
// Make newform 650.6.a.m in Magma, downloaded from the LMFDB on 10 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_650_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_650_6_a_m();".
// 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_650_6_a_m();".
// 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 := [22320, -1614, -547, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [5232, 2330, 3, -5], [-2928, 394, 15, -1]];
    Rf_basisdens := [1, 1, 288, 72];
    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_650_a();"
function MakeCharacter_650_a()
    N := 650;
    order := 1;
    char_gens := [27, 301];
    v := [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_650_a_Hecke( : Kf := Kf)
    return MakeCharacter_650_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 := 6;
    raw_aps := [[4, 0, 0, 0], [-5, 1, 0, 0], [0, 0, 0, 0], [-46, -2, 1, -3], [-74, -4, -5, -1], [169, 0, 0, 0], [-454, -9, 38, -5], [124, 46, -12, 16], [421, -5, 50, 8], [-1099, 4, -28, 74], [1450, -188, -45, 91], [-2868, -290, -24, -162], [3208, -16, -112, -200], [-5173, -11, -170, -22], [-2136, -522, -21, 117], [-13059, 40, -230, 28], [-10968, -1532, -7, 359], [2443, -1096, 282, 52], [-1030, -358, 345, 711], [300, -2078, -648, 20], [-28782, 1588, 180, -158], [-26357, -1603, 1502, 156], [-34116, 2652, 977, 411], [-18958, 678, -1328, -1530], [-25036, -182, 2308, -922], [-25147, 2764, -1758, 1520], [-88303, -901, -724, -422], [49685, 4127, 1110, -1010], [-5522, 6502, -1260, -60], [-67205, -1609, -2220, 2549], [5077, 455, -2630, 3908], [-15571, 4143, -5584, 1216], [35528, 9404, -6980, 78], [28475, 9085, -2266, 378], [548, 1642, 6800, -5682], [-24376, 2768, 483, 489], [5398, 14429, 5528, 1303], [-146496, -5158, 4148, -2096], [-81648, -770, 4780, 3382], [-148394, -8655, -5410, 10717], [215219, 19261, -2884, -5544], [4190, 2737, -7340, 10807], [-200927, -22961, 4906, 668], [-469908, 6404, -3660, -790], [-244120, 25352, 12276, 592], [218043, 25377, -5686, -3900], [183188, 23164, -2962, 8254], [-414722, 14948, -7840, -1490], [-174792, 14734, -7901, 4881], [493930, -26760, 19836, 1496], [-185869, 21275, 11292, -13123], [316516, 40694, 12603, 17847], [289946, -11754, -6644, 2432], [-313340, -16808, 20580, -35488], [-312073, -51700, 2148, -15916], [-17892, -15632, -28508, 1512], [10385, -15040, -7992, -27870], [343316, -56744, -24057, 17339], [240323, -55661, 20404, 19199], [406454, 44468, 18592, -834], [-1221460, -8444, 25650, 2114], [112002, 58604, 17232, -7666], [508854, -30492, -13732, -13034], [568439, -61507, -33826, 11360], [-619229, 67000, -11078, -10538], [1037606, -53344, -19336, -3810], [737930, 43912, 39136, -38162], [839093, -56816, -4218, 28346], [-876157, -7231, 64226, -49874], [291614, -77964, 16936, 956], [-356892, 30788, -3260, 27002], [-1429368, -65922, -399, -71667], [195331, -81419, 47604, -32310], [-528497, -129652, 20664, -43090], [24850, -22130, -45809, 31091], [1069640, -49774, -9960, 1962], [-752785, -32465, -3852, 38699], [862626, 129716, -23132, 65930], [1073574, 95508, -50356, 65390], [920310, -49090, -84372, 64714], [-2604457, 65817, -4128, 17288], [2038430, 136882, 24672, 32076], [-111254, 77476, 60368, -69036], [-727669, 74715, -7652, 26393], [-2208856, 168856, 56298, -16150], [914908, -132732, 7358, 48506], [-3298080, -278520, -35608, 89860], [3315652, 59340, -22536, -16910], [2454942, -48600, 50944, 19926], [-2805458, 35144, -79837, -68493], [2036355, -114343, 720, -57904], [423726, 376754, 36339, -18171], [1631200, -25338, 31705, 54463], [914255, 255737, 10718, -54202], [-3179474, 374084, -108879, 30863], [6031527, -240151, 93912, -39510], [1551032, 52868, -37676, -41506], [4934169, -256547, -102156, 14231], [-294044, -151760, 109078, -58530], [1806564, 118478, 31980, -181076], [-1262580, -54728, 12276, -138464], [-1683986, 3878, 84316, -67072], [4543161, 50895, 86546, -123734], [-2348191, 55796, 964, -136200], [1035908, -456236, 7698, -116418], [-1553066, -237932, 6220, -104736], [3394398, -224436, 181803, 9885], [190886, 536680, 117072, 57014], [-147423, 181023, 231328, -147182], [-2111679, -388264, 30278, 39578], [-11760, -396928, -239454, -10530], [-3099868, 485854, 151572, 47692], [2041612, -216650, 112852, 158534], [4603352, -107802, -250964, 17240], [-1598480, 270922, 288956, -165162], [-2443125, 84012, -100574, -19414], [64498, 16908, -210036, 328058], [-4019256, 342948, -242712, 48012], [972206, 338705, -197768, 40035], [2868573, 98267, -159342, 57974], [1130594, 381398, -59364, -297166], [-2637026, 902487, -43342, 70743], [-1513175, -380903, 370548, -70379], [426516, 676130, -315383, -43367], [5521284, 437678, -8171, 172967], [-12494057, 77124, 286152, -96950], [2593828, 19892, -22556, -195024], [5699511, 389409, 94204, -266726], [4527703, -379711, 190836, -136274], [-857940, -975364, -258440, 16854], [593294, 291686, 392361, -147123], [-2625330, -416874, 362563, -164243], [-4844249, 782681, -60586, 208428], [2759522, -564925, -73874, 26447], [-14878528, 733290, -62708, 43174], [-8927486, -21834, 165988, -216680], [-5963294, -282940, 192608, -337252], [-19651784, -226984, 285767, 16815], [-5758275, -908348, -74844, -198150], [5635530, 835940, 57408, 50224], [2880878, -383396, 17905, -278099], [-27196350, -21568, 48796, 59234], [1145135, -295439, 69152, -458794], [-8395204, 592646, -364895, 309243], [-8551471, -816264, -141502, 263800], [-3320112, 146882, -189832, -331456], [-9510092, 530168, -403916, 267984], [9631639, -757317, -249892, -141335], [-2625068, -260852, -23174, -114614], [3151626, -1057350, 15795, 187389], [-7544678, -1329010, -326284, -278840], [-25686195, 827652, -97648, 7264], [-56189, -154035, 265888, -372480], [6907632, -644500, -267070, 390794], [-16542853, -649515, 596136, -64944], [-10693593, 38845, -349786, -243020], [-1413964, 1862644, 164122, 325558], [-270244, 371894, 165112, 105112], [-648607, 541844, -685956, -318560], [-779258, -1748040, 476132, 229040], [-5350148, 466950, -102139, -425397], [10426218, -373767, 235228, 80519], [-25203248, -218590, 111105, -161401], [-16529028, -527500, -352166, -248642], [22127046, 777966, 248420, -337944], [-777588, 2142304, 16873, 94235], [2688667, -1903895, 248198, 2648], [-1960207, 95904, 715410, -5480], [-5799126, -1097734, 704592, 313948], [19281251, -602440, -166550, -129792], [7761412, 248238, -320156, 693630], [22992704, -612612, -803780, 462704], [-4741191, -736397, 561576, -36558], [2408082, 2073664, 474632, -271928], [17213873, 1161927, -502078, -81604], [-18274709, 1058084, -738764, -214036], [21873220, 857832, 203033, 450477], [-26030962, -544636, 257112, -353526], [-13510489, 895105, 575782, 195140], [-32459150, -1699405, -208632, 295725], [-32429266, 2124408, -774768, 689616], [-7035384, -2435744, -696374, 209834], [-795810, -1654524, 177800, -833660], [-38187892, -592356, -398396, 738742], [40397922, 204944, -423905, 105761], [26942467, -944177, 628500, -273025], [22409669, -1002692, -239624, 501118], [12714326, -2138300, 301871, -839773], [-12297088, 2096398, -974268, -36990], [9548718, -2391166, 1241903, 76109], [14051534, -1006788, -1316720, 381056], [33118102, 1316894, -595851, 478641], [39032128, 37944, 604202, -701238], [5994912, 1855030, -1143104, 187936], [811348, 3833484, 377022, 297378], [23218578, -1000487, -1040348, -53053], [40616676, -432834, -1178396, 327972], [28627937, -2162896, -4016, 267614], [-33687636, 745678, 1088660, 177644], [41398620, 157976, 60242, 964546], [-15063192, -680678, -640180, 255762], [44224080, 1166172, -511692, -54148], [34400346, 2117136, 459376, -723780], [-78667463, 193568, -209178, 61290], [50549296, 1906054, 566115, -353695], [23942773, 967189, 127628, -636911], [2171422, 559218, 643877, -397753], [8631953, -412777, 1131204, 561368], [9995842, 2077818, -559708, 1449016], [38101756, 234896, 549460, -626424], [1453406, 4115161, 159200, -213705], [-15456446, -1311032, -90328, -783704], [3859408, 741568, 782092, -1297416], [11616828, -1949156, -465717, 599701], [3460986, -863486, 839657, -1114459], [34284010, 1578842, 822720, 1635752], [-58049335, 332567, 408262, 361040], [-16400478, 730545, 2059740, -1550589], [31598666, -1434124, 1854365, 41987], [22858008, -2833852, 694696, -1343748], [-3955789, -3784469, 230156, -1231033], [-10257692, -3017618, 871864, -531572], [-48400644, 327864, 547064, 300198], [36107068, -1168672, -1861489, 6893], [40629351, 723201, -2070830, 1822622], [-48854607, -1677648, -1687458, -1266532], [6080877, 752029, -1796220, -273489], [72584384, -748244, -1389486, 608326], [-33967224, -505604, -76700, -860128], [27781690, -1900618, -1716167, 639645], [-46658459, -2822095, -571676, 2370009], [-13985579, 46779, -1176364, -1701040], [-69165018, 275860, -929897, -334679], [-27926706, -4534132, -748336, -86988], [-44936752, 1749428, 1401158, 340658], [-36230340, 970382, 1827048, -842550], [-14980368, 478018, -246448, 1877572], [40553468, -4646482, 233920, -720960], [110526248, 399192, -65484, -431032], [49936221, -407509, -2341352, 1605070], [-10382120, -3113914, 903268, 1058204], [65040507, 1119265, 1047566, 11558], [6144967, -509431, -1319036, -784374], [-60588904, -2802486, -961824, 450988], [-34956706, -2479208, 1542716, -1011968], [104492451, -862851, 839180, 366770], [-32880194, -5063944, 244280, -952240], [16155618, 2467884, 77280, -1324696], [14958288, 484132, 1524719, -127907], [-58572279, 251775, -396800, -2035122], [-93086390, -790858, -955624, -577390], [-49642122, 122364, 997048, 281050], [-66763110, -229068, -1443960, -1582686], [-113851009, 4111784, -585650, 723590], [-45869633, 2315659, -1018340, 2219563], [11317776, -3402516, -35199, -1950135], [17179757, 5090585, 516028, 124781], [-4478300, 3507232, 362585, 484565], [44469537, -2053040, 1105096, -1694330], [-40824080, -2196698, 247428, 795894], [-94701171, 3961687, -942070, 2025772], [90116285, -4270413, -778246, -310590], [-77726586, -3814574, 475324, -1949256], [-104612435, -2969319, 1311620, 30957], [35137860, 5435252, 2220700, 1306784], [12213268, -6788928, 1739262, -2263382], [34372770, -3397710, -2757096, -129300], [9336600, 7103682, -33588, 734604], [-37837490, -5129624, -2276847, -143973], [21496367, 5283264, -1430186, 2169816], [-34529203, 7208925, 2568772, 44733], [-112401178, 1670160, -689685, 2212855], [28193794, -1767548, -1082200, -581774], [126462541, 698799, 1755876, 1106206], [-120395977, 3159524, 1728566, 306778], [-28663162, -5521852, -2098019, 2623547], [-15410406, 2908428, -2646796, 3633258], [84749444, 5528566, 1686532, -1577326], [45355008, -4595876, 1679100, -2939344], [79065909, -490697, 1875282, 1235382], [-56388279, -7155765, 423578, 1883264], [-4776370, 4939012, 1354776, -2190752], [14909548, 4572784, -158916, -2178776], [-39962888, 6603840, -265440, 715130], [-162364086, -5130052, -1461387, 788397], [-32891573, 3454045, 70550, -15792], [-160959487, 3596088, -2107620, 1102760], [-12169652, -3508676, 2014748, -2232], [13536366, -4046096, 674769, 1169725], [167008689, 4668216, 1379828, -437428], [-75819378, -1793355, -631354, 2387881], [21011156, 2202236, 3506160, 206036], [-97394012, -2998442, 781340, -4286190], [116236032, -7016950, 3162995, -3151683], [-51974169, 11018340, 1947140, 25978], [137362890, -8415975, -1113768, 109311], [-76952856, -4050252, -2194756, -1580656], [-22881986, -10490948, -2830784, 701020], [38480604, -4729544, 1471486, -1175642], [-116419048, 990070, 4051425, 439505], [42775539, 899352, 2795324, -1809984], [-103342470, -7287838, -818516, 2180536], [-101278461, 8587801, -4086720, 3598202], [-39911839, 5980935, -1551408, 2318924], [89055283, -6006491, 1373068, 1719600], [-34425856, -4771322, 4544544, -2127234], [-12413712, 1627144, -3216748, -2360368], [-105611953, -8461137, -1761084, 5722575], [170367520, 1437770, 2527320, -4221210], [-158920783, 2476084, 372668, 1828970], [21917431, 9693665, -806502, 5268192], [-187221336, -7293984, 714948, 686730], [48987117, 1383500, -2718954, 2111646], [-184042961, -753703, -2135534, -3583734], [23731362, -6946030, -1029899, 244209], [-30256999, -4600591, -2112684, -282513], [136220184, 1750996, 4485175, 1717943], [33539430, 7117192, 1756748, -4337060], [-129418386, 2100862, 2126676, -3436444], [-256906905, 3382300, 1567418, -1416394], [-94376695, -2767813, 1823448, -6290512], [-15272894, 8336886, -5664928, 4250994], [-62720142, -1575696, 161256, -2340012], [201141406, -1145330, -1810011, 1672513], [13740092, 3298434, 2664700, -1621460], [-56974429, -3518244, 332844, -3728904], [137200334, 6797536, 329955, 2106727], [60068480, -7134354, 2850019, 2781947], [144297505, -4099313, 2971448, 563422], [-70391709, 8943980, -2257916, 2351902], [79193970, 5844772, -1103116, 7777238], [171283322, -7704920, 2623776, -705048], [-141139887, 16143107, -2841316, 4503034], [223620086, -2392810, -943205, 255643], [45657556, 1199916, -2079029, 1001129], [71965480, 3384232, 1034797, -2476559], [-204178952, 2827542, -2414528, 391310], [-2939478, -1124526, 3741720, -9370296], [70286824, -2895192, 353136, -2361030], [46831736, 779536, 1212918, 3525046], [76139132, 2947544, -1358196, 2853032], [11722126, -2288864, 156848, -7312], [72508786, 5707414, -656988, -3342200], [-13749663, -6195985, 670346, 4374700], [132396906, 13077900, -1266248, -1507178], [-102758187, 5186327, 2740664, 672212], [46364437, 3960240, -360844, -167010], [-59007024, -4940254, -2408395, -1832503], [17364222, -7553230, -1804221, 4479671], [-110739355, -2402968, 2463644, -5289514], [-5716668, 12557980, -4734025, -2830421], [-95860378, 12702630, 282308, 16626], [-6836758, 8117870, -1711888, 2064534], [268070472, 588204, 3670310, -6450566], [-280620759, 1053689, 5740180, -2821235], [-29336605, 13807876, 552734, -1509438], [116233762, 9409068, 453353, -3892105], [348350346, 3812346, 1265979, 2534047], [5631807, 5002880, -5743804, 8606864], [86450538, -10692932, 4859513, 499337], [61332480, -6852048, 6808240, -2177654], [-158237802, -6759236, -2383728, -6324192], [-121723057, -1658271, -41232, 3028970], [-193733186, 4012876, -3890596, 7497258], [168436819, 17329013, -4774674, -5141750], [-200876513, -12755341, 185940, 3306169], [117116990, -20357518, 3062476, -1956674], [155236436, 5065720, 1364789, -5121847], [-24354505, 7947732, -6510330, 8412862], [-226476091, 5205383, -6269772, 6676984], [153807581, -6729381, -7643592, -67752], [124099752, -11842466, 1543536, -3385854], [116200299, -363164, 3178986, -509000], [-257721784, -4642704, 7557302, 1574850], [-194557998, -11544025, 1169754, -1320141], [130392294, -20379224, -3478547, 2233887], [229065792, 11780800, -4260070, -3193294], [216075694, 11256562, -1889212, 1384072], [93460874, 7283681, 4709392, 5122539], [-295443651, 10456393, -906276, 1670699], [-141005882, 13164494, -8740672, -3164442], [203534860, 2521456, 4167968, -8240268], [-258744942, -13988476, -4362889, 3046589], [3104810, 2554302, -2953160, -3712950], [-201497466, -8261064, -1555692, 3038920], [206448392, 3901214, 3950920, 20640], [-84245428, 6203050, 5727047, -1271517], [-74960153, 5417400, -3919398, 6491964], [131600470, 7438936, 5971772, -2565782], [-13787185, -1456063, 1267642, 7916444], [85887830, 23549430, 7418236, -4391372], [-121748772, -16202412, -7243176, 918852], [475926138, 2693182, 3586269, -3911965], [112553389, 17594303, 4674610, -6820662], [103380588, 3817254, -12433907, 6710615], [-156102359, 7959320, 2203172, -3669552], [-407728802, 4382832, -4478873, -4960229], [134421210, -2000055, -13752158, 3108905], [-134465383, -20041785, 2337542, 767302], [-167915480, -437568, 8247820, -4979278], [224917150, -2928160, 2089056, -5736044], [-31753827, 889748, 10155894, -1202538], [-307716378, -14797700, -1290473, 7080803], [-181229132, 7416354, -7946088, 2420968], [24874876, -13452626, 11561156, 4425888], [-50587963, -1185285, 2165178, 145460], [-119612492, -12715386, -1613584, 12559300], [-24601468, -14562600, 11685168, -4959838], [-65412080, -23108094, -4176879, 4286997], [542138888, 12055542, 3018700, -1686704], [-170834526, -2685463, -7409856, -4072529], [-157916783, 10148400, -2498338, -5876272], [16577535, 17949861, -8658364, 4660492], [366369576, -5374004, -2456622, -5047106], [-25427778, -10710520, 1259648, 5919304], [194724210, -3598783, 5624834, 10189997], [128817942, 7603368, -9489004, -440850], [230915075, 11291757, -10001372, -4391434], [-358100356, -18378512, 2001988, 7963878], [170593947, 20798809, 2445786, -8589108], [128907646, 8795579, -5695142, 9994235], [159346606, -3976524, 3753452, -840962], [-12467572, -2157006, -5764931, 3807423], [202441165, -1700205, -6903260, 3525068], [-141905416, 26820022, -8487088, 6149860], [-192844218, -16881520, -3380852, 8165338], [-335762712, -21263812, 7504784, 4569232], [-29694060, 16583220, 9100928, -503934], [-378182184, -7176374, -9103709, -7368003], [208651431, -18237551, 12348492, -3109754]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_650_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_650_6_a_m();".
// 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_650_6_a_m(:prec:=4)
    chi := MakeCharacter_650_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_650_6_a_m();".
// 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_650_6_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_650_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![1200, -6584, -397, 20, 1]>],Snew);
    return Vf;
end function;