
// Make newform 1950.2.y.k in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1950_y();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1950_y_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1950_2_y_k();".
// 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_1950_2_y_k();".
// 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 := [2704, 208, 8, 36, 185, 30, 2, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-320632, 258164, 5488658, 204731, -21770, -47714, 27304, -2225], [-62031216, 34618460, -5277648, -784473, -2370014, 800522, -125666, -1777], [2274116, -1640788, 360425, 80915, 84154, -39200, 7801, -299], [94417440, -18392236, -2633192, 3714597, 3350390, -318186, -71918, 43733], [-3328, -46500, 37828, -7135, -2554, -2018, 822, -151], [-117816036, 3493156, 7562729, -10547568, -5793110, -209286, 362933, -123974]];
    Rf_basisdens := [1, 1, 20561424, 6853808, 395412, 20561424, 51792, 10280712];
    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_1950_y();"
function MakeCharacter_1950_y()
    N := 1950;
    order := 6;
    char_gens := [1301, 1327, 301];
    v := [6, 3, 5];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1950_y_Hecke();"
function MakeCharacter_1950_y_Hecke(Kf)
    N := 1950;
    order := 6;
    char_gens := [1301, 1327, 301];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, -1, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 1, 0, 1, 0, 0], [2, -1, 0, -1, 0, -1, 0, -1], [1, -1, -2, 0, 0, -1, -2, -1], [0, 0, 0, 0, 0, 4, 0, 0], [1, -1, -2, 1, 2, -1, 0, -2], [1, 0, 1, -1, -1, 0, -1, -1], [2, 1, 1, -2, -1, 1, -2, 2], [4, -1, -1, 0, -1, -1, -6, -1], [2, 1, -2, -1, 0, 3, -2, 1], [0, 2, 2, 0, -2, 2, 0, 0], [8, 1, 1, -1, -1, -3, -4, 1], [1, 2, 0, 0, -2, -2, 0, 2], [-1, -1, 3, -2, -1, -1, 4, 1], [-4, -1, -1, 1, 1, 1, 2, -1], [0, 0, -4, 0, 0, -2, -4, 0], [4, 0, 0, -2, -2, 2, -4, 2], [-2, 2, 0, -2, 0, -4, 2, 0], [-4, 2, 6, 0, -2, 10, 0, 2], [4, -3, -3, 0, 3, -3, 0, -3], [2, 0, 2, 2, 0, 4, 0, 2], [-2, 0, -1, -1, -1, -2, -4, -1], [-2, 2, 4, 2, 0, 4, -8, 0], [0, 2, 2, -2, 0, 0, 0, 2], [-3, 1, -9, 0, 1, 1, 4, 1], [4, 0, 10, 0, 0, 10, 4, 0], [2, -2, 6, -2, -2, -2, 0, 0], [3, -3, -4, 3, 4, -7, -1, -4], [-4, 0, -9, 1, 1, -8, -2, 1], [-5, -1, -1, -1, 1, -1, 0, -2], [1, -1, -12, -1, 0, -7, 7, 0], [-1, 1, 0, 1, 0, 1, 8, 0], [6, 1, -1, -1, 1, -3, -2, -1], [4, -2, 0, 2, -2, -2, -4, -4], [-5, -1, -5, 1, -1, -1, 12, -2], [-4, 1, 1, 1, 3, 3, -4, 3], [-11, 0, -8, 2, 2, 6, 11, -2], [5, 1, 0, -1, 0, -5, -2, 0], [10, -1, -5, 0, -1, 5, -10, 0], [-10, 2, 6, 0, -2, 10, 0, 2], [2, -2, -12, -2, 0, -8, 2, 0], [-10, 2, 4, -2, 0, -2, 10, 2], [8, 0, 1, 1, 1, -2, -8, -1], [2, -2, 8, -2, 0, 2, 0, 0], [-4, -2, -3, 1, -1, 2, 4, -1], [6, -2, 5, 1, -1, -6, -6, -1], [0, 0, 20, 0, 0, 10, -2, 0], [6, -2, -10, -2, -2, -2, -8, 0], [0, -3, 9, -2, -3, -3, 6, -1], [-2, 4, 0, 6, 4, 4, -4, -2], [-2, 0, 4, 0, -4, -4, -1, 4], [-5, 2, 13, 2, 3, 10, 4, 3], [-5, -2, -5, 1, 3, -4, -3, 1], [-9, 4, -4, 2, -2, -2, -5, 2], [4, 0, 12, 0, -4, 4, -4, -4], [11, 0, 5, 1, 1, 6, 13, 1], [11, 0, 1, 0, -1, 6, -6, 1], [4, 0, -4, -2, 0, 0, -8, 2], [-4, 1, 9, 2, 1, 11, 0, 2], [3, -1, 14, -1, -2, 5, -8, -2], [-4, -4, -6, 0, 4, -8, 0, -4], [0, 6, 6, -2, -6, 6, 0, 4], [2, 2, -2, -4, 2, 2, -8, 6], [3, 5, 7, -4, -5, 9, 0, 1], [6, -6, -4, 6, 4, 6, -4, -4], [2, -4, 4, -8, -4, -4, 4, 4], [-12, -2, -2, 2, 2, 8, 6, -2], [-10, 0, -2, 0, 0, -2, -10, 0], [18, 2, -4, 0, 2, 4, -18, 0], [4, 0, 14, -2, 0, 0, -8, 2], [-10, 2, -2, 4, 2, 2, -2, 4], [-12, 3, 7, -3, -7, 5, 4, 7], [4, -8, -7, -1, 7, -8, 2, -1], [-5, 3, 6, 3, 2, 7, 11, 2], [-4, -8, -2, 3, 8, 4, 0, -5], [3, 0, -1, 0, -3, -2, -6, -3], [0, 4, 15, -1, -5, 14, -2, -1], [3, -7, -6, 7, 6, -7, -2, -6], [-4, -2, -6, 4, 2, 2, 4, -4], [0, 2, 18, -2, 2, 2, -4, 4], [6, -2, 12, -2, 0, 10, 2, -2], [18, -4, -2, 4, 2, -4, -10, -2], [12, 2, -8, 4, 6, 4, -12, -4], [0, -2, -4, 4, -2, -2, 4, -6], [-29, -1, 0, 1, 0, -3, 14, 0], [-4, -4, -2, 8, 4, -6, 4, -8], [20, -1, -1, 1, 1, 3, -10, -1], [16, 2, 6, 0, -2, 10, 0, 2], [-2, -2, -4, 0, -2, -2, 8, -2], [6, 4, -6, 0, -4, -6, 6, 0], [3, -1, 10, -1, -2, 3, -14, -2], [-2, -4, 5, 5, 1, -10, 2, -5], [-2, 0, 4, -2, 0, 0, 4, 2], [-5, 3, 0, -3, 0, -19, 4, 0], [2, 1, -13, 0, -1, -13, 2, 0], [17, 1, 7, 0, -1, 13, 0, 1], [-12, -3, -7, 0, 3, -7, -12, 0], [-4, 4, 6, 6, 4, 4, 0, -2], [-18, 2, -6, 0, 2, 2, 32, 2], [0, -6, 3, 5, -1, -8, 0, -5], [16, 0, 4, 0, -4, 2, -10, 4], [18, 0, 5, -3, -3, -2, -18, 3], [3, -3, -3, 2, 3, -3, 0, -1], [12, -10, -12, 2, 10, -14, 0, -8], [4, 2, 4, -6, -4, 2, -4, 6], [2, -1, -5, -2, 1, -9, 0, -3], [-2, 2, 10, 6, -2, 18, 0, 8], [-5, 4, -10, -2, 2, 12, 5, 2], [3, -3, -6, 3, 6, -3, 0, -6], [6, 5, 12, -7, -2, -5, -6, 7], [-9, 5, 8, 5, 4, 11, 7, 4], [11, 1, 7, -2, 1, 1, -24, 3], [-6, -2, 0, 2, 0, 6, 2, 0], [3, -1, 6, -1, -2, 1, -4, -2], [0, 2, 2, 2, -2, 2, 32, -2], [10, -4, -5, -1, 3, -6, 8, -1], [-14, -4, 1, -3, 1, -2, -20, -3], [2, -5, 25, -5, 3, 9, -22, 3], [-18, -2, -6, 0, 2, -6, -18, 0], [8, 0, 0, -4, -4, -4, 0, -4], [-4, -2, 2, -2, -2, -2, 12, 0], [4, -2, -2, -2, -2, -4, -4, -2], [-2, -2, 31, -1, 1, 30, -4, -1], [10, -6, -10, -3, 6, -14, 0, -9], [14, 0, 2, 0, -2, 6, -8, 2], [0, 0, -32, 0, 0, -16, 16, 0], [-9, 3, 1, -2, 3, 3, 12, 5], [1, -3, -3, 0, 3, -3, 0, -3], [-14, 8, 23, 5, -3, 28, -4, 5], [17, 2, 9, -7, -5, -2, -17, 7], [-19, 2, 9, -5, -3, -4, 19, 5], [8, -8, -15, -9, -1, -24, -10, -9], [15, 7, 4, -7, -4, 9, -6, 4], [1, 2, 13, -3, -5, 10, -5, -3], [1, 1, 14, 1, -2, 7, -10, -2], [-2, 1, 31, 1, 1, 17, -20, 1], [-4, 12, 8, -12, -8, 4, 4, 8], [-14, 4, 4, -4, 0, 0, 14, 4], [17, -3, 7, -2, -3, -3, -28, -1], [2, -3, 15, -4, -1, 11, -6, -4], [3, -7, -10, 7, 10, -7, 0, -10], [-10, -2, -18, -2, 2, -34, 0, -4], [0, -4, 12, 6, 2, -18, 0, -6], [22, -4, -2, 4, 2, 12, -12, -2], [18, -4, -6, 1, 4, -8, 0, -3], [-20, 5, 5, 6, 5, 5, 30, -1], [-9, -1, 3, 2, 1, 7, 0, 1], [14, -5, 7, 6, 5, 19, 0, 1], [2, 1, -9, 1, -3, -5, -20, -3], [-4, 2, 3, -3, -1, 0, 4, 3], [-18, 2, -16, 5, 2, 2, 32, -3], [-15, 8, 14, 0, -8, 14, -15, 0], [8, 5, -7, 0, -5, -7, 8, 0], [16, 4, -4, -2, -4, -12, 0, 2], [8, -6, -18, -6, -2, -16, -4, -2], [-24, -4, 4, 4, -4, -16, 8, 4], [-16, 4, 6, 0, 4, 4, 24, 4], [-4, 0, -8, -6, 0, 0, 8, 6], [-2, 8, -4, -2, 6, 6, 2, 2], [21, 7, 4, -7, -4, 19, -9, 4], [-19, 11, 15, -6, -11, 19, 0, 5], [-1, 1, -20, 1, 0, -9, 10, 0], [7, 8, -3, -7, 1, 10, -7, 7], [19, -6, 4, 4, 6, 14, 0, -2], [-3, -6, -1, 9, 3, -8, 3, -9], [-3, 1, 0, -1, 0, 25, 2, 0], [-11, 2, -14, -4, 2, 2, 18, 6], [-16, 8, 0, 10, 8, 8, 16, -2], [10, 2, -22, 4, 2, 2, -24, -2], [-34, -2, -4, 2, 4, 4, 18, -4], [8, -6, 26, -6, -2, 6, -14, -2], [12, -4, 20, -4, -8, 2, 28, -8], [-8, -5, 7, 0, 5, 19, 0, -5], [-5, 9, 16, 9, -4, 15, -12, -4], [-8, -2, -36, -2, 0, -38, -12, -2], [4, 0, 20, 10, 0, 0, -8, -10], [0, -12, 18, -6, 6, 12, -12, -6], [-10, 6, 2, 2, 8, -4, 10, -2], [0, -2, 10, 2, 2, 22, 0, 0], [-17, -1, -5, 5, 1, -9, 0, 4], [19, -5, 27, -9, -5, -5, -28, 4], [6, -1, 15, 2, 1, 31, 0, 1], [23, 0, -18, 0, 0, 18, -23, 0], [4, -1, 31, -1, -3, 13, 18, -3], [-9, 7, -33, 5, 7, 7, 4, 2], [30, 8, 8, -3, -8, 8, 0, 5], [-9, -8, -2, 2, 10, 0, -5, 2], [12, 2, 6, -2, -6, 16, -8, 6], [2, 8, 2, -8, -2, -2, 2, 2], [-2, 6, -16, 6, -4, -4, 4, -4], [-17, 1, -23, -6, -1, -47, 0, -5], [-22, 1, 7, 2, 1, 9, -18, 2], [-2, -10, -8, 10, 8, -30, 0, -8], [0, 10, -20, 4, -6, -16, 8, 4], [8, -3, -13, 6, -3, -3, -10, -9], [35, -2, -1, 2, 1, 16, -18, -1], [6, -5, -9, 2, 5, -13, 0, -3], [-18, 1, -1, -4, 1, 1, 34, 5], [2, 12, 10, -12, -10, 20, 0, 10], [7, -1, -14, -1, -6, -11, -1, -6], [32, -4, 2, 2, -2, -4, -32, -2], [5, -5, -17, 4, 5, -29, 0, -1], [0, 12, -5, 7, -5, 2, 14, 7], [-6, 2, 4, 2, 0, 6, -2, 2], [0, 2, 38, 8, 2, 2, -4, -6], [-8, -14, -10, 14, 10, -4, 2, -10], [-4, -6, 5, 3, 9, 8, 2, 3], [-1, 3, -6, 3, -2, -1, 6, -2], [-6, 0, 6, 4, 4, 10, 2, 4], [28, -12, -12, 3, 12, -12, 0, -9], [-30, 6, 5, 1, -5, 6, -28, 1], [6, -2, -28, -2, -4, -18, -8, -4], [-18, -4, -6, 4, 6, 26, 10, -6], [-23, 10, 22, 4, 10, 10, 26, 6], [22, -6, -2, 2, -6, -6, -32, -8], [-9, 9, 12, 9, 0, 15, -10, 0], [-6, -10, 1, 11, 1, -12, 6, -11], [-35, -7, -13, 4, 7, -19, 0, -3], [20, 2, 14, -6, -4, -8, -20, 6], [8, -2, -28, 6, -2, -2, -12, -8], [-3, -1, -13, 8, -1, -1, 8, -9], [8, -12, -4, -12, 4, -12, 12, 4], [4, 10, -12, -2, -12, -14, 0, -2], [-2, 4, 14, -6, -4, 24, 0, -2], [-5, -8, 27, 1, 9, 28, -3, 1], [-6, 2, -4, 2, 4, 2, -24, 4], [-4, 8, 0, -8, 0, -12, 6, 0], [-17, -3, -1, -3, -3, -3, 40, 0], [20, 7, -4, -3, -10, -7, 14, -3], [8, -12, -7, 15, 3, -8, -8, -15], [-18, -2, 40, 0, 2, 40, -18, 0], [1, -9, -3, 8, 9, 3, 0, -1], [4, -4, 10, -1, -4, -4, 0, -3], [17, -3, -1, 2, 3, 1, 0, -1], [1, 1, 6, -1, -6, -11, -3, 6], [-16, -2, 12, -2, -4, -10, 16, 2], [-5, -3, 12, -3, 8, 7, 22, 8], [-4, 4, -18, -6, -2, 24, 4, 6], [40, 0, -2, -6, -6, 8, -40, 6], [-24, -2, 6, 4, 2, -10, 24, -4], [-1, -1, 0, 1, 0, 7, 0, 0], [-6, -10, 22, -2, 8, 20, -10, -2], [0, -8, 24, -8, 8, 8, -10, 8], [-18, 4, 2, 2, -4, 0, 0, 6], [-4, 0, -4, 0, 4, 0, -28, 4], [4, 0, -9, -1, -1, -10, 2, -1], [9, -3, -4, 3, 4, 49, -4, -4], [-9, 12, 14, 4, -8, 18, -1, 4], [4, -1, 7, 4, 3, -11, -4, -4], [0, -4, 9, 5, 9, 14, 10, 5], [23, -8, -6, 8, 8, -4, 0, 0], [13, -8, 13, -5, -13, -8, -13, 5], [-13, 9, -27, 10, 9, 9, 8, -1], [31, -1, -4, 1, 4, 17, -14, -4], [-4, 2, 26, 2, 2, 16, -8, 2], [0, 4, 30, -8, -4, -22, 0, 8], [3, -9, 3, 2, -9, -9, 12, -11], [-12, 4, 40, 4, 8, 28, 30, 8], [-5, 1, -13, 14, 1, 1, 8, -13], [-6, 8, 10, 2, -6, 12, -2, 2], [-12, 2, 22, 6, 2, 2, 20, -4], [8, -6, -22, -12, -6, -34, -16, -12], [-13, 15, -6, 15, -2, 11, 13, -2], [2, -6, -20, -6, 4, -14, 10, 4], [26, -8, -18, 8, 18, -24, -8, -18], [15, 1, 37, 8, 1, 1, -32, -7], [2, -7, 19, -7, 5, 5, -16, 5], [6, 12, 10, -12, -10, 8, -2, 10], [-26, 2, 2, 8, -2, 2, 0, 10], [6, 0, 16, -5, 0, 32, 0, -5], [-24, -2, -14, -2, -4, 16, 24, 2], [7, -5, 6, -5, -2, -3, 21, -2], [-8, 8, 8, -8, -8, 32, 4, 8], [-20, 4, 16, 0, 4, -16, 20, 0], [12, -8, 28, -8, -4, 4, 2, -4], [-26, 4, -20, 8, 4, 4, 44, -4], [25, 1, -2, -1, 2, -7, -11, -2], [2, -3, 35, -3, 1, 15, 26, 1], [10, 4, 1, -7, -11, -6, -4, -7], [9, 4, -21, 3, -1, -18, 15, 3], [6, 3, 5, 12, 3, 3, -18, -9], [-4, 0, 4, 8, 8, 12, 12, 8], [16, -6, 6, 1, 6, 18, 0, -5], [-20, -4, -14, 2, 4, -24, 0, -2], [0, -2, -20, 4, -2, -2, 4, -6], [5, 3, -7, 0, -3, -17, 0, 3], [-15, -3, 6, 3, -6, -23, 3, 6], [9, -3, -30, -3, -6, -21, -14, -6], [-14, -3, -7, 0, -3, 7, 14, 0], [8, -10, -10, 10, 10, -42, -4, -10], [25, 6, -23, -3, 3, 26, -25, 3], [5, -6, -3, 6, 3, -10, -4, -3], [20, -10, -6, 10, 6, 18, -12, -6], [34, -2, 16, 3, 2, 34, 0, 1], [0, -21, -9, 16, 21, 3, 0, -5], [6, 0, -15, -3, -3, -18, 0, -3], [-30, 5, 13, -4, 1, -9, 30, 4], [-8, -5, -7, 12, 7, -5, 8, -12], [-40, 2, 2, -2, -2, 26, 20, 2], [18, 8, 0, -2, -8, -8, 0, 6], [-39, 0, -13, 1, 1, 12, 39, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1950_y_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1950_2_y_k();".
// 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_1950_2_y_k(:prec:=8)
    chi := MakeCharacter_1950_y();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1950_2_y_k();".
// 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_1950_2_y_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1950_y();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2704, -1040, 1388, 172, 349, 2, 23, 2, 1]>],Snew);
    return Vf;
end function;