
// Make newform 1336.2.a.b in Magma, downloaded from the LMFDB on 20 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_b();".
// 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_b();".
// 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 := [1, -17, -4, 25, 5, -9, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [-3, -1, 1, 0, 0, 0, 0], [1, -4, -1, 1, 0, 0, 0], [1, 2, -4, -1, 1, 0, 0], [-2, 4, 3, -5, -1, 1, 0], [-4, -7, 11, 9, -6, -2, 1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1];
    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], [-1, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 1, 0], [-1, 0, 0, -1, -1, 0, -1], [0, 0, -1, 2, 2, -1, 1], [0, -2, 0, 0, 1, -1, 0], [-1, -1, -1, 0, 2, 0, 1], [-1, 0, 2, -2, -1, -1, -1], [0, 0, 0, 2, 1, -1, 2], [-1, 0, 0, -1, 1, 2, -1], [-4, 1, 1, -1, -1, 1, -3], [3, -1, 1, 0, -1, 1, -1], [-4, -2, -4, 0, -1, 1, 2], [-5, -1, 1, -4, -3, 1, -3], [1, 4, 2, 4, 2, -2, 3], [1, 3, 1, 0, 0, 0, 1], [-2, -2, 0, 0, -1, -1, 2], [5, 1, -2, 2, 4, -2, 4], [-5, -3, -1, -4, 1, 1, 1], [-6, 2, 0, 2, 0, 0, 0], [-5, -3, -5, -2, 3, -1, 3], [-4, 0, 0, -4, 1, 1, 2], [1, 3, 3, 2, -2, 2, -3], [0, 4, 3, 2, -3, 5, -5], [3, 1, 0, 2, 4, 1, 0], [-4, -2, 0, -4, -1, -3, -2], [1, -1, 3, 2, 4, -2, 1], [-2, 5, 3, 5, -3, -4, 0], [-4, 0, -2, 0, 4, -2, -2], [-1, 7, 5, 0, -5, 1, -1], [-4, 0, -3, 2, -2, 1, -1], [-2, -2, -4, 0, -1, 1, 4], [2, 2, 3, -4, 2, 1, 1], [2, 2, 2, 6, 0, -2, 0], [-4, 0, 2, -4, -4, -2, 2], [-2, 0, 2, -6, -3, 3, 0], [7, 0, -2, 5, 8, -4, 4], [-3, -7, -5, -2, -4, 4, -3], [-1, 0, 0, 0, 0, 0, 0], [-5, -5, -4, -8, -3, 8, -4], [0, 3, 3, 0, -3, -1, -2], [5, 6, -2, 4, 1, -1, -1], [-6, -5, -5, -1, 2, -4, 3], [-1, 1, -7, 2, 2, 2, 5], [3, 3, -5, 6, -1, -3, 3], [-8, 0, -1, -2, -6, -1, -3], [1, 0, -6, 1, -1, -1, 4], [6, -6, 3, -2, -2, -4, 1], [10, 0, 2, 0, 4, 2, 0], [3, 6, -2, 1, 1, 4, 3], [1, 4, 6, 3, -4, 4, -4], [3, 5, 2, 0, -6, 2, -6], [3, 3, 5, -2, 2, 0, -1], [7, 2, 6, 5, 0, -7, 1], [-3, 1, 3, -2, -14, 6, -3], [2, -2, 5, -2, 4, -6, -1], [-11, -5, -7, -6, 1, 5, -3], [2, -2, -2, -2, -3, -5, 2], [8, 4, 0, -4, -9, 5, -8], [-12, -4, -3, -10, -7, 5, -7], [7, -5, -2, 6, 3, 0, 2], [4, 7, 7, 7, 2, -4, 3], [-5, 3, -7, 0, 7, 7, -1], [-5, 5, 3, -2, -6, 8, -7], [-4, -6, -2, -4, 1, -3, 2], [4, 2, 1, 4, 2, -2, 1], [9, -3, 5, 2, 6, 2, 3], [2, -9, -3, -2, 1, 5, -2], [7, 7, 5, -6, -9, 3, 1], [10, 0, -2, 0, 10, 0, 6], [2, 1, -1, 9, -4, -7, -2], [-8, 3, 3, -3, -3, 3, -5], [5, 0, 2, -2, -5, 7, -7], [-1, -5, -5, -2, -4, 4, -5], [-9, -1, -3, -8, -2, 0, 1], [4, 1, -1, 3, -4, -1, 0], [-5, 3, -3, 0, -5, 5, -5], [8, 0, 1, -6, -11, 1, 1], [-6, -2, 0, 0, 3, 5, -4], [-2, -2, 9, 0, 4, -11, -1], [0, 11, 7, -3, -6, 8, -5], [0, 3, -3, -3, -11, 0, 6], [12, -2, 3, 8, 5, -13, 5], [-7, 4, -8, 10, 10, 0, 1], [-6, 0, 0, -6, 3, 5, 0], [-9, -7, -11, 6, 13, -3, 5], [-15, 4, -4, 8, 3, 1, -1], [-3, 3, 1, -4, 3, 3, -9], [1, -10, -2, 10, 10, -8, 9], [0, -4, 4, -2, 2, 2, -6], [-7, 4, 0, -7, -8, 1, -5], [4, 10, 0, 0, -12, 2, 2], [-5, -9, -9, 0, 10, 0, 11], [-13, 7, 9, -12, -13, 9, -13], [-1, -1, -1, 0, 6, 2, 7], [-10, -6, -5, -6, 0, -2, 1], [-8, 6, 7, 6, -9, 6, -7], [-4, -8, -2, 2, 5, -3, -4], [-12, -9, -5, -7, -9, 5, -9], [13, -11, -3, 10, 8, -2, 1], [16, 2, 6, 0, 4, -2, 6], [-4, -4, 13, -8, -8, -5, -9], [-5, -4, -2, -14, -3, 13, -1], [-11, -7, -1, -2, 1, -7, 5], [-2, 0, 2, 8, 4, -10, 10], [8, 9, 5, 1, -3, -5, 5], [-7, 5, 9, -6, -16, 2, -5], [-11, 5, -1, 4, 6, -4, -3], [0, 1, -9, 15, 15, 0, 2], [-6, -7, -1, 5, -7, 4, -4], [8, -12, -2, 4, 11, -5, -2], [8, -1, 3, 3, -3, -1, -5], [-8, -1, 5, -11, -13, 14, -6], [-4, 8, 6, -2, -15, 13, -6], [-2, -12, 1, -4, -9, 5, -1], [0, -2, -12, 2, 12, 0, 8], [6, 10, 10, 2, -10, 0, 2], [7, -7, 5, -4, 0, -8, -5], [-2, -21, -5, -11, -1, 5, -5], [-12, 2, 4, -4, -3, -9, -6], [0, -12, 0, -2, 5, 1, 2], [-20, -2, -8, -2, 7, -5, 0], [-3, 7, 1, 8, -3, 3, -5], [0, -2, -2, 18, 10, -4, 12], [0, -18, -6, -4, 9, -9, 2], [0, -4, -3, 10, 13, -3, 17], [0, 10, 0, 12, 6, 0, 12], [8, -10, -6, 2, 9, 1, 2], [8, 4, 8, 4, -13, -5, -8], [14, -5, -7, -2, -3, 5, 2], [-10, 2, -2, 8, -11, -5, -2], [-6, -4, -4, -6, 7, -5, -4], [-11, -11, -13, -10, 10, 0, 11], [6, 2, 1, -12, 2, -1, 3], [17, -3, 2, 10, 10, 0, 4], [-5, -11, 7, -10, 1, -3, -13], [-14, 2, -10, 6, 7, 3, 6], [4, -12, -4, -6, -7, 5, -4], [1, -9, -9, 8, 0, 0, -5], [-12, 10, -5, 2, -17, -2, 9], [-16, 6, -8, -4, 5, -3, -2], [-6, -2, -2, 14, 4, -8, -2], [1, 1, -3, 10, 7, -7, 11], [-3, -1, -9, 14, 14, -12, 5], [-12, -4, 0, -18, -4, 12, -4], [-11, 1, 4, 6, 17, -7, 8], [-10, 2, 15, -12, -10, 1, -15], [-1, 4, -4, 4, -7, -1, 1], [-3, 18, 6, 7, 7, -4, -3], [-9, -9, -4, -2, -5, 6, -4], [12, 13, 7, 6, -15, 5, -16], [7, 3, 13, 10, -6, -12, 3], [-12, 3, 5, 5, 4, 5, -8], [17, -13, 1, -2, -2, 2, 5], [-14, 12, -3, -12, -1, 5, -1], [-6, -1, -5, -4, -5, 13, -8], [-5, -16, 6, -4, -6, 2, 1], [4, -7, -3, 9, 11, -3, 5], [22, 10, 10, 12, 2, -10, 8], [-12, -20, -10, -10, 1, 7, -4], [-5, -5, -6, -14, -1, 15, -2], [-5, -13, 11, -8, 2, 4, -3], [-12, 3, -1, 9, -3, -5, -5], [-7, -1, 11, -4, 3, 13, -5], [-5, -13, -15, -8, 19, -5, 11], [-2, -8, -4, -8, -3, 5, 8], [-11, 23, 15, 0, -13, 13, -7], [16, -1, -3, -7, 5, 0, -6], [-10, 0, -1, -6, 1, -2, -11], [2, 5, -5, 1, -8, 1, -2], [-3, -1, -1, 2, 5, -9, 5], [9, -11, 0, -16, -9, -8, -4], [5, 1, 0, 14, 7, 4, 0], [10, 13, 1, 11, 8, -2, -7], [28, -4, -10, 18, 20, 0, 16], [-2, 3, 3, 13, -7, 3, -1], [-4, 13, 5, -5, -15, 15, -15], [29, 7, 7, 18, 19, -13, 11], [-14, 0, -11, -8, 3, -10, 3], [-9, -13, -13, -8, 17, -15, 7], [-29, -2, -20, -8, -2, 6, 1], [15, -8, 2, 3, 20, 4, -4], [10, 22, 8, 18, -4, -4, 8], [5, -11, 3, -18, -5, 1, -11], [-24, -6, 2, -20, -6, 14, -8], [-8, 8, 0, -6, -3, -7, -8], [18, -5, -21, 13, 28, -2, 15], [-5, -14, 6, 5, 8, -11, 5], [8, -9, 9, 13, 9, -8, 6], [25, 11, 1, 0, 0, 2, 9], [-16, 0, -12, -8, -1, -9, 2], [4, 6, 12, 12, -3, 5, -2], [10, 8, 7, 16, 1, -1, -3], [12, 1, -5, 7, 15, 0, 26], [-7, 16, -2, 6, 9, -3, 1], [18, 13, 11, -12, -20, 16, -4], [5, -10, -14, -6, 24, -6, 13], [18, 9, 9, -5, -4, 6, -11], [11, -4, 0, 0, 9, -13, 1], [-11, -4, -2, 5, 7, -1, -4], [9, 17, -1, 0, -5, 3, -15], [5, 23, 6, 16, 2, -10, 10], [36, -4, 2, 6, 11, -5, 6], [24, -12, 0, 0, -8, -4, -2], [-12, -18, -8, -8, 11, -3, 10], [35, -9, 4, 4, -1, -2, 4], [-3, 7, 3, 6, 10, 2, -9], [9, -18, -4, -6, 1, -11, 9], [3, -3, 9, -16, -14, 2, -15], [-10, 1, 19, -15, -11, 0, -24], [-32, -4, -9, -12, -2, -1, -11], [15, 7, 18, -4, -11, -2, -6], [8, -14, -14, -4, 22, -14, 8], [-14, 22, 2, -2, -6, -4, -4], [-18, 17, -5, 1, 6, 3, 10], [35, 0, 12, -1, 1, -2, 3], [8, -2, -2, 14, 19, -27, 12], [19, -7, 4, 6, 18, -13, -4], [-9, 19, 8, 14, 17, -16, 0], [25, 3, 15, 14, 12, -14, 7], [-12, 12, -2, 8, -9, 1, 14], [-11, 8, 2, -15, 0, 16, 0], [-20, 10, 8, -10, -6, 2, -6], [-8, 11, -1, 6, 12, -10, 0], [5, 11, -19, 6, 17, 1, 11], [4, -3, -3, -7, 9, 11, 9], [31, -16, 0, -2, -4, -2, -5], [-3, -7, 1, 10, -3, -11, -7], [-3, 11, 3, 14, 23, -15, 3], [-35, -3, 8, -14, -13, -1, -4], [-6, -8, -16, -2, 11, 7, 2], [16, -10, 0, -4, -20, -6, -10], [-1, -15, -13, -14, 5, 5, 7], [-16, -6, 4, -18, -23, 5, -28], [-24, 6, 1, 4, 5, -10, -5], [2, 12, 8, 6, 7, 5, 0], [-18, -8, -6, -2, -1, 1, -2], [7, -4, -12, 4, 7, 5, 5], [-8, -10, 2, -4, -12, 12, -12], [-11, 29, 10, 2, -13, 2, -2], [8, 24, 10, 2, -8, 0, 2], [4, -13, -13, 10, 14, -12, 16], [-21, 15, 5, -22, -18, 26, -15], [4, -6, 2, 6, 23, -7, 4], [-24, -18, -11, -4, -8, 8, -3], [5, -4, -12, 13, 15, -6, 9], [2, 8, 12, 6, -1, -9, 12], [29, 9, 7, 8, -3, -9, -9], [-26, -9, -7, -5, -6, 5, 10], [-8, -2, -18, 0, -9, -1, 4], [7, -11, 2, -12, 3, 13, -4], [9, -10, -18, -4, 13, -13, 15], [-10, 6, 0, -26, -18, 14, -22], [-9, 1, 9, 0, -3, -7, 3], [1, -5, -11, -8, -22, 18, -1], [6, -1, -1, 19, 12, -14, 15], [12, 14, 12, 16, -10, 2, 8], [14, 14, 7, 2, -2, 4, 9], [4, 20, 6, 2, -7, -1, -4], [9, -6, -18, 4, -1, -7, 3], [-18, -10, 4, 0, -20, 12, -2], [27, -13, 3, 10, 10, -6, 5], [8, 10, -2, 6, 7, 11, 14], [-26, 12, 22, -8, -13, -3, -10], [2, 29, 9, -1, -1, 1, 5], [0, -16, -4, 0, -2, 10, -16], [-11, -15, 17, -26, -12, -6, -11], [-16, -14, -16, -14, -17, 17, -14], [-8, 0, 22, -12, -2, -2, -14], [-13, -9, 16, -10, 4, 2, 2], [-20, 24, 12, -6, -9, 1, -2], [-2, 19, 3, -13, -4, 6, 3], [22, 10, 8, -4, -10, 6, 6], [8, 0, 12, 8, -4, -20, 4], [-6, 20, 9, -4, -20, 12, -19], [-14, -20, 2, 10, 21, -17, 14], [12, -8, -4, 20, 9, -19, 12], [2, -10, 2, -16, -10, -8, 0], [-9, 27, 5, -10, -13, 7, -11], [-10, -3, -15, 13, -3, -9, -3], [27, 7, -7, 18, 23, -7, 11], [-23, 3, -7, -2, -2, 2, 5], [-45, 1, -3, -30, -19, 21, -9], [-24, -15, -13, -24, -22, 18, -14], [17, -5, -5, 16, 23, -23, 13], [-8, 0, 16, -8, -7, 9, -2], [-10, 4, 1, 4, 2, 6, 9], [7, -3, 1, 10, 6, 0, -7], [17, -1, -2, 20, 18, -21, 14], [0, -12, -6, -12, -14, 12, 0], [8, -7, -3, 5, 12, -22, 5], [-11, -13, -9, -12, 2, 14, -17], [6, 15, -7, 23, 28, -9, 6], [-8, 6, -6, 14, 27, -13, 6], [12, 3, 13, -3, 10, -17, -8], [-4, -9, -7, 24, 14, -10, 10], [-5, 9, 8, -12, -24, 5, -8], [6, 8, 2, 8, -2, 18, 0], [9, -9, -5, -10, -15, 7, -1], [-12, 10, -2, -2, -9, 15, 2], [-20, 8, 2, 0, 19, 3, 2], [9, 15, 5, 24, 11, -7, 21], [-36, 10, -11, -8, -5, 17, -1], [21, -5, 3, -2, 10, -14, 5], [-25, -14, -16, -22, -11, 1, -9], [0, 4, -12, 14, 13, 1, 6], [-30, -10, 0, -4, 8, -10, 16], [-24, 7, -3, 17, -4, -7, -4], [-3, -13, 13, -12, -16, -6, 5], [-19, -1, 4, 2, 16, 11, -12], [16, -14, -10, -10, -8, 18, -10], [-2, 15, 15, -7, 5, 7, -13], [17, 0, 16, 5, 7, -14, -11], [-10, 10, 8, -6, -29, 17, -16], [4, 2, -14, -6, -7, 9, -10], [-28, 3, 17, 4, 16, -12, 10], [14, -8, 4, 0, 3, -19, -10], [-14, 15, 3, -8, 1, 11, -4], [-16, -10, 10, -22, -19, 9, -10], [-34, -18, -8, 8, 33, -15, 8], [-19, -11, -10, -14, -29, 19, -10], [-28, 25, 3, 3, -14, -3, 4], [-16, 13, -11, 9, 10, 8, -3], [7, 19, 19, 4, -10, 12, -3], [21, -27, -7, -14, 14, 0, -7], [-4, -5, 5, 1, 7, -10, -6], [-13, 13, -13, 18, 4, 14, 9], [3, 26, 14, 5, 4, -7, -15], [-14, 3, -11, -9, -12, 27, 6], [5, -7, 0, -8, 9, 6, 8], [7, -6, 10, -8, -11, 11, 1], [1, -11, 16, -16, 1, 2, -4], [28, -6, -22, 6, 26, 2, 28], [9, 7, -10, 4, 12, 1, 10], [36, 8, 14, 26, 4, -10, 6], [27, 4, -22, 18, 15, 7, 27], [-8, 5, 15, 2, -21, -1, -32], [0, 10, -4, -6, -19, -5, -4], [14, -6, 4, 14, 16, -2, 24], [-5, -9, -11, 0, -5, -3, 15], [5, -7, 10, -18, -12, -6, -16], [16, 4, 3, 10, 3, 15, -21], [-16, -6, 4, -2, 18, 2, 16], [12, 10, 10, 10, -15, -11, 2], [15, 6, 16, 7, 15, -17, 2], [-4, 2, -11, -8, 7, 8, -13], [29, -4, 8, 9, 16, 5, -1], [-13, -1, 10, 10, 15, -12, 14], [6, 8, -14, -12, 0, 18, -4], [31, 0, 2, -3, 1, 21, -8], [-8, -16, -13, -18, -8, 5, -19], [16, -30, -26, 14, 21, -17, 24], [9, -17, -3, 4, -3, 15, -5], [-31, -29, -23, -16, 19, 1, 11], [-30, -6, -20, -2, -2, 14, 16], [22, 16, 20, -2, -25, 3, 4], [25, 14, 0, 7, -12, -8, -2], [-18, -10, 22, -14, -8, 16, -14], [24, -6, 6, 12, -1, -15, 26], [4, 39, 11, 34, 2, -12, 12], [12, 30, 17, 20, -8, -12, 13], [-2, 22, 16, -26, -19, 7, -8], [27, 7, 9, 18, 6, -20, 13], [9, -12, 6, 8, 18, -2, 3], [-3, 9, -11, 26, -1, -13, 9], [-26, 8, -14, 4, -5, 19, -4], [-28, 26, -6, -10, -17, 1, 6], [1, 11, 13, 4, -27, 23, -13], [4, -40, -11, -10, 13, -3, 1], [-35, 1, 7, -16, -9, 9, -9], [-13, -1, -9, -4, 9, 11, 5], [11, 6, 6, 21, 17, -32, 7], [-8, -17, -23, 20, 10, -6, 18], [29, -15, -5, -8, 13, 3, 1], [22, -18, -10, -6, 1, -1, 20], [59, 11, 19, 16, -1, 1, -3], [12, -28, -18, 0, -1, -5, 16], [1, 10, 10, -5, 11, -4, -11], [10, 2, 10, -16, -24, 24, -8], [-26, -37, -5, -27, 2, 8, 7], [-3, 0, -4, -12, -31, 31, -5], [18, 19, -7, 18, -7, -3, 6], [-1, -13, -3, -20, -13, 5, -5], [39, 4, -20, 34, 34, -12, 19], [22, 4, 4, 2, 3, 11, 0], [-50, 10, 8, -20, -19, -9, -20], [-10, 20, -12, -6, 10, 8, -10], [17, 23, 11, 4, -19, 5, 5], [-19, -4, -4, 9, 9, 0, -3], [53, 13, 15, 28, 10, -10, -5], [-4, 30, 22, 12, 6, -12, -6], [22, 12, 4, 16, 8, -8, 16], [-24, 1, -21, -17, -12, 15, -12], [-16, -14, -14, -8, 17, 3, -14], [-3, -7, -13, 6, 33, -19, 15], [-36, 26, 14, -20, -15, 5, -8], [-1, 25, 0, 16, -8, 17, 0], [-23, -1, 14, -16, -11, -1, -20], [-18, 12, 6, 2, -15, -1, -8], [-14, -14, -8, 30, 14, -18, 20], [-14, -4, 23, -16, -10, 10, -13], [14, 4, -16, 4, 12, -8, 12], [-10, -14, 8, -18, 2, -8, 4], [-24, 2, 10, -38, -17, 29, -16], [-8, -10, -12, 36, 12, -4, -2], [8, -12, -4, -6, -2, 8, 2], [43, -3, 15, 2, -3, -15, 5], [20, -18, 4, -26, 1, 9, 4], [4, 8, -10, -10, -17, 25, 4], [6, 5, -7, -11, 26, -4, 25], [-8, 16, 0, -2, -25, -9, 8], [5, -11, -5, -20, -17, 1, -5], [19, 11, -14, 6, 3, 20, 18], [20, 3, 11, -27, -22, 8, -5], [-9, -10, 14, 5, 14, -13, -3], [4, 36, 9, 26, -7, 5, 1], [12, 22, 10, -26, -31, 25, -14], [-13, 4, 14, -42, -55, 13, -17], [29, -19, 7, -14, 14, 16, -3], [32, -2, 19, -8, 5, 4, -7], [-23, 3, 7, -18, -11, 21, -7], [21, 21, 15, 6, 18, -6, -7], [48, 5, -5, 31, 20, -9, 16], [-3, -5, 6, 8, -12, -15, -22], [-27, 1, -6, 22, 0, -6, -8], [-36, 8, 4, 2, 13, 9, 2], [-40, -7, 9, -5, 8, -14, -7], [0, -7, -17, -1, -2, -19, 4], [-12, -12, -17, -22, 4, 10, 9], [-12, -12, -18, 2, 1, 3, 8]];
    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_b();".
// 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_b(:prec:=7)
    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_b();".
// 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_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1336_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 26, 17, -30, -20, 6, 6, 1]>],Snew);
    return Vf;
end function;