
// Make newform 435.4.a.j in Magma, downloaded from the LMFDB on 11 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_435_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_435_4_a_j();".
// 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_435_4_a_j();".
// 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 := [638, -767, -167, 329, 18, -35, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [-10, -1, 1, 0, 0, 0, 0], [-94, -131, 128, 67, -23, -4, 1], [-1262, 157, 814, 1, -99, -2, 3], [744, -304, -363, 43, 38, -1, -1], [-8898, 3183, 4946, -381, -581, 2, 17]];
    Rf_basisdens := [1, 1, 1, 20, 20, 10, 40];
    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_435_a();"
function MakeCharacter_435_a()
    N := 435;
    order := 1;
    char_gens := [146, 262, 31];
    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_435_a_Hecke( : Kf := Kf)
    return MakeCharacter_435_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, -1, 0, 0, 0, 0, 0], [3, 0, 0, 0, 0, 0, 0], [-5, 0, 0, 0, 0, 0, 0], [-6, 0, -1, 0, 1, -1, -1], [-2, 8, 1, 0, -1, 5, 1], [-18, -4, -4, -2, -1, -2, 3], [1, 12, 1, 1, 3, -5, 2], [-22, 8, 1, 2, -6, 11, 0], [-10, -8, 1, 0, -2, -13, -8], [-29, 0, 0, 0, 0, 0, 0], [-69, 2, 12, 7, 8, 2, 1], [-138, -26, 4, 0, -20, 0, -10], [-105, 14, 17, -5, 18, 9, -7], [-76, 24, -6, -2, 6, -26, 14], [-33, 18, 11, -21, 9, -19, -6], [74, 40, 6, 18, -30, 10, 2], [-32, -72, -10, 10, -20, 30, 20], [-143, -66, 51, 11, -34, -1, -5], [-239, -118, -36, -11, 45, -8, 24], [-172, -64, -40, -14, -10, 8, 34], [-154, 90, -12, -24, 28, 96, 10], [45, 22, 26, 11, 12, -32, -13], [87, -114, -6, 5, -8, -92, -7], [12, -112, -42, -12, 7, 32, -11], [-482, -126, 30, -6, 44, 22, -14], [-535, 30, -45, 35, 59, -35, 0], [-461, -10, 40, -77, 28, -22, -23], [107, -310, -58, -1, -10, -60, 15], [-211, -114, -57, -23, 13, 93, -62], [-181, 168, -23, -1, 107, 3, -24], [-452, -284, 96, 32, -14, 4, -10], [-230, 384, 31, -80, -101, 171, -23], [132, -138, 22, 46, 18, 26, -26], [-131, 142, -5, 113, -57, -63, 54], [166, 276, -20, -30, 20, 60, -100], [-924, -376, -46, 12, 84, -2, 64], [-64, -318, -38, -24, -120, -130, 96], [-1192, 104, -112, 50, -84, 84, -80], [862, -432, -11, -144, 34, 111, 12], [-300, -40, -154, -30, 50, -150, 20], [1086, -196, -166, -16, -10, 58, -16], [-503, 338, 121, 113, 199, -45, -92], [432, -132, -247, 38, 18, -93, 102], [-1160, 158, 88, 146, -34, 20, -78], [584, -480, 98, 66, 52, 178, -46], [-679, 158, -113, -75, -11, -111, 4], [644, 476, 49, -154, -220, 243, 16], [-1852, 196, -73, 2, -73, -69, -197], [999, -262, -152, -15, -220, 446, 161], [-211, -26, -259, -115, 138, 17, 19], [935, -326, 131, 91, -334, 15, -103], [1492, 344, 66, -40, 198, -278, 178], [939, 134, 197, -253, -99, 103, 4], [422, 740, 261, 164, -125, 165, 201], [935, 6, 377, -63, 206, -51, -103], [478, 696, 262, 106, -136, 38, -18], [1545, -726, -365, 55, 3, -239, -12], [-1347, 170, 116, 149, 158, -510, -39], [684, 608, 538, 232, 77, -76, 43], [-54, 28, -188, 98, -42, -96, -366], [-198, 464, -123, 132, 334, -125, -64], [-3511, -248, 291, -163, 97, -115, -56], [-1150, -324, 92, 294, -172, 96, -354], [-2275, 14, -264, -83, 73, 508, 244], [-651, 1646, 7, -129, 57, -435, 258], [293, 696, 425, -167, -127, -205, -340], [-1504, -784, -447, 58, 568, -373, 100], [504, 346, 210, 130, -262, 450, -22], [-11, 774, 522, 129, -72, -368, 87], [-790, 644, -442, -328, 144, 342, 88], [-1817, -182, -109, 163, 358, -193, 133], [880, -1040, -293, -178, 450, -239, 170], [-1628, 380, -108, -52, -132, -788, -192], [-1713, 1414, 565, 71, -194, 597, 293], [1498, 1152, 19, 32, -596, 725, 178], [894, 800, 1089, 154, 358, -397, -264], [-2959, -1202, -573, -37, -227, -415, 138], [-2975, 318, -521, -47, -234, 459, 663], [-468, 1512, -428, -224, 310, 388, 240], [164, -1028, 384, 544, -350, 340, -400], [-4382, 156, -624, 22, -422, 808, 48], [2099, 970, -395, 481, -432, 685, 123], [-1734, -324, 306, -206, -560, 98, -546], [-4622, 1370, -424, -218, 652, -68, 438], [-3681, 558, -277, -445, 693, -611, 170], [3031, 1386, -59, 691, -613, 59, 480], [-728, 276, 310, 100, 393, 52, -489], [2871, 1414, 169, 53, -367, -153, 20], [1788, 2664, 170, 138, 8, -470, 18], [-5068, 864, -91, 322, 75, -383, -557], [-1848, 700, -406, -704, 1072, 138, 544], [2896, 1188, -187, 434, -550, -25, 102], [-4659, 158, -588, -295, -492, -350, 235], [1963, 914, 342, -429, -604, 188, 541], [7487, -434, 303, 211, -173, 213, -372], [5401, -1550, -65, 41, -315, 457, 288], [-6104, -328, 396, 788, -626, 516, -428], [-1954, -1284, 572, 722, 732, -756, -444], [3389, -1402, 640, 77, 371, 92, 222], [1405, -254, 17, -883, 466, 157, 423], [-2849, -2762, 376, 171, 479, -256, -500], [4280, -3412, -560, 40, 582, -2036, 248], [7131, -322, -643, -597, -45, 1055, -40], [-2644, -1712, -290, -1184, 567, -532, 301], [4878, -1896, -338, -306, 148, -1190, -250], [-4342, 1466, -838, 596, -616, 242, 522], [3867, 754, -208, 597, -484, -210, 37], [-2673, 154, -1675, -303, -1036, -371, 611], [2427, 1754, 1228, 47, 625, -16, -386], [10, -1916, -1008, -344, -820, 1408, 288], [-11342, -1656, -294, 232, 1020, -818, -214], [476, 1432, 282, 308, -921, -120, -331], [5436, -3826, 840, -250, 222, 144, -982], [8600, -1236, 87, 218, -244, 85, -636], [5625, -3170, -317, -151, 893, -435, -396], [-1058, 984, 1228, -398, 747, 1658, 83], [7839, -1490, 900, 87, -903, -716, 406], [13522, -2800, 843, 118, 124, -323, -742], [-751, 2120, -1125, -259, -719, 857, 16], [7692, -3980, 137, 554, 625, -895, -987], [-1495, -938, -911, 181, -505, 2271, 960], [15957, -2270, -801, -45, 705, -1919, -98], [-3977, -2308, -583, 191, -433, -49, 1408], [7087, 3814, -44, 63, 952, -46, 769], [3931, -2266, -1381, -585, 353, -1435, -514], [8198, -516, -868, 342, -432, -1244, -232], [-7136, 1992, -1976, 338, 516, -776, 526], [-2244, -824, 2056, -314, -1194, 1024, -518], [8194, 2424, 1354, 258, -640, -726, -602], [1570, -3682, 1182, 54, 1810, -1422, -944], [11434, 1576, 1237, 762, -632, 739, 310], [11331, -570, -299, 663, 537, 871, -46], [-8173, 2994, 10, 175, -1068, 1260, 661], [354, -3506, 1190, -80, -444, -194, 434], [2262, 1684, -2068, 202, -1314, -1304, -98], [3420, -1584, -1624, -504, -1540, -208, 374], [-1374, -1262, 1260, 224, -1942, 1072, -324], [-17282, -780, -265, -320, 1310, 1161, -296], [-11946, -3726, 1290, -972, -618, -838, -444], [-3626, 4540, 1792, -350, 873, 246, -331], [4257, -1558, -1949, -747, 205, -1231, -308], [-4922, 984, 762, -60, -920, 2550, 308], [4474, 3160, 6, 340, -534, -1554, -376], [16998, -1324, 1382, -454, 1568, -290, 502], [-8005, 1374, 439, 3, 724, -33, -781], [-9209, -2402, -554, 1439, -735, -1430, 1110], [-7314, 854, 950, 518, -10, 122, -704], [-3701, 22, -3517, -125, -866, -217, 21], [-11538, 2380, 1959, -490, -406, 2733, -200], [-4036, 1580, 187, -1124, 416, 793, -364], [-12075, -26, 263, 553, -590, -1105, -1377], [-2320, -1984, -1822, -1804, -861, -20, 365], [14080, 816, -533, 770, -2650, -223, 494], [-1825, 3710, 1229, 1227, -2045, 3883, -120], [6516, 1152, 1682, 194, 664, 3410, -168], [13925, -2678, 1506, 549, 1225, -1778, -1280], [6111, -3358, -11, -653, -719, 1623, 154], [-13284, 6196, -2090, 294, -780, 134, 1602], [-1671, -5854, 911, 1255, -685, -2199, -1020], [-2294, -4068, -516, 586, 1940, -532, 504], [9437, -2458, -833, 17, 661, 1201, 1716], [-8473, -2382, 2005, 57, -926, 193, 1521], [-8270, -248, 3754, -56, -944, -1546, 6], [9, -4918, -2194, -479, 472, 2396, 567], [14633, -4934, -35, -367, -792, 637, 1321], [6092, 1948, -34, -900, -1264, 1634, -472], [1605, 3426, 787, -1311, 1303, 997, 806], [-25266, 470, -318, -1616, 1916, -886, 422], [-753, -3114, 569, 1727, -823, -1813, 196], [-2799, 284, -299, 41, -1275, 7, -1168], [-8932, 1472, 2638, -924, -268, -1758, 40], [4887, 7586, -1729, -73, 523, 1741, -842], [17010, 428, 1986, -934, -860, 2306, -806], [242, -2146, -3834, -1176, 648, -242, 150], [-6506, 3956, 576, 1342, 944, -932, -1270], [-1532, -5416, -1806, -184, -764, -3118, -234], [16477, 6010, -2123, 361, -279, 2215, 1116], [-3265, -2570, -1647, -987, 1463, -3625, 34], [7380, 3052, -962, -116, -338, 2062, 2090], [-12585, 2110, 981, -669, 821, 2687, -964], [-24460, 6668, 2418, -1658, 874, 2310, 142], [28202, 2924, 3347, 1136, -1381, 627, -243], [2272, -5392, -946, -1480, 1551, 232, -335], [19445, 5610, -111, 789, -288, 1073, -1427], [12318, 7964, -113, -830, 1968, -1279, -962], [12998, 6868, 1252, 1750, -1332, 1708, 148], [2128, -2786, 3768, -48, 322, 2180, -530], [8258, -1388, 1806, -206, 410, 482, -1916], [-18862, -2496, -2370, -1622, 0, 870, 272], [20524, -3552, -1282, 1710, 952, -554, -84], [-2213, -1602, 1433, 545, 255, -2713, 1230], [10729, 926, -2707, -1407, -1555, -1917, -120], [8568, 4480, 2963, 1218, 1854, 1693, -374], [4225, -726, 7, -2769, 2101, -1611, 1830], [-13133, 982, -629, -185, 615, 113, -2188], [-14873, -794, -2151, 589, 4394, -3795, -1923], [-5100, 3904, 1998, -1270, 50, 5570, 180], [2506, -5408, -4168, 1554, -1005, 5326, 2111], [-1611, 4458, -309, 1155, -240, -525, -1219], [-3318, 116, -1301, 516, -2628, 5325, -66], [-1044, 2192, -318, 506, -4468, 6222, 1798], [25441, 7242, 2975, 709, -1189, -1015, -2210], [-33390, 4746, -2700, 280, -3046, 3100, 1300], [7422, 1596, 62, -1222, -2450, -2766, -218], [-13740, -4696, 243, 2330, 299, -6641, -1061], [7310, 6476, -1106, -3534, -1290, -3614, 2326], [-4075, -11938, 1096, 5, 278, -1906, 1749], [20183, 8194, 858, -359, 972, 884, 969], [-4640, -216, 2062, 2812, -1200, 862, -34], [-3152, -5124, -4163, -522, -738, -437, 414], [-7774, -6790, -1052, 214, 1204, -728, 2150], [-21316, 2572, -3064, 498, 1342, 808, -1012], [-2740, -5724, -4328, 372, 1364, 880, 1264], [-19919, 2390, -6173, 1481, -345, -4367, 1186], [15891, -1982, 3144, 2419, 927, -3696, 428], [14181, -10570, -4627, -239, 991, -1437, 888], [-8206, -4796, 1513, 504, -739, -2911, -1809], [4872, 8284, -1054, 1516, -2405, 276, 1257], [-18194, 8636, 861, -1684, -4000, 4299, 1590], [16873, 80, 3463, -295, 105, 4021, 984], [16125, 1702, -1913, 55, 1774, -3869, 559], [6653, -446, -1001, -1919, 2879, -2903, 238], [11190, -1224, 4808, 538, 1053, 1450, -2283], [-32850, -2888, -4090, -1816, 1150, 5134, -680], [-2195, 3198, -3298, 369, 3686, -2488, -1475], [23441, -12518, -2283, -23, -2782, -3087, 995], [-24420, 18962, 618, -482, -306, -446, 2414], [25422, -2252, 6483, 1400, 2470, -4911, -1608], [-27654, -6544, 4684, -558, -2032, -1144, 2046], [-6690, 8948, 2516, -942, 2352, -1384, -3158], [-25500, -4690, -1712, -704, -3230, 4420, -318], [13186, -4124, 4526, 222, 1816, -722, -126], [-19537, -3614, 458, -215, -10, 5192, -621], [-13665, -7530, -3807, -33, 862, -839, -387], [-10419, -3466, -1134, -783, 2743, -4034, -174], [-14637, 14898, 671, 23, 2955, -4099, 1776], [-15426, 8544, -678, -166, 1744, 2498, 1172], [-18710, -13138, 2126, -488, -338, -5882, 556], [-8272, -15304, -4332, -1564, 4064, -4500, -520], [-13459, -12966, -68, 797, -3991, 860, 2968], [17199, 4242, -803, 2479, -2483, -1409, 738], [5456, 208, -564, 3284, 696, -812, 772], [5348, -7344, -2443, -3182, -4231, 2289, 905], [-12323, 4974, 2889, -1115, 1908, 2049, -1923], [-43595, 5206, 135, -429, -4168, 2335, -471], [-35290, -548, 202, -654, 5424, -118, -1742], [-13915, 9162, 6184, 969, 3768, -2694, -2965], [-18708, 4024, -3324, 560, 1830, 4984, -4854], [455, 8886, 3491, 1963, -2153, -191, 1168], [4437, -10410, 1365, -699, -307, 575, 2392], [-33468, -2134, 4552, 2600, 788, -2840, 224], [-24088, -7964, 3248, 1194, 2338, -3512, -1844], [-21131, 16826, 4229, 1013, -3843, 5235, 988], [-5654, 3248, 386, -4330, 3206, -1050, -1706], [-12122, -7362, -3992, -1224, -490, 4156, 200], [32476, -7432, -3794, 530, -326, 3410, -194], [-6567, 5146, 3291, -297, 1240, -1741, 2473], [2116, 832, 2750, 1928, 438, 6, 1134], [-26864, -108, -970, -3214, -1952, -1706, -742], [-9363, 2434, 1395, -2721, 7037, -4907, -2982], [42848, -2172, 1736, 56, -1992, 1040, 512], [-24595, 9306, 3483, 261, -2881, 1405, -3302], [-15521, 3206, -1253, -3217, 491, 605, 1010], [-8252, 1502, -8520, 1308, 2018, -7740, 794], [-37104, 4766, 1388, -1074, 1574, 180, -2794], [-9354, -948, -1879, -2258, -1804, 2999, 194], [-2077, 7558, 7665, 4165, 943, -4245, -738], [-30619, -3494, -2761, -379, 3418, 1867, 2571], [-32600, 792, 134, 1356, 1496, 594, 4308], [-36816, 3284, 6424, -656, -1890, -580, 2088], [-30401, -5370, 4481, -597, -1473, 4663, -1186], [-1977, -5282, -756, -777, -3541, -1760, 1952], [-22060, -1904, 378, 2342, 1316, -3158, -786], [-17477, 11934, -914, -545, 1638, -11660, 115], [10542, 5342, -1616, -3674, -4696, -2220, 2842], [-19582, 3928, 4168, -54, -3402, 6024, 2476], [21226, 3244, -1102, -3198, 1500, 3970, 1446], [30153, 6054, -2511, 3085, 1285, 5203, -50], [24184, 7016, 1376, -732, 398, -1796, -3252], [-24082, -5860, 4426, -3130, 3852, 1314, -2466], [5292, -13196, -3959, 2464, 130, -4153, 2510], [-36161, 3234, -5539, 1251, -203, -3633, 5014], [-3872, 5256, 6997, -830, 2192, 6615, -1452], [-4873, 23922, -1967, -1021, -2587, -4769, 3072], [13656, 5992, -4438, 128, 3258, 2962, -4306], [33711, -16090, -936, -565, 4001, -6964, 3198], [-3214, 3110, 4316, -2868, -112, 2016, 278], [-15873, -9048, 1651, -781, 2231, -6839, -1740], [7581, 14770, 3615, 1017, 4661, -931, -2756], [3186, 2536, 3086, 3336, 2118, -2210, 918], [-7406, 1108, 4710, -1502, -2852, 3194, 1616], [12811, 7594, -2372, 2887, -4214, -282, 339], [-20309, -7378, 2879, 531, 2268, -9553, -1357], [-24319, 24266, 2858, -683, 5364, -6060, 763], [-41150, 814, 2396, -298, -582, -2076, -488], [43876, -15820, -2704, 278, -734, -5048, 3184], [35161, 3802, 7406, 2667, -3694, 6944, -2127], [-2626, 7832, 4828, 300, -154, 6616, 4850], [-20888, 12224, 1306, -2078, -1126, 7806, -2886], [15512, -4280, -4098, -3438, -760, -2522, -996], [-24582, -1330, 2080, 5080, -4016, -588, 2110], [-27346, 13472, -1574, -1438, 2592, 1658, 5036], [-9727, -6238, -6808, -3473, 1134, 7598, -3675], [-35514, 17500, -3342, -14, -2852, 2682, 2858], [17268, 10912, -4535, -1842, 6582, -2633, -2834], [5110, -9112, -2480, -3256, 1568, 1052, -8364], [1693, -22178, -1207, 1281, -967, 4335, -3760], [-43932, 3720, -8252, 1006, 996, -4836, 698], [40708, 22312, 1550, 3922, -7428, 894, -232], [2870, -5664, -856, 734, -2247, 538, 81], [-24236, 13512, 7107, -1896, 78, 9109, -2034], [35695, -5570, -6815, -1655, 465, 8527, 476], [-14544, 10500, 4278, 2806, -2800, -42, -446], [9857, -13598, -1748, -4167, 5923, 1728, 1982], [-29924, -14348, -303, 2196, -8472, 3291, 1116], [96421, 14562, 3125, 3829, -2865, 1935, -708], [31331, 13094, 7130, 1339, 2640, -2160, 21], [-23576, -19008, 680, 3860, -1512, -5360, -2048], [-64185, 5050, 1061, 237, -4161, 6351, 2094], [2202, 14660, -1688, 2226, 6379, -4994, 907], [38666, 14404, -6081, -262, 1870, -3531, 1936], [62141, 9358, 1255, 2051, 1957, 13, -602], [49722, 13172, 1286, 4250, -128, 7442, -756], [-86446, -13408, 312, -1476, 10, -6092, -704], [24181, 6446, -7101, -117, 4878, 6295, -745], [81972, -9136, -3820, 1796, 4230, -368, -736], [-59090, -8028, -654, -658, -556, -11622, 3090], [10334, 5356, 133, -5880, 5910, 5699, -592], [361, -7422, -5329, -4367, 3715, 5045, 5542], [11038, 17520, 808, 5952, -1156, -200, 232], [-3091, -17626, 1065, 3237, -404, 8601, -3099], [-17758, -1420, -5218, -106, 2184, -2642, -1060], [4168, 24672, -8388, -3328, -1552, 5356, 3400], [10137, -29406, 4355, 5253, -541, -867, -6862], [-32340, -988, -6181, 948, -2774, 405, 2166], [43187, -18702, 5744, 1531, 9082, -9006, -1317], [25899, -10250, 2889, 351, 1913, 4411, 1916], [-9107, 11948, -4195, -4423, -8003, -3833, -392], [-20944, 25680, 3170, -38, 1812, 5050, 6130], [-43612, -11760, 3297, 2762, -7609, 6121, 5295], [-2838, 21598, 8092, -910, 5598, -2588, -6604], [-18247, -24134, 2045, 1701, 3964, 9421, -3563], [4657, -6558, -4341, -161, -1521, -7959, -4792], [18205, -2462, -8167, -5627, 5383, -4153, 2206], [-28786, 4556, 4512, -6140, 3050, -11012, -1646], [-42519, 18818, 2244, 4021, -332, 1170, -6845], [-68477, 7010, 6961, -3711, 3578, 3781, 1749], [37750, -4516, -584, 456, -8284, -8776, 2562], [-12849, -9970, 3642, 387, -2393, 1618, -484], [-12347, -7420, 1595, -3339, 5009, -5831, -4176], [-29552, 16048, 11234, -4696, 6580, -10730, -6432], [64535, -21198, -4299, 1097, -2487, 735, -3936], [-40566, 23740, -5352, -2684, 3476, 2412, 6344], [8637, -13134, 9780, 5437, -4144, -8926, -3877], [112825, -1610, -1499, 1793, -3694, -1711, -2353], [-28622, 9078, 50, 1524, -10042, 4586, 2824], [-41827, -5098, -1234, -799, -2909, 6170, 1998], [-89298, 9276, -486, -2590, 4112, 5370, 154], [-7233, 2512, 11783, 5215, -4869, -787, 3028], [83600, -868, 2377, -756, 2070, -3017, -1706], [-5238, -6988, -3444, -4718, 6793, -718, 321], [-83058, 5632, 276, 2072, 3890, -296, -6178], [35773, 6626, -713, -2051, -5215, -4439, 768], [-15666, -10716, 1334, 1718, 9086, 8466, 5056], [-10008, -29928, 3428, 6406, -5710, -5244, -522], [13522, -28134, 3368, 1222, -1512, -2492, 1066], [-97131, 15508, 2881, 1845, -3595, -4945, -6684], [24348, 9552, -3955, 2086, -3065, -1255, -4713], [59910, 6236, -1590, -4514, 5210, 8518, 1730], [46261, 14534, -7908, 1573, 2236, 970, 3051], [-128827, 10014, 327, -2047, -1085, 7773, 138], [95088, -3520, -6319, -1544, 1400, 5771, 212], [-14873, -11270, 3765, -2615, 3277, -17341, -3524], [91501, 6258, -5481, -855, 4247, -3407, -346], [30241, 6478, -11081, -4947, -2094, 1807, 3115], [-68826, 30624, 4517, -940, -3793, 16549, -143], [4707, 5098, -820, -1453, 845, -16184, -1366], [26810, 29518, -10530, -6994, 92, 12142, 4674], [47926, -6836, 9890, -346, -6208, 7246, 1192], [91470, 6284, -10160, -4608, 3820, -13024, 188], [-25828, -48, 5746, 4002, 4400, 11882, 866], [-57237, -28922, -10453, -1913, 8580, -8917, 3595], [3966, 1600, -5856, -6770, 5858, -1616, 1068], [11721, 2884, -6491, 2649, 3157, -5881, 1412], [-56282, 26028, -9712, -1586, 4440, -4476, 4750], [87590, -4916, -3837, -110, 11654, -13735, -764], [-5475, 14130, 6622, -9275, 7436, 6964, -3953], [58660, -7044, 5330, 3904, -4345, -9752, -2287], [30140, 18464, 222, -2648, -5214, -2458, -2646], [26561, -9358, 9907, -751, 11379, -7815, -6618], [-30312, 30532, -3048, 352, 6156, -18544, 4982], [11346, 7936, 7138, 7866, -268, 9050, -7380], [19500, -10472, 7485, -2614, -917, 3941, -2749], [4770, 17156, 8478, 11454, -6032, -2018, -1710], [57635, 27450, -7134, 8123, -8675, 4562, 2946], [3745, 12786, 995, -3609, 1635, -935, -1564], [-13372, -25240, -8116, 1272, -9548, 3460, 56], [-33239, 610, 2507, -4147, -654, -5301, -3769], [15685, 19686, 2015, 497, -4577, 19333, 5062], [50647, -12334, -2607, 1345, 707, 2679, -2030], [14444, 9416, 5730, -4450, -1136, -2018, -3634], [-32276, -2498, -2146, 4406, 10466, -14534, -1394], [39474, 27768, -1470, -4364, -3504, 14054, 6766], [-24605, 16286, 7627, -2201, -1572, 15987, -941], [-24726, 2660, 10528, 3438, -2096, -2552, 2860], [-230, 13220, -2754, -502, 3544, -4746, 3358], [56332, -21328, -5858, 1380, 5961, 3016, -529], [-45176, 4576, 6126, 3084, -11957, 8908, 2245], [-36180, -15904, 11116, -2972, -2248, -4724, -5672], [4722, -20284, 6592, 2006, 1334, 664, -132], [15779, 12614, 721, 5833, -7327, 179, 864], [99688, 7332, -88, -2644, -3198, -5492, 324], [-8577, -11938, 882, 9625, -3486, 2360, -8165], [-40232, 264, -11930, -9640, 3380, 7218, 1000], [85112, -19546, 3632, 218, 3346, 10156, 1990], [-68403, 12434, 4051, 1647, -609, 5369, -6100], [-37455, 3054, -9280, 1641, -655, 608, -7048], [-83256, 28544, 8541, -1762, 4235, -439, -5129], [20183, -1104, -4159, 2543, 5221, -8813, 1494], [79024, 32172, -4512, -988, -4664, -1568, -2372], [-48218, -13996, -1134, -3310, 2364, -10970, -4164], [15466, 8202, 4792, 590, 7300, 9500, -434], [-82972, -8528, 4784, 4780, -5128, 2604, 3644], [-74007, -25394, -7726, -5999, 5980, -15840, -3425], [-12929, -4514, 3525, -3171, -10205, 855, 1014], [-104040, -20972, 5588, 4744, 9792, 7772, -2030], [63911, -2966, 11530, 2075, 2642, 12592, -2721], [62270, -19932, 8060, -7442, 9257, -2410, 6589], [-28975, -15110, -4701, 4813, 11725, 937, 9164], [3444, -20472, 3397, 3374, -7826, -4041, 2898]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_435_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_435_4_a_j();".
// 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_435_4_a_j(:prec:=7)
    chi := MakeCharacter_435_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_435_4_a_j();".
// 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_435_4_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_435_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![-638, -767, 167, 329, -18, -35, 1, 1]>],Snew);
    return Vf;
end function;