
// Make newform 8040.2.a.t in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8040_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_8040_2_a_t();".
// 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_8040_2_a_t();".
// 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 := [6, -29, -6, 43, 3, -15, -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], [-4, -1, 1, 0, 0, 0, 0], [112, 143, 42, -43, -73, -4, 6], [39, -484, 104, 234, -76, -23, 7], [-123, -242, 497, 197, -213, -29, 16], [267, -132, -773, 62, 347, 16, -24]];
    Rf_basisdens := [1, 1, 1, 55, 55, 55, 55];
    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_8040_a();"
function MakeCharacter_8040_a()
    N := 8040;
    order := 1;
    char_gens := [6031, 4021, 2681, 3217, 5161];
    v := [1, 1, 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_8040_a_Hecke(Kf)
    return MakeCharacter_8040_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, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 0, 0], [0, -1, 0, 0, 0, 0, -1], [0, -1, 0, 1, 1, 0, 1], [0, 0, -1, 0, 0, 0, 0], [1, 1, 0, 1, 1, 0, 0], [1, 0, -1, 0, 1, -1, -1], [0, 0, 0, 0, 0, -1, -1], [1, 0, 0, 0, -1, 0, 0], [2, 1, 1, 0, -1, 2, 0], [1, 0, 0, 0, 1, 0, 0], [-1, -1, 1, 1, 0, 1, 2], [2, -1, 0, -1, -1, 0, 1], [3, -1, 0, -1, 1, -2, -2], [0, -1, 0, -1, -1, 2, 1], [0, 2, 0, 1, 1, 1, 1], [1, 0, 0, 0, 0, 0, 0], [0, 2, 0, 1, 0, -1, 2], [1, 3, 0, 1, -1, -1, -1], [4, -2, 0, -2, -1, 2, 1], [-5, 1, 1, 2, 1, 0, 1], [2, -4, 0, -1, 0, 1, 2], [6, 0, 0, 1, 0, -4, -1], [-1, 0, 0, -1, -2, 3, -1], [6, 0, 0, -4, 1, 0, -1], [-6, 4, 0, -2, -3, 1, -2], [5, -1, 1, -2, 1, 2, 1], [-1, 2, 2, 1, -2, 0, 0], [3, -2, -1, -2, -2, 2, -3], [3, 3, 0, -1, 1, -5, -1], [2, 0, 2, 1, 0, 0, 3], [5, 0, 1, 0, -1, -2, 0], [-1, -3, 0, -1, -3, 2, -2], [-1, 2, -1, 1, 1, -2, -3], [7, 5, -1, 1, 1, -2, -2], [8, 1, -1, -2, 4, -3, 2], [-1, 1, 1, -1, -2, 2, -1], [7, -3, 1, -4, 0, 2, -2], [-2, 1, 1, 3, -1, 3, 4], [-4, 4, 0, 2, -2, 1, 3], [3, 1, -2, -1, -1, 1, 1], [2, 2, -1, -2, -3, 0, -1], [7, 0, 0, 0, 2, -5, 0], [11, 4, -2, -1, 2, -2, 0], [-2, -2, -1, 4, 5, 1, 4], [5, 0, 2, -2, -1, 0, 2], [-2, -7, 2, -3, 0, 3, 1], [-3, -1, -1, 4, 1, 1, 2], [1, 3, 0, -4, 1, 3, 2], [5, 1, 0, -1, 1, -4, 0], [2, -3, -4, -2, 1, -3, -3], [-2, -4, 1, -1, -4, 1, -2], [0, -5, 0, 3, 2, 2, 2], [0, 3, 1, 3, 4, -1, 5], [0, 1, 1, 3, 4, -7, 3], [2, -1, 2, 1, -4, 2, -2], [6, -6, -1, 1, 2, 1, 2], [-2, -1, -1, 1, 3, -1, 2], [-1, 1, -2, 0, -3, 0, 1], [-4, -5, 2, 0, -1, -1, -1], [-1, 1, -2, 2, 3, -1, -4], [1, 5, 3, -1, -4, 7, 0], [3, -5, -1, -4, -2, 4, 2], [-3, 9, -1, 3, -2, -3, -2], [1, -4, 0, 1, 0, 3, 3], [0, 4, 2, 0, 0, 3, 1], [-11, 6, 1, -2, 1, -1, -1], [4, -2, -1, 0, 1, 1, -4], [3, 3, -1, -3, 2, 2, -3], [9, 2, -3, -3, 3, -2, -3], [8, 6, 1, 0, -1, -3, 4], [5, -7, -1, -5, -2, 2, -1], [11, -1, 2, -5, -3, 3, -5], [4, -3, 0, -2, -3, -3, -5], [-5, 0, 0, 0, 0, 1, 2], [4, 4, -1, -5, -2, 2, -7], [0, -2, -2, 2, -3, 2, -3], [7, 2, -1, -2, -1, -3, -5], [10, -8, 1, -2, 3, -1, 0], [-10, -1, 0, 3, 1, -3, 2], [6, -6, 2, 1, 0, 0, 7], [3, 8, 0, 0, 0, -8, -1], [19, -4, 0, 1, -2, -2, 2], [-7, -2, -2, 4, 0, -6, -3], [-2, 1, 1, 0, -4, 5, 4], [-2, -1, 0, -1, -5, 3, -2], [-1, -2, 4, -4, -4, 3, 4], [13, -5, 0, -5, -1, 0, -4], [4, -7, 1, 2, -1, -1, -1], [-1, 6, -1, 3, 7, -1, 0], [13, -2, 1, 3, -1, 6, 3], [-7, 1, 1, 7, 4, -2, 7], [12, 6, -3, 1, -3, 1, 1], [8, 7, -2, 2, -1, -4, -2], [1, -2, 4, -4, 0, -2, -1], [-9, 4, 0, 6, 3, -5, -3], [8, 7, 0, 0, 1, 2, -2], [-10, 6, -1, -2, -4, 2, 0], [-1, 2, -2, 0, 8, -1, 4], [5, 6, -3, 2, 5, -1, -3], [3, -12, 0, 0, -2, 2, -5], [-1, 11, 0, 0, -3, -2, -1], [-2, 3, -1, -3, -2, 0, -6], [-5, 2, 6, -2, -6, 9, 2], [3, -4, -4, -4, 2, -3, 6], [-2, 1, -1, -2, -4, 4, -3], [12, 1, 0, -3, -1, 6, 1], [-24, 9, 4, 0, 1, 2, 0], [-7, 4, 3, 4, 0, 3, -2], [-1, -2, -4, 2, 7, -5, 3], [8, -5, 1, -1, -1, -4, -1], [16, -1, 3, -5, 2, -2, 4], [7, 0, -3, 2, 6, 0, 1], [-11, 2, -2, 2, -3, -3, -3], [10, 6, -1, -8, -1, -1, 0], [14, 0, 3, 7, 5, 4, 10], [-8, -1, -1, 7, -2, -5, -1], [-1, 7, 1, 5, 0, -4, -5], [-6, -3, 2, -3, 0, -1, -5], [-8, -2, 2, 2, -2, 9, 9], [-1, 4, 0, 0, -2, -1, 4], [-8, 2, -2, 0, 4, 5, 7], [-8, -2, -5, -2, 3, -2, -13], [-5, 2, 0, -5, -6, 8, 2], [6, -3, -4, -3, 1, -9, -4], [2, -2, -1, 4, -5, 2, -1], [15, 2, 2, 0, 4, -5, 4], [-5, 8, 2, 6, 0, -6, -3], [5, 4, 2, 2, 0, 3, 10], [3, -6, 0, -6, -8, 1, 0], [0, 3, 5, 3, -2, -3, -3], [-12, -2, -1, 0, -5, 4, 1], [-21, -4, 2, 5, 2, 1, -1], [-10, 4, -4, 6, 0, -5, -3], [-14, 2, 2, -3, -6, 6, 1], [-7, 6, -4, 0, -4, -6, -5], [-6, -3, 3, 2, -6, 7, 6], [0, 5, -1, -1, -3, -1, -2], [12, -6, 3, -5, -5, 2, 6], [-14, 6, 2, -4, -4, 8, 6], [0, 9, 2, 11, 2, -5, -5], [-18, -4, -1, -5, -4, 0, 1], [-7, -12, 0, -4, -2, -1, 0], [14, -7, 0, 1, 1, -1, 8], [5, 1, -3, -7, 3, 1, -3], [19, -15, -1, 0, 4, -2, 6], [18, 8, -2, -2, 2, -4, 0], [13, -6, -4, -6, 10, -9, -2], [-4, -9, -1, 5, 3, 0, 5], [-6, -7, -3, 0, 8, -6, 1], [0, 3, -3, -7, -2, -1, -5], [-3, 7, 0, -1, 5, -1, 1], [-14, 8, 5, -3, -3, 6, 6], [-17, -6, 0, -7, -2, -1, -3], [7, -5, 0, 3, 0, 3, 2], [-3, 2, 0, 0, -3, -8, -6], [5, -1, -1, -6, 2, -6, -8], [-1, 11, -2, 3, 7, -9, -5], [-8, 3, 8, 4, 0, 1, 8], [0, -1, 1, 1, 1, 2, 9], [16, -11, -2, -5, 0, 5, 3], [-11, -3, -2, 1, -1, 9, -3], [-6, 7, 1, 1, 3, 7, 6], [6, 3, 0, 3, 1, 0, -3], [-10, -10, -4, -14, 3, 4, -3], [-4, 2, -2, -3, 7, 4, 6], [-5, 0, 4, 1, 0, 12, 6], [-4, 3, 7, 3, -2, 6, -4], [-1, -3, 5, -3, -2, -3, -4], [-11, -8, -1, -6, -1, -1, 1], [-11, 3, -1, -5, 4, -12, -3], [3, -5, 5, -2, 0, 9, 5], [18, -1, 1, -3, -1, 8, 1], [4, -2, -6, -2, 5, -10, -7], [-12, 4, 2, 1, 6, -3, 6], [15, 1, -2, 5, 5, -7, -9], [30, 1, 1, 2, 4, -4, -5], [-15, -1, -2, -3, 5, 2, 0], [-3, -12, 1, -4, -4, 2, 9], [-6, -4, -3, -12, 1, 2, -1], [22, -2, 0, 0, 5, -2, 1], [-13, -5, 0, 1, 0, 6, 1], [-4, 8, 0, -1, 2, 3, -6], [1, 10, 0, -4, -6, 0, 5], [9, -5, 1, -1, 2, -9, -10], [-3, 3, 0, -5, -5, -2, 4], [3, -6, -1, 4, 9, 2, 4], [-15, 2, -1, -12, -5, -4, 0], [5, 0, 2, 2, 0, -1, -8], [-19, -1, 4, 6, 0, 2, -8], [9, -4, -4, -7, -6, 0, 2], [-7, 4, 2, -3, -10, 3, -1], [2, -1, -6, -5, -3, -9, -14], [-10, -6, -3, 2, 2, 0, 4], [-17, 10, -1, -5, -1, -13, -8], [-5, 9, -1, -1, 0, 1, -10], [-19, -1, -4, 7, 1, -1, -5], [-9, -2, 1, 10, 1, 1, 7], [-9, 0, -1, -4, 3, 9, 5], [-12, -3, 7, -4, -8, 15, 6], [-14, 6, 5, 2, -13, 11, -4], [-1, -4, -2, 6, 12, -7, 8], [9, 1, 4, -5, -2, -8, -3], [2, 0, 6, -8, 0, 8, 4], [18, 6, -3, -7, -1, 2, 2], [-4, -8, 1, -1, -2, 10, 3], [-17, -2, 7, 3, -5, 7, 8], [-9, 10, -3, 1, -5, -5, -4], [11, -1, 1, -11, -6, 1, -4], [4, -1, 1, 12, 2, 1, 6], [6, -3, 7, -3, -4, 15, 7], [2, -5, -4, -15, -1, 4, -9], [9, -2, -1, -10, 1, 3, -5], [-23, -2, 2, -8, -2, 1, 0], [26, -7, -1, -3, 1, -5, 0], [6, 3, 6, 0, 3, 6, 4], [18, 4, 1, -1, 1, -3, 1], [-13, 12, 1, 5, -3, -2, -3], [-9, 10, 4, -14, -5, 7, 3], [5, 0, 3, 12, -3, 6, 14], [-1, -1, 1, -1, 13, -1, 3], [12, -1, -9, -3, 3, 4, -1], [8, 6, -1, -8, -3, 2, -17], [-15, -3, 3, 9, -4, 9, 14], [-33, 6, 0, -2, -5, 6, 10], [-13, -2, 5, 0, -3, 3, -1], [-3, 2, 2, 0, 3, 5, -1], [-18, 5, -2, 5, 6, -5, 3], [-18, 1, 2, 11, -3, 5, 12], [-1, -6, -1, 7, -5, -2, -3], [-1, -6, 1, 5, -3, 5, 0], [-19, 6, -1, 0, -12, 16, -5], [15, -4, 1, -1, -1, -1, -4], [9, -8, -4, -8, -2, 0, -3], [-9, -3, 3, 2, 0, -6, -2], [9, 5, -5, 4, 6, -14, -2], [3, -12, -6, 6, 12, -6, 3], [5, 6, -5, 2, 3, 7, 5], [-7, 5, -1, 4, 5, 8, 9], [-6, 0, -5, 1, 5, -3, 1], [3, -3, 2, 10, -9, 1, 6], [-9, 15, -1, 0, -3, 5, 8], [-12, -9, 0, -8, -1, -2, -2], [13, -2, -3, -10, -5, 7, -3], [-17, 7, 1, -1, -11, 5, -9], [-2, 5, -2, 8, 0, -14, -3], [18, -6, 3, -4, 13, -5, 2], [11, 6, 3, 15, 5, -4, 5], [6, 3, -7, -2, 3, 3, 1], [7, -4, 5, 3, 3, -12, 3], [-9, -6, 0, -1, 2, -11, 3], [1, 0, 3, -11, -11, 6, -7], [-3, -5, 4, 9, 5, -3, 7], [-11, 9, -3, 2, 9, -5, 2], [-11, 13, 2, -8, -6, -1, -7], [0, -7, 3, -8, -4, 1, -4], [25, -5, 4, -6, -3, 4, 1], [14, -13, -4, -9, 7, -6, -3], [-2, 4, -4, 10, 11, -13, -2], [-21, 1, -5, 4, -3, -8, -9], [-7, 12, -4, 3, -7, 8, -3], [6, -9, 0, -3, -3, -1, 2], [-6, 2, 4, 10, -5, 1, -4], [2, -13, 2, -1, -7, -7, -10], [-21, 7, 3, -12, 0, -2, 0], [-7, 7, -1, -5, -3, 0, 2], [20, -5, 2, -17, -1, 10, -1], [10, -11, -3, -1, -2, 12, -2], [-14, 9, 1, 4, 11, 1, -1], [-11, -1, -1, 9, 12, -2, 5], [-8, -9, -7, -6, 6, -3, -4], [-4, 1, 0, 15, 9, -2, -7], [18, -6, -3, 2, 7, -2, -5], [30, -11, -4, -2, 5, -9, -3], [-12, -9, 5, -1, -4, 16, 6], [-1, -5, 3, 3, 6, -8, 1], [12, -1, 9, -4, -6, -6, -7], [-17, -16, 3, 2, 4, -2, 5], [-11, 6, 4, -16, -10, 5, 2], [-23, 15, 6, 0, -11, 6, -9], [10, 5, 10, 0, -5, 17, 3], [-7, -5, -1, 2, -11, 0, -5], [-1, 21, -2, 3, 1, -1, 7], [-18, 8, 0, 9, 8, -10, 7], [16, -3, -1, -11, -12, 0, -12], [22, -8, -7, 1, 3, 0, 8], [7, 6, 0, 12, 15, -6, 6], [24, 6, -6, -10, 3, -8, -11], [-18, -1, -4, 9, 1, -7, -2], [14, 10, -2, 2, 12, -17, 1], [-9, -3, -2, -15, -10, 5, -4], [-7, 5, -1, -7, 0, -1, 2], [20, 2, 4, 6, -2, -15, -7], [-14, 1, -1, 1, -6, -13, -13], [-12, 11, 5, -11, -9, 6, 5], [-8, 0, -5, -6, 9, 0, -9], [4, -4, -4, 0, 8, -8, 0], [3, -3, 6, 1, -3, -1, -13], [-8, -11, 8, -2, 3, 16, 8], [2, -11, -3, 0, -8, -11, -8], [22, 1, -4, 6, 7, 1, 1], [-6, 6, -5, -7, -4, 4, -15], [-3, 4, 7, 12, 9, -7, 5], [13, -5, -6, -11, 3, 0, 2], [-18, 2, -2, 4, -5, -6, 1], [-16, -6, 0, -6, 4, -2, -4], [-10, 18, -1, 6, 5, 3, 22], [-10, -12, 1, -2, -15, 11, -12], [-33, -8, 4, 7, -14, 12, -6], [0, -8, 0, 10, 14, 3, 13], [-10, -4, 2, -7, -10, 3, -4], [-13, 0, 3, -1, -9, 13, 6], [7, -16, -1, 4, 5, -3, -5], [-22, 11, 0, -3, 5, -5, -2], [-9, -6, 2, 2, 8, 2, -7], [-4, -3, 3, -11, 6, -11, -7], [-9, 1, 0, -15, -9, -4, -8], [-21, 1, -3, 15, 7, -6, -6], [5, 9, 5, 5, -6, -3, -10], [-4, -8, -1, 4, -3, 1, -2], [0, -5, -4, 1, 15, -14, -1], [7, -4, 0, 3, 0, -5, -9], [14, -8, -1, -14, -6, 3, 5], [-15, -3, -1, -4, 0, 5, -5], [-10, 3, 3, -3, 2, 6, 10], [8, 8, 0, -3, -4, 1, -8], [6, -17, 1, 8, 6, 1, 10], [-23, 5, -6, 11, -3, -1, 9], [-18, -4, -7, 6, 13, -7, 14], [1, -2, 5, -3, 1, -11, 4], [2, -3, -6, -1, 10, -8, -6], [-34, 11, 10, 3, -5, 12, 13], [-25, -1, 1, -1, -8, 9, 12], [-3, 4, -2, 7, -3, -1, 10], [1, -3, 11, -7, -1, 12, 8], [-18, -2, -3, -20, -4, 6, -2], [-39, 5, -6, 3, -7, -4, -14], [25, -5, -2, 15, 10, -8, -13], [15, 13, 0, 3, -8, -3, 0], [12, 12, 2, -4, -7, 11, 2], [1, -1, 5, 10, -2, 11, 7], [1, 5, 2, 10, -5, -10, -3], [12, 9, 0, -2, -1, -5, -1], [3, -5, 11, -13, -11, 9, 7], [-5, 8, -5, -6, 2, 3, -2], [-11, 7, -4, -9, -9, 8, 6], [-31, 6, 3, -2, -3, 14, 10], [0, -7, 7, -5, -4, 2, -8], [-34, -3, 11, 5, -4, 10, 10], [-1, 2, -7, 9, 5, 2, 9], [12, -2, -5, 6, 3, 7, 4], [15, 10, -3, -10, -7, -1, -7], [-18, -8, -1, -12, 2, -8, -8], [-26, 16, 4, -9, -4, 1, 6], [1, -1, -2, -8, -7, 4, 7], [-9, 5, -1, -4, -11, 12, -5], [24, 3, 1, -6, -8, -2, -9], [3, 11, -3, -12, 2, -14, -2], [-8, -19, 4, -12, -4, 8, -1], [11, -15, 6, -6, -3, 11, 8], [-9, -8, 4, -2, -6, 16, 7], [-9, -4, -10, 2, 4, -11, -10], [-14, -2, -7, -1, 1, 14, -2], [21, -11, -6, -2, 11, -1, 10], [7, 11, -8, 0, -1, -3, -8], [12, -2, 1, 4, 10, 0, 2], [-1, -2, -7, -6, -3, -12, -8], [-24, -9, 4, -14, -6, 10, 5], [5, 1, 1, -7, 4, -14, -3], [-6, 20, 2, 8, -6, -3, -3], [-6, 12, 1, 2, 17, 2, 3], [-13, -1, 0, -5, -2, -4, 17], [3, 3, -2, -7, -7, -11, -11], [-7, 10, 3, 10, -3, 2, 18], [-39, -4, -7, 2, 7, 2, -10], [37, -17, 2, -3, 7, -1, 11], [-16, 5, 3, -5, -13, 0, -3], [13, -3, -2, -3, 15, -8, 2], [24, 3, 0, -18, -3, -2, -14], [-6, 2, 11, -1, -7, 13, 19], [5, 8, -3, -16, -7, -6, -14], [25, 2, -2, 5, 10, -6, -12], [-29, 10, -3, 7, -3, 2, 7], [25, 11, -7, -8, 2, -16, -14], [6, -6, -3, -4, -10, -5, -17], [-22, -2, 8, 17, 5, -7, 7], [-10, -4, 3, -2, 7, 5, 6], [13, -3, -8, 14, 11, -13, -2], [-4, -12, 4, 10, -4, 17, 3], [2, 7, 3, 1, -14, -3, -11], [4, 10, 9, 0, -17, 5, -8], [-14, -5, 1, 13, 4, 10, 8], [8, 19, -7, 5, 6, -15, 3], [-24, 7, -1, 7, 16, -3, 9], [25, -14, 5, -9, 5, 5, 12], [22, -9, -3, -1, 8, -14, 8], [-20, 0, -1, -1, 1, -7, 5], [25, 9, -1, 3, 4, -15, -4], [-41, -9, 3, 7, 3, 3, -7], [-31, -3, 6, -9, -8, 1, -2], [14, 6, 10, -1, -4, -8, 5], [26, -8, -6, -7, 0, -2, -3], [15, 2, -6, 3, 6, -11, -1], [10, -12, 3, 10, 2, -11, 5], [3, 9, 5, 11, 6, 8, 15], [4, -19, 2, 7, -3, -4, -3], [-12, -5, 0, 6, -3, 5, 15], [-8, 12, 3, 5, 2, -3, 0], [10, -1, 4, -5, 3, -1, -10], [23, 4, -5, -1, -1, -2, -11], [6, -1, 6, -12, 0, 6, -1], [7, 8, -10, 6, 16, -16, -7], [29, 10, -6, -17, 0, -1, -7], [5, -13, -3, 1, 6, 5, 4], [29, -1, -2, -9, -1, -7, -23], [-25, 17, 2, 3, -4, -2, -5], [25, 6, 7, 4, 2, 7, 0], [-6, -21, 0, -15, -3, 9, 6], [-19, 20, -4, 15, 2, -10, 0], [-7, -4, -5, 1, -6, -1, 13], [-2, 14, 7, 12, 1, 7, 18], [30, 4, -5, 3, -4, 2, -1], [40, 9, 5, 16, 8, 14, 17], [32, 10, -3, 4, -5, 4, 9], [10, -11, 5, -6, -1, 12, 10], [-1, 21, -3, 15, -4, 3, 12], [8, -10, 7, -16, -13, 6, -7], [3, 0, -1, -18, -5, 2, -8], [-25, 19, -5, 5, -8, 4, 9], [13, 16, -4, 16, 9, -8, -2], [7, -1, -6, -11, 3, -15, -3], [33, 3, -4, 1, 5, 1, -1], [12, 3, 0, 10, 6, -5, 4], [7, 6, -6, 4, 6, -5, 4], [-19, -4, 1, -14, -9, -2, 8], [15, -8, 5, 11, -2, 16, 22], [-12, 10, -4, 6, 8, -15, 1], [55, -5, 8, -4, 5, -3, 20], [0, 1, -4, 18, 18, -4, 1], [35, 5, 4, 7, 8, -3, 12], [5, -12, -9, -9, 10, -9, -9], [-7, 9, 9, -4, -16, 6, -4], [7, -9, -2, 1, 1, 9, 3], [23, -10, -3, 8, 16, -22, 3], [-15, 3, 6, -5, -7, 1, 7], [19, 14, 7, 15, -7, 4, 15], [37, 5, -6, -7, -11, -4, -4], [-32, -9, 1, 5, -5, 11, 0], [38, -6, -1, -2, -5, 4, -13], [36, -20, -1, 2, 5, -2, 3], [-2, -17, -5, 12, 5, -8, -4], [34, -1, -7, -5, 0, -1, -7], [16, 31, -3, 9, -8, -7, 3], [30, -8, 4, -5, -6, 5, -16], [-4, 5, 10, 2, -10, 1, 4], [-4, 7, 3, 3, 0, -3, -9], [14, 4, -14, -5, -3, -19, -9], [13, 7, -5, -3, 10, -12, 1], [3, 12, -5, 4, 13, -17, -3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8040_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8040_2_a_t();".
// 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_8040_2_a_t(:prec:=7)
    chi := MakeCharacter_8040_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(3259) - 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_8040_2_a_t();".
// 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_8040_2_a_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8040_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![8, 52, 17, -223, 74, 19, -10, 1]>,<11,R![1530, -2000, -377, 525, 18, -41, 0, 1]>,<13,R![-10152, -5580, 1494, 971, -69, -55, 1, 1]>,<17,R![3332, 1834, -4766, 1653, 51, -76, 2, 1]>],Snew);
    return Vf;
end function;