
// Make newform 4026.2.a.w in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4026_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_4026_2_a_w();".
// 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_4026_2_a_w();".
// 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, -13, 16, 8, -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], [6, -15, -1, 9, 0, -1], [-14, 39, 13, -25, -2, 3], [-14, 59, 13, -29, -2, 3], [-26, 59, 17, -29, -2, 3]];
    Rf_basisdens := [1, 1, 2, 4, 4, 4];
    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_4026_a();"
function MakeCharacter_4026_a()
    N := 4026;
    order := 1;
    char_gens := [1343, 1465, 3235];
    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_4026_a_Hecke(Kf)
    return MakeCharacter_4026_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, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0], [1, 0, 1, 0, -1, 0], [1, 0, 0, 0, 0, 0], [0, -2, 1, 0, 0, 1], [-1, 1, -1, 0, 1, 0], [0, -1, 0, 1, -1, -1], [-1, 3, -2, 0, 1, -1], [-2, 0, 1, -1, 1, 0], [-1, 1, -1, -2, 1, -3], [-2, -1, 2, 3, 1, 1], [-3, 1, 0, 1, 2, 1], [1, 3, -1, -1, 0, 1], [0, -2, -2, -3, 1, -3], [-3, -1, 2, 2, -1, 2], [-4, -3, 3, 1, -3, 3], [1, 0, 0, 0, 0, 0], [3, -3, 0, 1, 2, -2], [-5, -1, 3, 2, 1, 1], [4, 2, -3, -1, 3, -4], [-4, 4, 1, 0, 2, 3], [0, 0, 1, -3, -1, -2], [-4, 2, 1, 2, -2, 4], [0, 3, 1, -3, -1, 1], [-4, -4, -1, 0, 0, 0], [5, -2, 3, 1, -4, 0], [0, 2, -3, 2, 4, -5], [-1, 4, -6, -2, 3, -2], [-3, -5, 6, 0, -3, 2], [9, 1, -2, -1, 2, -3], [-3, -3, 0, 0, -5, 2], [4, -1, 0, -4, 4, -3], [2, 2, 0, -3, -1, -2], [-1, 7, -4, -3, 0, -1], [0, 0, 5, 1, -5, 7], [1, 2, -1, -2, -5, 4], [-4, 0, 1, 1, -1, 2], [-5, 1, -1, 5, -4, 1], [0, -6, -1, 3, -1, 4], [-12, -2, 1, 4, 2, 0], [-3, 1, -1, -2, -3, -2], [-2, 4, -2, -3, 3, -8], [4, -1, -1, 5, -1, 1], [-1, 2, -3, 2, 1, 6], [-1, -4, -4, -1, 2, -2], [-6, -4, -2, 4, 2, 0], [-11, 1, -1, -1, 4, -3], [-2, 2, 3, 1, 1, 6], [3, -7, 2, 0, -5, 1], [-3, 6, -4, 1, 0, -4], [-8, 0, 3, -4, 2, 2], [-7, -1, -2, 6, 7, -1], [-10, 4, -4, -4, 6, -7], [-1, 5, -2, -1, -2, 0], [-8, -2, 3, 5, -5, 2], [-9, -5, 6, 5, 4, 7], [-7, 1, 2, 8, -1, 5], [-2, 2, -8, -7, 1, -4], [-3, -4, 9, 0, -7, 4], [0, -7, -3, -8, 0, -5], [-4, -7, 5, 2, -4, 3], [-11, -1, 0, -4, -1, 7], [-13, 7, 6, -2, -5, 7], [7, 3, -2, -2, -3, -3], [-1, 3, 1, 3, 0, 3], [-2, 1, -3, -5, -7, 1], [4, 2, -1, 1, 1, -5], [3, -12, 3, -3, -2, -4], [-6, -2, 4, 4, 4, 4], [-3, 4, -4, -4, 7, -8], [3, 11, -7, -6, -3, -8], [3, -3, 3, 2, -3, 6], [-5, 3, -3, 0, -3, -3], [-1, 1, -2, -5, 0, 3], [-1, 3, 2, -1, -4, 3], [-13, -4, 5, 10, -3, 4], [-11, 5, -1, 8, -1, 3], [0, 2, 4, -2, 4, -8], [-12, -11, -1, 3, 1, 3], [-9, 10, -4, -4, 3, -4], [-7, -3, 7, 7, -2, 0], [3, -1, -8, -9, -4, -9], [3, -3, -6, -1, 2, -9], [2, -2, -6, -3, 3, -8], [-19, 5, 1, 0, -3, 5], [-3, -9, 15, 6, -11, 14], [5, 1, 4, 5, -2, 6], [-4, 4, 1, -3, 1, 1], [0, 8, -7, 0, 2, 3], [-3, -7, -3, -3, -2, 2], [5, -5, 1, 7, 0, -5], [-7, 8, -3, -4, -1, -4], [-1, -5, -4, -2, -1, -10], [-4, 0, -2, 1, 5, 2], [-9, -7, 13, 4, -1, 5], [-11, -1, -1, -3, 0, -4], [-6, -2, 9, 1, -1, 8], [7, -1, -1, 1, 0, 5], [1, 1, 5, -1, -6, 7], [8, 6, -4, -9, 7, -10], [11, -9, 5, 0, 1, -9], [-22, -5, 4, 10, 6, 3], [11, 3, -1, -4, -7, 1], [2, -6, 3, -4, -4, -6], [9, -3, -6, 6, -5, 5], [-5, 9, 1, 0, 3, 11], [22, 7, -3, -10, -2, 1], [10, -4, 6, 3, -11, 4], [3, 3, 0, -3, -8, -1], [4, 6, -5, -2, 4, 6], [3, -1, -1, 5, -6, 10], [-2, 0, 11, 9, 3, 12], [-20, -4, -5, 0, 6, -4], [11, 11, -3, -2, 5, -7], [-6, 8, -4, -6, -2, -10], [-4, 0, 0, 7, 7, 1], [-1, -1, 7, -1, 6, -1], [9, -5, -11, -6, 9, -12], [-2, 0, -7, -5, -5, -3], [19, 3, -4, -8, -5, -3], [-14, 4, -1, 0, 6, 8], [14, -2, -7, 0, 10, -8], [2, 0, -4, 0, 6, 0], [-13, 1, 5, 8, 3, 7], [9, -3, -6, 2, 7, -11], [-5, 2, 0, -1, 12, 2], [-10, -8, -3, 6, -2, 7], [10, -9, -5, 1, -5, -5], [10, 7, -7, -9, 7, -11], [12, 6, -16, -5, 7, -4], [16, 4, -4, 2, 2, 4], [-16, 12, -14, -3, 5, -9], [32, 2, -5, -7, 1, -4], [8, -4, -3, 4, -8, 3], [-15, -7, 10, 0, -3, 7], [2, -6, -6, -13, 3, -12], [14, -6, 3, -6, -6, -2], [4, 0, 10, 11, -7, 12], [8, -4, -8, 4, 10, -10], [8, 12, -3, -1, -3, -6], [-25, -3, 10, 3, 2, 9], [4, -4, -5, 6, -4, 0], [-4, -10, 17, 4, -14, 13], [-11, 5, -1, 6, 5, 9], [-13, 24, -20, 5, 12, -8], [15, -5, -2, 4, 7, -11], [0, 4, 3, 0, 4, 0], [6, -18, 14, 3, -7, 8], [5, 18, -5, -3, 8, -6], [-8, 10, 3, 8, -4, 6], [-17, -4, 5, 5, -8, 6], [-12, -9, 14, 11, -11, 7], [-1, 0, 10, 0, -5, 10], [9, 0, -2, 4, 9, -16], [22, 7, -17, -8, 10, -13], [16, -4, -11, 3, 9, -16], [8, -1, -18, -4, 12, -7], [4, 4, 8, 1, 7, 4], [-1, -3, 8, -1, -6, 9], [6, -12, 3, 2, -6, 9], [-2, -13, 5, 3, 7, 7], [22, -4, 1, 5, 1, -12], [-3, -5, -9, 6, -5, 7], [35, 6, -3, -1, 2, -4], [8, 2, -8, -7, 9, -2], [8, 4, 3, -12, 2, -9], [1, 1, 15, 2, -19, 15], [-4, 14, -13, -6, 6, -10], [-14, -12, 14, 5, -11, 10], [-21, -9, 4, -4, 1, 5], [5, -11, -4, 1, 6, -7], [7, -2, 4, -2, 1, 0], [18, 2, -7, -4, -6, -12], [-2, -6, -15, -6, 10, -6], [13, -1, -10, -7, 2, -14], [-6, -10, 11, 2, -6, 5], [18, 6, -18, -13, 7, -13], [-14, 18, 4, -3, 3, 0], [-3, 1, 0, 4, -3, 3], [12, -6, -4, 4, -8, 8], [-1, 7, 12, 2, -5, 3], [14, -4, 5, 9, 7, -6], [10, 12, -8, -9, -5, -2], [-2, -11, 15, -11, -9, 7], [-2, -4, -5, 3, 9, -14], [21, 14, -22, -14, 1, -12], [4, -8, 2, -7, 1, -6], [-35, -6, 20, 4, -5, 16], [1, 5, -5, -15, 14, -6], [4, -14, 18, 1, -9, 6], [7, -8, -6, 0, -1, -10], [1, -1, 10, -1, -6, 19], [-13, -9, 8, 7, -10, 15], [27, -5, -1, -11, -12, -7], [26, -17, 14, -2, -8, -7], [35, -3, -9, 5, 10, -13], [-13, -14, 10, -12, -7, 8], [-11, 7, -7, 15, 6, -1], [9, 5, 11, 0, -15, 11], [3, 13, 6, 0, 3, -4], [9, -12, 10, -9, -12, 6], [8, -1, 3, -12, -6, 3], [-1, 19, 9, 4, -7, 9], [6, 12, -16, 0, 10, -6], [20, -6, -7, 5, -3, 0], [-35, 5, 11, 0, 9, 11], [-19, 5, -9, -1, -12, -3], [-10, -2, 24, 9, -15, 11], [-2, -10, 0, 8, -14, 15], [-15, -1, -2, 13, 2, 3], [29, 11, -8, 13, 6, -11], [-30, 23, 1, 5, 3, 11], [-27, 5, -3, 1, -6, 2], [18, 6, -12, 1, -5, -16], [-23, -3, 9, -1, -2, -3], [18, 0, 5, 6, 10, 2], [-4, -24, 9, 8, -4, 8], [-11, -5, 4, 4, -1, -3], [-5, 7, 2, -5, -6, -5], [6, 9, -8, 5, 3, 11], [2, 8, -18, -4, 2, -10], [3, 11, -20, -9, 6, -21], [12, 8, -16, -7, 17, -10], [1, -1, -19, 6, 3, 9], [-11, -9, 11, 0, -5, -1], [12, -14, 10, -17, -7, 2], [-12, 18, 1, -5, 9, 3], [14, -7, 4, -2, -6, -1], [8, 19, -16, -9, 13, 1], [-2, 4, 1, 1, 7, 10], [-11, 15, 6, 1, 8, 8], [9, -13, 18, 6, -19, 13], [15, 0, -2, 10, -1, -8], [-28, 16, -19, 10, 12, -2], [10, -13, 8, -1, -5, 5], [12, -14, 8, -5, 5, 9], [10, 12, 8, -8, -8, 4], [-1, -1, -6, 6, -5, -8], [-15, -21, 8, 3, -10, 3], [-7, 4, 1, 15, 10, 4], [2, -12, 9, -9, -9, -2], [-25, 15, -14, 0, 13, 1], [-16, 14, -5, -2, -6, 5], [-14, 0, -13, -9, 5, -6], [19, -7, 5, 2, 9, 1], [12, 24, -13, -10, 0, -8], [-18, 8, -6, -2, -12, 10], [12, -14, -6, 6, 10, 0], [15, -9, -2, 5, 2, -13], [30, -6, 6, -7, 5, -2], [16, 8, 0, -6, -8, -10], [-10, 10, 8, 6, -8, 2], [-16, -6, -11, -10, 10, 0], [-6, 24, -2, -5, 7, -2], [-1, -9, -6, -9, 8, -11], [-2, 16, 4, 1, 13, 5], [14, 6, 4, 2, 16, -10], [13, 1, 3, -1, -2, -7], [-14, 4, -6, 5, 13, 1], [-14, -4, -3, 7, 15, -4], [12, -19, 6, -7, -13, -5], [-20, -2, 0, -3, 3, -3], [-19, -11, 2, 13, 6, 8], [5, -1, 4, 5, -10, 17], [7, 5, -8, -10, -13, 7], [13, -19, 6, 6, -5, 11], [7, -19, 2, 7, -2, -6], [-17, -23, 1, -2, -11, -1], [18, 10, 10, 10, -2, 3], [-42, 3, -10, 9, 15, -3], [33, -1, -22, -2, 5, -13], [-24, 2, 16, 1, -9, 21], [4, 4, -4, -12, 4, 1], [-1, 2, -5, 5, 24, -22], [-46, -2, 18, 4, -10, 18], [18, -27, 22, -3, -19, 3], [12, -9, -19, 9, 11, 1], [37, 12, -7, -17, 0, 0], [-19, -3, -4, 9, 0, 0], [26, -14, -1, -11, -3, -12], [18, -2, -12, 18, 8, -7], [39, -9, 8, -1, -8, -6], [-15, -7, 5, -6, 1, -21], [-22, 22, -18, -1, 21, -8], [-10, -2, 7, 5, 17, 4], [12, 18, -27, -16, 0, -6], [-16, -2, 13, 8, -8, 10], [-20, 1, -5, -7, 17, 13], [32, -15, 14, -2, -26, 7], [-6, 2, -2, 22, -10, 9], [-29, -5, -1, -8, -5, 9], [10, -2, -5, -1, -11, -18], [-38, 8, -17, 0, 20, 0], [-3, -6, 8, 4, -25, 16], [-13, -4, 12, -2, -13, 12], [-1, -15, 11, 7, -14, 10], [-31, 2, 17, -1, -8, 24], [5, 30, -7, -11, 16, -14], [-7, -1, 10, -6, -19, 17], [-8, 20, 2, 0, -6, 16], [15, -11, 22, 3, -2, -2], [10, -4, -8, 6, -16, -14], [2, -24, 3, -6, -2, -21], [1, 12, -21, 2, 1, 0], [-7, 6, -1, -18, -9, 0], [-19, 11, 1, -12, 5, -9], [-8, 10, -15, -5, 7, 2], [-8, -12, -1, -14, -8, -12], [-16, 12, 7, -9, -13, 12], [-16, -10, -3, -16, -2, 7], [12, 6, -15, -8, 20, -20], [1, 6, -21, -8, 21, -6], [28, 4, -1, 9, 3, 0], [14, 6, -21, -4, 4, -24], [-17, 25, -25, 3, 12, -11], [-6, 10, 3, -3, 9, -8], [5, 12, 2, 4, 13, -6], [-20, 14, -10, -6, 4, 3], [12, 14, 0, 8, 10, -2], [12, 8, 2, -2, 10, 1], [-5, -21, 9, 20, 5, 15], [-9, -27, 29, 11, -16, 25], [-10, -6, -9, -1, 15, 0], [1, 7, 14, 20, -7, 12], [-22, -20, 18, 9, -7, -2], [-5, 1, 6, 14, -9, -4], [-41, -2, 0, 3, 2, 8], [9, -3, 11, 7, -12, 6], [-5, 5, -11, 2, -9, -17], [5, -10, -1, 3, 0, 10], [-10, 14, -16, -10, -6, -4], [9, -21, 7, 6, -17, 9], [11, -20, -1, 5, 6, -12], [28, 0, -3, 8, -2, -16], [2, 8, -12, -21, -3, -3], [-8, -10, 10, 8, -14, -14], [18, 20, 0, -1, 11, -8], [9, -21, 21, -15, -12, 5], [42, -18, -1, 1, -3, 1], [11, -5, -1, 10, -13, 5], [-40, -2, 5, 16, -8, 2], [36, 4, 6, 3, 1, -7], [-16, 10, 12, 10, 2, 22], [-20, -30, 33, 10, -18, 18], [-19, 1, -3, -12, 9, -13], [9, 0, 15, 6, 7, 14], [4, 0, 3, -6, -4, 0], [-6, 8, -12, -11, 11, 7], [25, 13, 1, -5, -2, 11], [-1, 25, 2, -4, -7, 7], [42, 6, -5, 9, 7, -16], [16, 17, -8, -21, -9, -3], [5, -19, -9, 6, 11, -21], [-2, -22, 11, 5, -5, -14], [-9, -5, -10, 3, -2, -19], [7, 6, 6, -7, -4, -14], [22, 12, 7, 6, -2, 2], [5, 7, 10, 2, -5, 0], [6, -24, 34, -1, -11, 4], [3, -20, 17, 14, -3, 18], [21, 25, -31, -6, 17, -11], [47, -13, 13, -2, -11, 5], [-21, -9, 13, 6, -3, 12], [3, -11, 15, -15, -12, -11], [-39, -9, 27, 0, -5, 29], [-45, -5, 1, -5, -8, -1], [-17, 5, -6, -3, -6, 20], [22, 6, -26, -11, -3, -12], [8, -7, 24, 11, -19, -1], [0, 12, 5, 8, -14, -7], [-10, 14, -14, -1, 9, 2], [15, -14, -6, 1, 6, 0], [14, 10, -4, -18, 12, 9], [16, 4, 14, -3, -9, 4], [12, -14, 12, 2, 6, -15], [-38, -9, 13, 16, 2, 7], [-14, 4, 2, 8, -12, 16], [-43, 11, 4, 8, 17, 15], [6, 5, 4, -10, 12, -11], [11, -3, 15, -13, -8, 0], [-42, 4, -8, 17, 13, 6], [41, -17, 19, -6, -13, -8], [32, 0, -7, 12, 4, 10], [40, 3, 0, -17, -11, 7], [-23, 9, 1, 15, -8, 17], [-8, 9, -16, -8, 18, -1], [34, 12, -5, -11, -5, -11], [-9, -3, 30, 1, -18, 12], [23, -9, 12, 9, 2, -4], [17, 18, -20, 1, 16, -18], [-7, 3, 4, -5, -8, -11], [22, -8, 16, 8, 10, 6], [4, -16, 13, 6, 4, -3], [-16, 8, 5, 12, 6, 0], [11, 11, -12, 3, -2, -16], [-11, 7, -8, -1, 10, -17], [16, 11, -6, -17, 3, -9], [-12, -2, 34, 20, -28, 26], [11, 13, -4, -8, 15, -12], [-8, -2, -6, 8, 2, -2], [3, -17, -2, 6, 13, 7], [-5, -3, -7, 8, 15, -17], [28, 8, 15, 17, 1, 4], [24, -8, 19, 2, -10, 0], [25, -3, 0, 8, -13, -7], [10, 14, 3, 8, 6, 14], [5, 7, -26, -2, 3, 3], [-6, -12, -14, -16, 2, -32], [-14, -30, 4, -1, -11, 6], [-49, 21, 10, 3, 2, 15], [-34, -28, 26, 14, -10, 12], [-8, 28, -1, -7, -1, 10], [28, 0, 8, 1, -3, 9], [-41, 12, -11, 11, 14, 0], [-5, -10, 24, 1, -6, 16], [31, 1, -10, 10, 7, 7], [17, -13, 2, -17, 6, -14], [24, 8, -10, 10, 8, 2], [10, -14, 21, 5, -23, 30], [-19, 3, -3, -2, -11, 17], [35, 7, -10, 1, 4, -5], [-7, -6, 1, 13, 0, 10], [-17, 19, 22, -6, -9, 3], [16, -13, -16, 7, -9, -7], [-4, -6, 7, -20, 0, -2], [0, 12, -15, 7, 1, -26], [1, -7, 16, 7, -4, 29], [19, 18, -25, -6, 15, -14], [-14, -23, -8, 4, 0, -3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4026_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4026_2_a_w();".
// 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_4026_2_a_w(:prec:=6)
    chi := MakeCharacter_4026_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_4026_2_a_w();".
// 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_4026_2_a_w( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4026_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-8, 8, 32, -3, -14, 1, 1]>,<7,R![356, -281, -48, 82, -10, -5, 1]>,<13,R![-94, -457, 215, 104, -44, -2, 1]>],Snew);
    return Vf;
end function;