
// Make newform 200.14.a.l in Magma, downloaded from the LMFDB on 02 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_200_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_200_14_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_200_14_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 := [1405679570677464988623046875, 25030138583264107436718750, 105082623154370946015625, -94224520282915730000, -919603429545956750, -30081688120700, 2615193085270, 94050736, -2792237, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [-2233829, 190, 4, 0, 0, 0, 0, 0, 0, 0], [16585219512084466660258107556215270001904876219623501875000, 232943455391214529111476640088184521276834896612876593750, 917677170376849535083996398486178794303282422929120000, -285155561827508616212513049288145826789856331580500, -6193174932738815719193897629063750896117828595200, -4011193772629459223941950374766758541052041080, 10985615633492767236613564931283114083181056, 8897629401489351642408889953726677003448, -5816919680307091706708454281680212992, -4957743578444329340225626531621504], [175956482775384508803465864108265595544241336736094120703125, 1239124349848621068314707041055311987486710143505679187500, 3897785358707532368932237962023668906358501555886230000, 5152995760152712217044997381664768487554905488621750, -30168799102625357531517269585637835129877133115800, -64463485222088765888683070138496991636970464570, 61255152369973549941381382997643018203629024, 111092715066058668525122656093857045975942, -35363950915238546695336072674023888068, -53918652852930585357753611998270466], [-33298980790303146918333981978304908191942107347809763046875, -165947701968326529556158407838308185088753189477707218750, 1343639836651376098399679339150072000819922505793573750, 4990525789033215765126449498971044121038670865765500, -7719843159278098696276746328318411925496603727050, -25585281883446353895068839108243058983429145320, 15464686860938665897858476257196808075391974, 37297701892000827571425181616243793396792, -9110595458378544609612623117177081968, -16825971935826480542524217589259016], [124343712601669903264538893974139419322543314956534869687500, 2319972688722095438253611577044747350503416510546811000000, 12753409196616576040461286381016280128956575375230558750, 4879744312178815648240061025589354480194740010930250, -81353091542430258009927603531873449834381870212650, -79463960610251383634033995791610206292087264810, 139799439779184068682562838050859826286761742, 143315032572861136119154973269362724975686, -71826551359454587606010757674009063644, -72086708548511248862093747073005678], [631265309105219795059347750490129059223219488447217083515625, 10652658144273158270250085460106492403174907521282426312500, 45247267810486047773537441394692878113314901750436577500, 5362457679102756748459260125339847808936357086666500, -291516774607154554304580392712244294877645379644400, -300954643290932222167011925865859915447471823760, 522955776008331136998100501142619852308254232, 578902375525277446126436599886167637333856, -280533275445588043436343368866794687824, -299962590960225460282468075044792088], [290889181571822468109657890783002251161009883223271679375000, 5467111973459721113367404287368630486613975002790495218750, 24895413019635701472992386269297699607198404100793927500, -951942181968448813603496520950294361357028280204750, -160408576566083085163191953414282961665829850220400, -130768578521022102340487478727750497791032582910, 281045612714425877969983888957939316835252712, 263627250869635877678837270164818060206946, -147649182443580705304490375379537065684, -140133619337133646139507425087552658], [-166293005405198463180075205261037961930313778959049430121875, -5023879083282676254457823561504434379006628095358468093750, -38636385297379623213268041366898649920528507827239995000, -47396525260612083432038011097597020969940386596839000, 243971833682554203797112122901704703019549301484900, 403051419954152975758053165117866503699584478160, -452891385184021362974554391990340719358455804, -671758512674066393380707671391581021121072, 250590165913626171323728082823763107008, 327410779670359983217331619954715516]];
    Rf_basisdens := [1, 1, 1, 9734651153416859837263491458465287559662956399140625, 9734651153416859837263491458465287559662956399140625, 1081627905935206648584832384273920839962550711015625, 3244883717805619945754497152821762519887652133046875, 9734651153416859837263491458465287559662956399140625, 360542635311735549528277461424640279987516903671875, 389386046136674393490539658338611502386518255965625];
    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_200_a();"
function MakeCharacter_200_a()
    N := 200;
    order := 1;
    char_gens := [151, 101, 177];
    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_200_a_Hecke(Kf)
    return MakeCharacter_200_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 := 14;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [128, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [34104, 8, 0, -1, 0, 0, 0, 0, 0, 0], [-407324, -321, 1, -3, 0, -1, 0, 0, 0, 0], [-772011, -165, 0, -6, 0, -1, -1, 0, 0, 0], [2906010, 8202, 17, -58, 3, -4, 0, 0, 1, 0], [-8270498, 38139, -7, -131, 5, 3, -2, 4, 1, 1], [44921828, -86461, 185, -21, -8, -77, 11, 4, -5, 2], [412035614, -396044, 98, -4630, -22, 8, -32, 42, -18, -6], [-16128874, 34496, -56, -2724, -145, -16, 46, -48, -25, 19], [3780640419, -1652168, 6174, 5484, -725, 140, -412, 684, -18, 48], [3120942361, -1004709, 529, -28298, -1036, 2640, 856, -2094, 108, -36], [4887853744, 1915793, -3056, -13346, 2476, 5252, 516, -1968, -144, 96], [-5407409060, 1265667, -10901, 29771, -5612, 2949, 2141, 4412, 185, -178], [-11614962605, 7910831, -32250, 78734, 15824, 5839, -2009, -10584, -378, -32], [-65856120262, -12841199, -116773, 98455, 25647, -13623, -3382, 28092, -909, -285], [40574214775, -129288881, 17134, 200727, -24484, 33392, -7880, -41467, -832, 664], [-29127175200, 9860547, -26748, -118690, -46844, -13896, -6216, 68736, -1764, 2328], [-95512932038, -432164262, -100538, 1030914, -67035, 2810, 32418, -36600, 2709, 849], [26245400012, -13895036, 163961, 2138258, 88343, -26578, 19714, 4272, -1503, 2424], [-317539906502, -187909204, -532812, 3016056, 104841, 80652, 19002, -23896, -855, 6309], [512486264992, -274194211, 89868, -4598926, -196780, 207280, -53744, -97632, -13884, -584], [45942224318, -1493201300, 28034, 848534, -487108, -33776, -14616, -14190, -23436, 6660], [-1148813105762, 1982405104, -1592789, 1676126, 216167, 286526, -10710, -845496, -11781, 32640], [727885464298, -5986113500, -416390, -14987094, 1424638, -556424, -45760, 355074, 24570, 26910], [-1012506679284, 670817673, -884841, -8322909, -1374984, -245123, 298309, -69444, 26757, 27054], [-1587184198528, -962477095, -4327528, -8651020, 1693776, -419896, 289896, 337376, -45896, -1632], [-3387441430919, -1271796123, -4592736, -23489285, -1810724, -485344, -13264, -64519, 61896, 11912], [-420350169780, -323805168, -484776, 6695408, -77524, -1973696, -150848, 2249584, -81624, -23600], [9422163207608, -10310688862, 8210636, 19688553, 1955324, 1610288, 648256, 4041312, 231804, 66288], [-7673737923554, -3617193551, 1703035, 29389407, -4486487, -4834583, -420370, 99276, 250317, -54027], [14457271455678, 17444478998, 22332694, -14214592, -7441682, 1548810, 1397714, 14704, -109234, 155096], [-9305788258396, 6849899705, -7563521, 88767275, 8171500, -3613911, -2505760, 2076616, 83252, -39652], [9811446537085, 13166132877, 36615070, 84115865, 4540588, -7922160, 1720312, 10582731, 629352, 18192], [-9161989376576, 48253068886, 7335858, 27805938, -9980604, -1973658, -1838928, -6436232, -227412, -70956], [-17873445495195, 89496570826, -31791552, 7262364, -7533803, 13450, 2208490, 3100380, 536112, -306576], [26911423312920, -25272035811, -33460038, 121475716, 3740240, -13895506, -7647490, -67992, -685314, 693828], [-26037261902780, -92536649537, -53361603, -85091999, 1311568, -26118649, 4870271, -6083148, 1671591, -308182], [55165618651711, 70527252884, -39022890, 111217060, -19885733, -16479448, 646496, 931084, -1901082, 976112], [-53310716532672, 127133805319, -92132031, -439630755, -27280274, 19420567, 7839652, -6136824, 16038, 765270], [-8991409226550, 156185381308, 47025134, -344340410, -7733530, -45482184, -7505456, 35308406, -1678094, -93290], [121021955027652, 148561229520, 183540720, -314889416, 76073354, 7218832, -10848860, 13180176, 2558538, -119838], [269657367142214, 143234029432, -70964701, 902123134, -65682945, 48806998, -8559198, -9574776, 1315251, -1016640], [201669999821301, 277177656769, 231716000, 465354446, 76340274, -71295139, -13537347, -544808, -297008, -2094672], [64035194881134, -66455190538, 230577754, 443634462, -83288401, -30877258, -36787850, -6254232, 888911, -416525], [-210133402469658, -121846891947, -236962377, -643835557, 69662021, -77898419, -3105338, -22723892, 1363881, -2091359], [1190545594466776, -232287888242, -256528472, -1426964525, 78579132, -36198652, 23954100, 66690768, -10421424, 3112680], [159302026731496, -535034357061, -192236810, -1736657420, -44352816, -293825150, -30943950, -64296, 10708178, -402468], [-162206943519002, -247530845460, -228613162, 369166854, 107822578, -122802872, -27438448, 63224318, -7326362, 5017490], [1289262607674692, 477228992404, -358986799, -1378020254, 50944151, -224287698, 11334370, -110853040, -2198471, 1530808], [401900244170186, -672290285124, 583828892, -275408936, 47906969, -28085684, 28508322, 165068688, 8553249, -3951531], [310825987356125, -1103292115089, -249559675, -3340251250, -121839048, 152571872, -48912168, -157310190, 8590600, 2475216], [-1333376347463912, -838138424045, -1883747723, 1148845585, -237510270, 93227683, -74436436, -25977240, -5836806, 10533402], [1353876646199648, 309668629250, -65775814, -7876425152, -251556676, -292602354, -16151018, -5788632, -16326614, -740336], [4041070863430464, 430609948238, -902848494, 5418020925, 63071208, 210356526, 20709150, -73762648, 9446646, 4644708], [1169241999118177, -1384550394967, 1697898658, -2475344431, -376079004, -339088944, 54115096, 153694275, -39502224, 6405000], [1327363521977994, -1655132140812, 1806465276, -9110846576, -147955631, 230856308, -20102662, -16110184, -1065951, -8392051], [2887331764821607, -454477983776, 843177526, 8505342796, -69654729, 296104484, -99980580, -333216420, 30171078, 9933168], [1442806869801913, -817495270977, 781326307, 5954190708, 431924424, -214204736, 148987712, 84786540, -65846232, -243336], [6615386522282344, 252750714053, -693592538, 12667670628, -750913672, 53492506, 185230058, -477098568, -27474318, 951756], [8198104402898913, -841342986628, -692011848, -796278328, -260202937, -1048791884, -190923500, 1000220628, 37309944, -10309952], [3557000380461104, 4215319159523, -3035507224, 3743495634, 1012838588, 274466092, 29456268, 660676272, -97691256, -109536], [-369691740887632, 5121693326038, 4968701066, 21481169506, 879598404, -23449002, -52252456, 1185891840, -25732620, 48237876], [11580889490614694, 635447043126, -3385484110, -11902030440, 139268586, -250552446, 189439610, 780459216, 79051914, -9411528], [1649985504866875, -4485392692665, -5012425160, 38042403794, 563620414, 6584571, 154245179, -1023807352, -75673048, 60357104], [-5554139728197646, 7313264825147, 601264049, -34388546571, -1905203489, -1387922957, -67648438, 619618876, -152355389, 29104595], [6896617947559136, -5116058292902, -5123113972, -35329481788, -586744174, 1592876090, 733509802, 1088282472, 13660524, 93518112], [8383688589645112, 9356565234125, 727632782, -18502693990, 696123852, 254005550, -36391074, 773121416, 22642426, -106830996], [5163923111253662, 5533839596476, -9712418234, -44552887274, 1828275618, 4965822472, 551581920, 1427935774, -106194682, 6431106], [17420419688944324, -4328565776768, -5047169834, 15374894588, -669448442, 746415708, -1074696652, -1542775248, -36626314, -136381696], [-2474307899612800, 2813743195278, 2264152274, -28821731990, -2409896260, 384181646, -771545720, 2900009664, 228218220, -11980596], [8119686889768736, 7573771028880, 12393464426, 10367367225, -6740177508, -1958197574, 347715834, -1945109736, -92673882, 42901884], [13111174162216121, -4348123810444, 3159588050, 28556202012, 2273370653, -2599205744, 934312632, -4095398796, 130491810, 150826608], [-2003135559598288, -6248544496987, 2404491531, -61119504393, 2845853226, -1573999035, 1988255892, 2285287712, -146738214, -40268862], [16256914710153728, 447267741024, 4715244462, -8020331353, 3143004740, -2638393586, 2248913230, 7091637832, -91102782, -49382444], [11767477536884965, -5518692984635, 7722763318, 10935363337, 2021563596, -55244080, -535493496, -4782452181, 394819416, -96105552], [-7519129232578889, -9790370280609, -2072738648, 57377846338, 1156563818, -7317053965, -1058918541, 3802667352, 160215768, -129113808], [25452258128943898, 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[8003043871411417968123, 2141127003579413947, -4339422094742412, -2067200699779070, 352844433314718, 563303026286295, -32609151166329, 69050636626968, 12199710633444, 9612495875376], [-10760101419841700146658, -4378757320311919453, 234496393531993, 17060936059075589, 780832553945997, -1048623947977733, 47940086283494, 962197624150620, -112546790701695, 23939283359625], [-1082315745207483066064, 71174629841036720, -2539599329599432, -9379663729510048, -917329988701584, 1230045805267648, 305049346901496, -856291639371336, 54215717044968, -21461003332272], [3355549231816112221576, -1320795560999066348, -7276509890164316, 16943562045645184, 630230552466096, 874566768892492, 251341626763868, -868283886685776, 9292587124644, 43422953503296], [4568140338230676683315, 4145015675522630681, 3520579371770183, -32011499837796826, 38636947126704, -757861306365216, -12049633899352, 336401128624026, 25165193398848, -64257948215736], [8179403427692901055494, 585967847998697572, -3448623926076666, 6418096419371844, -71066021726672, -741920402146576, -66148772874952, 689518775184264, 63267572520990, -25174622987592], [3298091309804281396642, -1930941481001801908, -603437433092858, -13552061956732802, -714746586376034, 761356341970520, -278917016055312, -1580787470025330, 34724868567930, 52563011132814], [3312083174628200343654, -2353383449109221320, 2165981473756394, -40051939335347580, 1563200346015112, -549601204578692, -449743343912812, -764205980944232, -21857739376670, 39421020698072], [8376261600783164392756, -4933652319728103729, -4551592228145159, 14804543742228189, -1351420472962920, -7900294999165, 221836064971899, 203745333839556, -42939145623861, -2643650900718], [-6951237165843382499552, 366943957377644542, -3614602513606638, -19009325143860616, 5474910024920, -632322435992534, 345002520281938, -39893673151720, -54869474251038, -10874183147408], [-6755936288075899457020, 4126881655629722547, 1951332085776445, -7859100316776647, -414511015726200, 816875555266387, 398636311005312, 1872210939101520, -46991298697480, 29619026065704], [-5308988576893588082924, -518833988482605697, -9502593769126883, 6347865699814859, 1730002169562952, 176060499064815, -210965677487641, -885756821156396, -74315642785273, 16337669416106], [-620124218161819161600, -528806993238484156, -7088868067451468, 5745171332175388, 1859887099942608, 463873310990836, -632253693051000, 800263838292584, 82949698197464, -69257310405264], [12204725636536765790439, 3431826248867439706, 617752285545732, 1062190013590468, -129172720915689, 540114132820658, 181415618248418, 1680301796790180, -37686825553644, -24255608623152], [2891759052733394830704, -2138596822621495781, -1017541979183220, 745554483442972, 614918430198360, -1695592594001620, -329223705000052, 1036622048350416, -80397356240124, -19713680477832], [-4931922716384944094876, 2294729172102486685, 2979192743617897, 25161182941539239, -2411588339732132, -33556968367113, -315601784113633, 984281461207188, -94113407214253, 19914953271914], [-1357856780212591451357, -4595351201449134155, -1247852828725099, -16369116422590968, 1488664712791156, -922817696483984, -217095340643440, -665187822213216, 33509467336828, 56633804323172], [-5303172738099436823235, 4680324935379711510, -984871630611956, 19669786864702588, -199440093515883, -682881929267522, -66119639918706, -482356404570324, -107135415106084, 8754678992400], [-11116229136292718713340, 1455346136925708857, 198090729454591, -7672059472876789, 288072952200444, 426714560682665, 54806928218400, -773624596874424, 111690628132068, -57956713560852], [10575526671764996135184, -8072237030562836375, -3067050357842748, 1587096779761916, 570573297264728, 893478933922260, -315231168031884, -1949675444388432, 50493218883372, 27820768821864], [3788328833114557359678, 5102953634993064774, 6262888040272770, -8346205666646466, 79282185699999, -2303135995350602, -223251481895138, -1140343546093872, -45680380723161, -35096378287101], [4401016758796870613278, -3221366864467884188, -836414271902072, 13395264483395864, -2297803733096458, 1795659332264644, 843133336759028, 220930341096648, -18499927179024, -36041274617688], [8274747556188027142416, 3783752252821484920, 2970473895278216, 15479032625204488, -390765378452584, -2897854866980232, -37700898211664, 1487014679416256, 12400128020056, 5610089762104], [4087396815518791678534, -7419473316401938672, 445308678241720, -36678882643368392, 683453942226664, -1055744148179296, 218816775174048, 1691134198261368, -76842057894696, 12100484401128], [-2227439344897826553712, 7065229828016771735, -512641694507751, 7776947971014173, 739011737873582, 1136579274397111, -44253168142532, -939407493572160, -186073784466210, 59191453126966], [1206118207836728579875, 3459227711585819467, 1142571904653358, -3489621744637989, 363460539308524, 853188071258544, -352573619829320, 1651570511159985, 71583229684512, 55872184466232], [-792492822523181721511, -5610726288955111, -617683460176618, 30649101728165509, -551309762416396, 2350833835660848, 420683629541416, -1734747406443729, -70062133836704, -2603004511064], [-3727710633755364860718, 1688722903516843782, 1856094210015599, -47438644604488902, -83497443931103, 707576863325140, 191553338737904, -745180424163472, 94837542672079, -50052075557680], [12033529816612766068680, 227237343341412694, -1165157524866024, 4374873602701977, 206493493930972, 961309205159220, -36762791576220, -1885871875746288, -103763986924896, 18705739404744], [-7464359698920062525262, 4057788198376418716, 2065786878588680, -34928259393552632, -47422045136086, 198158184948060, 70704114099436, -467241560707720, -131484294151328, 60073612572952], [-2446829890595552736315, -3472665862575111139, -10096515518048886, -3985626544170459, 1499710897795388, 1188115183778352, 87280877995272, 202649450001279, 46383538462200, -14167785961968], [9320267036841129910690, -684027380380192318, 3101282623955326, 15240576245488074, 1917838269096665, -347635608108766, 159666646199626, 944766449811336, -43428050219671, 96784026740917], [13598959165578103354395, 3914334159994874403, 3244446976906122, -13246337905790770, 3448003256929888, -455513485045253, -508207299683933, -2018920056777000, -32441711129622, -24554639000832], [-7698461841986282881990, -1069895625004526020, -5855262180961126, -14013297767464658, -905957832534332, -673691862164624, -525715086266040, 144352155071850, 142679517621228, 6455619203724], [1029193735830772025664, -9536766780404402587, -1999677750407868, 1973817630387042, 728698818005348, 2457764100220328, -244028300796280, 791878101246144, 98844627812364, -64630771777848], [-9038208422936954411986, 1087893111732956271, 4275291367647749, -18456514419909991, 842033166585877, -690667272648201, 157159260791582, -1844630089164540, 49565606856849, -33175000076847], [-8940245811198192036272, 4678695274392843826, 366273052453552, 26117347597502652, -3204384446597210, -1314359192294758, -128099246145894, 580062636175176, 112643605426896, -14961095998944], [-13120683404892538667609, 3978201266045052218, -2472825631466792, 45933989983741964, 525677903719939, 507082837518618, -232224275831782, 7615321935076, -52823338022680, -38391110651632], [-10618728565624936222688, -1255613972190568185, 4026515997909984, -12391802843290700, -292043203684760, 326370021761432, 635580384277976, -2045034928272160, -7348826527680, -76752749376160], [6636999270285221586094, -12783209592591688095, 4207384672314963, 19496615481748511, 1743514103283785, -1041393259763735, -410145184402298, 1842701864980324, 4182163685925, -3042747839435], [8855600118505373714920, 6405792818716935070, -4036834596342137, -17087051872835894, -209288693247669, 556693496860860, -102866844297080, 18955333925736, -23332832687889, 7974374307864], [-3360962810324928518078, -1757538354736310716, 6339445155307138, -14476084314027390, 1522182664095134, -2439720961485384, 359859527244016, -268193579064006, -53753130426006, -23671519106082], [8654284245303571188350, -15970612255939163008, -2524394536848544, -9643203685906564, -1106863596944229, -1246080900274400, 379760764804110, 871110876230328, -80052814881189, 60059742035439], [8228779808658058507292, -8414643979188897023, -8338796674619905, 9466515986627481, -92528741433728, 806433550089165, 871087790148629, -315319520315844, 63955685013021, 37319001974238], [-12569808155119047051110, 475583627490426040, 1877510182869087, -24582354039042058, -1428806372506705, -1162465425164882, 263094103256266, 1510392137076632, 84102234726495, -33306302813840], [-13557423547007413992336, -11452397781523532654, -18354191165402, -17541270935181355, 386373144320200, -1132769476251126, -1390823264704294, 1098307727232888, 15775565256434, 33563267937356], [1802028269323562902337, 2971028798879146093, 4446652302226496, -66516715269506525, -2409940132285476, -1569556654414496, 610413437359680, 81610935953217, 11856684514680, 60342039766248], [11680125244981050657505, 5654893119599264501, 3678979331786494, 9637899030016722, 75313445268884, -2772470891248851, 587015610632293, 928547395961976, -151498022993154, 104608788025056], [-12178449021523073002332, 6240167236257503773, 6093595028468169, -20539181278468017, -2218068129084468, -383736248521785, -659915938923345, -25401700083500, 117188730600147, -40152614190966], [-6653989570928792859170, -17834864814882292103, -5320648684884053, 14466944907696103, 4003382521301889, 2001737558662241, -1930278522618, 169863468572532, -108917537138451, -33047749844403], [-143818294453002238554, 7952738205795256858, 4614174590764044, -7799055646322348, 4418919684050556, -33433794192710, -1525438358376518, -431170693142208, -36259494681708, -16684289597688], [-14062406359671565776607, 1566154138815721134, 615257095248384, 73437542154814484, -3155718037444363, 3498869043788062, -46634388651650, -242798562872228, 129867895566192, 4922521106416], [-13186170851239239893728, -4026426622166285999, -3578787351289852, -37315259556741642, -1042884105161036, 1843807491453928, 184661743634216, -3505006293405888, 168271965699004, -56197806054856], [4226347443922866502615, -6903763027537136055, 788440370143585, -32998070485678248, -597005790983592, -85645982053696, -166219430259792, 3081038500340904, -92954915891784, -44761982718120], [1281243229590921239332, -9547633026248236477, -10895889232797395, -14353983218498487, -2659044137606632, -1022272949606941, 657443429738944, -2927085059251248, 107493178571048, -64327083259784], [3378248147797234768826, 1944427625599149818, -6444653518829050, -68391146518336670, 597389516900485, -3244646932321958, 83806583809602, 2665459124864808, 11832914787669, -18620923556943]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_200_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_200_14_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_200_14_a_l(:prec:=10)
    chi := MakeCharacter_200_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 14));
    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_200_14_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_200_14_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_200_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,14,sign)));
    Vf := Kernel([<3,R![3479598213094465980216236928000, -18622915250230103032475827200, 16631649104482782704666880, 40611347654132910532608, -49341071631529026432, -30128590782361728, 37449134609760, 10435297920, -10436276, -1276, 1]>],Snew);
    return Vf;
end function;