
// Make newform 230.6.a.h in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_230_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_230_6_a_h();".
// 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_230_6_a_h();".
// 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 := [-8930700, 680040, 297325, -2857, -1168, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [92904390, 1317025, -1670407, -51558, 1939, 51], [-166695750, 14649059, 889995, -57654, -843, 49], [-222262350, -171142265, 251391, 731214, 2613, -643], [-336035700, 34179795, 4409948, -319078, -7861, 416]];
    Rf_basisdens := [1, 1, 1380040, 414012, 4140120, 1035030];
    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_230_a();"
function MakeCharacter_230_a()
    N := 230;
    order := 1;
    char_gens := [47, 51];
    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_230_a_Hecke(Kf)
    return MakeCharacter_230_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 := [[4, 0, 0, 0, 0, 0], [2, -1, 0, 0, 0, 0], [25, 0, 0, 0, 0, 0], [61, 2, 0, 0, 1, 0], [26, -9, -1, 0, -1, -1], [78, -6, -1, -1, -1, 1], [109, -13, 1, -3, -4, 1], [569, 17, 4, -1, -4, 3], [529, 0, 0, 0, 0, 0], [985, 14, -15, 7, -14, 1], [1700, 81, 25, -8, -8, 2], [2839, 122, -2, 19, -13, -4], [126, 90, -58, -13, 59, -11], [2217, 195, -78, 12, 54, 27], [4073, 143, 115, -22, -55, -18], [2866, 185, -22, 1, -57, -45], [10348, 657, 20, 51, -49, -37], [5322, 855, -73, -50, -37, 33], [8967, 236, 66, 103, -59, 16], [-2485, 257, -242, -11, 11, -76], [5376, 1250, -33, -90, 121, -69], [12417, 435, 104, 28, 212, 141], [10468, 1805, -36, -59, -305, 55], [-10763, -3059, 14, 128, 52, -65], [-2078, -65, 247, 152, 333, -93], [-8458, -1851, 248, -129, -445, 263], [-28109, 812, -461, -250, 293, 66], [12791, 1384, 718, 213, -543, -200], [-33209, -4793, 112, 1, 714, 53], [26315, 2764, 294, 609, 155, -60], [-5633, 7211, -397, 52, 669, -48], [-47272, -2302, -373, -426, -1119, 245], [-5118, 5740, 754, 59, -216, -18], [-32058, 2085, -333, 113, -1230, 104], [-75458, 94, 608, -587, 592, -258], [33325, 8398, -872, 208, 600, -205], [-104245, -2848, 1172, -121, -633, -320], [-26289, 17765, 1523, -39, -2115, -118], [-64886, 6750, -2388, 456, 1068, 798], [-159045, 6040, -1169, 424, 747, 506], [-87958, 2210, 479, 1654, 1689, -525], [-62021, -14115, 902, -1137, -610, 29], [96576, -3170, 2796, -284, -1108, -800], [-306392, -11144, -105, 1176, -793, -879], [-1515, -4498, 2020, 578, -2404, 537], [-371413, -5463, -170, -792, -68, 1159], [91128, -2255, -2912, 301, 3003, 599], [-329782, 4328, -1352, -518, -118, -124], [82275, -9495, -2870, -1166, -698, -583], [-446805, 16205, -1376, -1114, -6958, -201], [-182474, 17778, -675, -1716, 3417, 417], [-141184, 24531, 3183, 1151, 3852, -268], [327722, -14326, -3106, 2186, 1674, -758], [131450, -31716, -3552, -483, -444, 1008], [628046, -13506, 5161, 200, 1153, -1815], [33666, -124, -2249, 3103, -3286, -2926], [-70387, 48356, -1703, 2385, 1724, 1615], [469866, -22383, 3524, -926, 3815, 447], [439170, -21556, -7561, -2500, 1701, -191], [182825, -57461, 2658, -504, -5410, 105], [301474, -28991, 1422, 927, -9085, -1023], [337944, -33871, 3640, -3309, -4009, 1441], [1505277, -15088, 5467, -2502, 1151, 1510], [316296, 11432, -7247, -4312, -2209, 3591], [663688, -22687, -8192, -1524, 5825, 2017], [1334326, -30784, -2610, 1861, -3554, -152], [194314, -29293, 8031, -2745, -188, 918], [437333, -63099, -9952, 1459, 9122, 2643], [1829997, -4416, -1579, 800, -4873, -2822], [-306553, 970, 3851, 1687, 13564, -5139], [561686, -76840, 7189, 4196, 4937, 1269], [766533, 2417, 5106, -2874, -2786, -309], [1813580, 28665, 6082, 5609, 6931, -4449], [1051060, 10462, -8156, 8322, 9420, -4442], [71387, -103598, -4191, -1186, -4429, 4040], [145415, 9698, 12448, -6723, -823, 184], [-820615, 26432, -17043, 3916, 1387, 1012], [-24643, 52300, 6494, -5472, -15092, 809], [-1217460, 124942, 2612, 4726, 10072, -4146], [-2269671, -109140, 7297, -6118, 3418, -4617], [720827, -49253, 21126, 618, 1808, 6735], [-1246901, -33077, 7156, 9333, 15678, -1571], [-1396183, -109865, 3806, 978, 1990, -709], [-718734, 112449, -10062, -5096, -4467, 307], [-260814, -34640, -20579, 2937, 26697, -215], [-1847837, 92191, -16787, -7735, -7863, 1494], [-2026896, 38975, 14096, 4394, -147, -803], [527094, 174419, 5652, 2239, -30797, 1375], [-895097, 191063, 1029, 6742, 22795, 40], [-1273274, -60790, 5590, -11560, -25156, 9114], [1417386, -86853, 20202, 2451, -1587, -3441], [527294, 36232, 13618, 9974, 35166, -11052], [2235365, 81633, -12677, -3256, -20771, 5166], [1480690, -86289, 27809, 8603, -24508, 4054], [309630, -241133, 18359, 1361, 6250, 700], [-1495468, -33955, -6336, 4942, -27833, -5315], [-4658121, -237051, 1387, 14980, 10621, -13822], [-2039672, 47190, 402, 3368, -13746, -1996], [2320497, -197587, -1180, -2626, 26772, 3721], [1421207, -37491, -21653, 6714, -15683, -10034], [3165517, -242471, 1379, 9125, 44731, -38], [2554771, -138246, -3924, -25509, -1251, 7458], [789466, -470769, 2194, 10165, 1759, 3653], [-4677676, -236210, -11336, -20528, 8388, 11918], [7263005, -4090, 6103, 38, -13327, -14844], [217178, -381170, 1299, 4674, 38207, -5361], [1681372, -26353, -3644, -20990, -4726, 21932], [-895478, -176226, 30228, 13828, -2622, 454], [-5971, -20944, -13207, 10792, 14547, -25854], [-17841, 40224, -10719, -21588, -29748, -3345], [-1461280, 1420, -25388, -2856, -30864, -10424], [4042630, 195520, -11094, -8052, 38426, 3618], [-1728080, -164474, -21747, -9841, 41582, 12926], [3978060, -172016, -10100, -27429, -5478, 9400], [7418391, 265163, 704, 18996, -2676, 1679], [-2368537, 386805, 59218, -6974, -55394, -5053], [6657678, 33465, -27196, 5085, -14713, -6475], [-6829975, -70337, -11643, 24064, 27869, -14834], [-3113294, 314054, 22651, -7569, 79943, -899], [-5191760, -164066, -42508, 13294, 17842, 7686], [2843001, -249910, 13573, 29340, -34273, 28], [-2481154, -81386, -18507, 13914, -37279, -14835], [-684774, 67647, 39906, 44987, 33225, 8453], [232961, -20567, -21485, 8911, -82185, 7694], [9414726, -58530, 15116, -3238, 29504, -7106], [-4020673, -57843, 2280, -4211, 28812, -7397], [5610182, -74391, -1573, 28946, 64871, 5077], [-12806902, 147771, 48772, -15090, -48383, 16813], [-2003965, 341285, 11147, 21379, -2166, -14361], [3028406, 189057, -35624, 14637, 72457, 26449], [7340288, 92635, 15149, 32999, 202, 22582], [-3441128, 191216, 19316, -54567, -4974, 7100], [9951369, 357639, -34080, -8138, 8622, -19805], [5510881, 24814, 54254, 14141, 5197, 1576], [-7943765, -265110, -71555, -41170, 54358, 35325], [6337192, 354936, -58452, -44382, 6954, -1368], [-107845, 112521, -21282, -24374, 28062, 3979], [4341824, -668651, -35692, 11143, -44237, -23757], [-1477683, -228148, 95708, -15607, -86409, 10450], [488805, 100287, -46315, 40789, 36170, 26391], [1363546, -8039, -33585, -2839, 83876, -5482], [-4815396, -242166, 28436, -53816, -40, -10098], [4883455, -107045, 18161, 48202, 118651, 28866], [-10777148, -118073, 9964, -45743, -51447, -15475], [15477574, 405141, 44858, -20795, -39073, 30447], [-14225709, 151367, -81322, -40028, 22104, -23985], [-1255354, 100886, 64362, 26257, -126512, -36434], [-8881768, -92582, 8837, 42700, 19063, -24675], [11372676, -396497, -18393, -13287, 67520, 19452], [533955, -172475, 21287, 40782, 24655, -3730], [-14990053, -652092, -1219, -74062, 115859, 32662], [-13686578, 112800, -16088, 32760, -11150, 37870], [1833106, 737496, -83026, -17775, 105596, -30280], [-19557155, 85797, 12947, 37014, 15587, 28760], [3503003, 409536, -22354, -3519, 22805, -57040], [-15971641, 381181, -82796, -15922, -187716, -18993], [8576655, 493709, -110146, -47584, 26538, -24719], [-9551297, 249918, -13823, -15589, -62186, -29325], [-607003, -351515, 100222, 2367, -2556, 12373], [-1957411, 464472, -71917, -30242, 82811, -38652], [4636339, -62373, 108265, 13461, 96109, -11918], [4258613, 504996, -113989, 26728, -96331, 46302], [11308313, 710613, -39901, -5880, 66281, 36782], [17200929, 661484, 55055, 9038, -106203, -7808], [11150403, 673458, 17768, 71112, 44081, 29756], [7798789, 159261, -31393, 62265, 267008, -72841], [-2426997, -27079, -88429, 47391, 188976, -37219], [14222220, 198304, -113240, -30694, 74334, 24804], [4681763, -1102235, 51786, -103368, -111472, 68109], [11170225, 100232, 128642, -41100, -188172, 42437], [-8070029, -976571, 74696, -34252, -10772, 17899], [-7739667, 178208, 92133, 73305, -51176, 35235], [1702448, -462734, -102704, 26208, 131562, 32686], [11361445, 816493, 71279, -101463, -71534, 8483], [24161437, -1083938, 34570, 13784, 58818, 18117], [-10618469, -503305, -158366, -38830, 83864, 74493], [-3606112, -399702, -2879, -89918, 91003, 53063], [-4220517, 1677900, -146277, -31035, 197616, 19365], [2283738, -504970, -135123, -44957, -283304, 33940], [-1437186, 1109320, -104878, -63241, -956, 34138], [-8006451, 288551, -158465, -5590, -157345, 60072], [15994100, 1844733, -40541, 20406, 116341, -27653], [-22190994, -80882, 127169, 82827, -10463, -53557], [24088334, 905510, -99582, 67512, 278722, -64086], [7067685, 556219, -92109, 50347, 65978, 32449], [17990636, -255654, 102722, -101909, -87268, 40032], [-10190502, 615674, -171182, -167253, -89308, -48254], [-17809483, -787508, -59934, -120743, -31729, 51952], [22493398, -278534, 94796, 146723, 113389, -13649], [12423005, -1244949, 104497, -88755, 18052, -61457], [-32441520, 239238, -48439, 3998, -335569, -77855], [291480, 259946, -85224, 49702, -59174, -57890], [-850270, -224260, -78798, -85870, -249848, 111014], [6385414, -892093, -233812, 144547, 111849, 39729], [-13401254, -15096, 193356, -30810, -281364, 50658], [22762581, -1480072, 4942, 17842, 387311, 21752], [11541109, 1559243, -128892, -136760, -129692, -32597], [-11673960, 331346, -127834, 135568, 325854, -146400], [31030018, -1758884, -38470, -83756, -79610, -77394], [-849861, -1728361, 55845, -57092, 357373, 81532], [3933115, 547062, 220867, -151518, -186953, -23426], [-28653454, 499436, 125774, -45791, 408324, -23388], [-27794083, -314254, -295496, 63410, 131768, -11163], [-9007389, -982363, -247068, 77290, 35860, 26557], [-17666866, 548774, 136848, 24584, -197972, 62558], [6113784, -2309866, 229019, -18484, -54435, 89191], [-41053705, -396945, -77306, -55598, 116290, -67951], [-2455718, -169581, 368258, 176886, 21044, 1736], [35556321, 1064960, -134677, -109604, 48, -116511], [-32981758, -1477886, 216791, 160870, 48589, -49785], [28070812, -2736484, 306743, 21608, -73203, 52147], [-22511125, 1136167, 166947, -84036, -28399, 61614], [9772583, 5961, 218714, 17769, 291560, 165065], [-13453460, 184504, -538, 164026, 210934, -157806], [-4423786, -970915, -253066, 93351, 66705, 49223], [-11198977, 1101099, -170262, -84950, 246432, 79417], [-24822323, -982441, 49629, 98694, 13873, -798], [-36152851, -1327106, 25675, 71409, -222570, -149423], [2821352, -2007047, 63552, -128240, -291625, 64837], [-9821450, 1713372, 132441, -57724, 301335, -57769], [-10828593, -1053122, -268653, -103191, 358490, 153135], [-28335323, 947967, 82462, -52592, 420280, 82157], [-7873312, -150348, 234950, 204685, -138967, -125487], [-20966819, -2276904, 237684, -60104, 469287, -104462], [-12733639, -2350807, -169621, -137606, 330243, 170544], [-17825569, 2243632, -77357, -27445, -188712, -22755], [-12215057, -914173, -3254, 6605, 603776, -72393], [-7788404, 1002589, -376809, 26948, 547898, 39164], [25348650, -2939902, 167410, 107075, 69260, 79524], [26457060, 744849, -386538, -137883, 26391, 121623], [-45233870, -2125470, 121062, 170929, -199836, -210122], [16528374, -3464113, -137147, 119600, -304519, -2235], [7497264, -893470, -97998, 1818, 233500, -756], [-2665439, -253815, 349387, -65511, -401744, -71075], [-22170548, 4085951, -407118, -5883, 93901, 43353], [11309005, -2224601, 363194, -126260, -37970, -149277], [16129728, 3338954, -151912, -185890, 241074, 26692], [77745565, -443349, 27140, 215820, 192992, 50747], [-25579120, -1001625, 480115, 138069, -77552, 27976], [13527629, 1712201, 51958, 402645, 387731, -195422], [55029104, -1603389, 488282, -121567, -287359, -29087], [11656036, 1876837, -89399, 112517, 118146, -148686], [6821639, -1949679, 119613, -260600, -376419, -19508], [52782717, -1170107, 252207, -247394, -95047, 40420], [-38605022, 770752, -45682, 140538, -191576, -183004], [-51010670, -415944, -434405, 125287, 201427, 190415], [-32502855, -978757, 159011, -124026, -462123, 279194], [6590932, 1960742, -416106, -110905, 291358, -106984], [20318319, 3664444, -312609, 161736, 1115, 17004], [-6738429, -1598723, -267735, 17253, 650990, -221841], [72745620, 1768018, -132410, 60866, 361482, -160572], [-22532948, 3029790, 230308, 260466, 134596, -175022], [-51507748, 4063673, 233076, -39996, -779237, -3027], [32211567, -1448295, 619234, -83602, -277466, -198597], [13813826, 1283008, 319628, -128968, -687344, 277732], [32847554, 1161523, -134105, -324573, 128304, 92506], [-6709852, 648051, 286905, 135562, -435255, -56363], [-15991310, -2075504, -261392, 8764, 275148, -219058], [-41977548, 857804, 339149, -164181, -24159, 264185], [22525498, -1730460, 250891, -20036, -432119, -150007], [-24866097, -395305, 438831, 131871, 142054, 153579], [31953875, 823792, -541210, -154633, 238735, 203854], [22503236, -3202129, -130404, -219807, -264365, 39369], [-99371555, 450034, 154228, -206769, 559779, 28918], [-38262506, -1698822, 554376, 140558, -335658, -219382], [-31275400, -1415605, 234535, 178162, -326587, -131653], [64835130, -4342953, 189975, -126838, -386553, -75211], [-7811687, 3479653, 458800, -60436, 152856, 259455], [19999243, 1197132, -174732, 87049, 348045, 159304], [51312378, -107972, -141635, 243291, -16977, -296495], [-70781831, -455148, 375433, 5729, -223712, -19353], [-38116294, 5234997, -297004, -48449, 173855, -7539], [48044033, 46690, -193522, 106542, 865801, 62858], [37101067, -804489, 103734, 6750, -13584, 31083], [84700566, 2065946, 25047, 274094, -306737, -10909], [-156485483, 3228731, 462869, 137314, 163641, 198282], [-60717159, -4623926, 254187, 111080, 64661, 24128], [-32654120, -2601323, 510084, 85841, 183797, -69613], [30630386, -28910, 811156, 39026, -630354, 202374], [-33722826, 726182, -160068, -338955, -1088822, 210654], [6631695, -2919025, -179139, 71926, -758783, -251480], [-69552460, 2873120, 663698, 223172, -444342, -177146], [-26224472, 2718183, -736501, 261038, 438545, -242213], [-2775868, 2733036, 45992, -144490, -91270, -63296], [-42154708, 1388292, 712150, 549335, 793990, -192012], [80259661, -790605, -84331, -206131, -601300, 115399], [-86191966, 618997, -196766, -159271, 742995, -86037], [-79662073, -1395785, -136094, 352570, 533862, -8173], [-51445359, 4616036, 944737, -61684, -475477, 194358], [28290487, 2809374, 93103, -44398, 1077601, 134424], [-56177546, 5205940, -212456, 261028, 856848, -40888], [56740346, -3513975, 381456, -238919, -191943, 176395], [-45538658, 512498, -704584, 4676, 758982, 391750], [-41585289, -2533959, 181972, -27425, -283370, -146425], [-35958639, 3894566, 510262, -639300, -171604, 4705], [2882975, 3027510, -400199, 591366, -187787, -521840], [-6659334, -3016412, 149040, 332532, 34406, 248028], [-84255423, -4279004, -101992, -321278, -626540, 97617], [13001230, 4306821, -520845, -381361, -148518, -17356], [34037298, -6937291, -496798, -330614, -283344, 203928], [10730849, -848560, 563798, -509960, -741189, -199932], [-120135161, 715470, -178150, -471583, -483493, -169856], [-31043414, 2903068, -173022, -172052, -114850, 282412], [60252057, 2168563, -524355, -92556, -790009, -193740], [38993870, -4970306, 192688, -50873, 97746, -432262], [-19929186, -5763786, 279489, -344300, 542985, 341689], [-19719703, -705988, -233129, 222750, 1053563, 134206], [48221528, -4346102, -245394, 98691, -802768, 120288], [83901050, 2164686, -82514, 24686, 20878, 77964], [-22975881, 3991860, -799053, -424208, -806217, -181286], [-23067229, -1894095, -13189, 299654, -374107, -87644], [-8168199, -1399615, 178634, 610809, -151332, 53225], [-39748256, -1746548, 816426, 37416, -539296, 246522], [27241780, 1592889, -37130, 229180, 322330, 453212], [-26340673, -9704646, 640778, 54216, -908947, 103466], [-22914289, 2623307, 102950, -170250, -1163124, 10733], [-62608610, 2721979, -298913, 86813, -620712, -48870], [-96367186, -3576065, 318181, -50958, -666792, 340960], [73820938, 5444653, -279152, 625045, 41973, -299827], [105808012, 4821255, 435476, -502478, -549436, -67164], [108907706, 795827, -496873, -89938, 677481, 259549], [93937234, -4521850, -1095411, -460025, -15458, -24620], [-150440973, 475131, 958463, 44620, -620143, 369606], [37220450, 470848, 1496336, 283044, -271010, -455884], [-37171729, -2946639, 697155, -382379, -939834, 241237], [23692366, -746680, -215572, 519428, 16158, 68770], [28727294, 3890510, 653326, -248159, 156056, 131084], [2284479, -2750223, -388185, -500863, 951057, 241886], [-64844249, 6241561, 1090432, 438957, 104922, -364469], [164005676, -5245185, -747138, 120291, -1647, -78525], [-787411, -4004084, -327689, 113449, -211476, -729721], [-161413503, 8886252, 1277116, -433326, -1050224, 64837], [186595682, -2928805, -75275, 325837, -356488, 151754], [-28423349, 7209097, 141615, 148701, 1425557, -349338], [-35569817, 4658125, -65560, -189876, 1358716, 207143], [28316035, 542359, -443365, 113763, 140815, -269056], [131950773, 1557554, 1066517, 201808, -521577, 192072], [-27469091, 1731417, 254872, 319320, 188920, 170395], [-795226, -7002565, -206082, -221178, -177578, -232116], [-59266962, 1275698, 6841, 311385, -796942, 281374], [-84034466, -5802656, -365331, 72916, -156469, -89789], [-179445990, 9440653, -693255, -4148, -1109389, 41149], [17458228, 1052620, 718446, -52834, -1076686, -547792], [99631434, -10425762, 32698, -613830, 127882, 49072], [-73207762, 5096143, 228194, 4511, -714383, 370009], [3270436, 1220593, -809126, -160127, -517665, -485953], [-198232053, -5866284, 567125, -292751, -1870624, -62427], [58970268, -653676, -269218, -541071, -707194, 267464], [100791182, -7991298, -358620, -442904, -208374, -145064], [-56748928, -7576073, 467984, 219755, 648715, 86473], [30097212, 7580953, -431809, 65339, -1087598, 792622], [109463340, -679972, -1086797, -59494, 17903, -323031], [-131353790, -959385, -735518, -357783, -818279, -29423], [-54176807, -1324747, -1273227, 194443, -1198580, -187229], [151936132, 1371120, -376354, 122790, 661028, 563906], [198637574, -593382, -455370, -161724, -370554, -216858], [142837924, 57312, 1759186, -271578, 668620, -86372], [106450334, -9588064, -1574134, 564167, 364218, 77374], [-65222028, 1825310, 1578379, 623060, -163483, -311793], [126112294, -11332842, 685914, -547272, 448860, 233130], [76386870, -2891387, 681494, -578865, -524657, 165707], [-40377199, 4768370, -27433, 596589, 451542, 517541], [45936551, -10272399, 65931, -80142, 79383, -62526], [-21050531, 4782848, -723313, -106018, -951015, -235088], [-115193824, -11300307, -662588, 741426, -413042, 106528], [-122901918, -4122588, -1070189, 460384, -216167, -445447], [102853853, -11412404, 1315450, 623628, 1095384, 206893], [-85284105, -8584488, 243530, -237061, 491045, -232148], [-24267360, -13409910, 910558, 17434, 631552, 275282], [-141506106, -6296335, -44005, -1391417, 234462, 26256], [35520110, 9196091, -214764, -843125, 535219, -280727], [289399577, -5895557, -705643, 71164, 880693, -64818], [-174490981, -1948412, 188901, 624138, 1440701, -473640], [142173696, -4347640, 272224, 465432, -2550448, -161528], [-111576222, 9928214, -1182149, -28827, -1068877, 207829], [-39872674, 11106821, 88653, -686303, -878762, 320020], [214867649, -404157, -163478, -95982, -1436480, -320579], [-100133511, 8695101, 568261, 24353, -315824, 182133], [7510758, 3867300, 926176, -46810, -2010158, 22314], [168472476, 2005568, 1150359, -33950, -1670039, 439939], [-91509007, 1683864, 1615347, -406396, 2217, -320830], [121859508, -7470189, -850313, -491424, -1874399, 53725], [-260381114, 730108, -1432098, 537916, 250754, 57724], [13021986, 8345352, 102053, 301632, -798577, -273889], [120270539, -6673315, 767986, 194716, -536278, -170695], [262221737, -4652795, -1375479, -348358, -261955, -283522], [-46255518, 11747043, 608367, -156156, -2081490, -47730], [287939552, -8810070, 1907370, -53319, 1108098, 7950], [-11304102, 22341419, -29950, 75077, -1434435, -57077], [72113318, 4713089, 750534, -473352, 1607911, 695487], [-100416426, 1711628, 61082, -826816, -1220778, -296822], [11742482, 6562874, 669578, 133329, 1448352, -725830], [-23379490, -9534501, 680599, 879345, -1340858, -217472], [-59289314, -8588958, 1761428, 93232, 755864, -570078], [-100565513, 13563921, -943204, -375276, 357206, -393667], [-57305268, -6437837, -1126142, -441972, 1345401, -287], [-660557, 5842425, 1028116, 171068, 2028310, 306393], [-109724477, -776589, -706834, -58624, 507404, 664717], [111511246, -6282802, -830920, -733957, 1357290, 564634], [-173127750, -4857083, -2151546, 39681, 1073135, 254187], [36498157, 17028520, 484261, 249869, -871638, -473547], [313471916, 10176490, -768253, -645314, 1619557, 994419], [-40842665, 25876286, -1366064, 111401, 1209161, -131016], [201775096, -1143911, 361964, -500344, 606685, -51371], [-277345399, -11065230, -1728797, 211045, 388324, 865103], [-228159261, 10399351, -1668344, 375765, 763269, -395774], [91402361, 6352430, -1534194, -113821, -709559, 517280], [-161536197, -1510031, -464661, 204741, 832391, -54402], [79943067, 11887709, -585254, -165962, 164654, -279175], [-136508284, -24644180, 674836, 537564, 286020, 335212], [-71434222, -2225926, 2335801, 449944, -1382383, 262829], [-76081961, -742180, -697082, -835878, 1406846, 528253], [390533287, -7336715, 289768, -164364, -450780, -250409], [419965539, 14300756, 116885, -987918, -42345, -245944], [22684619, 10449514, -2070976, -822002, 2661421, 339876], [-158408135, -5679871, 279653, -348216, 1346283, 599174], [-16535656, -190014, 2454466, 331355, -2205272, 68298], [-68329352, -1579568, 769644, -1392594, -105136, 464748], [233273650, -2247346, -2995817, -1542703, -664654, 786828], [395271531, -4749907, -1961678, -212347, -830188, -224895], [-207670742, 13563247, 2314163, 237640, -1301723, 380967], [67828847, -1105087, -1927130, 948592, 1165898, 340559], [57714382, -11141600, 760004, 380164, 397690, -614172], [154532232, -4227688, 648212, -81634, -2259600, 341368], [136166746, 1032944, -123647, 646102, -3001675, 149465], [-122125948, 3307172, -2339071, 569706, -120435, -12565], [11726725, -17407165, -1261444, -687300, -364944, 182177], [-215144698, -16038143, 1962964, -191592, -3166515, -442793], [266397150, -9364100, -466645, -262116, 2459317, 214787], [-310903767, 33194581, -2151680, 1186349, 1598973, 197352]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_230_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_230_6_a_h();".
// 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_230_6_a_h(:prec:=6)
    chi := MakeCharacter_230_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_230_6_a_h();".
// 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_230_6_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_230_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![-6422832, -1797792, 252311, 12081, -1118, -11, 1]>],Snew);
    return Vf;
end function;