
// Make newform 5265.2.a.bf in Magma, downloaded from the LMFDB on 24 April 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_bf();".
// 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_bf();".
// 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, -19, 66, -70, -1, 31, -8, -3, 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], [1, -5, 0, 1, 0, 0, 0, 0], [1, 15, -51, 20, 21, -10, -2, 1], [-4, 43, -73, 9, 31, -9, -3, 1], [1, -59, 130, -29, -53, 19, 5, -2], [-7, 92, -179, 40, 74, -28, -7, 3]];
    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+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, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 1, 0, 0, -1], [1, 0, 0, 0, 0, -1, -1, 1], [-1, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 0, -1, -1, 2, 0, -1], [-2, 1, -1, -1, -1, -1, 1, 1], [2, 0, 0, 0, 1, -2, 1, 0], [2, -2, 0, 0, -1, 0, 1, 0], [-5, -2, -1, 0, -1, 2, -1, 2], [1, -2, -1, 1, 1, 1, 0, 0], [-1, -1, -1, 1, -1, -1, 2, 0], [0, 0, -1, 1, -3, 2, 0, 0], [-3, 2, 2, -2, 2, 2, 0, -2], [2, -2, -1, 2, 2, 2, -1, -2], [2, 1, -1, 1, 1, 1, 0, 2], [2, 0, 1, 3, 2, -4, -1, 0], [-3, 1, 2, -2, -3, -2, 1, -1], [2, -2, -2, -1, 0, -1, -1, 2], [1, -2, 1, -2, 0, 1, -3, 0], [3, -6, -1, 1, -1, -1, 0, -2], [0, 1, 0, 1, 2, -1, -2, 0], [0, -1, 3, 1, 2, 1, -1, -3], [-6, 3, 2, 0, 1, 0, 3, 1], [1, -2, 1, -2, -3, 4, 1, 1], [-2, 0, -1, 0, 0, -2, -2, -1], [-3, 3, 3, -2, 3, 0, 1, 2], [0, -4, 0, 0, 1, 2, 0, 0], [-1, 0, -1, 0, -3, 0, 1, 1], [-4, 0, 1, 2, 0, 3, -1, -1], [3, 0, -2, 1, -3, -5, -2, -1], [0, -2, -2, 3, -2, -1, -2, 0], [-7, -2, 2, -1, 2, 2, -1, 0], [-3, -2, 3, -2, -1, 3, 1, -1], [-9, 4, 1, 1, -1, -1, 1, 2], [-5, -2, 3, -1, 0, 4, -4, -6], [-6, 5, -2, -1, -1, -3, 3, 3], [1, -6, 0, 2, 1, 3, -3, -1], [-3, 1, 2, -1, -2, -3, 1, -3], [2, -5, 2, -2, -2, 5, -1, 1], [-5, -3, -4, 3, -1, 1, -3, 1], [6, 2, -3, 4, 1, -5, 4, 1], [-2, 3, 2, -1, -3, -1, 4, -2], [-4, 1, 0, 0, -1, 2, 0, 1], [-5, 5, 3, -1, 4, -3, 3, 4], [-6, 4, 2, -4, 2, 0, 4, 3], [-4, 5, 0, 0, 7, -3, 3, -1], [0, -4, 0, -1, -4, -3, 1, -4], [1, 4, 0, 0, 0, -2, 3, 1], [2, 6, 0, -1, 0, -1, 1, -3], [1, 5, 2, 1, 3, -2, -2, 4], [-4, -9, -5, -2, -6, 2, 1, 5], [0, 10, 1, 1, 3, -4, 7, 0], [0, -2, 2, -2, 0, 0, 0, 1], [12, 1, -2, 5, 0, -2, 2, -5], [-7, 3, -3, 1, -5, -5, 4, -1], [-3, 2, 0, 1, 1, -1, 3, -5], [0, -8, 0, -3, -4, 5, 1, 5], [-1, -1, -3, -3, 0, 4, -1, 2], [-2, 4, 1, -4, 1, 4, -4, 2], [-11, 3, -1, -6, 3, -1, 1, 3], [-2, -1, 3, 0, 0, -2, -3, 0], [2, 0, 1, -5, -3, 6, 1, 1], [-7, 7, 4, -4, 3, 5, 1, -1], [14, 3, 3, 2, 1, -2, -3, -6], [-6, -4, 6, 3, 1, 6, -2, -4], [15, -6, -1, 0, -1, 2, 2, -1], [0, 0, 0, 4, 5, -1, -2, -2], [-5, 1, 0, 1, 0, 4, -1, 1], [2, -3, 2, 4, -4, 3, -8, 1], [13, 3, 5, 4, 5, -3, -4, -1], [-13, -2, -3, 3, -2, 2, -3, 6], [1, 4, 2, 1, 5, 1, 4, 1], [6, -2, 2, -1, 7, 0, -4, -3], [1, 2, -5, 1, 2, -2, 4, 9], [-9, 5, 1, -4, 5, 4, -1, 0], [-12, -7, -2, 2, 2, 3, -1, 2], [-15, -1, -2, 1, -3, 6, -2, 7], [-11, 4, 1, 3, -3, -3, 1, 2], [4, -3, 2, 1, -7, 5, 1, 1], [-14, 6, 0, 2, -1, -3, -2, 8], [6, 11, 0, -2, 5, -2, -2, -4], [1, 3, -1, -2, 0, 0, 3, 10], [-7, 4, -2, 4, -3, -2, -3, -2], [-12, 1, 2, 0, -3, 3, -4, 1], [15, 5, 3, -2, 1, -7, 1, -8], [-5, 1, 2, -1, -2, -3, 3, 4], [-1, -1, -4, -2, 5, 4, 1, 1], [-2, 2, -2, 5, 6, 1, 0, 1], [-8, 10, 2, -3, 0, -1, -1, 5], [4, -4, 3, 5, 1, -6, 3, 0], [10, -2, 4, 2, 13, -1, 2, -7], [3, 2, -2, -1, 3, 1, 1, 1], [3, 9, -3, 0, 0, -8, 2, 4], [-9, -8, -3, 2, -3, -3, -5, 4], [1, 4, 4, -3, 5, -4, 0, -5], [7, -8, 1, 4, -1, 5, -7, -4], [9, -1, 3, -2, -4, -1, 2, -7], [-10, 4, 0, 2, -5, -4, 3, 3], [0, -4, 1, -2, -2, 3, -5, 2], [-13, 7, 2, 0, 2, 4, 4, 6], [11, -1, 0, -2, 4, 5, -6, -6], [4, 6, -4, 4, 0, -8, 7, -1], [0, -3, -5, 8, 7, -4, 0, -1], [5, -10, -6, 4, -6, -5, -1, -1], [-14, 7, 1, -5, -1, 4, 5, 1], [-9, 3, -4, -1, -2, 5, -1, -3], [-4, 4, -5, 4, -7, 3, -2, 0], [7, -2, 6, -3, 7, 1, -3, 5], [-5, 4, 6, -7, 10, 3, -5, -2], [-8, 7, 3, 0, -4, -3, -1, 2], [-1, -2, 1, -6, -5, 2, -4, -2], [-15, -1, -4, 5, -3, -2, 12, 0], [-6, -7, 0, -2, -3, -4, -2, -9], [7, 4, 2, 4, 1, -3, -7, 1], [-4, 4, 6, -4, 4, 5, 7, -2], [-13, 5, -2, -1, 0, -4, 1, 1], [-5, 3, 4, 0, 0, -4, 1, -3], [-7, -2, -6, -2, 3, 4, 6, 9], [-2, 5, -3, 0, 4, 1, -8, -2], [1, -5, -2, -1, -4, -1, -5, -3], [-4, 15, -2, -1, 2, -6, 6, 7], [-1, -9, 1, -3, -4, 15, 1, 3], [-11, -3, -7, 1, 0, -3, 9, -5], [10, -2, -3, 7, -5, 1, 0, -5], [1, 6, 1, -1, 13, -8, -5, -3], [3, -1, 7, 1, 3, -4, 5, 8], [-13, -2, -4, 4, 5, -7, -2, -3], [-10, -2, -2, -3, -5, 8, -5, -1], [4, 8, -5, 0, 10, -4, 1, -2], [5, -5, 8, 0, -2, 3, 3, 9], [-9, 8, 0, -6, 1, 2, 9, -1], [19, -4, -1, -1, -3, -3, -4, -10], [6, 0, -5, -8, 1, 0, 8, 2], [-4, 4, 2, 0, 5, 0, -1, -6], [8, 10, 6, 4, 9, -3, 3, -7], [-9, 3, -6, 5, 0, -9, 3, -4], [1, 3, -4, 7, -1, -12, 0, -4], [8, -2, -6, -5, 7, 2, -4, 1], [1, 4, 0, 10, 2, -1, -10, -8], [10, 2, 4, 2, 8, -13, 5, 1], [-5, 7, 5, -3, 4, 6, 5, -4], [-8, 2, 8, -3, 11, 3, 3, -6], [-5, 8, 5, -4, 11, -5, -2, -7], [-1, 2, 6, -10, 6, 5, -2, 0], [6, 8, -3, -2, 9, -3, 4, -5], [-10, 6, -1, -4, -1, -3, 5, 13], [-2, -4, 0, 3, 2, -3, 3, 1], [-14, 10, -2, -10, -8, -3, 8, 7], [19, -3, -4, 5, 12, -7, -2, 14], [-6, -5, -5, 0, -3, -5, -6, -2], [2, 5, 3, -9, 6, 5, -2, -1], [-6, -9, 4, 10, 3, 6, -1, -5], [18, -4, 8, -1, -2, 6, -1, 4], [4, -4, 0, -7, 7, 6, 0, 9], [-9, 0, 2, -4, -8, 11, 4, -7], [13, -6, 1, 4, 2, 3, -9, -6], [0, -8, 3, -4, -2, 4, 4, 10], [5, 2, -3, 9, 3, 6, -1, -13], [-17, -2, -8, 3, -7, 2, 3, -8], [-4, 0, 1, -5, -4, 3, -9, 5], [-19, -1, 0, 1, 2, 4, 5, 6], [7, 9, 1, -4, 11, 3, 0, -5], [9, 1, 1, 1, -1, -10, -5, -3], [-24, 6, 5, -7, -4, 8, 3, -5], [9, -11, -8, 1, -3, -4, -3, -10], [-6, -8, 5, 3, 3, 8, -3, -10], [-15, -9, 3, -13, -3, 5, 1, 3], [-2, -6, -4, -11, -10, 1, -3, 4], [7, -10, -2, 2, 1, 6, -3, -15], [-3, -2, 0, -1, -4, 3, 6, -5], [2, 7, -3, 3, 12, 0, 1, -6], [5, -12, 1, 3, 1, -5, -1, 10], [-7, 1, -7, -2, 2, 6, -5, -1], [-22, -7, -2, -2, -10, -1, 5, 1], [4, -3, -2, 5, 12, -4, 1, -3], [5, -6, 0, 9, 3, -5, 2, 2], [-19, 6, -11, -3, -8, 1, -4, 8], [8, 3, -6, -4, -11, 1, 3, 4], [-14, -8, -9, 0, -7, 7, -3, -7], [-13, 5, -4, -5, -7, -4, 6, 18], [17, 3, 0, 12, -2, 0, 5, -13], [11, 2, -4, 0, -3, -4, -1, -6], [-12, -12, 3, 0, -8, -1, 1, 13], [-7, 2, -11, 2, -16, 0, 6, 14], [12, 4, -2, 0, 6, 2, 3, -9], [-14, 7, -1, -3, -1, 4, -1, -7], [28, -2, 1, 2, 0, 5, 0, -11], [2, -7, -3, -2, -10, 1, -5, 1], [-4, 13, -3, 1, 5, 1, 1, 6], [0, 10, 6, 2, 5, -14, -2, -8], [24, -6, 0, -4, 7, -5, -1, -5], [4, -17, -3, 3, -15, 6, -2, 10], [1, 1, -6, 6, 9, -4, -7, 8], [-2, -5, 2, 1, -2, 16, 0, -6], [-14, 9, -1, -8, -4, 6, -2, -2], [-1, 11, 0, 0, -10, -10, 2, 0], [2, -17, 4, -3, -16, 10, -9, 8], [-9, -4, 10, -3, 0, 15, 8, -14], [2, -8, 8, -2, -11, -3, 6, 6], [-5, 6, 9, -1, 6, -8, 2, 9], [-21, 7, 4, 0, -13, 1, -2, 6], [19, 10, -3, 1, 5, -14, 6, -6], [3, -23, 7, 0, -1, 15, -7, 2], [-13, 22, 4, 0, 1, -6, 9, 3], [-5, 9, 2, -4, 5, -15, 12, -1], [-17, -7, -5, 6, -3, -4, 1, -6], [26, -4, -5, -1, 4, -8, -3, 2], [7, 13, 3, 10, 6, -6, -4, -7], [10, 2, 3, 8, 16, 0, -5, -11], [30, -4, 0, 10, -4, -10, 3, -7], [7, -16, 3, 4, 1, 9, -5, 7], [-16, -8, -6, 11, 3, -6, 1, 6], [-10, 14, 5, 6, 0, -2, 2, 4], [2, 7, 2, 0, 8, -14, 2, -2], [2, -6, 6, 4, 2, -11, -7, 4], [-41, -2, 7, -2, 4, 1, -4, -3], [-12, -8, -6, -1, -6, 6, 1, 2], [5, 0, -1, -4, 7, 4, 5, 3], [16, 1, 3, 2, 3, -5, -1, -4], [-17, -14, -3, 3, -8, 2, -4, 1], [17, 9, -3, -1, 8, -14, -7, 12], [2, -6, -1, 5, -6, -2, -5, -14], [-34, 1, 7, -1, 0, 13, 4, -13], [-1, -6, 0, -8, -9, 18, -3, 2], [4, 1, 6, 6, -8, -17, 3, -4], [-3, -15, -14, 6, -9, -5, -9, 10], [10, 8, 3, 1, 15, -21, -6, 7], [25, -3, -5, 1, 8, 5, 1, -2], [-3, 7, -8, 7, 5, -8, -11, 4], [24, 8, -1, 6, 8, 4, 4, -9], [2, -14, -2, 5, -4, -7, 10, -1], [-15, 14, 9, -8, 14, -12, 3, 6], [-1, 5, -9, 0, -11, -12, 12, 17], [13, 1, 8, 7, 11, 1, -6, -2], [-9, -5, -5, 3, -10, 8, 7, -3], [-21, -7, 1, 1, 0, 5, -5, -8], [20, -23, -5, 7, 0, 18, -11, -16], [9, -6, -11, 4, 2, 9, 1, -5], [10, 0, 11, 5, 5, -7, -15, -9], [6, -9, -7, -1, -5, -1, 8, 0], [-17, -9, 4, 5, -11, -13, 3, 6], [-31, 5, 1, -5, -13, 5, 3, -10], [-13, 8, 2, 2, 22, 1, 4, -5], [2, -3, -8, 6, -2, -7, -1, 7], [-23, 10, 4, -1, -13, -4, 7, -1], [28, -6, -8, 8, -4, -12, 10, 3], [-7, 8, -7, -12, -5, 4, 2, 10], [-1, 0, -6, 13, 4, -7, -6, 1], [14, 4, -1, -1, 14, 5, -1, -13], [6, 0, -12, 2, 2, -3, 10, 9], [-23, 2, -9, 1, -5, -1, 9, 25], [-16, 23, 1, -1, 3, 0, 3, 13], [-1, -7, 0, 10, -6, -6, -3, -1], [21, 8, -1, 6, 14, -11, 5, -10], [15, -12, 5, -3, 10, 8, -17, -12], [-22, -6, -5, 6, 0, 5, -12, 6], [11, 5, 7, 12, 10, -2, -9, -15], [-12, -6, -6, -8, -4, 8, -2, 9], [3, 7, 8, -8, 11, 8, -1, -10], [30, -5, 3, -6, -4, 10, -5, 0], [14, 15, -5, -1, 4, -4, -3, -14], [17, -5, -2, 7, 18, -11, -3, 11], [-15, -11, -1, -13, 0, 5, -8, 15], [-25, -6, -1, 10, -2, 1, 4, 14], [-12, -11, 3, -8, -14, 13, -9, 5], [-7, 16, 2, -6, 2, -16, 10, 7], [3, -4, 6, -8, -7, 5, 1, -5], [6, 12, -3, 2, 0, -6, -1, -12], [21, -16, -11, 0, -5, 4, -8, -11], [10, -15, -5, -2, -10, 2, -16, -4], [10, 0, -2, -12, 9, -3, 2, 6], [11, 2, -4, -9, -2, -4, -2, 4], [0, 6, -5, 9, 2, -3, -4, 5], [-26, 0, -2, 2, -5, -12, -3, -6], [-1, -10, -5, 6, -2, 0, -16, 8], [0, 10, -4, -3, 1, 4, 3, -9], [-2, -1, 5, 1, -17, -12, -5, -2], [6, 0, -2, -8, -6, 0, -1, -16], [-1, -9, 2, 6, 9, -1, -6, 5], [3, 13, 4, -2, 0, -8, 18, 5], [-20, -23, 0, 1, 1, 3, -6, 2], [13, -6, -8, 7, -11, -10, 0, 1], [-20, -20, 9, 1, 1, 9, -13, -9], [-3, 2, -6, -1, -7, 3, 7, -2], [-8, 6, 10, -6, -15, 3, -4, -5], [-1, -8, -1, -15, -19, 15, 1, 8], [-9, 12, -11, -7, -11, -15, 3, 23], [-8, 12, 9, -9, -3, -1, 2, 13], [26, -16, -2, 8, -4, -6, -3, -16], [4, -23, 2, -1, -16, 10, -3, -1], [0, 4, 4, -3, -3, -10, -8, 3], [3, -5, 9, -1, 13, -2, -13, -22], [6, 10, 9, 1, -12, -11, 11, -8], [15, 10, -16, 4, 5, -7, 13, 12], [-22, 0, 11, 3, -13, 8, -4, -9], [18, 4, 17, 1, 27, 6, -8, -16], [-26, 1, -11, -14, -11, 0, 7, 10], [-12, 13, 4, -9, 1, 1, 14, -6], [5, -2, -7, 8, -10, -3, -16, 11], [17, 0, -2, 4, -6, 1, 6, 7], [2, 10, -7, 0, 3, 9, -11, -6], [3, -13, -4, -3, 7, -2, 6, 6], [-21, -9, -6, -14, -5, 11, -10, 5], [5, -11, 3, 16, -3, -5, -2, -2], [-5, 17, 14, -4, 11, -4, 6, -1], [27, -4, 6, 11, 18, 3, -5, -10], [-2, -3, 2, -1, 4, -6, 9, -6], [33, -10, -6, 3, -10, -14, 8, 17], [-6, 16, -2, 9, 4, -3, 6, -3], [-7, -12, -3, -11, -5, -2, 5, 15], [-11, 5, -16, -6, -9, -1, 13, 14], [-5, -13, -9, 7, -12, -4, -2, 13], [-24, 0, -8, 9, -8, -9, -7, 6], [8, -13, -11, 3, -14, -15, -14, 13], [-2, 5, -4, -4, 0, 1, 16, 17], [7, 10, -3, 5, -2, -8, -10, -6], [-38, -6, 3, -4, -12, 0, -9, 11], [8, 7, -6, -9, 4, -8, 3, -2], [29, 4, -1, 6, 3, -10, -6, -10], [-3, -2, -2, 12, 0, -11, -3, -15], [-16, -11, -11, 10, -1, 4, -6, 13], [-19, 11, -8, -7, -6, -5, 8, -1], [-6, -2, -3, 2, -7, 3, -13, -18], [10, 4, 7, -5, 6, 4, 6, 6], [5, -7, 6, -8, 8, 6, -2, 3], [-20, 4, -7, 5, 3, 7, -8, -1], [5, 22, -7, 6, 3, -7, -4, 11], [8, 1, 7, -5, 17, 7, -3, 2], [-25, 5, 7, -6, 4, 7, 1, -5], [15, -4, 6, 3, 8, -14, 7, 5], [13, 6, -1, 13, 24, -2, 0, -10], [-10, -9, -6, -1, 6, 4, 12, 12], [2, -1, 18, -1, 4, 8, 7, 5], [3, 15, -1, -8, 3, 12, 2, -24], [-39, 11, 8, -10, 4, 19, -8, -2], [-30, -11, -2, -6, 2, 11, -2, 9], [-18, 4, -2, -4, 6, -3, 0, -3], [5, 21, 5, 10, 13, -9, 1, -3], [0, -3, 2, 2, 11, -6, -1, -5], [-10, -4, 9, -4, -11, 3, 1, 11], [-1, -18, -11, -4, -11, -5, -7, 6], [-12, 9, -2, -4, -6, -7, 3, -12], [1, 1, -10, -3, -5, -12, 1, 0], [18, 4, 12, 11, 27, 0, -8, -21], [17, -4, -12, -4, 8, 5, 1, -10], [5, -12, 2, -9, -3, -4, 8, 7], [-12, -14, 2, -3, 9, 0, 1, -1], [-11, -2, -17, 3, -9, -3, 4, -2], [13, 4, 12, -8, 18, -1, -4, -7], [34, -13, 4, -11, -3, 13, -1, -13], [19, 17, 7, -10, -4, -3, 1, -6], [-19, 0, 0, 4, -13, 1, 14, -14], [17, -5, -8, -3, -1, -11, 15, 1], [-8, -7, 3, -1, 8, 10, -4, -7], [18, 9, 11, 10, 15, -6, 10, 0], [-22, -10, 10, 5, -6, -15, -5, 12], [19, 19, 18, -10, 3, -5, 6, -8], [2, 1, -13, 4, -8, 2, -4, 7], [10, -2, 4, 7, -9, -14, -5, -14], [-22, -10, -11, 1, -26, 4, 4, 7], [-23, -2, -6, -4, -3, 14, 9, -3], [22, 13, 12, 2, 1, 4, 12, 4], [-21, 0, -16, -10, -16, -1, 11, 8], [-10, -6, -4, -3, -7, -4, 10, 0], [6, -20, -8, -5, -13, 0, -6, 4], [-15, 9, -2, 6, 11, -10, 8, 9], [-5, 1, 4, 3, 0, 17, -2, 19], [-8, -10, 1, 2, -6, 2, -13, -20], [41, -15, -4, -8, -4, 2, -6, -7], [-17, 14, 0, -16, 6, 8, -2, 1], [17, -12, -9, 9, 1, -10, -3, 13], [-5, -12, -9, 6, -2, -3, -6, 12], [-23, 13, 21, -21, 7, 18, -1, -13], [-30, 6, -5, -3, 11, 14, 1, -1], [-3, 14, -13, -1, -4, 0, -2, -2], [-20, -20, -8, -5, -12, 5, 12, 11], [-3, 1, -8, 11, 0, -10, 4, 5], [14, -29, 0, 8, -4, 15, -17, 7], [-14, 20, -4, -15, -11, 5, 6, -2], [9, -12, -7, -1, -12, 8, -10, -14], [-20, -4, 6, -3, -3, -1, -5, 13], [-15, -9, -19, 8, -12, -2, -2, 22], [17, -5, 4, 16, 0, -10, 0, -4], [-27, 9, 1, 4, -12, -4, 15, -2], [13, -4, -12, 14, -10, -9, -4, 7], [-34, 22, -2, -6, -6, -20, 9, 17], [-21, -14, 11, 3, 9, 14, 1, -17], [22, -11, 10, -1, -3, 6, -4, 1], [7, 10, 15, -14, 15, 13, -8, -11], [-12, -1, 9, -12, 0, 13, 9, -9], [44, 7, -12, 8, 1, -6, 3, -1], [30, 0, -5, 4, 18, 8, 7, -22], [-15, 7, 5, -11, 12, 15, -3, 3], [-25, 2, 13, -4, 6, 11, -13, -7], [-16, 2, 7, 14, -1, 13, -7, -9], [41, -2, 1, 18, 3, 0, -2, -3], [-1, 5, 3, -11, -9, 8, 1, 19], [14, -20, -14, 6, -10, 18, -7, -2], [-22, 11, 4, -5, 2, 9, 21, -10], [-14, 13, -12, -5, -17, -15, 12, 15], [21, -13, 2, 4, 2, 12, -5, -3], [-3, 2, 23, 5, 4, 8, 4, 11], [23, -10, 11, 0, 8, 2, -1, -11], [-22, 14, 3, -3, -11, -16, 3, 11], [-14, -9, 19, 4, 6, 11, 1, -11], [5, 15, 3, -5, 12, -18, 20, -3], [19, -12, -8, -9, -22, 7, -10, 1], [13, 13, 13, -5, 15, -14, 18, -3], [10, -5, 1, -12, -11, -2, 5, 12], [-28, 12, 6, -3, 4, 9, 3, 11], [-1, -5, 6, -15, -12, 6, -10, -3], [13, 6, -14, 13, -3, 1, -14, 3], [-7, 11, 8, -14, 7, 9, -13, 4], [-21, 0, 7, -17, -14, 1, -1, 17], [6, -1, 12, -1, 2, -2, -14, 0], [9, 15, 4, -9, 2, -23, 14, 5], [13, -14, -12, 18, 19, 0, -4, 1], [-41, 1, 0, -1, -14, -6, -4, -5], [7, 14, -3, 5, 7, 12, 3, -15], [-25, 18, 10, -10, 19, 14, 0, -1], [-16, -18, -23, 6, -18, 10, 4, 3], [8, -1, -13, 3, -7, -23, -3, 6], [-4, -11, 7, 18, 1, 0, -13, -6], [-1, 15, -9, 1, 3, -2, -1, -18], [-11, -4, -2, -12, 2, 4, 0, 5], [21, 5, 3, -14, 2, 10, -16, -15], [-10, 8, 5, 24, -4, -2, 4, -8], [-16, -15, 1, 0, -5, 4, -17, 11]];
    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_bf();".
// 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_bf(: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_bf();".
// 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_bf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5265_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -19, 66, -70, -1, 31, -8, -3, 1]>,<7,R![-629, -30, 1360, 232, -385, -106, 24, 11, 1]>,<11,R![634, 11999, -1002, -3220, 413, 250, -38, -6, 1]>,<17,R![892, 1275, -5408, 748, 1532, -73, -78, 2, 1]>],Snew);
    return Vf;
end function;