
// Make newform 387.6.a.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_387_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_387_6_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_387_6_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 := [-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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-51, -1, 1, 0, 0, 0, 0, 0, 0, 0], [33312422351204, 3431882125456, -6223544561327, -301260007075, 277010815095, 8269172671, -4078413669, -40404153, 19231001, -192531], [167478232916, -762684015384, -64822510771, 20833059243, 6546797939, 397671097, -116173337, -7832247, 594245, 24107], [-912070606876, -418161182816, 287212616929, 54235511681, -15768164889, -2591602757, 213763899, 35600547, -820999, -144543], [4700999183556, 1612360020176, -981675569783, -220849424171, 39727817087, 8979038055, -421392509, -116809809, 1144977, 474917], [-19301122595508, -25027510353488, 4539319373339, 3248986969295, -194780377283, -122844007131, 1540088777, 1552893885, -47565, -6235841], [-12026382560220, -7514795536912, 2509398609001, 953163113617, -106024108753, -33961389861, 1148308531, 411977523, -3158223, -1603183], [-56109167945572, -25999649489104, 6054797547951, 4112939498419, -139815344567, -158825858063, -264140251, 2044433577, 11663207, -8421725]];
    Rf_basisdens := [1, 1, 1, 197066179584, 24633272448, 24633272448, 21896242176, 197066179584, 49266544896, 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_387_a();"
function MakeCharacter_387_a()
    N := 387;
    order := 1;
    char_gens := [173, 46];
    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_387_a_Hecke(Kf)
    return MakeCharacter_387_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 := [[-1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-14, 0, 0, 0, 0, 0, 0, -1, 0, 0], [7, -3, -1, -1, -1, 0, 0, 0, 0, -1], [-73, -15, 3, -1, 1, -5, -1, -2, 3, 0], [191, 11, -3, 1, -1, 5, 5, 2, 9, 4], [-383, -56, -11, -8, 0, -7, 7, 4, 0, 4], [-249, -23, -9, 19, -11, 20, -16, -15, -2, 7], [-183, 21, -6, 4, -8, -17, -10, 9, -20, 4], [-608, -25, -69, -52, 60, 20, 9, 42, 24, -6], [-423, -52, -17, -20, 40, 27, 35, 34, -22, -36], [147, 234, -54, 57, 33, 30, 11, 84, -84, -21], [-1038, -460, 120, 112, 40, -37, -161, -45, -155, -31], [1849, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4824, -27, 100, -79, -25, 82, 72, 124, 139, -56], [-12898, 116, 109, 13, -223, 101, -52, 50, 236, 45], [-9965, 803, -32, 50, 108, 310, 293, 136, -49, 179], [2015, -2743, -37, 161, -443, 264, -30, -206, -80, -7], [463, -1269, -163, -525, 541, -477, -171, 208, -297, 208], [-998, -440, -214, 156, 168, -482, 14, -110, -40, 64], [4663, -479, 863, -63, -207, 312, 174, -78, 108, 213], [-8866, -124, 553, -169, 809, 328, -319, 161, -81, -132], [11270, -974, -57, -453, 483, -299, 636, 570, 28, 283], [6109, 147, 1099, -503, 237, -230, 50, -156, -480, -423], [10723, -1265, 108, 700, -14, -1075, -266, -171, -1316, -434], [-58332, -150, -971, 1081, -1243, 609, -534, 154, 442, -101], [-3886, 2688, 320, -2, 276, 1309, 1019, 1343, 1133, 117], [-12511, 4993, 1022, 268, -854, -362, 37, 432, 1463, 409], [-24621, 5469, -527, -629, 293, 689, -1067, 36, 1139, -120], [-39092, -6856, 532, 402, 1198, -1038, -1204, 44, 758, -404], [-38202, 17989, -213, 702, -1400, -2241, 448, -232, 905, -405], [4115, 6412, 1906, 1453, -675, 1546, -2317, -1478, -1200, 861], [-15318, 2728, 568, 1068, -456, -3484, -12, -338, -1860, -832], [-33504, -2612, 621, -299, 193, -617, 2690, -2088, 14, 211], [-13651, 26940, -266, 1261, -23, -622, 3065, 1248, -1254, -2599], [-215979, -15625, -889, 2431, -2117, 1456, -3602, -1620, -1546, 601], [106821, 923, 4253, -1221, 1207, 662, 3528, 1709, 2082, 729], [-102854, 20332, -2916, 3954, -1440, -680, 2138, 233, -4230, -1490], [-2624, 7408, -859, -5859, 3981, 677, -384, 968, 2164, 65], [-144402, -22462, -368, 1838, 3202, 2748, -2484, 3628, -1010, 1268], [-3362, 15560, -916, 2730, 616, 2510, -3508, -3953, -2968, 2628], [75173, -353, 4396, -310, -404, 1802, -495, 1206, -1377, -2337], [111882, -546, -3388, -44, -1740, 1280, 5150, 2826, -3838, -410], [159073, 16089, -230, -4918, -1736, -1575, 1314, 4885, 1264, -256], [-312365, -39101, -694, -7138, 4044, -3178, 5149, 1434, 213, -1071], [144154, -9694, -792, -6644, 4200, -4026, 5260, -730, -4260, 1152], [-95065, 43314, -1574, 223, 1863, -3824, 6189, -714, -3728, -937], [-161547, 42571, -2187, 2193, -1983, 7640, 1788, 836, 6752, -83], [64549, 39343, 2703, 10543, -9967, 11482, 140, -4951, 12030, 563], [258343, 17247, -8259, -3635, 1711, 5271, -1725, -5150, -2437, 3402], [550, -17046, -2666, 5224, 6512, -4004, -3902, 4024, -8232, -1396], [-8470, 7184, -7163, -437, 7317, -2112, 6993, 12685, -129, 4448], [88496, 7540, 3078, 13344, -14476, -3810, -5214, -7888, -1980, 1244], [278065, -32655, -3087, 11815, -4535, 5721, -4575, -13938, -3797, 4920], [-218583, -23691, 783, -2301, -4149, -11670, 11238, 4788, -1866, -4383], [149952, 38206, -5610, 6300, 560, 9994, 15216, 1414, 6866, 3702], [-386391, 55777, 1153, -8527, -8469, -5157, -1771, -1758, 12601, 5056], [-41522, -79492, 18386, -1856, 4342, -14657, -12825, -5127, -10591, -2645], [121400, 21826, 3972, -12080, 8164, -16716, -13878, -5169, -5130, 898], [-190396, -54979, -20787, -3584, 4078, -3175, -7962, 3390, -10113, 3821], [10220, 48074, -16507, -3209, 14903, 5229, 5820, -7248, 11050, 645], [-358621, 95891, -10814, -4402, -896, 20622, -3395, -2160, 4073, 2777], [-385850, -69002, -7815, 18553, -23359, 10569, -6274, -8142, 17734, 147], [283893, -90529, 6273, 16613, -8677, 17871, 7719, -16998, -15553, -4770], [-79369, -91351, -3387, 8855, -5005, -4360, -684, 4728, 1186, -2887], [179157, 50839, 943, 2065, 2399, 11961, -5775, -7308, -5861, 3550], [-1276631, -60703, 16617, 11131, 7923, -19566, -11450, -16090, 780, 6399], [-160384, 62361, -13535, 19264, -17562, 12861, -1314, -1806, -1333, 2201], [493236, -83342, -15822, 14456, -20702, -16630, -39192, -22479, -5872, -1202], [368834, -34197, -22479, 18740, -10632, 15148, 16733, 9146, 13786, 19076], [286357, 149225, 10021, 1861, -20869, 9353, -18041, -5710, 11371, 5418], [-59055, -131178, 21613, -3340, 2204, 15973, 16883, 12750, -15264, 5002], [141504, 159821, -4866, 34211, -6779, -6262, -8642, 3408, -19455, -2076], [329891, -15315, 12971, -30223, 23865, -18612, 7422, 35499, 5800, -7559], [-1376919, -12709, 4409, -25127, 28335, 11581, 6321, -3096, 9457, 1592], [512250, -125694, 16910, -6382, -2126, -13988, 17356, -6516, -16712, -24906], [-53742, 199772, -15788, -30432, 5236, -6828, 21684, -18260, -34740, -17684], [402209, 147179, 7722, 23976, -36366, 13754, -56389, -16802, -37923, 4463], [-657854, -221909, -43327, -41120, 5718, -9941, 28658, 18922, 5671, -15843], [-541919, 121047, -25257, -11591, -29483, 9862, -8202, -2612, 3352, -2355], [-1964491, -57527, 29549, 17851, -9793, 7700, 4136, 23522, 3306, -7311], [1524917, 244015, 24877, -21961, 48859, -1186, -54072, 11553, -8202, 16521], [-1203856, -133519, -635, 3878, 30460, -28185, -19388, -14436, -20281, 2057], [-2634215, -14227, 27495, -9577, -4157, -12002, -33246, 3660, -19814, 7569], [1447727, 204676, -12729, -16926, 40242, -55439, -12389, 14378, -20628, -23108], [-1863363, -27695, -13282, 19720, 1134, 4898, 16037, -43390, -8149, -5735], [-271961, 6475, -37159, -29615, 17229, -4140, 24068, 35176, 17832, 5213], [-375115, -399023, -7987, 43429, -33023, 5616, -2498, -15452, 34478, 27951], [1134821, 267535, -26088, 23810, -6688, -17618, 66925, -14622, -23885, -2201], [-219193, -299595, 2993, 37785, -59715, 4196, -24710, -10166, -31326, 5711], [-2359211, 28234, -19556, -25417, 33667, -53760, -9047, -970, -8242, 1577], [-1499221, -22697, -46040, 55966, -78482, 36021, -30984, -78303, 43034, -1224], [1009841, -385375, -27861, -21839, -5135, -66018, 52, -25417, -25382, -33301], [-1015984, -184594, 71016, -20058, 33928, -53700, -34848, -9507, -20368, -31988], [1814384, -284433, -57937, -16050, -1386, 15542, -17033, -2960, 22920, 39326], [645398, -251486, -48176, 30416, -41620, -1162, -15056, 27812, -5292, -9964], [-2952250, -383232, -21573, -7009, 11831, 863, -18556, 6386, -17772, 19775], [883859, -181715, 73369, -32345, 32903, -98356, 15984, 37384, -24714, 20493], [-2548128, 156107, 22453, 29616, -38420, 2590, 45433, -23304, -33738, -23326], [123402, -404916, 1579, -39229, 3803, -64399, -12562, -32000, -48162, -5987], [-131218, -377980, 83073, 12167, 25267, -32909, -43324, 60222, 76546, -10611], [100867, -147015, -5471, 111803, -8031, 85325, -9063, 14368, 62563, 10080], [-2306727, -211557, 97279, -63217, 105109, -101059, -23853, -23610, -10449, -21668], [-442200, -332374, -57779, 57769, 4733, -79407, -13506, -44440, -56498, -33341], [-2152426, -429455, -29655, 16104, -42236, 61348, 43961, 1508, 65506, -4338], [-2257705, -269441, -29179, -37979, 76413, 20574, 83406, 60490, 38122, -19241], [-1759324, 226224, -79444, 1796, -38440, 3934, 43746, -27860, 79730, 44674], [-2169599, -585495, 173145, -9087, 10397, 112774, 18244, -57778, -22080, 2193], [-1021970, -692635, 97533, -39720, 3046, 41635, -54310, -49836, 63611, -15661], [3362994, 171460, 69328, 16804, 41480, -126930, -65554, -40414, 2546, -36518], [1845998, 1086, -109476, -100696, -21108, 556, 12674, -45148, -60674, 70838], [-2951825, 390123, -14124, 81400, -142814, 102838, 31983, -142688, 59959, -6531], [-2469405, 565363, -73360, -33192, 102218, -14849, 58912, -7827, -46498, 11928], [4868628, 567728, 7612, -51916, 107476, -108928, -20196, -12064, -81452, -49136], [-884622, 524938, 4224, 131652, -128960, 31194, -67532, -26392, -35164, 15372], [2043707, 667811, -41463, -77155, -27607, 49646, -28310, -67768, 6326, -13537], [-3005519, 212945, -131640, -15986, 83400, 58342, -58625, -1444, 52313, 56517], [2342690, -1092698, -125852, 45732, -71120, 83736, 4682, 3410, 79250, -4802], [661307, -67595, -76679, -22621, 80219, -163470, -160094, 14087, -91952, 12627], [-738989, 1480195, 7401, 85627, -52439, 31941, -31873, -32852, -136611, -21822], [-753932, 182896, 46965, -6485, -4865, -6725, 66280, -31830, 830, 33025], [-3673057, -69753, -30219, -119971, 70401, -9044, 46016, -62302, 20308, 11321], [-3892169, 1800029, 49239, -60475, -3727, -27830, 41640, 23249, 28626, 4411], [3380479, 720059, -4151, -54169, -87475, 36679, -107915, -47116, 25907, -4508], [1668221, -840743, 73009, 20799, -90293, -102768, 6304, -195686, -82182, -11723], [-2774759, -205425, 121942, -122292, 59958, -230602, -29653, -4320, 66317, 44999], [-2418827, 485731, 56303, 57297, -2605, 194181, -17383, 101304, 67631, 62998], [318639, 478952, -107512, 8087, -87381, 254168, 78537, 106322, 101834, 13035], [190958, -114434, -51154, -76078, 23838, 86260, 66460, 89318, -52652, -63262], [7503653, 251950, -22084, 41041, -147563, 37290, 62995, -133184, 122918, -20187], [-3448353, -11158, 12818, 36183, 112607, -45228, -22895, 232178, -40404, -24613], [2879766, 14458, -107156, 59044, -8896, 7932, -107534, 54490, -14062, -12506], [4990161, 1268397, 17939, -145603, 3161, 7370, -144644, 54438, -7214, -13873], [9021111, -80930, -95146, 85821, 99173, -61438, -11529, 16296, 16056, 22939], [-312548, -1015752, 83440, 84382, -58362, -193614, -102152, -129556, -225146, 12716], [90822, -847865, -107030, 37029, 36395, -170644, 84754, 25378, -97897, -137646], [-6575328, -212817, -183155, -142354, 161126, -162786, 241777, 138278, -31952, 37308], [296121, 718817, 94512, 53526, -226940, -110822, -42553, -102560, -9907, -13175], [-5398657, 912155, -158758, 19026, 75572, -82638, -2111, 18336, -65999, -117847], [11183550, -296177, 66980, -41773, 145421, 10484, -48500, -39990, -1595, 9098], [2465786, 440832, 24620, 76222, -6972, 238252, 50510, 231641, 2282, 9426], [-8680725, 960331, -120115, 66625, 63359, 107709, -54267, -35398, 56465, 58588], [-1083695, -732489, 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29795261, 846187, -1351356, -1134156, 1280152, 2315145, 702232, 1800896, 1210516], [183037865, 7116527, -2402986, -565678, 599300, 458302, -1014695, 269374, -153467, 1373629], [35903416, -5875584, -254821, -3081489, 1167007, -4838445, 1957388, 2884746, -1919076, -621965], [-17412612, -3593482, -3072638, 1460862, 1634062, -152450, 1561390, -545580, 1058838, -475792], [52651930, -4204384, -873650, -2251638, 268250, -379312, 635346, -2420408, -241478, -603272], [-51562350, 12770768, 1163588, -2083740, -1965792, -107486, 919514, 1595396, 146490, 268942], [15495461, -5205381, -673641, -2312893, 2531651, -958322, 1823596, 1089854, -1978968, -2493], [46168496, 6383408, -1431341, 3274365, -4358547, 1862763, 1316158, -2170830, 3706608, -532431], [-77487411, 8867473, -2026225, -1001741, 573503, -2802884, -1981304, -981060, -382064, -805861], [18739716, 4418086, -2828690, -363002, 938574, 960062, -1532266, 263868, 978514, -806084], [33830289, 26199551, -171375, 1674885, -36491, 1697370, -1870126, -96326, -739206, -440657], [85813799, 6708163, 1061897, -1150495, -1779845, 1161665, 698001, -3067252, 1565149, 1512532], [26065972, -12284560, 178597, 2224501, -731727, 1357445, 1147332, 2620826, 2578988, 759601], [-79297162, 10418837, 4421381, -241018, 649802, 1182306, 2392759, 359170, 796982, -1191914], [144015096, 23788298, -1516314, 1073476, -844340, -1278540, 765342, -48190, 2640952, -158512], [-10206594, 20073232, 941800, 2309782, 958974, 1269992, 2663978, -83040, -704948, 250798], [-90659026, -13341666, -25949, 1316725, -2615399, 2533695, -1905360, -3431486, 1035338, 931035], [-49498907, 30636935, 1702810, -1327022, -395836, -1944242, 552341, -1774976, -1247251, -322471], [-68419765, -7292015, -504839, -3054441, -1069005, -2048560, 944436, -1303748, -729090, -2014867], [44757071, 5056213, 49792, -2146894, 1944848, 1098582, -673829, 2802608, 1286029, -438243], [122174279, 4432875, -2741033, -790843, 1411645, -1335420, -221116, 3551530, -1585934, 399983], [-158287023, -10267609, 563776, -829510, 1796200, -1786618, -4081131, -486256, -4609709, -450425], [-2907687, -12230963, 3829969, -194707, -1223279, -595098, -3193608, -1012558, -3455144, -656215], [26951880, -5676316, 1123616, 2375948, -315884, -1616016, -509044, 1441298, -2605592, -1071088], [98849423, -2896444, -338826, 2514205, -1406691, 3146006, -513497, 553350, 2158136, -312203], [40492741, 29220714, 349031, 2331104, -2093212, 1662311, 139143, 197552, -2776542, 910346], [31161725, -96602, 152351, -5518, 221530, -733081, 799967, 2997574, 2712, -94928], [-29014151, -4732002, -1342662, 4756523, -1016057, -1797012, 1814059, -2189848, -368262, -173909], [-11312525, 2150719, -490681, 1572575, -811369, 3294858, 1442784, -4063262, 621184, 504759], [-50664965, -7720629, -2386887, 3243571, -516105, 4429412, -65670, 874496, -407996, 336367], [68250023, -8116007, 2523379, -2188473, 2772429, -52596, -2733108, -268189, -42770, 1573911], [30257109, -17240507, 2730525, 1807619, 355881, 4146269, -1521571, -336986, 224203, 965784], [23390630, -17425050, -1423434, -2533904, 1899064, -3393810, -911464, -1749434, -5748622, -558546], [-40041781, -3036279, 2636983, -196907, 1822789, -2038608, 1046882, -1713783, 41188, 675777], [-116842002, -2552679, -227762, 1181333, -1900753, 2389218, -890896, -2095112, -2517415, 1238770], [-9719717, -12127339, -4997005, -84675, 275085, -4651010, 71212, -2170420, -2465490, -1801613], [61196385, 8121043, 1805195, 1750779, -1532051, 681028, -2002898, 542279, -1821420, -1054723], [236169814, -32364149, -95139, -43906, -1512550, -2654930, -410529, 1362748, 2293318, -5888], [-30356930, -19653736, 3003357, -3332321, 429161, -2095592, -443867, -121887, 1343615, -198100], [179351644, -17197894, -2894212, -1819472, -1998716, -3250832, 96686, -2450759, -325158, -1086674], [11094645, -7070718, -1603859, 1978196, -1630388, 6015021, -1924025, 243754, 826276, 1814318], [99377934, -25672904, 2280282, -485126, 1011286, -2855366, -4514116, 754860, 3203432, 86698], [-28722109, -2435379, -2046541, 965103, 1349973, 1482108, -1167728, -861669, 989298, 598491], [-43854105, 6172857, 2207030, -3395706, 1624876, 3089106, 69143, 747562, -1772793, 1380943], [96866362, -26682037, 3999045, -175440, 756810, -1256311, -2532816, 1311106, -55023, -948085], [56267991, 18987223, -313803, 5162953, -3856587, -111776, -4130628, -277096, 231516, 664449], [134668343, -27494095, 5210235, 370365, 732613, 768674, -4147068, -3059030, -4396778, -194789], [-43460414, 28446674, -1193003, -2459921, -3367281, -391143, 2069064, 1709750, 2345862, -330991], [-97406610, 1911640, 2135866, -5187366, 2379454, -741973, -426207, 1729787, 55679, -1648527], [-26003542, -2253795, 1127423, -2950930, 1549316, -3699903, 3055034, 4793256, -1677441, -2450083], [-57480437, 38025237, -2350687, -2192553, 4051719, -2367782, -892746, -940596, -1431648, -729881], [-145268725, -6142583, -4861201, -1811865, -2093687, -975689, -2862357, -477042, 1172379, -528880], [-81738414, 2110045, 2077525, 4907060, -2171250, 1233477, -1757400, -2519156, 382445, -2359547], [-120594955, -26033805, 813899, -110561, 250071, 575392, -3961618, -548776, -5904418, -314031], [-73872931, -30760635, 2164829, -1829991, 1249991, -1568429, 1891101, -550064, 3191673, 574392], [192076711, -21920259, -3013790, -682262, 4031632, -3386094, 1210775, 256440, -1915305, -580349], [48840059, 18371009, 2616101, 1306065, -131473, -1826091, -1169385, -3285994, -2294145, -22294], [-60588334, -5577949, -1337793, -2362382, 1422088, 4732193, 2214478, -351094, 1003535, -845497], [12064446, -13553663, -2936017, -962360, -659704, 746094, 602037, -1539070, -4020244, 535210], [118858616, -21869560, 3193762, -1972634, 406098, -6531168, -1333470, 829972, -2967282, -321664], [-26667126, -20590392, -1044201, -500947, -2110687, -4152813, 1062450, -91174, -4734932, -2169675], [-24136627, 6996173, 312255, -1256507, -1340771, -376798, -865288, -725890, 3183518, 3743147], [-132072284, 22382382, 1581849, 1602561, 2108209, 583640, 2940233, 1546449, -2300785, 1315292], [121636122, 33637729, 885827, 4859382, -4351088, 1287347, 1097456, -875472, -828799, 125123], [33392939, 18707045, -7186405, -2249943, 3830673, -2986944, 824202, -997452, -1564828, -1927875], [77673928, -32001410, -1290254, -476040, 626974, 505934, -904864, 2084131, 208948, -972314], [180269062, 6491532, -2728900, -2533948, -1002976, -1891970, 801158, 3343297, 1363318, 1787686], [-80003348, -16943114, 3784347, -1331377, -2367437, 6950301, 1641656, -2446580, 3010324, 140207], [-136623812, -17218774, 4772953, 1908353, 2040641, 5067324, -4943423, -2506423, -3420017, 1856676], [-113979752, -9753478, -2128404, 250834, -910428, -1626803, -1066091, -2948839, -3249961, -859017], [-100836046, -25426488, 399165, 2146489, -1442503, 4007016, -636497, 982471, 4079365, 2121538], [-11932755, 7477712, 2477106, -41659, 2558741, 1559112, 3111819, 2797908, -4402720, 1481331], [19070737, 19561459, -2898653, 4150147, -1914433, 883938, 3498496, -272950, -1661836, -144557], [79519851, -30702102, 718646, 1722269, 2351857, 3506884, 3512229, 1825536, -3020450, 57813], [12425560, -28827577, 7323878, 2577255, -3846731, 2514184, -50344, -2681930, -3690617, 966428], [-195503885, 3880021, 3862555, -3662801, -1834927, -2080438, 1377550, 2013227, -787392, -2653187], [-46965346, -22960228, 7953639, -2239763, 2412913, 2340161, -4290504, 210744, 178734, -19621], [-77476772, -6752058, 4602867, 787579, 1129039, -4042584, -5132505, -3165767, -1813315, 540696], [-127572792, -9596094, -2490802, 1925966, -1260438, 1802780, 6114688, -2595460, 3654588, -931782], [-24875665, -2079952, -1383120, 3210771, -4136213, -1972364, 2611219, 364642, -676520, -560869], [-60627358, -1754108, 3800693, -3079129, -872705, 2280701, 3482318, -3880126, 1516744, -499121], [-56002515, 25421245, 2057997, -2183, 678613, 2899102, -3845356, 1625052, 1638500, 3164161], [20193448, -6120252, 4150529, -5939829, 1656563, -5305453, -2301916, 2100136, -4967246, -2319023], [171394259, 4100013, 1276241, 5575177, -1426989, -1535553, -5444267, -1267386, -2006911, -438568], [-19301246, -2005512, 1893459, 1666441, -891049, 5040040, -3195245, -4935287, 5458889, 2349596], [-159591205, 16182039, 3941820, 297336, 2804382, -7810742, 2518551, 989182, 3353875, -354347], [-103851197, 9624737, 3103671, 651775, -542973, 587258, 784510, 1512898, -7503946, -14231], [-43165288, -14068542, 226350, 4966686, -1985670, 6226742, 1837894, -2436793, 3902718, 2902016], [175838289, 1135153, -1020494, -796266, -1349592, 547758, 3696887, 5327992, -1994185, 905231], [-66724241, 11257433, -1668105, 1695019, -1972313, -4181558, -5688442, 106344, -3126062, -760115], [1399483, 8698035, -2774739, -4636817, 2687005, 3435753, 4522431, -712830, 6429013, 2556198], [72383200, 32898974, -2530970, 731802, -4058730, 3329736, 3841700, -91254, -538836, -2676546], [-254469599, 2135827, -1741819, 1142475, -4813621, -2039160, -1360830, -2302523, 3018492, 219695], [-50368759, -1435474, -5547788, 232039, -3674269, 2431080, -3458883, -5124390, 249538, -1316267], [-62077245, -6093903, 604353, -4046629, 4785511, 293196, 331088, 5876592, 2404094, -3646171], [38123294, 3577092, 1075293, -2535301, 1125447, 3097345, 4485242, 4696388, 2979028, -1778353], [131285963, 1581943, 409447, 6827821, -2554051, 7568508, 2577220, -783922, -4995022, 57187]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_387_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_387_6_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_387_6_a_e(:prec:=10)
    chi := MakeCharacter_387_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_387_6_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_387_6_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_387_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![-20373120, 9477216, 5497616, -2163208, -450596, 112716, 16722, -1734, -229, 8, 1]>],Snew);
    return Vf;
end function;