
// Make newform 1075.6.a.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1075_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_1075_6_a_b();".
// 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_1075_6_a_b();".
// 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], [-40045363612, -996337120808, 224666692021, 59540816201, -14422814397, -1005840413, 239748231, 3451179, -1188163, 18033], [-192111505364, 762684015384, 64822510771, -20833059243, -6546797939, -397671097, 116173337, 7832247, -594245, -24107], [-912070606876, -418161182816, 287212616929, 54235511681, -15768164889, -2591602757, 213763899, 35600547, -820999, -144543], [2423789997252, 1590463778000, -937883085431, -220849424171, 39727817087, 8979038055, -421392509, -116809809, 1144977, 474917], [-5839603776708, -1656152504528, 1003571811959, 220849424171, -39727817087, -8979038055, 421392509, 116809809, -1144977, -474917], [-7896805510540, 768287694064, 1606414562725, -180644132719, -78913681917, 9333970747, 1407118071, -155012637, -7898035, 808641], [-30463422181444, -17120027667600, 5229545463959, 2673777652347, -189530468575, -100861721975, 1411070749, 1269871233, 307343, -5132245], [-18517909728476, -11776641778800, 3165045014041, 1432035207537, -122440956545, -50464104709, 1209002243, 612520467, -2695583, -2389487]];
    Rf_basisdens := [1, 1, 24633272448, 24633272448, 24633272448, 65688726528, 65688726528, 98533089792, 197066179584, 49266544896];
    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_1075_a();"
function MakeCharacter_1075_a()
    N := 1075;
    order := 1;
    char_gens := [302, 476];
    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_1075_a_Hecke(Kf)
    return MakeCharacter_1075_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], [-3, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-8, 4, -2, -1, 0, 1, -1, -1, 0, 1], [66, 16, -3, 3, 5, -4, -11, -3, -5, 2], [-192, -12, -1, -3, -5, 0, 11, 3, -7, -6], [-402, -55, -8, 4, -7, -8, -25, 4, -4, 8], [-241, -15, -16, -4, 20, -10, 10, -4, 13, -8], [-214, 44, -28, 17, -17, -25, -23, -13, -29, 13], [749, -2, -36, 18, -20, 102, 47, -10, 18, -36], [-500, -89, 50, -6, 27, -16, -87, -14, -56, -2], [87, -286, 42, -51, -30, 127, 280, 141, 168, -63], [1037, 375, 93, 85, 37, -4, -251, 67, 110, 76], [-1849, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4628, -31, 251, 149, 82, 48, 106, -45, 15, 68], [-12425, 263, 146, 273, 101, 7, 341, -63, 186, 95], [10469, -825, 407, -28, -310, -125, 1032, 186, 185, -315], [2554, -2926, -66, 237, 264, 139, -5, 45, 126, -213], [954, 682, 713, 333, 477, 542, 9, -317, 505, -416], [1374, 500, 168, 278, 482, 90, 392, 46, -70, 46], [-5706, 518, 318, -129, -312, -1115, -551, -141, -186, -135], [-9232, 224, 183, -648, 328, 73, 1250, 8, -242, 29], [9905, -757, -538, 87, -299, 9, -1701, -117, -542, 853], [-7070, 482, -366, 393, 230, -1305, -1601, -659, 324, 579], [-10134, 1604, 448, 157, 1075, -13, 771, 529, 1145, 605], [57671, -437, -644, -1397, -609, 1659, 595, 1235, -288, -53], [4169, -3129, -899, -1067, -1309, 4, 2043, 1341, 210, -1460], [-10013, 5797, 645, 1286, -362, 627, 1302, -700, 1031, 841], [-24218, 6452, 1379, -257, 689, -1630, 3459, 593, 1103, -84], [-38640, -6012, 1566, -1154, -1038, -716, 3656, -446, 714, -360], [35473, -16904, -1715, -1168, 2241, -467, 269, 470, -1137, 637], [-6113, -8112, 2922, 803, -1546, -1067, -4504, -25, -278, 617], [-14984, 1570, -196, 118, -3484, 894, -1166, -730, -1522, -1170], [-31185, -7179, -408, -2281, -617, 5399, -2569, 2387, 2102, -1877], [13797, -22524, -3944, -1271, 622, -1551, 7560, 2509, 2502, 1351], [-215278, -13042, -2748, 497, 1456, -2871, 3357, -811, 74, -1019], [-109511, 167, -624, -502, -662, -6072, 1938, 488, -373, -2438], [107303, -16937, 1250, -1673, 680, 1011, 14119, 4187, 4463, 1257], [-6333, 8825, 2034, -3013, 677, -2211, 6899, 4891, 1196, 1033], [-153878, -15082, -3546, 426, 2748, -6480, -3144, -5466, -4638, 4896], [6537, -17743, 8224, 4569, -2510, 471, -1727, -1223, -985, 1325], [79727, -989, 3297, 1610, 1802, 2695, 5450, -896, -2583, -1131], [-103980, 3280, 3018, -4566, -1280, 1064, 19898, 2782, 6664, -2416], [-157214, -19404, -1776, -6621, 1575, 3801, 1305, -33, 3621, -4629], [-317253, -43887, 2355, -2610, -3178, 3021, -4004, 5704, -1221, 363], [134480, -14532, -6564, -4930, -4026, 5198, -9350, 7374, -3530, 422], [-100579, 35474, -1854, -2577, -3824, 5329, -16252, 491, -3014, -1651], [155752, -41536, -6918, -2819, -7640, 1235, 1957, 3029, -5916, -753], [93637, 34815, 10904, 5016, 11482, 7794, 8298, -5592, 16981, -4388], [247684, 17224, -9241, -6861, 5271, -4834, -1863, 8785, 2713, -1748], [-19488, -10516, -5440, -2488, -4004, -10592, -10946, -9248, -12256, 2628], [40212, -17324, 9025, -5368, 2112, 12855, 37974, 12248, 12814, -17133], [110714, 6110, -4468, 6588, -3810, 5752, 4826, -5456, 5908, -6644], [-284734, 37098, 13637, 9403, -5721, -6276, 531, -2123, -10141, 9018], [-215922, -34524, 6900, 8937, -11670, 7233, -21663, -2487, -6654, 405], [145532, 28106, -538, 854, 9994, 8192, -40592, -7714, 5452, 5116], [367466, -60846, -2489, -6711, 5157, -1140, -22525, -10285, -14359, -3298], [-30297, -74439, -5301, -9469, -14657, 10688, 43073, 6983, -5464, -7772], [-116349, -31433, 6926, 13333, 16716, 4737, -42949, -17249, -39, 4271], [232585, 39806, 17755, 688, 3175, 32139, 18603, -194, 13503, -7211], [3537, -35651, -9760, 22151, -5229, 3439, 7285, -10457, -18298, 6603], [-365083, 99903, -1481, -1264, 20622, -12049, 3834, 6562, 6233, 617], [344577, 70723, -17440, -15217, -10569, 5947, -11909, 10411, -25876, 7995], [-305058, 120016, 6013, 8321, -17871, -30990, 19563, -385, -1445, 21768], [78406, 88826, -6960, -9733, 4360, 8799, 11529, 13583, 3542, -1841], [175598, 52856, -12961, -9707, 11961, 2476, 8325, 5243, 1447, -3758], [-1257466, -58944, -12018, -24013, -19566, 21257, 38857, 4959, 16870, -9691], [158085, -64070, 5735, -15756, -12861, 13043, 31899, 17458, -473, -395], [515925, -67831, -3468, -1777, -16630, -32535, 65915, 8023, 16607, -23681], [342551, -22708, -24366, 19778, 15148, -14892, -89121, -27886, 4640, 28222], [328910, 166974, 535, 15159, 9353, -2310, 55905, 3849, 17081, -292], [72662, 130309, 25268, -10546, -15973, -25746, 41829, 9410, 28014, -17752], [-122630, -169795, 15303, -10187, 6262, 16916, 42580, 37619, 22863, -1332], [-296857, -5203, -20918, -11634, 18612, 15106, -6210, 5276, 29699, -27940], [-1389884, -20534, 6273, -31431, 11581, 13826, 27855, 28223, 12553, -1504], [545996, -174472, 33100, -4390, -13988, 40782, 3290, 12898, -10196, -31422], [73562, -142144, -628, 23496, 6828, -24156, 27520, -48692, 16480, 35944], [-420713, -191229, 46849, -19564, -13754, 31865, -70940, 7174, 21121, 12339], [704307, 247488, -37357, -13204, 9941, 33591, 56931, -22198, 13251, -3079], [-529374, 124282, 8062, 26871, 9862, -30847, 11057, 14203, 5964, -4967], [1938054, 45452, -17928, -33315, -7700, -10163, 9479, 41373, 20216, -16211], [1464659, 326161, -41244, -37306, -1186, -40748, 118706, 10408, -19755, 28074], [1242853, 126510, 24395, 44896, 28185, 5587, -26361, -10558, 5845, 12379], [2633606, -30248, 34952, -7817, 12002, 9411, -72521, -5917, 23474, -11229], [1382858, 208335, 25588, -25864, -55439, -39496, 17077, 2548, -35006, -8730], [-1836803, -92857, 3321, -44524, 4898, 46145, -24100, 23670, 35241, -49125], [348906, -22796, -7406, -17947, 4140, 48267, 78237, 5561, 17344, -40389], [328100, 384026, 21424, -17571, -5616, -4967, 24441, 27977, -49930, -12499], [-1108359, -183787, 19483, 7934, 17618, -55459, 190810, 9188, 9263, 16823], [183356, 279340, 42748, -49549, -4196, 11551, 12243, 27619, 21160, 4455], [-2421283, 37888, -11396, -34637, -53760, -27633, 15106, 26387, -7272, 607], [1344218, 71240, -45482, -179, -36021, -1279, -82523, -22337, -121337, 79527], [-1020451, 444145, -41220, 20282, 66018, 2392, -27210, -47256, -35, 58718], [957769, 191241, -43608, 43435, 53700, -45675, -181897, -29565, 10861, 41495], [1744387, -231034, -55732, -1574, 15542, -72010, -21267, 19010, 25880, 36366], [615702, -218582, 14636, 69432, -1162, -91044, -71620, -58228, -33104, 17848], [3029587, 338515, 57322, 5445, -863, 46515, 21385, -623, 24158, -26161], [-841734, 139822, 65700, -4481, 98356, -51969, 8845, 5039, 62098, -57877], [2474363, -64044, -12914, -15116, -2590, -91190, 85137, 6312, 10434, 46630], [111003, -430341, -36188, -35803, -64399, 21017, 55757, 71229, -16162, -37987], [46477, 285045, -97768, -34955, 32909, 20473, -184489, 72389, -16324, -49611], [141542, -113504, 42403, 22399, 85325, -28902, -61033, -126171, 48195, 24448], [-2279728, -232982, 32887, -128719, -101059, 97036, 243031, 86827, 13161, -45278], [483429, 396649, -10184, 49173, 79407, 26845, 67253, 13329, 12058, 77781], [-2071509, -476246, 74182, 43744, 61348, 12798, -65345, -17612, 63998, -2830], [2366424, 311598, -76604, 15923, -20574, 6263, 161139, 22511, 22368, -41249], [-1737412, 199292, -9618, 10580, 3934, -7838, -105816, 26064, 107590, 16814], [-1973346, -659324, -26466, -68175, 112774, 249167, 179249, 66865, 35698, -55585], [798375, 703822, -94933, 52882, -41635, -93059, -374981, -89556, -113447, 65497], [3470320, 160082, 75582, -81894, -126930, 44188, 263110, 23610, 42960, -76932], [-1671266, -14102, 202350, 24040, -556, 51654, 120492, -145844, 15526, -25690], [2613957, -208921, -73021, -126, -102838, -160547, -49688, -61288, -202647, 149219], [-2695582, 510552, -70354, -110045, -14849, -6621, -208591, 41019, -38671, 4101], [4748512, 526724, 16820, -119540, -108928, -520, 79668, 63980, -69388, -61200], [-775582, 581450, -65908, 102568, 31194, -36916, -16508, -105260, -8772, -11020], [-2117482, -614816, -33400, 40161, -49646, 2005, -179943, -144923, -74094, 81305], [3223319, -326643, 60721, 84844, -58342, 188821, 1164, -17430, -53757, -55073], [2368600, -1098772, 88854, 74530, 83736, -124580, -100306, -49142, 75840, -1392], [385743, 119213, -117206, -66132, -163470, -250860, 147464, 8534, -106039, 26714], [761086, -1457394, 92967, -19587, -31941, -8380, 96417, 52775, 103759, 54674], [-702467, 117811, -65220, -26965, -6725, 145075, -79475, 38315, 32660, 1195], [3702388, 166750, 2334, 132703, 9044, -78099, -69009, -182273, -82610, 50981], [-3838237, 1786049, 19804, 26976, -27830, 67630, 27400, 37226, 5377, 27660], [3541334, 776350, 34923, 40359, 36679, -64950, 343379, 101285, 73023, -51624], [1956750, -1054456, -58736, -105393, -102768, 274999, 164829, 174887, 113504, -207409], [2687137, 135093, 23681, 64278, 230602, -96609, -329178, -126612, -70637, -40679], [-2456590, 667416, -58365, 103909, 194181, -62384, -62899, -158601, -33673, 164302], [-280985, -519772, -75764, -193703, -254168, 135297, 290196, 114409, 4488, -119357], [-37258, 154838, -73872, -65480, -86260, 74012, 186704, 13240, 141970, -26056], [-7905421, -35584, -163292, -14379, -37290, -174095, -87174, -92143, -256102, 153371], [-3796111, 219302, 8822, 119571, -45228, -242255, -225544, -268361, -272582, 207565], [2725460, 163976, 10950, 63386, 7932, -269180, -7178, -113534, -68552, 41984], [4957160, 1453606, 20532, 51277, 7370, -181143, 405051, 91165, -61652, 40565], [-8803613, 30166, 29822, 82877, 61438, 122971, 181088, 102117, 240, -39235], [279052, 1030440, 250578, 71194, 193614, -110844, -95056, -45174, 95590, 116840], [-38232, -1044887, 177395, -11017, -170644, -47654, -299196, -62407, -123275, -112268], [-7127147, -279008, -106568, -22848, -162786, -79656, -731893, 4076, -170230, 175586], [-723401, -645635, -16443, -124380, 110822, -154519, -231920, -49034, -92653, 115735], [-5599475, 814755, 169695, -57236, -82638, -179205, -140050, -37362, -84335, -99511], [-11134406, 278569, 19791, 185411, -10484, -58470, -235050, -81763, -38395, 30892], [2258593, 631389, -16570, 238613, 238252, -156511, -391407, -307863, -229359, 241067], [8830924, -1037788, 60711, 98757, -107709, 138984, 44931, 31227, -91863, -23190], [971648, 688858, -153463, -103389, -231733, 142304, -53671, -44551, -217197, -66466], [-583776, 1144682, 264428, 301470, 259572, 1790, 348616, 12258, 115898, 110742], [-2395145, -369734, 5706, -365235, -173726, 124991, 155220, 289373, 62200, -165247], [-10052778, 1099252, -340438, -29611, 236400, 91715, -275965, -54771, -68216, 163615], [-5933829, -515249, -168670, 13339, 230457, -12687, -1081513, -204317, -191154, 220085], [-2755913, 556283, 83362, 321440, 32322, -398576, -179154, -180636, -6255, 120742], [5427325, -49273, -90592, 52084, -43714, -82978, -92274, 63860, -1759, 221760], [6104608, -1243502, 86004, 34804, 278042, 316160, 181266, 193968, 173252, -234152], [-5181279, 799829, -108650, -114317, -255257, -70123, 136059, 81379, -363994, 15977], [928725, 1140808, -108977, -50620, -47447, -116973, -4231, -50510, 148759, 148293], [4520130, 213096, 105439, 5511, 337611, 16578, -498115, -207403, 86909, -93640], [12342852, 302304, 34986, -43422, -37986, 206220, 1311066, 262914, 119622, -143988], [1246641, -1430235, -21000, 289993, 174257, -300035, -159689, -514331, 13348, 48761], [-2861008, 1468534, 37126, -130493, -35506, -229155, -437647, 203235, 153156, 14005], [-6941411, -1542678, 65951, -106642, -113723, 30485, 10977, 91200, -7083, 13229], [-14573764, 519386, -202402, -30090, 156328, -72300, -104682, 197046, -39890, -44828], [5996194, -1131770, 99426, 156683, -155494, -488303, -626035, -28865, -139332, 141993], [7825278, -101390, 27275, 421755, 47363, -241050, -1026607, -275311, -45665, 164536], [3910187, -574753, -353926, -73725, 187125, -160071, -580763, -102857, 24094, 363169], [-6236972, 1004184, 383248, 403925, 171540, 131613, -628197, -172543, 10024, 241281], [-3651125, 679957, -324747, -168161, -7021, 231426, -133455, 362707, 229828, -141538], [-8199338, 1323396, 247911, 87159, 157053, -446242, -560427, 67709, 89215, -70084], [-6784670, 315801, -19489, -103293, 34588, 189642, -36342, -228963, 262117, -2636], [-1737486, 594992, 147726, 445547, -171088, -366939, -743719, -319409, -369016, 229945], [6869594, 1267446, -146082, -133373, -174858, -22187, 323483, -253041, -304432, 116305], [6546867, -1919846, -137242, -56210, -178326, 177084, 543257, 231070, -36600, -138262], [-3970926, 40834, 45170, -44948, -216832, 809406, 205942, 270416, 36314, -57498], [-5823480, 1396086, 12672, -5912, 179140, 174196, -1093964, -56064, -179406, -34932], [1638901, -1531414, 74778, 70203, 444088, 520773, -253528, -272473, 87786, 20817], [5764445, -934285, -119480, -402554, -106186, 307724, -20438, 453300, 48051, -298658], [-9054730, 2058598, -187686, -133455, -372210, -290337, 596433, 143829, -437760, -182409], [8832102, 82200, -556586, -241359, -468963, 147501, 520583, 215945, -215293, -320893], [-6062908, 1593240, -155890, -197654, 99746, -404432, 58682, 418526, 390086, 69576], [-2181170, -3655386, -141533, -131266, -275812, -373581, -305702, -375468, -621266, 365583], [-7754415, 1193268, -302182, 275755, -34314, -366107, -560430, -180641, 44958, 459593], [6551393, -801491, -9642, -78795, -501764, 442413, 349199, 483059, -240165, -293649], [-105855, 861073, 203183, 101852, -283206, 282391, -1312178, 80722, -159661, -118455], [-9689455, 1790527, -79662, -429928, 240164, -139712, 396476, 353002, 369633, -402546], [-4491986, 1544096, -222486, -203615, 101668, 322175, -73181, -201247, -100080, 114411], [11898869, -2381035, 701657, 321324, -5430, -88683, -656810, 223726, 494671, -322049], [-6583116, 1926800, 444460, -26941, -50228, 417999, 253253, -471737, 131632, -57573], [-6462264, 168486, -629274, 4695, 129762, -660735, -310581, -395601, -148632, 283761], [1679534, -93634, 528416, -76927, -294598, 600545, 350757, 349093, 245608, -476231], [-5084278, 3641980, -422870, 637936, -126210, -910878, -1311332, -1034848, -495798, 829066], [5724073, -2131567, 235638, 146861, 80428, -753603, 140559, -656029, -220777, 123367], [-14577157, -141268, 118194, 154706, 6842, -74172, 207031, 396478, 484410, -47712], [-17459202, 2734111, -221135, 7517, -148880, 28294, 596550, -57797, -356729, -94212], [-6947052, 567952, -224966, -155455, -128646, -920997, -583335, -126727, -180902, 397227], [-7927949, 2583168, -44387, 6150, -125321, -482709, -378063, -128240, -283215, 169585], [-21817277, 1277863, 147760, 419855, 326154, 183449, -83085, 22759, 86741, 264923], [-2290989, 3795645, -28643, -219272, 248906, 732829, 1121700, -58302, 662343, -217721], [-2843295, 3102611, -83444, 171012, 573524, -271014, -1156416, -618786, -388759, -120544], [20249374, 1933040, -633564, 259049, -15088, 632333, -290497, -312519, -19800, 376725], [-858840, 1273368, -187830, 302160, 624330, 177330, -246726, -300696, 111210, 230226], [5634866, 2997836, 47838, 244509, -766865, -197723, -1105693, -47731, -428147, 96503], [-327846, -2022336, -162883, 136215, -865727, 329472, 515651, 151665, -587329, -74934], [203052, 2197944, 658105, 210058, -374660, 245973, -831030, -395852, 48240, -239171], [-1300586, -19578, 487068, 268363, 130710, 69399, -355897, -635721, -211220, 286487], [2163205, 606967, 482884, -287469, 251791, 709923, -591527, 119263, 535784, -252661], [-185242, -5371194, 117576, 90960, -117942, 76104, 714864, 305988, 207864, -102084], [-6513458, 1455096, -350450, 283422, 289822, -507708, -1226082, -480194, -472650, 357316], [11940140, -2151106, 664748, 228916, -615900, -649852, -329108, 164644, -508326, -75244], [-7713251, -1280556, -400658, 58298, -286026, 1009032, 736821, 253790, -391754, -57924], [-46149953, -2313992, 747156, 551457, 509826, 237733, 193242, 9013, 868958, -33689], [-2021136, -5009134, -156668, 352410, -412944, -13886, -2407728, -551518, -261502, 705482], [19503139, 3557431, 175828, -179951, -86971, 220357, 197411, 487285, -79246, -164815], [-28022468, -4439650, 652992, 148782, 148548, 36954, -856470, -555750, 865278, -152610], [23439043, -1658377, -523148, -77867, 575959, 233933, -540393, 396829, 650564, 217793], [-24316924, -5897034, 117706, -495602, -472192, 778504, -291786, 656838, 485898, -415040], [-2744297, -5793223, 526476, 509277, -163773, 699945, 403023, -480579, -198102, 233337], [-16188631, 4335875, -899136, -123895, -32335, -25959, -974735, -260147, -29502, 807153], [17416842, 1698804, 470023, -34987, 460561, 324724, 635843, -638317, 425907, -555098], [-8402636, -2754808, 48746, 371529, 266054, 402355, -1062199, -1235187, -113238, 604451], [8779646, -4051123, -35584, 162666, -212895, -968042, -745459, 73042, -771128, 216718], [-14502040, -2794864, -449952, -381099, 173642, 609769, 1357761, -667247, -479134, 181141], [-14495750, 1631544, -881332, -290166, -223454, 24666, -1992006, 272274, -723050, 530866], [45305184, -2650552, 182014, 144997, -145636, -383153, 351041, -635039, -13924, 692111], [-1181204, -408930, 753319, 31589, -251845, -535884, 803257, 179223, -593225, -231438], [-39563887, 510805, 420984, -210899, 840800, -33237, 354795, 190723, 177619, 251861], [-3383401, -4688037, -148227, 1184330, 407250, -779365, -1893644, -827700, -270779, 670049], [-3514954, -2157646, -1145292, -513725, -833780, -660837, -818915, 1088831, -354806, -392437], [-8734112, 1366646, 633384, 897981, 927457, -344393, -2252803, -210673, 478459, 314333], [-4077049, -2620447, 747728, 468984, 330222, 1210854, 2902906, 514816, 1492435, -648332], [-13857933, -241717, 115486, 327973, -288603, -85849, -2192155, 103289, -38030, 688047], [2723612, 212088, -1144843, -340523, -125855, -1055514, -278089, 639915, -809413, -233652], [-2329204, -2120698, -903574, -198593, -324022, 849733, 789457, 178851, 630508, -326387], [15679580, -3084314, -463816, 119642, -666142, -401582, -1173012, -438634, -802688, -79754], [-42615703, -158321, 164298, -410232, 409824, 169140, 1699494, 208620, -309657, -324774], [18104965, 981776, 627474, 1235372, 1060372, 314478, 339015, -334540, 380108, 140860], [-18812277, 1424141, 318904, 20033, 216561, 50901, 1080025, 202433, -911832, 160333], [10691904, -4598058, 461750, -763353, -843699, -118725, 1446187, 294649, 406219, -565607], [8003864, -959246, 393224, -331462, 63738, 499170, 1570694, 830962, -112152, -114178], [-3988787, -3019879, -148992, -524595, 442921, 1125097, -681241, -268543, 27040, -666199], [-7289116, 1054499, 148421, -84593, -799726, 1575340, 826102, 422577, -413543, -191432], [-28327426, -2004400, 862298, -39802, 753414, -147192, 146778, -928394, 39374, 523404], [-26259499, -6125041, 155316, -501243, -379869, 180149, 1807719, 418289, -834668, -554587], [-42399021, 2668579, -168528, 1277317, 197392, 135507, 501697, 34901, -151781, 38729], [17788284, -887394, -644262, -185343, -1111717, -1224531, -393371, 177871, -1563015, 590127], [-3712023, -5907794, -142340, 90245, 491478, 309993, -3066846, 130253, -385784, -598743], [-13313273, -505035, 20396, -291694, -240902, 915140, 338760, -957654, -289797, -525598], [5024529, -886699, 875036, 853006, 1342886, 1366100, 785442, 133172, 1804301, -509718], [-27439643, -1152985, -1225283, 535394, 42838, -1695525, -2626084, -799244, -1407899, 174285], [20154056, -5682244, 523034, -242214, -395534, -13092, -931380, -325398, 22846, 688120], [-62969702, 1008626, -934871, 32634, -1021312, 1099781, 964056, -798946, -757584, 56341], [11316228, 3676908, 1127672, 456992, -154304, -226240, -318000, -877920, -477156, -164224], [17537301, -3413193, 1045452, 241295, 95093, 297635, 3639767, -796533, -573582, 512127], [57744762, -7998022, -90527, 432599, 667719, 341436, -250997, -330739, -816897, -420650], [-17841185, -1012295, -1051871, 70275, 429015, 185154, -64555, 838559, -749524, 764534], [8979383, -3041015, -107219, -94534, 692038, -505265, -339108, -852944, -1267973, 550713], [30743574, 2771846, 136497, 1273021, -69165, 475658, -1108813, -890437, -745417, -48512], [-60099473, -1114787, -244036, -716513, 977099, 1500315, -907937, 868499, 714338, -921717], [67677534, -4539736, -20372, 568806, 350508, -1048214, -1677318, -335374, -638656, 808874], [-16731743, 2685202, 1041802, 827414, -841482, -138880, 636111, -733394, -87460, 363482], [-46458907, -1833990, 544788, -505095, 299270, -264035, 981286, -57055, 798644, 325549], [-8470615, 4818983, 1084260, -164806, -434024, 154056, 4346370, 1186460, 1711913, -1330706], [12044502, -7454686, 111051, -432627, -1070793, 836440, -1043411, 667503, 678975, -870090], [-36466260, -1758808, -396866, -1350777, 453934, 1291917, -192743, 411103, 543878, -189355], [33575219, 6052801, 216143, 1198229, 1241687, 77938, -1332987, -427825, -111290, 157234], [-39259596, -7978146, 730012, 802412, 314946, -1033360, -82842, -2046688, -337344, 681916], [-4268913, 7617819, 1257626, 435257, -421726, -2586087, 1335129, -107179, -695223, 831347], [34175090, 63760, -53784, -108236, -187408, 1365828, -3577260, 116692, 762760, -200008], [-18825941, 2459281, -937779, 852226, -907570, -1945641, 116208, -624524, -2067137, 1300865], [-41078266, -3098906, 53020, 1062715, -648384, 3219, -580883, -1583893, -774954, 635467], [-10547861, -8558636, -153280, -38319, -871380, 274409, 5075862, -459791, -295352, -250655], [-28982235, -4189664, 1136462, 569753, -527328, -402241, -5628940, 264681, -595664, 194905], [110395159, -3426491, 786761, -274247, -538825, 812436, -595583, -285193, 410282, -366780], [14933012, 1966816, -1142910, -1716342, -591786, -1237876, 1377314, 1454670, 893466, -1589424], [-79874094, -7063548, 1355575, -1026103, -826163, 1365544, 5112185, 1059779, 1093611, -1358738], [-89704349, 1902547, 1228820, 479607, 1125585, 392531, -3317445, -112685, -614678, -104093], [-7094549, -5277813, -1038834, 2349585, 267030, 598389, -1589091, -1319355, -1476987, 902079], [-2500324, -7959428, 496084, -388699, 204478, 620917, 1311147, 541873, 432226, -917019], [18292248, -10028748, -563594, -1678377, -1638268, 107217, -541985, 1662171, 660520, -1489671], [-33493190, 5011245, 711759, 881433, 721548, -2994192, -2659980, -774885, -97575, 713418], [-2754676, 7648178, -2490118, -1437563, -444482, 1394843, 1449383, 646845, -217304, -1359545], [32872806, -6302204, -955817, 163711, 521207, 1332050, 3106913, 1106897, -149399, -409092], [-19123179, -2684827, 1336816, 726965, 256146, 1587543, 1677163, -883657, 1210293, 289477], [-16159395, -1220392, 903259, 2611996, 776133, 13799, 1850729, -1189294, 372331, 1255785], [-52954115, 2830493, -397416, -607647, -1679440, -2211573, 6665413, 912817, 619341, 568677], [-50850276, -1002789, 43850, -912462, -2448249, -730112, 2360485, 509282, -1397112, 724022], [7516957, -1578532, -772588, 1097719, 202136, 713111, 4810200, -211489, 843512, 650685], [30334489, 9603538, 1298819, 1254940, 763385, -125505, 1371715, -933778, 221993, 348327], [-31693167, 3347772, -279056, 1548682, 2075198, -517118, -3961743, -1405842, 193734, 1503796], [-13601823, 1696104, -81668, -1059098, -1011000, 13498, -4047647, -204130, -69560, -821358], [58019816, 1375734, -470976, -272161, 1930120, -340533, -619899, -938269, 318290, 347023], [-38757115, -1604289, -192659, -241211, -468637, -1712264, -1900515, 87867, 442680, 80308], [53973919, -4269784, 1848468, -398760, -192190, 440100, 279181, -780068, 276486, 1013370], [-19573240, -5339848, -887578, -1437741, -194334, -91679, -4535413, -1443425, -671906, 916861], [116289659, 6244725, -1534559, -875133, -142393, -1621458, 1135071, 555993, -430054, 543662], [79322743, -5614923, -226922, 26615, 659591, -374631, 5024161, 739275, 108706, -476295], [22566287, 507549, -1696048, 902713, 192997, -1144611, 724805, -227187, -2099584, -311667], [29670842, -1233258, -384401, -422734, -1273872, 932867, 387390, 412840, -1250158, -394633], [59741113, 6659406, 3398486, 857939, 1299900, 2032097, -2036440, -1545537, 966106, -1053371], [22848805, 9845389, -2515940, -1413939, 389409, -426019, 1324081, 438509, -846424, -1137835], [-11927948, 9683788, 1371869, 209099, 661551, -2332472, -1755319, 116897, 948041, -561522], [36240055, 8275365, 1085189, -926923, 1437271, 63704, 4522179, -197415, 502776, -1543084], [-36797634, 8896702, 1141163, 870691, 237367, 1026454, 223097, -4967, 860575, -1354328], [-12460207, 7302665, 6643, -1980610, -980762, -68715, 7880516, 1558384, 692995, -1481217], [43370998, -3982960, 495636, -165458, -1741520, -4635646, -4211636, 249350, -1539288, 1217730], [-61173722, 18210297, 1557640, 1291748, -231263, 929960, 4444111, 41068, 2765644, -811680], [-6941981, -3560485, -4082186, -714779, 1409142, 2671149, -2545061, 1407131, -875169, -870809], [8703081, 932171, -1657118, -2452453, -388605, 800949, 3750547, 913403, 1783632, -577415], [-74198251, 5715481, 2013574, 711837, 1240189, 1145363, 774823, -1485291, 1471692, -2479289], [-109644847, -1054590, -807344, -1294198, -242942, -707854, 1282387, 631402, -1971584, -1623702], [-18885194, -4449758, 1213428, -1207058, -765556, -417498, -456048, -806138, 915046, -1639990], [12263056, 16615238, 285471, 164301, 241849, -193512, 114827, -500617, 979329, -177582], [-37799703, -8990453, 1376962, -977060, -2235860, 2267292, 6144316, -142154, -298123, -3592774], [-86374546, 6639106, -1290554, -1064201, -1833996, -1754355, 4480507, 1081751, -1479318, 221677], [29011279, -8717011, -1487886, -552490, -28328, -1148070, 9388230, 759452, -341665, -1036724], [21417830, 3132492, -1909514, -1863670, 326202, -797848, 6514620, 160690, 1225670, 89312], [13881478, 1612722, 1306640, 1593927, -1532674, 903687, -1075831, -3038869, -1059704, -159317], [125517248, -12170918, -105498, -90843, 1835509, -494067, 2937719, 1148765, 2361813, -1378929], [78836715, -2228207, -1186960, -498810, 1616268, 614864, 4835712, 2026558, 680845, -1716878], [158503400, -8787985, 946186, -2471716, 154403, 2508882, -1198985, 1460796, 1102840, -323784], [58356215, -4143299, 1251489, 575127, 2942225, 1418054, 495115, 609157, 2439224, -2126558], [-97467711, -13160771, -2400225, 876974, 2247746, 2257393, -2625436, -1484476, -139193, 836207], [77862129, 6454701, 987999, -678338, -1048434, 317641, 9910508, 2172700, 2005635, -2252661], [21070253, -15228392, 1581747, 674294, -871887, -3576647, -2709657, 454980, -2375153, 762347], [24961978, 5358626, 1432082, -122194, 67804, 1331616, -7789448, 1185246, 1109946, 670136], [-34385968, 2928974, 2129811, 83765, 370765, -2045220, -1278873, -230629, 511961, 281434], [98376074, 11619434, -4048274, -2209512, -2834766, -114566, 3985026, 1042760, -2270428, -1589338], [-49315541, 10452274, -1248284, 1374147, 1984048, -2498433, -1499186, -740489, -473374, 633869], [-78668979, 4703265, 815223, 1090348, 1522934, -1116593, -3710270, -1001174, -948569, 1273289], [26098150, 698986, -1020344, 1098214, 439978, 1330390, 874056, -1565518, 294308, -407302], [16926076, 1534209, -109306, 1692806, 445771, -4149760, -4188633, -1370166, -435486, 1748372], [37344890, 14170576, -883956, -1103039, 460340, 1891413, 3314691, 1565713, 3491380, -422995], [-120238090, -5850020, -1362157, -2441119, -1350127, 172012, 1577343, 1009727, -704549, -2320574], [-48399661, 10312791, -1700456, 964242, -1142878, -6489528, -3496482, -2938092, -3232753, 4392482], [-15369073, -5490470, 2342734, 555034, 2402946, 497520, 6310417, -733294, 3173928, 920086], [-82907110, -8144898, -44656, -6503, -1139794, 1162849, -5599957, -961747, -2186452, 1323261], [-139956954, 2096298, -4710795, -663387, -362829, -1635810, -4609485, 109731, -2161509, 1737408], [-16281182, -7845110, 2851542, -662743, -1973594, -2316965, -2668523, 1538717, 24932, 755191], [44082423, 29392864, -620136, 1836388, 1280152, 2459100, -2544599, 649124, 1098664, 1912748], [178239109, 9774225, -2900725, -329926, 458302, -3687055, -1604388, 296304, -422841, 1643003], [30299731, -5570191, -675146, 1717739, -4838445, -1182179, -5269965, 196743, -4803822, 2262781], [-20039102, -6176244, 2010422, -2179642, -152450, -965668, -5576070, -915282, 1604418, -1021372], [54292232, -7863410, 965066, -2688658, -379312, 2182104, 2889498, 4672046, 2178930, -3023680], [-50150818, 13715592, -391394, 3561188, -107486, 487706, -309548, 488344, -1448906, 1864338], [-9223840, 5941616, 1973982, 1441797, 958322, -60101, 5074269, -1223039, 3068822, -1087361], [-55505571, -2363989, -4771470, -2187717, -1862763, -2055647, 928153, 1103535, -5877438, 2703261], [-78682282, 9061856, 1229658, -1554563, -2802884, -3026469, 4342981, 1982801, 598996, -1786921], [-16493182, -5408136, -2590682, 674706, -960062, 4624824, -2119574, -99134, -714646, 542216], [33493182, 27532694, 142108, -59835, 1697370, -1945175, 1691769, -1578559, -642880, -536983], [91775410, 4455442, -1459915, -1287407, 1161665, 4827150, 3936259, 4217747, 4632401, -1554720], [-26174857, 10051465, -1059786, -3352553, -1357445, 1294897, 5142007, 4845327, 41838, -3380427], [-73895857, 7193334, 3180810, -290632, 1182306, 6454970, 1506607, -118152, 437812, -832744], [-146190776, -22816254, -2957976, -796150, 1278540, 702782, 1272820, 1025286, -2689142, 206702], [11096474, -17577012, 1206544, 1042014, -1269992, -3688818, 7568644, 2226742, 621908, -167758], [-84533787, -13936757, -826732, -816087, 2533695, 1500177, 6003835, 2114761, 4466824, -2500451], [-46550065, 27987147, -602309, -1379140, -1944242, 4030127, 2775346, 3101998, 527725, -2097447], [65266074, 11555066, -3300644, 234743, 2048560, -1743345, -3250255, -4358189, -574658, 3318615], [-41783679, -8094407, -2162515, -857760, -1098582, 3426645, -2466008, 655714, 1516579, -2364365], [-112484154, -8605504, 2385900, -2139885, 1335420, 6513679, 7045405, 2760687, 5137464, -3951513], [-163192475, -7123159, -3708859, -2282456, -1786618, -3031099, 5882520, 1315766, -4123453, -936681], [359190, -10706128, -2142714, 210721, -595098, 1648919, 8625521, 1207265, -2442586, -1668773], [-25415578, 4797062, 463416, -1757182, 1616016, 826726, 3210046, 3817246, 4046890, -370210], [-101834277, 2141800, -2782542, -1960041, -3146006, 1405673, -90952, 3067555, -1604786, -241147], [39050544, 30189469, -4597234, 2290764, 1662311, 290622, -6144799, -2528656, -2974094, 1107898], [28192612, 2006077, 192568, 2776044, -733081, -2045256, -4341999, -2992056, -2994862, 2902646], [27345261, 8909818, 20444, 1173791, 1797012, -2661245, 7731808, 2566675, -1821586, 2363757], [6812150, 2850568, 388334, 3251893, -3294858, -5015365, 769137, -2490687, -4684446, 3558503], [-54154606, -6444096, -1080730, 1390601, 4429412, -3327053, -7117977, -4118067, -1282492, 1210863], [-66652481, 4077177, 3190592, 3040618, 52596, -58460, -8829576, -2456662, -225419, -1305722], [-32227158, 15090138, 1707365, 692867, -4146269, -1545940, -3561453, 1470633, -561189, -628798], [-16627490, 18821566, 4631530, 3648498, 3393810, 585464, 507244, -4283338, 3999188, 2307980], [-38148393, -5121167, -1310366, -3536572, -2038608, 5397648, 1819320, 1910690, 1754971, -1038006], [116830084, 2518125, 4994955, 194359, -2389218, -976454, 1288376, -913779, 422303, 856342], [-13485264, -16170584, 1137736, -2445505, -4651010, -2755373, -3548211, 2255095, -295070, -3972033], [-63224655, -9611497, -288026, -2074330, -681028, 739982, -2751236, 2293058, 2363699, 512444], [238523683, -30596760, 2305094, 2875298, -2654930, -1868416, 1706283, -1318842, 930570, 1356860], [-26119132, -19529856, 1739815, -551048, -2095592, 2681377, 8887014, 3454208, 1465502, -319987], [-181668423, 20832013, -1848190, 452043, 3250832, 346767, -1944163, -4270231, -2125601, 3537433], [9889378, -3088621, -2802360, 1874142, 6015021, -3771638, -965801, -2221950, 582522, 2058072], [-103008804, 20317230, -3030036, 256426, 2855366, 2988694, -14155514, 269734, -2448572, -841558], [30866147, 1530829, 207684, 2211642, -1482108, 2352600, -576288, 103434, -1850967, 263178], [47173249, -8232219, 4534679, 877314, -3089106, -1528611, -1563152, -2648144, 2520355, -2128505], [-99690553, 23786200, -1841147, -554296, 1256311, -155123, -8822073, 1135666, 1366129, -363021], [-59654938, -23505204, 1097382, -3579491, 111776, 4167335, -2628663, 4885857, -508612, -387353], [-138003006, 26600846, 4007200, 3791643, -768674, -4122197, -11991047, -2688665, 1337748, 3253819], [40319033, -27756369, -3007844, -5077031, 391143, 833689, 1904107, -750171, -636112, -1378759], [-97675779, 2419107, 3352733, -649667, -741973, -20128, 8150065, 3457579, -1674108, 81260], [30446049, 2965656, -3222725, -3243940, 3699903, 610799, 6052329, 1842326, 6470697, -2343173], [-63644014, 37247506, 28114, -4992315, -2367782, -2302837, 1866187, 3133149, -491052, -1670477], [-145857938, -4286148, 2230139, 1616645, -975689, -7246516, 4853679, 2288907, 1649421, -1005922], [-72228491, -1011258, 5101539, -347906, 1233477, 2839281, 5946413, -2387904, 2901601, -4878703], [-125072738, -22934994, -5276356, -798847, 575392, -2598943, 3806085, 659337, -5355642, -862807], [69790748, 32627408, -2042889, 1800055, 1568429, -4605994, -3379963, -2380055, -3741737, -24328], [-183439893, 23454943, 754607, 3775192, 3386094, 2059455, 6344474, -425822, 2171745, 323909], [52601776, 16232106, -2249557, -3154521, -1826091, 4732710, 4662949, 1979929, 991849, -3308288], [-61148089, -8989018, 2694529, -1773182, 4732193, 1227779, -1204241, 2713476, 1354629, -1196591], [6771749, -15159560, -5090664, -879366, 746094, -794910, -6194115, 2501430, -2481174, -1003860], [-118170690, 20027782, 2323954, -423874, 6531168, -1030320, -4357746, -1142662, 3797254, -508308], [28074723, 23913691, 395582, -2019513, 4152813, -109423, 5142237, -592121, 4643758, 2260849], [22317340, -10878718, 4302776, -614881, 376798, -172857, -3465599, -1982397, -3909408, -3017257], [137761170, -22303890, 4931369, 561760, -583640, -2975633, 11063704, 3149010, 3847234, -2861741], [126794587, 31789924, -1079045, 3475616, 1287347, 2858755, -6246917, -3983910, 46673, -750349], [-23736360, -14957516, -2290922, 4828125, 2986944, 5364751, 5224367, -3247395, 567376, 2925327], [74853831, -29984729, 2153576, 1457157, 505934, -4279249, 92645, -1608091, -1875183, 1111817], [-174775473, -10821357, 2212054, -4346273, 1891970, 5271039, 5564933, 809349, 1979979, -5130983], [-68534867, -20891143, 2729910, -79143, 6950301, 7872583, 7149109, 3777957, 5456904, -2306373], [-136661770, -12925098, -7133369, -4547064, 5067324, 2335953, 9981176, 598070, -913594, -649747], [114639833, 12495243, 1531927, 2038411, 1626803, 245656, -310839, -2698005, 301122, 3807856], [-98019022, -21685982, -163711, 2424974, 4007016, -1219803, 501026, -3128960, 3096894, 3104009], [20696349, -8645132, 7365382, -239167, -1559112, -2791017, 12386832, 2756249, 7200628, -4279239], [16842188, 15645456, -1372722, 1641483, 883938, 872793, -15290121, -3877197, -1388886, -417507], [-72982841, 32330982, 3136076, 526321, -3506884, -2405339, 12931880, 3547805, 4845986, -1883349], [-21621054, 30492735, 5623473, -1164801, -2514184, -9955464, -2872196, -104675, 1008687, 1715502], [189953835, -1862511, -4518982, -3848154, 2080438, -3226878, -4622824, -1649574, 2800619, 639960], [41437009, 18478601, -217976, 2202169, -2340161, -3452391, -18762021, -2029019, 32010, -191123], [73191188, 4244624, 2894707, 4294806, 4042584, -2636129, -14892054, -2378188, -1352452, 2625071], [-121621326, -19238024, 5518152, -1335022, 1802780, 6219346, -9464314, 669494, 6250048, -3527242], [-22602865, -4887398, 445218, 4500855, -1972364, 863457, -10296622, -3575413, -1041162, -196227], [-50057969, -9615673, 2514986, -3007421, 2280701, 11163137, 5811177, 6959255, 5396870, -4379247], [57773844, -34055814, 4689822, -946439, -2899102, 3412411, -6600179, 1622869, -13448, -4789213], [-17939055, 4037223, 329200, -443573, 5305453, 251523, -9945831, -3839693, 7067382, 218887], [-173796532, -7838326, 1129775, -159603, 1535553, 2900640, -6288641, 4307791, 739525, 1705954], [8472158, 1395958, -759697, 4044238, -5040040, -3633501, -14662088, -3268846, -10394176, 2585691], [-155734727, 14298323, 4062569, -1815200, -7810742, 5471189, 1326422, -1286518, 2364693, 634835], [-109207166, 10338894, -7475484, 2055871, 587258, 2375283, -8119737, -2164673, -9016844, 1498667], [37515773, 15441213, 1901314, 451123, -6226742, -4501037, 4978629, 2529893, -6339511, -465223], [-167939979, -3671489, 3804647, -6677584, -547758, 2651599, 15845410, 4531726, 7322177, -6233223], [-68892324, 16292104, -1605832, 2078657, -4181558, -7462891, 5541469, -1801363, -3232406, -653771], [-523384, -6018972, -1316617, 3399835, -3435753, -2460522, 2597139, -5349647, -7141843, -1843368], [-76139924, -26289474, -4814256, -3967476, -3329736, -1401984, 8717208, 640548, 447582, 2767800], [-246296477, 1413829, 2579102, 2511098, -2039160, -800126, 4938686, 1160048, 5321015, -2082828], [-45552083, -4417248, 2882072, -1450121, 2431080, -3882281, 7828134, 4892351, 5373928, -6440657], [66084730, 4194570, -9696436, -1091081, -293196, 4941151, -4162479, 1829963, 3472498, -2230421], [38134133, 2009885, 6535734, 3570941, 3097345, 864147, -5298897, -2161087, -1717360, 2918035], [-129981174, 1722012, 5109396, -1770129, -7568508, -3770589, 15841101, 6043899, 4211100, 726735]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1075_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1075_6_a_b();".
// 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_1075_6_a_b(:prec:=10)
    chi := MakeCharacter_1075_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_1075_6_a_b();".
// 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_1075_6_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1075_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;