
// Make newform 688.6.a.h in Magma, downloaded from the LMFDB on 10 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_688_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_688_6_a_h();".
// 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_688_6_a_h();".
// 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 := [-22734604, -5752252, 8437093, 384582, -719484, -10450, 21986, 266, -256, -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], [-2, 8, 0, 0, 0, 0, 0, 0, 0, 0], [27676046268420, 1758751326736, -6241757248231, -177037969291, 318622394671, -1235557305, -5215675021, 154132815, 26922273, -1300283], [24355668723972, -2373830719216, -1609930003559, 442286142037, -16891402705, -23691741465, 479070067, 334667631, -1596255, -1397467], [-5839603776708, -1656152504528, 1003571811959, 220849424171, -39727817087, -8979038055, 421392509, 116809809, -1144977, -474917], [15723616018348, 7380178820016, 397753048627, -1348681107321, -48836433947, 46987493645, 1117284305, -569012379, -7177205, 2282743], [-40534043519788, -25558198123248, 7733935811885, 3011920741641, -310996161541, -113680795613, 3217063567, 1437405579, -8152363, -5726311], [-19301122595508, -25027510353488, 4539319373339, 3248986969295, -194780377283, -122844007131, 1540088777, 1552893885, -47565, -6235841], [81663371814556, 59454523056496, -17271162848625, -6867015632269, 724329792617, 236840256305, -8018172347, -2828120727, 23722567, 10838435], [-87327465633820, -62809921330800, 17296111236689, 9030426367437, -699986633545, -345300273137, 6072341467, 4377440727, -6000871, -17656675]];
    Rf_basisdens := [1, 1, 197066179584, 197066179584, 65688726528, 197066179584, 197066179584, 197066179584, 197066179584, 197066179584];
    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_688_a();"
function MakeCharacter_688_a()
    N := 688;
    order := 1;
    char_gens := [431, 517, 433];
    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_688_a_Hecke(Kf)
    return MakeCharacter_688_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 := 6;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 0, -1, 0, 0, 0, 0, 0], [14, 0, 0, 0, 0, 0, 0, 1, 0, 0], [-6, 0, 1, -1, 0, -1, 0, 1, 0, 0], [-77, -4, 0, -2, 5, -1, -3, -1, -2, 0], [191, 1, 0, 0, -7, -3, 1, -1, -2, 4], [411, 10, 4, 8, 27, 0, -2, -5, -2, -3], [229, -5, -4, -1, -17, -9, -3, 13, -10, -12], [-168, 0, -2, -16, -12, 6, -2, 28, 16, 3], [728, 0, -4, -4, 48, -33, -9, 21, 45, -3], [508, -4, -10, -44, 62, -9, -11, 2, 27, 38], [163, 41, 45, 16, -141, 32, -2, 109, 15, -29], [982, 47, -76, 17, -276, -36, 42, -62, -42, 54], [-1849, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4822, 21, -52, 139, 176, -58, 25, -12, -85, -30], [12563, -50, 105, -70, -417, 51, -13, 25, -16, -123], [-9921, 144, 79, -11, -589, -220, 257, 133, 60, -15], [1827, -194, -141, 176, 75, 37, 125, -184, 258, 104], [484, 307, -8, 593, 196, -63, -130, -290, 18, 22], [-1458, -281, 162, -356, -664, 68, -318, 210, -32, 30], [5257, 29, -135, 112, 819, -625, 409, -44, 156, -74], [9043, -94, -320, -214, -1383, 51, -141, 195, 430, 347], [10050, -356, 15, -417, -1364, -431, 220, 636, -114, -117], [-6365, -72, 133, -362, -1161, 363, -219, 210, 90, 678], [10615, -395, 343, -286, -871, 296, -629, 975, 448, -503], [57213, -254, 81, -428, -155, -803, 485, -141, -350, -747], [3678, -509, -137, -812, 280, 728, -422, -454, 646, 62], [-10004, 323, -293, -168, 1582, -734, -420, 642, -100, 519], [-23330, 990, -168, 371, 2802, 857, -64, -1346, -776, 550], [41124, 1340, -800, 254, -2662, -538, 1440, -44, 1180, 144], [33492, -1377, 349, 398, 1608, -85, 1809, -902, -45, -315], [10063, 1616, 389, -205, 5439, -225, 960, -611, 1368, 402], [14454, 704, -306, 1016, 2304, -138, 1868, -2346, -742, 906], [35797, 909, 53, 128, 4625, 1761, -331, 1439, -2, 767], [9311, -3078, -619, -2484, 6469, 100, 1418, -1713, 661, 2307], [215022, 878, -157, 363, -4264, -1945, -96, 549, -2132, -1566], [106231, -427, -7, 684, -103, -4255, 1347, 1173, -1280, -1362], [98695, -2420, -1257, -704, 11325, -1134, 1393, -3474, -2090, 1248], [-6102, 1184, -939, -109, 3844, 589, 1836, -3732, -3040, -111], [149907, 2129, -2520, 777, -3033, -792, -711, -3429, 462, -294], [3947, 2949, 1673, -2374, 2191, 898, 1687, -2046, 2490, 1524], [78456, 1247, -357, 3604, 7288, -782, 942, 36, -278, -2387], [108618, 522, -892, 1008, -14782, -676, 2334, 3836, 2584, -1170], [161804, 1962, -3327, -21, 84, -414, 1422, 2964, 108, 351], [332204, 6229, 1547, 440, 8292, 1692, -1416, 1110, 2664, 2607], [-132426, 2685, 1222, 3664, 13208, 2810, -3516, -168, -606, 2656], [91387, -3529, -1043, 1097, 17493, 1839, 750, -2583, -464, 3020], [150060, -7103, -105, -4945, -1420, -241, -348, 4263, 2426, -2242], [88851, 7560, 288, 8283, 11731, -1703, -2159, -8049, -750, 5070], [258638, 4468, -962, -2533, -5042, 3291, 5606, -8450, 2064, 3160], [7652, 3408, -5868, 3270, 5558, -3982, 3860, -9392, -162, 2742], [-21433, -2168, 3440, -6456, -25227, -2139, 499, 14859, -1370, 405], [111172, 399, -566, -1854, 4612, 446, -5670, 3596, 8850, 4280], [280888, -2415, 3640, -2925, -2208, 1371, 446, -7398, 6626, 4560], [231765, 7155, 3225, -336, 23331, 3819, 4155, -10002, -816, 4626], [150138, 4443, 3430, 3284, -30624, -6654, 2254, -26, -1862, 1026], [366941, -5177, 8498, 5078, -12709, 2219, -469, 4659, -462, -7950], [49153, 12486, -1243, 5855, -32491, 1458, 3599, -4075, -1802, 6446], [124536, -787, -1958, -12156, 23496, 4504, -2204, -2737, -1068, -398], [215195, 6343, -247, 10776, 16993, -12464, -1804, -4785, -4928, -5550], [5471, -4319, -5847, -152, 20543, -5021, 2801, 18487, 12654, -2031], [334261, -19950, 2649, -6379, -7981, -5716, -10709, 16475, 200, -6183], [378524, 5512, 2403, -8513, -12910, -6971, 9410, 9568, -2886, -13869], [277322, -4031, 3968, 14595, -13038, -4611, 6910, -11346, 9730, -7602], [-99263, -13237, 1925, 972, -6325, 3479, -8379, 9564, 466, 1122], [-199093, -11293, -2232, -794, -11099, -641, -8017, 9187, -3506, -918], [1281804, 16773, -9527, 16429, -21112, 5783, 12786, -1821, 1152, -122], [-128317, 12461, 851, 4402, -15703, 6324, -1262, 3335, 10548, 8870], [505011, -10090, -3123, -12136, 39661, 28108, -18231, -9624, 12686, 16860], [356598, -5807, 4054, -9168, -58052, -6665, 3183, 15921, 7961, 17079], [-362199, -24693, 9504, -2286, -55853, -1301, -3797, 9405, -7458, -12774], [-84505, -11093, -568, 5613, -10001, -21749, 22418, 15855, 9031, -11490], [-198651, -17972, -13716, 4328, 19781, -7792, 14820, -29061, -12079, 830], [293825, -8807, -3179, -1456, 10903, -6417, -583, 23305, -18612, -13394], [1410255, -182, -1604, -3180, -13035, 6161, -13403, 26401, 18100, 6606], [473178, -16861, -4254, 20372, -15048, -4680, -5440, -4350, -3034, -24666], [41194, -29403, 12598, -10010, 60142, -5894, -16070, 25794, -5910, 20310], [526182, 36335, -6195, -3864, 87918, 22102, 9804, 3224, 39376, 14009], [780760, 29352, 17701, -9033, 76136, -15256, -2993, 9264, 13616, 4266], [-497959, 24864, -20537, 7208, 4649, 17907, 7795, -16270, 11888, 15396], [-1950274, -5327, 4029, 17009, 27358, -13187, -4536, 29137, -412, -8802], [1591881, 32414, 13449, -27896, 120265, 6337, 11967, 2745, -12068, 1294], [-1255195, -26850, 17169, -24995, 2095, 8983, -13708, 3919, -4061, -6651], [-2597197, 1124, -6867, -16246, 76407, -909, 9803, 15888, 15770, 1308], [-1442182, -8896, -11658, 15360, 16716, -24113, 31245, -21646, 21361, 20492], [-1883864, -4501, 10427, 7368, -54790, 1490, -7020, -35588, 640, -5877], [333049, 5084, 6193, 8844, 67809, -3939, -4867, -14568, 14924, -6030], [-434467, -56564, 5203, -18016, -11507, -8733, -15043, 7318, 13248, 33472], [-1153407, -19590, -8561, 3653, 188539, 19318, 6917, -19889, -14186, 15609], [208688, 26844, 10965, 2779, -17934, -8003, -2558, -25981, -48376, -15382], [-2406549, -15698, 4125, -37271, -29357, 13513, -18076, 9297, -11924, -2538], [-1428322, 12132, -11258, 24796, 33646, 39124, -26540, -84898, 17112, 49023], [-873428, 62228, 13487, 5715, 52102, -30555, 32056, 13294, 1392, 14760], [-1049545, -34045, 6935, 2872, 127435, -2090, -24897, -762, -22774, -38610], [-1725763, 33198, 8718, 37888, 22611, -17822, -17666, 16886, -8896, -44482], [603728, -26682, -5602, 6020, -34880, 36598, -21488, 37902, 18474, 19908], [3054667, 46114, -2411, 28060, 8015, -10177, -13993, -12507, -12858, -12519], [-821043, 61138, -279, 65704, 10241, 54923, 6435, -96960, -6152, 26496], [2483409, -23039, 4402, -33692, 78061, 22280, -764, -4486, -24416, 27464], [6300, -61997, -17713, -34081, -14838, 8485, 634, -6196, 19186, -13239], [186230, 67094, -18717, -5535, -202140, 14569, 54738, -50560, 56212, 2423], [-117367, 7299, -51886, -39590, -32665, -2227, 32473, -8373, 22226, -26322], [-2421059, -62582, 20946, -34128, 126407, -25879, -29797, -15565, -58164, -58080], [563788, 66219, -31251, 22927, 127498, -52313, 73574, -19162, -9212, 14157], [2258753, 40551, 13066, -50268, 44777, 4984, -2956, 40780, 13804, -21190], [-2476219, -43555, 19217, 31520, -136041, -17493, 2587, 4524, -66888, -32750], [-1743492, 13315, -18322, -32398, -138030, -4488, -6150, -36148, 1692, 60804], [2199027, 30855, -655, -74058, -205315, 96791, -92931, 86708, -6938, 54870], [-1114280, -65587, -18337, 49452, 299440, -13781, 14015, -117176, -41159, -3495], [3416886, -21011, 29142, -31342, 143706, 16372, -95794, 7526, -40272, -28936], [-1788654, -12715, 60902, 87606, 167000, -12982, -99336, 47064, -76310, -49728], [-2721217, 84330, -30707, 85529, -6547, -5664, 5053, -159073, 38162, 42963], [2503391, -53431, -34513, 39748, 231621, -1260, -17989, 8181, 21896, 27603], [-4891585, -42021, -27780, 30517, 21849, -30860, 37537, -7411, 37580, 68654], [546750, -87894, -1346, -4036, -67602, -23968, 21150, -52916, -89748, -46962], [-2215543, -119629, 52381, -33880, -84009, -41655, -48989, 145580, -3872, -19158], [3052713, -33794, -33707, 55015, 6997, -66874, -11597, 64175, 39598, -68053], [2071422, -111904, -12694, 50320, -70538, 62986, -19374, -53968, -6466, 65484], [-594479, 47023, -28799, 145480, -69677, -112073, 69253, -85683, -12180, -22464], [-257497, 216642, 16594, 24212, -26671, 23147, 16971, 37805, 83614, -25836], [-711173, 9242, -5675, -21520, -86945, -73135, 30845, -745, 18530, -10375], [-3831705, -15620, -24785, -20130, -66581, -13559, 46683, -126546, -39694, -10812], [3458703, -213258, 32101, 2922, -9793, 47645, -19619, -10223, 10244, 12426], [3693149, 132282, -70822, 23012, 272475, 58287, 30381, -107583, 28530, 54108], [-1565582, 123994, 34747, 28733, 25684, 33795, 15626, 87745, -80376, 23662], [2869095, 114570, 31167, 147969, -150083, 68644, 52483, -51507, 40638, -1017], [-2178896, 112542, 27346, 47737, 175472, -50959, 79770, 44190, -1014, 39328], [454648, 140918, -39647, 134079, -77116, 7970, 98671, -41088, -709, 61065], [-61294, -30755, 26120, -98744, 105248, -23978, -68216, 4804, 17224, 54580], [7641755, 38077, -53261, 77756, -39473, -10362, -34504, -163937, 2229, 47299], [3530717, 25965, -74395, 18387, 10379, -17345, 53196, -260393, 48408, 48040], [2906776, 15276, 25074, -11010, 54096, 113598, -52260, 53342, 5226, 46128], [-5368984, -199835, 71221, -423, -369908, -70289, -56022, 42513, 4910, -22146], [8871665, -59713, 92497, -77382, -172883, 41868, -93912, 69643, -46145, 17359], [395410, 163504, -13010, 132710, -6628, -2998, 43254, -105186, -148112, 30660], [-328433, -160304, 36712, 32753, -380071, 79961, -143282, 89769, -38072, -96827], [7098064, 120885, -9386, 135508, 728362, 47965, -7607, -166317, 45933, 44271], [-743684, -78175, 86707, 8612, -132702, 19392, 50184, -11604, -111836, -34681], [5371763, -94450, -47299, -38723, 217249, -139358, 114561, -5439, 63710, 54099], [-11049089, 40922, -51824, 19923, -168695, 13789, -15408, 76839, 67364, 36825], [-2646603, -128015, -34625, -120248, -11591, 42278, -95135, -164746, -7058, 11756], [8486064, -136586, -64992, 27393, 14760, -57897, 1056, 77538, 30618, -71166], [1277756, 24481, 73970, -97635, -85532, -37919, -76412, 199872, 58270, -131656], [-525034, 237468, -156864, 210194, 475838, -56266, 139480, -132350, -36850, 66762], [-2597925, -87310, 37931, -88992, -163935, 20128, -29782, -145429, -102847, -92883], [9529886, -183132, -42191, 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-109379, 896751, 5051868, 573731, -1245440, -980723, 277588, 1416390], [50548738, 1070693, -542091, 2457559, -1656192, 323807, -1831574, -2590802, 59754, 1778869], [76905297, -507036, 214119, 1215968, -627323, -2406695, 1732857, 912774, 498650, -360060], [-18038150, -1010317, -287786, -745988, -1249546, -2583520, 770894, 1830090, 1361586, -697182], [-42299126, -4087907, -819197, -946315, -3087818, -1241079, 182510, 301321, -167520, -114368], [-89460921, -1168477, 1465170, 293588, -1486395, 1636215, -1767883, 3611529, -863614, -1152540], [23792647, -1773865, 746387, 498702, -1060321, -1042765, -763641, 2724241, -543830, 875997], [-76155420, 1444893, 204392, 2653232, 2212138, -2811113, 587379, -476775, -1319349, -2262697], [-148853920, -2059470, -114568, 138968, 2481484, -454742, 2233950, -69876, 977562, -979810], [-6057988, 2300578, 1634378, 626930, -5141192, -1928288, 200710, 1157460, -1614, -1192026], [89465390, 562865, 649337, -550741, -4660336, -500137, -771262, 3877194, -1255548, -1861029], [41516801, -3443056, 861429, 474183, -2595, 966340, -329247, 828509, -660308, 937877], [-71901517, -939341, -2061723, 1088026, -1150443, 787635, -42509, -2352138, -108386, -1032354], [-44239253, -1047506, 101023, -827177, -3721967, -531140, -600245, -157377, 1834560, 227817], [-120239032, -151610, -310401, 1812659, 4309014, -1281083, 24296, -4071811, -287136, -191134], [-159712958, -931611, 483345, -1683292, 4927434, 1983890, 360680, 831532, 1181542, -947351], [434935, 661663, 708993, -159228, -8418087, -9927, -999831, -387664, -2011104, 1045164], [27077522, -1049838, 1030032, 53688, -830922, 228258, -1438122, 3995140, 925194, -1261614], [100532571, 512703, 553757, 2224269, 2578301, 582263, -580206, -658657, -531824, 948186], [-51333668, -4462101, -118946, 103585, 2131490, 699260, -1484653, -2350998, -2890050, -162297], [30330255, -402451, 108006, -179925, -1670857, -428695, -320914, 3404223, -159585, -340248], [-30538619, -1891924, 1870233, -584698, -8204825, -148744, -3039692, 1323995, 605205, -42855], [10007913, -1088170, -380603, -1440752, 2158813, 728431, -1101011, 4684534, -347472, -372492], [-51540954, 372180, 1363733, 1585283, -2327864, 1058095, 672046, 360701, 630234, 808350], [66919007, -956973, 291978, -1978325, 7871077, -682091, 1498163, -905903, -577874, 135588], [-29995039, 829202, -1081750, -1242010, -6706861, 1087369, -1674775, 1000861, -192850, -147734], [16149718, -2469126, -317420, -2386182, -3873320, 1446722, 917262, 27734, 1645506, -1607470], [41797061, 1567729, -812941, 1457298, 457143, 2179821, 365773, 237855, 594100, 1028478], [113885547, -937992, 359710, 261076, -62965, 60056, -2265120, 1037877, -2828469, -744884], [-17486025, -2759361, 95205, -1482188, -8712831, 3363703, -2909525, -513610, 194396, -354502], [67519197, 1853055, 109364, 1334353, 4831883, 626213, -212411, 1399619, 1149374, -550278], [-228636531, 5081173, 778228, 909052, -982535, -255778, 2042590, -1931182, 393328, -1074940], [-35065820, -2967375, -1451580, -415453, 5714556, -1180695, 281750, -1076866, -985438, -510309], [-173433136, 2727132, 1909094, 679238, 3563002, -1942100, 1236934, 2256333, -618600, -574434], [-13302258, -1238746, -173904, -1059290, -3634680, -856783, -2378531, 1414540, -1309215, -2403296], [95107454, -4192647, 263080, -1519592, 11676550, 1073568, -2748698, 376402, -2064010, 1149844], [-30423675, -90524, 1157538, -304305, 295953, 1307889, -383769, -1762443, -870390, 1012644], [-41906293, 2191714, -885415, 23423, 3594095, -1828558, 4679087, -1058163, 764430, -727989], [93027976, -3437052, 290685, 401729, 9052572, -259072, -905167, 1486810, -824936, -1043558], [-68174793, -2572524, -653183, 764878, -5049081, -1889991, 2465439, -2532194, -2144760, -2465358], [133787757, -2194052, 551489, 70848, 11094253, -434189, 1791871, -1665468, 1734688, -1645534], [36529913, -3627064, 2913601, -959070, 2134665, 272535, -519613, -160585, -345214, -1043775], [99310853, -564086, 1403956, -1200776, -5986315, 30566, -1293829, 1421777, 2041830, 1832622], [31192948, 1359925, 700807, -719594, 4851348, 866187, 641345, -4900884, 1160979, 3077345], [53574193, -4014625, -929583, 1544860, 2094543, -1986657, 827581, 1592948, 1674976, 924924], [147197328, 368155, 1952776, 516151, -1575044, -4126219, 989728, 2302946, -196214, -2482966], [-76016789, 851146, 1367905, 3335513, 5485757, 1019711, -2848560, -1694133, -285371, -621381], [123561768, 1783384, -69755, 561049, -5122142, -1730875, -1975394, -740665, -2670158, 908708], [-83590718, -5067886, -290000, -608905, 571750, -2315161, -428158, -1239784, -1933636, -241366], [-179505852, 4267879, -1674685, 2081442, 8423478, -1191682, 1653182, -1307090, 1348338, 1326155], [56064967, 1900496, 587296, -980246, 2163373, -712211, -519647, -673485, 1201662, -913494], [-65191681, 898301, -470147, 3268860, -1237679, -15594, 2680038, -4765631, -1635560, -746558], [-7624254, 851290, 811032, 873978, 6275406, -899385, -2420139, 827385, -2607579, -10965], [114029954, -4409264, -783268, -3133056, 2069490, -769314, -1145594, 4242912, 1119760, -1469250], [32398522, 3074380, 1305817, 232481, 6603412, -1075713, 869054, -2691638, -2337972, 1745461], [-20348924, 300210, -1298639, -3435139, 2699278, -1595057, 1216076, -389523, 950200, 3140904], [-128007390, 2308474, 1855385, -1009336, -6601980, -2918687, 1312714, 3878084, 753934, -1575123], [-135392051, -4574712, -254147, -1206747, 2883867, 1054203, 620860, -2075703, -2652505, -447313], [-28115645, -1354248, -790365, 1035310, 10491499, -4145039, 2019737, 2280188, 2096860, 722262], [-68622381, 3620789, -75467, -898638, -1274449, -1583716, 426287, -675870, 904118, 41884], [-179733290, -50048, 1768462, 2678852, 4167592, -70028, -587992, -3289311, -713672, -1967352], [-79764945, 351802, -1849407, 4685414, 7427403, -2783105, 4056251, -6672007, -251340, 58043], [134005741, -67621, -1974497, 26905, -12897047, 843103, -3513941, 2151453, -1618026, 394209], [116437944, 1099903, 329797, 646138, 2822266, -2026756, 435100, 1506302, -1650686, 215768], [103724349, 2459229, -351993, -542537, -4241903, 942103, -611917, 266139, 257662, -2500569], [11398885, -1180942, -1258541, 873342, 6059739, 3535128, -3570844, -4767981, -1662655, 2396931], [-23068124, -2285682, -1117857, -693081, 12180834, 227643, 767910, -2225349, -1715856, 159090], [69739759, -3100646, 2037063, 1552950, -7095315, -2144344, 2288760, 2329681, 363203, -2371889], [-15169238, 2155127, 634738, -1219370, -5993214, 4119981, -3225693, -606834, -4251888, 1296141], [193375873, -649933, 2748864, -2398923, -3617489, 1293459, -644239, -1075103, 15434, 2374736], [-47166260, -1497688, 86575, 1368321, 19102406, -1821757, 2369404, -1999776, -1305634, -1484139], [-74520001, -1798913, 958309, -2995987, 10841791, -5529, -1672075, -7679, 612486, -332831], [134004946, 1711347, -332764, -2900536, 11421222, 1346052, 1287414, 3759688, 1662966, 143106], [-24018979, -1055679, -462721, 160661, -8668977, -333615, -1947768, 3600009, 2125932, 277464], [60617264, -567486, 1975917, -2848529, -1732962, 3391945, -2881040, 6358916, 571580, 1472951], [-45240937, 4060602, 338215, -1818044, 10333087, -688401, 2110039, 833806, 423524, 2268892], [-18476500, 1020645, 2141633, 930603, -7150538, -235205, -1044400, -2510160, -495830, 3277617], [179019323, 404364, 2074094, -922702, 7231745, 2303297, -3364687, 2478657, 1497666, 677742], [14260858, -965747, -387696, -801443, -12156414, 523905, -763172, 7545714, 1109122, -3087729], [159496159, 1983748, -1550859, 2674791, 2418977, 3053012, 5031917, -4065343, 3256302, 1654893], [-98276242, 2251422, 54401, -76239, -2626564, -1936975, 3328500, 4905915, 4016344, -2719404], [42916156, 876495, -1490508, -1205434, 1324980, 2483130, -1123756, 2573093, -492484, -2407044], [174768061, 289544, -1072929, -377377, -8937989, -1790812, 2375695, 6359663, 2124504, -377631], [-60092304, 1070584, -138647, -2394451, 4523730, 4058331, -2819102, 3163819, 2169130, 1170642], [-3802712, 930933, -974906, -152109, -5160696, -2151945, 2786090, -5999310, -3279910, 1776678], [80590016, 5698599, -1663464, 4945242, -7242720, 682908, 833940, -1855248, 960510, -785982], [250508401, 273483, 1835573, 648092, -1379605, -1441477, 2399033, 1519921, -1007412, -2843538], [-48512687, 1728201, -1721115, 1896449, 2617043, 5161469, -318740, -7642171, 1054232, 2574486], [69767665, 630276, -369441, -3612022, -4956475, -1355365, 1035967, -500800, 5417888, 2652300], [-35225127, -1074133, 704927, -4253790, 1773413, 1891091, -1364397, -428139, 2941414, 1815915], [136799142, 2796789, 2136885, 3795279, -6832542, -1521927, 2964660, 1440051, 3803130, -1328316]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_688_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_688_6_a_h();".
// 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_688_6_a_h(:prec:=10)
    chi := MakeCharacter_688_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_688_6_a_h();".
// 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_688_6_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_688_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![316744901280, 209262527352, -7322683500, -4269535704, -53106408, 23720286, 644833, -45584, -1501, 28, 1]>],Snew);
    return Vf;
end function;