
// Make newform 825.4.a.w in Magma, downloaded from the LMFDB on 21 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_825_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_825_4_a_w();".
// 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_825_4_a_w();".
// 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 := [-200, 294, 3, -41, -1, 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], [-16, -1, 1, 0, 0], [48, -23, -4, 1, 0], [44, 31, -27, -3, 1]];
    Rf_basisdens := [1, 1, 1, 2, 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_825_a();"
function MakeCharacter_825_a()
    N := 825;
    order := 1;
    char_gens := [551, 727, 376];
    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_825_a_Hecke( : Kf := Kf)
    return MakeCharacter_825_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 := [[-1, 1, 0, 0, 0], [-3, 0, 0, 0, 0], [0, 0, 0, 0, 0], [7, 1, 0, 0, 1], [11, 0, 0, 0, 0], [-10, 3, 2, 5, 0], [-31, 5, -2, 1, 2], [-65, -12, -2, -4, 2], [37, 6, 6, 1, -1], [-34, 11, 0, -8, 3], [52, -5, 4, 11, 0], [77, -31, -4, -9, 2], [-29, 45, 6, -17, 0], [-34, -23, 14, 13, 2], [-93, 11, -22, -7, 4], [-116, 18, 2, -36, 0], [71, 71, -12, 31, 4], [-218, 38, -44, 9, 5], [-226, 38, -12, -47, -7], [229, 114, -6, -65, 1], [-4, -32, 38, 20, 10], [433, 105, 2, -25, -8], [210, -35, -46, 2, 17], [80, 57, -50, -20, -5], [443, 144, 0, 10, -2], [-61, -74, -24, 63, 11], [-506, 181, 30, 12, 3], [-74, -139, -12, 0, 13], [134, 181, 16, 71, 14], [44, -128, -30, -68, 18], [-5, -237, -30, 93, -4], [-922, 65, -98, -54, -39], [574, -239, -44, -154, -9], [-620, -227, -16, 40, 17], [521, -251, -112, -89, -6], [1022, -230, 90, -1, -23], [932, -88, -106, 3, 27], [-18, -310, 62, -3, -45], [-716, -292, 2, 210, -24], [-733, 108, 132, -33, 25], [823, 93, -72, -211, -28], [-203, 203, 122, 50, 71], [1517, -266, -46, -37, 21], [298, -140, -134, 15, -37], [-2217, -182, 44, -57, 19], [418, 169, -10, -7, 130], [1338, 390, 14, 123, 13], [-366, -418, 70, 255, -43], [-2498, 31, 96, 90, -91], [425, -241, 150, 78, -101], [-785, -225, 14, 101, 104], [2372, -250, 194, 76, 44], [920, -52, -248, 9, -41], [-292, -204, 232, 258, -58], [1382, -779, 220, 314, 83], [-846, 86, -300, -216, -46], [-1430, -776, -94, -258, -96], [1901, -103, -150, 87, 64], [414, 86, 72, 359, 43], [1947, 787, 106, 385, 88], [3289, -917, -250, 98, -9], [181, -39, -272, -373, -54], [4762, 38, 214, -165, -67], [-292, 1089, -20, -410, -53], [3687, 301, -364, -128, -3], [-4096, -56, 64, 34, -46], [-224, 648, -418, -190, 44], [-418, 724, -128, -393, 33], [-880, -344, 72, -172, -8], [-5282, 1181, 360, 120, 41], [-352, 95, 276, 186, 201], [-450, -1954, 258, 444, 16], [-856, -67, -470, -669, -32], [6232, 42, -118, -187, 3], [1752, 82, 316, -21, 259], [-76, -565, 76, 154, 117], [-2798, -1074, 178, -246, 190], [-660, -840, 114, 79, 331], [3424, -1185, -8, -46, -143], [-510, 962, 336, 747, 59], [1833, -1319, -104, -559, -8], [8665, -1973, -264, -77, 28], [4702, -26, 6, 334, -150], [-2112, 597, -298, -317, -60], [2023, 357, -28, -392, 1], [-4181, 2025, 92, -687, -108], [3424, 871, 174, -698, -69], [-2334, 1118, 188, 388, 222], [-958, 592, 20, 120, 340], [4634, -439, 344, 158, 143], [5824, 88, -104, -82, 182], [-2352, 504, -824, -80, 284], [-12, -1139, -342, -1145, -188], [5754, 867, 332, 1184, 47], [4826, -1680, -60, -1025, -203], [9740, 1382, -370, -284, 152], [-4136, 340, -210, 638, 192], [-2854, -891, 44, -912, 209], [7187, -18, 598, 176, -80], [-2670, 2979, -114, -399, -168], [1477, 1208, 908, 421, -177], [-3866, -1873, 400, -394, -247], [-1310, 1108, -118, -726, -168], [21, -1965, -666, -487, 248], [6134, 2849, -386, 219, -92], [4089, 1343, -144, 370, -127], [-11121, -2135, 404, -307, 276], [-230, -1206, -144, -668, 318], [6127, -681, 1058, 909, -40], [-1478, -198, -1010, -981, -223], [-1376, -654, 360, -864, -262], [4740, 964, -522, -156, -50], [-7514, 1009, -934, 768, 291], [10816, 2372, -256, 581, -473], [7760, -1326, -8, 807, 443], [11308, -145, -18, -1412, 17], [-4534, 2158, -254, 732, 180], [12543, 231, 434, 625, -60], [-9394, -370, -26, -336, -720], [7260, 1443, -884, 292, -421], [3243, -3695, -244, 1189, 12], [6176, 1802, 634, 664, -140], [-10558, -27, -320, -1480, -219], [117, 2133, 328, 427, 178], [-772, -4026, 266, 602, 566], [1565, -2766, 628, -587, -111], [-1527, -447, -942, 754, -563], [7308, 3044, 620, 50, -734], [-5370, 1907, -1234, -237, 354], [1208, -621, -330, -1512, -291], [-4325, 2753, -148, -737, 90], [-12648, -360, -868, -1106, 26], [-1332, 4039, 324, 448, 343], [-4964, 844, 2, -1343, 61], [3726, -2334, -300, -360, 18], [7370, 1884, -876, -1403, 543], [-12080, 2000, 202, -564, -682], [10259, 2101, -174, 204, -485], [-4326, 4838, -62, -244, -228], [17777, -3221, 362, 1339, 2], [5967, -1227, 22, 836, -121], [12402, -1045, 988, 1186, -123], [-11574, -1040, -472, 541, -37], [4524, -2263, -516, -66, 219], [-12174, 1864, -348, 600, 28], [17132, -588, -1756, -1398, 362], [3708, -2802, 694, 97, -697], [2673, -1217, -1010, -1715, -392], [14672, 4106, 40, 148, 318], [4336, 4731, 608, -852, -781], [-12476, 2941, 2040, 235, -230], [-5984, 701, 384, 286, -737], [-732, 1494, 300, -2053, 31], [10188, 4130, -752, -1990, 272], [682, 3736, 1088, -162, -286], [1161, 3089, -2326, -1783, 366], [17495, -405, 348, -50, 9], [-12200, -4579, 1798, 62, -175], [-10644, -916, -868, 803, 1329], [2487, 3819, 816, 675, -272], [-9461, -961, 2020, 2377, -774], [14633, 6779, -1402, 537, -40], [8776, 3704, -1112, 228, 772], [15061, -1857, 200, 2539, -40], [2622, -5042, -138, 14, -106], [-18089, 2908, 938, 105, 565], [-7198, -7424, 632, -1391, -97], [-8390, -6420, 1000, -1188, -112], [-9298, -5150, 288, -2057, 407], [11476, -416, -1618, -1068, -142], [-12590, 739, 1340, -1588, 171], [-6780, 6354, -482, -1981, -363], [-10230, 3414, -1480, -486, -592], [-21384, 5444, 990, -545, -329], [25648, 2521, -1226, -1229, 216], [-4072, -3369, 2034, 1124, 125], [-29701, 3374, 458, 2196, 576], [-6706, 3702, -1776, 762, 652], [-1491, -205, -2388, -1506, -175], [-160, -3061, 1662, -643, -28], [-2016, -6273, 1668, 91, 52], [2702, 9556, 178, 364, 506], [-16261, -751, -1498, -2220, -57], [-25122, 4574, 692, 2106, 92], [5208, 3460, -1232, 1132, -820], [-22822, 450, -932, -2676, 2], [-24608, -3475, 260, 1793, -280], [21966, -2216, 1806, -531, 241], [-30636, 3032, 2722, -253, 35], [-5606, -2350, 2712, 1594, -48], [29376, -1473, 1006, -1079, -688], [-2916, 39, -3128, -212, 187], [-11168, -6136, 2118, 3449, -619], [-1322, -6400, 1742, -368, 14], [2032, 1455, -698, -2786, 739], [-6498, -1705, 1146, -228, 997], [-5072, -7068, 1834, 3932, -310], [-16190, 836, 1728, 3460, -472], [-20071, 1019, -570, -1457, -806], [-23788, 30, -936, -2014, -1032], [-43383, 8160, 3460, 1303, -147], [23488, 6634, -788, -2864, 6], [-729, 2983, 1556, -663, 158], [-10260, -4626, 534, 2316, 0], [-137, -1741, 1912, 3437, 398], [1890, 1824, -32, 550, -414], [44792, -1244, -878, -521, -833], [2054, 3425, -2708, -144, -299], [-6943, -4497, 3034, -447, 1360], [-28606, 6436, 2918, -1357, -33], [44924, -3924, 908, 1015, -371], [-24816, 5744, -498, 2636, 538], [-27584, -2142, 288, 1367, -589], [13965, -2467, -2376, -1143, -1120], [3714, 3392, -3618, -720, -966], [-23819, 6424, 3246, 57, 717], [-357, -4959, 1776, -872, 1185], [16911, -10641, -1570, 1881, 1376], [16909, 6834, -2546, -175, -193], [2054, -1288, -2684, -1736, -1768], [-32644, 6846, -2274, -137, 745], [-28530, 140, -2554, -4401, 415], [-6980, -2377, -2732, -5238, -1555], [23762, -3244, -2558, -3212, -422], [26986, -1088, -1894, -2460, 762], [-43312, 3435, -4606, -5607, -548], [-41280, 5696, 2002, 2860, 650], [21169, -5794, 1114, -545, -915], [-7457, 4897, 1684, 494, -2193], [18548, -4579, -1146, 1094, 441], [-13213, 6617, -2634, -1040, -709], [-8701, -9162, -28, 475, 1639], [-14063, 4791, 1976, 3508, -933], [-12495, 2589, 1378, -2003, -1394], [-25008, -15454, 1582, 1746, -1138], [-30876, 1198, 568, -3198, 640], [-15887, 5630, -1194, -1687, 1043], [-12387, -4028, -2272, -1609, -843], [62665, 10167, -2248, -2041, 1388], [-5315, 10045, 106, 4441, 1642], [-22915, 6029, 4160, 4603, 378], [2147, 7077, -1090, -714, -1547], [-25246, 4743, 3844, -1489, 392], [35041, -9229, 250, 3280, 1337], [41279, -5911, 2262, 4317, -206], [8368, -11042, 2116, 2298, 208], [-17898, 6849, 128, 987, -894], [743, -385, -2064, -8245, -1240], [33850, -8133, 2798, 2321, 1672], [-13876, -6142, 1322, 1014, -350], [17746, 2307, -2374, -5822, -969], [1515, 1271, 2290, -3387, -880], [12624, 4093, 4052, 746, -2321], [-24476, 9149, -634, 3921, 534], [-13530, -12167, 52, 470, 1195], [12538, 6487, 410, 4568, 405], [-27357, 2417, 866, -2276, -1533], [33384, 3936, 668, 1908, 288], [27190, 1778, -6404, 198, -276], [3788, -10482, 862, -844, 472], [20379, 5621, -4292, -1968, 897], [-10986, 5115, -1632, -2424, -93], [26040, -6437, -1278, -3380, -2099], [31776, -5871, 1260, 287, -1178], [13750, 9265, -2206, -5386, 241], [-15386, -1182, -806, -956, 2272], [-19178, 4148, 1160, 3062, 1250], [21962, 3261, 142, 3212, 1827], [1433, 3166, -3524, -4231, 385], [-16893, -18057, 2616, -585, 256], [8561, -5425, 2134, -5768, -359], [47992, 18496, -2874, -1250, 1264], [43166, 5481, -3336, -820, 513], [54727, -1241, -480, -2032, -2177], [53996, -10044, -1774, -101, 439], [-19220, -4533, 392, 5671, 368], [13945, -4241, -28, 1385, -2142], [28056, -1582, 1846, -2075, -305], [28880, -2270, 2632, 571, -1605], [-40000, 12718, -354, 1766, 1502], [-3766, 2080, 3414, 6730, -1876], [3052, -7979, 3592, 132, -1683], [44556, -4724, 2256, 5652, -152], [-2424, 1447, -662, -4975, -1248], [47930, -1459, -1442, -941, 208], [19161, 7902, -2726, 1379, -183], [83219, 9488, -492, -1458, 2702], [-61429, 4052, 2824, 255, 1081], [31022, -647, 1398, -1876, -905], [-36233, 9699, 2070, 6545, 1424], [29240, 9722, 792, 2196, -1590], [-11227, 11541, 636, -3897, 1610], [12011, -5103, -410, 647, -528], [-41186, -2359, -2540, -1978, -2765], [55576, 10853, 1744, 415, -602], [-11114, 16099, -2568, 1576, 1551], [-26802, -2679, -4784, -6731, -808], [18694, 11984, 2182, 8420, -1122], [1652, 16851, -4132, -1382, -763], [51, -1934, -2892, -1757, 1627], [2006, 2200, -1200, 6132, 1092], [-62127, -10908, 2328, -1923, 1143], [-27235, 4407, 3092, 2774, -1107], [-44116, -1916, -2292, 1664, -1772], [-29160, 24824, 1652, -2516, -1536], [-28124, 6146, -650, -5183, -1345], [-50289, 9179, 256, -5403, -562], [1003, -3921, 674, 2208, -1523], [-8555, -5664, -2150, 2219, 4347], [5284, 25915, -4696, -4582, 413], [1738, -3776, 3384, -832, -1204], [44648, -1500, -2412, -476, -2828], [6779, 2019, -2746, 1249, -252], [-35356, 21354, 896, 3063, 419], [54246, 11554, -1426, -934, -606], [5282, 8448, -1122, 4617, 2049], [-61709, -4289, 4740, 2471, 360], [54894, -6336, 5858, 3990, 2688], [57855, -9579, -2640, -2570, -1701], [49750, -15538, -4928, -6886, 268], [-24437, -25319, 32, -4793, 454], [2312, 16073, 1882, -4841, -1416], [19353, 2353, 1376, 2357, -1684], [34556, -3141, 6206, 7753, 2068], [-31306, 19474, -3312, -6430, -1804], [-13781, -946, 3522, 147, 3221], [4224, -2858, 5350, -433, 1749], [-40128, -3428, -3250, -2265, -997], [-56446, -4620, 3740, 3268, -1296], [12383, 12272, -9140, -1691, 87], [55870, -11313, -6316, -2977, -1148], [-79894, -814, 960, -388, -130], [48507, -17569, -4728, -4992, -1201], [-28298, 4245, 5678, -756, 2539], [13643, -2658, -3874, -564, -2572], [-100951, -8705, -1580, -3383, -2554], [-34180, -6376, 4398, -4113, -1705], [13298, -26116, -1776, -344, -860], [10693, 13411, -6016, -2800, 1279], [76741, -8707, 5018, 12503, 972], [47133, 5215, 1714, -6968, -267], [22664, 6877, -6792, -1516, -1119], [-39818, 8776, -4832, 4694, 1010], [111534, 15551, 634, 2701, -2456], [17934, -4806, 7272, 3114, -2000], [23492, 15892, 2188, 4642, -3986], [-61483, 11211, 4846, 8526, 1991], [-940, -9508, -3438, -5085, 3923], [68560, -1278, -3790, -3904, 2204], [-83572, 7808, -1342, 3900, 3562], [-51595, 4270, 1826, 414, -1514], [1515, 14025, -6062, -5445, -964], [-33390, -6252, 7062, 7866, -1620], [76739, -16831, 9152, 7914, 1935], [-83462, 6899, 7566, 2079, -694], [-59299, -22029, 1070, 5269, -2552], [57639, 13291, 1970, -997, -2418], [73932, -10813, 8698, 2952, 485], [23293, 3817, -3558, 1519, -4144], [3984, 4935, -2248, -10140, -937], [64892, 10365, -574, 3433, 598], [13359, -1571, 4042, 383, -2476], [-53403, 8990, -12038, -7989, -315], [36604, 3231, 4142, 7196, 977], [59972, 14934, -2658, 2614, 1374], [-39359, -9692, -6692, -9174, 2598], [-61415, 1669, 3312, 6691, 1026], [-9624, -6162, -7616, 2856, -354], [73934, 13586, 916, -8539, -315], [34272, -2524, -1604, -6734, -94], [67218, -6176, -3870, 67, 7], [70514, -10376, -1720, -1036, -1296], [-43773, -7309, 9572, -1375, 1390], [-1495, 19961, -3974, -11365, -1476], [15075, -18931, 1638, -6108, -373], [-75804, 6440, -9360, 3978, 1566], [-52450, 24550, 4970, 10902, 2562], [-49778, 6468, -766, -661, -5073], [-66782, -7748, -7484, -4778, -1390], [-18922, 14070, -6470, -11731, -2905], [48244, -15812, -9764, 1948, 536], [62700, 5230, 4824, -3840, -3474], [-47315, -4907, 4238, 2939, -1612], [50862, -9554, 6412, 742, 2048], [83976, 17008, 844, 712, -540], [6823, 23414, 714, -7361, 329], [31572, -16089, -1150, -11075, -4360], [-45335, 2277, -268, 315, -1022], [-301, 32890, 3280, -57, 411], [115944, -18472, 4680, 8076, 2548], [-29997, 17250, -7652, 2222, 4176], [71672, -7508, -4322, 3486, 2200], [-22801, 8893, -4308, -9379, 2152], [-109680, 17116, 3832, 5599, 1365], [5712, -10573, -11188, -10180, -1709], [-62274, -800, -1946, 2633, 537], [26707, 4225, 3724, -8472, -1515], [-63402, 21014, 2338, 4828, -1936], [-95924, -26744, 482, 9829, -1059], [-45017, -14748, 6708, 8307, 1597], [-30808, 873, 184, 4106, 6539], [105032, 19113, -4682, 4602, -1947], [10150, -13848, -11058, -9777, 2547], [16170, -19117, 5112, 7572, 2459], [60101, 11458, 1604, -14381, 971], [-18813, -6295, 6062, 1569, 3138], [-84084, -328, -2590, 4621, 2677], [107754, 27350, -6460, 566, -88], [-26638, 13876, -6068, -2773, -4875], [77585, -10077, 13184, 1425, -1938], [-27720, -22778, -1154, -5123, 2043], [-106155, 18236, 2968, -2645, -4659], [-32591, 12931, -13876, -329, 1252], [6689, 3546, -6690, 4018, 3578], [-36896, -15340, 700, 12849, -1413], [53362, -13416, 1792, -9340, -704], [64280, 17022, -1064, -17106, 840], [-42548, 4896, -8570, -6885, -2989], [-10934, 3517, -8778, 1108, -4841], [85786, 622, 3790, -2814, 3754], [-22450, -3300, -1714, 7506, 596], [75128, -14451, 1854, 5861, 1398], [-56269, -26600, 4190, -357, 2163], [64058, 20987, -12082, -7364, 973], [53221, 15668, 8390, 21853, -795], [-14400, 11680, 3128, -514, -646], [4593, -31119, -1620, 4879, 4682], [-2357, 22837, -6686, -14443, -1406], [-30656, -14542, 3300, 13243, -517], [106882, 11188, 6116, 7634, -386]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_825_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_825_4_a_w();".
// 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_825_4_a_w(:prec:=5)
    chi := MakeCharacter_825_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_825_4_a_w();".
// 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_825_4_a_w( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_825_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![56, 178, -116, -35, 4, 1]>,<7,R![-10623494, 106197, 55764, -1326, -38, 1]>],Snew);
    return Vf;
end function;