
// Make newform 245.6.a.m in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_245_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_245_6_a_m();".
// 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_245_6_a_m();".
// 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 := [-22888100, 5935300, 7583055, -722920, -639206, 25380, 20336, -192, -246, 0, 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], [-2613859980, -258750600, 837690357, 111936823, -43954077, -6721179, 554383, 98973, -1947, -433], [-107486394420, 113744633960, -50966745557, -27571140327, 4236045917, 1395513643, -57068271, -19567597, 200027, 83137], [-590033063220, 103896742760, -41118854357, -27571140327, 4236045917, 1395513643, -57068271, -19567597, 200027, 83137], [621408182180, 252933052440, -140160241487, -45551323789, 6994915287, 1760844265, -98631325, -21012719, 438897, 76219], [-737664945060, 53779123840, 187758838379, 23203382625, -9503152979, -1626243037, 118409649, 23439115, -431909, -95695], [-1980608051620, 31720273720, 688500755943, 55560393629, -38975931263, -3997103913, 579871301, 61777919, -2579273, -271659], [-489826503660, 436039731880, 97163101973, -44108638233, -4622163965, 1507316885, 77430063, -20161043, -419483, 91631], [64943739940, 16776991880, -24282359591, -6897894681, 1347458171, 381410769, -16286853, -5627811, 50621, 25091]];
    Rf_basisdens := [1, 1, 69843200, 9847891200, 9847891200, 4923945600, 4923945600, 9847891200, 1969578240, 205164400];
    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_245_a();"
function MakeCharacter_245_a()
    N := 245;
    order := 1;
    char_gens := [197, 101];
    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_245_a_Hecke(Kf)
    return MakeCharacter_245_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], [6, 0, 0, -1, 0, 0, 0, 0, 0, 0], [25, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [80, -7, -8, 1, -3, 0, 0, 3, -1, -1], [47, -13, -4, 9, -2, 3, -3, -6, -1, -3], [76, -102, -1, -13, 17, 8, 2, 11, 5, -2], [727, 17, 29, -38, 23, -7, 5, -1, -6, 7], [391, 90, 25, -66, 4, 25, -23, -14, 5, -14], [-916, 160, 136, -9, 15, 42, 60, -3, 9, -24], [1341, -336, -103, -22, -46, 15, 71, 12, -3, 36], [1671, 505, -274, -120, -63, 15, 47, 15, -38, 63], [3522, -626, 265, 98, -128, 42, -58, -24, -2, -62], [-2888, 44, 110, 16, -94, 22, 110, 58, -28, -56], [1797, 245, -69, 359, -175, 33, 81, -15, 92, 35], [3601, 1031, 700, 354, -71, -99, 113, 63, 36, 161], [1756, 415, 132, 850, -157, -88, -124, -91, -99, -83], [6720, 1790, 556, 146, 68, 270, 226, 324, 118, 2], [8881, -1072, 1165, 420, 590, 279, 119, 168, -31, -100], [512, 580, 1052, 190, -40, 372, 60, 336, 276, -52], [16973, 2573, -1437, -832, 477, -269, -45, -29, -118, 43], [2802, 243, -1724, -449, -233, 424, 340, -575, 121, 65], [35461, 2313, 1835, -316, -173, -485, -161, -449, 48, 63], [8465, 117, -3375, 336, 369, 279, 607, 327, 314, -509], [22336, 5170, -1885, 3957, 295, -308, 578, 685, -257, -250], [644, 4786, 604, -3376, -598, 790, -394, 526, 364, -962], [14635, 7779, 289, 1657, -669, -221, -341, 763, 20, 429], [2934, -5982, -4838, -2930, 814, 770, 234, 350, 564, -994], [-34612, -4312, 2680, -213, -1313, 254, -1080, -379, 589, -192], [24597, 9647, -3688, -2850, -911, -1033, -549, -181, -74, 321], [45465, 3540, -6081, -2004, 1414, -1233, 863, -252, -975, -224], [-18236, -1265, -10696, -1452, -1629, -372, -1288, 333, 313, 429], [27920, -7284, -1356, 564, 3312, -264, -1624, 1608, 1144, 308], [59472, -3187, -984, -1552, 2689, -2080, -2156, -1513, -185, 1271], [-26848, -18110, 12552, 3722, 380, -166, 3430, 1244, -342, 1022], [-45472, 14841, 2574, 7239, 3869, -1198, -2330, -1441, 245, 667], [113482, -9024, 7774, 4784, 5584, -220, -396, -1048, -604, -464], [63457, 26096, 22739, 7972, 520, -879, 25, 810, 1329, 720], [-83911, -2339, 15655, 1859, 529, 1137, 1641, -2343, -1356, 1091], [-48521, 15111, 17536, 3487, 3044, 63, -1091, 2052, -967, 2361], [-65686, -22154, -2006, 12290, 3754, 1038, -1770, 3138, -204, 154], [-41291, 81, -6086, -12956, 7317, 1825, -1199, 751, 224, -1377], [-174716, -16573, 11802, -493, 2103, -3258, 2290, 381, 351, -727], [159788, 27060, 8996, -3064, -812, 508, -2172, -4100, -1040, 812], [-106437, 11617, -29208, 8298, -5373, -889, -2541, -1891, 500, 1143], [201897, 4880, 3377, -9226, 5538, 1899, -4989, 2964, 2929, -428], [-74786, 29091, -34100, 19207, 1429, -564, 100, 1419, -789, -863], [69371, 12503, 7701, 167, 5047, -3789, -1493, -489, -328, -4927], [16030, 25464, -48, 3573, -1524, 4060, 7540, 13708, 3864, -1576], [25583, 20803, 12602, -15932, -4073, -2553, -665, -5979, 276, -3747], [392682, 27138, -6848, 2840, -8006, 1858, 8962, 1966, 432, -2194], [140436, -51833, -12486, 36277, -4477, 4814, -3750, -1543, -3373, 2141], [111012, 50288, 39417, 31640, 1880, 2880, 9376, 6360, 1936, -400], [262282, -9643, -14286, 30032, -7747, -1402, -1990, -5881, -3871, 567], [102651, 4439, 29033, 3406, 2661, 3605, 1329, -397, -3656, 209], [553128, 4666, 28252, 33774, 718, -8584, 2912, 1898, -1030, 2342], [708759, 315, -550, -29470, -2983, -4437, -4281, -7989, -2446, 2613], [469716, 66576, -42692, 27512, -8948, 7904, 7136, -2044, -1620, 3288], [-296059, -50849, -1104, -11276, 5251, -12655, 2129, 2141, -1358, 1529], [-172688, 44036, -26762, -14045, 5609, -17948, -14990, -5045, -3987, 5100], [922286, -14636, -117316, -16361, -5224, -1704, -4728, -7512, -2712, 4604], [840081, -156471, 17018, -5713, -8974, 1749, 4595, 9318, 705, 1559], [938272, 78470, 35378, -4845, -18170, 4430, 8182, -3850, -4080, -2630], [247576, -185442, 100196, 83174, 15694, 1968, -16, 9042, 2002, -10734], [640948, -85814, 60685, 37199, -12191, -3992, 1290, 2467, 133, -1634], [-590740, -68506, -57322, -15408, 20418, -4882, -10890, -16546, 3644, -4278], [-983982, -46388, -4168, 12768, 10858, 3008, -5320, 11510, -1858, 6520], [456255, -76085, -19056, -46676, 1767, 12455, 21615, 4661, -212, 613], [686077, -76768, 106567, 5944, 22728, -13859, 2637, -17342, -5595, 14192], [836468, 7230, -62958, -31500, -36126, -6514, 13318, -6178, 1588, 11746], [461790, -148208, -140709, 15895, 10423, 2194, -16764, -20051, 4913, -88], [-385500, 183320, 26572, -26176, 13100, 10556, -2100, -12028, -440, -13064], [585155, -134715, 139993, 44261, -47735, 9291, -12949, 6765, 7294, -5565], [1574602, -83644, -9690, -11410, -28650, -1312, -972, 37946, 5238, 2700], [-584120, 100008, -29510, -7448, 12382, -17746, -18250, -8674, 13864, -5732], [-710148, -222346, 79196, 91216, 946, 4656, 50320, 36558, 2238, -5286], [-792854, -26474, 63660, -78339, -52681, 180, 21090, 26205, -185, 12450], [1077065, -15759, -117152, -3727, 5664, -11471, -509, -18248, 6427, 9311], [-1502504, -201746, -51658, -103469, -7149, -5906, -2140, -4991, 2773, -6398], [555495, -107061, -52463, 4894, 9429, -1387, -22799, -3013, 9984, -32499], [-450146, -176527, 28546, -35226, 24033, -2546, -14318, -38981, 3765, -18293], [241842, -64986, -19630, -173665, 20821, -1204, -23690, -6481, 4141, 9442], [-713012, 196003, -44014, 19397, -3413, -19450, 7930, -12511, -7013, 737], [-1759001, 111655, 141539, 31830, -17955, 14797, -16911, 40555, 3448, -10095], [2263099, -93080, -22397, -62856, 19494, -4767, -28207, 32652, 10651, 9596], [1116418, -185246, 174106, 71242, 12974, 35078, 38566, 5054, 17440, -9234], [526068, 448818, -139218, -66005, -17173, -20426, -46340, -17711, -3027, 11742], [-342019, 122315, 243416, -16088, 59379, 22427, 30435, 42257, 9364, -3739], [-994512, -37206, -149646, -117642, 26780, 17934, -13382, 44460, 6894, 9350], [-652806, 3300, 164454, 76644, -45368, 20134, 26174, -4196, -9098, 24308], [-889758, 54478, -53404, -105615, 12282, 6152, 11936, 27566, -6714, 30218], [715771, 242874, 51191, -154516, 33072, -25775, 17337, 3586, -25495, 20178], [-456241, -151910, -385419, 78406, -24812, 20849, 7369, -26758, 2981, 4402], [-1203310, 94501, -171820, 127865, -83281, 32644, 26820, 41281, 13689, 10183], [-1118042, 112589, 275960, 105013, -19389, -23936, -23840, 6301, -17431, -19057], [-836361, 116997, 36357, -47679, 21033, 5403, -389, -22371, -5386, 18907], [-2004645, 57971, 133838, -27998, 26369, 4847, -43685, -33493, 10654, 9421], [-2244315, 116293, 196061, -100686, -14909, 2791, 11723, 25213, -11924, -17901], [1303909, 35079, 110027, 54716, -101959, -46637, 10927, 5233, 10226, 16169], [-375746, 2422, 148978, 53513, 12883, 16564, 57210, 29713, -26765, 25794], [-381162, -657362, 145200, -36170, -17360, -83064, -28064, -12408, -20544, -10690], [-348778, -475712, 311994, 5202, -80930, 46868, -52328, 20114, 19506, -42080], [-2112739, -216945, -139649, 44026, -80495, 13487, -57693, -39655, -3402, 1745], [1617116, 74130, -140820, -150202, 10940, -12742, -58050, -86548, -15542, -7394], [-1466162, 410652, -416464, -75076, 30358, -23032, -114120, -50014, -29830, 4848], [150780, 59334, 74877, 237363, -12619, 59580, 32110, 35223, -31131, 25474], [682787, 291401, 195637, -212832, -5041, -28267, 32753, -5833, -6722, 40231], [-1122096, 378898, -160885, -7017, 109257, -43392, -4254, 7611, -42315, 3958], [166182, -545137, -208732, -8829, 83787, 52896, 17660, 110901, 18757, -25987], [2390203, -22745, 158575, -57346, 13441, 28381, -44823, 22783, 11500, 10849], [6606797, -105825, -162045, 116411, 7775, 1217, -48807, 19355, -9242, -49295], [2001362, 140072, -120700, -46412, 2408, -55628, -30044, -22904, -33108, 55992], [868391, -370469, 229548, 174332, -215137, 40611, -36997, -76395, -8448, -44763], [3418212, -95905, -82528, -77820, -35789, -16836, -65768, 3573, -4823, 8029], [-615302, 414265, 684112, 85769, 87905, 52716, 44760, 72927, 14555, -48269], [1290062, 830680, 369612, 251022, -77070, -32056, 27020, 42494, -38262, -5128], [2709154, -247906, -465220, -22175, 8546, 14656, 47176, 59038, 37638, -34806], [-1704612, 312438, 162676, 24016, 134202, -10960, -13312, 20486, 25238, -34102], [-1264382, 209416, -689424, 118456, 59828, 62584, 30264, -44828, -3884, 13592], [-3138310, -589261, 611822, 174571, 169131, 17074, 33250, 39169, 43523, 4677], [-605136, 73370, -214868, -45258, 12788, -74642, 73786, -92636, 7990, 10102], [-2764978, 41246, 194980, 283620, -100070, -117062, 9338, 115606, -17672, 33010], [-6353307, 292445, -400968, 206669, -100020, -35779, -85193, 1100, 41339, 3715], [-2023530, -126516, -34162, -195072, 87804, -67146, -18906, -65400, -41670, 9516], [9269387, 89493, -5407, 282564, -59653, -2775, -28875, 38571, 4318, -23037], [-2264422, 1246434, 352028, 137123, -130779, -91928, -121070, -63545, -17087, 950], [-3155832, -505864, -396720, 18899, -41521, -99322, -8840, -96163, -89067, 28256], [-1461133, 657882, 141719, -76708, -38052, -48707, 48405, 20134, 65369, -17478], [264906, 1824, 68746, -131926, -253584, 79394, 148722, 33188, -25110, -30056], [758401, 897225, 103510, 227443, -213250, 68661, 14299, -44550, -15331, 100215], [-3302554, 276805, -741798, 352285, -208515, 76102, 20950, -116249, -12019, 39283], [2298745, 364692, 383831, -292576, 293086, -88865, -56881, -26564, 22937, 42144], [2301640, -549147, 294666, 194511, -139723, 148574, -1550, -3985, 16121, -140105], [-1405828, -569218, 440860, 373362, 219332, 132186, 51742, 185940, 43626, -46142], [-8885630, 252810, 437611, -154076, 54982, -28230, -105566, -116214, 71772, -35402], [2092841, -820543, 299487, -293978, 157235, -95269, -135321, -88435, -50992, 52055], [-5935031, 494909, 803326, -255819, 229116, -14071, 60307, -412, 28375, -19981], [6263918, 143872, 16632, -226457, -72072, 5872, -183648, -143456, 61080, -1648], [-7236901, 726931, -249840, 7531, 144464, -21885, 72393, 33432, 6561, -101379], [-5434244, -981788, 336002, 431143, -68507, 17908, 105730, 49015, 33577, -36660], [-1755412, -1044519, 981840, 450464, -159451, -51132, -108680, 43707, -73577, 93611], [1731992, 1335544, -103746, 461035, -157809, 43390, -28560, 41341, 106653, 80288], [487613, 47338, -1693585, -319440, -119520, 7591, -167129, -82598, -3369, -49830], [8604965, -155350, 568561, -401162, 124820, 200807, 10263, -15814, 102179, -97510], [1555675, -1041665, -499664, 32476, -10157, -2265, -16089, 37569, -41912, 76337], [-10402293, 823470, -577493, 208384, -140052, -115111, -122319, -94066, 44869, 25182], [5189152, -987770, -671038, -683208, 61990, -118206, 63994, -213390, -93164, 102010], [-9718497, -570681, 35589, -370578, -268459, -21179, 123977, 81163, -20472, 145489], [4733435, -1011803, 208685, 531860, 22651, 130489, 133861, -24077, 82206, -127021], [-5739771, -87926, 698343, -520368, 174068, -98893, -10301, 163622, -93457, 113842], [4336750, 336352, 220032, 971728, 87092, -32032, -138704, -58012, 26812, -112992], [-285633, 1387755, 223219, -601436, 314083, -141591, -75399, -35211, -138446, 28797], [-4891854, 49024, 604610, 164580, -171480, 48530, -167710, -199420, 68010, -169480], [4935126, 1134704, 390194, -677654, 414116, 140194, -67486, 37408, 62630, -10736], [5193813, -540894, -1441171, -171106, 207332, -60121, -133521, -227326, 6667, -7102], [-2520248, 1310264, -480296, -644680, -95444, -76320, -50880, 128316, 59148, -33752], [5193866, 320577, 1110108, 640353, 274953, -82096, 86820, 124223, 39455, 29699], [-25197010, -593658, 1314159, 669534, -71772, 126334, 83570, 194524, 5198, -7878], [-6461606, 1028740, 1447031, 227975, 22095, -67370, 308344, -17555, -52715, 67940], [-7346190, 675452, -998852, 207822, -90178, 97164, -19752, -165534, -143318, -70332], [11732441, -1788618, 1002671, 400442, -309826, 101809, -191983, -57044, 65851, -83094], [-10915207, 782149, -406956, 178646, 306119, -45491, -13215, -6563, -156978, 101131], [19745492, -90500, 856284, 168346, 2644, -112216, -99640, -294028, -17708, 46676], [1099480, -302465, 967752, 119410, -299845, 221180, 77560, -119395, 17425, -51795], [1402846, -1285434, 1001624, 33054, -186248, -95318, 210382, 5920, -11770, 58858], [2854099, 1596463, 1159865, 99897, -143673, -72561, -195065, -129729, -9916, -83047], [8548818, -1078360, -1592586, 234842, -261500, 4646, 64790, -101320, -60966, 280], [-3113109, 1135407, 884334, 420297, 169678, -123237, -380611, -117846, -72321, 27377], [-12296168, 1835872, 158994, -423185, 411153, 219352, -68850, 102979, 99585, -37968], [-14455570, -254736, -1480290, -1015662, -111154, 69452, -23600, 337538, 69242, -134336], [-201684, -1156670, -845860, -1054178, 560330, 114292, 44660, -62042, 29882, -211466], [10640570, -931780, -602358, -599956, -378488, -288180, -468236, -139224, 108092, -2812], [653940, 2460129, 1129262, 784669, -362859, -199078, 17950, -179929, 28341, 45123], [-9664347, -1905341, 426004, 694262, 310605, -167145, 173379, 204903, 76978, -81795], [11256650, 145756, 621910, 169022, -14216, -120578, -147290, -180428, -40026, 164756], [8609405, 319689, -1088457, 905652, 191193, -323117, 154099, 12679, -159042, 68687], [294876, -1273015, 883966, 326361, 199115, 25922, 45250, 54857, 130979, 155451], [-7933564, 1303168, 2000460, 1469467, 292587, 6034, 420, -174839, 62829, -197992], [19152944, 266086, -2005960, -832382, -291026, 49968, 283480, 108042, -186102, 209946], [-3218008, -2863166, 1288324, 1112882, 31464, -269638, -261714, -124768, 105150, 152926], [1004874, 2028938, -51606, -631576, 164902, 257142, -189178, 167910, 133000, -139642], [2147951, 1801597, 770325, 471232, -164445, 185461, -6991, 266155, 117178, -136405], [-7864364, -2572678, -184952, 420252, -215142, -83458, -184506, -256258, -44092, -32618], [9415415, 289123, -752601, -71546, 86177, -236383, 365013, 4631, -245464, 404213], [-705482, 272034, -923434, -140840, 77122, 227166, 388238, 174642, -88972, 40198], [-11156323, 2799101, -961758, 966974, -67569, 30461, 81409, 22373, 60182, -130061], [-11103571, 251455, 1843304, 1430454, -656523, 160369, 310669, 414379, 96372, -11847], [19981156, -1482552, 1286294, 127299, -104168, -16654, -99646, 172316, 94322, -43552], [-12009094, 4288894, -1939818, -162491, -187679, 64902, -145440, -268149, -108149, -24302], [28977588, 393474, 237352, -1293152, -20590, -69196, -96612, -62890, -46246, 253030], [-15961779, 135337, 2301849, -572542, -147197, 113307, -132425, -347331, 111920, -421153], [-3009102, 398326, 1850798, 874694, 403050, 215270, 323878, 234266, -342364, 62330], [9867270, -2891293, -340636, 527947, 345433, -367956, 84700, -262473, -152913, 336801], [-4377971, 187737, -1430122, -1072540, -694867, 31577, -4935, -58561, 25000, -168553], [-523302, -16872, -808718, -174696, -272192, 29098, -203886, -2468, 136722, 41712], [574662, 3666124, -409533, -700583, 339521, 93034, 175256, 322483, 128315, 231084], [7971840, -487926, 4898118, 104277, -66459, -180934, -118540, 35831, -44373, 246118], [-22964989, 757793, -2206994, 2703342, -107481, 29195, 180783, 162353, -85348, 156391], [-9410880, -812258, 932513, 1794585, -686597, 135472, -223794, -42911, 95055, -90998], [28261947, -1684315, 229481, -947327, 365545, 180379, 414315, -195875, 84926, -159885], [-15068196, 332394, 3070860, 1099434, 411516, -339070, -200410, -343420, 15330, 313670], [5147840, 590876, -2041556, 562754, -772360, -135508, -113788, 438560, -78820, 246900], [-29084518, 3651148, -2412076, -634848, 564612, 453452, 252308, 152516, 40536, -262012], [-13035863, -3185763, 943503, 1343294, -253121, 183103, 221627, 1777, -41920, 157851], [11363227, -1555793, 2080781, -154806, 714629, 485893, 607257, 241657, -78760, 92721], [2514079, 2195791, 2358630, -142463, 96574, -707181, 250829, -252558, -94633, 203841], [14313905, 5102514, 3834541, 394692, 438408, -151257, 211199, 445494, -48737, 339962], [229842, -3737202, 1176872, 1335638, -181068, 255376, 238904, 121924, 266820, -328602], [-27630020, 1287780, -3329004, -344140, -595480, 250676, -92860, 359888, 3908, -157956], [7367627, 1531411, -53495, -141848, 1085705, 106161, 463797, 377205, 17136, -371635], [36871741, 2306593, -2313088, -1588542, -269433, -380335, -782307, -447283, -63286, -198417], [-2923719, -2593395, -951984, -286266, -501585, 630697, 788045, 291685, 66938, -46605], [403201, 284016, -768977, -1101608, -251782, -79129, 183903, -180880, -168643, 37012], [8117152, 720506, -2335126, 202765, -227866, -93526, -794270, 56102, 253012, -408074], [26796124, 2202058, -2611680, 1148146, -90106, -108376, 585064, 226202, -33494, -73794], [-6988141, -1342829, 150173, -2128996, 707627, -211335, -447079, -494619, -141966, 164373], [6841041, 3231220, 292483, 626174, 489810, -383113, -896945, -325840, -251707, -213480], [1141942, 1977338, 1672451, 1498246, 193968, 195150, -262478, 516144, 186874, -130538], [12761222, -4019406, 3075326, 367434, 265046, 56574, 993678, 533934, 44176, -160386], [-15179673, -133375, -2004626, -595480, 3553, -148517, -1074189, -476729, 11326, -51113], [-16486064, -5517602, -1753826, 720620, -463438, 356306, 1004730, 582830, 214384, 440370], [10994364, 987978, -1381816, -1114070, 908846, -52416, 129408, -295422, 70218, 184334], [-11035708, -5456842, 1194590, -2317641, 649207, -349754, 76060, -26651, -160639, -380438], [25173922, -601188, 2014638, 825114, -739360, 229570, -445342, -393884, 59586, -312020], [15140976, 4748926, -569106, -2800667, 699014, -272318, -382918, 71902, -120796, 177426], [-24298580, 751442, -3951122, -121008, -405982, 258478, 338830, 377854, 312964, 160862], [-34211314, -553176, 1146987, 259971, -1044269, -750854, -462436, -351647, -272987, 213824], [-23143478, 726935, -4211148, -2345499, -872425, -335192, -441580, -728015, -31383, 135365], [-6386453, 1760845, -5804087, -2394755, 72905, 142855, 559527, -272315, 104770, 196115], [-33075540, 3483884, 4474942, -1582552, -240934, 206146, 124370, -483278, -34160, -459144], [25365500, 3451026, 4064836, 847750, -1225348, -470330, 111402, 98140, 109342, 558078], [21068283, -579443, -2821107, 1292086, 789123, 276485, 111905, 448979, 82802, -303893], [-21741684, -3021024, -2197180, -1016650, 251984, -289076, 65820, 172336, -152796, -53192], [-22218722, -3362908, 588790, -2842253, -513427, 140520, -986250, -523185, 198885, -476740], [169814, 4304382, -39388, -617751, 1161918, 545500, -379716, -499406, 321742, -622518], [-2926113, 8643749, 3585109, -2769747, -621119, 210091, 92667, -332507, -64242, -436981], [-50087326, -2532802, 737619, 1930326, -198236, -134926, 289574, 37052, -50134, -440974], [-5170158, -147268, 18306, 724930, -182718, 487288, 721100, 862606, 141250, -594812], [-44378832, -1179035, 1685306, -2604031, -167245, 119510, -500890, 34049, -36553, 364767], [-23768899, -1021896, 1416869, 1526216, 1521478, -124805, 245099, -854096, -72019, -117668], [860576, 620954, -5918086, 1198082, 290880, -509202, -146578, 365364, 393326, 29850], [-9026143, 489965, -1575129, 1067352, 1033567, -547621, -453481, 570851, 305468, 416483], [-7732209, -2824879, -2254771, -451751, -412291, -105929, -393657, -585023, -246890, 109511], [-25669516, 603762, -2825376, 2668266, -354052, -103382, -1249610, -756692, 24586, -363666], [-13908800, -6921178, 1122887, -98687, 385003, 76520, 648494, 638249, 210967, 161202], [4321962, -1689580, 4282498, 422658, -228760, 653414, 1364990, 503180, 421246, 47500], [42262844, -3743480, 30952, -3633888, 222220, -534928, -814304, -299020, 88548, -491600], [9498725, 5898197, 1719127, 4135590, -1100209, -272581, -88569, 13163, -414324, 335339], [12422478, 3956219, 1534758, 2686427, -1267429, -576854, 237330, 360721, -365373, 542853], [-4153166, 1760802, 1509086, -717298, -2094866, -494278, -1041510, -733018, 94004, 92206], [34878548, 2216814, -2030478, -495936, -211982, 957574, 989590, 574414, 369788, -596558], [21790833, 6653309, -1855515, -1113718, 1384587, 394799, 367427, 88061, -35084, -112837], [4084925, 2946174, -2292737, 1586176, -261088, 6307, -94629, -88502, -423557, -252082], [7403645, 2120689, -2128131, 2428020, -1204423, 748683, 331867, -405129, -302538, 330823], [37062620, 3745842, -2592184, -1720432, 184658, -209654, -209086, -1330466, -704384, 84702], [15405846, -2185481, -457796, -1377533, 1087381, -93548, 1347420, 436579, -383845, 819197], [3187362, -11361892, 7611314, 3008782, -244798, 48168, 215980, -443562, -520654, 197684], [8300081, -12940925, 4850539, -422166, 731965, -408029, 61575, 277165, -254706, 385485], [6739323, -5491313, -4165662, 1839007, -152252, 897187, -1252479, -329876, 426605, -78463], [217758, -8615612, -6481767, -242393, 1332747, -396418, -366136, -328399, -410359, -112892], [23560520, 4849612, 3697952, 2024166, -906060, 1115756, 1686708, 779036, -294400, 120900], [26373551, 3344256, 4557933, 1349744, -156340, -507557, -5149, 65362, -243401, -14760], [6296535, 3944475, 1170172, -1777328, 1112371, -584325, -282173, 79833, -822324, 338229], [12485239, -13562825, 1920378, 478879, 372660, 598619, 593553, 1204340, 568981, -516535], [-49032379, 647477, 3760301, 3512698, -1926617, 874655, 1008995, 1044569, 270592, -643773], [8218088, -3370317, 4364300, 346292, 553599, 1227352, 738244, -141023, 246313, -1023319], [-36752144, -2378412, 6192186, -1467073, 1257447, 212330, 892780, 616109, -237703, 688532], [-12845153, 3633931, -4785671, -4982314, -1745123, -187879, 326821, -330829, -13892, -56227], [-16673239, 7214232, 4048981, 174772, 2008134, 589319, 445047, 806576, 479161, -864444], [-34207874, 5451952, 2375632, -1345208, 165356, -197732, -204852, -978476, -174368, -169216], [6134345, 6071705, 3470810, -2134988, -92379, -175355, -1186763, -1166417, -90004, 772359], [-65800179, 9568794, 4123175, 2139962, -394398, -134043, 259653, -8568, 72187, -369802], [-23203954, -7351740, -4693901, -1734203, -988335, 693658, 1055528, 745195, 271627, 432340], [-4616940, -6807446, -9298560, -1192094, -189554, -367888, -457120, -145102, -270278, -133666], [-7461113, 4838403, 6433763, 2712058, -208643, -904063, -1405451, -383029, -76684, 222693], [-1975941, -14842309, 2293401, 6412435, -545481, -304185, 189007, 40527, 55516, 108981], [49204654, -5780270, -5689512, -2449822, -518216, -3236, 92124, 474880, 337060, -16574], [-4841350, -3182300, 3118272, -1998134, 378510, 319928, 242540, 330530, 842942, -908100], [-42156716, -9052340, -8897494, -3098013, -2008395, 979728, 612390, 708111, -329835, 196468], [-1642425, -5453121, -3508959, 4089330, -1492763, -1436447, 316837, 216721, -160468, 710353], [-50675833, -5259507, -6876163, 420065, -534703, 980771, 309611, 339821, 225062, -539837], [-23839512, -11094921, 2476290, -856197, -1012309, -740426, 23610, 473161, -571021, 874013], [-76565126, -38398, 645954, -462736, -1646142, -582662, -2171286, -1692446, 311624, -13258], [39654147, 713199, -5134980, -3497894, -807779, 274247, 562995, 1076183, 617562, -155551], [53469912, 13432168, -5218630, -4187289, -420368, 343742, 743462, -316564, 282566, -263152], [73088492, 4658751, 1473358, -2855251, 3630519, -1367334, -397410, -392619, -272453, 351133], [27894808, 1648126, 411364, -1905004, 2052594, 113006, 594222, 1258646, 132804, -793294], [93281424, -4783150, -4891174, 1580228, -872950, -983734, -1015758, 1089254, -25952, -10130], [-100044221, 6811940, -647765, 576654, -1484966, 62353, -276087, 95512, 492111, -65064], [1806066, -8784558, -1628855, 1193962, 2313580, -460714, -218206, 1294340, -149882, -196610], [-7863904, -5074576, -5441234, -71663, 125801, -1004546, 265700, -404765, -154009, -187400], [-16991778, -350922, 2557016, -311737, 1257942, -660536, 896184, 1453786, 74594, 484818], [-5580483, -3013903, 3324500, -386226, -381981, -552107, -1144215, -386423, -141302, -354129], [24165162, 9915881, 901154, 703510, -507031, -1168614, 899310, -598533, -417375, 362371], [-42364928, 293926, 4778126, 4428024, -1805654, 314566, -7290, 719878, -569772, 613594], [37735347, 5245075, 512050, -1306432, -749485, -23841, -1289705, -2233575, -546824, -312595], [-2119589, -8130248, 2375423, 3989744, 2268926, 1848881, 428001, 146108, -166781, -10436], [-82229949, 13802699, 2630832, 1690619, -199684, -228557, 137737, 178828, -661219, 390229], [64163023, -7499879, -1701551, -4135432, -222849, 236973, -375831, -408057, -15170, -64641], [79509868, -9225712, 8660942, -1966407, 1304532, 281102, 86270, 882176, 95946, 139928], [5267098, 12537980, -4558933, 566113, 2859765, -1469718, 489536, 307695, -108657, 459580], [51039661, 1484543, 6086082, 2416436, 203995, -746467, -639715, 410525, -791898, 587145], [-50758334, -150247, -4271524, 552865, -527003, -734832, -784620, 728571, 515515, -156277], [-38444908, -1249802, 1066994, 3657730, -143672, 591566, -1501506, -1773748, 635574, -866698], [51673006, -12737449, -6986658, 2441831, -1586561, -1115358, 453050, -527379, -773369, 232793], [-25365091, 1022303, 4101348, 1302806, 1327217, -1018285, 64599, -618301, -1694, 797233], [113790566, 14066172, 255556, 2491692, -466666, -421016, -178208, -528854, -1029854, -199904], [-9513650, -7265608, -4914404, 794518, -2418750, 153164, 551464, 350270, -312690, 369240], [-39073888, 12078, 8218832, 5776710, -2720298, 96704, -515840, 296234, 497938, -256238], [-33661701, -9672467, -2444668, 283134, 1770003, -1164661, -770505, -392371, -509992, 408987], [-47600530, 6524692, -7527098, -2121292, -2390192, -838470, -2127886, -1708836, 1125082, -508748], [-102297714, 1786974, -5683796, 1026739, 793221, 681224, 327850, 119207, -301863, 408906], [36517936, -4286420, 4453292, 333716, -3935560, -938372, -332020, 498880, 733612, 940500], [69624266, 8688428, 8985780, -551560, 2365110, -2016996, -2293316, -1026102, -679986, 1289120], [112681248, 7584466, -2524296, 1117806, 679614, -145224, 633592, 320946, -462766, -597794], [-10314665, -8437073, -1915437, -2872128, -2601253, -889671, -973887, 116421, 517642, -29847], [-1906777, 5006849, 894209, -7691330, -631969, -517111, -923355, -44017, -318790, -184521], [31403714, -14233161, -4977536, -3126257, -1071969, -256692, -1502960, -955047, -202379, -200571], [-18969518, -2341344, 2546783, -2278527, 477789, -1134458, -2142052, -2684633, -467221, -289144], [-43513286, 7078514, -8002230, -425991, 887601, -847390, 1071980, -1255165, 428175, -202450], [-12426173, -19765491, -725735, -48322, 484667, 767637, 331857, -1715469, -3622, 327643], [25429347, 11713893, -391396, -3643348, 2761665, -479347, 1007757, -255301, -384024, -865205], [-124174315, 601665, 2222240, -4494634, 2561403, 1344005, 30041, -1131031, 601258, -413313], [29774091, -11986930, 786437, -8168456, 1794188, 914373, 2002309, 180162, -82423, 912902], [-111559378, 16824558, -2765772, -1549438, 1392508, 581794, 377286, -369524, 264490, 533122], [41346412, 10140632, 11536313, 3076294, -4112946, 115128, -581428, -438678, -397530, -107384], [12492598, 7640949, -14596754, -7669331, 43921, -839654, -579550, 541123, 351933, -183861], [-36741013, -20753273, -3912267, -1186548, -1074347, 82093, 2360625, 986329, -918576, 25137], [113478367, -7771864, 7403073, -4741076, -1266490, 1049369, -688783, 1290464, 197475, -258860], [-127006135, -17022357, -2170790, -382110, -2344115, -750687, 1581853, 1272171, -250772, -150515], [62546520, -16309638, -5625412, 4613102, 619208, 1967938, 1876710, 2679424, 1054014, -514778], [137355923, 1858401, -7264034, 7438574, 246751, 446287, -21789, 1376065, 765784, -535721], [-11883191, -8446366, 1153077, -10426856, -217524, -1622829, 28715, -727902, -689817, 177154], [-21287936, 62912, 4427138, 7974517, -3808972, -1648014, 347042, 2322744, 227086, 652632], [-16245258, -7837346, 8054230, -1067890, -3091374, 991506, 799290, 2048514, 288496, 292482], [-4479352, 16946612, -7375978, -10322435, 1556668, 248250, -403246, -554616, -204778, -163868], [-17098834, 19860766, 11153008, 526319, 435749, 658396, -1572330, -992609, 787677, -64342], [32297907, 2721347, -5919475, 1125304, -55997, 1100469, -744923, -2222571, 98478, -935163], [194800419, -8491849, -5933213, -337094, 1304293, -1289715, -933327, -1800621, -1077320, 977], [65419297, 2287429, -9653061, -10998761, 481101, 93325, 236813, -594707, 153124, -593261], [-101519573, -12595711, -4104812, 3539928, -872031, -2218929, 53191, -1976961, -1523518, 986631], [35475961, 7436781, 921716, -2149266, 73711, 2866225, 3089149, 1585501, 658082, 188419], [-244953, 6492319, 7665478, -5395954, -148159, -1517353, -1447525, -320917, 547862, -579151], [67527196, -9871261, -3982054, -6374933, -18509, -1236346, -2210190, 338585, -233109, -73175], [-142967279, 7809763, 7648214, 2648780, -2515497, 712465, 1922233, -984095, -861002, 1963157], [-868796, -765410, -1257862, -5936570, 1373480, -276562, -452330, 785204, 921094, -1530738], [92912725, 3258681, -12808398, 8581818, 76967, -928663, -693899, 814421, -156358, -1447337], [48233256, -1214850, -20253758, -7041818, -2885084, -572994, 121846, -365520, -219630, 187174], [-101153247, 8683172, 13593113, 16033216, -1711402, -261085, 715875, 1677704, 255077, -232968], [27886473, 13986841, 6568426, 13586081, -934616, 335925, 1171047, 1224552, 958311, -41209], [111039470, 10446095, -5254114, 8591410, -1164705, 25990, 93250, 186565, 1368575, -152955], [-40659936, 18715186, -1517996, -337894, 1672144, -2818218, -415822, -875832, -454430, 427246], [-13761598, 9467750, 4519358, 2524193, -283845, -1442844, -644350, -3527679, -1940533, 1138818], [105827135, 8606731, 18429021, 6560341, 2337779, -819177, -768801, -1021053, -1095432, 204101], [-31031640, 15302972, -7454438, -155205, -4428357, -318350, 466940, -1296647, 642389, -982516], [30437135, 16228935, 3097021, 2162128, 3252735, -1550799, -510847, -936735, -75426, -394335], [-131931954, 8200164, -4933466, -7685834, -1106592, -56590, 143754, 411044, -429014, 438972], [-114736866, 10929519, -3536942, 1567192, 1687495, -1825454, -1780242, -1589555, -1635013, 969925], [30717858, -4576636, 10614848, 8109044, 116640, -1560792, -2172984, -4047048, -754376, -523460], [22501839, 15678702, -6343579, -14266714, -3025232, -943559, -165119, -372914, -464127, 2609342], [-55515382, -7125520, 7245266, 6481090, -3394178, 597872, -1381236, 1292234, 427734, -713552], [117006694, 8871754, 5600451, -2350286, -2808164, -1543854, 1322350, -408012, -525582, 1437974], [117550427, 10908732, 8024821, 1549344, -4299574, -1395043, 140021, -698380, -834221, 1088624], [94777082, 5726587, -7745496, -4498177, 3207893, 2040600, -682480, 354771, 235863, 11433], [4357757, 22458985, -4167069, -10203920, -4466405, -764453, -1106833, 93655, -528792, 1257055], [-9467752, -7882862, -5471596, -2426268, 4959070, -1745106, -1962402, -226974, -697528, 381870], [124510635, 18002007, -13522960, 1034483, -158652, 838407, -194171, -2042676, 255825, 1756137], [28552336, 6562304, 4690796, -7840862, 1612760, 1515412, -1839172, -2380784, 2152484, -1911200], [77701148, 4545508, 12397202, 326151, 3185357, 770420, 1228650, 1698023, -891711, -1258916], [2668362, -1070715, 23462372, -643139, -987625, 1212628, -1033940, -450335, 1342977, -826185], [156894969, 8631223, -444885, 4696924, -4341767, 772419, 430487, -565311, 18770, -194783], [59797337, -5318046, 2234907, -19120820, 2017104, -1182741, -176957, 41298, -527837, 1413906], [-49682034, -13491546, -7423430, 3513161, 1761531, 450932, 239370, 49193, 391947, -7406], [3716272, 19240230, -2525544, 6345414, -1462922, 164984, -1299784, -3537526, 326714, -377158], [2506170, 6938736, 1425538, 8964292, -1689072, 1181960, 779672, -964072, 56216, -956128], [148214593, -7340441, -11225429, -6524320, -2724679, 4179, -4535705, -3154407, -249590, -1089151], [-110410116, 6541493, -11964930, 1646095, 1099197, 254402, 2741190, 1603583, 725077, -1090681], [256447458, -15534120, 5397431, 4697617, 1767825, 1137934, 341340, 2390635, 597111, -332080], [-152450802, -14416134, -5152783, 2872664, -2804726, -685962, -3483210, -3540258, -248652, -1182794], [219090528, -11683314, 10271539, 990775, -3208111, -192092, 2868166, 803107, 803433, 720906], [14452168, -27988210, -12228066, 5326184, -384674, 2900746, 2818762, 1089778, 1500900, -445246], [-122617537, 19091453, -12794728, 3335126, 2149375, 3999945, 3353317, 2000109, -226006, 42995], [-165807929, 6876798, -12758455, -4501828, -568728, 1239381, 1949997, 3693150, -452459, -933922], [43008861, -18050131, -682868, 18518344, -8486027, 433277, 1398413, -265945, -1068524, 31507], [164041758, -1594587, -3543326, -963866, 4560285, 1845982, 945426, 585175, 1240209, -1080265], [-211152575, -20278159, 582327, -5698123, 1336889, 921601, -1556975, -921383, -366100, -443369], [-46556768, 5179011, -6346506, -10277181, 337679, 2588122, 1126230, 2090749, 1575035, -3779623], [141699053, 19689, -6911066, 10505539, 2678386, 1632029, 2379539, 3917558, 1047145, -848441], [-106085442, 19803484, -5675374, -4206624, 254188, 2622622, -352618, 1806448, 31114, -999588], [-80541049, -23852373, -10274075, -292199, 258203, 2715363, -1296197, -2047101, 1967924, -199203], [121821068, 14904906, 12488742, 4927101, -2031551, -486378, 3049840, 2535411, -1808909, 2195238], [1052227, -9853021, -9675794, 7832337, 5603066, -2045665, -4067519, -2737522, -1003321, 1182429], [-49503426, -4697973, 19584228, -6047749, 4482353, 986772, -1077580, 191775, -292497, -1006455], [29449831, 3092245, 10778913, 6885864, 897595, 261257, 2745645, -1434205, -1422602, 2875515], [74461812, 2302333, 13176554, 4178791, 3026917, -747338, -1577430, -781481, 807065, -2445433], [-124981124, -16081884, -13409448, 3078468, -3946008, 1796628, 1851356, 1865688, 1938292, 2236508], [102312804, -15424738, 7654308, 5109152, 3430894, 824132, 3818092, 3142802, -2788274, 1429946], [-76084290, -29377722, 15588215, -1974850, 151204, -471434, 1950314, 2376652, -1724146, 1737946], [-33405988, 22517877, -3523066, -13882699, -5372877, -1236382, 1071650, -341047, -218189, 2161839], [3621296, 4684654, 628594, -4508094, 1084296, 120006, -243870, 2246160, -1520406, -163950], [376757150, 27115936, 192224, 1468006, -1337814, -1696288, 1010044, -254602, -1255374, 149184], [-8929545, 11335311, 12319820, 2376114, -5654411, -1450945, -3081053, -3329929, -1255350, -2168879], [-59147313, 4854280, 15871393, -516478, 830418, -2937719, 141553, 936340, -917513, 2584412], [137390644, -24414158, 9950923, 6465191, 3028473, -3902848, 700706, -1864741, -1245515, 2063542], [86927684, -8631338, 7865238, 1156654, 549640, 2100782, -1576006, -1158280, 2904082, 283850], [-185480312, 62212, 7401280, 1288098, -1824656, 1163060, 1915692, 1787312, -679172, 1473276], [-2836287, -23198113, -26384692, -4814456, 1762295, 2659701, 2310589, 1692753, 407402, -2357335], [81861946, 13775138, -19030762, 11523481, -2606363, 1941766, -1144240, -1935689, 1035927, -1927922], [41092803, 8050723, 3296693, -14826956, 5063017, -384355, -1477919, -642379, 988988, -1958667], [-152684561, -24735889, -2317775, -7521456, 5320357, 2939005, 1041361, -3941639, 834824, -2154207], [230885086, -23757348, -3542779, 10508181, 646293, 581402, -3053952, 1371959, 546335, -1292388], [-45453708, 7916054, 20860016, -2035756, 2599086, -1555546, 1539638, 3037394, -15824, -483846], [110278657, 21317684, 2671531, -473100, 4743632, 2611685, 1134597, 2159090, 332277, -554412], [122267330, 20284558, -9465590, 7431638, -66038, -2371970, -2349650, -787190, -811500, 1188610], [-38817009, -58053943, -1994364, 4564754, 2481711, -2209699, -431815, 1253933, -322322, 230279], [86402350, -17219596, 953402, -4893434, 2531510, 652008, -474332, -1950150, 302630, 621020], [-64379104, -19118917, 18194374, 1490523, 2493767, 4599314, 2406070, -397843, 2432987, -2372159], [199939068, -16500642, -19172228, -1577384, -2561198, -2300936, -2484832, -4311194, -348906, -1449182], [-215289161, -20520593, -34364527, 2524530, 1942393, -532787, 3591145, -1606241, -638476, 777737], [-291172951, -39364143, -12699969, -12302691, -3410487, -2933439, 421409, 1292889, -55844, 1879847], [15493836, -2090580, 639100, -6672121, 2462155, -1092186, -2136360, -6735999, -559431, 1033468], [161044906, -18308140, 3728440, -1260300, -1895080, -631316, -5526244, -1294400, 676236, -2618020], [-54248716, 27406098, -22026160, 1621166, -7051014, 2501048, 1239776, -1911130, -1260002, 883014], [127574807, 4559493, 32883685, 3709168, -4506061, 4794885, 4346609, 3448107, -14750, -541949], [71340768, 8069630, 19982083, -6258561, 6679985, 116260, -2348378, -1436045, 966025, 552730], [50698757, 373157, 25935758, 1372375, 3571058, -3253895, -2106217, 3299014, -1456483, 2986507], [138341183, -293986, -11456751, 6279126, 11967914, -2506741, -807245, -1645108, 100577, -1937174], [-47307654, -21074962, 11739208, 13676868, 948082, -2020922, -6370, 1769974, -1513436, 2171042], [-16272372, 26348985, 4985408, 3747850, -4152091, 1577976, 1060428, 5165307, 102723, -1066549], [-190092210, 13107302, -2928966, -2010904, 1372094, -3563426, -1399498, -2140538, -2568536, 3558386]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_245_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_245_6_a_m();".
// 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_245_6_a_m(:prec:=10)
    chi := MakeCharacter_245_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_245_6_a_m();".
// 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_245_6_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_245_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![-10686592, -16621440, 2127024, 2639600, -230996, -129840, 12314, 2040, -201, -10, 1]>,<3,R![-12031198596, 23098421592, -11335365576, 285902184, 168931223, -9277842, -571489, 44916, 117, -58, 1]>],Snew);
    return Vf;
end function;