
// Make newform 4030.2.a.d in Magma, downloaded from the LMFDB on 24 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_d();".
// 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_d();".
// 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 := [-1, -3, 7, 6, -5, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 4, 4, -4, -2, 1], [1, -1, -7, 2, 3, -1], [-3, 0, 8, -2, -3, 1], [2, -6, -6, 5, 2, -1]];
    Rf_basisdens := [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_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], [1, 0, 0, 0, 0, 0], [0, -1, 1, 0, -1, 0], [0, 0, -1, -2, 0, 1], [1, 0, 0, 0, 0, 0], [-2, 1, 0, 1, 1, 0], [-1, 3, 0, 0, 0, -1], [-2, -1, 2, 0, 1, 2], [-3, 0, -1, -1, -1, -3], [-1, 0, 0, 0, 0, 0], [-3, 2, 0, 3, 0, -3], [-1, 0, 2, 1, 3, 2], [1, -1, -1, -1, 2, 1], [-2, 4, -1, 0, -1, -2], [-2, 1, -3, -3, -3, 1], [0, -3, -5, 0, 2, -2], [0, 1, -1, -1, -2, 3], [0, -2, 1, 4, 3, 0], [-1, 2, -2, -2, 2, 0], [-1, 0, -1, -2, -4, -4], [0, 0, 1, -2, 2, -1], [0, -2, 1, 2, 3, 0], [-1, -5, 1, -4, -1, 2], [-3, -6, 0, 0, -1, 4], [-4, 3, 6, 0, 1, 0], [2, -7, 1, 0, 1, 1], [-8, -1, -1, -3, -4, 2], [-1, -2, 7, -4, -2, 4], [-4, 4, -5, -4, -6, -3], [3, -2, -5, -2, -2, -5], [6, 0, -6, -1, 2, -4], [-1, 0, 1, 4, 1, 1], [2, -2, 6, -5, -4, 3], [0, -4, -2, 0, 0, 5], [5, 5, 0, 4, -3, -1], [-2, -4, 4, 1, 7, 7], [-1, -1, -8, 3, 1, -5], [4, 1, 3, 2, 6, 2], [0, 3, -6, -2, 3, -1], [-1, -2, 8, -2, -2, 2], [-5, 3, 1, 2, -2, -5], [1, -3, -3, 5, -3, 2], [3, -5, 5, -4, 0, 5], [-12, -1, 3, 3, 0, -2], [-1, 10, -2, -2, -2, 0], [4, -9, 3, 4, 1, 5], [-3, 8, -9, 2, 1, -3], [-5, 5, 4, 6, -1, 2], [11, -8, 0, 3, 7, 9], [-7, -3, -3, -1, -3, -1], [0, -3, 6, 1, -1, -5], [-1, -7, 1, 10, 2, 1], [0, -1, 5, 8, -1, 0], [-4, -8, -2, -2, -2, 3], [-7, 0, 3, -1, -7, -5], [-2, -3, 8, 6, 4, -1], [4, -2, -2, -1, -3, 0], [-1, -1, -8, -5, 1, 0], [-7, 4, 3, 11, 1, -5], [-7, 3, 0, 0, -2, 4], [-8, 0, -6, -8, 0, 1], [4, -2, -1, -8, -7, 1], [4, 0, 5, 4, 5, -1], [-3, 0, 8, 1, -1, 2], [3, -1, -4, 1, 2, 4], [4, 10, 5, 9, -3, -5], [-5, -7, -4, -2, 5, -2], [-17, -1, 5, 3, 3, 3], [-8, 11, -3, -7, -8, -6], [-6, -9, 0, 0, 7, 2], [7, -2, -10, 7, 2, 0], [-7, -1, 4, 9, 6, -2], [3, 0, 6, 4, 3, 0], [2, 7, -1, 8, 1, 4], [3, 3, 3, 5, 4, 1], [-5, 5, -6, 0, 3, 1], [-11, 3, -4, 11, 7, -5], [-3, 1, -12, -5, -10, -5], [12, -9, 2, 7, 10, 9], [-7, -1, 1, 6, -6, -12], [-2, 3, 0, -7, -2, 4], [-1, 5, -9, -1, 2, -3], [5, -2, 3, 6, 8, 1], [-2, -6, 2, 2, 5, 4], [-9, 10, 6, 2, -5, -9], [2, 6, 9, -1, -6, 9], [-22, 10, 13, 6, 4, 2], [-4, -3, 3, 7, -3, 2], [9, 0, 0, -11, -3, 11], [-20, 2, -7, -3, -1, -1], [-3, -2, -4, -6, 6, -5], [-4, 13, 0, -4, -1, -1], [-14, 10, -2, -2, -9, -7], [11, -1, -4, 2, -5, -6], [-3, 14, 0, -1, -3, -11], [-11, 1, 2, -10, -12, 3], [-1, 0, 11, 11, 7, 5], [5, 4, -11, 1, -5, -6], [-6, 12, 3, 3, 1, 3], [2, -4, 2, 10, 19, 3], [-11, -1, 1, 2, -9, -8], [-13, -1, -4, 1, -2, -2], [-5, 5, 12, 3, 1, -5], [18, -6, -3, -2, -2, -5], [-1, -12, -1, -9, 1, 2], [-10, 2, 9, -4, 13, 8], [-4, -4, 7, 5, -14, -4], [-8, 3, 0, -3, -6, 0], [11, -4, 1, 1, 1, 9], [-13, 8, -8, -5, 1, 7], [0, -10, -8, -6, 7, -5], [2, 10, 0, -2, 6, 3], [11, -10, 3, -10, -3, 10], [10, -10, 10, -4, -2, 9], [12, -1, 9, 3, 6, 11], [21, -15, 1, -5, -5, -1], [-14, -9, 0, 3, 1, 8], [-11, 8, 5, 1, 7, 5], [4, -14, 14, -8, 0, 9], [8, 1, -12, 0, -4, -14], [-13, 8, 8, 4, -3, -2], [0, -19, 5, 1, 10, 9], [8, 0, 6, -2, 2, 3], [6, 6, -10, -5, -7, -4], [-21, 0, 8, -9, -4, -3], [4, 8, -17, 14, 7, -6], [-6, -1, 0, 1, -7, 3], [5, 6, -5, -17, -3, 0], [-2, -10, 10, -2, -3, -4], [-8, 1, 16, 1, -2, -4], [-1, -11, -18, -9, 5, -1], [10, -9, -1, 9, 5, 4], [-1, 11, -2, 4, 4, -6], [18, -8, -13, -5, 1, 7], [-15, 11, -3, -9, -10, -6], [14, -5, -6, 2, 2, -9], [-1, 2, 4, 4, 2, -6], [-1, -10, 3, 7, 0, 0], [-12, -14, -3, -1, 0, 1], [3, 5, 5, -10, -4, 1], [-13, 12, -7, -15, -3, -1], [15, 14, -3, -4, 0, -6], [7, 1, -9, -15, -10, -4], [4, 4, -9, 3, 10, -2], [12, -3, -12, 9, 5, -6], [2, -9, 7, -10, -3, 14], [-16, 17, -5, -2, 4, 2], [12, -6, 4, -10, 7, 20], [-8, 5, -8, 0, 6, 8], [15, -11, 8, 6, -13, 2], [-3, -6, -1, 8, 0, -10], [19, 2, -7, -15, -15, -3], [-7, -14, 13, -3, -9, 3], [9, -6, -12, -20, -10, 8], [12, 0, -11, 5, -5, -11], [-8, 10, 9, 12, 5, -4], [-5, 21, 7, 14, -5, -14], [0, 11, -13, 7, -7, -15], [-6, -4, -15, -14, -1, -2], [-1, 5, -1, 9, -10, -17], [-11, 8, -6, -11, -7, -9], [3, -7, 1, 14, 6, -4], [0, 0, 7, 13, 10, -7], [-20, -6, -3, 4, -1, 5], [11, -4, 14, 19, 1, -2], [24, -13, -3, 7, 7, 8], [0, 2, -2, -3, -8, 5], [5, 7, -6, 15, 11, -3], [15, 5, -4, 14, 5, -12], [9, 13, 20, 0, -9, 11], [16, -6, 12, 8, 7, -13], [10, -8, -2, 13, 0, 1], [21, -12, -4, 1, -14, 0], [8, -5, -2, 13, -1, 9], [-6, 2, -13, -8, -8, 4], [3, 4, 10, 1, 10, -3], [-1, -8, 11, 17, 11, 10], [-8, 11, -1, 3, 4, 1], [19, -4, -9, -5, -4, -6], [19, -13, -6, -1, 12, 12], [0, -9, -1, -10, -10, 5], [16, 4, 16, 2, -4, 3], [-18, 22, -7, -9, -13, -13], [3, 3, -5, -13, -2, -13], [-1, 2, -17, -8, -2, -22], [8, 11, -4, -10, -9, -2], [5, -5, 15, -6, -6, 11], [15, -6, 11, 0, -6, -3], [13, -6, -7, 4, -13, 3], [3, 4, -21, -5, 9, -9], [-11, 8, 12, 26, -2, -14], [2, 0, 16, 6, 14, 0], [10, -5, 8, -3, 8, 22], [-6, 6, 4, -3, 10, -7], [-11, 0, 6, -4, 5, 17], [-8, 13, -20, -5, 0, -4], [19, 16, -5, 4, 6, -14], [23, 5, -8, -2, -8, -6], [-30, 10, -1, -18, -23, -6], [-5, -2, 19, 5, -3, 10], [1, -2, -5, -3, -11, -2], [-4, 1, 4, -18, -11, -5], [-6, 10, 3, -1, -10, 8], [18, 5, 3, -1, -9, 10], [2, -16, -3, 16, -4, -1], [15, 16, 8, 1, 3, -6], [19, 11, 12, -2, 6, 1], [-1, -3, -19, 10, -6, -3], [-5, 0, 9, 6, -15, -2], [-12, -12, 10, -4, -17, 10], [-16, 12, -7, 8, 12, 5], [6, 6, -15, -8, -18, -22], [-13, -6, 16, -1, -21, -6], [-14, -5, 13, -11, -8, 18], [4, 2, -22, -3, 2, -14], [-27, -4, 10, 0, -7, 5], [-8, 9, 6, -5, -8, -12], [-9, 5, 10, 18, 3, 2], [22, -3, -13, 10, 13, 3], [16, 2, 8, 11, -1, -5], [11, 22, -7, 18, -1, -25], [9, -4, 23, -1, 9, 13], [10, 4, 0, -2, 10, 17], [-3, 20, 12, -8, -10, 8], [4, 10, -10, 15, 6, -7], [-20, -8, -5, -13, 1, 6], [15, 1, -3, 7, 4, -9], [6, 0, -17, -14, -15, 10], [20, -11, -18, -3, 1, 4], [-10, -3, 6, 15, 15, 1], [-13, 12, 1, -7, 8, 15], [-26, 5, 12, 4, 20, 12], [-23, -9, -4, -6, 4, -5], [-12, -18, 3, 11, 12, -3], [-4, 9, 10, -1, -14, -1], [4, 15, 2, -19, -20, 0], [6, 3, 1, -11, -1, 13], [-7, 6, 11, 4, -1, 4], [9, -13, 22, -3, 6, 31], [0, 13, -6, -5, -11, -22], [15, -16, 10, -7, -6, 9], [-4, -4, -19, -17, 17, 2], [-3, -12, 11, -2, 1, 9], [-9, 4, 4, -15, -2, 12], [-9, 10, 4, 14, 13, -4], [2, -6, 14, 15, 8, 15], [2, 11, 9, -8, 1, -7], [-14, 18, 15, 16, -6, -18], [-25, 15, -4, 6, 1, -13], [4, -9, 6, 21, 22, 7], [-7, -6, -18, -8, 15, -7], [3, 11, 7, -20, -8, 1], [0, -2, 0, -5, 1, -12], [-24, 11, -31, -4, -12, -22], [0, -4, 1, -2, -15, -7], [14, -22, 7, 4, -6, 6], [6, 8, -5, 14, 22, -13], [-23, 5, -23, -17, 12, -2], [-30, 13, 21, -3, -4, -7], [-11, 9, 4, -1, -13, 0], [-15, 16, -23, -11, 4, -16], [-23, 0, -9, 6, -1, -11], [17, 9, 7, -17, -4, 1], [-8, 4, -9, -15, -13, -8], [6, -6, -24, 9, 11, -4], [1, -24, 2, -2, 12, 10], [0, 10, -17, 9, 11, -23], [15, -13, 22, 3, 11, 20], [-2, 6, 24, -5, 5, 2], [3, 18, -20, 13, 6, -16], [-17, -12, 6, -6, 6, -2], [-10, 22, -12, -14, 8, -10], [-1, -3, -17, 16, 13, -1], [-20, 22, 14, 1, -10, 0], [0, -2, 16, 2, 12, 4], [10, -12, -22, 4, 16, -13], [2, 2, -4, -9, 9, 0], [6, 24, -10, 2, -9, -27], [35, -29, 10, 24, 18, 20], [2, -13, 7, -1, 5, 18], [-16, 9, -11, -6, -22, -24], [-4, 6, -4, 9, 5, -14], [-7, -4, 14, -22, -17, 14], [9, -3, 3, -22, -1, 9], [-22, -13, 10, 6, -1, -6], [-14, -19, 23, -15, -18, 14], [42, -2, -5, 2, 4, 8], [1, -2, 13, 29, 5, -15], [-8, 39, 3, -4, -10, -9], [-28, 23, 3, 22, 8, -15], [-35, 21, -11, -2, -5, -5], [-10, -9, 12, -2, 13, 28], [15, 16, 10, -25, -13, 5], [-2, 7, 28, 1, 7, 13], [-32, -10, -2, -7, 3, -3], [8, -3, -7, -4, -8, 18], [-12, 2, 20, -14, -7, -5], [-22, 8, 28, 1, 14, 3], [38, -12, -13, 7, 20, 24], [-25, 13, 18, 28, -2, -13], [48, 2, 6, 9, 1, -3], [-7, 22, -6, -8, -3, 0], [7, 6, -10, 16, -1, -15], [19, 4, 7, 22, 0, -6], [-20, 14, -19, 4, 10, -12], [-10, -23, -2, -4, 1, 10], [-33, -12, 18, -21, -6, 7], [23, 17, 8, -15, -2, 15], [-4, 0, 3, 9, 2, 11], [23, -19, -10, -4, 3, 5], [-39, 4, 17, -2, -1, 7], [-11, -2, 23, 11, 7, -8], [4, -3, 2, 5, 8, 3], [-28, -15, 9, -9, -22, 20], [-20, -2, -6, -23, -1, 5], [2, -16, -25, -12, 12, 1], [-2, -29, 0, 17, 31, 6], [-5, 10, -1, 9, 18, -10], [5, 0, 1, -18, -16, -5], [8, 8, 12, 1, 1, 7], [-5, -14, -1, 31, 18, -9], [-8, -1, 19, -3, -21, 3], [26, -21, 24, -13, -9, 13], [-21, 16, 6, -3, 1, -12], [-28, -16, 5, 16, -12, -5], [21, -13, -3, -23, -7, -9], [-5, -21, -1, 13, 8, -3], [11, 10, -10, 21, -9, -16], [-11, -11, 0, -10, 28, 24], [2, -9, -22, -12, -9, -3], [-5, -35, 3, 10, 12, 18], [2, -3, 2, -6, -17, -26], [-3, 6, -9, -1, -15, -34], [-4, -18, 6, 1, 2, 17], [-3, -10, 3, -15, 19, 29], [-11, -21, 1, -14, -10, 22], [-6, 13, -13, -35, -13, -6], [50, -21, 8, 5, -2, 0], [-19, 13, -2, -15, 2, -9], [31, -2, 9, -10, -17, 7], [20, -15, 1, 1, 9, -6], [19, -22, 17, 1, 7, 8], [-2, -9, -8, -7, -4, 0], [16, -7, 20, 10, -23, 2], [-10, -21, 1, 7, -12, -1], [7, 3, -23, -13, -20, -5], [13, -8, -10, -16, 5, 25], [-19, 27, -16, -11, -14, -3], [5, -31, 21, 0, 9, 24], [-18, 1, 27, -7, 17, 17], [20, 15, -17, 4, 0, -17], [-35, -6, 14, 6, -1, 21], [13, 3, -3, 16, 7, 6], [-33, 17, 14, -14, -26, -8], [-13, 5, 20, -1, -14, -2], [-11, -10, 25, -5, 3, -3], [-35, -1, 10, -23, -3, 4], [8, -11, 20, 12, 16, 21], [38, -25, 1, -1, 14, 24], [19, 22, -4, -20, 3, -14], [-10, 13, -25, -9, 23, 2], [2, 33, -15, 22, 4, -29], [-20, 15, 1, 18, 5, -18], [-15, 18, -6, -12, 17, 6], [-14, 18, 7, 12, -1, 4], [8, -1, 20, 8, 4, 0], [-18, 4, 4, 3, 38, 9], [-4, -34, 8, 6, 13, 0], [10, -13, 4, 11, 0, -1], [11, 6, -17, 15, -9, -2], [9, 15, 4, -7, 3, 29], [-7, -14, 13, 4, 12, 8], [28, 0, -7, 5, 18, 11], [29, -22, -7, 8, 1, 10], [8, -29, 9, 1, -8, 10], [2, 9, -13, 0, -15, -1], [-12, -10, -11, 8, 4, -15], [12, 21, -7, -6, -11, -17], [22, 6, -10, 21, 20, -22], [21, -2, 14, 10, -22, -10], [13, 7, -32, 11, 21, -23], [16, -3, -1, -6, -14, -17], [23, -3, -9, 15, 0, -16], [-43, 27, -14, -15, 3, -5], [4, 3, -33, -4, 0, -21], [-18, 24, -11, 15, 16, -3], [15, 12, -17, 11, 2, -21], [17, 14, -17, 12, -22, -19], [44, 0, 1, 10, 19, 9], [15, -16, 5, -26, 1, 25], [0, 9, -5, -22, -10, -16], [-8, 13, -8, -20, -25, -10], [-37, 14, -24, 4, 11, -8], [4, -13, 10, -10, -16, 13], [-6, -7, -1, -3, 6, 34], [-12, -12, -26, -22, 15, -3], [8, 22, -5, 0, 17, -2], [9, 22, -8, 4, 1, -7], [5, -22, -4, -7, 5, 15], [-40, 21, 13, 17, -2, -16], [-14, 0, -27, -28, -2, -4], [3, -6, 30, -13, -4, 10], [45, 3, 2, 13, -3, 12], [3, -22, 20, -4, -3, -10], [-9, 1, 28, 14, -19, -2], [31, -3, -6, -11, 14, 6], [-10, -12, 5, 1, -2, -27], [-26, -1, 9, -17, -27, 23], [11, -11, 0, -30, 10, 31], [0, 1, -21, -8, -9, -7], [2, 13, 30, 20, 2, 8], [26, 16, -17, -2, -14, 7], [33, 28, 6, -4, -8, -10], [-22, -3, 9, 11, -10, 4], [-4, -2, 1, 40, 22, -11], [-3, 25, -6, 1, -15, -36], [-31, -2, 1, -13, -20, 8], [23, -22, 0, -7, -13, 3], [-1, 4, 41, 9, -7, 30], [10, 6, -9, -6, 8, -25], [0, -12, 8, 3, 21, 31], [17, 12, 12, 16, 3, -1], [-10, -28, 0, 12, -3, 10], [-33, 15, 15, 18, 4, 11], [5, 15, -1, -5, 4, 13], [28, -37, 19, 27, 28, 23], [21, 2, 6, 1, -8, 0], [5, 10, -12, -14, 17, 21], [6, 2, 24, 27, -11, 9]];
    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_d();".
// 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_d(:prec:=6)
    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_d();".
// 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_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4030_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-9, 4, 16, -4, -8, 1, 1]>],Snew);
    return Vf;
end function;