
// Make newform 825.6.a.u in Magma, downloaded from the LMFDB on 23 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_u();".
// 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_u();".
// 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 := [-7036608, -12856224, 7443872, 1367872, -822204, -33410, 24456, 309, -271, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-54, 0, 1, 0, 0, 0, 0, 0, 0, 0], [75, -90, -1, 1, 0, 0, 0, 0, 0, 0], [9482457408, 3221579520, -1955331720, -594332024, 79330102, 19464294, -1041837, -218253, 4245, 793], [3384141600, 982974192, 402736312, 284331468, -41925694, -16419288, 667197, 241401, -2837, -1039], [-9246211872, 1095458256, 1358850856, 284331468, -49960270, -16419288, 667197, 241401, -2837, -1039], [-109330622496, -70047031776, 23826735712, 8919161552, -900724178, -285383202, 11342781, 3331917, -45041, -12773], [31465564896, 216228317088, -20512414832, -29318748608, 1150777794, 926492130, -16937505, -10562481, 72205, 39585], [-92954209248, -256225969248, 29699694960, 35398857040, -1298267594, -1138140594, 16796133, 13136229, -65505, -49661]];
    Rf_basisdens := [1, 1, 1, 1, 32138304, 16069152, 16069152, 64276608, 64276608, 64276608];
    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, 0, 0, 0], [-9, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-19, 1, -1, 0, -1, 0, 0, 0, 0, 0], [121, 0, 0, 0, 0, 0, 0, 0, 0, 0], [107, 22, 2, 1, -1, 0, 0, 0, -1, 0], [-112, 11, -6, 1, -4, 0, -1, 1, 0, 1], [268, -82, -2, -1, -2, 1, -2, 0, 0, -1], [-223, 34, -2, 1, 2, 2, 0, 4, -1, -1], [589, -24, -30, 2, -9, -2, 6, 4, -2, -3], [-466, 13, 10, 10, 0, -9, 3, -3, 0, 0], [1774, 281, 129, -10, -5, 7, -8, 4, 1, -4], [-539, -269, 61, 17, 9, -4, -4, 10, -1, -3], [-2072, -28, 47, 15, -20, 12, -19, 7, 6, -2], [1514, -96, 36, -39, 5, -10, 8, 10, -9, 2], [-6163, -129, -170, 13, -47, 3, -3, 3, 7, 6], [5978, 819, 181, 52, -19, -17, -22, 10, -11, -20], [721, -521, 332, 8, -123, -3, -17, -13, 28, -1], [910, 2105, 85, 32, -42, -3, 44, -12, -13, 17], [6076, -1254, 237, -53, 65, -20, 53, -3, -38, 15], [2168, -419, -133, -25, -40, -49, 58, -24, -76, -12], [23174, 2827, 12, -7, 8, 56, -11, 23, -18, -17], [-18262, -757, -804, 29, -148, -51, -5, -35, -51, -9], [17411, 8915, -123, 63, 303, -57, 46, -40, 10, 17], [-30544, 880, 577, -168, 165, 22, -165, 31, 71, 26], [-579, 3138, 361, 62, 19, 93, -157, 49, 126, -28], [-2737, 1083, 1167, -199, 175, -29, -60, -52, 116, 59], [26958, 9730, 733, -100, -64, 34, 81, -67, -85, -91], [2355, 329, -1965, -262, -69, -63, -102, -92, 31, -60], [-2231, 7759, 8, -87, -127, 95, -69, 165, 49, -64], [469, 14923, -213, 25, -757, -92, 38, 92, -139, 13], [-3796, 4141, -1374, 307, 26, 39, -169, 61, -67, -203], [-85818, 10964, -641, -438, 122, -126, 165, -19, -7, -69], [23709, -2222, -1848, -592, -103, 218, -266, 222, 242, 81], [21701, 4404, -824, 238, -1151, 5, -234, 162, 123, 83], [25244, 7168, -1242, 449, 76, -60, 382, -104, -99, -49], [-48307, 12430, -707, -647, -97, 342, 29, -141, 92, -63], [167899, -7290, 2025, 241, -503, 272, -131, 223, 198, -53], [-209166, 11935, -2145, -817, -582, -235, 0, 78, 32, -34], [16941, -16396, -5891, 112, -477, -315, 341, -225, 2, 62], [-70971, 11017, -4626, -314, 1042, 11, -93, 19, 314, 36], [126240, -29137, 891, -259, -629, -213, -76, 302, -526, -139], [153806, 10938, -4821, -617, 471, 64, -113, 263, -56, 199], [184393, -38244, -3163, -981, -963, -228, 475, 165, 206, 245], [-312588, 28664, -2934, 280, 1890, 333, -28, 174, -121, -622], [17959, 9412, -7418, 929, 1771, 102, -158, 294, -105, 40], [101420, 8670, 588, 955, -652, -262, 238, -1156, 261, 131], [131209, -35006, -4697, -1497, -533, -120, -405, -635, 150, 149], [-35194, 38138, 11441, 432, 290, -88, 337, 445, 89, 639], [218857, -34779, -2168, 933, -562, -257, -551, -29, -979, -175], [164309, -38701, -5859, -73, 57, 766, -400, 638, 1187, 39], [290514, -1667, 1397, -2563, 1168, -419, 804, -2, -314, -100], [228899, 7373, 6844, 822, 789, -333, -107, 573, -682, -247], [346772, 37950, 7926, -2320, -1360, -286, 272, -236, 882, 486], [-23140, 6014, 9389, 2098, -1542, -298, 477, -331, -15, -131], [171412, 29904, 1212, 4424, 1998, 228, -620, 88, -440, -138], [336807, 61591, 2182, 729, 643, -479, -191, -1001, -103, -876], [149001, 19595, 7155, 3361, -33, 400, -410, 1420, -351, 369], [54817, -60213, 9628, 624, -4679, 1225, 451, 103, 516, 675], [327596, 12883, -8958, -2577, 2528, -364, -417, 493, 158, -237], [-269702, 31810, -9007, 2691, -1524, -483, -653, 1185, -735, -1068], [-231930, -104284, -7729, 2054, 1382, -889, 747, -13, -1448, 195], [-664186, -81376, -6248, 2695, 6738, 36, -1180, -230, 257, 901], [599890, 30012, 4459, -2618, 1832, -1236, 693, -1293, -1119, -693], [134022, -42348, -14496, -5400, 2459, 582, 1356, -1220, 308, 679], [-645984, -45240, 19310, -856, -1490, -886, -490, -734, 424, -168], [34995, -144043, -5262, -6185, -2683, -25, -1157, -555, -229, -908], [-370155, -20077, -4394, 2032, -351, -607, 425, 1301, -1534, -1211], [401680, 44516, 6398, -5586, -2866, 3442, -1422, -474, -334, -62], [1062210, 31344, 2469, 510, -11624, 1518, -485, 1097, 353, -227], [-3398, -241242, 19109, -2688, 3956, 24, 1033, 429, 365, -1891], [203458, 48325, 4949, 3867, 4316, 951, -2496, -882, -162, -1370], [-46517, -135724, -9860, 1471, 7251, 1024, -328, 2736, -223, 636], [561603, 94176, 11883, 2953, 2941, 668, -895, -1317, 2514, 845], [274622, 175985, -7519, 7774, -5410, -2197, -236, 2036, -417, 1327], [-181838, -207158, 17729, 3424, 7784, -114, 2061, -1761, -1819, -185], [322769, 121853, 1682, -3451, 7291, 129, -1233, 405, -2137, -376], [-127105, -149360, -21583, 79, 8933, 2208, -381, -1659, 1014, -171], [810026, -44810, 3955, -4044, -2728, -428, -1605, -1937, 4903, 611], [-153023, 81477, -23202, 2272, -8599, 863, 1379, -1121, 2222, 393], [-787415, 166445, -12900, 7654, 9744, 835, 273, 2569, 884, -1296], [-612319, -304045, -20602, -4916, 2846, 3659, -1879, 341, 1134, -230], [960158, 60091, -38445, 2435, 5176, 889, 320, 858, -2272, 1008], [-1180077, -11868, -22078, 657, -499, -2644, 34, 1180, -3617, -2200], [-20994, -139681, -8072, -13576, 4288, 284, -2999, -4807, 2651, 718], [-689700, -96611, 36103, -4254, 3437, 2245, 1218, 3866, 1413, 2516], [-490759, -18169, 34611, 4167, 1727, -647, 80, -974, 3318, 3263], [501162, 68244, -101552, -7018, -1930, 98, 1224, -1028, 2368, 2668], [282587, -158816, -61361, -8354, 13071, 4602, -2931, 2057, -3089, -1468], [-435598, -6842, 9793, -4524, 14518, -1138, 3257, 635, -2935, 1617], [1180426, -7894, 20764, -8616, 22086, -1734, 4510, -2138, -3224, 366], [-520938, 70284, 56980, -5802, 7048, 4690, -1196, -176, -1304, -2078], [1550927, 37546, -32452, 2491, -9239, 1124, -1102, 4970, 4459, -166], [2196830, -244074, 92023, 122, -4966, 1512, -1777, -3305, 2029, -1385], [-537862, 125829, -15367, -14330, -14660, -2129, 3832, -932, 2911, -455], [-1537072, -220693, -85983, 311, 2602, 1275, -3074, -660, -796, -1336], [-606571, 524995, 29928, -13069, 1705, 2459, -1593, -1875, 5991, 266], [-204186, 125322, -13125, 12042, 15390, -8488, 6839, -1713, -3803, 2599], [2106887, -59870, -17605, -223, -10495, 1291, 1217, 291, -4715, -2595], [-1210919, -67868, 46278, 1623, 21891, -2504, -4474, -1042, -1745, 630], [4493766, -559808, -41262, -14136, -7644, -1211, 2188, -3928, 1837, 2450], [-209349, 835238, -45052, -9006, -31951, 784, 3586, -1972, 5128, 2889], [-571508, -692279, 32679, -10365, -17224, -3331, 1190, 1152, -2330, 3568], [-2657108, -34417, 108682, -9019, -4444, -5218, 1167, -267, 1230, 1849], [-1449288, -275240, -122963, 9339, -7192, 9616, -9413, 3693, 8084, 804], [4308681, 304352, -28039, -924, -20494, -1356, -1781, -1721, 8743, 1727], [3839104, -127109, 38077, -7176, 25501, 4323, 1378, -4630, 277, 1544], [-48297, -529962, 35175, -7620, -8581, -10232, 11755, -2593, -7179, 734], [-641039, 99010, 92267, -30029, 848, 3886, -3217, 1953, 7154, -658], [-1848535, -690066, 77063, -3307, 7443, -4636, 3407, -2931, 3240, 691], [5214481, -339850, -119287, -15154, 5813, 422, -7051, 3305, -4461, -6744], [-3613482, -212809, -49707, 775, -23880, -69, -4746, 1796, 328, -3878], [584217, -259491, 57583, 9251, -16951, 13103, -14150, 11300, -4742, -3131], [934978, 212815, 2411, -11244, -10800, -5209, 6604, -1072, -2291, -4019], [944498, -26363, 137531, -3500, -10776, 6957, -8262, 462, -2051, -6831], [417081, -62765, -22745, 12221, -4665, -10635, 2192, -7458, -10684, 3457], [-299537, -108115, -64088, 643, 20193, 1081, -6111, -8337, -797, -9442], [2963463, -371646, 16677, 657, -3140, -8222, 16829, 10291, -7930, 410], [-2418143, -451547, 50680, 2873, -9515, -7577, 12501, -8257, -17545, -2128], [-221202, 66396, 23075, -4868, 49424, 4046, 3153, 8069, -13341, 1311], [-4484349, 512685, 7742, -2572, 14766, 5801, 7387, -2261, -370, 3546], [-239332, 519227, 93273, 29279, -21982, -557, 8550, 7944, -8480, 2444], [3152767, 296258, 17418, 1668, -39007, -320, -11824, 2970, -10392, 4381], [-5028127, -241823, 104928, 14438, 14282, -9685, 24417, -9299, -9132, -5222], [4737749, -204235, 179566, -29053, 35711, -12285, 14767, -11023, 4539, -1218], [5305470, -65556, 114196, 4436, -13400, 2153, -8354, 9016, -7275, -5986], [3354571, 261535, 60926, -18601, -3476, 10333, -20587, -8973, 4525, -3933], [-656378, 271760, 97338, 52978, 11622, -9268, 9254, -6158, -20076, -6798], [3906687, -1095460, -191220, 12499, -20751, 1748, 7232, -11392, 3437, 2022], [-2824074, -477359, 211624, -30813, 2744, -9065, 13341, 7321, -21555, -6901], [7682865, -77114, 9156, 3084, 54155, 18283, -3140, 9000, -7467, -2345], [-1161392, 159762, -10664, 42362, 7926, -9164, -7442, 3898, 4376, 2868], [-278262, 845864, 43817, -5668, -55344, -2696, -7149, 7777, 9345, 14859], [8811755, -702024, 86049, 36837, 3661, -8136, 2595, -787, -9556, -353], [272785, 224152, -139613, -15082, -80455, 19506, -18183, 7881, 15059, 3374], [1644155, 1292151, -50618, 13410, -16823, 21487, -28447, 15037, 12434, -1709], [-12341007, 246971, -44274, 8075, 41523, 5589, -19163, 9303, 6895, -3082], [10978011, -144974, -30852, 36139, -15383, 9463, -11372, -810, -6678, 6008], [-2731439, 508605, -71186, 24847, -10417, -11447, 11307, -9087, -7251, 4734], [-3758542, 33367, 66300, 13471, 14430, -20142, 14437, 3671, -6574, -12369], [-1468628, 973036, 104789, -26273, -3694, -22265, 14057, -1169, 877, -7738], [-2380185, -906758, -162354, 20902, 7149, 15484, -6748, 9114, 9166, 13935], [94606, -782333, -244655, -34026, 51186, -6401, -2182, -16542, 12357, 3879], [4532484, -869930, 167309, 31398, 19004, -1322, 7773, 5069, -7383, -6037], [2295994, -633750, -197966, -6910, -10404, -13422, 17136, -12904, 3200, -4080], [4907158, -906164, -7374, 53962, -11918, -4004, -10398, 2414, 19624, -2042], [5223635, 310468, -53275, 26363, 30451, -6046, 8447, -3211, -11416, 8087], [550498, -44119, 5036, -29401, 1374, 14586, 3427, 11545, -13624, -18089], [2298720, -246798, -214742, 16168, 24578, 1340, -15764, 15936, -4916, 6260], [8016074, 1482026, 241341, 14958, 3312, -2170, -6615, 23015, -2757, 7521], [14746195, -570895, -113127, -2338, -29297, -25557, 30128, 13594, 467, -4366], [-4205888, -995898, -129063, 6462, 48168, 1398, -12047, -24167, 11965, 4403], [6511724, -1212655, -19759, -12438, 99954, -13503, 18456, -9548, -6119, 4203], [3287606, -55230, -140500, -33338, -51860, -18888, -5750, -6522, 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[-65528930, 10563891, -585362, 89627, -546646, 132205, -158967, 101871, 199937, 122357], [48948092, -13460721, -1647830, -85701, -892084, 185028, -286325, 286501, -26334, -198137], [98791942, 3375376, -4283297, -27028, 528532, -22812, -113163, -132145, 254315, 20657], [-28652721, -10697656, -534878, 268213, -275265, 170920, -142370, 96182, 225845, 67960], [-75835192, -18685681, -2678860, 175501, -541452, 182744, -94199, 137123, -103874, -155169], [-2501585, 5335678, 1695332, -88893, 954268, -24278, 533494, -121570, -348649, 59585], [105377840, 13461293, -982236, 126483, -7706, -146115, 265009, -11193, 66501, -10309], [-127209903, 7731225, 4304910, -116665, 297099, 76467, -134933, -41407, 135471, 132746], [100883715, -19497602, -3700718, 205392, -50912, -145142, 66436, 30084, -178174, 12006], [38382282, -10526700, -1578865, 397942, 616538, -260977, 244315, 25739, -218024, 59231], [92746945, -22805860, 2392935, 105312, 1092985, -303052, 86085, 180193, -168585, -31692], [-177253295, -844595, -2988136, -59692, -702101, 100433, 122173, -134307, -33018, -3], [16922618, 5180798, -452296, 701020, -200326, -18738, 49574, -104858, -261968, -74166], [82490817, 8050096, 981243, 525123, 328299, -292082, 94765, 145051, 44442, -56107], [-35442062, -15634516, 3114404, -453228, -125996, -231608, -134688, 71180, 289520, 91456], [-110004954, -14697966, 2443177, 282844, -685990, 262305, -474577, 413411, 188028, 169907], [27123185, 32138289, -3386935, -1511537, 736743, 64990, 72542, -78492, -106199, 36629], [11427564, -20188993, -2648433, -420209, -1424037, 77235, 108008, -6586, 120596, 23411], [17920978, -1198598, -1164428, 186032, -170874, 466570, -269642, -159002, 159208, -72666], [39738392, 1315819, -1596437, -752123, 735440, 57215, -160476, -215666, -59564, -128082], [6689433, -6019342, 1020819, 282349, -22719, -37694, 236779, -214411, -99230, 105161], [-216559299, 15597712, 2682205, -586840, -527839, 258402, -233433, 45227, 126583, -90030], [-21250953, 7100164, 1275271, 3899, -1014007, 185018, -277779, 53491, 498666, 349791], [-16234464, 7630026, -3022354, 262082, -1544344, 178090, -312460, 121364, 135428, 84624], [-55907646, 7811360, -947566, -849434, 677588, -61578, 172754, -122986, -325006, 62340], [81322198, -8755063, -3107360, -358381, -425752, 74612, -361821, 296165, -31026, -365563], [-132887497, 2785622, 602727, 1058870, 294015, 145248, -164269, 208475, -81245, -19884], [-59396677, 10222980, 1202703, -1435324, 403031, 460744, -213195, -83119, 262131, 18414], [-25804460, 11539536, -127785, -218449, 387173, 89038, 129689, 10401, -206456, 3293], [71656051, -4621294, 5453266, 259865, -87569, 168790, 143544, 396924, -4599, 242348], [35525290, 7191851, -5463465, -360600, 209383, -322179, 177556, -345668, -53947, 109916], [272011937, -23853496, 2270719, 586012, 264603, 300363, -210341, 56923, 72144, -213306], [-148805144, 31626668, 7042648, 303820, 916108, -240744, 198988, -219052, -135528, -27720], [-19448084, -30807222, 3536167, 673430, -367153, 120032, 24399, 232183, -346385, -27556], [-78672327, 8111805, -2745448, -1002673, 414035, 293671, -181395, 273467, 298013, -147380], [-80503289, -2255531, 3192992, -767750, -90040, -20513, 467965, -9183, 78388, 99276], [-9070156, 7122459, -1854861, 164622, -296552, 38909, 337172, 11876, -93639, -212017], [-60487738, 7824632, 3616585, -591948, -596696, -224684, 86539, -108759, -86235, -26245], [-32910293, 14919484, 2971385, -75777, 852445, -174894, 366747, 236761, 117148, 89169], [-88845165, -19816677, 1918593, 13784, 1447717, 452064, -524924, -139444, 290836, 112924], [-144430411, 24910509, -2610438, -51945, -1115165, 54373, -532265, 161229, 475885, 156262], [14436691, -386506, 143249, 529883, 112015, -405628, 298557, -310629, -40236, 464027], [-158818258, -705602, 4608972, -46610, 367860, -73407, -188676, 14618, 75733, 24776], [32957190, 15216336, -1313573, -640920, -33050, 108862, -16675, -495877, 542073, 315223], [5389355, 16298217, -3620983, 936899, -751105, 54545, -333392, 153994, -50522, -130355], [-106878536, -31654734, -775750, -206655, 498040, 285198, 103796, -154838, -22619, 231847], [74000142, -1051640, -714115, -21016, -891258, 28602, -484397, 391959, 149929, 70619], [-276213924, 19913416, 324158, -335750, -744826, -191355, -158008, -261394, 175345, -23060], [-49726641, 739853, 4787247, 545739, -379415, -459200, 144664, -213742, 78113, 65717], [61415269, 5256038, 3050657, -313649, -637425, -366314, 11689, -339537, -152818, -64849], [-164354319, 27050954, 518597, -241934, -243627, -88816, 70849, -156291, -84347, 135904], [147785854, 19681837, 456713, -645968, 336540, 138067, -236288, 146976, 59521, 322659], [296178087, -27573611, 2478368, -196702, -1694246, 53883, -531587, -76463, 679540, 168622], [117661241, -176376, 2487055, -132669, -702427, 142786, -257503, 1803, 231282, 63263], [-236529759, -6299518, 4840639, -634884, 1038655, -182763, 125517, 4723, -212666, -108328], [-17850425, -3009971, 814012, -570142, 862196, -95469, -309919, -231027, -94988, -302060], [-113126376, -20547104, 2442414, -207043, 63242, 48433, -446812, 343866, -212108, -153811], [346031935, 32163427, 1601856, 490352, 101435, -14045, -95845, 312155, -191538, 100693], [45545892, -8769274, -3945816, -199802, 1333634, -3410, 514294, 45242, -170930, -69820], [-35775958, 2603128, 1451142, -71844, -1516964, -341610, 452750, -8862, -65036, 157258], [16612497, -8129200, 3023233, 714313, 1234937, -190958, 158375, -93947, 61418, -18461], [97728767, 17156239, -822547, -588345, 939017, -153857, -166462, -201068, 164138, -312621], [63073970, 44779015, 1976501, 830181, 136774, -584805, 597510, -111276, -235686, 310354], [-156529751, 6585911, 3612452, 292953, -621575, -384167, 24707, -37015, 34543, 94472], [44780169, 37561556, -5704124, 110661, 97473, 15904, -470064, 286664, 334363, -16428], [-71545324, 18872928, -387851, -891005, -1134503, 455022, -309741, -267441, 209482, 41663], [1412922, 28637245, 1665816, -289921, 734134, 600797, -49107, -46541, -81587, -201549], [46007988, -2338046, -315135, -272425, -2539569, 256190, -26707, -313, 135230, 151371], [261940044, 22097566, 6990896, -524174, 1133910, 11400, 306194, -402114, 9256, -63316], [-423089871, 31642191, 5016454, 480416, 449198, 125517, -227511, 56621, -8940, 54646], [-21141308, -22612969, 869600, 790535, -448008, 377312, -107331, 795247, -466020, 36397], [164086394, -47817127, -1325021, 1067204, -291788, 22457, 185126, -62630, -462923, 217037], [-99261676, 4555520, 4268054, 625806, -221522, 497000, -364350, -313962, 272584, -124222]];
    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_u();".
// 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_u(:prec:=10)
    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_u();".
// 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_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_825_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![-7036608, -12856224, 7443872, 1367872, -822204, -33410, 24456, 309, -271, -1, 1]>],Snew);
    return Vf;
end function;