
// Make newform 2230.2.a.p in Magma, downloaded from the LMFDB on 20 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2230_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_2230_2_a_p();".
// 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_2230_2_a_p();".
// 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, 24, 30, 0, -11, -1, 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], [6, 16, 7, -8, -2, 1], [12, 28, 2, -11, -1, 1], [3, 21, 9, -9, -2, 1], [-7, -22, -8, 9, 2, -1]];
    Rf_basisdens := [1, 1, 2, 2, 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_2230_a();"
function MakeCharacter_2230_a()
    N := 2230;
    order := 1;
    char_gens := [447, 1341];
    v := [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_2230_a_Hecke(Kf)
    return MakeCharacter_2230_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, -1, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [-1, -1, 0, 0, 0, 1], [2, 0, -1, 1, -1, 0], [1, 0, 0, -1, 0, 1], [1, -1, 2, -1, 1, 2], [1, 0, 0, 1, 1, 0], [1, -1, 1, -1, -1, 0], [-1, 0, 0, 1, 1, 2], [1, 1, 1, 2, 0, -1], [-3, 0, 3, -1, 0, 1], [2, -1, 1, -3, 2, 1], [-5, -1, 1, 0, -2, 0], [4, 0, 0, 2, 2, 0], [4, 0, -1, 2, 2, -1], [2, 1, 3, -1, -2, 0], [3, 2, 3, 0, -3, -2], [0, -2, 0, 0, 2, 0], [5, -3, -2, -2, 0, -1], [-3, 1, -2, -2, 0, -1], [0, 0, 0, 1, -3, 1], [6, -1, 1, 2, -1, 3], [1, -1, 1, -1, -2, -1], [-1, 3, 3, 0, -1, 2], [9, -2, -2, -1, 3, 0], [1, -2, -3, 4, 0, -2], [6, 1, 0, 2, 0, 0], [6, -1, 0, -3, 2, 0], [1, 4, 1, 2, -1, -1], [-1, -1, -4, 0, -4, -3], [0, -5, -2, 0, -3, 1], [-6, 3, -3, 4, -4, -5], [-1, 0, 0, 0, 6, 3], [1, 2, 7, -6, 3, 4], [0, -4, 0, -2, 6, 4], [-6, 2, -2, -1, 1, -1], [3, 0, 6, -3, -1, 0], [-4, 1, 3, 0, -5, 0], [2, 1, -4, 3, -3, -1], [-2, 5, 4, -3, 2, 2], [11, 0, 0, 0, 2, -1], [7, -1, 0, 5, -1, 2], [2, 4, -1, 0, 0, 0], [4, 2, 1, -2, -4, -7], [-1, 4, -1, 1, -2, -2], [-1, 0, -2, 3, 0, -1], [1, 0, 0, 0, 0, 0], [-1, 1, -1, -2, 0, 0], [9, 7, -3, 7, 1, -3], [7, -2, 0, -6, 8, 3], [4, 1, -5, 3, -3, -4], [-11, 4, -1, 7, -5, -7], [3, -4, 4, -5, -4, -3], [-1, -3, -2, -2, 0, -1], [8, 3, -2, 2, 1, -5], [1, 5, 1, -1, -3, 1], [-4, -6, -2, 0, 2, 0], [9, -2, 4, -1, 5, 2], [12, -4, 5, 1, 1, 8], [1, 7, -5, 4, 0, 0], [5, 6, 2, -1, -3, -4], [1, 4, 0, -3, 4, 5], [15, 3, 2, 3, 3, -4], [-2, -7, -4, -4, -3, 3], [7, -4, -3, -1, 6, -1], [0, -15, 1, -5, 4, 6], [-1, 3, 5, 2, 0, 2], [-6, 9, 5, 6, 1, -3], [-8, -11, 0, -1, -4, 4], [6, 6, 4, -1, 1, 3], [7, -7, -3, 0, -4, -1], [-13, 1, -2, -3, -5, -4], [-12, -3, 4, -2, 0, 10], [-6, -5, 2, 1, 3, -1], [5, 2, 10, 0, 2, 1], [-11, 3, 6, 3, 1, -6], [1, 2, 6, 1, 0, 3], [-7, -1, -7, -1, -4, -1], [1, -5, 6, -2, 2, 5], [0, -3, 4, -4, 6, 4], [-3, 1, -1, -9, 9, -3], [11, 1, -8, 8, 0, -7], [2, 2, 2, 0, 6, -2], [-2, 0, 12, -7, 3, 5], [-6, 5, -5, -2, 1, -7], [11, 7, -4, 3, -5, -4], [-9, -5, -5, 2, -5, -4], [3, 5, 4, -6, 4, 7], [-16, -2, -4, 6, -6, -4], [-3, 4, -8, 0, 6, 1], [15, 1, -3, 2, 6, 3], [-21, 5, 0, 1, -1, -2], [-8, -4, -2, -4, 4, -4], [-6, 5, 0, 4, -2, -10], [-6, 8, -5, 0, 0, -1], [-1, 2, 8, -4, -7, -2], [8, -6, -2, 1, 5, 1], [-4, 1, -4, 5, -8, -2], [3, -7, -1, -5, 5, 3], [-14, 3, 5, -2, -3, 3], [-14, 3, 2, 1, -2, 4], [15, 9, 4, -1, -5, -4], [-11, 5, 0, 8, 0, -7], [-8, 8, 10, -2, 4, 6], [-10, 2, -4, 1, 1, 1], [-9, -6, -6, 0, 0, 5], [-1, 4, 1, 6, -3, -9], [9, -2, -1, 13, -2, 2], [6, 4, 10, 4, 2, 2], [5, -2, -3, -1, 2, 5], [-1, 6, 12, -4, 2, 7], [-8, 0, 2, -12, 6, 0], [10, 7, -13, 11, 4, -2], [-1, 3, 0, 1, 3, 6], [8, 8, 6, -9, 3, 3], [26, 7, -1, 2, -3, -5], [-15, 11, -2, 3, -1, -4], [1, -7, 4, -4, 16, 11], [5, 5, -8, 8, -6, -3], [19, 5, 1, 1, 1, -1], [-4, 4, 10, -5, 7, 9], [17, 6, 5, -5, -6, -3], [3, 7, 5, -6, 10, 0], [14, 6, -6, 6, 2, -8], [-1, 10, 4, -4, -3, 2], [-3, -14, -5, 2, -1, 0], [2, -3, -9, 3, 3, -2], [10, 8, 2, 1, 5, -3], [6, -2, -3, 4, -8, -5], [2, 5, 9, -5, 12, 4], [5, 11, 10, 7, 1, 0], [-1, 2, 3, -7, -4, 4], [12, 1, 10, -2, 8, 4], [-25, 5, -2, 3, 3, 4], [6, 8, -1, -7, 5, 3], [6, 11, 8, -7, -2, 0], [-6, 11, 6, 6, 1, -5], [2, -16, 7, -6, 4, 9], [23, -7, -2, -7, -1, 2], [-2, -8, -5, 1, -1, -4], [4, -9, 4, -6, 5, 3], [-10, -3, -15, 4, 1, -8], [7, 4, 2, 4, 1, 2], [11, 15, -1, 7, -3, -9], [13, -5, -2, -9, 3, 4], [6, 7, -12, 5, -5, -11], [-9, -7, 2, 4, 0, 7], [-12, 6, -9, 3, -5, -2], [14, 0, 6, 0, -2, 0], [2, -3, -10, -1, -7, -3], [1, 6, -3, 5, -1, -15], [-13, 10, -10, 12, -5, -10], [-15, 15, -10, 6, -2, -9], [8, -9, -5, -8, -5, 1], [-10, -4, 3, 4, -9, 1], [-4, -8, 10, -9, -1, 11], [-6, -7, 3, -3, -2, 4], [-2, -9, 1, -2, -7, -3], [-15, 2, 6, 8, -13, -8], [15, -3, -11, 10, 2, -14], [-10, -7, 1, 6, -5, -7], [14, -4, 1, 6, -6, 3], [2, -9, 1, 1, -10, -4], [7, -5, -17, 6, -5, -12], [13, 1, 3, 3, 9, 8], [2, 2, 20, -5, -3, 7], [32, 0, -1, 8, -6, 3], [15, -1, 0, -2, 12, 9], [1, 0, 7, -9, 2, 9], [5, 2, 14, -6, 0, 9], [33, 4, -11, 6, 3, -4], [-6, -9, 1, -5, -9, 4], [-4, -1, 6, 4, -7, -7], [-14, 2, -12, 3, -3, -13], [5, 3, 0, 16, 0, 1], [2, -6, 2, -2, 6, -4], [2, -5, -12, -4, 6, 10], [-11, -5, -2, 2, -12, 1], [22, 9, 2, 7, 2, -4], [-8, -1, 11, 0, 5, 8], [4, -2, 2, 13, 3, 1], [1, -6, 1, 1, 4, -5], [7, -8, 1, 6, 9, -1], [-4, -12, -14, 1, 7, 3], [7, 6, 10, 2, 3, 2], [2, -4, -4, 0, 4, -2], [-12, -6, 0, 0, -8, 4], [-2, 9, 1, -1, 6, 9], [-5, 12, 9, 0, 7, 6], [1, -9, -14, -6, 2, -5], [-6, 4, -12, 7, -3, -1], [28, 5, -12, 7, -11, -3], [13, -12, -16, 2, 3, -8], [-10, 8, 4, 12, -4, -2], [-3, 9, -2, -6, 6, 7], [7, -11, 13, -5, 2, 5], [20, -4, 10, 6, 6, 2], [-4, 4, -3, -10, 4, 2], [10, 3, 4, 10, 3, -5], [5, -12, -17, -2, -9, -4], [-6, -6, -11, 4, -11, -9], [12, 3, -2, 11, -2, -6], [-4, 14, -8, 6, 6, -4], [3, 9, -2, 12, 4, -7], [8, 10, 2, 6, -2, -8], [-6, -18, 3, -2, -5, 15], [1, -1, 12, 3, 3, 10], [-11, -3, 12, -2, 0, -7], [10, -10, 6, -20, 2, 8], [-5, 11, 15, 0, 6, -2], [11, -18, 9, -14, 3, 8], [-2, 4, -4, -8, 12, 4], [17, 0, 10, -13, 2, 3], [0, -2, -16, 3, 11, 3], [-2, -10, 16, -9, -3, -1], [-2, 22, -4, 5, -11, -17], [-40, -8, 8, -5, -5, 5], [-14, 16, 18, -1, 3, 7], [3, -8, -10, 0, 11, -6], [20, 5, 6, -9, -7, -9], [-3, 1, -2, 6, -8, -13], [-12, -6, 0, -1, -17, -9], [10, 1, 5, -4, 13, 6], [-15, 9, -3, 6, 6, 0], [-13, 6, 5, -14, -3, -6], [-8, -13, -18, 8, -7, -3], [7, 9, 7, 11, -7, 5], [-1, -3, -8, 4, 0, -9], [-6, -1, -17, 5, -4, -6], [-12, -2, -14, 10, -12, -8], [-8, -3, -18, 3, -2, -8], [14, 9, 11, -13, 13, 10], [-1, 3, -13, 21, 0, 1], [-8, -11, -8, 0, -1, -7], [-16, 8, -5, 13, -2, -14], [-2, 6, 0, -7, 3, 17], [10, -5, -12, 7, -9, 3], [14, 6, -1, 5, 5, 4], [-4, -4, -6, -11, 3, -9], [4, 1, -2, 3, 17, 11], [-14, 8, 11, 5, -13, 4], [-28, 0, -5, -9, 0, 0], [21, -16, -16, 7, -16, -7], [0, -11, 1, 6, -19, -5], [4, -18, -4, -7, 15, 9], [6, 24, 5, -5, 14, 0], [12, 18, 0, 6, 2, -4], [3, 20, 4, 4, -7, -8], [6, 4, 5, 1, 2, -14], [3, 6, -7, 3, -8, -25], [-25, -5, 0, 9, -13, -12], [21, 8, -5, 4, -6, -16], [2, 2, -6, 18, -4, 2], [7, -8, -6, 4, 3, -10], [30, 7, 6, 3, -6, -2], [-31, -16, 4, -9, -12, 5], [-4, 16, 12, 8, -14, 4], [-2, 2, 17, -10, -6, 5], [4, -5, 7, -2, 20, 13], [-9, 2, -6, -6, 2, 9], [12, 5, -5, 0, 7, -7], [31, 11, 5, 3, 7, -5], [12, 2, 6, 10, 8, -4], [18, -6, -4, 5, -15, -7], [-15, 4, 8, -21, 3, 2], [5, -4, -2, -12, -7, 0], [18, -14, 0, -20, 4, 4], [4, 9, -5, 14, -7, -11], [-24, 12, 3, -2, -8, -3], [27, -4, -1, 0, 7, 8], [-2, -17, -8, -7, 16, 8], [-3, -6, 19, -2, -7, 1], [-11, 3, -4, -1, -3, -2], [25, -10, 3, 2, 13, 1], [-3, -9, 11, -17, 3, 8], [21, -6, -14, 14, -5, -4], [-22, 10, 0, 6, 6, 12], [2, -6, -11, 19, 1, 5], [-22, -5, -7, 5, -6, -4], [-12, 0, -8, 2, 8, 10], [38, 5, 7, -11, 5, 4], [48, 12, -8, 13, 13, -5], [9, -18, 13, -12, 9, 14], [-13, -11, -7, 4, 0, 8], [-22, 6, 6, 6, -2, -12], [-16, 4, 10, 3, -5, -5], [19, -10, -18, 8, -15, -4], [-9, -9, 2, -6, 0, -1], [-1, -1, -3, -7, 14, 7], [-7, -27, 1, -5, 13, 11], [-9, -20, -6, -2, 15, 6], [-43, 8, 0, 2, -2, 5], [16, 2, 3, 9, -11, -8], [-14, -9, 22, -16, 13, 11], [-1, 1, -5, 7, -19, -3], [32, 12, -20, 21, 7, -11], [35, -4, 8, -11, 12, 13], [14, -5, -9, -13, 8, 12], [-4, 19, -11, 0, -5, -7], [-8, 9, 11, 6, 6, -3], [6, -4, -12, 4, -12, -4], [2, -24, -6, -1, 7, 7], [-1, -6, -6, 9, 16, 13], [-6, -11, -6, 11, -16, -14], [18, 4, -8, 12, 0, -16], [-36, 14, 0, 6, -4, -2], [44, -4, 18, 10, 8, 8], [9, -1, 5, 12, -4, 13], [-22, 9, -6, 2, -3, -19], [-10, 4, -8, -15, 11, -3], [18, -3, -22, 12, 7, -11], [-23, 10, 1, 5, -7, -11], [-14, -4, 6, -10, -10, 2], [-7, 4, 15, -1, -8, 3], [0, -7, 3, 11, 8, 19], [8, -19, 11, -1, -2, 2], [21, -5, -10, -6, 0, -9], [9, -9, -14, -7, 17, -2], [-15, 13, 0, -6, -18, -9], [-15, -10, -24, 1, -2, -3], [-1, 1, -8, -4, -8, -3], [-29, 9, -13, -1, -9, -17], [17, -9, -20, 8, 4, -13], [-10, -2, -14, 11, -3, -21], [-11, 11, -4, -2, 8, 5], [12, 11, 8, -14, 15, 5], [17, 1, 9, 12, 12, -6], [-40, -4, 12, -17, -3, 7], [-9, -17, -20, 1, 5, 8], [13, 5, 5, -3, 5, -7], [27, -6, -1, 9, 14, 5], [-43, 8, 0, 0, 4, -13], [-6, 9, 16, -12, -7, 9], [22, -6, -8, 2, -10, 2], [14, -6, -14, 6, 4, -22], [-8, -1, -10, 3, 7, 11], [13, -3, 10, 2, -14, 7], [10, 2, 2, -14, 0, -6], [-12, 24, 5, 13, -26, -8], [8, -12, 23, -22, 6, 19], [7, -15, 7, 8, 3, 2], [7, -11, -7, 7, -15, -1], [-4, 19, 0, 4, -13, -13], [-43, -12, 0, 1, -5, 0], [-12, -1, 16, -12, -10, 8], [17, -10, -3, -14, -3, 0], [21, -4, 6, -8, -10, -5], [-18, -9, 20, -12, 1, 11], [11, -16, 11, -15, 4, 23], [54, 4, 1, 7, -3, -6], [-2, -8, -5, -14, 2, -2], [-3, 4, 1, -6, -7, 18], [-21, -25, -8, -7, -3, 6], [2, -4, 0, 2, 24, 18], [1, 1, -19, 0, 17, 2], [1, -6, 2, 16, -4, -3], [22, -20, -6, -7, 15, -1], [43, 1, 7, -2, 7, 0], [51, -4, 13, -10, 18, 18], [20, 1, -14, 12, 0, 6], [-42, 1, 7, 7, -12, -14], [9, -11, 0, 1, -1, 18], [-12, 5, -1, -11, -8, -20], [-16, -6, -10, 10, -12, -4], [-23, -2, 12, 2, 0, 9], [10, 17, 8, -5, -19, -13], [29, -3, -8, 11, -7, -2], [6, 15, 5, -1, 2, -12], [-11, 14, -4, 15, 12, 11], [5, -11, -2, -4, -2, 15], [4, 3, 13, 2, 7, 4], [3, 2, -11, 8, -9, 4], [47, 21, 5, 13, 11, -11], [-27, -10, 4, -5, -21, -6], [-24, 7, 16, 5, -9, -1], [-11, 14, -2, 0, -2, 9], [-21, -6, -13, 6, -9, 1], [-14, -15, -5, 1, -12, -9], [-15, -13, 14, -14, -8, 5], [7, -8, -17, 4, -1, 16], [-4, 13, 5, -8, -10, -15], [5, -20, -2, -12, 30, 13], [-29, -2, 9, 0, 9, 23], [-28, 7, 7, 9, -8, 10], [-25, 19, -16, 5, -17, -30], [31, 14, 8, -16, 2, 5], [21, 8, -15, 11, 4, -29], [-52, 4, 0, 4, -8, -8], [-2, 1, 11, -20, 21, 8], [17, 5, -4, -4, 20, 3], [28, -2, 3, -12, -10, -9], [26, 5, 27, -11, -4, 2], [14, -11, 3, 2, -9, 13], [-22, 8, 16, -5, -9, 13], [-21, 13, -4, 2, 6, -3], [-25, 13, 5, 17, -17, -19], [0, 5, 11, -1, 14, -4], [10, -13, 8, 0, 3, 3], [11, -10, 22, 0, 0, 5], [0, -6, -4, -6, -6, 0], [28, -4, -7, 0, 18, 0], [43, -8, 1, 20, 6, 4], [-1, 19, -10, 14, -14, -9], [-41, -7, -5, -11, -13, 5], [24, 22, -7, 14, 5, -5], [24, 8, 26, -4, -2, 12], [-13, 17, 24, -16, 28, 9], [-16, 7, -3, 6, -3, -7], [4, 3, -4, -1, -23, -31], [8, 11, 7, -8, 17, 3], [-8, -5, -18, 6, -28, -20], [-2, -5, -12, -9, 19, -5], [19, -17, -20, 2, 8, -3], [-19, -11, 14, -5, -21, 0], [-31, 25, -3, -1, 5, -7], [15, 9, -19, 9, 7, -11], [25, -15, 14, -8, -6, -3], [10, 3, 5, 4, -8, 7], [-17, -25, -6, -6, -6, 7], [30, 18, 8, -12, 12, 0], [-24, -14, -17, -6, -16, -3], [-4, 10, -24, 14, 2, -4], [4, -28, -9, 7, -11, 4], [2, 11, -3, -16, 10, -1], [-15, -12, -29, -5, -6, -5], [-4, -29, 3, -26, 3, 9], [-22, 20, 2, -6, 20, 6], [9, -9, -14, -11, 5, 18], [17, 4, 23, -27, 8, 22]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2230_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2230_2_a_p();".
// 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_2230_2_a_p(:prec:=6)
    chi := MakeCharacter_2230_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_2230_2_a_p();".
// 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_2230_2_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2230_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, -24, 30, 0, -11, 1, 1]>,<7,R![112, 208, -36, -76, -6, 6, 1]>],Snew);
    return Vf;
end function;