
// Make newform 6036.2.a.g in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6036_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_6036_2_a_g();".
// 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_6036_2_a_g();".
// 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 := [-44, 543, -1381, -1041, 5356, -134, -7100, 1501, 4076, -1105, -1004, 266, 106, -27, -4, 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], [-546137578798, 4026629840681, -3528723902040, -8590765226022, 11397811685951, 4454535017866, -9186271236513, 706301147372, 2268619321365, -551873907794, -197604821838, 78821293300, 1639898571, -3455243067, 358340218], [400272956308, -2298192379866, 1460693112978, 6569919538899, -5530760533108, -5774695912957, 5206469459211, 1834247254936, -1737416359970, -288553642002, 253795233195, 21911983318, -16928063565, -628785145, 423712226], [-198959454251, 3937809774856, -13294732750773, -234681574773, 32094202063099, -16261330615534, -22290712534461, 16869229948300, 3855492281382, -4887018761755, 74099057337, 551620921145, -62131751904, -21649170771, 3567036137], [31492118482, 92994480428, -2545481553708, 3759925861648, 6053346531801, -9166061389940, -3740738899709, 7086723160006, 103512865275, -1906796199932, 200675102334, 208529518638, -33188007874, -8034154554, 1533791708], [-117449952898, 1272169711354, -3637991875353, 53788357764, 8186083588136, -5686955517688, -4622078754486, 5955137864209, 43750977609, -1793151763368, 247221177892, 207446618172, -38548196723, -8250353036, 1731551174], [319535662097, -5018365640773, 4020848205666, 22577656847343, -15219609456415, -31919893118456, 18163440635214, 18554016837980, -8834705170920, -4681899694094, 1780883206311, 502149347719, -162031083048, -19093754208, 5568687106], [-142936166389, 2460004392646, -6663639693128, -665985805523, 15997332900914, -8457780235689, -10999683311240, 8924962826124, 1813047829279, -2588425841213, 68286852293, 292039374175, -34583644911, -11454090727, 1924356769], [-157050312484, -1781773064236, 2349937306503, 11347474643385, -8733916171558, -21508962230990, 12916624866327, 16007511236417, -7918515647748, -4611036393497, 1768601832687, 529474125973, -170476323909, -20946481836, 6083709676], [-54571509779, 1781074947118, -8214201190113, 8297232990042, 16205144774653, -28904743145578, -5863573811574, 25024944501244, -3323137051056, -7184392173223, 1410838550580, 814474288403, -175831046238, -32010037530, 7260291644], [-461279859325, 965571711005, 9683158812840, -19020179732652, -18546626159971, 40974759929053, 7854912941457, -30237803769097, 2542527461046, 8109490478674, -1307054951988, -888312598286, 171462500097, 34249747695, -7262151155], [92979418225, -98458986287, 4844271601440, -15180056486778, -8610542857280, 38258309514041, -737955429000, -30533855283224, 6103608989484, 8575689751328, -1943112254400, -963258026887, 223317906615, 37650120882, -8881419661], [716312928815, -5480104040764, 15978620346393, -6976084915473, -39366359011501, 40592532372535, 24162681101964, -37488998318524, -669825226899, 10729352393770, -1246495383216, -1208912683499, 204278546022, 47363234112, -9385042712], [164727575441, -2704514891878, 8383557145413, -2788505274183, -18525088631636, 17503280621078, 10599496006485, -16011768064319, -202805385751, 4576721419850, -556047690094, -515653560454, 89153456075, 20201794850, -4057390393]];
    Rf_basisdens := [1, 1, 58444065093, 19481355031, 58444065093, 19481355031, 19481355031, 58444065093, 19481355031, 58444065093, 58444065093, 58444065093, 58444065093, 58444065093, 19481355031];
    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_6036_a();"
function MakeCharacter_6036_a()
    N := 6036;
    order := 1;
    char_gens := [3019, 4025, 2017];
    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_6036_a_Hecke(Kf)
    return MakeCharacter_6036_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], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 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, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [-2, 1, 0, 1, 0, -1, 0, 0, 0, -1, 0, 0, -1, -1, 1], [-1, -1, 1, 0, 0, 0, -1, 0, 1, 1, 0, 0, 0, 0, -1], [1, -1, -1, 0, -1, 0, 0, -1, 1, 1, -1, 0, 0, 1, 0], [-2, -2, 1, -1, 0, 1, 0, 0, 0, 0, -1, 0, 1, 1, -1], [-2, -1, 0, -1, -1, 0, 2, 1, 0, -2, -1, 0, 0, 0, 0], [0, -1, -1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0], [1, -1, 0, 0, 0, 2, -1, 1, 1, 1, 1, 1, 2, 1, 0], [1, -1, 0, 0, 1, 1, -1, -1, 1, 2, 1, 1, 1, 0, 0], [0, 1, 1, -1, 0, 0, 0, 1, -1, -2, 1, -1, 0, -1, 1], [-2, 1, -1, 1, 1, 0, 0, 0, 0, 0, 0, 1, -1, -1, 0], [-3, 1, 0, 1, 0, 0, 1, 0, -1, -1, 0, -1, 0, 0, 0], [-2, 1, 2, -1, 0, -1, 0, 1, -1, -2, 1, -2, 1, -2, 0], [0, -3, -2, -1, -1, 1, 1, -2, 0, 2, -2, 1, 0, 2, -1], [0, -2, -1, 0, 0, 0, -1, -1, 0, 2, 0, 0, -1, 1, -2], [-3, -2, -1, 0, -2, -1, 0, -1, 3, 3, -3, 1, -2, 3, 0], [-1, 1, 1, 1, 1, 0, -2, -2, -2, 1, 2, -1, 1, -1, -2], [-4, 3, 2, 1, 2, -1, 0, 1, -3, -3, 2, -2, 0, -4, 1], [-4, 0, 0, -1, -1, -2, 1, 1, 1, -2, -3, 0, -2, 0, 0], [-1, 3, 0, 0, 1, -1, -1, -1, -1, -1, 2, -1, 1, 0, 0], [0, -2, 0, -1, 0, 2, 0, -2, 0, 1, -2, 2, -1, 2, 0], [-4, 0, -1, 2, 2, 0, -3, 0, 2, 3, 1, 2, -1, 1, 0], [-3, 3, 0, 1, -4, -3, 3, 0, 0, -3, 0, -3, -3, -2, 3], [-6, 3, 1, 4, 1, -3, -2, 0, 2, 2, 0, 0, -2, -1, 0], [-5, 3, 2, 1, -2, -2, 2, 0, -4, -2, 0, -4, -1, -2, 0], [-2, -1, -2, 0, -1, -1, 3, 0, 1, 0, -2, -1, -2, 2, 1], [0, 1, -2, 1, 3, 1, -3, -2, 2, 4, 1, 3, 1, 2, 0], [2, 2, 0, -1, -1, 1, 0, -1, -2, 0, 1, 0, 0, -1, 0], [1, 3, -1, -2, 0, -2, 3, 2, 0, -2, 4, -2, 2, -3, 4], [0, 1, 0, 1, 3, 1, -1, 1, 0, -2, 1, 2, 0, -1, 0], [-2, -3, 0, -1, 1, 1, 2, 1, -1, -1, -1, -1, 0, -1, 1], [-2, -2, -1, -1, 0, 4, -2, -1, 2, 2, -2, 3, 0, 4, -2], [-1, -2, 1, 1, 1, -1, 1, 1, 0, 1, 0, 2, 1, 0, 0], [-4, -1, 2, 0, -3, -1, 2, 0, 1, -1, -2, -3, -1, -2, 0], [-2, 4, -1, 0, -4, 0, 3, 2, 1, -2, 2, -1, -1, -1, 4], [0, 2, 0, -1, 2, -2, -2, 1, -1, 0, 3, 0, 3, -2, 2], [1, -1, -3, -1, 1, 2, -1, -1, 2, 4, -2, 3, 1, 5, -1], [0, -1, 2, -2, 1, 1, -3, 1, -1, 2, 2, 0, 2, 0, -1], [-5, 3, 0, -1, -1, 0, 2, 1, -3, -2, 1, -3, 2, -3, 0], [2, -5, -3, -1, -4, 1, 3, 2, 3, 2, -4, 0, 0, 3, 0], [0, -1, 0, -1, 0, 0, 2, 1, -2, -1, 3, 0, 1, -1, 0], [-2, 0, 1, -1, -5, 1, 2, 0, -1, -1, 0, -2, -1, 1, 1], [-4, 4, 3, 3, 0, -3, 3, 2, -3, -4, 2, -4, -3, -3, 3], [3, 1, 3, -3, 1, -1, -2, 2, 1, -1, 2, 1, 5, 0, 1], [-3, 1, 1, 0, -1, -1, 2, -2, -1, 0, -1, -2, 0, -2, 3], [-3, -6, -1, -5, -4, 3, 2, 0, 1, -2, -4, 1, -1, 2, -3], [2, 1, -2, -1, -5, 1, 5, 1, 0, -3, 1, -1, 0, 0, 2], [-8, 3, 1, 1, -1, -5, 3, 1, 0, -2, -2, -1, -1, -1, 2], [0, 2, 0, 1, 1, 3, 0, 1, -1, 0, 2, 0, 2, 1, 0], [-3, -2, -1, -2, -2, 2, 0, -1, 0, 1, -5, 2, 0, 1, -5], [1, 0, -3, 1, 2, 0, -3, -4, 0, 5, 0, 1, -1, 4, -1], [-4, 4, 0, -1, 2, 1, -5, 1, 2, 1, 4, 3, 5, 1, -2], [-1, -1, -1, 3, 3, 3, -5, -2, 4, 6, -1, 4, 1, 1, 2], [2, 1, 1, -2, 1, 1, 0, 0, -4, 1, 0, -3, 0, -1, -1], [-1, -2, 0, -4, 1, 3, -1, -3, -2, 0, -5, 0, 0, 2, -2], [-7, 1, 0, 2, 0, -2, 3, 0, -3, -4, -5, 0, -4, -3, 2], [6, -1, -4, -3, -5, 2, 7, 1, 3, -1, -1, -1, 1, 0, 1], [-1, -5, -2, -4, -7, -2, 4, -1, 1, 0, -9, 1, -3, 4, -3], [3, -2, -2, -1, -3, 4, 2, -2, 0, 2, -2, -3, -1, 2, -2], [-7, 5, 2, 0, 0, -3, -1, 2, -3, -1, 1, -4, 2, -1, 0], [-6, 3, -3, 8, 1, -5, -3, -6, 4, 6, -3, 1, -8, -1, 2], [-5, -3, 2, -2, 2, 2, -4, 2, -1, -1, 3, 1, 1, 0, -3], [6, 2, 5, 0, 5, 4, -7, -1, -4, 2, 8, 0, 5, 0, -3], [7, -3, -1, -3, -1, 3, 3, 3, 1, -1, 1, 0, 2, 0, 1], [2, 1, -2, 2, 1, 4, -4, -4, 1, 6, -2, 2, -1, 5, -2], [-3, -1, -3, 1, -4, 1, -1, -1, 7, 6, -3, -1, 1, 4, 0], [-14, 6, 4, 4, -1, -5, 4, 1, -6, -8, -2, -3, -7, -5, 3], [1, -9, -1, -2, -3, 1, -2, -2, 7, 8, -4, 4, 1, 7, 0], [-1, 2, -4, 2, 2, 2, -2, -1, 1, 3, 4, 1, 0, 2, -3], [9, 1, -3, -2, 6, 3, -3, 1, -1, 2, 6, 1, 5, 0, -2], [5, -1, -2, 2, 4, 2, -4, -2, 3, 1, 1, 6, -1, -2, -2], [-1, 6, 0, 0, 0, 3, 0, 3, -4, -3, 7, -2, 0, -4, 1], [-10, 3, 6, 3, 4, -2, -3, 1, 0, -3, 2, 2, 1, -3, 3], [0, 2, -1, -2, -7, 1, 6, 4, -2, -5, -2, -4, -1, -1, 0], [-4, -1, 0, -1, 2, -6, 4, 2, 0, -5, -2, 2, -1, -2, -1], [6, -3, -2, -1, 3, 3, -5, -4, 2, 5, 4, 2, 4, 1, -2], [6, 1, 2, -2, 4, 1, -2, 1, -6, -2, 1, 2, 4, -2, -1], [-8, 8, 2, 2, -2, -7, 4, 6, -1, -7, 4, -3, -2, -5, 5], [-9, 9, 5, 2, -2, -5, 6, 7, -3, -10, 2, -4, -1, -7, 6], [-3, -1, -2, 2, -3, -3, 3, -3, 2, 0, -9, 2, -9, 3, -1], [-5, -3, -3, 1, 0, 0, 1, -6, 2, 0, -8, 2, -4, 4, -3], [-4, -4, 3, 1, -4, 1, 4, 0, -2, 0, -1, -2, -1, 1, -1], [-11, 0, 2, 5, -2, -5, 0, -2, 3, -1, -7, -3, -6, 1, 0], [5, -2, -5, 0, -2, 2, 3, -1, 3, 0, -5, -1, -2, 4, 3], [-16, 6, 6, 7, 2, -6, -2, 3, -2, -3, -1, -2, -1, -7, -1], [-1, 5, 1, 5, 5, -3, -2, -5, -2, -1, 0, 0, -5, -2, 1], [-3, 8, 1, 7, 6, -5, -4, -2, 1, 1, 7, 0, -4, -4, 4], [-5, -4, -4, 1, 5, 3, -2, -3, -2, 3, -2, 1, -2, 3, -4], [5, -3, -2, -4, -2, 0, 1, 1, 0, 0, 2, -1, 1, 1, -3], [3, -3, -1, -2, -1, 5, 0, 3, -2, -1, 3, -1, 0, 0, -1], [7, 0, 1, -2, 0, -3, 4, 2, -3, -5, 2, -4, 1, -3, 2], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -7, -1, -1, 0, 5, -1, -2, 1, 8, -3, 1, -2, 4, -1], [7, -7, -3, -2, 7, 4, -9, -5, 2, 8, 2, 7, 3, 2, -5], [10, -1, -5, 3, -3, 0, 2, 1, 9, 3, -4, 3, 0, 3, 1], [-2, 1, 6, -5, 2, 0, -1, 3, -2, -7, -1, -2, 2, 0, 2], [16, -4, 0, -9, 0, 3, 3, 1, -2, -4, 2, 1, 7, 2, 0], [6, -5, 0, -2, 0, 0, 2, -5, 1, 1, -8, 2, -4, 5, 1], [1, -1, 3, 0, -3, 1, -2, -1, -3, 2, 1, -2, 2, 4, -4], [-4, 1, -1, 4, -4, -3, 0, 1, 6, 0, -3, 1, -3, 0, 2], [9, -5, 0, 4, 4, 2, -10, -2, 5, 13, 1, 6, 0, 9, -4], [-3, 6, 4, 4, 5, 0, -9, -1, -2, 0, 3, 1, 1, 0, -2], [8, 1, 0, 4, 6, 0, -8, -2, -2, 3, 3, 6, 1, -1, -1], [8, 0, 1, -4, 1, 1, 5, 5, -1, 0, 4, -2, 8, -2, 0], [-1, -12, 0, -1, 3, 6, 0, -4, -3, 5, -1, 2, 0, 3, -7], [-5, 8, -1, 2, -1, -1, 6, 2, -5, -7, 4, -4, -4, -3, 5], [-1, 2, 2, 3, 1, -4, 1, 0, -1, -6, -3, -2, -3, -3, 0], [5, -4, 2, -5, 1, 3, -2, -1, -1, 1, -4, 1, 7, 3, -5], [0, -2, -2, 2, 0, -5, 3, 2, 2, 0, -1, -3, -1, -1, 1], [2, 0, -1, 4, -1, -4, 4, 0, 3, -4, -6, 5, -6, 1, 6], [0, 1, 3, -2, -2, 1, 0, -1, -1, -3, -2, -2, 2, -1, 2], [0, -3, -4, -6, -2, 5, 3, -4, -3, -2, -8, 3, -1, 4, -3], [12, -3, 5, 2, 6, 4, -9, -2, 0, 5, 9, 2, 6, -1, -2], [13, -2, 0, 0, 5, 2, -5, 0, 1, 4, 7, 1, 9, -3, 1], [8, -9, -3, 2, -1, 3, -2, -5, 2, 9, -3, 6, 0, 8, -3], [-1, 0, 3, -1, 0, -3, 5, 5, -5, -8, 1, -1, 5, -6, 4], [-4, 1, 1, 1, 0, -1, 2, 5, 0, -4, 3, -3, 4, -4, 6], [-13, 18, 3, 3, 1, -8, -1, 5, -1, -8, 5, -4, -1, -8, 3], [4, 1, 2, 3, 8, -3, -7, -2, 1, 4, 9, 3, 0, -2, 0], [-1, -1, 1, 2, -2, -2, 5, 2, 3, -2, -4, -2, 0, -2, 1], [-3, -1, 0, 4, 3, -1, -3, 1, -3, 1, -2, 2, -4, 2, -4], [2, -1, 1, -1, 2, 3, -8, -4, 1, 5, 1, 3, 1, 3, -5], [8, -3, -6, -5, -7, 3, 5, 1, 0, -1, -2, -3, -3, 3, -1], [1, 0, -4, 1, -1, -5, -2, -11, 3, 8, -3, -1, -4, 3, -1], [5, 7, 4, 8, 3, -2, -3, 1, 0, 2, 6, -1, 5, -4, 4], [1, 3, -1, 3, 5, 3, -2, 1, -3, 3, 4, 4, 1, 0, 1], [-1, 15, 6, 6, 1, -5, 0, 6, -6, -8, 7, -4, 3, -10, 1], [2, 3, 2, 2, 8, -2, 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-9, -3, 1, -1, -6, 3, -5, -1], [-19, 11, 6, 3, 6, -3, 4, 14, -4, -19, 0, 0, 2, -5, 4], [9, -3, -4, 0, -4, -4, -5, 3, 16, 2, -1, 7, -3, 7, 5], [7, -7, -9, 7, 9, 0, -9, -6, 21, 20, -3, 14, -3, 14, -1], [5, 4, -3, 2, 0, 1, -9, 7, 3, 1, 4, 1, 2, 0, -1], [23, -19, -8, 3, 0, 5, -4, -2, 14, 19, -1, 12, 2, 9, -7], [-26, -5, 4, 2, -3, 2, 2, -8, -4, 2, -9, 2, -8, 4, -7], [-18, 6, 5, 2, -8, -10, 11, 0, -4, -14, -10, -10, -11, -7, 3], [5, 1, -1, 5, 13, 4, -4, -2, -9, -2, 8, 5, 3, 3, 2], [5, -16, 0, -3, 4, 10, 1, -6, 0, 7, -5, 6, 2, 6, 0], [15, -4, 13, -12, 2, 5, -10, 3, -1, -4, 14, -4, 14, -5, 0], [10, 5, 3, 8, 15, -2, -13, -1, -7, 7, 8, 6, 4, -2, -2], [4, 7, -4, 7, 4, -6, -3, -3, -11, -5, 4, -3, -7, -8, 5], [5, 1, 14, -2, 12, 3, -12, 10, -5, -5, 14, 2, 14, -1, -3], [-5, -3, 4, -1, 12, 8, -7, 6, 8, 2, 6, 7, 9, -5, 0], [15, 8, -2, -7, 0, 4, -5, 2, -5, 6, 0, -2, 7, 4, -3], [-4, -9, 0, -1, 2, 0, -6, -2, -2, 6, 5, -1, 0, -3, -9], [14, -5, -14, -4, -2, 5, -4, -4, 1, 11, -5, 2, 5, 11, -10], [7, -8, -6, -1, 4, 2, -2, 3, -6, 1, -1, 4, -1, -1, -7], [9, -3, -7, -2, -4, 0, -5, -4, 9, 12, -12, 6, 1, 8, -10], [-21, -6, 1, 4, 2, -16, 3, 2, 5, 3, -5, -6, -6, -6, -5], [1, 10, 8, -6, 15, 0, -5, 11, -11, -12, 17, -2, 19, -9, 3], [4, -14, -7, -2, 5, 14, -5, 1, 9, 16, 5, 10, 7, 11, -8], [0, -15, 0, -7, -14, 4, 6, -7, 1, 6, -8, -1, -1, 9, -3], [6, -6, 0, -2, -3, 12, -7, -3, 7, 11, -2, 7, -2, 11, -4], [18, -4, 2, 0, 10, -1, -15, -5, -2, 16, 3, 1, 1, 2, -8], [-29, 12, 1, 5, -10, -5, 1, -1, -3, -10, -2, -3, -9, -10, 4], [21, -4, 0, -4, 4, 7, -4, -1, 0, 0, 11, -3, 9, -5, 0], [5, -12, 5, -5, 0, 2, 0, 2, 15, 9, 0, 5, 12, 6, -1], [-7, 10, 2, 0, -7, -4, 13, 5, -10, -14, 1, -5, -9, -10, 6], [7, -5, -5, 8, -5, 0, 12, 2, 5, -1, -12, 1, -10, 4, 8], [13, -14, 3, -5, -13, -1, 6, 4, 4, 4, -4, -6, 3, -1, -6], [0, 9, 2, 6, 2, -14, -5, -11, 0, 6, 6, -8, -8, -2, 4], [-16, 8, 3, 2, -15, -15, 5, 5, 11, -12, -10, -10, -8, -1, 10], [-5, 2, 0, 2, 7, -15, 9, 1, -4, -15, -6, -2, -9, -1, 3], [-3, -13, -5, -4, -7, 12, 4, -2, 6, 5, -5, 4, 6, 10, -7], [-7, -8, -8, 3, 1, 6, -6, -15, 7, 14, -3, 7, -3, 12, -11], [-11, -3, -7, 4, -6, -11, 13, 8, 7, -6, -14, 0, -13, -3, 3], [-1, 8, 7, 3, 4, 1, 2, 2, -14, -10, 17, -7, 6, -14, -1], [6, -4, -7, -2, -2, 15, -1, -7, 9, 4, -4, 8, 1, 12, 1], [-3, -8, -1, -4, 8, 11, -14, -2, 9, 7, 1, 9, 8, 7, -11], [8, -12, -10, -2, 1, 1, 6, 2, 15, 1, 2, 10, -6, -6, 7], [-2, 3, 4, 5, -1, -11, -3, 1, 7, 1, -2, -5, -6, 1, 8], [-8, 6, 9, -1, 0, -5, 9, -7, -9, -7, -6, -6, -3, -3, 0], [-12, -11, -4, 8, -4, -6, -6, -11, 13, 19, -6, 5, -9, 5, -3], [4, 14, 11, -3, 8, -6, 7, 2, -16, -15, 9, -6, 2, -11, 4], [-15, 11, -4, 3, 0, -12, 1, -3, 1, 4, -6, 1, -9, 4, 9], [8, -3, -5, 9, 2, -10, -4, -7, 7, 13, -5, 2, -5, -1, 1], [7, -25, -5, -18, -11, 18, 9, 6, 16, 1, -23, 11, 5, 23, -4], [2, -5, -4, 0, 10, 14, -9, -2, 7, 17, 15, 8, 13, 7, -8], [-12, 24, 0, -2, -2, -4, 0, 4, -12, -16, 7, -12, -10, -11, 6], [-1, -5, -5, -6, 1, 2, 7, 10, -6, -6, 10, -2, 6, -6, 2], [7, -11, -10, 0, -3, -8, 11, -6, 8, 3, -15, 2, -11, 6, 6], [-6, -3, 9, -12, 8, -1, 8, 15, -8, -22, -6, -2, 5, -5, -1], [-14, 6, 0, 4, 0, -10, 1, 1, 1, -3, -2, -7, 0, -7, -1], [6, 5, -5, -2, -10, -3, 15, -6, -8, 2, -7, -9, -2, 1, 5], [-21, 1, 9, 7, -1, -8, 1, 3, 5, -4, -11, 4, -5, 6, -2], [-1, -11, 10, -3, 8, -2, -9, 3, 10, 3, 5, 5, 13, -4, 1], [-7, 27, 0, 3, 7, -2, 0, -4, -11, -4, 19, -7, 1, -4, 4], [13, -18, -6, -19, -10, 16, 9, -2, 1, 1, 0, -1, 7, 9, -7], [-4, 11, 9, 2, 2, -2, -6, -2, 2, -4, 3, 0, -3, -4, 2], [-9, 27, -2, 22, 5, -15, -3, -8, -7, 2, 6, -5, -17, -16, 7], [-6, -8, -14, 7, -7, 2, 4, -14, 9, 13, -18, 7, -12, 12, -6], [9, 6, -3, 2, 8, -7, -5, -4, 1, 5, 11, -4, 6, -5, 0], [2, -17, 8, -12, 0, 8, -2, 6, -7, -6, 1, 1, 13, -4, -10], [-22, 16, 17, 1, 11, -5, 3, 8, -19, -19, 13, -8, 1, -21, 2], [5, 4, -6, 2, -2, 11, -5, 3, 7, 4, 11, 4, 10, -6, 7], [6, -12, -5, 3, 0, 12, -3, 1, 6, 10, -1, 2, 12, 9, 3], [10, -13, -4, -8, -16, -3, 9, 0, 9, -5, -17, 3, -10, 9, 2], [-27, 18, 7, 12, 11, 0, -5, 5, -11, -16, 14, 1, -8, -19, 6], [-13, -6, 2, -4, 6, 11, -5, 5, 5, -6, -4, 14, 4, 7, -3], [12, -25, -5, -6, 5, 15, -13, -3, 20, 24, -2, 11, 14, 14, -17], [-17, 5, -1, 4, 4, 5, -8, 1, 1, 1, 8, -3, -3, 2, -3], [14, 4, 6, 3, 6, 8, -16, -3, -4, 12, 5, 1, 7, 5, -7], [5, 2, 9, -12, 1, 0, 3, -2, -5, -6, 3, -10, 7, -5, 4], [-10, 14, 1, 6, 6, 2, -1, -1, -19, -12, 4, -3, 2, -6, 0], [-2, -5, 6, 3, -4, 12, -3, -2, -2, 8, -1, -1, 9, -3, -12], [5, 12, 8, 5, 10, -1, -6, 1, -17, -8, 8, -4, 11, -9, -5], [5, 3, -4, 10, 7, -3, -8, -3, 3, 6, 5, 8, -9, 6, 2], [-13, -17, 4, -11, -4, 6, 9, 1, -7, -6, -15, -8, 8, 1, -9], [1, -7, 4, -7, -12, 0, 11, -6, 0, -2, -4, -9, 1, 1, 0], [3, 16, 0, 1, -1, -15, 0, 1, -6, 1, 4, -4, -2, -6, 6], [-23, 6, 6, 5, -1, -14, -3, -12, -4, 3, -8, -4, -17, 5, -2], [1, 11, 1, -2, -6, 0, 5, 2, 1, 0, 7, -8, 3, -11, 2], [23, -12, -11, 4, -10, 4, -4, -8, 18, 20, -5, 7, -1, 15, -4], [-11, -5, -6, 0, -2, 4, -5, 3, 5, 9, 6, -1, 3, 4, -1], [3, -6, -5, -7, -10, 3, 4, -5, 0, 8, -13, -6, -4, 8, -9], [9, 8, 2, -1, -1, 9, 6, 8, -11, -7, 12, -4, 12, 1, 4], [-3, -11, 3, -8, 3, 7, 10, 3, -3, -6, 0, 1, 4, 0, 9], [-20, -17, -2, -6, -11, -4, 3, -4, 15, -1, -17, 3, -12, 11, -1], [-21, 0, -4, -7, -13, -5, 15, 11, 1, -17, -21, -2, -2, -3, 4], [-3, 6, -2, -2, 0, 9, -3, -2, 2, 2, 1, 8, 8, 12, -6], [-21, 13, 2, 10, 5, -10, -2, -6, -7, 4, 4, -7, -3, -1, -4], [-14, 20, -9, 9, 1, -11, 11, 4, 4, -11, 10, -4, 0, -7, 12], [-5, 12, 6, -13, 1, -4, -2, 10, 2, -10, 13, -1, 16, -9, 4], [-8, 10, -2, -1, -9, -3, 10, 4, 3, -13, 2, -4, -4, -7, 11], [-12, 21, -1, 17, 15, 1, -19, -12, 0, 10, 9, 9, -4, -6, 2], [-5, 3, -8, 0, -7, -6, 14, 6, -8, -21, 4, -7, -1, -8, 8], [11, -11, -15, 10, -1, 9, 3, -9, 7, 20, -4, 10, -7, 9, 1], [-28, 16, -3, 7, 3, -10, 1, 10, 0, -6, 4, 4, -4, 0, 5], [6, -1, 2, -2, -6, 5, 0, -2, 8, 3, 5, -2, -7, 3, 4], [-6, -24, 1, -8, -12, 8, 4, -6, 5, 10, -5, -1, 6, 6, -3], [-18, 14, 1, 7, 0, -5, 7, 9, -2, -16, 7, -9, 2, -11, 12], [-34, 0, 6, -1, 4, 9, 4, 7, 1, -6, 6, -1, 5, 0, 2], [-2, 17, 7, 5, 13, 0, 2, 7, -5, -13, 14, 5, 4, -10, 6], [30, -10, -10, -12, -2, 7, 5, -3, -4, 7, -9, 1, 11, 10, -4], [-22, 1, 4, 9, 2, 6, -8, -8, 5, 9, 6, 5, -2, 0, 3], [3, -3, -14, 7, 0, 2, -8, -5, 0, 11, -5, 6, -5, 11, -1], [9, 3, 5, 9, 7, -13, -9, 0, 16, 10, 8, 6, -3, 1, 3], [-1, 5, 2, 1, -5, -5, 0, 4, -1, -7, -8, -4, -11, -2, 9], [-2, 2, -2, 4, -18, -1, 3, -4, 13, -7, -12, 0, -2, 3, -1], [-20, 19, 6, 11, 8, 0, -20, -13, -1, 12, 3, 7, -3, 3, 1], [1, -20, -1, -4, -5, 12, -13, -3, 4, 12, -9, 4, 6, 16, -9], [-1, -5, 4, 5, -3, -3, -1, 0, 9, 4, -7, -1, 8, 1, -1], [-13, 7, 10, 7, 13, 2, -11, 6, -1, 0, 10, 4, 9, 5, -4], [-25, -2, 17, 5, 2, -3, -1, 6, -20, -8, -3, -9, -1, -9, -7], [-5, -24, 0, -5, 2, 10, -16, -10, 11, 21, -8, 5, -2, 15, -16], [-6, -3, -12, 3, 8, 11, -5, -14, -5, 16, -5, 6, -5, 15, -10], [-23, -4, 20, 3, 8, -1, -6, 6, -12, -4, 1, -3, 3, -7, -13], [-4, 7, -6, -3, 5, -9, 5, 6, -7, -14, -2, 5, -6, -7, 5], [7, 0, -9, -4, 2, 8, 1, 0, 0, -5, 5, 0, 5, -5, -4], [13, -10, 3, -4, 4, 6, -5, -7, -1, 9, 6, 1, 11, -8, -2], [-21, 27, 3, 3, 4, -8, 0, 6, -7, -8, 18, -9, 0, -19, 4], [3, -9, -11, -3, -4, 8, 0, -2, 8, 7, 1, -4, 4, 12, -3], [-6, -21, -13, -9, -15, 3, 9, -9, 6, 3, -18, -3, -12, 13, -7], [5, 3, -7, -3, -3, -7, 11, -5, 5, -3, -8, 0, -11, 5, 12], [13, 16, -3, -1, 15, 2, -2, 3, -20, -8, 22, 3, 8, -4, 6], [11, 5, -3, 3, 6, 3, -9, -3, 10, 9, 12, 7, -2, 4, -2], [-16, 7, 7, 5, -17, -5, 8, 13, 7, -11, 2, -7, 2, -6, 12], [-9, 22, -5, 4, -8, -11, 5, 8, 1, -12, -5, -3, -15, -4, 11], [2, -4, -1, -9, -19, 2, 9, 9, 5, -1, 2, -7, 1, 1, 0], [-18, 13, -9, 8, -6, 1, 10, -6, -5, -4, -4, -6, -5, 1, 0], [4, -5, 8, -13, 9, 3, -8, -4, -6, 0, 11, -5, 15, 2, -12], [-17, 2, -1, 8, -14, -11, 8, -2, 5, -4, -11, -9, -14, -1, 9], [-10, 2, 5, 2, 14, -6, -11, -5, 3, 13, 6, 7, 2, -5, 0], [5, 1, -5, -5, 7, 5, 3, 6, -10, -1, 1, 5, 15, 0, -6], [-17, -1, -6, -2, -5, 1, 0, 4, -1, 2, -5, 1, -8, 8, 5], [12, -16, -3, -7, 8, 7, -7, 3, 14, 8, 8, 14, 9, 8, 0], [-16, 2, 8, -5, -2, -7, 2, -10, -13, -11, -5, -10, 0, -12, 2], [-21, 11, -3, 2, -2, -9, 10, -4, -10, -12, -3, -3, -1, -12, 10], [29, -10, -10, -13, 1, 19, -2, -15, -3, 5, 2, 4, 8, 13, -7], [2, 19, 3, 2, -2, -6, 10, -2, -10, -14, 13, -15, 3, -16, 13], [25, -7, -2, 2, 9, 10, -30, -4, 9, 23, 19, 11, 7, 0, -2], [-5, 0, -16, 11, 10, -4, -3, -8, 0, 7, -1, 2, -16, -6, 0], [10, 7, 7, -10, 6, 9, -9, -6, -18, -6, -7, 0, 2, 0, -6], [-6, -11, 1, 0, 7, -17, 5, 6, 9, 1, 3, -3, 5, -10, 9], [-31, 8, 0, -2, -3, -17, 9, -2, 6, -9, -6, -2, -14, 1, 11], [-24, 5, -7, -2, -8, -8, 11, 2, 2, -12, -7, -3, -18, -4, 3], [-7, -21, -10, -1, -10, 2, 2, -10, 0, 9, -20, 3, -1, 12, -7], [17, 6, -6, -1, -2, -1, -2, -6, 4, -2, 10, -1, 1, -1, 5], [-10, -29, -5, -7, -3, 1, 4, -5, -2, 6, -12, 0, -2, 8, -7], [-11, 2, 5, 2, 10, 3, -6, 8, 8, 4, 20, 3, 9, -8, 6], [-27, -3, 1, 3, -5, -12, 9, -6, 10, -10, -16, -8, -11, 2, 11], [0, 8, 9, 6, 2, -3, -2, 15, -4, -10, 11, 2, 6, -18, 5], [2, 1, -8, 4, -8, -4, 1, 1, 7, 17, 1, -5, -3, 6, 5], [-37, -9, 9, 2, -5, -16, 3, 8, 0, -2, -12, -2, -4, -4, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6036_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6036_2_a_g();".
// 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_6036_2_a_g(:prec:=15)
    chi := MakeCharacter_6036_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_6036_2_a_g();".
// 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_6036_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6036_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![9, -252, -258, 3021, 4277, -8105, -18628, -8044, 4985, 4497, 195, -659, -154, 22, 11, 1]>,<7,R![18812, -69742, 38341, 113161, -106403, -62003, 67247, 20887, -18592, -4662, 2530, 609, -164, -40, 4, 1]>],Snew);
    return Vf;
end function;