
// Make newform 177.6.a.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_177_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_177_6_a_a();".
// 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_177_6_a_a();".
// 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 := [-14846072, -74443300, -23890558, 13849341, 2364885, -813818, -79077, 20782, 1067, -238, -5, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-45, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-46744666460014620, -277273564078787976, -97185495955214979, 54695618337973207, 5261841092179714, -2430423048764513, -68202867113046, 37422999014637, -3169078538, -187297311739, 2778460829], [49137760341652876, 436511974211010068, 96919307362486299, -58563201370458676, -2949746727927086, 2276300832143201, -63472531727643, -33222926157014, 2207753333560, 162977595845, -14193213478], [218108849645718264, 208390806737984508, -46718511586843450, -79788139597326609, 11268268678038488, 4067068942117866, -557963364451955, -67011465294593, 9131611480310, 351855671964, -47676035497], [82872747608509216, 444545767039652472, 91261763313963386, -97755073144838957, 164851990846319, 4579736170630205, -225728352893616, -73166083554387, 4935210416071, 377512348731, -28858453840], [33416666008470872, 222471726912464284, 14790555254723370, -45698521971969343, 2217875051857707, 2082506296044915, -159340236044106, -32727493696379, 2866062504113, 166942662163, -15568094860], [-537772628306245264, 457773222420002388, 290581103124887564, -168561666697995505, -1080392737069084, 8505381357970468, -545370177084889, -138996549095733, 12116309216206, 724092912446, -70699645381], [-289556126907450288, -1114371303608622340, -139765268110871872, 223364021183059031, -5726984781855596, -10099241323288128, 677824163285755, 158556548973571, -13430535220258, -810327211110, 76278029747], [949786364217219244, 3475120040752796400, 449709576651727347, -790733613061331263, 33519765392353652, 37113529280203671, -3010055469617252, -591234225816113, 56749599552340, 3040305309961, -314311388515]];
    Rf_basisdens := [1, 1, 1, 4205126743692352, 4205126743692352, 4205126743692352, 2102563371846176, 1051281685923088, 4205126743692352, 4205126743692352, 8410253487384704];
    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_177_a();"
function MakeCharacter_177_a()
    N := 177;
    order := 1;
    char_gens := [119, 61];
    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_177_a_Hecke(Kf)
    return MakeCharacter_177_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], [9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-17, -1, -1, 0, 0, 0, 0, 0, 0, 0, -1], [-31, -3, 2, 2, 0, -1, -1, -1, -1, 0, 2], [-50, -24, -6, 3, -5, 0, 3, 1, 4, 2, 2], [-140, -8, 3, -20, -4, -4, -1, -4, 0, -7, 5], [-446, -10, -18, 8, 14, 19, -1, 22, -5, 18, 0], [-344, 9, 3, 32, 23, 9, 29, 7, -1, -16, -6], [-632, -89, 3, -53, 8, -7, -1, -45, 7, 3, -8], [-1395, -94, 72, -85, -26, 17, 25, -19, -13, 33, -29], [-414, -190, 17, 74, 42, 19, 122, 80, 29, 3, -71], [827, -184, 42, -99, -20, -11, -60, 90, -59, -74, 43], [-802, 311, 23, -8, 59, 169, -90, 56, -63, 87, 66], [-2076, 804, 84, 72, -70, 82, 11, -283, 102, -177, -14], [-4382, 252, 2, 207, -113, -273, -152, 135, 49, 130, -18], [-8133, 2217, 441, -54, -224, -420, -154, -116, 16, 112, 367], [3481, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-7752, 882, -53, 259, 23, -331, -188, -28, 217, -2, 107], [-14956, 193, 299, 581, -282, 70, 131, -62, -364, 264, 328], [-17230, 2462, 229, 797, -297, -318, 227, -242, 204, -330, 133], [-3417, -914, -709, -937, 442, -53, -45, 392, -365, -449, -342], [-24411, -1578, -603, 890, 183, -760, 502, 20, 172, -170, 68], [-25698, 4445, -167, -496, 83, 601, -658, 556, 125, -189, -440], [-9660, 2220, 14, -4, 36, 983, 1084, 677, 33, -119, -830], [-2523, -819, 688, 529, -1265, -672, -264, -1281, -688, 385, 1106], [17730, 809, -129, -1147, -242, -798, -1734, -1485, 250, -1297, 2], [-6149, -470, -1072, 1202, -125, 120, 943, 1162, -556, -207, 25], [-13007, -2315, 20, 858, 72, -1779, 96, -1330, 1399, -29, 1694], [17371, -3977, -1587, -4130, 1008, 941, -1658, 111, -659, 1191, -1193], [2097, 2157, -2146, -1010, 920, -588, 216, 571, -42, -300, -1804], [17901, 6039, -1644, -1037, -35, 1792, 1410, -1673, -840, -943, -1086], [-15168, -4018, 310, 3194, -1384, -84, -2346, 332, -1138, 3216, 1372], [-61322, 8694, 292, 3991, -357, -1298, -3005, -793, 980, -652, -1820], [71053, -5629, -4858, 1365, 3635, 780, 3230, 4805, -2, 4015, 396], [-99467, 2954, 373, 7987, -512, 2093, 4466, -977, -1693, 2464, 1680], [56376, -5204, 5605, 8306, 1994, -683, 4892, 266, -2429, 3135, 3393], [70702, 11719, 1756, -3145, 916, -986, 3979, -1551, 858, -1364, -1607], [53859, 4292, -1905, -733, 874, 471, 79, 4514, -2443, 3059, 726], [-191497, 2108, -3532, -1746, 2167, -1347, 3749, 1499, -1823, -7954, 1077], [-45636, 4567, -4127, 1039, 7374, 7571, -4072, 5096, -925, -18, -3346], [24048, 3480, 1096, -8147, -2885, -1956, -1869, -1951, 3670, -1636, 438], [62443, -6516, 1961, -4946, -1645, -6881, -215, -2042, 1309, 3554, -694], [62376, -23005, 13153, -3712, -3945, 4188, -1545, -7460, 3088, 2575, -1748], [49316, -7153, -9319, -2401, 4366, 8164, -2052, 1069, 950, 807, 4372], [-377053, -12728, -6922, -5158, -1369, 152, -1787, -7440, -1248, -1597, 2859], [-152407, 47492, 4615, -7435, -1542, -2563, -3525, -7374, -995, -11965, -3340], [-3769, 18906, 7451, -5440, 4991, 11381, -946, 2813, -4837, -2347, 2102], [-25408, -44377, -4393, 1741, -756, -2712, -4454, -8123, -2832, -783, 3808], [-174980, -58072, 10402, 8446, 3382, -459, -3593, 4332, -7291, 4942, 2394], [-239344, 24735, -3712, 8386, 5015, 1469, -4268, 3365, 2685, 5303, -5329], [40292, -77294, 6170, 5092, -5792, 3108, -1370, -5198, 7214, 6114, 6646], [-13338, 1569, -3622, -2818, -7473, 2138, -1819, 13, 6376, -11021, -3137], [-129850, -51903, 1115, 6363, -4222, -7606, 12314, 20045, -2990, 6911, 3390], [-29953, -4379, 2758, 8344, -7822, 9942, -7460, -7939, 14344, -11524, -10236], [-635172, -26983, -4913, -9543, -3542, 4276, -8768, -17289, 6338, -3151, 1208], [-78529, -54415, 325, 10083, -9979, -8563, -5627, 7924, 3269, 8417, 2933], [63893, -78612, 15309, -9351, 8582, 1268, -5382, -664, -12954, 947, -10244], [-194919, -45442, -7883, -27676, 849, -5146, 42, 1562, 12774, -12358, -15092], [-192104, 69357, 4669, -45279, 3480, 11740, -24207, 8770, -3798, 3744, -15784], [-152314, -96929, 18495, -807, 286, -10790, 1402, -11671, 7800, -12891, -9132], [90705, 32840, -10099, 15330, 3057, -569, 11192, 18461, -2427, 14087, -17968], [-334217, 79685, 1771, -14375, 2481, -10451, -21232, -8335, -5021, -13794, -1629], [-603386, 46906, 2556, 22982, 466, 8082, 19318, -2714, -9196, 12294, -1784], [-270020, -13654, -30799, 3457, -3965, -6191, 13407, 2761, 1575, 4539, 4325], [269076, -15333, -49899, -10322, 15481, 23553, -5627, 8807, 10309, -8390, -9072], [-277743, 33749, 18411, -3950, 6526, -7632, -12228, -4214, -21656, 19110, 15671], [176912, -25488, 23000, 1254, -32146, -4676, 5906, -7354, 7598, 10808, -17788], [143592, 92519, 13895, -17098, 4969, 7581, -30623, -1027, -13999, 8490, 3422], [-150112, 159648, -52090, -68443, 1799, 16953, -7662, -16159, -3307, -29352, -3368], [144152, 37948, 4207, 66614, 22402, -17650, -13861, 26490, -6706, 22265, 13293], [840919, 88413, 17006, 32430, -26872, 1303, -13324, 5970, -3719, -1591, -16640], [459293, 57973, 8467, 42437, 8155, 2218, 35867, 16597, 8938, 12872, 7047], [-226787, 75289, -8721, -36317, 9053, -3222, 1746, -36402, 4692, -23175, 7785], [122257, 21861, 40920, 73737, 19105, -19661, 25224, -3392, -10501, -25126, 9980], [-198821, -232455, -3366, 75239, -7023, 11284, -452, 34357, -16216, 9543, 720], [-1171972, -90900, -34089, 77617, -22833, 10572, 14709, 3686, -19482, 18258, -5113], [699101, -132930, -20340, -3314, 11639, 13446, 76163, -11604, 3250, -38033, -11071], [1728036, -122995, 30146, -15521, -7312, -31566, -1789, -27263, 32070, -11208, 3155], [442673, 54127, 1476, 17224, -9086, 7025, -2780, -30124, 2209, 18209, 41642], [-86050, 128420, -73126, -57056, 9512, 63180, 39568, 13610, -4072, 11196, -47114], [547616, -174469, 18239, 51611, 31132, 9692, 59894, -33053, -36538, 36249, 26108], [31346, -25752, -20813, -22388, -25082, 35706, -92479, -27778, 25862, 2581, -3641], [-860852, -149034, -65886, -109985, 4139, 59661, -67878, 22429, 10895, -238, -42442], [532655, -80248, -29909, -14630, 19009, 17159, 2846, 15357, -21377, 27425, 40890], [-1126216, -288538, -13946, -31315, 27297, 37597, 15343, 39984, -33303, -19179, -37804], [725942, 58894, 100790, 79914, 26786, -1999, 28433, 23220, 14309, -16824, -3362], [1190572, -357056, -3012, -2381, -18325, -20844, -4259, -13595, 29078, -10252, -32856], [445911, 424196, 47523, 17349, -29802, 6263, 30057, -21274, 21555, 39179, 19676], [-1070265, -67419, -46199, -603, 28941, 82237, 16240, 44639, -18557, 49354, -363], [-910790, 378711, 110660, -43519, 17462, -43179, -62359, -42066, -26211, -11975, 47021], [49946, -69351, 10631, -31529, -6028, -49002, 18110, -78213, -27150, -74705, 28536], [-1578144, 100125, 44692, -75766, -44417, -5851, -61265, -19704, -4433, -42, -34315], [-483468, 196392, 57604, 52725, 21883, 16959, 47953, -47630, -26693, -38011, -35420], [39158, -7770, 70644, -73916, -60842, 3485, -91802, -17525, 5609, 22709, -20742], [1522249, 204489, 26012, 25153, -10491, -79412, -90564, -25551, 5820, 18945, 47384], [-1957242, -135810, -72160, 59711, -8423, -21043, 9432, -27089, -4121, 10292, 25250], [1275226, -346650, 7958, -15462, 10030, 42545, 44530, 61911, 22169, 23039, -32794], [842236, -387740, -8582, 66259, 5443, -14299, -18816, 126149, -43617, 112108, 69586], [1098287, -261685, 44240, -71801, 25091, 9952, 3592, 5087, -62068, -51141, 39904], [2749148, 123602, 61097, -46976, 7058, 6876, 26597, 9376, -23784, -72205, -55083], [402908, 329979, 119901, -26242, -66213, 22673, 13137, -45535, 31405, -93982, -8614], [-1403209, 15036, 19352, 15632, -77539, -48666, 83001, -37492, 50462, -53113, 54693], [-3766284, -98639, -42503, 49010, 60087, 4780, 30021, 98894, 6522, -5657, -74798], [1434402, 66374, 15056, 50526, -54914, 11551, 54646, -5841, -69553, -3219, -11620], [980692, -343615, 28781, 166529, -53430, -49988, -38489, -5890, 29334, 24494, 33058], [2970578, 210388, 19602, -140820, -65810, -32142, 28645, -103559, 18516, -84103, -8548], [-429736, 97424, -11176, 97433, 84391, 22654, -30765, 76659, -2694, 41966, 17934], [371250, -179000, -39434, -135643, 118143, 69909, 96018, 40229, -54957, -1452, -108864], [730955, 150505, 83555, 74375, -76405, -150640, -16143, -24601, 73924, -49634, 49101], [6806424, -194787, -59855, -107134, 11875, 30947, -66655, 393, 38035, -60556, -32826], [-2747355, -33546, -173569, -128986, 79963, -47022, -60902, -10252, -54790, -37032, -41674], [3407357, -464255, 17003, 129313, 20597, -64288, -48989, -16783, -58224, -25830, 53685], [-484428, 297166, -199750, 105250, 44194, 229, 85407, 122990, -4579, 29562, -8970], [1273953, -221113, -4190, -260949, 88813, 71086, -120754, -4081, -17002, 30587, -21688], [4199473, -456284, 89155, 121504, -39081, 32540, 112996, 39108, 53644, 37626, -18348], [-1416926, -511429, -70867, 207554, -7783, 32288, 177479, -7776, 79152, -68431, 12336], [2598992, 405955, -41175, -168256, -145207, -21405, -282113, -79255, 74667, 44064, 3314], [-4188321, 356394, -74806, 36706, 71903, 27366, 27067, 7442, -18702, -5421, 103783], [-1373433, 737068, -18662, -338807, -31824, 90595, -227629, -58579, -3447, -85265, -32559], [-588954, -334571, -25423, -217105, 43340, -55963, -13883, -17023, 2015, -82695, 45042], [2940847, 163910, -193853, -109462, -12147, 63039, 156775, -11270, -48207, 8588, -85054], [6473216, -334288, 3428, 181366, 8808, -43540, 142670, -61484, -39808, -27512, 66018], [543329, 695586, -27044, 158114, -105641, 70892, -100403, -154080, 74268, 83335, -50087], [392712, 584186, 184126, -52657, -117649, -3938, 83317, 48791, 58570, -77438, 105214], [5252310, -900825, 38611, -140275, -37296, -85558, 23328, -149663, 182148, -154633, -54818], [-4665042, 148959, -131981, -52561, 34464, -153239, 43508, 154194, 44217, -13952, 19846], [6670226, -655552, -224232, 100140, -120516, -213810, 187236, -17902, 47226, -13206, 103488], [3044546, -365951, 287519, 363521, -122518, -113453, -66587, -131775, -32471, 115733, 169148], [-3546830, 575173, 148781, -55483, -41644, 11528, 261210, 3197, -5504, -67287, -83826], [-1560918, -570269, -23031, 236007, -73394, -82888, 136681, -88960, 3918, -66410, 169440], [-5532549, -315394, -165083, -242768, 44443, 53801, 81218, 2399, -147977, -104127, 55132], [-2474290, 52381, 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498846], [55025241, 7419091, 918128, -1065564, 292562, -135220, -1409890, 70385, -750546, -1443100, -460058], [35938568, -4509263, -1125942, 734784, 960375, -1401699, -1928500, 1227671, -1170123, 1328081, 3148743], [25124650, -37130, 1453138, -718443, -1059657, -2474767, -869724, -157979, -515813, 46474, 316356], [85219597, 11702358, 3597576, -1369108, -87951, 492708, -607401, 1244202, -1369272, 1478815, 919089], [-19331734, 1767388, 1921317, 2393711, 1645883, 337426, 2216685, -1012720, -508144, 1864938, -351537], [-32931415, 9720267, 3280411, 2946479, -624325, -712293, 3607468, 690477, -141491, 1046302, -1237901], [-122740910, 1344118, -178712, -3841585, -859457, 617268, -2893755, -1229293, 1567922, -678044, -2259452], [122359989, -3964925, 4095402, 2572793, 1726211, -384065, 1252477, 650265, -196465, -401811, 13460], [70700568, 17465028, 369131, 3878071, -319465, 1267193, 2525011, 1390243, -486033, -574087, 971081], [-3801977, 13449165, -3208248, 852753, 198077, -1294340, 1887154, 2126395, -947260, -1180389, 1355044], [86418766, -2455802, 383346, 6145153, 93831, -704719, 1055054, 2331527, -992821, -411564, -71184], [77532266, -3728500, -3912739, 2695527, 2301263, 1986932, 1608297, -971278, 56718, -1356196, 192075], [36105604, -6935878, 1495034, 2247974, -26898, 934044, 40060, 2209388, -1083354, 525642, -1303550], [65686672, 25910374, 260608, -3209265, -305027, -1412262, -2090247, -2072425, 1096188, -8530, 46842], [20005538, -4255640, -2834822, 610186, 1418580, 871880, 268880, -6940, 627188, -2080710, -1736048], [-17497364, 11749576, 1422660, -661186, 528440, -462014, 1106282, 2874934, 773610, 2522814, -317174], [-45392091, -8492944, -292946, -3295001, -796312, 1053643, 431597, -1950035, -159915, 1672267, 2238663], [-22463728, -1503626, 3701208, 1410224, -2031758, -58588, 1776162, -1755278, -746484, 1981930, 1574978], [-49172686, 2676251, 529589, 1981406, 374725, 2410612, 3329751, -253972, -402704, 1650737, -158864], [-17190146, -7943322, 1057288, 3793372, -10726, -727520, 2080316, 1273984, -283108, 4002428, 1223034], [27921964, -11038050, 657094, -992112, -613472, 463974, -78681, -4015777, 944538, -697757, -1957598], [-56465346, -11753700, 1754530, -1587258, -713756, -1026866, 1646332, -729282, -1562198, 1456540, 469188], [49903352, -2840772, 2834022, -543680, -2462864, -4497120, 1568552, -1627238, 636216, -104076, -942866], [-27013434, 15628352, -3235886, -5468644, -678986, 1879163, -4324804, -2701905, 1172319, -2552983, 405088], [-135616289, 12193892, -1023086, -714113, -3342294, -876677, -446140, 1509900, 82631, 321454, 339127], [28970848, -6439422, -2169128, -3674124, 1276276, 2541338, 949384, 663440, 1331698, 2066874, -1520220], [25333941, 2862384, -861144, -5202986, -339937, -2148226, -2840515, -1844924, -320930, -252655, 887237], [27017933, -19661529, -366826, -164424, 763906, -94128, 975000, 2246377, 917054, 2721114, -1289444], [-92354824, 11313647, -2204519, -329668, 703401, -1087839, 901403, -667253, 713165, 1505598, -1745406], [4619630, -8592328, 2799458, -2611248, -985560, -3275813, -1088950, -2959445, 1040415, -1855881, -1815734], [111398331, 6206, 2441913, 2148848, 941543, -285501, 5803889, -554046, 595513, -2723388, 591332], [-204067884, -12036174, -470678, 2466952, 593576, 589190, 3001758, 2626368, -340500, -968964, -2284030], [-38346297, -8249368, -1545703, -836479, 3248738, 5349659, 3074813, 2025744, -1633649, 2243043, -1353110], [-128397405, -16116433, -2150070, 2560163, 240171, 1545250, 771488, 1643707, 485722, -674831, 719726], [50517820, -34663846, 1224632, 2512120, -1017400, 2471758, 2947546, -1664902, 20082, 1078520, 3225260], [5745047, 16388389, -1555026, -2118464, -2278184, 1718718, 1066510, -1088291, -146642, 1361982, 689994], [-88594494, -13195431, 399435, -553901, 856460, -2570734, -1069474, 1003487, 1132086, 492773, -1684298], [84579233, -11572438, -1586058, 1131136, -1540793, 921782, -641987, 823710, -695578, 1072775, 650421], [-108363410, -5875927, 1506587, 1127247, 1765780, 1197726, -1729482, 824791, -1310866, -1755163, -1946638], [-24979778, 8023392, 7627746, 966739, 179895, 2076462, -1269031, -148651, -1179560, 3072806, -244618], [33062020, -3554918, -3460189, 316296, -1240654, -1510094, -6041169, -1704770, 451734, 2347667, 3899947], [82416468, 4909218, -3735778, -4121086, 2602876, 759444, -108092, 5067254, 740604, 2001742, -3082796], [121873660, -4465973, -1273371, -3222376, -2514953, -3041504, -3191685, -3770496, -1772508, 916883, 1879804], [-11103189, 6986347, 291380, 2515454, 642048, 658040, 1553468, -1418981, -1097982, 4570720, 3054982], [261073, 8663953, 4122586, 3719403, -3106395, -515023, 3589268, -3747870, 2659049, -2739570, 1136520], [68823803, -9906981, 1660416, 2288917, -3917589, -3712803, -133487, -864571, 2564309, -1625361, -79902], [42390659, -16472340, 1255566, -395739, 1322216, -3275321, 355201, 1917283, -21903, 391205, 398813], [-89976060, -20100918, -1304419, -2050196, 1229498, 3464563, 3868, 3305104, -2203355, 3708647, -1110065], [42215329, 2744616, -2954219, 3763290, 2654677, 471133, 4458889, 793118, -425065, -684672, -1244184], [10595800, 2119140, -5628754, -1635320, 3656138, 5052544, 1608336, 943776, -577548, -3283848, -961560], [-92027106, 4440748, 5115400, -4722912, -1466394, -2290420, -1966064, -222886, 657312, -4651056, -3194114], [-14615984, -3828626, 5274470, 535868, -2352004, -2985477, -917554, -1237763, -24021, -898545, -350838], [-67352868, -2989777, 4181785, -3155269, -2330450, -1224807, -1347877, -2237041, 645979, -61237, 3769048], [-17108, -9873954, -2083211, -2226545, 1070567, 2350776, 1053649, 3823924, 1147842, 4723228, -522709], [90263300, 7391386, 7118019, 9206668, 1104258, -3530596, 2682625, -514990, -2336148, 1805967, 4055251], [95986080, -3575258, 1223910, -4114018, -46870, 1215827, -820795, -3044378, 2441603, -5586430, -3038602], [3825675, 9325123, 3357827, 5469215, 935653, -3002142, -3171255, 2894053, -143326, 2053576, 1207123], [-64966486, 5738054, 3456012, -1071793, -4983987, 1111458, 1975971, -4503799, 327660, -1675952, -1004254], [165679, 791260, 3470251, -1484737, -1485562, -1128799, -2699115, 196530, -594129, 707699, 3894414], [105744631, -28175056, -2413819, 2254447, 3731692, 5334805, 2792395, 4836920, -1568171, 248097, -2583802], [-68169458, 10915673, 399561, 6046089, 3615654, 2657356, 1440452, 3609865, -34048, 1564135, 527980], [-81472252, -4859679, 206719, -131877, 3665720, 2681998, 356858, 895287, -1883942, 274081, -3734950], [115925296, 20942100, -627592, -3968786, 207536, -798432, -6435250, 256344, 1861548, -133824, 722466], [166442296, 18675972, -3920500, -7373148, -1419822, -1324275, -4386440, -2012161, 3164717, -4005085, -2158638], [116196256, 11947248, -705972, 922346, -2247812, 5533404, 490973, -1265435, -767992, 2012681, -585766], [-111802750, 1517317, 7305123, -774018, -1693335, -881414, -1769353, -2507998, 313574, -6230041, -1406266], [-107692613, 3030025, 3564687, -6038651, 1047917, -5244067, 2135224, 2191257, -280597, -3437742, 2040911], [-70680841, 9003158, 4037440, 5412556, -4142635, -3294908, -485061, 540722, 2058248, 2790071, 2501809], [32908925, -12293662, -1442689, 4613690, 3577075, 1010745, 4512071, 2375414, -3170273, -215566, -670840], [116087440, 11112595, -7423107, 953835, 5349460, 1696811, 1108961, 4518885, 840999, 492399, -960966], [61780394, 7422017, 757669, 1879213, 3761998, -290843, -195153, 1479175, -1149473, 1865613, 2862902], [171297685, -7302638, 5146537, 5132723, -1544672, -1627693, 1669723, -3689528, 4090319, 816315, 1819274], [-120094928, 17377114, 2590474, -7958930, -3979466, 113010, -6785358, -3556260, 817360, -1091798, 1588446], [-4792974, 1872009, -7056057, 2564851, 811564, 1942500, 1026067, 154352, -1215592, 2873404, 863964], [-20407509, 13019932, 7794758, 789940, -5006295, -3534688, -2733853, 918538, -4898908, 4853147, 8558475], [33932570, 30219644, -2872240, 4345582, -1377724, -1320663, -2009790, 259889, -1321933, -312481, -369030], [-89974036, -8372996, -2293450, 2909171, -2769299, 227257, -2455961, 4204492, 1493221, 5372535, 2404582], [106943921, 14872715, -2993946, -756282, -3196838, -4312000, -7471896, -5134185, 455900, 816600, 4610990], [-70493805, -2247593, -7367740, -6899059, -1027899, 4747238, -895214, 3009763, 1621842, 3652217, -2553676], [-249103230, 12181963, -1566321, -693468, -1465215, -2425730, -2620157, -4255502, 1811270, -1067851, 291816], [-64496981, -4446566, 6116653, -7371014, -2699515, -2059942, 2136990, -7519996, -1651846, -3480372, -148986], [175254285, 8015978, 3714685, 4523278, 624839, 846493, 5198059, -447004, -3844693, 1544158, 2886570], [193137992, -19669809, -3057931, -3083344, 2477909, 2476181, -3848817, -3796489, 264287, -5454026, 915986], [-87541998, 47302827, 2621369, 994208, 640149, -341302, -1402263, -228470, 2033210, 1115611, -619654], [128679151, -36416711, -2398028, -3828305, -1846221, -1841982, -1982972, -1186637, 1693718, 216121, 2047380], [-68179117, -5067220, -2725653, 2266617, -349894, 1565027, 1963200, 1896435, 1678881, 3377162, 1402350], [48378574, -6920322, -4273790, 1926802, 1804942, 56901, 4212369, 970168, -589957, -271796, 3961094], [-35072250, 2273283, 1800835, -4697759, 1313280, -1731450, 606160, 5246013, 744412, -3667697, -1972358], [125268174, -18414982, 1398454, 2533219, -2455323, -2313063, 3332174, 3285575, -1097625, 1236382, -629348], [11046744, 23365898, -3576070, 7753940, 2848588, -1808530, -87140, 6004010, -613878, -1606138, 1115014], [-54718153, -20477550, -3961881, -8384879, -3051030, 1156495, -1602789, 258546, 797535, 1957633, 2790264], [-23646198, -1843428, 3595574, 5787800, -637368, -2534684, 3090908, 4131786, 894880, -1303500, 611566], [156641112, 39200285, -2253309, 4193056, -965077, -2223289, -1507807, -154583, -976987, -212878, -2187582], [79639513, 26958897, -3406623, -1230644, -4172848, -301796, 496948, 1134644, 1308388, 1867588, -284139], [40217944, -7824977, -3479445, 3489741, -4800890, -8373196, 8583685, -3257752, 1827058, -1567552, 2881476], [-135824378, -5390438, 9539148, 4241044, -3174394, -5219266, -1797464, 489374, -647652, 5041378, 3610018], [-60151748, -10929031, 1866169, -5642197, 667404, 991162, -208992, -3269261, 1086734, -1065769, 144492], [-66064958, 3895695, 3515687, 164102, 1305505, -2413207, 5408284, -5961146, 1194413, -905517, -993334], [-211671451, 14705733, 5636597, -3068955, 1196839, 1465861, 446428, -339523, 1827531, -2514742, -3850405], [167944523, 7907349, -4253037, -1585155, 1760701, -12096, -111642, 4684286, 921914, 860725, -794881], [149142031, 9945954, -7389008, 613042, 958325, 3631304, -3846747, -2090782, -1003248, 3296051, -2765761], [98884, -17364669, -3046921, 5420181, 4898192, -1537278, 7552150, 1537227, -4932152, -1025815, -2701462], [201726178, -14488396, 1266232, 3913224, -388260, -1396956, -3075219, 2336647, -130636, -2333201, 3427768], [168613493, -22018835, 209622, 6240108, 2470660, 4223984, 10130126, 3166615, -937314, 3158860, -1280930], [-35546647, 22858298, 1214927, -7679578, -2640031, -5451129, -8426388, -3037483, 1427327, -7622669, -2931318], [78322815, -2499580, -8567418, 2924885, -778992, -1168859, 7448473, 6697473, -1381905, 29479, -304867], [35001404, -8706573, -3991111, 575615, -1848112, 2702691, -3529559, 4290201, -1905151, 3747827, 372030], [183795321, 15813519, 12293614, 930147, 1078069, -4822336, 7051450, 553481, -750330, -1798467, 1496196], [-19932144, -4526052, 3741224, -6352600, -2634420, 3284520, -11969611, -1441069, -1629994, -1214553, -1521300], [177354615, -13828438, -1298312, -4369696, -2166529, 3977534, -5391293, -2785342, 693282, -7303875, -3457953], [41457072, -16140688, 2977772, -2344944, -1712756, -3588584, -7064448, -8131400, -1033184, -4237796, -2423148], [-30414832, 1647158, 5179885, 10659924, -998158, 4988607, 1048460, 1340968, -2473827, 2288435, -2082529], [-228532181, -19257229, -4374034, 1391156, 767838, 203760, -943440, 1520925, 3075766, 5615610, 1306192], [-155301337, -54273307, -3105017, -1609665, 1288589, 4225299, 1204615, 4141954, 1403131, 761725, -2293017], [-35285989, -5896493, -10218252, -6564118, 3413146, 88501, 8395828, 152762, 653155, -5569349, -1625508], [36771680, 35520989, -3600125, 4101725, -91680, -304402, -4177193, 3267904, -2668816, 3405296, 2728506], [102043108, 1339173, -3364422, 4596808, -3650971, -1938675, -3858609, -5513086, 4109835, 149684, 485795], [-96986964, -15378964, -1372498, -15061890, -6230650, 1026446, -10252912, -5339716, -1529314, 3484716, 3894834], [80713784, -11669713, -2656699, -977455, 2799700, -3400832, 3150660, -66563, -2848010, 3584257, 6225740], [-122990131, 7787314, -1242062, -3555592, 6119029, 1457254, -4140447, 3390264, -1260334, -2082717, 569275], [90496968, -42873943, -3804105, 6470901, 777496, -4014905, 6255431, 1447073, -104191, -4716337, -771418], [-384386872, -18397743, 4677561, -6088576, 1901411, 346189, -1932976, 2729154, -130599, -14767, 2333908], [-259205083, -10270140, 5750649, 5437425, 369442, -2955621, 3156149, -325892, 89859, 1934499, 681474], [-452156694, -5753959, -6556240, 567720, 783573, 81872, -2298903, 3037981, -2637690, 2798107, -206047]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_177_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_177_6_a_a();".
// 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_177_6_a_a(:prec:=11)
    chi := MakeCharacter_177_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_177_6_a_a();".
// 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_177_6_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_177_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![-97836992, -75608576, 22980000, 14355856, -2119008, -822892, 75693, 20480, -1135, -233, 6, 1]>],Snew);
    return Vf;
end function;