
// Make newform 69.8.a.c in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_69_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_69_8_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_69_8_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 := [28965760, -9348928, -33920, 167184, -474, -775, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [-221, -1, 1, 0, 0, 0, 0], [99283584, 3642512, -1938192, -3278, 4145, -52, 1], [-63110496, 641136, 2595192, 60982, -16039, -192, 25], [299476832, -51923504, -5892416, 349718, 27199, -500, -33], [-164228248, 19840408, 4149624, -125532, -21619, 154, 29]];
    Rf_basisdens := [1, 1, 1, 301376, 150688, 150688, 37672];
    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_69_a();"
function MakeCharacter_69_a()
    N := 69;
    order := 1;
    char_gens := [47, 28];
    v := [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_69_a_Hecke(Kf)
    return MakeCharacter_69_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 := 8;
    raw_aps := [[0, 1, 0, 0, 0, 0, 0], [-27, 0, 0, 0, 0, 0, 0], [-74, -10, 0, -1, 0, 0, 0], [143, 3, 6, -1, 0, -2, 1], [1291, 157, -8, 0, -4, 4, -11], [533, 255, 17, -14, 6, -25, -7], [-5833, 625, 4, -27, -12, -36, -5], [11668, 510, -24, -47, 16, 20, 72], [-12167, 0, 0, 0, 0, 0, 0], [22138, -880, -88, -326, 376, 12, 112], [34686, -3982, 98, -698, 324, -366, 22], [82589, -1511, 1312, 590, -260, 300, -155], [17544, -13362, -2694, 716, 340, 1050, 262], [-94122, -18040, -1010, -541, -548, 166, 786], [-84727, 21469, -253, -300, -438, -47, -949], [30550, 25762, -1620, -1125, 1344, 1660, 1740], [64293, -6147, 2703, 1472, 2250, -287, 1031], [321777, -26607, 426, 3974, -1344, -2062, -1231], [639922, -29556, -7424, 6383, -7192, 1436, -142], [-734368, -114856, -6512, 1128, -504, 2412, -3836], [-1891514, 2608, 7414, -320, 5236, 462, 6288], [1372849, -38903, -1628, -3245, 9260, 4672, -6005], [-84733, -181271, 1006, -8468, -2712, 5814, 14493], [-555995, -24657, 11712, -3471, -13364, -1440, -695], [7068666, 177124, -10036, -17662, -6920, 2740, 11400], [2857912, -449906, 7732, -4240, -5672, -10272, 13186], [2723967, -169325, -20872, -6537, 15948, -13804, -11435], [707393, -221369, -16116, 3496, 1324, 9320, -20585], [9091543, -67033, -20030, 6324, 23640, -134, -6909], [5787969, -585461, -18620, 3349, 11220, -33700, 4933], [5666078, 638638, -69712, -29450, 17016, 32728, 53010], [7103626, -218390, 107352, -60964, 44392, -65640, -12854], [-867495, -62661, 58196, 65939, -16212, 104532, 22617], [12734180, 811600, -144512, 35252, -85296, 21064, 4784], [-9479404, 1041552, 117376, 22209, -35728, 29856, 58758], [32031726, -1751822, 888, -22692, 14096, -63212, -18082], [-25761275, 707561, -110676, 69994, -11596, -50632, 23453], [8288882, 447090, 193966, -19138, -63612, 127066, 94850], [-28608204, 2115692, -115672, 82476, -19720, 79940, 159488], [-6503678, 131320, 52928, 94710, 90984, -28980, -97736], [-813895, -1040923, -343163, 140776, 34118, 28203, -10965], [-12991235, 221997, 266476, -8804, 97876, -37240, -132835], [25907486, 1643450, 378886, 33834, 89956, -232918, -42710], [-30208439, 3843003, -217817, 50610, -151902, -6695, -201847], [-22432262, -2785612, -90830, -147138, -11556, -262822, 73480], [-21329183, -3253707, 443226, -78155, 66096, -192414, -191361], [9198922, 574778, -87626, 85402, 150836, 204266, 346890], [28708400, 3746196, 344758, -289820, -189724, -129386, -147260], [83103261, -7006801, -471352, 249378, 122244, 75556, 132731], [-57290009, -8490705, 185982, -174852, 174784, -485194, 256203], [-25574564, -5620534, -260090, 228312, 47300, 229146, -129922], [73687692, 636404, -862484, 156668, -429144, 469116, 418988], [-119639300, 9664494, -429850, 498242, -260940, 307002, 84206], [78373681, -1079881, -104338, -80898, -422736, 51462, -458177], [-144021296, -5640602, -831534, 189564, -736492, -83998, -315682], [105830608, -5228192, 475766, -157262, 630588, 210066, 144400], [-133742066, -13735936, 1529846, 2246, 387268, -405298, 83316], [157454210, -15406382, -1415768, 188102, -755640, 331952, -247402], [-58034048, 15068910, 1091940, -733136, 384192, 24588, 244142], [-43767421, 3387097, -1295224, 24459, -318100, 77448, -867473], [118852336, 18139702, -355728, 170203, 118704, 3972, 387172], [-122334368, 9321424, -631696, -569699, -292392, 177720, 307022], [41392124, -6261908, -792420, 823638, -1421240, 751692, -449776], [-256256807, 2390997, -2642545, 250224, -7510, 669457, -91025], [-38586702, 15135716, -317034, 69784, 23308, 1216402, 822120], [-140401456, -12662490, 1730014, -628522, -1168532, 553742, 608938], [260904532, 3719780, -1084228, 327638, 502440, -1237196, -408864], [-127233436, -10893614, -457504, 1614794, 188248, 1510808, -235554], [141162189, -82087, 4648081, 793108, 176398, -272721, 781663], [419625631, 5545973, 365945, 1536414, -1984514, 1197431, -1216121], [-438204948, 21191530, -469252, -595642, -569216, 758084, 597598], [-170903262, -21459418, -2735674, 908810, -1395756, -219630, 171518], [367556813, -18745411, -2683388, 165843, -165972, 629856, 430783], [-120476527, 48382573, -4814032, -147874, 1850332, -1941948, -924783], [490955092, -10861038, -2921742, -466133, 1900236, -1728110, -646532], [295947662, 8935574, 346880, 1754770, -1117640, -1084432, -2737926], [-283479266, 13922106, -280580, 2794857, -3001352, 2619004, -331456], [637966294, -2546024, 4078004, -1798184, -84328, 1188004, 266088], [124726299, 34572797, -3555232, -2635383, 1777820, -1538976, 1114727], [700187151, -7342767, 4839201, -849326, -1038370, 3359, 1206907], [318135435, 75905393, 3264990, -390744, 555608, -159050, -1268467], [290316831, 91191411, 3781148, 651980, 2646268, -2512088, 1394219], [500421632, 48528420, 1322126, -829668, 1408668, 1873322, 1337632], [17118952, -8188974, 3860088, -1822852, 1208152, 1246432, 1191754], [390800256, -124468704, 2478944, -106308, -792384, 543848, 1725824], [372344460, 56255560, 872368, 1593204, 2533552, -2737784, 371000], [-1131889246, -5695032, -10591776, 4424292, -3379336, 2006276, -2299996], [212541748, -75741762, 3108682, 1914302, 1565292, 1102686, 3879766], [-35130932, -25745938, 863736, 2918210, -1611376, 2241260, 285078], [-352854286, 53558090, 6414294, -792112, 619556, -3351230, -1448842], [619234513, -23346929, 7698262, 4186102, -3681200, -2299698, -4314057], [-71125234, -15729162, 10561490, -3425388, 2845228, -2299722, -2134766], [1383948234, -87460782, 6395918, -4697496, 3686708, 1798194, -2426410], [627601059, -27761905, 4493343, -313516, 1172626, -1837647, 890953], [-749809806, 8649594, -3133966, 2491950, 1223740, 251806, -7108678], [-454897888, 46368644, -7999738, -1803948, 1054060, -6140918, -44184], [342354868, 62987066, -3736770, -1767930, -4524324, 6513382, 6370710], [93340339, 73639737, 8624356, -444079, 5396508, 3844292, 9535051], [792237650, 78621436, 1195556, 5774395, -419168, 2002784, 6729306], [-1703818947, 12483659, -393325, -5807098, 4355626, 6968757, 106945], [2867842394, -9179158, -5906786, 5650262, 4959716, -3086518, -2216214], [2844996650, -59786946, -4003064, -4729011, 3109760, 1479904, 2508052], [-866535863, 107549707, -15719208, -3537634, 4521604, -6326604, -9170145], [1120960021, 55668491, 9084856, 6446687, 12316, -3922992, 756577], [-429180514, 133369988, -18278582, 10823733, -7611980, 5233642, 195666], [-2264441329, -234355387, 1721563, -6394982, 554538, 2887049, 7479239], [1623139494, 93819646, -8347884, 2188324, -9044032, 9490276, 10561534], [-706107406, -85447216, -15507560, -2536748, -4169512, -131508, 4173164], [1570376804, 56917884, 12795224, -9539316, 19050976, -1815232, 12004732], [-1829774605, -71116671, 13270959, 1227498, 5373282, -10935355, -4402645], [3470665326, 96557002, -4564060, -574620, -2549208, 2697616, -8459530], [3549876219, -20004605, 505356, 16440050, -6323828, -3517448, -7142609], [1094624105, 64498699, -21221548, 183693, -7446060, 7429284, -3134755], [-218267016, -114189742, -5072850, 5472073, 12940452, 2454238, 6329312], [3106581803, 260843867, 9801522, -15538139, 11010600, -2277222, 14948253], [-1552018221, 86583981, 78400, -3110531, -2780644, -9994056, -9124297], [3140233910, 21399440, -13939210, 9674363, 6668156, 10450542, -1495366], [-2859361685, 45113263, 4865705, 4866108, -2010634, -269481, 15606141], [-3984692342, -183887304, -3289882, 4467352, 8166852, 13942982, 6747984], [3958829657, 205306439, -3896562, -1755390, -15436720, -2341754, -2465081], [2153571707, 300602531, -25887328, 3332978, 3850724, 18174228, 10341687], [122263365, 314640363, 5634493, 1765342, -7774538, 23897095, 7657681], [-3228998104, -146186324, 5269480, -5983551, 4755168, 897224, -1584094], [-1103597582, -129112678, 23533204, -8963312, 25291184, -4687516, -13355182], [-1762174054, 89314810, -9404150, 9166174, -19053828, 8414982, 15495994], [768608548, -165725488, -59576520, 17439383, -7625464, 18507352, 10630546], [-4501232947, 128455569, 7023098, -7325424, -3954480, 5307946, 11617909], [3423095111, -250074893, 18746573, -13176020, 31548766, -7041957, 18674497], [3579460923, 304496531, 25769944, -11803751, 13459740, -9414828, 1469073], [-211167349, -543114169, 26545232, 13879008, -11797404, 6025036, -8671641], [2690168948, 397108960, 32530140, 6821608, -1839400, 4371676, -15133072], [-3643549410, 44565262, 21139144, 25100294, -15564696, 32383264, 9489122], [-924655251, 404466069, -56745418, -7008293, 17011976, -13294514, -3127437], [-3105363607, 138677409, 4818208, 11445636, 1351804, 3162092, -22205295], [788523952, 130126998, 36519024, 19816594, 784760, 7552744, 17935154], [8851608038, 195409672, 7104278, -18054472, -8362660, -226734, -10633380], [206919790, 78398358, 28191340, 11045745, -22665568, -16569196, -7203636], [-4017443610, -302587720, -16927586, -14555285, -16596068, -12246770, -4709862], [2314408428, 165004892, -33202144, 20410739, 7145248, 6628328, -15122970], [4673825558, 146709756, 7296942, 19053070, -10542708, 17513418, 23097748], [4926735774, 29468710, -15349962, 17262234, -28904668, 19185154, -34418002], [-2747371940, -265525614, 23455492, -20229736, 9351192, -44486752, -26294298], [-5785946600, 77078220, -25225452, 23108626, -6077968, -17447512, -16701732], [9176552321, 112664567, -67267342, -22893654, 14973208, -34530894, -13609441], [-1067387, -211374605, 2090499, -24345442, 20850970, -47654803, -52574495], [-2361128220, 151190508, -14431998, 50142630, -16401276, 24577046, -16966260], [-539380260, 168587406, 8583878, -14313100, 11799092, -36757014, -4224934], [-5151176528, 868164886, 37590442, -19169488, 18745852, 12790926, -3101606], [-5391457790, -815127458, -19092402, -14566076, -23007116, -7049750, -36234830], [9072603605, 720772745, -1770761, -1043892, -37587734, 22856089, -14225925], [-529508506, -408703428, -64468594, 1852880, -21636564, 33264474, 6284352], [6888863731, -281871591, 6841460, 20956235, -15958812, -16778140, -8108897], [-4358675344, 334483312, -76531848, -29800562, -2718672, -241776, -25109588], [-10264614657, -763468053, -8902963, 28788984, 2711526, -19664329, -38324355], [-226665792, 180231494, -70088220, 48490063, 3974952, 20009696, 33432020], [-5928910012, 238335952, 22769362, -20617212, 51612228, -32879850, 3593236], [11926407933, -286131183, -15905310, 45788329, -49722896, 22317594, -228237], [7500274880, 17230670, -7195782, 29275960, -85110172, 56707802, 40332990], [16665916394, -964779384, -7973480, 21340016, -42211456, 10464312, -37334664], [1020997814, 89349422, -19172876, -45377809, -33369336, 16371980, 5518240], [10366689509, -846492087, 105730677, -31569872, -5530634, 34458427, 7579343], [9923588501, -325905045, -6254636, -33793573, 45192420, -21200548, 6289161], [-927100302, -53498250, 125700774, 35020390, 14716060, 10332822, 20940426], [1300852817, 78382611, 21786412, -36483838, -31148180, 331968, 52553511], [-1406470165, -1427535247, 74905596, -28988355, 49269004, -12404580, 4357707], [2939028788, 644574984, -9692166, 14808588, -42246844, -14669786, 3944740], [3859429068, 452080776, -146294664, 23253842, 36514712, -38143524, -32809368], [4253328707, 287349065, 28218265, 25188558, 10296350, 59060447, -28516557], [-4831455542, -606284924, -33327812, -25997046, 65891816, -26156004, 34935632], [-456981594, 215132460, 55150166, 2788352, -39299292, 24111286, 35904244], [-1936814547, 144045987, -82300862, -17261754, 22221320, 49135330, -12730173], [32477695898, 118805784, -6799274, 33486376, -17617876, 24176546, 11700540], [9112949314, 195543522, -35744092, 2136454, 32928144, -60039020, -24239762], [-19196552962, 307609716, -134271074, 53716622, -30537716, 38499578, -25434836], [-14620567074, 1101314678, -6723552, 11748058, -20963128, -52424000, -40558166], [13356441245, 499728723, -106636012, -54811419, 94612748, -22981356, 20149949], [-772635250, -1809651982, -74423654, -11971836, -74151044, 23701446, -3940122], [-309859040, 1729292642, 73387790, -87635540, 15627004, 31572382, 24878938], [-9410037817, -582829833, 85235374, 69131093, 29050432, -18747162, -70057655], [12774203159, -2053249885, -29783234, -86630526, 73982912, -27638402, 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[610634537484, 11005871372, 3127881268, -702125421, 717718920, 245850100, -847714], [36034009653, 41744936887, -1089931548, -955203729, -711287108, 2625030084, 280903469], [-475318611998, 3081218962, 2988272332, -2025645502, -1410858080, -557693108, 720552182], [-434383304406, -20536575168, 2068327300, -130939636, -270619408, 1055229120, 1572418660], [-75690255216, -33821586066, -555303682, -254731597, -1109103228, -1056296770, -848352248], [-1012097447792, 4652676768, -525375404, 438458928, -92326520, -317160236, -1394250320], [-54634385912, -20387950056, 2175590514, 1461794690, -1158897724, -1642407322, -1851522272], [296023308500, 10367587858, -2324784880, -889096672, -1483453016, 334689792, -2845111110], [-83316469478, -63105041016, 1854195290, 11083233, -720528364, -1239101326, -1205630842], [61323600498, 27868734542, 1710717404, -529022260, -1339526560, -2051237564, -1744871386], [-422752721566, 28747197840, 1069961682, -1331792534, 838516964, -2093176882, 392833792], [215860238926, 23586990320, 3623799854, 576010762, -1596793100, -1972346666, -2400311540], [501183828851, -50509110213, 1380153101, 359016712, -295974162, 1327301507, 1964944313], [-50377786310, -27953357598, -2534442858, 1516278526, 273056036, 3076920194, -208596990], [66601702278, 8182620402, -3177162156, -2674445876, 713790688, -1754039796, -999709398], [220943356268, 14191456222, -537539410, 28596710, 2129351172, 914525106, 71757734], [-753085424195, 4810165629, 1375791654, -1813895017, 1115507704, -787143834, 546696267], [237583130807, -46476849779, 1324659976, 706304033, -2547603804, 1682021824, 1325880103], [87832570998, 67036586060, -5029588844, 962883605, -104753136, 1132056752, -240783018], [205812855148, -29431720634, -723294642, -1290394904, 1533668828, -3289130658, -1135178362], [231217946640, -46074803942, -3832002998, -123110380, -1671090084, 1331473934, -1504677946], [-1089394998962, 9934579508, 1420998314, 1799774846, -886510180, 301198930, -775540088], [-753359534219, 6499375285, 4315402220, 1554469971, 793177916, 627688944, 2029855695], [145160057517, 21875111427, -2079850768, 1986315023, -1923083348, -1420856928, 411965409], [-337016657910, 2446794972, -4712310172, -856582776, 1692712432, 385977424, 133378792], [-734394086562, -10462430598, -4899617628, -582593624, -443915800, 208994960, 316049822], [422491629845, -9637063315, 834537916, -272150972, 1544729428, -1087118912, -287938875], [368622286284, -49435740022, -3489145400, 448573674, -194075488, 268523988, 704180042], [341105084264, 41177323934, 2135428564, 271019063, 1111964968, -2009847016, 919202388], [161783213599, -17321026709, 2070767755, 1574955960, -676406838, 606064589, 1705734093], [478992354605, -5791218541, 2106495645, 3135269470, -892171194, 3542034415, -1224992767], [509763439948, -9721067578, -261203514, -682241912, 412540412, -924153730, -582901394], [-864571610697, -1112708219, -2457181774, -577913024, 1755359360, -1545067278, 1728868681], [1455354789638, 9947426384, 1163550360, 134883359, -160300328, -101399476, -1474795938], [-329521439426, 39061504868, 4528953934, -1165550850, 576034604, -2428248342, -990147876], [-86393754378, -50140837268, -508448804, 138081002, 2376501680, 240983464, 2159821908], [-212615605239, -47628301803, -3376552913, 700845900, 1634907826, 2349475357, 3012611], [136113469920, 14627833436, -2385552736, 197838030, -1906291512, 792087892, -333494156], [-240668909285, -18692451359, -3604241544, -804980159, 2450858628, 1712532104, 968207203], [-839110553488, 33033517244, -3029335608, 992466558, -2028103296, 1276618792, -627860960], [936763576880, 15108132456, 2027500700, 1087438547, 714606904, 1571144956, -1715193574], [59414861879, -37606606045, -356096894, 1767086094, 1063886648, 1706389418, -1916067093], [-416104568664, 13161027128, 1257617620, -1758286276, 1391433896, -1406110340, -1968369032], [55034437639, -34703734953, -869799339, 1463158040, 1664180622, -1136653253, 1364934901], [557641758050, 18847205952, -3346637124, -1920113448, -200607176, -1379793852, 1265519592], [432942578582, 57379430420, -883502998, 1282168940, 131435100, -2682925350, -2519552092], [-333830331117, 45654911741, 197839246, 1793482282, -2768067608, 1798792158, -568901219], [393784222102, 60440960420, -5216456768, 371539196, -2831491584, 4710306432, -1326934292], [-704772088528, -8426549896, 3343134020, -1455893418, -763520424, -2259046508, -982088772], [-172048780401, -2680153949, -4940528051, 3522111004, -330949450, 29931519, -1364067211]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_69_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_69_8_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_69_8_a_c(:prec:=7)
    chi := MakeCharacter_69_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 8));
    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_69_8_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_69_8_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_69_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<2,R![28965760, -9348928, -33920, 167184, -474, -775, 0, 1]>],Snew);
    return Vf;
end function;