
// Make newform 4030.2.a.f in Magma, downloaded from the LMFDB on 19 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_f();".
// 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_f();".
// 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, 12, -1, -7, 0, 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], [-2, 0, 1, 0, 0, 0], [0, -4, 0, 1, 0, 0], [1, 3, -4, -1, 1, 0], [2, 8, -1, -6, 0, 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], [-1, 1, 0, 1, 0, 0], [-1, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, -1], [-1, -1, 1, -2, -2, 1], [1, 0, 0, 0, 0, 0], [-1, -1, -1, 1, 1, 0], [0, 0, 0, -2, 1, 0], [-2, -2, 1, 0, -1, 2], [-1, -3, -1, -1, 1, 1], [-1, 0, 0, 0, 0, 0], [-1, 0, 3, 1, 1, 0], [0, 2, -2, 3, 1, 0], [1, 3, -2, -3, 0, -1], [-2, -3, 2, -2, -2, 1], [-2, -2, 0, -3, 1, -3], [-2, -3, 5, -2, -1, 1], [-2, -1, 2, -1, -2, -1], [-2, 5, 0, 0, 0, 3], [-3, 0, -2, 4, 0, -4], [3, 0, -2, 2, 5, -5], [-5, 3, 1, 4, 0, 1], [-2, 5, -2, -2, 0, -1], [-1, 2, 1, 6, 0, -1], [-1, 3, 0, -2, -1, 4], [-6, 0, 1, 0, 1, 4], [-1, -5, 4, 0, 1, -1], [-5, -2, 5, 1, -1, 1], [-1, 6, 0, 4, 5, -1], [1, 1, -7, 4, 6, -1], [-4, 1, 1, 2, 4, -5], [-7, -1, 0, -5, -2, 4], [-4, 2, -1, 4, -3, 7], [-2, 2, -3, -1, -5, -2], [-5, -3, -3, 0, 3, 2], [2, 1, -6, 4, 2, -2], [-4, -5, 1, 1, -4, 2], [-1, -2, -2, -1, 4, -4], [-2, -1, 1, -2, -5, -1], [-1, -1, 2, -4, -8, 2], [-15, 6, 2, 2, 0, 0], [6, 6, -4, 4, 6, -3], [-10, 1, -5, 9, 0, -5], [4, -2, 4, -10, -4, 7], [-1, 4, -1, 7, 7, -7], [-1, 2, 4, 2, 4, -2], [-3, 1, -6, -2, -5, 1], [-2, -10, 1, -2, -3, 7], [-3, 0, 3, 0, -3, 6], [-3, 7, -5, 9, 2, -6], [7, 2, 2, -3, 5, -3], [-10, 4, -4, 5, 4, 0], [2, -4, 6, -8, 4, -1], [2, 0, -1, -2, -4, 3], [-13, 7, 3, 4, -1, 4], [-1, 1, 3, 1, -5, -1], [-1, 0, 0, -6, -5, 2], [13, -4, -8, -3, 1, -2], [4, 1, -3, 7, -1, 6], [-3, 3, 5, -3, -9, -1], [1, 7, -7, 2, -2, -10], [-1, -3, -5, -4, -9, -2], [-5, 4, 1, 16, 3, -1], [13, -6, 5, -16, 3, 3], [4, -6, 4, 1, -1, 6], [-10, -2, 5, 1, 0, 2], [6, 17, -9, 11, 5, -3], [13, 6, -3, 2, 7, -10], [3, -8, 4, -17, -3, -1], [-13, -12, 5, 1, -5, 5], [4, -2, 5, -4, -5, 8], [-12, 9, -4, 9, -2, -6], [10, -2, 5, -9, 0, 4], [7, 1, -6, -2, 3, -2], [-16, -4, 3, -2, -10, 7], [-3, -9, 2, -9, -10, 9], [-12, -5, 4, 4, 0, 0], [3, 0, 2, 7, 4, -10], [7, -7, 12, -11, -1, 4], [2, -9, 6, -9, 3, 0], [-1, -5, 15, -2, -5, 3], [1, 2, -5, 9, 6, -6], [13, 19, -8, 5, 4, -5], [24, 5, 1, 2, -2, 1], [-22, 13, -6, 14, 3, -6], [-10, 4, 9, 0, -6, 8], [12, 4, -1, 5, 4, 5], [12, -6, 2, -4, 3, 3], [7, 1, 1, -7, 2, -1], [5, -7, -7, 1, 4, -2], [6, 1, -1, 1, -1, -1], [4, -3, 5, 8, 5, -2], [1, -9, 4, -8, -4, -10], [-13, 2, -3, 12, 12, -12], [-17, -2, -7, 12, 5, -8], [5, -5, 1, -7, 6, -2], [8, 16, -6, 2, 3, -4], [3, -3, -11, 7, -1, 3], [8, 10, -14, 7, 4, -3], [0, -9, 5, -11, -7, 15], [17, 4, -5, -2, -4, 0], [5, 2, -7, -2, 2, 1], [-2, 0, -9, -1, -8, -6], [-7, 6, -10, 1, -2, -14], [1, -9, -3, -8, -6, 9], [6, -14, -10, 3, 6, 1], [6, 9, -6, 4, 2, -9], [-7, -15, 10, -9, -11, -3], [-13, 8, 3, 1, -6, 0], [15, -3, 9, -11, 7, 1], [1, 7, -15, 5, 8, 2], [-1, 10, -3, 16, 10, 2], [-1, 3, -3, 2, -5, -6], [-3, -1, 14, -4, 4, 1], [-9, -9, 5, 2, 5, -10], [-4, -9, 10, 3, -4, 1], [13, -10, 4, -7, -3, -3], [3, -1, 9, -3, 1, 12], [-13, -3, 1, -5, -7, -5], [-21, 5, -3, 10, 3, 0], [8, -5, -5, 10, -2, 2], [-3, 7, -6, 14, -5, 4], [4, 3, -8, 3, 2, -9], [1, 7, 3, 6, 13, 4], [-5, -10, -4, 13, 9, -10], [11, -14, 1, -7, 1, -8], [0, 5, -6, -2, -10, 3], [10, 10, 6, -9, 4, 4], [-2, -20, 4, -13, -10, 7], [24, -11, 6, -2, -1, 6], [-5, 14, 3, -7, -4, -4], [-3, -16, 16, -13, -6, 6], [-21, -3, 1, 7, 4, -7], [-11, -9, 7, -18, -22, 12], [-2, 5, -17, 7, 11, -3], [-4, -10, -9, 3, 3, -11], [7, -12, 2, -20, 7, 2], [-1, 6, -18, 0, 6, -2], [10, 7, -10, 7, -7, -5], [-4, -6, -5, -7, -10, 15], [6, 8, 6, -14, 0, 11], [3, 13, -13, -1, -1, 3], [3, -8, 8, 4, -7, -3], [-8, -18, 7, -3, -7, 11], [5, -1, -12, -9, 1, -11], [3, 3, 7, -11, -3, 6], [4, 10, -9, -2, -4, -9], [2, 9, -13, 12, 3, -11], [-8, 3, 12, 16, 5, -4], [4, -17, 3, -24, -6, 6], [13, 4, 5, -16, -11, 12], [-5, 4, 6, 14, -3, 3], [5, 3, 5, -5, 1, 7], [11, -9, 1, -17, 7, -11], [-3, 6, 10, -6, 4, 0], [-12, 1, -11, 1, -1, -15], [10, 1, -20, -2, 10, -9], [1, -18, -1, -4, 6, 9], [18, 6, -4, -7, 7, 3], [14, 5, -10, 6, -4, 1], [1, 9, -4, -1, -10, -3], [9, 1, 3, 9, 0, 14], [-3, -1, 5, -18, -19, 11], [-2, 6, 13, 1, 8, -1], [-11, -6, 5, 12, -3, 1], [0, 8, -8, 5, 9, 4], [1, 3, 1, 11, 14, -7], [-6, 4, 21, -1, -7, 4], [13, 20, -4, 3, 0, 0], [1, -2, -9, -14, -13, 0], [-18, -9, 8, 8, -12, 2], [13, 2, -7, 6, -12, -8], [2, 4, -17, -3, -7, 4], [8, -7, 18, -3, 0, 20], [-14, -22, 4, -13, -12, 22], [2, -6, 12, 12, -1, 15], [5, -8, -1, -11, -9, 16], [-2, 2, 6, 3, 10, -7], [-8, -5, 10, 9, 14, -1], [4, -25, 4, -15, -9, 13], [-8, 6, 3, -9, -2, -12], [-1, -8, 6, -14, -8, -9], [1, 3, -3, 4, 13, -8], [12, 15, -11, 5, -1, -9], [-25, 7, 6, 7, -8, 9], [-5, 0, 10, 22, 11, -11], [-6, 8, 7, -14, -5, 12], [-2, 2, 8, 4, 6, -7], [-8, 1, 3, 8, -8, 1], [10, -6, -9, -18, -1, -1], [3, -21, 1, -7, -7, 15], [-25, 2, -16, 16, 10, -14], [-28, -16, 0, 2, -14, 0], [-13, -6, 3, -21, -8, 0], [10, -2, 3, -21, -1, 2], [-6, 10, -9, 14, 8, -12], [31, -12, -17, -11, 8, -16], [35, 16, -4, -2, 13, -9], [5, 11, -5, 18, 2, 0], [-14, 7, -8, -10, 0, 7], [10, -10, -4, -1, -8, -7], [-6, -2, 4, -3, -10, -1], [-15, 15, 4, 20, 6, -2], [-1, 7, 4, 5, 9, 5], [5, 17, -9, -5, -6, 1], [-1, 9, -11, 6, -6, 3], [-28, 12, 4, -3, -1, 0], [-12, -2, 10, -12, -7, 6], [-14, 4, 0, 2, -2, -5], [31, -3, 4, -4, 14, -1], [-20, 17, 12, 8, 1, 10], [-31, -5, 9, -8, -8, -1], [-6, 4, 26, -8, -15, 7], [-20, 12, -2, 3, -7, -10], [9, 0, 3, -5, 3, 3], [5, -13, 2, -23, -4, 6], [-4, -8, -7, 0, 6, 0], [-15, 0, 9, 17, -2, -6], [-21, 8, -7, 16, 15, -10], [-13, -3, 12, 6, 9, -5], [-30, 3, 9, 5, -8, 8], [-14, -12, -1, -20, -17, 7], [7, -19, 15, -7, -11, 7], [-21, -1, -9, 10, -5, 8], [-3, 4, 0, -6, -12, -8], [8, -6, 9, -1, 9, -10], [-19, 14, 10, 11, 10, 9], [-1, 17, -14, 19, 4, -1], [28, 7, -12, 18, 20, -5], [-7, 5, -9, 7, 9, -2], [4, 10, 10, -3, 8, -8], [-27, -4, -7, -13, -6, 5], [16, -13, -13, 14, 12, -12], [-16, -4, -10, -26, -3, 16], [-18, 4, 21, 7, -12, 7], [-14, 17, -8, 1, 7, 4], [17, 4, -13, -23, 0, -4], [-20, 22, -8, 35, 9, -9], [19, -17, 2, -36, -12, 9], [-21, -5, 0, -5, -5, 12], [-7, -21, -3, 1, -11, -2], [21, 20, -21, 5, 15, -2], [-13, 16, 2, -7, -12, 9], [-10, 2, 9, 6, 7, 13], [-12, 7, 10, -1, 4, -8], [17, 5, -8, -4, 13, -16], [8, 4, 3, 9, -15, -6], [19, 5, 8, 2, -11, 5], [2, -18, 16, -22, -5, -1], [-6, -5, 6, 8, 4, 8], [4, 9, 2, 13, 11, -22], [18, 0, -13, 14, -4, -2], [-28, 4, -4, 6, 14, -3], [-7, -14, 16, -7, -15, 12], [-26, -11, -5, -8, -7, 13], [-9, -26, 23, -16, -13, 11], [-10, -12, -8, -8, 15, -9], [1, 3, 13, -12, -14, 5], [-24, 14, -1, 19, 3, 11], [6, 7, -16, 3, 10, -1], [44, -17, 9, -15, -5, 4], [24, -5, -24, 5, 11, -9], [-6, -4, -7, -18, -7, 17], [-5, 7, 12, -7, -10, 33], [39, 2, 18, -15, 4, 7], [-21, 4, -4, 3, 7, -16], [19, 4, 4, 18, 20, -8], [-14, 1, -9, -9, -9, -11], [-20, 13, -9, 3, -13, -4], [25, 10, -8, 13, 19, -18], [-24, -11, 2, -11, 8, 4], [-19, -10, 4, -18, 8, 4], [22, -2, -4, 14, 6, -4], [24, -9, 0, -6, 19, 5], [15, -5, 4, -9, -2, 12], [-18, -8, 6, -22, -4, 16], [29, 23, -9, -4, 3, 0], [-19, -10, 0, -3, -13, 8], [3, 6, -3, -22, -12, 18], [7, 5, -1, 22, 18, -26], [-19, 11, 15, -5, 4, -3], [20, -3, 13, -2, -9, -3], [-29, -10, 8, -13, 7, -6], [-25, -3, 14, 10, 5, 10], [-30, 13, -12, 8, -3, -7], [12, -32, 1, -10, -7, 4], [41, -2, -7, -1, 9, -1], [-8, 8, 0, 12, 3, -21], [27, -17, -1, -21, 13, -3], [21, 20, -20, -2, 2, -5], [-23, -8, 8, 10, 18, -13], [-16, -7, 8, -8, 3, 21], [0, -8, 11, -6, 7, 4], [13, -25, -7, -21, -14, 6], [-2, 10, 4, 27, 22, -6], [18, -5, -13, -13, -8, 6], [0, 1, 7, -14, 9, 13], [-9, -14, 19, -22, -6, 4], [4, 4, -1, 13, 3, -24], [7, -15, -24, 13, 17, -9], [26, -4, -22, 2, -1, 0], [2, -13, -1, -3, -2, 6], [-1, -17, 0, 8, 11, -16], [-30, 16, 19, 14, -12, 16], [-29, 26, 4, 12, 9, 25], [-6, 26, -6, 18, -5, 5], [-32, -16, 7, -20, 5, 4], [9, -10, -1, -19, 7, 2], [-25, 1, 12, 3, -11, 2], [-24, -12, -15, 25, 10, -23], [4, -7, -2, -8, 8, -10], [8, -30, 31, -26, -15, 15], [-22, -10, 28, -3, -2, 13], [12, 1, -18, -5, 9, -20], [5, 18, 5, 13, -19, -3], [8, -3, 13, -5, -2, -18], [7, 21, -19, 16, -2, 5], [-21, -25, 11, -25, -31, 6], [10, -5, -10, 3, 9, 11], [-2, 5, 21, 0, -16, 19], [16, -1, 5, -25, 8, 4], [-11, -8, 3, 25, 8, -5], [-16, 24, -12, 11, 15, 5], [0, -16, 6, -29, -28, 22], [12, -32, 8, -41, -25, 8], [-9, 23, 9, -12, 0, 3], [7, 4, -4, 23, 13, 11], [29, -3, 26, -15, 0, -7], [0, -12, -2, -37, -5, 14], [-21, -27, 7, -12, 10, 2], [23, 17, -14, -14, 0, 6], [-19, -17, 19, -30, -15, 3], [-24, 8, -15, -8, -1, 8], [10, -12, -6, 5, 18, 3], [2, -14, 17, -17, -5, 6], [-5, 19, 3, 29, 17, 5], [29, 11, -9, 12, 31, -1], [-3, 5, -1, 9, -4, 13], [-33, 18, -13, -3, 0, -14], [9, 17, -4, 27, -3, -10], [46, -14, -5, -24, -15, 5], [-23, 13, -1, 7, -4, 1], [28, 2, -12, 17, 6, -1], [13, 2, 1, -17, -18, 26], [-6, -22, 29, -8, -7, 18], [24, -9, -14, -13, 18, -25], [-17, 39, -6, 31, 4, -19], [14, 16, 1, 30, 24, -8], [-5, 21, 8, 3, 11, -2], [-7, 16, 5, 10, -22, 17], [22, -18, 30, -11, -1, 17], [31, -6, 4, -30, -4, 3], [-8, -14, 7, -26, -16, 18], [15, -8, -3, 8, -10, -7], [-39, -11, 9, 0, 0, 14], [-12, -16, 5, -19, -24, -8], [-9, -17, -5, 1, 11, 1], [12, -13, -13, -7, 7, -16], [-9, 22, 2, 26, 17, -12], [-17, 0, 11, 21, 33, -19], [25, 7, -12, 8, 19, 6], [11, 7, -1, 7, 18, 11], [13, -8, -18, 2, 26, -19], [-14, -36, 25, -8, -6, 9], [7, -15, -12, -12, -1, -30], [-38, -15, 14, -14, -32, 3], [18, -5, -11, 6, 6, 2], [26, 6, -9, -7, -5, 0], [-10, 7, 14, 18, 13, -8], [2, -3, -12, 17, -11, -2], [24, 36, -18, 9, 12, -17], [11, 4, -20, 11, -2, -12], [-21, 10, -2, 28, 21, -7], [-22, -14, 11, -13, -8, 5], [-3, -1, 12, -2, 2, 5], [7, 16, -17, 23, 5, -15], [-21, 15, 12, 6, -7, 31], [35, -13, -9, 2, 8, -17], [-33, 25, -8, 44, 1, 1], [25, -7, 22, -29, 14, 12], [19, 4, -2, 20, 22, 4], [23, 0, -16, 19, 8, 6], [-41, 6, -6, 24, -4, 11], [-28, 10, 11, 29, -11, 17], [-15, 8, -14, 23, 10, -10], [-15, -20, 6, 18, 12, -17], [2, 4, -11, 5, 8, -5], [-13, 8, 15, -5, 4, -5], [38, 13, -17, 14, -2, 11], [-1, -16, -1, -2, 5, -15], [2, -14, 13, 4, -7, 17], [-18, 35, -9, 28, 1, -9], [-2, 12, 9, 12, 15, -2], [3, -15, 10, -12, 13, 20], [-13, -14, -2, 0, -5, 2], [-3, -4, 17, 5, -1, -9], [13, 12, -9, 2, 18, -4], [-2, -23, 4, -20, 2, -15], [22, 0, 11, -34, 4, 10], [-13, -21, 3, -9, -6, -2], [-19, -16, 33, 19, -3, 9], [-28, 32, -12, 30, 11, -9], [28, -23, -8, -43, 10, 0], [-16, 11, -13, 7, -11, -6], [5, -13, -4, -18, -27, -14], [-33, -26, 3, -10, -21, 2], [-48, 6, 21, 25, 12, -4], [-34, -32, 13, -1, -12, 21], [0, 6, -16, 21, 29, -17], [-10, 8, 2, 6, 1, -2], [7, -23, -10, 8, 1, 11], [-14, 5, -13, -18, 6, 14], [15, -3, 13, -30, -24, 11], [25, -4, 8, -48, 0, 14], [-39, -16, -5, 3, -25, 17], [-33, 17, -17, 42, -14, -7], [8, 19, -29, 19, 19, -26], [12, -11, -2, 11, 5, -5], [-9, 1, -27, 3, 12, 4], [32, 10, 4, 17, 30, 13], [-29, 13, 13, 14, 24, 9], [-26, 5, -1, -17, -18, -4], [8, -12, -1, 20, 10, -10], [-20, 31, 12, 2, 13, -6], [28, 6, -2, -8, 24, -3], [-17, 11, 6, 11, -24, -9], [-32, 3, -30, 9, -4, -17], [-12, -5, -14, -13, 4, 12], [-6, -46, 9, -16, -10, 16], [12, 15, -1, 5, -12, 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_f();".
// 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_f(: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_f();".
// 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_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4030_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 2, -12, -18, -4, 3, 1]>],Snew);
    return Vf;
end function;