
// Make newform 280.6.a.i in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_280_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_280_6_a_i();".
// 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_280_6_a_i();".
// 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 := [39040, 24832, 280, -791, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [-12926, -19234, 729, 28, -1], [-316, -2, 1, 0, 0], [49360, 4052, -2335, -10, 3]];
    Rf_basisdens := [1, 1, 222, 1, 74];
    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_280_a();"
function MakeCharacter_280_a()
    N := 280;
    order := 1;
    char_gens := [71, 141, 57, 241];
    v := [1, 1, 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_280_a_Hecke(Kf)
    return MakeCharacter_280_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 := [[0, 0, 0, 0, 0], [-1, 1, 0, 0, 0], [-25, 0, 0, 0, 0], [-49, 0, 0, 0, 0], [-54, 3, -1, 0, -1], [-152, 18, -1, 0, 2], [-225, 59, 4, 0, 0], [-517, 34, 5, -3, -6], [-241, 86, -7, -3, 10], [695, 0, -8, 0, -11], [1477, 262, -15, 3, 10], [2674, -117, -12, 12, 15], [4683, 226, 31, 15, -22], [544, 595, 2, -12, -45], [1860, -116, 33, 0, 22], [523, -167, 19, -63, 23], [-3070, 86, -102, -60, 56], [1159, -448, -49, -75, 70], [1169, 903, -15, -51, -123], [6311, 477, 27, 21, -87], [38163, -839, 93, 39, -95], [9260, -2271, 171, 33, -30], [11521, 669, -167, 21, 199], [47579, -3178, -17, 33, 132], [48859, 1551, -196, 108, -76], [32367, 1616, 177, -219, 2], [-30126, -1050, -37, 36, -130], [20405, -317, -41, -21, 35], [32517, 2114, -88, -324, -119], [61127, 5944, 81, 33, 142], [-65770, -6494, -258, -168, 268], [-29983, -1636, -229, 213, -410], [26969, 2608, 589, -231, -40], [-95026, 4888, 404, -294, -228], [104936, -15952, 134, -60, -518], [20228, 2795, -237, 741, 1064], [18448, 4154, -398, 948, 1176], [-219565, -5709, 209, -477, 461], [-194184, 4176, -559, -132, 218], [29202, 5206, -953, 36, 68], [102937, -12001, -559, 369, -1055], [-24088, 2972, 1596, 696, -1606], [-186196, -7395, 1453, -483, -818], [90195, -702, -329, -1089, -398], [-258246, -14873, -954, -210, 133], [-208870, -14316, -486, 528, -78], [-251066, 6027, -287, 1074, -515], [-428304, -8998, 281, 168, -296], [-508431, 32121, 958, 2328, -266], [273341, 15640, 3345, -267, 1672], [290866, -6234, 2938, -168, 1066], [437120, -18841, -1805, 513, 2076], [-64111, -27098, 27, -1929, -2252], [-34820, 41026, -2498, 444, -1138], [19313, -12933, -905, -519, -3533], [369296, 11310, 2310, -1050, 2364], [-425220, -8068, -1566, -2256, -2734], [-97153, -22028, 2327, -3405, -834], [-599957, 16893, -3167, -1329, 4561], [16975, 1888, -2056, 1083, 5838], [-175651, -29195, -2616, 546, 2248], [142692, 26664, -243, 1140, 3996], [-877017, -28129, 330, -282, -724], [136749, -3352, 2591, -3579, 1342], [1381039, 40961, -4806, 984, 1538], [-72445, -50715, 6561, 2349, -5781], [340041, -11413, 1281, -4305, -5473], [-783696, 18998, 778, -1476, 5556], [1050474, -48999, -2932, -4092, 4031], [-518991, 33508, -625, -5949, -4122], [497835, 37997, 6260, -2736, 3406], [2881551, 4677, -1821, 15, -2793], [-1252392, -44860, 5749, 948, -6642], [802152, -42861, -8104, -606, 7055], [1667175, 62669, 2127, -5853, 3551], [1363010, -146876, 502, -3972, -4354], [1034437, 113846, 2404, -3624, -1743], [-866512, 67506, 1189, 4416, 9358], [688251, 76128, -6650, -7527, 1624], [405964, 52468, 62, 7416, 6498], [1011854, -175806, -4774, -1704, 5564], [-1545235, 197588, 3616, 6606, -6183], [3064236, 132367, -1159, 1947, -1110], [14447, -117861, 505, -7587, -14711], [2898854, 79896, 1848, -3420, -3030], [1350278, 31113, 4856, -10830, 845], [-340577, 190828, 10620, 1731, -10238], [-1468287, 185026, -1831, -4221, -6168], [-639116, -54048, -1868, 8106, 17200], [-2435657, 115816, -9241, -6135, 17652], [-200265, -46797, -4594, 6234, -2620], [2393765, 163816, 8679, -3909, -3872], [501730, 4468, -4992, -4602, -5720], [231340, -111201, -1629, -2838, -18501], [1927482, -94465, -10757, -11250, 4005], [-839598, -108936, 6985, 8418, 9958], [-1346978, -98496, 788, -2802, 4094], [1109403, -27616, -3603, 1695, -8956], [-1614059, -35329, 7629, 9621, -571], [-3073675, 290522, -17514, -1692, 2501], [-4691314, -107693, -5686, 8298, -4917], [84861, -135849, 8481, -5283, -6153], [-417965, 331689, -9997, 6195, -20605], [5232578, 65004, 15608, 9570, -8014], [2598337, -186919, 753, 4017, -6517], [-7746047, -104961, 5080, -8736, 5710], [-7426865, 28243, 1979, -11757, 9369], [11219869, 43195, -2428, 1182, -3744], [6627348, -50181, 12409, -12081, -2054], [1568808, 580618, -19766, -4404, 9320], [-3224354, -346104, 7949, 1728, 5492], [3725615, 315259, 7169, -10911, -18351], [1049382, 353658, 21924, -18, 8760], [1119387, -456816, 8933, -549, 6938], [-4148398, 259371, 21599, -10437, -10600], [13134338, 213428, 2904, -9606, 3002], [-9836379, -1833, 422, 6720, 4736], [-249814, -86916, -1950, -13146, 19908], [5585658, -321903, 24396, -1614, -17817], [5566052, -545957, 741, -5850, -14225], [-7618174, 633800, -3494, 7536, -20292], [3856153, -253008, 7565, 11373, 8780], [8153310, 198214, -17523, -9678, 3502], [9836016, 110915, -33492, 9168, 5483], [-7358198, -161156, -9746, -2322, 33188], [-2330021, 132584, 35600, -9570, -1421], [-990023, 397296, 11570, 10854, 22877], [14379943, -29256, 1193, 8889, 56], [-8106298, -230720, -42714, -16992, 18898], [5219778, -189652, 17269, -25788, 27390], [7160046, -40989, 25723, 10482, 8881], [5328994, -164576, 29318, 1752, -32298], [-9831248, 405325, 27455, 3945, -11190], [-64977, -701745, 2139, 5379, -33777], [19629629, 176580, 22615, 11367, -18410], [-7260448, -164918, 53786, -30888, -33132], [1729630, 304396, -64843, -17484, 60546], [-3117327, 346195, 17334, -13020, -10772], [12520870, 209194, 13603, 27954, -35626], [4654931, 500032, -35242, 20703, -27294], [-4104582, 479970, -5828, 26394, 77986], [6681297, 612684, -16728, 4446, -17163], [-8361737, 440664, 7395, -22677, -18078], [16984329, 281977, -71741, 15045, 62915], [-15161623, -18388, 47709, 20805, -40864], [-877461, 889010, -38183, 37029, 52266], [10704712, 129266, -23318, 1530, 7530], [9929535, -722301, 35817, -12351, -59307], [-8041484, 75984, -37968, -29352, -23076], [-139635, -1240154, -5635, 8781, 17208], [10517270, -342779, 47248, -11982, -3409], [12914926, 70738, 18286, -33888, 27962], [-7625634, -1055907, 50030, 11682, -56539], [-912022, -130806, 34426, -51138, 12418], [12824792, -105107, -94570, -7554, 38739], [10276409, 64729, -59483, -5265, 69713], [-15990364, 358039, 19999, -40749, 28214], [-9280270, -798904, -30666, -21606, 108026], [10413495, -626711, -32768, -690, 50888], [-6157569, 422856, 29823, -9075, 53502], [7440554, 993781, 55014, 18492, 15703], [-11120749, -293674, -20221, 7629, -21368], [16309342, -944638, 60408, 17016, -19576], [-8242843, 346502, 84341, 28947, -97022], [-1167445, -169044, 14293, 2031, -48656], [1729598, 891472, 37501, 25914, 53522], [-9620561, 167513, 59875, -72621, -68649], [-2193992, -1064784, -29727, -30708, 61320], [-22192691, -266648, 2940, 74517, 19426], [-15171062, 137674, -18552, -71220, -99266], [-4470409, -927675, -33581, -31881, 67315], [14657956, 717320, -44334, 63834, -113716], [-13509308, 1381727, 16549, 37401, 88044], [594390, 1106098, -16132, 33510, 132936], [-4929225, -386000, 14697, 29763, 19894], [-20968194, -1408650, -28086, 7848, 18534], [11886874, 429681, 39995, -5382, 22907], [-19129695, 542018, 73680, 37746, -8809], [-8665116, -1847314, 13912, 5856, 38706], [-8147911, -1662168, 136657, 9867, -83906], [5007053, -733028, -25039, -8289, -8844], [-9206011, -783168, -104631, -56907, 11610], [-16185418, -761957, 172972, 17550, -100783], [8028121, -2075359, 43607, 13911, -18797], [15453565, -1223458, -24655, 14271, 78766], [-9563145, -884432, 61285, 90471, -73720], [9107615, 585323, -112885, -23355, 114025], [-16901737, -260551, 61400, -105438, -82250], [-10721093, 1348488, -102666, -25155, 144468], [19269143, -986804, -85457, -114561, 42986], [-4883009, 1704323, -174975, -24993, 62615], [27907656, -121887, -25830, 10044, -33279], [-17089196, 1924969, -96829, 51168, 48087], [-15140787, -1911592, -71395, 12789, 131800], [22655183, -299633, 194311, 88701, -32665], [-3024407, 1717467, 181544, 46938, -153376], [-578733, -2959864, 78528, -14085, -172846], [-1289944, -170419, 14048, -46344, 38539], [26433377, -2926555, 37806, -37314, 19490], [6295622, 556050, 16167, 157842, -128592], [-46009516, 705838, -60122, -17196, -63724], [6517354, 1562532, 122284, -58770, 10024], [24744184, -797932, -19146, -14610, -132592], [-5547029, -2807898, -124695, 56679, 39138], [3867029, -2877400, 37595, -3549, 76096], [28600246, -1816748, 71903, -6300, -26538], [-11285796, -234234, 66558, -14430, 167124], [-9435482, -929999, 143398, 52890, -51775], [-14242346, 641484, 59816, -70674, -256246], [14264765, 648906, 14141, -142995, -166420], [11883023, -588376, 102187, 13629, -82074], [4447910, 1550092, -48716, -158082, -129646], [-15589001, 2161050, -232205, -75645, 54892], [-13674799, -2257827, -90514, -16920, 46850], [6328844, 3653104, 39446, -19020, -175740], [44875995, -723598, 73151, 43173, 24580], [-20336195, -647428, 44391, 5439, 87680], [17113675, 129928, 58447, -90345, -32584], [-38008755, 494260, -97825, -126567, -123450], [-3853947, -2157923, -37077, 27213, 289105], [42002926, -2183170, -175388, 25770, 204516], [-58416350, 72004, 59188, -109062, 34724], [-8163827, 336560, 205172, -6987, 82042], [-18352442, 2428910, -9336, 132696, 185492], [-5136379, -1422771, -134958, 197850, 110052], [74588004, -2058144, 28006, -3552, 124258], [-32906017, -11327, -16160, -70560, -242938], [-36968622, 73301, 30359, -72255, 131092], [-57840940, -1880340, -8179, -65910, -26956], [-7838463, 2807017, -153831, 155709, 13855], [-2539646, 1476525, 220498, 52908, -343787], [-6549488, 551996, -171586, -9870, 157894], [32663881, -491919, 120485, 113703, 87881], [-60211255, 56996, -73284, -61083, 10562], [26031425, -1997075, -76354, -59478, 90528], [13811160, 423206, -28839, 112110, -189034], [26560173, 122592, -111507, -37911, -163092], [-25120155, -2518315, -210903, 32817, 30185], [47710698, -3331413, 59057, -51312, 16955], [-4800129, -554392, 59769, 205689, 13328], [-56523769, -1215871, 7387, -74451, 34617], [-177036, -2496398, -44396, 223452, -4798], [-21594888, 1594366, -12611, 9468, 172064], [57524784, 2942314, 224770, -114654, -45562], [37157549, -2280005, 214436, -85470, -43236], [-47378759, 901056, -219697, 28857, -24922], [18462322, 1892756, -112886, 213486, -59172], [2987514, -548218, 145164, -35328, -27208], [-59401076, -1751381, 36655, -2424, -219751], [8554155, 752346, 231449, 9039, 215720], [27716912, -3590663, -294214, 31236, -5289], [-63989956, 127848, 22210, -15768, -9842], [-90509024, 1521962, 144932, 31722, -60920], [10543726, 356096, -58210, 212832, -293192], [-10686436, 5537708, 7622, -71958, 118456], [4404568, -449079, 153761, 210564, 44219], [35384156, 1715652, -156114, 206448, 251790], [-169151, 2255083, 64825, 139821, 141893], [-7866154, 1988084, -115499, -259470, -129228], [-41052347, 3474269, 182148, 152502, 206282], [11048293, 5009190, -131973, -201867, -50700], [-7714061, -1980543, -45575, 15531, 41563], [54635682, 5531332, -37010, -163350, -119674], [93474304, 761700, -255501, 31758, 272166], [-94675635, -3662301, -23409, 44385, -173721], [-20039385, -6973794, 182497, -26001, -428336], [48176395, 3687456, -34942, -347988, -97519], [-3246326, 2093928, -352112, 84078, 81514], [6842621, -3982969, 207517, 148917, -447015], [3071846, -1155747, -281520, 31434, -129993], [-31933619, 1744468, -129995, -114531, 272012], [-85223970, -4737593, -210850, 114576, 322929], [19100991, -7576319, 36920, 72978, 10644], [35247029, 4004979, 28909, -23841, 301999], [-21215055, -1222581, 171583, -218367, -164453], [-39265658, -8027568, 243515, 214440, -128152], [-87393719, 1215123, -138871, -124029, 324707], [86766632, 4959082, -279818, 45696, -42856], [62619703, -1511936, -172998, 34965, 40888], [-107886546, 709850, 276320, -199944, 300352], [112631822, 871858, -30881, 324186, -342808], [57963730, 5217767, -168599, 35235, -150792], [-146904938, 764262, 123294, -190194, 217938], [38464736, 6178564, 68074, -96216, 318038], [59288815, -5880029, -73316, 124608, 172658], [48444314, -1916983, 215695, -58197, -110202], [-25727267, -2341006, 357935, -5067, -199250], [-78896800, -3619897, 380602, -99372, -540123], [-41540814, 350762, 545182, -136200, -311352], [-79741828, -105744, 174392, 38916, -144370], [-37076491, 2001140, 21948, -85656, -314569], [-19581091, 5359837, -398692, 145008, -413502], [-91296081, 1369322, -131049, 13197, 363488], [107796489, -4945240, -217995, -345837, -82672], [-57491792, -2830699, -292142, -105468, 264339], [-13532901, 4294052, -467829, -330765, 300848], [140124890, 681746, -161580, -155340, 8480], [41512196, 8995682, 28291, -162834, 283680], [62365470, -7125616, -585670, -41538, 38260], [53947519, -653455, -8912, 25602, -391026], [-9730711, -2362733, -413202, 227898, 504820], [23241529, -1177999, 128039, 168729, 44053], [89626190, -1372427, 730583, 145479, 121746], [58132717, -4064165, 19217, -29337, -399981], [29251092, -126845, -188981, 21234, -48985], [-20813943, -504236, 151847, 406299, -197682], [37470612, 848005, 49598, 215184, -417363], [-43019600, 9498264, 318328, 197190, -365144], [54049129, -1559139, -233299, -29685, 254417], [-116924985, 6695983, 507843, 21405, -535163], [37408942, 634212, -176898, -53496, 551046], [88033417, 2209884, 13950, 342930, 158997], [-99761918, 3098316, 43036, -83334, 376762], [15377695, 3730799, 578337, -69783, 333887], [-20326427, 3885130, 269771, 249561, 624900], [93486781, -3565390, 476705, 24243, 171988], [-179463388, -4744946, -315066, 189522, -110396], [161928570, -278965, -430305, 103005, 128054], [57190441, 4565245, -410234, 104124, -229900], [75968295, 2133916, -494986, -31161, 433074], [30648602, 675646, -126168, 525636, 536428], [5387459, -820784, 347979, 173133, -795542], [-110248480, 4891612, 158854, 180108, -277462], [-152304013, -5412804, 401031, -230745, -299274], [23750372, 1409906, -104690, -236010, -311670], [110450532, -5422206, -742480, -60804, 152432], [49114466, -2253737, -311967, -10452, -537821], [-38466709, -6717733, 350051, -309831, -431633], [26962937, 1793996, -489791, 347661, 642930], [3032542, 11031915, -328994, 173358, -336107], [-45840784, 1450626, -527822, 118146, 382510], [135301018, 5180005, 890122, -69342, 398159], [-146663634, -12215246, -206404, 335964, 277380], [-59689209, -2568101, 192858, -716112, -601226], [106305685, -125459, -937459, -36159, 939363], [-27302363, 4093387, 218230, -562278, 152666], [88081939, -3596482, 449046, 50550, -13483], [-97583351, -1477507, 270443, -171621, -453749], [68994305, -6728686, 163171, -296175, -644202], [159069892, 509658, -110855, -418146, 109486], [-51441705, 4202999, -423679, 174291, 806875], [24431192, -9844048, 654256, 265668, 283458], [-211635605, -12992660, 312377, 334671, -474984], [-174194150, -10121239, -339197, 67293, -387156], [-144953946, 10286231, -349460, 460956, 184791], [1603337, -2606317, -338959, -83829, 589651], [-103629248, -5722528, 364554, 233952, -852814], [142892576, -11858981, 707136, -84420, -83993], [22813169, -6866446, -11141, 281223, 551922], [-39641486, -11264700, 281287, -393156, -875870], [15640164, 7316872, 144324, -15738, -236666], [133876481, 7948490, 395121, 135033, -1201894], [-202563029, 49032, -489022, -412266, -60145], [-116308018, 4774036, -273013, 452850, 184602], [64344133, 13271060, -215820, -260616, 248111], [85611153, -3539785, -182265, 389961, -440857], [61574074, 9314978, 1053472, -375246, 227922], [11861989, 28152, 1411115, 587469, -827098], [-115370649, 5600426, 325989, -225609, -908470], [-201427160, -1808902, -126926, 346284, 514746], [135497762, -12735645, -900866, -53076, 62107], [-119518484, -6646644, -219484, 319116, 768338], [62553991, 4432806, -158081, -1039593, 232612], [-77123380, 871769, 497325, -715950, 28721], [-21873979, 8101291, -491163, 367293, 113761], [-180181241, 4899682, -310003, 430791, 808410], [89656648, 3428942, -1155163, -31038, 768976], [123005711, -14819060, 572033, -613197, -136110], [112792899, 9097172, 638258, -535731, -911840], [-84283504, -2047415, -622855, -32574, -409851], [115496595, 12906668, 134971, -222837, 525978], [36979402, 9992690, 1101490, -51540, -678708], [341760319, -501782, 384524, 394008, 723471], [-154793798, -2733892, 194792, -263466, -185600], [76097045, -2971695, -354913, -755625, -531829], [-290245858, 18361064, 506998, 624114, 183552], [-40658990, -5457345, -750973, 717381, 739994], [11501727, 5804995, -486434, 449208, 1544324], [-91208986, -1805348, -1326496, -133692, 682404], [25597425, -1075719, 946796, 121710, 543722], [-32665576, -11766808, 394386, 78294, 642308], [290618826, 1422430, -826092, 281982, -60560], [-128657010, 14282682, 624658, 706980, -766664], [361064144, 156200, -804174, -44064, 581738], [-305056769, 10883588, -187313, 224901, 407220], [63543078, -4703960, -914553, 683520, 536800], [237616676, 4899069, 172593, -83781, 726432], [187854020, -1164360, 1281539, 332682, 786026], [-80138933, -6379153, 369155, -534531, 65395], [47968890, 3548926, -317811, 102708, 779746], [170079087, 5224188, 350258, -341145, -897982], [49716990, -3420840, 82911, -313140, 510786], [-134434114, 10798395, 726041, 6738, -779155], [234037083, 58881, 730563, 474915, -224301], [-102842530, -5740621, -944165, -476637, 1549380], [-72652102, 2450044, 220842, -898788, -610388], [-72707409, 4176172, -89173, 869391, 333834], [-312528369, 4585338, 110863, 7971, 122098], [10600430, -14568477, 699003, 40128, -830349], [274893092, -7025262, 695770, 460122, -606374], [57352363, -4274864, 1677135, 582693, -671462], [-138067430, -5564810, -305042, 119502, 384428], [284728669, 10442636, 9271, -473865, 287736], [-70981645, 4023883, 229156, -127356, -206920], [137913292, -987024, -122968, 316878, -1215484], [185913544, 5608210, 1092210, -158376, -158762], [327424613, 2282125, -730841, -458919, 789461], [276017149, -6697288, 659274, 640719, -868276], [-173190051, -14228913, -625803, -381015, 1129929], [42976932, -14559684, -209088, -73884, -1434300], [37540959, -2806251, 1082816, -366210, -1155436], [189744505, -6287185, 561455, -94101, -779327], [-104106504, -3921829, -373166, -84906, 1088241], [113609437, 4620059, 1470469, 706863, -655989], [-67695295, -580458, -1110189, -1180233, 629748], [-475621420, 1791048, 82420, 428250, 169960], [467599408, 1735817, 241877, -1401, -1242668], [-37851376, 18115234, -636420, 54708, -259250], [-32396267, 10513487, 560687, 181509, -1396361], [-164175396, -3213700, 927621, 383022, 459272], [-126520141, 7921382, 606638, -588660, 118327], [-149040998, 16198356, -1293580, -966780, 58712], [124177888, 12835678, 931058, 138558, -1734204], [-133055104, 10347342, 648268, 1006596, -1064246], [71233453, 14088129, 821088, -128052, 454536], [276709900, 2170796, -68799, -375540, -546160], [-118277749, 20504519, 264791, 424215, -417191], [318555864, 21480028, 476470, -274284, -726718], [164490753, -11744482, -1946335, -767265, 1034938], [-11712857, -1512314, -118729, 1552179, 138720]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_280_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_280_6_a_i();".
// 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_280_6_a_i(:prec:=5)
    chi := MakeCharacter_280_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_280_6_a_i();".
// 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_280_6_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_280_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![63360, 23016, -2095, -789, 3, 1]>],Snew);
    return Vf;
end function;