
// Make newform 8022.2.a.z in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8022_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_8022_2_a_z();".
// 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_8022_2_a_z();".
// 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 := [4704, -13616, -16144, 62828, 7822, -84842, 2130, 45390, -615, -10139, 45, 1068, -1, -53, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-7, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1152392517613515277584, 216349037206284972520, -5379972760527430385824, -1454391463816287278438, 6192320267897190525666, 1401466170449747084734, -2701455319595076197340, -397820650922556789157, 542030391828967189447, 47291815758556009753, -53924977414688377018, -2513103468751495587, 2589151861807955487, 49332284762810190, -47844143301035099], [972715425559024637664, -1863587837228183355760, -5416320835690325800040, 5432623156253615496098, 9152779826696919384090, -4340534606751413151214, -5832120207138219564504, 1033349294454819909079, 1395329695881849133439, -107643012000915815299, -151656890480087505806, 5190919725719812869, 7649328658641877611, -95336249374922946, -145674275134685923], [-2002669376503714503696, 2868338026195500538272, 10486974637156346458736, -8917630560881720595370, -16967789826021060630834, 7188181795410538835814, 10398114673415142018212, -1726900772962635860003, -2451587671951550009307, 181506316507140562235, 264823325479108909458, -8822558076321033065, -13320607269945514819, 163121527367248846, 253373654766014459], [1206422294142049350504, -834595104935603406168, -6578574159891447888460, 3578981562674026030742, 10365451786874507600046, -3104114478921165345438, -6113609103065956762018, 742126355596970318929, 1417576896961580856189, -76830397287487824811, -151809858709144529196, 3677196111640734007, 7596232450925625803, -67158442937616560, -143980564676940343], [4542537945343916384832, -3899901115490818892152, -22952407479971842707792, 13200028444408522075990, 33373539998586804849870, -10495753003077381214318, -18940119918113541544108, 2476016431170807843125, 4330733628662455297817, -256299807939580952225, -460925788792527955646, 12311307959306123203, 22992219471492275225, -225793991413084982, -435077682637100285], [805397881609767914556, -1280281318799657216572, -4170727637139449111910, 4534350434211526941760, 7087919805467433423612, -3712164418355826904708, -4519906876304426386669, 902828136570275154188, 1080789454132306352144, -95940771488732908961, -117429187152098562389, 4708706949542762884, 5922298344475101917, -87742709231470043, -112790344931326241], [-895775762206901213472, 1190538423485802479440, 5076173451478455403366, -4313146196078542278522, -8534637097252352622594, 3499047092328247027912, 5277326861489579230277, -834512276572118834799, -1246618106742559952105, 87023924027426484918, 134696803396025775311, -4200761904910892459, -6773705946064420810, 77237585175435763, 128794562939054244], [-1957215294219116942160, 3343798898748498202936, 10809145508320427944858, -11445320208127070346804, -18187811553138106782456, 9092146572248850432838, 11355123931290340017479, -2182757191771485703410, -2692389644334917119448, 229919971336010242191, 291401389817549796611, -11207020184843952140, -14667014841434591587, 207687442650093487, 279026151355819425], [-1195388994843941406456, 1848534323084673928088, 6702141628972424751542, -6502986235248961530102, -11331099879512724663330, 5250954159974349563444, 7056391543742806488717, -1263088163896430650861, -1671375884765616054283, 133039038295354289916, 180824512333209673471, -6483381202731471853, -9100209681746782816, 120176375325262075, 173119505841442118], [-8797899846853478728656, 11318351887192471778592, 50352317577861990590464, -44167667913375731550710, -85392155189455265874222, 37115891207988086050122, 53424516372419043995356, -8980266604861904662237, -12684048256174737262341, 945479263628513599765, 1374118258779025758558, -45976513730063865319, -69211945835863404749, 850209309626005346, 1317384423576065221], [-5217773626313525601528, 7313700939316505024480, 28033124759210837863734, -25997785502318239929068, -46170514381419896585760, 21133358892345838770122, 28565379640083352955801, -5089691294902940317042, -6753625554358573827100, 536023255394080824133, 730099435097944633141, -26104614174926647544, -36727261839951130129, 483435293977683721, 698489089123427251], [-10648217680809741481920, 12890322571316985517456, 57679223018953714068464, -46806338252758402728074, -94498052084908879726242, 38424449791032430708966, 58031340399936060374532, -9209169621689696787019, -13685037087157198369715, 963630109770286975987, 1477960742755837792178, -46651004556455924913, -74317793613973453179, 860002258497780510, 1413151433182310851]];
    Rf_basisdens := [1, 1, 1, 71344467406408743984, 71344467406408743984, 71344467406408743984, 35672233703204371992, 71344467406408743984, 17836116851602185996, 17836116851602185996, 35672233703204371992, 11890744567734790664, 71344467406408743984, 35672233703204371992, 71344467406408743984];
    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_8022_a();"
function MakeCharacter_8022_a()
    N := 8022;
    order := 1;
    char_gens := [2675, 6877, 2311];
    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_8022_a_Hecke(Kf)
    return MakeCharacter_8022_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 := [[-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, 0, 0], [0, -1, 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, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [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, -1, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0], [2, 0, 0, 1, 0, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0], [1, 0, 0, 0, -1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1], [1, 0, 0, 0, 1, -1, -1, 0, 0, 0, 0, 0, 0, 0, -1], [1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 1, 0], [3, 0, 0, 0, 1, -1, 0, 0, 0, 0, 1, 0, 0, 1, 1], [2, 0, -1, 0, 0, -1, 1, 0, -1, 0, -1, 1, 1, 0, 0], [-1, 0, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, 1, 0, 0], [1, 0, -1, 1, 0, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0], [2, 0, 0, -1, -1, 0, 1, 0, -1, 1, 0, 0, 1, 0, 0], [1, -1, 0, 0, 0, 0, 0, -1, -1, -1, 1, -1, 0, 0, 1], [-1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 1, 1, -1, -2, 0], [2, -2, 0, 1, 1, 0, -1, -1, 1, 0, 0, 0, -1, 0, 0], [1, 0, 0, 1, -1, 0, 0, 0, 0, 0, -3, 1, 0, 0, 0], [2, 2, 0, 0, -1, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0], [1, 0, 0, -2, 1, 0, 0, 0, 0, 0, 2, -2, 1, -1, 0], [4, 0, 0, 0, 1, 0, 0, 0, -1, 0, 1, -1, 0, 0, 1], [2, 0, 1, -2, 0, 0, 1, 0, -1, 0, 1, -1, 0, 1, 0], [2, 0, -1, 0, 0, 0, -1, 0, 1, 1, -1, 1, 0, 0, 0], [3, 2, 0, 0, 1, 1, -4, 0, 1, 0, 2, -1, 0, 0, 1], [-1, 0, 0, 0, -1, 0, 0, -1, 0, -1, 1, -1, 0, 0, 1], [2, -1, 0, 1, 0, 0, 1, -1, 0, 1, -2, 0, -1, 2, 0], [-1, 0, 0, 0, -1, 1, 1, 0, 0, 0, -2, 2, 1, -1, -1], [1, -2, 0, 0, 0, 1, 0, 0, 1, 0, 0, -1, 1, -1, 0], [5, 0, 1, -2, 0, 0, -1, 0, 0, 2, 0, 0, 0, 0, -1], [0, 0, -1, -1, 0, 1, 1, 1, 1, -1, 0, 1, 1, -3, 0], [0, 0, 1, 1, 0, 0, -2, -1, 1, 1, 0, -1, -1, -1, -1], [0, 0, 1, 0, 0, 1, 3, 0, -1, 0, -1, 0, 0, 0, 0], [-1, 0, -1, 1, 1, 1, -3, -1, 1, 1, 1, -1, 0, 0, -1], [2, 0, 0, 1, -2, 1, 0, 1, 0, -1, -1, 1, 0, 0, 2], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -2, 0, 0, -1, 0, 2, 1, 0, 0, -3, 1, 0, 2, -1], [0, 0, 0, 1, -1, 0, 0, 0, -1, 0, -2, 0, 0, 2, -1], [4, 0, 0, -1, 1, 0, 0, 1, 0, -1, 1, -1, 0, 0, 1], [1, 0, 0, 0, -2, 0, -1, 0, 1, 0, -3, 1, 0, 0, -2], [3, 0, 1, 0, 1, 0, -1, 0, 0, 0, 3, -1, -2, -2, 1], [1, 0, 1, -1, 0, 0, -2, -1, 2, -1, 2, 0, -1, 0, 1], [2, 2, 0, 0, 0, -1, 2, 0, -2, 0, 0, 1, 0, -1, 0], [7, 0, 0, 0, 2, 1, -1, 0, -1, -2, 3, -3, -1, 0, 0], [-1, 2, 0, 1, 1, 0, 1, 1, -2, 0, 0, 1, 1, -1, -2], [2, 0, 0, 0, -1, 0, 1, 0, -1, 2, 1, 1, 1, -2, 2], [-2, -2, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 1, 2], [-1, 2, 1, -1, -1, 0, 1, -1, 2, 1, 0, -2, 0, 0, 1], [0, 0, -1, 1, 0, -2, -2, -1, -1, -1, 0, 0, -1, 0, -1], [-3, 1, 1, -1, -1, 3, 1, 1, 1, 1, 2, 2, 0, -2, 0], [5, 3, -1, 2, 2, -2, -3, 0, -2, -1, 1, -2, 0, 1, 0], [3, 0, 1, -2, 0, 1, -2, 0, 2, 0, 4, -1, 1, -1, 2], [0, 2, 0, 0, 1, 1, -3, 0, 1, 2, 2, -1, -1, -1, -1], [3, 2, -1, 0, 0, -1, -3, -1, 0, -1, 0, -1, 0, -1, -1], [3, -3, -1, 0, 2, -1, 0, 0, 0, 0, -2, 0, 1, 2, 0], [2, 1, -1, 1, 2, -2, 2, -1, -1, 1, -3, 0, 1, -1, -2], [2, 2, 0, 1, 0, -1, 0, 1, -1, -2, -1, 0, 0, 2, 1], [-1, 3, 0, -1, -3, 0, 1, 1, 0, 1, 0, 2, -1, -1, 0], [1, 2, -1, 0, 0, 1, 1, 0, 2, 0, 3, 2, 0, -3, 2], [6, 0, -1, 0, 3, -1, 0, 1, 0, 1, 0, -1, 1, 3, 2], [2, 0, 1, 2, 0, 0, 0, -2, 1, -2, -1, -1, -1, 2, 1], [-1, 2, 0, 0, 1, -2, 2, 0, 0, 1, 1, -1, 0, 1, 1], [4, 0, 2, -1, -1, 1, -2, 0, 2, 0, 0, -1, 0, 1, 0], [2, 0, 1, 0, 2, -1, 1, 0, 1, 2, -2, -1, 0, 0, -1], [5, 2, 0, 0, -1, 1, -3, 0, 0, 1, 2, 1, -1, 1, -1], [2, 0, -1, 0, -1, -1, 0, -1, -2, -1, -1, 1, 1, 0, -1], [10, 0, 0, 1, -1, -1, 2, 1, 1, -1, -3, 1, 1, -1, -2], [1, 2, 0, 0, 0, 1, -3, 0, 1, -2, 2, -3, -1, 0, -1], [-2, 3, 0, 1, -1, -1, 2, -1, 1, -1, -5, 1, 0, 1, -2], [-7, 2, 0, 0, -1, 0, 3, 0, 0, 0, -2, 1, -1, -1, 1], [0, 1, -2, 0, 1, -2, -2, -1, 1, -1, 1, -1, 0, 2, -1], [2, 0, -1, -1, 0, 0, 1, 1, -1, -1, 0, 1, 0, 2, 2], [4, 0, 1, 1, 2, -1, 2, -1, 1, 0, -1, -2, -1, 3, 2], [1, 3, 0, -1, 1, 1, 1, 1, 0, 1, 2, 1, 1, -2, 2], [5, 0, 1, 0, 2, -1, 1, 0, 0, 0, 0, -2, 2, 0, 1], [-9, 2, 2, -2, -3, -1, 1, 0, 0, 0, -2, -1, -1, -1, 1], [1, 4, -2, 3, 0, -1, -2, 1, -1, 1, -2, 1, 0, 1, -1], [0, 4, 1, 1, -1, 0, -2, 0, 0, 0, 1, -1, -1, -2, -2], [4, 2, 2, 0, 1, -2, 3, -2, -3, 2, 0, -1, -1, 1, 1], [0, 2, 0, -1, -1, 3, -4, -1, -1, 2, 2, 0, -1, 0, -3], [10, 0, -2, 2, 2, -1, -4, -2, 2, -1, 4, 1, -1, -2, 0], [-5, 1, -1, 0, -2, -1, 3, 0, 0, -2, 0, 0, 0, -2, 1], [-2, 2, 0, 1, -1, 1, -1, 0, 1, -4, 1, -1, -3, 0, -1], [0, 4, 0, 1, 0, 2, 2, -1, -4, 1, -1, 0, 0, 1, -2], [-3, 1, -1, 1, 0, 0, -4, 0, 4, 2, 0, 1, -1, -1, -1], [-4, -2, -1, 2, -2, 1, -1, 0, 1, 0, 1, 3, 1, -2, 0], [-3, 4, 1, 0, -1, 1, 1, 2, 1, 2, 2, -1, -1, -2, 1], [7, 2, 0, -2, 3, 1, -2, 0, 0, 0, 1, -2, 1, -3, -1], [3, 4, -1, -1, 0, 1, -5, 0, -1, 0, 2, 0, 0, -4, -1], [4, 4, 0, 0, -1, 0, -3, 1, -1, 3, 0, 1, 1, 1, 1], [-5, 0, -1, 2, -1, 2, 0, 0, 0, -2, 1, 1, -3, -2, 0], [2, -1, 2, -1, 0, 1, 4, 1, 2, -1, -1, -1, 0, 0, 2], [5, 0, 1, -1, 0, 0, 3, -1, 0, 1, 1, -1, 0, -1, -1], [3, -2, 0, 2, -1, -2, 2, -2, 0, 0, -6, 3, 0, 0, 0], [-4, 0, 1, 1, 2, 1, 3, 1, -1, 1, -2, 1, 2, -1, 0], [0, 2, 1, 2, -1, 3, 2, 1, 1, 1, -3, 1, 1, 0, 2], [3, -2, -2, 2, -1, 3, 1, 0, -2, 0, -2, 2, 1, 0, -1], [-8, -4, 0, 0, -2, 1, 3, 2, 0, 2, 0, 2, 3, -2, 1], [1, -2, 0, 3, 0, 0, -2, -3, -1, 0, 3, 0, -2, 1, 2], [2, -1, -2, 1, 3, -2, 0, 1, 3, -3, 1, -2, -2, -1, 0], [1, -2, 0, 0, -1, 0, -5, -2, 0, -1, -1, -1, -1, -1, -4], [-8, 2, 1, 1, -2, 0, -1, -1, -1, 0, -4, 0, -2, 4, 0], [3, -2, 1, -2, 0, -1, 4, 2, -3, 0, -1, 1, 1, 1, -2], [3, 1, 0, 1, -1, 0, 0, -1, -2, -1, -2, 3, 0, -1, -1], [3, 0, -1, 3, 1, -3, -6, 1, 1, -1, -2, 2, -3, 0, -3], [-6, -2, 0, 1, -2, 0, 2, 0, 2, 2, -3, 1, -2, 2, -2], [2, -2, 0, 1, 3, 0, -2, -1, 3, 1, 2, -3, -2, 4, 1], [1, -6, 2, -3, 1, 0, 6, 1, 0, 1, 1, 0, 1, 3, 0], [-15, 0, 0, 3, -2, 0, -2, -1, 3, -1, 0, 2, -1, 0, 3], [2, 2, 1, 0, -3, 5, -6, 0, 1, 0, 1, 0, -1, -1, -2], 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-3, 0, 4, 2, 4, 0, -7, 5, 0, -4, -1], [5, -4, 2, -2, 3, 1, 3, 1, -6, 1, 3, -5, 0, 0, -2], [8, 4, 1, -2, -1, 3, -4, 2, 0, 0, 5, -4, 1, -1, 5], [6, -5, 0, 0, 3, 1, -1, 0, -3, 0, 4, -2, -3, 2, 3], [-14, -2, 3, 0, -6, 4, 6, 0, 1, -2, -7, 5, 1, 0, 1], [3, 8, 3, 2, 0, -2, -2, -2, -2, 0, 5, -6, -5, 3, 1], [-11, 3, -4, 0, -3, 3, 3, 2, 0, 0, 0, 1, -1, 3, -1], [-8, -2, 0, -3, 2, 1, -7, 1, 0, -1, 5, -3, 3, -4, -3], [-2, -4, 2, 2, 2, 3, 2, -1, 2, -3, -1, -1, -3, 4, -3], [7, -9, -1, 1, 6, -3, -5, -1, -2, -1, 6, -4, 2, 2, 0], [8, -4, -2, 5, 0, -1, -3, -1, 5, 1, -5, -1, -3, 2, 0], [-9, -4, 1, -3, 1, 1, -3, 3, 3, 0, 0, 4, 4, -6, -4], [-18, -6, 0, -2, -1, 3, -3, -1, 1, -2, 0, 4, 1, -2, 1], [5, 4, 3, 0, 3, 6, -2, 0, 3, 2, 7, -2, -3, -3, 7], [-2, 0, 1, 0, -3, 7, -5, -2, 2, -2, 4, -4, 0, 2, -1], [-2, 0, 3, 0, -2, 2, 0, 0, 7, 4, 2, -3, 1, -1, 4], [-9, 2, -2, 2, 1, 3, -5, -2, 6, 0, 8, -1, 5, -4, 3], [-16, 1, -2, 1, -2, -1, -5, -3, 2, 1, -5, 2, 1, 5, 3], [0, -6, -1, 4, -2, 1, -2, -6, 1, 1, -1, 2, -1, -1, 5], [-7, -4, 2, -4, -2, 1, 9, 0, 1, 0, -1, 1, 3, 4, 4], [-3, -6, 4, -2, 3, 2, 3, 2, 2, 2, 0, -1, 0, -3, 3], [-7, -2, 2, -3, 3, -1, 6, -1, -2, -1, 4, 0, 2, 2, -1], [9, -6, 2, 1, 4, -3, -3, 1, -3, -1, 3, -4, -5, 3, -1], [-1, 0, 0, -3, -2, 1, -2, -1, -1, 3, 3, 0, 2, 2, -4], [-12, 2, 0, -5, -2, -1, 3, 1, 0, 3, 0, 5, 1, -3, -2], [-1, 2, 1, 4, -1, -2, 2, 1, 1, -1, -6, 0, -5, 2, -2], [4, 0, 0, -2, -2, 1, -5, 3, -1, 3, 0, 0, -1, -2, -4], [-30, 1, 3, 2, 1, -2, 0, -4, 0, 1, -4, 2, -5, -2, -2], [-22, 0, -5, -1, 0, 3, 0, -2, -1, 2, 1, 0, 5, 3, 0], [1, 4, 3, 2, 2, -8, 3, -4, 0, 1, -8, 2, -2, 5, 3], [-17, 0, 2, 3, -2, 2, 5, 5, 3, -1, -1, 6, -1, -5, 3], [-8, -4, 1, 1, -1, -1, 2, -3, -2, -5, 0, -1, -1, 3, 5], [-12, 2, -5, -1, -4, 3, -4, 1, 4, 0, -2, -1, 1, -1, 0], [-19, 4, -1, -3, 7, -5, 0, 1, -3, -2, 6, -2, 3, -2, -3], [15, -5, -2, -2, -1, 2, -2, 0, -2, -4, 7, 0, 2, 2, -5], [-12, -6, -2, 4, 1, -4, 1, 0, 3, -2, 0, 3, -1, -1, -5], [-13, 4, 0, -2, 4, 1, 1, -2, 0, 2, 5, -3, -3, -4, -3], [-17, -4, 2, 1, -3, 4, -4, 1, 0, 1, 2, -1, -4, 0, 1], [-9, 8, 1, -3, -4, 3, -8, -1, -2, -4, 7, -3, -3, -1, -2], [9, 4, 2, -2, 2, -2, -2, 0, -3, -4, 0, -5, -2, -2, -2], [-16, 2, 0, 3, -9, -1, 6, -1, 0, 2, -12, 3, 0, 1, 0], [4, -4, 1, -3, 6, -7, 4, 0, -3, -4, 5, -2, 3, -5, 0], [-10, 6, 1, 3, 1, 3, -3, -3, 2, 1, 0, -1, -3, -5, 2], [7, 6, 1, 1, 0, 1, 2, 0, -5, 4, -7, 0, 5, 3, -3], [-2, 6, 2, 1, 0, -6, -2, -1, 0, 2, 2, 0, -3, -2, -3], [4, 1, 1, 2, -4, 4, -4, 0, 3, 4, 2, 1, -5, -2, -2], [1, -6, -4, 1, 5, 1, -1, 1, -4, 1, 7, 3, 1, 0, 1], [0, 4, -1, -2, -5, 3, -2, 2, -3, 2, 0, 6, 3, 0, -1], [-8, -2, 0, 2, 6, -1, -6, -1, 0, -1, 7, -4, -4, 5, -1], [-5, -4, 0, 2, -1, 3, -3, 2, 6, -2, 1, 6, -1, -7, -4], [0, 4, -1, 3, 3, -5, 0, 5, -2, 1, 4, -1, -3, 1, 3], [0, 2, 1, -6, 4, -1, 6, -3, -7, -1, 4, -3, 3, 1, 2], [2, 3, 5, -6, 3, -3, 0, -6, -2, -2, 7, -6, -3, -1, -3], [-1, -2, -5, -2, -2, 0, 4, 0, 0, 2, -9, 4, 2, -1, -1], [-12, -4, 0, -3, -2, -2, -3, -5, 4, 4, 3, 1, 3, 0, 1], [-11, 4, 0, 4, -1, 4, -2, 2, -4, -2, 2, -3, -2, 1, 2], [-12, 4, -4, -2, 0, 4, 4, 5, -6, 3, -1, 0, 8, -3, -3], [17, -2, 0, -1, 6, -8, -11, 0, 1, -2, 3, -7, -5, 0, 1], [-4, -4, -1, 7, -1, 3, 1, -2, -4, -4, -3, 3, -1, 0, -1], [-5, 0, -5, 1, -2, 1, -5, 1, 2, -5, -2, 1, 0, 1, -4], [-13, -4, 1, 5, 7, -2, 1, -3, -5, -6, 3, -3, -4, 2, 7], [16, 8, 0, 2, 4, 1, -3, 4, 4, 0, 7, 0, -2, -5, 2], [-9, -1, -3, 2, 1, -2, 8, 2, 1, -4, -8, 1, 5, 1, 2], [-8, 2, 4, 1, -6, 2, -3, -1, 0, 1, -4, 2, -3, 2, -6], [-16, 8, 1, -1, 5, -1, -3, -2, 2, 0, 2, 0, 0, -2, 2], [-12, -6, 1, 1, 4, -5, 10, -5, -1, -5, -8, 1, 5, 3, 1], [-13, -4, -1, 5, -1, 4, 2, -1, 7, -4, -3, 2, -3, -1, 0], [7, 4, 0, -3, -3, 2, -6, 0, -1, 4, -3, 1, 0, -6, -5], [7, 2, 3, -6, -1, -6, 4, 3, -2, 3, -2, 2, 3, 4, -1], [-17, 2, 3, -3, 1, -4, 5, -5, -5, -1, 3, 0, 0, -1, 3], [-11, -2, -1, -4, -2, -4, 0, 6, 0, 2, 3, 4, 3, -3, 1], [-7, 0, -1, 2, -5, 3, -3, 0, -3, 0, 3, 6, -4, -6, -2], [-6, -4, 0, -2, 0, 1, -5, -3, 2, -3, -5, -1, -3, 2, -6], [2, 6, 3, -1, -5, 1, 2, 0, 6, 3, 0, 5, -1, -3, 3], [5, -1, -3, -3, 1, -2, 2, 1, 3, 5, -3, 0, -1, -3, -6], [-19, -10, 0, -1, -1, -1, 1, 1, 4, 4, -9, 0, 3, -1, -1], [8, 6, -2, 3, 0, -1, 1, 3, -4, -1, 3, 0, 2, 3, 1], [-6, 8, 2, -2, -1, 1, 4, 2, -7, -1, 2, 3, 0, -3, 0], [-20, 2, 2, 3, 0, -3, 2, 1, 2, -3, -5, 3, 1, -2, 2], [-25, 2, 3, -2, -2, 8, 2, -3, 2, 1, 7, -1, -1, -4, 5], [2, 6, 1, -4, 4, -5, 0, 4, -3, 4, 6, -1, 5, -6, 2], [4, 4, -1, 1, 1, 1, -10, -1, 2, -3, 6, -6, 0, -4, 3], [-5, 6, 2, -2, -4, 0, 0, 4, 5, 8, -2, 2, 2, 0, -4], [6, -2, -2, 4, -1, -2, 0, 6, 1, -6, -1, 0, -6, 2, 1], [-2, -2, 2, 4, -1, -1, -4, 0, 2, 2, -1, -2, -4, 5, -4], [-7, -7, 1, -5, -3, 7, 10, -1, -1, 5, -7, 1, 5, 3, 4], [-6, -6, 1, 4, 0, -3, -5, -2, 3, -1, 0, 1, 0, 1, 5], [13, 0, -1, 1, 2, -5, -2, 1, -5, 5, -3, 7, 1, 6, -3], [-1, 6, -1, 5, -5, 3, 4, 1, -7, 7, -10, 5, 2, 6, -5], [-20, -6, 1, -1, -2, 5, -2, 1, 3, 2, 7, -1, -1, -3, 0], [-2, -10, -4, 4, 1, -11, -1, 0, -3, -1, -4, 10, 3, 0, -5], [-14, -2, 5, -1, -8, 0, 2, -1, -3, 1, -8, -1, -1, 0, 1], [-19, 8, 1, -1, 2, 0, 2, -1, 0, 1, 2, -2, 5, -2, 3], [5, 2, -1, 3, 3, 2, -10, 1, 1, -3, 4, 5, -4, -3, 1], [3, -4, 2, 0, -4, 4, -3, 0, -2, 0, 5, 6, -3, -3, -7], [-14, -4, -3, 6, 0, -1, -2, 2, -1, 2, 1, 7, -2, -2, 3], [0, 2, 1, 1, -4, 0, 1, -3, 1, 6, -5, -4, 0, 1, 1], [3, -2, -2, -4, 1, -3, 0, -6, -2, -4, 5, -1, 4, 0, 3], [1, -3, -5, 2, 6, -2, -2, -1, -4, -3, 1, 1, -1, 0, -5], [-15, -2, 2, 2, -6, 1, 7, -5, -1, 1, -4, -1, -1, 1, 3], [-11, 0, 0, 6, -1, 5, -7, 2, 5, -1, -1, 2, -3, -5, -1], [1, -1, -3, -1, 1, 7, -7, 1, 3, -9, 2, -2, 4, -6, 0], [-6, -6, 2, -4, 3, -2, -4, 0, -3, -6, 11, -10, -2, 2, -1], [2, 8, 1, 1, 0, 4, -5, 5, 1, 0, -3, 2, -1, 0, -7], [9, 4, 3, 4, 2, -3, -2, -4, 2, 6, -6, 0, 0, 2, 2], [-15, 4, 3, -2, 0, 2, 5, 7, 2, 1, 11, -1, 0, -2, 8], [16, -4, -2, -3, -2, 0, 1, -1, 0, 1, -1, -5, 1, 8, 3], [7, -7, 0, 2, 6, -2, -1, 0, 3, 0, 2, -1, 1, -5, 7], [1, 5, -4, 2, 5, -4, 2, 1, -6, 0, -3, -3, 4, 0, -5], [4, 2, 2, -4, -1, 1, -7, 0, 0, -4, 6, -4, -1, 1, -2], [-14, -10, 1, 1, 2, -1, 3, 5, 3, 0, 0, 2, -3, 2, 2], [-15, 0, 1, -1, -1, 5, 10, 1, 3, 0, -8, -1, 7, 2, 1], [5, 12, 3, -2, -6, -1, 1, 3, 2, 1, -6, -2, -1, -4, 1], [-11, 0, -5, 4, -3, 4, 1, -1, 1, -3, -6, 3, 0, 3, -3], [14, 4, -1, -3, 2, 6, -4, -2, -1, 6, 3, 2, 2, -5, -4], [-22, 4, 0, 0, 2, -1, -7, 0, 0, -4, 8, -5, 3, -1, 1], [2, 4, 3, 4, -7, 4, -2, 0, -1, 2, -4, 4, -1, 4, 1], [18, 4, 1, 4, 3, -1, 0, -2, -8, -2, 1, -3, -1, 2, -3], [-3, -4, -1, 1, 3, -2, 9, 2, -3, 0, -2, 5, -2, -3, 0], [-7, -2, -1, 3, -2, 3, 2, 0, -2, -4, -3, 2, 1, -3, -6], [1, 0, 4, -2, 5, -2, -1, -4, -4, -4, 5, -4, -5, -3, 0], [-17, 0, -2, 4, -2, 3, -8, 1, -1, -2, -9, 2, 0, -1, -3], [-10, 5, -1, -5, -3, 1, 1, 3, -2, 7, 3, -1, 6, 0, 1], [-12, -6, -1, 4, 5, -7, -3, -4, 8, -3, -9, -2, 0, 5, -4], [-10, 2, 0, 4, -3, 10, 0, 0, 1, -3, 2, 0, 0, -2, 0], [-3, 5, 1, -9, -5, 7, 3, 1, -7, 2, 2, 1, 4, -1, 0], [8, 2, 1, 6, 1, -3, 1, -6, 2, 2, -4, 0, -6, 6, 9], [3, 0, -1, 1, 3, -5, -6, 1, 5, -3, 0, 0, 3, 4, 1], [4, 6, 1, -7, -3, 6, 5, -1, -2, -1, 1, -3, 4, 0, 3], [-2, 2, -1, -1, 6, 7, -3, -1, 0, -1, 2, -8, 2, -5, 1], [-6, -2, -4, -3, -7, 6, 4, -3, 6, 7, -3, 3, 4, -4, -3], [12, 0, 2, -7, -1, -6, 8, -3, -5, -1, -2, -2, 2, 12, 3], [3, 6, -1, -3, -6, 2, -4, -3, -7, -3, 5, 2, -3, -3, 1], [-8, -10, -2, 4, -2, 6, 0, -6, 8, -1, -7, -2, -1, 3, 1], [5, 4, -1, -2, -4, -2, -10, -4, -2, 2, 6, 0, -2, -2, 0], [-1, 4, -1, 4, -3, -3, 12, -2, -3, -2, -9, 4, -1, 6, 7], [-13, -9, -2, -3, 3, -1, -5, -3, 4, -3, 12, -7, -1, -3, 4], [3, -8, 3, -1, 0, 2, -2, 0, 1, 3, -1, 0, -3, -1, -3], [-16, 10, 1, 1, -2, 1, 3, 7, -1, 1, 0, 2, 1, -4, -2], [5, 5, 2, 3, -3, 4, 0, -1, 4, -7, -6, 0, -4, 2, -1], [-1, 6, -1, 4, 5, -1, -8, 2, 7, -2, -3, 0, -5, -1, -1], [12, -4, -2, -2, -7, 5, 0, 0, -2, 0, 3, 3, -2, -4, 2], [-9, -2, 2, -2, 5, 6, -2, 1, -4, -6, 7, -3, -2, 2, -1], [19, 0, 1, -3, 3, 5, 1, 8, -7, -4, 7, -1, 2, -4, 1], [-12, -4, -1, 2, 1, -4, 1, 0, 2, -6, -2, 0, -2, 1, -3], [2, 0, -3, 5, 2, 1, -6, -5, 6, -1, -9, 0, -1, 3, -3], [-14, -2, 2, -1, -1, 0, 9, 0, -5, 6, -10, 1, 5, 5, -4], [-3, -6, 2, -3, 2, 7, -1, 6, -1, 1, 3, 4, 2, 1, 1], [-14, -14, -1, 4, 1, -4, -1, -2, 3, -1, -3, 1, 2, 4, -1], [1, -3, -3, 7, -2, 0, -6, -1, 0, -5, 0, 3, 1, 3, 7], [7, 6, -3, 5, 2, -9, 0, 0, 4, 4, -2, 3, -5, -3, 1], [11, 0, -2, 2, 1, -1, 1, -5, -4, 5, -6, -4, -1, 8, -7], [20, -2, 2, -3, 1, 3, -8, -1, 6, 1, 8, 1, 2, -7, 0], [6, -8, -3, 3, 3, -5, -4, -3, 6, 4, -4, 0, -3, 4, -5], [22, 0, -1, 3, 3, -1, -11, -1, -4, -3, 7, 0, 3, -1, 1], [39, 6, 4, -1, -3, -3, -3, -5, -2, -3, 5, -5, -7, 3, 1], [3, 3, 1, 2, 0, 2, 0, 0, 4, 0, 3, 0, -3, -3, 11], [0, 5, 5, -2, -3, -1, -1, 0, 4, 5, -5, -3, -8, 4, -8], [5, 4, 0, 0, -4, 2, 9, 2, -3, 5, -11, 5, 5, 1, 2], [-2, -4, -3, -1, 2, 3, 5, 6, -3, -2, -1, 3, 0, -2, -1], [-10, -4, 0, 5, 0, 2, 11, 0, -4, -4, -1, -2, -3, -1, -1], [-20, 7, 0, 0, 3, 1, -1, 1, 3, -7, -3, 0, 3, -3, 0], [-1, -12, -5, 4, 2, -3, -3, -2, -3, -6, 3, 0, 2, 6, 1], [19, 8, -5, -1, -2, 6, -6, 5, 0, 5, 4, 0, -3, -3, -1], [5, 4, -5, 3, -4, -8, 6, 0, 4, 0, -14, 7, 1, 1, -5], [-30, -2, 0, -1, -8, 7, 2, -3, 6, 1, -9, 1, -3, 2, -2], [5, -2, -1, -2, 4, 2, -4, 2, 4, -2, 11, -6, -1, 1, -3], [10, 2, 1, 0, 2, 1, 4, 3, 3, -1, -1, 1, 1, 0, 1], [-6, -12, -2, 2, 2, 0, -7, -2, -4, -6, 9, 0, -4, -1, 0], [-16, -6, -2, -8, -3, 3, 7, 5, 3, -1, 2, 7, 7, -7, 3], [8, -12, 4, 1, -6, -4, 9, 2, 0, -2, -3, 2, -1, 7, 5], [9, -12, 0, 6, 1, 1, -3, -2, 2, -6, -4, 1, -5, 5, -3], [-17, 6, 0, 1, -1, 1, -6, -5, 6, 5, 2, 0, -4, -4, 5], [32, 2, 0, -2, -1, -1, 2, 0, -4, 4, 3, 1, 0, 2, 0], [-7, 4, -4, 7, -8, 7, -2, -2, 1, -1, -8, 6, -2, -2, 3], [21, -12, -2, -2, 4, -5, 2, -2, 2, -4, -1, 2, 2, -1, 2], [1, -4, -2, 5, 1, -4, 1, 1, 7, -1, -1, -2, -3, 6, 0], [0, 9, 0, -4, -3, -1, 12, 0, -7, 2, 6, -2, 4, -4, -2], [-3, 8, -3, 1, -1, -3, -6, 0, 1, -4, -6, 2, -3, 0, -3], [2, 4, -1, -4, 3, -3, 5, -2, -8, -2, 8, -5, -1, -3, 1], [-13, -2, -3, 0, 0, 2, -9, -4, 1, -2, -1, 0, 2, -3, 3], [8, -5, 2, -1, 5, 1, -3, -1, 1, -4, 11, -5, -3, -5, 9], [20, -10, -2, 0, -3, 5, 0, 0, -1, 1, -4, 5, 0, 3, -6], [-8, 0, -3, 4, 2, -1, 0, 2, -5, -2, 3, -1, 3, 2, -5], [0, -2, 2, 0, 0, 4, 3, 2, -6, 2, 2, -3, -1, 5, 1], [20, 5, -1, -4, -7, 5, -2, 6, 4, -5, -3, 5, -1, -5, -1], [6, -1, -1, 1, 0, -8, 9, 3, -3, -1, -2, 3, 2, 3, 4], [17, 5, 2, 1, 1, 2, -2, 1, -2, 3, 6, -4, 0, 2, 5], [-12, 2, -2, 8, 3, 2, 0, 4, -5, -3, 3, 4, -2, -2, 3], [2, 16, 1, -1, -4, -1, 3, 1, -7, 5, 5, 1, 0, 0, 3], [2, 4, 5, -4, -3, -2, 1, 8, -2, -2, 5, 1, 0, 3, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8022_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8022_2_a_z();".
// 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_8022_2_a_z(:prec:=15)
    chi := MakeCharacter_8022_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(3067) - 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_8022_2_a_z();".
// 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_8022_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8022_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4704, -13616, 16144, 62828, -7822, -84842, -2130, 45390, 615, -10139, -45, 1068, 1, -53, 0, 1]>,<11,R![7362048, 46393088, 53215104, -7640768, -25004496, -1535720, 4648764, 513990, -444934, -56697, 23255, 3135, -632, -88, 7, 1]>,<13,R![-125200, -1865528, -5737732, -2717238, 4567808, 2307434, -1675638, -476405, 302354, 26843, -24556, 421, 845, -60, -10, 1]>],Snew);
    return Vf;
end function;