
// Make newform 4008.2.a.g in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4008_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_4008_2_a_g();".
// 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_4008_2_a_g();".
// 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 := [3, -6, -14, 12, 11, -7, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [-1, -8, -2, 7, 1, -1, 0], [-1, 8, 3, -7, -1, 1, 0], [1, -1, -9, -2, 7, 1, -1], [-1, 3, 9, 2, -7, -1, 1], [6, 0, -15, -3, 8, 1, -1], [-5, -2, 23, 6, -15, -2, 2]];
    Rf_basisdens := [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_4008_a();"
function MakeCharacter_4008_a()
    N := 4008;
    order := 1;
    char_gens := [3007, 2005, 1337, 673];
    v := [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_4008_a_Hecke(Kf)
    return MakeCharacter_4008_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], [0, -1, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0], [0, -1, 0, 1, 1, 0, 0], [0, -1, 0, 1, 1, -1, 0], [1, 0, 1, 0, 1, 0, 0], [1, -2, 0, 1, 0, -1, 1], [2, 0, 0, 0, 1, 0, 0], [-1, 0, 2, 1, -1, -1, 1], [2, -1, 0, -1, 0, 0, 0], [-4, 0, -1, -1, 0, 2, 1], [1, 1, 2, -1, -2, 0, 1], [0, 0, 2, -2, -1, -1, -2], [4, 0, -1, 1, -1, 1, -1], [1, 1, 2, -1, -2, -3, -1], [1, 1, 0, -1, 0, -1, -1], [-3, 1, 2, 2, 0, 1, -1], [0, -1, 0, 1, -1, 1, 2], [4, 0, 0, -1, 1, -1, 0], [-2, -1, -3, -1, 0, 0, 3], [0, 0, 0, -1, 0, 1, 2], [1, -1, 0, 2, -2, 0, 5], [3, 2, 0, -1, 0, -1, -3], [1, 2, 2, 0, 1, -2, -3], [-3, 3, 1, -1, 1, 3, 0], [0, 4, 4, 2, -3, -2, -2], [3, 2, 2, -2, 2, -2, -1], [1, -1, -1, 1, 1, -5, -2], [-1, 1, 3, 4, -1, -1, 0], [4, -3, -3, 2, -1, 1, 3], [2, 4, 1, 1, -3, 0, 1], [4, 1, 1, -3, -6, 0, 1], [2, 3, 1, -4, 1, 2, -5], [1, -4, -6, -1, 4, 0, -1], [3, 4, 3, 2, -4, -2, 0], [-3, 1, 2, 2, -3, 3, 5], [3, -5, -5, 1, 0, 1, 4], [-1, 0, 0, 0, 0, 0, 0], [-1, 1, 2, -4, 3, -1, -1], [6, 1, 1, -2, -1, -1, -3], [-5, 2, 1, -1, 3, -1, 0], [8, 4, 3, -1, 0, -1, -3], [1, 1, 5, 3, -2, -2, -4], [0, 3, 0, 3, -5, 5, 4], [0, 7, 7, -1, -4, -2, -3], [1, 5, 7, 0, 0, -1, -6], [6, 1, -5, -3, 3, -2, -3], [6, 1, -2, 1, 3, -1, -2], [-4, -2, -6, 0, -1, 6, 2], [0, -4, -5, -2, -3, 6, 3], [11, -1, -1, 1, 2, 0, 2], [-3, -2, -4, 0, 2, 2, 5], [4, 1, 4, -1, -3, 4, 4], [6, -3, 3, 4, -1, -3, -3], [11, -3, -5, 1, 1, 0, 0], [1, 4, 5, 2, -2, -1, 0], [9, -4, -1, -3, 0, 1, 0], [-3, 7, 6, -2, -5, -2, -1], [2, -5, -1, 2, -3, 1, 1], [9, 3, 2, -4, -3, -3, -1], [-3, 7, 6, 0, 0, -1, -5], [0, 6, 7, -3, 0, -2, -5], [10, 2, -3, 2, -1, 6, 1], [-4, 1, 0, -2, 5, 7, 0], [-2, 0, 1, -4, -1, -2, 3], [-1, 1, 10, 2, -2, 0, 1], [14, -2, -2, 0, -1, -4, -4], [7, -2, 1, -4, 4, 3, 0], [-1, 1, 3, -3, 4, 1, -6], [4, -7, -1, 5, 0, -4, 3], [10, -2, -6, 3, 1, 1, 4], [-2, 0, 0, -2, -1, 2, 4], [3, -2, -5, -2, 2, 3, -4], [-9, 5, 6, 1, -4, -3, -1], [6, 0, 3, 0, -2, -4, 1], [-4, -4, -1, 3, 1, 0, -5], [-1, 1, -2, -1, 0, 4, 3], [0, 0, 3, -4, 1, -2, -1], [0, 0, 5, 4, -6, -2, 7], [0, 3, -1, -1, -4, 0, 5], [-12, 3, 1, -5, 3, 2, -1], [10, 1, 8, -2, -4, -7, 2], [3, 4, -5, 2, -2, 2, 2], [-1, -2, -8, 0, 2, 2, -7], [-2, -2, -2, 0, -1, 3, -4], [-13, 1, 4, 3, 7, 0, -1], [10, -4, 1, 7, 3, 1, -3], [-2, 1, 6, 2, -4, -5, -2], [-5, 1, 4, 7, 2, 2, -5], [5, -1, -2, -5, -2, 0, -5], [4, -6, -8, 3, 6, 5, 2], [9, -5, -7, -3, 8, -2, -6], [2, 2, -6, -6, 6, 2, 0], [-11, -1, 5, 0, 0, -4, 6], [-6, 4, 0, 7, 5, -1, 2], [-2, -7, -6, 4, 3, -5, 0], [11, -3, 2, -5, 1, -6, -7], [-3, -7, -3, 5, 4, 0, 2], [-3, 4, -1, 2, 5, 3, 0], [-15, 0, -6, -5, 0, 2, 1], [14, 1, -8, -6, -2, -1, -4], [-8, 6, -8, -1, 1, 9, 0], [3, -4, 3, 3, -5, -1, -2], [-19, 0, -4, 3, 6, 6, 3], [9, 0, 1, -3, 4, -7, -8], [10, -10, -5, 8, 5, -3, -3], [-5, -2, -5, 2, -2, 0, -2], [2, 6, -5, -9, 2, -1, -5], [1, -2, 4, 6, 4, -6, -3], [3, -1, -5, 2, 4, 5, 4], [-3, 5, 9, -5, -9, -4, 6], [6, -1, 5, -3, 3, -4, 1], [-12, 10, 6, -4, -5, -3, -2], [4, -10, -3, 4, 1, -8, -1], [13, -7, -10, 1, 4, 2, 1], [-9, 7, 7, -6, -11, 0, 0], [15, 5, 12, 1, -6, -6, 3], [-6, 4, 5, 2, -6, -6, 1], [-14, -5, -1, 7, 3, 3, 7], [3, 2, -1, 1, -11, 2, 0], [-15, 8, 1, 4, 5, 0, -4], [-4, 4, 7, 0, 4, 4, -7], [5, -2, 6, -6, 2, 1, -1], [-8, 2, 3, 8, -3, -3, -5], [-13, 2, -6, -3, 5, 11, 5], [-4, -2, 3, -3, -2, -2, 5], [0, -11, 4, 5, 4, -4, 2], [-6, 1, -2, 10, 0, 1, 4], [8, 5, 4, -7, -3, -10, -2], [-20, 7, 6, -3, 3, -3, -4], [1, 5, 2, -5, 5, 4, -7], [-12, 9, 11, -4, 2, -1, -1], [9, 5, 1, 0, 3, 3, -4], [-11, 6, 1, -13, 1, 3, -2], [-1, 0, -10, -2, 6, 6, 1], [-5, 2, -3, -9, -5, -4, 2], [-1, 5, 3, 8, 9, -2, -4], [4, 9, -5, -4, 5, 4, -11], [3, 12, 11, -1, -10, -5, -8], [-7, -2, -1, 0, -1, 1, 10], [-15, 13, 4, -6, -8, 6, 3], [0, -3, -2, 0, 1, -5, 4], [3, 3, -1, 4, -7, -2, -4], [8, -3, 0, 12, 2, -4, 4], [-9, -2, 2, 2, 0, -4, 7], [-2, -1, -12, -5, 8, 8, -2], [12, -8, 0, 7, 11, 1, 2], [-1, 5, 0, -12, 0, 3, -3], [0, -15, 1, 10, 9, 1, 7], [3, 5, 1, -12, -9, 5, 6], [-2, 7, -9, -4, -1, 7, -7], [-19, 2, -3, 0, 7, -2, -6], [18, -6, -2, 6, -1, 2, 2], [-5, -2, 4, 8, -6, 0, -3], [-10, -7, 0, -6, 0, 5, -2], [0, 6, -4, -5, -2, 1, -6], [-12, -5, 0, 4, 9, -3, 8], [17, 2, -5, 0, 3, 0, -2], [5, -3, 0, 6, 9, -4, 1], [-18, 3, 6, -1, -1, 2, 0], [2, 1, -7, 2, 6, 7, -7], [0, -5, 3, 6, 1, -1, 1], [-1, -6, -2, -4, 3, -1, 3], [9, 2, 1, 5, -8, -3, 8], [10, -7, -3, 5, 9, 2, -1], [-2, -2, 5, 7, -1, -7, -5], [12, -4, 0, -3, 1, -9, -2], [-5, 8, -9, -2, 2, 10, -2], [7, -12, -6, 3, 6, -3, 3], [5, -5, 1, 2, -3, -4, -8], [-9, -4, 0, 3, 8, -1, 1], [4, 9, 17, 0, -3, -7, 3], [-18, -13, -7, 2, -1, -3, 11], [6, 5, 4, -6, -14, 1, -6], [3, 2, -4, -11, -8, 1, -3], [-14, -5, -3, 5, 4, -8, 1], [-15, -2, 12, 7, -1, -4, 11], [6, -1, -8, -3, 1, 10, 2], [3, -11, 8, 4, -1, -4, 3], [-17, 16, 16, -10, -5, 0, -1], [-4, 5, -13, -9, -1, 8, -9], [2, -5, 5, 13, 13, -1, -5], [14, -8, -8, -6, -9, 0, 6], [10, 4, 4, 0, 4, 2, -10], [6, -10, -1, 2, -5, -8, 9], [33, -16, -14, -4, -2, 0, 3], [-19, 9, 1, -5, -5, 14, 8], [-1, -22, -10, 6, 1, 4, 7], [11, 2, 0, -1, -11, -1, 5], [4, -4, 0, -4, 7, 2, -8], [-13, 1, 9, 8, -4, 6, 8], [-20, -2, -11, 3, 4, 7, 3], [9, -9, -9, 11, 2, 6, -2], [13, 1, -3, 0, 10, -1, -8], [-7, 11, -3, 7, 1, 8, 2], [-19, -5, -11, 2, 0, 9, 2], [6, 6, 13, -2, -4, -2, -3], [11, -5, -14, 3, 16, 4, 3], [8, 9, 10, -8, -1, -3, -12], [-10, 5, -10, -16, 1, 10, -6], [-12, 13, -2, -11, 3, 6, -6], [9, -7, -1, -8, 3, 0, -4], [7, -3, -3, -11, 1, -4, 0], [-3, -11, -9, 0, 9, 2, -2], [21, 9, 13, -1, -9, -12, 2], [3, -4, 0, 5, -5, 7, 9], [-13, 10, 4, 11, 6, 1, -5], [6, -3, -4, -11, 3, -4, 0], [4, 4, -7, -7, -1, 1, -3], [-8, 16, 9, -11, -11, 5, 3], [-3, -2, -1, -1, 11, 3, -12], [-1, -6, -2, 0, 0, 0, 15], [-5, -2, 14, 2, 6, -5, 5], [2, 17, 10, -5, -17, -2, 0], [-22, 1, -3, -1, 3, 4, -7], [13, 10, 9, -8, -1, 0, -6], [8, 1, 8, -1, -8, -4, -8], [19, -16, -14, 1, 0, 1, 13], [-14, 5, -3, 1, -14, -3, 1], [-4, -6, 2, 4, -5, -1, -4], [-1, 6, -4, -8, 1, 4, 3], [-36, 3, -7, -2, -1, -1, -3], [-2, -4, -3, 9, -6, -1, -3], [-7, 13, -15, -2, 1, 6, -4], [1, -3, 1, 11, -5, -14, -2], [3, -9, 7, 5, -1, -10, 10], [-8, 5, -12, -7, 9, 20, -4], [14, -15, -1, -1, 4, -6, -1], [-34, -1, 2, 1, 1, 2, 0], [11, -12, -15, -1, 12, 8, 8], [-6, -1, -18, 0, 6, 8, 6], [-1, -7, -4, 6, 14, 5, 9], [-13, 22, 4, -6, -4, -2, -9], [12, 10, 7, -5, 3, -13, -5], [-2, 13, -2, -17, -4, 2, -14], [2, 10, 5, -8, -7, 8, -5], [-11, -3, -9, 7, 3, 0, -4], [8, 0, 3, 7, 7, -2, -3], [4, -7, 4, 3, 8, -8, -10], [1, 10, 8, -3, -4, -6, 7], [-10, 11, 27, -1, -10, 0, -1], [26, -7, -9, 4, -8, -7, 13], [23, 16, 20, 0, -2, -3, -15], [2, -5, 11, -2, -4, -9, -1], [-15, 12, -5, 4, -6, 6, 2], [13, 1, -6, 5, -5, -2, -3], [-9, -1, 2, -1, -1, 5, 1], [-8, -7, 7, 13, 1, 8, 3], [12, -15, -7, 9, 5, -4, -3], [18, -8, -3, 11, 8, 1, 7], [-1, -3, -8, -8, -6, 3, 7], [24, -7, -7, -2, 10, 13, 9], [11, 8, 9, -10, -12, 2, -4], [0, -1, -11, 9, 8, 5, 11], [-3, 7, -11, 12, 8, 1, 4], [-8, -2, -8, 1, 13, 2, 12], [2, -13, -2, 6, -11, -9, 6], [15, 5, 6, -5, -17, -3, -1], [15, -17, -22, 0, 5, 7, 3], [-2, 8, 4, -12, 5, 4, -4], [3, -14, -11, -17, 13, -2, -4], [20, 6, -9, -2, 6, 17, -5], [27, 4, -11, -8, -7, -5, 0], [18, 5, 13, 2, -2, -15, -3], [-17, 7, -5, -10, -1, 13, 6], [-3, 19, 7, -3, -7, 3, 6], [-5, -4, -6, 6, -12, 8, 21], [-16, 22, 5, 0, -1, 5, -7], [-14, -2, 11, -6, -2, 8, -1], [7, 10, 2, 6, 1, -7, 1], [5, -2, 5, 10, 15, -4, -10], [18, -6, 9, 1, 1, 3, 1], [-3, 5, -7, 3, 2, 0, -2], [8, 1, 7, 0, 20, -1, -15], [28, 5, 2, -6, -1, -1, -8], [-23, -5, -3, 8, 6, -4, 10], [30, -7, 3, 4, 0, -12, -3], [6, -5, -10, 4, -7, 5, -4], [6, -1, -17, -1, 3, -2, -7], [12, 9, -2, -10, 6, 1, -6], [-1, -12, -5, 7, 19, -3, -12], [-28, 9, -11, 2, -4, 11, 9], [5, 5, -19, -7, 7, -4, -6], [30, 10, -2, -8, -17, -3, 4], [-23, 10, 14, 1, -2, 9, 3], [18, 14, 2, -7, 0, 11, -8], [3, 11, -12, -6, 6, -2, -7], [14, -4, -9, 2, 7, 4, 7], [27, -6, -17, 0, 9, 4, 0], [18, -4, -10, -15, 4, 9, 0], [-25, 10, 3, 3, 3, -8, -10], [5, -15, -11, 4, 4, 9, 6], [-16, 5, 9, 5, -19, -8, -3], [10, 26, 9, -10, -17, 2, 3], [2, 8, 15, -4, -13, 8, 1], [-3, -16, 4, 11, 9, -9, 3], [-18, -7, 4, -1, 6, 5, 4], [-12, 3, -1, -16, 0, 6, -15], [-12, 10, -8, -14, 2, -10, -4], [-31, 6, -4, 10, 4, 8, -7], [21, -2, 3, -11, -9, 0, -10], [32, 11, -1, -8, -13, 11, -7], [16, -11, 3, 7, -8, -1, -5], [-14, 7, -13, -7, 4, -6, -5], [3, 7, 17, -9, -15, -2, 8], [-14, -13, -21, 4, 7, -2, 13], [15, -4, -4, -16, -7, -6, 9], [-20, 5, -7, -9, -13, 4, 1], [2, -5, -19, 11, 9, -2, 1], [-8, 5, -2, -9, -10, -6, 4], [7, -4, -13, 10, 12, 0, 2], [-6, -5, 2, 9, -10, 0, -8], [7, -10, -6, 18, 6, -1, -7], [24, 5, 1, -3, -13, -6, -15], [13, 5, -13, 0, 0, -3, -4], [-15, -9, -2, 5, 5, 3, -5], [-3, -12, -3, 0, -1, 6, 2], [-21, -11, -2, 12, -2, 7, 17], [14, 14, -5, -3, -19, -1, 3], [-7, -7, -4, -15, 0, 8, -3], [25, 5, 0, 15, -10, -10, 7], [21, -9, 5, 8, 0, -9, 2], [18, -15, 4, -11, -5, 0, 2], [22, 4, 20, 2, -3, -2, -14], [-28, 9, 1, -3, -10, 0, 1], [1, -16, -12, 3, 14, -1, -3], [-10, 17, -12, -24, -11, 9, -8], [-27, 13, 11, 4, 10, -7, -2], [-4, 13, -5, -14, -2, 5, -3], [-3, -1, -29, -6, 12, 11, -8], [-3, 1, -9, 9, 9, 4, -8], [8, -5, 11, -5, -11, -10, 7], [3, -5, -6, -14, 8, 3, -1], [35, -10, -1, 8, -4, -7, 2], [-17, 1, 13, 8, 0, -1, 2], [38, 7, 7, -2, -6, -7, 3], [-26, 11, 4, -2, 0, 9, 4], [-19, 0, 6, -2, 1, -14, 9], [40, -11, 3, -5, 2, -12, -11], [-19, 7, -4, 12, -10, 1, 1], [46, 0, -7, -5, -1, -1, 5], [-34, -8, 8, 12, 4, 3, 2], [24, -16, -5, 4, -2, 10, 15], [-35, -7, 1, -2, -5, 3, 0], [-27, -5, -17, 1, 12, 18, -4], [23, -1, -10, -5, 3, 2, 1], [-25, -10, -18, 0, 18, 16, -5], [-11, -6, 2, 5, 1, 7, -3], [13, -12, -15, 12, 10, 0, -10], [-13, 2, -2, 5, -17, 5, 15], [21, 12, 1, -7, -15, -15, -2], [1, 4, -3, -17, 4, 0, 10], [-5, -2, -4, 7, -8, 11, 11], [32, -7, 3, 2, -7, -10, -7], [-14, -12, 9, 9, 17, -3, -3], [26, -6, -10, 5, 7, -9, 0], [-27, 2, 20, 5, 1, 0, 5], [-6, 11, 9, -15, -5, 10, 1], [1, -1, -3, -9, 11, -6, 0], [-19, 10, -9, -5, -1, -1, -4], [10, 2, 8, 13, 3, -11, -4], [8, -9, -15, -1, -17, 8, 9], [-18, -14, -7, -1, 4, 3, -11], [-2, 2, 3, -19, -17, 2, 7], [1, -2, -10, -1, 17, -11, -13], [8, -12, 3, -5, 6, -6, 9], [23, 5, 17, -2, -9, -15, 8], [-6, -12, -9, 3, 11, -9, -9], [5, -1, 7, 21, 1, -9, 16], [7, 18, 7, -5, 1, 10, -18], [9, -9, 20, 3, -2, -2, 5], [12, -9, -10, -6, 14, -3, -4], [23, 0, -9, -9, -14, 7, -6], [34, 13, 2, -3, 0, 5, -2], [-1, 17, 27, 4, -4, -4, -14], [-23, -6, 18, 9, 5, -3, 13], [23, -14, -3, 3, 7, -1, 4], [-10, 6, 10, 6, 0, -2, 2], [0, -11, -8, 7, 5, 10, 14], [-7, 29, 38, -2, -17, -9, -7], [-26, 9, -13, -20, -5, 13, 9], [20, 3, -3, -19, -9, 12, 5], [-24, 6, -15, -6, -2, 22, 7], [11, 2, -10, -4, -9, 0, 1], [5, 5, 26, 0, -1, -5, -5], [27, -29, -22, -2, 5, 5, 13], [21, 9, -18, -8, 9, 8, -5], [24, 11, 4, -12, -11, -9, -10], [27, -1, 8, 1, 0, -10, -21], [-10, -20, -5, 5, 22, 3, -3], [7, 16, 17, -3, -3, -3, -24], [28, -18, -12, 2, 5, -8, 2], [-35, -9, -15, 7, -6, 3, 16], [-14, 10, 10, -3, 16, -5, -10], [0, 6, 1, -3, 4, 11, -15], [17, 12, 10, 4, -9, -8, 7], [11, 18, -4, -17, -2, 11, -3], [2, 2, -19, -3, 7, 14, 5], [0, -8, 2, 0, -19, -3, 0], [13, -21, 5, -2, -6, -3, 2], [-26, 5, 25, -3, -5, -2, -3], [-19, -14, -5, 5, 13, -21, -6], [25, -24, -7, 4, 10, 2, 6], [-12, 0, 3, -6, -15, 7, 1], [29, 15, 4, -3, -4, 6, -7], [19, 26, 4, -14, -1, 9, -5], [-20, 27, 4, -8, -11, -1, -12], [32, -10, -1, 19, 2, -8, 11], [19, -6, -11, -7, 1, 11, 14], [-30, 4, 6, -15, -1, -1, -10], [4, -27, -33, 1, 10, 13, 9], [-12, -12, -4, 4, 1, -4, -2], [-23, 7, 5, 5, -22, 4, 6], [20, 7, -10, 6, -10, -3, 0], [14, 11, 16, 3, 2, -18, -14], [-11, -9, 5, 15, 13, -2, 8], [-28, 0, 7, 13, 3, -1, 3], [28, -6, 3, -10, -8, -6, 1], [-13, -10, -18, -8, 18, 8, 19], [8, -17, -27, 1, 8, -5, -1], [0, 2, 6, 14, 9, -15, -4], [-11, -1, 11, 16, 6, 11, -2], [-3, -25, -26, 0, 3, 9, 11], [-11, -20, 9, 18, -1, 0, 12], [-24, -1, 3, 4, -17, 2, 13], [-32, -7, 6, 3, 6, -10, 8], [-3, 29, 29, -4, -19, -13, -20], [19, -5, 8, 2, 5, -5, -1], [-9, -4, -7, 5, -18, -7, 12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4008_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4008_2_a_g();".
// 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_4008_2_a_g(:prec:=7)
    chi := MakeCharacter_4008_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_4008_2_a_g();".
// 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_4008_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4008_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4, 56, 133, 23, -50, -15, 3, 1]>,<7,R![-192, 336, -37, -147, 55, 11, -8, 1]>],Snew);
    return Vf;
end function;