
// Make newform 525.4.a.x in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_525_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_525_4_a_x();".
// 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_525_4_a_x();".
// 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 := [-10, 30, 7, -18, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [10, -20, 27, 3, -2], [70, -320, -84, 24, 4], [155, 140, -138, -12, 8], [-785, 100, 582, 18, -32]];
    Rf_basisdens := [1, 15, 15, 15, 15];
    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_525_a();"
function MakeCharacter_525_a()
    N := 525;
    order := 1;
    char_gens := [176, 127, 451];
    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_525_a_Hecke(Kf)
    return MakeCharacter_525_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 := 4;
    raw_aps := [[0, -1, 0, 0, 0], [-3, 0, 0, 0, 0], [0, 0, 0, 0, 0], [7, 0, 0, 0, 0], [15, 8, 0, 0, 1], [-1, -6, -1, 0, 1], [18, -6, 1, -8, -2], [33, -2, 1, -6, 1], [-27, -12, 3, -2, 1], [72, -8, -2, -6, 0], [65, -26, -1, 8, -1], [-74, 24, 2, 10, 8], [171, 32, 3, 20, 7], [-122, -16, 2, 2, -8], [138, -36, -6, -10, 8], [15, -14, 11, 14, -1], [144, -28, -12, 28, -8], [118, -36, 16, 18, -10], [52, 64, -12, -2, -6], [291, -16, -8, 2, -17], [-87, 102, 13, 24, -1], [-182, -96, 14, 36, -2], [312, 48, -4, -20, 4], [81, 72, -15, -8, 25], [83, 162, -1, -20, -7], [507, 24, -1, 40, 35], [-338, 148, -2, -60, 10], [261, 24, 7, -66, -39], [794, -88, 40, -52, -28], [-27, 394, -27, 102, -7], [-494, 336, -2, 36, 38], [294, -44, 56, 12, -46], [-1365, 186, -19, 34, -1], [1103, 118, 41, 106, 7], [-630, 200, -48, 54, 6], [92, -168, -50, 0, 56], [357, 66, -61, -96, -5], [-784, -376, -40, 30, 50], [738, 324, 22, 206, 28], [978, 150, 19, 2, -20], [1791, -128, -32, 92, 29], [-960, 396, -82, 86, 12], [393, 792, -12, -62, -23], [-1930, -516, 2, -68, -70], [969, 342, 53, 186, -15], [-241, -114, 27, 64, -11], [-382, 188, 20, -68, -6], [-108, 392, -16, 4, 120], [2610, 336, 54, 76, -54], [500, -412, 14, -14, -24], [3153, 254, 11, -62, -23], [1875, -584, 60, -322, 91], [16, -160, 138, -124, -6], [1560, 220, -76, -380, -8], [912, -130, -129, 16, 72], [1527, -748, -27, 74, 3], [-99, -944, -39, 180, -83], [727, -30, -11, -336, -7], [-3842, -608, -2, -378, 4], [-1860, -440, -82, -112, 58], [1294, 436, -22, 308, -10], [1650, -166, 89, 426, 20], [4058, 732, -30, 48, -90], [1380, -1492, 102, 172, -40], [-1365, 222, 29, -124, 33], [303, 74, -81, -254, 107], [3078, 2188, -96, 124, -66], [-3204, 16, -84, 244, 192], [-1587, -720, 191, -342, 13], [-14, 1476, -4, 278, -186], [2778, 38, 95, 48, 94], [2061, 1104, -148, 350, 89], [3420, 104, -140, -192, 60], [-4638, 16, -66, 90, -64], [3212, -8, -178, 360, -132], [1164, 1028, -216, 222, -94], [888, -720, 246, -414, -92], [-3937, -2002, -35, -56, -215], [3120, -680, 142, 140, -186], [5312, 416, -122, -468, 150], [2718, -376, 14, -88, 142], [-426, 700, 32, 664, -8], [2307, 936, 44, 986, 191], [-245, 2070, -143, 392, -131], [5103, 414, 55, -156, -103], [-3807, 536, 79, 226, 33], [1566, -1272, -120, 140, 184], [-7690, -364, 302, 312, 194], [-735, -1352, -279, -224, 201], [1482, -1152, 254, 216, -14], [1290, 672, 206, -208, 54], [3408, 196, 110, 380, -244], [-3300, 384, -156, -896, 40], [4767, -1072, 232, -436, -203], [2632, 920, 94, -16, -16], [-780, -2844, 176, 250, -10], [495, 712, -237, -596, 63], [687, 2248, -65, -28, -185], [1156, -360, -84, 84, 228], [-2790, 2260, 328, 96, 40], [4360, -1272, 24, 158, 254], [-81, -1686, -229, -862, -181], [-1374, -2936, -106, 0, -126], [-2076, 64, 50, 1324, -182], [-1644, 796, -300, -668, -168], [-1637, -3234, 269, 876, -47], [4446, -752, -122, 184, -442], [-9354, -230, 37, -608, 202], [-711, -2568, -324, -518, -303], [-10154, -152, 300, -256, -224], [-672, -1528, 40, 148, 196], [-544, 1740, -344, 322, 594], [153, 1054, 419, -578, -547], [4039, 742, 469, -130, 3], [-350, -1088, 442, 644, -90], [-2760, 928, -342, -236, 410], [-2674, 3588, -258, 340, -170], [7026, 2028, -66, 810, 256], [3117, -318, -173, -1454, 125], [-2067, -2536, -52, -352, -5], [-5874, -52, -4, -518, -498], [-4488, -784, -480, -452, -280], [-3474, 390, 447, -670, 256], [-2619, 3024, 51, -42, 361], [7993, -302, -77, -962, -679], [-13596, 1320, 318, -1414, -12], [-4594, -1576, -20, 16, 508], [7110, 4676, -454, -960, 242], [-6854, -1460, -522, -796, -50], [10115, -550, -277, -916, -479], [-5744, -3780, -188, -92, -12], [-11187, 2764, 375, 170, -787], [-9690, -5188, 332, 112, 390], [13234, 360, -50, 346, -432], [-27645, -1040, 55, 8, 163], [-2406, 128, -4, -1700, -432], [6174, 466, -587, -734, -376], [3760, 5296, -544, 640, 52], [17298, 1574, 371, -318, -20], [-6942, 2368, 424, -1324, -360], [14203, 86, 105, 58, 515], [-6810, -6256, 24, -674, -362], [802, 4152, -346, 920, -294], [4101, -960, -37, 778, 485], [-1908, -4484, -658, -14, 100], [690, 1904, -756, 356, 330], [711, 4894, -35, 1712, 369], [20964, 3550, 231, 264, 44], [-9641, 3042, 827, 786, -725], [-7095, 5260, 271, -558, -335], [268, 1132, 556, -186, 322], [-4533, 4112, -365, 216, -545], [-8250, 1208, 74, 2730, 788], [11238, 2236, -878, -342, -96], [-6394, 4104, 266, 730, -460], [-4743, -6120, 92, 914, -15], [-11738, 6420, 528, -80, -818], [-14157, 3400, 507, 16, -397], [5005, 5090, -813, 16, 227], [3111, -1040, 271, 732, 31], [-18573, 408, 461, 626, -633], [5973, -5022, -155, 890, 181], [-16294, -1832, -114, 500, -638], [-9840, 9224, -18, -1520, 460], [573, -4778, -349, 386, 533], [252, 4140, 804, -54, 114], [4030, 1532, -260, -952, 86], [-10001, -5154, -211, 240, 409], [-19142, -5444, 648, 508, -216], [24054, -2806, -323, 1846, 364], [4755, 1160, -516, 2072, 1053], [-20046, -2868, 180, 1810, 118], [39351, -312, 376, -10, 359], [-6074, 3548, -838, -1844, -102], [-4439, 5274, 497, 1780, -289], [-13179, -648, 757, -84, -427], [20762, -912, -450, 512, 446], [4080, 2944, -134, 1562, -804], [8270, 2988, 378, -1692, -1614], [-2078, 8812, -148, -474, -870], [-15502, -2864, -314, -4280, -970], [-744, 3880, 578, -816, -492], [12692, -1300, 956, -2246, 666], [11322, -498, 131, 604, -1054], [14667, -2620, 1489, -1414, -1001], [6321, -5112, -351, -2376, 873], [28856, 4620, 60, 978, -6], [-6104, 1616, -160, 204, 1304], [-2222, -4196, -820, -860, 244], [44586, 3852, -786, -144, -250], [-12485, -1662, 151, 3108, 473], [9525, -2312, -921, -1714, 209], [9548, -3976, -1310, -48, 108], [-12705, -456, 687, -5136, -457], [15315, 4208, 1021, 3210, -513], [35970, -990, -1111, 816, -530], [-18606, -7028, -780, 300, -1144], [-19792, -52, -200, 1798, 130], [15492, -2354, -117, 72, -220], [-1134, 8212, -54, 2986, -192], [14370, -3536, -808, 3562, -170], [34039, -274, 107, 2360, -275], [26033, 7390, 557, 1680, 15], [-13700, -4008, -1450, -2752, -294], [-12144, 2968, -814, 56, -92], [4470, -9338, 1123, -450, -1076], [21457, -966, 29, 2804, 735], [-23955, 1392, -1169, 2922, 989], [-32778, 1784, 792, 1944, -1672], [7459, -3846, -53, -2398, 775], [17378, -6004, -410, 2548, 322], [-30777, 4016, 187, 2156, 391], [-40868, -3880, -428, -904, -508], [24048, -4128, 392, 1420, -292], [1446, -12224, 448, -524, -930], [-12214, -3360, 552, -2076, -744], [18168, -416, -272, 5528, 300], [-2463, -472, 1973, 1176, -219], [6207, 1876, -623, -2126, 1299], [7365, -118, 1123, -1006, 1029], [-4402, 1192, -1204, -1736, 1156], [-17351, 4406, -777, -1884, -29], [13098, -120, -1028, 3300, -312], [-32400, 3456, 1248, 1156, 100], [-41964, -9704, 684, -1444, -404], [35626, -2892, 1164, -2140, -1120], [25932, -2314, 1175, -608, -1420], [-15327, 1488, -780, -150, -791], [-37914, -6244, 174, 400, -394], [25491, 10256, -1212, -1156, 105], [18462, -8024, 1466, -1018, -160], [-16299, -13410, 1137, -310, -763], [-1316, -11152, 262, -1384, -1312], [15150, -768, -296, 2384, 2224], [-50506, 4548, 1106, -2000, -646], [-7092, 3820, -108, -4154, -322], [220, 1312, 222, -1804, 874], [2049, -8650, 1209, 722, -1959], [18099, -12928, -132, -1156, 993], [-13446, 6828, 1322, -80, -930], [11643, -2224, -971, 474, -1161], [43147, -5410, -3, 2782, 23], [17040, -16168, 348, 1400, 312], [-13114, 8716, 2644, -2120, -364], [-21528, 1190, 1655, -820, -884], [-22206, 14600, 404, -684, -430], [-23806, -11260, -246, 1660, -1258], [-12594, -13568, 902, -5912, -482], [23266, -14192, 1242, -1520, 174], [-8751, -6004, -1653, -1214, 121], [-12404, -5428, -2984, -3794, 850], [-41127, -6656, -255, -3088, -971], [14667, -2692, 621, 6086, 371], [-4352, -12864, -422, -3224, 114], [14394, 10186, 1729, 442, 692], [60231, 12344, -624, -1012, -851], [-46178, -13604, 40, -5000, 248], [-8782, -6476, -126, -2044, 442], [-28122, -4846, 81, 4266, 352], [39099, -8370, -27, 5224, -539], [38130, 6072, -490, 1168, 578], [-7491, -15000, 547, -2310, -491], [10288, -4548, -1282, 4606, 800], [-25259, -8406, -149, 2684, 1255], [-55098, 9018, -1431, 4908, 1398], [43897, 434, -741, 2854, 2009], [25902, -2552, -244, -4950, 42], [1611, -2720, -3129, -5748, 563], [53160, 14792, 504, 1170, 82], [30993, 7506, 299, 4654, 885], [9006, -7236, 1200, -1038, -246], [52990, 12456, 3058, 3406, -300], [-23633, -3790, -169, -1472, -3567], [-15778, -13804, -1632, -784, -762], [36987, -5894, -1501, -2376, 453], [42494, 5036, -1906, -2040, 1118], [30909, 14368, 1227, 2878, -1383], [-17578, -572, 2512, 204, -860], [-5038, 19908, 992, 1804, -1948], [2364, -14972, -1656, -100, 1780], [5472, -15524, -1496, -518, -1194], [39350, 488, 2206, 3892, -998], [-63060, 11092, -428, 2654, 726], [12706, 3236, -2732, 4446, -366], [88468, -5528, 1632, -3732, -820], [34905, 5296, -1048, 4798, 1789], [-7490, -764, 814, 3696, -1094], [-15297, 5586, 1219, -2098, -3217], [13829, -3962, 783, -3424, -821], [-10353, -10120, -953, -5976, -2785], [33006, 3200, -500, 5722, 754], [-11772, -5296, -648, -984, -860], [-24099, 86, -3725, -1302, 829], [47154, 2096, -538, 408, -726], [-70282, 1616, 834, 4774, -2608], [-21639, -8896, -2111, 3008, 2553], [26345, 12058, -279, 3156, 551], [6576, -12450, -2461, -2618, 466], [68388, -6204, 292, 5596, 356], [-28904, 1960, 288, -5072, 1116], [44053, 9634, -973, -3808, 1907], [-69519, -2550, 283, 2410, 2601], [-16248, 148, -3024, -676, 880], [-68001, 16232, -299, -3230, 1283], [61144, -3180, -372, -2364, -1956], [-63344, 17120, -1232, 4552, -8], [-11361, -6408, -2163, 1914, -2733], [2412, -2444, 798, 7830, -676], [38487, -15320, -212, 458, 2979], [-11056, -5300, -4432, 2314, 1894], [-32856, 5772, 4184, -166, 558], [27408, -12944, -18, 7674, 2192], [13428, 480, 670, 5980, -1278], [31990, 5896, -94, 2418, 564], [3729, 1476, 359, 930, -1919], [25400, 10624, -998, -4008, 886], [4182, -10188, 2688, -4076, -2014], [40305, -7224, 1408, -2642, -3587], [-30157, -14910, -265, 3364, 2297], [55092, 2360, 1362, 1152, -602], [-27793, -11094, -685, -1798, -1125], [-32688, -4656, 2648, 9372, 64], [-24501, -5968, 2411, 136, 1499], [18844, 16312, -1788, 6608, -1220], [60831, -4218, 1075, -2138, 3015], [-4012, 31104, -1598, 7524, 614], [12128, 7204, 1684, -884, 1820], [40298, 1316, -318, -3608, -682], [-993, 12572, 469, -3770, -401], [-27753, -3522, -283, -4854, -1485], [-35438, -16960, -1452, -10672, -12], [-17499, -4198, -2249, -862, 1973], [-16581, 6946, 1525, 4876, 185], [-28362, 2312, 1810, 4172, -1806], [38962, 21936, 30, -1520, 2974], [258, -4936, -2994, -8060, 590], [-27170, 24296, 1816, 2636, -3532], [14688, -14974, 1665, -568, -2288], [38944, -9920, -3854, 2616, -482], [29290, 14372, -566, 9432, -102], [4246, -4152, 1602, -1198, 1772], [-57015, -2922, 1631, -3610, 525], [-71415, -7504, -463, 2212, 945], [-27588, 9104, 1170, 4728, 568], [-61815, 4642, 907, -8994, -1187], [37125, -464, -3496, -11196, 319], [-880, -14268, -4322, -3950, 1020], [-19950, -2968, -4442, 2198, 1028], [-17802, -13004, 326, -1888, 2218], [-7926, -262, 1397, 4130, -1940], [-16979, 9062, 1253, -1730, 369], [37132, 29352, 744, 2440, -464], [4350, -1944, 1644, 4510, -2426], [40800, -32184, -628, 64, 584], [-8422, -20880, 668, 1060, -1768], [-21882, 2942, -977, 6280, -270], [-48618, -11748, -2510, -928, -1158], [-69534, 468, 1000, 5252, 3318], [-47751, -33038, 901, -2738, -2219], [-68319, -42652, 1131, -5146, -51], [-10930, 18712, 50, -8158, -1132], [-61683, 10472, 793, -9340, -615], [-42987, -4692, 1131, -12874, -1983], [-35811, -13192, -884, 2848, -1973], [77060, -19640, -784, 7176, 1896], [-57032, -15400, -2400, -3640, -612], [22140, 16890, -1303, -6214, 594], [34222, -23840, -398, -6952, -1254], [-88246, 25444, -2132, 6164, -420], [-2505, 384, -3884, 5520, 1721], [46547, -9318, -425, -12742, 55], [-73803, 7220, -4473, -338, 3617], [-59886, 3728, -1264, -3958, -430], [587, 38602, -1983, -5832, 1349], [-5498, -10736, -1018, 910, -3380], [12174, 7212, -2148, -1012, -3894], [1893, -22448, 2084, -12714, -2143], [-1397, -2802, 1109, -10128, -35], [-68913, 8816, -1197, -2860, 1167], [-63239, 7206, 237, 12644, -997], [-35793, 6112, -2229, -3320, 239], [11685, 14410, 73, -3618, 1225], [131464, -5032, 504, 1192, -2284], [-43305, 5670, -2745, 2706, -965], [17171, 46, -731, 3866, -1029], [-54816, 17324, 1372, -4530, 1046], [-75622, 17708, -1928, -1000, -502], [51707, -15814, -101, -1380, -1431], [-70476, 20536, 2404, 1030, -1774], [-6762, 6108, 1814, 10554, 3208], [-41078, 24300, 744, -1636, 3716], [65004, -12258, -2525, -3822, 1314], [-18903, 104, 3872, -800, 1143], [-84338, -43628, 194, -8032, -1006], [-41673, -4360, -1296, -3662, 1259], [-22396, -1848, -2228, 7696, -1452], [-64363, 1706, 329, 9664, 4191], [-8469, -1400, 2967, 5896, 279], [-11756, -14644, 4624, 2940, -2520], [46644, 14272, -3210, 198, 3320], [-73902, -1404, -2436, -5602, 1634], [8361, -2374, 4125, 3418, -4551], [-21618, -28152, -2918, -7176, 3578], [-1596, 14834, -3163, 4320, 1244], [-7425, 12960, -2657, 12620, 3055], [-28303, -21446, -451, -4592, -6125], [-54302, -15272, -4054, -15546, 288], [22146, -5120, -3856, 2844, 2700], [15200, 4800, 1040, 11268, 1744], [138480, 12444, -2608, -4796, -24], [-47847, 562, 3427, 2776, -625], [-20877, 3782, -379, 4574, -221], [64041, 20752, 5659, 2122, -2015], [-238, -17196, 2500, 6348, 3458], [-21170, 3084, 4762, -2604, -3838], [-3591, 8944, 2329, 8940, -2127], [-121737, 9320, 1092, 10086, -1849], [92694, -19884, 2846, -9736, -4914], [18786, 11798, -5021, 1440, -1074], [33054, 2428, -3298, -4414, -2376], [-47016, 17944, 1538, 850, -5420], [17779, -3314, 3441, -5932, 4081], [-48567, -34308, 2319, 2450, -1639], [-47760, -19584, 986, -4168, -420], [-78357, 12590, -715, 4100, 1129], [30018, 10530, -715, -4774, 2148], [-102141, -3928, -1611, -194, 395], [-11508, 1904, 1030, 6760, 1966], [25785, 30242, -3989, -1382, 2197], [-204042, -4720, 1276, 1076, 1358]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_525_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_525_4_a_x();".
// 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_525_4_a_x(:prec:=5)
    chi := MakeCharacter_525_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_525_4_a_x();".
// 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_525_4_a_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_525_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![80, 200, 17, -33, -1, 1]>,<11,R![-55852416, 1472448, 140456, -2100, -66, 1]>],Snew);
    return Vf;
end function;