
// Make newform 4016.2.a.l in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4016_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_4016_2_a_l();".
// 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_4016_2_a_l();".
// 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 := [4, -747, -7607, -13777, 33209, 64793, -72526, -80907, 76260, 43551, -41921, -10845, 12681, 926, -2109, 90, 179, -21, -6, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [166591551288078888, -1027269288097940330, -8517014376848128207, 2970308817404871287, 26640229615794321610, -9681015167402184673, -30398145885453740584, 13024341630912266379, 16553976937434110332, -8221073571569711829, -4580166278975943512, 2706294034633503240, 595092550551867088, -475669164938282307, -17818217143386024, 41912041347671166, -2924913574354401, -1441579400369817, 196521312495034], [-242077884568730940, -1800728050905473725, -2070932208707001660, 7316444036690675014, 14446277403237981944, -17067609862621199939, -20363135569951304978, 18475628716806910011, 11898633774999893396, -10210653079904899581, -3266269544593699975, 3079098801819725010, 372341170757747395, -509405238497308593, 3243544109696637, 43016014683542135, -3871628649572152, -1435803919056848, 209544026239557], [-47302041294770420, -12176117032348777, -3803526816531730763, 912217879033491144, 20551698355449594859, -11837322142659575168, -26919821315765788637, 18022917578774451206, 15060880540333249297, -11513787143158348190, -3952782696839577594, 3769404406780378010, 394842138098307975, -658008988812589996, 19676691355848852, 57670999367330371, -6464820675496617, -1976289568186404, 325315462336008], [-498446106796647004, -5219928908967227585, -308192843555754662, 29568840637183398650, 2855882369841072346, -49510080137058458545, -2411607128241522824, 39545581195445075145, -194355797236659890, -17538233581202119839, 1011790432893158971, 4531249610863146126, -480330669882914705, -672831196084720603, 99268216390112767, 52690311990224015, -9554492206151678, -1668795238793348, 346551847676251], [-4071618367057092, 2667131240989337419, 4920039105746828385, -11312765533213992939, -20413894654710609824, 29170695205661226350, 23418204013206537144, -32825601884278571158, -11013728035806195900, 18353616170262585332, 1903876238659999871, -5531017263463239106, 150593332231996867, 909421304377374380, -98542977570888683, -76188368510367967, 12303917053161873, 2523550044875629, -498188113457669], [10329328802699356, 701698033358479469, -8788558770665667400, -7789723814283011947, 58885576401540912769, -16139376886382327193, -82552732261583313803, 41554386482378362657, 47778169234604056599, -30953673001625573677, -12827190398504637216, 10860911091044278840, 1333989630839258891, -1963532947901539733, 51557910709593326, 175301510143739749, -19728860249241516, -6067612651914337, 1007706538158390], [-90972821278178156, -817380982643208987, 14887804797132335270, 15105377852532462783, -88948367973291215153, 9494499845645210107, 119852409134664945067, -41967676115055920471, -68537331180212418241, 33399622787542986827, 18873883305931959538, -11959308405924989292, -2280762022625764797, 2178765600486528667, 22140646155625798, -195121779192913155, 17708158431050788, 6762123352898153, -1027594418075442], [-355943311226414196, -6540496659385454301, -21158349312845396762, 26968630910803904880, 92460718390550759524, -67030073201629306155, -114320497483603387644, 77126164322063243227, 62515975677943484674, -44788609308024169219, -16481112740348660517, 14027991543956850112, 1798484996667138455, -2387368572651253167, 29617048668770011, 205820671089989371, -20471590021611702, -6973741260199556, 1083690820042657], [316411219035010592, -1314890913361810684, -23331520877142669999, -1609513481442969668, 105275876872953735805, -31609719806534699185, -134698131321362958305, 58998155312108487979, 76099498049635194623, -40699585786793027837, -21008885863502095005, 13809479980337519780, 2587612671357407948, -2452948852753917143, -38084178623578787, 216809473764291880, -18319998171235509, -7459601496024570, 1092816357613181], [-12617454281865112, 4862472104716683620, 20358214123365373471, -17474895021582544010, -90005221911718988253, 56413798233430700797, 114267928088836612117, -72903731379051850569, -63643830928625667821, 44531006101484114293, 16964033962802824291, -14289017657424015228, -1860936011562635530, 2462130918680185093, -31574624618047029, -213680552404619654, 21427225666173599, 7266804211212438, -1134567769505457], [-364829756993871316, -5320072813767842773, -21757429321116481975, 20562863803007876548, 104965677147146286421, -67877444561900174468, -134126430474056569771, 87637830612841723000, 74236047532850061909, -53342262959495678406, -19537063246727999042, 17062702713254432978, 2068849535716815029, -2933166057645453758, 54680083798284962, 254137595755781765, -26488322998174035, -8632704946027114, 1372031837712026], [266086065397816924, 7656742856427456895, 22114795187908175950, -36001730803095990114, -100607382747545578412, 90812240101347596701, 126833847987152230770, -102777671504280390989, -69370865243696314662, 58672937274694536375, 17826983466581622477, -18126754975236576178, -1727735731011966637, 3052987536609974455, -93169367682922185, -261180023056501289, 28760184662207652, 8799653434478100, -1426213381500197], [-18145595185768772, 3058107440142454636, 12445583835862380927, -12687570570163284801, -55645495433960070451, 39555619296920917818, 69525859366441445447, -49289123370662958684, -37923471534717328802, 29356222080806208834, 9800513948355364761, -9260428297436121382, -983893018901486730, 1577439360532800966, -41251315050697064, -135847265665372052, 14706488707551162, 4595827585193675, -741068081655316], [-577666401596282284, 1617997536471123617, 32758860467229390110, -4144258394197102016, -153155685352928992056, 61210305129251803963, 196461996303947941470, -100149415988085444129, -109891497445168112760, 66273975827091356685, 29610043641973866985, -22042316492135969886, -3397394588597605237, 3869531900522449111, -11582361937006627, -339399271749954761, 32536276431610078, 11616156032253848, -1788992130708019], [-440523358723489168, 6903621981002853558, 46303300320837429364, -26845110033970579257, -204811156944413783707, 107332204391096892346, 259387225697679758869, -147891536233421275612, -144639030879757924651, 92817108697767086806, 38921551280019068667, -30218043560971088396, -4439755889786174314, 5253377279220361316, -24696620256853403, -458604356413412660, 44007385279792924, 15658637998627299, -2405904627279449], [-498882579668991336, -12531931956461062566, -37801683907989238741, 57037782759786383952, 175208861191622404069, -157164230605325737913, -219246910241231539663, 185420740613861372835, 117984898012449172935, -107791343874420349483, -29233897625988081233, 33581661759727533114, 2420943149231987866, -5678792202416026997, 264505957024420359, 486780419195914490, -59062739523594601, -16416910358581216, 2792699668717361]];
    Rf_basisdens := [1, 1, 1, 77904810267289528, 77904810267289528, 77904810267289528, 77904810267289528, 77904810267289528, 77904810267289528, 77904810267289528, 77904810267289528, 77904810267289528, 77904810267289528, 77904810267289528, 77904810267289528, 38952405133644764, 77904810267289528, 77904810267289528, 77904810267289528];
    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_4016_a();"
function MakeCharacter_4016_a()
    N := 4016;
    order := 1;
    char_gens := [2511, 3013, 257];
    v := [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_4016_a_Hecke(Kf)
    return MakeCharacter_4016_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, 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, 1, 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, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 1, -1, 0, -1, 0, 0, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0], [-1, 1, 0, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0], [1, -1, 0, -1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0], [1, 0, 0, 1, 1, 0, -1, -1, 1, 1, 1, -1, 1, 0, -1, 0, 0, 0, 0], [0, 1, 1, 1, 0, -1, 0, 1, 0, -1, 0, -2, -1, -1, 0, -1, 0, 0, 1], [3, -1, 1, 0, 0, -1, 0, 1, 0, -1, 0, 0, 0, 0, -1, -2, 1, 0, -1], [-1, 2, -1, 0, -1, 1, 1, 0, 0, 0, -1, 1, 0, 2, -1, 1, 0, 0, -1], [-1, 0, 1, 1, 2, 0, 1, -2, 0, 0, 1, 0, 2, 0, 0, 1, -1, 0, -1], [2, 2, -1, 0, -2, 0, 2, 1, -1, -1, -2, 1, -1, 1, 0, 0, 0, 0, 0], [-1, 1, -2, 0, 0, 1, 1, -3, 0, 2, 0, 1, 1, 2, 0, 2, -2, 1, 0], [-3, -1, 1, -2, 2, 0, 1, -2, 0, 1, 0, 1, 1, -1, 0, 0, 0, -1, -1], [3, 0, 0, 2, 0, -1, 0, 2, -1, -2, 1, -2, -2, -1, 0, -1, -1, 1, 0], [0, 2, -2, -1, -1, 1, -2, 1, 0, 0, 0, 0, -1, 2, -1, 1, 2, 1, 0], [2, 0, 1, 0, 1, -1, 1, 0, 0, 0, 0, 0, 1, -1, 1, 0, -1, 0, 0], [2, 0, 2, -1, -1, 0, 0, 0, 1, -1, 0, 0, 1, -2, -1, -1, 1, -2, 0], [1, 1, 0, 0, -1, -1, -1, 1, 1, 0, 1, -2, -1, -1, 0, -2, 1, 0, 0], [0, 4, -2, 0, -2, 3, 1, -2, 1, 0, -1, 1, 1, 3, -1, 1, 1, -1, 0], [4, 1, 1, 0, -1, -1, -1, 3, -1, -1, 1, -2, -3, -1, -1, -3, 2, 0, 0], [-1, 1, 0, -2, 1, 1, -1, -3, 1, 1, 0, 1, 1, 2, 1, 2, 1, -1, 0], [3, -3, 1, 2, -1, -2, 2, 4, -2, -2, 0, -2, -4, -2, 3, -1, -2, 1, 2], [-1, -1, -1, 2, 3, -2, -1, 0, 0, 0, 1, -2, 1, -1, 1, 1, -2, 2, 0], [0, 0, -3, -3, 1, 2, -2, -3, 1, 3, 0, 3, 3, 3, 1, 4, 0, -1, 0], [2, -1, 2, 0, 1, -2, 0, 0, -1, 0, 1, -1, 1, -4, -1, -2, 0, -1, 0], [-1, -3, 2, 1, 1, -1, 1, -1, -1, -1, 2, -1, 0, -2, -1, -1, -1, -1, 0], [-1, 0, -1, 1, -1, -1, 0, -1, 1, 1, 1, 1, 1, 1, -1, -1, 2, 0, -1], [1, -2, 0, -1, 3, 0, -2, -2, 1, 2, 3, -1, 1, 0, 1, 1, -2, 1, 0], [3, 1, -2, 3, 1, -1, -1, 2, 1, 0, 1, -1, -1, -1, 1, 0, -1, 2, 0], [-2, 1, -2, 1, -2, 1, 2, 1, -1, -1, -2, 0, -1, 0, 0, 1, -1, -1, 2], [-1, 3, -3, 0, -1, 2, 0, -2, 2, 1, 0, 2, 1, 1, 1, 2, 0, -1, 1], [1, -2, 3, 0, -1, -1, 0, 3, 1, -2, 0, 0, -1, -3, -1, -4, 3, -2, 1], [2, 1, 1, 3, -2, -1, 4, 2, -2, -2, -2, -1, -2, -1, 0, -1, -3, 1, 0], [2, -3, 0, -1, -2, -1, 0, 2, -1, -1, 1, 0, -3, 2, 2, 1, -1, 1, 1], [3, 1, 2, 2, 0, 1, -1, 2, 0, -2, 1, -3, -1, -2, -2, -2, 0, -1, 1], [-1, 1, -2, -2, 1, 1, -2, -4, 0, 3, 1, 2, 2, 3, -3, 3, 0, 1, -2], [-1, 0, -1, 1, 0, 0, -2, 1, 0, 0, 0, -1, -1, -1, 0, 0, -1, 1, 3], [6, 0, 2, 0, -3, -2, 0, 3, 1, -1, -2, 0, -1, -1, 0, -4, 3, 0, 0], [2, 2, -4, 2, -3, 1, -1, 2, 0, 0, 0, 0, -1, 5, -1, 2, 1, 3, 0], [2, -1, 0, 1, 0, -2, 2, 1, -2, -2, 0, -1, -3, -1, 2, 2, -1, 1, 2], [0, 0, -1, -4, -2, 4, 0, -5, 1, 2, -1, 4, 2, 1, 0, 2, 1, -3, 0], [-4, -2, 0, -2, 3, 2, -1, -3, 0, 1, -1, 3, 3, 1, -1, 3, 0, -1, -2], [3, 0, 0, 2, -2, -1, 0, 3, -1, -3, -2, -2, -3, -2, 2, -2, 1, -1, 2], [0, -1, 1, 0, 4, 0, -1, -2, 0, 2, 1, -1, 1, -1, 1, 2, 0, 0, -1], [1, 3, -2, 0, -1, 0, -4, -2, 3, 2, 3, -3, -2, 4, -1, -2, 2, -1, 0], [3, -3, 4, -1, 1, 0, -2, 1, -1, -2, 2, 0, 0, -2, -2, -2, 1, 0, 1], [0, 2, 2, 2, -2, 1, 2, 2, -3, -2, -2, -1, -1, -2, -2, -2, 0, 0, 1], [2, 5, -5, 5, -3, -1, 3, 4, -2, 0, -2, -1, -6, 3, 4, 1, -2, 3, 2], [7, -5, 3, -1, 4, -3, 0, 3, 0, -2, 1, -3, -1, -3, 2, -1, 1, -1, -1], [7, -3, 3, -2, 1, -2, -3, 1, 1, 1, 1, -2, 1, -6, 0, -4, 2, -2, 1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 1, -3, 1, 4, 0, -1, -1, 0, 1, -3, 1, -1, 1, 0, 1, 0], [-4, -1, -1, -5, 0, 4, -2, -5, 0, 3, 1, 4, 4, 1, -2, 2, 0, -2, 0], [-1, 0, 0, -3, 4, 0, -5, -4, 3, 1, 2, 1, 4, 0, -2, -1, 3, -2, -2], [4, -3, 1, 2, 1, -2, -1, 4, -3, -3, 0, -2, -2, -2, 0, -2, 0, 2, 0], [2, 1, 1, -1, -5, 2, 1, 2, 1, -3, -1, 0, -2, 1, -3, -3, 4, -2, -1], [-2, 1, -5, -1, -1, 1, 3, -1, -2, 0, -2, 3, 0, 4, 3, 6, -4, 2, 0], [-2, 1, -1, 3, 0, 0, 2, 2, -1, -2, 2, -2, -1, 1, 2, 1, -3, 1, 1], [-6, 2, 1, -4, 1, 2, -1, -3, 3, 2, 1, 1, 2, -1, -3, -2, 1, -1, -2], [1, 2, -1, 0, 1, -1, -3, 3, 0, -1, 1, -2, -1, 0, 1, -1, 1, 3, 2], [-7, -1, 1, -3, 1, 3, 0, -6, 3, 3, 2, 3, 4, 2, -2, 3, 1, -1, -2], [7, -2, 3, -3, 1, -2, 4, 3, -2, -3, -3, 0, -2, -5, 3, -3, 0, -1, 0], [-7, 0, -1, 1, 2, 4, -3, -1, 3, 2, 1, -1, 3, 3, -2, 2, 1, 0, 0], [-2, -1, -1, 2, -1, 2, 3, -5, 1, 2, 0, 2, 3, 3, 2, 5, -3, 2, -1], [6, -4, 0, 1, 5, 0, -4, -2, 1, 0, 3, -1, 3, 1, -3, -1, 2, 1, -2], [2, -5, 2, -2, 5, 0, -2, 0, -1, 0, 0, 2, 3, -3, 1, 1, 1, 0, 1], [-3, -4, 2, -4, 3, 2, 1, -4, 0, 1, 1, 4, 4, 0, 1, 2, 1, -1, -2], [-3, 5, -4, 0, -2, 1, -1, 2, 2, 0, 0, -2, -1, 1, 0, -3, 1, -2, 0], [1, 1, -4, 7, -3, -2, -2, 1, 1, 1, 0, -4, -1, 1, 0, 1, -1, 2, 2], [4, 1, 1, 1, -5, -3, 1, 5, -2, -4, -1, -2, -6, 1, -1, -3, 3, 1, 1], [3, 1, -2, 3, 2, -1, -1, 0, -1, 1, -1, -1, 1, 2, 0, 4, -4, 3, 0], [0, -4, 2, -4, 0, 2, 0, -4, 0, 1, 0, 2, 1, -1, -4, -1, 4, -4, -1], [-2, 0, -5, 2, -1, 1, 1, 1, 0, 0, 0, -1, -1, 2, -3, -1, -2, 4, 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1, 2, -3, 2, 0, 3, 1, -2, -2, 0, 1, -1, 1, 1, 4], [-8, 6, -1, -2, -6, 1, -6, 1, 0, 2, 3, -1, -3, 1, -6, -4, 3, -1, -3], [6, 6, -1, -1, -8, 0, 1, 7, -4, -3, -9, 3, -5, 1, 2, 1, 4, 3, 0], [-10, 3, -7, -2, -4, 3, 7, -7, 1, 4, -3, 8, -2, 8, 4, 5, -2, 3, 0], [13, 6, -5, 10, -8, -4, -1, 11, 1, -4, -2, -8, -9, 0, 0, -6, 0, 5, 4], [2, 5, -8, 3, 0, -1, -2, -3, 3, 3, 1, 1, 5, 3, 2, 4, -4, 2, 3], [8, -8, 3, -2, 1, 2, 1, 0, -3, -1, 0, -1, 1, -2, -1, 3, -1, -1, -2], [-4, -6, 8, 2, 3, -3, -1, 0, -1, -5, 3, -3, 6, -4, -7, -4, 1, -1, -3], [-1, -1, -3, 1, 5, 0, -5, 0, -3, 1, 0, -1, 1, -4, -3, 0, -1, 1, 0], [-10, 7, -2, -1, -5, 5, 7, -4, -5, 0, -4, 4, -2, 4, 1, 5, -2, 0, -1], [4, 1, 5, 1, -5, -3, 1, 2, 0, -1, 0, 0, 0, -3, 3, -4, 1, -5, 3], [0, 4, 0, 0, 1, -3, 1, 4, 1, -4, 2, -4, -7, 4, 3, -2, 2, -2, 1], [-3, 8, -6, 12, 1, -1, -2, 4, 3, 1, 6, -15, -4, 1, -1, -4, -5, 4, 1], [-4, -7, 2, 3, -3, -3, 5, 0, -5, -3, 1, 2, -3, -4, 2, -3, -4, -1, 1], [-13, 2, -3, -4, -4, 0, 2, -4, 3, 3, -2, 2, 6, -2, 4, 1, -1, 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3, 8, -4, 4, -4], [-8, 14, -5, -1, -5, 2, 0, -2, 0, 1, -4, 2, 5, -1, 1, 1, 0, -1, 1], [-1, 3, -2, 8, -5, 0, 6, 2, -5, -2, -3, -3, -7, 5, 3, 4, -6, 5, 2], [9, 8, -4, -4, -2, 0, 0, 3, 2, 0, 0, 5, -3, 5, 2, -1, 7, 1, 2], [-5, 6, -8, 1, -5, 5, 0, 4, 0, 2, -4, 1, 0, 5, 1, 6, 1, 2, -2], [11, -11, 6, 6, 5, -5, 0, 12, -2, -6, 4, -11, -7, -11, 5, -3, -5, -1, 4], [-2, 6, -6, 0, -5, 7, 2, -3, 4, 1, -1, 1, -2, 9, -4, 4, 4, 2, -2], [15, -6, -3, 1, 5, 1, -3, 12, -3, -6, -1, -3, -2, -5, 2, -1, 3, 0, 0], [-2, -6, 0, -3, 5, 1, -3, 7, 3, 2, 6, -7, -4, -9, 7, -4, -3, -3, 5], [-4, 3, 5, -8, 3, 5, -4, -16, 5, 9, 2, 7, 13, 7, -3, 6, 4, -3, -6], [15, 2, -6, 3, -13, -6, 0, 8, -5, -4, -1, -3, -11, -2, 1, 1, -2, 5, 4], [3, 1, -6, -1, -5, 2, 3, -7, -2, 6, -1, 5, 2, 10, -3, 3, -2, 0, -3], [0, -2, -5, -4, 7, 5, 5, -4, -2, 4, -3, 7, 5, 2, 3, 5, -6, 5, -5], [-2, -11, 1, -9, 8, 4, -2, -9, -3, 5, 0, 5, 10, -7, 2, 7, -2, -4, 2], [6, 1, 2, -11, 9, 6, -5, -6, 0, 3, -1, 8, 6, 8, -3, 9, 7, -5, -2], [-3, 5, 5, 6, -2, -4, 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-4, -2, -3, -4, 0, -3], [-1, 6, -1, 4, -7, -5, -7, 12, 0, -3, 2, -7, -5, -8, 0, -5, 6, -1, 5], [2, 10, 1, -7, 0, 1, -5, -1, -1, -1, 1, 4, 2, -1, 2, -1, 2, -5, 3], [-1, 1, -3, 4, -7, -1, 7, -3, 2, -2, 1, 0, 0, 5, -1, 2, -1, -1, -3], [14, -7, 4, 2, 7, -6, 5, 0, -7, 0, -1, -4, 1, -8, 9, 1, -10, 0, 5], [14, 9, -6, -1, -4, 7, -3, -6, 1, 2, 2, 2, -3, 14, -8, 4, 5, 4, -6], [3, 5, 2, 5, 1, -3, -1, 3, -3, -4, 2, -5, -6, 2, 1, -4, -5, 3, -3], [3, -1, -7, 4, 8, -1, 9, -9, -1, 2, -6, 7, 3, 11, 4, 6, -5, 4, -3], [8, -3, 7, 3, 3, -5, -7, 5, 4, -2, 5, -6, 2, -5, -5, -11, 1, 1, -3], [-12, 4, -3, -2, -6, -1, 0, 3, 1, 0, -3, -1, -7, 0, 2, -6, 2, 2, -2], [5, -6, -3, 1, 7, 1, 8, 1, -3, 0, -2, 2, -2, -1, 5, 3, -9, 8, -1], [8, -13, 9, -4, 4, -6, -3, 8, -8, -4, 1, -2, -5, -5, 4, 0, 1, 0, 5], [2, 3, 7, 2, -2, -5, -3, 7, 5, -5, 2, -9, 3, -8, -8, -13, 7, 0, -4], [10, -11, 2, -2, 5, 2, -2, 1, -1, 0, 1, 2, -3, 5, -5, -5, 1, 2, 2], [2, 2, 1, 9, -8, -4, 11, 8, -2, -7, -6, 1, -5, 1, 0, -7, 2, -3, 1], [-10, 13, -18, 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1, -2, -1, 0, 1, 8], [-2, 1, -6, 13, -1, 0, 7, 10, -6, -1, -5, -2, -10, 6, 5, 5, -7, 7, 2], [0, -14, 9, -7, 12, 5, -11, -6, 5, 6, 9, 1, 15, -9, -10, -7, 3, -2, -2], [5, -10, 21, 5, 12, -8, 2, 2, -5, -9, 2, -6, 10, -19, -3, -10, 2, -3, 0], [21, -1, 4, -1, 5, -3, -2, 7, -5, -3, 3, -5, -5, 1, -5, 1, -1, 2, 0], [7, 4, -2, -1, -5, 1, -4, -1, -2, 2, 1, -5, -6, 1, -5, -6, -3, -2, -3], [7, 2, 3, -6, -3, 7, -7, 1, 2, 4, 1, 2, -6, 3, -7, -6, 11, -5, -3], [11, -5, 5, 7, 2, -1, 4, 0, -4, -3, 0, -2, -4, -6, -1, -4, -2, -1, -3], [5, 6, 8, -9, 2, 3, 3, -2, 4, -1, -2, 4, 1, -4, 0, -3, 5, -8, -3], [7, 14, -9, 6, -6, 3, 0, 5, 6, -1, -5, -6, -5, 8, -1, -1, -2, 3, 2], [-22, 13, 3, -1, -4, 1, 2, -6, 2, 2, 0, 1, 2, 3, -2, -1, 1, -1, 2], [-7, -4, 6, -7, 7, -1, -1, -2, -4, 3, -1, 3, 5, -1, 4, 2, 0, 1, -2], [-11, 2, 1, -1, 8, 4, 8, -14, 2, 2, -1, 9, 11, 1, -7, 3, -1, -8, -3], [0, 13, -5, 3, -15, 0, -7, 2, -1, 0, -1, 0, -3, 4, -9, -2, 4, 4, -4], [20, 8, -9, 3, -10, 6, 0, -6, 0, 5, -1, -3, -1, 14, -4, 5, -5, 3, -1], [0, 3, -1, 18, -2, -5, 5, 7, -1, -2, 3, -9, -4, -4, 1, 1, -7, 11, 4], [-2, 13, 1, 4, -4, -3, 3, 7, 1, -4, -3, -2, -10, -10, 2, -7, 4, -3, 4], [17, 2, -2, 2, -8, 1, -1, 14, -3, -5, -8, 2, -6, 1, 2, -4, 1, -2, 5], [-1, 16, -8, 11, -2, 2, 7, -1, -1, -5, -4, -2, -2, 3, -6, -2, -4, 5, -4], [20, 3, -3, 0, 6, 2, 3, 3, -2, -3, -2, 2, -2, 4, 12, 7, -1, 0, 3], [-9, -2, -1, 1, 5, 5, -5, 4, -1, 2, -1, -3, -3, 5, 3, 8, -4, 4, 5], [4, 14, -2, -8, 1, 6, -3, -2, 2, 0, -5, 8, 4, -1, -3, -1, 3, -7, -1], [-8, -4, -3, 7, -4, -1, 5, 10, -9, -3, -1, 1, -6, -6, 8, 4, -9, 4, 5], [5, 5, -3, 2, 1, -5, -2, 5, 1, -2, -1, -5, -2, -5, 7, -1, 0, 5, 6], [18, -3, 2, -1, -7, -3, -8, 9, -2, -5, 4, -3, -9, 3, -10, -11, 9, 2, 1], [11, 5, -9, 7, -7, -6, 1, 15, -5, -6, -2, -2, -16, 0, 3, 0, 0, 4, 9], [-7, 4, -2, -2, 8, 1, -5, -10, 2, 1, 0, 3, 9, 2, 7, 11, 0, -1, 1], [3, 10, -13, 2, 5, 3, -3, -1, 6, 5, -2, 3, -1, 5, -1, 4, 2, 3, -1], [1, 9, 0, -2, 9, 0, -1, -7, 4, 5, -2, 1, 7, 0, 6, 3, -2, 2, 1], [0, -17, 2, -3, 1, 2, -3, -8, 4, 4, 4, 3, 13, 0, -3, -3, 2, -4, -2], [4, 3, -2, 4, 2, 2, 3, -7, 3, 0, -1, 3, 11, 5, -2, 5, -3, 4, -2], [5, -6, 5, 2, -4, -5, -3, 3, 2, 0, 2, -3, 0, -12, 1, -13, 4, -8, 0], [0, 11, -7, -3, 2, 1, 4, -2, -5, 4, -6, 5, -4, 2, 6, 3, -4, -2, 3], [-27, 0, 2, -7, 10, 1, 0, -6, -1, 4, 0, 3, 2, -3, 5, 5, -1, -2, -1], [5, 4, -2, -9, 5, 1, -7, -8, 1, 4, 2, 6, 2, 7, 2, 7, 1, 6, -1], [13, -11, 11, -1, 8, -1, 0, 0, -2, -2, -1, 0, 4, -9, 8, 4, -2, -2, 1], [15, 7, -7, -3, -15, -1, 2, 12, -1, -2, -9, 2, -12, 1, 5, 3, 5, 2, 8], [-2, -13, 11, -10, 12, 2, -4, -8, 7, 5, 3, 1, 5, -9, 2, -2, 6, -12, 1], [10, -15, 5, 2, 4, -3, -8, 4, 0, 4, 1, -7, 4, -4, 5, 0, -3, 1, -3], [-9, -6, 6, -5, 6, 3, 8, -11, 0, 5, 3, 9, 11, -4, 1, 3, -1, -4, -4], [3, -5, 4, -2, 1, -2, 2, 7, -3, -11, 3, -2, -1, -12, -3, -6, 5, -2, 3], [-18, 6, -8, -2, 3, 5, 1, -16, 5, 5, 0, 12, 12, 5, -1, 11, 4, 0, -4], [14, 11, -6, 0, -3, 3, 6, 6, -6, -1, -4, 4, -3, 6, 0, 4, -2, 2, -1], [-1, -1, -1, 1, -6, -3, 6, 5, 0, -4, 0, -2, -3, 4, 0, 0, 6, -2, -4], [10, -1, -4, 1, 3, 3, 4, 2, -7, -6, -7, 4, 0, -1, -4, -2, -1, 1, -2], [16, 11, 1, 4, -6, -6, -3, 5, 4, -6, 6, -6, -11, -3, -2, -4, 5, -2, 2], [8, -1, 2, -6, 4, 4, -13, -7, 5, 5, 5, 3, 4, 2, -8, 0, 7, 1, -3], [-3, -5, 1, 6, -4, -9, 11, 10, -5, -9, -2, -4, -6, -9, 0, -8, -2, 3, 0], [3, -3, -7, 4, 6, 0, -5, -9, -2, 9, 6, -4, 6, 1, -2, 5, -7, 6, 3], [-9, -11, 3, -2, 3, -4, -8, -12, 8, 6, 4, 2, 14, 0, 1, 3, -2, 1, -4], [-15, 2, 14, 5, 4, -9, 5, 9, 0, -7, 1, -10, -1, -26, -2, -17, -1, -2, 0], [8, 7, -4, -6, 1, -4, -6, -10, 1, 4, 0, 8, 7, 7, -5, 5, 3, 6, -9], [-7, -4, 9, -8, -2, 4, 6, -7, -1, 3, 3, 8, 10, 6, -6, -1, 3, -7, -5], [15, -9, -2, 5, 8, -4, 0, -2, -2, -2, 0, 2, 2, -8, 2, -1, -6, 3, 2], [7, -2, 9, -1, 19, -5, 4, -5, 5, 0, 3, -1, 12, -12, 5, 0, -4, -7, -4], [10, -4, 8, 5, 6, 1, -7, -3, 2, -2, 11, -12, 7, -7, -8, -8, -3, 0, -3], [-9, 3, 0, -6, 1, 4, -6, -9, 11, 4, -1, 4, 14, -5, -5, -1, 7, -8, 2], [3, 3, 1, 10, 9, -6, 2, 1, 0, -1, 3, -12, 1, -7, 1, 0, -10, 4, -4], [8, -5, 1, 0, -6, -1, -2, 6, -3, -1, 1, 0, -1, -8, 0, -6, 1, -3, 4], [12, -1, -5, -5, -2, 3, 6, 2, 2, 4, -7, 10, 0, 6, 6, 4, 2, 2, -3], [-15, 2, -5, 2, 4, -3, -2, -1, 0, 5, 6, -3, 3, 1, 3, 8, -5, 6, 1], [6, 1, -3, -7, 4, 8, 6, -3, 1, 6, -7, 14, 7, 5, 6, 7, -3, -1, -2], [-5, 1, -4, 3, -2, -1, -3, 7, -2, 3, 0, -7, -1, -13, 2, -2, -6, 1, 7], [-15, -1, 5, -8, -4, -1, -2, -1, 0, -2, 2, 1, 1, 2, 3, -4, 7, -5, -4], [-11, 0, -6, -10, -8, 3, 0, -1, -3, 0, -7, 12, -4, 12, -3, 8, 9, 2, -2], [-12, 0, -8, -1, 0, 6, -1, -9, 3, 5, 7, -5, -3, 6, 0, 3, -7, 2, 1], [-5, 14, -11, -10, 4, 12, -3, -16, 4, 8, -5, 14, 8, 11, -4, 14, 7, -4, -5], [6, 7, 7, -10, 2, 4, -3, -1, -1, 1, 1, 3, -2, 2, -4, -4, 7, -8, -1], [-4, 10, -4, 6, -3, 0, 9, 0, -4, 0, -4, 5, -4, 6, 7, 10, -2, 6, 4], [3, -2, 5, 6, 1, -6, 10, 10, -3, -8, -6, -4, -9, 1, 7, 1, -5, 4, 2], [-18, 14, -10, 7, -2, 3, -1, -1, -3, -1, -2, -3, -10, 4, -4, 0, -3, 2, 2], [4, 4, -2, -7, -12, -2, 4, 8, -1, -6, -6, 10, -2, 1, -4, -1, 11, -5, 0], [15, 5, 2, 4, -9, 2, 7, 4, -3, -7, -4, 1, -13, 6, 2, -2, 9, -2, 4], [-8, -7, -3, 8, 1, 0, 8, -2, 0, 0, 1, -5, -1, -1, 3, 0, -3, 2, -1], [5, -4, 3, -1, 8, 1, 1, 3, 1, 3, 1, -4, -1, 3, 0, 1, -4, 5, -3], [-20, 10, -7, 3, -11, 2, -8, 8, -4, 0, 4, -8, -13, 1, -1, -2, -6, 5, 10], [-11, -4, 2, -14, 6, 1, -10, -7, -1, 10, 1, 4, 8, 0, -7, 0, 5, 0, -6], [-5, 0, -3, -2, -10, 2, -2, -7, 7, 4, 0, 9, 9, 4, -6, -1, 6, -8, -7], [7, -5, 17, -5, 5, -5, -10, -2, 1, -1, 6, -7, 2, -12, -10, -9, 3, -5, 0], [0, 1, 10, 11, 7, -5, -1, 1, 1, 0, 2, -1, 4, -5, -7, -8, -1, -1, -1], [-5, 6, -5, -7, -3, 4, 2, -4, 4, 3, -7, 5, -2, 6, 3, 2, 5, -4, 2], [-5, 6, -2, -2, 14, 3, -10, -12, 5, 11, 6, 1, 13, 3, -2, 7, -6, -4, -6], [14, 2, 1, 2, 3, 0, 5, 7, -7, -5, -4, 0, -9, 3, -3, -2, 4, 0, -2], [-10, 1, 0, 2, 6, 0, -6, -8, 3, 6, 6, 2, 11, 3, 7, 11, -3, 0, 3], [-2, -12, 3, 16, -4, -7, 5, 1, -1, 0, 10, -12, -9, -1, 9, -1, -12, 5, 9], [11, -10, 9, -6, 3, 0, -8, 2, -2, -2, 1, -1, 0, -19, -5, -10, 5, -8, 3], [-7, 1, 0, -8, 6, -3, 1, -10, 2, 7, 2, 0, 4, 6, 2, 4, -3, 2, -5], [9, 2, 0, 6, 12, 1, -2, -10, 7, 5, 1, -2, 3, -2, 4, 6, -7, 1, 1], [19, -6, 11, -3, 0, 1, -4, 3, 2, -7, 5, -5, 1, 5, -11, -10, 6, -1, -8], [17, 10, -4, 4, 3, 2, 3, -2, 0, -1, -1, 2, 4, -2, -7, -4, 1, -3, -2], [35, -3, 4, 3, 6, 0, 10, 1, -2, -8, -5, 6, 4, 2, 2, 7, 2, -2, -3], [-1, 0, -2, -1, -4, 1, -14, 0, 5, 5, 5, -2, 2, 2, -8, -9, 5, 3, 1], [7, 12, -3, 7, -7, -3, 1, 7, 0, -4, -7, -3, -6, 7, 8, 10, 8, 4, 7], [-15, 13, -7, -1, -14, 2, 8, 1, -1, -3, -6, 3, -7, 5, 0, 5, -2, 0, 2], [-5, 6, -6, 16, 1, -5, -4, 8, 2, -3, 9, -16, -10, -1, 1, -9, -4, 3, 6], [-16, -6, 1, -1, 4, 7, -2, -11, 4, 7, 0, 7, 14, -3, 7, 12, -5, -3, 3], [-15, -10, 11, -10, 12, 6, -8, -11, 0, 9, 3, 4, 8, -9, -2, 1, 8, -6, 2], [-3, -15, 5, -12, 16, -1, -3, -12, 1, 8, 9, 1, 5, -1, 0, 2, -1, 0, -10], [-20, -3, -3, -5, -4, 10, 2, -9, -1, 9, -5, 8, 10, 4, 2, 10, -7, 1, 3], [2, 4, -7, -6, -5, 5, -3, -2, -2, 3, -4, 4, 6, 2, 7, 11, 5, -1, -1], [-34, 2, 2, -14, -8, 3, -7, 2, 0, 1, -3, 7, 2, -4, -1, 1, 5, -5, -2], [-6, 14, -8, 0, -5, -1, 3, 6, -5, -1, -9, 3, -4, 0, 1, 2, 1, 5, -1], [-8, 5, -8, 2, -1, 4, 3, -10, 6, 9, -3, 5, 2, 8, 2, 7, -4, 4, -2], [9, -2, -1, 5, 5, -12, -6, 12, 2, -2, 6, -10, -7, -8, 7, -10, -4, 9, 4], [-7, 10, -1, 0, 9, 8, -3, -10, 4, 0, -1, 2, 7, 5, -5, 1, -1, -3, 3], [12, 3, -2, 11, -7, -7, 6, 14, 1, -4, 0, -3, -3, -10, -3, -6, -1, 0, 0], [-9, -11, 2, -5, 6, 0, -4, -6, -4, 3, 2, 2, 9, 5, -8, 0, 0, -2, -5], [16, -1, 0, 5, -6, -8, -7, 12, -3, -5, 5, -12, -16, 4, -1, -4, 1, 1, 5], [-5, -7, 4, 1, -2, 1, 2, 3, -3, 2, -7, 3, 1, -2, 4, 1, -2, -4, 6], [2, -3, 6, -9, -6, 1, -2, 5, 3, -3, 5, 0, -9, 1, -7, -12, 11, -6, -3], [14, -2, -3, 0, -8, 3, 0, 2, 2, 0, -1, 2, 5, -2, 2, 1, -4, 2, 3], [15, -3, -1, 10, 8, -6, -8, 1, 2, -4, 6, -15, 2, -11, 5, -4, -7, 4, 8], [22, 1, 2, -1, 2, 1, 5, -4, 2, 3, -2, 3, -1, 0, -1, 0, 0, 1, 3], [13, -12, -11, 11, -1, 3, -8, -3, 0, 4, 2, 0, 8, 12, -3, 7, -4, 5, -4], [6, 4, -8, -12, 4, 4, -10, 4, -2, 0, 2, 0, -6, 10, -2, 4, 4, 4, 2], [6, 0, 6, -7, -13, 0, -1, 12, -7, -10, -2, -1, -6, -11, -2, -12, 5, 0, -1], [-5, -6, 8, -6, 4, 1, 17, -8, -7, -1, -10, 16, 12, -9, 8, 6, -7, -2, -1], [12, -11, 10, -6, 11, -3, 5, 0, -7, -11, -2, 4, 1, -6, -6, -3, 9, -4, -2], [18, -3, 0, 9, -6, -2, -4, 7, -2, -6, 4, -11, -9, -1, 6, -2, -5, 5, 6], [5, -8, -1, 5, 1, -4, -6, 8, -6, -2, 4, -6, -3, -2, 1, 0, -2, 9, -2], [-13, 4, 4, 8, -5, -3, 1, 10, -1, -3, 5, -13, -8, -5, -3, -10, 1, 3, -1], [-6, -7, 1, -6, 8, 1, 1, -6, -1, 6, 3, 5, 4, 1, -2, 2, 5, -2, 0], [-22, -1, 6, -4, 0, 1, 2, -12, 6, 6, 2, 3, 9, 1, -2, -4, 0, -4, -1], [16, -8, 4, -8, 3, 0, -5, -3, -1, 0, 1, 5, 4, 1, -11, -6, 11, -1, -8], [2, 2, -1, -7, 9, 7, 6, -5, -1, 5, -2, 8, 4, 2, 7, 10, -9, 0, -2], [-16, 17, -14, -1, 1, 4, 4, -10, 3, 4, -6, 6, 2, 5, 5, 12, -1, 2, 1], [-4, 13, 4, -6, 3, 8, -4, -3, 7, 4, -2, 5, 7, 4, -13, -6, 6, -5, -3], [-6, -8, -9, 2, -7, 4, -3, -5, 1, 3, 3, 3, -3, 14, 1, 9, -3, 9, 1], [-6, -1, 1, 2, 5, -2, -5, -4, -2, 3, -4, 3, 14, -5, -3, -7, 4, 0, -6], [5, -5, 0, 2, 1, -2, -2, 10, -1, -3, 0, -2, -1, -6, 1, -14, -3, 0, 2], [8, -14, 3, 7, -5, -10, 8, 11, -6, -8, -2, -7, -12, -7, 5, -5, -3, 0, 8], [-3, 0, -11, 0, 1, 1, 19, -5, -2, -1, -9, 14, 1, 10, 11, 9, -3, 1, 4], [-1, 0, 0, -10, 3, 3, -5, -4, 5, -1, -5, 9, 9, 4, -7, 6, 9, -7, -5], [16, -2, 1, 3, 0, -2, 1, -3, 2, 3, -1, 3, 7, -1, 8, 9, -4, -1, 0], [12, -10, 10, 4, 7, -3, 2, 4, -2, -6, 1, 3, 3, -2, 2, -3, 7, -2, -3], [19, -8, 0, 8, 8, -1, -6, 7, -3, 2, 2, -13, -10, -3, 1, 1, -4, 7, 5], [12, -5, 3, 5, -18, -9, 12, 19, -10, -14, -4, -6, -22, -4, 0, -12, 6, 4, 4], [-7, 0, -5, 5, -4, 5, 3, 5, 1, -4, -3, 4, -4, 8, 0, 2, -4, 5, -2], [18, 1, 5, 15, 3, -10, -4, 1, 1, 3, 4, -15, 1, -7, 1, -4, -13, 6, 1], [25, -1, -4, 14, -5, -3, 0, 9, -6, -4, -1, -15, -17, 5, 7, 1, -7, 8, 5], [-1, -4, 12, -5, 8, 0, 2, -3, 1, -7, -1, 6, 6, 2, -6, -2, 11, -5, -8], [31, 2, -11, -6, -10, 2, -4, 9, -5, -5, -4, 3, -10, 7, -2, 2, 10, -2, -2], [-4, 5, -6, -2, -10, 8, -9, 4, 7, -1, -6, -1, -6, -2, 0, 4, 7, -6, 10], [-17, 3, -3, -10, -5, -2, -8, -1, 12, 1, 3, 0, 3, -7, -5, -12, 12, -6, -2], [9, -8, -1, -5, 1, 0, -18, -10, 8, 7, 7, -1, 10, 8, -7, 2, 9, 3, -2], [-18, -14, -4, -3, 5, -1, 0, -11, -6, 10, 5, 1, 1, 3, 3, 6, -11, 8, 3], [-18, 24, -9, -7, -6, 5, -1, -9, 3, 4, -8, 7, 1, 15, 6, 13, -3, -1, 5], [-7, -2, -5, -1, 3, -2, 6, -6, 0, 10, 2, 8, 5, 10, 17, 10, -12, 7, -2], [-30, -4, 4, -14, -3, 3, -7, 5, 5, -3, 0, 3, 0, 0, -5, -8, 15, -5, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4016_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4016_2_a_l();".
// 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_4016_2_a_l(:prec:=19)
    chi := MakeCharacter_4016_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_4016_2_a_l();".
// 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_4016_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4016_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, -747, -7607, -13777, 33209, 64793, -72526, -80907, 76260, 43551, -41921, -10845, 12681, 926, -2109, 90, 179, -21, -6, 1]>],Snew);
    return Vf;
end function;