
// Make newform 273.12.a.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_273_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_273_12_a_c();".
// 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_273_12_a_c();".
// 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 := [45555354651594891689197568, 1287697466669064032092160, -963518272786595267739648, 6711995200063092490240, 3737994671267475603456, -25041700255043457024, -5703452025696599040, 21355761647276288, 4072481535854976, -6951601887328, -1473489774048, 940095168, 279954048, -38138, -26640, -1, 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, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3330, -4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [176834599502592400698457516408918434121722277635046241051425343307811291598422016, 240897147968156598813537970936088374909801485020930669720575954614345765224448, -3745295559122432085210806503983519550067294080245748411140629304725213858824192, 126479005998672795277442521811489885289790712627804410888097831841872254009344, 11122503103109230453548815277603709110657036351178299336577046414005640474624, -395307570649520004140376482808457530565156859267187140382663397886713860096, -11561934393622871089439544864431278848419638721498390775127915731560469248, 392695581959005879795133506604889928448108179351560539435927670265942272, 5303191883484463394946802781135341731977446239820327486154662619110688, -169055082079650808243187465400928071820689465466348080912585408590432, 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    Rf_basisdens := [1, 1, 1, 24831738498029602485719145138738681664385431148220399263587878522637516800, 12415869249014801242859572569369340832192715574110199631793939261318758400, 2483173849802960248571914513873868166438543114822039926358787852263751680, 6207934624507400621429786284684670416096357787055099815896969630659379200, 775991828063425077678723285585583802012044723381887476987121203832422400, 12415869249014801242859572569369340832192715574110199631793939261318758400, 6207934624507400621429786284684670416096357787055099815896969630659379200, 365172624971023565966458016746157083299785752179711753876292331215257600, 2483173849802960248571914513873868166438543114822039926358787852263751680, 3103967312253700310714893142342335208048178893527549907948484815329689600, 6207934624507400621429786284684670416096357787055099815896969630659379200, 12415869249014801242859572569369340832192715574110199631793939261318758400, 24831738498029602485719145138738681664385431148220399263587878522637516800];
    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_273_a();"
function MakeCharacter_273_a()
    N := 273;
    order := 1;
    char_gens := [92, 157, 106];
    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_273_a_Hecke(Kf)
    return MakeCharacter_273_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 := 12;
    raw_aps := [[-4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-243, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-199, 20, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [16807, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-29142, -758, -38, -9, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [-371293, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [109068, 8475, 46, -26, 21, 5, 0, -7, -7, -2, 1, -3, -2, 1, 5, 1], [270037, -81702, 458, -53, 1, -7, 2, 2, 3, 4, 5, -3, -3, -1, -5, 2], [-9400775, -71794, -2458, 1107, 51, 1, -9, -33, -2, -6, -4, 3, 0, 19, 12, 4], [-6954128, -157359, -5841, 1582, -50, -59, -24, -36, -62, -20, 41, -12, 6, -44, 45, -17], [-5661422, 51704, 12366, -7033, -685, -152, -29, 57, 120, -36, 2, 47, 36, -33, -115, 11], [-109829746, 850542, -18748, -6901, -210, -357, -4, 182, -103, -113, 59, 75, 25, -95, 1, -16], [40667506, -1105331, 17958, -12645, -1734, -297, 91, -8, 378, 9, -204, 123, 229, -231, 172, -233], [-118087957, -569963, -79465, -45635, -716, -953, 533, 13, 199, 841, -141, -91, -407, 344, 304, 378], [-252462946, 3128996, -28244, -30615, -272, -357, 804, 1072, 174, 230, -566, 1318, 466, 518, -293, 363], [31412868, 8632700, -31278, 39511, -3995, 2240, 415, 2527, 1742, 1018, -1750, 799, 1530, -2091, -2273, 575], [232522531, 6287539, -660525, 107873, -10606, -5049, -953, -3513, 234, -1738, 362, 3092, 1560, -605, -1392, -2687], [1638380044, 11203266, -457491, -159846, 10776, 3971, -4820, 603, 834, 879, 1300, -3261, -1591, -4056, 4619, 825], [-331594140, 5189100, -1701177, 163249, -234, 7702, 1282, 1453, 320, -18084, 2047, 1194, 1133, 1722, -9412, -6611], [1066178697, 21303921, -1308991, 425629, -21446, -3749, 1822, 4386, 4609, -21370, -2339, 5576, -873, 6911, -8955, -16363], [1318718484, -22521913, 474162, -176543, 6710, -9081, -4322, 6227, -2844, -11312, -5878, 5960, 12203, 5648, -283, -6056], [-5228324044, -41212575, -4640952, -597540, -40538, 4454, -14794, -6985, 13111, -4297, 1273, -4169, -5080, -5139, 7132, 7374], [-4557373084, 46686629, -3588050, -924532, -21053, 24197, -4611, 4486, -4273, -2337, 9705, 1160, 6876, -3268, 1930, 22084], [2012719387, 33129976, -249676, -559722, 111998, -11475, -446, 35512, 35676, 45683, 7635, -29235, 18274, -46164, 8917, 19960], [455290903, -327830240, -5542011, 136449, -182617, 19554, 19296, -8935, 10863, -35807, 9597, 42600, 4166, -45853, -7314, -24949], [11445719638, -618105166, 4686977, 4268373, -160474, 538, 3333, 58, -34937, -127396, -20758, 13746, 50888, 120218, -43073, -100063], [2562543757, -231025148, -2970174, 384186, -68902, 49961, 87535, 26821, -36946, -23818, -19363, -17848, -60448, 132723, -34150, -59565], [42566210155, -660225144, 13739173, 1265766, 225696, -40504, 103754, -1099, 32234, 31417, 22704, 21011, 79815, -13304, 30602, 53282], [-2995251865, -512875065, -9565814, -1017217, -126539, 123671, 38350, 39247, 38249, -107247, 43614, -22124, 17848, -85419, -75427, 69266], [1720084263, 23934935, 10644682, 10159207, 601666, 170280, -113551, -73774, 83754, -80444, 27681, 40998, -68335, -90195, 100709, -22789], [-20391595873, -2790159452, 19505356, 7394621, -592828, 15620, 17785, -147387, -145627, 82009, -119728, 113279, 21791, 104384, -187567, 76991], [-28093874752, -504286773, 2249822, -11960174, -373313, 363095, -138413, 71436, 130073, 419504, 212704, -142191, -153370, -126110, 42792, 213677], [-228118717802, -983854995, -48592985, -16892437, 1336498, 9234, 324874, -319904, 59595, -60610, -41893, 63404, -344329, 255785, 3280, 339968], [31023502520, -1499806968, 87381563, -35597059, 2252276, -100622, -11954, 316997, 414236, 262474, -160437, 352888, 147757, 45820, -199198, -24969], [-111165970127, -3587553971, -47854718, -25017452, -1402678, 614923, -8637, -600692, -1514798, -645932, 168575, -49128, 39071, 483521, 174898, -528868], [-177908667881, 1726021002, -31935311, -22277367, 2545360, 310783, -100697, 324160, 474027, 103642, -89552, -642606, -284812, -314012, -257998, -66380], [245380718195, -9921823327, 200793299, -32286092, 83723, -1622499, 584924, 695290, -445188, 341169, -374075, -23972, -195448, 202696, 123565, -66229], [-1314832435, -8372890935, 95764700, -64743486, 639173, 933126, -411518, 42193, 1261274, -31236, 35693, 436559, -252657, -383544, 335162, 614292], [-125039874580, -6451609263, 16937169, -73940310, -660483, 1245408, -47103, 283277, -76200, 1105020, -775031, 355063, 1149096, -594189, -1214251, 196879], [-437767303074, -4437205427, -202304114, 8271387, 3381427, -1538548, 1076518, 812699, 1193672, 728645, -381638, 86994, -853755, 53436, 869360, 405811], [-436830483246, -4564847486, -330665554, 56620029, -926487, 2396430, -1229782, -1357172, 587613, -2096364, -358587, -453723, 173007, 181325, 864094, -21515], [-642107205595, 7961737984, -64777261, -57795840, 8663878, 2502842, -910145, -404036, -267605, -1204645, -642455, -252609, -1256362, -30424, -485617, -1534466], [-599837872350, -7637628732, -452571103, -19740470, -5772810, -953899, -534041, 711908, -958531, -129403, -998695, -14393, -95854, 2378800, 550190, 141093], [-1339889064643, -5848554072, -5948493, -41736882, 15345766, -583010, -1026629, 157074, -4086899, -891937, 267413, -377797, 1472608, -130646, 1635197, -757078], [-1119471536047, -24503905932, -35444421, 33531622, -3289585, 602633, 270584, 1902113, 2488391, 1449031, 728153, -644834, -2183072, -641277, 522095, 1594404], [-453000597847, -9696781149, -32824887, -54975064, -1409665, 1578751, -2037878, -1136184, -254711, 2389437, 2510555, -3869650, -136123, -1259915, 33587, 730049], [-766262683486, -36315138822, 1175726197, -377723283, 9889789, -960531, 2653015, -462376, 2068385, -2510413, 2166192, 1732756, 966266, 1406554, -1990804, 293568], [-1205458604931, 15235390275, 661167722, 212723785, 33982790, -8787598, 2325701, 9836110, 6766442, 2140562, -1759817, 3470404, 3413009, -1139151, -8361221, 609659], [-437424769169, 9639018577, 28166803, -131399043, 5313296, -11355019, -2743927, 2559571, -103336, -91482, -2384950, 1699174, 4079144, -111817, 503296, -5575069], [-483503189393, -13577645152, -50591962, 114932325, -9393890, 4342794, -7443369, 1140363, -7726475, 3860403, 1713964, -5844967, -1019451, 2333596, 1806933, -4366615], [-2236456502353, 30625511889, -1537774378, -249595399, -28662478, 3038246, 3810483, -6509350, -2049388, -845520, -386721, -1415122, -4886275, -229647, 4635185, 3032279], [1381391306417, -2448497215, 411068841, 560159565, -40838970, 2579823, -2484948, 7649224, -369681, 3459906, 1308095, -3699796, -543999, -2026893, -5534705, -4401871], [-41781432076, 48276918250, 238241261, -158805470, -1657657, 1231516, -1064987, -3232066, -2051940, 2156642, -2662603, -252895, 6364155, -5793199, -3348967, 2971768], [965145668472, 15114268937, 1242997988, -531001104, -17153885, -22371413, 6929157, 2356678, 9827593, 5085962, -2771338, 11383047, -1993726, 7950740, -4938446, -5483317], [-525548071520, 40590262593, -66681732, 319061741, -19085211, 16525136, 1992436, -16506765, -938884, 1474723, 6383538, -93466, -3914529, -5484594, 3110154, 2858709], [3864299379937, -5034333273, 2262498446, -327339093, -29256288, 25185958, -5522964, -8004191, -3047579, 2979944, 3582164, -10308060, -13135288, -894571, 7008796, 15407975], [758563396370, 82934968521, -136989954, -1452481833, -25221921, 25274056, 6797632, 490457, 18184, -12306455, -6313556, -9544870, -1876211, 4947850, 1109738, 6733267], [-5055056493515, 527786075, -817673002, -1789297696, -50670891, 1441062, -1072746, -20680769, -16759468, 7832832, 470359, -4038175, -19862231, 1998912, 14825154, 17248110], [1697127653735, -62789876249, 4768385501, 988264798, -2641580, 5786140, -8414327, 3232611, -14339204, 3159736, 4587606, -3249302, -1730612, 10781485, 5754185, -4410956], [1641722176652, -7771427728, 1495571237, 62246113, -15855383, -12049389, -2123401, 16941890, -273750, -19668884, -13743281, -1327553, 26361583, -23202665, -14707360, -1206755], [6203092223616, -28747790334, 815939015, 1378738245, 42322098, -9245530, 1991286, 14176313, -8212816, -21895350, -13359191, 188822, -2645529, 6867160, -11383044, -11894741], [-3306356357923, 58916770025, 680465356, -732862949, 78126829, 6866071, -16321749, 4525444, -6812279, 23800438, -6110266, -125979, -13186750, 3585170, 5465572, -15546451], [-8749049920313, -106166611225, 647448994, -2425257617, 85597432, 37975610, -11940969, -56192828, -13138126, -42146382, 12352267, -9339844, 5023297, 33415041, 29147055, 13038039], [-8055357579840, 290251182099, -5038793213, -1034931436, 52172899, 43788376, -15590447, 37566739, -9870407, 11904757, -1727964, -15403862, 10561995, -12613762, -13461725, -43952192], [-14925686509939, 259082692299, 3007424734, 1770005434, 97710365, 561032, -6605446, -4117559, 1810126, -40565164, -3100613, 891661, 24495795, 12259490, 24921802, 11273756], [-3285097604129, 632601129765, 6512995866, -982203724, -58641762, -13726059, 36455917, 3992866, 28950840, -14836226, -13267645, 8986142, 10734929, -5902087, -30843314, -178264], [-7968390305271, 369737642366, 3959841181, 2731074636, 103435118, -67212448, -2758823, 13623050, 30986369, 20210617, -1557005, -20063733, -10545784, -6961346, -17921791, -15914320], [-25447675228035, -50958920388, 661989925, -1151872346, 213117251, 31957499, -4496510, -60060701, -46165, 18112342, 19965000, 6642337, 177008, 5780655, 8830637, 22287827], [5249981016045, -269519722760, 6917300117, -170974014, -430577674, 93868060, 36816112, 31153003, 33783142, 6415121, -18594072, 56057257, 15298083, -23754094, -29965424, -24056804], [-22683303283205, -260850534908, 4782526059, -2685041459, 28848617, -63906680, 29168750, -29173883, -21188509, -49083599, 10368955, 18300652, -22406948, -6015897, 52574428, 22273751], [-9074930359916, -450311888145, 1422214622, -4535065871, -367756584, -8395247, -45147479, 22412444, 40038784, 38259531, -7198934, 31200947, 30082633, -55445591, 28474736, -35597165], [-20672459174685, 423813518878, 5482911364, 4444572573, 157078091, 10603383, -63443939, 43056533, -46988496, 3227029, 54508103, -38212618, -14745350, -13593423, 53751350, 37238823], [16624191271147, -334192910915, 5889692108, -1067626170, -180459065, -28021014, -31496392, 96873069, -23973812, -47615806, -5974065, 32013905, 114814161, -87750206, -30411104, -64195714], [-9872825391696, 108503016791, -8668404020, 3678204269, 101462711, 214713010, -13849161, -9094822, -113897681, -122842707, 20017055, -82246292, 50350736, 10190868, 25944393, 32681343], [-1957477935070, -931227773263, -7281186154, -2611156709, 246660049, -131362854, 35075581, -112580216, -71380769, 17012381, 8412881, -6608512, -94660104, 130752188, 36412901, 15617583], [-10696006127271, -183198122667, 13311789990, 3002137851, 58320080, 22787720, 13163105, -134855294, -34755943, 178225965, 75109354, -20319027, -104427204, -20941070, -17010605, 97274406], [-22701563073229, 474698732765, -11106605174, 10571918228, 268124859, -16708870, 3511584, 45204845, 26234270, -7166190, 7299725, -204083, -12818403, 3111032, 19241870, 26533584], [15728475794148, 822694716394, 17527609893, 16315916172, 622677565, 97040954, 9422415, 39713020, 25072644, -139165082, -119307879, 130600449, 107543019, -1431623, -77144577, 37470718], [-8305605039490, -63227668843, 25515972268, -2195496233, -458536954, -411702699, 94944114, 103008257, 111495021, 149677959, 34100729, 17930057, 10666250, -31281875, 28481675, -60203127], [14621736566089, 812241643441, -9505343016, -1090730542, 1243332430, 58684679, -12137877, 14518842, -33689593, 89310627, -10807074, -174910317, -141192824, 2568464, 63129906, 98445531], [28097904663902, -473041480258, 12449303351, 211119046, 27004544, 171267881, -168230568, -169081819, -96082804, 133061511, -39310434, -151625017, -258548131, 17270322, 116515745, 112123757], [23306051233508, -676380619348, -2630422624, 2941069678, 1284894632, -60576892, -88859006, -28092120, -211130852, -6413260, 32703994, -70304382, -39411648, 29853196, 19888670, -146126334], [-31947762112697, 985468708474, -29230601460, 7125279057, -459187357, -230756297, 96125821, -112196019, -4726010, 32129875, -14693439, 96920680, 12974800, -3590705, -56424666, -11133903], [17332685764965, -3935608750118, 37160703687, -2262904330, -132206678, -80276152, 99043775, 54201198, 10372935, 78732503, -195224347, 147609387, 128477640, -68440296, -168325631, -25526934], [151541800623267, 1677077705984, 19829608577, -2540854979, 1015096852, -123779161, -18339909, -194287586, 112624680, -251787552, -70113968, 56269718, -69172197, 113622031, 71223012, -86658696], [17828473065602, 284620644486, 46089382988, 12181388793, -639155585, 43217038, 16782720, 262565440, -42223925, 25952266, -126574947, -70735547, 186673159, 68080017, -56668112, 36794239], [46603460964712, 1030593646005, 10342656435, 1740054693, 163260792, 345361196, -284705016, -366500970, -85154485, -65991772, 279929605, -255707496, -228152637, 220595, 84321000, 19160738], [71833358553288, -1526667746483, 4782194495, -9413525140, 380509089, -308267476, -163902417, 389026623, 316486591, 149264613, 23948492, 48257932, 144938659, -205805018, -159105615, 1877452], [163448537040326, 169843734931, 52048775702, 6650137360, -319739686, -294287284, 145792395, 37084888, 440858180, 180370682, 76668113, 69225736, -272789573, 8007805, -24751373, 309525845], [-9140077693775, -69461073145, 19256283934, -6352295759, 2635429321, -319794163, 195729392, 221992607, 98065300, 4033957, -14301082, 191992932, 275308703, -73171362, -330928205, -331741690], [75919245792178, 4050755772144, 47346619393, -18998271328, 354762834, 106401051, 1786926, 75554727, 96696822, -76675965, -41070908, -20658659, -6887605, 154673530, -43005399, -292755439], [18152003680060, 989428092719, -33585294557, 10561450028, -1730756307, 238153520, 13182513, 294745897, -159085744, -26670772, -318469925, 66492555, 222976904, -21925953, -130350935, 43892955], [-130141729011615, 1444032463337, -115076965257, -500886205, -1403941277, 236037680, -269532884, 80820226, -412978080, 188188140, 42981432, -58235459, 175225194, -203811392, 130677494, 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66224770857178030, -2454055078780457, -772427362123575, -9542109065417, -13074757292535, -436987551589, 9672356048794, -6991648024230, -6484385796296, 2492094646939, -797471105051, 5379699922715, 891629710227, -6129547173146, -11426890658403], [-1471581953276659522, 15000282669226447, 903168945548192, -6658000027267, 11044893009946, -3075875020275, 1650394912488, -1285094795241, -3286964611713, 2403275674948, -8972597935176, 3309506041326, 24008639524, 9745643960043, 676059694431, -6273255443664], [-31897213313087204, -24060437804562483, -251624509674261, -66729309494820, -56519932748829, -9045610635924, -859160025587, -12173087188503, -192490131243, 10351913999337, 992675133844, 5033758844290, -1823708165675, -1395180670686, 7993766471355, 4681294245362]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_273_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_273_12_a_c();".
// 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_273_12_a_c(:prec:=16)
    chi := MakeCharacter_273_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 12));
    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_273_12_a_c();".
// 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_273_12_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_273_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,12,sign)));
    Vf := Kernel([<2,R![36627955869967928900190208, -5207303533036789699706880, -561136899669412766810112, 55632436776472648089600, 1986997724310628532224, -140752306565502345216, -3396107610598334464, 136659347406252288, 2806337209265664, -61183583231328, -1143989124064, 13713811200, 239619832, -1495818, -24780, 63, 1]>],Snew);
    return Vf;
end function;