
// Make newform 338.10.a.m in Magma, downloaded from the LMFDB on 16 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_338_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_338_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_338_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 := [204373529570400572304, 28730203732505382432, 637502596296640360, -2585100275453696, -99239916178943, 59076086702, 5434866119, -182380, -124193, -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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [10712544021035256335362975993620260592, 698570827374493151356699765397702064, 4636770254883426205309565880305864, -106463854475195758981372331054672, -535963111195592387285636644361, 6299130733952694445613280049, 16064740459513950942453490, -156093106470142102202186, -141061838860705515505, 1338793984138194057], [5833095590736099363020032612747676070712405399968, -149543953833138080724894991372733632006535410644, -13513234125945841407118605806009093457842633680, -61593962897390318829079991675045319312283839, 1441572307487614768003426806809994486239798, 5900838237327394816061727980666283825755, -47849552301365567594508564727139157196, -175860173053325915345135192408224057, 495979904803493897592925592147334, 1732112549155915806788876084177], [139047349227521002060018381392856117452422752163088, 6243813856032148605331068166230822977426329678896, 117158740566492275619548203915567555736355844088, -1116916352837780920544210883914146930135716552, -15077456847775607646669786430166636740722295, 75072377749142375544274871272601548082039, 510335807985925752071845481594734418798, -2093338084817110741731289596418966750, -5031318881463997111017729653562159, 19679123173032298137601950451631], [-322880802086728634410256861829969575566850765757744, 25126302623814547245488647951338728606178003630576, 18709848738581557308987651879186393270000629944, -4222242808945478849160724210645327002871380264, -1365903704852788184418386696852404733262707, 240614436578286718431036817273995423281107, 60734875852891638098018863211465161222, -5704887806854105727989472665155270614, -545967808929286743926846269216779, 47647342402150924749993399868075], [-151471270432346361913431748563612697493852082408288, -7066447632251523060539408506173705609095591829348, -15373735831694679766602723362071419284076049424, 978297535647991511387698380731526281072119053, 2273985232055505917322446580718108925543356, -49945661275796528821888866121679035798943, -75087586624598582925265301579709138284, 1084864757287873034106686255607737419, 745371654070794626988620586184656, -8320896039519098673943461283493], [409394882564528126798040248725909032329160478946928, -28817319529482012055109136051988583758673999630640, -639216274657906318699887264474658225896343583512, 1028775411660588141751821187797262251007183944, 58702902202634378505058887796478987386705751, 109060902931446298387319269020355362155369, -1795459873421569019653142932254347618542, -5699554094302813250609499428443330498, 17917839265984990121715139731720687, 68357914067100722501385138448913], [3700946244258230868523331051444726601442965534360, 253791220888165536032012425963409190892556362824, 2695910467935605328961994596897791778382174378, -62683227920007401114482829347800508907720334, -383036804278688539144309615523522192918629, 4608524173492016764034837793036577225901, 13431442539580440592927173014350879436, -123805406561530158831148489141771900, -135453160338474396008817699357621, 1080461942356322965976401715225], [1864613606315146652692016113690588171200408433022832, 38456067811457741900407071416771544696598361772240, -169885982592581238474878446478185974217260486808, 298586611044248305683735144622481728660687560, 3575023307424946286763861154739622686638391, -374935491935439061375215702353450566218391, 137493110494523539237170172332792090002, 15724925162005894708750420550006302334, -4032846104872819634108055490030833, -174224309324281704012499883991727]];
    Rf_basisdens := [1, 1, 16811513967774159533012745201613056, 9766058175524677432463251831818132651152110944, 78128465404197419459706014654545061209216887552, 78128465404197419459706014654545061209216887552, 9766058175524677432463251831818132651152110944, 78128465404197419459706014654545061209216887552, 1220757271940584679057906478977266581394013868, 78128465404197419459706014654545061209216887552];
    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_338_a();"
function MakeCharacter_338_a()
    N := 338;
    order := 1;
    char_gens := [171];
    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_338_a_Hecke( : Kf := Kf)
    return MakeCharacter_338_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 := [[-16, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-16, -1, 0, 0, 0, 0, 0, 0, 0, 0], [309, -2, 0, -1, 0, 0, 0, 0, 0, 0], [1136, 6, -1, -1, 1, 0, 0, -1, 0, 0], [-875, -19, 21, -6, -3, 1, -1, 2, -10, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-625, -447, 252, -59, -2, 47, -37, 44, -17, -7], [151923, 806, -86, 291, -99, 77, 19, 13, 39, 36], [-131428, 2228, 1252, -287, 158, -109, 183, -64, -341, 1], [-235853, 7224, -465, -770, 763, 74, -267, -364, 333, -116], [271371, -9224, 5105, 736, -542, 399, 225, -56, -703, 72], [3291814, 28403, 11762, 835, -32, -296, 676, -231, 567, -1064], [3267177, 17053, 2665, -2467, 6198, 455, -335, 987, -3718, 532], [-832254, 47027, -13841, 7480, -1405, 620, 1211, -1600, 4951, -206], [6752614, 7771, -1505, -9749, 756, 180, -2244, 845, 163, -932], [8775387, 60449, 51946, -27624, 10450, 480, 2626, 1440, -6477, 980], [15823188, -495508, 92339, -18179, 5715, -6902, 7142, -3201, -2240, -1540], [6781773, 165016, -11113, -21964, -1369, -3308, 12975, -1636, -12507, -2806], [46269125, -495646, -193804, 1483, 2045, -5881, -3639, 2741, 15073, -5612], [12456162, -217393, -281084, -3872, 23625, -10864, -10976, -15774, -16699, -11348], [38334880, -473214, 143897, 79591, 5368, 2777, 4923, -26026, 32031, 11296], [-38794132, -521153, 86084, 46493, 54382, 17431, -4421, -25772, -59395, 13553], [-17283454, -854594, 66910, 10140, 8682, 23974, -17686, -7136, 78420, -6128], [172717957, 1853421, 593042, -205318, 27690, 40052, 9088, 623, -36333, -7204], [269781973, -2249627, -1329986, -240198, 101526, 19936, 111892, -20461, -33089, 48460], [28369079, -2530085, -212293, 86021, -169619, -77345, 13958, -63128, 70506, -14721], [-287106730, -220882, 1003459, 657878, 86243, 69382, 57129, -41640, -2707, 58912], [-348838484, -165593, -1323472, -548, 90456, 97220, -124348, -10672, -201468, -91252], [157123505, -3232787, 1787038, -498803, 374326, -64161, 78361, 21691, 54274, 13748], [-408556875, 3025423, 937202, 401059, 91176, -124591, 166659, -184468, -14425, 80395], [1296228992, 1825463, -773264, -126102, -239140, 104958, -235874, 3172, -178824, 81022], [-513557114, -1816800, -2129649, -704800, -376957, -288332, 299155, -188864, -47329, 84538], [-1110640688, 6126530, 3160195, -355968, -755400, 174666, -70242, 458678, -12380, 38056], [-1505596566, 15080524, -2007925, -1470655, 442801, 256231, -376718, 349636, -340108, 48479], [1664113137, 2835578, -6029764, 2068282, -1071390, -155603, 885431, 83610, -151067, -19564], [906214825, 21413604, 2337119, 567212, -511090, -435179, -345413, -125276, 1182455, 210520], [-1524058067, -25831155, -6609738, -676034, -415950, 519002, -590816, -77600, 946111, -296046], [-1061742139, 50589481, 1578105, 3009614, -545287, -27763, 377891, -389718, 1314990, -337960], [2894072120, -6359738, -31089, -1170705, 1126449, 722984, -240104, 483631, -1395416, 474288], [-4421659340, -6310184, 1415129, -1289307, 2457923, 665655, -1041116, 464280, -3156971, 588419], [-642075770, -32195775, -24873783, 256896, 3035397, 694764, 880613, -186888, -754919, 591022], [-450729797, 8919917, 33985960, -625564, -618536, -474572, 688060, -851992, 3284365, 96124], [2032004856, 11594410, -12025174, -199245, -2438944, 204081, 266267, -736704, -2448849, -619865], [5103897989, -32941391, 22766039, 6282089, 3366802, -1662857, 1173737, -935309, -1419450, -422508], [12495225197, -12116001, -584842, 1297774, -1124604, 138852, -3072168, 282355, 544853, -596200], [6361668020, -40915381, 33614716, 1027640, 497188, -628760, 1075012, -2207620, -2641994, -834116], [5262493800, 34100349, -33667878, 5689822, 6220398, -970510, 3387532, -2947832, 3394636, -464526], [19059198679, -23313230, 21445860, 8329603, 3321595, -440589, 2796029, 1557427, -8862831, 1897864], [15195579871, -30490278, 40370202, 18073175, -2666787, 1294417, -1018945, 156233, 5946335, 320364], [30571985391, -7537045, -8846014, 8011643, -5328302, -3296959, -998361, 438333, -3016994, -1073660], [3837193226, -80222009, 154466132, 10648437, 790022, 1319631, -1957389, 632892, -1619803, 524937], [17595614362, -5165762, -32267770, -637820, 6296478, -1934830, 4471150, 10196, -213624, 913040], [-31130059965, -98351301, 6450011, 12454141, -3416866, 256539, 352357, -2187671, 1165976, -1937572], [21346225184, 115147527, 70260258, -4441312, -12824326, -428792, -7049766, 3018088, 15690126, -1933036], [-26384518179, -1028479, -28267460, 17839971, -7497830, -7395015, 8410405, 3295092, -1565517, -1751713], [41717163298, 18069270, -194070627, -5061579, -8208261, 2379, 3525272, 4822680, -13156364, 1911955], [428150520, -179208869, -44945149, -22699724, 8360091, 390236, -4863837, 596372, 146252, 563434], [12874613270, 16680270, 35940249, -21087965, 3570109, 7704002, -3782482, 9388991, 7514542, 3675296], [28919141987, -43713973, 197252239, 16489893, -16191543, 650247, -4104438, 1142120, 17628490, 1338535], [47574652378, -110747192, -109370649, 86431, 20485572, -4220591, 1981955, -2709112, -1513735, 4879896], [2252914730, -209609365, -220636967, -22110110, -4251351, 11425442, -3959333, -1121280, 7472771, 882952], [-59408181763, -42000372, -230264806, -13489910, 23316810, -1091421, 327153, -7527344, -21086683, 858908], [103116571539, -54788374, 84002639, 33985058, 6597510, -8130521, -43431, 3626170, -17498669, -13122448], [17270068946, 130598998, 420121301, -72952978, -2497571, -3190394, -2101573, -4402616, 7862631, 9761460], [-9269043014, 242830513, 579620, -3194161, 32053770, -6862307, 10408625, -10380612, 7643227, 3207299], [12846105028, 324179009, -75294036, 55668240, -13086258, -6878165, -3255307, -9734743, 47392904, 2911884], [-21393997685, -32392727, -350076507, 52642592, 11660511, 15781539, -18163059, 26150468, -1160068, -2065656], [86647627285, 40785777, 243410505, 96608012, -52021943, 9959716, -7879079, -20499312, 15536970, -7520910], [-57510903836, -209179401, -197174496, 7753938, -3209052, -12739338, 11796470, -22841448, 49231966, -134822], [114178693386, -448632882, 269089578, 23787571, -28482934, 21073545, -21280853, -16842212, -14211481, -349180], [149281608232, 769132356, -230833263, 641678, -21459768, 5046418, 1982126, -14834956, -26124466, -1578792], [117585525228, 263853522, 361204828, -45203172, 46754010, -1156624, -14305424, -33687624, 2442686, 13212088], [62028646996, 582126972, 38446872, 72460319, -11906690, 30825901, -29302187, 23476752, -6355491, -4158441], [19730256395, 7024999, -292450561, 112641635, -76969331, 22611193, -7495292, 16048888, 59050598, 17640677], [122866782780, 561387909, -290415114, -65306486, 31381535, -14619516, 19818604, -5840222, -44693101, 21471608], [124955585726, 993846216, -644293459, 18423835, -62046969, -37937598, 54000750, -53931137, -31384156, 2505248], [-193897080825, -989116398, -160576182, -73733721, 37182792, -33919107, -6766497, 28514604, -1458294, 2671167], [102881471890, 1499709250, -270924738, -13607183, -5376670, -34543877, -18330991, -20693760, -10399323, -16517820], [-169378775907, -520096151, 1037746529, 124282631, 57331002, 21252853, -17503909, 13947531, 39903628, -2495836], [-116131166729, -436593069, -1186232549, -102022721, -5997694, -15790347, 18508107, 14488501, 112215738, -11410356], [-26852247488, 281593512, 241460118, -92392117, 63749252, -74886791, -7096145, -5088164, 108871973, -16972589], [27357530365, 108222730, 507437910, 272628569, -76221952, -3061930, 9282018, 8901324, -72418750, 5865400], [-301340459035, -669225583, -512338571, -14537464, -30929631, 37732075, -76873579, 39235150, -30623832, -35396564], [-112340258055, -646371375, -620656254, 237247915, -65878840, -23763463, -19320517, -24487116, -7107663, 461691], [111613870966, -1740091023, -229189185, -181325092, 171966371, -22698632, -49350633, -26962340, 111527785, 6882114], [-219986302660, -1817926198, -145434458, -237898022, -73931374, -33599338, 19280988, 17768576, -166607588, 29722030], [-65985771580, -1240964318, -783350094, 175096065, 141912648, 27005615, -76114259, 15501406, 43586773, 10873920], [-397182742008, -1215518474, -584002059, -131718866, 41151392, 13592596, 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1684130677626, -1194353811829, -268787179011, -12924226863, 7462045302, 12873462666, 56977848855, -71671806042, 10129578156], [270080153766121, 3541172678256, 3875954715599, 213537523796, -86303989962, 27251402153, -56722264409, 106186453400, 139060753551, -43269819664], [59606185528583, -1841662287398, 1768007849740, 87571432154, 145475477194, 32115133159, -99324820387, 41560389406, -153756299925, -35456284140], [651272794690246, 995823302612, -20240556750, 274464941937, 291072240252, 32033381563, -45346900303, -24732058852, -124656062497, 28043762568], [531524457100899, 417215539985, -596090995449, 59473197023, -160837033306, -60235329717, -74969133051, 79363474715, 262632108718, -19965450444], [213473843672656, 2917038990568, 268298315486, 26948322316, -144247219474, -58088244340, 59611139882, 40921113688, 143944309382, 71062606904], [520177449913905, -2310662482859, -2786229465418, 619030230965, -112374923154, -1405703673, 78210983329, -50884824997, 242705095858, -56358374284], [-300604342174143, -2599679522636, 705534758228, -105281070866, 118768215888, 155091708458, -41927419266, 100657112048, -137585226496, 77891488342], [354727436230742, -3722765183443, 1488461676464, 84591168890, 25159378806, 63594686703, -95279877615, 36956960233, -103704066988, -20109706684], [-199796442144898, -4193530257346, 878528289055, -338934310513, 88983017753, -57942970607, -47601499488, 25528088184, -149627928124, -5293399607], [223133398380729, -1711886408081, -1477853174698, -236220792460, 163678147752, 34368846226, 129562946834, 114105262983, -11549021405, 15119778320], [66965402250900, -1270589343870, 3530821751166, 181897128850, 31063524250, 92227995950, -40684175780, 33763074536, -334073984956, 78999406398], [-344417045315203, -2832585390885, -2981570362194, -239997668464, -29567088138, -69904964600, -37849248658, 101084161688, 95888669705, -139463290116], [379227250535525, 2853323485451, -1087724356761, -385796891356, 286037318585, 38639928621, -40303403261, 71878495328, -71057026344, -15257470264], [192827882544706, 1425330195861, -3360588186316, -374711178256, 206877915, 44705699156, -103087634420, 59878191758, -365034039621, -12656837668], [-130266529885536, -6636039117639, -419973611151, -192563030984, -140351646239, -79460226568, 18182426821, 109350721964, 327272144984, -100568959398], [288054521184955, 557060759489, 1337909728426, -167209154957, -67505889392, -107621488451, -75506960093, -23199793323, 307262196252, 2784571396], [-469075971791366, 6010378431296, -547314083390, -327442457127, 261612382584, -107091211969, 36742372389, 52419683524, -356855848193, -25902524912], [85128673466222, -1152787892049, 3673977748652, 307639157215, -49555687368, 101405120664, 46577633580, -526559027, -224090685253, 29394741128], [-15478607201520, 802939900899, 5183492657328, 385068726308, -215860471104, 47697454780, 29874888044, 129501903068, 290058456974, -16863509184], [-386612361801497, -565596158792, 2138458635779, -161137363112, -93182490965, 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-198150496002, 52737401702, 84444143466, 27104521420, -29689278072, -37476456866, -50399042494], [816458925761416, -428363875324, -1790763429884, 103695598443, 363010289218, -188447304447, 46505424501, -234584992776, -102453413615, -19455470445], [-839103899257035, 770229208389, 1246887841795, 201104511660, 24098013183, -36189837679, -80437187105, 42159537348, 273003890882, -18301703120], [-384113898022690, -2071455459160, 3427464161215, 140225327725, 69485667757, -9550308613, 164750719194, 16753677076, 341753890992, 8232193171], [28906858068236, 3777744497702, 1043461453651, -954075852951, -145437442555, -94696270629, 213608546242, 21768168724, -109615196913, -20976263153], [69688396665053, 1459028615963, -2915537647886, 74372982470, 125803611300, 67538137264, -53839293364, -45178061345, 166431454821, -41591621648], [786207003152998, -4485412855216, 4242474986567, -258369633543, 175360472565, 18301665938, -35897263314, 105501175429, -337385949440, -28406010104], [-1003448290939631, -1963819385217, -110034470601, -670518865554, -17523404277, -178888299414, 11787126405, -33433859952, -576101996910, 31373485800], [-234167458254184, 3380863094386, 1321987044098, -262084592793, -11798861368, -122310831507, -16273616097, -69029877300, -81333532495, -45965947161], [-385093658739136, -456862243521, 2780101554114, 989565657776, 581363034, 60693690856, 21152413418, 12278400560, 199482304742, 201722412], [-1067515912271237, -745146932112, -624667365075, 170651156804, 131618564430, -131744469497, 162968928569, -37666392268, -436965956095, 80467941264], [437969447154471, 2547672169963, 4420735054208, -623418346986, -151768858516, -86826944094, 109853084618, -19190753112, 51501423663, -66135561266], [526717534721868, 5649043973212, 1316868280166, -15949375141, 90701530316, -30759256631, -55970793921, 87832838384, -106315272489, -71439962713], [-285446600201347, -720991553539, 254690523259, -791460901321, 549949198225, -54494004875, 13252054876, -3114873016, -486712595540, -119161805999], [-505874242560894, -3597354772019, 2247851305873, 586452924821, 357255288792, 86258872064, 41088260848, -112742240361, -140967611215, -59894561564], [439776080154642, -5436486101228, -1865944988965, -738513644429, -229440083811, -167102181763, 177453376536, 32555500424, -415558478416, 12350965093], [994066171140220, 1725742078466, 1823330103074, 546766633395, -191225480496, -44871292663, 297725841723, -208258489106, -108055603553, -44161392680], [-747981547305977, 4179068882382, -513522121878, 137582282822, -524901028234, -10825280270, -250382265788, 195566576056, 163298982236, 61169603482], [367481353736231, 3525920869821, 6298258449948, -385384444127, 156860804414, -72001674565, -2497450033, 41537150100, -344484223089, 91356109677], [-17678733028627, 1024169778833, 3082941773046, 138571386497, 4059169254, 158844702323, 86785638853, -156563052329, 293901067642, 77459073324], [-794742518111289, 6586133457927, 1619971443701, 665110463523, -332204279094, -170935050155, -160384631173, 110899101033, 178126118590, -141575447092], [390742279176286, -3776484681966, 1225579775534, -795719362544, 210432291002, 202406986290, 51026293374, 36370336756, -411863808180, -80713707264], [-324546756732081, 4010375689713, -1539868630236, -304146428223, -72557361654, -293934067779, 207129891771, -276398520297, 363383657064, 57675878244], [2426710523227918, 8460318722482, -1207184228613, 251037705783, -182985659397, -98249979536, -33563329232, -90551262427, -119133092144, 66077831860], [-536727886142337, -5041585102530, -2324416305882, -232374540063, 148047543969, 193462689279, -395990768751, 301584559389, 378071972349, 13289663472], [1342366222433939, 4506201340107, -747125557971, -1364775974046, 546279813, 255459105933, -201936952637, 217630966380, 110324166704, -5830764964], [464544300930714, -3895567225111, -6463763197172, 286557985807, -832317376214, 293934416645, 41720069241, 6394664956, 254241657387, 94581634099], [-479596001603652, 5226337405813, 553528336191, -387582318915, 96412399392, 90299039590, 226290412586, 111899557211, -21172268829, 118736943924], [147805964476308, -7475665064717, -420452515186, 628823209486, 372012914922, 66541122962, -6068187504, -129253094812, -233781540118, -53387182826], [-506066038242650, -10825458021314, 5184287849617, 357966187810, 7388491460, 161565611110, -53703053182, -136832461350, 15527922048, 49517648760], [-1006139009108687, 4821003635114, 1041713038512, 168736186056, 132800076474, 230944272307, -228385161919, 145783476818, -123302692285, -27320754612], [-839611624766626, 1866929299086, 3310054652398, -112922726720, -198080969994, 220708086202, -145455603322, 380103983308, -419550542976, 13027299736], [1170633600951970, -3011115452262, 2323741187763, 1044723023101, -302464980239, 101645531531, -245999417394, 90965892284, 71521841240, -21654263133], [-17409263141647, 2230735166645, -4063522220893, 757834776374, 72820305831, 90128501235, 234553144797, -107992982000, 174566942312, -41846745700], [382132323562112, 2203827126412, -795193790833, 1052199737343, -242479347763, -39661845123, 123876894064, 56101904244, 407386495189, -63069347827], [178059199858635, 7568102190593, -2729264718985, -277779982808, -330620547959, 111916640581, -319089280885, -43599027342, 285170377598, -18664281540], [-293168034571330, 12880742603744, -4192240137524, -42632805576, 125323789444, -187930625328, 115727761220, -1605231888, -104392247092, -63131887040], [-1751652855151936, -2109796942752, -2432267270334, 564563814224, 75701253930, -417872507096, 308464656362, -213791683096, 463603955402, 4467227652], [-412943779332345, -2149175047370, 918198613106, -868994857330, -255164241594, -210534355805, 321890880881, -286961128986, 573139666675, 20658833420], [-393406335951787, -4611536285057, -1792476495342, 726395492692, 702358791896, -164581071266, 266115528070, 42711567639, -804526198433, 66448516984], [785204032313404, -5362293587533, -1737052662839, 632987689758, -405642439259, -263886157518, -25528648785, 204321069744, 165502697560, -46777221036], [-818961644380847, -112195377793, 458968862345, 809599023692, -829028408161, 148099245267, -383055343939, 218301110438, -351811720198, 591017212], [-1347082025257768, 10278022754184, 1694210536715, -37046221331, 123019773516, -53918350239, 115416720363, -217351916036, 29874239261, 162854536328], [10002515741553, -4369696605242, -2694057991176, 252073661096, -178072661154, 74449328235, 173541628521, 118292043738, 40793292783, 276670136844], [-262772074348036, -708270506429, -1552189471228, 1473883811946, -255171670528, 190795527198, -93269952158, -196572255780, 328467424060, 110243429006], [796067769691402, -6464691942754, -111617269338, -298217379687, -57355925658, -20758471645, 101776938873, -247367999624, -284287631951, -93482351948], [-272937616876151, 697751338861, -4326411533493, -108347981591, 55301212554, -100798844585, 243768535793, -243331222297, -41794762678, 74415717708], [-45419549482926, 2864255427072, -2693168819508, 110919941678, -124584610222, -298638570350, -92700250786, -256614709658, 336131052418, -7319875128], [1270639048613467, 6730123081253, 2314454079147, -507765730422, -25809445647, -171914900705, 172572175937, -5392994398, 38115183416, -130966304088], [914001412010763, 2003666812403, -3574769065697, 127773303292, 73335614063, 141739911236, 324250561799, -4997319832, -808327833302, -21205256546], [1286518714843409, -6061353464139, 1109824623375, -482214968689, 62355218645, 161712056509, 25396672488, -174420211688, 167927146560, 164590212209], [1210015066445566, -851254443196, 3325793880121, -65382650255, -342318918169, 213525120719, -58829036648, 517245506972, 520391802282, -88616749701], [-758339555930232, -6889633197912, 1997055566628, -976102484589, 305333080442, -206029359207, 22026685789, -268339794508, 201873132539, -89480787417], [-147666303655863, 9116847926119, 719102394596, 1564479964481, 382364843870, 115226807491, 131912322663, -417857772628, 203821356633, 56697266885], [-573647650313501, 6060438045054, 1979590514754, 153772826659, 31790762584, 366161973809, -8095947189, 397750231148, -575545736490, -49075214357], [-1066166740403542, -7367217957637, -4988258512696, -290672124976, -90996282963, 65849527896, -15286245336, 365653031074, 172075473161, -48594916252], [-210327058869042, -3467099278858, -1601327330753, -1389544660234, 308324865327, 343380696358, 141976484789, 47405597152, 334043645721, 25922554000], [394300428715274, 1833867712556, 4944431631162, -759597161587, 223027234716, 145931942699, 271694196457, -85386982724, -839045105729, 162023623536], [614790313855082, -2984711248598, -2278832384521, -304367446731, -188697839107, 82811871987, 226398252810, -186021981816, -387110329030, 232609776927], [-63420912089085, -5035992842280, 2444655044703, -588924787316, 464417699887, 32646192236, -428932720481, 290692159292, 430381225805, -32200658478], [958455575883861, -3872902480696, 5467357595758, -678374735359, -7208960596, -384578088264, 218106676800, -227881088710, -246616744862, -38344768440], [-1306239847256437, -14093792206645, -4382169588835, -113940604606, -116127170475, -58605178323, 279476179059, -15062311244, 233989774616, -116136234268], [-594655113323210, -5367669818556, 369681738309, 860168867640, 635836071729, -317978601324, 224146257401, -366122334696, -487063088255, -61202249330], [142581058235505, -4391524707599, -4365474271502, 552722642668, 508269549658, 293920697934, -203289272946, 31147432627, -180413635327, 160481961228], [-1199333050259401, -3125202636622, -3727557325156, -311914693672, 467155896174, 71129374905, 189964876155, 141096004042, -1263633578299, -114753913636], [2471211435985392, -15235422748178, -223385051332, -369279144570, 103750889960, -30717116478, 164984504318, -374608834320, 326734307310, 189034844974], [-231416631725498, -21029495651464, -450984557659, 190808351781, 85680031092, -112090407013, 91652765257, -65616606544, -229733496381, 34579592720], [554843080511632, -7486575294399, -9247413546536, 1261730068176, -193246819875, -195419343108, 44668092596, -48286810380, 162476747707, 65931600280], [-138583420153224, 5392369797334, -1399846718716, -218147501733, 297624792098, -128339992479, 16951294565, -240602878896, 525013238361, -76263645701]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_338_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_338_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_338_10_a_m(:prec:=10)
    chi := MakeCharacter_338_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_338_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_338_10_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_338_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,10,sign)));
    Vf := Kernel([<3,R![-87995528970734921328, -7956119776309125408, 612023720013831528, -3479104187464896, -83635921636191, 435445398258, 4579527943, -15204372, -112385, 162, 1]>,<5,R![-2957278787230750538398064640000, -12482104189978530618071040000, 4794496741565555513299200, 51355697453393447623680, -209882615848322784, -64885743921426048, 11664060879169, 25886451056, -7137486, -3088, 1]>],Snew);
    return Vf;
end function;