
// Make newform 207.10.a.b in Magma, downloaded from the LMFDB on 29 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_207_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_207_10_a_b();".
// 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_207_10_a_b();".
// 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 := [-13379024, -5741296, 321544, 114552, -1455, -640, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0], [-14876298328, -2328138428, 273707722, 30366663, -1488724, -79358, 3199], [-2521122692, 269497648, 87308078, -269473, -682136, -6842, 1391], [6770868456, 2265197396, -360613014, -32964761, 3328828, 105666, -7713], [-10950324936, -2333713076, 383451574, 32964761, -3328828, -105666, 7713], [17652094632, 10082064652, -1009430098, -154091107, 11250276, 497782, -29851]];
    Rf_basisdens := [1, 1, 11419280, 2854820, 11419280, 5709640, 11419280];
    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_207_a();"
function MakeCharacter_207_a()
    N := 207;
    order := 1;
    char_gens := [47, 28];
    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_207_a_Hecke(Kf)
    return MakeCharacter_207_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 := 10;
    raw_aps := [[0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [341, 13, 0, 1, -2, 2, 0], [-1410, -146, 7, 4, 9, -8, -6], [11195, 324, 42, -58, -67, -32, 35], [-42363, 160, -103, 8, 52, 22, 102], [160894, -2491, -567, -307, -582, 308, 219], [-185237, 6878, 393, -46, 1080, 14, 595], [279841, 0, 0, 0, 0, 0, 0], [-404997, 9993, -239, -2365, -5835, 72, -629], [1051361, 86663, -6096, -1721, 14536, 2092, 4353], [-1918501, 52245, 10318, 289, -47826, -22236, -10346], [2248985, 78681, 5629, -7997, 60259, -136, -18821], [-6663521, 100856, -10615, 13248, -50766, -12756, -28937], [-1138081, 387979, -46784, 34871, 128230, -28576, -615], [7720557, 77638, 6991, 101854, 106474, -57146, -11331], [48732788, 1248234, 62290, 4170, 4, 57836, -94242], [-39611819, -543760, 119657, 102520, -325738, -22884, -6825], [12534231, 1224662, 172306, -289716, -514573, 10106, -120427], [40551199, 2587738, 332868, 9224, -1438665, 202688, -18882], [-90939599, -3013328, -255533, 242496, 155010, -181070, 130468], [39480892, -690906, 116700, 152002, 706738, 277466, -68728], [-166374325, 1747014, -312732, 661784, -398399, -502018, -201201], [72810440, -2159802, 543928, 170286, 1292004, -332478, 445786], [-502801630, 18725941, -144601, -19647, 478880, -500594, -97337], [-1757354, -4940481, 181180, -422567, -2355303, -694998, 476881], [-478373768, 10741948, -972205, -1265678, -2951663, -149164, 260646], [-16082919, 8586906, -1569387, -1303430, -900542, 401148, -186959], [165056911, 3752254, 301879, -1452902, 1924656, 1629236, -1046967], [201889112, 17114337, 1867823, 70641, 2214994, 1350308, -738159], [7550351, 49878987, -3264254, 3053115, -585524, -245780, -1997955], [-1597889277, -89330173, -1529456, -14381, -8905334, 2150936, 4087573], [-671262210, 45070652, -1962888, 1327756, 6718448, 2417956, 2236590], [-119522569, -48676875, -1806536, 346497, 16267908, -318600, 3234087], [-73855959, 55287318, 5723345, -6178990, 10678392, -8630308, -1943167], [1688251057, -69229124, -5124576, -7837718, -4785137, 13345288, -2253508], [-2531228823, -156040726, -7367741, -1930766, 32815942, 8236350, -75229], [-985766611, -16041014, 9208624, 8930688, 24978249, 3377424, -5765328], [936201848, -265509014, 7564246, -4720102, 36502776, 121480, -5896990], [-682296896, -437651679, -4165512, -8664181, 31911197, 3547062, -10877007], [-141005773, -219306191, -10358764, -7887971, 39178052, 23585412, 13871297], [-3332492363, -158955641, 9687792, 7860255, -31057280, 1203374, 2262046], [-8433456916, 29654280, -8318367, -1465342, 45143875, 2937138, -16034120], [-8450137725, -13742435, 14321389, 1206515, -8002003, -37726292, -17462715], [-1469115449, 347198405, -22981549, -19038797, -63068557, 1305052, 4147383], [-9205302528, 326644656, 5252163, 557558, -1647139, -18730154, -5724480], [-11653559908, -209723200, -18111158, 38475660, 178212118, -23031868, 11818412], [-14699620072, 617660107, 41074854, 7488241, -243585485, -31680880, 26279153], [-5817920187, -535121682, -42614984, 16233548, -77948117, -34695376, 45495357], [-5272914133, -54676972, -28701657, -47350116, -58347122, 94487932, -11716085], [13227660835, 236879450, 21825977, -9301026, -124724824, -50767658, -19638556], [-17483860829, 749532690, 9857970, -26144620, -20033587, -4417780, 9393846], [-27323783340, -975642003, 10184601, -17675719, 172100668, -23071362, 32986515], [-2254765649, 1778735984, 19795344, 47994302, 41889863, 45067122, -5930631], [20096896093, 1328518122, -70133501, -10039018, -59059600, 36634886, 31666226], [-25776599562, -43180620, 51885918, -38932868, -58264532, 149832802, 21461580], [8680754903, -1486941448, 57601599, -32850392, 267760524, -89531434, 21019552], [42502963576, -1580223376, 41362832, 76935040, -228604616, -42788708, 18827808], [11014302677, -652885427, -40866327, 108385803, 69461083, -94226964, -37927953], [18844750160, -1181653231, 15832433, 107487613, 210519012, 10287142, 1099757], [65783399665, 1973130688, -119983506, -79571090, 213935247, -21978346, 125335713], [21714368335, 100968326, -110388723, 25832862, 705466490, 1594298, 10897269], [52995075490, 465299788, 119167432, -75188056, 595373074, 7283032, -93982622], [-130841144205, -1154539831, -56396022, 44839949, 292502242, 16912148, -63128977], [12464135854, -3717758024, -66910366, 265387780, -279609646, 72082378, 17092858], [51224083200, -4072299907, 214955922, 4703499, -281436097, -193567246, -36970019], [159674404779, -1578888157, -17971004, -1765669, -275163426, -369894436, 162505329], [-139246269516, -3759737442, 136249716, -161718482, 362237160, -132682462, 105288502], [23449314922, -6351656731, 65350396, -148088809, -1123319855, -64692364, -156013163], [-62822647387, -1605774134, -228048633, 260808722, 559134864, 33243982, -132606842], [-13849708999, -4498230958, -49533069, 91115622, -199182056, -216213354, 3743296], [-105594473188, -1811598932, -380827332, 65356508, 1148255716, -12438762, -35187486], [-122040181994, -32697764, 381181149, -5008270, -949790583, -15773050, 6856020], [-222737023801, -10251214363, -371067914, -325054975, 1557786982, -127763046, 206603302], [-214886882595, 4235255434, -301884191, -222486646, 190606930, 835157154, 329447439], [56934548104, 2628607298, 228095736, 439488110, -61779482, -1177110572, 242852314], [199246427757, -11365801474, -782113837, 240174414, 495450034, -224755650, 283572571], [-467805339059, -5934464968, 30724599, -66873364, 1659955118, 531538058, 206681810], [-143141777636, -5271084525, -78991021, -339253065, -30224492, 983832068, -22015879], [-180367618073, -4233536764, 777639193, -130255204, 619400382, 216162554, 31779686], [183659245411, 16938568776, -701180889, -353603180, -3127971740, 496071190, 149234855], [331618774537, 4346450589, 144574076, -249851903, 3262681282, -632880984, -229710734], [-123315466362, -4778775778, -643882852, -903012266, 2685311008, 1381843178, 597663948], [-340427092384, 7863252539, 570113285, 543715859, 2277241250, -422308824, 32562325], [-171024651613, -3530080160, 285994418, -1428154098, -1531503169, 337478220, 252339264], [218953987949, 4848689758, 450114326, 1258408632, -440941745, -346056600, -430085240], [-648088960180, 4518864328, 1549567942, -316485448, -543769024, -479573956, 422963330], [-441240363962, 5296204359, -47464125, 784607487, 3082506142, 354113890, 431300769], [-800681096577, 10853223688, 11233273, -31309012, 103915234, 1625276570, -486706300], [-467562231896, -9172067397, -668817474, 1058712113, -1460491453, 397495056, -111874279], [-229847209043, -3648277934, 812368219, 1173879846, -2153395668, 125546722, -507361163], [668674094996, -21945214198, 1131717661, -99469968, -826168595, 799361696, 556208656], [-362465215241, 33861733706, 369720314, -1686449100, -1577241263, 1797439232, 944486246], [-323699975917, 2872872085, -441046754, -471236507, -4912948058, 425173004, -311348807], [-984015821637, -10497463281, -1017454864, 864253479, 1220284074, 228215332, 1330182967], [266828741772, 21276626622, 822371036, -171935110, -7719584102, 279622058, -643659812], [-668289876441, 15318633123, -1897037069, 393225469, -6245483723, -1139625116, 706035621], [-470750351818, -14745787220, -2390988946, 1071560432, 6491733150, -990511206, 1121395410], [-480439329743, -8243253606, -316684848, -1169722360, -1156268405, -608593076, -1437705899], [343811207401, 50948273328, 314059507, 33339988, -6283531510, -2928339698, 43997272], [-420698456185, -3531202153, 273054232, -2526110881, -2944998054, -787277712, -1535641399], [-7933419003, -12254402768, -869026077, -2851647988, -3362537440, 3252523750, 1908967397], [365041482677, -28435360188, -105609593, 454235660, 935346990, -2796077264, -1599355859], [-204934576544, 11125861301, 3967120027, -923965907, -3768120378, -1467759792, -3429749885], [86348178557, -32908982440, -197373622, 636025142, 1470508883, 2212075282, 1193252515], [-485245223683, -37189837151, -1814855113, 1349922143, -8722688191, 4292255424, 687478515], [918337203431, -16228540396, -3522424644, -252211022, 5545416749, 3274408276, -2193014716], [837370339648, -4544035231, 492985724, 2358290483, 6625067701, 1355453582, 1376490569], [1017222196164, 54832816188, 1513201620, 1814062964, 15374166060, -234101840, 248139232], [399375304865, -35581285936, -2910109423, -4746693132, 13848376828, 2596885282, 2021743488], [-1766803595580, -46622162438, -902132150, 3439477898, -4500342848, 1786647908, -610011462], [627039119119, 143350163035, 2402815946, -1158157861, -8173882348, 1285384032, 522949402], [-829424569254, 19843861718, -17107518, 4376048110, 4506841008, -4560458786, 72613986], [-1648718857989, 7277857630, 1287568639, 1095465694, -18557606650, 1989810958, -728908935], [-302539563404, 70184104498, -3073193022, 4885425770, 7707493700, 4395712378, -255963582], [-1216018460074, -19822161422, 3986328398, 23784842, -32517795680, -3279139330, -1165888060], [-490537947899, 12769632198, -2571792352, 344372788, -17656279993, -1142637044, -1761966505], [-66054160677, 83934195948, 2315361026, 1030132874, -16162215709, -6504495228, 2399378984], [1686125909583, 12698975031, 5825562035, -3746939711, -4610681563, 2814625724, 2041603263], [-1514826560629, 13322487238, 78434580, 1979082536, 24822868533, 805512324, -1161688057], [1322691314633, -115747731880, 938194373, -3357827184, -4486639338, 4705923464, -743328925], [459620660883, 83899206288, -2966545345, 3727787844, -13323447700, -7125802770, -2506728740], [-1910242554903, 38734638296, -3409207531, -1641808296, -6160705086, -7932056236, -238063437], [-1561546837295, 35697465201, 6820570174, -4020947459, 8805715728, -9137528324, -4464095137], [-2195154179268, -145956471598, 1755278928, 5554277278, 3659281646, 3556992060, 1334002686], [-2941273893763, 120642917359, -2972799912, -6167683161, -10593286820, 1216505360, 235945484], [1137247939177, 85341552795, -4686038672, 1591532979, 16685387436, 283866214, -417906120], [-3320755553288, -115132180382, 1966241680, 9863307494, 5068435638, -720379368, -3570128954], [-1798165287588, -138319982068, 284645097, -11071207234, -3773372885, -9095067280, 4622705880], [1186024608585, -60623752813, 4819134844, 8007066859, 7726597796, 5562744732, -5075127108], [-1315521207955, -2382368096, -4150979178, 7079959086, -29314320915, -15170113180, -7571926966], [3908836807632, -42687316080, -9775767689, -5144710314, -28959901875, 16835965854, -4665977046], [817790642568, -93221935248, 2778857010, 3723468116, 13940937818, -12903514, -273152500], [2532752325709, 6221849415, -6884703204, 10075132431, 9545081856, -3796521556, -2857593710], [2562437237737, 64880640047, -9014270487, 9876436245, 26376587901, 11596041020, 5587166747], [-3067687239652, 102379645227, -12728377483, -8790287853, -48956934, 13650572210, 10616172365], [5810244994557, -1682475014, 8318518327, -6543217286, -36218752990, 3846902866, -2615518981], [3610994764715, -305014999696, -5298125281, -638916740, 26561592388, 2029023314, 2674742779], [-5484969625541, 4710191426, 5352148797, -8884316582, -1953747546, 14279019194, 7869258427], [-4889801198950, -268660721149, -364955670, 3955735105, -49782362241, -1773110314, -13007261931], [6931683021841, 98888416384, 2881015158, 14894621278, 2398629205, 7577971540, -2659907020], [-152414506118, 77120380727, -15428856622, 14852316429, 81075959, 13383977258, 12597681921], [5427666834937, -250541419374, -3967923956, -12483627612, 42076836701, 14195636932, 1870807790], [-4156070460429, -427704037366, -2785484669, -871488694, 58696344182, 13211525190, 3553485637], [-4259976950456, 312235934457, 1950865950, -17755783305, 6968761275, 27587980834, 4865442013], [-10614459668984, -186752625158, -9280689739, -17912086840, 9260836805, 10289791862, 13351423284], [-2802212699878, -262479520819, -17983421176, 16417280755, 15699859151, -3834967850, 6596012711], [-2856726684869, 199575614682, 4098082801, -255719858, -24113314584, -6880857954, 7376192116], [-2075557766925, 114598543654, -2377405018, -21483778116, -34002170367, 1795151720, 6633049276], [3350687320841, 80195505128, 5939923070, 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-792009848102, -285007310519, 1044999796714, 3323857709392, -745222910674, -143596087764], [123873347124412, -10051017681570, 724934173622, 135420366126, -2976010995144, 1036534205928, 6397215858], [-59875243875920, 15865212093803, 422589310347, -1350230745297, -1458505669252, 744270731046, 976523807825], [-293837905985970, 26056713071140, 222319358718, 694796029056, -1716179084174, -1031996392934, -72902775562], [229325054738133, 29278963156268, 148017174682, 445201258322, -1393707205597, -1888608414542, -517446909251], [-161844171602441, -1441959915244, -1121243009186, -655549191090, 6868408205803, 497125841028, -789934919808], [236315015314714, -14417639086788, -182707445041, -535637863706, -152439726025, -1142885741390, 220335555502], [178663453477865, 27972939572081, 609744402684, -627677877595, -2754607525566, -1065110518436, -743252875262], [-59656669083739, 10067621137369, -920597425627, 281974364591, -5096155249077, -533879506560, 223180424595], [319872645309039, 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547530528301, 687691191835, 1725798066235, 980010918244, 583642029453], [295676550503274, -12127690902650, 680943930588, 1824504053202, 597011195446, -1301504326976, 1290262073316], [-509339295294842, -358663388723, -372646479468, 1181524481523, -4756839444685, 914168873758, -1429886651169], [-132278923160984, 9123458988382, -1339620023128, -839836674726, -3653707199774, 989076016766, -1215577479120], [79322942396203, -27412229920529, 710002917832, -1246433659757, -2195468982010, -1546119940840, 1539387287488], [-382228144464342, -3048449122364, 1590652055856, 165927268160, -3969848344974, -2045641109140, -398998020210], [-897785331605699, 847902536031, 1009501495751, 262014991933, -190792321681, -2010275342780, -1062218325293], [895145079530414, 21051598960156, -794339070794, 1071225689500, -2997925189528, 742224213564, -1286821609098], [456960684990252, 21410020673196, -1728870045604, -743556943668, 7143385567804, -152728534258, -97818512974], [615021915426249, 2758849692289, -2703437351278, -118188067655, -5493791755140, 114721200694, 825087276184], [125371421777295, -38230845890648, -84756593504, 434596145254, 4322426328825, -1752160713728, -919867401434], [168086059089417, 19812147747639, 361644817361, 1120334641821, -6030238511291, -42327278060, -336165999087], [14166608400244, -2089084607043, -996485568596, -1521094359089, 960639821785, 4042738001170, 1220102595777], [85298547857640, -2218129401316, 712016996126, 952854797608, 3835473645074, 45641114352, 303585045562], [-354105645367549, 8962479631287, 1313343589376, -1550617198213, -1247325436662, 1569292174598, 919507603882], [-629402408466730, 6422616578467, -1509367867054, 32543666361, -1095714582917, 261871464738, 610323266329], [207781124730612, -35535647991742, 87849213589, 506141488504, 1209983331089, -1431661693870, 1965319465428], [653123999166625, -11228490444614, -275670247746, -93522271564, 477882564985, -675112793528, 1462527435587], [524820871919113, 5062813540588, -920236944545, -1315540478164, -3287203674150, 780473888870, -113875935846], [570313924209028, -31066944600800, 861150990698, 836438072060, -4127922994390, -964521669188, 474144324660], [-284500821681515, 8100389533829, 1306858534872, -1591794443831, -4918704645482, 1649657220532, 640502318244], [458814872737559, 27023401520238, 1665826197371, 737138462046, -1504679309526, -2622049141142, -3336436531414], [-1107937163765279, -15217178758891, 1058531750740, -222968209423, 4659115077334, 240413810592, 261768848191], [-373247854720005, 7192461012574, -79172977092, -459563858672, 823844634005, -249489289674, 1266179019417], [-866883058048149, -1270711832026, -551878813418, 1382828106468, -9514434983315, -2039856671848, 280211038762], [-216863110847299, 34227760940408, 1345306589359, 2390315461000, -1958033999598, -574313135238, -1222853647220], [374531342633747, -16365991811122, 2279122691085, 2773414614934, 4814717147640, -1299070616370, -235452976388], [755355791344204, -5624568221548, -296950726379, -2161590439890, -5996848376801, -1216572079964, 622990889104], [-200595187741664, -58413096437698, -499007169710, -125909168046, -5544988476658, -745982720592, 775189927880], [-268796635295473, 26503545480076, -1782445900296, 986586407802, 834929925197, 3189966065988, 1517138479982], [172616849199114, -17662161497068, -227729063870, 359259156356, -1622336486184, 1821854030984, 616309334698], [-19861389949421, 6324822031305, -1833127733676, 3201757485633, 3012796098184, 8470911148, -913670788485], [635455600105293, -9126476562501, 915731205576, 1319356987659, 2972905553088, -2242583333460, -443889310904], [-94854114927128, -763604135876, 284402941546, 2389877585912, 6116008060814, -1299177700090, 651333431744], [-262595758618579, 6719209604559, 714722007687, -2530344887651, 1004337124655, 2192116301432, 1111600553627], [716203349595949, 35454494253938, -673140532662, -308623961212, 571209346547, 2161322625088, 2954863537086], [-558473572341752, -17018264321064, 2343593255715, -1925020163714, -7732470248847, 2606146241920, -1340635054888], [-1413786656089894, 2116223786724, 3484813842302, -1230256293424, -4851074280510, 603730741354, -1520882860812], [486580928608962, 30848733279616, -1466966373800, 306881094928, -8449060036400, -623857144584, 1825383371638], [-689810961124567, -25642025213318, -2156194780355, 539013294970, 1425740580842, 1319471994946, 80756325116], [1020653538093827, 18147270860213, -2113768090718, -1018673959183, -3793671671578, 1119858443236, 961285269402], [-1411370990192308, 42499548329254, -2958744874808, 1030449002558, -1302121600340, 1801829840938, -407130530528], [654948886427968, -31749444877589, -686746323226, -3152048467727, 4413215341307, 4110078047468, 1299051858637], [-868536113690499, -64728125388016, 1023200673257, -196362149860, -6960325794052, -1264868621074, -884157610552], [64985780138273, -35920546541746, -2268268376326, 2850063007640, -2737815700975, -2699995800110, -1974743356767], [-382061527365523, -1159640025156, -3300953847652, -1020472072310, -12040159221633, 460094528792, -809066404904], [423099110977519, -36481931511106, 1144288768798, -3441887886320, -6212832509815, -1549280826192, 2122875095246], [-1751230114308196, -14664226371295, 92632840600, -1252200167917, 11404339591909, -2513543936454, 63066564457], [621203732377779, 34166317764402, 2104808864350, 1479782636960, 3786272283435, -603628018376, -808576575511], [51736719330694, 15078449806886, 3525916754481, -2972457322384, -963501528801, -518600184856, -965547890710], [-146572939930764, 31289937335573, 3915390457573, -74793800479, -10246520051604, -6453372413482, -3651114596731], [107813378948048, -559018509028, 2873007864122, -1489339227420, 3826059180640, -1229826417608, 268149943642], [-79813992209481, 2837128917253, -437884277548, -1792510394383, -4653596121176, 2304643530712, -2048737037265], [226428391106380, 16230635225694, -1109646357932, 1023426121242, -14374592849098, -1283915666816, 360177464450], [-556759999281790, -19824711841700, -234960697973, -16166525418, -9341575055333, -4247263701360, -32376553316], [-52794127334144, 2048206735679, -3588101756181, 1753716475307, -1656512016440, 3310628254896, 898226156025], [997978062187892, 19267446986380, 1289650621858, 1274558300384, 2248158446130, -960110147162, -22896004952], [-888873407738270, -4024288755743, -3072622914725, 1497624086005, 10568503350740, 2038762587818, 807518968695], [277038047491299, -26897798951526, 1894113530888, -892839137028, -13061635366711, 4441373090582, -1257058244217], [-986965143888358, -36332712935053, -2505820908028, -3299897225827, 1095021958097, 664943170006, 3529305860669], [-1484124024756403, 12803121150005, 3444323223293, -2380711470461, -12047642082485, 2473184721712, 2800205677641], [-1866837130818833, 7743532232541, -1522225698681, -3659675511561, 8156793235247, -787621440128, 824684942411], [393148484855250, -17831057716836, 4189101746561, -1018823559310, -11586659271787, -5791081005732, -1731567367024], [782800736927742, 50750291532597, 1877836623793, -3053716890331, -5478581998106, 4641299394714, -433875815621], [-1035149105367169, -9416524006859, 2142683399015, -416269233029, -753055619413, -1844586942420, -4194307004291], [-261168861916639, -36125397277619, 4814492965420, 902422446601, 14611104994126, -3860031039276, -285516423361], [1810735304427613, 22731410589476, 2377393985207, 2770396468936, 16660637120392, 1951510665242, -548288141729], [692464477238031, -38602284386089, 4252927604507, -2046132635283, -7200760368583, -4428635593388, 372240312929], [773160553234775, -46088783648558, 2447664298182, -1127465330504, -10356135876353, -3590360399546, -815688519977], [2051034162556902, -45744696826637, 4771868591948, 2649154468673, -6434410602209, -3901437327516, -418522000077], [-1902975661579789, 34815659475457, 91257796525, 437290274167, 3820871807237, -2658258493820, -1531208872221], [168815613048431, 579741870885, 162923966533, 1325688359975, -11664648081819, -4998477888028, -665806715443], [609516774250901, -41916804671512, -3402749862004, 3882983881098, 13623048117017, 1064268456080, -983762359075], [85103350268839, 36461549702268, -2771132316015, -899156400596, 22220294293478, -1154083624746, 917909654665], [-1310349167119714, -77327859391634, -973817377822, 334445969078, 11171307864360, -3875914177306, 1598511790418], [-1429559640389525, 32228918865592, -835873315107, -1405413903020, -10779041646982, -3509508167662, 1154274228332], [1031005688446472, -38595221248909, 2818705720910, 1109013017529, -7928553648197, -5559279212232, -4640887035575], [-2043630526616836, -3123086754248, -3394793777810, -2577946274004, 3164284087278, -800423654084, -2646849749940], [963024541670978, -2981362330570, -2924561524570, -4745773920066, 3108506861576, 7173504632710, 825505852008], [2357654644250618, -5144314942354, -295303730180, 1815875890638, -4253587182316, -1945093770238, 4989852574284], [-263385421425963, -39199403762871, 530916607412, -2032977443843, 9523946602570, 4286335699158, 3158985789142], [963187939039573, 90529963864934, 1385734640857, 2141277002242, 24484817265596, 1912915226200, 1140917341943], [814753703646920, 35530771601303, 1347616602594, -6036146677155, -6010415746077, 4342826620980, 1393791966149], [2126381650611812, -5121528003308, -4616322188612, -3064011560948, -5007594875808, 1084792276072, 3515128127988], [-1569136669256089, -33058382623925, 1930776285301, -5998265064371, -7647439669045, -1211693642272, 698781195415], [1074688796950771, -21703771971683, -3453970860879, 2805608613911, -1133085321251, -4238503813164, -2194701214203], [695953237415023, -17852469525682, -4396247962079, 4291882496426, 7589472541468, 4756988178710, 546782677447], [-613966154781278, -92091162976205, 667533857604, 1753444223589, 14693359173349, 1266530156782, 1105324566477], [8037298848777, -15695747613103, -3597815659670, 243586564669, -17200995630508, 9719122690164, -1602241683213], [2536317299632067, 70321625872389, -2080520890882, -142777403311, 8379472514170, 2778329679980, 2733787699007]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_207_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_207_10_a_b();".
// 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_207_10_a_b(:prec:=7)
    chi := MakeCharacter_207_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 10));
    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_207_10_a_b();".
// 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_207_10_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_207_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,10,sign)));
    Vf := Kernel([<2,R![1712515072, -367442944, -10289408, 1832832, 11640, -2560, 0, 1]>],Snew);
    return Vf;
end function;