
// Make newform 177.6.a.c in Magma, downloaded from the LMFDB on 24 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_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_177_6_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 := [49172480, 21512960, -75006848, 6775664, 15565376, -720516, -1009736, 8539, 25384, 143, -269, -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, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-45, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-17699688792576, 17723236279616, 5866887507152, -827674808048, -433160653132, -84243883332, 5505304837, 4536395590, 91864075, -70855647, -1428132, 350941], [1009628811466240, 320260391715264, -631726297910064, -134622184486576, 59174525644532, 10557691149516, -1674621922747, -268943328324, 18376727045, 2640430613, -67427634, -8192145], [16317679113344, 13945651094144, -9713875467792, -4941116508272, 775699987236, 346024848588, -19612654933, -8690291434, 187648889, 88399319, -524416, -299377], [-324693088418048, -292465352655168, 206836035149408, 98729745120848, -21595833573480, -7467607930832, 677570013916, 191946287467, -8347251655, -1903226618, 35029081, 5842853], [-94789227549184, -85679977174848, 80769880387280, 25092970187984, -7354011389740, -1840580628180, 202161478261, 46570135316, -2148263267, -465055595, 7464662, 1506791], [-4701282618368, -2979411689408, 3716852794288, 1335646794992, -313727728564, -108469681492, 6891575767, 2821805338, -48656271, -28964453, 53452, 100767], [-156500452428544, -669401779958912, 262460919500528, 222176054432320, -26224218394372, -15987697561584, 624811840007, 404519679498, -4582591287, -4146949925, 3300860, 14438755], [-172136514045440, -745180880860416, 458277283491552, 275119055854096, -46080636431880, -21677470373468, 1132584010846, 593347027999, -8306848553, -6527695920, 1545329, 24199755], [849314392654336, 788942374823680, -576938518125872, -275046633380496, 44920811814580, 21825571577712, -781324808245, -598161796452, 346810155, 6681309643, 47327234, -25955403]];
    Rf_basisdens := [1, 1, 1, 277202304512, 3049225349632, 108900905344, 1524612674816, 277202304512, 11773070848, 1524612674816, 1524612674816, 1524612674816];
    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 := [[2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [13, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [35, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [122, 10, 1, -1, -1, 1, 1, 2, 0, 0, 0, 1], [40, -2, 1, 0, 2, 2, 3, 2, -1, -1, 1, 2], [126, 44, 5, -5, 6, 6, 5, 1, 4, 1, 0, 4], [320, 17, -2, -4, -4, -2, 5, 1, 1, -8, 0, 11], [614, 27, -13, -15, 1, 13, -2, 6, -6, 4, -3, -7], [823, -34, 1, -16, -3, -27, -10, -4, 4, -18, 5, -5], [632, -160, -46, 5, 4, 18, -22, -9, 7, 12, 5, -12], [515, 60, -81, 44, -3, -13, -12, 9, -1, -10, -6, -10], [2882, 121, -36, -6, -18, 82, -35, -18, -34, -4, -15, -28], [-636, 336, -88, 60, 26, -6, 12, -7, 55, 16, -13, 1], [5314, 1064, 76, 54, 37, -43, 16, -93, 10, -51, -12, 39], [5107, 283, 38, 107, -48, -88, -42, 2, -22, 82, 22, -60], [-3481, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [5590, 1042, 107, -104, -49, 29, -13, 28, 14, 20, -22, 16], [9382, 639, 104, -2, 115, -33, -29, -72, -109, -29, 70, -52], [6574, 3160, -128, -319, -19, 161, 174, -15, 118, 25, 54, 234], [11241, 1034, 46, -92, -149, 57, -49, 179, -66, 195, 5, -14], [1631, -804, -398, 22, 56, 484, -9, -121, -137, 146, -90, 56], [9008, 2199, -182, 114, -74, 106, -499, -168, 106, -38, 51, -94], [14756, 6178, -195, -3, 80, -242, -41, 372, -235, 247, 57, -403], [-4947, -5921, 536, -748, -163, 563, -190, -484, 209, -177, 91, 311], [12800, 687, -458, -382, -231, -49, 81, -245, 16, 237, 129, -113], [-15255, -1792, 1433, -12, -108, -96, -126, 109, 158, -335, -39, 492], [4471, 4161, 1470, -886, -378, -532, 156, 19, 305, -586, 93, 364], [-18129, 2311, 2369, -352, 544, -28, 217, -176, 275, -43, -1, -377], [1145, 3057, 987, -1039, 452, 818, -159, -189, 112, 237, -60, -589], [-12915, -1163, 906, 874, 255, 1257, -996, 142, -297, 995, 89, -79], [-12928, -7666, -112, 1062, 542, -716, -100, -524, -494, 56, -84, -208], [-12080, -8042, -1667, -793, 765, 981, -541, 440, -1028, 772, -492, -983], [-59483, 9861, 830, 1010, 825, -35, 474, 2, -19, -125, 223, -161], [112151, 2502, 3618, -1562, -591, 197, -931, 224, -649, 471, 14, -473], [-65708, 4076, -276, 759, -1154, -152, 154, 557, 23, 1440, -149, 66], [-29724, 18539, 497, 1298, 25, -589, 794, -1811, 327, -1088, -154, 1431], [-157565, 9270, -2524, 1584, -725, 295, 631, 727, -32, -945, 327, 476], [92649, 7672, 132, 1733, -126, -406, 219, 601, -31, 402, -282, 847], [58126, -2813, -2949, -2953, -297, -91, 908, 1101, 261, -1152, 720, 1916], [21916, 350, -3877, 1615, 1061, 3225, 559, 1702, -1648, 258, -338, -2031], [-759, 30744, 1045, -2217, -986, -2476, -530, 432, 1799, -1879, 316, 1714], [208442, 2207, 427, 2637, 1692, 3624, -396, -2481, 648, 1151, -329, 610], [1854, 10419, 294, 898, 753, -2743, 107, 271, 390, 1229, -205, -775], [151859, -10742, -2565, -614, 424, -640, 1630, 1141, -1108, -1633, -335, 1346], [-63615, -2638, 730, 966, 869, -349, 713, 1369, -1402, 1433, -149, -600], [56237, 940, 1825, -4219, -1492, 3914, 516, -871, 1466, -823, -1403, 3219], [50066, 1671, -956, -4410, 539, 2021, 455, -113, 476, -1195, -2243, 859], [242876, -71886, 597, 4837, 896, -2836, 35, -837, -820, 2987, 114, -2668], [116096, -36015, -3653, -290, -1222, 4478, -1138, 309, -1444, 893, 549, -1307], [169324, -69862, -6820, 3344, 280, -1126, 1078, 1418, -1286, 734, -538, -1526], [431972, -54125, 5670, 2723, -384, -5150, -1013, -4934, 3774, -1520, -737, 2267], [174916, -15239, -10144, 164, 977, -3213, 267, -519, -378, 2689, -1805, -5355], [200601, -19875, -7777, -5435, 72, 1022, 4779, 4867, -1768, 421, 1806, -1387], [354242, -50423, 50, -230, -2225, -2617, 1065, -6035, 4074, -3049, -63, 4207], [-99187, -11281, -16364, -3097, 4103, 7517, 2162, 2548, -1071, -2391, -745, 1374], [322379, 15298, -1567, -3011, -4905, -6537, 4088, -2506, 5374, -2528, -1435, 6892], [65151, -86660, -8302, -2354, 3750, -9430, -1637, 153, 731, 376, 2810, -4038], [273050, -36041, 5374, -13440, -1539, 9241, 1339, 982, 1661, 2707, -2510, 4118], [314388, 82991, -10770, 1750, 5415, 775, -3887, -2651, 782, -809, 713, 453], [291919, -49498, 625, -913, -8042, -6584, 2500, 1065, 24, -1445, 571, 3971], [397983, -33413, 19302, -10937, 61, 7833, -3438, 5889, -7694, 2647, 784, 169], [-3738, -2310, 16758, 7920, 3778, -7228, 3308, -710, 262, 1802, -4214, 2442], [207674, -77610, 2121, -7254, -7925, -961, -7689, -4165, -2615, -204, -5075, -2641], [86298, -39627, 8162, -4292, -2458, -8798, 1275, -14275, 4843, -2628, -1574, 4669], [-368241, 59857, 22492, -11919, -3734, -2786, 4864, 6284, 6370, 146, 2342, 2926], [-220148, 114160, -12562, 6580, 2882, 7588, 3476, 7568, -8652, 8078, -536, 2858], [-78838, 54105, 4736, 21248, -2974, -6026, -10483, -4509, -2437, -4270, 1870, -1849], [-109452, 36316, 11578, 19212, 6183, -11979, 8578, 1735, -510, -1173, 2038, -7867], [-642794, 102016, -5425, 5310, 5842, 6282, -5529, 10938, -5085, 3393, 3091, -11344], [-198213, 69499, 23310, -1372, -4352, -9366, 5456, 8241, -3231, -3600, 4009, 11398], [48345, -109799, 6659, 1952, -7213, 793, -6527, 512, 2174, 5384, 1276, -1323], [-117407, 59633, -4653, -11492, -1801, -2777, -10109, -8509, -7539, 2650, 1895, -2978], [-494711, -51973, 7019, 13681, -6931, 407, 3957, -5756, 2832, -5194, 2778, 19876], [-1046813, -84893, -16654, -13148, 2795, 12313, 11360, -1096, 5601, -4137, -1237, 17511], [-331130, -150220, -6096, 10683, 3227, -2257, 5604, -1215, 2322, 1867, -6012, -10518], [233951, -199470, 1707, 16820, 3474, -12826, -408, -14743, -1016, -719, -2473, 13646], [-60568, 51731, 939, -3706, 15375, 57, -3642, 8047, -5141, -332, -4600, 2955], [-486871, -138855, -18686, -11004, 12142, 15650, 13030, -6233, 7353, -5588, -1391, 8180], [235650, -21444, -386, 2712, 3508, -9720, -6786, 8882, -19204, 5290, -3984, -13386], [1146878, -141343, 57718, 8382, 7083, -14181, -1201, 885, 8202, -9229, 4103, -10223], [48408, 238512, -52565, 12280, 6160, -11456, -1739, -3574, -3331, -4957, 2559, -11900], [1059176, -90602, 32080, 2702, 4565, -21259, -11840, 6559, -2634, 18025, 7852, -771], [-105899, -188106, -18767, -9029, -9180, -6564, 16998, -3947, 20434, 6791, 353, 10773], [-148470, 190838, 16662, -17050, 15971, 25825, -4464, 2334, -2701, 9879, -7511, -9770], [-652034, 39492, -45209, -2269, 12124, 5052, 3277, 20767, -2038, 4193, -3980, -18064], [-1216458, -18376, 65881, 1559, 18617, 9869, 1887, -2672, 12012, -18812, 4792, 9171], [-1339991, 123024, -34420, 16000, -14495, 2615, -1537, 7741, 3180, -309, -4019, -12168], [-18115, -8159, -48502, -9151, 4651, 31263, -15344, -16095, -15830, 2449, -1730, -10977], [-295552, 9909, -26278, -7029, -2961, -15771, 16863, 3963, -16100, 1239, -11957, -7068], [668076, 235611, 5446, -8156, -789, 9317, -12847, -693, 6188, 8265, 10229, -8987], [11020, -337431, 3225, -564, -12546, -33808, 654, 1816, 10019, 4621, 4280, 19780], [511022, 49140, 10458, -70786, -22197, -32807, 9910, 1916, 10577, -12725, -1315, -176], [298486, 1382, 17885, 11559, 3256, 3952, 10809, 4616, 4065, -20539, 13913, 15677], [100305, 131735, -18728, 41218, -13279, -32825, -16464, -3044, 6841, 6667, 11453, 31367], [-1092886, 189542, -34768, -16310, -5863, -3343, 1098, 4923, -13874, -24179, 3466, 10323], [-1836066, -254764, 25343, -44951, 18270, 5414, 20491, -8466, 7027, 2921, -4633, 3831], [-1718090, 195458, -110460, -30282, -2393, -11837, -2712, 7331, -3320, 31311, -5894, -28107], [363511, -198699, 50212, -36354, 7167, 5573, -27440, 2240, -8589, 22145, 7863, -29107], [-2140100, 224312, -62617, 14538, 2318, 19630, 18599, -21740, 33799, -18959, -2205, 23698], [728280, 500467, 27590, 21782, -26910, 15090, -14235, -38589, -8651, 6356, -7526, 17541], [-1900683, 303264, 64915, -94700, -13000, 652, 29356, -4245, 8228, -17105, 3683, 20750], [1558350, -32779, 24077, 60839, -11686, 10788, -2422, -13411, 9810, 2369, -16365, -13984], [-803118, -317928, 3707, 12907, -15386, -50706, 1207, -9520, 27525, -27139, 11201, 29455], [506158, 367103, 46152, -5556, -6413, -24965, -15221, 12678, -30547, 1871, -9516, -41456], [-1092430, 68850, 9530, -23634, 6018, 56684, -16090, 15477, -28855, 39268, 955, -38943], [3555068, 42186, 71473, 68407, -5835, -38901, 18851, -28058, 40438, -12986, -8452, 42407], [-876732, -223554, 80468, -49692, -1279, 36141, -27326, -18661, -14670, -11543, 17230, 23961], [3175523, -726543, 95293, -8384, 12949, 40075, 12909, 4412, -20706, 1484, -4626, 9043], [-1624778, -617189, 17244, 127392, -6838, -39966, -14637, -7301, -16189, 15766, -5192, -54041], [-2641589, 85528, -97134, 67496, -31804, -13764, -5485, -28491, 9673, 20850, -13176, 6944], [-1692145, -868697, 13705, 37286, 60497, 58515, 2401, -2396, -4310, 17436, 430, 1055], [-1623408, 156692, -175641, 29203, -5948, -1664, -13415, -33593, 31694, -7631, 880, -17332], [-1782099, -792999, -18402, 23136, -15417, -25607, -2752, -17628, 31153, -53689, 19683, 26601], [1012083, -1225678, -8174, -34922, 3774, -22086, 11213, 49435, -9423, 20272, 11494, -28616], [-795314, -460697, 26819, -51295, 12798, -38810, 31798, 10413, 5956, -435, -23239, 6], [810592, -823393, 97270, 30186, 9540, 29060, -5323, 7283, -16389, -7590, 13628, -8855], [1466919, 223996, 52079, -23106, 20132, -38056, -35294, 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-1191385], [27768610, 101484, 197820, -1418194, -269906, -2786766, 521286, -167413, -170575, -755020, -14813, -646445], [-18324389, -1261727, -6137873, 2107019, -745048, -1207278, -2737767, -2407703, 213330, 52493, -269698, -99297], [-117751325, -9521836, -2459417, -1064395, -982436, -1912524, -2335656, -1207359, 85984, 934817, 306403, -2256691], [-15749433, -26166902, -4778703, 288586, 505137, 938437, -1977484, -453972, -1442830, 2294918, -714731, -1792161], [-138873138, -2315813, 3872003, -1506893, -1152919, -505379, 1222766, 710890, 303402, -727540, 407229, 3894855], [-179471227, -36768999, 1593264, -632092, -1318261, 915943, 436128, -345084, 705947, -463893, -393099, 483319], [-94570992, 11972146, -1241240, 351592, -591870, -1051736, -817746, -1438265, 459417, 760658, -198819, -55389], [192922279, 8717356, 1998229, -555196, -1068006, -701598, -27638, -204191, 760804, -655229, -873837, 1110700], [309268680, 14770220, -6057716, -1552436, -691928, -972448, 1484944, 387408, -120780, -1536464, -252184, 1563948], [141178612, 13395742, -3233780, -3642375, -343560, 462702, 1611740, 458101, 1259369, -939630, 937539, 2625504], [-114239769, -14963129, 11060323, 842941, -347754, -1791652, -413403, -133197, -88426, 60859, 434930, 601429], [-31945411, -41380453, -1067248, 3657685, 1111163, -796951, -10794, -370848, -1096047, 804865, -567505, -2039348], [54107195, 8082213, -3517086, -2179140, 1907560, 2185598, 1471702, 1846449, 553601, 270004, 33443, 491214], [213144046, -11466293, -5765894, -1576346, -726199, 1815505, 699771, 35330, 446569, -716093, -627554, 1703832], [150455362, 12562979, 4936527, -3899492, 1246478, 953048, 2600902, 1651082, 246381, -509865, -412916, 778666], [-22911468, 5608220, 401414, 272440, 1083346, 3357790, -1057018, 1514620, -1944844, 1191120, 22428, 26532], [-44642406, -15162197, -6171992, 3262508, 437459, 3691, -1271523, 1869421, -1348996, 1453881, 130325, -2924075], [89961, 840052, -880187, 2519052, -1150058, 1398494, 197968, 1296501, -261000, 233387, -424607, -1801790], [-16451350, -18795231, 2461275, -3895425, -1832409, -48397, 416356, 1100242, -773704, 11554, 725081, -1501825], [-19398272, 3910163, 1914120, -2662220, 764722, 1726930, 2243073, -217432, 1289112, -1046532, -157825, 370314], [28914477, -23811650, 4506954, -443268, -1761507, -3139653, -159775, 2089693, 1439086, -1024627, 288131, 1537030], [9772000, -11457245, 523036, -568413, -1543682, -133044, -1481707, -2668918, -325162, -504616, 110755, 469131]];
    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_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_177_6_a_c(:prec:=12)
    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_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_177_6_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_177_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![15131776, -78147904, 88035904, 12144672, -24399488, 2586728, 1174642, -191819, -15182, 3917, -49, -22, 1]>],Snew);
    return Vf;
end function;