
// Make newform 5265.2.a.be in Magma, downloaded from the LMFDB on 08 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5265_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_5265_2_a_be();".
// 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_5265_2_a_be();".
// 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 := [9, 7, -38, -14, 33, 7, -10, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0], [2, -3, -1, 1, 0, 0, 0, 0], [4, 3, -5, -1, 1, 0, 0, 0], [2, -4, -11, 10, 10, -6, -2, 1], [-3, -6, 20, 0, -16, 4, 3, -1], [-6, 2, 24, -6, -17, 5, 3, -1]];
    Rf_basisdens := [1, 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_5265_a();"
function MakeCharacter_5265_a()
    N := 5265;
    order := 1;
    char_gens := [326, 2107, 2836];
    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_5265_a_Hecke(Kf)
    return MakeCharacter_5265_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]];
    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, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 1, 0], [1, 0, 0, -1, 1, 0, 1, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, -1, 1, 1, 0], [-1, 0, 0, -1, 0, 1, -1, 0], [0, 1, -3, 1, -2, -3, -2, 1], [2, -1, 2, -2, 1, 2, 0, 1], [-3, 3, -2, -1, 0, -1, 0, -1], [-2, -1, -1, 2, -2, -2, -2, 3], [-2, 0, 1, 0, 2, 0, 0, -3], [-3, 1, -2, 0, 1, -1, 0, -2], [1, -2, -3, 1, -1, -1, -2, 2], [-1, -2, 3, 3, 2, 1, 0, -1], [1, 0, -1, 0, -4, 0, 0, -1], [-3, 0, 0, -2, 0, 3, 2, 3], [-2, -2, -2, 3, 0, -1, -3, -1], [5, -4, 1, 3, 0, -1, -2, 0], [-2, 0, 1, 2, 0, 3, -2, 2], [-6, 3, -4, 1, -1, -1, -2, 1], [0, -3, 0, 1, 1, 3, 0, -3], [-2, 3, 0, -2, -1, 2, 1, 0], [-1, -4, 4, 1, 2, 1, 1, -3], [0, -3, -1, 5, 0, 1, -3, 3], [-5, 2, 3, -1, -1, -1, 1, -2], [-1, 0, 1, -2, 1, -4, 0, -1], [3, -5, 5, 2, 4, 0, 2, -4], [2, -5, 4, 0, 3, 2, 3, -3], [-4, -2, 1, 1, -1, -2, 1, 0], [-1, 3, -4, -1, -3, -3, -3, 1], [0, -2, 8, -1, 1, -1, 0, -3], [-4, 2, -3, -4, 0, -5, -2, 1], [2, 1, -1, 2, -4, -1, -3, 6], [1, -5, -4, -2, -1, 0, -2, 4], [-3, 0, -2, 1, -4, 2, -2, 2], [-1, 2, 2, 0, -1, 2, 3, 0], [0, 1, -2, 2, 0, 1, -1, 6], [1, 1, -4, -1, 2, -5, -1, -6], [6, 1, 5, 0, 6, -3, 3, -6], [4, 0, 6, -2, -1, 0, -1, -2], [4, 1, -2, 0, -3, 1, -3, 5], [3, 8, -2, -2, -4, -4, -2, 3], [-4, 2, -8, 0, -4, -2, -1, 4], [-10, 1, 4, -1, 4, 5, 6, 0], [-11, -4, 7, 4, 6, 4, 3, -4], [-1, 6, -2, -2, 0, -1, -5, 0], [1, 2, 3, 7, -4, -1, 2, -2], [-3, 4, 0, -2, -3, -6, 1, -3], [-4, 0, 0, -3, 0, 5, 7, -2], [6, -4, 6, -1, 0, -2, 0, 0], [8, -3, 7, 2, 1, 2, -1, -1], [-6, 3, -8, -1, 3, 4, -2, 1], [-5, -2, 5, 2, 4, 6, 5, 0], [-5, 3, -2, -1, 4, 0, 1, 0], [3, 4, 1, 1, -3, -1, -1, -5], [-4, -5, 11, -1, 4, 5, 3, 2], [-6, 2, -2, 3, -4, 3, 5, -4], [4, 1, -5, -5, -5, -2, 0, 3], [-9, 3, -3, 3, -7, -3, -4, -2], [-4, -2, 3, 6, -4, 7, -1, 6], [3, -9, 10, -2, 1, 4, 2, 1], [1, 1, 5, -5, 3, 0, 9, -3], [4, -6, 3, 1, 5, 0, -7, 2], [-12, 0, 0, -3, -2, -5, 0, 1], [-11, -1, -4, -3, 2, -2, -4, 0], [-7, -1, 2, -3, -5, 1, 3, 2], [-3, 13, -5, -9, 0, 0, 4, -5], [1, 3, 7, 1, -1, 0, 1, -1], [-7, 3, -2, -2, -5, 5, 3, 7], [9, -2, 11, 1, 8, 10, 3, -7], [-12, -6, 1, -3, 1, 2, 8, 3], [-3, 7, 0, 1, 1, -5, -3, -7], [0, 7, -3, -9, 2, 0, 9, -2], [-6, -2, 1, 1, 6, -4, -3, -4], [13, -6, 2, 3, 0, 1, -8, 5], [1, -3, -2, -2, -2, -1, 2, 0], [13, 10, 0, -4, 3, -1, -5, -4], [-3, 1, 2, 0, 5, 6, 6, -6], [1, 6, -4, -4, 5, -8, 5, -6], [5, -7, 4, 4, -1, -5, 2, 1], [5, 0, 3, -2, 10, 2, 0, 4], [4, -5, 3, -1, 8, 5, 4, 1], [-3, -3, 0, 4, 3, -6, -10, 5], [0, 12, -9, 2, -5, -1, -7, 5], [1, 8, -1, -1, 3, 2, 0, 6], [2, 3, -11, 1, 0, -7, -8, -10], [-2, 6, 0, -1, 0, -7, 5, -7], [-9, 2, 2, 2, -6, -4, 5, -5], [0, -4, 6, 3, 10, 1, -1, -4], [4, 13, -7, -6, 0, -5, -6, -3], [-10, 3, -7, 0, -7, -1, -1, 3], [10, 5, -2, -2, -3, 0, 7, -2], [-7, -11, 11, 8, 6, 10, 6, -2], [1, 1, 5, -3, 3, 4, 6, 2], [-3, 9, -7, -5, -4, -2, -5, 0], [3, 3, -4, -8, 0, 5, 4, 2], [-4, 3, 2, 3, -4, 8, 5, 3], [-5, -5, -6, 1, -2, 5, 5, 9], [-5, -2, 5, 0, 2, 1, 4, 6], [-5, 11, -5, 1, -5, -5, -6, 4], [7, 1, 6, 5, -6, 2, -6, -1], [20, -4, 16, 0, 7, 6, 11, -7], [-12, 6, -12, 9, -9, -11, -15, 8], [5, -4, -4, -3, -3, -6, -11, 2], [-9, -2, 10, -4, 8, 3, 13, -5], [-1, 3, 1, 0, 5, -2, -1, -6], [-5, 1, 11, -4, 1, -3, 2, -1], [-12, 5, -1, 1, -8, -5, 3, -4], [-12, -6, -14, 6, -1, -6, -2, 2], [1, 7, 9, -6, 6, 5, 6, -2], [2, 3, 0, 4, 4, 2, -2, 4], [2, 0, 2, 3, 12, 0, 4, 2], [-8, 2, -6, -2, -2, -10, 3, -4], [-12, 5, -3, 3, -7, -10, -7, 4], [-7, 2, -7, -9, -1, 2, 0, 12], [-7, 9, 2, -4, -8, 7, -1, 3], [-1, 1, -1, -3, 0, 5, -7, 5], [5, 7, -7, 1, -5, -1, -5, 14], [-6, -3, 7, -3, 10, 8, 4, 1], [-13, 7, -6, -3, -6, 3, -3, -2], [1, 7, -8, -1, -10, -14, -7, 0], [-7, -3, 1, -7, 4, 7, 11, 0], [-3, -3, 0, 0, -1, -6, 7, -8], [-4, -1, -3, 1, 1, -1, 1, 1], [8, -7, 12, 8, 2, 3, 1, -11], [-1, -4, 14, -1, 9, 12, 12, -1], [1, -7, 10, -1, 12, 12, 5, -1], [11, 17, -1, -6, 2, -1, 9, -1], [15, -4, -2, 0, 3, -14, -6, -7], [13, 9, -14, -7, -15, -4, -11, 2], [2, -13, 5, 6, 5, 2, 3, 5], [-2, -13, -14, 5, -5, -11, -14, 1], [-7, -1, 2, 0, 0, -4, -2, 10], [-4, -3, -1, -1, -4, 3, 0, 15], [-7, 5, -2, 0, -14, -9, -1, 0], [12, -1, -5, -5, -2, 7, 0, 4], [-4, -12, 9, 6, -1, 7, -2, -2], [6, -7, 13, -1, -4, -6, -1, 2], [3, -2, -6, 1, -8, -2, 0, -3], [-14, 10, -4, 7, -5, -8, 3, -12], [-16, 1, -4, 2, -6, -5, -8, -1], [14, -13, 2, -1, 0, 3, 6, 2], [3, 13, -7, -7, -6, 2, 1, -9], [-23, 6, -9, -6, 1, 5, -6, -1], [4, -1, -8, 6, -5, -3, 1, -1], [-15, -7, -6, 3, 1, 4, -5, 0], [4, 14, -6, -5, -2, -7, 5, -4], [-1, -8, 8, 0, -3, -1, 1, 13], [-5, 1, -6, -3, 5, 13, 8, -1], [-3, -4, -1, 4, 9, -7, 2, -9], [-1, 5, 11, 0, 0, 0, 5, -5], [-17, 6, -10, -1, 2, -5, -7, 5], [-1, -6, -5, 0, -2, 7, 2, 7], [0, 17, -18, -4, -1, -9, -3, -12], [2, -10, -1, -3, 0, -12, -5, -5], [10, -12, 9, -2, 6, 3, 0, 0], [-18, 2, -9, -2, -2, 4, -8, 4], [12, -11, -14, 4, -2, -3, -15, 5], [-15, 7, 2, -2, 0, 5, 2, -1], [-2, 2, -15, -1, -12, -7, 1, 12], [4, -7, -4, -13, 9, 6, 0, -1], [-3, 14, 4, -12, 1, 1, 5, -7], [2, 4, 10, 6, -6, -7, -1, -3], [2, -12, -4, -3, 7, 0, -1, 7], [-1, 8, -7, -5, -11, -6, -2, 13], [10, -3, -10, -7, -7, -4, -6, 9], [10, 8, 4, 3, 8, -11, -1, -8], [-13, -6, 5, -5, -2, -7, 2, -4], [4, -9, 1, 0, 1, 2, 9, 3], [6, 0, -13, -1, -1, 1, -1, -1], [-9, -11, 6, 4, 8, 7, -2, -5], [23, -5, 13, 5, 0, 13, 0, -4], [-17, 1, -6, -3, -2, -3, -7, -5], [14, -3, 18, -5, 9, 16, 21, -4], [-10, 13, 12, -6, 4, 1, 13, -7], [12, -5, -9, -3, 5, -1, 4, 1], [-13, 0, 1, 11, -11, 11, -10, 17], [-6, -18, 12, -5, 9, 0, 12, -10], [7, 1, 14, 3, 5, 10, 3, -10], [16, -18, 3, 6, 1, -7, -6, 0], [-1, -15, 3, 1, 2, 1, -3, 11], [7, -7, 1, 3, 0, 7, 6, -1], [8, -2, 9, 3, 3, -4, 3, -18], [-21, -10, -1, 12, 0, 6, 6, 6], [6, -4, 11, 7, -6, 3, 5, 1], [-1, 6, -12, -10, 2, -3, 3, 3], [7, -2, -5, -6, -3, -1, -9, -1], [14, 1, 13, 5, -5, 4, 3, -6], [8, -4, 4, -2, 15, 14, -2, 0], [-19, -9, 13, 0, 4, 12, -2, 4], [-1, -9, 5, 5, -1, 12, 13, 8], [-21, 10, 5, -5, -4, 10, 2, 10], [-17, 14, 3, -5, -2, 1, 10, -11], [6, 3, -5, -17, 4, 0, -4, 4], [28, 3, 4, 3, -15, -1, -8, 0], [5, 7, -4, 2, 0, -14, -4, 6], [-3, -15, -10, 5, -7, -12, -8, 7], [1, 4, 9, 0, 12, -2, -4, -17], [-9, 7, -14, 2, 3, -5, 4, -9], [10, -30, 27, 13, 8, 6, 3, -4], [10, -18, 12, 14, -7, -7, -4, 1], [-8, 23, -7, -13, -8, -2, 2, -4], [15, 20, -10, -2, -6, -1, -4, 2], [2, -19, 11, 10, -5, 8, -5, 7], [-1, 4, -7, 8, -15, 1, -7, 9], [3, 8, 3, 2, -3, 4, -12, 7], [-17, 6, -13, 3, -5, -10, -4, 5], [-18, -1, -20, -2, 4, -2, -1, 2], [6, 0, -2, -3, 10, 5, -7, -1], [10, 4, -1, 9, -4, 9, -9, 1], [-16, 17, -5, -10, 2, -15, 1, -12], [20, 1, -1, 3, -7, 4, 0, 9], [-16, -8, -1, 8, 6, 12, 6, 0], [4, 15, -19, -15, -5, -3, 2, 4], [-5, 16, -4, 2, 2, -19, -8, -10], [34, 4, -6, 3, -12, -8, -7, -1], [-7, -12, 0, 7, -9, -8, -6, 17], [32, -17, 16, 12, 3, -4, 1, -11], [4, -2, 17, -1, 17, 20, 20, 0], [2, 18, -4, -4, 3, 7, -7, 6], [2, -17, 9, -5, 13, 14, 10, -1], [-7, 4, 3, 3, 7, -2, -2, -15], [-9, -5, -6, 3, -3, -3, 1, 5], [-3, 9, 10, 2, 13, 12, 10, -3], [5, 9, 10, -14, -6, 7, 12, 4], [9, 10, -10, 5, -9, 4, 2, 0], [10, -19, 5, 15, 15, -1, -7, -11], [6, -5, 11, -4, 7, -4, 2, -18], [1, 4, -5, 1, -9, -8, 2, -9], [-27, -6, 2, 0, -4, 6, -1, 12], [19, 12, -1, 1, 7, 5, -7, -13], [2, -18, 7, 3, 0, 11, 18, 3], [-17, 12, -23, 2, -14, -2, -9, 14], [-10, -16, 0, 6, 2, -2, -12, 3], [7, 17, 2, -1, 7, -8, 7, -5], [12, 7, 13, 2, -7, 0, 14, 0], [-13, -1, 0, 0, 14, 1, 8, -15], [-11, -8, -12, 12, -6, -5, -9, 8], [13, -17, 0, 7, 4, 7, -5, 8], [-3, 14, 0, 7, -1, -13, -10, -3], [7, -14, 7, 4, 5, 18, 8, -8], [11, 8, -5, -3, -6, 9, 14, 4], [-18, -10, 1, -9, 18, 12, 9, -7], [-16, -5, -1, 2, 4, 7, -7, 2], [14, -3, 9, -4, 0, -5, 11, -7], [1, 6, 11, 0, 0, -2, 17, -8], [-6, 3, -8, -6, 2, -12, 0, 6], [-20, 12, -20, -4, -12, 2, 3, 7], [0, -6, 6, 14, -1, 10, -3, -4], [7, -12, 0, -4, 3, 11, -5, 7], [-12, -19, 1, 6, 3, 6, 1, -6], [-1, -2, 15, -11, 5, 10, 1, 7], [11, -15, 15, 3, -2, 15, 7, 23], [-6, -8, 0, 13, 3, -4, -2, 6], [-4, 6, -5, -5, 3, -4, 0, -11], [18, -4, -10, -18, 9, 1, 0, 1], [-3, -11, 16, -7, 4, 5, 23, -7], [13, 6, -10, 13, -5, -1, -7, 2], [7, -20, 2, 4, 9, 10, -2, 3], [-19, 1, -1, -1, 4, 1, 14, -7], [-5, -11, 0, 8, 14, 11, -4, -9], [-11, 17, -14, -7, -2, -11, 3, -2], [10, 7, -6, -11, 1, -11, 1, 1], [7, 8, -1, 7, -4, 6, 0, -7], [2, 11, -12, -8, -4, -9, -3, -4], [-12, -24, 12, -3, 9, 9, 17, 0], [-18, -11, 6, 6, 0, -5, -11, -16], [-5, -6, -17, 1, -8, -5, -10, 15], [3, -11, 17, -10, 20, 16, 15, -14], [-16, -7, -14, 11, -1, -9, -14, 8], [-23, 1, 9, -1, 8, 20, 4, -3], [-9, 8, 5, -3, -2, 2, 14, -12], [19, -8, 21, 6, 4, 12, 19, 3], [-8, 7, -5, -3, 2, -5, 4, 1], [23, 0, -2, -11, -7, 3, -4, 16], [7, 25, 0, -1, -3, 0, 1, 5], [22, -6, 21, -10, 16, 7, 22, -21], [11, -12, -18, -7, -10, -7, -16, 11], [29, 4, 0, -1, -8, 10, -1, -3], [-24, 11, -3, -10, -4, 4, -3, 4], [-7, 12, 0, -6, 8, -20, -6, -9], [13, 9, 23, -3, -4, -2, 19, -2], [-1, -7, -4, 1, -14, 9, -3, 10], [5, 9, -21, 6, -1, -2, -2, -2], [8, -1, 7, -2, -3, -1, 5, -23], [15, -23, 3, 1, 2, -5, 2, 2], [-30, -5, -12, 9, 0, 1, 5, -8], [-36, -5, -35, 13, -11, -15, -17, 13], [17, 0, -13, -6, -5, -18, 2, 1], [-14, -7, 4, -6, 12, -11, 1, -11], [-28, -3, -3, 3, 20, 4, 11, -10], [13, 2, -20, 3, -9, -14, -6, -1], [-5, -4, 2, -3, 0, -7, 0, -10], [13, 3, -9, -20, 0, -3, -8, 2], [0, -5, -19, 20, 1, -3, -7, 6], [-25, -3, 20, 6, 17, 7, 16, -6], [-2, 8, 11, -8, 3, -6, 8, 11], [-23, 0, 9, -3, 3, -1, 4, -7], [16, -18, 18, 13, 2, -6, -5, -11], [-32, 8, -6, -8, 11, 17, 7, 9], [-20, 17, -11, -3, 5, -14, 0, -16], [-30, 16, -11, -12, -5, 1, 4, 6], [3, 4, 3, -10, 9, 11, 5, 21], [-4, -4, -13, 12, 1, 19, 0, 15], [-1, 4, -21, 3, -3, 11, -5, 6], [8, 0, 11, -5, 18, 11, -10, -2], [-1, 17, 6, 5, 1, 10, 0, 5], [-2, -6, -10, -1, -4, -10, 17, 0], [-3, 12, 10, -14, 6, 2, 3, -21], [-2, -9, 12, 0, 16, 5, 5, -21], [-9, 18, 8, -13, 3, 10, 18, -10], [-18, -11, 18, 4, 7, 1, -7, 8], [19, -10, 27, -1, 4, 3, 20, -20], [19, 11, -8, 5, -9, 9, 5, 11], [3, 7, -4, 2, -9, -17, -11, 7], [-9, 0, -3, 8, -15, -17, 2, 8], [-4, -6, -21, -6, 5, 2, 11, 15], [-3, -5, -12, -9, 17, 8, 0, -5], [-30, 1, 11, 19, -7, -1, -10, 8], [15, 10, -10, -1, 7, 6, 3, 10], [20, -2, -10, 7, 1, -7, -23, 2], [-24, -7, 14, 6, 14, 6, -3, -1], [-4, 5, -6, 10, -16, -21, -4, 8], [-16, 6, -2, -11, 2, -2, -4, -12], [-2, 25, -24, -7, -17, 1, -7, 8], [-6, 9, -4, 6, -14, 8, -3, 25], [-9, -17, -6, 16, 3, -7, -19, 1], [12, -12, -2, 0, -17, 4, 2, 22], [5, 5, -5, -5, 19, 12, 1, -12], [-3, 0, 14, -8, 10, 9, 11, -15], [-13, -14, 0, 9, -12, -6, -4, -4], [0, -1, 13, 7, 10, 12, -2, -2], [16, 9, 3, 13, -13, 0, -17, 1], [-2, -4, -4, -8, 2, 2, -4, -4], [-17, 11, -9, 4, -25, -9, -14, 3], [-11, 9, -8, 8, -3, -7, 3, 5], [3, 12, 8, -8, 13, -17, 17, -13], [-36, -8, 9, -8, 8, -1, 11, -2], [-11, 8, 2, 9, 18, -7, -1, -9], [-7, 13, -28, 9, -11, -12, 1, -14], [5, 11, -4, 10, -8, -1, -18, 4], [29, -1, 24, -14, 4, 3, 4, 2], [25, -10, -9, -11, 1, 10, -8, 25], [4, 9, 20, 2, 1, 9, 11, -2], [8, 0, -1, -3, -15, 0, -2, 10], [-3, -1, 13, 1, 22, 16, 15, -14], [-35, 17, -23, -4, 4, -15, 1, -5], [2, 5, 8, -5, -7, 3, -2, -13], [-26, 8, -10, -8, 1, -13, -9, -5], [-5, -12, 29, -7, 22, 19, 17, -10], [-6, -9, 28, 5, 17, 12, 8, -24], [-16, -9, -8, -10, -1, 3, 2, 9], [-4, -6, 6, 12, -3, 12, -9, 0], [-3, -1, -3, -7, -6, -14, -7, -18], [-13, 2, -16, -4, -12, -24, -2, 0], [7, 8, -10, -8, -3, -6, -2, -4], [18, -4, -12, 0, -15, -21, -8, 7], [-1, -9, 15, -3, 7, 3, 5, -6], [-2, -13, 30, 0, 8, 1, 10, -13], [-9, 2, -7, -1, 12, 16, -5, 6], [12, -13, -16, 1, 10, -3, -5, 1], [-5, 10, 3, -2, 10, 4, 0, -12], [10, 6, 5, -3, -5, 0, -2, -8], [-5, 15, 5, 0, -3, -1, 10, 2], [21, -19, 8, 3, -14, 4, 0, 4], [1, -10, 6, -2, -14, 14, 11, 26], [-30, 3, -9, 17, -9, -7, -5, -1], [1, -2, -9, -1, -12, -1, -16, 25], [-24, 17, 5, 6, 8, 12, -9, 12], [18, -16, 38, -11, 21, 11, 17, -16], [-16, -1, -4, 18, -4, 7, 7, 7], [-5, 18, -3, 5, -16, 8, -10, 15], [-14, 12, 5, -19, 11, 14, 29, -17], [37, 13, -4, -4, -14, -9, -13, 9], [-9, -10, 6, 1, -4, 16, -4, 4], [-21, 20, 15, -9, -3, 1, 17, -11], [44, 9, -5, -6, -13, -11, -3, 6], [0, 25, -9, 4, -34, -1, -11, 8], [-9, -19, 8, 2, 34, 2, 4, -17], [-11, 16, 2, 5, -4, 16, 10, -2], [19, -9, -5, -3, -4, -23, -17, 12], [13, 17, -27, -4, -22, -20, -16, 6], [3, 3, -1, 7, 5, 5, -2, -12], [8, 1, 5, 17, 4, 7, 4, -1], [-12, 24, -4, -12, -1, 9, 5, -1], [-32, 2, -17, -7, 2, -7, -5, -3], [-6, 13, -16, -8, -8, -17, -1, -3], [0, -32, -16, 10, 5, -11, -5, 8], [-15, -23, 21, -1, 8, 16, 13, -7], [-12, -1, -12, -2, -4, 19, -9, 10], [-5, -28, 7, 0, -3, 22, 15, 9], [-5, 14, 2, -18, 7, 6, -10, 1], [-7, 15, -17, -1, -2, 13, -1, 20], [25, 4, 2, 12, 4, -21, 1, -14], [17, -25, -7, -2, 4, -3, 1, 2], [23, -7, -5, 12, -6, -4, 5, 12], [2, 0, 1, 13, -15, -16, -5, -7], [-19, 0, -4, 9, -6, -7, 2, -5], [21, -17, 8, 2, -9, 2, 6, 0], [-20, 8, 6, -17, -1, -7, 5, -19], [-9, -5, -5, 5, -2, -15, -11, 23], [18, 8, -11, 3, -18, 0, -9, 14], [5, 12, -9, -4, -7, 8, -3, 7], [-17, 13, -9, -15, -1, -20, -3, -5], [-2, 3, -18, -20, 0, 1, 3, 20], [0, 23, 0, -12, 3, -3, 1, -16], [0, -6, 1, -5, 11, -1, 9, 19], [29, 4, 1, -12, -7, 1, 19, 0], [-22, -14, 11, 30, 7, -2, -6, 3], [-14, 10, -2, -1, 8, -17, -7, 2], [6, 9, 0, -9, 4, -2, -9, 16], [11, -9, 2, 1, -11, -3, 3, 5], [15, -14, 4, -15, -3, 16, 4, 8], [-25, -6, 17, -9, 4, 9, 19, -8], [-22, 7, 10, 0, 5, 25, 26, 2], [-40, 1, 3, -3, -7, -1, 11, 1], [-19, 13, -29, -3, -5, -15, -5, 14], [-18, -1, -15, 10, 15, -7, -9, -4], [12, 17, 9, 1, 7, 6, -15, -7], [21, -7, -16, 2, -14, -6, -7, -2], [-15, 16, -12, -16, 12, 10, 19, -4], [22, -38, 10, 4, 13, 10, -4, 6], [-10, 4, -3, 2, 13, -16, 6, -19], [11, -4, 5, 2, 6, 9, 4, 7], [-18, -4, 2, -5, 7, -3, -5, -10], [-10, 1, -18, 2, -8, 10, -7, 12], [-14, 18, 7, 16, 6, -14, -2, -10], [-1, 8, 6, 2, 1, -14, -5, -9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5265_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5265_2_a_be();".
// 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_5265_2_a_be(:prec:=8)
    chi := MakeCharacter_5265_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_5265_2_a_be();".
// 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_5265_2_a_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5265_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, 7, -38, -14, 33, 7, -10, -1, 1]>,<7,R![6, 51, 129, 75, -49, -50, -2, 6, 1]>,<11,R![81, 497, 316, -503, -181, 158, -2, -9, 1]>,<17,R![822, -1441, -1684, 1014, 466, -159, -36, 6, 1]>],Snew);
    return Vf;
end function;