
// Make newform 570.6.a.l in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_570_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_570_6_a_l();".
// 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_570_6_a_l();".
// 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 := [111960, -8028, -1748, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3399, -1592, -3, 1], [14808, -1466, -24, 1]];
    Rf_basisdens := [1, 1, 147, 294];
    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_570_a();"
function MakeCharacter_570_a()
    N := 570;
    order := 1;
    char_gens := [191, 457, 211];
    v := [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_570_a_Hecke(Kf)
    return MakeCharacter_570_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], [9, 0, 0, 0], [25, 0, 0, 0], [-7, 2, 1, -1], [-85, 3, -5, -2], [-183, -10, -1, 1], [123, 8, 17, 6], [-361, 0, 0, 0], [-470, -23, -2, 41], [-2205, -159, -23, -50], [-17, -64, -61, 86], [-3589, 70, -151, 135], [-2819, -45, 51, -92], [-4205, -220, 155, -21], [-8760, 545, 116, -115], [-4528, -643, 294, 37], [-9352, 216, -468, 646], [-3191, 1006, 111, -342], [-21276, -1384, 92, -444], [770, 682, -258, -256], [-10454, -442, 28, -318], [3998, -952, -570, 360], [33582, 1757, -418, 611], [33015, -235, 421, -822], [-39961, 1572, -931, 875], [13806, -2026, 1256, -344], [-11020, 1454, 464, -1510], [-22170, 3186, -198, -1682], [20664, -530, 1002, 48], [32150, 461, -992, 103], [-35242, 5134, 2246, -1870], [-47691, -2203, 621, -2430], [26930, 4544, -256, 3008], [-24048, -1804, 2412, 1582], [-38026, -2630, -4844, 254], [-195799, 6140, 3589, -3070], [-127622, -5960, -308, 1102], [-255025, 1164, -1921, -2023], [-175353, -8904, 1279, 568], [-112092, -8229, 5274, -1247], [-120186, -2608, -3718, -1866], [-451766, 7718, -1004, -792], [-78161, 16269, 7311, 1770], [-252539, 1782, -9181, 4407], [-99303, 2864, 3723, -5126], [-16786, 14042, -7626, 6664], [-570890, -6512, 1242, 1072], [-789916, -9122, 3232, 7658], [24059, 13556, -3161, 5104], [144255, 1684, 3829, -12768], [118690, 6446, -2136, -2926], [128437, -821, -15003, 5054], [-558676, -22934, 1954, 5686], [-110705, -23997, -3009, 8098], [295728, -17393, 6466, 8081], [101918, -10929, 20674, -15435], [792233, 8793, 15871, -15566], [-427472, 56342, -7640, -5908], [-716342, 45182, -4864, -4762], [450365, 20479, 13011, 1810], [-1295185, -4342, 9463, 17131], [112286, 40199, -12012, 23709], [-427436, -32048, 15312, -26430], [-128009, -13371, 5943, -34314], [-485898, -65694, -13056, -15804], [594124, -14525, 17546, -5709], [-390095, 15112, 22521, -40122], [-413917, 75306, 7849, 9913], [636434, -44993, -43294, -12987], [-33038, -10648, -13244, 33764], [-412414, -35076, 26792, 13216], [708623, 47297, 35127, -26810], [365437, -58898, -9387, 12965], [-1676735, 36600, -25937, 4163], [1589977, 34424, -10839, -6438], [273444, 78706, -22060, 32066], [2810272, -71440, 21730, -6464], [-1422440, -64370, 10198, 29382], [1517375, 27623, 10997, 38126], [904512, 1014, 4782, -68352], [5310815, 33947, 3943, -2782], [1424260, -99250, -49578, 14674], [529696, 67710, -1168, -8752], [415097, -26182, 13727, -45581], [2038688, -6132, -62296, -28224], [-844930, -7369, 21374, -41005], [2128117, 62609, -41533, 85666], [1976128, 23102, -10030, 10576], [1361448, -94806, -64146, 33326], [-748675, 116122, -8595, 25905], [-3602, -26251, -62706, 53049], [1356087, -67865, 17503, -54016], [-702586, 119346, 86534, -48380], [1918641, -29485, 43209, -81600], [-1656760, -118196, 6540, 26206], [-366990, 50007, -60346, 64505], [4957737, 13339, -41041, 9686], [1738501, -7763, 100779, -85996], [-3521704, -136780, 42104, 13366], [2796944, 195158, 12530, 34466], [1854216, -66768, -47052, 140498], [53158, -3338, -129120, 28454], [-2387033, -174168, 1955, -18756], [-1045331, 4197, 39003, 49438], [371716, -171718, -62496, 46548], [-4690350, -79854, -57300, -60444], [-92782, 304901, -87366, 5309], [-7123910, 63314, -112224, 20674], [-185376, -140504, -119204, 113424], [3794816, 129368, 118094, -36038], [-2250816, -270750, 26288, -91656], [-7252220, -32616, -48034, 11678], [4238659, -227004, 119601, -35294], [-406130, -34416, -147590, 147790], [-1396676, 21848, 26708, -163420], [1383763, 213885, -150607, 88642], [3750041, 404788, 45129, 108621], [28248, 27997, 106788, -57717], [3822753, 52454, 21939, 81192], [-3021288, 331298, 127764, -137718], [12125420, -79076, -32666, 12594], [1347067, 112980, 189313, 44071], [-349020, -149203, -24334, -134059], [-2347458, -283938, 135650, -212170], [331108, -462666, -103672, -32016], [2503194, -156770, -256040, -266], [4584381, -174958, 228595, -126030], [-1497459, -189707, -158755, 49204], [7145671, -190464, -72369, 187539], [-8470904, -104456, 134922, -174036], [-997708, 137400, 265856, -88088], [-6890228, -68718, -162724, 328722], [8519887, 225440, -64493, 219290], [11232128, -165722, -64842, -336], [4267962, -64370, 13996, 40582], [9177072, 7074, -67594, 102994], [-2068400, 154287, -184666, -62981], [3803348, -384268, 46404, -52734], [9216424, 489351, -105058, 98029], [1236516, -307044, 143490, 157628], [-6337088, 138872, 10300, -243152], [-5804760, -508670, -188134, -8076], [4036153, 304560, 207073, 154367], [-12382173, 93514, -85841, 8842], [5382754, 69750, -148276, -192556], [5946066, -210278, -138658, -125776], [-11291200, 122598, -14746, 173504], [-23864346, -110807, -140040, 38065], [4429222, 21696, -236814, 556808], [-15860370, -255102, -63330, 16318], [20515581, -178446, -98277, 91301], [9127636, 358522, 84930, -118780], [-5106037, -117532, 164195, -449551], [-13685084, -366972, 245884, 274500], [10870230, 214392, 376718, -294246], [2271694, -725364, 27454, 172620], [-1492458, 182152, 183750, 37904], [447814, 110724, 98392, -3462], [7982526, -661920, -319276, 48126], [26304829, -173379, -39477, -211066], [-10258362, -1136037, 201374, -265943], [8403912, -407697, -156406, 360089], [-2602505, 791242, 355191, 81645], [-21556376, 516394, 257516, -100862], [6581704, 673377, 217002, 14961], [-6043983, 623268, -510599, 191060], [27506200, 203864, -16824, -104796], [13666522, -301448, -146356, 335670], [4041886, 392186, 670380, -252742], [11469266, -328950, 118404, -368586], [5902824, -329296, -232296, 345046], [21323794, 25332, 455668, -486768], [-757741, 316399, 152523, 486580], [4027674, -483084, -514472, 139248], [-34344274, -996760, 151494, -159608], [-7560110, -745600, 421252, -86218], [2765722, -297364, -441202, 986182], [14988664, 617402, -129830, 652820], [-27988956, 470730, -311292, 329396], [13418182, 817140, -138032, -429808], [10809363, -636058, 85771, 137731], [453280, 990746, -55564, -595218], [10014753, -360360, 503135, -110649], [11414770, -482469, 139924, -715287], [-279552, -992403, -135876, 43725], [35289480, -276322, 259110, 9118], [27612943, 309168, 435965, 33077], [-9087844, 735048, -497476, -154238], [3148730, -438638, -522680, -35172], [-20536371, 491075, 721973, 60838], [-7506439, 948550, 562031, 211179], [-2510960, 633063, 474176, -528889], [-15878310, 60728, -698418, -66404], [47565465, 224667, 383751, -646716], [-18301932, -529425, 29244, 362895], [17596934, 343573, 413376, -93149], [18916023, 1318112, -94879, -63304], [19247258, -1048580, 295420, -686400], [23619327, -889098, -455579, 1080112], [26640552, -498981, 466140, -462297], [-16926371, 583863, -668373, 148676], [-36637950, -125144, -649566, -22512], [47327869, -848820, -128661, -227921], [9890848, 679718, -132330, 786], [21826749, -593183, -16675, -868022], [23459999, 523542, -414679, 932044], [20617186, -1406958, 109034, 252894], [-37623289, 1156322, -858293, 476654], [-735102, 2409486, 1488, 14472], [-44795300, -941204, 21008, -435292], [87926792, 55714, -291974, -300432], [-23432580, -1099556, 601522, -194554], [-25077099, 398398, 334909, -309571], [-2081199, 1864138, 1230261, -333438], [-29745366, -1691278, -104454, 308276], [-35978024, 1202412, 308766, 50966], [50375589, 1336966, -436475, 800625], [7448585, -2591451, 152567, 611272], [-65078247, 1258722, 598257, -14014], [53338535, -114368, -776895, -293418], [-70250406, 1108818, -41852, -100422], [-20904777, 1026160, 794355, -641582], [-9787065, -2091563, 500581, 278106], [-24949969, -2283612, 557775, -272723], [-46674737, -1221848, -604229, -308268], [59394392, 223694, 438314, -681402], [6604198, 1496931, -476216, -540827], [-43050784, 1229374, 988760, -375852], [12285412, 479742, 1282180, -487242], [-44952047, -115531, -758455, -387962], [-15419314, 961534, -1366044, 148060], [16417547, 2259376, -270509, 1115294], [-22101948, 1190256, -1085188, 159260], [-71403441, -1000419, -1119667, 191962], [-44237443, -1868112, -426491, -248633], [50035876, 124813, -555608, 1425069], [48821305, 571778, 1366115, -726562], [-44721306, 2796032, -760216, 141164], [-29073267, -282379, 863845, 941974], [-81754046, -953754, 367746, 875320], [6027744, -1980610, -1021588, 999478], [-53950838, -417326, -904026, 765022], [22612105, -212016, -445119, -304971], [-16698744, 752348, -1746982, 548262], [-89429304, -2128873, 353186, -1248641], [40434245, 2174713, -171867, 1745212], [-25107733, 1362672, -1162957, -207443], [-101380098, 644594, -195050, 323394], [24402787, -1123648, 1262835, -225634], [-60433316, 2015681, 323608, -13017], [445252, -3554726, 724898, 1090308], [14630670, -417484, -703020, 90648], [-68493610, -2079327, 671962, 161063], [9970502, 460938, -713748, 463464], [18581463, 244260, -1081175, -473426], [-44319785, 3346441, -920233, 34104], [9674375, -1840946, 859349, 79490], [-17090402, -549020, 1156778, -248960], [-27997666, 3676001, -1003692, 359501], [30694756, 600190, 22254, 253262], [-33632532, 618878, 385584, -992534], [-36204072, -211580, -772668, -793080], [41724344, -1562562, -402502, -900090], [-25495319, -118586, 94487, 825277], [-102866485, 1836498, -1503343, 901052], [549925, -956516, -926891, -267490], [-43627475, -1566757, 325347, -1880536], [-17171314, -1143220, 843640, -1291214], [-85689108, -1639680, 639288, 1256360], [-30777, -599112, 756769, 1236938], [29119598, -625432, -757472, -81220], [62836588, -642656, 1514508, -458172], [1921126, -1378276, -2331292, 2587598], [35874428, 1563204, 400548, 277586], [-60007853, -239702, 1851505, 123269], [14224139, 1858934, -519885, -443427], [-9507102, 1323893, 1100986, -1021601], [111162582, 685636, 43968, 67414], [-25796310, -1538232, 406360, -1454706], [-93148387, -2744331, 438925, -764254], [56099610, 2819010, -1415862, 1297834], [-32984578, 262816, 1226390, -1271120], [-21346280, -3883361, -1418484, -472085], [-37362342, 2506088, -724740, 1046672], [56166769, -2356254, 3097241, -1669843], [-57923629, 1782401, -1005509, 117510], [-5535038, -1492348, 1318240, 607392], [-30631918, 2151119, -760, -1364889], [48323328, 821704, 1461280, 493416], [57548529, 3607017, 13239, 1781490], [13538468, -5261186, 1048230, -1394736], [5066634, 1005149, -710934, -2618617], [-19678422, -266205, 722404, -2650385], [11510892, -113640, -2079860, 1609518], [-192320891, -363392, 829539, -990721], [158457588, 733880, 1693466, -470638], [-220328312, -449548, -763064, -128432], [127323757, 2189736, 1070847, 671426], [153651928, -1067754, -1836544, 1122818], [-116483315, -1301836, -3914891, 1272743], [41596318, 5185466, -648992, 116706], [69611772, -2569475, -709054, 459313], [142993616, -3528162, -1014304, -178714], [139687551, 2356826, 411723, -1874162], [-19960200, 5798708, -2029188, 1071998], [-33841217, 1552078, -887219, -2559091], [-15201056, 3605092, -476100, 1989624], [93496202, 2249746, -2228292, -2639070], [19290219, 810257, 3971947, -1011798], [-70214236, 2816710, -305174, -24272], [297309252, -2091137, 517216, 60331], [189967534, -1465656, 1307272, -946704], [1251951, -1382429, -2502787, 3256646], [-58589520, -3962388, 1802564, -897446], [15290372, -2462807, -980984, 2366427], [172691568, 161258, 695006, 194642], [11152197, -1649727, -2058051, -229870], [103752516, -3400668, -911036, 384976], [41484740, -1444670, 1470334, -799732], [-12048466, -1210762, -3576456, 328862], [-153596739, 3163168, 1119453, 825370], [-16477904, -2762720, 2907830, 90706], [-30892479, -1738257, 58487, -435428], [93263716, -1404114, -2368128, 1206788], [54570855, -4368872, 19829, -992202], [-134911344, 2461026, 2532198, -949514], [-85110766, -1698092, 1358574, -3808352], [34860428, 5310864, 470868, -2627146], [111906463, -9616178, 204767, 1612666], [146610683, -4533598, -2187435, 1711988], [182328266, 1468312, 363556, -894988], [131619896, 5070645, -1765626, 1433877], [180076962, -1521444, 622906, -724970], [200875498, 500523, 1113914, 560661], [-146975222, -5658598, 2756926, 2542768], [-51515274, 4497693, 603294, 1525899], [232862600, -772696, 517186, -1539240], [213785624, -1036999, 1620458, 55929], [-110400029, -1059120, -1414371, 1749880], [-99689183, -5571636, -1233871, -2012635], [35462725, -131194, -2736917, 1063681], [134359294, -5845120, -662060, 960464], [-69457759, 3631401, 298831, -2628144], [-54838393, -3062868, 3845539, -555690], [249169392, 5547345, 856046, 432333], [-281435918, -3373962, 344742, -754422], [-35876383, 9638192, 4916719, -214388], [-34005582, 5082732, -1595878, 673232], [-176092880, 4066860, 3545428, -1470220], [-37030686, 3254758, 1094572, -5373074], [-143117265, 369436, 2475159, -972146], [-62033099, -2679352, 576739, 3282829], [-252963340, 1479876, 1111362, 1258446], [-56511856, 2832994, 3939944, 1545524], [-213211472, -1379524, 3375874, 1590518], [-171524948, -3256647, -2249210, -2135643], [-76689063, -4962479, -4279687, 88772], [114825739, 7961441, 4233235, 1109180], [-115461619, 4373260, 1421019, -4384298], [157078334, 2472334, -3745882, 218050], [9906488, 3266930, -5202634, 1126238], [-97129656, 9402098, -2382326, -1517610], [45892030, -6129062, -2684042, 4317786], [-89566710, 1907872, -430478, 107186], [-132162053, -4201520, 2047483, -2921571], [-163588235, -1892960, -3351037, 3674921], [69654014, -167122, 3484624, -6623302], [-134057702, 5429118, -725766, -2759148], [-352604952, -957440, 1773942, -1929322], [-167415443, 1297835, -1668059, 150924], [-114288154, 3017344, -1836390, -3238736], [9088248, -9940475, -1058348, -2430321], [46225017, -2390163, 3340423, -2447520], [-334163912, 3833698, 1833864, -4439910], [18003500, 3087432, -2004314, 932110], [-412851528, -5250094, -1450312, -1512718], [-187092207, -4000301, -1112551, 633406], [-86639302, -1024688, -3902152, 3336332], [-320238566, 5448480, 2345308, -1036302], [-110477041, -8737826, 979853, -2968775], [128468877, 212633, -777397, 4861112], [114536127, 6305074, 2753917, 824340], [-173678761, -718136, 2182679, -2463293], [-135565078, 9109078, -644712, 1666882], [57536965, 846796, 805941, -898646], [47098401, 7880690, 5005873, 1731558], [-21762674, 3858922, -5306234, 160464], [29620082, -11881090, 852304, 1178254], [-83884983, -4269066, 3592345, -677105], [207077287, 289208, 3958743, -1646724], [-107049890, 386974, -1318572, 2027132], [-184861568, -3120307, -1573786, -3129023], [-20054365, 786987, 3433259, -871878], [96057401, 2894710, 1426641, 2544747], [-135123216, 896414, 6312416, 3170408], [16030701, 6765298, 3358611, 1907079], [-169888237, 349044, -3315649, -1200338], [-87148971, 1185935, 1757539, -1702658], [-512758541, 1588460, -2679413, 2100434], [93520780, -130352, -450602, -8593636], [-139033496, -277272, -1179686, 7626552], [180207011, -12192350, -4305351, 6357804], [-220442918, 6185854, 3818758, -4048924], [129418216, 1516863, -2508038, 5510961], [-192781893, -13997761, 777337, -10436], [469285068, -4343650, -1049908, -2023118], [-40294730, -1417104, -736960, 4144070], [-158647311, 8297421, -5760129, 5650914], [-338840708, -1728392, 2799424, -856106], [-224564913, -4590825, -3223777, 952742], [-469951597, -2257402, 398841, -1647651], [-23028367, -8776672, -365141, 976310], [-94890644, 2342836, -2069456, 6320560], [-97697672, 206538, 2896356, -5457468], [-98264522, -3565618, 4886272, -1106458], [40402914, -6790998, 60192, 7024734], [-166515246, 7846348, 2042614, -1240960], [386902100, -6094162, 255164, 1266718], [61083550, 1350116, 270900, -7444284], [156611604, -1190178, 3354348, -7888986], [-106064937, 213963, 3491421, -2970282], [-190878223, 6611530, 1547991, 6110345], [133776982, 3059845, -6740734, 5936345], [-310401288, -3158542, 4342992, 2773738], [-116479663, -9645490, 611879, 6341463], [-217614024, -6427013, -7666362, 904149], [100790774, 7235466, -1162638, -4516674], [-174804546, 13677736, 5455140, -372980], [177890298, 3002916, 1491518, 4862926], [-222112191, 5870783, -1295351, -866270]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_570_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_570_6_a_l();".
// 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_570_6_a_l(:prec:=4)
    chi := MakeCharacter_570_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_570_6_a_l();".
// 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_570_6_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_570_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<7,R![-17197800, -1049380, -16260, 26, 1]>],Snew);
    return Vf;
end function;