
// Make newform 1083.4.a.s in Magma, downloaded from the LMFDB on 10 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1083_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_1083_4_a_s();".
// 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_1083_4_a_s();".
// 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 := [-49519296, 62731584, 157450608, -101642904, -174155400, 53300700, 83975016, -10997190, -18914886, 1134069, 2264487, -62307, -153482, 1731, 5904, -19, -120, 0, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-13, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [94530880531366662033440640, -185365873461562580091461184, -144912373651251796633203600, 271104798748890857455451880, 86980910391913607574356160, -138392093127307278046801764, -25570291536860751514126548, 31592082084162345576996066, 3842491917734064653108736, -3786526904907536036476407, -315240186501889998402156, 256031797374478212599573, 14319176607925840100007, -9832706509469680329582, -337893096504576417590, 200103027669781441191, 3227886941243287071, -1675795684475211901], [66620058876368290708926336, -153128892194898504009739776, -78703002673014581526292992, 242681175574389196849187640, 37498179154705131413748864, -133387621589081099627508804, -10775290901300570761064136, 31961857385100170674372506, 1639826014956842258908764, -3971250545703801807195933, -138089685959601448311944, 276209835287657570052847, 6556063518166108034625, -10837901369501368602298, -164137564449233853898, 223922407083267790569, 1675795684475211901, -1892919116870925647], [-69004634313298394085894720, 135642005657411079245304960, 78703002673014581526292992, -241886317095412495723531512, -37498179154705131413748864, 133387621589081099627508804, 10775290901300570761064136, -31961857385100170674372506, -1639826014956842258908764, 3971250545703801807195933, 138089685959601448311944, -276209835287657570052847, -6556063518166108034625, 10837901369501368602298, 164137564449233853898, -223922407083267790569, -1675795684475211901, 1892919116870925647], [982050314858458579680464869056, -2029006711518015404487588310080, -1275032107058504193848559221424, 3110033917976577367438798444224, 704236198685078716234324080456, -1644916582637072189257070713344, -222749660236838849561756014272, 377681113667367714538800958128, 35339447639199871496464195650, -45061096815576576615820684938, -2981862361403060777875865879, 3022825815247970924083040647, 137256097422207250494821472, -115042428711603457835384707, -3258109593242800347243115, 2318995939796269278498129, 31200867242791706027464, -19231443507404990330747], [924649547343861848026562975808, -1960343571472638313507603786944, -1372475665923588302023769629680, 3013810313379450097853697815952, 777138412537674857311795595952, -1628019425698769708064402460512, -211289935160763680009109276384, 391809115466684616942086123376, 29516052173115675221047213116, -49160400479964876409293115056, -2303891164239905770423789149, 3453714707317585027558786027, 102791358388856183621726478, -136716309765226184397638859, -2452187645228474523584557, 2845430807939770406433189, 24154193263578028480800, -24196682468341223646005], [208907599493669922685810865856, -490682536366395169687955840160, -470605130751408957276587617632, 871124397036591988952987980056, 355461070378064407626428311728, -522043362078996239884688100348, -109342729963399567739004313764, 131400200719466910186977245194, 16071796957611591113877535536, -16825917606913217482494256847, -1272853763731167330812833953, 1194183901832106996501976720, 56236341159768335782888469, -47556047360069745766712949, -1307917052060285790227299, 994177095802194157518676, 12465150992505306364947, -8488754619667774637414], [-569773091541529426928285140704, 1058695943675968701256948636896, 956819040968418565098226928760, -1806136925202578794518342102456, -583859369367079916901724068612, 1045204396992949734669156580200, 165012785998527603108041318250, -259344318718154323194036333420, -23474981973155693112268150893, 33019572237364293203583356685, 1842982451491297408766055022, -2337708928365814430270226899, -81940333525769741751105099, 92957295014023769829239597, 1936044803159730341480498, -1940491507000960398668025, -18833076302305641721190, 16538948590780380835918], [-131573681088342341719788288, 283846797052614532634017824, 201335791929830167318376880, -457705115263768003469917608, -117387395961774826279269216, 255004905975568566615492468, 32815315729995927403716912, -62014062756387096847013886, -4665435786905225924686280, 7795846746588849505980581, 367977510942695538487243, -546885810137594540013316, -16498103219855751256223, 21591645914765437051055, 393880866260662450597, -448105114662113121808, -3872833206562461641, 3800457820555636626], [188460948577935021593220672, -369982420096985187397183872, -314537597472225351956750784, 587625361340615967553778784, 204165136658227115385762792, -319954458117864647390707416, -64138600312916147943043548, 74609672469632168950797852, 9871163667231226591200750, -8975417187004936197497474, -810240582107722410320281, 604772636368055589743425, 36416057076925680618416, -23076695707237558255829, -846257057795041588849, 466089564851219546491, 7949823804563273312, -3872833206562461641], [603402087203665883966600112096, -1176697552199808468780660823488, -564765286386419651660940277488, 1958120725466507300099285359392, 169065169059776416454922620868, -1122203182874445873034367669340, -23896961949801233126885574420, 281984095482478272731713803156, 1622604623156794494040402881, -36460416315909579206216234265, -86312258147054271832765530, 2615086720124751061045403761, 6185262327292678084849503, -104957778017315895117928329, -276028260438392440951420, 2203343708720545410344487, 4331886349604524862070, -18824539178091576947054], [-1596320181163845991380176323776, 3887036542406476904106170816640, 1839030650423213989515158492928, -6269378643365735903770234306584, -830137027699214879676652032312, 3501175906289145087450621339684, 229200340192242414675355614564, -850067958003576402887335884750, -34199681212436685282461032818, 106646292097288495858194789813, 2875806030718395341529347069, -7467258501540132854223394438, -138476867905850213630388291, 294273449524313197086073255, 3548274070305653353194835, -6095448275012375770115922, -37171341563397819153991, 51588236747326861525694], [686639740073727305099904945216, -1461660199738556774738572748640, -1205676915794068188469517599776, 2411501496187892280789221975424, 800065328779796421341470338744, -1347588693985676868931090124592, -243866334244677162940684642644, 323790085462393380922481427660, 36554295524571152684084595042, -40105017022101384986489015374, -2958297521658583056375659049, 2774844892378549368375702757, 132723624486621948742770316, -108316995503149579078567353, -3109632257643411536348761, 2228920535069829711192647, 29652258821077861377552, -18795114767797047536033], [-1056923247560579439365596436160, 2168109507153305606835933681120, 1628198970356866607305601320032, -3379482712015839128022632253744, -1027427747986939586562708453384, 1798979919600187958263467912768, 325322534397576972759078217884, -415164072618716486812582189056, -50552761287066806670936766002, 49718910096980086637284491494, 4181127499980924537546850345, -3344261891026248207803516469, -189099186863374953659250680, 127550896196703634867376225, 4418767630274235268905585, -2576473060035508995022819, -41722247627413976377700, 21414967151085380832609], [1143307505726713290468194562528, -2688866940781705356662439953520, -1744157967185248644378251187792, 4481179721497436891540304135240, 1102723733847734286528932469720, -2550113280054890114687094094548, -358523378218061921062893432096, 616292481153606899210392481766, 56888702174415174915894932256, -76417944345395770688567839320, -4801470828009330228754311783, 5283551414866684611317573917, 222242471499441115342658364, -205960681856729213542370781, -5330635250249708795627611, 4231538740925880861197973, 51742121635327035520350, -35625929264464381764455], [7254395403484733137199219109696, -14727258258860671858107952384032, -11056059054598569589802785327200, 23200124755285379068919394687792, 6423405347261024447651960867016, -12718674735276346398926569655544, -1823581668871000094636580429348, 3053608207102500754821475714716, 263682408697684875698814039150, -380010931804535550740535423750, -21008237882825181498311925997, 26446722358324193867885825633, 942335946485991836297625660, -1037771064845475744820410977, -22322339474495944037695325, 21439875984292368469933923, 216667807383614616395420, -181251166910180823313681]];
    Rf_basisdens := [1, 1, 1, 794858478976701125656128, 794858478976701125656128, 794858478976701125656128, 2418754351526101525371597504, 2418754351526101525371597504, 806251450508700508457199168, 1209377175763050762685798752, 264952826325567041885376, 264952826325567041885376, 1209377175763050762685798752, 2418754351526101525371597504, 806251450508700508457199168, 806251450508700508457199168, 1209377175763050762685798752, 2418754351526101525371597504];
    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_1083_a();"
function MakeCharacter_1083_a()
    N := 1083;
    order := 1;
    char_gens := [362, 724];
    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_1083_a_Hecke(Kf)
    return MakeCharacter_1083_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 := 4;
    raw_aps := [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [6, -1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [-2, -1, 0, -1, 0, 0, 0, -1, 0, -1, -1, 0, 0, 0, 0, 1, 0, 0], [17, 1, 1, 0, 1, 0, 1, -1, 1, -1, 3, 0, 0, 1, 1, 0, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, -11, 1, 0, 2, 1, -2, -1, 1, 0, -1, 0, 0, 0, 1, 0, -1, -1], [5, 2, 12, 1, 3, 0, 1, 1, 0, 2, 0, 3, 1, 1, 0, 0, 3, 1], [-58, 2, -3, 8, 14, 0, -1, 1, -2, -1, 4, 1, -1, 3, 1, -1, 0, 1], [1, -3, 12, 0, -4, -4, -3, 9, -1, 5, -5, 0, 0, 5, 0, 0, 0, 1], [5, 7, 0, -5, 2, -3, -1, 2, -2, 0, 1, -4, 0, 3, 1, 1, 3, 1], [28, 3, -1, -13, -14, 3, 2, 5, 3, -1, 6, 0, 0, -2, 2, 1, 0, 2], [42, -39, 6, 4, 26, -4, 1, 0, 0, -1, -15, 1, 3, 3, 3, 3, -1, 5], [-5, 12, 0, -8, -12, -7, 0, 0, -4, -4, 6, 5, 3, 0, 1, 3, 3, -1], [32, -11, -14, 13, -10, -3, -3, -8, -5, -12, 10, -1, -3, 1, -1, 0, 0, 1], [74, 20, 7, 17, -5, 5, -4, -3, 0, -4, 8, 3, 1, 8, 4, 0, 0, 5], [-168, 13, 2, 0, 22, 5, 11, -5, -6, -2, 5, 0, 0, 1, 11, 0, -2, 4], [88, -1, -5, 0, 15, -3, -13, -11, -6, -15, -32, 0, 0, -5, -3, 0, -2, 1], [86, 61, -21, -2, 32, 1, 2, 22, -5, 9, -4, -2, 2, 2, -5, 0, 1, -3], [-159, 16, 2, 17, -28, -5, -12, 13, -6, 4, -26, -9, 1, -2, 4, 0, -2, -14], [260, -5, -18, 16, -13, 1, -2, 5, -6, 0, -23, -11, -1, -6, -7, -1, -4, -16], [52, 37, 12, -5, 24, -8, -3, 19, 7, 4, 0, 4, 0, 3, 14, 9, 3, 16], [92, 13, 5, 9, 20, 0, -14, 38, -2, 19, -4, -14, 2, 10, -11, 1, 0, -5], [135, -2, -72, -31, -30, 4, 23, -6, -1, -5, -6, 0, -8, -3, 17, -9, 0, -3], [-121, 22, 0, -26, -65, 4, 4, 0, 16, 16, 80, 6, 2, 10, -3, 0, 0, 16], [19, 43, 27, 35, 54, -14, -19, 15, 2, -7, 17, 5, -9, -5, -10, -12, -2, 1], [53, 16, -6, 20, 28, 9, -7, 13, 7, -6, -48, -15, -1, -1, 18, -1, 0, -5], [47, 83, 27, 27, -76, 4, -5, 2, -15, 13, -26, 2, -2, -9, 6, -5, 3, -5], [-373, 130, -16, 10, -47, -2, 24, 29, -4, -7, 3, -12, 0, -10, 8, -22, -2, -4], [120, 19, -81, -51, 88, 11, 5, 21, -5, -31, -58, -3, 7, 17, 10, -2, 8, 0], [268, -4, -35, -16, -149, 24, 22, 22, -13, -14, 12, -9, 9, 0, 10, -3, 19, 1], [-9, -40, 3, 15, 62, -2, 0, 8, 18, 1, -29, 18, 22, 22, 1, 23, 1, 22], [166, 102, 32, -25, 121, 36, 11, 20, -4, -8, 18, -7, -1, -21, 8, -4, 9, -25], [-484, -61, -25, -14, -14, 8, 24, 3, 30, -20, 28, 28, 0, 14, -13, 2, -1, 8], [1043, -17, -47, -14, -17, -26, -9, 21, 13, 12, 15, -12, 0, 15, -16, -22, 2, 8], [762, -56, -65, 26, 61, 19, 31, -26, 0, 9, -23, 10, -2, 13, 0, -24, 1, -10], [895, -182, -71, 39, -248, 28, 41, 14, 4, 22, 39, 3, 9, 9, 22, 2, -3, -8], [528, -66, -58, -17, -12, 10, 41, -57, 51, 21, -18, -8, 0, -5, -31, 9, 4, 3], [244, 32, 63, -9, -137, -6, -36, 17, -16, -1, 71, 2, 22, 32, -9, 19, -5, 12], [-524, -233, -16, -105, 226, -9, -15, 54, 8, 18, 41, -34, -2, -27, -15, -1, -25, -11], [-58, -106, 15, 19, -311, 58, 42, 34, -10, 25, 19, -4, -12, 0, 4, 17, -1, -16], [213, 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-1225, -459, -67, 386, -148, -1486, -286], [33637, -8764, 1714, -2692, 4258, -642, -256, 1675, 856, 308, -3624, 354, 1054, 24, -2536, 734, -286, -1184], [37973, 4913, -3423, -1163, -9855, 1451, 1539, 1536, 1470, 2340, -536, 120, -768, -1275, 88, -573, -271, -608], [7810, 11050, 6361, -2088, -7046, -352, 311, 620, 368, 1621, 531, -183, -489, 797, -679, -585, 170, -377], [17203, 838, 1708, -310, -840, -337, -1280, -564, 263, -814, 2409, 632, -484, -648, 919, 890, -922, 1053], [9586, 205, -926, 2016, -920, -114, 245, -2824, 910, 1263, 1131, -183, 463, -221, 255, -917, 1519, -37], [5832, 3016, -6532, 2344, 5225, -2467, -332, -1462, -313, 995, -1402, -4, -24, -806, -494, -1444, -660, 835], [596, 9093, 1203, 2139, 2277, -838, 217, 3995, -511, 3270, 3700, 62, 1054, 2505, -1057, 383, 307, 1507], [7712, -3617, 2653, -2628, 1586, -1905, 749, -93, 854, 703, 2926, 818, 554, -421, 711, 454, 1600, -45], [19608, 2900, -130, 1803, 12440, -567, -2682, -3148, 52, -1109, -4274, 845, -217, 262, 699, 656, 930, 54], [11166, 3385, 2122, -863, 1509, 7, -1851, 5791, -556, 91, 4604, 774, -234, 2035, -1585, 485, 597, -879], [-37744, 10151, -2197, -1087, -6378, -663, -889, 83, -109, -59, -4158, 179, 45, 2235, -502, 976, -552, 814], [19510, 94, 4501, -1132, 5921, -130, -534, 2116, 451, 574, -1100, 209, 1399, -680, -1176, 783, 1283, -947], [-28572, -5980, -2948, 1973, -8846, -2667, -361, -2004, -1343, -2535, 2876, 27, -739, 159, 644, -1312, 185, -540], [-7035, 7192, 2435, -2446, 8743, 308, -511, 485, -998, -162, 4684, 55, 625, -365, 385, 763, -850, -561], [41599, -335, 1208, 3340, 3123, 343, -1382, 3080, 1135, 1966, 628, -1678, -118, -322, 748, -866, -5, -1014], [27453, 8522, -523, -399, 3639, -258, 130, 1018, -2233, 11, 3004, -1945, 81, -232, -858, 876, -204, -1062], [33177, 16120, 2133, 3080, 2400, -25, -921, 512, 19, 3563, -3080, -117, -743, -165, 175, -531, -514, 668], [14867, -2872, 2064, 1388, 3586, 1127, 47, 844, -296, -1110, -2195, -2172, 572, -787, 1087, -1052, 89, -1345], [9480, 1844, -2374, 2114, 850, -224, 233, 2157, -1671, 1782, 526, -1908, 988, -427, -666, 30, 889, -1524], [59901, -6270, -1009, 1301, -4133, 111, 1246, 1653, -913, 413, -3692, -1558, 322, 556, 117, 157, 1581, -1711], [34988, -3172, 726, 660, 7721, 4, 2655, 771, -621, -777, 2051, 124, 200, -887, 1230, -260, 871, 96], [4454, -6547, -3797, 281, -5767, -132, -367, 3350, -1467, -1122, -5577, -618, 286, 1253, 519, -483, 833, -25], [36650, -10696, -2074, 2701, -13400, -2470, -1004, -564, -1482, 2443, 1562, -270, -858, 278, -341, -1091, -649, 1749], [24652, -7944, -2123, -73, 72, -149, -908, -2677, -833, -604, 536, 870, -890, 1072, 1987, -1077, -979, 1805], [-2251, -378, -2829, 1792, 6946, -599, -1268, -1750, 273, 1113, 2868, 1288, -608, 894, -2388, 832, 978, 822], [-43098, 466, 1914, -1208, -4694, -572, 656, -2142, 1224, -582, -2996, -1848, 120, -1808, -36, 36, 578, -1580], [23552, 5545, -1752, -2422, 4261, 828, -698, 3856, -716, 953, -787, 1028, -604, -1318, -245, 122, -555, -1389], [-42659, 10610, -4776, 1379, 3638, -243, 429, 628, 36, -1523, -5939, 418, 318, 2753, 1436, -585, 618, 1226], [1328, -10402, 3468, 79, -12921, 1386, 1048, 2199, 477, 2644, 278, -1363, 403, -2024, 424, -256, -296, -745], [12821, 2937, -1449, 4062, 1485, -155, 165, -2260, -1092, 846, 770, 110, 194, 553, -1026, -188, 2010, 789], [53808, 17715, -1106, -542, -1329, -1780, -1342, -1645, -106, -1221, -3802, 1662, 26, 1944, 772, 24, -421, 2015], [67554, -7194, 1582, 145, 3395, 418, 2307, -1963, 849, 1480, -2935, 1584, 536, -49, -1638, 187, 818, -767], [-9542, 3151, -290, -598, 7807, 214, 2246, -2242, 757, 802, 1436, 955, -683, -238, -1659, -1125, 166, -7], [-17094, 5005, 2159, -1199, 14317, -628, -959, 2044, -1503, 154, 711, 1086, -250, 2301, -563, 141, 817, -577], [13714, 15071, -3026, 1219, -12028, -908, -1339, 4215, -1623, -666, -2626, -1538, 14, 183, -1616, -609, -109, -2172], [-17817, 22332, -324, -931, 7543, 323, 1570, -1093, 1127, 629, 3976, 532, -172, -320, -1376, -553, 901, 494], [-2040, -92, -5388, -2700, 10199, 2724, 84, 467, -91, -323, 5784, 1089, -477, 954, -135, 251, 1311, -548], [-16149, 4886, 1111, 3105, 14083, -858, 501, 709, 707, 1973, -3859, 1264, 368, -933, 681, 19, 970, -382], [-30377, 9921, -792, -1408, 4395, 2471, 2064, -2042, 10, 268, -979, -283, -257, 1108, -1444, 351, -416, -850], [5793, -9855, -2553, -570, -12096, 333, -294, -588, -262, 1830, -2976, 37, -1193, -1628, 674, -359, 166, -307], [-59215, -7397, 2526, 4684, 4098, 957, 1234, 3891, 2611, 2836, 3783, -388, 1204, 1768, -707, -332, -779, 526], [21750, 14888, -1856, -3445, 2521, 1057, 1715, 2389, 579, 354, -991, 1193, 255, 681, -1262, 1772, 1591, 380], [-58741, 6947, 504, -3822, -9169, 606, 982, -15, 988, 743, -483, 252, -720, -1084, -673, -810, -831, -1085], [17122, 5411, 1888, 1305, 15759, -464, -494, 2434, 1021, 1935, 2202, -1213, 1781, 386, -761, 940, 2063, 544], [-8559, 4856, 5068, 310, 13792, -1166, 786, -4503, 912, 2526, 908, -410, 126, -1508, -238, -712, -280, 680], [34356, 18090, -5075, 3197, 1212, 338, 86, 1385, 836, 2734, -4426, 286, 482, 478, 1528, 405, 181, 1126], [-15058, 19407, -2310, 104, -11941, 1422, 856, -1227, 1335, -2307, 473, 137, 699, 14, 981, -377, -554, -333], [17470, 10810, 1096, 2130, -1602, 1173, 782, 571, -1050, -168, 4244, 489, -433, -94, 1777, 869, 453, 2199], [20839, 3446, 1699, -2629, 3220, -120, 162, -1236, 539, -1118, 169, -417, -647, -652, 1993, -570, -2150, -968], [-9161, 16365, -2679, 327, 9656, 1318, -404, 2581, -93, 1601, 1208, -1291, -749, -428, 521, -484, -1161, -1295], [18522, 18283, -5617, -5795, 6534, 980, -304, 4242, -867, -1293, -7515, 25, 563, -982, 2069, -58, -395, -467], [-38492, -14145, 626, 2034, 194, 1510, 79, -1646, -1428, -1037, -2833, -663, -961, -1449, 1111, -1499, -97, -1421], [7216, 274, 1968, 3032, -2320, 1366, 496, -1259, -547, 2041, -2318, -1683, 51, 236, 35, -1313, -913, 1528], [17156, 3932, -55, -5913, 1292, 926, 3546, -1197, 2737, 4624, -1809, -929, -403, -180, 659, -206, 212, 1064], [-38299, 19013, -3996, -2726, -9962, -177, -1963, -341, 1386, -1572, -5519, 220, -1532, -153, 1228, -878, -286, -130], [32097, -10776, -4920, -515, 7920, 129, 233, 191, -595, 1569, 564, 177, -409, -461, -1946, -22, -1709, -890], [27283, -1126, 625, -1204, 10934, 1055, 115, 4098, -1249, 464, 6082, -1305, 5, -2003, -835, 77, -1150, -1992], [-14930, 3269, -6044, -5020, 7184, 1193, 13, -2022, 79, -1611, -4104, -993, -959, -1171, -1150, -755, -923, -556], [-44188, 2056, -495, -669, -28091, 467, 1349, -2914, 410, -3255, -1594, 174, -1110, -79, 450, -597, 1460, 497], [-21494, 12907, -380, -1733, -6284, 1690, -232, -208, -600, -1528, 6714, 1266, -654, -1358, -1466, -1, -1012, -2614], [6577, 4076, 4234, 2432, 7992, -944, -2346, 7699, 474, 4250, -8362, -1248, -124, 3358, -548, 640, -150, -418], [-3759, 15169, 1096, 1199, -12188, 2258, 1451, -1735, -1471, 2151, 2218, 348, 80, 1253, -142, 833, 1153, 514], [30857, 2853, -313, -3911, -6056, -2370, 3499, 971, -133, 982, -1550, -1328, 412, -163, -168, -209, -93, 2197], [-26442, -2112, -1531, 4072, 14455, -1602, -2743, 1765, 454, 426, -1864, 691, -1683, -497, -2620, -1315, -1360, -3295], [-20228, 2844, -4276, 5129, 3200, 2806, 1195, 1321, -820, -1219, 3259, 1981, -49, 3833, -13, 434, -753, -419], [37989, 7564, 1133, 1768, 11553, -354, 1134, -834, 2167, 1774, -2312, 1229, -545, -164, 537, 555, -767, 1477], [-28686, 5931, -6241, 82, -7052, 2685, -614, 731, -585, 1002, 4408, 1512, -1476, 852, 1493, -358, -17, -1769], [21802, 3702, -460, 311, 1650, 2801, 3851, -2038, 1011, 3109, 7296, -791, 227, 1545, 942, -344, 129, 470], [11331, 5502, 1258, -2340, -1200, -585, -1893, 3741, -66, 1146, -2187, -1900, 1200, 2169, -1620, 120, -661, -1125], [-16898, 2769, -1246, -4207, -4261, -1305, 1661, 1113, 1185, 2928, 2831, -1807, -241, 1023, 362, -1198, -88, 975], [30704, -524, 238, -1120, 1201, -404, -796, 73, -83, 2261, -10482, -321, 797, -1176, 2516, 365, 663, -288], [-14057, 14312, 8506, -2532, 16502, 1481, 1826, 1171, 750, -824, -1966, -1863, -41, -1672, 1202, -437, -1733, -2533], [-7213, 6378, 4792, 1322, -2356, -1640, 182, 2502, -563, -1148, 5183, -515, -517, 1198, -2748, -263, 79, -2498], [-6556, 1628, -1893, -3576, -4062, 521, 1403, -2019, 563, 939, -1322, -619, 715, -1059, 149, -193, 890, 528], [31639, 12529, -4484, -3646, 18339, 2325, 2848, -1016, 589, -552, 5540, -1916, 560, 232, 1258, 194, -643, -1122], [18450, -2538, 4821, 704, -2239, -791, -80, 2399, -2130, -2294, 3411, -175, 1151, -250, 1316, -109, 739, -2452], [-29665, -5790, 2176, 7561, 15963, -3047, -2281, -2314, -160, 1084, -7674, -6, -558, -793, -409, -51, -1430, 232], [2070, -15112, 4330, 2164, -16808, -1709, -688, 548, -55, -2530, 2024, 1618, -818, 500, 1061, 838, 912, 567], [-1564, -12028, -1625, -2375, 816, 1538, 2301, 2762, 1233, -168, 7954, 1468, 1680, 1245, -1406, 1767, 1414, 658], [23129, 6294, 6094, -3631, 7124, -2042, 179, 3242, 2154, 1543, -1707, 1063, -227, 1853, -324, -488, 411, 3453], [18416, 5962, -1265, 2029, -8206, -2434, 1957, -7130, -255, -1448, 4369, 434, -38, -1493, 1654, 1009, 272, 2980], [6409, 626, 3596, 5252, -4728, 3308, -1596, -3572, 744, -548, 3622, -1192, 704, -1662, 1175, 1460, -8, -1464], [84516, 1231, 1867, 5032, -5708, -1488, 2995, -6320, 1063, 972, -2930, 408, 152, -405, 1512, 256, 1576, 3026], [-10868, -722, 208, -3707, 733, 716, 362, 2503, 824, 146, 3523, -1914, 802, 870, -29, -287, 193, 1241], [-70210, 15806, -769, -3529, 6981, -1207, -1885, -3255, -1868, -856, 1033, 874, -586, 1723, -302, -97, -1816, 2123], [-43499, -4777, 6047, -1256, -4076, -335, -4519, 1296, 925, -440, -6674, 1379, 693, 1883, -367, 1213, -775, -127]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1083_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1083_4_a_s();".
// 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_1083_4_a_s(:prec:=18)
    chi := MakeCharacter_1083_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_1083_4_a_s();".
// 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_1083_4_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1083_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![-49519296, 62731584, 157450608, -101642904, -174155400, 53300700, 83975016, -10997190, -18914886, 1134069, 2264487, -62307, -153482, 1731, 5904, -19, -120, 0, 1]>],Snew);
    return Vf;
end function;