
// Make newform 8036.2.a.r in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8036_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_8036_2_a_r();".
// 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_8036_2_a_r();".
// 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 := [3381, -399, -22584, 8429, 38415, -22309, -20311, 13971, 4929, -3853, -616, 537, 39, -37, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-53909708702913, -33588397215636, 315331689321588, -34154729402677, -518478043052718, 125086901622927, 290964255861732, -63368265036347, -78350006830010, 12550722332099, 11058300703621, -1076896328346, -780431594905, 33366538546, 21568415171], [964719687046731, -382307010614964, -2504020310672772, 1086025882031975, -14256388065710, -337765061235149, 1061905945918156, -26134945978879, -471109920542042, 19997951254183, 84630564241705, -2264494596922, -6864511642829, 75537732154, 207367915591], [-394220916659787, -925571855654028, 3676894843258596, 2957473201007065, -7504847710757362, -1062369256372771, 4597551204850484, 1604986959871, -1297715569983766, 41971019614937, 187920186390263, -5416632772358, -13458627125155, 197307815606, 375675670889], [-1413258738046953, -1583632566957636, 8604345326302092, 4090409019404467, -13752243630567190, -1280840055975121, 7768694051421020, -64560226435499, -2097712145040178, 65601128365283, 294326513071613, -7746814774274, -20576643380593, 271104011042, 563847518531], [-123529112322861, 51742333507248, 916971510040800, -134303162335843, -1587383470520204, 359635444491391, 943989351011626, -191585391007741, -268865569213064, 39661374923485, 39739641971875, -3508288802422, -2903659812299, 110890419958, 82318075711], [-132293240921499, -36292597607964, 743133409360908, 128904705950189, -1127435686751094, -12151856350959, 661794329691888, -21783985515293, -188732383570406, 6543610028837, 27895029041707, -636604082118, -2038445185159, 19902931966, 57933555161], [-369589303008177, 395328021911436, 2549675888785068, -1188084234706517, -4158684124869190, 1337367111898247, 2574917570991548, -576510242603315, -772667003836402, 110071063789547, 119443657067957, -9319581543506, -9037992446569, 284951510738, 263299916363], [2285567947233123, 2212465125784428, -14659545336879396, -7066331684287153, 22280485010412706, 3575923666457563, -12199681840975508, -708242318071255, 3202662100683478, 64143993774511, -437414324713775, -2773231651450, 29873873602171, 55554951370, -803771605457], [1300628388320541, 1033679062742580, -8114892589426428, -3195696128916367, 12894254808423982, 925907072139781, -7259458623838364, 105751315240871, 1958752009116682, -63910333187519, -274960302342017, 7202431283834, 19237736393941, -247737754298, -527518007087], [5020217462879745, 2619386748959364, -31166858944757532, -8617048669258075, 49218218809824022, 1482781049982937, -27720511397395436, 870143948269859, 7472488513327378, -291365982179867, -1045949435856725, 29524283624978, 72904083529129, -970794976610, -1991383769819], [-312802565393409, -196898900419467, 2062137225183060, 660872437651219, -3386733041584858, -139122061304389, 1963368044552309, -55206873249419, -543496021568218, 20374588096121, 77954228306660, -2103794805215, -5550942035752, 69132074966, 154414062065], [3416523056642787, 1854323154805428, -20547001848439524, -6160094617298017, 32538031186207330, 1233561484274971, -18440308332790964, 533869176269825, 5008459392835390, -191670870598865, -706794387141431, 19718267961758, 49657832878051, -650669688638, -1366326190169]];
    Rf_basisdens := [1, 1, 1, 8954190428232, 53725142569392, 53725142569392, 53725142569392, 6715642821174, 4477095214116, 17908380856464, 53725142569392, 17908380856464, 53725142569392, 3357821410587, 26862571284696];
    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_8036_a();"
function MakeCharacter_8036_a()
    N := 8036;
    order := 1;
    char_gens := [4019, 493, 785];
    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_8036_a_Hecke(Kf)
    return MakeCharacter_8036_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, -1, 0, 1, 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, 1, 0, 0, 0, 0, 0], [-1, 0, 1, -1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 0, 0, -1, 1, 0, 1, -1, 0, 0, 0], [1, 1, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [1, -2, -1, -1, -1, 0, 0, -1, 0, 0, 1, 0, -1, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1, -1, 0, -1, 0, -1, 0, 1, 1, 0], [2, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, -1, -1, 0], [1, 0, -1, 1, 0, -1, 1, 0, 0, -1, 1, 1, 0, 1, 0], [1, 1, -1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, -1, 0], [0, 0, -1, 1, 0, -1, 1, 0, 1, 0, 1, -1, 0, -1, 0], [1, 0, 0, -1, 1, 0, -1, 0, 0, -1, -1, 1, 2, 0, -1], [0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, 1, 0], [1, 2, 0, 2, 1, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1], [1, -3, -1, -1, -1, 0, 1, 0, -2, 0, -1, 2, -1, 1, 0], [-1, 0, -1, 0, 0, 0, 1, -1, 1, 1, 0, -1, -1, -1, 0], [1, -1, -1, 1, 0, 1, -1, 0, -1, 0, 1, 0, 0, 1, 1], [0, 1, 0, 1, 0, 1, 0, -1, 0, 0, 1, -1, -1, 0, -1], [1, 2, 1, 2, 1, 0, -3, 0, 1, -1, -1, -1, 2, -1, 0], [0, 1, 0, -1, -1, 0, 0, 1, -2, -1, -1, 2, 0, 0, 0], [4, 0, -1, 2, 1, -1, 2, -2, 0, 2, 1, -2, 0, -1, -1], [2, -1, 0, -2, -1, 0, 2, 1, -1, 1, 1, 1, -1, 0, 1], [1, 0, -2, 0, 0, -1, -1, 0, -2, 0, 0, 0, 1, 2, -2], [-4, -1, -2, 0, -1, -1, 0, 0, -1, -1, 0, 2, 1, 0, -2], [1, -1, -2, 0, 0, 0, 1, -1, 0, 0, 1, 1, -2, 2, 0], [4, 0, -1, 0, 0, -2, 0, -2, 0, 0, -1, -1, 1, -1, -1], [-6, -2, -1, -3, -1, 0, 0, -1, -1, 0, -2, 2, -1, 1, 0], [3, -1, 0, -1, 1, -1, -1, -1, -1, 2, -1, -1, 0, 0, 0], [0, 1, -3, 1, 0, -2, 3, -1, -1, 1, 2, 1, 0, 0, -1], [3, 0, 1, 0, 1, 0, 3, 0, 1, 1, -1, 1, -2, -1, 0], [3, 1, 0, 1, 0, 1, -3, 0, -1, -1, -2, 1, 3, 0, 0], [-2, 1, 0, -2, -1, -1, 0, -2, -1, 2, -1, -2, -1, 0, 0], [-2, -1, 1, 0, 0, 1, 1, 0, 0, 1, -1, -1, -1, 1, 1], [1, -1, -1, -2, -3, 3, -2, 1, -2, -2, -1, 3, -2, 1, 0], [-3, -1, 1, -1, 0, 1, -1, 2, 1, -1, 1, 0, 0, -1, 0], [2, -2, -3, 1, 1, 0, 2, -1, 1, 0, 1, 0, -3, -1, 0], [3, 0, 0, 2, 0, 0, -3, 1, -1, -1, 1, 0, 2, 2, -1], [-1, 0, 0, -2, 0, 2, 2, 0, -1, -1, 0, 2, -2, 0, 0], [3, -1, 0, -4, -1, 2, -1, 0, -2, 0, -1, 1, 1, 0, -1], [-1, -2, -2, -1, -1, -2, 2, 0, -2, 2, 1, 1, 0, 2, -1], [-2, -1, 1, -2, 1, -1, -2, 0, -1, 0, -1, 1, 0, 1, -2], [7, 1, 1, 2, 0, 0, 2, 0, 0, 1, 1, -1, 1, 0, 2], [-2, -1, 0, 0, 2, 0, 2, -2, 1, 0, 2, -3, 0, 0, 0], [6, -1, 0, -2, -1, 3, 2, 0, -1, 0, 0, 0, -3, 0, 1], [1, 3, 0, -1, 1, -1, 0, 0, -1, 1, 0, -2, 0, -1, 0], [1, -2, 0, -3, -1, 0, -1, 0, 0, 0, -2, 3, 0, 0, -2], [-2, -3, -1, -2, -1, 1, -4, 2, -1, -2, -1, 3, 2, 1, -2], [2, -2, 0, -1, -1, 2, -2, -1, 1, 0, -1, 1, -2, 0, -1], [-2, 0, -2, -1, 1, 0, 1, -2, -2, 2, 0, -1, -2, 1, 0], [1, 0, 2, 4, 1, 0, 0, -1, 3, 1, 1, -4, -1, 0, 2], [0, -1, 2, -2, 0, 2, 2, 0, 3, 0, 0, -1, -2, 0, 1], [6, 1, 1, 1, 1, -1, 1, -3, 3, 3, 1, -3, -2, -2, -1], [3, 2, 2, -3, -1, 1, -5, 1, 0, 0, -3, 0, 0, 0, 0], [8, 1, 0, 0, 2, -1, -2, -1, 3, 1, -1, -2, 0, -2, -1], [-2, -3, 0, -3, -2, 1, 0, 1, 0, 2, -1, -1, -3, 0, 0], [-2, -2, -3, -3, -3, 0, -4, -1, -1, 0, 0, 0, -1, 1, 0], [2, -1, 1, 1, 1, 2, 0, 3, 0, -1, -1, 1, -1, 1, -1], [2, -3, 1, -4, -4, 2, 0, 2, -3, -1, -1, 4, -1, 3, 0], [3, 0, -3, 5, 2, -4, -1, -1, 2, 0, 2, -4, 2, 1, 0], [9, -1, -1, 1, 0, -1, -3, 2, 1, -3, 2, 0, 0, 1, -2], [6, 4, 0, -1, 1, 0, -1, 0, 2, 1, -1, -2, 1, -3, 0], [7, 1, 0, 3, 1, 0, 0, 3, -1, -1, -1, 1, 2, -1, -1], [0, -4, 0, -2, -1, -2, -3, 0, -2, 0, -1, 0, 3, -1, -1], [-7, -1, 0, 1, 1, -1, -2, -2, 0, 0, 1, 0, 1, 3, 0], [2, 2, -2, -1, -2, 2, 0, 0, -3, 1, 0, 1, -1, 2, -1], [0, 0, -2, 0, 1, -2, 2, -1, 0, 0, 1, 0, -3, 2, 0], [3, -3, 1, -4, 0, 2, 0, 2, 1, 0, -1, 4, -3, -1, 1], [9, 0, 0, 3, 1, -3, 1, 1, 0, 0, 1, 0, 2, -2, -1], [3, 2, 0, 2, 1, 0, 0, 0, 1, 1, 1, -2, -1, -2, -1], [3, 3, -1, 4, -1, 0, -2, 1, 1, -1, 2, 0, 0, 1, -2], [0, -1, -2, 5, 2, -4, 2, 0, -2, 2, 2, -2, 3, -2, -2], [6, 1, 0, -4, -1, 2, -1, 0, 0, -3, 1, 2, -2, 1, 0], [-4, 0, 0, -1, 1, 1, 2, 0, 0, 0, -2, 1, 1, -4, 0], [-5, -4, -1, -3, -2, 5, -1, 2, -2, -2, 1, 1, -2, 3, 3], [12, -3, 1, 3, 0, -1, 0, -1, 0, 1, -2, 0, 0, 1, -1], [2, 2, 0, 4, 0, 0, -2, 0, -2, -1, -2, 0, 4, 0, -2], [2, -1, 1, -2, -1, 2, -1, 2, -2, 0, -1, 0, 0, 1, 0], [-2, 1, 0, 1, 3, -3, 1, -2, 0, 3, -1, -3, 4, 0, -1], [2, 4, 0, 2, 1, -3, 0, 0, 0, 1, 2, -3, 1, -2, -3], [-1, 2, -1, -6, -1, 0, 0, 1, -1, -1, 2, 1, -4, 3, 2], [3, 3, -1, 4, 1, -3, 1, -1, 0, 1, 0, -1, 2, 1, 2], [4, -4, 3, 2, 0, 0, 1, -1, -1, 1, 1, 0, 2, 0, 0], [0, -4, 2, 5, 1, 1, -2, 2, 0, -2, -2, -1, 4, 0, 0], [-5, -2, 0, 0, -1, 1, -1, -3, 1, 3, 1, -3, -3, 0, 3], [-5, -5, 1, -3, -1, 0, -2, 0, -1, -2, -3, 5, 1, 1, -1], [9, 2, 5, 2, 0, -2, -1, 5, -1, 2, -2, 1, 3, -1, 0], [7, 1, 0, 4, 3, 0, 1, -2, 4, 0, 0, -5, 1, -2, -2], [6, 2, -1, -1, 1, -1, -1, -3, -2, 2, -2, -3, 1, 0, -1], [1, -2, 2, 2, 2, 1, 2, -3, 1, 1, 1, -2, 0, -3, 1], [10, -3, -2, 0, -2, 1, 0, -1, -3, 1, -1, -2, 0, 0, 2], [-1, -6, -2, 2, 0, 2, 3, -3, 1, -1, 3, 0, -2, 0, 1], [-6, 2, 1, 0, 0, 2, 0, -2, 4, -2, 1, -1, -3, 1, 1], [4, -1, -1, 0, 1, 0, 1, 0, 1, 1, 2, 0, 0, 0, 2], [-8, 0, 4, 4, 2, -3, -3, 0, 1, -3, -2, 1, 4, 1, 0], [0, -3, 3, -3, -1, 1, -4, 2, 2, 1, -3, -2, 1, 1, 2], [4, -2, -3, -4, -1, 0, -3, -1, -4, 0, 1, -1, 2, 3, -1], [1, -2, -1, 1, 1, 0, 2, -1, 4, 0, 3, -2, -5, -2, 0], [4, 4, -4, 2, 3, -2, 1, -5, 2, -1, 6, -6, 1, 0, -1], [4, -3, 4, 1, -2, 0, 0, 2, 2, 0, 0, 0, 3, -2, 1], [-1, -1, 3, -6, -2, 3, 0, 0, 0, -1, 1, 1, -3, 3, 2], [13, 2, 0, 1, 1, -1, -3, 1, 0, 1, -1, -2, 0, 0, 0], [-4, 0, -1, 0, 0, 4, 0, 0, 0, -2, -5, 3, 1, -1, -1], [-1, -2, 1, 2, -1, -1, -1, 1, 3, -1, 2, 0, 2, 1, 2], [-5, -6, -2, -6, -4, 4, -1, 3, -3, 1, -3, 6, -4, -2, -1], [5, 1, 0, -5, -1, 5, -3, 3, 0, -3, -1, 4, -3, 0, 0], [2, -1, 0, 0, -1, 0, 5, 0, 2, 1, 4, 1, -1, -2, -1], [8, -6, 1, -5, -3, 1, -4, 2, -1, 0, -1, 3, 1, 1, 1], [-5, -2, -2, -4, 1, -3, -5, -3, -1, 0, -3, -1, 1, 0, -3], [1, 0, 3, 0, 1, 0, 1, -2, 3, 3, -1, -5, 0, -5, 2], [-1, -1, 0, 8, 2, -3, 2, 0, 0, -1, 3, -2, 4, 2, -1], [2, 2, 3, 9, 2, -2, 6, 0, 3, 2, 1, -4, 0, -1, 3], [0, 1, 3, 0, -3, 1, -2, 2, 1, -3, -5, 3, 2, -1, 0], [2, -11, -4, -4, -3, 0, 1, 2, -2, -2, 0, 7, -1, 3, 0], [6, 0, 6, -5, -1, 8, 0, 3, 2, -1, -3, 0, -3, -2, 6], [-4, -3, -2, -2, 2, 1, -2, -1, 1, 0, -1, 0, -2, 0, 2], [3, -4, -4, -2, -2, -3, -2, -1, -4, 0, -4, 2, 1, 1, -5], [-3, -2, -2, 4, 2, -5, 0, -1, 2, 4, 1, -3, 2, -1, 0], [-4, 3, -3, 0, 0, -1, -2, -1, 0, 2, -4, -4, 5, -1, 0], [-1, -3, -3, -1, -1, -3, -5, -3, -4, 3, -2, -3, 0, 1, -1], [11, -4, -2, 2, -1, 3, 1, 1, -1, -1, -3, 3, -1, -2, 0], [1, -2, 5, 2, 2, 1, -4, 3, -1, -1, -8, 3, 3, 0, 1], [-5, -1, -5, 1, 0, 1, 2, 1, -2, 3, 2, 0, -2, 0, 0], [-2, 5, 0, 2, 0, -2, 0, 2, 3, 0, 2, -3, 2, -2, -1], [11, -6, -2, -5, -2, -1, -1, -4, -4, 2, 2, 0, -1, 4, -1], [-4, -6, -4, 0, -1, -1, 0, 0, -2, -2, 0, 1, 1, 2, -2], [0, 4, 2, 1, 0, -1, -3, -1, 3, 0, 1, -2, 1, -2, 0], [7, -1, 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-2, 3, -3], [5, -4, 5, -5, 2, 2, -3, -1, 0, 1, -6, 2, 4, 1, -2], [16, 4, 2, -5, -2, 6, -1, -1, 5, 0, 5, -5, -11, 0, 2], [-4, -3, -1, -8, -3, 3, 0, -2, -1, 3, -1, 3, -6, -5, 0], [7, 7, -2, 7, 0, 0, 7, 1, 3, -1, 3, 4, -1, -4, 3], [10, 4, -3, -5, -8, 2, 1, 1, -5, 2, 2, 2, -2, 1, -3], [2, -6, -5, 5, -1, 0, -4, -1, -3, 2, -2, 0, 5, 3, 1], [-17, -1, -5, -3, -3, -3, -1, -3, -1, -3, 4, 0, 5, 1, -5], [5, 0, -7, -3, -3, -2, -3, -6, -2, -3, 1, 0, 3, 3, -2], [-2, 11, 1, -3, 0, 1, -4, 7, -2, -2, -8, 6, 0, 1, -2], [-9, 11, 4, 10, 1, -5, -5, -3, 5, 1, 1, -7, 10, -2, -1], [-23, 3, 2, -2, 1, 0, 11, 0, 0, 7, 2, 1, -1, -2, 0], [-16, 5, -3, -3, -4, 1, -2, 1, 0, -3, 0, -2, -2, -1, 3], [3, -7, 8, 1, 3, 5, -3, -3, 3, 4, -5, -7, 2, -2, -1], [-17, 0, -4, 4, 3, -7, 0, -6, -2, 1, 0, -3, 3, -1, -7], [8, -6, 0, -2, -1, -4, -6, 3, -6, 2, -3, 4, 5, 0, -5], [7, -3, 0, 6, 3, -4, 1, -4, 2, 5, 0, -1, -1, 0, -2], [6, -4, -3, 0, 0, -6, 9, -5, 2, 4, 4, -2, -1, 0, -5], [2, 5, -1, 7, 1, 3, -1, -1, 3, -1, 0, -6, -1, 4, 3], [7, -12, 1, -2, -2, 6, 0, 2, -5, 0, -5, 6, -3, 2, 1], [6, -3, -4, -6, -2, -4, -4, -2, -7, 3, -2, -3, 0, 0, -4], [4, 7, -2, 5, 6, -8, -5, -1, 3, -3, 0, -5, 8, 1, -3], [-12, -9, -4, 3, -4, -3, -6, -5, -4, -4, 0, -1, -1, 8, -2], [-4, -11, 0, -9, -2, 5, -5, -3, -4, -3, -3, -1, -5, 4, -1], [5, 1, 2, 0, -2, 3, 5, 4, 2, -3, -4, 4, -4, 0, -1], [0, -7, -5, -6, -5, 5, -5, 1, -4, -4, 3, 2, 1, 8, 1], [1, -3, -2, 3, 2, -4, 0, -6, -2, 2, 10, -4, 5, 1, -5], [17, -2, 4, 9, 4, -1, 1, 0, 8, 1, 6, -8, -1, -2, 2], [-7, 9, 0, 3, 3, -2, 7, -2, 3, 1, 2, -6, 4, -4, 0], [-2, -5, 7, -1, 3, 2, 2, 3, 2, 0, 0, 1, -3, -3, 2], [6, -1, -7, 8, -2, 1, -2, -3, 3, 0, 8, -5, 1, -1, 1], [-6, 4, 1, 5, 3, -4, 6, 1, 1, 2, 0, 0, 3, -3, -5], [-1, -12, -6, -1, 0, 3, 5, 4, -6, -1, 6, 4, -7, 4, 4], [-4, -8, 5, 5, 0, 0, -4, 0, -9, -2, 1, 2, 4, 9, -2], [-21, 5, 0, -1, 0, 1, -1, 0, 2, 1, 0, 6, -7, 2, 1], [2, -5, -1, -4, 5, 5, 1, -3, 1, 1, -2, 1, -4, -3, -2], [4, 0, -3, -5, -1, -4, -9, -5, -7, 1, 0, -2, -1, 3, -5], [22, -5, -1, 2, 0, 2, 9, 4, 1, -2, 7, 2, -7, -1, 5], [0, 1, 6, 10, -2, -6, -9, 7, 4, -4, -8, 4, 11, 1, 1], [-14, 9, -1, 0, 5, -1, 2, -4, 3, 1, -3, -5, 4, -5, 1], [-9, -5, 4, -3, 5, 1, -5, -3, 5, 0, -3, -5, 2, -6, -2], [-21, -3, 3, 5, 2, 1, 5, 2, 3, 0, 2, -2, -4, -3, 4], [0, 0, -2, -5, -1, -1, 0, -2, -3, 3, -2, -2, 2, 2, 8], [-12, -8, -2, -13, -5, 0, 4, -1, -3, 5, 1, 3, -4, -2, 4], [5, -15, -8, -5, -4, 4, 3, -3, -7, 2, -3, 4, -1, 4, -2], [-5, 2, -2, 6, 4, -1, 0, 1, -2, 3, -6, -2, -1, -1, 1], [-23, 2, -10, -1, -8, -6, 2, -4, -2, -1, 4, 2, -5, 5, -2], [-5, 10, -4, -4, 0, 1, 1, -4, -1, -2, 6, -1, -4, 0, -3], [-7, -6, 2, -8, -6, 9, 1, -2, 3, -3, 5, -1, -6, -2, 6], [10, 2, -2, 5, 4, 0, 14, -2, -1, 5, 6, -3, -1, -2, 0], [-10, 4, 5, -8, -1, 6, -5, 2, 2, 2, -3, -1, -8, 1, 3], [12, 6, 2, 0, 3, 4, 6, 7, -4, 0, -1, 6, -1, 0, 2], [-12, 1, -6, 5, 5, -6, 4, -1, -4, 0, 1, 4, 2, 0, -9], [15, -1, -2, 1, 0, 7, 6, -3, 0, 1, -2, 0, -2, -3, 0], [-7, -17, -6, -13, -3, 7, -3, -3, -9, -1, -3, 5, 0, 6, -5], [2, 9, -8, 4, -3, -4, 4, 3, -1, -1, 7, -1, -1, 0, -5], [6, 4, 7, -1, 2, -1, 14, 1, 5, 4, 2, 1, -6, -9, 4], [-14, -1, -3, -5, 0, 2, 2, -10, 4, 2, 7, -7, -6, 3, 4], [-15, 1, -1, 13, 2, -6, 1, -3, 5, 3, 0, -7, 10, -3, -2], [2, -12, -1, 2, -5, 3, 9, 3, -2, 1, 0, 6, -4, -1, 4], [-2, 9, 0, 2, 0, 3, -3, 2, -1, 5, -2, -4, -2, -2, 9], [5, -2, 6, 3, -2, 0, 8, -1, 0, 7, -2, -3, -1, -6, 3], [-19, 11, 10, -6, -4, 3, -1, 4, 2, -1, -2, -4, 2, -4, 5], [-5, 10, 0, -1, 0, -3, -1, -2, 4, 0, -4, -2, 5, -4, 1], [-2, 8, 0, 3, 0, -3, 3, -1, 5, -3, -4, 2, 3, -4, -3], [-11, -5, -4, 4, -5, -6, -2, 0, -8, 3, 0, 3, 5, 2, 1], [-1, 0, 7, -5, 1, 1, -8, -5, 2, 3, -7, -3, 1, -1, -2], [16, -2, 3, 0, 0, -1, 3, 0, 1, 2, 2, -1, 6, 2, 2], [3, 1, 1, -2, 0, 4, 7, 3, 0, -1, -12, 8, -3, -1, 2], [-11, -8, -3, -7, -2, 0, 2, -4, 0, 3, 2, -4, -12, 4, 5], [18, -2, -1, -5, -8, 2, 0, 0, -3, 1, -5, 6, -8, 5, -2], [-11, -12, -9, -9, 0, -5, -4, -10, 0, 8, -2, -3, -2, 1, -4], [5, 3, 2, -6, -3, 2, 5, -2, -2, -1, 0, 3, -3, 6, 3], [0, 2, -1, 6, -2, 5, -5, 8, -1, -2, -3, -2, 3, -6, 5], [6, 2, 6, -2, -1, 10, 6, -1, 10, 1, 3, -6, -9, -6, 5], [8, 15, 1, 5, 9, -5, -6, -4, 4, -1, -3, -6, 5, 1, -6], [13, 8, 7, 12, 4, 2, 1, -3, 1, -1, 0, -7, 1, -3, 3], [-16, -10, 1, 0, 0, 2, 0, 2, 4, -2, 7, -1, -1, -3, -2], [8, -8, -5, -5, -6, -1, -8, 1, -7, 5, -6, 3, 0, 1, -5], [-16, 3, 2, 0, 3, -6, -9, -6, 9, 1, 4, -15, 3, -2, -4], [10, -1, -1, -4, -2, -2, -4, -2, 1, -2, 7, -4, 1, 3, -2], [13, -2, -2, 12, 0, 2, -10, 8, -1, -4, -2, 3, 4, 1, -3], [-3, 0, -3, 10, 6, -7, 3, -6, -3, 1, 7, -6, 3, 3, -5], [7, 7, -1, 0, -3, 3, 11, -1, -2, 0, 8, 3, -2, -5, 0], [-4, 4, 5, -8, 1, 3, -3, 1, 1, 4, -1, -3, 3, 0, 0], [-16, 7, 3, 8, 1, 2, 6, -1, -3, -4, 2, -2, 0, 3, 3], [11, -2, -3, -1, -1, -3, -4, 2, 3, 1, 3, -5, 5, 4, 0], [15, -10, -3, 14, 6, -7, 4, 1, 1, 2, 9, -6, 7, 2, 3], [9, -10, 3, 10, 2, -1, 2, -9, 4, 1, -2, -6, -1, 2, 3], [2, -4, 5, -13, -3, -1, -5, 1, -3, -3, -3, 9, 1, 8, -4], [6, -6, 1, 7, 4, -4, 0, -4, 5, 5, -1, -12, -2, -3, -4], [12, -3, 0, -4, -2, 4, 0, 8, -7, -6, -4, 11, 0, 2, 0], [-9, -10, -1, -6, -3, -2, -3, -2, -9, -1, -1, 13, 0, 11, -3], [11, -2, -2, 7, -3, 2, -3, 4, -8, 3, -2, -1, 0, 8, 0], [10, -5, 5, 0, 7, -3, 6, -4, -1, 3, -3, -1, -2, -3, 1], [-3, 3, 1, -5, 1, 8, 3, 4, -5, 2, 1, 5, 1, 1, 2], [-7, 5, 7, -13, 1, 9, -8, 0, 1, -2, -14, 4, -1, -4, -2], [-6, -11, 1, 2, -5, -2, -6, 1, -5, 1, -8, 8, 2, 3, -1], [20, -7, 7, 11, 3, 1, 3, -1, 3, 5, 0, -10, 1, -5, -4], [15, 3, 3, -7, 1, 1, -9, 1, 1, 1, -10, -2, 7, 1, 0], [-13, 10, 5, 7, 3, -2, 2, 4, 8, -1, 5, -2, -3, -5, 6], [16, -7, -2, 0, -2, 1, -8, -3, -1, -3, 1, 0, 2, -2, -3], [19, -6, 4, -8, -2, -1, -5, -2, -3, 5, -7, -5, 4, 0, 0], [-7, 0, 6, -10, -1, -1, 3, 1, -1, 7, -1, 1, 3, -6, 6], [-6, -5, 0, 15, 2, 4, 6, -2, 4, 3, 10, -4, 3, -2, 0], [2, -7, -3, -17, -5, -2, -8, -7, -4, 2, -4, 5, -1, 5, -8], [-7, 8, 3, 9, 4, -1, 4, 3, -2, -5, 0, 5, 4, -1, -3], [3, 19, -4, 1, 0, 5, 4, -2, 5, -4, 9, -3, -3, 2, 1], [5, -2, -1, 5, 6, 3, -2, -6, 8, -2, 2, -3, -4, -5, -5], [-2, 4, 6, -3, 1, 5, 4, 0, 1, -5, -4, 0, -2, 2, 2], [7, 5, -2, 22, 4, -7, 7, -4, -2, 2, -2, -2, 10, -2, -4], [-5, -8, -4, -2, -5, -1, 1, -5, -1, -5, 5, 1, -11, 6, -3], [14, 5, 8, 5, 3, 5, 8, 2, 2, 4, -2, -1, -2, -4, 8], [0, -9, -1, 2, 1, -4, -4, 1, 1, -2, 4, 2, 4, -1, -4], [-12, -2, 6, -9, -3, 10, -6, 9, -1, -3, -5, 5, 0, -2, 0], [-6, 9, -3, 1, 2, -9, 1, -6, 0, 3, 5, -4, 2, 3, -3], [9, 6, 0, -2, 0, 6, -3, -3, 7, 1, 7, -8, -8, -4, 2], [9, 7, 1, 12, 6, -4, -3, -7, 2, -4, 2, -6, 5, 3, -8], [0, -8, -1, 2, 0, -2, 4, 4, 0, 2, -3, 5, -3, -1, 3], [1, 7, -3, 14, 0, 0, -2, 4, 3, -7, 3, 0, 1, 5, 2], [-4, -7, 4, -4, 6, -7, 2, -5, 1, 6, 3, -4, 0, -2, -3], [-9, 4, -5, -9, -1, -3, 8, -6, 3, 8, 4, -4, -4, -2, -2], [9, -9, 4, 7, 6, 10, -2, -1, 5, 1, -1, -2, -5, -4, 1], [-3, -5, 4, -1, 3, 2, 1, 0, 5, -1, -6, 6, -2, -2, 0], [-9, 11, 6, -3, 1, -6, -14, -2, -3, 0, -9, 0, 10, -2, -6], [23, 0, -5, -2, 4, 6, -1, -5, 3, 4, 8, -13, -3, -1, -1], [11, 7, 3, 14, 2, 1, 7, 4, 2, 4, 7, -5, 1, -3, 7], [-28, 10, 0, -6, 2, 1, 0, 5, -9, -3, -7, 10, 8, -2, -2], [14, -2, 0, -1, -3, 0, 4, 7, -7, 1, -1, 7, 4, 2, 2], [7, -5, 1, -9, -4, 6, 7, 1, -1, -1, 2, 11, -12, 5, 1], [-1, 6, 0, -2, 2, -2, 9, -3, 0, 1, -1, 1, -2, -6, 9], [-11, -2, -2, 13, 0, -3, -3, -6, 8, 0, 8, -8, 3, 6, -1], [7, 9, 6, 1, -3, 0, -9, 10, -7, -5, -8, 12, 8, 4, -1], [8, -2, 1, 1, 6, -4, -6, -4, 7, -2, -1, -6, 0, -1, -1], [-21, 4, 3, -5, 1, 0, 9, 0, -1, 5, -3, -3, 1, -5, 3], [4, -1, -5, 3, 5, -7, -3, -9, -3, 6, 3, -13, 4, 0, -3], [5, 7, 7, -1, -1, -4, -9, -8, 7, -1, -3, -7, 1, -3, -3], [-8, 3, 6, 2, 3, 8, 2, -1, 7, 1, -1, -1, -7, -8, 6], [5, -7, 2, -1, 4, 5, 0, 1, 4, 1, 4, -2, -2, 3, -3], [0, -5, -8, -4, 1, 2, -4, 9, -5, -11, 1, 7, 1, 4, -1], [2, 4, 9, -1, -3, 7, -8, 6, 1, -4, -5, 5, -7, 3, 0], [-4, -13, 2, -7, -2, 3, -2, -3, -4, 3, 3, 1, -5, 4, 0], [-21, 6, 6, 6, 7, -4, -7, 0, 3, -2, -6, -6, 9, 4, -2], [-13, -4, -1, 1, 8, 0, -2, -10, 7, 0, 3, -6, -2, -2, 2], [10, -6, -10, -6, 0, 0, 10, -4, -3, 4, 4, 1, -9, 2, -5], [29, 7, -4, 7, -2, -3, 7, 8, -11, -2, 2, 5, 1, 2, 6], [-10, -4, 0, -7, -6, 3, -4, 3, -2, -2, -1, 1, -3, -4, 2], [-20, -8, -5, -9, -6, 1, -2, -1, -7, -1, 4, 3, 0, 3, -5], [2, -8, -8, -1, -2, 1, 13, 2, -5, 1, 0, 10, -1, 2, 3], [-9, -8, -9, -13, -3, 3, 3, -3, -1, 5, 8, 0, -10, 3, 5], [7, -4, -6, 7, -5, -1, 0, 6, -2, 4, 8, 3, 2, 3, -1], [8, -11, 2, -6, -5, 1, -6, 2, 2, 2, 4, 1, 5, 4, 1], [-6, -3, 2, 12, 3, -7, 0, -2, 3, -1, -2, -2, 3, 4, 1], [-6, -5, 0, 1, 0, -8, -8, -2, -8, 2, -6, 0, 7, -4, -9], [6, 3, -2, -9, -2, -9, 4, 1, 2, 5, 1, 1, 3, 2, 4], [24, -11, 7, 1, 1, 4, 4, 5, 7, 2, 6, -4, -1, -3, 2], [1, -6, -1, -11, 0, 2, -9, -13, 2, -1, 0, -4, 0, -3, -3], [-9, 4, 9, -1, 5, 7, -3, -1, 4, -6, -6, -5, -5, 1, 4], [3, 12, 4, -11, -4, -1, 7, 4, 0, 2, -4, 8, -5, 2, -1], [2, -13, -8, -8, -4, -4, -10, -6, -15, 2, -4, 5, 2, 8, -10], [7, 6, 9, 3, 4, 7, 1, -2, 4, 4, 1, -9, -4, -1, 4], [4, 8, 5, 1, 2, -2, 3, -2, 3, 0, 12, -8, 2, -3, 2], [-18, -4, -13, -2, -2, -4, 9, -7, -1, -1, 1, -2, -6, 2, 0], [-5, -7, -13, 5, 0, -4, 8, 0, 2, -4, 7, 5, -2, 3, 0], [12, 8, -6, 3, -4, 0, 5, -2, 3, -4, -3, 1, -1, -4, 0], [3, -7, -10, 2, 0, 2, 1, -5, 0, 1, 3, -1, -6, -2, -6], [20, -3, -7, -1, -5, 1, -5, 0, -6, -5, 3, 8, 1, 1, 1], [-12, -1, -1, 0, 0, 1, -2, 9, 0, -6, -6, 6, 1, -3, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8036_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8036_2_a_r();".
// 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_8036_2_a_r(:prec:=15)
    chi := MakeCharacter_8036_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_8036_2_a_r();".
// 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_8036_2_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8036_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3381, -399, -22584, 8429, 38415, -22309, -20311, 13971, 4929, -3853, -616, 537, 39, -37, -1, 1]>,<5,R![1237, -10789, 6878, 55903, -58391, -60329, 56453, 32765, -17135, -7949, 2258, 911, -135, -49, 3, 1]>,<11,R![470097, 3230766, -1888083, -10332549, 1114080, 6700194, -1920865, -1037563, 437463, 34098, -31027, 1161, 888, -76, -9, 1]>],Snew);
    return Vf;
end function;