
// Make newform 4030.2.a.o in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4030_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_4030_2_a_o();".
// 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_4030_2_a_o();".
// 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 := [4, 30, 25, -118, 24, 40, -12, -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], [-4, 0, 1, 0, 0, 0, 0, 0], [-6, -31, 38, 13, -13, -1, 1, 0], [-13, -25, 139, -61, -52, 25, 4, -2], [-64, -217, 484, -94, -172, 58, 13, -5], [-30, -87, 192, 2, -66, 10, 5, -1], [-116, -353, 854, -110, -302, 80, 23, -7]];
    Rf_basisdens := [1, 1, 1, 1, 3, 6, 2, 6];
    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_4030_a();"
function MakeCharacter_4030_a()
    N := 4030;
    order := 1;
    char_gens := [807, 1861, 2731];
    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_4030_a_Hecke(Kf)
    return MakeCharacter_4030_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, -1, 0, 0], [1, 0, 0, 0, 1, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, -1, 0], [0, 0, -1, 0, 0, 0, 0, -1], [1, -1, 0, 0, 1, 1, 1, -1], [1, -1, 0, -1, 0, 1, 1, -1], [1, 0, 0, 0, 0, 0, 0, 0], [2, -1, 1, 0, 1, 1, 2, 0], [0, 2, -1, 1, 0, 0, 0, 0], [4, -1, 0, 0, -2, -1, 0, 1], [1, 0, 0, -2, -1, 0, 0, 1], [2, 0, -1, 1, 0, -1, 0, -1], [0, 0, 1, 0, -1, 0, 0, 1], [3, 0, -1, 0, -1, -2, -3, 2], [-1, 1, -1, 0, 2, 3, 1, -2], [3, -1, 0, 1, -1, 1, 0, 1], [1, 0, -2, -1, 0, -1, -1, -2], [1, 0, 2, 0, -1, 4, 2, 0], [1, -2, 0, 0, -1, 0, 0, -1], [0, -1, -2, -1, -1, -2, -2, 1], [2, 0, 0, 1, -1, 1, -1, 1], [-3, 1, -1, -1, 1, 3, 2, -3], [4, -1, 1, 1, 0, -1, -4, 1], [0, -2, 1, 1, 0, 1, 2, 2], [-5, 2, -1, 2, 2, 1, -2, -2], [-1, -2, 0, -2, -1, 0, 2, 2], [2, 1, 0, -1, 0, 1, -2, 1], [0, -1, -2, 2, 2, -2, -2, 0], [1, -1, 1, -1, -1, 2, 2, -2], [-1, -3, 1, 0, 3, 1, 2, 0], [1, 1, 1, 3, -2, -2, -4, 3], [1, 1, -2, 1, -2, -4, -2, 1], [-2, 3, -2, 2, 4, 2, -2, -1], [1, -6, 3, 0, 0, 4, 3, 1], [2, 0, 2, -1, 1, -2, -3, 1], [-4, 2, -2, 2, 1, -3, -4, 2], [1, 1, 1, -4, -3, 1, 1, -1], [2, -2, 4, -2, -2, 1, 0, 3], [2, 1, -2, 0, -2, -3, 0, -2], [4, -7, 1, -4, 1, 2, 3, 0], [0, 2, 1, -1, 2, -1, 0, 2], [-1, 1, 0, 3, 1, -3, -2, 3], [4, -5, -1, -3, 0, -5, -2, 1], [5, -3, 1, 1, 1, 0, -4, 0], [2, 0, 1, 0, 0, 4, -2, 0], [1, -1, 3, 1, 0, -2, 3, 1], [3, -3, 5, -2, -1, -2, 0, 2], [2, 2, -2, -1, -2, -2, -2, 1], [-2, 0, 3, 3, 0, -1, 1, 3], [11, -3, 0, -1, -3, -4, 0, 1], [-7, 3, -2, 0, -2, 0, 3, -1], [1, -7, -2, 1, 2, 3, 1, -1], [-1, -2, -1, 2, -2, -6, -4, 1], [3, -2, 0, -2, -4, 4, 3, 2], [14, -1, 1, -2, -2, 0, 0, 2], [5, -4, 3, -1, 1, -2, 4, 2], [3, 3, -1, 6, 0, -2, -3, 2], [-1, -3, 6, 0, 0, 4, -1, 3], [0, 4, 3, 0, 0, -1, 0, -1], [0, -2, 0, 3, 0, 1, -1, -3], [-2, 0, -1, 3, -4, -4, -6, 4], [-1, -3, 2, -4, -3, 3, 6, -1], [6, -1, -1, 4, 0, -7, -4, 1], [-2, -2, -2, 1, -2, 6, 5, -2], [3, -1, -1, 6, -1, -4, -2, -2], [4, -2, -1, -3, -2, -3, 0, -2], [5, -3, 1, -3, -3, -7, 0, 3], [-5, 8, 2, 1, 1, 3, 2, -1], [-9, 0, -3, 2, 0, 0, 1, -2], [-4, -1, 0, -4, -4, 2, 3, 0], [-7, 2, 4, 0, 0, 3, -2, 4], [-4, -1, -4, 2, 2, -1, 4, -3], [-1, 5, -1, -4, -4, -2, 1, -1], [-3, 2, 1, 4, -2, -8, -7, 3], [0, 3, -3, 4, 4, 0, -6, -1], [-13, 1, -2, -3, 4, 6, 3, -1], [-7, 5, 6, 4, 2, -1, -1, -2], [-8, 3, -1, -1, -3, -1, -1, 1], [6, 1, 4, -4, -4, 3, 4, 5], [-6, 3, -1, 0, 0, 9, 3, -3], [-11, 4, 0, -5, 3, 5, 9, -5], [-5, 0, -3, -3, 2, -4, -1, -1], [-7, -2, -1, 0, 2, 1, 3, -5], [2, 9, -2, 4, -3, 0, -5, 1], [7, -3, -1, -3, 0, 7, 3, 0], [-1, -1, -1, 1, 6, 4, 7, -3], [0, -7, -2, -3, 0, 3, 5, 1], [0, 0, 2, 7, 1, 1, -3, 4], [4, 0, -3, -3, -1, 1, 5, 0], [-2, 5, 2, -3, -5, -1, 0, 2], [-4, -2, 5, 2, 0, 2, 2, 0], [-7, -1, -3, 5, 6, 2, -3, -1], [-14, 3, 0, 4, 1, 5, 3, -2], [-3, -7, 2, -1, 0, 1, 9, -3], [-6, -1, -5, -1, 1, 2, -1, -1], [-18, 7, -2, 1, 2, 5, 1, -7], [4, -7, 2, -7, 1, 7, 4, 2], [6, 3, 2, 1, 3, 2, 4, -1], [3, 9, 0, 0, -5, -3, -8, -1], [-13, 0, 3, 4, 8, 6, 1, -1], [7, -4, 0, -2, 3, 4, 2, 0], [-8, -3, 0, -8, -1, 4, 0, -1], [17, -3, 2, -1, -6, -3, 0, 2], [4, -2, -3, -7, 1, 0, 2, -1], [4, 2, 5, 0, 4, 2, -1, 2], [-13, 2, 3, 1, 1, 10, 6, -2], [-7, 3, -7, 3, 6, -2, -1, -3], [6, 1, 4, 1, -1, -7, -2, 2], [9, 7, -1, -3, -6, 0, -4, -3], [0, 8, 5, -3, -1, -4, -1, 1], [-9, 0, 3, -6, -3, 7, 6, -4], [11, 2, 2, -3, -3, 4, 2, 6], [19, -8, 0, 0, -4, -3, 1, 1], [1, 6, 1, 5, 3, -3, -4, -1], [-3, -4, 1, -1, -1, -2, 0, 6], [-1, -1, 0, 0, -6, -2, -7, 1], [0, 1, 1, -2, 0, 8, 3, -4], [2, 5, 2, -2, -2, -4, 3, 2], [-7, 0, 2, 5, 5, 2, -2, -2], [-1, -5, 4, 0, -2, 8, 1, 3], [-1, 1, 6, -1, -1, 3, 1, -1], [10, 1, 2, 1, -5, -7, -5, 4], [-9, 1, -1, 4, -4, -7, -7, 2], [8, -9, -1, -3, 1, 3, 5, -2], [6, -7, -3, -1, 3, 4, -1, -5], [-7, -6, 3, 2, -1, 3, 4, -3], [-12, 0, 2, -3, -6, -2, 6, 2], [7, 0, 1, 2, 2, -8, 1, -1], [-13, 4, 0, -3, 3, 4, 2, -1], [1, -2, 2, -4, -1, 3, -4, -1], [2, 5, 4, -1, 6, -1, 3, -5], [-11, -4, -4, 2, 0, -1, 3, -6], [1, 9, -1, 1, 3, -1, -2, -2], [1, -6, 5, -1, -4, 0, -1, 2], [3, 7, -2, -1, 2, 1, 3, -4], [-9, 5, 2, -4, 1, 4, 10, -4], [-8, 10, 0, 4, 4, -1, -2, 3], [9, 2, -1, 1, -1, -2, -4, 6], [9, 10, 1, 4, -4, 3, -4, 2], [5, 4, -3, 5, 4, -1, -2, 4], [6, -5, 7, -2, 0, -5, 0, 4], [-17, 6, -9, 5, 1, -5, -8, -3], [1, 8, 1, 1, 2, -1, 1, -2], [8, -2, -2, 5, 5, -12, -3, 1], [-7, 2, -4, -1, -1, -1, -1, 1], [-12, -3, 0, 1, 0, -6, -4, -2], [8, 0, 9, 2, -4, 6, 6, 4], [-7, -2, 2, -3, 4, -11, -1, 2], [7, -4, 1, 3, 5, 2, 2, 2], [-1, -4, 1, -9, -5, -4, 4, 4], [-11, 7, -9, 2, 3, -1, 1, -9], [4, 1, -5, -2, 0, -5, -8, -1], [-11, -5, 2, -3, 2, -1, 4, -2], [8, -2, 4, -2, -2, 1, 6, 0], [-16, 0, -1, -5, 6, -5, 4, -1], [-11, -1, -1, -8, -2, -3, 3, 0], [-4, 13, -6, 0, 2, 7, 4, -5], [11, -12, -2, -7, 5, 3, 4, -4], [9, -3, -6, 2, -2, 1, -3, -2], [13, -3, -4, 0, -7, -6, 0, 6], [2, 4, -8, -1, 6, -7, -5, -1], [16, 8, -1, 1, -6, 0, -8, 0], [-3, 9, 1, 9, 4, 7, -3, 0], [5, 7, 1, -10, -7, -5, 4, 0], [-9, 7, -4, 2, 3, 3, -6, 2], [0, -12, 5, -4, 0, 0, 2, 0], [17, -9, 3, -2, 6, 2, 1, 3], [-6, 3, -11, -2, -5, -7, -3, -2], [9, 4, 3, 3, -2, 6, -2, -1], [9, 7, -1, -5, -12, -4, -7, 2], [-14, 4, 0, 5, 2, -2, -4, 5], [-16, -3, 8, -2, -3, 6, 1, 1], [0, 7, -4, 7, 7, 5, -5, -10], [4, -4, 0, -7, 0, 11, 2, 0], [1, 2, 2, -5, -5, 0, 0, 0], [9, -1, -1, 2, -2, -9, 0, 2], [9, 0, 9, 2, 0, 4, 5, 0], [7, -5, 2, -1, -5, 8, 0, 2], [-17, -1, 1, 1, 4, 8, 4, -5], [10, -13, -1, 4, 2, -1, -6, 1], [0, 4, 7, 0, -3, 8, 1, 8], [3, -4, -9, 6, -2, -5, -6, 2], [9, -12, 6, -3, 0, 6, 3, 6], [-22, 10, -2, 6, 0, -1, -6, 3], [-10, -8, -5, 1, 1, 0, -9, -4], [11, 1, -1, 5, 7, -10, -2, 4], [-13, -5, 2, 3, 8, 5, 3, -3], [13, 1, 4, -9, 1, 7, 10, 1], [-6, -3, 1, -6, -1, 3, 5, -3], [-2, 12, 4, 5, 2, -4, -2, 2], [19, -3, -1, -4, 1, -7, 4, -4], [7, 2, -2, -2, -6, 10, -8, 5], [12, -16, 6, -1, 6, 4, 6, 4], [27, -6, 0, -5, 0, -9, -7, 2], [-3, -2, -1, 5, 9, 1, 1, -1], [1, -5, 9, 2, -6, 3, 5, 2], [-2, -12, 0, 3, 8, 3, 12, 0], [18, 11, -1, 3, -5, -2, -9, 1], [16, 10, -3, 11, 3, -1, -3, -2], [0, -10, -3, -5, -1, -4, -8, -1], [25, -9, 0, 2, 0, 1, -6, 4], [-9, 4, 4, 2, -1, -6, 0, 12], [2, -8, 1, -5, -4, 6, 2, -2], [-20, 9, -4, 6, -1, 7, 1, -6], [0, 0, 4, -4, -2, 17, 2, 1], [16, 0, -9, -11, -3, 2, 1, -7], [-11, -1, 2, -2, 2, 2, -1, 2], [5, 1, -5, 4, -12, -7, -9, 3], [-12, -9, -4, 2, 15, 4, 3, -9], [10, -2, -3, -3, -8, 0, 0, 0], [2, 7, -1, -4, 5, 16, 12, -5], [-14, 5, -2, 2, -2, -18, -6, 4], [-1, 10, -1, 7, -2, -2, -5, -1], [-2, -7, 9, 1, 3, 19, 13, -5], [12, 10, -7, 0, 2, 0, -2, -4], [8, 5, -9, -1, 4, -1, -6, 3], [4, -2, 2, -5, -3, 9, 12, 6], [-11, 13, 1, 9, -3, 0, -4, 2], [7, -16, -1, 1, -5, -4, -6, 2], [19, 0, -5, 0, -3, -13, -6, 10], [-7, 2, 7, -5, -8, 18, 5, 4], [5, -6, 0, -2, 2, -16, 1, 3], [-9, -12, 3, 0, -4, 7, 7, -2], [4, -7, 12, -10, -2, -7, -2, 9], [19, 11, -6, 1, -6, -11, -4, 2], [7, -1, 0, -3, 0, 0, 5, -2], [8, -1, -1, -3, 1, 1, 3, 2], [-20, -4, 2, 1, 2, 9, 8, -5], [2, -1, 6, 1, -10, -2, -4, 10], [-12, 11, -12, 10, 1, 3, -3, -8], [5, -7, 12, -4, -1, -8, -4, 8], [11, -3, 6, -9, 8, 4, 11, -9], [-24, -10, 0, -9, -2, 6, 6, 0], [12, -2, 1, 3, 4, -1, -2, -1], [22, -4, 7, -7, -3, -6, 1, 1], [24, -8, 2, -8, 1, -7, -3, 2], [11, 8, -9, 3, -3, 13, 0, -7], [-8, -11, 5, -2, -3, -9, 0, 6], [-3, 7, -2, -2, -1, 8, 2, -3], [-3, 3, 1, -3, -7, 2, 6, 2], [7, 4, -4, 3, 5, -17, -4, -5], [4, 6, -6, -1, -5, 5, -3, -3], [-15, 16, 4, -4, 6, 4, 1, 3], [4, -11, -9, -5, -2, -13, 0, -3], [0, 7, -4, -2, -1, 0, -13, -1], [-9, 1, -3, 2, 4, 0, 5, -9], [15, 17, -4, -1, -6, -8, -3, 1], [3, 8, 3, -3, -8, 4, -9, 1], [-4, 10, -8, 8, -4, -1, -8, -3], [3, -1, 4, 11, 7, -7, 1, 3], [4, -16, -4, -1, 3, -12, -5, 1], [-8, -1, 2, 6, 3, -6, -1, -4], [-16, -5, -6, 6, 11, -2, -3, -1], [17, 2, -3, -3, -3, -4, -5, -2], [1, -2, 3, -9, -6, 3, 2, 0], [3, -14, 9, -2, -7, 0, 9, 4], [10, -9, -3, -10, -4, 0, 8, 2], [11, -5, 12, 5, -2, 1, -3, 8], [-25, 1, 3, -1, 13, 6, 2, -2], [-2, 1, -5, 1, -6, -1, -7, 5], [17, 6, -7, -4, -2, -15, -9, 6], [-13, -1, 7, 2, 15, 19, 12, -7], [23, -5, 7, 6, 1, -1, -7, 9], [-12, 3, -3, 4, 4, -1, -6, -5], [2, 11, -7, 8, -4, -2, -4, -4], [9, 14, -4, 12, 2, -2, -5, 2], [-19, 8, -4, 5, -3, 11, 4, -5], [5, -8, 13, 9, 2, 0, 5, 8], [12, 6, 0, 6, 8, -8, -2, 2], [25, 0, -11, -8, -7, 0, 0, -4], [-2, 13, 2, 2, 6, -4, 2, -6], [2, 13, -3, 8, -5, 3, -5, 5], [27, 7, -6, -6, -10, -8, -7, 7], [5, -1, 5, -8, -3, 3, 4, 1], [-20, 14, 6, -6, -4, -2, 6, 1], [21, 0, 2, 4, 5, -9, -8, 7], [-1, 3, -5, 3, -6, -13, -7, 10], [4, 6, -4, 7, 3, -8, -1, -1], [-17, 11, 1, 2, 9, 9, 6, -5], [-22, 4, -8, 5, 7, 9, 11, -5], [1, 0, -5, 2, 5, -6, -8, 0], [-13, 3, -1, 1, 7, -1, 6, 1], [-2, -9, -2, 1, 17, 16, 11, -13], [6, -1, -6, 2, -7, 12, 5, -1], [-21, -6, 5, -3, 1, 8, 12, 0], [7, -5, -2, -9, -1, 14, 4, -2], [17, 1, -4, 5, -7, -2, -12, 10], [-7, 22, 2, -7, 7, -4, 1, 2], [21, 5, 3, -3, 3, 7, 10, 3], [23, 1, 7, -3, -4, 4, 5, -1], [-8, -11, -7, -7, 3, 1, 7, 0], [-10, 15, -2, 0, 14, -2, -4, -3], [-8, -8, 6, -1, -1, 16, 15, -3], [18, -5, -2, -1, -1, 1, 4, 0], [-3, 13, -11, 2, 1, -19, -10, -2], [2, 10, 7, -11, -1, 14, 8, 5], [18, -13, 4, -6, -3, 13, 15, -4], [20, 4, 4, 3, 5, 3, 2, -6], [2, 19, -5, 1, -4, -10, -16, -4], [19, -16, 12, -10, -4, -16, 6, 9], [9, 11, -11, 1, -3, -8, 2, 1], [6, 1, 2, -10, 3, 2, 11, -1], [11, -11, -3, 9, -5, 3, 2, -5], [-26, -5, -6, 5, -3, 1, -7, 4], [6, -2, 0, -2, -11, -9, 5, 6], [-9, -2, 1, -2, -4, -5, -11, 1], [-3, 7, -6, 1, 8, -2, -4, 3], [17, 3, 1, -1, -3, -8, 8, 8], [10, 0, 10, -3, -8, -5, -10, 10], [5, -5, 6, -15, -2, 8, 13, -7], [30, -1, 6, 7, -7, -6, -7, 3], [-20, 4, -1, -2, -3, 1, 5, -2], [11, 2, -10, 2, 17, -4, -6, -12], [13, 10, 5, 1, -9, -7, -4, 7], [7, -1, 14, 3, 8, 1, 9, 2], [-2, -3, -4, -5, -16, 5, 4, 3], [26, 0, 0, 3, -9, -9, -18, 12], [16, -2, -12, -9, 0, -9, -4, -5], [-16, 10, -1, -5, 0, -1, 4, -11], [30, -1, -3, -11, -5, -1, 1, -2], [10, -7, 10, 3, -11, -7, -7, 11], [-3, 8, 4, 2, -3, -2, -6, 4], [-11, 10, 2, -4, 0, -1, 7, 5], [-2, 19, -6, -2, 4, 3, 4, -1], [10, 1, 2, -5, -3, 3, 1, -5], [-11, 6, -16, 4, -3, -9, -6, -1], [22, -1, -6, 7, -6, -12, -8, 3], [-13, 11, 9, 3, -4, -3, 2, 0], [-27, 10, 12, -5, 2, 2, -1, 0], [6, 11, -4, 8, 1, -14, -14, 5], [-11, -8, -2, -4, 0, 18, -3, -3], [21, 2, 3, -7, -9, -4, -2, 10], [1, -19, 2, 2, -10, -7, -7, -2], [11, -1, 0, -8, 2, 1, 0, -4], [-32, 0, 6, 3, 14, 12, 0, -8], [-28, -4, 7, 5, 9, 7, 2, 4], [-7, 7, -11, -3, -3, -12, -14, 0], [13, 15, -7, 3, 12, -1, -5, 2], [38, -5, 11, -7, 3, 10, 11, 7], [-14, 0, 0, -7, 0, 8, 10, -16], [5, 5, -7, -9, -13, 1, -2, -5], [1, -8, -4, -3, 3, 0, 8, -12], [-22, -7, 2, 2, 8, 13, 4, -3], [-7, 3, 4, 5, -1, 11, -8, 10], [-12, 10, 1, -1, -1, 15, 10, -4], [10, -3, -10, 5, 11, -4, 6, -5], [18, -2, 3, 3, 2, 3, -8, 5], [9, 11, 0, -1, 3, 6, 6, 4], [-25, -11, 0, -3, 7, 5, 0, -12], [12, 16, 12, 4, -4, 5, -4, 0], [3, -12, 0, 6, 15, -3, 6, -1], [-17, 9, 1, -1, 5, -5, -5, -5], [21, -4, 4, 2, -1, -2, -3, 2], [4, 5, -13, 0, 2, -8, 8, -8], [23, -14, -5, -10, -2, -18, -4, 1], [6, 1, 7, -4, 11, 6, 12, 3], [-32, 1, -1, 3, 8, -6, 8, 0], [-27, -21, 1, -3, 10, 3, 15, -4], [-3, 2, 3, 7, 6, -1, -7, 1], [33, 10, -4, 10, 4, 5, -14, 2], [-14, 5, -6, 4, -4, -12, 1, -4], [3, -8, -2, 6, 9, 4, 0, 1], [20, -1, -8, -5, -10, -10, 4, -2], [21, 4, -2, 0, 8, 4, 9, -5], [-7, -3, -4, 2, -10, -4, -9, 2], [25, -9, 8, -1, 0, 2, 7, 5], [10, -1, -15, -11, 3, 4, -1, -7], [17, -8, 6, -11, -14, -2, 10, 1], [-11, -10, 8, -7, 6, -7, 9, 0], [9, -5, 5, 3, -9, 6, 9, 6], [-6, 18, -7, 5, 5, -2, 3, -9], [6, 3, -14, 9, 13, 8, 11, -15], [26, 1, 0, 0, -6, -27, -5, 7], [-11, 6, -12, 6, 7, 14, 10, -10], [-6, -11, -2, -6, -4, -15, -1, 9], [10, 1, -6, 4, 4, -2, 0, 4], [3, -7, -10, 3, -3, -1, -6, -1], [15, 8, -3, 10, 6, 8, -17, -3], [-9, -5, -10, -1, -1, 10, -6, 0], [7, 2, -16, 0, 6, -1, -3, -10], [-29, 0, -3, 4, 2, 10, -1, -1], [-3, 1, -12, 1, -9, -18, -6, 2], [1, -7, -8, -16, 1, -2, -2, -4], [-14, -2, -6, -7, -14, 7, -4, -5], [26, -7, 3, -2, 4, -1, 15, -3], [-18, 19, -16, 4, 9, 2, -3, -12], [15, -13, -1, 1, 11, -14, -2, 2], [30, -4, 3, -10, -3, -6, -4, 7], [-1, 13, -2, -2, -5, -3, -19, 1], [-40, 8, -4, -11, -1, 3, 1, -7], [33, -4, -4, -1, 6, -4, -1, -3], [4, 3, 12, -1, -1, 2, 1, 3], [-20, 5, -22, -1, 3, -11, -2, -8], [-9, 9, 4, -11, 4, 1, 6, 0], [-13, 4, -8, -2, 8, -4, -4, -7], [-1, -13, 3, -17, -4, 6, 7, -9], [-16, 1, -2, -5, 3, 18, 25, -11], [-4, -13, -6, -4, 3, 6, 7, 3], [23, 10, -1, 10, 7, -9, 1, -3], [-6, 0, 10, 8, -1, 1, -11, 15], [26, 13, 5, -3, -16, -2, -12, 10], [34, -3, 0, 6, 13, -13, -13, 5], [-7, 7, 10, 2, -9, -17, -2, 7], [9, -24, 21, 2, -8, 3, -3, 17], [20, 16, 3, 5, 4, -11, -16, -3], [22, -19, -10, -2, 5, -9, -3, -2], [2, -4, 0, 7, -7, -22, -15, 6], [-6, 16, -19, -3, -7, -21, -1, -9], [11, 22, 8, 6, -13, 12, -4, 10], [9, -7, -1, 0, 3, 5, -5, -3], [-38, -3, -4, 11, 13, 2, -11, -13], [15, -14, -12, 4, -1, -4, -2, 0], [48, -5, 8, -7, 2, 6, 13, -4], [-29, 5, 0, -9, 0, 1, 7, 0], [25, -5, -5, -5, -4, -6, 5, 1], [18, -3, 3, 11, 9, 8, -3, -1], [1, 1, -3, -12, 6, 5, 17, -13], [38, -4, 6, 11, -13, -17, -8, 10], [1, -4, 3, -13, -4, 8, 5, 1], [19, 10, 1, 8, 13, 3, 2, -13], [-34, 24, -13, 5, -4, 1, -7, -19], [-8, -2, 3, 8, 15, 4, -11, -6], [-31, -8, 8, -7, -13, 0, -2, 6], [23, -1, 5, -5, -4, 0, -5, 4], [-3, -14, 3, 15, 2, -8, -7, 5], [4, 4, -1, 12, -5, -11, -5, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4030_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4030_2_a_o();".
// 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_4030_2_a_o(:prec:=8)
    chi := MakeCharacter_4030_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_4030_2_a_o();".
// 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_4030_2_a_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4030_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, 30, 25, -118, 24, 40, -12, -3, 1]>],Snew);
    return Vf;
end function;