
// Make newform 4900.2.e.s in Magma, downloaded from the LMFDB on 03 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4900_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4900_e_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_4900_2_e_s();".
// 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_4900_2_e_s();".
// 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 := [9, 0, 22, 0, 9, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 5, 0, 1, 0, 0], [0, 7, 0, 6, 0, 1], [0, 13, 0, 9, 0, 1], [-2, 0, -9, 0, -2, 0], [6, 0, 6, 0, 1, 0]];
    Rf_basisdens := [1, 1, 3, 3, 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_4900_e();"
function MakeCharacter_4900_e()
    N := 4900;
    order := 2;
    char_gens := [2451, 1177, 101];
    v := [2, 1, 2];
    // 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_4900_e_Hecke();"
function MakeCharacter_4900_e_Hecke(Kf)
    N := 4900;
    order := 2;
    char_gens := [2451, 1177, 101];
    char_values := [[1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 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]];
    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, 0], [0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 1], [0, -1, 3, 0, 0, 0], [0, 0, -3, -1, 0, 0], [-1, 0, 0, 0, 1, 0], [0, 1, 1, -1, 0, 0], [-1, 0, 0, 0, 1, 2], [0, 0, 0, 0, 1, 2], [0, 3, 1, 0, 0, 0], [-3, 0, 0, 0, 1, -3], [0, 1, 3, 1, 0, 0], [0, 1, -2, 1, 0, 0], [0, 3, -6, 1, 0, 0], [-2, 0, 0, 0, 0, 1], [-6, 0, 0, 0, -1, 2], [0, 2, 6, 0, 0, 0], [-7, 0, 0, 0, 0, 2], [0, 0, 4, 0, 0, 0], [1, 0, 0, 0, 1, -3], [0, -4, 11, -1, 0, 0], [-4, 0, 0, 0, -1, -1], [0, 1, -6, 2, 0, 0], [0, 0, 0, 0, -3, 0], [0, -3, 4, -2, 0, 0], [0, 1, -11, -1, 0, 0], [5, 0, 0, 0, -2, 1], [0, 2, 2, -3, 0, 0], [0, -3, 1, 1, 0, 0], [-2, 0, 0, 0, 3, 4], [0, 5, -1, 1, 0, 0], [-13, 0, 0, 0, 2, 3], [-9, 0, 0, 0, 3, 6], [9, 0, 0, 0, -1, -2], [0, -3, -4, 2, 0, 0], [0, -1, 10, 1, 0, 0], [0, 1, 1, 2, 0, 0], [0, 2, 14, -1, 0, 0], [2, 0, 0, 0, -2, 2], [-5, 0, 0, 0, 1, 0], [5, 0, 0, 0, -2, -4], [0, 5, -11, 1, 0, 0], [0, 3, 0, -2, 0, 0], [-2, 0, 0, 0, 1, 5], [4, 0, 0, 0, 3, 2], [0, 3, 10, -2, 0, 0], [0, 2, -4, -3, 0, 0], [-8, 0, 0, 0, 0, -1], [0, -4, 5, -2, 0, 0], [-11, 0, 0, 0, 1, -2], [5, 0, 0, 0, 3, 1], [-6, 0, 0, 0, 3, 6], [0, 0, -18, 2, 0, 0], [0, 1, -8, 4, 0, 0], [11, 0, 0, 0, 4, 2], [-8, 0, 0, 0, -1, -3], [0, -5, 2, -2, 0, 0], [-8, 0, 0, 0, -2, -2], [0, 7, 2, 2, 0, 0], [0, 3, 18, -1, 0, 0], [0, 1, -3, -5, 0, 0], [2, 0, 0, 0, -2, -10], [0, -3, 1, 0, 0, 0], [0, -4, 5, 0, 0, 0], [-7, 0, 0, 0, 2, -6], [0, 6, -20, 1, 0, 0], [0, 1, -2, 4, 0, 0], [8, 0, 0, 0, -1, 3], [0, -4, 8, -5, 0, 0], [0, 0, 0, 0, -5, 0], [0, 0, 20, -1, 0, 0], [0, 0, -10, -3, 0, 0], [0, 0, 0, 0, 3, 2], [0, -7, -1, 4, 0, 0], [13, 0, 0, 0, -3, 7], [0, 1, 6, 1, 0, 0], [-2, 0, 0, 0, 0, -5], [3, 0, 0, 0, -6, -2], [-5, 0, 0, 0, -1, -2], [2, 0, 0, 0, 0, -9], [11, 0, 0, 0, 4, -1], [0, 2, 3, -5, 0, 0], [-7, 0, 0, 0, -1, -9], [0, -4, 8, -1, 0, 0], [-11, 0, 0, 0, -4, 4], [0, 3, 10, -5, 0, 0], [12, 0, 0, 0, -1, -6], [0, -6, 2, -6, 0, 0], [0, 0, -24, -1, 0, 0], [-13, 0, 0, 0, -2, 5], [0, 0, 1, -3, 0, 0], [-14, 0, 0, 0, -4, -2], [5, 0, 0, 0, 4, 4], [0, 0, -15, 3, 0, 0], [16, 0, 0, 0, -6, -8], [0, 0, 0, 0, -2, 6], [0, -2, 11, 0, 0, 0], [2, 0, 0, 0, 1, 3], [0, 3, 10, -4, 0, 0], [0, -1, -7, -3, 0, 0], [0, 0, 9, -2, 0, 0], [17, 0, 0, 0, 7, 2], [-28, 0, 0, 0, -2, -3], [0, 14, -8, 3, 0, 0], [0, -3, 6, -1, 0, 0], [0, -4, 8, -2, 0, 0], [6, 0, 0, 0, 0, 6], [11, 0, 0, 0, -5, -5], [0, 0, 23, -1, 0, 0], [0, 0, 5, -5, 0, 0], [0, 4, 22, -3, 0, 0], [-17, 0, 0, 0, 4, -4], [-2, 0, 0, 0, -5, -4], [10, 0, 0, 0, 1, 10], [0, 6, -11, 4, 0, 0], [0, -11, 10, -3, 0, 0], [0, 7, 4, 7, 0, 0], [16, 0, 0, 0, 4, 7], [4, 0, 0, 0, -2, 0], [0, 15, -4, 2, 0, 0], [0, -6, -21, -3, 0, 0], [0, 3, 36, -1, 0, 0], [-11, 0, 0, 0, 4, 9], [5, 0, 0, 0, 5, 2], [-18, 0, 0, 0, 2, -1], [38, 0, 0, 0, 1, -4], [0, -1, 19, 2, 0, 0], [0, -6, -14, 1, 0, 0], [-17, 0, 0, 0, -2, 6], [0, 15, -12, 0, 0, 0], [2, 0, 0, 0, -6, 3], [0, 8, -21, 5, 0, 0], [17, 0, 0, 0, -2, -10], [26, 0, 0, 0, 5, 0], [0, 5, 17, 7, 0, 0], [0, 1, 15, -5, 0, 0], [0, -3, -24, 0, 0, 0], [3, 0, 0, 0, 3, 0], [1, 0, 0, 0, 3, -9], [0, 0, 0, 0, -3, 12], [0, 8, -8, -3, 0, 0], [0, -8, 22, -3, 0, 0], [-16, 0, 0, 0, 3, 15], [22, 0, 0, 0, -2, 4], [0, -9, -23, 6, 0, 0], [0, 1, -41, -1, 0, 0], [-25, 0, 0, 0, -2, 3], [0, -3, -18, -5, 0, 0], [0, -4, 6, -2, 0, 0], [30, 0, 0, 0, -2, -3], [0, 3, -7, 10, 0, 0], [0, -7, -7, 3, 0, 0], [0, 6, 25, 0, 0, 0], [-24, 0, 0, 0, 3, -3], [18, 0, 0, 0, -5, 8], [22, 0, 0, 0, 0, -11], [0, -4, 13, -5, 0, 0], [15, 0, 0, 0, 4, 3], [0, 12, -12, -2, 0, 0], [0, -5, -17, -7, 0, 0], [0, 0, -11, -5, 0, 0], [25, 0, 0, 0, -4, 4], [0, 9, -27, 0, 0, 0], [0, 11, -28, 2, 0, 0], [-11, 0, 0, 0, 6, 2], [0, 3, 8, -3, 0, 0], [14, 0, 0, 0, 9, 1], [0, 4, 13, 1, 0, 0], [2, 0, 0, 0, 11, 2], [10, 0, 0, 0, 0, -15], [12, 0, 0, 0, 9, 6], [0, 0, 32, 1, 0, 0], [-3, 0, 0, 0, -1, -14], [15, 0, 0, 0, -5, -9], [42, 0, 0, 0, 1, -5], [-4, 0, 0, 0, -8, -13], [0, -5, -16, 2, 0, 0], [6, 0, 0, 0, 4, -14], [0, -9, -14, -6, 0, 0], [-3, 0, 0, 0, -6, 6], [0, 4, -12, -4, 0, 0], [0, -3, -27, -4, 0, 0], [0, -14, 1, -7, 0, 0], [26, 0, 0, 0, -8, -7], [0, 1, 11, 9, 0, 0], [0, -3, -17, -3, 0, 0], [14, 0, 0, 0, 0, -5], [12, 0, 0, 0, 3, 3], [0, 0, -26, 5, 0, 0], [0, 8, -28, 5, 0, 0], [39, 0, 0, 0, -5, -17], [31, 0, 0, 0, -3, 7], [0, -6, -18, 0, 0, 0], [0, 5, 20, 7, 0, 0], [-22, 0, 0, 0, 5, 18], [0, -6, 8, 8, 0, 0], [0, 19, -14, 8, 0, 0], [0, -8, -23, 0, 0, 0], [7, 0, 0, 0, -2, 7], [31, 0, 0, 0, -3, -6], [0, 0, 17, -2, 0, 0], [32, 0, 0, 0, -2, 3], [26, 0, 0, 0, -2, -10], [0, 9, -27, 8, 0, 0], [6, 0, 0, 0, -4, -17], [0, -6, -3, -6, 0, 0], [3, 0, 0, 0, 0, -12], [-46, 0, 0, 0, 2, -5], [0, -4, 12, -3, 0, 0], [-9, 0, 0, 0, -6, -6], [0, -13, 10, -3, 0, 0], [0, -2, -20, 1, 0, 0], [-4, 0, 0, 0, -2, 11], [-29, 0, 0, 0, -3, -3], [0, -6, -5, 4, 0, 0], [38, 0, 0, 0, 1, -4], [0, -1, -46, 3, 0, 0], [0, -8, 4, -1, 0, 0], [-14, 0, 0, 0, 1, -7], [22, 0, 0, 0, 10, 8], [-29, 0, 0, 0, 2, -5], [0, 0, -40, -5, 0, 0], [0, 7, -2, -5, 0, 0], [2, 0, 0, 0, -1, -8], [0, -14, 19, -5, 0, 0], [-20, 0, 0, 0, 0, 4], [0, 12, 5, -10, 0, 0], [1, 0, 0, 0, 4, -11], [0, 3, 8, 11, 0, 0], [-16, 0, 0, 0, 7, 6], [-36, 0, 0, 0, 7, 10], [30, 0, 0, 0, 0, 12], [0, 12, -11, 1, 0, 0], [0, 3, -15, -3, 0, 0], [-1, 0, 0, 0, -7, 12], [0, 3, -12, 5, 0, 0], [-2, 0, 0, 0, 0, 7], [25, 0, 0, 0, -2, 16], [0, -5, 34, -6, 0, 0], [-35, 0, 0, 0, 0, -9], [0, -21, -5, 5, 0, 0], [-10, 0, 0, 0, -3, -2], [0, 3, 27, 6, 0, 0], [50, 0, 0, 0, 1, 5], [0, -4, -14, -11, 0, 0], [-6, 0, 0, 0, 9, 6], [-17, 0, 0, 0, 6, 12], [0, -13, -1, 3, 0, 0], [0, 0, -8, -11, 0, 0], [32, 0, 0, 0, 1, -7], [0, 2, -28, -8, 0, 0], [0, 0, 0, 0, -3, 19], [0, 7, 7, -1, 0, 0], [-2, 0, 0, 0, -2, 19], [1, 0, 0, 0, 7, 2], [0, -9, 11, -4, 0, 0], [0, -10, -16, -2, 0, 0], [0, -3, 29, 4, 0, 0], [0, -6, -19, 7, 0, 0], [0, -1, 41, 0, 0, 0], [-16, 0, 0, 0, -1, -9], [0, 1, 20, -4, 0, 0], [0, -16, -19, 4, 0, 0], [-19, 0, 0, 0, -6, 12], [-14, 0, 0, 0, -7, -14], [-43, 0, 0, 0, 2, 5], [0, 12, -13, -4, 0, 0], [0, -17, -26, 2, 0, 0], [7, 0, 0, 0, 2, 9], [0, -21, 10, -4, 0, 0], [0, 15, -26, 0, 0, 0], [3, 0, 0, 0, 3, -10], [0, -15, -22, 0, 0, 0], [0, -4, 0, 4, 0, 0], [0, -5, -2, 3, 0, 0], [6, 0, 0, 0, 13, 2], [-11, 0, 0, 0, 2, 2], [13, 0, 0, 0, -5, -11], [0, -13, -25, 3, 0, 0], [-4, 0, 0, 0, -9, -6], [0, -17, 1, -7, 0, 0], [15, 0, 0, 0, 3, 5], [0, -7, 16, -2, 0, 0], [64, 0, 0, 0, 2, 1], [0, -14, 0, -5, 0, 0], [0, -18, 18, 3, 0, 0], [-10, 0, 0, 0, -6, 18], [1, 0, 0, 0, -4, 13], [43, 0, 0, 0, -1, -5], [0, 13, -44, 4, 0, 0], [0, 11, 2, -4, 0, 0], [4, 0, 0, 0, 0, 7], [0, -22, 7, -4, 0, 0], [15, 0, 0, 0, -9, 3], [-7, 0, 0, 0, 6, 4], [0, -12, 51, -1, 0, 0], [-16, 0, 0, 0, 7, 17], [0, -3, 23, 12, 0, 0], [0, 6, 7, -4, 0, 0], [0, -1, -28, -13, 0, 0], [43, 0, 0, 0, -11, -33], [0, -11, -29, 5, 0, 0], [-37, 0, 0, 0, 9, 12], [0, 13, -1, 13, 0, 0], [0, -20, -2, -6, 0, 0], [-13, 0, 0, 0, 7, 3], [17, 0, 0, 0, -5, 14], [0, -3, -7, -12, 0, 0], [0, 12, 3, 15, 0, 0], [9, 0, 0, 0, 3, -9], [-25, 0, 0, 0, 2, 20], [0, 13, -42, 2, 0, 0], [0, 7, 55, 2, 0, 0], [-39, 0, 0, 0, 2, -11], [29, 0, 0, 0, -5, -7], [31, 0, 0, 0, -1, 1], [0, -5, -16, -14, 0, 0], [-32, 0, 0, 0, 0, 1], [25, 0, 0, 0, -11, -3], [0, 9, -8, 1, 0, 0], [-50, 0, 0, 0, -6, -5], [0, 11, 52, -1, 0, 0], [0, 15, -27, 0, 0, 0], [4, 0, 0, 0, 9, 15], [-3, 0, 0, 0, -8, 5], [0, -5, 47, -5, 0, 0], [0, -3, 15, -7, 0, 0], [0, 9, -30, 6, 0, 0], [-21, 0, 0, 0, 2, 19], [0, -5, 31, -1, 0, 0], [20, 0, 0, 0, 11, 0], [0, -2, 43, -7, 0, 0], [6, 0, 0, 0, -1, 25], [0, 7, 4, 11, 0, 0], [15, 0, 0, 0, 14, 8], [0, 22, 31, -10, 0, 0], [-12, 0, 0, 0, -7, -13], [0, 3, -16, 0, 0, 0], [0, -1, -17, -3, 0, 0], [0, 23, -13, 0, 0, 0], [-29, 0, 0, 0, 8, 16], [-31, 0, 0, 0, -6, 3], [0, -3, 30, -6, 0, 0], [14, 0, 0, 0, -3, -13], [-49, 0, 0, 0, 1, 1], [0, 9, -38, 7, 0, 0], [31, 0, 0, 0, 14, 7], [0, 25, -23, 2, 0, 0], [-33, 0, 0, 0, 7, -7], [0, -15, 16, -16, 0, 0], [-25, 0, 0, 0, 13, 17], [0, -3, -26, 0, 0, 0], [-6, 0, 0, 0, 6, -19], [0, 2, 38, 1, 0, 0], [41, 0, 0, 0, -5, -28], [31, 0, 0, 0, -2, -2], [0, 21, -33, 3, 0, 0], [0, -13, 32, -8, 0, 0], [0, -16, 0, -8, 0, 0], [-33, 0, 0, 0, 7, 21], [0, -11, -11, 6, 0, 0], [-6, 0, 0, 0, 2, -21], [0, 15, -40, 2, 0, 0], [0, 16, 24, -2, 0, 0], [0, -5, -32, 0, 0, 0], [0, -12, -22, -7, 0, 0], [-5, 0, 0, 0, 3, 5], [11, 0, 0, 0, 10, 2], [-26, 0, 0, 0, 9, -13], [0, -8, 34, 3, 0, 0], [-17, 0, 0, 0, -4, -11], [25, 0, 0, 0, 9, 15], [0, 31, -19, 6, 0, 0], [1, 0, 0, 0, 16, -8], [7, 0, 0, 0, 2, 13], [0, 7, 14, 17, 0, 0], [27, 0, 0, 0, -3, 15], [0, -9, 5, -11, 0, 0], [-37, 0, 0, 0, 6, 20], [0, -10, 14, -4, 0, 0], [0, -12, -17, 2, 0, 0], [0, 2, 11, 10, 0, 0], [-7, 0, 0, 0, 16, -15], [0, 8, -16, -5, 0, 0], [22, 0, 0, 0, -4, -7], [0, -12, 2, -6, 0, 0], [0, -24, -19, 6, 0, 0], [0, 6, -12, 4, 0, 0], [29, 0, 0, 0, -1, 1], [0, 4, 13, 4, 0, 0], [37, 0, 0, 0, -11, -35], [0, 10, 0, 0, 0, 0], [24, 0, 0, 0, 1, -18], [0, 22, -18, -2, 0, 0], [-17, 0, 0, 0, 9, 11], [-84, 0, 0, 0, 1, -3], [46, 0, 0, 0, 2, 2], [54, 0, 0, 0, -6, 12], [-49, 0, 0, 0, -11, -3], [0, -5, 34, -4, 0, 0], [0, -18, 28, 4, 0, 0], [0, 2, -19, -14, 0, 0], [14, 0, 0, 0, -7, 5], [-5, 0, 0, 0, -8, 0], [0, -16, 9, -11, 0, 0], [-55, 0, 0, 0, -5, 5], [0, -7, 6, -3, 0, 0], [-2, 0, 0, 0, -13, 16], [0, 6, 10, -3, 0, 0], [0, 0, -54, 0, 0, 0], [0, -3, -6, -3, 0, 0], [-20, 0, 0, 0, 6, 2], [0, 1, -28, -10, 0, 0], [21, 0, 0, 0, 17, 9], [33, 0, 0, 0, 9, -3], [0, 12, 17, -2, 0, 0], [0, 26, -28, -2, 0, 0], [0, 5, -10, -10, 0, 0], [18, 0, 0, 0, -4, -9], [0, 1, 45, 5, 0, 0], [0, -24, 12, -3, 0, 0], [12, 0, 0, 0, 2, -9], [0, -12, 10, -16, 0, 0], [0, -7, -10, -19, 0, 0], [0, -12, -12, -16, 0, 0], [-20, 0, 0, 0, 14, 13], [-50, 0, 0, 0, 7, 0], [68, 0, 0, 0, -6, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4900_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4900_2_e_s();".
// 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_4900_2_e_s(:prec:=6)
    chi := MakeCharacter_4900_e();
    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_4900_2_e_s();".
// 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_4900_2_e_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4900_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![9, 0, 70, 0, 17, 0, 1]>,<11,R![9, -3, -4, 1]>,<19,R![21, -23, 2, 1]>,<31,R![-37, -42, -3, 1]>],Snew);
    return Vf;
end function;