
// Make newform 1352.4.a.k in Magma, downloaded from the LMFDB on 01 November 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1352_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_1352_4_a_k();".
// 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_1352_4_a_k();".
// 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 := [144, 804, 112, -83, -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], [96, -66, -3, 1, 0], [432, 118, -77, -2, 1], [-176, -66, 5, 1, 0]];
    Rf_basisdens := [1, 1, 8, 24, 8];
    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_1352_a();"
function MakeCharacter_1352_a()
    N := 1352;
    order := 1;
    char_gens := [1015, 677, 1185];
    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_1352_a_Hecke( : Kf := Kf)
    return MakeCharacter_1352_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]];
    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 := 4;
    raw_aps := [[0, 0, 0, 0, 0], [1, -1, 0, 0, 0], [-2, 0, 0, -1, 0], [4, -1, -1, 0, 0], [2, -2, 0, 1, -1], [0, 0, 0, 0, 0], [-5, -2, -3, -4, -1], [10, -4, -2, -3, 3], [-22, 6, 0, -8, 0], [-44, 4, 6, -12, -2], [26, -22, 4, 5, -1], [78, -36, 4, -5, 0], [96, 8, -8, 4, -4], [23, -11, -4, -24, 4], [-124, -29, 11, -20, 0], [-142, 40, 4, -8, -4], [86, 76, 10, -25, -7], [20, -60, -2, 8, 2], [214, 62, 20, -1, 1], [-284, 5, -3, -44, -12], [-140, -72, 0, 46, -16], [190, -22, 32, 0, 16], [146, -80, 10, 1, 7], [564, -104, -8, -22, -8], [-256, 32, 24, 20, 24], [24, -68, 14, 16, 2], [-314, 26, 0, 0, -48], [-204, 120, 8, 80, -16], [102, 4, 28, -25, 8], [-350, -152, 16, -20, -12], [238, 82, -56, 88, 8], [595, -119, -20, -8, -60], [1676, 8, 32, -22, 32], [787, 53, -104, -16, 24], [-400, -76, -112, 0, -2], [-872, 109, -47, 2, 34], [-1014, 216, -36, 4, 40], [1974, -96, -18, -17, -39], [966, 306, 24, -141, 1], [-714, -176, -92, 64, -20], [-337, 313, 72, -176, 8], [438, 168, 12, 68, 0], [972, 228, 48, 72, -80], [2564, 120, -80, 162, -44], [2034, -12, 76, -27, -48], [556, 60, 16, -160, -64], [-537, -147, -108, -136, 28], [1732, 121, 41, -264, 20], [-802, 286, 92, 311, 25], [-1894, 312, -88, -139, -64], [803, -246, -23, 0, -89], [232, 151, 27, 90, 22], [1340, 232, 160, -50, -76], [3284, -16, 56, -16, 32], [-781, -370, 13, -312, 19], [-1632, -392, -80, 24, 48], [-3116, 300, 78, -12, 38], [1564, 63, 63, -432, -12], [-3556, -172, 210, -200, 78], [-1644, 96, 32, -206, 144], [1788, -344, -72, 224, -96], [2166, 116, -212, -217, 16], [622, 44, 114, 107, 197], [-858, -126, 8, -336, 72], [-3507, -738, 119, 204, -19], [664, -852, -224, -20, -54], [986, 24, -214, 485, -117], [1981, -210, 171, -44, 113], [3271, -447, 152, -80, -8], [1438, -624, 176, -17, 16], [-740, 832, 72, -282, 108], [-1090, -130, -4, -201, -83], [-3960, -560, -8, 320, -24], [-2648, 236, -318, 28, 18], [-3318, 476, 62, 49, -89], [-1004, 529, 289, -696, 140], [-226, 1032, -124, -532, 24], [-3096, 868, 80, 492, 94], [-1492, -1096, -56, -134, 116], [-620, 768, 136, 70, -80], [945, -729, 200, 496, -136], [2294, -1088, 128, 291, -248], [-1480, -569, -133, -102, 174], [-4387, 254, -221, -80, 125], [-1306, -270, 104, 232, 136], [4989, -177, -52, -264, -108], [-3848, 456, -240, -72, -24], [-5332, -192, 16, 782, -304], [230, 1172, 28, 463, -440], [2998, 1438, 180, -309, -111], [-6260, 0, 424, -112, -256], [-7236, -201, 103, 40, 108], [2858, 78, -280, -187, -57], [5395, -479, 36, -536, 204], [4974, -850, -532, 231, -207], [7886, 66, -16, 240, 160], [-8248, 60, 240, -28, -38], [-6331, -538, 391, -692, 61], [1256, -300, -88, -528, 96], [7194, 1156, 332, -519, 280], [4069, 435, 88, -464, 8], [-10270, -68, -396, -579, -176], [5095, 1161, 288, -608, -272], [205, -2182, 95, -408, 241], [-4399, -49, -32, 368, 16], [4532, 440, 296, 334, 332], [-3878, -1516, -730, 81, 55], [1424, 2504, -352, 604, -240], [-594, 714, 504, -776, 488], [-733, 1502, 193, -508, 131], [2902, 1154, 848, 408, 192], [-9032, 660, 240, 84, 462], [5632, 512, -776, -344, 300], [11058, 1478, -232, 889, -153], [-14920, -315, -7, -366, -198], [-2846, 1840, -124, -540, 296], [4294, 1174, 508, 231, -79], [-6218, 1114, -288, 616, -624], [-912, -212, 290, 204, -334], [-1848, 2044, -424, -144, -288], [4616, 1020, -256, 1444, 50], [263, -2002, 689, 252, -245], [-1246, 2080, -420, 560, -300], [-6858, 656, 206, 1027, -67], [-9746, -1638, 376, -781, 349], [-762, 1000, 116, 508, -296], [-14512, 780, 384, -336, 34], [-7136, -2824, -872, 1056, -264], [5140, -588, 880, 24, -224], [5978, 1672, -408, -835, 64], [1730, -1390, -380, -1583, -57], [-2200, 2035, -209, 1094, -418], [-12296, 1720, 560, 664, 304], [1582, -1232, 384, 1400, 192], [-2500, 1088, 88, -558, -160], [-5680, -864, -1008, 304, 280], [-482, 616, 584, -281, -136], [-7934, -38, -612, 297, -1], [6742, -296, 420, 692, 40], [-2095, -1474, -925, 572, 609], [14998, -2838, 768, 719, 313], [6846, -2696, -536, 119, -480], [-21922, 1202, -272, 104, -144], [-7186, -900, 162, -105, -207], [-3640, 836, 1230, 960, -894], [8814, 1326, 508, 375, 573], [13162, 496, -736, -83, 704], [-4802, 928, 88, 224, 792], [-1232, -3012, 424, 192, 240], [-14680, 1071, 403, -502, -450], [98, -88, -216, -1151, 624], [-4177, -562, 481, 1384, -529], [-6619, -1977, -548, -552, 132], [-6236, 148, 272, -1208, 656], [919, 2533, 332, -360, -1004], [-16324, -3372, 320, -600, 176], [-8236, 3164, 456, 528, -88], [-2040, -1776, -480, 564, 624], [6854, 1016, 124, -160, 308], [-8398, -4168, -120, -2143, 208], [-486, -1250, 1016, -891, 531], [6249, 1050, 483, -3144, -3], [19100, -3191, 393, -696, -368], [-1011, -1641, 4, 1816, 636], [19944, -1960, 680, -444, 432], [-16532, 2683, 259, -900, 756], [-12522, -1862, 136, 400, -104], [18276, -340, 734, 1256, 322], [3636, 528, 1536, 1886, -252], [-6008, 1436, 526, 1296, 450], [278, -1462, -1088, 1079, -71], [5394, 788, 572, 2109, -384], [-42892, 44, -120, -1208, -424], [6888, -5368, -128, 1276, -788], [21656, -552, 32, 48, -16], [13253, -458, -1217, 2048, -143], [37450, 1278, -264, -1591, 343], [-11640, -4484, -288, 2796, -326], [-21408, -344, 448, -752, 1184], [-8678, 3280, 548, 100, -928], [1774, -734, 80, -1937, -691], [6468, -3480, -488, 1168, 976], [-1080, 796, 298, -2920, -1034], [-11652, 1048, 992, -2558, 224], [12454, -1762, 1524, 1987, -39], [20730, -2184, -1112, -1972, -316], [-8398, 2016, 1232, -1108, 28], [22570, 2872, -646, 2609, -273], [28056, 1488, -200, -2676, 548], [28198, -3578, -1188, 435, -935], [19027, 2706, -587, 284, -49], [5478, -3878, -720, -537, -231], [-13631, 3071, -432, -368, 608], [872, 2708, -400, -1780, -306], [3621, 5047, -1156, -984, 52], [-8302, 3512, -420, 2736, -316], [25952, 784, -1176, -516, 680], [29802, -744, -176, 2964, 524], [-10956, -2240, -696, 2330, 1316], [-23558, 2558, -904, 2360, 952], [40222, -3944, -1624, 1735, -816], [4586, -6554, -792, 3568, -1160], [4202, -2916, 644, -1023, 200], [6375, 4994, 1201, -2660, -269], [-16966, 5158, -1264, 1017, -1481], [-9690, 2888, 424, 2712, 472], [41700, 3857, -79, -8, -340], [-10692, -152, -552, 1232, 240], [7728, -696, 232, 1252, -924], [26849, -5569, 176, 880, -640], [15754, 584, -1284, 2432, 388], [24432, -876, 1818, 856, 998], [-11838, -7154, -1472, 3968, -624], [33614, -2160, -122, 867, -403], [-15596, -427, -1747, -468, 676], [15740, -3560, 264, -2058, 496], [-24296, -1864, 1776, -1064, -496], [-27113, 4794, 2257, -2128, 479], [15090, -1966, 932, -1167, -245], [-18702, 4028, -1884, -2235, -360], [-11084, 5228, -2754, -432, 562], [26424, -1651, -943, -2654, 1610], [-15592, 5088, 1856, 3428, -720], [27380, -8527, -335, 3416, -676], [38901, 5303, 1212, -72, 196], [5002, -6184, 288, 4248, 352], [-31951, -2202, -2045, -96, -1299], [28386, -3162, 1168, -1032, 1872], [-35406, 2278, -1304, 2680, 1480], [-20894, -2302, 2772, -1171, -685], [11954, -3288, 392, -2076, 2308], [32324, -2608, 280, -80, -1648], [-25430, -998, 1364, -2531, -1001], [-27803, 5098, 671, 428, 125], [793, -4733, 1340, 72, 116], [-14272, -5189, -729, 2574, -934], [-25876, 976, 48, -5206, -948], [10954, -7128, -1368, 3629, -56], [6700, 1176, 1368, -5568, 176], [-3348, -11004, -24, -2072, 296], [12842, 2332, -82, 661, -13], [28591, -5379, -1748, 1400, 324], [-6956, 6036, -1808, -1320, 1472], [-32870, -1488, 864, 1205, 976], [3352, 1160, 2424, -3204, -2408], [39306, -9002, -864, 2224, -768], [-35192, -2007, 709, 4390, -1326], [-27206, 1666, -1524, 1289, -313], [-16154, -5010, 916, 2067, -1395], [-17270, 830, -608, -2392, 256], [20054, 904, 232, -2229, -1168], [-15128, -4800, 1768, -1544, -744], [20480, -9991, 1301, 3862, 238], [12197, -566, -769, -1172, -1923], [22622, -2408, -2328, -4628, 412], [-11230, -570, -1848, 229, -1261], [31042, -9808, -1404, 268, -928], [-11814, -1116, -986, 5153, 1895], [-22166, -3200, -1808, 4, -1220], [31220, -3160, 1328, -1074, 456], [28208, 280, -360, 5184, 1352], [65087, -2083, -2020, 2840, -956], [50994, -5228, 1308, -83, -224], [28772, 124, -762, -2912, 330], [-20368, -1144, 3352, 1128, 736], [-24285, 7609, 556, 6760, -1820], [-30746, 884, 812, -2713, 408], [-16404, -14464, -200, -1038, 688], [-55886, -942, -860, 2381, -117], [17360, -2108, -2722, 456, 834], [17482, -6424, -216, -3240, 1560], [19098, 6480, 3042, 1073, 311], [40040, -2000, -1632, -4056, 292], [-14838, -11026, -1456, 1264, -272], [5538, -2088, -2656, -1396, 2340], [-35998, 2402, -2924, -2215, 163], [43834, -8176, 130, 6725, 979], [60960, -6684, 1456, -56, 918], [-55952, -8528, -3536, 148, 1364], [63313, -1365, 1756, 4568, 1604], [-19634, 5586, 632, 5792, 200], [41776, -7559, 1205, 1662, 894], [-28320, 5872, -840, -112, -1448], [-59552, 36, 2656, -480, -1450], [66062, -7264, -138, 1643, 5], [30986, 158, -2160, -2256, 656], [-31677, 9274, 2277, 832, -421], [-43262, 684, 804, -307, -2448], [-51948, -2609, 175, 3184, 312], [-934, 888, 844, 7232, -1020], [17844, 596, -3922, 2912, -1118], [28943, 9741, -1140, -472, -1692], [3584, 8704, 2712, 5632, 1588], [-15374, -5674, -2568, -5263, 1111], [12496, -13268, 482, -1156, 2258], [9184, -1140, 5190, 532, -1658], [38708, -9420, 1368, 552, -200], [-39472, -6468, 446, 228, -1282], [52921, 5043, -3556, 1272, -348], [32330, 8346, -3708, 5113, -1073], [-35071, 6202, 2235, 3128, -651], [-7154, -1256, 2984, 4079, -576], [-44908, 3660, -1408, 4480, -1264], [-6548, 9000, 4344, -4160, -752], [117, 2239, 3460, -1256, 3180], [-49508, -1568, 176, -4638, 1380], [-12757, -4655, -4452, -4728, 212], [2748, -5004, -538, 5904, 1226], [-86858, 74, -1776, 1547, 1017], [-25980, 372, -942, -7348, -1670], [66454, -5334, 2328, 664, -648], [-15078, -16564, -2700, 857, 856], [18909, -2678, -2413, 1172, -1135], [-14284, 4576, 4008, -3536, 1216], [-3784, -9635, -863, 634, -1406], [-28177, 7126, 3885, -8188, -1033], [-42654, 11672, 2, 1049, 1815], [-52254, 6486, 280, 8425, -1621], [-44592, -11272, -1808, 1732, -1400], [-48458, -6792, -3656, -4828, -508], [-13700, 4832, 1656, 10624, 256], [28056, 2736, 1128, -2812, 2308], [-88494, -8704, -1948, 2168, -68], [85448, 2689, -4099, 5566, -750], [12840, 5536, -4424, 860, -844], [53962, 7456, -1048, -4000, -552], [47242, -94, -1380, -2263, 335], [64598, -7762, 2356, -3229, -2091], [-1670, 4014, 2752, -3968, 1568], [-46526, 280, 460, 2876, -392], [45928, 9012, 640, -4964, 718], [-9924, 4156, 2550, -3316, -2930], [-95850, 1946, -1672, -4136, -1368], [-9170, -3300, 2082, -1853, 2485], [27382, -18646, -592, 8711, -1663], [-25034, -1790, 776, -601, -3823], [8768, -7308, 1632, -3848, 1598], [8992, 1144, 3704, 1336, -1896], [43196, 544, -4296, 1490, -40], [8276, -156, 3192, -2504, -392], [-23586, 16080, -1632, -2193, -504], [-77255, 3942, -2077, -996, -1983], [-29266, 1732, 1644, -1501, 1464], [-36010, -21550, 568, -6848, 2376], [38154, -9384, 168, -6275, 3152], [-16549, 597, 64, -1040, 1472], [-81542, -1120, -4260, -540, 88], [502, -8404, -4038, 4119, 513], [-7488, 11717, -1831, -3758, -1282], [-520, -13148, 526, 3940, -594], [64882, 4284, -2042, 7437, -1701], [83032, -11392, 2576, -1752, 3572], [57942, -3636, -3596, 1439, 1824], [-26320, 9264, 4656, -6240, 3232], [17604, 10612, 442, -2944, 2646], [-13150, -20552, -2280, 2364, 380], [91872, 7805, 3537, 322, -930], [-38306, 11390, 1004, 5155, -3963], [15913, 10758, 2415, 3136, -2415], [-23204, -2815, 593, -2528, 3736], [-17064, 15476, 432, -3788, -346], [-50338, 5416, 1560, -1612, -1836], [12412, 6844, -4024, 1408, -1176], [-26654, -652, 2638, -10655, 15], [-35932, 6984, 6232, -256, 96], [-19889, 10254, 5397, -8396, 1487], [73986, -7644, -3412, -3027, 120], [-41906, 22982, -348, 3271, -819], [-41297, -14542, 1341, -2092, -673], [88341, 8267, 1616, -8992, 3040], [85897, -7097, -1504, -2944, 3408], [-3492, -4297, -1625, 7024, -4236], [-18986, -2176, -648, -672, 2680], [-51780, -12644, -2352, -2040, 1232], [-48584, 13828, 2538, -7888, 3878], [17510, -12342, 4608, 3099, 1613], [20374, -11774, 2272, -3440, -1536], [22988, 12040, 4776, 3610, -44], [-128902, 13312, -968, 720, 1576], [-18686, -4736, -2100, 14804, -240], [-55792, -1564, 4448, -2376, -2762], [66232, -56, 3064, 4368, -3524], [-36202, -8174, -1656, -6357, -1319], [-25728, 1664, -184, 4604, 2032], [82150, 5068, 1754, 5491, -171], [96940, -1389, 667, -2148, 1276], [25744, -13333, -1737, 9430, -534], [-55810, 2864, -428, -7160, 972], [-838, -10184, -5526, 3437, 123], [122406, -20846, 192, 2592, -2528], [11828, -1888, -1016, -5110, 2228], [108514, -5444, 1028, -2275, 2456], [-66666, -2600, 2182, 1779, -651], [-29129, 2317, -572, 6472, 236], [29428, 2711, 1847, 5432, -4568], [43727, 16302, 2781, -3416, -1933], [-596, 2461, -4491, 7220, -6168], [-50145, 3818, -4507, -2140, 823], [3228, -5568, 504, -9328, 1520], [50430, -24276, -1932, -6437, -2440], [-33866, -18796, -6468, 4623, -2256], [-94422, -9808, 188, -860, 3184], [107872, -8875, 601, 7530, 1022], [-81436, 5856, 1056, -7526, 856], [73466, 9512, -4264, -4075, 2288], [-64274, 12394, 2136, -15472, -2904], [32994, 7504, 7648, -1535, -5176], [35624, 6320, -3232, -6724, 120], [15338, 7142, 5456, 9149, 763], [-34246, -21238, -4940, -9631, -809], [136111, -7298, 241, 6072, 2703], [-71186, -22408, -5320, 1496, 8], [21801, 4643, 4380, -9096, 1076], [83527, -11335, -1824, 11184, 1056], [60671, -13566, 437, -3600, -4629], [128346, 2168, -6396, 8848, 716], [-162644, 1573, 221, 3660, 1064], [21668, 9636, -1728, -4384, -816], [21816, 18048, 2816, -7364, -712], [-7730, 5402, -7512, 928, 4632], [-105090, 17320, -1164, -496, -868], [89440, -5020, -2720, -6352, 3126], [77450, 3994, -6828, 4453, 1903], [-33707, 6399, 5156, 10792, -452], [37340, 21248, 3056, -922, -5408], [36040, 1468, -10576, 1468, 4122], [3127, -6279, -1824, 3056, -5744], [114392, -3832, -2544, -668, 1924], [-9262, 2270, 2416, 2633, -3417], [70796, 1564, -1152, -984, 3344]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1352_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1352_4_a_k();".
// 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_1352_4_a_k(:prec:=5)
    chi := MakeCharacter_1352_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_1352_4_a_k();".
// 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_1352_4_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1352_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![-976, 776, 139, -81, -3, 1]>,<5,R![30288, 12824, -841, -265, 9, 1]>],Snew);
    return Vf;
end function;