
// Make newform 338.8.a.p in Magma, downloaded from the LMFDB on 10 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_338_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_338_8_a_p();".
// 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_338_8_a_p();".
// 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 := [-82422805113727, 7261452181419, 282726076566, -26856295558, -155625075, 24977559, 24639, -8757, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0], [11550979557658684610226163934612199, -1703661586745643626356181106738618, -36380562058697058325326515133020, -146249358678924135870030828970, 32193670136977269388671908473, 217940764079933098729173592, -8364266098951067159283743, -29472704939638137011040, 593725760621328914155], [-534315876112954130087494590403708, 29912580576306800376117235431395, 1855412721203291873168878022395, 50365523207910716622651918489, -2466007799099287816451934915, -41214929903128015168039916, 941675736511074827530586, 6345443057435389860181, -109930983837339971745], [-4068813005234296410780894911811171, 79720755475761812830106900340522, 14115609404257555847299978864280, -458136919816004463601792192898, -2601576547785815591248689493, 276638394757603083287305400, -2935842739181830771083289, -46672611749586541311132, 725446045426750746785], [14664758269486986323062640291791636, 9456054075439151392533660016521118, -108186555800000239025026293377243, -20356315057775628234487202086061, 217953384978441566141399431388, 10655415648787133105654495461, -117753451529575973353438137, -1617062885830763237473861, 18885702666514120289309], [-102003194783580101310443134594113211, -22782617342684725338043476518109680, -214892202078559329320797442415364, 120298852575824724242548712777312, -781351914827486010877763766271, -83293886114085966329562407766, 629681975884613441590635143, 14348258252622129119332038, -123790017648657623417907], [192397794177973425557094196616112439, -3996013858468481706914532333654104, -2663965762054326939535236556690946, 3276577705070676797296085521816, 4204219213313992420639529016731, 822279641830755434628132724, -1963449699351327432252802947, -1403314484839361147828662, 266155550122288885225313], [-1287951472263390355405602618078936390, 45162397365834181987586717243487157, 5098159283762418776638734013170367, -110492750311527163873989971774653, -4799251644454742178863016859887, 58032165081523860618245067426, 1798823274752572210017253718, -7356983451442439628186791, -214996958391096760490929], [-310654953112950908601594283164096241, 10498945628351042587851084388739194, 1389435682014786157761492640743015, -23850402523999995045100516642754, -1829233536286007189991465251277, 13125832789051965864186524815, 824084002214475565852968972, -1533025932768705375349122, -107539551773813700380864]];
    Rf_basisdens := [1, 1264989864354200209012667381221356, 48653456321315392654333360816206, 97306912642630785308666721632412, 316247466088550052253166845305339, 632494932177100104506333690610678, 1264989864354200209012667381221356, 632494932177100104506333690610678, 316247466088550052253166845305339];
    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_338_a();"
function MakeCharacter_338_a()
    N := 338;
    order := 1;
    char_gens := [171];
    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_338_a_Hecke( : Kf := Kf)
    return MakeCharacter_338_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 := 8;
    raw_aps := [[8, 0, 0, 0, 0, 0, 0, 0, 0], [-8, -1, 1, 1, 0, 0, 0, 0, 0], [-37, 1, 3, 0, -1, 0, 1, 0, 0], [-156, 3, -8, -2, 0, -1, -3, -1, 0], [-139, -2, 9, 62, 7, -2, -7, 6, 10], [0, 0, 0, 0, 0, 0, 0, 0, 0], [-3590, 43, -386, -450, 115, -20, -10, 20, -11], [3362, 111, -93, -590, 76, 55, 28, -32, -68], [10527, 177, -1386, -924, 45, 24, 53, -29, 52], [-15799, 930, -853, -1421, -956, -34, -79, -50, -13], [46318, -589, 1280, 4908, 410, -47, -267, 285, -292], [-93113, -446, 2248, 548, -1136, 215, 104, -264, 849], [-114510, 3963, -5096, 9271, 1631, -52, 1123, -12, 138], [-138619, 8004, -1907, -4782, -833, 265, 697, 139, 1032], [65283, 1366, -5392, -20818, -1599, 327, -3658, 52, -1084], [8881, 7389, 21444, 9035, 4055, -573, 2798, 1166, -1926], [-225458, -15598, -34647, -35665, -332, 1263, 569, -3633, -230], [-613939, 28788, 11858, -39314, -574, 905, -1876, -3720, 517], [-1215332, -1405, 90398, 11341, -1658, -2446, 5208, 3345, 4035], [-1165712, 14290, -83511, -80258, 5990, 652, -6818, 4059, 127], [-51026, -22834, -21549, -6464, -3302, 844, 7206, 1428, -1616], [155592, -1754, -73657, 11340, -18126, -3322, 5884, 1649, -4721], [-2153426, -1416, 238787, 257046, 5954, 3024, 13535, 1917, -9546], [-3027627, -44023, 44028, -57408, 1769, -422, 17331, -4472, -10370], [-1408445, -154872, -390762, -291787, -920, -741, 14444, -4548, -8429], [61432, -34046, -5689, 167471, -3826, 5859, 31299, -10134, 13562], [-4216631, -64364, 36551, 156600, 43281, -5819, 11192, 10953, 21825], [-2400115, -88027, -237783, -645542, -31739, 10718, 29660, -3440, 10890], [-3604174, 52850, 543816, 400312, -8922, -2202, -29634, 5996, 310], [-6637071, -11357, 143816, 254852, -45037, 372, -59919, 18876, 1376], [-9761599, -167712, 304202, 769898, 26771, 3719, -38960, -25000, 14178], [-7751202, -338121, -45134, 546748, 29978, 23795, 56794, 2463, -35777], [13270335, -285203, 178057, 284367, -20791, -9441, -131212, -9008, -4712], [-13789757, -5495, -137760, -129416, -927, 3615, -47014, 14896, -30663], [-24128286, -513295, -796805, -842766, -41531, -28739, 113073, 28636, 17261], [-4758129, -71595, -1215242, 1206970, -109179, 15770, -35379, 13995, 5380], [-35742571, 86600, -1015855, 1310093, -141266, -2942, -226107, -19246, 40877], [32864007, -62291, -468090, -947592, -57423, 17807, 371218, -28510, 29875], [5359106, 76756, 3970056, 1267758, -142750, 34446, -197610, -58594, -61650], [-32737966, 471818, 3705052, 511834, 307594, -18028, -186180, -13448, -21470], [-38494974, 806368, 1813894, 1154888, -243864, 75785, -144787, -16269, -38543], [-1951638, 93849, -2993721, -246878, -199987, 60313, -256627, -17856, 89685], [-29138199, -495330, -4748353, -4174166, 382789, 24757, 265794, -11485, -38099], [34320118, -544924, 2426992, -799408, 98952, -8228, 140923, -11148, 220251], [27963856, -27333, -2781496, -2681703, 185903, -182832, 321552, -71822, 51995], [-42610846, 321943, 4592585, 2001270, -57296, -21415, 321803, 64200, 65261], [5988075, 1428698, 2082123, -1667110, 46297, -27600, 584833, 164032, 55188], [44864112, 796855, 1543028, -3508866, 1056988, -206959, 309241, 300033, -29896], [-21601898, 838221, -8063256, -2599531, -195996, -46622, -533642, -200247, 127713], [48766771, -703161, -2231300, 3731033, 574977, 82255, -227588, -53858, -395594], [32124041, 576771, 3322450, 1983466, 528447, -80090, -650181, -13056, -275056], [13642759, -481801, 2268295, -1287494, 727387, 127484, 255517, -177866, -112927], [40305609, -714949, -4207977, -1933262, 108215, 49680, -92081, 389656, 314478], [-113177758, 3938534, 2368573, 1683646, -816830, 27523, -139163, -43460, 426528], [2428279, -1369320, -5750685, 2549359, -431436, -107022, 717302, 74224, -91212], [-27936126, -1552901, 95499, -4750058, -1153828, -151257, -738421, 365252, 772503], [35765385, -3526057, -2867599, 2682556, 580473, 231888, 471461, -484688, -514946], [81858327, 4367970, 5870379, -2088754, -440165, 104047, -733376, 328975, 4247], [-110518209, -839373, -6903854, 10131783, 962281, 31301, 151048, 339694, -407416], [-22122290, -1088842, -16936020, -16996729, -367130, 100862, 585576, -59228, 973760], [-41310728, -5169196, -13374516, -6770424, 1192582, -204474, 969923, 122406, 461057], [7236060, -2280485, -2157341, -1761464, -331729, -236063, -531293, 516340, -840443], [32709388, 1786958, -1374291, -17484078, -663256, -283813, 1998039, 71834, -725214], [128218210, -2338539, 23116718, 23598178, -2129222, -229459, 164657, -590079, 439676], [-109627977, -2648642, 14882927, -14727538, 344382, -220936, -297313, 363704, 762341], [277350380, 4466822, 6146006, 2882306, 1507166, 177652, -1636596, 8820, -342110], [144595229, 2391964, 9470567, 24321462, -1717175, -157221, 189348, -235721, 1644391], [-158190096, 1685990, 26580817, 20811667, -343918, -390409, -308794, -290028, 1677479], [744421239, 322466, -9759567, -3065910, -612201, 451675, -1326358, -666357, -730727], [81070703, -1762076, 3299758, -6554100, -2294166, 27497, -231606, 87520, 548631], [-288958245, -5858884, -8746718, 3297359, 4122620, -204489, 3677964, -1184240, 306683], [217254833, -982021, -13535632, -17404952, 220903, -940806, -745469, 1434141, -359466], [-97690329, 4436771, 20326965, 43217418, -942925, -542020, 3580379, -480508, -94659], [253265321, -6575497, -20210152, -9856627, 1434365, -247739, 2829812, 906066, 2673288], [-55445548, 9293070, 22470426, -2782087, -2858320, 355009, -2152613, -1064068, -1160009], [570351582, -4774050, 37998112, 56048848, 1589726, 297572, -607430, -597522, -1416572], [557018860, 613468, 12363698, 30936382, 1811000, 205250, 150772, -130296, 1982498], [-169796312, -17198818, -21477927, -44133231, 1141858, 631637, 1450571, -1049922, 935096], [458029195, -26987157, -6817220, 40978874, -4999133, -131103, -4106302, 577012, -310466], [726195402, 7616522, 55726340, 67321799, 918286, 1128619, -1000885, -274220, -310752], [-16743672, 7954272, 31049659, 40189218, 1682642, -1629164, 3588442, 891451, 1776991], [622750021, -17701201, -33709406, -5634469, 2587045, -75807, 2866824, 1311478, -713868], [511942391, -13288963, -37307886, 21025508, 1720129, -328532, 3726171, 139399, -4157002], [-357173812, -878428, -11742248, 8741231, -7003624, -1262473, -789303, 1005940, 935812], [-202541553, -9673821, -1266385, -68708860, -4724741, 251986, 235189, 227218, 480361], [-845839094, -6329522, 21170763, 10152926, -6248174, -2306071, -1629805, 1035292, 292024], [-309603499, 4834635, 23191188, -51971205, 1048671, -1515155, 4165877, 2552900, 1668401], [-262467694, 1180626, -21403009, -32886431, -2703078, -969037, -4225867, -753256, 4857902], [72075587, -4144241, 84768696, 40999715, -7603987, 747541, -6257498, -506106, 1095754], [262180182, 26214407, -31493635, 20226946, -1990172, -982901, -460583, -964674, 379015], [-435321020, -14139636, 18288552, 21244989, -5759700, -2514997, -1449507, -212796, 3003553], [-319650205, 51309723, -10018517, 27906512, -4281953, 793308, 4294809, -399732, -1310275], [-9036388, 35696977, -63004392, -67038076, -5352648, -448421, 978243, -2213023, 826096], [-1010531227, -20123559, -23616830, 41400535, -2413003, 161753, -5316818, -2299881, 4509233], [732402434, 36925442, -29851886, -35290108, 2788888, 580147, -6095789, -768871, 1814077], [302721415, -38495851, 10032037, -11260958, 2149367, -2061382, -1573697, -269894, -5535473], [-745899651, -322291, 3541144, -19491039, 2906231, 1566545, 4777610, 867150, 5814312], [-347138793, -9052732, -42876071, -105336044, 1627488, 3296989, -9915102, -4333868, -3280921], [-1273570015, -24018202, -2865686, -26737784, -4143399, -1513362, -5524217, -2580597, -7536569], [-789167238, -14533667, -15235178, 47543855, 1405293, 544360, 4155460, 3846074, 15152281], [532711599, -80326438, -76143508, -1746115, 9963061, -332726, 5288001, -3002882, -5883479], [-943940436, -23576005, 14108499, 41700996, 3019495, -1310765, 782593, 2555492, -7441651], [-1796542619, 7594474, 197651724, 175535412, -264849, -2063655, 2315044, -4196678, 5923720], [177131194, 43590010, 43754895, 219457576, 6650, -545188, -1100370, -1204636, -6848772], [-1015607233, -35895297, 42344314, -52920001, 429695, -1507759, -9285728, -1129147, -6669475], [-2246590745, 20945056, 89338662, 77712962, 8435848, -331815, 3827781, 3800092, 10835246], [-489879, 53778898, -12455272, -177142788, 10297501, -917517, 4237755, -1260364, -4349391], [-2279099638, 79337124, -31313661, -29598512, -9567276, 2848093, -19716407, -3442840, -1459614], [1371975908, 769365, -171288241, -66306828, -2331978, 4269127, -23394825, 1546512, 1643601], [-1281186788, -5953607, -118968510, -209014737, 16868197, -3168264, 677917, 9237184, -6257342], [-1821034270, -13862838, 67819193, -4346636, 7237404, 3171128, -5493856, -6372637, -372263], [-1994765241, 14112087, 23753316, -32170079, 22514257, -7049505, 10375870, -2218510, 1126916], [-1561681755, 38209494, -213798309, -154571256, -16461390, -369119, 3648316, 4483224, -1761033], [-569835705, 52888102, 44869069, 101142178, -6691581, -2658317, -6171766, 2858719, -3503379], [225191642, 48655494, -3072033, 262537026, 17289724, 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142504095, -4404113912, 1888459846, -522006170], [111843038990, 2237488638, 44826884664, 19836898512, -2006498896, -413175705, 2803356407, -1510716497, -207022901], [8671461992, 24600693486, -44829556495, -40747912532, -1473232682, -340740694, -1630581690, -389559304, 566232006], [101459196247, -12596043064, -18335739618, -1719573948, -2784955926, -135102407, 1344291982, -1034325800, -976107229], [433263326865, -631905662, 41571362673, 39833441540, 2234871033, -1447921537, 1521077710, 799604813, -2205203669], [-519206420911, -6128913550, 6705246161, 52316535496, 2400452349, 938613329, 2054344132, 1464523811, 1327837297], [412274010815, 13916162542, 3605741322, -8778288881, -3081036874, 799607647, 699695580, -147675112, -485198939], [185368088942, -877491167, 18486570818, -31591069784, 1229750590, -353222699, 4429305321, 151245971, -267831912], [-1053039442066, -6335066475, 16429358580, 35610923827, -96105331, 762275544, -2775831410, 670161234, 365074531], [-229696850327, 1034829269, -29847120996, 43795358455, 1095297071, 101882843, 23022048, -984246078, -2810455944], [269152433598, 7987050168, -9434923235, 22296283834, -387900824, 619556730, 246482650, 961258023, -105834455], [-152150061809, -4218956315, 25467139452, -31558503400, -313844541, 1294826806, -4043624961, -636996635, -1947960700], [513469528344, -13475380955, 1797752542, -13281222308, -2269765491, -1073068253, -187309951, -569299780, -1670839579], [164953754858, -11709615867, 11811528361, -15557463782, -3813238511, 865696195, 618594893, -1619727612, -446633613], [-32325541756, 316848560, -78730606718, -16885054361, 117981876, -698348839, -240112757, 446818918, -3565274943], [-313901302626, 9917315725, -1223191216, -30371584423, -3115819831, 144518080, 3192064485, -1409700048, 4753727334], [127469307822, 25804900218, 54725599831, 6326118374, -3882140122, -78211641, -4813250383, -331570314, 396287714], [-338565870528, -2269981127, 16825027343, -8434795302, 232770246, 1793429677, -790515649, -160882724, -3170739831]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_338_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_338_8_a_p();".
// 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_338_8_a_p(:prec:=9)
    chi := MakeCharacter_338_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 8));
    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_338_8_a_p();".
// 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_338_8_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_338_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<3,R![233327013346323, 3992765117796, -1272574768572, -25873130781, 1361835243, 25538085, -534745, -8682, 69, 1]>,<5,R![-98431045530076819000, 2275821795816417800, 11144249096309350, -469930945307711, 1101471280068, 17743146285, -45527490, -221282, 318, 1]>],Snew);
    return Vf;
end function;