
// Make newform 105.10.a.h in Magma, downloaded from the LMFDB on 10 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_105_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_105_10_a_h();".
// 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_105_10_a_h();".
// 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 := [-126461952, -8110848, 1722956, 14856, -2759, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-922, 5, 1, 0, 0, 0], [-5899, -1432, 12, 1, 0, 0], [-8372464, -2957236, 61632, 6725, -50, -3], [12322200, -1422524, -50796, 2783, 22, -1]];
    Rf_basisdens := [1, 1, 1, 1, 464, 232];
    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_105_a();"
function MakeCharacter_105_a()
    N := 105;
    order := 1;
    char_gens := [71, 22, 31];
    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_105_a_Hecke( : Kf := Kf)
    return MakeCharacter_105_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 := 10;
    raw_aps := [[4, 1, 0, 0, 0, 0], [81, 0, 0, 0, 0, 0], [625, 0, 0, 0, 0, 0], [2401, 0, 0, 0, 0, 0], [15746, 351, 9, 1, -1, 0], [-1380, 658, -71, 1, 2, -1], [134986, -5869, 86, -14, 7, 1], [-15634, -3551, 21, -15, 5, -14], [338692, -14657, 690, -46, -9, -11], [322470, 26109, -2338, 18, 33, 35], [1393566, 67698, -353, 259, -42, 49], [683110, 63045, -3364, -76, -59, 45], [-1920938, -40652, -584, -892, -184, -224], [-511868, 44749, -19484, -972, -251, 357], [-433140, 15961, -8496, -1196, 69, 25], [22456688, -160175, 43793, 2565, -35, -554], [21428172, 671728, 4058, 2886, 2076, 598], [45713458, -2092249, -10530, -2474, -1217, 1473], [66829280, -1654241, -5450, -1450, 1439, 2505], [-26633046, -1630156, -249111, 1365, 912, -2075], [-24531996, 3620662, -155161, 7335, -5826, -1759], [56090048, -6708358, -420398, 5414, 3750, -840], [280549032, -6816436, 199182, 15878, -2468, 3690], [196394910, -17420420, -635724, -5560, -3616, 3116], [-10089832, -438662, -618055, 19465, -14942, -11489], [-41905698, -10877838, -1071480, 3432, -2374, 7394], [98886896, 18095224, 762400, -41752, 13200, -3440], [66693128, -34375927, -1008884, -616, 5149, 2829], [687485790, 11590832, 1366144, -34328, -1024, 24520], [554785328, -2154377, 2886953, 40137, 2607, -10608], [-319769780, -22049078, 1610706, -102806, -16150, -22060], [-1356257632, 22820612, -3272664, -11348, 103432, 20712], [764081444, -76566437, 3964155, 174763, -18317, 54574], [655819366, 42366689, -1176087, 178813, 61245, -30274], [-2734424438, -83128003, -5731944, -45700, -118719, -76683], [-1872789388, -50854236, 3094704, -95124, -30792, -35400], [-1094793472, -273605730, 86471, 2479, -11458, 124161], [-3612599044, 21688675, 1703036, 128284, -72005, 149947], [458043432, -235376309, 3661226, 218558, -89073, -56635], [-3621896438, -211452070, 11687542, 64142, -158590, -58056], [50478458, -223412273, 12150473, 453777, -47457, 98400], [-2982347850, -48940283, 8200792, -367616, 313389, -100503], [7556248254, 269821132, 9513347, -709249, 59096, -30545], [-14271937614, -470243686, -5941424, 780496, 45434, -209582], [10174497708, 248947141, -1744453, 578671, 3825, -25524], [-380103762, -493936022, 9152379, 750671, -236882, 218789], [5536444564, 346816532, 10850230, -1367118, -268136, 61670], [5383123892, -341974174, 3719130, 2118066, -281366, 614228], [25235210752, 350019952, -23009510, 20626, 500928, -421350], [-9137996190, -340594605, 44084250, -1559998, 865419, -468775], [12790898960, 932254137, -10794903, -2285967, 220633, 172818], [-6865233598, 893251384, -35706263, -1381595, 736388, 67305], [19372856806, 586370818, 12249882, -443630, -1205718, 121620], [27291606140, -833907018, 19689962, -683530, -382942, 326308], [15805323614, 423888175, -47443908, -727640, -235309, 330587], [27128269296, -1787352275, -1826544, -3718968, 165469, -455079], [19827958046, 823018768, 16774592, 3946200, -710464, 260024], [-3148601902, 1264833102, 31319449, 626261, 322810, -883641], [21251483030, -793317925, -23994776, 2674760, -627981, 479535], [6386464918, 2239901018, -45586882, 3145718, -1876494, 1163840], [-13054442148, 1335996608, 47703640, 3075256, 113488, -65832], [12687025866, -1243100784, 39240918, 4797910, 1413216, -911074], [2911306672, -1509194920, -110112022, 7637978, -1703544, 136506], [68590800832, -2378169320, -26845904, -7114824, 2710032, 88384], [-29706212456, -3617805784, 106729617, -6450983, 1171112, 734821], [73837124928, 230848393, -153146379, -4428647, 663237, -1144526], [-24319182308, 5257387932, -69772374, -3250946, 2037960, -1265702], [53035564186, 5952008578, 83175868, -5952996, 1012594, -631530], [123739111692, 6094704021, -168375454, 8296494, 2374857, -365161], [-4963645806, 658159985, 288347910, -1636114, -4684823, -127541], [143940167978, 6826605945, 216714234, 3558478, 30533, -3277989], [96747079358, 1191241898, 57317707, -295825, -4305986, 3978405], [14867549292, 7023348984, -306629370, -6268354, 151440, 1326974], [14248105482, 5258294417, 1062778, 378802, -817247, -2495745], [-54640473384, 3929592720, 205906732, -20076992, 4918348, -4435000], [-2196714696, 2472839155, 219644914, -4491058, -7120817, 5238413], [46144040154, 4788620347, 28489812, -3396336, -5825385, 2840351], [81153212684, -3121832648, -288455, -13585759, -10430328, -395171], [-139540940422, -894747134, -265876684, 10931684, 4044002, -7656354], [-43922312766, -6364356286, -358152424, 22684896, 6242554, 3176090], [-285845572184, -794127664, -296167320, 15731172, 7162988, -698228], [127831301262, 3215619640, 204790920, 26075552, -8373696, 256840], [248719527306, -1067720260, 242068417, 9052245, 16340048, -8838659], [6355257672, -8352379440, -20370675, 15104549, 2765280, -6233911], [-543376510562, -795954572, -349775405, 19321575, 9687352, -718449], [139754263828, 12384140761, -300578414, 8517110, -15993211, 17185699], [14354635890, 3905161766, 766990268, -12594716, -4273746, -8279326], [584526815966, -14207019476, -601498938, -15698682, 13580716, 2562522], [64812666822, -7624742646, -44755568, -14464952, 11234082, -10605078], [384239871220, -13402502116, 1059415946, 31212266, -14348796, 8412702], [-86178567200, 12970816586, 495219746, 11397002, 13012810, -14546412], [200536221264, 12366364, 696888584, -14586680, -23108932, 1202420], [34433329812, -3972852952, -1429883098, 3445926, -4661072, -7631098], [-544790417534, -12006024929, -1048404271, 9580681, -5387761, 21244200], [-806062116992, -28470769016, -109184396, 12515784, 4918724, -23404472], [-311874412904, 37293787861, 673502478, -18769798, -1760767, 6952755], [-691664543314, -12918525392, 1563865728, -28923032, 12077952, 21081288], [755627701310, 23528643670, 114236584, 8784076, -13184942, 20359682], [-1354586166816, 3512984768, -387534994, 57224046, -15063920, -14985850], [982032947590, -18668949502, 425411672, -19671088, 18461850, 999786], [-260024129872, -17626622565, 1331126914, -13089670, 44439763, 4970745], [212104554672, 68618035645, -115099775, -18801651, -33900831, 9894762], [19811287100, 65943829432, -305114916, -32678676, 2525432, -37931628], [-454824871830, -34884978034, 1247277136, 11408496, 41901454, -28180746], [-310975777728, 16239629340, -716604412, -44906640, -19490368, 26990468], [957732709856, 18967639772, -666026903, 56555057, -10675252, -34119199], [-485125915292, -22140129560, -901985032, 59347568, -5004480, 28505544], [-563828573182, 20913140431, 1261868094, -64560702, 79831291, -2379091], [-386791365570, 43810139536, -1280466341, 85165951, 1812292, -29323725], [140187295414, 82363848344, -296722174, -128931790, 14790456, -54737550], [-1997112285476, -18696703578, 1157050594, -22277646, -16724938, 64154552], [-15324839682, -9258651399, 1428149352, 259336064, -41166103, -12259667], [-1492591497400, -49339292703, 1157289325, 136985269, -12721791, -5766802], [165445539534, -33737824017, 489500333, -68699295, 36864203, 15799352], [-2573169698920, 31055674444, -2576481710, 14816054, -67988360, 40420610], [4631374046, -1679179080, -2527505390, -122192334, -81635464, 44437106], [-1835552946824, -36497546282, 1291789030, -124569066, 11527462, -24202404], [2247391005872, -12926814835, 4781441210, 126643206, 6407193, 1832327], [-437941821548, 129564140411, -3395207693, -59985833, -46868697, 61403522], [2319035342910, -40995311913, 4232238739, 114855091, 13619471, 34946786], [1709434518850, 110868011835, -417811470, -101045942, 56962067, 13380201], [-2132766675338, -3692323134, 52410118, -260435466, 36228242, 19522136], [-4523413006134, 46906555782, -3763117758, -67301606, -82009810, 30875920], [3506733372308, 78676721747, -4031933806, -99272250, 62197407, 26398709], [749866954682, 108074474059, 8784507279, -159182421, 36424831, -40300794], [-3481691346970, 59895305501, -476792586, 245736298, -5908623, -70745029], [-3222920197146, 66675799562, -5718074268, -333583588, -61384686, 24468750], [3933465885612, -109479494930, -761433594, -219648610, -61310666, -45109652], [-5847894751652, 67526392540, -571132222, -61290886, 71376916, 71819486], [-3949139680812, 54436311914, -2411273483, 308656381, 74449018, 58131171], [1209226131288, -272587395082, -3711554268, 24988448, -113578406, -19522538], [4612727004264, -9597333739, -6781637808, 122174608, 35209357, -35088455], [-3655916034560, 180484416762, -3513120214, -51851650, -19891114, -34958672], [-5630688448990, 109899390849, 9530276434, 140544242, 24437673, 6443231], [-646034213370, 234937777462, 3279678644, 201295456, 172127970, 95366166], [-3908203448106, -47002014390, -2636892130, 175031070, -29164886, 93661096], [-1117586355358, -20223262172, -5297082142, 168121122, -107803260, 38719230], [-2894040832608, 144360214186, -302446238, -460640606, 122515338, -66442404], [-6104141951350, -104596029218, 611388110, 403737990, -15256410, -113282460], [3927535344294, -205340850896, -2318835954, -457747322, 68265592, 15037502], [-1761197766906, -50797758623, 5636094657, 785486069, -159877939, -4124778], [-5079137217170, -173140256561, 15475874282, -654086090, 140230907, -79282047], [4408627656972, -156156196180, 5981389986, 403768130, 265309684, -36442586], [5668080235048, -84419807889, 15291400764, -176366440, -261765725, -89387925], [-2409147825010, 96592905227, 6543957808, 471270776, 65531203, 17173663], [3426874717124, -538111733862, -4490960178, 110577030, 45807826, -26991408], [-12751249047636, -72391755896, 10645858445, -598324251, 30838232, -49098527], [-4796414221090, 134800462279, 5410460612, -576777544, 216511379, 27542547], [3042332175786, 100567756737, -1957151529, 544628219, 218886541, -166290348], [9185663368924, 232575381113, 5282540282, -435285286, 4107009, -291128257], [-14785405059034, 83843590999, -9601160960, 230484536, 270602631, 144778651], [-1788816064442, -37848995200, 4042947968, 1047220620, -199436036, 294108260], [10880803318168, -225202117087, -2606529106, -582774154, -128707271, -204621917], [11831736269020, 310664153203, -6742548636, -493195144, 13715615, 227851119], [3186695950112, 246966189629, 2176762854, 404292798, -17917291, 90844215], [-4695655142106, -462233039860, 8107435711, -559742981, -256409272, 88094811], [-2841454381248, -128556061510, 2979623814, -1053787318, 50273414, 56376908], [-350721986386, -180019161186, -29771302460, 76704304, -340882262, 1689390], [-10059145682412, -620836498308, -7634369021, -311182133, -174943380, 158534947], [-6443899690434, -91628809840, -1792669856, 372420880, -421002016, -206885568], [7122515664324, -100687253719, -18345712034, 255295746, 209620293, -6667153], [8432367274456, -303188800009, -3603545435, -666960851, 316298423, 367017076], [7155753663784, -157344184674, -759178960, 72369776, -153740498, -175369530], [14719199160580, 138116728532, -2257743302, 766385454, 159510872, -197668990], [3610206509744, -104638654803, 3805671393, 212260977, -476352027, -299728762], [16395536992108, 201487265239, 13619370252, 1431178112, 267003171, -142441781], [66465316808, -584339708254, -20875171778, 702682322, 919990270, 65287516], [5499945259412, 868198673640, 6275198649, -445820271, 138883272, -77620323], [1262785611370, -215070543044, -25027707752, 934447992, 83674740, 157420964], [29111662756010, -51419047946, 1929327242, 575725874, 53244478, -266296360], [-2521670619590, -27149560489, 24166539805, 1150569877, -286873209, 519676220], [-2314480789802, 196589765719, -36035725924, 41941508, -546443569, -265268649], [-3329000854518, 78766821598, 36196230821, -1355806295, 609927538, -105283925], [-13102676151358, 411305069426, -20802610364, -177657292, -681499294, -174881314], [13462798582538, -218963705216, 28177601889, -339284235, -1118824460, 135608625], [9234239392190, -544174606980, 1263411356, 443714712, 234336848, 270633996], [9681571539036, 766127480180, -48960237842, 570483610, -517853624, 228949614], [-12331302074734, 422575291599, 42409853408, -7391700, 64409115, 603918127], [4232860325164, 459925958556, -6147064270, -420831622, 808255988, 545433790], [-5679774030002, 585765955293, -24217825232, 1446738408, 1082965237, -17421815], 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-24916957063758, -38553940663, -25536945599, -7784161822, 7750389951], [899414563195000, 16153970776607, 322527304542, 61338265966, -5017572369, 4613861729], [823055738496956, -14350371980317, 1902215890144, -15094613612, 3228565887, 25069770219], [2119254207101858, -35784488738368, 36315518676, -81594657468, -4875679752, -21380147948], [1042936321343886, 45761641418448, -2480198979012, 32098966396, 11587064048, -19146156356], [-2071715172449746, 29325715429079, 261875392787, -59218694469, -12130128201, 16465531514], [1518724100416752, -43787595348508, 1995478994158, -20326863002, 35117906988, -14508834910], [453650544342394, 9972790746228, -836347348723, 6947080945, -6175162328, -18681541887], [517875793872530, -27777424288896, -1262173395068, -58065057532, 16328257856, -39433896620], [1206071675716570, -83273300375478, -87340736347, -59891299327, 16667300358, -47216590129], [-594698129288698, -37499635984152, -706779768320, 56888116388, -22047893124, -42930522956], [-891061451318280, -11587763889098, 1916377536540, 17902624256, -8463843214, -30290758362], [1670369922356390, 47242851276391, -3646222788450, -14277915462, -3692496909, -1170751763], [-185846880145290, 23778482831937, -272454904420, -29335682412, 34338905929, -5986920743], [3129380558935280, -18400720540385, 2048267491781, 44964372061, 44554734031, -14115308148], [639201989933904, 78469874585428, -1654150100148, -151298885404, -475250052, 23280754024], [824391397769742, 28464687719765, 609171563316, 64642547448, -20174215903, 58047820281], [1956509303297982, 27871358140000, -2343357366744, -42479837792, -31342473608, 5604433424], [169102837960330, -52338349478174, -3098112770923, -20595968791, 15538021174, -23330381857], [308833180734710, -36511284379533, 603550085220, 65596596948, 34010140747, -4928757437], [2861904969315250, -33782070170542, 1678095006524, 70928961668, 56121913322, -7151541026], [-1399558147039180, 11298032734774, -233501360628, -121954345980, 91750537614, 6688833842], [-1371201327437348, 31408615649896, -137511497306, 71213775026, 14905694324, -8787801782], [74104761685548, -44504896136880, -571308208725, -81711801901, 1239304032, -43038635089], [240097058733664, 461810611955, -400593894511, -44260034219, -5283787553, -48948460688], [-2521630460388, -1640463899411, 1207633673854, 129220019042, 17319894201, 33472807651], [268507893773692, -37533365944994, -2729489861946, 162562980778, 237404986, 22031044960], [-2008352182726742, 7939032548030, -1182068715616, -78011219056, -23262597682, 27621283046], [1485618642604222, -32493052300168, 1140305631176, -28034233360, -10985066728, 2946110488], [-328313783459694, -20654966892444, -738874977319, 93891677061, 75961099632, -40432081531], [-1608173643665084, 53446989602322, 1084797980694, 174215777902, 38984567482, -34144534864], [-1163145627005846, 81754759635957, 843112422294, -206119668094, 11686614137, -33066231845], [524483264013316, 22600411950951, 154923122256, -169097281756, -25341066221, 9981634383], [215235563894130, 70019526890335, -526657872484, 58990097032, -45215683893, -8945820597], [-2406707478403644, 19817085676396, 2875370251289, 159456650769, -33419904452, 18029154993], [193552594288536, -61537824655327, -1334403723332, -78290717444, -18467926727, 64414765265], [388695273433292, 104235142172540, -1835048925070, 8352824334, -21122357864, 34588241946], [-2049489945480024, -24500187378156, 3264216391809, 2463333881, -8180998572, 39254204417], [-307031313494238, -22066185180892, -1712227877482, 44440322358, -9701652700, 21456321890], [1596191069999296, -20715138390651, 783240237724, -162038296, -19106700599, 19384441033], [1656015970337898, -13088548852836, 189872150868, -36990433860, -33744124956, -40622463192], [-1458693919936630, -22258610676523, 2352570042095, 44185155331, -21208310863, 333369856], [3081742668698756, 94347321338974, -2225678863442, -8556121290, -8764427866, 26386835956]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_105_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_105_10_a_h();".
// 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_105_10_a_h(:prec:=6)
    chi := MakeCharacter_105_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_105_10_a_h();".
// 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_105_10_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_105_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,10,sign)));
    Vf := Kernel([<2,R![-68102208, -20483808, 1284940, 57400, -2479, -26, 1]>],Snew);
    return Vf;
end function;