
// Make newform 8470.2.a.df in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8470_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_8470_2_a_df();".
// 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_8470_2_a_df();".
// 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 := [-2, 22, 31, -4, -11, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [-2, -6, 0, 1, 0, 0], [-4, 6, 7, -1, -1, 0], [-4, -8, -5, 1, 1, 0], [-8, -2, 7, 1, -1, 0], [6, 32, 3, -17, -1, 2]];
    Rf_basisdens := [1, 1, 2, 2, 2, 2];
    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_8470_a();"
function MakeCharacter_8470_a()
    N := 8470;
    order := 1;
    char_gens := [6777, 6051, 7141];
    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_8470_a_Hecke(Kf)
    return MakeCharacter_8470_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 := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1], [0, 2, 0, -1, 0, 0], [0, 1, 0, 0, 1, 0], [1, -1, 0, 1, 0, -1], [1, 1, 1, -1, -1, -1], [1, -1, 0, 1, 0, 0], [1, -1, -1, 1, 1, 1], [2, 0, 1, 0, 1, 1], [-1, 1, 1, 0, 1, 1], [2, 2, -1, -2, -1, 1], [2, 0, 2, 0, -1, 0], [4, -2, 0, 1, 0, 1], [0, -2, -1, 2, -2, -1], [4, 2, 0, 2, 1, 0], [3, 3, 2, 2, -2, 0], [3, -1, 1, 0, -1, 1], [4, 0, -2, -4, 1, 1], [-2, -4, -1, 2, -1, 0], [0, 0, -1, -2, 3, 3], [1, -1, 2, 1, -3, 0], [-1, -2, -1, -3, 1, -1], [3, 1, 1, -5, -2, -1], [-2, -2, 1, -1, 1, -1], [1, 1, 2, -3, 2, 2], [3, -1, -5, -1, 3, 2], [-6, 1, -1, -4, -1, 1], [0, -3, -1, 2, 3, -1], [5, 3, 3, -1, 1, 0], [4, 3, 4, -4, -3, 0], [-3, -1, -3, 3, -3, -1], [-2, 0, 3, -2, -3, 0], [2, -1, -2, -4, -2, -2], [0, 4, 3, -2, 2, -1], [0, 4, -4, -1, 4, 2], [2, 0, -2, 8, 0, -2], [4, 0, -3, -2, 1, -3], [10, 0, -1, 5, 1, 2], [-1, 0, -4, -3, 0, -2], [-7, 6, -1, -3, -1, -1], [-8, -2, 3, -5, -1, 1], [-5, 1, 5, -1, -3, -3], [3, 1, -5, 5, -1, -1], [7, -1, -7, -1, 0, -1], [0, 0, 4, -6, 2, 2], [-3, 3, -2, -6, 0, -2], [-7, -4, 1, 3, 3, -1], [-6, 4, -5, -6, 1, -2], [-9, 3, 3, -5, 1, -3], [-7, 1, 8, -2, -2, 0], [7, -3, 1, -1, 2, 3], [-8, 5, 3, -2, -1, 1], [-4, 0, -1, -3, 5, 2], [-2, -4, 4, 0, 0, -2], [0, -2, -9, -2, 1, -1], [-8, 6, -3, -8, 0, 2], [2, 2, -7, -4, -1, 0], [4, 2, 6, 6, -2, -1], [-10, -4, 4, 0, -4, 0], [4, 4, -5, 2, -1, 1], [2, 2, -2, -2, 6, 4], [1, -9, 3, 11, 0, -1], [-2, 0, 1, 4, 7, 3], [9, -2, 9, 3, 1, 1], [-7, -9, -2, -2, 0, 0], [1, 2, -1, 3, 5, 1], [-7, -3, -1, 1, -2, 3], [4, -4, 1, -4, 0, 1], [0, 0, 4, 4, -1, -2], [-16, -2, 1, -1, 1, 1], [-11, -1, -1, -5, -5, 1], [5, -3, -1, -7, 0, -1], [9, -1, 2, 3, -4, -6], [0, -8, 4, -6, -2, 0], [10, 0, 0, 4, 0, -2], [-3, 5, -1, -7, -3, -3], [-12, -4, 4, -3, 4, -1], [2, -2, 0, 6, 4, 0], [4, 0, 2, 4, -7, -1], [11, -3, 12, 3, 0, 2], [17, -5, 0, -1, 0, 0], [18, -4, 2, -2, 0, 0], [-7, 0, -2, -6, -2, 0], [-9, -4, -1, 3, -5, -3], [-12, 6, -5, -2, 4, 3], [-13, -7, 6, -1, -2, -1], [3, 0, 4, -7, 0, 0], [8, 0, -7, 2, 1, -3], [-5, -7, -8, 1, 4, 5], [6, -2, 5, 6, 3, 5], [1, -1, -5, 1, -7, -3], [7, -3, -5, 4, 1, -3], [16, -8, 1, -1, -5, -4], [1, -3, 9, 1, 1, 3], [8, 2, 7, 8, 3, 4], [-10, 4, -2, -8, -2, 6], [3, -1, -11, 0, 5, 3], [-10, 6, 3, 4, -3, -1], [-2, -8, 11, -4, -5, 0], [0, -4, 5, 12, 2, 4], [12, -10, 2, 6, 2, 0], [-4, 4, -6, -12, 3, -2], [-3, 8, 0, -7, 4, -2], [-1, 1, -14, 0, 2, -2], [0, 3, 6, -1, 2, 2], [5, -1, 6, 11, 0, 2], [7, -5, -4, 4, 0, 0], [0, 8, -8, 5, 2, 2], [-10, -2, 4, -8, 4, -2], [-5, -3, 6, -1, -4, 5], [-5, -7, -8, 0, 4, 0], [1, 8, -8, 4, 4, 2], [-2, 8, 0, -2, -4, -4], [-11, 13, -7, -7, 0, 1], [15, 5, 1, -7, 0, 3], [18, 4, 3, 10, 1, -3], [-1, 7, 9, 1, -3, 0], [-3, 2, 3, 15, 5, 7], [6, -2, -4, -4, -4, 1], [1, -7, -8, 3, 5, 0], [-15, -1, -8, -8, -2, 0], [-3, 5, 8, 1, -8, -4], [1, -7, -15, 1, 5, 1], [8, 0, -6, -6, 6, 2], [1, -1, 9, -11, -3, 1], [-6, -8, -10, 10, 0, 5], [18, 2, -3, 8, 1, -3], [-10, -8, 10, 2, -4, -6], [-1, -4, 6, 1, -6, 4], [-3, 7, 1, -13, -6, 3], [-4, -10, 5, 4, -5, 3], [-19, 5, -1, 3, -3, -1], [18, 7, 0, 0, -8, 2], [8, 8, 0, -10, 4, -2], [10, -5, -10, -4, 4, 0], [2, 6, -7, 10, 6, 5], [4, -4, 8, 12, 2, 4], [-20, 2, -8, -4, 0, -4], [20, -2, 2, 2, -6, -2], [-2, 4, 0, -5, -6, -8], [-11, 7, 5, -5, -3, 0], [22, 2, 7, 4, 5, 1], [-4, -2, 6, 4, 2, -3], [0, -10, -2, -6, -2, 0], [3, 5, 4, -6, 10, 2], [12, 4, -10, -6, -2, 2], [3, 7, 3, -4, 7, -1], [5, -1, -5, 7, -5, -3], [-19, 9, 4, -5, -1, -2], [-6, 6, -3, 6, -5, -5], [8, -4, 3, 14, -4, 5], [-2, -7, 8, 3, -8, 0], [-16, 0, 0, 0, 2, 6], [12, 0, 10, -4, -10, -10], [-9, -3, -5, 13, 1, -1], [-6, 4, -1, 0, -2, 5], [4, 2, 9, -8, -3, -1], [-1, -2, -5, 19, -3, -5], [-22, 4, 10, -16, 6, 4], [14, 4, -4, -3, 0, -1], [1, -1, -5, -5, -9, -6], [15, 5, 1, 3, 9, 3], [6, -3, -2, -9, 6, 6], [-6, -8, 16, -8, -4, -3], [6, -4, -5, 2, -10, -7], [22, 1, -10, 4, -4, -2], [0, -12, -7, 12, -1, 7], [-20, -4, 1, 9, 7, 0], [-18, 8, 16, -2, 2, 4], [3, -10, 7, 11, -1, -3], [-9, 1, -9, -5, 1, -3], [9, -3, 5, 13, -5, 9], [5, -13, 2, 13, -4, 4], [-6, -4, 17, 0, 1, -1], [16, -2, -4, 1, -8, 0], [-4, 4, -3, -2, 2, -1], [-13, 7, 14, -7, 0, 3], [8, 6, -3, 6, 10, 3], [-2, 2, 7, -10, 5, -3], [2, -8, 5, -4, 11, 5], [18, 6, 2, -2, -6, -6], [-5, 1, -15, -3, 5, -4], [4, -10, 2, -2, -2, 0], [5, -10, 0, -1, 0, -2], [24, -6, -10, -4, -2, -2], [-5, -1, -4, -3, -2, 6], [31, 7, -12, 7, -1, 4], [-13, -13, -22, 6, 4, 8], [-7, 1, -11, 1, -5, -1], [-2, 0, 13, -3, -9, -6], [-4, 4, -6, -7, 8, 8], [10, -4, 8, 14, 1, 4], [2, -4, -5, 10, -10, -2], [-8, 4, -3, -2, 6, -1], [17, -9, -3, 15, -5, 5], [-5, -1, 14, -6, 6, 0], [-1, 1, 3, 13, 3, 3], [-7, -7, -7, 1, 3, -1], [10, 1, -8, 4, 8, -2], [22, -12, 9, 12, -1, -1], [19, -3, -8, 8, -10, 0], [-12, 5, -8, -6, 2, 10], [-26, 20, -2, -12, 4, -2], [-19, 5, 2, -19, -2, 0], [20, 6, 1, 20, -2, 2], [-30, -4, 4, 14, 0, 0], [-34, -8, -6, 14, 2, 2], [8, -12, -5, 15, 5, 4], [-27, -3, -10, 3, 4, 9], [5, -8, -10, 11, 0, -2], [40, 6, -3, -4, -1, 1], [5, -5, 7, 3, -11, -5], [24, 11, -3, 0, -11, 3], [-26, -4, 1, 2, 13, 3], [-7, -1, 18, -1, 2, 3], [12, 10, 1, -14, -2, -1], [-30, -4, 9, 6, -3, 1], [1, -9, 8, 7, -6, -8], [31, 0, 0, -4, 0, 2], [11, -3, -14, -17, 2, -5], [-4, 6, 13, -12, -1, 0], [0, -8, 1, 28, -5, -1], [5, -13, -14, 5, -1, 2], [-7, -3, -16, 4, -4, 0], [4, -2, 3, -19, -7, -1], [-17, -3, -9, 9, 7, -3], [9, -1, -8, -3, 0, 0], [12, 1, -4, 4, 5, 2], [10, -14, -6, 18, 1, -1], [-10, -14, -18, 6, 0, 3], [13, 2, 10, -3, 6, 0], [3, 7, 6, -21, -6, 2], [-11, -3, 11, 5, 5, 5], [-11, 1, -5, -16, -1, -3], [-3, -13, -3, 11, -7, 1], [-23, 5, -18, -3, -6, -4], [13, -3, 13, 11, -2, -5], [8, 10, -9, -6, -2, -1], [6, -20, -1, 18, -10, 3], [5, -3, 24, 5, 8, 2], [9, -3, 16, 19, 4, 2], [-7, 11, 11, -17, -3, 3], [8, -2, -18, 8, 5, -2], [-20, -8, -14, 13, -2, 3], [13, 9, 10, 7, -6, -8], [2, -1, -13, 12, 1, 1], [-27, -7, -2, -16, 0, 6], [5, 1, -11, -5, -9, -5], [27, 13, 6, 3, 4, -1], [-11, 3, 3, 17, 7, 3], [-6, 8, 8, -2, -6, 0], [3, -3, 8, -1, -10, 4], [-35, -1, 3, 1, -7, 1], [10, 10, -6, -22, -2, -2], [-16, -2, 4, -4, 2, 2], [19, 7, 5, 1, -1, 3], [18, 4, -2, -8, 10, 2], [-23, -9, -1, 0, -1, 1], [18, -14, -10, 10, 2, -2], [-24, 14, -8, -12, -2, 3], [-4, 10, 0, 8, 4, -2], [0, -4, 8, -4, -12, 2], [-24, -2, -15, 10, 13, -1], [10, 0, 5, 8, 5, -7], [1, -5, -2, -2, -4, 0], [24, -16, 4, 13, -6, -8], [-18, -2, -5, -23, 1, -2], [0, 10, -6, 4, 0, -2], [19, -9, 10, 11, 7, 8], [-28, 10, -4, -4, 7, 1], [-13, -7, -14, 5, 6, -4], [28, 4, 9, 8, -6, -7], [3, 3, -22, 1, -5, 2], [12, -4, 1, 18, -7, -13], [-8, -8, 5, -4, 0, -2], [10, -6, -5, 16, 0, -7], [-9, -11, -6, -2, -10, -4], [-20, 5, -2, -11, 2, 2], [8, -10, -20, 4, -8, -2], [-12, 0, -12, -8, 8, -8], [16, 6, 11, -4, -11, -10], [17, 0, -8, -17, 12, 10], [-19, 7, -17, -3, -3, -4], [-18, 2, -11, -17, 5, -1], [3, -13, -1, -7, -5, -3], [5, -7, 0, 17, 4, 2], [-23, 7, 17, -17, 5, -5], [-29, -2, 4, 15, -12, -4], [-21, -8, -21, 15, 5, 7], [2, -3, 4, 10, 7, 6], [9, 15, -12, -6, -10, 0], [17, -10, 6, 17, 1, 0], [-15, 9, -12, -5, 8, 12], [-8, -1, -14, -6, -10, -2], [30, 2, -11, -4, -2, 3], [40, 2, 8, 16, 4, 0], [6, -6, 11, 9, 1, 5], [-3, -7, 9, 12, 1, 9], [26, -16, 7, 14, -2, -3], [10, 10, 8, -8, -3, -6], [-35, -13, 20, -3, -2, -6], [-5, 5, 23, -13, -5, 2], [-4, 10, -10, 18, 5, 4], [-30, 8, -17, -13, -7, 0], [16, 13, -4, -5, -4, -4], [8, 6, -16, -11, 2, 10], [20, -10, 19, 13, 5, 5], [-11, 7, 0, -11, -6, -8], [-32, 2, -1, -8, 0, -11], [25, 3, -2, -11, 4, 2], [-10, 11, 23, 4, -13, -3], [1, 15, -2, 9, 2, 0], [-27, 21, -12, -19, -2, -1], [6, 16, 2, -10, -7, -5], [6, -10, -2, -6, 1, -2], [-22, -14, 1, 14, 12, -1], [-14, -7, -13, 12, 15, 5], [-2, -2, -4, -6, -2, 0], [5, 8, 12, -11, -7, -6], [15, 17, 10, -7, 4, -3], [13, 6, -5, -7, -1, -7], [-2, 4, 2, -18, 4, 10], [-19, 5, -10, -17, 8, -14], [-17, -6, 14, 13, 0, 2], [16, -3, -11, 0, 3, 9], [15, -1, 14, 13, 6, 0], [-36, 10, -6, -2, -4, 0], [-18, 4, 27, 0, -2, -1], [-13, -1, -12, 1, 10, 4], [16, 16, -20, 0, 12, 8], [24, 2, -14, 2, -2, 2], [11, -16, 14, 1, -8, 2], [-26, 4, -1, 2, -9, 3], [-2, 2, -23, 13, -1, -5], [-13, 1, -4, 7, -10, -4], [11, -7, 7, 11, -11, -5], [-12, 6, 10, -14, -1, -6], [-31, 9, -11, -9, 3, 0], [-5, -22, -24, 7, 5, 8], [-22, -5, -18, 9, -2, -2], [12, -2, -20, -4, 8, -4], [8, 12, 13, 0, 3, -5], [0, 8, -9, 11, 1, 8], [9, -3, -11, -7, 1, 9], [15, 7, 15, 7, -5, -11], [16, -7, 4, 4, -6, 0], [0, 0, -19, 6, 0, -5], [19, -23, 7, 5, 5, 4], [15, -13, 21, 1, -1, 1], [24, -4, -9, -1, 5, 1], [-17, -13, 14, 11, -12, 2], [-13, 1, 8, -3, 12, 2], [18, -12, -20, -10, 10, 0], [50, -9, 5, 8, 5, 7], [29, -4, 23, -9, -5, 1], [-14, 10, 10, -20, -2, 8], [-21, 3, -19, -6, 9, -7], [9, -7, 16, 5, -10, 4], [19, 5, -10, 13, 0, -9], [14, -8, 0, 24, 4, 0], [-3, 6, -6, 1, 22, 4], [8, -4, 18, -14, -8, 4], [10, 10, 10, -4, -20, 1], [22, -11, 5, -4, -3, 7], [-30, -14, -4, 2, -4, 2], [13, -13, -19, -11, 1, -7], [-7, 7, -4, -15, -6, -3], [-38, -8, 4, 4, -4, -2], [-28, -2, 15, -16, -5, -1], [13, -7, 4, 1, 2, 8], [-33, 11, -9, -3, -10, -7], [12, 10, 22, -19, -10, 1], [6, 10, 0, -10, -15, 1], [23, 17, -15, -2, 7, 3], [-17, 1, 1, -13, -11, -5], [-19, -7, 10, 10, 6, -2], [1, -7, -37, -9, 3, 0], [-10, -8, -2, -14, 10, 10], [2, -4, -2, -16, 8, 4], [-5, 20, -7, -13, 3, -1], [14, -1, 8, 14, -7, 6], [-8, 2, 9, -4, -4, 5], [-5, -7, 18, 6, -10, -10], [26, -6, 6, 18, -2, 6], [-17, -17, 15, 27, 1, -5], [22, -6, 19, -2, -10, -7], [-5, 11, -8, -28, -4, 2], [26, -13, -6, 18, -6, 0], [-2, 6, 29, 0, -3, 3], [5, 6, 10, 25, -6, 4], [8, 5, 0, -17, -8, -4], [34, 6, -8, -2, -6, 0], [19, 5, 13, 13, -7, 3], [-9, -5, 2, -17, 12, 2], [-9, -1, 26, -11, 14, 6], [-21, 5, -1, -25, -5, 7], [-23, -2, 15, -5, 1, -3], [33, -19, 7, 23, 9, -6], [-18, 1, -29, 2, 1, 1], [17, 13, 6, 9, -6, -2], [28, 4, -5, -2, -4, -11], [-14, 2, -10, 10, 14, 8], [14, 4, -13, 20, 0, -7], [-10, -8, 4, -18, 6, -3], [26, 16, 4, 0, 10, 2], [21, 5, -16, -7, 6, 5], [21, 9, 26, -16, -2, -2], [-39, 21, -9, -10, -1, -11], [22, 4, -12, -6, -2, -6], [1, -3, 1, 9, -9, -3], [3, -10, 3, -19, -13, -5], [-2, 6, -37, -9, 1, 4], [28, -14, -5, -4, 13, 12], [2, 6, -1, -6, -1, 3], [-10, -10, -2, 26, 5, -4], [18, -10, -12, -9, 0, 4], [24, -16, -1, 12, 9, 5], [-12, 14, -16, 12, 10, 8], [7, -19, 14, 25, 12, -5], [-12, -1, 7, -14, 3, 1], [-23, -13, -15, 13, 7, 5], [-6, -1, 2, 24, 0, 10], [37, -1, -14, -1, -1, 8], [-16, 14, -5, 0, -8, 1], [-28, 2, 10, -12, 14, -2], [-6, 8, -12, 6, 8, -6], [25, -13, -22, 27, -4, -6], [-12, 2, 3, -12, 15, 3], [-44, 8, 3, -14, -9, -5], [-31, -5, 19, -3, 3, -5], [-36, 19, 14, -4, -10, 4], [15, -3, 26, -15, -4, 4], [-32, -10, -6, -16, 0, -5], [-42, 12, 12, -8, 0, 6], [6, -32, -1, 6, 3, 1], [-30, 2, 1, -26, -3, -9], [40, 2, 24, 12, 8, -2], [13, 14, 26, 8, -6, -8], [-9, -1, -9, 7, 15, 9], [24, -12, -12, -12, 1, -5], [-4, 4, -4, 0, -2, -12], [-31, 6, 1, -37, 1, 9], [-10, -10, -4, 38, -6, -4], [47, 5, -13, -22, 9, -1], [12, -6, -17, 2, 15, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8470_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8470_2_a_df();".
// 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_8470_2_a_df(:prec:=6)
    chi := MakeCharacter_8470_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(3169) - 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_8470_2_a_df();".
// 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_8470_2_a_df( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8470_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, -3, -2, 1]>,<13,R![-108, 0, 441, 0, -42, 0, 1]>,<17,R![-38, -24, -1, 1]>,<19,R![709, -944, 283, 80, -41, 0, 1]>],Snew);
    return Vf;
end function;