
// Make newform 6042.2.a.v in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6042_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_6042_2_a_v();".
// 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_6042_2_a_v();".
// 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, -4, 6, 9, -4, -3, 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, -3, 9, 0, -4, 1], [-3, 5, 9, -4, -3, 1], [6, 1, -12, 1, 4, -1], [-7, 1, 19, -4, -7, 2]];
    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_6042_a();"
function MakeCharacter_6042_a()
    N := 6042;
    order := 1;
    char_gens := [2015, 4771, 2281];
    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_6042_a_Hecke(Kf)
    return MakeCharacter_6042_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, 0, 0, 1, 0], [-1, 0, 1, 1, 0, 0], [-1, 1, 0, 0, 1, 1], [-2, -1, -1, -1, 0, 1], [1, -2, -1, -1, -2, 0], [1, 0, 0, 0, 0, 0], [2, 0, 1, -1, -1, -1], [-1, 0, 0, 2, -1, -1], [-1, 3, -2, -1, -2, 0], [0, -3, -3, -2, 0, 0], [-2, 0, 0, -1, -2, -1], [-3, 3, 1, 2, 1, -3], [0, 1, 0, 1, -2, -1], [-1, 0, 0, 0, 0, 0], [2, 1, 2, 1, -3, -3], [-4, 1, 2, -1, 3, -1], [1, -2, 1, 2, -2, -1], [3, 1, 2, 1, 0, -2], [-5, -2, -1, -3, 0, 2], [-2, -2, -3, -1, -1, 3], [-5, 3, 1, 0, 3, 3], [4, -1, -4, -4, -3, 2], [-1, -1, 1, 4, 3, -3], [-1, -1, 1, 3, 2, 2], [-1, 0, -2, -1, 1, 4], [-3, -6, -3, -4, 0, 5], [-2, -2, 0, -5, 1, 1], [-6, -4, -3, -2, 1, 4], [-9, 6, -2, -1, -3, 0], [3, -6, 0, -1, 2, 4], [-2, 2, -1, 5, 2, -1], [-4, 3, 3, 8, 0, -2], [-4, 0, -1, 4, 1, -2], [-5, 3, 0, 3, 4, 4], [2, -1, 0, -3, -3, 1], [0, 4, 6, 4, 2, -4], [0, 0, 5, 6, -1, -2], [-4, 3, 4, 5, 3, -1], [-5, 7, 0, 3, 4, -2], [-7, 1, 1, 0, -2, -7], [4, 8, 1, 0, -3, 0], [-14, 3, 1, -1, 1, 3], [1, 2, 2, -5, -1, 0], [-2, -7, 4, 0, -1, 2], [-8, -2, 5, 1, 2, -1], [-1, 0, 5, 3, 1, 0], [-3, -4, -1, 7, 4, 0], [8, -4, -1, 4, -3, 2], [-5, 0, 0, 1, 2, 0], [2, 3, 5, -1, -2, -5], [2, 1, 2, 5, 3, -1], [0, 3, -2, 0, 9, 0], [4, 3, 0, -7, -1, 1], [5, -2, 2, -2, -3, -5], [-3, 10, -1, 5, 2, -2], [-3, -1, 1, -6, -1, -1], [0, -2, 0, 3, 4, 7], [-5, -5, -7, -4, -5, 1], [1, -6, -5, 0, 0, 1], [-5, 1, -2, 4, 0, 1], [-9, 3, 6, -1, -3, -6], [4, 4, 4, 2, 5, 0], [-4, -1, 8, -4, -1, -4], [15, -8, -1, 2, 0, -5], [1, 4, 4, 9, -4, -6], [5, -9, -6, -6, -5, 5], [-2, 1, -1, 8, 1, 2], [-5, 4, 3, 5, 2, -6], [-7, -7, 0, 5, 1, 2], [17, -6, -4, 1, -5, 0], [-5, 1, 3, 0, -3, 3], [-4, 4, 4, -2, -1, -8], [1, -2, -1, 6, 4, 7], [-2, 16, 3, 1, 3, -7], [-5, 8, 5, 3, 7, -6], [5, 4, -7, -4, -2, -3], [2, 0, -2, -9, 4, 3], [3, 0, -3, -12, -2, 1], [4, 9, 4, 3, -3, 1], [4, -1, -6, -3, -4, -1], [7, -11, -3, -4, 3, 1], [-1, 6, -5, -4, -2, -3], [1, 7, 5, 6, -2, -7], [7, 5, -3, -6, -5, 1], [11, -6, 0, 3, 3, 4], [11, 3, 5, -3, -5, 0], [-2, 1, 1, 3, 6, 1], [-12, 6, 6, 9, 2, -5], [-16, 6, 5, -4, 10, 2], [-2, -7, -11, -12, 0, 6], [-10, -4, 6, 0, -5, 4], [-2, 1, 2, -3, -3, -7], [1, -15, -6, -2, -2, 1], [1, 2, 0, 7, 3, 2], [0, -8, 2, 5, 8, 7], [0, 17, 5, 7, 4, -7], [-6, -8, -1, -1, 4, -3], [3, -7, 5, -6, 1, -1], [-17, 10, 6, 6, 4, -3], [-3, 6, -5, 2, 15, 1], [0, -3, 3, 3, -10, -5], [9, -3, 9, -6, -7, -7], [5, 2, 4, 7, 3, -8], [-13, -2, -5, 5, -2, 0], [-3, -1, 6, -7, -5, 4], [-4, 1, 0, -2, 1, -8], [17, -8, -11, -7, -10, 10], [-2, 8, 1, 0, 2, 6], [-13, 2, 5, -2, -11, -3], [-25, 8, -3, -13, 8, 4], [-14, -1, -1, -2, -5, 4], [3, -13, 1, -3, 5, 0], [-2, 2, -3, 2, 7, 4], [-5, -6, -5, -5, 6, 2], [7, -4, -5, -8, -10, 3], [7, -18, 1, 7, 5, 10], [-7, -7, 0, 2, 4, 1], [-12, -3, -3, 3, 2, -1], [1, -8, -8, -5, -1, 0], [7, -8, 3, 3, -4, 2], [-11, -8, -3, -9, 2, 0], [-8, -2, -6, -9, -2, -1], [-18, 4, 0, -5, -8, 3], [7, 15, 11, 7, 8, -8], [-6, 1, 2, -4, 1, 8], [-11, -1, -10, -8, 8, 9], [-3, -12, -1, 0, 2, 11], [15, 2, 2, 0, 3, 9], [-4, 3, -6, -10, -5, 2], [12, -11, -10, -7, -13, -3], [-2, 3, 2, 8, -2, -12], [1, 2, -7, 6, 6, 5], [20, 2, -1, 0, -8, -4], [-16, 12, 0, -3, -6, -1], [4, -12, -16, -16, -4, 10], [-11, 6, 7, -3, -2, -8], [3, -4, -13, -1, -10, 0], [4, -9, -3, 4, 7, 4], [0, -1, 3, 3, 11, -3], [31, 5, 11, 8, -2, -11], [-20, 4, 6, -4, 2, -2], [-9, 7, 2, 6, 6, -1], [-8, 6, 10, 18, 7, -8], [7, -7, -9, -21, 1, 10], [-2, -4, -16, -6, -10, 4], [-10, 11, 1, -1, -4, -1], [2, 9, -8, 8, 9, 4], [11, -11, -5, -4, 5, 13], [-10, -14, 3, -2, 5, -2], [1, -3, 2, 4, 3, -5], [1, -12, -6, -4, -5, 11], [-1, 6, 4, -1, -3, -4], [-4, -8, 1, -12, 2, 8], [2, 4, -8, -11, -8, 7], [-10, -4, -6, -10, 4, 12], [-1, -3, 11, 3, -3, -6], [-23, 0, -11, -4, 10, 11], [12, -2, 5, -17, -3, 3], [-4, 14, 11, 12, 12, -10], [-19, -8, -3, -9, 2, 12], [15, -17, 5, -1, -3, -4], [-3, -8, -11, -11, 8, 18], [-13, -1, 1, 15, 7, -4], [-8, 7, 0, -5, -11, -5], [10, 2, -2, 3, -7, -15], [-5, -3, -8, -4, 2, -7], [-5, -14, 10, -3, 4, 0], [-5, -8, -14, 7, 9, 14], [0, -5, -6, 9, 11, 7], [14, -1, 0, -1, 4, 5], [10, -21, -8, -8, -14, 2], [7, 3, 5, 7, 7, -4], [-9, -2, -8, -8, 1, 15], [17, -15, 5, 8, -1, -1], [19, -2, -5, 3, -6, -8], [10, -5, 5, 1, 0, 3], [22, -12, 3, -7, -1, 1], [-16, 4, -1, 5, 9, 9], [-16, 13, 3, 7, 20, 5], [-6, -26, -3, -10, -3, 10], [-11, 13, -9, 1, 0, 4], [-6, 7, 9, 10, -6, -8], [19, 0, 1, -5, 13, 6], [-8, 6, -5, -13, -7, 3], [-8, -11, -11, -19, 0, 13], [-5, 6, 8, 10, 4, 1], [-1, 12, 1, 0, -4, -3], [10, -2, -2, 8, -9, 2], [-11, 9, 13, 9, 21, 10], [1, 3, 11, 8, 5, 1], [3, -6, 1, 5, 8, 6], [-9, 21, -1, -5, 3, -2], [4, 8, 8, 7, -6, -11], [-20, -10, 2, -8, 0, -8], [-3, -2, 3, -12, 4, 1], [19, 10, 6, 6, -9, -5], [12, -4, 5, -15, 0, 5], [-5, 12, -5, 6, -11, -5], [13, -9, 9, 4, 1, -1], [2, -11, -9, 0, 12, 14], [11, 20, -1, -2, -1, -11], [-15, -10, -6, 22, 10, 3], [21, 10, -10, -1, -6, -6], [-12, 5, 4, -5, 12, 13], [21, -11, -5, -7, 1, 2], [9, -5, 3, 2, 11, -3], [1, 7, 23, 9, -9, -18], [16, -3, -10, -8, -17, -4], [-3, 20, -3, 4, -4, -1], [19, -9, -2, -2, 0, 7], [6, 5, 11, 20, -2, -10], [-49, 2, 3, 7, 8, 4], [22, -18, -2, 0, -4, 10], [8, -18, -2, -3, 0, 9], [5, 0, 3, 16, -2, -7], [13, 5, 15, 11, -7, -8], [3, -11, 1, -2, -13, -7], [9, 11, -5, 5, 8, -8], [8, -1, -2, -2, 0, -8], [11, -13, -6, -8, -12, -1], [20, -9, 4, 7, -7, 3], [19, -1, -9, -12, 9, 7], [-3, 23, 7, 10, 15, -9], [-16, 16, 0, -4, 0, 8], [19, -7, -13, -6, 1, 11], [29, 2, 5, 4, -8, 5], [-20, -7, 0, 7, 0, 3], [-6, 4, -8, -3, 4, 13], [27, -20, -11, -9, -10, 10], [-23, -3, -4, -4, 5, -3], [-25, 2, -17, -8, -1, 15], [-15, 24, 7, 8, 10, -5], [27, -2, -13, 9, -12, -6], [18, -14, -7, -17, 3, 19], [37, -14, 3, -5, -6, -2], [13, -2, 0, -2, -3, -3], [15, 13, 20, 13, 3, -14], [17, -10, -1, -2, 0, -1], [-14, 0, 10, 13, 20, 5], [1, -2, -1, 2, 2, -15], [3, 4, -9, -2, 8, 3], [9, -16, -11, -20, -1, 5], [4, 11, 8, -5, -13, 3], [12, -3, -7, -6, -16, -14], [-11, 15, -3, -12, -9, -15], [-20, 8, 0, -6, -4, 6], [-2, 18, 13, 14, -6, -22], [27, 6, 13, 10, -3, -1], [45, -26, -7, -16, -18, -1], [-24, -13, -18, -14, 1, 16], [-10, 2, -2, -12, -5, 14], [18, -21, -15, -16, -7, 16], [-16, 7, 4, -6, 2, -8], [-2, -3, -13, -4, 5, 2], [17, -7, -9, -9, -15, -14], [-18, -2, -11, -10, 0, 8], [-16, 1, 4, 8, -11, -10], [3, -3, -11, -9, -7, 8], [-18, 8, 13, 17, 3, -15], [-7, 6, 7, 0, -2, 9], [34, 6, -4, 3, -8, -3], [13, -11, 3, 4, 12, 1], [10, -5, -16, -16, 3, 16], [-11, -9, -7, 8, 15, -7], [20, 0, 7, 11, 17, -1], [5, 18, 9, 3, -2, -12], [-18, 10, -6, -1, -8, -5], [3, 4, 16, 6, 1, 3], [18, -17, -4, 12, 0, 8], [-20, 12, 8, 15, 8, 5], [1, 1, -6, 9, -10, -8], [0, 6, 8, -2, 4, 4], [-7, 2, -17, -13, -10, 4], [4, 9, 15, 2, 12, -2], [26, 11, 5, 7, -7, 3], [22, -3, 11, 21, 2, -5], [4, 19, 10, 14, -5, -20], [-12, -13, -11, -8, -5, 6], [15, -9, -1, -8, 1, 7], [3, 9, 7, 3, -7, 4], [-12, 9, -3, 1, 0, -9], [17, -19, -6, -6, -18, 9], [0, -4, -19, -14, 1, -4], [-8, 6, 8, 0, -9, 0], [35, -11, 9, 6, 9, -1], [-50, 13, 4, 4, 15, 18], [2, 5, -2, 19, 9, -5], [-16, 2, 10, -5, -2, -15], [34, -20, -1, -2, 1, 2], [-39, 10, 10, 9, 22, -10], [-25, 2, -5, -2, 12, 17], [-20, -13, -8, -9, 3, 1], [19, -11, 4, 4, -6, -7], [24, -4, -2, -17, -19, -11], [-33, 9, 3, -4, 0, 15], [12, -26, -12, -26, 11, 20], [0, 8, -2, -3, 0, -5], [19, 1, -8, -3, 8, 0], [-24, -1, 4, -6, -7, 4], [-32, 6, 0, -11, -6, -3], [9, 15, 9, 4, -3, -7], [-17, 24, 5, 18, 12, 5], [-7, -5, -9, -25, 5, 14], [25, -13, -22, -18, -17, 25], [8, 18, -4, 2, -4, -12], [43, -1, 0, 6, -8, -1], [34, 0, -4, 5, -3, 1], [-21, 4, 19, -1, -2, -2], [-32, 18, 4, 15, -7, -9], [20, 0, 6, -13, -3, -1], [-49, 14, 7, -5, 18, -10], [15, -2, 0, -15, -20, 10], [-4, -12, 9, 16, 9, 0], [26, 8, 15, 17, -12, -15], [-8, -9, 14, -2, -9, 2], [18, -21, -22, -28, 9, 24], [34, -11, 5, 4, -6, -2], [33, 16, -1, 18, -2, -11], [-10, 12, 13, -8, -8, -10], [0, 23, 4, 0, 9, 2], [-13, -5, -11, -15, 5, 28], [-35, 14, 0, 16, 21, 5], [7, 6, 6, 4, 28, 15], [18, 30, 3, 3, 6, -9], [-20, 29, 13, 15, -5, -13], [-14, 12, -10, 8, 20, 14], [-35, -3, 9, -4, 13, 1], [16, 7, 1, 12, -12, -10], [12, -19, 8, 8, 7, -10], [-10, 38, 8, 21, 2, -1], [3, -8, -3, -17, 14, 16], [27, -13, 4, 10, -17, 5], [-28, 2, 12, 24, 26, -8], [6, -28, -14, -12, 2, -2], [22, 2, -1, 22, -15, -6], [-4, 1, -20, -16, -17, 2], [12, -20, -4, 8, 0, 0], [5, 18, 12, -2, -13, -25], [-16, 8, 4, 12, 1, -18], [-24, 8, -11, -14, -12, 6], [-9, -9, 5, 20, -3, -3], [-19, 15, 6, 11, -6, -22], [16, -1, -1, 10, -6, -24], [-23, 25, -13, -13, -13, 4], [-12, 3, -4, 3, 21, 27], [41, -30, -12, -10, -32, -3], [-19, -5, -29, -17, 11, 18], [21, -39, -6, -5, -2, 8], [-24, -1, 2, -23, 5, 7], [-12, 9, -9, -17, 11, 7], [24, -23, -3, -18, 0, 8], [-15, 9, 16, 15, 24, 22], [5, -3, 6, 11, 8, 8], [10, -8, -20, -28, -10, 12], [-12, 8, -15, -15, 7, 11], [17, -22, -17, -9, -26, 6], [9, 17, -2, 4, -3, -11], [46, -4, 10, 1, -16, -7], [-19, -6, 12, -11, 7, 10], [-43, -3, -12, -4, 1, 5], [-33, 8, -3, 25, 16, 8], [13, 2, -5, 6, 12, -1], [-1, -29, -9, -9, -5, 6], [-13, 11, 18, 11, 29, -6], [2, 5, -30, -18, -20, 2], [26, 18, 7, 11, -1, -13], [-22, -14, -3, 10, 2, 4], [7, -16, -14, -27, -10, 22], [-10, 0, 4, -10, -10, -8], [-13, 15, 1, 4, 0, 1], [22, 4, -10, 7, 0, -9], [-16, 9, 26, 24, 15, -2], [-4, 19, -6, 3, 27, 3], [1, -19, 4, -9, -12, -4], [0, 5, -12, -9, -16, -19], [-29, -9, 6, 4, 2, 7], [51, 9, 8, 7, 13, -4], [12, -33, -3, -6, 3, 4], [16, 5, -9, 7, 16, -1], [12, 6, 0, -12, -10, -10], [-1, -21, -27, -11, -7, 18], [-12, -10, -10, -19, -8, 5], [-17, -6, 2, -15, -12, 14], [0, -9, 2, -4, 13, -8], [-23, -6, -8, -4, 19, 1], [-25, -2, -5, -5, -10, 4], [6, 7, -3, 11, 22, -3], [-13, -2, -13, -10, 24, 1], [29, 0, -4, 9, -2, -2], [-24, 9, 1, -11, -10, -11], [55, -9, -8, -15, -19, -8], [-5, 2, 3, -1, -12, -12], [-16, 8, 4, 3, -20, -19], [3, -9, 7, 3, -17, -2], [-40, 3, 2, 2, -4, 8], [6, -36, -18, -24, -6, 30], [6, -8, 1, -17, 0, -3], [-4, -18, -23, -17, -5, 3], [23, 4, 6, 4, -3, -19], [-31, -16, -8, -8, -2, 21], [-18, -19, -28, -17, 2, 21], [-17, 2, -7, 17, 14, 4], [-13, 15, 9, 3, -23, -28], [-39, -3, -14, 2, 14, 11], [7, -9, -19, -5, 13, 14], [-1, -5, -21, 7, -5, 10], [-8, 18, -9, -7, -24, -15], [-4, -23, 5, -2, 9, -2], [-7, -42, -13, -17, -10, 8], [-9, 9, -6, -8, 26, 5], [-2, -7, 19, 15, 12, -7], [-24, 27, 4, 17, 21, 15], [19, -12, -15, -8, -20, 5], [-13, 0, 20, 5, 19, -4], [27, -3, 15, 16, -15, -11], [10, -1, 26, 20, 24, -8], [-40, 18, 9, 6, 0, -10], [-28, -11, -3, -12, -2, 12], [14, 15, 37, 22, 6, -30], [-34, -7, 10, 3, -16, 1], [-5, 18, 14, 8, 27, -1], [-17, 5, 15, 12, 31, 1], [18, 15, -4, 4, 3, -4], [-27, 13, -2, 5, 2, 10], [-27, -29, -2, -15, 2, 6], [11, 12, 19, -3, -4, -4], [25, -21, -19, 2, -23, -1], [-8, -4, -17, -3, -13, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6042_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6042_2_a_v();".
// 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_6042_2_a_v(:prec:=6)
    chi := MakeCharacter_6042_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_6042_2_a_v();".
// 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_6042_2_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6042_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![10, -24, 2, 23, -8, -3, 1]>,<7,R![200, 148, -90, -69, 2, 7, 1]>,<11,R![-18, 142, 102, -31, -28, -1, 1]>],Snew);
    return Vf;
end function;