
// Make newform 1336.2.a.e in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1336_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_1336_2_a_e();".
// 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_1336_2_a_e();".
// 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 := [224, 1224, 148, -2399, -433, 1804, 129, -641, 36, 100, -15, -5, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-7879652, -43596932, 3377831, 72747321, -4221695, -38734849, 6214148, 7226582, -1750092, -343875, 110987, -4134], [-23226840, -112484744, 34242297, 181435679, -37971272, -91240819, 25075571, 15580284, -5651176, -500007, 333527, -26659], [3831112, -2303608, -17935139, -3899557, 22278164, 3606689, -11991941, 500524, 2394592, -387059, -134357, 28693], [13360032, 53347564, -33729889, -91641495, 42238968, 48730959, -25143667, -7428812, 5338476, -114673, -307651, 41307], [-173194, 1275146, -1861465, -5950306, 6497698, 4244680, -4950043, -358916, 1111702, -134173, -65032, 12411], [5796342, 17581492, -13839681, -27880538, 16669301, 13627972, -9086400, -1744024, 1828134, -108909, -102912, 15928], [-18920800, -90263824, 42811521, 149898571, -62809378, -76436047, 39588421, 11005132, -8486560, 295481, 489671, -70349], [13689440, 42986204, -112352907, -86413565, 168228596, 40411013, -89411357, 1949556, 17268420, -2586835, -951441, 194845], [12190868, 63733948, -28079769, -111353595, 42956708, 58720147, -28321401, -8855778, 6201508, -146621, -360853, 49825], [26939384, 121867592, -112277337, -215847827, 171582340, 109890519, -100394311, -10923772, 20591672, -2032305, -1165859, 207127]];
    Rf_basisdens := [1, 1, 243358, 973432, 973432, 973432, 243358, 243358, 486716, 973432, 243358, 973432];
    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_1336_a();"
function MakeCharacter_1336_a()
    N := 1336;
    order := 1;
    char_gens := [335, 669, 673];
    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_1336_a_Hecke(Kf)
    return MakeCharacter_1336_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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [1, 0, 1, 0, 0, 0, -1, 0, 0, 0, 1, 0], [-1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 1, -1, 1, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, -1], [2, 0, -1, 0, 0, 0, 1, 1, 1, 0, -1, 0], [0, 1, 0, -1, 1, -1, 1, 1, -1, -1, 0, 0], [0, -1, 0, 1, 1, 0, -2, -1, -1, 0, 0, -1], [-1, 0, 0, 0, -2, 1, 1, 0, 0, 0, -1, 0], [0, 0, 1, -1, 1, 1, -1, -1, -2, -3, -1, 0], [3, 0, 1, 1, -1, 0, 0, 1, 2, 0, 0, 0], [3, -1, 1, -2, -1, 1, 2, 0, -1, -1, -1, 2], [-1, 0, 1, 1, 1, 0, -4, 1, 2, -1, 2, 0], [1, -1, 1, 2, 0, 0, -3, -2, -3, -1, 1, 0], [-1, 0, 1, -1, 1, 1, -1, 0, 0, -1, 1, 0], [5, 0, 0, -2, 0, 1, 3, 2, 2, 0, 1, 0], [2, 0, -2, -1, 1, -1, 0, 2, 1, -1, 0, 0], [-1, 0, 1, 0, -2, -1, 0, -1, -1, 1, 0, 0], [2, -2, -1, 1, 1, -1, -3, -1, 0, 1, 1, 0], [4, -1, 0, -1, -1, 2, 1, -1, -1, -1, -3, 2], [1, -1, -2, 2, 0, -3, 0, 0, 0, 1, -1, -1], [-1, -3, 1, -1, 3, 0, -3, -1, -1, -2, 0, 1], [4, -2, 0, -2, 0, 2, 2, 0, 2, 1, 0, 0], [-1, 2, -2, 3, -1, -2, 1, 2, -1, 2, 1, -2], [3, 1, -2, 3, -1, 0, 2, -1, -1, 2, -1, 0], [-4, -4, 0, 2, 0, 0, -4, -2, 0, 2, 0, 0], [3, -2, 1, -1, -1, 0, 0, 1, 0, -2, 0, 2], [1, -2, 2, -2, 2, 2, -3, -1, 1, -2, 0, 1], [2, 2, -3, 1, 1, -1, 1, 3, 4, 3, 1, -2], [-2, -1, 3, 2, 2, 0, -4, -1, -2, -2, 1, -1], [4, 0, -1, 0, 0, 0, 3, 1, 3, 5, -1, -2], [3, -3, -2, 2, 0, -2, 0, 1, 4, 3, 0, -2], [-4, 0, 3, 0, 0, 0, -3, -1, -3, -2, 1, 0], [-6, 0, -1, -2, -2, 2, 1, -1, -1, -2, -2, 2], [-1, 0, 1, 1, 1, 0, 0, -3, -4, -1, -2, -2], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 4, -3, 1, 2, -1, -1, -4, -4, 1, 2], [4, -3, 1, 2, -1, 2, -4, -4, 1, 2, 2, -1], [-4, 2, 2, -5, 0, 2, 4, 2, -1, 0, 2, -1], [-1, 0, 2, 3, 3, -2, -6, -4, -3, -2, 0, -1], [1, -2, 0, 3, 1, -2, -1, 0, 1, 4, 3, 0], [-1, -4, 1, -2, 0, 1, 0, -3, -3, -3, -2, 0], [-1, 2, 1, -3, 1, 1, 1, 2, -1, -7, -1, 0], [-1, 3, -3, -2, -2, 0, 5, -2, -1, 0, -4, 0], [0, 0, 0, 0, 0, 1, 0, -1, 0, -1, -1, 1], [3, 1, 2, -1, 1, 3, 2, -1, 1, 0, 0, 2], [-3, -2, -1, 0, -2, 2, 2, -1, 1, -2, -1, 2], [7, 1, -1, -1, -5, 3, 4, 0, 2, 1, 0, 1], [3, 0, -1, -2, -2, 4, 3, -2, 1, 2, 1, 0], [-1, 0, -2, 3, 1, 0, -3, -4, 1, 2, 1, 0], [-3, -2, -2, -2, 2, -2, -1, -2, -2, 2, 0, -2], [-3, 0, -2, 1, -1, 0, 1, 0, -3, -2, 1, -2], [3, -1, -1, -1, 3, -2, 1, 1, 0, 0, -2, -3], [8, 3, -3, -3, -1, 2, 6, 2, 6, 1, -2, 2], [-5, 1, 0, 1, -1, -1, 0, 3, 1, -1, -2, 2], [-12, 0, 0, 3, 1, -3, -4, 0, 3, 2, 2, 0], [6, 0, -3, 2, 0, 3, -1, -4, 3, 1, 0, -1], [-3, 3, -1, 3, 3, 0, -2, -3, -5, 0, 2, 0], [0, -1, 0, 0, 4, -5, -1, 1, 2, 3, 2, -2], [1, -8, 2, 1, -3, 4, -3, -6, -1, -3, -3, 2], [-7, 4, -5, 4, 4, -7, -1, 5, -1, 5, 2, -5], [-9, 1, 2, 0, 2, -2, -2, -1, -4, -6, -4, 4], [-2, 2, 1, 2, 0, -3, 0, 2, 5, 5, 4, -2], [2, 1, 3, 0, -2, -1, -2, 2, 3, -1, 2, 0], [1, -2, -1, -3, 0, -1, -3, 1, 2, -1, 2, 0], [-2, -3, 5, -3, 1, -2, -2, 2, -2, -5, 0, 0], [-5, -1, -4, 5, -1, -1, 2, -1, -1, 4, -2, 0], [-4, 2, 1, 2, 4, -7, -8, 3, -1, 1, 4, -3], [0, -3, 1, -5, -1, 0, -2, -4, -2, -2, -1, 0], [-1, -1, 3, -1, -6, 10, 1, -4, 2, -2, -1, 5], [-6, 1, -4, 2, 0, -3, -1, 3, 0, -4, 1, 0], [2, -3, -2, 3, 1, -2, 1, -1, 3, 9, 3, -2], [-4, 2, 2, -2, -2, -3, -2, -2, -2, -1, 3, -2], [5, -4, -1, -5, 1, 2, 0, -1, 2, 0, -4, 2], [-14, 0, -2, 3, 5, -4, -1, -1, -1, 4, 1, -4], [8, -2, 1, 2, -6, 4, 1, -3, 1, 0, -3, 4], [8, 3, 2, 3, -1, 1, 0, -2, 2, 1, 0, 2], [-7, -2, -1, 6, -2, 1, -4, -3, -1, 1, -1, -2], [-4, 2, -1, -1, -3, 2, 5, 3, 2, 0, -2, 4], [7, -1, 0, 2, -2, 1, 4, 0, 4, 1, 1, 1], [3, -1, -3, 3, 4, -2, -2, -2, 0, 4, -1, -8], [8, 0, 3, -5, -1, 1, 7, 1, 2, -3, -3, 2], [8, 1, -1, 1, -3, 2, 4, 6, 6, 5, 2, 0], [8, -4, -1, 3, -2, -2, 4, 3, 0, 6, -1, -2], [-2, 1, 0, -2, 0, -1, 3, 3, 0, -1, -3, -2], [6, -4, 5, -5, 2, 2, 1, -1, 0, 4, -1, -3], [1, -1, 3, 0, 6, 0, -7, -2, -1, -4, 3, 2], [-2, 1, 3, -7, 1, 3, 2, 0, -4, -7, -3, 6], [-8, 2, -4, 0, 4, -6, 0, 4, -2, -2, -2, -4], [-4, -1, -3, 4, -2, -7, -4, 2, -1, -3, 2, -2], [7, 0, 3, -5, -1, 4, 0, -1, -2, -8, -2, 2], [6, -5, -1, -3, 1, 2, 0, -8, -4, -4, -4, 0], [-4, 0, 1, 2, 2, -1, -4, -6, -5, 1, -2, -4], [9, 2, 6, -6, -2, 6, 0, 1, -1, -6, 0, 4], [5, 11, -4, -7, 1, -1, 10, 5, -3, -3, -2, 0], [8, 5, -1, -3, -7, 2, 8, 2, 2, 0, -3, 4], [6, 3, 2, 1, -1, 4, 3, 3, 3, -3, 1, -2], [1, -5, -2, 2, -2, -2, 2, -1, 4, 5, 2, -4], [-6, 5, 2, 4, 4, 0, -4, 2, -1, -2, 0, 2], [-7, 5, 1, 4, -3, 1, -1, -4, -5, 1, -1, -3], [9, 2, -7, 7, -3, 0, -2, 1, -2, 4, -2, -4], [-5, -1, 0, 8, 4, -6, -10, 3, 2, 3, 4, -6], [2, 3, 6, 4, 0, -1, -8, -7, -6, -3, 2, -3], [12, -7, 2, -4, -2, 7, 1, -1, 2, 4, 1, 2], [4, -1, 1, 4, 0, -1, -4, -2, -3, -7, 0, 6], [1, -3, -1, -4, 0, 1, 0, 6, 3, -5, 0, -3], [0, 4, 6, 0, -2, 5, -1, -6, -4, -1, -1, 3], [-5, 3, 2, 4, 0, -6, -4, 1, -4, 0, 2, 0], [-5, -2, 6, -4, 0, 1, -1, 0, 2, -3, 0, 4], [15, -7, 2, -1, 5, -2, -2, -1, 1, 4, 1, -4], [-1, 1, 6, -3, 5, -1, 0, 1, -1, -1, 4, 0], [10, 0, 1, -1, -1, 0, 5, 0, -4, 0, -2, -1], [12, 0, 0, 1, 1, -1, 6, -2, 1, 7, 0, 0], [4, -2, -6, 1, 1, -7, -2, 0, 1, 3, -2, -2], [0, 11, -3, 1, -3, 0, 10, 4, 6, 12, 2, -2], [9, 0, -6, 10, -6, -5, -3, -2, 4, 3, -2, -2], [-5, 1, -1, 2, 4, -2, 1, -2, -3, 3, 1, -2], [-10, 2, -1, -6, 2, 0, 1, -1, -3, 0, -3, -2], [11, -5, 6, -7, -1, 7, 2, 1, 5, -3, 0, 0], [8, -1, 2, 3, -1, 0, 3, -5, 5, 6, 1, -4], [14, -6, 6, 3, 1, 3, -10, -6, 1, -3, 0, 4], [-6, -4, 5, -1, 1, 1, -7, -9, 0, -3, -1, 4], [3, 5, 1, -4, -2, 1, -2, -3, -3, -1, 2, 4], [-6, 4, -3, 0, 0, -2, 5, 5, 3, 3, -3, 0], [-3, -3, 2, 4, 4, -4, -8, -1, 2, 8, 6, -8], [10, 2, 5, -8, -4, 8, 3, 1, -1, -10, 1, 10], [-7, 2, -4, 1, 4, -1, 3, 0, -1, -1, -3, -7], [-8, 4, 0, 1, 1, -5, 0, 6, -1, -2, 2, -2], [9, 9, 1, -6, 2, 4, 6, -2, -7, -8, -3, 1], [18, 3, 3, -4, -4, 7, 4, 4, 1, 3, 0, 8], [-10, -3, -4, 3, -1, -3, 2, 2, 4, 7, 2, -10], [4, 5, -1, 0, 2, -2, 6, 3, -3, 2, -5, -3], [9, -4, -1, 5, -5, -2, -2, 3, 2, 6, 2, -2], [7, -1, -3, 4, 0, 0, 5, 0, 5, 13, 1, 0], [15, -3, 4, -5, -3, 7, 4, 5, 11, 3, 4, 6], [10, -1, 3, -2, -2, 1, -2, 2, 7, 3, 0, 8], [7, 4, -3, 2, -2, -2, 0, 8, 6, 2, 1, -3], [10, -6, -4, 0, 4, -2, 6, 0, -4, -1, -6, 2], [-7, 3, 5, -1, 1, -5, -3, 6, -4, -1, 3, 0], [4, 5, 0, 0, 0, 7, 3, -5, -4, -5, -5, 4], [2, -9, 5, -3, -1, 4, -10, -6, -6, -10, 2, 2], [0, 12, -5, 5, -5, -1, 13, 3, 2, 12, -1, -4], [7, -8, 5, 0, 0, 6, -2, -8, -1, 6, 1, 1], [-6, -5, -4, -1, -1, -3, 2, -2, 2, -1, 2, -2], [7, 3, 0, 2, -5, 7, 11, 1, -2, 3, -2, 2], [3, 4, -1, -1, 3, -3, 6, 5, 0, 3, 1, -4], [-2, -2, 6, 0, 0, -3, -5, -1, -5, -3, -1, 2], [-14, -3, 5, 0, 11, 0, -10, -6, -7, -6, 2, -7], [9, -4, 6, -4, -2, 5, 3, -2, -2, -7, -3, 0], [8, 2, -4, -3, -3, 2, 10, 0, -3, 0, -3, 2], [0, 1, -1, 2, 2, 1, -6, -2, 1, -5, 0, 6], [5, -9, -1, 2, -6, 3, -4, -7, -5, -3, -4, 0], [3, 4, -1, -1, 6, -3, -3, 1, -2, 1, 4, 0], [12, 0, 4, -3, 0, -2, 1, 2, 3, 0, 8, -4], [6, -5, -2, -2, -6, 1, 6, -3, 0, 3, -2, -1], [6, 2, 2, 0, -4, 6, 2, -4, -2, -6, -4, 10], [12, -10, 3, -1, 1, 3, -3, -7, 4, 1, -1, 2], [-3, 0, -6, 3, -1, -5, 0, 7, 1, -1, 4, 2], [0, 5, 3, -3, 3, 6, -2, -8, 0, -2, 0, -2], [5, -2, -4, 6, -6, -1, 1, -2, -2, 11, 0, 0], [3, -6, 2, -8, 6, -3, -1, 0, 0, -2, 1, 0], [-12, 1, -8, 4, -2, -3, 5, 5, 2, 11, 3, 0], [7, 3, 10, -9, -3, 3, 6, 5, 1, -9, -4, 6], [0, 3, -2, 3, -7, 4, 1, -1, 9, 2, -1, -2], [-14, 4, -1, -3, 7, 0, -1, 1, 0, -2, -2, 2], [9, -10, 5, -3, 1, -1, -8, -4, -7, -5, 1, 1], [11, -7, 3, -7, -5, 8, 7, -4, -2, -10, -10, 10], [6, -3, -1, 8, 0, -7, 0, -2, -1, 2, 0, 0], [-4, 6, 0, 3, -1, -6, -1, 11, 4, 4, 7, 0], [22, -2, 1, -2, -10, 5, 11, -2, 6, -1, -6, 5], [-10, -1, -5, 4, 4, -3, -5, -6, -7, -5, -3, 1], [-23, 1, -7, 4, 4, -12, -1, 0, -9, 4, -5, -4], [-1, 2, 10, 3, -1, 0, -9, -4, -5, -10, 4, 2], [-1, 2, 0, 5, 1, -4, -7, 2, 3, -4, 6, -4], [4, 5, -7, -4, -6, 3, 12, 4, 7, 10, -4, 4], [17, -2, 0, 5, 1, -1, -6, -1, -2, -3, -4, 4], [3, 2, 0, -7, -3, 6, 11, -4, 7, 5, 1, 6], [-10, 8, -4, -7, 2, 2, 11, 2, -1, -2, -2, -6], [17, -7, -2, 1, -3, 1, 2, 1, 3, -1, -3, 4], [-19, -3, -2, 3, -1, -1, -6, -5, -7, -8, -6, 0], [-4, -11, 8, -4, -2, 3, -3, -5, 4, 3, 3, 2], [-6, -2, 2, -2, 2, -2, 4, 2, 2, -2, 2, 4], [12, 6, 5, 0, -4, 6, 3, -1, -1, -2, -1, 4], [-9, -2, 5, -1, 3, 2, -4, -1, 4, 0, 3, 6], [1, 2, 6, -4, 0, 0, -3, -2, -4, -6, 0, 8], [4, -10, 6, 1, 7, -4, -9, -4, 3, 2, 3, 3], [-2, -1, 1, 4, 2, -3, -12, -4, 3, 5, 8, -2], [-14, 0, -11, 5, -1, -3, 9, -5, 2, 15, -5, -8], [-6, 6, -8, 10, 8, -12, -4, 10, 4, 9, 8, -12], [7, 5, -3, -3, 1, 0, 8, 3, 6, 2, 6, -2], [1, 7, 2, -2, 0, 3, -2, 1, 6, -1, -3, 2], [-7, -3, -5, -1, 6, -10, 5, 6, 2, 2, -1, -7], [-4, 7, 1, 6, 1, 0, -5, -2, -3, 4, 6, 0], [-7, 11, 3, -1, 2, 6, 2, 0, 2, 2, 3, -2], [-20, 5, 2, -4, -4, 5, 1, -7, -8, -7, -6, -4], [-24, 0, 3, 1, 6, -6, -16, -5, -6, -2, 7, -8], [-13, 7, 4, 3, -1, -3, -11, -5, -11, -5, -1, -5], [11, 10, 3, -3, 3, -2, 0, 1, -2, 0, 0, -6], [-5, 2, 0, 5, 9, -9, -7, 1, 3, 13, 6, -9], [-9, 5, 3, -2, -2, 4, 9, 6, 5, -1, -1, 6], [19, -1, 1, -10, -4, 4, 7, 4, 7, -8, 3, 2], [2, -8, -1, -4, 6, 2, -5, -7, -5, -8, -1, 2], [-11, 7, 3, 10, -6, -1, -6, -1, -2, 3, 6, 2], [4, -3, -3, -3, 3, -4, 6, 6, 4, -4, -2, 6], [-3, 3, -7, 1, 2, 0, 9, 6, -2, 2, -9, 3], [5, -4, -6, -7, -3, -2, 11, -2, 1, 10, -7, -4], [6, 0, 3, -1, -5, 4, -5, -3, 8, 4, 0, 10], [-6, 3, -3, 9, 9, -7, -10, 3, -5, 9, 7, -5], [24, -4, -2, 2, -2, 7, 5, -5, 7, 9, 1, 4], [1, 7, -5, 3, 5, -5, -1, 0, -6, -1, -5, -8], [-10, 10, -2, 6, -2, -2, 4, 4, 4, -2, 0, 4], [-17, 3, 1, 10, 6, -7, -9, 8, 5, 11, 10, -8], [-8, -7, -7, -2, 0, 2, 7, 0, 1, 2, -7, 3], [-15, 5, 4, -9, -3, 5, 10, 7, -3, -7, -6, 2], [-17, 9, 8, -6, 2, 7, -3, -4, -5, -15, -1, 8], [-4, 16, 3, 0, -4, 1, 5, 8, 8, 1, 2, -5], [4, -7, -6, -5, -1, 2, 7, 5, 7, -5, -1, 0], [0, 6, -7, -3, -5, -1, 11, 7, 0, 1, -9, 6], [25, -7, 3, -8, -4, 12, 8, 4, 9, 0, 0, 7], [-6, 8, -4, 3, 1, -1, 2, 6, -1, 11, 8, -12], [18, -5, -2, 9, 0, -3, -7, -1, -3, -1, 3, -3], [-24, -3, 6, 6, 6, -9, -23, -3, -2, -7, 9, -2], [15, -4, -4, 13, 1, -8, -5, 4, 11, 10, 0, -2], [-12, 0, 2, -9, -2, 0, -1, -6, -3, -6, 0, 4], [-15, 2, -1, 1, 5, -2, -2, 7, 2, -6, 4, -6], [-7, 10, -1, 1, 3, 0, 2, -5, 2, 0, 4, -4], [9, 2, 6, -10, 2, 8, 0, -7, -14, -12, -4, 6], [-10, -4, 2, -2, -4, 2, 4, -6, 0, -3, -2, -6], [3, 7, 1, 0, 0, 6, -3, -4, -3, -10, -1, 6], [14, -3, 10, -9, 1, 10, -5, 3, 9, -6, 7, 6], [-14, -4, -1, -3, -3, 9, -3, -5, -2, -5, -1, 6], [7, 4, -6, 8, 4, -8, -4, 1, -1, 4, 6, 0], [-15, 3, 4, -6, 6, -4, 0, 5, 0, 4, 2, 6], [-10, -6, 7, 2, 6, -2, -11, -7, -11, 0, 5, 2], [-7, 1, -3, 0, 1, 1, 11, -6, -11, 3, -3, -3], [-14, -6, -11, 8, -2, -4, 1, 5, 7, 17, 5, -4], [24, -6, 2, -2, -2, 5, 9, -5, 9, 5, -7, -2], [7, 11, 3, 0, 2, 4, -1, -6, -3, 0, 3, 8], [4, -3, 6, 5, 0, 5, -5, -3, -3, 5, 11, -1], [1, 4, 2, 9, -3, -6, -7, 10, 9, 6, 5, -4], [7, -1, -1, -12, -2, 2, 1, 2, 9, -9, 1, 0], [1, -15, 6, -5, 5, -2, -7, -12, -3, -2, -1, 2], [15, 0, -1, -11, -11, 9, 10, 1, 2, -9, -1, 8], [16, -16, 2, -5, -5, 5, 1, -8, 1, 1, -4, -1], [3, -2, 4, 2, 0, 3, -1, -2, 0, -4, -3, -6], [29, 5, -7, -2, -6, 3, 13, 6, 5, 9, 0, -2], [4, -10, 1, 7, -1, -5, -3, -7, 0, 11, 5, -8], [-2, -1, -5, -4, 6, 0, 5, 3, 3, -6, -5, 6], [-5, -11, -7, 3, 0, -6, -3, -6, 2, 2, 1, -1], [0, -4, -14, 2, 0, -8, 12, 16, 12, 14, 0, 2], [22, -17, -4, -4, 0, 3, 7, -1, 4, 1, -7, 0], [-10, 1, -1, -2, 6, -9, -2, 6, -1, -3, 2, 4], [11, -4, -3, -7, -11, 8, 8, -3, -2, 2, -11, 6], [4, 6, 3, 3, -1, 9, -1, -5, 8, 4, 5, 4], [-10, -2, 15, -3, 1, 6, -4, -6, 2, -8, 6, 4], [-2, 6, 6, -6, 0, 2, 0, -8, -12, -5, -4, -4], [13, 3, -14, 3, -4, 2, 11, 7, 3, 10, -4, 0], [-6, -12, -6, 9, 7, -5, -16, -4, 5, 5, 0, -2], [-9, -2, 3, 7, 3, -8, -10, -3, 6, 7, 6, -4], [8, 2, 12, -5, 7, -5, -6, 4, -7, -14, 2, 0], [9, -5, -5, -11, -7, 11, 9, 2, 8, -8, -5, 2], [-15, 14, -2, -7, 1, 4, 5, -4, -15, -14, -8, -2], [3, 3, 11, -10, 6, 6, -1, 0, 1, 0, 7, -2], [1, 2, -6, 10, 0, -13, -1, 8, -2, 1, 3, -8], [-10, 0, -3, -1, -1, 1, 7, 3, 0, -3, -7, 0], [7, 7, -16, 7, -7, 1, 12, 7, 7, 16, 0, -4], [-12, -9, 4, -6, 0, 2, -4, 0, -12, -8, 0, 0], [-18, 8, -5, 10, 4, -10, -3, 3, 1, -2, 5, -8], [3, -16, -1, -2, 10, -7, -8, 3, -5, -1, -3, -2], [10, 10, 2, -4, 4, 2, 10, -4, -4, 2, 0, 2], [24, -12, -1, 6, -2, -2, -6, -7, 1, 2, 1, 3], [16, -6, -3, -11, -5, 4, 8, -4, -10, -4, -10, 4], [6, -4, 3, 10, 6, -4, -13, -1, 9, 0, 5, 4], [-3, -7, -3, -6, -6, 8, 9, 0, 7, 5, -1, 0], [-4, 0, -3, 15, -5, -9, -3, 5, 0, 6, 3, -4], [23, 2, 3, 0, -10, 5, 8, 11, 13, 13, 4, 8], [4, 7, 5, 4, -4, 3, 4, 2, 9, 7, -1, 6], [-17, -14, -5, 2, 8, -7, -2, -1, -3, 1, 2, -8], [-13, 0, 6, -2, 0, -5, -7, 0, -2, -7, 5, -4], [11, -2, 1, 0, 0, 3, -4, 3, 13, 13, -2, -8], [26, -3, 2, -2, -5, 2, -1, -9, -6, -2, -3, 13], [18, -9, -1, -6, -8, 3, 8, 8, 1, -6, -6, 4], [19, -3, 2, 0, 2, 6, 4, -3, 2, 12, 0, 2], [-1, 1, 11, -6, -4, 1, -3, -3, -13, -17, 0, 3], [14, -11, 3, -12, 8, 3, -4, -8, -17, -15, -6, -4], [10, -16, -6, 12, 0, -1, -4, -7, 3, 7, 3, -3], [10, 0, 2, 3, -1, -3, 0, 0, -5, 7, 0, 2], [7, -16, 3, -6, -2, 3, 6, 7, 7, -5, -3, 2], [-22, -3, 0, -10, 6, 3, -1, 1, -8, -4, -7, 2], [17, 7, 4, -3, -1, 2, 1, 5, 11, 6, 3, -5], [15, -7, -10, -2, 2, -2, 0, 1, 2, -12, -6, 0], [-2, 2, -6, 1, -5, -2, 0, 10, 5, -4, -1, 8], [8, 1, 0, 5, 8, -7, -9, 1, 1, -3, 3, -7], [-2, -10, 2, -6, 2, 11, 0, -5, 1, 1, -3, 3], [-18, 8, -7, 14, 6, -6, -3, 11, 5, 5, 3, -10], [2, -1, 11, -8, 4, 3, -6, -8, -3, -8, -4, 4], [-13, 5, 0, -10, -6, 6, 12, -5, -8, 8, -4, 10], [-28, 10, -1, 0, 4, -4, -5, -3, -13, 4, 1, -10], [10, 1, -3, -1, -3, 2, 0, -2, 6, -1, 4, 2], [-2, 6, 3, -2, -2, 7, 9, -2, -9, -1, -6, 3], [-16, 11, -9, 15, 1, -10, 2, -4, -8, 4, -6, -2], [0, 2, -2, 0, 7, -3, -2, 4, -8, -5, -2, -7], [15, 13, 4, -3, -3, 3, 6, -1, -1, 7, 2, 8], [10, -14, -5, 4, -10, 0, 15, 3, 1, 9, -3, 0], [7, 11, -9, 10, 0, -1, 7, -2, -3, 1, -2, 2], [0, -1, -15, 12, -2, -5, 8, -4, -3, 17, -8, -4], [33, -1, 2, -4, -4, 3, 11, -2, 5, 3, -9, 6], [12, -2, -8, 5, -7, 3, 6, 4, 9, 9, 6, -8], [11, -2, -6, 9, -7, -6, -1, 4, 5, 16, 8, -10], [8, -5, 6, 11, -3, -1, 1, -7, 1, 9, 2, 0], [13, -3, 4, 1, -3, -5, -2, 5, -3, -1, 0, 8], [-10, 4, 1, -3, 3, -5, -7, 7, -10, -1, -1, -2], [1, 10, -5, -6, 1, -6, 14, 7, -5, -2, -8, 9], [-8, -2, 4, 8, -6, 8, -6, -4, 2, -11, 2, 10], [14, 11, 6, -3, -12, 11, 4, 1, 3, -7, 3, 14], [-17, -15, 0, 1, 9, -13, -6, 1, -5, -3, 6, -14], [6, -12, 9, -1, -1, -5, -9, 3, 4, 3, 5, 8], [-20, -3, 3, -3, 9, 1, -3, -3, 0, -9, 1, 4], [6, -2, -5, 2, -8, 1, 4, -3, 1, 13, -2, 9], [17, 2, 1, 7, -9, 0, -4, -5, 4, 2, 1, -4], [16, 1, 15, -2, -8, 15, -12, -12, -11, -13, 2, 0], [-13, 6, -2, 6, 0, -3, 3, 8, 10, 5, 11, 0], [-11, 7, -5, 7, 5, -10, -8, 8, 6, -2, 4, -15], [26, 13, 3, -1, -9, 12, 12, -6, -4, -8, -2, 8], [1, 2, 0, -9, -5, 7, 15, 0, -1, -5, -12, 6], [6, 6, -3, 15, 7, -12, -6, 11, 4, 10, 16, -13], [-6, 0, 14, -2, -2, 1, -7, -5, -13, -7, 3, 2], [-6, -12, 2, -10, -3, 7, 0, -4, -12, -7, -4, 7], [4, -2, 2, -9, -5, 6, 15, 1, -2, 2, -3, -2], [-29, 5, -3, 13, 5, -7, -17, 0, -4, -7, 3, -8], [13, 3, 2, -6, 2, -7, 11, 4, -3, -7, -3, 2], [4, -6, -2, -7, 3, 1, -10, -4, -1, -1, -4, -4], [25, -11, -3, -5, -4, 4, 7, 4, 12, -2, 1, 3], [7, -10, 2, -11, -8, 5, 3, -2, -11, -11, -7, 1], [-23, 5, -8, 1, 3, -7, 4, 1, -11, -3, -8, 4], [9, -11, -7, 11, 1, -3, -11, 0, 2, 7, 1, -12], [-3, 10, 0, 3, 3, -4, -6, 8, 3, -2, 5, 3], [-10, -1, -6, -7, 3, -1, 10, -2, -1, 1, 2, -12], [-26, -4, 10, 4, 6, -5, -11, -3, -2, 7, 13, -6], [2, -4, 15, -12, 6, 6, -3, -5, -11, -21, -5, 0], [10, -6, 7, -13, 11, -3, -3, 3, -2, -15, -3, -4], [20, 0, -14, 19, -5, -5, 0, -2, 7, 9, -9, 4], [4, 13, 1, 6, -6, -2, 13, 7, 4, 12, 9, 0], [0, -7, -13, 4, 2, -14, 4, 12, 7, 10, -5, 0], [-24, 6, 8, -6, -2, 3, -9, -1, -11, -19, 1, 4], [26, -14, 4, -1, -3, 3, 0, -8, 5, -9, -6, 6], [4, -2, -8, 0, 4, -2, 2, 2, -8, -6, -9, 4], [-16, 3, -4, 20, 8, -16, -8, 0, 3, 12, 0, -12], [3, -7, 8, 3, 5, -1, -12, -1, -3, -21, -2, 8], [-9, 2, -12, 22, -2, -7, -15, -2, -2, 18, 7, -8], [-12, 3, -8, -13, -7, -2, 21, 21, 13, 6, 1, 4], [-1, 1, 10, -6, 6, 10, -4, -5, -6, -11, 0, -6], [-2, 0, 5, 8, 0, 6, -7, 1, 7, -6, 1, 14], [0, 6, 0, -2, 6, -4, 7, 12, -4, -14, -3, 5], [-4, 10, -5, 4, 0, 4, -1, 3, 1, -11, -3, 0], [18, 0, -5, 4, 4, -14, 1, 15, 15, 14, 9, -8], [-10, -9, 3, -2, 5, -2, -1, -2, -11, -2, -4, 4], [-15, 9, 4, 7, 1, -6, -7, 6, -9, -2, 5, -12], [-26, 2, 11, -12, -2, 2, 1, -1, 5, -2, 5, 2], [-2, -5, 7, -14, -2, 13, -6, -10, 3, -15, -4, 12], [-5, 15, -2, 1, 2, -4, 6, 7, 5, -4, -4, 3], [-5, -16, -6, 4, 2, 1, -3, -4, 6, -2, 1, -2], [-4, -20, 3, -1, 1, -3, -1, -7, -8, -5, -3, 10], [2, 2, 4, 2, -12, 8, -4, -6, -2, 5, 4, -2], [12, -3, 2, 5, 5, -6, -11, -3, 3, -4, 7, -6], [35, -15, 10, -4, -6, 13, 2, -8, 6, 9, 1, 3], [10, -19, 13, -4, 10, 1, -26, -20, -11, -10, 2, 6], [-6, 3, -2, 1, 5, 8, -9, -3, -5, 0, -3, 0], [7, -14, -17, 3, -5, -3, 8, 3, 8, 19, 1, -8], [-3, 18, -5, -4, 7, 0, 0, 7, 1, -8, 0, 3], [-9, -4, -13, 4, -8, -7, 8, 1, -7, -1, -6, -10], [10, -12, 2, 8, -4, 0, -4, -2, -2, 9, -2, 0], [-36, -5, 2, 1, -3, 0, -5, 3, 11, 12, 3, 0], [-8, 8, 9, 7, 7, 1, -5, -3, -6, 5, 13, -2], [25, -12, -4, 4, -10, 6, 2, -8, 6, 10, -2, -1], [18, -2, 2, -5, 1, -1, -4, 2, 13, 3, -4, 6], [-2, 1, -8, -1, 3, -6, 7, 3, -9, -6, -6, -8], [14, -18, -2, 4, 7, -9, -12, 6, 6, 9, 6, -3], [-9, 0, 7, -14, -15, 20, 3, -5, -3, -20, -8, 16], [0, 5, 4, 2, 2, 5, 1, 3, 2, 7, 11, -6], [5, -11, 3, 5, -4, -2, -12, -6, -4, -10, 9, 10], [26, -2, -5, 3, -13, 9, 9, 1, 14, 5, 3, 6], [-36, -10, -3, 1, 9, -3, -15, -9, -2, -3, -3, 6], [-13, 9, 1, 6, -12, 6, -1, -2, 7, 4, 3, 2], [3, 14, 4, -6, 4, 3, 9, 4, 2, -12, 3, 4], [12, -15, 6, -13, 11, -1, -3, 3, -5, -11, -2, 6], [-22, 5, 5, -11, 15, 0, 4, 4, -12, -2, 4, 0], [-5, 11, 3, 7, -3, -5, 7, -2, -6, -5, 3, -2], [-20, -2, 8, -17, 3, 11, 4, 2, -7, -11, 2, -2], [7, -7, -5, 4, 0, -1, 3, 0, -3, 7, -8, 0], [-27, -13, -1, -1, 9, -5, -9, -4, -8, 12, 3, -2], [1, -14, 9, -11, 1, 2, -4, 3, -4, -6, -4, 0], [-1, 3, 20, -12, 8, 12, -4, 1, 2, -2, 12, -6], [-22, 14, 6, 10, -6, -1, 0, -5, -7, 9, 5, 7], [-18, 5, -4, 12, 10, -8, -1, 2, -4, 12, 6, -5], [20, -12, 4, -14, 2, -2, 8, 6, -10, -9, -8, 12], [15, 7, -4, 1, -1, 13, 12, -5, 5, 12, 0, -2], [13, -5, 2, -12, -2, 1, 7, -2, 0, -3, -1, -4], [10, -4, 4, -6, 2, 6, -8, 0, -2, -18, -4, 8], [-10, -4, -2, 4, 10, -12, 0, 10, 6, 12, 8, -14], [-41, 7, 0, 9, 3, -13, -12, -1, -17, -1, 4, -12], [-15, -11, 4, 14, 4, 2, -10, -3, 10, 9, 4, -2], [6, -6, -8, -8, -4, -4, 20, 8, -2, 4, -10, -2], [-57, -4, -6, 1, -5, 4, -3, 0, -1, -7, -1, 2], [-1, 5, 8, -4, 6, 0, -6, 5, 4, -2, 12, 2], [-13, -7, 9, -14, 2, 12, -1, -10, -9, -5, 5, 6], [-16, 7, -8, -3, -11, -2, 19, 9, 1, 2, -8, -2], [2, 6, 5, 3, 9, -3, -13, -3, 2, -9, 5, 0], [6, -15, 0, 5, 1, -8, -17, 9, 3, -2, 5, -2], [-29, 1, 3, -4, -8, -7, 0, -1, -2, 1, 0, 0], [11, 0, -9, 4, 0, -5, 2, 3, 1, 7, 7, -4], [6, -7, -3, -14, -4, 2, 22, 6, 7, 4, -5, 2], [13, -9, 13, -17, 1, 0, -6, -1, 2, -8, 8, 2], [-2, 8, 7, -17, 13, -3, 3, 1, -16, -15, -3, 2], [-30, 6, -5, 11, 4, -4, -4, -5, -6, -6, 1, -4], [3, 12, -9, 18, 2, -1, -3, 1, -3, 5, 6, -3], [-18, -3, 0, -10, 14, -6, -5, -6, -3, -8, 2, 3], [-6, 5, -6, 8, -8, 7, 5, -5, -8, -5, -5, 8], [-21, -12, -3, 10, 10, -11, -24, -7, -3, 8, 12, -10], [14, 12, 3, 2, -10, 13, 12, -1, 3, 1, -6, -1], [-24, 2, -3, -4, 4, -13, 8, 4, 2, -3, 4, 2], [0, 1, 9, 7, -3, 1, -16, -7, -4, -11, 7, -1], [-19, 5, -12, 3, -9, -3, 12, 7, -5, -1, -6, -6], [8, -1, 14, -13, 3, 8, -3, -5, 1, -2, -4, 2], [30, -2, -11, 11, -19, -2, 12, 7, 6, 14, -6, -3], [1, 15, -4, 9, 5, -14, 5, 12, 7, 12, 3, -10], [-6, -10, 7, -17, -7, 3, 19, -7, 2, 3, 3, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1336_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1336_2_a_e();".
// 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_1336_2_a_e(:prec:=12)
    chi := MakeCharacter_1336_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_1336_2_a_e();".
// 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_1336_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1336_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![224, 1224, 148, -2399, -433, 1804, 129, -641, 36, 100, -15, -5, 1]>],Snew);
    return Vf;
end function;