
// Make newform 4576.2.a.p in Magma, downloaded from the LMFDB on 31 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_4576_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_4576_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_4576_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 := [-2, 1, 15, 7, -9, -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], [-3, -6, 1, 1, 0, 0], [1, 9, 7, -8, -1, 1], [-2, 8, 8, -8, -1, 1], [3, -7, -14, 8, 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_4576_a();"
function MakeCharacter_4576_a()
    N := 4576;
    order := 1;
    char_gens := [287, 4005, 2081, 353];
    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_4576_a_Hecke( : Kf := Kf)
    return MakeCharacter_4576_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]];
    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, 0, 0, -1], [1, 1, 0, -1, 1, -1], [-1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 1, -1], [0, 0, -1, 0, 1, -1], [3, 0, 1, 0, 1, -1], [0, -1, -1, -1, 0, 1], [3, -1, -1, -2, 0, 0], [-1, -1, -1, 1, 1, 1], [1, 1, 0, -2, 0, -2], [-1, 1, 1, 0, -2, -1], [-1, 2, -2, 0, -1, 1], [0, -2, 1, -1, 0, 0], [1, 0, 2, 1, -2, -1], [-1, -3, 1, 1, -3, 0], [2, 2, 0, 1, 1, -2], [3, -1, -1, 0, 2, 0], [1, -1, -2, -2, 1, 0], [6, -2, 0, 1, -1, 1], [-1, -2, -1, 3, 0, 1], [0, 3, -5, -3, 4, 0], [1, -1, -1, -2, 0, 0], [2, -2, 0, -1, 1, 1], [4, -1, 2, 1, -2, -2], [2, -1, 1, -1, 2, 1], [-3, 2, 1, 2, 2, 0], [-1, 0, -3, -2, 4, -3], [6, -2, 0, 3, 1, -1], [5, 3, -1, -3, 1, 2], [0, 5, 1, 0, -1, -1], [3, 2, -4, -3, 4, 2], [-5, 2, -3, 0, 1, 0], [4, -2, 0, -2, -4, -2], [-9, 3, -2, 1, 1, 2], [0, 3, -5, -3, 4, -1], [2, -1, -2, 4, -4, 5], [1, 5, -1, 2, 0, -3], [7, -3, -1, 1, -3, -3], [-1, -5, 0, 4, -1, 5], [6, 2, -1, 0, -1, 4], [-1, -6, 5, 4, -9, 0], [-1, 6, 1, -4, 0, -8], [5, 7, -4, -3, 6, -3], [-4, 0, 2, -1, 1, 3], [2, 1, 1, 6, -3, 1], [2, 4, 1, -4, -3, -3], [-7, 0, -4, -5, 4, 1], [4, 2, 4, 0, -2, 2], [8, -7, 4, 2, -2, 5], [-5, 6, -1, -2, 3, 4], [-1, 3, 2, -2, 0, 1], [-2, -3, -2, 4, -3, 5], [9, 0, 0, -2, -1, -3], [6, 2, 1, -2, 2, 1], [12, 0, 1, 4, -5, 1], [5, 3, -3, -5, 5, 0], [1, 3, -2, 2, 3, -4], [-8, -3, 1, 6, -3, -2], [4, 10, -4, -6, 10, -2], [-6, 0, 4, 0, 0, -8], [1, 0, -3, -1, 3, 7], [-3, 6, -3, -3, -2, -2], [5, 12, 2, -8, 7, -8], [-8, 12, 2, -5, 3, 0], [2, 4, -2, 3, 5, 1], [5, -12, 0, 6, -5, 7], [7, 0, -3, 4, -5, -2], [3, -1, 5, -1, 1, -3], [1, 5, 0, 3, 5, -3], [4, 5, 0, -2, 4, -4], [-1, -3, 7, 2, -2, -3], [5, -3, -1, -3, 1, -3], [6, -5, -5, -1, 0, 0], [3, 3, 0, -4, -3, 3], [-3, 7, 1, 6, 4, -3], [4, -6, 0, 1, -1, 3], [2, 9, -1, -4, 3, 2], [2, 10, -3, -3, 1, 4], [-9, 7, -3, -6, -4, -1], [3, 0, -3, 3, -2, 7], [1, -2, -1, 0, -10, 1], [8, -6, -2, 0, 4, -4], [5, -7, 4, 2, -6, 3], [3, 5, -1, -4, 0, -4], [9, 7, -4, -6, 5, -4], [-7, 6, -3, 0, 6, 4], [2, -1, 1, -1, 0, -8], [-9, -1, -5, -3, -1, 3], [16, -3, -1, -2, -5, 2], [2, 1, 5, 0, -9, -5], [2, 5, 1, -3, 2, -7], [-7, -3, 3, 3, -9, -2], [-1, 0, -4, 2, 7, 1], [-6, 13, -1, -2, 3, 3], [-1, -2, -3, 5, -4, 10], [-11, -12, 0, 2, -5, 7], [-1, -10, -1, 4, -5, 8], [-11, -1, 3, -2, -8, 1], [5, 2, 7, -2, -9, -2], [-6, -8, -2, 6, 0, 6], [-4, 4, 2, 1, -3, 1], [-19, 3, -7, -3, 7, 0], [-5, -13, -1, 0, -6, 4], [-10, 7, -3, 4, 5, -4], [1, -10, 3, 2, -1, 2], [7, -7, 4, 5, 0, 5], [7, -16, -2, 8, -5, -1], [6, 4, 2, -8, 2, 2], [5, -2, -1, 6, -2, 0], [2, 2, 2, -7, -3, -5], [-28, 8, 2, 0, 2, 3], [0, 5, -5, -8, 7, -9], [-19, -4, 1, 5, -6, 4], [-9, -1, 1, 1, 5, 5], [4, -9, 8, 2, -4, 4], [5, 7, 3, 5, -5, -5], [19, 0, -4, -5, 2, 0], [-3, -11, -3, 8, -10, 10], [4, -14, 4, 5, 1, 3], [4, -6, 2, 2, 8, -6], [-9, 7, -3, 1, 1, 2], [-8, 9, 1, 2, 9, -3], [15, -19, 1, 3, -7, 2], [1, 12, 2, 0, -9, 6], [-1, 12, 4, 1, 0, -3], [-17, 11, 2, -2, 0, -5], [-5, 14, -7, 0, 9, 0], [15, -4, -3, -7, -6, -1], [-4, 3, -1, -2, 3, -2], [2, 14, -3, -2, -1, 8], [-4, -4, -5, 2, 0, 3], [9, 3, -12, -4, -2, 6], [-7, 3, 2, -2, 10, -6], [-7, 2, 4, 4, -5, -1], [-7, -4, 7, 3, 0, -1], [9, 11, 5, -1, -7, 1], [7, -8, 6, 7, 0, -2], [-13, 4, 3, -5, -2, 7], [10, -2, -4, 4, 0, 2], [3, -10, 7, 10, -7, 9], [-4, 8, -7, -14, 11, -13], [11, 3, 4, 1, 7, -8], [-7, 7, -1, -5, 7, -7], [-17, -16, 5, 2, -4, 4], [-19, -2, 2, 2, -11, 9], [2, 13, -6, 2, -2, 6], [-6, 0, 12, 5, -5, 1], [21, 18, -1, -4, 6, -4], [-9, -4, -7, 2, 2, 1], [-29, 4, -3, -9, 7, 1], [-3, 6, -8, -6, 9, 1], [-22, -8, 11, 9, -7, 0], [16, -6, -5, 5, -9, -4], [-12, 0, 4, 4, -6, 10], [4, 8, 2, 3, 7, -11], [9, -2, 4, 13, -4, 2], [5, 8, 7, -6, 8, -4], [7, 7, -3, -6, 0, 4], [13, -6, 8, 14, -5, 3], [-26, -3, 2, 2, 4, 7], [18, -9, -4, 0, -6, -2], [-3, 1, 1, 13, -3, 5], [-14, -11, 7, 14, -13, 15], [3, -11, 2, 9, -8, -1], [9, -13, -1, 3, -1, -4], [21, 9, -2, -4, 4, -6], [-3, -5, -5, 1, -9, 3], [3, 12, 3, 7, -4, -1], [-19, 5, 3, -3, -5, 6], [-7, -2, 12, 3, -4, -4], [-8, -13, -3, -2, 3, 6], [-1, -10, -1, 4, -3, 1], [13, -10, 9, 3, -8, 2], [-9, 0, -7, -5, 10, 3], [10, -18, -8, 3, 3, 2], [7, -4, -10, -7, 0, 6], [14, 11, 11, 3, -10, -9], [-19, -2, -2, -12, 9, -1], [0, -10, 10, -3, 1, 1], [13, 7, -6, -14, 5, -3], [0, -1, -1, 2, -13, -4], [28, 13, -9, -7, 12, 4], [-26, -2, -7, 2, 0, 1], [-9, 2, 4, -2, 7, -5], [-7, 13, -11, -5, 13, -4], [15, 13, -10, -12, 9, 0], [0, -7, 6, 12, -6, -7], [-7, -5, -2, 11, 3, 2], [-15, 6, -3, -7, 8, -11], [-2, 3, -2, 10, -4, 12], [24, -1, -1, -8, 7, -10], [12, -10, 4, 19, -7, 1], [-15, -7, 2, 4, 7, -13], [13, -14, 1, 2, 4, -2], [-5, 9, 0, 6, -3, 1], [-8, -13, -3, 14, -13, 18], [11, 4, -8, -3, 0, -8], [-4, -4, -2, 14, -6, 6], [-1, -6, 14, 4, -15, -5], [-11, -5, 5, 11, -5, 12], [25, 12, -6, -7, 14, 0], [16, 10, -8, -11, 13, -2], [-5, 17, -6, -10, 7, 5], [-3, 1, 11, 3, 3, -11], [51, 7, -1, -5, 7, -6], [23, -2, -3, 0, -10, 6], [-6, 8, -4, -12, 12, -10], [-15, 4, 4, 2, -7, 13], [-3, -25, 5, 3, -5, 7], [1, -20, -6, 3, -4, 7], [-13, -18, -10, 6, -3, 11], [-9, -8, 1, -3, -10, 9], [-17, 5, 5, 0, -12, 2], [12, -2, 4, 6, -18, 2], [-11, 8, 6, 7, -14, 14], [-21, 20, 2, -9, 10, -4], [-24, -11, 9, 13, -10, -4], [-22, 3, 1, -1, 2, 11], [-12, 37, -3, -16, 13, -2], [2, -21, 5, 9, -10, 10], [13, 9, 11, -10, 8, -6], [9, 5, 1, -8, 4, 5], [-6, 8, -4, -1, 15, 5], [-24, -6, 5, -6, 6, -1], [16, 5, -15, -8, 15, -4], [-1, 6, 0, 3, 10, -10], [-6, 18, 5, 0, -3, 5], [13, 17, 0, -14, 17, -19], [-17, -1, 3, 6, -8, 9], [-6, -13, 0, -4, -8, 3], [17, 0, 2, -3, 0, 4], [13, -11, 1, -15, -7, 1], [-12, 0, -7, -10, 11, -5], [-18, 37, 1, -7, 8, -4], [7, -2, -1, 10, -14, 4], [16, -9, 4, -2, -6, 2], [7, -15, 6, 10, -4, -1], [20, -8, 0, -3, -9, 4], [0, 21, -1, -2, 11, -8], [-4, 3, -5, -8, 13, 4], [-22, -5, 9, 3, -12, 4], [15, 3, -4, -6, 3, -14], [-1, 4, 2, -6, -5, -11], [-19, -8, 2, 8, 9, -7], [22, 3, -9, 0, 1, 14], [-6, -4, 1, -4, -15, 7], [22, -4, 2, 5, 13, -3], [8, -4, -2, -8, 12, -14], [19, -2, 3, 6, -8, 2], [-7, -17, 5, -1, -5, 13], [-24, 7, 8, -2, 1, -3], [5, -13, -7, -1, -1, 18], [16, -8, -7, 2, -5, 21], [0, -18, -13, 4, -2, 15], [1, -6, -5, -1, 0, 9], [-7, -5, 12, 1, -3, 6], [23, -19, -4, 6, -14, 6], [17, 10, 4, -1, 18, -14], [-8, 5, -3, 8, -3, 14], [15, 4, -7, 2, 1, 16], [-10, -3, -3, 2, 1, 4], [-39, -2, 2, 3, 10, 2], [27, -6, 3, 17, -13, -1], [0, 8, -18, -2, -10, 5], [-5, 1, -1, -5, 11, -9], [-10, -16, 6, 14, -12, 20], [-23, -14, 2, 0, -11, 1], [9, -1, -2, -15, 9, -16], [-9, 1, -1, -8, -16, 8], [5, 0, 10, 5, 2, -10], [6, -16, -6, 1, -9, 13], [1, -2, 4, 8, -11, 11], [17, 3, 5, 1, 15, -3], [-14, 18, -6, 3, 13, -4], [21, -8, 11, 6, -3, -2], [-7, -11, 2, 18, -12, 8], [17, -5, -9, -9, -1, -2], [7, -22, -6, 11, -12, 8], [19, -11, -5, 10, -6, 12], [-2, 22, 2, -3, 11, -9], [-19, -8, -11, 4, 7, 6], [-29, -6, 7, 7, -2, 11], [-2, 0, -3, 16, 1, 12], [-38, 14, 7, -1, 10, 0], [21, 2, 3, -8, -7, 4], [6, -23, 5, 6, -7, 11], [4, 8, -6, -21, 11, -1], [-9, 7, -2, -5, -1, -12], [30, -2, 0, 4, -2, 4], [-5, -9, -2, 6, 13, -16], [12, 20, 1, -8, 9, -1], [33, -11, -3, -9, -3, 0], [14, -21, 5, 13, -6, -3], [27, -6, -5, 4, 7, -3], [3, -11, 16, 10, -5, 5], [33, -5, 1, -9, 3, -14], [-20, 2, -11, 4, -13, 11], [15, -8, 4, -3, -2, 0], [-33, 0, -9, -6, 11, 4], [32, -20, 8, 6, -10, 6], [19, -2, 3, 5, -5, 7], [37, 3, -6, -8, 14, -17], [-6, -1, 11, 5, 2, -6], [-7, -17, 0, 2, -2, -5], [5, -7, 6, -11, 9, -6], [7, 12, -4, 5, -4, 12], [8, 15, 13, -3, -8, -5], [28, -18, 4, 13, -13, -3], [-14, -17, 11, 15, -18, 21], [11, -2, 2, 2, -11, -15], [-9, 3, -4, 12, -8, 17], [-9, 4, -11, -7, 6, 1], [-19, -6, 4, 3, -12, 10], [1, -24, 4, 20, -11, 0], [25, 5, -8, -1, 11, -15], [1, -32, -6, 12, -1, 3], [-1, 14, -14, 1, 0, -6], [-45, -4, 0, -3, 4, -6], [-18, 3, -5, -5, 18, 14], [14, -17, 1, 14, -3, 2], [56, -9, -5, -9, -4, -12], [10, -25, 7, 18, -9, -2], [-19, 16, -1, 1, -2, 10], [-3, -28, 0, 11, -8, 17], [5, -11, 5, 6, 6, 10], [6, -26, -8, 5, 5, -3], [13, 24, -2, -15, 18, -4], [-6, 2, -4, -4, 6, -14], [0, -10, 11, 17, -26, 4], [-15, -4, -2, 19, -10, 26], [31, -8, 13, -1, -8, 3], [3, -24, -7, 13, -14, 13], [10, 21, 11, -4, -13, -5], [-10, 8, 6, 2, -18, 17], [10, -11, 3, 2, 3, 22], [-10, -35, 7, 17, -24, 12], [16, 8, -4, -10, 0, -18], [8, -12, -6, 9, 3, -8], [15, -12, 6, 5, -14, 15], [-19, -6, 7, 6, -4, -2], [-1, -3, -5, -3, -11, 11], [-21, 24, -2, -15, 20, 3], [21, 13, -11, -14, 22, -13], [-15, -13, 2, 2, -15, 3], [-5, 26, 2, -15, 12, -23], [34, -7, -6, 2, -14, 1], [19, -19, -2, 8, -1, 5], [29, -4, 1, -7, 0, -15], [9, -29, 9, 6, -8, 0], [2, -24, 0, 3, 3, 1], [-1, -10, -4, 9, 6, 26], [0, 5, -13, -16, 19, -2], [25, -2, -4, -7, -4, -14], [-9, -33, 9, -2, -6, -4], [-8, -6, 12, 4, -26, -8], [-13, -24, -4, 23, -18, 4], [0, -5, 1, -14, -3, -9], [-7, -4, -1, 4, 15, -12], [-8, 2, -6, -17, 3, -3], [35, 4, 4, -5, -8, -9], [45, 4, -1, -7, 16, -7], [23, 9, -2, -18, 4, -11], [19, -24, 15, 16, -14, 1], [5, -10, 11, 16, -13, -6], [14, 9, 8, -12, -8, -3], [-2, 4, 6, 12, -12, 18], [59, -3, -4, -2, -2, 5], [15, -2, 11, 9, 3, 3], [19, 17, 8, -17, 17, -15], [-8, 12, -4, 12, 4, -2], [20, 8, 0, -9, 7, -5], [-19, 23, -8, -10, 7, -1], [2, 24, -4, 3, 7, -5], [2, 24, -8, -14, 8, -12], [-10, -28, 6, 12, -12, 22], [9, 22, -2, 8, 3, -3], [-22, 2, 16, 13, -9, -9], [32, -5, -3, -19, -6, 3], [28, -1, 5, -9, -6, -8], [14, -13, 1, 4, 19, 0], [-3, -15, 4, 6, -10, -4], [14, 16, -3, 0, 13, -15], [29, -3, 1, 11, -17, -7], [-3, 16, 4, -12, 13, 3], [15, 17, -8, -10, 11, -5], [10, 20, -8, 4, -4, -4], [-19, 3, -2, 4, -6, -13], [-28, -15, -9, 24, -5, 17], [-6, 18, -8, -11, 5, -8], [20, -11, -11, -8, 11, -1], [-33, 7, 2, -12, 2, -15], [-6, -7, -1, 18, 3, 7], [6, -6, -14, -1, 13, -3], [-16, 17, 8, 2, -14, 3], [-20, -9, 3, 32, -5, 6], [-6, -20, -12, 4, -2, 2], [11, 14, 7, -16, 5, -8], [7, 4, 9, -10, -11, 24], [-35, -4, 9, -9, 14, -4], [22, 9, 0, 10, -6, 1], [-9, 18, -2, -6, -11, 13], [-3, 16, 1, 2, 0, -20], [-5, 12, -6, -7, 2, -6], [2, -21, 11, -1, -2, -3], [19, 3, -6, -4, 8, 14], [-25, 7, 1, -7, 7, -30], [36, -22, 6, 24, -10, 9], [22, -10, 8, 26, -12, 16], [-5, 16, 4, -19, 0, -16], [45, 10, -3, -5, 15, -11], [-1, -39, 11, 26, -22, 5], [-9, 4, -2, 13, -2, 18], [-13, -19, 10, 13, -17, 8], [8, -21, 6, 16, -18, -3], [-1, 13, 15, 5, -15, 3], [7, -6, -8, -7, 0, -12], [3, 22, -20, -21, 20, -18], [9, -1, -6, -9, 9, -12], [-29, -12, 1, 8, -11, 18], [-19, -6, 12, 7, -2, -10], [21, -21, 7, 11, -17, 16], [-16, 2, 16, -6, 8, -10], [-6, -4, 2, 9, 11, -12], [-31, 5, 8, 6, 4, -3], [0, 6, 18, -6, 6, -18], [21, 4, 14, -11, 14, -7], [19, -4, 18, -4, -9, -13]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4576_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4576_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_4576_2_a_p(:prec:=6)
    chi := MakeCharacter_4576_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_4576_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_4576_2_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4576_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, -1, 15, -7, -9, 1, 1]>,<5,R![-12, 32, 10, -34, -15, 2, 1]>,<7,R![128, -147, -49, 73, -9, -5, 1]>],Snew);
    return Vf;
end function;