
// Make newform 171.10.a.f in Magma, downloaded from the LMFDB on 01 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_171_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_171_10_a_f();".
// 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_171_10_a_f();".
// 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 := [-16080668672, 6645763712, -758043152, -1801192, 3186388, -1330, -3356, -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, 1, 0, 0, 0, 0, 0, 0], [-839, -2, 1, 0, 0, 0, 0, 0], [-393975956593088, 84408693326008, -1013778819240, -437043316606, 754417652, 500071971, 155494, -155435], [2773878385421312, -1030920702682528, 111871348958616, -1936993853096, -218132022350, 5717983110, 89863001, -2442607], [138044266497152, -66804468512032, 11505217469640, -575295885096, -24198670538, 1092447946, 10482859, -420885], [-3468382549222912, 1192076208860480, -125928181294728, 2507333239456, 249515611234, -7003343598, -103632415, 2957621], [78261663501824, -485451580589632, 132548764856328, -7542983885792, -277519681442, 13443010254, 120563999, -5088661]];
    Rf_basisdens := [1, 1, 1, 22333949760, 89335799040, 14889299840, 89335799040, 89335799040];
    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_171_a();"
function MakeCharacter_171_a()
    N := 171;
    order := 1;
    char_gens := [20, 154];
    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_171_a_Hecke( : Kf := Kf)
    return MakeCharacter_171_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 := [[-2, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-489, 16, 0, -1, 1, 0, 1, -1], [-887, -38, -6, 2, 1, 5, -1, -4], [-21605, -56, -28, 5, -5, -20, 23, 1], [13602, 148, -82, -15, -64, 43, 28, 39], [-72468, -3549, 182, -87, 62, -221, 35, -9], [130321, 0, 0, 0, 0, 0, 0, 0], [287842, -9492, -314, 391, -396, -781, -498, 1117], [-561151, -61375, -2604, -1659, -1856, 120, 8, -2015], [-1042529, 67915, 3050, -1106, 4156, 3675, -182, -2128], [3076577, -94009, -1406, 2394, -1896, -4685, -11758, 5980], [-4399197, 180473, -2198, 12796, -7084, -49, -14882, 9366], [4592919, -133178, -5240, -9229, 5111, -34, 8397, -20421], [-1174005, -405470, 16008, 20209, 19705, -1464, -12159, 25849], [-1782299, 226687, 33424, -2513, -8286, -6718, 38034, -16737], [-2501813, -1039731, -510, 2376, 71617, -12117, -33394, 24554], [22915517, -2781624, 84728, -6431, 3813, -5048, 37271, -15055], [534604, -2718928, -18328, -3342, -265133, -27872, -30558, 42194], [-34198728, -369562, -141096, 176408, -53174, 97392, 31126, 119176], [-32999138, 73739, -74682, -67629, 117496, 218499, 25825, -112967], [92868029, 779517, -15786, 131098, -26058, 196101, 200814, 39136], [-102981500, -6897812, 173228, 19598, 295672, -56726, -302692, 63202], [36337581, 1350405, -126378, 434548, -160808, -427887, -64512, 175466], [-436309190, 4425006, -524540, -49482, -523154, 88938, 178126, 490882], [52028104, 18150630, -342604, 575736, -774560, 43378, -851364, 649404], [-578239126, -3640240, -65664, -14998, -2164032, -178240, 16510, 208258], [218912869, 22810403, 1414662, -693212, -163821, -410343, -117710, 353034], [-147108053, 9728985, 1005604, 230063, 131578, -717600, -433826, 561675], [-424550355, -817057, -52250, -464932, 68010, -655687, -233102, -584494], [-343950085, -13824707, 1698150, 199276, 2304660, 1045357, 325220, 1024346], [-220365211, -32315132, 2797692, -1459049, -3885993, -389796, 1146553, -1282621], [-777950890, 35408349, 2035626, 557241, 6631910, 4498257, 1688047, -1355405], [-1796651481, -28571198, -2347784, -314089, -1722289, 563162, -2454711, 1802679], [-1980626361, -190519078, -1073448, 1628519, -864211, -1226772, 464199, 663887], [1423648996, 91431794, 8899332, 1689330, 13017028, -575466, 886690, 2042854], [2503216192, 22668758, -1667872, -145214, -6465830, -1093304, 5348844, -860502], [-3986618084, 107896756, -2668860, -4089394, -1310906, 4315502, -1173112, -5324158], [3818045269, 64991819, 6159934, -2682566, 8328776, -1669363, -3529316, 2798296], [-5831341742, 1267772, 1806780, 834930, 7746106, -1622262, 4951232, -6877098], [-6744126809, -76652427, -4610726, -1594290, -17313896, -3551893, 4958260, -331568], [1103521278, 51698792, -12237716, -11845246, 1951670, 4734642, 12285764, -14636474], [-1225273783, 106605716, 11902402, -8931098, 15183087, -644119, 10813881, -7457064], [-678223373, -231601597, 14259802, -442896, 14778406, 4674263, 815444, -5530878], [-2816484316, 286725190, -8039980, 10119516, -28683282, -8836346, -21693476, 15524216], [-4917531033, 33626092, -24392226, 6660978, -10578703, 11545335, -10367425, 3885272], [3182148829, -658217305, 23092470, -9856636, 12159199, -9830655, 12590446, -1018062], [-16549689311, 178040579, 9311686, 11077574, -33287486, -24422719, -6070612, 17362508], [26860817413, -13991019, 20083018, -2217594, -43986345, -36061885, 5486768, 14702640], [3132702389, 787832918, 8222292, 2642723, 67572735, 28043462, -1171691, -6251633], [-11931123155, -48820974, 12581308, 21905429, 5599053, 16767308, 1745881, 25208681], [-6956049483, 792839908, 28622186, -22545266, -17866345, 3203341, 214605, -3637728], [-1443975775, 1116533343, -6267042, 17486708, -31420500, -16686771, 23789454, -4167886], [41124364527, 574153390, -16546564, 4472165, 48521747, 5773084, 6189277, -15589727], [-9890218114, 407128094, 35149192, 6559196, -37544058, -34655764, 4970350, 6283660], [-4183072827, -421671376, -2512612, -19014135, 113014677, 74274766, 9187009, 911189], [34728359789, -875526735, -67295490, -10432232, -12117038, 43371285, -288036, -37939130], [28212646068, 2972034770, -45640338, -45345815, -9586384, 7913983, 15962974, -31213769], [23068297533, -1300317506, -61270892, -65421869, -1787875, 39429050, 14126001, -20437489], [14600982616, -1019297386, -187929036, 4771220, -2674014, 41147118, 16523852, -30761048], [-57140086501, 1144609126, 81418008, 53327151, 98854533, -34986254, -35159323, 2207479], [-13864546224, 733819850, -28603750, -36346739, 30438156, -90301283, -51233644, -1918137], [-87745088783, -677144367, 3918698, -48288840, -39033266, -85519489, 4346354, 44138154], [-9867437169, 1795190608, 36536602, 9599144, 149753177, -38213047, 41424947, -781942], [82836408175, 899125965, 121114802, 13318106, -130007021, 43902107, -86273908, 99043460], [-42248746576, 292358264, -32364418, -27786281, 300854158, 74146195, 3576850, -69860859], [77546066806, -3728372174, -163388652, -18515854, 39669649, 49112978, 29408138, 6049118], [-26801725408, 2167042512, -80185492, 65938072, -6045774, 158793730, -16106386, -84760804], [-163420006529, -2475470674, 217444964, -26770727, 89625555, -159978824, 97995341, -27901595], [19109614415, -5106258604, 43271064, 161057531, -1110969, 61770452, -169653539, 41882083], [47407174141, -2689804237, 144550686, 50926690, 121684325, 54979341, -65550544, -74188192], [-106055679521, 5529640434, -123181302, -121088988, -57383817, -101681835, 15046629, -87366194], [24185986128, -4028384312, 142575640, -80524260, -5557768, -125629276, 256302024, -44451908], [-133435284504, -233331170, 70025134, 70507807, -417711268, -8268433, -104452980, -110794435], [-75964962545, 3353637241, -197212606, 185235976, 422660107, -66063237, -322049438, 67913666], [-136648707561, -10652292129, -164872138, -352421328, -167077296, 128520001, 78540094, -297787658], [-132594912147, -696030314, 80983624, 102390381, 204765865, 177383204, 242927293, -219568771], [-12255481139, 10600317674, 1992956, 86360639, -410201723, 39158782, -19712931, 294140851], [-121662323831, -10069747699, 89709474, 146113710, -61591242, 70060047, -153462912, 412339928], [-108461727711, 8048801705, 452240598, -81124228, 479245952, -59238911, -150522020, -87358918], [-105679157040, 10224941280, 404787668, -13294910, -426873254, -596991338, -61170200, 397721742], [-57634361789, -7037480379, -153930584, 116190333, -107665226, 230811810, -290851406, 114558085], [-349839501474, -13669368142, 629861680, -15999306, 959702220, -253566772, -201092008, 372154998], [-112447620741, -16818200039, 578319270, 143201202, -244458112, -417751683, -286798174, 244330712], [-86711120001, -2202704015, 447586338, -571164062, -279635284, 241548075, 268410332, 121631196], [-26592433267, 17749303136, -286022044, 418536331, 310930165, 136351276, -264030935, 342781335], [186235969052, 8804205116, 147308436, -660516592, -538200668, -244256274, -299388050, -15100332], [-201899692070, -6665199167, 77719786, -210486939, 126969694, -207476123, 196440099, 345065279], [23330460797, -12274374660, -391850668, -427526845, -418834239, 544086970, 429589839, -424872337], [-132288322323, -12019789038, -36936520, 17043295, -1183576837, 90128832, -360898837, 472155063], [-303166421281, 4260735910, 152398364, -624978975, -256608597, 384695344, 1010357365, -514190131], [-689441955284, -2495090744, 4829800, -88005104, -359404732, -215788636, 323739988, -670508896], [235786819510, 6722849192, 1141979688, -736148118, 466991236, -562631004, 24699106, -774124934], [-872917674122, -6452041910, -1589405764, -584829656, -1262354860, 268280182, -702488964, 71320420], [-312692044417, 8720582172, 594294724, -429703167, 2130856317, 1216677536, 670851811, -460370875], [89659116596, 20533478010, 607488362, -74591057, 807208896, 915169993, 520565822, -242743151], [-896827893940, 1876765820, 493719172, 815917964, -307786716, -419672434, -307429358, 323635576], [-11046457805, 30667929955, -727503938, -616252350, 1086514274, -597174199, -239751720, -1016572784], [466311454925, 23899753815, -39238642, 334389404, 901967589, 251727733, 1093409258, -515404110], [-103588655549, -46094278678, 31736160, 257373771, -2556926129, 565903812, 238945747, 1237961515], [390424064604, -26212136960, -1160782968, -850454572, 1108192892, -219080196, 1227144176, -1654303172], [-933044384973, 19207028450, -1831547980, -212871331, -3614338989, -7008410, -788242449, 1293322753], [686182687225, -25639167145, -1557500082, -1057294082, 181428448, 397422537, 1081428264, -1685746268], [-260750741826, -37861599886, -1325877788, -1271501062, -749123192, 530438522, 987890550, -1208161146], [89568917644, -17881559144, -2341427816, 453092280, -1655810672, -774708512, -1784633136, 831506936], [60598961330, -39764566383, 806478914, 314269241, -1385894964, 73315117, -1095084721, -37715181], [-1490580743665, -18757501906, -2783799436, -979084231, -762337951, 849571732, -511991159, -1699949035], [-1059602214524, 12983407514, 1809732700, -784478276, -921478904, -2108026546, -1116434464, 348843912], [102385604135, -47430714249, 1587483478, 1186895780, -2890762322, 198630425, -663384798, 256298914], [-1467793090774, -29236022140, 253110356, 1453947790, 2201758752, 1126754442, -807597084, 2351239474], [-1878777458436, 1939437312, -4355754844, 752882362, -785943172, 1507034346, -964963900, 569698854], [1571105560955, 7486543960, -960368060, 1409515229, 112154613, -55693326, 1245510177, 754171649], [2261446043145, 22284253864, 1197545336, 1686267863, 2507427291, -206638490, 475654323, -790400137], [-487217866780, -73681980760, 1790870392, 496426500, -2761570242, -2693127968, -715636276, 970411900], [424782968177, 31933424640, 4325149812, 211422957, 1076925105, -1985340186, -863401995, 1035831777], [310522799656, 11807854216, 2217753796, -1548214492, 915447408, -2082902698, 471656014, -301828688], [-2416883899155, 63531639700, 804631456, 630008865, -866256541, -767220526, -1060693311, -696302543], [-576867176027, -59513323152, -1446960398, -2488210050, 6388243159, 833374133, -707818751, -1045291304], [-2477238850155, -56571071200, 2466330272, -71303207, 5399074779, 1176185248, 546961943, 844490961], [661166808485, -117521667097, 1771624318, -2252494312, 1156785283, -2813277259, 769204230, -2595915442], [-319044133382, -86638649652, 3656257098, -1247353381, 2248144888, -264470303, -188979732, 1478867749], [-413402508340, 82519819454, 3172342076, -3624014744, 1513316094, -875276694, 82994100, -2095858340], [804212564839, -71372038237, 1974234356, 2233004387, 272232044, -5028838724, -2292245072, 4711707295], [-610970513246, -19758416442, 4385811652, 1248533844, 4498634272, -1775355394, -1364393084, 4356944584], [-161956991757, -21320397080, -1181885768, -415074341, -1294472815, -876169750, 2434160115, -1408598869], [-3617783471882, 20826554724, -4344043784, 963117776, -5468804214, 1451553488, -105790196, 4371807320], [2432882354328, 2147415178, -8125139968, -385277606, 3267764726, 5420916832, 2417544496, -4197948950], [-1850851310429, 150586625362, -1278626722, 2437757398, -1418711249, -2174713601, -2052432431, -556880796], [1035564748491, 114843328618, -1833334454, 1907625428, 1446208751, 739053689, 764859513, -2268020778], [4284984951180, 85825882294, -342756584, -1140200426, -5309350348, -2273710468, -1634417892, 2724509894], [2048772064997, 64682127060, -562459520, 698831545, 3092376441, -1448059618, -3553989755, 5284072017], [-1517787271063, -63641138359, 7313925046, 159403506, 1860276580, 3136983965, 3123697234, -5092633248], [-1435622720618, 168711398846, 5546977092, -2928801404, 8330991208, 7146230502, -389211692, -3315883888], [-1996795723737, 66367269298, -5331404828, -1486698047, 8944270709, 1061460974, 2088269595, -232900099], [-4216036852670, 151398448173, -6782183830, 468420809, -5699955166, -3481076243, -6716495837, 2822049459], [-4151151807562, 123910389753, 2530566042, 1574726017, -8080680954, 2981141181, 3263090199, -3610996133], [-3874016717572, 24235004536, -4484734090, 314490991, 650329474, 554218951, 2742410230, -4941764075], 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36543735070, 527018501380, 921913058992, 492171771088, -2906745258682], [-362571402475116, -28604810262564, -654755340056, -244460485212, -4137459746498, -399559788676, -428867650604, 2631190481508], [-1086219888329581, -32815540969377, -1420800145558, -849324254494, -1786983745800, -3238710953453, -21891508654, 1184339851668], [-381494289888859, -46916659494325, -2748974853794, 537976400042, -2876879551909, 1926654034985, -827910649924, 963306929880], [-974571109843878, -83158168942240, 290416969392, 1806401302630, -5650012935890, -1552238248920, -2525398193838, 3700024278414], [806130600188679, -10303641557493, -4538760001302, -322670053708, -476426774150, 386601415887, -209028519728, -849518955186], [2346832087617501, 18863405584309, 1774509481188, 282079749909, 2096771513504, -24844690968, -1187171078116, 1267007599601], [-1321441644933253, -11999336423900, -4586936559306, -939886636030, -1720795146667, 1862849909643, 2505517618131, -2645571021880], [-2025634925515996, -63427223979862, 4463105240396, -612916728224, -243671550939, -419130611438, 458975360636, 482972610788], [-86448333181372, 57097430462420, 1936010013488, -742314001166, 1306846449762, -3042308420580, -668617476858, -337312006310], [-2009387789245012, 21509646912906, -711472623414, -1194168826231, 3515806183212, 3221031669477, -227090284048, -1824176175173], [-956300558850486, -15393063140428, -888189709656, 68290408116, -2086744442885, 2671502331756, 1564976528336, 907060451132], [-334835237485764, 84709258445445, 1298164225322, 465327845619, -165017379916, -511218176999, -502781465275, 1326861199317], [-1940417437022733, -68796899768766, -905940224052, 1893663787853, -4420379445271, -1093701687940, -1801909082943, 2369369378601], [-362059896382031, -26336369813862, 1003264780174, -2097459335656, -3876854968459, -400515829105, 792047118167, -3833261410610]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_171_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_171_10_a_f();".
// 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_171_10_a_f(:prec:=8)
    chi := MakeCharacter_171_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_171_10_a_f();".
// 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_171_10_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_171_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,10,sign)));
    Vf := Kernel([<2,R![-5784998400, 3693191040, -693287712, 23100984, 2972568, -41238, -3258, 15, 1]>],Snew);
    return Vf;
end function;