
// Make newform 289.10.a.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_289_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_289_10_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_289_10_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 := [245077910606835, 1064197876989579, -214257360688838, -13493392282513, 1825619566875, 45662797046, -5553197164, -58373994, 7481825, 30079, -4542, -5, 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], [-760, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [701509941388414409433140763459, 3732897309622961792946904076232, -548628192819923458648856575006, -65957615113789675276347461075, 3292956958038732608477108606, 275013179932437482820616952, -6948299449742092042655364, -454514350612185104303014, 5896655418155773172983, 322634003582572989064, -1668432767667957782, -80093781578284239], [5255414047772828734012062681015, -47137456301623675542202638476736, 6271988632058581672051154664066, 830170664689432011658635188065, -38853988970844220928548203898, -3495269006865732661897190648, 83912267684529093307762012, 5809816055731970105835522, -72280913916115925890469, -4134688441827015236896, 20639885694648035978, 1027318723442988917], [-18986969898323996210182528522605, 49335026553834213842769693626862, -4669017400385352154334333734772, -847340873690671014742472820745, 27949877987499825416074974046, 3420432139412872864132007476, -59342598944998734836732304, -5451799779840212435870594, 50614426288897207070903, 3735766330517180425382, -14416752093429009316, -900897902017065789], [2359739161006408280825862502365, -34989710991916834158048917072022, 4032336743372817742689793040572, 543713607213283367639166505025, -25024295530584749935814767166, -2277685280233752642897121796, 54248826451219204927502304, 3789219103533746198997874, -46839866164392227951623, -2697723826892045191342, 13394373311401079756, 670371815609386389], [8662793208503295878078855808663, 33639153956546129459008461216096, -5342321987414437803190005836510, -649788465738338760770811241119, 31899187465172393995042989510, 2787598575616195980365790920, -67510941766707768483402212, -4675560810984620609410430, 57625014826991253144123, 3346313128168059926272, -16404894253827117782, -834216383833978891], [-275526238849183879649282172772305, -1256908941343437190429417964976912, 277924747615285773947753802361058, 28969767384961347880393591130361, -1734729304056625276164990597834, -129717785895488546113318450904, 3737836836902791769863653596, 222729126452520365073830066, -3206928289437307271662221, -162162795112870331791024, 911977503281285182058, 40942056061913583085], [-82703258449303166667783527020605, -100970716386893921192277805163952, 19831728501896212918730559222682, 2091695195385783247141789314085, -116468461958859652257548430466, -9069757075530163387757868216, 242917792176288379859012204, 15248916558194711495917834, -204676297414330842841673, -10919117377568207322192, 57664270417532406786, 2723145386634939529], [-522763747975699712741618218931985, -1330219526532410422035111109153512, 218075646266251424503402000602490, 24648431426288915422541328320273, -1309508399666758654746154761290, -105181967259297219159697054280, 2754592608886413282124299660, 176792970112448756785443490, -2331463978248493959698317, -126988627419959499901288, 658974985043379944162, 31760766357754896357], [276147846796300786476702103266405, -1837825967478208821502769004876384, 273838509849131543772109488838054, 33472142953519634570880217210115, -1656373188096760002436379918702, -141066887150315321200515021992, 3492878412547823975070240468, 234354542123187532885421638, -2960241598858705747480111, -166686752186959380648064, 837346354213994513342, 41393373062460576063]];
    Rf_basisdens := [1, 1, 1, 5652204141514324784999104512, 9420340235857207974998507520, 7065255176892905981248880640, 3532627588446452990624440320, 2826102070757162392499552256, 28261020707571623924995522560, 1487422142503769680262922240, 14130510353785811962497761280, 9420340235857207974998507520];
    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_289_a();"
function MakeCharacter_289_a()
    N := 289;
    order := 1;
    char_gens := [3];
    v := [1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_289_a_Hecke(Kf)
    return MakeCharacter_289_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 10;
    raw_aps := [[1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [38, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0], [-483, 56, -1, 4, -1, -1, 0, 0, 0, 0, 0, 0], [12638, 251, 28, 62, 2, 2, 0, 1, 0, 1, 0, 1], [1979, -88, 36, 3, -15, 1, 0, -1, 2, 0, -2, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [87781, 185, -118, -135, 51, -16, -8, 18, 2, 19, -12, 2], [166367, 2670, -262, -2015, 159, 116, 48, 35, 2, 30, -12, -9], [1048103, 37430, -305, -4506, -547, 74, 148, -580, -28, 114, -32, -4], [476824, -34385, -657, 4299, 1132, 143, -212, 208, 54, 73, -24, -8], [54005, 135341, -91, -16277, 1549, -813, 76, -2319, -22, 91, -58, 63], [3419849, -32575, 906, -21846, -7415, 204, -32, 4170, -96, 63, 56, -146], [-1197488, 67637, -20189, 20087, -6754, -277, -368, -3052, -86, -53, 176, -172], [11038704, 196269, 25308, 1882, -5406, 606, 100, 2911, 272, 215, 388, -5], [-25116189, 363140, -51183, -246398, 12765, 2770, 292, 1076, 84, -1012, -64, 276], [-8989231, 251099, -75148, -224128, -15811, 1190, 856, 2300, 650, 945, 420, 516], [21273672, 1262695, 98093, -10009, -1415, -4689, 912, 475, 1330, -255, -190, -1387], [-40458963, -943697, -20013, -275695, -32953, -11273, 1640, 20543, -24, -3463, 644, -17], [15447911, 3699041, 23272, -167696, -2665, 7614, -4280, 34028, -878, -125, -892, 1748], [56758499, -1303328, -20709, 505037, -22892, 7519, 8444, -6869, -1618, 5624, 3458, -127], [35103295, 2611553, 94798, 257582, 61371, 11532, 10168, -39406, 982, 3043, -2412, 2850], [33221712, 2114014, 130117, -1000638, 14584, -10525, 15752, 25437, -4106, 7838, -2144, 3325], [81741027, -13441449, 220732, 1357408, 147947, -29406, 8512, 52808, 1156, 8137, 1364, 5000], [-303924296, 11857348, 46905, 783622, 60763, -18180, -7952, -3690, 680, -28712, 6920, 1122], [102444760, -7205095, -348425, -1500223, 847, -2579, 6416, -41367, 2774, 19159, 14886, -2609], [-168704673, 2469353, -864754, -1055150, -286817, -95696, 18312, -226530, -8274, -22605, 4844, 5534], [-147920883, -28980140, -627712, 1537754, 11025, 21732, -7800, -3427, 10798, 7636, 3652, 16677], [217559952, 20082797, -1143948, 574740, 402379, 130718, -45092, 207622, 3360, 32731, 6000, 9218], [445620317, -12260971, 502102, -1809755, -292528, 14719, -54444, -100637, 5158, -32721, 19082, -7823], [-111600588, 6412951, 847547, 3507615, -297448, -107257, 28188, -94320, -32706, 86625, 25496, 14744], [-226680011, 43849219, -1235067, -3741161, -37293, -152243, -56784, 175461, 50756, 63037, -33476, -3555], [930607857, -13174159, -3043639, 9329599, -992851, -203113, 4324, -199431, 12502, -2401, -32262, -17989], [78203575, 98863064, -856460, 7967759, 1537029, -124424, 10760, 58185, 21670, -98776, 26780, -36943], [-279826322, -14571666, 966613, -7328053, 949162, 130569, -108316, 69185, -58382, -64406, 84002, -99097], [-887589799, 29681112, -153238, 14432876, -2061245, 152570, 164988, 364985, -20470, 5280, 97000, -8299], [2000615877, -22845981, -5821298, -4030655, -243870, -128023, -99524, 1427235, 76598, -33299, 98002, 29669], [-1798071686, 13078545, 7518029, -10425511, -1830112, -1399, 112408, -68608, 125938, -45401, 34472, -74720], [1800666958, -80153302, -619732, -11461049, -990720, 265670, -206340, 342014, 59048, -243774, 109932, 70494], [3323658021, -91430538, -4057149, 23369049, 1210328, -275037, -8272, 474261, -122706, -47454, -44294, 11411], [2205670270, -38509455, -7716607, -18289926, 859722, -277263, 280992, 462334, 34550, 19223, 120040, -52058], [2573425912, 41990723, -8955678, -20999798, 420105, 251656, 380956, -1208276, -10044, -210995, 62364, 57772], [4532995551, 318619773, -9146519, -7832872, 3206271, -588521, -19732, 902015, -131928, -449401, 67628, -143541], [9154266183, 52268882, -14893132, -15230848, -1464878, 61852, 119096, 1249276, 34448, -424906, 126032, -82036], [-8233514286, -170518656, 4119267, 27015640, -4799401, -760100, 211712, -434724, -135184, 94168, 335144, 45932], [-273377439, -351998504, 2742415, -29944845, 4061293, 287921, -237196, -2385834, 82972, 635372, -264092, 48254], [5736027371, -376648286, -4600876, -90714367, 1496073, -866012, -600896, 2730367, 167022, -360674, -9452, 284231], [5903480903, -907228612, 15634404, -91433415, 2222659, 1332794, -153120, 1392491, -96058, 355056, 702788, -220417], [21370234869, 72275808, -36752301, -84588921, -666309, -1475799, -316392, 1507284, 157908, -49308, -495708, 62676], [-10994504212, 156259496, 50175004, -159128812, 1860924, 1136520, 164432, 1628488, 120224, 1109432, -134624, 370824], [-7253176190, -550380346, 2708709, -26285509, 3851238, 2069185, 1047252, -3153847, -272598, 293714, 503998, 163003], [6469258284, -587761051, -65337088, -53211774, -11774830, 854874, -1231036, 3142291, -40576, -1133881, -642844, 34887], [18497838433, -404649127, 18831937, 65787428, 5024034, 1735260, 2834752, 3643412, -361112, -428213, -210888, 343908], [906414600, -106880982, -672738, -89672157, 54358, 3012038, -1289272, -4902158, 509568, 1024414, -225928, 1092938], [-5945378542, -752193342, 61758248, 103223829, 3531023, -6950507, 737780, -2723773, -448254, -493534, -664746, 192737], [-34061187047, -1045849589, 19072986, -170861057, -11583961, 3998806, -1105276, 12759084, -712834, -87155, 648944, -362908], [16154358408, -20987481, -66330989, -60549386, 15688726, -6647744, -1078036, -9866564, -646940, 882537, 941460, -274620], [-5377790394, 1323746065, -45029621, 81408178, 7880300, 7094825, 2945976, 3506214, -528450, 1155483, 1098780, -745490], [-9225380424, -1091416516, 6811026, 265617858, 9703780, -4465982, -2833704, 2843490, 617052, -1236996, 611308, 114270], [24174852378, 337165790, 84570896, -109329109, 10335209, 1436975, -1062196, 14312821, -842010, -1145354, 469570, 989223], [11291001792, 329442212, -45038474, -221174954, 47635212, -2976382, 2478144, 8532394, -636380, 586092, 1220040, 151594], [23895550011, -171685501, -14399529, -340313651, 21243395, -2738659, 3619940, -7437197, 1176198, 1468045, -2129718, 1185757], [257205670, -761712175, 108046033, 189183322, 34194966, -3666891, 5040920, -14285414, 1644470, -4370137, 428400, -2229270], [72872531577, -3594081313, 48590277, 186018568, -37926187, -378737, 134724, 16376555, -1009932, 4948925, 125772, -1206289], [13997777295, 1454726103, 43833801, -35048101, 43007839, -8552821, -4309956, 18215597, -548674, -1779999, 585034, -2246625], [-15873063352, -2319549683, 48665617, -428412089, 12983191, -1595977, -1540312, 17816039, -221998, -3707853, -1138502, -231367], [-29067470785, -2559688004, -82313951, -360301394, -4924281, -13649793, 6233936, -1759826, 400472, 12800, -1835052, -1532194], [38927942308, 2167529979, -7096797, -407013800, -33255838, -4837596, -4349800, -8092848, -1121168, 3406577, 2368848, 712552], [87417904276, 4646601802, 145481345, -218513688, 22479168, 13490799, -2103880, 9869957, -439034, 6959674, -72576, 4374349], [-62279133740, 4912949609, 123442891, -65355183, 20842431, -5986707, -420960, 19949013, -1460154, -4797969, 114294, 1851555], [-21536616737, 2678631904, -205573628, -673024092, 63007884, 4846352, -3115384, 16069352, 1007984, 1329120, 192336, -3427512], [-11724380066, -2226165215, 157569616, 491920753, 111853906, 9267796, -6014392, 11619799, -2466680, -8036577, 4777560, -1383261], [11416835649, 3283594487, -72442669, 671892225, 104185959, -1898489, -8554264, 23777415, 2513284, -6638311, 4179096, 903243], [97992875534, 4875108097, 56677918, 395087070, -40898865, -3200416, 1302812, 20660984, 1110660, 6163247, -7354460, 4062792], [-66695131668, -3495440581, -262420751, 845477561, 100089278, -11490883, 1045032, -1763100, 3612654, 5004189, -2274224, 2114828], [-28921291076, 2123002940, 32778333, 598145228, -72996380, -24334467, -4735156, 71305491, -1843122, 2776836, 191472, -6852801], [13233425577, 1084190976, 227382040, -805450616, -16392990, -30112200, -6402076, -12917062, -1325000, -12446840, 5997940, 534078], [-303125822467, -7385621862, 95908937, -799037681, -33482920, 23171453, -2925600, 27920895, 231618, -959226, -3749418, 2987929], [65135777903, 1980739075, -420445663, 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1141040180546, 666549528713, -19418371032, 22221267860, -59092589287, -32863908542, 19728832336, -9281546488, 5855938589], [489058243179129, -3006786805255, 1492211896018, 5538663395414, 97478487065, 25237883288, 75194221920, 145628693226, 19114903296, -8696246049, -9181522736, 4753306094], [-1110273050713469, 2055923826364, -1704469201655, -5945418090864, -104808759503, -13489205578, 23552133696, 99284520608, -17545507372, 3742785864, 3488021252, -3172895096], [-342505022950449, 19296856350616, -1505139774740, -1609849871131, 440799322793, 161976406421, -16841918140, 114342088651, 2529790442, 26358266828, -7292714442, 10307000257], [438813370020688, 26529802013308, -867748863757, -6871995199631, 194624461280, 7841174127, 72831360740, 273387691143, -20255516058, 2896265748, 21606776318, -16817400047], [-696647686489246, -52409200287193, 548698299916, -3614348594120, 71681364495, -29476365866, -56314809968, 273618939100, 9182384020, 32127869049, 9245297740, 9906219284], [870322231661150, -7064574991759, 2721051889783, 2879899637655, -32193209358, 49596945043, -2182926004, 205644007704, -27202839614, 10126656359, 9879000144, -12667015864], [1417118096249721, 10902597592397, 2469135259263, 958910651672, -40408044840, 52174120140, 69042189768, 55526973232, -26702251696, 30673431335, 39849088896, -19153450840], [1058844542869645, -29899358708005, 705113726958, -2967682515218, -194538874979, -116301953532, 95790993776, -100594222702, -33390936066, 76677162233, 9220938516, -2108154518], [1774268646070838, 16525503867743, 216519136640, -4747348672709, -195476940566, -86785562424, -79177968800, 159398856381, 4407307200, -14872766679, -7890332040, 24306232569], [-956694475584801, -31154643729019, 941304594747, 3878928209669, 887322078717, -55634022745, -12885147840, 114251178163, 6332457136, -26242196565, -2921075576, 16747651623], [182005494596166, 18981296537819, 2734686109351, -8343102695868, 288600636744, -117397935830, -14703351924, 90680006258, 4881430880, 63308909397, 6849800800, 20278574078], [-165378001017531, -51092304632857, -821431350119, -426084874805, 1214204475159, 87172432293, 3284593936, -80113051759, 24609508900, 60317345641, -53897989900, 21927623169], [-194010184402638, -6348358642683, -1280559214826, -1261159976808, -23359613332, -137413653076, -35328634196, 234359131967, -22481325184, -961837377, 6229929484, -3099565309], [-132027500413491, -24997638175766, -187064275105, -9036178342414, -497926411569, 49303919080, 32799531680, -38849027154, 11692323160, 3688760090, 5946818000, 25787171906], [1303735658895460, 23370696003729, 511405260947, 2278663462400, -364841204654, -11958479502, 18739345212, 114011593290, -25638921432, -67091772897, 29940734312, 1660065238], [634058359809188, -2121566165212, 794589760856, 4027487841480, -343620350748, -75970542352, 34767915312, -43084767192, -24399907520, -7521074980, 15896039760, 16612188488], [595211860006180, -19445258198895, 12901618545, 3019538644920, 885585774568, -146820742347, -88805868632, -42367723980, -10511862838, -48266511377, -30092562272, -9012256684], [-40399720702446, 909667551422, -636911563425, -3489043190584, 513149737330, 171036507531, 47024311228, -239939155319, 23993299670, 83896341890, -52888339960, 34562386485], [-48943321396831, 23910256330144, 2819732945648, 1701533822431, -153911556585, 51519559395, -123360507740, -28614982327, 20452900750, -12695306372, 9877649810, -17605394277], [209809601033128, -61893636027685, 1549262317630, 3709553679092, 770937015770, -110682497572, -51979837100, 200789876339, 20894372332, -127360919783, 3622750388, -12009305329], [1437548218466574, -1257219081845, -1416391498096, -3092162522155, 143560904552, 15173648935, -17781251996, 135996515371, 6206508094, 28821657849, 6536319986, -19054428167], [1510787084145886, -15419370782336, 356190341783, -11012491401314, -21586481322, 55768246645, 48894733936, -224532589445, -5909170434, -60189970628, 22468815744, -4757925037], [504073098670034, 62768928975530, -119830247634, -6496971664142, -42898823334, -121412378838, -14012242616, -277684033250, -1195797868, -88370248810, -7368766892, -8471885006], [-754233449313476, -24784320125892, -263171836585, -2095860710831, 971524696744, -248565873461, 32236520756, -284254752389, 5183201798, -36323807428, 15669973342, -18812939523], [21556213725266, 59290368339603, 1078913899651, 900657386570, 14680670388, -92174296978, -28389075888, -6979518850, 25663083548, 47753801985, -6564297428, -4389754462], [870899976772261, 7619161485578, 4399871958268, 8317739504474, 971561533167, -114599846172, 3700655668, 66242193821, 7624596310, -2890119530, -10675685520, -8579907495], [723883011253466, 24677477742218, -1658767683932, 511926627932, -593037402350, -43788007840, -82172508640, 40187781772, 5657384976, 81953334934, 21116722624, -33225560700], [649697139648445, 16185084404898, 346213579216, 1280612132787, 1278428907355, 67836496249, -34789470304, -26288014957, 24837136482, 22113336758, -25137361002, 50818501493], [360320785823150, -43697630403410, -1908244242795, 6367604734702, -592387912382, -124203552763, 52591029388, -137582490941, 43254843186, -44635766078, -34984353536, 18399923455], [459999255464145, -42408658006495, -544171052069, -8525124783859, -134967611651, -81649151183, -15252943788, -310859795469, -36293938986, -44259618825, -19278425094, 22010176525], [-486948185890231, -46044952333103, 2415931462139, -4356162143299, -216222866983, -152093464503, -155528645780, 26534413191, 18927280458, 52087844607, -10597205498, 16913671805], [-1123276725420600, 19425159956075, 541350233453, 1023219860962, -585086916820, 35563306765, 143665882568, 104010819182, -26844050670, 57808378301, 22869980200, 19743103462], [755434751880863, 43693683632264, -3011443640332, -4499901744662, 435067861267, 135916316100, 48175015288, -256448386257, -21270169542, 17672450688, -14927551308, 21052346319], [-206186656980318, -20289738713746, -2358773684703, 1121190240424, -454502660249, -17797786798, -105525508488, 610636724492, -13347996340, 42226908438, -46015951500, -14579642884], [-379493887613865, 28625532353064, -1386487092996, 9552806890096, 574162830426, -162515790744, 76743658924, -484366276850, -40590797584, 92228347832, 58879618084, -1081148534], [-345119132247985, 74704850625139, -394502914854, -6666012281313, -568639826587, 71341259092, 26156472152, 49504912942, -15377521610, -118195759839, -17422603780, -2872068434], [-196556771288426, 32250414676362, 1958133645803, -5917005280083, -1864006964, -4194339595, -81897586328, -763672004489, 21462173610, -83616660422, 20820755520, -3755903977], [490361196241728, -10886285884416, 1400198948741, -10292180523638, -294492471261, -173212784640, -36557357240, -224546250706, 2039249656, -105283350092, -26859867976, -29737130542], [1169967159227957, 9066903055264, -943307712934, 563455551126, -455308782398, 55822185550, -46405107756, 472485658332, -17129423212, -8346629096, 49336711632, -18957406508], [519973720433085, 36053020633135, 1750144267313, 1975775185971, 1921286703731, 97475060813, -72464011688, 150879000445, -18282349776, -118324319447, 21536212896, 3249655849], [1925095282914004, -28598582037193, 1487559150962, -1403508790172, -112887439822, 292131836396, 61786807948, -322788769549, 39885604132, -9537883835, 28586619628, -3442127985], [1659452984308542, 7511894161211, 757086911964, 3341296636401, 571770788188, -173908014765, 106530284876, 29647885571, -758975954, -71107108367, 38799708954, -20114593631], [1067163004291689, -36768797443224, 655362584691, 15016294185231, -606908024847, 220409510001, 60112260348, 186879961470, -4959391164, -66661267236, 1198738884, 18053582118], [281704615898654, 36496414956411, -939150585672, 3085042024184, -605960127415, 44912981562, 111688540212, 216006335418, -19123961712, -74798145843, 90887732432, 5005596622], [-641404901111367, 107459352746807, -1498894911013, -11467998984230, -900723629764, 381983420048, 49621250688, 92966024418, -935025996, 10666577169, -602186304, 43844801862], [208443040991462, -39022823493130, 641204449201, -11194675151741, 844576482564, 6697228399, -106229786144, 443210488417, -25883950298, -128407408242, -7664704052, 1527110837], [139584759903011, 31602260886471, 886265261528, 7307709048876, -1790173115797, 124465944894, -125072815152, 146103085434, 1872904926, 19936676205, 26827259124, -13051414734], [-640237755800803, 123997362218573, -3420924006628, -2642162657867, 96132315686, -181830384937, -25490769316, 57369483499, -31222062154, -38656187425, -24662966206, 20649935577], [-1191050467724052, -38979180489819, -816108704728, 7265223488180, -146303104553, -37231529862, -91812219076, -147204101498, -16326227680, 9350804939, 17010954280, 27900543346], [-1105142340562155, 16940077732588, 2147765263297, 519972447133, 833312915791, -36252602729, 68268684160, -418620875578, -19319297904, -63976309720, -64599878716, 17062231326], [-564599907791469, 45598199379594, -638433446082, -6387999958902, 262184377338, 108547044794, 49342577968, 233546079454, -63544382484, 18608109958, 112142354844, -39993086486], [162766966275091, 37549601914348, -2792288915105, -6027966396493, 757619146033, 338428552385, 42261730168, -260769090422, -37513379808, 65997149056, -11809461532, 12370112042], [-24539205312274, -88055905091687, 844920893249, 4929791875632, -179371157588, 16682872459, -93453187584, -352658132808, 37669505954, 36704680843, 9639972276, 44795412712]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_289_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_289_10_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_289_10_a_e(:prec:=12)
    chi := MakeCharacter_289_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_289_10_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_289_10_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_289_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,10,sign)));
    Vf := Kernel([<2,R![-1018109465395200, 1445191108263936, -163361958309888, -20230637637632, 1516569700096, 77342075392, -4938565328, -116603152, 7008044, 75004, -4421, -17, 1]>,<3,R![3604544866952434779806439, -15084709192180256275152, -4892673972957993072945, -336930405417511938, 1857421710435299328, 5576249697859260, -197610553835829, -392253170814, 8307631332, 9283700, -151309, -74, 1]>],Snew);
    return Vf;
end function;