
// Make newform 8004.2.a.i in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8004_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_8004_2_a_i();".
// 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_8004_2_a_i();".
// 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 := [-208, 72, 14352, -17844, -138024, 172218, 60168, -121665, -137, 32512, -3483, -4048, 634, 234, -43, -5, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [52042966114613282035144, -265212571902372090704692, -505057097539218663987944, 2820087633149276715197960, -345737863611574611652196, -2206166321766519707780335, 327739783995970876724887, 652807380563049726096164, -85086576675455429657055, -92816762444366568951910, 10014744839360784183052, 6701952172673579845542, -552866610872686710043, -232887392825327605241, 11451848272286613279, 3040780320993915014], [-181736188397342579200, 20345563209993771194972, 28814439094940890866468, -217538448934933635766102, 162630129553180087835490, 118176839691438596424822, -133459508425382606930219, -15054873801417991803487, 37694840050022447013852, -1931243557888597965827, -4841993240933967346590, 594553806911671718002, 285577742288708681086, -45962507843656667153, -6183981961343391225, 1133155627719690365], [39550714308552523443072, 75547605968724778430888, -871645844830868408937972, -572372392754759088010752, 2076442970619250704115952, 15498431467411784950720, -1253648947739701211892215, 160070546734796190670629, 317505626992755431744994, -55997241963193925731391, -38562172136739934951652, 7701026401458349054308, 2199331736699944090170, -468648207638402451527, -46623029354138300703, 10327108820298630749], [-48618750485435144983000, -28435539046608856852172, 637548574623595049191988, 1224989810538650711651380, -2745160541549121158528824, -268495880029625550510797, 1833305564855172683452274, -187404565471993343666351, -480110131920749519232459, 79549194381055089354751, 59064258354862485012260, -11475349920655983031686, -3385320715110873005891, 710848605201316221158, 71891299794190027998, -15783173244444101129], [-115448866564292248468160, 9531422201946669156536, 2965875364568320786841260, 2968897987381385534458088, -8528134675336330855060592, -669736051113647402565376, 5334598757017222508511037, -481998571182840860681887, -1369734242025953674106622, 209050863408559362327029, 167641296806600010566452, -30738946990350785496204, -9609952842332358293734, 1933384284718598061661, 204436784173408593213, -43409510040342958591], [430803475028090582512, -32576648691634350903024, 246595002574247516330436, 799934920651196090914288, -2074495686819045701186224, 27305100885580799650810, 1494626491148611763355897, -252413267047417095561809, -405344514391113240217288, 84152959587324240424423, 50934833529316434627484, -11324818386317160387376, -2960246772843543741264, 680444530030849926369, 63444477320010917489, -14876422501216426365], [-27182643118546951377232, 25492596062475603023440, 538400774762995113598740, 189737204442907705092132, -1376527448283916327600720, 154277444344167389409560, 849163607976476235455277, -165337223101262763914851, -217058424842370164675842, 47517289986971177956637, 26465002897883635403568, -6050691028509434703612, -1511395690869302031178, 353456303235461540053, 32046172073721707913, -7603870522084452615], [-31358839502922458309168, -83620908096978852457952, 1071305771301748204569172, 910802804643754812688216, -3016049778386644474401072, -43320182299082538137646, 1878957650573029373491257, -244010727226114098109473, -480661070940613526614664, 86125788536977874427823, 58582378625148598351460, -11838796361473972966656, -3344745531914834911664, 719012486847903488913, 70916958369897065433, -15813712956832157237], [-65047851739085236716416, -387205328131515237238792, 1130638678095749386304068, 4671490927199216453556224, -7825684947137813046148304, -1501117875210636465270160, 5502983380583967143683615, -369662096234080721052685, -1476019645175755861347138, 215733079036281931399607, 184493534161900674094564, -33512675596998272085780, -10697144276185392156634, 2151043186106097123631, 229053636521692224375, -48750376286856419701], [83494722494782046960, -256144554369799347906592, 124510648830211096930172, 3297458934296158318389232, -3441103630357287161444048, -1670543280670879404138622, 2503760282664111920367379, 145894747281510593499017, -675789922002448975704604, 49699417845194961077353, 84635925481199523199764, -11381973623510737504984, -4912947016947780053268, 825924130302220654971, 105296243094269658907, -19819065957461023811], [4443088667932115690440, 121576357609022372496944, -426428181856744945912580, -1934597359389313500952252, 2755519788258439982119666, 833413740406888323847562, -1894731945704220803403863, 17345193658302384622799, 501622126176053457192078, -55554094321534472370211, -62181408508523380600868, 9839491871465361076638, 3585945259827232385072, -662035163879610543401, -76517980267269701535, 15344073927299171897], [-31359800394432217976480, -662086086998971531518376, 2151780320120874407422252, 5541104129287872654006968, -11349378748818973102168016, -1086937561234508913856240, 7797419486270366874993913, -869371686203195894627251, -2075255844526952664807366, 358664574332552237611265, 258321554986471115357812, -51494181033152023858860, -14936160860061056201230, 3190335724091810475001, 319190212777976757561, -70933075913422809427], [-12820290579032488312704, -386578636465775801362264, 543054530270901314400956, 4274698916700503360392296, -5468684705975812969273176, -1830412119817231334118652, 3874435369925228060759257, -23473862923459495877375, -1037020682429358346607922, 115592584583412047647733, 129273187698150131786804, -20617773136724243749644, -7479591954295351894242, 1389566703547393943857, 159924296878592499289, -32225813913671340095], [39675257796285985779040, 319835708049430771492988, -1480438220894389050010748, -4618652122090728228850804, 7880544085015915696563274, 1602889278627538618860671, -5367934888868512540382867, 293196484616290703750216, 1419614721106134505598337, -195236339850544654368244, -175989519979563028243244, 30918959212470415787136, 10150483883027433779483, -1997551184811081394793, -216603173312831897727, 45395358841873163252]];
    Rf_basisdens := [1, 1, 8244411418171060528944, 1374068569695176754824, 5496274278780707019296, 8244411418171060528944, 16488822836342121057888, 5496274278780707019296, 2748137139390353509648, 5496274278780707019296, 16488822836342121057888, 5496274278780707019296, 4122205709085530264472, 16488822836342121057888, 5496274278780707019296, 4122205709085530264472];
    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_8004_a();"
function MakeCharacter_8004_a()
    N := 8004;
    order := 1;
    char_gens := [4003, 2669, 3133, 553];
    v := [1, 1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_8004_a_Hecke(Kf)
    return MakeCharacter_8004_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+1)]];
    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 := 2;
    raw_aps := [[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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 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, -1, 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, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [1, 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, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [-1, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, -1], [1, -1, 0, -1, 1, 0, -1, 1, 0, 0, 0, -1, -1, -1, 1, 0], [0, 1, 0, 0, 0, 1, 0, 1, -1, 1, -1, 1, 0, 0, 0, -1], [1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, -1, 1], [-1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [0, 0, 0, 1, 0, 1, 1, 1, 0, 1, -1, 1, 0, 0, -1, 1], [-2, 1, 1, 0, 0, 0, 1, 0, 1, -1, 1, 0, 0, 1, 0, 0], [-1, 1, 1, 0, 0, -1, -1, 1, 1, 1, 1, 0, 0, 1, 0, 1], [0, 0, 0, -1, 1, 0, 0, 2, 0, 1, -1, 1, 0, -1, 1, -1], [1, 1, 0, -1, 1, -1, -1, 1, 1, 1, 0, 1, 0, 1, -1, 0], [0, 2, 0, 0, -1, 1, -1, 0, -1, 0, -1, 1, 1, 1, 0, 0], [0, -1, 1, 0, 1, 1, 0, -2, 0, 1, 0, 0, 0, 1, -1, -1], [-1, -1, 1, -1, 1, -1, 0, -1, 1, -1, 1, 0, 0, 1, -1, 0], [3, 0, -1, 0, -1, 1, 0, -1, -1, -1, 0, 0, -2, 0, 1, 1], [2, -1, 0, -1, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 1, 0], [3, 1, 0, 1, -1, 0, -1, 0, -1, 0, 0, 0, 0, 0, 1, 1], [-2, -2, 0, -1, 3, -2, 1, 0, 1, 0, 0, -1, 2, -2, -1, -2], [1, 1, -1, 0, -1, -1, 1, -1, 0, -1, -1, 0, 0, 0, -1, -1], [2, 1, 1, 1, 0, -1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 2], [-4, 2, -1, -1, 0, 0, 0, 2, -1, 0, -1, -1, 0, -2, 2, -1], [2, -1, 2, 0, 1, -2, 1, 1, 2, -2, 1, 1, 0, 1, 0, 1], [-1, 2, -1, 0, -1, 0, 0, 2, -1, 2, -2, 1, 2, -1, -1, 0], [0, 1, 0, -1, 0, -1, 0, 1, 1, 1, -1, 1, 3, 0, -1, 0], [3, 0, -2, 0, -1, 0, 0, -2, -1, 1, -1, 1, 0, -1, -1, -1], [1, -3, 1, 0, 0, 1, -1, 1, 1, 1, 1, 1, -2, 2, 1, 2], [2, 1, 1, 0, 1, -1, 1, 1, 0, 0, -1, -1, 2, 0, -1, -1], [3, 0, 0, 1, 3, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1], [1, 1, 2, 0, 2, 0, 0, 2, 0, 2, 0, 0, 2, -1, 1, -2], [2, -1, 1, 0, 1, 0, -1, 1, 0, 3, 1, 1, 0, 1, -2, 1], [-2, 0, -1, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 1, -1, 1], [2, 1, -2, -1, -2, 2, 0, -2, -3, 0, -1, -2, 0, -2, 1, -1], [-2, 1, 0, 0, 1, 1, 0, 2, -1, 0, -2, -1, 2, -1, 1, 0], [-2, 0, 2, -1, 1, -2, 0, 3, 1, 1, 0, 0, 2, -2, 1, -1], [-2, 3, 0, 0, -2, 3, 0, 0, -3, 0, -2, 1, 0, 0, 1, 0], [5, -1, -2, 0, 1, 1, 0, -2, -2, 1, 0, -2, 2, 0, -2, -2], [2, -1, -1, -1, 0, -2, 0, 0, 0, -3, -1, 0, 0, 1, 0, 1], [5, -3, 1, -1, 4, -1, -1, 2, 1, 0, 0, -2, -2, -1, 0, -1], [1, -1, 0, 0, -1, 0, -1, 1, -1, 3, 1, 1, 0, 0, 0, 0], [-1, -1, -1, -2, -1, -1, 0, 0, 0, 0, 0, -2, 2, -3, 0, -2], [2, -2, -1, -1, -2, 2, -1, -2, -2, 0, -2, -1, 2, -1, 0, -1], [7, -1, 0, 1, -1, 0, -1, -2, 1, 0, 0, 0, 0, 1, -1, -1], [-3, 2, 0, -2, 2, -1, 0, 4, -1, 2, -2, 1, 3, -2, 0, -2], [2, -2, 2, -2, 3, -3, -1, 2, 3, 2, 0, 1, 2, -1, -2, -1], [-5, 3, 0, 0, -1, 1, 1, 2, -1, -2, -1, 0, 2, -1, 2, 1], [4, -3, 1, -1, 5, 0, 0, 1, 1, 1, 2, -1, 0, 0, 0, -1], [3, -2, 0, -2, 1, -1, 2, -1, 1, -2, 0, -2, 1, -1, -1, -1], [-2, -1, 1, 1, 1, 1, 0, 2, -3, 3, -1, 1, 0, -1, -1, -1], [4, 0, -1, 2, -1, -1, 1, -1, 0, -1, 1, 0, -4, 0, -2, 1], [-3, -2, -1, -1, -1, 1, -1, 1, 1, -1, 1, -2, 0, -1, 2, 1], [1, 0, 0, -1, -2, 0, 1, -4, -1, -2, 1, -1, 2, 2, -1, -2], [-1, 0, 0, 1, 1, 1, 0, -1, 2, 1, 1, -1, -2, 0, -1, 0], [3, 1, 0, -1, -1, 0, 0, 1, 1, -1, 1, 1, -2, 0, 1, 1], [-1, -1, -1, 1, -4, 3, 0, -2, -3, 2, -2, -1, 2, -2, 1, -2], [5, -2, 0, 0, -4, 2, 0, -2, -1, -1, 0, -1, -2, 2, 1, 2], [4, -2, 0, -1, -2, 1, -1, 2, -1, 1, 0, 0, 2, 1, -1, 1], [6, -2, 0, -1, -1, 0, 0, -2, 1, 0, 2, 0, -2, 3, 0, 1], [8, 3, -2, 1, -2, -1, -1, 0, -1, 2, -2, 3, 0, 0, -2, 1], [2, -2, 0, 0, -3, 2, -1, -3, -1, -3, -1, -2, 1, 2, 2, -1], [-8, 0, 1, 0, 0, 3, 0, -1, -2, 0, 0, 1, 0, 3, 0, 1], [5, -1, -2, 0, 2, 1, 1, -3, -1, -2, -1, -2, -2, -1, 0, -3], [6, 0, 0, 3, -2, 1, -1, 1, -2, 1, -2, 3, 0, 0, -1, 2], [4, 1, -3, 0, -3, 2, 0, -5, -4, 0, 0, -2, -2, -2, 0, -3], [6, 0, 0, -1, 0, 0, 0, 0, -1, 1, 1, -1, 4, 0, 0, -1], [7, 2, 0, 0, -1, 3, -1, 2, -2, 4, 0, 2, -2, 2, 0, 1], [7, -1, 1, -1, 1, -1, 0, 1, 0, -2, -1, -1, 2, -1, 3, 0], [0, 0, 1, -1, 4, 0, 0, 2, 0, 3, -2, 1, 2, -2, -2, -2], [2, 1, -2, 0, -2, 1, 0, -1, -2, -2, -2, -2, 4, -4, 1, -1], [1, -3, 0, -2, 2, -1, -1, 4, -1, 4, -1, -1, -2, -3, 1, -2], [1, 3, 0, 0, -1, 1, 1, -1, 0, 0, 1, 0, 0, 1, -1, -1], [-1, -2, 2, -1, 0, -1, 0, 2, 2, -1, -1, -2, 3, -1, 0, 1], [-1, 1, -1, 1, 0, 2, 2, -3, -2, -4, -1, -2, 0, -1, 3, -2], [10, 1, -1, 0, 1, -3, -1, -2, 1, 0, 1, 0, -2, 2, -1, 0], [4, 1, -2, 1, -3, 0, -1, 1, -2, 1, -1, 2, -2, 0, 1, 1], [3, 2, -3, 0, -4, 0, 0, 1, -3, 0, 0, -2, 0, -1, 0, -1], [0, 0, -1, -2, -1, -1, 0, 1, 1, 2, -1, 1, -2, -1, -1, 0], [0, 4, -3, -1, 1, 1, 2, 1, -2, -1, -1, -2, -2, -3, -1, -2], [2, 1, 2, 2, 0, 1, 0, -2, 0, -2, 2, 1, -2, 2, 2, 2], [4, 5, 0, 3, -2, 0, 0, 1, 0, -1, 1, 2, 0, 4, 0, 2], [2, 4, 1, 2, 3, 1, 0, 0, 0, 2, 0, 2, 2, 4, -3, -2], [6, -3, 0, -1, -3, -2, -1, 0, 0, 0, -1, -1, 4, -1, -1, 0], [7, -1, 1, 0, 2, 0, 0, 3, -1, 3, 0, 1, 0, 0, 1, 3], [2, -1, 0, 1, 0, 0, 2, -3, 2, -2, 3, 1, -1, 3, -1, 4], [12, 1, -1, 1, 1, 1, 1, -1, -3, 1, 0, 0, 0, 3, -3, 0], [9, 5, 2, -1, -2, -2, 1, 3, 1, 0, 0, 1, 1, -1, 2, -2], [-2, 3, 0, 2, -5, -1, -1, -1, -1, 1, -1, 1, 0, -2, 0, 1], [1, 1, 2, 0, 1, -2, 1, 1, 2, -2, 0, 2, 2, 4, -1, 0], [2, 3, -1, 0, -3, 4, -2, -2, -3, 2, 0, 3, -2, 2, 0, 2], [1, -1, 1, 0, -2, 3, -2, -1, 2, -1, 2, -2, -2, 0, 2, 1], [8, -2, -2, 2, -1, 1, 1, -2, -1, -3, 1, -2, -2, 0, 1, 2], [3, 4, 3, 3, -2, 0, 1, 1, -1, 0, 0, 0, 0, 2, -1, 2], [3, 0, -1, -3, 2, -4, -1, 1, 0, 0, -2, -2, -2, -4, 0, -2], [-1, -2, -4, 1, -2, 3, -2, -5, -4, 2, 0, -1, 0, 0, -2, -1], [4, 0, 0, -3, -2, -2, -3, 2, 1, 1, -3, 2, 2, 2, -1, 2], [-1, 4, 1, -1, 2, 0, -1, -1, 1, -2, 0, 0, 2, -1, 1, 0], [-9, -1, 1, -2, 2, 1, 2, 2, 3, 0, 2, 0, 2, 1, -1, 0], [6, -2, -1, 1, 1, 0, 2, -4, 1, -3, 2, -3, -4, 0, -1, 0], [4, 1, -3, 0, -2, -1, 0, -3, -3, 2, -1, 2, -2, -1, -3, -2], [2, 1, 2, -1, 1, -6, 0, 1, 3, -2, 3, -1, 0, 2, -2, 0], [6, -1, 2, 2, 3, -5, 1, 1, 4, -1, 3, 1, 2, 0, -1, 1], [9, 4, -2, 2, -1, -1, -1, -2, -1, -1, 1, 0, 0, 3, -2, 2], [-7, 1, 2, 1, 1, 1, 0, -1, -2, 1, 1, 1, 2, 2, -1, 0], [10, -2, 1, -1, -1, -1, -2, 2, 3, -3, 1, -1, -4, 2, 3, 3], [-7, -2, 2, 0, 6, 0, -1, 0, 1, 2, 1, -2, -3, -3, 1, -4], [1, 2, 2, 0, 7, -2, 1, 4, 3, 1, 0, 1, 0, -1, 1, 2], [5, -7, 0, 0, 3, -4, -1, 0, 4, -1, 2, 1, -4, 1, -1, 1], [9, 3, 2, 1, -1, -4, -1, 1, 4, -2, 0, 1, 1, 4, -1, 2], [-1, 1, -1, -1, 0, -1, 0, -2, 0, -1, 1, -1, -6, -2, 3, -1], [-4, 1, 0, -2, 0, 3, 1, 1, -3, 1, -3, 1, 0, -3, 0, -3], [0, 4, 1, 4, 3, 0, 1, 0, 0, 0, 1, 1, 0, -1, 0, 2], [8, -2, 0, -1, 2, 0, 0, -1, 2, 1, 1, 1, 2, 3, -3, -1], [-5, 0, 0, -1, -2, -3, -1, 4, 3, -2, -2, 1, 0, 1, -1, 3], [1, -1, -1, -3, 2, 3, 0, -1, -2, -1, -1, -2, -4, -1, 2, -2], [-5, 4, -1, -1, 1, 1, -1, 4, 0, -1, 0, -1, 2, -2, 2, 1], [-5, 2, -1, -2, 4, -3, 1, 6, 1, 1, -3, 0, 0, -4, -1, -2], [-16, -1, -3, -4, 2, 2, -1, -1, -2, 1, -1, -3, 2, -4, 0, -3], [-1, 5, -1, 1, -4, 1, -1, -5, -1, 0, -2, 1, -2, 1, 1, 0], [5, -4, 0, -3, 3, 4, -2, -4, 0, 2, -1, -3, -2, -3, 0, -6], [2, -4, -2, 0, -3, -1, 2, -5, 0, -3, 2, -3, -4, -3, 0, -1], [-6, 3, 3, 0, 5, -5, 0, 7, 3, 3, 0, 4, 0, -2, 0, -1], [10, 0, 1, 1, 4, -1, -2, 3, -1, 8, 1, 3, -4, 1, -3, 2], [1, -6, 1, 0, 4, 0, -1, 3, 0, 1, 3, -3, -2, 0, 1, 3], [-9, -4, 2, -1, -2, -2, -2, -2, 3, 1, 1, 1, -2, 1, -3, -1], [-3, 1, 5, -1, 1, -2, 0, 5, 5, -4, 1, 2, 2, 4, 2, 4], [-8, 0, 5, 0, 4, -4, 1, 6, 4, 0, -1, 3, 0, 1, 0, 2], [-7, -5, 0, -1, 0, 3, 1, 0, 1, 1, 0, 0, -4, -1, 0, 1], [0, -1, 3, 0, 4, 1, 2, 4, -3, 3, -2, 0, 4, -5, 2, -5], [5, 2, -2, -1, -4, 5, 2, -5, -1, -7, -1, -3, 0, 0, 2, -2], [7, 5, 0, 4, -3, 2, 1, -2, 0, -1, -1, 1, -2, 2, -3, 2], [-13, -5, 0, -2, 0, -1, 1, 2, 2, 0, 2, -1, -2, -2, 0, 1], [1, -5, 2, 0, 3, -1, 0, -1, 3, 0, 2, 0, 2, 2, -3, -3], [1, -1, -2, -2, -1, 4, -1, -1, 0, 0, -6, -2, 4, -4, 1, -4], [0, 0, 1, 0, 2, 3, 1, -1, 0, 1, -5, -2, 2, -3, 0, -3], [-12, -2, 1, -1, -1, -1, -1, 3, 1, 0, 0, 3, 6, 0, 0, 0], [-1, 8, 0, 0, -3, 4, 0, 1, -6, 3, -4, 2, 4, 0, 1, -3], [3, -1, -3, 3, -7, 6, 1, -7, -4, -3, 0, -1, 2, 4, -1, 0], [-7, 3, -4, -1, -1, 2, 0, -1, -4, 4, -5, 0, 0, -1, -2, -4], [7, 4, 0, 1, 0, 0, -1, 5, 0, 6, -1, 6, -2, 0, -1, 1], [12, 3, -3, 0, -1, -2, 1, 2, -2, -1, -1, 0, 2, 0, 1, 2], [-6, 3, -3, -2, -4, -1, -2, -4, -1, 1, 0, -1, 4, 1, -3, -2], [1, 0, 3, 1, 2, 3, -1, 2, 2, -2, 1, 3, 0, 6, 1, 4], [-11, 7, -3, 3, -6, 3, -1, -3, -4, 2, -4, 1, 0, 0, 0, 2], [2, -4, -2, -2, -1, -1, -1, -2, -3, 1, 1, -4, 4, -2, 1, -2], [-5, 5, -4, 1, -3, -1, -1, 0, -2, 2, -2, 1, -6, -1, 0, 3], [-3, -2, -3, -2, 0, 0, 0, 2, 0, -2, 0, -3, -2, -1, 0, 0], [-4, -3, 0, 2, -7, 4, 0, -3, -2, -1, 0, 0, -2, 1, 0, 3], [-2, 3, 1, -1, 2, 5, 1, 5, -4, 2, -2, -3, 4, -6, 5, -2], [8, 2, -2, 2, -3, 2, 2, -2, -2, -3, -3, 2, 4, 2, -1, -1], [-4, 0, -2, 0, -3, -1, -1, -1, 2, -6, 3, 0, -3, 2, 0, 3], [-10, 5, -1, -1, 2, 0, -1, 2, 0, 2, -2, 1, 2, 0, 4, -1], [7, -3, 0, -2, 1, -2, 0, -2, 1, -3, 2, -3, 4, -3, -1, -2], [18, -3, 1, 3, 1, 5, -1, -3, -2, 3, 2, 4, -2, 2, 0, 0], [-9, 11, -2, 2, -9, 1, 0, -1, -4, -3, -2, -1, -4, 0, 3, 1], [-3, -6, 2, -2, 2, -1, -2, 5, 2, -2, 2, 0, -2, 1, 4, 3], [5, -5, 1, 2, -2, 1, -2, 2, 0, -3, 1, 0, -2, -1, 5, 3], [-2, 5, -2, 1, -5, -2, 0, 7, -3, 2, -3, 5, 2, -4, 4, 2], [-8, -8, 4, -2, 4, 2, 0, -2, 3, -1, 3, -2, -4, 0, 0, 1], [4, -5, 1, -2, 4, 0, 0, 1, 3, 2, 2, 0, -4, 2, -1, 1], [-1, -1, -3, 2, -6, 1, 1, -4, -1, -4, 0, -5, -4, 2, 0, 1], [7, -6, -2, -1, -2, 0, 2, -1, 2, 0, -3, -1, 4, -1, 0, 0], [4, -10, 5, -6, 9, -1, -3, 3, 6, 3, 3, -2, -1, 0, 0, -3], [7, -4, 2, 0, -1, 0, -1, -6, 0, -2, 5, -4, 2, 3, 0, -2], [-4, 4, -3, -1, -6, 4, -1, -2, -1, 1, -2, 2, -4, 4, -2, 1], [15, -3, -5, 0, -4, 0, 1, -8, -3, -2, -1, -3, 0, 1, -2, 2], [15, -2, 0, -2, 3, 2, -4, 3, -1, 8, -3, 2, 4, 0, -3, -2], [-6, 5, -2, 4, -7, 1, -2, -2, -2, 0, 2, -1, 0, 1, 4, 2], [6, -1, 1, 3, 4, -3, 1, 2, 1, 3, 2, 3, 2, -1, 0, 3], [-10, 6, 2, 1, -2, 6, -1, 6, -2, 6, -2, 3, 2, 0, 4, 4], [5, 2, -1, 1, -3, 0, -4, 4, -3, 3, -3, 4, 0, -2, 0, 3], [3, -4, -1, 1, 3, -4, 0, 0, 4, -1, 5, -1, -6, 0, 0, 0], [3, -4, -2, -2, 3, 3, 1, -1, -4, 1, -3, -1, 6, -4, -2, -3], [-3, -2, 2, -1, -2, 0, -1, 0, 0, 1, -2, 3, 0, 4, -3, 3], [0, 2, 2, -2, -1, 1, 0, -3, 2, -1, 2, 1, -2, 2, 0, 1], [6, -7, -3, -1, 0, 0, 5, -8, -2, -6, 0, -5, 0, -2, -2, -3], [2, 10, -1, 4, -4, 3, -1, -4, -4, -1, 1, 3, 0, 6, 1, 0], [-1, -5, 0, 3, -3, 3, 1, -7, 1, -1, 0, 1, 0, 4, -3, 0], [0, 2, 1, -3, 5, 0, 3, -1, -1, 0, 1, -1, -4, -3, 2, -3], [-3, -6, 3, -1, -2, 1, -1, 2, 1, -3, 0, 0, 0, 4, 2, 4], [0, 1, 2, 2, 6, -2, -1, 6, 1, 2, -1, 2, 6, 4, -2, 0], [12, 3, 2, 6, -1, 1, 3, -1, -2, -1, 1, 1, 0, 2, 1, 3], [3, 4, 2, 1, 4, 2, 2, 1, -1, 1, 2, -1, 0, -1, 2, 1], [-7, 0, -3, 1, -5, -4, 0, 0, 0, -2, 1, -1, 2, 2, 0, 5], [4, -3, -3, -3, -2, 3, -1, -6, -1, -3, -2, 1, -1, 1, 2, -1], [-3, -5, 0, -1, -1, -2, -1, 1, 2, -1, 2, 0, -4, 5, 0, 2], [14, -4, -2, 4, 1, -3, 1, -3, 0, 0, -1, 2, 0, 2, -4, 1], [6, 12, -1, 2, -3, 0, 3, -1, -1, 0, -3, 0, 5, 0, -2, -1], [7, 0, -3, 3, 0, 0, 2, -3, 0, -1, 1, -4, -6, 0, -3, 1], [9, -3, -4, 4, -2, 0, 0, -10, 0, 0, 0, 0, -2, 2, -3, 0], [-10, -1, 5, -2, 1, -6, -2, 8, 5, -2, -2, 3, 6, 0, 0, 2], [-11, 5, 2, -1, -2, -5, 2, 0, 4, -6, 3, -1, 0, 1, 1, -1], [-3, -8, 1, -2, 2, 1, -2, 1, 2, -1, 0, 0, -1, 1, 2, -1], [1, 4, 1, 1, -2, 4, -2, 3, 0, 1, -3, 5, -1, 5, 2, 4], [-13, 6, 0, 2, -3, 5, 2, 2, -4, -4, -2, 1, -2, 1, 5, 2], [-19, 7, 1, 0, 0, 7, 0, 6, -5, 2, -6, 4, 6, -2, 5, 0], [-8, 1, 2, -4, 2, -5, 3, 4, 3, -2, 0, -3, 2, -5, 3, -6], [-7, -6, 2, -2, 1, 1, -2, 4, 0, 2, -2, -1, -2, -7, 3, -2], [7, -1, 1, 2, -3, 2, 1, 4, -2, 4, -2, 3, -4, -4, 1, -1], [4, 1, -1, 0, -8, 4, -3, 0, -5, 1, -3, 6, -2, 3, 0, 0], [-3, -2, 4, 1, 5, -1, 1, 11, 3, 8, 0, 2, 6, -7, 0, 0], [-14, 0, -1, -4, -3, 0, -4, 2, -3, 1, -2, 0, -4, 0, 3, 0], [-2, 4, -2, 3, -8, -1, 0, -4, -2, -2, -3, 1, 1, -1, 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0, 2], [10, -7, 4, 0, 8, -3, 0, 8, 1, -4, 2, -3, 2, -4, 5, 0], [20, -8, -1, -3, 1, 2, -2, -1, 1, -4, -1, 0, -6, 4, 1, 0], [0, 8, 2, 0, -6, 6, -1, -1, -2, 5, 1, 4, -4, 4, 3, 5], [6, -5, 2, -2, 10, 3, 4, 4, -4, 6, -2, -3, 2, -9, 0, -6], [6, -3, 3, -2, -4, -3, -4, 1, 4, -1, 1, 2, -4, 0, 1, 0], [16, -2, 0, 5, -2, 5, 4, -8, 0, 2, 1, 3, -6, 4, 0, 1], [2, 13, 2, -1, -1, -3, 0, 7, 2, 4, -3, 6, 4, 1, -3, -2], [-18, 6, -1, -5, 0, -1, -2, 3, 0, -1, -3, 0, 0, 3, -1, -4], [-13, 9, 6, 4, 0, 0, 0, -1, -1, 7, 2, 4, -2, 6, -6, -1], [-25, 4, 2, 1, 0, -5, 1, 5, 0, 0, 0, 0, 6, -1, -3, -2], [19, -1, -4, 0, -2, 8, 0, -2, -8, 4, -2, -2, 0, -2, 1, -4], [-16, 3, -4, -1, -10, 6, -1, -3, -8, -1, -6, 0, 8, 0, -1, -2], [10, -8, -4, -3, 7, -2, 2, 1, 7, -5, -4, -4, 0, -6, 1, -3], [-1, 4, 4, 4, -3, 1, 1, -1, 2, -1, 3, 2, 4, 10, -5, 6], [-3, 3, -1, 1, -1, 5, -1, -13, -3, -2, 0, -5, -7, 1, 1, -2], [-4, 1, -1, -9, -7, -1, -4, -1, -2, 0, 0, -5, 2, 3, 2, 1], [10, -2, 0, 9, -11, -2, 0, -4, 2, -3, 3, 5, -8, 5, -5, 9], [7, 14, -8, 6, -12, 9, 3, -4, -10, -1, -3, 4, -6, 0, 7, 4], [12, -9, -2, -6, 4, 0, -6, 1, -1, 7, -2, -4, 2, -6, -3, -5], [10, -3, 2, 3, 1, -4, 0, -2, 3, -4, 3, 2, 4, 8, -7, 3], [23, 0, 4, 4, 0, -4, 0, -6, 4, -2, 6, 0, -10, 6, -1, 6], [-12, 8, -3, 1, -9, 5, 2, 0, -9, -3, -5, -1, 4, 3, 1, 3], [-7, 2, -2, 3, -4, 6, 5, -2, -3, -10, 1, -7, 0, -2, 9, -4], [15, 0, -3, 2, -5, 5, 0, -6, -8, -3, -4, -4, 4, 0, 6, -7], [-21, 8, -2, 0, -8, -1, -5, -3, -5, 0, -3, -2, 2, 4, 0, 1], [-9, 6, 1, -5, -1, 6, -3, 4, -3, 2, 0, -1, -4, -3, 6, 1], [-6, -2, -1, -3, 1, -6, -2, -2, 4, -2, 3, -3, -1, -2, 3, -1], [-10, -3, -4, -5, 1, -2, -2, -1, -1, -4, -3, -3, 6, -5, -2, -4], [28, -3, -3, 2, -4, 6, -4, -6, -6, -1, 2, 0, 2, 5, 0, -2], [15, -9, 2, -2, 7, 5, -5, 5, -1, 9, -4, 2, 2, -3, -1, -2], [-27, 0, 0, -6, 3, -4, 0, 4, 1, -3, 1, -5, 2, -3, 1, 3], [10, 0, -3, 0, 0, -1, 0, -1, -2, 2, -1, -1, -4, 1, -2, 6], [-7, 1, 0, 1, -1, -10, 1, -7, 8, -12, 2, -1, 0, 2, -3, 2], [-19, -7, -1, -1, 6, 3, 0, -5, 2, -1, 3, -6, -2, -5, 2, 2], [-7, 7, 0, 5, -7, -7, -1, 3, 1, 3, 4, 5, -2, 2, 1, 6], [3, -13, 2, -4, 6, 2, -8, 7, 1, 6, 0, 1, 8, -1, -1, -1], [-6, 6, 5, -1, -3, 2, 0, -4, -4, 0, -1, 1, 2, 6, -1, -3], [-20, 5, 0, -5, 3, -4, 1, -1, -2, -1, 0, -1, 8, -1, 4, -3], [-12, -4, -1, -7, 13, -10, 0, 10, 6, 2, -1, -5, 0, -9, -3, -5], [8, -2, 0, -5, 12, -4, 2, 4, 0, 0, -1, -5, 6, -8, 4, -5], [-4, 14, -5, -2, -1, -6, 0, -2, 4, 2, -3, 1, -2, 1, -6, 0], [19, 0, -1, 0, 6, -5, 7, -4, 6, -4, 1, 0, 2, -5, -7, -2], [2, -2, 3, 1, -3, -1, -3, -1, 5, -4, 6, -3, -6, 8, 0, 4], [-15, 2, 8, -4, 9, -9, -2, 13, 13, 2, 4, 0, 1, -3, 5, 1], [-1, 0, 2, -4, 4, 0, -5, 3, -1, 2, 3, 1, -8, 3, -2, -3], [-14, -1, -4, 2, 7, -1, 4, -9, 2, -7, 1, -4, -6, 2, -4, 0], [10, -11, 7, -2, 2, 4, -4, -3, 4, 1, 5, 4, 0, 14, -3, 2], [13, -1, 2, 0, 1, -7, -2, 1, 1, -1, 2, 3, 0, 6, -2, -1], [-8, 1, -1, 5, -3, -2, -1, -6, 3, -4, 4, -3, 4, 1, 1, -1], [0, -7, -2, -1, -2, -2, -1, 4, 5, -5, 2, -3, -2, 1, 5, 5], [5, 0, -1, -2, 3, -4, 3, 1, -1, 2, -1, -2, -2, 1, -7, 1], [8, 0, -3, 4, -3, 1, 0, -6, 2, -6, 6, -6, -3, 1, -5, -2], [11, -6, 0, -2, -3, 2, -1, 0, -4, 0, -1, 2, 4, -1, 2, 2], [-2, 17, -2, 1, -6, 4, 0, 5, -13, 9, -10, 5, 6, -2, -1, -1], [23, -1, 7, 1, 8, 0, 1, 9, 3, 5, -2, 4, 12, 0, 1, -3], [11, 4, 2, -7, -3, 4, 0, 7, -5, 6, -5, 3, 0, -3, 1, -2], [-6, 7, -4, -1, -2, 7, 0, 3, -7, 7, -3, 3, -2, 1, -5, 2], [-7, -4, -1, 1, -1, 0, -4, 7, 3, -1, 3, 1, -10, 2, 5, 6], [6, 6, 0, -1, -3, 3, 2, 0, 3, -8, -3, 1, 6, 4, -1, 1], [-15, 1, 2, -2, 5, 6, -2, 6, 0, 11, -1, 4, 4, -1, 0, -3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8004_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8004_2_a_i();".
// 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_8004_2_a_i(:prec:=16)
    chi := MakeCharacter_8004_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_8004_2_a_i();".
// 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_8004_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8004_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-208, 72, 14352, -17844, -138024, 172218, 60168, -121665, -137, 32512, -3483, -4048, 634, 234, -43, -5, 1]>,<7,R![-1420576, -1621888, 2367696, 3019440, -1161776, -1972024, 124216, 599660, 48966, -91842, -14299, 7244, 1440, -277, -63, 4, 1]>],Snew);
    return Vf;
end function;