
// Make newform 1859.2.a.l in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1859_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_1859_2_a_l();".
// 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_1859_2_a_l();".
// 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 := [-12, 4, 22, -1, -9, 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], [-3, 0, 1, 0, 0, 0], [0, -4, 0, 1, 0, 0], [2, 10, -1, -7, 0, 1], [16, -10, -13, 7, 2, -1]];
    Rf_basisdens := [1, 1, 1, 1, 2, 2];
    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_1859_a();"
function MakeCharacter_1859_a()
    N := 1859;
    order := 1;
    char_gens := [508, 1354];
    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_1859_a_Hecke(Kf)
    return MakeCharacter_1859_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 := [[0, 0, -1, 0, 0, 0], [0, 0, -1, 0, -1, 0], [1, 1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [1, -1, 1, -1, 1, 0], [2, 1, 1, -1, -1, 0], [1, 2, -1, -1, 0, 0], [2, 0, 0, 0, 3, -2], [1, 0, 0, 1, 1, -1], [4, 1, 2, -1, -1, 1], [4, 0, 1, 1, 0, -1], [-1, -2, 0, 0, -1, -1], [2, 2, -1, -1, -1, 0], [2, 4, -4, 0, -1, -1], [0, 2, -2, 1, -2, -3], [-4, -2, 1, 0, 0, 1], [3, 0, -1, 2, 0, -1], [-2, -4, -1, 1, -3, 1], [2, -1, 0, 0, -1, 0], [2, -1, 0, -1, -2, -5], [1, -1, -2, -1, 1, 2], [1, 1, -4, -1, 3, -3], [2, 1, 2, 3, 0, -2], [-6, 1, -1, 3, 0, -3], [1, -2, 7, 0, 4, 6], [4, 1, -4, -2, -4, 1], [3, -1, -1, 2, -4, 0], [-3, 3, 3, 1, 1, 3], [3, -6, 4, 0, 3, 5], [-1, -3, 0, -1, -3, 2], [4, -3, -1, -1, 0, -4], [-6, -2, 4, -2, 2, 2], [5, 1, 4, -2, 6, 1], [6, -2, 1, 1, 1, -1], [-3, 4, -1, 3, -3, 1], [1, -4, 8, 0, 3, 6], [4, -2, -7, 0, 1, -3], [-1, -3, -1, -1, -4, 0], [2, 2, 4, -1, -6, -1], [3, -3, 2, -3, 2, -2], [-5, 0, 1, 1, 0, -5], [0, 0, 2, 3, 7, 4], [-3, -4, 6, -3, -1, 1], [0, 4, 4, 0, -4, 3], [-4, -1, 5, 3, 5, -2], [1, -4, 0, 2, -3, 0], [-7, 2, -1, -1, 0, -4], [4, -1, -5, -3, -7, 3], [6, 6, -4, 2, -4, -6], [-3, -4, 11, 2, 6, 6], [4, -1, 3, -2, 6, 3], [4, -4, 2, 4, 2, 3], [-3, -1, 11, -4, 5, 5], [-7, -4, -1, 1, 4, 0], [12, 2, -6, -3, 0, -6], [2, -2, 5, -3, 3, -3], [-1, 0, 7, -1, 3, 0], [5, -8, 3, -2, 5, -2], [7, 1, -8, 6, 5, 0], [4, 7, 4, -6, -1, 2], [11, -5, 4, 1, 5, -2], [5, -4, 3, 4, 0, 3], [12, -3, -4, -2, -4, -4], [-5, -5, -1, 2, 3, -3], [-6, 2, 0, 6, -4, 6], [6, 1, -1, 1, -6, -1], [-15, -1, -1, 2, 4, 1], [8, 12, -2, -1, 0, 0], [-4, 6, 2, -3, 3, -1], [5, 0, -5, -3, -8, -2], [-6, 6, -4, 5, 0, 6], [3, -3, 3, -1, 3, -2], [-11, 6, -7, -2, 2, 4], [-10, 0, 1, 0, -3, 7], [11, -5, 7, 1, 1, 3], [2, 3, -1, 7, 3, -1], [0, -2, -2, 8, -1, -1], [3, -7, 16, 2, 10, 1], [-1, -2, 2, -1, 3, 3], [7, -1, 0, 1, 2, 2], [-21, 1, 3, 3, -4, 7], [3, 5, -5, -2, 7, -5], [-1, -7, 3, 4, 0, -1], [-3, -2, -8, 4, 7, 0], [2, -5, 2, -2, 8, -2], [6, 0, -7, 4, 2, -5], [7, 6, 5, -3, 3, 2], [7, -4, -8, 2, 1, 2], [11, 4, -1, 0, 6, -5], [2, 5, 7, -1, -5, 6], [-11, 4, -4, 1, -7, -2], [7, -3, 8, 4, -3, 4], [5, -4, 9, -3, 4, -5], [-17, 3, 1, 5, 2, 9], [10, -4, 10, 0, 15, 1], [-5, -4, -8, 3, 4, 0], [-10, 3, 1, 8, -3, 4], [7, 3, 3, -1, 1, 2], [-6, -4, 15, 1, 1, 11], [4, -4, 14, -3, 5, 6], [9, 3, -5, -2, 0, -9], [1, 0, -2, -3, -1, -7], [-4, 1, -3, 0, 5, -6], [17, 3, -9, -3, 3, -9], [-12, 10, 6, 0, -6, 3], [3, 2, 1, -3, 5, 2], [11, 6, -6, 10, -1, -5], [-2, 16, -3, -5, 0, -1], [7, -9, 4, 2, 7, -6], [22, -5, -7, -7, -4, -11], [13, 2, -2, 3, 2, 6], [11, -6, 9, 0, 2, -3], [4, 2, 1, -7, 1, -4], [13, 8, 3, -4, 3, 7], [-4, 2, -11, -5, 3, -2], [8, -8, 6, 1, 14, 4], [16, -1, -5, -10, 1, -9], [-12, -7, 2, 1, -4, 7], [3, -7, -1, 7, 7, -1], [-16, -2, -1, -6, 8, 3], [-13, 0, 5, -2, 6, 8], [-6, -10, 9, -5, 7, 9], [5, -12, 0, 1, -3, -1], [8, -5, 1, 2, 5, -1], [20, 0, -6, -2, -3, -3], [4, 10, -17, 1, -5, -11], [-9, 10, -5, 2, -6, -4], [12, -2, -5, -5, 4, -3], [12, -10, 7, -2, 5, -9], [12, 3, 2, -8, 0, -3], [18, 12, -18, -2, -4, -10], [1, -7, 10, 8, 1, -1], [3, -6, -6, -2, -3, -13], [-2, 4, -7, -3, -5, -7], [19, -3, -3, 2, 4, 2], [-4, -7, 4, -13, 4, 3], [3, -11, -2, 4, -9, -5], [-9, -7, 10, 2, 2, -1], [-7, 2, 1, 10, -4, 4], [16, -2, -3, -6, -11, -13], [9, -14, -5, 4, 2, 4], [-12, 9, -6, 0, 2, 1], [-12, -5, 0, 2, -1, -9], [-5, 2, 11, 4, 10, 1], [26, 10, -3, 2, 4, -5], [-2, -4, -5, -8, -2, 13], [15, 0, 5, -1, 0, 5], [18, 10, -8, 3, -4, -4], [16, 3, -8, 0, -5, -2], [-4, 12, -2, -2, 1, -9], [-10, -2, 0, 1, -12, 10], [1, -10, 1, 2, -4, -6], [9, -10, -2, 4, 1, 0], [0, 2, -6, -5, -8, 3], [3, 18, 10, -5, -1, 1], [2, 10, -7, 1, 6, -4], [23, -8, 3, 2, -3, 2], [-8, -2, -14, 4, -12, -2], [17, 2, 1, 2, -10, 4], [2, 6, 2, 1, -2, -12], [-10, 1, 3, 1, -17, 10], [-1, 2, 3, -10, -4, 5], [5, 5, -2, 1, -8, 0], [-19, -2, -1, -2, -4, 6], [-13, 12, 10, 0, 1, 4], [-3, -3, 5, 8, -5, 4], [1, 8, -5, -4, -10, -10], [-7, 3, -1, -3, 3, -3], [-11, 5, 23, -3, 8, 9], [23, -4, -8, 3, -16, -6], [-9, 13, -14, -4, -5, -8], [25, -4, -7, -3, -5, -10], [-16, -2, 3, 0, -9, 1], [23, -2, 11, -10, 5, -1], [2, 4, -15, 5, -5, -1], [-17, 0, -10, 2, 2, 6], [23, -4, 15, -5, 14, 0], [5, -8, -2, -1, -13, 5], [-23, 4, -9, 9, -6, 2], [-6, 4, 11, -1, 15, 9], [5, -5, 1, 9, 5, -7], [14, -2, 6, -8, -2, 4], [-29, 2, -11, 2, 0, -16], [8, -7, -6, -7, -7, 9], [-13, 6, -7, 5, 6, -4], [7, 2, -4, 4, 11, -11], [-28, 1, -16, 7, -15, 2], [-5, 5, 1, 10, -2, -1], [14, 7, 15, -4, 4, -2], [-20, 4, -1, 7, -3, 7], [13, -2, -10, -4, 9, -3], [3, -15, -3, 0, 8, 8], [-4, 7, -6, 5, 0, 11], [-13, 11, 1, 5, 12, 8], [10, -15, 6, -1, 9, 2], [-10, -7, 1, -4, -4, -6], [9, 6, -16, 3, 0, -9], [21, -3, -7, -4, 8, -7], [6, -8, -1, -2, -2, -3], [8, 18, -11, 2, -3, -13], [8, 9, -4, -5, -3, 9], [12, 10, -5, -10, -4, -9], [-3, -6, 0, -8, 1, 1], [7, -2, -6, 8, -8, -10], [8, 12, -1, -2, 9, 4], [3, 15, 3, 2, 0, -5], [-3, 17, 9, -2, -1, 2], [11, 4, -1, 3, 2, 10], [-8, 14, -1, 6, -12, 7], [-33, -5, -8, 1, -3, 15], [-13, 4, -10, -4, -9, -12], [12, 12, 1, -4, -13, 5], [-7, -21, 16, -3, 9, 12], [4, -8, 9, -3, 12, 0], [-20, -6, -3, 3, 9, -1], [-12, -13, 9, -13, -2, -3], [4, 2, -14, 4, 12, -7], [-25, -1, -8, 7, -7, 9], [0, -2, -19, -5, 0, 1], [24, 10, -8, -7, 0, -2], [-16, -2, 4, 10, 1, 3], [-7, 4, 3, -4, -12, 3], [19, -13, 19, 2, 12, -5], [-9, -3, 6, 3, 0, -9], [-2, 4, -13, -10, -10, -6], [-3, -20, 7, 0, 16, -11], [-25, 4, -21, 3, -24, 0], [9, -13, 5, -1, 0, 15], [-12, -13, 8, -1, 7, 5], [-30, -1, 5, -4, 17, 2], [3, 4, 11, 7, -12, 2], [-5, -1, -7, -9, -5, -14], [24, 0, 15, 3, 5, -3], [-22, 14, -3, 6, -1, 13], [21, 8, 3, 5, 6, 6], [-25, 3, 23, -4, 13, 10], [-20, -6, 19, -3, -3, 9], [-12, -2, 5, 10, 1, 14], [-10, 8, 8, -9, 14, 4], [-3, -12, 9, 2, 20, -6], [2, 0, -4, 11, 16, 10], [21, 16, -7, 4, 5, -7], [5, 0, -10, 9, 2, 3], [-17, -11, -5, 0, -18, -3], [18, -10, -10, 3, -10, 1], [-8, -18, 5, 1, 13, -7], [13, 12, 10, 0, -7, 7], [-30, -4, 4, -10, 2, 0], [-8, -18, -10, 5, 0, 4], [-1, 12, -16, -2, -21, 3], [20, 12, -18, 8, -12, -15], [-4, 1, -23, 5, -4, 0], [8, -2, 1, 0, -27, 7], [-16, 14, -6, 1, -16, 0], [16, 7, -20, -7, 8, -20], [-20, 0, 21, -1, 17, 7], [7, -21, 3, 5, 6, 4], [-3, -2, 14, 3, 18, 19], [6, -2, 11, -7, -11, 11], [-6, 8, 27, -4, 5, 13], [17, 13, -1, -3, -9, -2], [-29, 5, -10, 4, -16, -1], [-10, 1, 3, 0, 6, -4], [19, -2, -18, 4, -11, -1], [5, 5, -12, -6, -5, -15], [2, -2, -14, 10, 9, -1], [-6, 20, 8, 1, 2, 0], [-32, 1, 16, 2, 1, 0], [5, 6, -2, 4, -14, -8], [-3, 12, -12, 6, 7, -9], [-26, -8, -14, 3, -3, -1], [-13, 13, -15, -8, -6, -3], [7, -13, 20, 10, 3, 3], [-28, -8, 2, -9, -8, 4], [-25, -4, 6, 8, 9, -1], [32, -1, 3, 0, 0, 11], [25, 13, 1, -13, -5, 2], [-30, -7, 5, -5, -5, -10], [-23, -15, -9, 0, -8, 15], [13, 10, -12, -9, 1, -8], [10, -3, 13, 4, 1, -2], [-21, -15, 4, 0, -4, 21], [-20, -11, 27, 4, -3, 18], [-2, -12, 0, -13, -6, -11], [2, 2, -16, -10, -11, -9], [-18, 5, 15, 0, 10, 9], [13, 12, -13, 7, 18, -7], [6, 1, -5, -1, -2, -5], [-7, 5, 6, -6, 1, 11], [-4, 4, -10, -5, -20, -13], [-25, -7, -7, 5, -27, 12], [1, -3, -2, -3, -11, -26], [8, -5, -3, -6, 9, -5], [-16, -2, 0, 15, 13, -14], [-36, -16, 16, -6, 0, 6], [18, 0, 5, 1, -2, 2], [2, 7, 1, -6, 5, -14], [10, -1, -10, 3, 16, -5], [17, 8, -5, 13, 7, 0], [0, 3, 2, -12, 9, 6], [33, 6, 10, -15, 3, 7], [30, -12, 9, -3, 21, 2], [-1, 10, -17, -2, -4, -9], [-12, -10, 1, -3, -8, -2], [3, 2, 7, 6, 14, 13], [-2, 10, 6, -1, 9, -7], [28, 10, -11, -2, -5, 10], [15, 0, 6, 9, 12, 7], [-35, 6, -17, -2, -24, 8], [-7, 5, 5, 6, 22, -2], [24, 25, -2, -8, 5, -4], [-39, 2, -10, 0, -3, -6], [-1, -8, 11, 0, -14, 20], [-21, -21, 20, 9, 4, 9], [-19, 16, -6, -6, 11, 0], [-3, 11, 4, 1, -13, -10], [-3, -3, 0, 6, -25, 9], [-23, -5, 11, 5, 7, 8], [5, 20, 7, -2, -6, 0], [6, -5, 7, -5, 7, -3], [40, 4, -4, 12, -7, -6], [-12, 12, -17, 4, -7, 9], [-28, -9, -6, -2, -8, 10], [4, -3, 8, -8, -15, -15], [11, -2, 15, -12, -2, 13], [-27, 4, -11, 14, -4, 0], [3, 0, -17, 5, 0, -18], [26, 5, 5, 4, -6, 7], [44, 1, 9, -2, 6, 7], [-27, -8, 13, -10, -5, 1], [-25, 6, -4, 6, -3, -17], [9, -2, 20, -5, 9, 15], [-21, 12, 0, 6, -1, 13], [-10, 4, 26, -2, 28, 17], [23, -13, 9, 1, 9, 7], [-12, 19, 11, 6, 11, 1], [7, -16, -18, -1, 0, -4], [-7, 15, -16, -8, -19, -6], [10, 3, -2, -17, -4, -8], [-11, 0, 2, 1, 0, 14], [10, 1, -15, 4, -4, -11], [-11, -12, 19, 3, -6, 14], [4, 0, 2, -14, -4, -4], [-21, -5, -4, 3, -17, -4], [-34, 8, 11, 14, 0, 2], [-8, 0, -18, 14, 0, -6], [16, -26, 8, 3, 10, 14], [6, -19, 21, 3, 13, 21], [-30, 5, -5, 4, 9, 6], [-2, -4, 25, -2, -1, 23], [22, -4, -18, 1, -21, -17], [47, 2, -8, -15, 1, -13], [10, 13, -11, -9, -2, 5], [9, -25, 0, 3, 18, -5], [-18, 16, 21, 2, -5, 9], [4, -16, 3, -2, 11, -21], [27, 10, -1, -19, 5, -10], [-25, -18, 1, 7, 6, -7], [24, 12, -2, -4, 1, -16], [-27, 0, 5, 7, -7, 2], [9, 4, 0, -5, -9, 18], [-3, 1, 15, 12, 26, -11], [23, 12, -25, -2, -4, -3], [19, 17, -42, 2, -13, -19], [36, -4, 18, -7, 8, 4], [-6, -1, -18, 14, 0, -9], [-36, -3, -7, -3, -6, -19], [-6, -8, -5, 8, 17, -7], [13, -10, -21, 2, -8, -6], [7, -17, -1, -15, 6, -13], [5, -6, -14, 10, -10, 1], [22, -1, -21, 2, -3, 2], [8, 28, -10, -15, -5, 5], [44, 0, 1, -8, 7, -8], [-6, 2, 18, -6, -10, 4], [6, 16, 12, 0, -12, 4], [-21, 9, -19, 1, -1, -6], [26, -6, -11, -3, -20, -9], [8, -5, 0, -3, 4, 12], [-61, -4, -13, 6, -1, -5], [-19, 19, -17, 7, -18, -7], [19, -6, 6, -12, -7, 9], [13, -13, 7, -2, 0, -23], [3, -22, 19, 4, 2, -6], [-37, 0, 4, -3, 11, 1], [-12, 11, 12, -3, 20, 12], [-24, -4, -4, 0, -12, 2], [18, 2, 20, -5, 16, 0], [-2, -18, 13, -1, -10, 16], [37, -18, 15, 10, 11, 3], [-6, -14, -15, 20, 7, -2], [-17, -20, 8, 1, 7, 3], [-8, -12, 3, -1, 13, 2], [-7, 4, -6, -10, -21, 13], [16, 14, 5, -1, -5, -3], [-15, -11, 37, -6, 22, 22], [3, 22, -17, 4, 2, 1], [23, -5, -10, -7, -18, -3], [31, -14, -9, 13, 16, 4], [-39, -13, 16, 7, 7, 7], [6, -9, -19, 5, -3, -4], [-15, -26, 0, -1, 16, 0], [25, 8, 7, 7, -9, -18], [28, 10, 3, 5, -13, -7], [-15, 4, -2, 4, 7, 7], [-29, 21, -6, 6, -16, -7], [11, 8, -29, 4, -8, 4], [5, -4, -18, -9, -11, -21], [-28, -7, -3, 2, 2, 23], [-11, 18, 7, 5, 29, 8], [-36, 4, -3, 12, -9, 3], [-18, -5, 9, 6, 15, 12], [-7, -9, -6, 22, 4, -3], [5, 0, -8, 6, -4, 10], [-1, 1, -10, -10, -19, -10], [-21, -10, 6, 4, 3, 9], [-27, -13, 4, 13, 3, 17], [-26, -11, -5, -6, 3, 2], [41, 3, -28, -2, -18, -12], [-25, -7, -6, 1, -10, -5], [-15, 4, -19, 8, -36, 3], [-20, 5, -2, 20, -14, 5], [16, 29, -17, 10, -7, -27], [-2, 10, -15, -7, -4, -18], [-15, 10, 0, -5, 21, 6], [18, 6, -15, 4, -18, -25], [25, -12, -4, 7, 19, 0], [17, -6, 14, 15, -1, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1859_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1859_2_a_l();".
// 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_1859_2_a_l(:prec:=6)
    chi := MakeCharacter_1859_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_1859_2_a_l();".
// 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_1859_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1859_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-3, -5, 24, 1, -10, 0, 1]>,<7,R![-122, -52, 82, 27, -16, -3, 1]>],Snew);
    return Vf;
end function;