
// Make newform 8281.2.a.co in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8281_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_8281_2_a_co();".
// 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_8281_2_a_co();".
// 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, 0, -36, 0, 172, 0, -197, 0, 88, 0, -16, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [32, 0, -248, 0, -127, 0, 242, 0, -66, 0, 5, 0], [27, 0, -223, 0, 360, 0, -209, 0, 44, 0, -3, 0], [0, 27, 0, -223, 0, 360, 0, -209, 0, 44, 0, -3], [0, -191, 0, 559, 0, -244, 0, -33, 0, 22, 0, -2], [-27, 0, 91, 0, 476, 0, -451, 0, 110, 0, -8, 0], [0, 12, 0, -76, 0, 60, 0, -14, 0, 1, 0, 0], [9, 0, 263, 0, -903, 0, 638, 0, -143, 0, 10, 0], [-24, 0, 274, 0, -903, 0, 638, 0, -143, 0, 10, 0], [0, 171, 0, -822, 0, 1147, 0, -605, 0, 121, 0, -8], [0, 194, 0, -1333, 0, 1923, 0, -1001, 0, 198, 0, -13]];
    Rf_basisdens := [1, 1, 11, 11, 11, 11, 11, 1, 11, 11, 11, 11];
    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_8281_a();"
function MakeCharacter_8281_a()
    N := 8281;
    order := 1;
    char_gens := [1522, 3382];
    v := [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_8281_a_Hecke(Kf)
    return MakeCharacter_8281_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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, -1, 0, 0, 0, 0, 1, 1, 0, 0], [0, -1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1], [0, 0, -1, 0, 0, 0, -1, 0, 1, -2, 0, 0], [1, 0, 2, 1, 0, 0, 0, 0, -1, -1, 0, 0], [0, -2, 0, 0, -1, -1, 0, -2, 0, 0, 0, 2], [0, 3, 0, 0, 1, -1, 0, 1, 0, 0, 0, 1], [0, -1, 0, 0, -2, 0, 0, -1, 0, 0, 3, -4], [1, 0, 0, -1, 0, 0, 2, 0, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 1], [1, 0, 0, -2, 0, 0, 0, 0, -1, 2, 0, 0], [0, -4, 0, 0, 0, 0, 0, 3, 0, 0, -4, -1], [0, 0, -2, 1, 0, 0, 0, 0, 0, -2, 0, 0], [0, -1, 0, 0, 1, -1, 0, -1, 0, 0, -1, 5], [0, 3, 0, 0, 0, 0, 0, -1, 0, 0, -3, 2], [0, -1, 0, 0, 0, -2, 0, 2, 0, 0, -2, 4], [6, 0, 1, 0, 0, 0, 1, 0, 3, 1, 0, 0], [0, -3, 0, 0, -3, 2, 0, -1, 0, 0, 1, -4], [0, 4, 0, 0, 2, -1, 0, -2, 0, 0, -1, 3], [0, -2, 0, 0, -1, -2, 0, 0, 0, 0, -4, 2], [-2, 0, 1, 1, 0, 0, -2, 0, -1, -2, 0, 0], [3, 0, -1, 2, 0, 0, 1, 0, -4, 1, 0, 0], [2, 0, -1, 4, 0, 0, 1, 0, 2, 0, 0, 0], [0, 2, 0, 0, 3, 2, 0, -1, 0, 0, 1, 2], [-3, 0, 0, -7, 0, 0, 2, 0, 2, 2, 0, 0], [-5, 0, -1, -2, 0, 0, 0, 0, -1, -3, 0, 0], [-6, 0, -3, 2, 0, 0, 2, 0, 3, 1, 0, 0], [0, -4, 0, 0, 1, -1, 0, 0, 0, 0, 0, -3], [-4, 0, -5, -1, 0, 0, 0, 0, 3, -3, 0, 0], [0, 3, 0, 0, 0, -1, 0, 5, 0, 0, 2, -1], [0, -1, 0, 0, 2, 4, 0, 4, 0, 0, 0, -1], [-6, 0, 3, 1, 0, 0, 3, 0, -3, 5, 0, 0], [0, 2, 0, 0, 1, -1, 0, -5, 0, 0, 1, 3], [0, 4, 0, 0, 0, 4, 0, -1, 0, 0, 3, 1], [-3, 0, 4, -6, 0, 0, -5, 0, -2, 1, 0, 0], [0, 0, 2, -6, 0, 0, 0, 0, -1, 2, 0, 0], [-2, 0, 3, -4, 0, 0, -1, 0, 1, 3, 0, 0], [-5, 0, -1, -2, 0, 0, 1, 0, -3, 2, 0, 0], [0, -3, 0, 0, 3, 3, 0, 2, 0, 0, 1, -6], [0, 3, 0, 0, 1, -2, 0, -5, 0, 0, 3, -4], [-7, 0, 1, 0, 0, 0, -1, 0, -2, 1, 0, 0], [-10, 0, -6, 2, 0, 0, 2, 0, 3, -1, 0, 0], [0, 5, 0, 0, 3, 8, 0, -3, 0, 0, 5, 0], [0, -3, 0, 0, -1, -4, 0, -2, 0, 0, -1, -3], [0, -1, 0, 0, 5, 1, 0, 0, 0, 0, -1, -2], [-3, 0, -2, -3, 0, 0, -2, 0, 9, 0, 0, 0], [0, 1, 0, 0, 3, 1, 0, -2, 0, 0, -1, 4], [0, -1, 0, 0, 6, 4, 0, 6, 0, 0, 3, 0], [-12, 0, -1, -8, 0, 0, -3, 0, 0, 2, 0, 0], [-3, 0, 0, 4, 0, 0, 1, 0, -4, -5, 0, 0], [-5, 0, -1, 0, 0, 0, -2, 0, -3, -5, 0, 0], [2, 0, 4, -1, 0, 0, -2, 0, 1, -6, 0, 0], [0, 1, 0, 0, -4, 3, 0, -4, 0, 0, 2, 0], [-17, 0, -3, -1, 0, 0, 3, 0, 0, -3, 0, 0], [0, 8, 0, 0, 4, -4, 0, 7, 0, 0, 2, -1], [-3, 0, 3, -6, 0, 0, -2, 0, -4, 6, 0, 0], [0, -1, 0, 0, 0, 4, 0, 8, 0, 0, -3, -4], [0, 1, 0, 0, -2, 3, 0, 3, 0, 0, 2, 9], [-13, 0, -3, 3, 0, 0, -1, 0, 1, -6, 0, 0], [1, 0, 5, 4, 0, 0, 3, 0, -6, 2, 0, 0], [0, -8, 0, 0, 2, 6, 0, -2, 0, 0, 1, -1], [0, 0, 0, 0, -2, 2, 0, -3, 0, 0, 0, 4], [8, 0, 3, 3, 0, 0, -5, 0, 2, 1, 0, 0], [0, 0, -3, 1, 0, 0, 3, 0, 6, -7, 0, 0], [0, -4, 0, 0, -2, 3, 0, -8, 0, 0, 5, -4], [0, 0, 0, 0, -3, -2, 0, -6, 0, 0, -1, 3], [0, -1, 0, 0, -2, 0, 0, -5, 0, 0, 3, -12], [-2, 0, 3, 2, 0, 0, -2, 0, -3, 0, 0, 0], [0, 0, 1, 3, 0, 0, -1, 0, -4, 4, 0, 0], [0, 0, 0, 0, 0, -5, 0, -6, 0, 0, -4, 8], [0, -2, 0, 0, -4, 11, 0, 2, 0, 0, 6, -4], [2, 0, 8, 2, 0, 0, -1, 0, 2, 0, 0, 0], [0, -4, 0, 0, -1, 5, 0, -5, 0, 0, 3, -2], [0, -5, 0, 0, 3, 2, 0, 8, 0, 0, -11, -1], [0, -6, 0, 0, 0, 5, 0, 7, 0, 0, 5, -6], [-5, 0, 4, -9, 0, 0, 6, 0, 3, 9, 0, 0], [0, -5, 0, 0, -2, -2, 0, 0, 0, 0, 1, 2], [0, -4, 0, 0, 0, 0, 0, -7, 0, 0, 8, 1], [3, 0, 3, -2, 0, 0, 4, 0, -8, 5, 0, 0], [5, 0, -4, 7, 0, 0, -4, 0, -1, 4, 0, 0], [11, 0, 6, 7, 0, 0, 1, 0, -4, -5, 0, 0], [0, -4, 0, 0, -2, 7, 0, 11, 0, 0, 4, -12], [0, 7, 0, 0, 9, -4, 0, 9, 0, 0, -1, 1], [0, 11, 0, 0, 3, 3, 0, 0, 0, 0, 3, -4], [0, -2, 0, 0, 3, 2, 0, 7, 0, 0, -4, 5], [-6, 0, -3, 4, 0, 0, -5, 0, 8, -6, 0, 0], [0, 5, 0, 0, -6, -8, 0, -5, 0, 0, -5, 3], [0, 10, 0, 0, 6, 7, 0, 6, 0, 0, 5, 6], [-16, 0, -1, 3, 0, 0, 1, 0, -6, -1, 0, 0], [0, 6, 0, 0, 6, -3, 0, 3, 0, 0, 4, 5], [-9, 0, 2, -3, 0, 0, -5, 0, -3, 1, 0, 0], [0, 11, 0, 0, 6, 3, 0, 3, 0, 0, -2, 6], [-7, 0, 2, -13, 0, 0, -5, 0, -9, -1, 0, 0], [-12, 0, 4, 2, 0, 0, 3, 0, -7, 0, 0, 0], [0, 7, 0, 0, 3, -2, 0, -8, 0, 0, 4, 3], [9, 0, 0, 5, 0, 0, -6, 0, -2, -6, 0, 0], [0, -10, 0, 0, -6, -1, 0, -2, 0, 0, -12, 0], [-1, 0, 0, -2, 0, 0, -3, 0, -11, 2, 0, 0], [8, 0, -8, -3, 0, 0, -3, 0, 2, 2, 0, 0], [-3, 0, -9, 8, 0, 0, 3, 0, 2, 10, 0, 0], [0, -2, 0, 0, -5, -6, 0, -7, 0, 0, -3, 2], [0, 11, 0, 0, -4, -9, 0, -3, 0, 0, -9, 5], [0, 1, 0, 0, -2, -7, 0, -12, 0, 0, 2, 5], [-10, 0, -7, 5, 0, 0, -2, 0, 4, 1, 0, 0], [2, 0, -3, 6, 0, 0, 10, 0, -1, 3, 0, 0], [-6, 0, -5, 8, 0, 0, -2, 0, -3, -7, 0, 0], [0, -1, 0, 0, -6, -4, 0, -6, 0, 0, 8, 1], [0, -1, 0, 0, 0, -5, 0, 0, 0, 0, -5, 1], [0, 10, 0, 0, 2, 1, 0, 4, 0, 0, -8, 1], [0, -2, 0, 0, -1, -3, 0, 5, 0, 0, -11, 1], [0, 0, 7, -1, 0, 0, 3, 0, -2, 0, 0, 0], [0, -2, 0, 0, -6, -2, 0, 0, 0, 0, 4, -4], [-10, 0, -8, 5, 0, 0, 9, 0, 5, 5, 0, 0], [-1, 0, 0, 2, 0, 0, 3, 0, 10, -6, 0, 0], [9, 0, -3, 7, 0, 0, -3, 0, -3, 5, 0, 0], [0, -10, 0, 0, -5, 12, 0, -2, 0, 0, 10, -12], [0, 0, -4, 0, 0, 0, -6, 0, -3, -5, 0, 0], [-8, 0, 2, -4, 0, 0, -6, 0, 5, -2, 0, 0], [0, 1, 0, 0, -3, 3, 0, -1, 0, 0, 8, -13], [0, -1, 0, 0, -3, 2, 0, 2, 0, 0, 6, -8], [4, 0, 5, 1, 0, 0, 1, 0, -4, 6, 0, 0], [0, 0, 0, 0, -13, -1, 0, -7, 0, 0, 0, -2], [-20, 0, 2, 0, 0, 0, 8, 0, -3, 3, 0, 0], [14, 0, 8, -1, 0, 0, -5, 0, -3, 2, 0, 0], [0, -2, 0, 0, 2, -3, 0, -7, 0, 0, -6, 11], [0, -2, 0, 0, -6, 8, 0, -4, 0, 0, 8, 1], [0, 0, 0, 0, -1, 10, 0, -5, 0, 0, 16, -1], [9, 0, 4, 18, 0, 0, 6, 0, -7, -2, 0, 0], [2, 0, -3, 3, 0, 0, 2, 0, 1, -7, 0, 0], [0, 19, 0, 0, 4, -3, 0, 3, 0, 0, 0, 13], [0, 7, 0, 0, 5, 1, 0, 4, 0, 0, 11, -13], [0, 5, 0, 0, -12, 1, 0, 0, 0, 0, 3, 0], [0, 10, 0, 0, -2, -15, 0, 4, 0, 0, -5, 2], [0, 0, -6, -6, 0, 0, -6, 0, 5, 7, 0, 0], [-8, 0, -4, 5, 0, 0, -8, 0, -8, -8, 0, 0], [0, 2, 0, 0, 5, 0, 0, -1, 0, 0, 4, -3], [0, 1, 0, 0, -3, -1, 0, 0, 0, 0, -3, -2], [8, 0, -1, 2, 0, 0, 8, 0, -5, 8, 0, 0], [0, -6, 0, 0, 5, 14, 0, 4, 0, 0, 11, -11], [-1, 0, -2, -20, 0, 0, -9, 0, 9, 2, 0, 0], [0, 5, 0, 0, 4, -6, 0, 7, 0, 0, -6, 7], [0, -3, 0, 0, 3, -6, 0, -2, 0, 0, -16, -2], [9, 0, 2, -1, 0, 0, -1, 0, 1, -7, 0, 0], [-17, 0, -2, 4, 0, 0, 7, 0, 0, -2, 0, 0], [0, -8, 0, 0, -11, -4, 0, -13, 0, 0, -2, -6], [0, 5, 0, 0, 8, 9, 0, 4, 0, 0, 7, 3], [-13, 0, -11, 1, 0, 0, 1, 0, 13, -3, 0, 0], [4, 0, 7, -2, 0, 0, 7, 0, 3, 2, 0, 0], [3, 0, 10, 4, 0, 0, 0, 0, 12, 2, 0, 0], [22, 0, 7, 4, 0, 0, 2, 0, -14, 2, 0, 0], [0, 0, 0, -4, 0, 0, 9, 0, 11, 10, 0, 0], [2, 0, -1, -14, 0, 0, -11, 0, 11, -1, 0, 0], [0, 7, 0, 0, -1, 0, 0, 7, 0, 0, 7, 7], [2, 0, 5, 7, 0, 0, 3, 0, 6, -1, 0, 0], [0, -1, 0, 0, -8, 11, 0, -2, 0, 0, 3, -4], [0, 2, 0, 0, 11, 1, 0, 6, 0, 0, 6, 8], [10, 0, 7, -8, 0, 0, -3, 0, -3, 0, 0, 0], [0, -13, 0, 0, 0, 1, 0, 17, 0, 0, -2, -4], [-16, 0, 2, 11, 0, 0, -2, 0, -7, -13, 0, 0], [0, -9, 0, 0, 15, 6, 0, 11, 0, 0, -3, 6], [0, -3, 0, 0, -4, -2, 0, -9, 0, 0, -8, 3], [15, 0, 1, 6, 0, 0, -9, 0, -10, -5, 0, 0], [2, 0, 7, 16, 0, 0, 0, 0, -11, -15, 0, 0], [0, -1, 0, 0, -2, -3, 0, -3, 0, 0, 5, 5], [-18, 0, -9, -3, 0, 0, -2, 0, -2, -7, 0, 0], [0, 12, 0, 0, 0, 0, 0, 5, 0, 0, 6, 8], [0, 0, 0, 0, 5, 12, 0, -4, 0, 0, -4, 3], [-12, 0, -5, 8, 0, 0, 6, 0, 0, 13, 0, 0], [0, -13, 0, 0, -8, -9, 0, -8, 0, 0, -3, 4], [-18, 0, 7, 3, 0, 0, -7, 0, 2, -3, 0, 0], [-10, 0, 3, -6, 0, 0, -7, 0, -5, 3, 0, 0], [0, -2, 0, 0, 8, 11, 0, 9, 0, 0, 13, -3], [0, 2, 0, 0, -4, -1, 0, 1, 0, 0, 3, -2], [-3, 0, 4, -14, 0, 0, -1, 0, 7, -3, 0, 0], [-18, 0, -12, 2, 0, 0, 0, 0, 1, 3, 0, 0], [0, -11, 0, 0, 8, 3, 0, 16, 0, 0, -16, -4], [-11, 0, -4, 1, 0, 0, 4, 0, -2, 9, 0, 0], [14, 0, -5, 1, 0, 0, 3, 0, 3, -10, 0, 0], [0, 2, 0, 0, -10, 7, 0, -13, 0, 0, 18, -20], [0, 8, 0, 0, 6, -4, 0, -8, 0, 0, -6, 7], [16, 0, 7, 21, 0, 0, 1, 0, -7, -2, 0, 0], [-1, 0, 4, -12, 0, 0, -9, 0, 4, 3, 0, 0], [0, 9, 0, 0, 8, 9, 0, -9, 0, 0, 5, 12], [0, 3, 0, 0, -10, -7, 0, -14, 0, 0, -9, -4], [0, 4, 0, 0, 1, 3, 0, -9, 0, 0, -11, 4], [26, 0, 10, -3, 0, 0, -3, 0, 5, 12, 0, 0], [0, 2, 0, 0, 5, 7, 0, -20, 0, 0, 0, 7], [-22, 0, 1, 5, 0, 0, 7, 0, -5, -7, 0, 0], [0, 4, 0, 0, -1, -16, 0, -6, 0, 0, -10, 9], [-11, 0, -12, -1, 0, 0, -6, 0, 8, -10, 0, 0], [27, 0, -2, 19, 0, 0, 0, 0, -4, 1, 0, 0], [0, -4, 0, 0, 4, -8, 0, 0, 0, 0, 1, -9], [5, 0, 1, -14, 0, 0, -7, 0, 3, -2, 0, 0], [0, 10, 0, 0, -4, -2, 0, -16, 0, 0, -2, -9], [-17, 0, -2, 0, 0, 0, -4, 0, 12, -2, 0, 0], [0, -4, 0, 0, -12, 5, 0, -25, 0, 0, 14, -7], [-14, 0, -6, 7, 0, 0, 10, 0, 5, -9, 0, 0], [0, 5, 0, 0, -5, -5, 0, 8, 0, 0, -6, 1], [3, 0, -9, 5, 0, 0, 5, 0, 5, -4, 0, 0], [0, -8, 0, 0, 4, 1, 0, 18, 0, 0, -1, 8], [2, 0, 1, -5, 0, 0, -2, 0, 17, -3, 0, 0], [0, -9, 0, 0, 0, 2, 0, 16, 0, 0, -14, -7], [-15, 0, -3, -8, 0, 0, -1, 0, 9, -6, 0, 0], [0, 6, 0, 0, -2, 4, 0, 2, 0, 0, -10, -7], [-8, 0, -2, 7, 0, 0, 0, 0, 20, -11, 0, 0], [-2, 0, -10, -4, 0, 0, 9, 0, -1, 6, 0, 0], [-16, 0, 1, -2, 0, 0, -3, 0, 3, -1, 0, 0], [29, 0, 7, 6, 0, 0, -1, 0, -9, 1, 0, 0], [0, -12, 0, 0, -1, 1, 0, 3, 0, 0, -11, 16], [0, 13, 0, 0, 14, 5, 0, 11, 0, 0, 6, -1], [0, -5, 0, 0, -5, -10, 0, -10, 0, 0, -2, -10], [-5, 0, 1, 4, 0, 0, -2, 0, 7, 0, 0, 0], [-52, 0, -8, -2, 0, 0, 5, 0, 10, -4, 0, 0], [0, -12, 0, 0, 3, 2, 0, -2, 0, 0, 4, -10], [0, -9, 0, 0, -11, 9, 0, 3, 0, 0, 4, -18], [0, 0, -6, 8, 0, 0, 11, 0, -11, 14, 0, 0], [0, 13, 0, 0, 7, -5, 0, -3, 0, 0, 12, 2], [0, 8, 0, 0, 1, -13, 0, 4, 0, 0, -7, -10], [0, 7, 0, 0, -5, -17, 0, 0, 0, 0, -1, 6], [-9, 0, 4, -2, 0, 0, 7, 0, -6, 1, 0, 0], [21, 0, -7, -2, 0, 0, -1, 0, 10, -1, 0, 0], [11, 0, -5, 6, 0, 0, -1, 0, 4, 5, 0, 0], [-2, 0, 7, 2, 0, 0, -1, 0, 10, -12, 0, 0], [-20, 0, -5, -3, 0, 0, -8, 0, -3, -4, 0, 0], [0, -1, 0, 0, 1, -3, 0, 19, 0, 0, -13, -14], [3, 0, 12, 3, 0, 0, -8, 0, -4, 9, 0, 0], [-12, 0, -9, 10, 0, 0, 14, 0, -6, 0, 0, 0], [0, -7, 0, 0, 5, 9, 0, -2, 0, 0, 1, 10], [1, 0, -4, 0, 0, 0, 4, 0, 16, -1, 0, 0], [43, 0, -4, 1, 0, 0, -7, 0, 0, -11, 0, 0], [0, -1, 0, 0, -10, -19, 0, -5, 0, 0, -9, -1], [0, 14, 0, 0, -3, 0, 0, -2, 0, 0, 9, 3], [0, -13, 0, 0, -12, 8, 0, 0, 0, 0, 18, -33], [-9, 0, -4, 12, 0, 0, -1, 0, 11, -4, 0, 0], [-14, 0, -9, 8, 0, 0, -2, 0, 10, -16, 0, 0], [0, -8, 0, 0, -8, -9, 0, 2, 0, 0, 1, 4], [0, 0, 2, -20, 0, 0, -6, 0, 9, 4, 0, 0], [-20, 0, -6, 10, 0, 0, 7, 0, -11, -4, 0, 0], [0, 10, 0, 0, -8, 9, 0, -11, 0, 0, 5, -11], [0, 4, 0, 0, -9, -13, 0, 9, 0, 0, 4, -8], [-29, 0, 1, 15, 0, 0, 12, 0, 2, 12, 0, 0], [0, -7, 0, 0, -12, 5, 0, -21, 0, 0, 18, -3], [0, -3, 0, 0, -12, 12, 0, 3, 0, 0, 0, -3], [0, -12, 0, 0, -3, 6, 0, 20, 0, 0, -7, -4], [0, 0, 11, -2, 0, 0, 9, 0, 3, 15, 0, 0], [0, -18, 0, 0, 0, 1, 0, -3, 0, 0, 6, -11], [0, -11, 0, 0, 0, 8, 0, 10, 0, 0, 0, 10], [0, -2, 0, 0, 6, 7, 0, 1, 0, 0, 10, -23], [19, 0, -1, -5, 0, 0, -5, 0, 1, 8, 0, 0], [-9, 0, -19, 11, 0, 0, 1, 0, 9, -16, 0, 0], [0, -5, 0, 0, -7, -10, 0, 6, 0, 0, -14, 13], [-11, 0, 8, 5, 0, 0, -6, 0, 1, -11, 0, 0], [0, -3, 0, 0, 7, -2, 0, 5, 0, 0, -8, 25], [-9, 0, -2, 16, 0, 0, 3, 0, 0, -17, 0, 0], [0, -10, 0, 0, 1, 10, 0, 24, 0, 0, -1, -15], [-1, 0, 16, -7, 0, 0, 4, 0, -11, -2, 0, 0], [-21, 0, 10, -5, 0, 0, -5, 0, 7, -3, 0, 0], [0, 1, 0, 0, 9, -4, 0, 4, 0, 0, 3, 14], [5, 0, -1, -16, 0, 0, 6, 0, 7, 28, 0, 0], [0, -1, 0, 0, 6, -1, 0, -4, 0, 0, -9, -2], [-18, 0, 9, -24, 0, 0, 6, 0, 8, 15, 0, 0], [0, -16, 0, 0, -15, -7, 0, -15, 0, 0, -2, -13], [0, 11, 0, 0, -11, -19, 0, -14, 0, 0, -12, 14], [0, 0, 0, 0, -2, 18, 0, -13, 0, 0, 26, -17], [-24, 0, 2, -24, 0, 0, 0, 0, 10, 6, 0, 0], [17, 0, -1, 6, 0, 0, -4, 0, -1, -8, 0, 0], [0, 4, 0, 0, -12, 15, 0, -12, 0, 0, 5, -11], [0, 3, 0, 0, 8, -3, 0, 14, 0, 0, -23, 13], [-5, 0, 12, -8, 0, 0, -10, 0, -4, 10, 0, 0], [11, 0, 12, 3, 0, 0, 3, 0, 4, 16, 0, 0], [0, 0, 0, 0, 1, -14, 0, 3, 0, 0, -5, 14], [0, -10, 0, 0, -2, -6, 0, -1, 0, 0, 1, -7], [0, 10, 0, 0, -9, -9, 0, -9, 0, 0, 18, 6], [0, 9, 0, 0, -8, 8, 0, -3, 0, 0, 13, -5], [-41, 0, 9, -1, 0, 0, 3, 0, 1, 10, 0, 0], [-12, 0, 6, 11, 0, 0, 6, 0, -17, 1, 0, 0], [0, -2, 0, 0, -11, -16, 0, -15, 0, 0, -5, 7], [-2, 0, -10, -4, 0, 0, 8, 0, -7, 7, 0, 0], [0, -14, 0, 0, 14, 1, 0, 10, 0, 0, -7, 14], [0, 2, 0, 0, 11, 3, 0, -3, 0, 0, 1, 1], [-16, 0, 6, 3, 0, 0, -9, 0, -5, -7, 0, 0], [-4, 0, -13, 9, 0, 0, 17, 0, 9, 3, 0, 0], [0, 19, 0, 0, 5, 6, 0, 13, 0, 0, -9, -4], [0, -7, 0, 0, -3, 2, 0, -11, 0, 0, 4, 0], [-12, 0, -3, 21, 0, 0, 3, 0, 0, 2, 0, 0], [-30, 0, 6, -6, 0, 0, -3, 0, 7, -10, 0, 0], [-2, 0, 13, -18, 0, 0, -3, 0, -2, 12, 0, 0], [0, 0, 0, 0, -4, -15, 0, -9, 0, 0, -9, -3], [0, -2, 0, 0, 1, -13, 0, 2, 0, 0, -8, 4], [21, 0, 8, -9, 0, 0, -4, 0, -11, 3, 0, 0], [-5, 0, -4, -17, 0, 0, -1, 0, 12, 8, 0, 0], [19, 0, -8, 21, 0, 0, -5, 0, 2, -7, 0, 0], [25, 0, 8, 18, 0, 0, -2, 0, -20, 7, 0, 0], [-24, 0, -6, -16, 0, 0, -1, 0, -4, -10, 0, 0], [0, -10, 0, 0, 5, 15, 0, 30, 0, 0, 9, -16], [17, 0, 0, 12, 0, 0, -8, 0, -1, -4, 0, 0], [0, -2, 0, 0, 2, -25, 0, -1, 0, 0, -15, 20], [10, 0, -10, -1, 0, 0, -10, 0, 6, -8, 0, 0], [6, 0, -5, 7, 0, 0, -5, 0, 5, 7, 0, 0], [12, 0, 9, 2, 0, 0, 3, 0, -1, 6, 0, 0], [0, -4, 0, 0, 3, -2, 0, 2, 0, 0, -10, 2], [34, 0, 2, -2, 0, 0, -6, 0, 0, -14, 0, 0], [-16, 0, -8, 23, 0, 0, 10, 0, 15, 2, 0, 0], [0, -9, 0, 0, -10, -12, 0, -9, 0, 0, -11, 22], [9, 0, 4, -4, 0, 0, 0, 0, -1, -18, 0, 0], [-4, 0, 10, -2, 0, 0, 1, 0, -8, 26, 0, 0], [0, 11, 0, 0, -4, 9, 0, 15, 0, 0, -8, -20], [-14, 0, 4, -23, 0, 0, -13, 0, 4, -2, 0, 0], [24, 0, 1, 3, 0, 0, 6, 0, 13, -7, 0, 0], [0, -4, 0, 0, -7, 17, 0, 0, 0, 0, 7, -13], [1, 0, 6, -17, 0, 0, 8, 0, 2, 26, 0, 0], [0, -11, 0, 0, -10, -1, 0, -15, 0, 0, -17, -7], [0, -25, 0, 0, -4, -1, 0, 10, 0, 0, -16, 16], [0, -16, 0, 0, -13, -5, 0, 2, 0, 0, -3, -34], [-9, 0, -1, 11, 0, 0, 11, 0, -5, -12, 0, 0], [16, 0, -4, -13, 0, 0, -3, 0, 8, 3, 0, 0], [0, -9, 0, 0, -22, -8, 0, -14, 0, 0, 0, -16], [-30, 0, -6, 1, 0, 0, 7, 0, -6, 4, 0, 0], [0, -6, 0, 0, -1, -12, 0, 7, 0, 0, -11, 6], [0, -12, 0, 0, 2, -6, 0, 7, 0, 0, -10, 14], [20, 0, 8, 6, 0, 0, -7, 0, -4, 1, 0, 0], [0, 16, 0, 0, 19, 3, 0, 18, 0, 0, 4, 19], [0, -11, 0, 0, 1, -2, 0, 8, 0, 0, 0, -19], [7, 0, -5, 14, 0, 0, 3, 0, -6, 7, 0, 0], [24, 0, -2, -21, 0, 0, 1, 0, -1, 17, 0, 0], [0, 14, 0, 0, 8, 8, 0, 2, 0, 0, -10, 2], [32, 0, 10, 0, 0, 0, -2, 0, -2, 2, 0, 0], [-5, 0, 0, 7, 0, 0, -12, 0, -11, 4, 0, 0], [0, 3, 0, 0, -13, 0, 0, -8, 0, 0, 1, 2], [0, 27, 0, 0, 17, 6, 0, 12, 0, 0, 2, 1], [0, -22, 0, 0, 0, -1, 0, -13, 0, 0, 6, -30], [0, 7, 0, 0, 4, -10, 0, 14, 0, 0, -4, -8], [0, 6, 0, 0, -1, 10, 0, -5, 0, 0, 4, 0], [0, 6, 0, 0, 14, 8, 0, 14, 0, 0, 14, -2], [9, 0, -6, 7, 0, 0, 0, 0, 7, -24, 0, 0], [0, -1, 0, 0, -2, 0, 0, -8, 0, 0, -5, -20], [-4, 0, 0, 20, 0, 0, -5, 0, -16, 14, 0, 0], [0, -9, 0, 0, 0, 18, 0, -11, 0, 0, 25, -30], [16, 0, 14, 16, 0, 0, 4, 0, 0, -11, 0, 0], [0, -7, 0, 0, -3, 3, 0, -3, 0, 0, 1, 35], [11, 0, -9, 6, 0, 0, -6, 0, 25, -4, 0, 0], [-2, 0, 2, -6, 0, 0, 13, 0, -20, 9, 0, 0], [0, 9, 0, 0, 11, 10, 0, 0, 0, 0, 2, -23], [0, 19, 0, 0, -10, 7, 0, 13, 0, 0, -2, 11], [-28, 0, 11, -22, 0, 0, 0, 0, -11, -13, 0, 0], [0, 22, 0, 0, 10, -2, 0, -8, 0, 0, 2, 16], [0, 3, 0, 0, -3, -4, 0, 28, 0, 0, -4, 12], [0, 9, 0, 0, -2, -28, 0, -14, 0, 0, -15, 22], [8, 0, -2, -6, 0, 0, -5, 0, 2, 15, 0, 0], [-7, 0, 18, -1, 0, 0, -2, 0, 1, -15, 0, 0], [1, 0, 16, 12, 0, 0, 0, 0, -1, 3, 0, 0], [0, -22, 0, 0, 12, 20, 0, 20, 0, 0, 7, -23], [0, -1, 0, 0, -7, -12, 0, -6, 0, 0, -17, 17], [-3, 0, 0, 2, 0, 0, 16, 0, -7, 16, 0, 0], [0, 1, 0, 0, -22, 12, 0, 0, 0, 0, 6, -19], [0, -2, 0, 0, -3, 1, 0, -8, 0, 0, 2, 27], [-27, 0, 9, -21, 0, 0, 1, 0, 11, -6, 0, 0], [8, 0, -12, 20, 0, 0, 7, 0, -4, 6, 0, 0], [0, -7, 0, 0, -3, -4, 0, -34, 0, 0, 9, 4], [15, 0, 3, -2, 0, 0, -8, 0, 10, -14, 0, 0], [-1, 0, -2, 1, 0, 0, 5, 0, -8, -6, 0, 0], [0, -20, 0, 0, 4, -6, 0, 29, 0, 0, -18, 11], [0, -15, 0, 0, -8, 10, 0, 21, 0, 0, 9, -32], [35, 0, 9, 21, 0, 0, -1, 0, -24, 5, 0, 0], [26, 0, 5, -8, 0, 0, 0, 0, -12, 13, 0, 0], [-6, 0, -1, -23, 0, 0, 7, 0, -8, 12, 0, 0], [0, 3, 0, 0, -7, -2, 0, 4, 0, 0, 14, -25], [5, 0, -1, 30, 0, 0, -1, 0, 3, -11, 0, 0], [-28, 0, -15, 13, 0, 0, -4, 0, 33, -17, 0, 0], [-5, 0, 2, -6, 0, 0, 6, 0, 31, 9, 0, 0], [0, 4, 0, 0, 20, -18, 0, -7, 0, 0, -4, 13], [8, 0, 1, 15, 0, 0, 1, 0, -26, -4, 0, 0], [0, -5, 0, 0, 16, -21, 0, 2, 0, 0, -3, 16], [-14, 0, 4, -4, 0, 0, 7, 0, -19, 15, 0, 0], [12, 0, 6, 15, 0, 0, 10, 0, 13, -5, 0, 0], [0, -7, 0, 0, -9, -18, 0, -18, 0, 0, -9, -1], [0, -4, 0, 0, -5, -5, 0, 5, 0, 0, -7, -11], [0, 20, 0, 0, -2, -15, 0, 0, 0, 0, 9, -1], [0, 15, 0, 0, 19, 5, 0, 5, 0, 0, 6, 16], [0, -12, 0, 0, 5, 6, 0, -13, 0, 0, 3, 4], [0, 22, 0, 0, 19, -3, 0, -8, 0, 0, 11, 18], [0, 1, 0, 0, 6, -4, 0, 11, 0, 0, 19, 4], [-8, 0, 2, -24, 0, 0, 13, 0, 12, -4, 0, 0], [0, 27, 0, 0, 12, 11, 0, 12, 0, 0, -11, 27], [-63, 0, -3, 7, 0, 0, -1, 0, 3, -5, 0, 0], [0, -28, 0, 0, 2, 16, 0, -11, 0, 0, 3, 7], [0, -24, 0, 0, -26, -6, 0, -7, 0, 0, -8, -24], [0, 12, 0, 0, -1, -5, 0, -5, 0, 0, -17, 29], [-28, 0, 5, -22, 0, 0, -10, 0, 18, 2, 0, 0], [0, -32, 0, 0, 8, 6, 0, 10, 0, 0, -16, -6], [-26, 0, 0, 9, 0, 0, 0, 0, -11, -2, 0, 0], [0, -15, 0, 0, 12, -10, 0, -10, 0, 0, -7, -10], [31, 0, 7, 37, 0, 0, 3, 0, -29, -4, 0, 0], [-2, 0, -3, -10, 0, 0, -6, 0, 12, -5, 0, 0], [0, -7, 0, 0, 12, 27, 0, 11, 0, 0, 17, -31], [0, -12, 0, 0, 14, 3, 0, 15, 0, 0, -6, 16], [49, 0, 1, 9, 0, 0, -7, 0, -2, -2, 0, 0], [0, -8, 0, 0, 12, 13, 0, 3, 0, 0, 16, 5], [0, 5, 0, 0, 11, -7, 0, -11, 0, 0, -1, 6], [0, -13, 0, 0, -12, -24, 0, -5, 0, 0, -14, 12], [-17, 0, -18, 19, 0, 0, 10, 0, 10, -5, 0, 0], [0, 0, 0, 0, 18, -1, 0, 11, 0, 0, -14, -9], [0, 37, 0, 0, -2, -2, 0, 9, 0, 0, -12, 7], [0, -3, 0, 0, 10, -20, 0, 14, 0, 0, -25, 16], [0, -13, 0, 0, -21, -2, 0, 1, 0, 0, -10, 6], [0, 0, 1, 34, 0, 0, -7, 0, -3, -10, 0, 0], [16, 0, -9, -33, 0, 0, -15, 0, 11, 0, 0, 0], [-32, 0, -9, 3, 0, 0, 13, 0, 2, 2, 0, 0], [-14, 0, -15, 6, 0, 0, -4, 0, 13, -1, 0, 0], [5, 0, 26, -5, 0, 0, 5, 0, 8, 11, 0, 0], [-1, 0, 10, -14, 0, 0, -8, 0, -6, 9, 0, 0], [0, 5, 0, 0, 10, 18, 0, 5, 0, 0, 2, 7], [26, 0, 3, -2, 0, 0, 8, 0, -2, 5, 0, 0], [0, -26, 0, 0, -18, -19, 0, -5, 0, 0, -13, -6], [-7, 0, 2, 21, 0, 0, -7, 0, 6, -19, 0, 0], [13, 0, -3, 6, 0, 0, 9, 0, -22, 25, 0, 0], [0, 21, 0, 0, -11, 1, 0, -14, 0, 0, -5, -22], [0, -6, 0, 0, 7, 12, 0, 3, 0, 0, 13, -16], [-11, 0, -4, -20, 0, 0, -13, 0, 3, 6, 0, 0], [0, -9, 0, 0, -22, -24, 0, -7, 0, 0, -16, 0], [0, 7, 0, 0, 16, 1, 0, 8, 0, 0, -1, -25], [-33, 0, -11, 10, 0, 0, -10, 0, 5, -20, 0, 0], [0, -6, 0, 0, -8, 16, 0, -25, 0, 0, 13, -12], [0, 15, 0, 0, 15, 3, 0, 3, 0, 0, 10, 4], [-15, 0, 8, -12, 0, 0, -15, 0, -5, -4, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8281_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8281_2_a_co();".
// 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_8281_2_a_co(:prec:=12)
    chi := MakeCharacter_8281_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_8281_2_a_co();".
// 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_8281_2_a_co( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8281_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 0, -36, 0, 172, 0, -197, 0, 88, 0, -16, 0, 1]>,<3,R![7, 19, 4, -16, -5, 3, 1]>,<5,R![121, 0, -539, 0, 878, 0, -636, 0, 201, 0, -25, 0, 1]>,<11,R![85849, 0, -122593, 0, 61227, 0, -13284, 0, 1355, 0, -62, 0, 1]>,<17,R![19, -146, 56, 198, 96, 17, 1]>],Snew);
    return Vf;
end function;