6701772942, 1164875215316, 3729621458, -21978261709, -27611348112, -41173777237], [986597316481609, 4708387019244, 38653168887, 316743204021, 332867728176, 36381435866, 922377948, -16377815182], [-1986628371753022, 1546117027233, -35076965664, -576157250883, -283024613415, -33135801642, -22029364386, 3598939451], [-754507819665900, 8953152740353, -22263678036, 19036789878, 57584533476, -4841987996, -14807175428, 53787799174], [232794334586143, -4649755831450, 43125670455, -1442423230454, -96755062752, -39752566702, 33891832224, 51443710678], [-704127212581928, 5232769406778, -52830336750, -1412545409378, -129424129204, 20081915346, 22134354188, -1171996974], [-994270100727341, -1668339518700, 16079236368, -181754535541, -395591116538, 13662018146, 4098086638, 35161132237], [1358653309509278, -4672061356848, 94010550624, -604988595060, 222168532832, -8715792538, 20580409576, -20661924060], [1101857179015934, 8227011194208, 16342010184, -738571335244, 342382832100, 43600762348, -14348583496, -19066307052], [-1958574832477068, 3243705800692, -71977901604, 203289781412, -192847095826, -3738790862, 16605488404, 51283215576], [-1319446362019268, 11204900480106, 33867989190, 607302976498, 319623673596, -25152734550, 1652857540, -31550863406], [-220123853170560, 4083650663180, -16670213142, 116906344150, 166272461944, 38731733056, 8508966008, -48665242560], [1474344360875276, -12415577314555, 72121821600, -777148941353, -329331130973, -5018529345, -57806841685, -27405649639], [-1290593577929098, 8682433592761, 20752073556, 615098290335, 20151440407, -15151470594, -9729415842, 24343507325], [-360841772377086, 7640335864362, -31124805348, -285772696776, 53774582488, 28416620034, -12645393928, -64715916338], [183254977255390, -14583583434764, 15693220632, -513511448376, -275393039232, -8129309020, -51506756480, -49768376148], [-3420849307122424, -6801676129589, -39527948916, 199781184150, 70114523782, 14107771822, 4084143036, 61568958230], [414715155442892, -5670081315410, -144863838726, -1126821311058, -35154364144, -8248111266, -24831678688, 50235474606], [1973024419781645, -11957690221523, -53596527552, -679886208283, -66926265962, 14025228496, 6296418338, 38433906131], [1457972317236234, -1546487798719, 51706633932, -341213262864, -195629316825, 117637741697, 8146035146, 16024264868]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_208_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_208_10_a_m();".
// 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_208_10_a_m(:prec:=8)
    chi := MakeCharacter_208_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_208_10_a_m();".
// 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_208_10_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_208_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,10,sign)));
    Vf := Kernel([<3,R![-11111737226899392, -1407962401337808, -34177629196692, 360203815197, 4560292185, -14336666, -126662, 141, 1]>],Snew);
    return Vf;
end function;