
// Make newform 71.8.a.a in Magma, downloaded from the LMFDB on 18 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_71_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_71_8_a_a();".
// 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_71_8_a_a();".
// 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 := [-12214728939560000, -30087511060852350, -22444897329077083, -4538991111669515, 881601559550273, 261484391866393, -12539030017071, -5322977548063, 78353550093, 54415399577, -193726721, -306314257, -21197, 955403, 843, -1541, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-181, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-100256907858953392599436152587890977538161846386035515447304000, -228144389785919050186097329657829615044557278693301728304256166, -100030479960525514600569529028364259185101296196242747990284145, 5033249459905330692234495590398677121903898104774504723197166, 3681506580498537954274606056688398248910894817348512146723831, -36138111117677484355886954214700992701615160821393477657222, -51356702386325391683046512351602070965954374151689537100501, 193149070950941330436577998547095410802480951945364076286, 354883747055721323500532639691413590608322713402478792579, -1559221284582520544637292919609399302970000387768350242, -1301874449296328381653044589842603643661556445759639059, 7818571917494617828091898421625553875986901156557146, 2504162695092594695531120059509109396955326546707781, -17473989625038954102182303428456214946834904124594, -2244851814323467996746630809401387110578303160807, 14094073153163722946541839567316934932566254778, 618813746878938386556008387461565516584700097], [212146746187069266337556432016150234469128908452740236477195200, 329531154277494530171737406803214895998654904417234319766707906, 111880006482010496058016255118601278878086045596929854478140363, -7871909462180292390739315364993585997603929267991597751262042, -4907993186364244229486287075491061209422625725571880841298477, 28255286759886465420328447021326977731724199339642577434466, 84463365567748755808461140496503334986241203493780349151959, 646059194662207551317847123618621149054046606335129856246, -759229772938164409837153636361751219588785094219507601521, -7316359831811633543247553856887258580979991716266725546, 3867453002264181630350104930872783198467778739857740961, 30819814675462889494483656374866319975022850668123842, -11176887095817359947969397469344012510881107419193255, -59082056050794909259510733011588373587517351085946, 17016679903677293073475845531314016470607068438909, 43219963651222878544958175572022471860464803234, -10573055489205991032987636409613053127029085755], [524552089663885908669305619840720203003135295526235741019860800, 1625792154317079380395720039198030514968972386846324222661896230, 1009527881516031282124711702627491147943942794868284268637460913, 107184782322975545793565907104335738144043775923808574190978834, -30484478730318404284814853670725999160843832895576098949267511, -4824024503196702098979369606711784143425922787300693239154426, 328644125298408876417801160544518945517395254527055118176533, 67441024554590226471319531397248769703512742164781250490882, -1599894263058790756812141136846050088626869397680525833667, -446997340081649364209498428265069035308470876524987476318, 3254745811945399303794668978862254456799295395756330707, 1533227334507444411766156065243859530225249651884716198, 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1500544155118674807957687934263914897206965279842119, -530492610923144018997379582760705712794575226983110, -6771510314513295404782959561483952192649724028637, 353405939608108201573461473925991772049757078878, 6291474058458368232038306669881128731582914523], [742734063861047462106092805794013057976223000972084035368180800, 1271643050186203613293113556923742617247919646361278379939744574, 592218131361138547080641402043106367697164600617537466161613877, 14229424920021013254640398694655962344522983161848727836428762, -27220914181078865326251614822805807491502174845845247232824403, -1670690349290586135005827825280910437195965970508136646609506, 514315808086402205412855759062823557993269995885713731161641, 29385931161125934381721040889798621488379463165974018018954, -5213293567766980700540349976590326028113864469244388100879, -224132907623317447260010148980932524183288954444141891414, 30050972395751984054336962589486020876813254239973839199, 846373154859929661509854292156029646862752679055462078, -96892429871663479638721474199788025401664274700199385, -1555358367489490186766727698126314674576875977301894, 161515121014213706440445391426747021402058033913731, 1112694141762179940612239006844053444815760377566, -108029680324296063565306290578826168162523030085], [-291533931117097870236040825618927411921771206830449805435960000, -482357432037718714378830075006079236294534636176649532314693002, -278606927295444617355463578035231059411593411762103198974815127, -61705575228287607174212089698937789828331414146963010666033534, -87602428947926647305645379583386289583334407075837867015999, 2230332887941654746540628422133439171140120523993371321631830, 208834700604585569922913840924512583821091041210464504926381, -27113576263118754750727532283836123112249175928534687391246, -3569805016972389943824430777326115586977222873557817457259, 158164130697678824173437269639574741435068404231935687698, 25519169902451686772364649540184916208990612602467634939, -485324900354589195234911053247457092040458777608272010, -90952863431500028230508682625691591457773262729084957, 753536013022853485068781900158509453224329762842402, 159200022402673470958363442607930870328311619931183, -462835236129646734770445054256074577290206523882, -108932737084021555608726268200872393398677703609], [3098617294435734197728216876395305904346329319645704209199083200, 5894023882341016510444804505140210705148056199058845812889537002, 2430671108441334328676337560913930086198588308482843855954186663, -78432437201063784259593485942505554535520995571554783829839458, -105025620603131408809652542363787028902431881989535043221805233, -1810085102980409100055336853096349634938939382020533372274422, 1819495563722787304136047245897239222195862606177954062960035, 44414298524908305185133227910854453577871577270662852350510, -16693061712753301512688573446713401531337172168888695725125, -358981653908810336356470188207587281452092984607658044914, 87400674330874730804053405576085196116764812550875619125, 1368041931809039422817800461286805908070623792934312426, -259618312272497341067555636018265387548934927241660851, -2504229336250052850448300134630002115175043787349762, 404870195192936363306776533531736093291805261942081, 1778709419991198571703077249603697678880933194826, -256595089925455044700418795586002756316622844887], [52684912948008042051495944632774777569235564415485508582676800, -99752437553748098793478435161474599012853330037761100614005802, -264241997236494101185616913452608696027383499605234170286815239, -129892172103667898420730592150333660868433742658898372081362270, -2710129439260462991798990449608711130214821960671645354817455, 4477120668862975008143401469824049904566020169008453580160694, 261203145101614062595991836999133716135904722667263773072509, -58106209002313763023970181473997042520308585195126040407534, -3874162477322434512643479848522851105937044074568541906331, 371747051112143288646061894829082095224218577132498366770, 25406312128946243416924390211324023184172791118900660651, -1250166671130114643115295016022157412016809301145578282, -84976710386497830509867370491367523657097130736877421, 2112603576854788322031535276812638047297763230445122, 141631506815294150286165796140579253773512527844895, -1409629435718609859722223376349451117768613096330, -93170187881256292545151117022536962462746034377], [-44107986007910568466898909070230873816762335409996911009211200, -86074464671291208998522463811643409748676878730903738813658246, -20736365115913934879689408183495537749286446776608694868827457, 10760021476909914403295702476027226201998216033966612270633230, 2393943066679917231009036583926300628226460775402413705817575, -281686426395129357617546136992140104109565417517051821184998, -66278443008964765238056744560144699339303559767943163333733, 3056033922297133888487702489059742040421888582595001248158, 798891801200964472779042141123584356304389066633524890547, -16704517651846974189821038943745439228548800947448064450, -4942686765742175511443300244844567738776762849136226947, 49032914305758289762755800015827426177400026700132794, 16320738442790885793001313716960099729985485936713237, -73573144566692901474920724612548520597988646497234, -27292156968110109253066924417908110532020824023575, 43698597459124729666531126339469902970510177690, 18134058187098851892667020502451013827887499729], [200915233032798304808591794029790061801829055514478762631211200, 530659882218649814802570090113766506369922259710923986462835482, 188607422846333057170146706907315056546295671791410489749028655, -54779809207478381669918213987052335394244850682373577452084562, -16235323938049422476377284576670814528141572088195396469717417, 1289438288986436737719744916728484518240823154809471185987514, 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-458308655287263997038319212775618536336776267892083774519874178, -190866084085216545338503936209762460454189541661470732955002753, 7827151838194165892020006591181524860876378564534639981959978, 4238003428542966093334167834104404663595269909810197290244755, -54537442937102837959120002381173267198269693464798761037810, -46239153849821993787999778639912902609156193600458165470805, 143017497478831818873338053046298203871203119718234495086, 272433686670912040094172144532620681374184968796049942725, 18644628375031512081970911327310045831013812596671626, -876960296360835323704115252734360302921350236113220131, -671873548793826336709808682353569843028098464757922, 1446128710577557373630782197817377562391328008392081, 816050667610329580534815935624798430872837920746, -953436121334382852723672455405343453224772127367]];
    Rf_basisdens := [1, 1, 1, 109224090338356950497779174683699009470821724299651658547200, 327672271015070851493337524051097028412465172898954975641600, 109224090338356950497779174683699009470821724299651658547200, 15603441476908135785397024954814144210117389185664522649600, 27306022584589237624444793670924752367705431074912914636800, 327672271015070851493337524051097028412465172898954975641600, 163836135507535425746668762025548514206232586449477487820800, 327672271015070851493337524051097028412465172898954975641600, 109224090338356950497779174683699009470821724299651658547200, 13653011292294618812222396835462376183852715537456457318400, 81918067753767712873334381012774257103116293224738743910400, 54612045169178475248889587341849504735410862149825829273600, 109224090338356950497779174683699009470821724299651658547200, 327672271015070851493337524051097028412465172898954975641600];
    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_71_a();"
function MakeCharacter_71_a()
    N := 71;
    order := 1;
    char_gens := [7];
    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_71_a_Hecke( : Kf := Kf)
    return MakeCharacter_71_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 := [[-1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-25, -4, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-100, -7, -3, 1, -3, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [-443, -46, -7, 3, -29, -1, 1, 0, -2, 3, 1, 0, 3, -1, -3, -3, 0], [-1000, -177, 3, -5, -20, 1, 12, -4, 3, -5, -2, 4, -3, 5, 2, -1, -1], [-3219, -442, 15, 3, -46, 16, 4, 2, -21, -10, -8, 7, 1, 3, -2, -5, 4], [-3321, -526, -27, 12, -19, 7, -19, 9, -36, -11, -6, -7, 8, 3, 24, 4, 15], [-3355, -191, 128, 55, -7, -4, 45, -21, 19, -25, -30, 23, -21, 8, 20, -15, -61], [-22201, -93, 6, 210, -45, 22, 134, -47, 22, 96, -65, 17, 18, -55, 47, -71, -104], [-37637, 2056, 276, 76, 152, 1, 139, 29, 134, -49, 61, 23, -1, -39, -148, 38, -23], [-70866, 6802, -357, -97, -1754, -182, -37, 60, -327, 116, -123, -54, 131, -78, -22, -53, -89], [-95366, 696, 374, 280, -1170, 151, -536, 243, 114, 192, 64, -163, -53, 168, 52, 348, 204], [-91098, 5549, 794, -223, -3571, 27, -106, 216, 380, -129, -246, 217, -14, 162, 22, -26, -217], [21342, -3486, 1059, 495, 71, -430, -434, 287, 486, 398, 108, -422, -102, -108, -119, 177, 616], [56401, 4964, 1553, 1072, 1352, -300, -18, -131, 313, 179, 1525, -847, -424, -407, 171, 916, 715], [113982, 1351, 166, -960, 6244, -852, -435, 19, 818, -1051, 124, -708, -448, 364, -189, 615, 430], [-431241, -15796, -4197, -1635, 3476, 85, -926, 639, 661, 68, -690, 32, 236, 957, -840, -603, -32], [100847, 28151, -1329, -3062, -8456, -696, 437, -698, 901, -541, 2490, -1566, -587, -946, 851, 360, 192], [357911, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-311823, 32533, 249, -429, -13188, -994, 1751, 801, -638, 251, 1442, 774, -1672, -537, -2148, 74, -1721], [-26653, 7244, -10722, -141, -10718, -1750, -3615, 3649, 1273, -90, -1974, -2151, -56, 871, 2788, 1849, -67], [1075405, 62441, 12014, 3288, -6892, 1943, 1404, 300, 4144, -4228, 3090, -1483, -2204, 188, 3416, 5256, -646], [-470862, 142034, 48, 1464, 31058, -1083, 2929, -1014, 443, -4956, -3464, 4576, -1493, 3867, -974, -3009, -689], [-2431992, 103245, -23108, 3297, 25870, 1597, -351, -1341, -610, 5872, -2207, -520, -56, -1754, 5282, -4598, -2022], [-2144509, 33753, -9184, -1617, 52990, -3113, -2914, -2991, -598, -50, 1480, -2246, -8012, 1555, 5188, 3562, -2736], [-616829, 122695, 1938, -8604, 100709, -338, -1504, -787, -790, 7224, -3605, 3376, 4736, -9227, -16555, -3645, 4097], [-4750719, -271195, -4230, -7660, -2075, 3777, 731, 5731, 1611, -4938, -3695, 6363, -629, 7618, 6557, 318, -8843], [-4751619, 186140, -26390, 8116, 115422, -1647, -9564, 2720, -10523, 2185, 10717, -923, 5819, -342, 2662, 4695, 7323], [-887329, -333671, 28207, 6895, 56648, -9092, -9601, 6431, 17217, -1683, 10060, -3230, -3675, -2319, 10115, 4264, 4985], [-1292049, 99644, 29321, 980, -46856, 7558, 6248, -19237, -18189, 2028, 1146, -9595, 5163, -15576, 8896, -9195, 4567], [-3087200, -820095, 12578, -41655, -108812, 828, 12797, -13388, -549, 15491, 7072, 4856, -5425, -11839, 10000, 2721, -3431], [-1379939, -622353, 24513, -32120, 139500, -19533, 15489, -8247, -2053, 3442, 5087, -3839, 5811, -9573, -23738, 3977, 3940], [-7402497, 12581, -2615, -44961, -14383, 24873, -18172, -4261, -4714, -19972, -21935, 16781, 4365, 9311, 15146, -4812, 9103], [-1003339, 392603, -50158, -5778, 77468, 13768, -15524, -13653, -10518, -13485, 12440, -686, 15631, -34156, -7146, -19754, 15923], [-13068685, 655633, -45548, -40166, -174992, -1835, 2519, -3861, -10031, -30660, 29535, -8335, -17657, 19318, 37531, 30114, 14637], [-7430098, 101623, 15437, 50894, 84987, 8791, -5294, 26255, 25603, -47776, -2975, -25349, -25146, -1129, 25205, 20760, 15359], [9232335, 452497, 108337, -17016, -63644, -42980, 29929, -13492, 5639, 6581, 35936, -29714, -8849, -20842, 25921, 29924, 19794], [-1233010, -1034933, -109451, 57459, -130557, 8744, 15067, 8804, -52087, 34095, 7224, -377, 16911, -25790, -34773, -11104, 7237], [5182065, -1100137, -16410, 32419, 326352, 10213, 48252, 4559, -49802, 34084, -58425, 70729, 53021, -4114, -73289, -38581, -26646], [3495425, -679714, -23603, -33045, 119160, 11665, -64456, -22223, 2720, -39768, 32078, -19553, -30543, 60259, -12561, 34715, 62497], [-15303815, -979196, 44858, -52751, -367920, 24510, 15063, 1537, -61290, 38333, -49409, 16285, -29812, 9000, 25122, -20502, -70245], [-3301296, -604830, -372865, -17612, -567583, 18383, -33183, 7996, 18901, -37766, -53868, 73580, 8585, 35848, -13529, -14556, -11976], [-5554517, -597778, 365813, -111314, 34892, 36179, -4322, 4166, 17695, -47387, -371, 3732, 46585, 79947, -91939, 2498, -1299], [1418481, -495326, -373206, 107019, -144628, 1822, -55055, 17529, -53614, -70993, 61997, -30547, 9542, -3112, 73746, 80470, 22751], [5506733, 575892, 80736, -66768, 69371, 96, 96012, -51337, -25155, 27221, 42588, 63309, 80879, -49387, -77085, -35752, 1819], [-9808894, 2140966, -16772, 175899, -375551, 64279, -18869, -25745, 83765, -163365, -48517, -14012, -108390, 51303, 82549, 87910, -14686], [-31607104, 644403, 428155, -105086, -676471, -91897, 150793, -32562, -57902, 75756, 102560, 62252, -29295, -1945, -37018, -67190, -13247], [24038743, 2147157, -353581, 191269, -720645, 23537, -168452, 16967, -146564, 101478, -143139, 59698, 88875, -157079, -44206, -164306, -74884], [-54933710, 2027470, 399719, -243033, 621575, -17673, 27932, -57916, -103783, 3042, 37166, 21150, -108440, -928, 63766, 6043, -73931], [-17717113, 5200081, 336893, -251755, -234359, 22089, -167426, -101037, -29816, -95690, -14518, 47387, -39482, 158734, 123319, -107171, -37644], [-16124119, 1213324, 23874, 120722, -544430, -38845, -20019, 136409, -43148, -72683, -531, -38069, -31495, -8723, 178470, -56234, 3595], [-111019878, 4577948, -185232, -516803, 583884, 17923, 78752, -9858, -99608, -105879, 19023, 23619, 70204, 117854, 49169, -44363, 81877], [38002871, 4090809, 246847, 83096, 1389617, 77997, 80200, -25440, 58846, 53395, 41835, -80943, 107313, -640, -90286, 135180, 87427], [-4986932, -3797852, 57760, 102741, -632328, -54050, 238192, -25997, 92464, 329359, 30893, -187282, -156409, -205232, -86185, -6735, -81309], [40408425, 1452919, -299710, 201429, 746544, -41688, -107174, 66462, -201643, 168168, -79917, -94085, 193673, 65928, -193296, 87369, 152533], [-57854585, -399495, 93454, 356892, 1032624, -40414, 112376, 80153, 96138, 115843, 133236, 12370, -87849, 126006, -249022, -198, -5797], [-85308498, -3398510, -164709, -228496, -585301, -19813, 394455, -77956, 194281, 304246, 27648, -19500, -37357, -57452, -211343, -213214, 41878], [-35638313, 192077, -543714, 359359, -1006786, 35957, -675428, 107979, 110676, -135869, 50263, -136587, -31295, 78722, 9045, 91263, 255607], [41505371, 5155067, 461743, 2216, -409136, 44682, -2699, -2962, 256297, -77018, -123213, 88536, -103266, 193413, 121362, 114945, -288924], [-30173589, 590058, -780203, 371044, -1139402, 134962, -291990, 29663, 114775, -90556, -242212, 98103, 153117, 256936, -175480, -4463, -111649], [179242590, 8448081, 70662, -350986, -404506, 55712, -404049, -174667, 245892, -128333, -287462, 257403, 104036, -79786, 16205, -129237, -96917], [27433970, -11848023, 20836, -883226, 386884, -123628, -362895, 156689, -77528, -339817, -130114, 139018, -84850, 232012, 132653, -87341, 39034], [54203178, -7631028, -302109, -287724, 851751, -186001, 433768, 335677, 274220, 15403, 127704, -74573, -369693, 103781, 167919, 285179, -66786], [-17392368, -14062800, 260906, 744383, -2394712, 416475, 593038, 1920, -420640, 265709, -291312, 133757, 187967, -312483, 50770, -339132, -240628], [72107982, 4722216, -1164534, -332733, -138696, -42425, -611908, -136562, 191522, 120977, 486661, -232931, -91532, 239490, 37995, 280673, 354867], [-33610476, 2136025, -637426, 76063, 2374441, -317389, 554496, -203252, -2889, 372648, 74591, -83052, 69730, 145969, -103964, -400727, 159702], [73522028, 5947982, -564882, 758082, 2331012, 235117, -1221804, 187293, -608594, -247464, -685082, 392333, 531173, 58412, -194998, -285606, 195266], [93920443, -2549508, -2215201, 85996, -1798618, 15598, -230502, 191801, 358513, 171932, 233832, -122568, -583703, 7242, 233856, 569243, 88170], [-43756157, 8851709, 1744470, 1133028, 2791380, 280882, 317330, -100429, 203138, -615959, 771028, 148518, 47277, -160640, -173920, 307200, 316633], [-16376169, 10232140, -2479895, 1411211, -574858, -229613, -165532, 395303, 105759, -102754, -311048, -306524, 420016, 131183, -165820, -53401, 167868], [9072967, 15234321, 699052, -653747, 5004586, 204912, -64306, 290262, 196419, -1085175, 55994, -213008, -192142, 522107, -146695, 295118, 86720], [24845479, -35537005, 1786078, -473735, 1188121, 257583, 1976947, -162832, -450184, 421699, 20404, -48619, 133847, -576582, -258423, 16617, -651430], [-53867368, -24439947, 2975239, 1348135, -2378871, -556755, 249305, 9429, 495755, 875253, 647886, -402465, -196590, -60286, -314781, 534428, 367110], [-242779494, -13802546, 1978841, 165424, 3703963, -153231, 433475, -207716, -191247, 371373, 243931, 273933, -10042, 312727, -72784, -472561, -909535], [-214601653, 9297855, -2952786, -2834431, 3226441, 667486, 131229, -137279, -1531491, -845265, -584150, 316501, 334809, 522128, -95232, -27757, 633059], [-85581092, -5695544, 229687, 3192395, 2731030, -124342, 545411, 81272, 381961, -870370, -20348, -404388, -2008, 527171, 1146653, 347984, 86526], [-137067623, -19782916, 1588439, 1190658, 6748764, 228924, -972788, -161423, 1252779, -720997, -241009, 419433, -191662, 344189, 19059, 219150, -350523], [-172180418, 13082709, 2453396, -1267296, -4018986, -684417, 1986917, 37188, 70154, 140203, 87002, 248596, 255511, -519092, -218435, -713309, -491559], [-660530884, -10029281, -4134536, 1086786, 4423564, -504523, -2143174, 318001, 274575, 142448, -893555, 257104, 364676, 544702, 310486, -711217, -562517], [-130419619, 7472468, -1854778, 780510, 10970866, 475821, -1013698, 361838, -792947, -720582, -385368, 967915, 686520, 324717, -1062501, -271394, 143055], [-157848688, 6500431, 6464069, 2540440, 3536432, 1025170, -134396, -84308, -321875, -89756, 606099, -232745, -231329, -449657, 1134517, 252532, 388522], [354457810, 11951466, 2213543, 3794692, 11217758, -194544, -831369, 582147, 708135, 535441, -621042, -987478, -638778, -339592, 961713, 427654, 75430], [-673395649, -26210964, -3450727, -2660944, 3460262, 1244342, -67440, 119113, -229667, -311507, -619025, -157771, 333882, -487635, 394771, 867048, -439091], [-140035102, 14574123, 2262706, -2256113, 2850853, -862461, -518482, -439106, -441019, -513254, 629199, 131062, -623000, -14479, -997072, -22489, 696404], [-323985868, 11250954, -8829560, -2356392, 1173694, -311074, 77130, -355780, -444310, 1628704, 556778, -330281, 200142, -1302762, -141938, -675036, -199503], [44368198, 46154202, 3882084, -938740, 9060098, -317861, -1571446, -297887, -740932, 291948, 1891648, -1324451, 797673, -1030086, 103874, 584748, 1171360], [-351979945, -12599003, -1198427, 1859606, -3177792, -774684, -1119519, 1234838, 1592435, 145852, 435841, -1380182, -498462, 473405, -168172, 626813, 1524690], [136799509, 71491225, 3467526, 1346737, 1221838, 1354645, -2019010, -433317, -1905836, 452380, -1131113, 449569, 317427, 313668, 677795, 385401, 325726], [266688586, 33069628, 2471357, 16860, -4577542, -273004, -1279377, 559527, -943659, -935241, -1453110, 1206422, 813078, 602040, 407805, -803696, -916524], [521152875, 21156503, 3968775, -2491849, -5858001, -189443, 1439110, 283423, -79030, -1314252, -170703, 544306, 758217, 480339, -356978, -575884, 60378], [258683128, 53186032, 5363775, -2401132, 11769868, 973489, -1011593, -1542052, -1793613, -107873, -711895, 809248, -852808, 575103, 269782, -791645, -1058224], [413554213, -16340128, -277105, 552509, -8491955, 1327349, 768375, 221810, -630924, -224583, 447999, -470191, -9583, 343405, 2255965, 1307075, -504259], [731237537, 42191320, 5956885, 2699610, -1524804, -2593452, 74148, 1431991, 3976297, 1373160, 114294, -1765279, -760181, 212274, 807422, 849357, -1048069], [-57118582, -6189772, 5349694, -636884, 6385067, -1048340, 2469074, -327296, -173820, 621942, 2055242, -788128, -963100, 132548, -1949418, 432514, 709976], [270400969, 28053597, -1717536, -2959800, 15011996, 244407, -1853017, 20351, 786615, -1454361, -1486090, 1024228, 318432, 1609867, 3152458, -58767, -1456864], [-1062754987, 7493415, -4416751, -2409770, -16564528, 1518195, 1281437, -182219, -2351115, -1510328, -2453903, 1029850, 426103, 735997, -1776778, -1715721, -151479], [-849522324, 66545756, -9087019, -1194969, 26666108, 372314, -2031391, 636412, 815616, -1519080, 1359300, 864609, -186093, -927567, 394512, 1285346, 1770039], [-671476288, -20648828, -10208970, -5730838, -21301080, 45350, 860834, -263790, -2130018, -5758, -850528, -1028297, 680422, 1128266, -1137386, -145772, -307061], [-672765420, 72664934, -7489942, 5760509, -7778880, -428127, 3007606, -764054, 935006, 1428571, -61441, -900047, 712088, -1588300, -3719221, -170863, -794209], [-254487851, -43140922, -10874047, 1000012, -15118685, 1205285, -974963, 1924873, -375286, -1649841, 768532, -647453, -27074, 306931, 878452, -377138, 2581129], [407848677, -55723635, 3065332, -2106944, -34946234, -552647, 740337, -1236439, -3292797, 40710, 3387149, -685456, 2766057, -908034, -2438163, 1151554, 2231504], [552128364, -69372319, 5296910, -5619765, -6932445, 544765, -1025586, -543752, -3291253, -954192, -3702053, 1537736, 1276902, 720535, 2061582, -1552517, -439544], [-1093137723, -74019714, -1858729, 973105, -46273472, 974778, 1437864, -398834, 1514526, 353721, 871957, 230208, -233001, -1127500, 463396, 1145702, 392245], [1044679701, 26800584, 13429024, 2003916, -20636118, 1857991, 5223133, 194339, -1946204, -1958916, -743465, 1905611, 1154007, -391849, 138750, 1033298, 1425788], [-1182362873, -58070913, -6566386, -8726265, -18136858, -669521, 3659024, -1953619, -1080169, 1219645, 3778256, -348966, -1589525, -799771, 1389761, -273820, 1995739], [572240640, -154831672, -6397533, -4795718, -28143986, -1838208, -555466, -103692, -484138, 972405, 1432999, 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-238247628], [-407850168777, 24154280489, 722358554, -6832500, -3902185539, 3256758, -15381514, 151986713, -248991424, -197673093, 41296114, -284519134, -254436027, -83871212, 653728036, 45970542, 327124863], [-63441357682, -6501188972, 501016905, 217895819, 4686289671, -204925700, 65238376, 123547459, 180503016, -462958386, 67566134, 215953448, 250701720, -110236552, -615895139, -45116977, 330919894], [-123514616339, 21187520481, -337708404, 963654782, -1924285449, 606243688, -593534084, 81226503, -157496982, -465819934, -473953137, 156017128, -185050904, -30345823, 528350783, 126444441, -233755251], [102461663378, 4726202515, 535029996, 994895657, 4240247449, -216443001, -600635562, -18657232, 155665025, -81993092, -468153227, 107243373, 55649962, 279946715, 291400384, -74792013, -117449133], [258115175654, -11404945259, -353615250, -213945777, 1014137760, 64019104, 415081169, -547621986, -352454584, 589968658, 228187649, 125520497, -125398045, -452762190, 250072142, -431611770, -432726592], [-93161115157, -12263577588, -1425561047, -81433752, 2859359442, -80549886, -93333026, -397090479, -583983505, 397987421, -248433095, 478086059, 211947628, -59678681, -1076740283, -978911328, 9399031], [27195801424, -8628307755, -1327035809, -1257606540, -3473242810, -155852650, -146056291, -108583161, 223340754, -193505852, 193400675, 21734463, -344592651, 548580870, 686815867, 105824695, 274130210], [-368644365373, -19065000537, -1310541684, -455011845, -3658357032, 275917583, -218587768, -38334231, -433375047, -183039043, -617871132, 563438608, 420223525, -124904353, -482111151, -354980350, -179993489], [-253473787807, -15700560185, 879211309, 252011612, -3703168800, 228536934, 135971963, 235787612, 537104223, -818418085, -37368726, 203301801, -413739489, 70199960, 639978681, 9639720, -390824811], [-269094617515, -9429031511, 525183949, -617310523, 2108135552, 11092793, 289301841, -388533022, 16062182, 122410042, -644535253, 213328289, 253126310, 94217386, -37753323, -595686501, -110650147], [-89709695691, 7116243556, -196861162, 997920452, 6358674547, -427186646, 317086171, -28738360, 293361528, 224782760, 256050956, 323016320, 217815817, -231959360, -545007827, -126682655, -235474449], [113216760943, -5565519497, 1091386600, 16442077, -2620401589, 112886276, -274739577, 125859031, 98196731, -657712510, -109309062, -182577529, -527323329, 406517514, -24092462, 130654031, -143653418], [-138221628846, 14342190932, 515152036, 55289497, 3225300643, 120609923, -663409059, 493392357, 327170817, -335246217, -528826703, 34584098, -35911004, 251628777, -208782499, -187065046, -297755540], [223876759488, 13894127952, -3833551926, -91834280, -8640180986, 33989216, 243539362, 426365808, -225369340, -382773452, 120681300, -245103182, 214907714, 437582710, 372913504, 298522424, 272784406], [-34661524598, -4905321995, -2976301180, -806771362, 1663299164, -467720638, -275230101, 324232791, 556213922, 132575625, -35712456, -222860771, -200742470, 307238926, 42669391, 257329399, -227549167], [249798215988, -15502252239, 241732969, 456724635, 1504877193, 259404628, -1084390423, -257594684, -687621058, -347841935, -370048200, 306854836, 380104520, -268583832, 51360436, -178123798, 172056846], [184987771158, 19893939036, -618826766, 179028789, 2123623507, -236303378, -532989623, 294286856, -113833985, 1297801367, -372460184, -264676582, -180529016, -373472652, 357335254, 66022225, -63805484], [-152600396782, -16963613768, 360337420, 104202009, -372659006, 152242990, -920448400, 142767645, 138714650, 180677297, 470980269, -767699724, -341486523, 133283566, 702783597, 521067921, 412954105], [39426987575, -27048443030, 1871822223, 1085774034, -9363027636, -516587557, -81598358, 269996846, 1354425771, 49772841, 521654447, -186631082, 65555919, -245204137, -269541563, 399982206, -7045737], [98166858551, -16816857309, -701122992, 271083809, -3829857118, 236921815, 106253602, -94602277, -706660652, -88043790, -208404377, -83347327, 142681119, -196086910, 863403757, 13060879, 72580768], [446611246318, -19837070059, -2492280774, -68741951, 2138600228, -18769797, -309602411, 1251271, 443120601, -313004132, -100145477, 29373437, -11608913, 139748528, 76856267, 129543186, 742104269], [244014179572, 29190643705, 1339688681, 351559190, 10131994820, 241041804, -68534009, 169763355, 138853144, 540361639, 392408610, 163407320, 141061152, -667848747, -85343382, -97435414, 18678443], [-9996889099, -21404700706, 1017192406, 205366388, -5213327306, 348097962, -1058226061, 159406532, 531563066, -477813545, 461991543, -134260083, -78652076, 506062289, 778646358, 543556850, -139960938], [393704193507, -4802695900, -2711490056, 407249699, 1248555699, 96145289, -479631586, 146482049, -125552564, -902464601, 301165585, 92842417, 131030256, 536079731, 17825233, 606096221, 490688472], [-233689383193, -17495581568, 1819105006, -1796789876, 2082870634, 328766109, 189996709, -213716267, -871953442, -430291515, -223862426, 278938700, 398490752, -39342964, 465356279, -358142589, -174170973], [174639856421, -4912009957, -3509514494, -571653710, -1358673118, -94685097, -491274426, 123948028, 422130424, -475982680, -779483527, 176703571, 235060807, 584725071, -266550455, -247005909, 42900577], [221873429174, 39070409418, 373161002, -155751582, 6871308346, -328676114, 117831470, -147945834, 475664752, 578484908, 284312834, -390324376, 56007024, -447304246, -141626258, 104045990, -265724222], [155298550689, 26818345998, -560563051, 644468656, -4129045830, -198103592, 527233884, 190699639, -271473915, 1162461330, -106548566, -328053808, 98953861, -239032496, 35673706, -248737815, 161009894], [207401101151, -8301965018, -1852801568, -659306230, -373863670, 170903979, -590735254, 338719116, 1277392213, -344703415, 303168301, -381186176, -917969041, 924336410, 559062180, 872917821, 328244360], [245587088202, -16819109891, -893211578, 289995538, 978625034, 113381433, 109343069, 393533712, -16274560, 73003101, 78714054, 86795674, -171575425, 16724496, 164282669, 183444913, -218751657], [-26705707974, 14780231382, -1207620197, -1493257629, 5644619162, -396917304, 236612847, 185220076, -300637905, 307574450, -236897760, 441188296, 273511138, 380739267, -352924105, -629573992, -374471872], [57051612268, 26166369374, 461504101, -1477117401, -3804035428, 80438665, 1709182329, -145029513, -194240388, 84046476, -610563810, 645347629, 158590230, 98514941, -533672008, -586597492, -95587844], [131930351310, -3123620567, 1502003740, -470289216, 3365662347, -164902238, 21780934, -142494466, -68876383, -944063818, 167974532, 328865614, -409498772, 341938665, 974356393, 206232560, -326613563], [-482210230563, -13011191834, -794857523, 1034665937, -2389608749, -483925141, 188620049, -340322378, 130405886, 1565755071, 205198811, -107558605, 29255889, -436340949, -48723989, -627866265, 127392567], [277066676031, 26207005981, -1932883805, -1656752991, 1255184439, 360116597, 421630070, -344698141, -388792920, -1211570674, 256098012, 488149157, -180164768, 352416418, 129580731, -308350319, -150026070], [8805408153, -5216296981, 988604390, -1321893628, 7247237908, 79307251, 417519308, -226617888, 559098972, -442822136, -398391365, 404142979, -413810717, 784282713, 727262469, -10992601, -575204751], [104845563849, -31256605424, 1303837763, -559727549, -3214346812, -387319884, 625273758, -475259640, 12573553, 114445174, 456971564, -207129047, 278913455, -525876009, -151369498, -152757699, 966539612], [49387303894, -5124188888, -1648013814, -141005828, -15746972192, -84289916, -258020570, -23849160, -161566712, 1227859898, -62597276, -132081917, 444156278, -578485226, -740701568, -378158108, -129027091], [110859035871, -19042957868, -3988998934, -785467865, -7236221647, -821624184, -421052820, 282610026, 16704584, -424089151, -113856753, -247151939, -547542585, 210336093, 900589173, -34860461, -411361161], [318633134538, 17714975794, -2389770361, 148604124, -6092071592, -338889081, -659892353, 701443186, 716908251, -202021321, -104300003, -605117759, 34452340, 790223623, -86447410, 310671635, -204432995], [299840281188, 2322940892, -327991724, 1168328904, 6786537832, -357051, 492466582, 177271885, 442076340, 85877960, -244476400, -406723895, -805722479, 351356406, 531277634, 178884636, -117900534], [330153297985, 18164591371, -2647935588, -806143558, -5900182408, 167366383, -710256200, 229105558, -1165265130, -245480436, 293246019, -672359509, -290229751, 272130273, 470389857, 75141593, 701913061], [246344457946, -26997724352, -2769911396, -2811836411, 4796390282, 424065733, 483518460, -71235356, -496076816, 55662271, -576754334, 151726455, 341455749, -193389033, 225260620, -161915534, -113053274], [277822207117, 19788583174, 3006634156, -229887893, 8324345733, -204752870, -1242440890, 29857656, -282013584, -213841455, -435798579, 230994437, 58193811, 633110643, 350652987, -105493079, -393553947], [236458104904, 25133613456, -2009740663, 546233886, -1872361286, -518707151, -366692929, -606979282, 264552831, 294360063, -228606509, 337653902, -46630850, 46428833, 512840422, -692840217, -878503476], [291953301367, -14686514766, 2251063294, -183816924, -4740280118, 176950567, -309916295, -1035108749, -607954590, 532699716, -8544130, -264259902, 184120974, -342174482, 318861481, -200282575, 431007160], [231552665966, -33674590776, 1733814970, -277583469, 1466311800, 290208313, -82356394, -421976666, 529059693, 604198004, -728728878, 557511917, 45017426, 424873145, -494870383, -336413952, 195965969], [2096958462, 25438691322, 2060446442, -2039072401, -6353705507, 328193503, 1491216933, -292828817, -376812955, 735322653, -268713741, 838993927, 377949040, -284636843, -853973495, -851028698, 38608223], [-259192354287, 28735168293, 2233614769, 415502893, -2192847665, -133521924, 71914566, -252479308, 1123118223, 369437218, 337504169, -259001904, 44036037, -259709051, -424088869, 43537808, 173175567], [292138981284, -41356546459, -1300076161, -723629776, -2094754673, -269638505, 33596096, 264259281, 622632997, 1259090156, 353507743, -621753477, -61349170, -79146337, -118897503, 14985098, 86879821], [-357655245771, 4923876468, 2544389562, 130614539, 4836102504, 687816156, 865693981, -348617577, 332916528, -642423775, -737884275, 692036797, -353535724, -189450882, 698795532, -193301954, -825362191], [-363435654730, -40218548974, -500567747, -1365299855, -1532750915, -161842732, 134455190, -208653659, 538488383, -1449861574, 267401370, -250155370, -126517783, 552936450, 91471954, 752548627, 526282660], [168454995807, -6195105325, 950846013, -404282717, -9378045531, -451550281, -38388532, 207034819, 164178684, 1161335514, 874899705, -558801830, -384001137, -402633905, -426002412, 817678180, -248689388], [455654118533, 23299993215, 4382839543, -1555827499, 267571976, 168679230, -201386725, 519867637, 453637357, -1210601235, 19386190, 108202112, -648382045, 1123210733, 1491496849, 274605102, -530404235], [63916654715, -4840698849, 2662227848, -541133090, 2174001092, 595705582, 701074072, -391342665, -1176743160, -1075008605, 442177036, 524781174, 340151575, 9186632, 87263162, 186832232, -204669823], [-592357070151, 5714892794, 1610124926, 775002405, 3827834613, 21519480, 76633942, 331764072, 292470921, -22006330, -286091442, -514082673, -8043967, -17746616, 427731995, 82950044, 227004883], [160384221583, -54646032518, 96883492, 523952625, 7046198789, -618702828, 159429710, 359462734, 417736679, -55082892, 422071556, -190518873, -423443159, 252457938, -362285479, -131144696, 264197759], [-320959444281, -30343669204, -2199768003, -1163550490, 2182282960, -361595940, -38776853, 497193234, -510478274, 72875337, -463953245, 413925249, 1006104936, 59232232, -257428913, -921457043, 583262717], [105689147941, -37853101991, -1373767913, 912694083, -2017460929, -166578111, -145409894, 515307945, 309967932, -869208454, -197119699, -520827507, -527307351, 343940235, 705218532, 660012840, -467150341], [484124116399, -18953028039, 1473981894, 965515196, -2691556650, -417784870, 2529456932, -185246651, 452620744, 1531949575, 408295364, -114585964, -26622927, -1513316586, -1650110424, -96650194, -149509131], [349840276314, 20699640803, -3377691728, -815758384, 6803895603, 272866989, 133428486, -533167239, -708566785, -1212521389, 49991646, 346845994, -662322531, 478334479, 757063827, -349443462, 659610507], [-253217215389, -16293433574, 333589200, 830632468, 5341981364, -262995523, -1587267974, -8851204, -588995799, 440033477, -367874671, 96850924, 146869893, -156382448, -445705204, -586168227, -724531500], [-228584043067, -20100661836, 5188447903, 2188074402, -1791574472, -19729444, -214749136, -21983523, 311509367, 847911202, 821673838, -65144594, 527459225, -1203509790, -800469642, 545930215, 715204510], [248373396197, 17989109780, -2115949507, -248459718, -700349410, 58851829, 766473652, -303782820, -507581725, 1360427511, -468170575, -211288826, 62351043, -985410727, 836910391, -534282600, -684370957], [261486535315, 9907448934, -2955361106, 1232893939, 2999056992, -314590792, -390045231, 170790629, 181224362, -795836339, 533807677, -447358687, -445189524, 292708958, 43975670, 301448870, 314162753], [-402467128094, 32788311267, -1831583278, -138886082, -2098841798, -1220938198, -657332081, -77736123, -61550574, 640932307, 482687474, 145646544, 295091146, -236549638, 525366827, -455029817, -158297898], [-454031836119, 5647248430, 3125412305, -681792090, 5094460766, -219869676, 1303102099, -324505616, 644421438, 312673867, -36116593, 176508211, 334956610, -31952728, 326891859, -556477635, -357519123], [136946303302, 23070306695, -723218403, -319869680, -7571459645, -365106951, -1225657745, 262291920, 959524826, 291189987, -60035019, -240884379, -460969760, 1152541954, 30334341, 642629563, -227006890], [-104448322924, 42077568736, 2954286345, 1811242839, 15242053513, -482525426, -1106229468, 670934759, 29372880, 740593876, 801460442, -924764540, 94282646, -690131214, -445991683, 678606259, -63486844]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_71_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_71_8_a_a();".
// 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_71_8_a_a(:prec:=17)
    chi := MakeCharacter_71_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_71_8_a_a();".
// 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_71_8_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_71_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<2,R![-68160772227465216, -83871834606993408, -28453158005047296, 1174059402170368, 1826833848342528, 85522695963648, -43215664975872, -2859910005504, 510044329216, 36303285888, -3295109632, -233835792, 11778968, 807220, -21712, -1421, 16, 1]>],Snew);
    return Vf;
end function;