
// Make newform 722.6.a.q in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_722_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_722_6_a_q();".
// 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_722_6_a_q();".
// 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 := [-34308015556724472, -19853455953893436, 5743342864926480, 2435236025483701, -166593470925336, -51993847360191, 1625551725606, 437045334939, -6060225486, -1680307373, 9081402, 3170019, -4674, -2871, 0, 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, 0, 0, 0, 0], [-12393023470267764093283924841219808863105377343259426568916595208674293981382960, 17955376184450829659894493518456176456174064452314034561820191514805777806167821, 741238053252024687699798249478063868598237050056344492872778175271091435325746, -317470832929184273607096080212045956210081476868250946079006935500862273263981, -74879260709363731083512438539835757001613304106823349297418581638572669444072, 7324310400115770730297555639990395103891185318313842539091455364231490085069, 1115986915532158891024729935440697827154512933892037267016661045356579830022, -36089727152540905306231732021351154601491796388975128115072181679422745379, -5435399152333585604396570883451160913751377261027912485358257669458124770, 67667588404002050136116005570846853168660930971521869280691748597143897, 11513933901434259974050510049759263843126410630659800501543066863428824, -51271901767815087872130034831977073713539679562842359914541451680333, 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-78932266384298523729344784998060691317516047395527464946328748563048331092203], [-168374119573160184517286967623634116355734395540539986206177678854189670254697177459443605700, -53750294467807347060971915656420618370994965825259066154308474937774504554886862083447561332, 5396751452946760361103267283935998871200734482192600851154698119471246995169279706302331631, 4072247228536580107815952694635795242225968181004260129107411031693161821059861874077916238, 230706205314171296280672699386263797075872597724145139395397724392206563280512252807602001, -56954938201075487875439935022849890633911032968795159027680140389034996065220830927495032, -4331329369260320250365442409318478702821287326452606595799266851129979367950308141798261, 262562071082758442988452631120651244000910901747906222543348032245118947427079425989932, 22046605848334917487132212872129088481599638155425688779784028330562766699901519651703, -508235364349582306477651699146149095492116528968813421133163390919634449235870247966, -47272786970056006777521103575004408664818659532899764897203274413986250265205373575, 428599751805844338615786237039009296172800496689216289631439529454936700539831614, 45667283469043024627841731914041337397604138764751035669405335085668576284673187, -127498156042329117255373297755452452440572696852065565023593655403883970124844, -16429674370372509658288176336772938494347217141927412319496016384007792566447]];
    Rf_basisdens := [1, 31907264485310175455659975890234025324781775019386655600486211367169490175703806, 10635754828436725151886658630078008441593925006462218533495403789056496725234602, 3358659419506334258490523777919371086819134212567016378998548564965209492179348, 1679329709753167129245261888959685543409567106283508189499274282482604746089674, 309565465215864996874625298849024968498786457594731540243546174944356521511806157242021967, 1238261860863459987498501195396099873995145830378926160974184699777426086047224628968087868, 206376976810576664583083532566016645665857638396487693495697449962904347674537438161347978, 1238261860863459987498501195396099873995145830378926160974184699777426086047224628968087868, 412753953621153329166167065132033291331715276792975386991394899925808695349074876322695956, 619130930431729993749250597698049936997572915189463080487092349888713043023612314484043934, 206376976810576664583083532566016645665857638396487693495697449962904347674537438161347978, 309565465215864996874625298849024968498786457594731540243546174944356521511806157242021967, 1238261860863459987498501195396099873995145830378926160974184699777426086047224628968087868, 206376976810576664583083532566016645665857638396487693495697449962904347674537438161347978];
    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_722_a();"
function MakeCharacter_722_a()
    N := 722;
    order := 1;
    char_gens := [363];
    v := [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_722_a_Hecke(Kf)
    return MakeCharacter_722_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 := [[-4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [7, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, -1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [8, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0], [7, 2, 1, 4, -6, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0], [275, -2, 0, 4, 1, -3, 2, 1, 2, 1, -3, 1, 2, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [254, -23, -5, 19, -35, 6, 5, 4, -1, 9, 3, -2, 1, -4, 1], [-974, 61, 2, -7, -3, 13, 2, -7, -1, 8, -18, 6, -8, -2, 0], [-453, 14, -7, -79, 56, 0, 12, 3, -5, 9, -21, 1, 3, 7, -7], [304, 83, -5, -55, 75, -56, 2, -1, -8, -16, -7, -7, -2, 1, 6], [-304, -30, -7, 85, -46, 27, -44, -8, -10, -12, -38, 4, 4, -12, -6], [6257, -97, 7, -9, -27, 62, 21, -6, -10, 24, -26, 2, 3, 8, 4], [69, -52, -10, 74, -258, -21, -19, 3, 25, -7, 1, 7, -1, -3, -16], [-3205, 187, -1, -61, -177, 105, 16, -11, -51, -6, 14, -4, 16, -4, 10], [574, -62, 35, -545, -11, 105, 12, -19, 27, -13, -15, 31, -17, 22, 1], [7957, 591, 3, -173, 41, -67, 34, -1, -20, -30, 33, -31, -22, 15, -20], [-4283, 1068, -26, -219, 180, 126, -4, -5, -16, -80, 46, 19, -16, 32, -2], [3578, 1255, 19, 895, -369, 120, 37, -26, 33, -57, -69, 14, 13, 22, -25], [-337, -314, 39, 105, -700, -70, 73, -5, -59, 15, -26, 42, 49, 21, -44], [7850, 186, -35, -215, -661, -191, -64, -48, -25, -30, -109, 56, -90, 72, 10], [13314, 1083, -24, -747, -367, -145, 107, 12, -63, -23, 314, -84, -47, -55, 24], [6734, 309, 15, -499, 946, -82, 50, 46, -148, -93, 108, -56, -26, 68, 41], [-19998, -306, -38, 773, -1909, 422, 65, -1, 85, 205, 28, -166, -99, -33, 36], [-4056, 1668, -40, -1338, 367, -358, 275, 63, 1, 3, 178, 4, 135, -61, -88], [-27968, -119, 121, 1280, 884, 38, -22, -103, 105, -322, 25, 55, -80, 151, -34], [-3419, -405, 19, 968, -570, 570, 198, -137, 44, 71, 54, -143, -228, 3, 115], [-53307, -83, -141, 877, -917, -141, -132, 153, -88, 164, -247, 59, -4, -49, 8], [-31488, 470, 56, 207, 2548, 146, -293, -49, -79, -349, -290, 66, -11, 65, -174], [56490, -4320, 101, -2105, 2543, 137, -76, -62, 25, 236, -85, 198, 72, -176, -154], [7276, 3298, -6, 6087, -4174, 748, 137, 73, 6, 417, -455, 65, 55, 1, 165], [7061, -2844, 95, 1540, 905, 26, 234, -268, -359, 108, -113, 227, 152, 285, -334], [-23231, -1673, -93, 1357, 645, -1038, -669, -124, -274, -330, -824, 264, 2, 127, 53], [60142, 5300, 636, -2570, -1139, 1612, -87, -149, 132, -483, 785, 179, -35, 100, -192], [101078, -6406, -251, 2427, 2883, -72, 270, 246, 504, 291, 553, -54, -165, -28, 230], [108069, -3438, -40, 3306, 1238, 893, 32, 246, 140, -114, 322, -418, -342, -236, 294], [94786, 6176, 443, 2362, -662, -293, -190, 165, -850, -275, -175, -133, 43, 265, -468], [25425, -280, 30, 1824, 866, 232, -671, -891, 54, -224, -131, 309, -830, 9, -24], [16591, -7158, 236, -42, -5135, -1575, 15, -277, -575, 421, -1038, -202, 81, 53, 2], [61989, 1260, -393, -6479, 6316, 1127, 61, 628, -71, 533, 151, -402, 27, 9, 31], [-173237, -2814, -108, 3292, 2169, -2019, -447, 331, -447, -1049, -520, -82, -67, -367, -300], [179615, 5475, 980, 3061, -3466, 1252, -278, -700, 732, -1705, 326, 118, -143, 220, -383], [-12809, -5468, 603, -3192, 328, -3538, 1379, -494, -213, 416, -979, -367, -147, -118, 57], [42388, 9863, -64, -7045, 1144, -524, -390, -654, 129, -1753, 1099, 21, -928, -281, 43], [89701, 1872, -699, -438, 2741, 891, -337, 474, 242, 563, -533, 206, -235, -278, -442], [173403, -3039, 468, 3015, -10686, -3507, 729, 185, 436, 341, 1858, -637, 448, 207, 52], [959, -2194, -1065, 465, 6991, -2698, 1875, -209, -608, 2179, 202, -117, 449, -359, 706], [50718, -864, 935, 21714, -4541, 965, -1309, -1150, 961, -312, 808, 248, -532, 559, 14], [287239, 2, 535, 8572, -15359, -2118, 2155, 237, 30, 1727, -1309, -741, 203, 214, 452], [161394, 9568, 342, -656, 8884, 1572, -1207, 489, 727, 905, -100, 874, 233, 193, 154], [278933, -2382, 820, -2085, -3843, -552, 1934, -194, 538, 1695, 1835, -312, 259, -366, 1119], [162095, 1869, 701, -29543, 14774, -591, 1768, 437, -112, 1748, 299, -501, 982, -63, -150], [211798, 11118, 530, -1247, 7144, 597, -667, -267, -2081, -956, 2793, 367, -242, -9, -667], [-219028, 658, 446, -7646, 4819, -7870, 2214, 76, 1729, 2920, -2057, 645, 512, -507, 1052], [467486, -18700, 1276, -30642, 16550, -691, -484, 1440, 1223, 1667, -1042, -1372, 731, -596, 88], [104134, 9171, 1893, 11803, -7500, 4042, -1959, -630, 1086, -431, -3118, 408, 25, 409, 810], [158509, 8166, 265, 7506, 6951, 1198, 2321, 434, -2095, 962, -2697, -918, 2086, -170, -788], [330214, 15327, -832, -3541, -18444, -386, -2712, 1330, 1142, -4751, -558, -1016, -418, 498, -1377], [29765, -10189, 1262, 22391, -11600, -822, -674, 61, -905, 2930, -3458, -228, 714, 204, -256], [612404, -10128, 1685, 10592, -2071, -3297, 834, -2688, -530, -2078, -1110, 826, 1270, 749, -307], [308629, 24542, -502, -15224, 20152, -3792, -859, -82, -889, -2204, -1675, -1191, -639, -98, -339], [-222784, -9423, -153, 4840, -6589, 2735, -3065, 1506, -469, 728, 652, -304, 920, -2473, 1088], [-75307, -27504, 2354, -6884, 22817, -2, 2626, -991, -2349, 2666, 3398, -1115, -494, -113, 127], [1243, 4218, 1305, 13006, 19876, -2148, -645, -487, -129, -793, -2252, 1066, -1349, 1065, -1716], [173710, 49177, 2349, -38645, 10230, -9038, 1721, 1202, 242, 3339, -1358, -1242, 1037, -855, -100], [459181, 37825, 2376, 14852, 6519, 7000, -877, -2682, 1220, -3633, -872, 1598, -3189, 463, -601], [39327, -38231, -1590, -15479, -34132, -4198, 2526, 289, 118, -1324, 311, -633, 720, 531, -2594], [-309701, 17804, -2415, 4326, -14725, -915, -5246, 3139, -887, 1032, 659, -453, 1348, -110, -1406], [720976, -5299, -2242, -6583, 14546, 1882, -2916, 216, 1907, 2111, -5567, -1441, -952, 797, 1443], [92867, 26913, 2366, 6330, -12506, 17123, 3904, -2156, -2911, -995, 2787, -1217, -1258, 731, 1137], [-112323, -17694, -3524, 30139, -28876, 12295, -657, 2104, -1858, -5154, 3416, -512, 2218, 584, -1742], [-225154, -21026, 1547, -38254, 5885, 7007, -1320, -5491, 2712, -5407, -5376, 2821, -5699, 1449, 2146], [295248, -23362, -1503, 9946, 14823, 4704, -7240, -559, -985, -3895, 5622, -666, 1426, -62, -3763], [-281809, 12441, 4519, -51904, 43311, 20080, 3936, -928, 1744, 832, 7369, -298, -1170, -1094, -2211], [708523, -25791, 5745, 3827, 30023, 9109, 8748, -1145, 1768, -426, 4554, -2289, 2044, 495, -2417], [-764146, 35878, -4197, -10962, 13850, 17271, 2526, 1008, 772, 2784, -620, -1412, 120, 370, 232], [1198556, -27711, -2266, -22321, 43221, -24011, -2178, -1011, 3932, 1508, 4379, 3317, -4090, 3525, 468], [137393, -10912, 1876, 20549, -1359, -1763, 1106, -3070, 1104, -2084, -9936, -650, -4830, 3428, 2468], [1027254, -39271, -1106, 1490, -15605, 10407, -693, 3800, 3499, -4663, -2233, 1127, -635, 900, 768], [-1833168, -13599, -4381, 19785, 40083, -5116, -7612, 4181, 7046, -3224, 801, -2421, -1905, 427, 348], [820033, 64523, -4886, -139, 10453, 17912, -2434, -423, 2063, 5514, -5342, 6248, -3264, -1440, 2434], [716424, -22214, -3006, 23770, -6316, 2681, -8110, -2890, -2959, -61, -16684, 3126, 4825, 3918, -4780], [763062, -73152, -657, 700, 3683, 11351, -5059, -8199, 3331, -7667, -4126, 3130, -9905, 1747, 5488], [474131, -15944, 201, -40491, -16777, 443, 6735, -1203, -2373, -5730, 7804, -81, -204, 1023, -4368], [307710, -5803, 2377, -73139, 25163, 11040, 2458, -2724, 1814, 7331, -8575, -310, -397, -1838, 586], [-1241881, -35272, -4559, -15402, 15275, 14359, -3461, 892, 8419, 7760, -9031, 87, 1023, -2228, 8007], [-543181, -29712, -1880, -38453, 111036, -12455, 16224, -1164, 2959, 2474, 15069, -947, -6766, -3447, 1206], [-56084, 1502, 4013, 4760, -15439, 5061, -4533, -7389, -1163, -517, -15602, -602, -1651, 5005, -4822], [1786173, -23950, -5059, -18124, 29934, -13436, -232, 608, -4028, -7462, -12680, 2282, -4712, -6626, -4561], [-987173, 25814, 3689, 84537, -65884, 4579, 5041, 4024, 1427, -20085, 16347, -1148, 3235, 4711, -1413], [578502, 22561, -2865, -62650, -43241, 8098, -6719, -8911, -3928, 454, -16158, 7479, -2496, 4509, 4727], [-975967, -1486, -13134, 13836, 49101, -1206, -12180, 2445, -7635, 1594, -15728, 2727, 1450, -6625, -933], [-3696459, -5974, 408, 35095, -49548, 15159, -6966, 3959, -5297, -16097, 6376, 2729, 2652, 3389, 403], [472425, -69021, 5615, 134624, -161931, 4686, 9068, -867, 11250, 12549, -1271, -4119, -4869, -778, 1192], [26372, -129785, 2969, 32160, -2021, 14927, 5499, -7383, 9127, -2019, 1004, 2905, -8199, 8755, 1061], [-1672392, 47860, -1015, 71272, -82325, -23183, 4421, -5379, -7285, 1427, 12948, 2042, 7703, 7105, -7950], [-1785909, -60161, 13048, -54689, 60276, 1801, 13743, 3217, 228, 7898, -7625, -4183, 9555, 6696, -2179], [1956489, -99686, 2336, 56016, -62196, 2971, 8641, 10991, -5691, 14649, -9769, -5807, 12791, -4131, -1682], [-1209473, -15378, 1964, 118152, -140111, -16055, 4365, -3015, 648, -9377, 10647, 4797, -10661, 552, -902], [696085, -28767, 1376, -68373, 43984, 37840, -3417, 4291, -6353, -10696, 2966, -1263, 3099, 10315, -4006], [1359782, 7364, -6124, -60180, -71181, -17298, -14897, -2247, -487, -8957, -16378, 11270, 8059, 1209, -1398], [-766977, 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650921, 184027, -208418, -261153, 904141, -516501, -23045, -175571, 28441], [-25505247, 5049857, 393155, 1144073, -1518963, -602761, -9478, 172724, 307536, -280827, 1023762, -195494, 20501, 22991, 90010], [-133380221, 2288028, 673599, 342179, -1290233, 831344, 55225, -508971, 282930, -157163, 139421, -45639, -154811, -5502, 128604], [10965649, -5021456, -76513, 484527, -5471457, 639263, -454793, 72429, 115503, -217094, 1197802, -32835, -61100, 28181, 292126], [99001163, 6313626, -589684, -2699666, 1942000, -1176173, -234056, 169404, -39105, 174052, 370071, 153693, -184034, -284155, 260020], [-75530854, -581424, 191580, -813962, -3934923, -1145626, 541488, 13164, -122769, 505696, -896839, 623373, 174868, 685345, -379524], [87154448, 5650717, 295353, -2704417, 4148805, 2240322, 413287, -9954, 10131, -248767, 373793, -204408, -60409, 121762, -119013], [9995774, -1015576, -82184, 3427883, -4080683, -835845, -865056, -80625, 156574, -176224, -158032, -31403, -302554, 178597, -234984], [98596436, -1282201, 109183, 6873849, 1392199, 2984063, 345984, -50849, 71015, 684425, -301483, 11329, 371292, 310665, 58408], [-1426920, -7861344, -100988, -1920354, 5934235, 1133386, 244986, -157661, 245542, -283513, 1126563, 145365, 58212, 6631, 230487], [-41154624, 9300652, -116501, 796218, -749011, 44648, -28374, 214383, 436424, -374315, 623369, 18353, 252544, 16327, -149849], [-25528961, -2885530, 250954, -2878648, 1744814, -34088, 381359, -269527, -21120, -256350, 197635, 329167, 22034, 158577, 141308], [-5561976, 5175835, -208728, 7491224, -7562885, 1733989, 639846, 102097, 438677, 478538, 1233482, 574, 442740, 34566, 146246], [-121587475, 5235001, 249759, -4476729, -2246881, 1139690, -337106, 284040, 80001, -262150, -75226, 95640, 346532, -120787, -206766], [-116469037, 1346688, 99641, -2064179, 3464772, 767984, -396954, 243853, 344613, 282235, 727749, -81865, -71761, -81707, 302713], [-71338166, -3809546, -290324, 640771, -1921286, -310969, -411354, -495957, -197326, 952600, -458803, 227063, -423320, 124505, 243052], [83286706, 2294213, 376173, -1550889, 191288, -599826, 19107, -243240, -452093, -172069, -675507, 368929, 647657, 133574, -45590], [-18889876, 2043519, 328655, -5292073, -3407479, 2275250, 863103, -270666, -364411, -45229, 869591, -354970, -142309, 22274, 182559], [-89386962, 6758275, 262270, -1992189, 1562442, 3718640, 322350, -104704, -391287, -271139, 567497, 5371, 110934, 501917, -328915], [39493002, -157218, 9154, 3745669, -5375428, -1725054, 2859, 307488, 324046, 656044, 698713, -285188, 60445, -185690, -53343], [-9572493, -3268188, 185372, 7998215, -5895579, 1100064, 428464, -114722, -463152, 286249, -185817, -207634, 263219, -162206, -78041], [121948390, -3339592, 668252, -2627, 842937, 1814252, 215137, -176845, 171976, 1023495, 741171, -367657, 336245, 52884, 316382], [-8970233, -3478870, -393953, -4286901, 6809124, -1578292, 524011, 216885, -228718, -276955, 643865, -120031, 85985, -221178, -139712], [-19856933, -5821223, -440403, 2823726, -2854671, 433123, -633581, 380426, -374902, 318543, -408614, 174630, 335597, -212793, -186328], [-136252475, -6982479, -56936, -211611, 1177650, -889500, -492463, 348150, -26494, 469427, 321444, -301900, 193367, -625453, -133430], [34377469, 10664166, -199050, -2730940, 5786242, -137842, -84294, -169794, 260499, 769332, 626337, -115791, -359790, -266035, 657522], [53304737, 8570249, -522162, -1281600, -3031620, 1421471, -291570, -276679, -179058, -416867, -1033121, 102301, -214641, -195731, 321195], [-69503336, 1403598, -178623, 7993439, -1299890, -576371, -1056648, -639518, -109597, -1259552, -291985, 701419, 3908, 155403, 43658], [-153954687, -5877946, 246511, -1227238, -3433655, -307992, -71381, 23015, -50445, 381, 279553, -228743, 233681, -127218, 142565], [-124725613, -4380251, 162134, -6792414, 9463294, -769165, -279718, 153107, -97346, 2333, -196179, -250405, 127753, 224051, 215063], [27703424, -3632541, -405344, 1721444, 1768001, -3412278, -367928, 237948, 364960, -602172, 513821, -548900, -89040, 278456, 263664], [45455067, 1802056, -394092, -1820392, -275631, -671741, 462061, 51241, 129206, 857673, 98359, -291875, -199739, 185612, 74906], [-8427287, -3947131, 397435, 3418025, 206993, 529739, -265231, -398039, 392723, 171298, -156186, -166325, -23188, -33953, 616654], [118080872, -518321, -570089, -3258491, 3283064, -1021250, 879480, 385936, -4888, 530963, -142028, -222470, 521337, -92778, 33220], [-31033994, -1468378, -94155, 5735361, -2343028, 1052749, -636603, -535129, 217169, 62103, -660502, 534792, -848485, 158391, 293792], [204074719, -1505379, 202399, 1049349, 1634375, 54947, 393216, -98415, 189956, 474998, -177465, -241285, 109484, -252177, 176920], [-19139941, 2166989, 82126, 4819373, 4320838, 540021, 556483, 1007, 48622, 890079, -636926, 17459, 343835, -236852, 90663], [64720420, 2006942, -129997, -1282638, -4541977, 103831, -22334, -389552, -162340, -313034, -313132, -25124, -885268, 165809, -98919], [48251105, -1327643, -197148, 3708505, 523116, 1554853, -998218, -452810, 568645, -413750, 568032, 290490, -493246, 193022, 437233], [-46485234, 6849604, -42716, 1452385, -1238766, -2595696, -196432, 215086, -43875, -308528, 69515, 300737, 221252, 504593, -448774], [-51948888, -4679215, 22469, 5343567, -1265133, 2074587, -501777, 232418, -203608, -889284, 9549, 263418, 64598, 418442, -345165], [-129375890, -8616773, -394363, -1732016, -1650267, -598419, 222429, 48631, -356309, 291210, 690428, -157284, 221899, -358473, 348071], [-64715581, -2502211, 181993, -2515167, 750577, -169292, 793124, -124302, -288657, 168198, -175310, -583956, -217764, -28675, 499094], [11184259, -9388101, 365477, 3573201, 35825, 597703, 641182, -163519, 119899, 433626, -312488, 374612, 574736, 221376, 339566], [-42397819, 1659981, -298136, -4875193, -2819845, -1180830, 544384, 278165, -20577, -320354, 75028, 66910, 673140, -191322, -148276], [-28636255, -8351120, -466425, -8788923, 4184672, -1841180, 123982, 1504, -576064, -644202, -1069982, 249880, -207710, 347880, -71596], [72383906, -5791791, 6355, 1024110, 113396, 2954180, 91826, -282361, -274153, 19972, 39133, -127623, 24216, -17903, 150892], [43717261, 2024212, 212975, -1013674, -2754586, 247686, -790811, 220584, -49111, -128928, -401503, 261131, -200631, 232924, -425895], [-31365530, -10797168, -306927, -3237674, 2880595, -523265, 751229, 215017, 169400, 49295, 339223, -599681, -176331, -561254, 174518], [-107354099, -6077684, 421933, -2086972, 5487451, 1708643, 92743, 114560, -65090, 942937, -717529, 257462, -392853, -183646, -141570], [-83952982, 9482134, -218369, -503752, 1755097, -1395813, -74225, 38837, -271653, -318947, 69026, -45746, 362427, 519129, -196186], [142675803, -10014329, 377639, 4638376, -2930839, -281810, 691716, -74088, 433651, -264979, -568377, 203935, 191138, 260087, 150023], [-32690210, -10925828, -482541, 922896, 1639699, -190083, 328365, 58002, 7405, -60526, -629566, 133348, 600509, 82438, -18635], [2861204, 1616209, -389543, -1320261, 2010418, -650167, 626315, 22117, 243556, -728307, -1587804, 593247, 172681, -44996, -105840], [-34891311, -3364119, -172783, -1559024, 7117644, 293075, 338062, -265561, 82930, 460938, 1937857, 132815, 4160, -184787, 368894], [-98355793, 4035623, 724324, -3878743, 4624469, -1074780, 292020, 144841, 371295, 549850, -1573980, -177160, 506354, -240340, 213608], [33081989, -5058304, -346238, -5436878, -3653067, -2342477, -31745, 145800, -1006203, 802028, -654766, -164694, 277034, -452645, 229445], [-74216835, -8170500, -53066, -1495255, 925804, -1674275, -83064, 95196, 675655, 545580, 60626, -368234, -2616, -330892, 452889], [-19683051, 3045413, -660746, 1413788, -286450, 181763, -1299778, -144432, -50958, -39555, -809102, 456744, -394140, 144348, 72431], [43708514, 7640244, 348745, -4455418, 1200913, -1082292, -31038, -17583, -74943, -316105, 1146492, 435918, 704924, 15374, -441763], [34981594, -1230423, 237871, 3425388, -6355815, -137871, 988895, -354399, -1009963, -945991, 382374, -183271, 382781, 587627, -273221], [-78906377, -2893727, -551991, -899859, 317285, 484406, 676342, 111575, 260201, 538947, -321704, -393777, 403631, -349769, -255937], [-93770015, 210126, -519708, -199536, -549361, -3742733, -10089, 593047, -204884, 1199749, 422255, -574467, 350651, -752576, 364066], [147867274, -2044157, -49521, -3387381, -1820033, -1790332, 1429299, 369204, 16997, 784453, 197967, -345970, 760531, -86996, -576427], [-138817336, 1301699, 295498, 10474197, -2545897, 1008597, -318452, -92551, 830061, -528660, -1619726, 123016, -404218, -6694, -37270], [-16624756, -1754162, 286691, -1865132, -2272598, -3072003, 90560, -59060, 459292, 565912, 729274, -219174, -432568, -407656, 278066], [58215715, 1302453, -134958, -1753388, 2384286, 466324, 722974, -595089, -251313, 534470, -925847, 640287, -162112, -129511, -340761], [-215046061, -3677432, -149092, -5083594, 4480056, -1921494, 814434, 95147, 376047, 471968, 159741, -55229, -192722, -148315, 172207], [-72916775, -11577530, -806976, -271848, -2336330, -1763908, 351771, 673708, 595483, 872184, 485395, -393267, -384749, -457240, 170215], [98477109, 494356, 687900, 1797374, 3410360, 19542, -544159, -9124, 69678, -855028, -57604, -104468, 480079, 31795, 233697], [-19458159, -542169, 400310, -824314, 746028, 2401277, 519417, -161178, 3863, 353206, 1504251, -18550, 273466, 399305, -358101], [-49065863, 12833914, -465368, 2896274, -2350580, 766215, -217880, -266170, 123987, -389650, -669521, 758197, -588074, 87543, -27796], [37873305, 3534157, -462454, 4798509, 1580367, 1119294, -532988, 594617, -70732, -978504, -488629, -57369, 136115, 539693, -703971], [-82764877, 1452001, -307510, -5595303, 2098424, -32525, 608266, 512894, 11237, 374896, -897692, -604062, 360474, -137870, 49803]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_722_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_722_6_a_q();".
// 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_722_6_a_q(:prec:=15)
    chi := MakeCharacter_722_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_722_6_a_q();".
// 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_722_6_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_722_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![-4922141548222971, -2028046240465461, 3989649082088241, 1690978504212342, -156214328638473, -46284758138952, 1645531637743, 434202352248, -6021674247, -1709279884, 8713875, 3213570, -4339, -2886, 0, 1]>],Snew);
    return Vf;
end function;