
// Make newform 825.6.a.n in Magma, downloaded from the LMFDB on 25 April 2024.
// 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_6_a_n();".
// 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_6_a_n();".
// 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 := [-8784, -218208, -1040, 12724, 137, -209, -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], [-60, 0, 1, 0, 0, 0, 0], [-78, -88, 1, 1, 0, 0, 0], [-34776, 346212, -70328, -11587, 2323, 71, -15], [-1252344, 291764, 104468, -7641, -1631, 45, 7], [889544, 1195020, -113220, -41409, 2097, 277, -9]];
    Rf_basisdens := [1, 1, 1, 1, 2344, 2344, 2344];
    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)
    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+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 := [[0, -1, 0, 0, 0, 0, 0], [-9, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [1, -6, 1, 0, -1, 0, 0], [121, 0, 0, 0, 0, 0, 0], [-207, 22, -7, 1, -1, -1, -1], [-79, -56, 14, -3, 4, -2, -1], [372, -42, -9, -2, -1, 3, -2], [-66, -74, 5, -1, -3, 4, -3], [-140, -14, -3, -18, 9, -13, -2], [266, 4, -26, -10, -6, -6, -6], [-3518, 314, -21, -5, -9, 12, 9], [-2588, 202, -251, -12, -31, 21, -8], [-1107, 510, 39, 16, -3, -14, -16], [-4910, 24, 128, -56, 36, 18, -8], [1516, -178, 265, 15, 73, 28, 13], [13706, -958, 427, 45, 3, 28, -1], [10096, -1280, -8, 94, 52, -48, 2], [-9814, 4278, 521, 49, 137, -70, -5], [9378, -3542, 179, -13, -5, -26, 57], [1459, 1782, 337, 67, -117, 47, -51], [2964, -1554, 151, -10, -129, -145, -50], [-2519, -820, -886, 90, 336, -71, 34], [3758, -556, 566, -36, -314, 66, -124], [24584, 1698, 1207, -89, -529, 134, -11], [-8814, -6918, 673, -98, 337, -163, -42], [-21876, -2074, -2163, 211, 121, 22, -31], [-9883, -4200, -2404, 28, 434, 133, 56], [-41654, 2164, 226, 94, -326, 384, 50], [21742, -7218, -2023, -39, -259, -68, 19], [15179, 4086, -1805, -126, -731, -110, 190], [-63998, -4268, 1470, 158, 58, -162, 274], [70786, 396, 1190, -594, -194, -420, 34], [494, -2970, -937, 334, -933, -547, -186], [-102078, -21846, -851, 700, 613, -81, 168], [127472, -4178, -1369, -614, 199, 327, -342], [104460, 16524, 1098, 670, -418, 702, -334], [-128612, -20866, 3409, -617, 901, -414, -99], [136907, 21820, -870, 38, 36, -445, -226], [-51131, -8960, -1942, 1625, -16, 542, 211], [-41180, -25144, 1364, 1186, -876, -562, 118], [460610, 934, 1977, -85, -1123, -102, -63], [256200, -14828, 1970, 560, 914, -242, -264], [-231157, -32754, -5023, -27, 35, -255, 379], [243121, -15140, -2468, -1141, -1534, -214, -311], [61344, 96132, 782, -396, -346, -186, -116], [231760, 67910, -3345, -1270, -2177, -285, 434], [-79538, -22744, 3872, -1662, -356, -392, -714], [57595, 43352, -10596, 1196, -2810, -677, -320], [290322, 72624, -3956, 1060, 3580, 388, 148], [-244377, -8880, 5598, 2463, -1952, 1158, 405], [162066, 26492, 5254, -104, -798, 1108, 792], [480084, 7612, -3818, -552, -4174, -348, 184], [130694, 33634, -13085, -1973, 7219, -338, -487], [-121856, 1072, -1152, -3114, 1708, 672, 10], [-156103, -1268, -4006, 2086, 616, -439, 518], [357194, 20184, -9088, 840, -4016, 164, -640], [79988, -18462, -1327, 702, 4569, 357, -1170], [91681, -89902, 2923, -2297, 2281, -957, 705], [-315692, 114446, -8517, 170, -2849, 605, 1690], [505897, -8226, -1209, -288, 3525, -610, 48], [-179465, 62168, -20926, -1181, -6724, 2130, -1295], [-567211, -4294, -25531, -4228, -2725, 456, 1420], [-520878, 8658, -3849, 4445, -4321, 540, 135], [-413710, 137374, -13631, -1683, -3971, 1796, -497], [369030, 79218, 3283, 3611, 4527, -112, 2769], [-592926, 54504, -22304, -4850, 2668, -4212, -758], [417499, 147266, -8437, -1555, 4333, 1767, -1125], [1255973, 188804, 28686, -2274, -1444, 221, 350], [-1019924, 310540, 25746, 3356, 1342, 1944, -196], [177196, -258206, 1339, -1431, 5539, -3528, 19], [-637718, -87084, -18738, 6542, 3810, 2290, -1166], [-116638, 41048, 600, 2850, 8300, 2996, 1510], [-349133, 20078, -2551, 391, 11443, -6569, 2529], [1415412, 229980, 5790, 1878, -1122, -136, -1862], [-263676, -77510, -22229, -63, 13023, 4302, -1125], [-529694, 41124, -39250, -3320, 654, -6194, -248], [-216500, 86450, 32335, 2399, -10633, -4266, 1437], [370338, -172740, -17474, -6138, 9118, 3520, 2778], [-1228638, 359224, -70672, -8558, -1688, -4614, 1534], [503456, 72196, -11938, 7796, -3234, 5418, 4412], [-286314, -149402, 12529, -909, -9603, -770, -3623], [220240, -169340, -48390, 1220, 2346, 11378, -2396], [166272, 4482, -36697, -2389, -13905, 7394, 1121], [422786, 55826, -65839, -6302, -10387, -1913, -5982], [381156, -156702, -2165, -167, 4655, 1970, -549], [1206386, -14352, 61992, 9302, 6568, -1298, -3982], [374945, 230262, -44207, 2971, 7391, -5107, 1933], [-1149838, -17090, 12539, 1970, 11755, -4073, -5782], [622494, 19056, 11212, -2438, 19192, 4340, 5438], [-597994, 283282, -14133, -951, 5563, 13364, -2749], [-137266, 19956, -28294, 1948, -22546, -8848, -2668], [189010, -55610, 5521, 429, -13911, -4182, -10881], [-3125592, -248688, 7844, -2682, -10268, 686, -8310], [2095156, 11288, -46812, -5838, 14084, 7886, 4310], [2014339, 451168, -1736, 4780, 7434, 2043, 6496], [-1381218, -281948, 66930, -4110, 30034, 3528, 558], [-1506750, -345436, 93802, 1510, -10390, -8468, 6522], [-1024385, -276490, -167069, 8968, -731, -7304, 432], [-522482, 270836, -52214, -6782, -4310, -21856, -10402], [-708217, 80938, -34991, -25342, 4347, -6388, 4318], [3798617, 265324, -29552, -12467, 5742, 5736, -3721], [-94245, -109468, -31878, -7376, -19244, 13513, -11340], [-121114, -153310, 124341, -11294, -23579, 9245, -11086], [5136282, -965462, -7823, 2098, -22451, 12987, 6394], [-1848184, -21928, 83884, 5884, 8, -5198, 12460], [-2109130, 204182, -66911, 34083, -19735, -3816, 4913], [3527193, -418384, -18314, -7961, 13988, -8456, -14619], [5620944, 804832, -24068, -1544, -10812, -8876, -768], [-3917336, -590740, 30274, -14462, -33298, 9150, -5794], [-3734981, -432786, 167207, 7206, 5129, 834, 3210], [9011819, 194970, 7019, 4053, -18915, 16515, 2635], [1822778, 105410, 22063, 19695, -27357, -168, 7541], [3720946, 638528, 130404, -15970, -17408, -14472, -6886], [30468, 549044, -8402, 13680, 43374, 8518, 13888], [-8111994, 665044, 69486, 10640, 526, 13486, 3408], [-2263188, -465814, 89067, -15867, 42151, -1474, -3049], [-483860, 260366, 36849, 15583, 10557, -10818, 8677], [2648168, -572612, 24622, -21020, -1262, 6412, 16604], [-1979208, 429556, -131250, 1498, 27566, 2504, 22974], [-4213566, -95720, -9500, 12792, -7972, 19984, 2104], [569883, 454130, -109457, -13009, 21417, -24043, 6857], [-435309, 61940, -275944, -7757, 29546, 12976, -807], [579594, 551824, -100760, -12144, 18476, -22442, -14712], [7948708, 239100, 105638, -4222, 27142, -38796, 686], [-9099116, -378878, 66649, 22316, -20371, 3637, 6640], [7008134, -589482, -11331, -10037, 36177, 5242, 13305], [6176636, -48688, 66284, -9178, 5276, 16630, -6454], [-5226204, -167764, 169730, -38742, -9678, 27500, 19022], [3393039, -502762, -227263, 18089, -8789, -25179, -6337], [6301838, -630070, -323039, -7514, 48197, 1019, 6630], [-8764459, 1247844, -3326, -920, 11184, -36097, 4628], [12284684, 222796, -56454, -18796, 48954, 10846, 8700], [-2040360, -532224, 235448, -20248, 54508, -11978, -17344], [522514, -273234, 370535, 10420, -69929, -20687, -4328], [3193616, -645688, -303388, 12728, -34792, -9238, -1104], [7010316, 437320, -6008, -24618, 18428, 19592, 1938], [5848433, -1447414, 47801, 29354, 8303, 25902, 15694], [-4110178, -772430, -33373, -18221, -2993, 36176, 6617], [-3320480, -1232694, -85263, -26350, -42835, -32077, 5770], [1603910, -588386, -370925, -24790, -43673, 29285, -11382], [-10534956, -173338, 275019, -29452, 14735, 22459, 13968], [-4959708, 1455926, 12893, 911, -12151, 4842, -12475], [3879957, 1082256, 144464, -24332, -42250, -13255, 1248], [-8909058, -218590, -67713, -27179, 73163, -10086, -3881], [6956886, 35206, -128275, -15757, 90885, -23128, 6441], [16127303, -835526, -235269, 4569, 60213, -6509, -22361], [6575989, -328332, -129952, 29431, 21686, -55718, -19139], [8262450, -53648, 219404, 22454, -68152, 28632, -3134], [749280, -1426450, 303137, 29615, 18165, 45050, 32997], [4296075, -519030, 145387, 52353, 20797, 22327, -193], [-8352902, 2310912, -71172, -24194, 16164, -21494, 22210], [-356532, 116676, -150810, -3966, -93554, -23484, 23174], [-4715629, 413716, -21622, -29504, -308, -34667, -21100], [10425734, -670552, -232168, -25618, 56748, -25184, -21062], [14221964, 95120, 41612, -37106, -105444, -1138, 866], [-3587794, -319798, -32011, -62934, -12119, 311, 22122], [12082074, 1634656, 161836, 14950, 57172, 67794, 18810], [-7487235, -1404474, -101731, -53063, -51397, -7669, -20017], [-8285478, -468550, -472439, 11338, 14473, 10305, 6954], [-2434682, -2070476, 158326, 48042, 18858, 15614, 20854], [14263587, 136828, 325732, -43497, -50994, -19316, -5787], [9044699, 3134242, -284907, 32012, 35695, 34730, 4924], [15840688, -1710004, -340958, -47474, 81970, -26472, -24038], [4745630, 546264, -291944, 64028, -114720, 108004, 23804], [10380872, 1995766, 605237, -68689, 34089, 25430, 3893], [6294522, -1108708, 77194, -9118, 40910, 71010, 56158], [-8752421, 341394, -89417, -52647, 135945, -61029, -3225], [13479216, -1547912, 141288, 25122, -38668, -11168, -10730], [7840104, 352418, -11601, 72821, -33969, 9456, 12735], [5218264, -782700, -203738, 30446, 99494, -6628, -19670], [12820750, -467842, -893567, 11443, -5963, 2206, -36895], [5915462, 2721196, -111062, 28606, -26874, 33794, 30146], [16447165, 751154, 70619, 67725, -599, -32135, -24525], [-142966, 1894424, 47936, -20782, -92044, -25484, -17130], [14259362, -2466036, 152134, -46500, 94934, -5610, 23468], [-7737994, -1703862, 38641, 67726, 102645, -40285, 29406], [-3707966, 1998500, 16994, 133042, 40970, 66660, 36862], [3935919, -3222670, 124885, 79756, 73047, 4590, -25156], [-6174290, 1407560, 77460, -57686, -52716, -55050, -18754], [17801452, -2248068, 16722, -27950, 233746, -115852, -15882], [9548028, 2521952, 99456, 94464, 36144, -13384, -14112], [16783684, 3392540, 154098, -97914, 5062, -66082, 31050], [-12691373, -107836, 136264, 19003, -46382, -103236, 54721], [-12667874, 818516, -251910, -5262, 94526, -28362, 29254], [30690, 813944, 498896, -32950, 63360, 23062, -1194], [9840503, 818102, -458431, 83917, 97947, 4081, 24523], [-1341926, 704484, -646950, 164620, 77214, -46768, 66324], [17630766, -1748868, -552666, -20320, 70862, 10086, 31768], [13720764, -4492580, 229350, 86816, -74306, -113730, 43800], [-10945252, 5161072, -264552, 1380, -171636, 62434, 16420], [-3304043, -1195548, 496966, 18674, 72484, 92505, -5846], [-10594200, -2195564, 89118, -19276, -113586, 111922, 18412], [-9891206, 75164, 6890, 51572, -66486, 110254, -23012], [14131707, 3320124, -196538, 43896, -100128, 38533, 49404], [-5538604, -5036336, -793048, 106906, 54700, -71672, -25642], [-2187636, 506028, -1216302, 10170, -185562, 48546, -1610], [5016758, -2983320, -68444, -45356, 152044, 43492, 26140], [-15576985, 1853364, 729080, -67165, -25990, 72688, 16025], [23024097, 39228, 99222, -4136, 19036, -24441, 35892], [1791440, -2747622, -1048267, -89828, -180175, -40715, 10112], [-8463894, -2580618, -891261, 131794, -29793, -38243, 29338], [5197174, -3822896, -121440, -92164, 129992, 21144, 22356], [996760, 4519432, 21056, 39094, 23292, -46956, 12050], [18197120, -1783892, 524450, -54320, 180626, -112534, -60352], [12879426, -4436838, 236399, 81523, -43389, -68792, -54519], [-6273664, -1181584, -36068, 26744, 102676, 27196, 86304], [-1562953, -3387308, -21686, 111184, -143396, -96139, -20620], [-8235212, -5279630, 1031741, 32818, -215455, -38213, -49382], [-34694310, 3113276, 498698, -6554, -17530, -16198, -29030], [-7431855, -1516102, -307069, -145745, -43999, 4879, 47921], [-16897862, 1473196, -563178, 2340, -201970, 47050, 26068], [20896828, -2730532, -897422, -136014, -114366, 18716, -20970], [-1093154, -1482778, -645847, -85461, 122977, 33936, -45047], [-9123490, -1941604, 868314, -51902, -177706, 24238, -34114], [-44716322, 3468672, 124068, -82648, 47460, 59848, -5536], [905661, -813378, 1429195, -39568, 30097, -39070, -75976], [-380622, 4119558, 807559, -84098, 60079, 33907, -738], [4694364, 2841200, -1233876, 26768, -118180, -57224, -10872], [5570764, 4208056, -650040, -113990, -57860, -129660, -3794], [-27990608, -920520, 1131752, -182368, -260, -105782, -70768], [-6251772, 3911406, -792361, 24162, 97487, 2883, -1934], [-7919606, 23400, 1467360, 169682, 188840, 27370, 38350], [36723756, 649610, -831797, 11933, -267489, 59762, 73679], [-67945359, -385664, 233836, 13650, -151022, -25537, -60798], [6653022, -6121256, 576084, -164330, 277044, 222058, 24178], [51205954, 1141172, 725290, -25170, 222050, -10472, 3922], [1479782, -1658282, -642003, 186331, -213851, 37736, -463], [78759591, -3815842, 40875, -10712, -87203, -63676, 17864], [1316602, -5336420, 795398, -94132, -71350, 141028, 92068], [7819032, 4776302, 621521, -102111, 362857, 112618, 2819], [66276146, -2064458, -112417, -40370, -52533, 51649, -12714], [-3291806, -3731186, -180545, -203922, -202677, 78261, -24418], [-58600500, 853098, -138743, 45338, 182877, -2273, -17502], [-31571506, 3206250, 1337367, -233925, 1159, 38650, -33431], [37087337, -6026484, 988830, -59448, 306564, 5407, 94916], [-375398, -472314, 205857, -305365, -75203, -31410, 57281], [-8431421, -1272456, -43934, -56201, 5700, 125554, -25283], [10626746, -3439762, -130267, 5257, 568261, -231546, 58595], [-1710672, -3062236, 484514, -454298, 58546, 74516, -28662], [-44251032, 1999398, -1069095, 4271, 145853, -227954, 49629], [-36366660, 4210298, 1610105, -32614, -81471, -82875, 29626], [-23317860, -4008496, 1093616, -72300, -357816, 35000, -151876], [49917334, 1559058, 39603, -112631, 447047, -40990, 61787], [-11217153, -5261956, -694064, -334435, -160022, -97462, -31185], [29697014, -10059472, -1554752, 156444, -58492, 32306, 71612], [31431402, 5920570, -432121, 68535, 223895, -134218, 66125], [44349526, 12887766, -739407, -39467, 211097, 69498, 71191], [46506586, 2690154, 395481, 266582, 109445, 148551, 4022], [65636381, 8094996, -475062, 274392, 291576, 53271, 8996], [-64768503, 1855654, -1498111, 84893, -185065, -37189, -46669], [-17023142, 5726626, -1104047, 60320, -79903, 28379, -75260], [1938116, 278904, 482400, -111744, 147648, -62148, -89368], [11687742, 2512172, -340210, 58962, 157206, 152672, -93554], [19038833, -3490476, 1041212, 212097, -395462, -14952, -60397], [40397804, 2370568, 872648, 44082, -163904, 280178, 38654], [-61688410, 8582586, 1865395, 19729, -54929, -12862, 54891], [59991495, -879062, -707131, 63260, 273831, -88534, 58052], [-94011962, 7180174, -829707, 28925, -125815, -94352, -22745], [-38251447, 6752802, 1004615, -29769, -47995, 114403, -59399], [3759777, 4138318, 1803955, -130684, -282895, 36210, 72356], [-29430177, 6499716, 1433946, 121796, -24500, 49377, -28128], [60010544, -372124, -1251694, 29858, 316270, -217638, -95898], [-11547259, -2903642, 891549, -225029, 256595, -138935, -58619], [-44208068, -5477610, -1679607, 294627, 161049, -166916, -79199], [-23879772, 2106340, 602734, -331924, -144906, -181770, 98836], [7451266, -1180608, -1722708, -97002, 756604, -105058, 44026], [69838978, -182728, 417752, 47366, 421216, -314554, 113578], [-97681291, -6172394, 1543983, 20142, -127539, -127408, 112930], [-13273933, -8880464, 405330, 536101, -97220, 198464, 36239], [15908770, -2856004, -2247690, 461906, -52066, 42760, -11506], [-22275226, -1527836, -279566, -3332, 482318, 184052, 100612], [-54678052, 1247036, -1091222, 224434, 144614, -120018, -217970], [-16775522, -5716502, 2669917, 121594, 224913, 41511, -1078], [-7476491, 5727354, 172119, -171559, 477397, 198553, -27081], [46717634, 12417486, 635053, -62703, 125653, 35010, -39605], [8042506, -2685292, 2798350, -91350, 7322, -54302, -208778], [68665548, 6921212, -1362630, -65176, 427526, -105220, -15968], [21511504, -8583624, 2258980, 256220, -213576, 81766, -9748], [51115254, -3819264, 2536328, 2652, 440356, 271038, 23436], [-87564103, -4068328, -1073452, 207592, 531962, -208639, 154460], [-2709114, -5013336, 635228, 240784, 35248, 283530, -37352], [30555505, 10711380, -147342, -15572, 108760, -118833, -21536], [-30296446, -12020656, 47512, -235702, 153536, -120402, -32034], [557647, 3977866, 2845993, -189666, 358755, -319424, 24450], [57509176, 6497344, 1383340, 28804, -52204, 123712, 4132], [-69349160, 3034638, -931751, 108493, -318007, 129986, -16569], [-34743741, 7620632, 449678, -34627, 670536, -336244, -69417], [-17783606, 5983836, -1175798, -62502, -54754, 55698, -66106], [-40227246, 3295798, -1851185, -246142, -669001, 97727, 90282], [166441074, -712116, 1228290, 178544, -145710, 223674, 86736], [-70537996, -3455038, 355827, -469811, 9015, 261030, -46585], [36471811, -7218792, -1405038, -385869, 629820, -121166, 153937], [4551858, 12189260, -887454, -335894, -316258, -302850, 93654], [-59434275, -1252778, -873275, -335725, -468917, -65695, -45235], [13054534, 7427118, 657523, -66742, 692411, 330603, -13022], [-38106280, -8926452, 3190762, -36020, -280750, -135214, -233484], [-44298480, 1661180, 191102, 123100, -580542, 380548, -102108], [-38724636, 10724408, 1420368, 563994, 108520, -6174, -107178], [33816725, 4944434, 2652929, -77528, -590185, -223752, -50144], [-703039, 1326374, 2726477, -97771, 812387, -65525, -102125], [-87939095, 18453108, -1331452, -78947, 488290, -753116, -157193], [-46810710, -1966410, -932997, 454274, -438361, 266133, 117562], [-19990610, -1752590, -41053, -663171, 219827, 134600, 7183], [64359726, -5311896, 952836, -38088, 964672, -279138, 136336], [-20199914, -2259422, -1469313, -209005, 69051, -365584, 7041], [59416634, -3763504, 614408, -160068, 79436, -157550, 214988], [67073530, 10952400, -601880, -102040, 742384, -118004, 10760], [-14761448, 18919944, 323592, -239620, 576400, 304164, -43316], [15067015, -10563874, 3225745, 502031, 122163, 141067, 143913], [88833323, 2000452, 2858582, -61022, 873664, -326629, 169970], [19890114, -292360, 2903204, 231894, -87844, 155614, 167106], [-101367758, 9052878, -862613, -283702, -766861, 310427, 87634], [-49227322, -11841278, 1108691, 355219, -296189, -76298, 211297], [29348343, -20112276, 1155242, 171990, -757876, 68223, 47598], [36541504, 16455792, 1451500, 556726, 144408, -447976, -85102], [43078752, -2028540, 3141766, -410338, 970774, 191932, -88854], [-29643406, 2243576, -5149404, 51612, -14176, 236994, 147452], [-85997496, 11386296, -986060, 333564, -279840, 348650, -868], [-122920018, 5870974, -363721, 551124, -468081, 42741, 153944], [62324068, 12768134, 3049543, 372658, -394017, -420905, -227830], [82795724, -6153340, -3466786, -263318, -208294, -222094, -90266], [-83678896, -5119874, 330427, 625870, -443425, 254585, 24566], [34985966, -7678358, -16505, 20487, 257727, -105638, -318659], [85094935, -12344376, -1731726, 308491, 83940, 5374, -85031], [62177762, -12077666, -439843, -510445, 822345, -218702, -98007], [-19891658, 156028, -422214, 16078, -415794, -77850, 111458], [-88469552, 8065600, -1121448, -332972, 893520, -575304, -77412], [44070365, 10629024, -1374496, -339480, -679090, 72985, 206828], [-34737503, -5071136, -224262, -357851, 408376, 572638, 188935], [101481788, 23435252, -3478206, -42270, -664954, 66206, 62118], [2172898, 14776388, 2747870, 366578, 278038, -20724, -195770], [-87723316, -7325986, 1706183, -1041070, -414865, 129379, -210798], [127580683, -3596308, 2008642, -334174, -546364, 728867, -286478], [119960788, 6294690, 1295061, -139582, -42243, 263785, -1198], [49368, 12688770, -2220053, -69815, -129305, -241402, -83301], [-23845494, -3617964, -1360262, -142562, 562594, -142024, -187118], [83006727, 11641020, -3295544, -106427, 968354, -47470, 69303], [-46271766, -7333514, -54959, -498961, -867731, 426362, 225565], [255160201, -2006422, -125631, 22128, -147601, -158560, -4608], [-90351816, 5966066, 7159071, -438941, -280897, -192754, -37831], [-56170332, 23814294, -1765607, -79145, -490199, -115616, 298845], [-156059298, 23372690, -877439, -103068, -765183, 325263, -93984], [30449558, 7664756, -801878, 548836, 93166, 317464, 180252], [169436084, 901164, 3862582, 203388, -130430, -616920, -156460], [-51207038, -7549692, -2580014, -599624, 688838, -84348, -86840], [111169294, -7842696, -2875396, -773332, -673172, -92168, 123380], [33212268, -3126898, 809913, -26085, 571197, 250742, 59753], [37894088, 11877084, 2223370, 4694, 406738, 705212, -25342], [96010920, 18711888, -6956860, -108082, 259440, 57328, 47466], [-14885638, -7265698, 118915, -645858, 823551, 22293, -98250], [170373502, 9887932, 11162, 63082, -490078, 507280, 250206], [-25224222, 1116752, -791156, -543726, -351220, 394426, -217986], [-163783157, 6824946, -2395963, -738880, -522177, 447334, 25992], [89336, -10403260, -6146450, 577508, -134126, -176328, -64140], [56951657, 4346312, 1779246, -722443, -759676, -462474, 427751], [-193776674, 11801338, -1964629, -184861, 715115, -356526, -152231], [87693366, -420364, 2236882, 122868, -134850, -45924, 69564], [-39593677, -22023972, -3415684, 74359, -1446938, 161660, -87499], [129539916, 16520268, -671002, -453862, 92270, -240688, -49818], [567509, 13159170, -401065, -423819, 1171445, -542791, 103987], [15190966, 2035818, 756229, 652328, 382317, 279683, -42980], [119270618, 13651618, 3456591, 369057, -796409, 454584, -89437], [-288036286, -3879812, 3105078, -404360, -20598, -230984, 10376], [149725136, 18908416, -2832312, -527352, -231920, -517660, -195272], [-148394776, 832232, 1017156, -149408, 648048, 106758, -60008], [10304817, -7909392, 4397350, 364659, -833732, -69128, 17961], [205626501, -3582162, 2339695, -413238, -869139, 141528, -409618], [11831688, 28397096, 1306748, 88772, 168512, -133314, -213212], [-32635630, 30817154, -2609597, 1277, 450219, -34508, -74113], [4040470, 28421492, 1680474, 492608, -921850, 95456, -433408], [25575913, -25605844, -3267906, 570112, 939300, -453897, 41644], [-117183422, 28450990, 2608255, -584516, -165609, 45637, -77840], [27519670, 12991050, -4174347, 223818, 172825, -598593, -389398], [7683028, 24215496, 1911812, 9768, 45856, 32334, 125232], [-25725260, -8745752, -2752780, 337458, 153108, 589390, -568250], [-135763578, -9921000, 1952312, 621918, 123604, 264724, -198774], [-10034371, -19781002, 1401325, -85283, 446283, -524329, -331349], [-16990494, -18446150, 5547585, -919422, 1140169, -43627, -526830], [20476821, -24052502, -458577, -29771, -306947, 644997, -282901], [21976722, -11383668, -3253334, -1297284, -719382, -71346, -14652], [63400828, 14900284, 99002, -262780, 122086, -250400, -341900], [-27147897, -15263606, -2044359, 114172, -88133, 681586, 42284], [169407033, -20715380, 3429860, 601143, 335570, 531234, 58797], [55069584, -14814070, 3411425, -199322, 1482609, -344091, -139826], [-113394172, -1885668, 69966, 433718, -1503546, 781452, -24742], [190537404, 11781344, -3439552, -248690, 223160, -406646, -32774], [-2727024, -33720518, -176205, -911533, 444355, -436078, 330809], [89834087, -26157190, -3601655, -154490, -938813, -350672, -56246], [40567472, -22097356, -2065838, -216582, 646962, 327408, 253142], [238790566, 12007662, 594657, -444537, -179523, 364902, 28061], [14750332, -13038086, 661147, -213727, 192979, -582636, -44861], [-15632286, -23809930, -3642807, -646159, 135961, -137422, 98187], [35252706, 20544566, -2907327, 376809, -765567, -538366, -643693], [105826150, 10776002, 4613151, -593811, 895103, -456792, 239239], [56997945, -27825954, 727357, 176667, -577429, -1113543, -14899], [-153958362, -17579330, 6000127, -1446802, 709075, 36905, 17470], [-207898970, 6319088, -4123456, -545368, -490472, -76812, 136504], [333164468, -10496880, -1233148, 550224, 227164, 493956, -34632], [-106266832, 199186, 5853397, 134102, 599305, -309449, -287434], [-29924318, -14664476, 5537166, -421564, 589662, 228194, 571196], [171771802, 10628242, 4045863, -268433, -908725, 520024, -322955], [351015817, -15852582, 5023769, 308684, 99783, 479396, -6636], [-68273472, -4889428, 1755354, -488040, 498062, 71192, 154320], [-57924410, -23947778, -3230173, 208458, 2261579, -787181, 217138], [26594800, 33851166, -2889353, 404346, -54329, 212007, -136582], [-13141520, -31499846, -474317, -599261, -865133, 1181306, -384183], [-353751282, -3050798, -1604095, -230628, -788839, -78949, -65832], [-41837186, -36414242, 6513397, 180029, -1532035, 480062, -6417], [-102685710, -19212298, 3465361, -75111, -2398623, 721962, 171795], [-26492779, -18360914, 752441, 182481, -256089, 937943, -106697], [-70718729, -41707084, -520820, -481655, 1568734, 265910, 138115], [-129151216, 20718400, 5411712, 968704, 1981800, 199836, 124880], [199591138, -27388438, 4421073, 316622, -677139, 359511, 479534], [-12319065, -12887934, 5363019, 1033221, -365051, 585203, 381779], [-199533962, -40825644, 3775950, 208258, 855494, -1612, -243674], [-23702644, -2179004, -5041598, 652224, 522890, -818334, -200296], [14797588, 20378902, 8068005, 890647, 1148097, -211894, 259245], [-27306184, 248382, 3607997, 463299, -2826859, 557524, 344321], [-69332465, -67976, 38156, 639560, -1090474, -345425, 328324], [166288558, -35159076, 3251534, 1184576, 471118, -468470, -178760], [-60671799, 31853942, -2982387, 122575, 263035, 368809, -74623], [-128792772, -2275624, -3264352, 1093252, -1391616, 984864, -114060], [227550666, -9300384, -4408236, 364610, 578160, -381740, 664182], [-24438426, 47349956, 2103590, -2051278, -46386, -115996, 8054], [-98082384, 240520, 6118120, -794018, 315812, 120940, 107802], [-325808712, -24974360, -3867132, 694396, -1681876, 833252, -273228], [-54643794, -2380942, 3098857, -595092, -1383999, 651931, -160968], [99354340, 8070076, -2717882, 661962, 1873046, -234284, 614054], [137936758, -9029976, -4284920, 46614, -2956508, -525644, -99806]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_825_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_825_6_a_n();".
// 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_6_a_n(:prec:=7)
    chi := MakeCharacter_825_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_825_6_a_n();".
// 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_6_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_825_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![8784, -218208, 1040, 12724, -137, -209, 1, 1]>],Snew);
    return Vf;
end function;