
// Make newform 8470.2.a.dg in Magma, downloaded from the LMFDB on 19 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_dg();".
// 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_dg();".
// 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 := [5, 10, -70, -10, 69, 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, 1, 0, 0, 0, 0, 0, 0], [29, 17, -70, -3, 34, 1, -3, 0], [-27, 140, 13, -103, -1, 19, 0, -1], [111, -237, -282, 271, 72, -63, -5, 4], [-26, 97, -7, -174, 14, 46, -1, -3], [42, 97, -24, -174, 14, 46, -1, -3], [-87, -512, 261, 559, -93, -139, 6, 9]];
    Rf_basisdens := [1, 1, 34, 34, 34, 17, 17, 34];
    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, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 1, -1, 0, 0, -1, -1], [-1, -1, 1, 1, 1, 0, 0, 1], [1, -2, 0, 3, 1, -1, 1, 0], [2, 0, -2, 0, 0, 0, 0, 0], [-1, 1, -1, -1, 0, 1, 1, 0], [-2, 1, 1, -2, -1, -1, -1, 0], [-2, 1, 1, -3, -1, -1, 0, -1], [0, -1, -1, 2, -2, -1, 0, 0], [1, 0, -1, 0, -2, 0, -1, 0], [-1, 0, 0, 3, -1, 0, 2, 0], [-3, 2, 1, -1, -3, 0, 1, -1], [-2, 2, 0, -2, 1, -1, -1, 1], [-6, -1, 2, -2, 0, 1, 0, 0], [4, -3, 1, 1, 0, 0, 1, 0], [2, 1, 1, 0, 0, 0, 0, -2], [3, 0, -1, 0, 1, 0, 1, 3], [-6, -2, 1, 0, 0, 1, 2, 2], [-6, -1, -2, -1, 2, -2, 1, 1], [-1, 1, -1, -3, 2, -1, 0, 1], [1, 0, -3, 3, 3, 0, 2, 2], [-1, 2, -3, 0, 1, 0, 0, -2], [4, 0, -1, 2, 0, 0, 1, -1], [0, -4, 1, 4, 0, 2, 0, 0], [0, 0, -1, 5, 1, -2, 2, -1], [0, -2, -1, -1, 2, 1, 1, 2], [0, 0, 3, 0, -1, 4, -5, -2], [-5, -2, 0, 7, 1, 1, 1, 0], [1, -1, -4, 0, 1, -3, 0, 2], [-2, 0, -4, -3, 2, -2, -1, -1], [-13, 0, 2, -4, -1, 1, 0, 0], [-1, 5, -2, -5, 3, -1, 0, 0], [0, 1, 0, 2, -1, -1, -2, -3], [3, -1, -3, -3, 2, -2, 2, -2], [-5, 1, 4, -4, -2, 5, -3, 0], [-1, -4, 1, -2, 0, 1, -1, 0], [-7, -1, 6, -5, -2, 4, -3, 1], [1, 4, 1, -2, -1, -2, 0, -2], [-3, 1, -2, -4, -1, -2, 0, 0], [-1, 4, -2, -8, 0, 0, 1, -2], [-6, -2, 0, 4, 2, 0, 0, -2], [12, 0, 1, 3, -3, 4, -4, -1], [-2, -1, 7, 0, -1, 4, -3, 0], [-3, -2, -5, -3, 1, 0, -1, 1], [-5, 2, -1, -6, 3, 0, 0, -1], [0, -1, -4, 4, 2, -3, 6, 2], [1, -1, 0, 2, 3, 1, -2, 2], [-12, -1, 1, -2, 2, 0, 2, 2], [-1, 3, -6, 2, -3, 0, -2, 1], [-5, -4, 4, -1, -2, 0, 0, 3], [-2, 2, 1, -3, 1, 2, 1, -2], [1, 1, 1, 3, -3, -1, -5, -1], [-9, 2, -1, -4, -3, 4, 0, -2], [-6, 5, 1, -8, -1, -1, 1, -2], [-1, -3, 0, 12, -2, 1, 3, 4], [-9, -4, 4, 7, -1, 0, -3, 1], [-7, 4, 2, -7, -1, 2, 2, 0], [-2, 0, 1, -6, 1, -2, 3, 0], [-3, 1, 4, -2, 0, -1, -4, 1], [-6, -4, 0, -2, 2, 2, 0, 2], [-1, -9, 3, 5, 0, 0, 2, 2], [0, 7, -1, -5, -3, -1, 0, 1], [6, 7, -3, -9, 1, 0, 0, -5], [-6, -6, -1, -5, 4, 3, 1, 4], [6, 5, -7, 4, -4, 2, -2, -1], [8, 4, 4, 10, -4, 4, 0, -2], [7, 5, -1, 9, -1, -1, -1, -4], [-15, 4, 0, -10, -5, -1, 0, 2], [-9, -5, 9, -5, 0, 4, -4, 1], [13, 1, -4, 8, -2, -1, -1, -4], [-1, -3, -3, 4, -3, -3, -1, -1], [3, -5, -3, -1, 2, 4, -2, 1], [-9, -1, -2, 1, 3, 0, 5, 3], [-3, 2, 1, 0, -2, -5, 5, 0], [15, 4, 4, 15, -2, 0, -1, -2], [-8, 4, -6, 7, -2, -2, 1, 3], [-16, -4, -2, -3, 1, -2, 0, -1], [-1, -6, 0, 0, 1, 3, -2, 4], [-5, 6, 2, -1, 1, -2, 0, 0], [7, -9, -1, 8, 3, -3, 4, 5], [2, -6, -1, 13, 1, 2, 2, 3], [2, -2, 1, -8, 2, 2, 2, 2], [5, 0, 4, 5, -2, -2, -2, 0], [0, -1, 0, 5, 1, -1, 2, 3], [-7, 6, 3, 0, -3, 2, 0, -4], [12, 0, -3, 18, -3, 0, 1, 0], [6, 0, 6, 5, -1, -1, -4, -1], [-6, -3, 1, 0, -1, -1, 1, -2], [-11, 1, -2, 1, 2, 1, 2, 3], [-22, 3, -1, -2, 2, -3, 2, -2], [8, -3, 3, 2, 5, 3, 0, 1], [-7, 4, 2, 2, 1, -3, -2, 2], [-7, 1, -7, 2, 5, -1, 2, 0], [-11, 1, -2, -2, 3, 2, -4, 1], [8, -4, 3, 3, -3, 1, 1, 3], [3, -1, -3, 2, 0, 0, 1, 4], [4, 1, -5, 2, 2, -4, -2, 1], [-3, -2, -7, -5, 3, -4, -3, 1], [-6, 3, 4, 1, -6, 1, 2, -2], [1, -2, 8, 1, -8, 1, -9, -3], [-15, 8, 6, -10, -1, 6, -5, -1], [-11, 1, 3, -13, -1, 2, -6, -5], [-22, 2, 9, -18, -9, 6, -3, -3], [-6, 3, -1, 0, 3, -6, 2, -2], [-4, -8, 1, 17, 2, 2, -2, 2], [3, 7, -1, -4, -8, 5, -3, 2], [1, 1, -1, -8, -2, -3, 4, -1], [9, -2, 7, -1, 1, -2, 3, 1], [-3, -4, 11, -3, -2, 3, -2, 0], [-19, 4, 2, -7, -5, -1, -5, -2], [9, 8, -6, 1, -3, 2, 0, -4], [6, -2, -4, -5, -2, 3, -1, -1], [15, -7, 0, 6, 2, 4, 1, 2], [2, 2, -11, -6, 0, -2, 3, -1], [1, 5, 2, -14, 0, -3, 3, -4], [-5, -7, 2, 9, 4, 7, 0, 6], [-6, 7, -4, -6, 6, -3, 2, 2], [15, 6, -7, 14, 2, -3, 3, -1], [0, 3, -3, 10, 0, 1, 5, -1], [10, 4, -6, 18, 0, -2, 4, -2], [-10, -4, 4, -18, 1, 1, 3, 3], [-8, 5, -8, -1, 4, 1, 1, 1], [1, 6, -5, -1, 4, -5, -4, 1], [-6, -3, 0, 3, -6, -1, -5, -5], [-7, 6, -4, -14, -3, 7, -2, 2], [7, 8, -8, 0, 3, -5, 4, 2], [3, 1, 6, 12, 5, 4, -2, -1], [14, 4, -2, 10, -4, 0, -6, -4], [10, -7, -5, -7, 2, -5, 2, 4], [-16, 7, 5, -1, -3, 3, -2, -1], [-8, -1, 2, 2, -1, 0, 3, 5], [-2, -8, 2, -1, 6, 0, 5, 9], [-11, 4, 1, -6, -4, -1, 1, -2], [10, -4, -1, 6, -1, -2, 1, -3], [-2, -5, 2, 11, -5, -4, 4, 3], [12, 6, 0, -5, -2, 2, -3, -2], [-1, -8, -4, 6, 3, -6, -4, 2], [-15, 4, 1, -12, 4, 4, -6, -3], [-5, -5, 9, -9, -1, -5, -1, 3], [-3, -13, 3, 15, 4, -2, 4, 10], [-21, 1, 5, 0, -1, -3, -2, -4], [-6, -5, -7, 6, 3, -3, 5, 2], [4, 3, 6, -9, -1, 4, -2, 1], [-7, 10, -1, -7, 0, 0, -6, 1], [3, -4, -2, 1, 3, -1, 5, 2], [7, -3, 7, 13, -5, -3, 5, -3], [14, -5, 3, 21, 5, -1, 2, 1], [-1, -5, -5, -17, 6, -1, 0, 5], [-4, 4, 1, -20, 0, -4, 6, -6], [-2, -5, 6, -3, -3, -1, -1, 4], [-3, 5, -5, -7, -6, 4, 2, -8], [3, -7, 10, -3, 7, -1, 6, -2], [-14, 2, 7, -23, -4, 6, -8, -2], [-11, 5, -6, -6, -3, 0, 10, -3], [5, 1, 1, -5, 9, -1, -3, 4], [-5, -7, 9, 14, -3, 3, -2, 2], [10, 3, -11, 1, 4, -6, -1, 0], [-15, -11, 6, 3, -3, 7, 3, 1], [-10, 10, 7, -12, 1, 4, -3, -4], [-6, -8, -4, 3, 6, -4, 3, 3], [9, 6, -12, 11, 12, -6, 4, 3], [-8, 1, 8, -11, -7, -1, -1, -4], [13, -7, -6, 4, 7, -4, 4, 8], [8, 3, 12, 12, -9, -3, 0, -5], [-3, -8, 10, 0, -2, -2, -2, -1], [-18, 5, 2, 5, 1, 6, -4, 5], [9, 1, -3, 16, 1, 5, 9, 3], [-13, 6, 1, -14, -9, 4, -2, -2], [14, -1, 5, 8, -4, 8, 0, -4], [4, -4, 1, 11, 10, 3, -1, 2], [-32, -7, 15, -18, 5, 1, -3, 2], [-13, -12, 2, -4, -2, 0, -5, 2], [6, 1, -5, 9, -1, -1, -1, -6], [8, -3, -1, 6, 3, -9, 1, -2], [5, 10, 0, -4, 7, -3, 2, 2], [-2, -3, -6, -2, -4, 1, 4, -6], [8, -2, 3, 2, 1, -6, -7, 0], [-6, 6, 4, -9, 8, -4, 5, -5], [1, 9, -10, -6, 10, -1, 1, 2], [-12, -10, 0, -2, 7, 0, -2, 5], [-4, 2, 2, -26, 4, 2, -2, 0], [-1, 9, -7, 6, -5, 3, -6, 2], [-15, -7, 2, -2, 8, -7, -1, 4], [16, -3, -4, 33, 1, -3, 8, 7], [-1, -6, 2, -4, 0, 0, -1, -5], [-8, -3, 2, 5, -4, 7, -9, 3], [-7, 9, -1, 2, -3, 6, -5, 4], [-16, 13, 5, -11, -4, 2, -1, -2], [39, 7, -2, 2, -2, 1, -3, -5], [-21, -1, 7, 4, -3, 9, -8, 2], [-22, 2, -2, 5, -2, 2, 7, 9], [-23, 6, 0, -17, 9, 0, 0, 2], [-7, 4, 2, 1, -4, -5, 0, -5], [-21, 2, 5, -17, -5, -2, -1, 3], [-17, 2, -10, -11, 9, -1, 5, -1], [28, -18, -5, 13, 6, -5, 9, 1], [-8, -8, 3, -25, 1, 0, -4, -5], [-14, 2, -6, -6, 0, 2, 2, -2], [-4, 13, -12, 11, 3, -2, 6, 7], [-11, 5, -3, -10, 2, -3, 5, 2], [-8, -4, -4, 1, 1, 4, -6, 5], [-4, 9, 0, -6, -11, 1, -6, -11], [-28, -7, 12, 1, 0, 5, 3, 1], [-27, 5, 1, -20, 10, 2, 3, 5], [-3, 6, -2, 13, -1, 9, -3, -3], [-7, -4, -2, 26, 3, 0, 6, -2], [15, 0, -1, -11, -2, 1, 2, 2], [6, 2, 12, 8, -4, 2, 0, 2], [1, 3, 0, 11, -6, 1, 2, 3], [11, 3, 8, -8, -6, 10, -9, -4], [-18, 5, -12, -13, -2, 3, -1, -1], [-2, 7, -3, -15, 0, 9, -1, 1], [23, 3, -17, 3, 5, -3, 3, -5], [-23, -1, 11, -29, 4, -3, -4, 1], [33, -1, 8, 9, 8, -1, 2, -1], [15, 6, 7, 21, -1, 6, -3, -7], [-12, 5, 0, 3, -4, 5, -3, -3], [4, -2, 14, -8, -18, 2, -4, 0], [18, -6, 4, -11, 5, 3, -6, -5], [18, 2, -2, -4, -4, -6, 6, 0], [-29, 10, 4, -12, -10, 4, 7, -8], [7, -5, -11, -6, -4, -6, 5, -4], [1, -13, 11, -17, -1, 5, -5, 2], [8, -8, 2, -4, -4, 8, 0, -4], [6, 7, 1, -7, -3, -5, 4, -1], [10, -1, -3, -12, 1, -8, 3, 8], [6, -8, 6, 12, -2, -2, -4, 0], [-4, -6, 16, 5, -7, 2, -2, 5], [-21, -8, 14, -16, -9, 11, -4, 0], [7, 8, 2, 23, -5, -1, -7, -1], [35, 7, -12, 17, -6, 3, -3, -6], [-21, 1, -1, -16, -8, 3, -4, -7], [-21, 9, 3, -7, -6, -1, -2, -5], [6, 19, -7, 17, 1, -3, 2, -7], [3, -3, 13, 4, 3, -7, 1, -1], [-4, -2, 1, -9, 5, -6, 8, -5], [5, 15, 5, -2, -8, 10, -5, -3], [-22, 14, 4, -12, -3, 4, 3, 0], [13, 6, 0, 3, 7, 0, 4, -2], [23, -10, -5, 32, 0, 0, 2, 1], [26, -1, -3, 26, -1, -1, 3, 1], [-28, -1, 10, -27, 0, -3, -3, 3], [9, 6, -7, -19, 3, -10, 1, -5], [14, 0, -16, 12, 9, -12, -3, 2], [8, 0, -6, 5, 1, -8, 7, -6], [19, -5, -7, -9, -1, -5, 5, 3], [6, -12, 0, 10, 6, 2, -2, -8], [-10, 2, 8, 2, -8, 4, 0, -8], [0, -6, 3, 20, 1, 8, 1, 0], [-2, -13, 2, 13, 1, -4, 0, 0], [-2, 8, -1, -12, 1, -2, -3, -10], [-2, -17, 4, -1, 10, -4, 0, 7], [-9, -3, 6, 3, 1, 6, -3, -7], [-16, 10, -1, -12, -3, -3, -4, -2], [23, 8, -2, 20, 1, 5, -6, -10], [17, 1, 9, 37, -4, 8, 0, 4], [8, -6, 3, 18, -1, -6, 1, 0], [-18, 14, -6, 13, -2, 2, -5, -3], [1, 8, 13, -2, -3, 12, -4, 0], [30, -5, 8, 4, 1, -9, -6, -5], [9, 14, -14, 2, -3, -1, 6, -4], [0, -2, 2, -15, 1, -2, -4, -5], [-22, -1, 1, -30, 3, -7, 3, 6], [16, 3, -2, 6, -5, 6, 1, -9], [-15, 5, 3, -15, -4, -6, 0, 4], [21, 14, -3, -5, -7, 4, -1, -3], [-16, 2, -4, 4, -4, 6, -12, -4], [4, 6, 11, 1, -14, 2, -7, -3], [20, -7, -11, 14, 1, -9, -1, -9], [18, -1, -11, 3, -4, -7, 6, -6], [-8, -1, -7, -1, -3, 1, 2, 10], [-15, 12, 2, -16, -9, -1, -6, -10], [9, 17, 7, -3, -2, -6, -4, -4], [17, -2, -8, 1, 5, -14, 2, -2], [11, 0, -16, 16, 8, -9, 14, 8], [-15, -2, -2, -18, 2, -4, -5, -8], [-9, -13, 7, -4, 0, -1, -4, -5], [12, 0, 3, 6, 0, -11, 14, 4], [24, 4, -5, 25, 8, -7, 7, -1], [4, -12, 4, 4, 8, 2, 4, 6], [13, -6, -3, -7, 8, 1, 0, -2], [3, 8, -19, -22, 3, 6, 2, 6], [-21, 4, -5, 10, 6, 5, 3, 9], [-18, 9, -3, -15, -9, 1, -2, 1], [-20, 7, -3, -6, 9, -5, 7, -2], [16, 2, 4, 13, -8, -6, 7, -1], [33, -4, 4, 5, -3, -6, -4, -6], [-10, 11, 6, -13, -7, -5, 4, -3], [9, 1, 10, -9, -5, -3, 3, 7], [17, 2, -4, 18, -1, 3, -2, 3], [15, -9, 6, 0, -4, 7, -3, 4], [-10, 11, -12, 14, 10, 5, 6, -2], [10, -8, 2, 42, 4, 6, 4, 8], [-9, -16, -2, -27, 5, 2, 0, 0], [-3, 14, -8, 16, 1, 2, -6, 0], [12, 11, -2, 23, -2, -3, 0, -2], [28, -2, 3, 16, 12, -4, 0, 10], [-10, 6, 2, 0, -4, 6, 0, 0], [5, -6, 3, -20, 1, 2, 0, 6], [16, 8, 1, 3, 4, -4, -5, -3], [33, -5, 2, 36, 0, -1, -1, -1], [-20, -5, 0, -5, -7, 5, -8, -11], [-2, 14, -7, -35, -2, -4, -7, 3], [-2, 1, 4, 12, 0, -5, 4, -6], [6, 3, -23, -21, 10, -5, 0, 4], [-11, -6, 21, -11, -15, 12, -3, 1], [8, -4, 7, 13, 10, -9, 9, 3], [-11, 4, -14, 11, -3, 0, 4, 4], [19, 6, -6, 5, -14, 6, 0, -10], [12, 3, 1, -12, 12, 2, 2, 3], [-15, 11, -15, -3, 7, 3, -3, 1], [-18, 7, 5, -25, 2, 5, 5, 5], [-1, 6, -6, 6, -1, -8, 6, 8], [-10, -10, 23, -9, -8, 8, -14, -14], [5, 0, 7, -3, 4, 2, 2, -3], [2, -14, -2, 12, 10, -12, 9, 13], [6, -12, 2, -35, 3, 0, -4, 7], [6, 3, -10, 23, -1, -5, 14, -1], [-1, 2, 11, 2, -11, -6, 2, 0], [-9, 9, 11, -5, -3, -3, -7, -1], [-21, 6, -7, -34, 4, -1, 5, 3], [-1, 1, -13, -8, 4, -9, -3, -6], [-9, -8, 12, -7, -1, 2, 6, 2], [-10, 4, -12, -6, 3, 3, 7, 6], [11, 7, 4, -19, -4, -1, -6, -5], [-10, 4, -6, 0, -8, -10, -8, -4], [5, 5, -9, 0, 14, 2, -1, 4], [-9, 14, 7, -23, 5, 4, -9, -3], [12, 3, 11, 6, -7, -1, -9, -10], [-21, 7, 22, -3, -12, -1, -7, 0], [21, -1, 20, 22, -13, 2, -4, -7], [3, 8, 6, 12, 0, -6, 1, 0], [-3, 0, -2, -1, 15, -2, 8, 0], [25, -13, 9, 12, -13, -1, -6, -7], [19, -13, 10, 29, -3, -2, -1, 0], [-4, -18, -2, 2, -2, -6, -8, -6], [-16, -12, -4, -2, 4, 2, -6, 4], [19, -7, -6, 20, -7, 7, -4, 4], [14, 8, 10, -10, -8, 8, -4, 8], [10, 4, 3, -5, 8, -4, -2, 0], [20, -8, -2, 24, 8, 0, 8, -2], [13, -8, 10, 7, 5, -3, 1, -5], [-10, -2, -2, 26, -10, -2, 2, -4], [33, 10, -8, 27, 10, -8, 6, 9], [4, -15, 5, -2, 3, 5, -7, 0], [-15, 11, 0, -1, -16, -3, 0, -7], [-2, 4, -6, 1, -5, -1, -4, 8], [-30, -15, 4, -7, 3, 3, -8, -3], [-9, 7, -9, 13, -2, 3, 1, -2], [22, -3, -2, 9, 3, 1, 11, 6], [-39, -5, -8, -18, 9, -1, 11, 9], [-22, 10, 1, -35, -7, -2, -5, 0], [22, -12, 0, 52, 10, -10, 10, 0], [5, -6, 2, 27, -13, -1, 3, 4], [11, 10, -3, 3, -8, 3, -10, -4], [36, 0, -6, 0, 0, -4, 0, -4], [-19, -11, 20, -14, 2, 11, -11, -4], [-13, -4, 2, 8, -2, -4, 3, -10], [7, -7, 6, -9, 4, -1, -2, -5], [-6, -13, 0, 31, 1, -4, 16, 2], [37, -4, -22, 17, 8, -12, 2, 3], [-15, -1, 6, -27, -1, 17, -3, 5], [-24, 11, 9, -14, -8, 11, -9, -1], [5, 1, 0, -19, 2, -1, 8, -5], [-2, 4, -14, 0, -4, -6, -1, -15], [-19, 23, 11, -5, -1, -2, -1, -9], [-7, 12, -4, -42, 2, 4, -9, -8], [17, -5, -7, -3, 7, -9, -1, 1], [-31, -11, 4, -1, 3, -6, 9, 5], [-20, -9, 1, 22, 11, 3, 7, 10], [-7, 12, 3, -19, -5, -4, -7, -9], [-6, 8, -16, -18, 9, -7, -1, 3], [21, 15, -16, 10, 3, -2, 10, -4], [4, 31, -5, -5, -3, -5, 6, 6], [-1, 11, 3, -10, -14, 7, -9, -10], [-31, 24, -5, -19, -3, 9, 1, 5], [-23, 7, -19, -4, 21, -3, 10, 6], [-3, -11, 6, 14, -3, -1, 16, -2], [23, 5, 4, 11, 2, -3, -10, 1], [-13, -4, 14, 2, 4, 2, 1, -6], [-4, 20, -14, 9, 3, -4, 6, -1], [20, 6, 3, 14, 2, -10, -1, -1], [26, -4, 3, 6, 8, 5, 8, 6], [1, -4, 11, 22, -6, 19, -1, 2], [7, 17, -7, 9, 2, -6, 4, -2], [37, -4, 1, 9, 7, 8, 5, -1], [12, 14, 8, 15, -4, 12, 7, 2], [1, -6, 4, 48, 5, -1, 4, 0], [12, -1, -9, 3, 8, 1, 5, -1], [-6, 16, 2, 3, -3, 12, 1, -6], [8, -7, -9, 35, -12, -1, 4, 6], [13, 4, 18, 3, 6, -2, -8, 13], [6, 23, 4, -19, -12, 9, -13, -11], [-13, -13, 16, -15, -1, -4, 7, 12], [20, 3, -11, 40, 15, -4, 11, 0], [3, 16, -12, -23, 3, -14, 4, 6], [8, 9, -6, -10, -4, -7, -4, 0], [22, -9, -4, 18, 9, -11, 1, -8], [8, -8, 5, 14, -3, 0, -3, 2], [-5, -15, -3, 15, 15, -11, 5, 12], [0, 18, 0, -2, -14, 2, 6, 2], [-6, 2, 20, 4, 4, 4, 0, -6], [-12, 8, 8, 0, 6, 6, -8, -6], [-22, -2, -2, 33, -1, 9, 8, 8], [22, -14, -10, -3, 7, 10, -1, 6], [-2, -22, -6, 5, 15, -5, 6, 2], [43, 7, -3, -25, 2, -8, -2, -8], [28, -3, -1, 19, 7, -1, 4, -5], [-7, 2, -4, -37, 8, -6, -16, 1], [17, 1, -16, -14, 0, -6, 6, 2], [20, 29, 2, 17, -17, -1, 8, -5], [6, -7, 16, 14, -2, 13, -8, 0], [-38, 4, -12, -4, 18, -14, 12, -2], [-3, -10, 4, -19, 9, 0, 10, 6], [17, 5, 9, -16, -5, -4, -7, -2], [29, 3, -7, -10, 14, 1, -4, -7], [-17, -12, 14, 15, 3, -12, 2, 2], [-29, -2, 6, -16, 1, 5, -6, 4], [-1, -3, 12, 23, -4, 9, 8, 5], [14, -4, 6, 16, 7, -4, 5, 4], [-5, -8, -3, -15, 7, -1, 5, -7], [-24, -4, 3, -52, 7, -12, -15, -4], [-4, 8, 9, -29, -4, -8, 3, 5], [-18, -4, -6, -4, 0, -9, 19, -2], [12, -16, 6, -10, -7, 6, -9, 6], [-2, -6, 3, -5, -3, -8, -10, -7], [-13, 1, -12, -29, 9, -14, 9, -8], [-4, 14, -14, 27, -5, -14, 9, -6], [56, -2, -12, 7, -1, -2, 3, -4], [33, 4, -10, 24, -1, 11, -2, -8], [-23, -6, 2, 1, 15, -6, 4, 2], [-25, -7, 2, 7, 3, -6, 3, 4], [-18, -8, 26, -19, -8, -4, -9, 2], [-11, -5, 18, -4, -19, 3, 1, -5], [-8, 6, -9, 12, 6, -10, 4, -6], [-34, -9, -2, -17, 8, -1, -7, 3], [-20, -15, 8, 17, -4, 6, -5, 7], [56, -6, -10, 33, 18, -10, 2, 0], [-33, -8, 18, -27, 9, 4, 4, 10], [44, 3, 7, 37, -8, 9, 4, 2], [-21, -3, 2, 15, -15, 2, 7, -2], [-29, -10, -4, 7, -6, -4, -4, -1], [-15, -6, -6, -9, -11, 8, 0, -8], [-1, -18, 4, 7, 5, 2, -6, -6], [6, -5, -6, 4, -5, 8, -5, -3]];
    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_dg();".
// 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_dg(:prec:=8)
    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_dg();".
// 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_dg( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8470_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![5, 10, -70, -10, 69, 0, -16, 0, 1]>,<13,R![-12764, 39548, -14908, -4850, 2216, 140, -87, -1, 1]>,<17,R![109, -556, -2152, 3264, 749, -350, -62, 6, 1]>,<19,R![-70900, -158450, -33635, 15235, 3534, -485, -106, 5, 1]>],Snew);
    return Vf;
end function;