
// Make newform 441.6.a.v in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_441_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_441_6_a_v();".
// 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_441_6_a_v();".
// 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 := [294, 7, -97, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-49, -1, 1, 0], [52, -89, -2, 1]];
    Rf_basisdens := [1, 1, 1, 2];
    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_441_a();"
function MakeCharacter_441_a()
    N := 441;
    order := 1;
    char_gens := [344, 199];
    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_441_a_Hecke(Kf)
    return MakeCharacter_441_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], [1, -2, 1, 1], [0, 0, 0, 0], [-107, 8, -9, -3], [-96, -76, 1, -9], [92, -108, -8, 4], [-72, -220, 1, 15], [-1804, 300, -8, -20], [-147, 254, 103, -5], [1523, 130, -94, -24], [3508, 1028, -95, -9], [1128, -328, -62, 142], [7142, 816, 93, -33], [-3818, 324, 56, -28], [-3095, -1338, -13, 239], [9011, -888, 163, -71], [1566, -796, 52, -240], [-2286, 2764, -193, 465], [-21390, -3660, 24, 180], [-14074, -3056, 143, 93], [-11635, 2286, -96, -786], [-50001, -6432, 1005, -129], [-20966, 9508, 1582, 622], [47705, -464, 863, 669], [42794, -11428, -340, -1264], [-36272, -1320, 765, 1527], [7979, 3008, -1057, -475], [16246, 11408, -575, -1917], [-38784, -5704, -518, -2594], [15991, 8210, -572, -2658], [-162257, 10216, -2711, -989], [111034, -3972, -2326, -502], [-132886, 6028, -1921, -1551], [-216046, -9708, -1142, -926], [97781, 17286, -1485, 855], [-134124, 34880, -3314, -1206], [158772, 3460, 44, 3624], [-33474, 11392, 716, -4900], [-300818, -38476, 3978, -2550], [-216140, 46552, -2810, 82], [31272, 25304, 10369, 3291], [-160096, 24536, 11306, 3230], [201845, -45720, -3030, 822], [-604506, 6500, -2966, 442], [-237412, 53824, 3644, 5796], [65622, -65960, -13246, 126], [-94047, -35890, 3271, -5733], [455117, -18216, -1937, -11387], [-75572, 92208, 8829, 8775], [-270940, 10980, 5806, 11878], [-71532, 56892, -17418, -15666], [-259031, -78384, 13875, -2007], [14259, -91808, -18313, -7843], [546658, 61220, -4572, -5736], [-576686, 44348, -9006, 7026], [-481465, 70506, -19925, -2861], [-444633, 143186, -10367, 1341], [-484176, 83968, 30143, 12237], [-392274, -215252, 18470, 15926], [-689502, -83584, -21317, 8073], [-610941, 91658, -5171, -4919], [-146604, 43244, -40853, -8907], [164408, -15456, 20806, 23410], [14959, 42576, 1062, -2718], [780725, 60774, -19637, 15151], [-775012, 67032, -47463, -1413], [-109623, -110720, -1732, 14244], [-936764, -337640, 15196, 33172], [-351162, 86192, 19960, 31752], [-1661632, -49232, -29850, -21510], [950450, -163424, -18498, -21462], [452945, 175102, 60866, -12492], [2346248, -18072, 43341, 42543], [-2114336, 59960, -7151, -23133], [-838166, 89616, 68798, -1390], [-2321654, 273804, 13640, 6620], [406460, -276704, 46559, 37725], [650864, 32784, 45196, 36316], [1286857, 229584, -76362, 690], [950592, 340992, -3942, -20802], [-81850, 288516, -36003, 1131], [1897420, -9456, 54626, -58258], [395586, -85624, 45661, -18033], [-551833, 290938, 45833, 7965], [2806793, -35808, 24025, -34061], [-575784, 23004, 21762, 52362], [-764433, 43120, -3982, 32454], [-1762494, 474644, -7502, -44582], [-3914808, 128464, -7339, -6009], [-3266660, -126192, -58930, 7010], [-3533566, 66620, 3962, 38594], [-5022309, 152146, 13970, 26472], [-2096679, 426424, 22409, 59267], [-2370786, -418724, 809, -15249], [-1741608, -575052, 137094, 28374], [4323007, 159554, 19359, 9423], [-2536738, -554064, -38198, 86014], [3617664, -208660, -38537, -39687], [-1830754, -217620, 91881, 52551], [1325796, 207964, -77812, -83568], [-1396145, 335258, 70359, 101847], [-1552393, -202328, -45651, -52521], [6342656, 255836, -48382, -59518], [-3182434, -957504, 22605, -50673], [1381425, -335512, -110360, -55872], [-7669203, -8792, -64807, -45709], [2821492, -1332180, 76022, 75662], [8134426, 314376, 43478, 81866], [-4762675, 53896, -228916, -70068], [2711413, -1515666, 21174, 98304], [10308588, -889556, -91018, -23346], [-7952346, -116112, 25104, 148896], [-4437510, -1078984, -39311, 69747], [-3778741, 14994, -67779, -62175], [-7776890, -597120, -97858, -143662], [2532814, 170748, 103923, -149475], [1109978, 321788, 24332, 40928], [-5232289, -407526, 237547, -4349], [4725312, 1464528, -182172, -72444], [4494822, 49052, 292933, 3723], [13360509, 423712, -80530, -1638], [2271853, -480234, 236831, 85451], [-1529153, 1336968, -21529, 84725], [7285096, -201936, 212823, 27261], [-11127675, -476826, -70941, -41793], [3276694, -1909552, 104440, 24024], [-9237818, 669960, -185248, -121576], [-6476633, -432678, 26832, 134298], [5954630, -1349468, 144671, 188625], [-1705562, 801188, -39251, 165363], [-467850, 479760, 344640, -72624], [9535683, -904694, -85108, 58926], [2746562, 721260, 158076, -73104], [-2747792, 931500, -381676, 38576], [11680915, 54816, 117354, -228882], [6354650, -1946620, -72568, -110212], [-1406826, -95320, 149134, -63006], [-3788955, 629686, -266245, -21865], [-13440416, -564552, -136162, -17062], [1872586, 2241084, -358122, -47394], [-6718903, -651098, -465855, -170391], [-9846008, -1086640, 134152, -112632], [2814321, -1916712, -61497, 104421], [12283192, -700836, -19047, 78279], [10392660, -1924724, -165676, -171304], [13283626, 897048, -228663, -46701], [15138038, 1614784, -115122, 34986], [-5808770, 916224, 216366, 334554], [1222538, -379920, 12100, 148636], [2996016, 3087184, -176530, -1974], [-2498970, 349668, -216126, -75198], [3318440, 1958760, -109719, 88059], [-3368564, -479356, -285678, -174006], [-19072660, 416820, -135591, -243177], [6525006, -1737828, -461070, 10746], [-8668696, 276048, -58995, 74031], [18437026, 680048, 181044, 331572], [24843437, 623392, -216580, -117612], [2572421, 1080638, 21791, -19093], [-3527248, 1605760, 209550, 428490], [-24928266, 1662712, 220946, 290918], [388221, -3875090, 318860, 254322], [-14480759, -1377000, 139121, -291037], [5314078, 2196936, -171148, -136252], [16747196, 2788848, -396632, -86960], [8391761, 1007118, 396954, 531612], [17801970, -2693752, 590209, 581595], [-34034265, -692504, 332282, 126390], [12206221, 2922182, 24267, -689841], [-2446829, 18504, 200567, -190387], [1556366, -2736812, 320936, 531228], [-16812544, -3570204, 357928, -380372], [2000045, -440376, 4368, 213480], [9175885, -3492762, -588300, 20658], [7074204, 1074704, -362132, -674972], [17190426, -4058888, -122149, 262737], [10828987, -149062, 877743, 585879], [-18883480, 2557000, 482024, 443280], [-13124358, 5062124, 607105, -131889], [-3602083, 359214, 208554, 231996], [8032173, 449328, -677343, 100899], [-4771330, -1236504, -773007, -273621], [710818, -1287796, -863766, -110910], [-18738914, 1951224, -807328, 98024], [-13015951, -5389786, 39587, -128617], [29123394, -3264716, -391936, 128148], [-3084654, -632212, 578113, -80121], [-18670173, 2019136, 820850, 62022], [18236480, 772036, -3702, -771630], [7253343, 4699280, -466016, 565344], [15517179, -3635328, 775785, -67053], [-22073272, 2154468, -446772, -488088], [16664134, 1453284, 1229006, 207302], [4168009, 366224, -381807, 69579], [-819754, 6260940, -126950, 614506], [-13433482, 5104872, -856617, -507699], [-23403902, -4553740, 59392, 383844], [-17558154, 468024, 632748, -434628], [-30226970, 7528116, 848750, 113654], [22620439, 2416810, -715073, -19361], [17605576, 997008, 1097859, 156753], [22952282, 929992, 121304, -907920], [1313886, 5941352, 765736, -582288], [-20237328, -4564720, 911122, -62714], [48164027, 2360114, -341389, 544595], [-17128600, -5340920, 478952, 239088], [-6586957, 718152, 349765, -66377], [-171276, 2614956, -1689180, 16488], [8792502, 7655372, 164326, 37998], [-22072814, -1180072, 371599, 522069], [18224325, 3876150, 305547, 1248399], [37975221, 4729654, -714337, 607131], [-28271315, -911216, 172141, -10481], [640328, -1354012, 1351016, -1852], [-7082610, -212584, 1230433, 69099], [-3129601, -3379158, 1019289, 1302765], [51245804, -5652164, 125796, 108216], [-51927706, 9060128, 310634, 66662], [-43998474, -8049972, -50964, 290904], [21761792, 2861040, 48855, 69141], [-27984048, 626840, 394801, -184413], [60184190, -10931912, -371784, 158760], [47079977, -3916050, -620769, 34995], [-21994287, -2774608, -973013, -790703], [-8683672, 8966952, 1003662, 872514], [33544054, -6830612, -1095350, -920534], [-39137324, -11561168, 284350, 868282], [-80843681, 1266450, -301140, 190374], [-39872980, -3645024, -1219154, -832046], [33263166, 8798572, -565663, 261135], [63187080, -1328152, -1407212, 270084], [74887493, -3719130, -331104, 92982], [39761710, -712780, 1595160, -248292], [26327894, 5287804, 425615, -1293615], [-22540384, -13029252, 374602, 691522], [-3466971, -12150184, 207778, 1106334], [-93544165, -1213310, -502077, 908835], [-1169868, -922176, -75240, 471000], [17430924, 1769660, 1324798, 784270], [-67518404, 5374272, -539902, -1184818], [-14783368, 6634432, -232675, -822465], [-20186239, -7862282, -1499752, -1962018], [43653066, -9481904, -786868, 119892], [47954088, -10661076, 319572, 1104312], [-58907204, -2786240, -1487504, -132536], [32230971, -11071870, 279076, 1203726], [33878588, 13517576, -1288294, -28210], [19271904, -4709744, -1148872, -1386744], [-35410032, 6875256, -2036496, -498360], [31767650, 6204600, 234657, -778149], [31733098, 4963800, 478706, -378394], [-8471990, 4937384, 1685484, -384228], [23701038, -6404728, 1750231, 1511037], [45887807, -11969198, 359865, 307509], [15763709, 2494144, -608199, -1916805], [28979462, -6623100, 515133, -1984965], [-23227804, -1754096, -1611169, -652491], [-257865, 7661610, -456417, 254199], [134719315, -2806536, 223581, 125487], [14910224, -18632160, 42513, -52293], [-9502335, -2473304, -2028187, -1476025], [-682526, -106344, 1051179, -499863], [-87918212, 1735632, -701397, 412929], [189628250, 2395808, 13340, -650524], [-30329150, -13094916, -847149, -960915], [-17289453, 15499098, 890199, 1548807], [-25566624, 1649764, 2227034, -252550], [86083500, -6314960, 190472, -2035176], [7930747, 12191786, 2518347, -15213], [107782896, 1759016, 2201335, -891795], [34487066, 1118460, -1165926, 70794], [27131859, -16789240, 384544, -500712], [-22583005, -4901214, -2008566, -1893120], [83708739, -5449168, 3067297, 1000803], [-112177520, -2657328, -1633260, -2044740], [-59313181, -19809832, 932441, 787835], [-42765236, -498844, -93299, 2683995], [-100461690, 18034408, 477839, 103605], [-51899912, -4863280, 188544, -481704], [-141933994, 5008796, -1485862, -1502854], [-35102903, 10527614, 1496476, -472302], [-92078736, -5644204, -325886, 2586586], [-140864364, -9157120, 981895, -308811], [-65117822, -5539260, 151818, 174690], [-9534844, -22687716, 1287238, 1997950], [-17452569, -27988016, 805526, 18738], [-54560061, 23265786, -1125597, -1860189], [-26923811, 7949526, -1660116, 993354], [-54621012, 3054720, -1709598, -1250754], [103649770, 7617764, -2532414, -266262], [-34443283, -16281056, -2445771, -2371713], [107871524, 1425384, 1249114, 474574], [-47741292, 23549136, 2656200, 365928], [-1230600, 2748580, 2454485, 239859], [-89092171, 20711206, 1785741, -278331], [-50651856, 6910208, -4495739, -1895625], [143199486, -19995892, -2006474, -976922], [-3918284, -17058136, 951318, 2888370], [-41770418, 11864448, 1180089, -1663101], [127944890, -2932208, 969944, -621624], [-28953637, -6044934, 3080767, 141343], [82236219, 14713810, 687209, 2028741], [36290841, 1856936, -1347215, -2001269], [-150710, 2108360, 531649, 2945235], [-94746734, 854984, -557447, 1431171], [-22366607, -28584336, 2741927, 387869], [-97320862, 4388656, -1374625, 606717], [84898850, -6568312, -2336206, -3442354], [109985070, 4642552, 226991, -142443], [-17842538, 13357320, -2801038, -2776426], [-33978890, -10720772, 2124532, 4336], [169637360, 2975292, 2876758, 3087742], [46233808, -15591040, 1956832, 46800], [2359908, -11105708, -1555192, -57124], [-116569094, 7052520, 1790007, -1191987], [-93232125, -26220480, 1070925, -699297], [-53936540, -6261492, -1578664, -520612], [31047723, -17825920, -3636140, 83004], [-161235624, -8922624, -32790, -2444562], [-121921040, -2483232, 663849, 3501075], [-143737830, -18201440, -345841, 2236317], [58661187, -5725542, 734583, -3191193], [33350744, 9842712, 3890397, 1249959], [200598928, -21304320, 2498174, 2108378], [-31004259, 5104904, 444142, -1655982], [-23421834, 28722808, -1392037, -1282863], [-32492246, -12643980, 1278111, -1165287], [-120128530, -4749872, 2788530, 2128566], [75297855, -35654838, -1904745, -1908369], [28906518, 33653164, 1828037, -559413], [14957707, -10377126, -3643357, 1904915], [89418568, 4549032, 1339533, 1242999], [-233065816, -4055088, -1389122, -1392662], [-115259382, -629948, -4204849, 678249], [181695180, 11417872, 3826970, 3692486], [840704, -11338212, -586158, 4353666], [332335726, 15010104, -191230, -156034], [107431495, 16492896, 522180, -619284], [-140474703, 15045710, 3665800, 3328950], [-9772974, -11666132, -1724077, -2453403], [39961512, 6764980, 1382948, 2136656], [-178832469, 30937362, 2773893, -1274271], [-155115993, -37162214, 2848952, 1301010], [54225629, 18504070, -6322335, -926151], [59081199, 2617784, 3072913, -1378949], [111522906, 3265928, -2238473, 1416621], [-8216346, -7244684, 1146614, -1576218], [-75929694, 7092984, -2212788, -3942084], [-5822957, 7516394, -2994717, 3727227], [-105078000, 40937756, 4297648, 4288524], [129385675, 22160408, 2530909, -2406753], [-77505546, 38367884, -3740180, -2039456], [45859877, 11831518, -309829, 2658999], [334802730, 11059324, 820523, -1270323], [-197680878, -6799456, -6203216, 2187592], [-297495626, -556596, -1000804, 1851128], [-50553541, -24709152, -556349, 1437169], [63155084, -39192608, -1875204, -5564100], [-24608838, 24665568, 3613584, 3548064], [39659974, 7532660, 1619608, -2691348], [49033312, -24229040, 3056626, -2154818], [-94755770, 23850408, -6707524, -1985620], [28083548, 18460600, 1569774, 1549410], [82396828, 7927140, -2809587, -6550365], [-21089349, -20004848, -4477783, -1331397], [409901754, 6977060, -3133226, -346394], [51917040, 15865528, 5353595, 3496737], [-158459039, 20696384, -3261137, -4125267], [12570307, 455448, 5409899, 4494305], [-32849944, 26669608, -3315199, 844371], [190969706, -11179824, 249076, 1345684], [-460731027, -17471058, 2179761, -298695], [-199514456, 28438680, -8724669, -5976327], [211851780, -18420020, 1914563, 4607589], [99261968, 35839936, -2280504, -5879520], [142190256, -27458044, 453316, 114424], [-268985723, -1778712, -475401, 2358333], [427478110, -5869332, 3085046, -208786], [201576578, 23563336, -2227969, 3499941], [-64602058, 9436244, 6066026, 3302954], [-13285374, 46639152, -684378, -3502926], [322125168, -16377104, -3678520, 287928], [-91133264, -39028700, 3067576, -2621132], [101466710, -58025000, -2900254, -1283538], [-198263674, 27392012, 10085534, 598598], [-206412808, -25207512, -2140971, 161535], [68195196, 5006936, -3577814, 985446], [-40737496, -38093412, 5870973, 4480299], [-95617992, -1012980, -1708242, 292638], [178028817, -17746328, 657680, -7214184], [188862382, 47300340, 513188, -4035376], [-255018935, 25578144, 528497, -3274429], [32960290, 7605624, -2654403, 2642343], [44335010, 12630352, 1895676, -1319436], [-93134601, -18001448, 2632649, -576669], [-271071363, -12645442, 8539927, -373653], [-332028210, 33826452, 3923256, 1725324], [32563780, 23386676, 2808973, 8680803], [37213339, -54222198, -3564853, -5263813], [-37035872, -40036476, 6106002, 5910090], [180572852, 10225612, -3819084, 4491312], [-22001316, 20919040, 13400588, -1129500], [254600542, -28200732, 565040, -4160284], [-143238505, -55456326, 761887, 4612687], [12278553, -16043818, 3274372, -2675022], [533931190, 21913400, 89344, 1483704], [102526374, -30184640, -5636158, -6529258], [-153690842, 36381600, -194529, -653547], [-222440297, 35255880, 2122319, 3864533], [40124233, 7435536, 412839, 2541861], [-107867353, 12185098, 4466469, -501759], [105157719, 53162992, -934045, 3892889], [190757706, 10656016, 800990, 2988090], [195508162, 19273184, -1193915, -1281873], [233173567, -64838046, -4318585, -2914369], [70684436, -25286592, -882338, -7462118], [-30665864, 13510512, -7615275, 1650039], [-144503676, -12490692, -4797156, -6902112], [-323296706, -37151220, -3321622, 5870498], [-96952457, 22889658, 1557521, -992995], [-206440494, -55014976, 3539437, -967017], [217519314, -13042172, -3281314, -5252530], [-20037129, 33427232, 2151529, -3658397], [70029478, 8374896, 12656700, 3920916], [32320802, 33351468, -10559900, 1331296], [102667763, 62435792, -1442959, -1486165], [-163762860, 18656148, 5470440, 979788], [-322879666, -8361404, -4470874, -8056314], [85883540, -60682608, 8416924, 9312820]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_441_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_441_6_a_v();".
// 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_441_6_a_v(:prec:=4)
    chi := MakeCharacter_441_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_441_6_a_v();".
// 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_441_6_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_441_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![204, -186, -94, 3, 1]>,<5,R![-3188244, 302700, -7657, 0, 1]>,<13,R![149501563456, -515112852, -1148423, 462, 1]>],Snew);
    return Vf;
end function;