
// Make newform 5292.2.l.f in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5292_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_5292_l_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_5292_2_l_f();".
// 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_5292_2_l_f();".
// 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 := [3, -12, 19, -15, 10, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [2, -1, 1, 0, 0, 0], [6, -18, 5, -8, 1, -1], [3, -5, 6, -2, 1, 0], [9, -21, 19, -16, 5, -2], [-9, 30, -22, 19, -5, 2]];
    Rf_basisdens := [1, 1, 3, 1, 3, 3];
    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_5292_l();"
function MakeCharacter_5292_l()
    N := 5292;
    order := 3;
    char_gens := [2647, 785, 1081];
    v := [3, 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_5292_l_Hecke();"
function MakeCharacter_5292_l_Hecke(Kf)
    N := 5292;
    order := 3;
    char_gens := [2647, 785, 1081];
    char_values := [[1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0], [-1, 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 := [[0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [-1, 1, 0, -2, 0, 0], [0, -2, 2, 2, 1, 2], [0, -2, 1, 1, -1, 2], [0, 0, 1, 0, 0, -2], [-4, 1, 0, 1, 0, 0], [1, 0, 1, 0, -1, 3], [0, 0, 0, 0, 0, 3], [2, 0, -2, 0, -2, 1], [0, 1, 2, 2, 1, -1], [0, 0, -4, 0, 0, -1], [0, -2, -1, -1, -8, 2], [0, -1, 1, 1, 2, 1], [-11, 0, 1, 0, 11, -1], [0, 7, -1, -1, 4, -7], [-3, 0, 1, 0, 3, -2], [-3, 5, 0, 3, 0, 0], [0, 2, 1, 1, 0, -2], [0, 4, -1, -1, 4, -4], [-7, 0, -1, 0, 7, -2], [-5, 0, 4, 0, 5, 1], [-1, 0, -5, 0, 1, 1], [4, 0, 0, -1, 0, 0], [4, 3, 0, -3, 0, 0], [9, 0, 0, 0, -9, 6], [0, -6, -3, -3, -4, 6], [0, 2, 6, 6, 0, -2], [3, 3, 0, 6, 0, 0], [1, -3, 0, -1, 0, 0], [8, 3, 0, 4, 0, 0], [0, -2, -1, -1, 2, 2], [-3, -3, 0, 0, 0, 0], [1, -3, 0, -3, 0, 0], [-4, 0, 3, 0, 4, 0], [-3, 0, -1, 0, 3, 2], [0, -4, 2, 2, 1, 4], [0, 1, 6, 6, -6, -1], [0, 0, -3, -3, 0, 0], [6, -3, 0, 6, 0, 0], [0, 5, -2, -2, 8, -5], [5, 0, -5, 0, -5, 1], [5, 7, 0, 1, 0, 0], [0, 5, 7, 7, 4, -5], [0, -4, 4, 4, 15, 4], [9, 0, 4, 0, -9, 4], [19, 1, 0, -1, 0, 0], [-11, -4, 0, 4, 0, 0], [-2, 0, -8, 0, 2, -7], [0, 2, 10, 10, 5, -2], [12, -1, 0, 7, 0, 0], [-7, -4, 0, 7, 0, 0], [-2, 4, 0, -10, 0, 0], [2, -10, 0, 1, 0, 0], [0, 5, -1, -1, -11, -5], [11, 0, 2, 0, -11, -1], [11, 0, 0, 3, 0, 0], [-23, 0, 1, 0, 23, 2], [2, 0, -2, 0, -2, 1], [0, 13, -1, -1, 7, -13], [7, 2, 0, 7, 0, 0], [-11, 0, 10, 0, 11, 5], [0, -9, 9, 9, 1, 9], [0, 3, 0, 0, 9, -3], [0, -6, -3, -3, 1, 6], [0, 3, -9, -9, -15, -3], [-8, 0, 7, 0, 8, -8], [-24, 0, 2, 0, 24, 2], [0, 9, 0, 3, 0, 0], [9, 0, 9, 0, -9, 8], [-6, -7, 0, 4, 0, 0], [-12, 5, 0, 4, 0, 0], [15, 17, 0, -5, 0, 0], [6, 1, 0, -6, 0, 0], [20, -6, 0, 1, 0, 0], [-6, 0, 6, 0, 6, 3], [19, 7, 0, 2, 0, 0], [-6, 0, -2, 0, 6, 4], [0, -5, 5, 5, 10, 5], [2, 0, -7, 0, -2, -4], [0, -11, 5, 5, -17, 11], [-21, 2, 0, 4, 0, 0], [0, -5, -1, -1, -5, 5], [0, -8, 6, 6, 12, 8], [21, 4, 0, 3, 0, 0], [0, -10, 1, 1, 4, 10], [13, 0, 7, 0, -13, -4], [0, -15, 9, 9, -4, 15], [4, 0, 10, 0, -4, -7], [14, 0, 0, -5, 0, 0], [0, 3, 3, 3, -11, -3], [0, -1, -2, -2, -28, 1], [16, 6, 0, 0, 0, 0], [-11, -5, 0, -4, 0, 0], [16, 1, 0, 11, 0, 0], [0, -6, 12, 12, 9, 6], [-13, 0, 3, 0, 13, -3], [-13, 0, 8, 0, 13, 2], [-2, 0, 7, 0, 2, -5], [0, -9, 11, 11, 1, 9], [-21, 0, -6, 0, 21, -19], [0, 6, -12, -12, -15, -6], [-11, 0, 7, 0, 11, 7], [0, -13, -5, -5, -6, 13], [3, 0, -6, 0, -3, 5], [0, 0, -3, 0, 0, 12], [-2, 0, -2, 0, 2, -2], [23, 0, 0, 0, -23, 3], [-3, 0, 0, 6, 0, 0], [0, -6, -6, -6, 9, 6], [0, -1, -4, -4, -23, 1], [0, 13, 0, -13, 0, 0], [-15, 7, 0, 2, 0, 0], [34, 3, 0, 5, 0, 0], [0, -15, 3, 3, -13, 15], [0, -6, -3, -3, -30, 6], [-17, -8, 0, -7, 0, 0], [-14, 0, 13, 0, 14, 8], [23, 0, -5, 0, -23, -5], [17, 0, -2, 0, -17, -5], [0, 9, -3, -3, -21, -9], [0, 20, -2, -2, 2, -20], [0, -3, 0, 0, 11, 3], [16, -4, 0, 2, 0, 0], [0, 2, 10, 10, -6, -2], [-5, 0, -11, 0, 5, 7], [-10, 0, -8, 0, 10, -11], [23, 2, 0, 10, 0, 0], [0, 5, 1, 1, -7, -5], [0, 12, -3, -3, -9, -12], [-28, -4, 0, -5, 0, 0], [3, -6, 0, -15, 0, 0], [-2, 2, 0, 14, 0, 0], [0, 9, -3, -3, -14, -9], [0, -8, 0, 0, -21, 8], [12, 0, 12, 0, -12, -3], [0, 4, -1, -1, -11, -4], [0, -18, -1, -1, -17, 18], [-22, 3, 0, -15, 0, 0], [0, 4, 9, 9, -6, -4], [-16, 0, 0, 0, 16, 0], [5, -19, 0, -5, 0, 0], [0, 0, 3, 3, -8, 0], [19, 0, -14, 0, -19, -6], [-2, 0, 11, 0, 2, 2], [20, -7, 0, 7, 0, 0], [25, -6, 0, 0, 0, 0], [27, 0, -3, 0, -27, -10], [27, 14, 0, 1, 0, 0], [-26, -5, 0, 2, 0, 0], [-18, -11, 0, -4, 0, 0], [-4, -13, 0, 7, 0, 0], [-9, 12, 0, 12, 0, 0], [1, 0, -5, 0, -1, -17], [-31, 0, 6, 0, 31, -3], [0, -25, 7, 7, -25, 25], [21, 14, 0, -5, 0, 0], [-12, 0, -12, 0, 12, 1], [0, -9, 5, 5, 34, 9], [2, -8, 0, -8, 0, 0], [0, 2, -5, -5, 15, -2], [17, 0, -7, 0, -17, 8], [5, 0, 8, 0, -5, 8], [22, 7, 0, 5, 0, 0], [0, 8, 7, 7, 30, -8], [38, -8, 0, -4, 0, 0], [-9, -8, 0, -13, 0, 0], [28, 22, 0, -7, 0, 0], [-9, -23, 0, -9, 0, 0], [0, 27, 0, 0, 13, -27], [-3, 6, 0, 0, 0, 0], [33, 0, 0, 0, 0, 0], [1, 0, -8, 0, -1, -20], [0, -19, -4, -4, -11, 19], [-4, 0, 8, 0, 4, -19], [0, 25, 0, 0, 0, -25], [-30, -13, 0, 1, 0, 0], [0, 9, -12, -12, 20, -9], [0, -1, 7, 7, 26, 1], [4, 0, 13, 0, -4, 12], [0, -10, -2, -2, -32, 10], [-18, 0, -9, 0, 18, 4], [2, 0, 17, 0, -2, -5], [44, -4, 0, 7, 0, 0], [0, -5, 2, 2, 22, 5], [13, 0, 0, 3, 0, 0], [-15, -15, 0, 6, 0, 0], [0, 0, 3, 3, -24, 0], [19, 0, -21, 0, -19, -3], [-6, 0, -12, 0, 6, -3], [17, 0, 9, 0, -17, 15], [0, -3, 2, 2, -23, 3], [0, 15, -6, -6, 18, -15], [-2, 0, 1, 0, 2, 15], [2, 0, -5, 0, -2, -2], [0, 22, -7, -7, 12, -22], [-27, 0, 15, 0, 27, 6], [17, 0, -10, 0, -17, -16], [5, 0, -10, 0, -5, 8], [-9, 0, 6, 0, 9, -24], [27, 5, 0, 13, 0, 0], [-29, -5, 0, -10, 0, 0], [25, 1, 0, -4, 0, 0], [0, -5, -16, -16, -23, 5], [-2, 0, -10, 0, 2, -10], [-26, 7, 0, -7, 0, 0], [16, 0, -2, 0, -16, -14], [7, 0, 8, 0, -7, 23], [1, 0, -12, 0, -1, -3], [0, -6, 13, 13, -28, 6], [14, 0, 8, 0, -14, 2], [0, 3, -2, -2, 29, -3], [-1, 0, 11, 0, 1, 8], [0, -21, 9, 9, -25, 21], [-38, 0, 11, 0, 38, -1], [-18, 7, 0, 12, 0, 0], [31, 0, -5, 0, -31, -10], [0, 17, -7, -7, 31, -17], [-23, -21, 0, 15, 0, 0], [0, 0, -12, -12, -2, 0], [24, -12, 0, 0, 0, 0], [0, 0, 16, 0, 0, 19], [0, -14, 2, 2, -14, 14], [13, 0, -3, 0, -13, -12], [0, 27, 8, 8, 1, -27], [0, -1, -9, -9, 0, 1], [0, -1, 4, 4, -1, 1], [0, 9, 0, 0, -12, -9], [-4, 0, -16, 0, 4, 29], [0, -6, 6, 6, -2, 6], [0, -10, 4, 4, -26, 10], [-38, 0, -5, 0, 38, -2], [-36, 0, 10, 0, 36, -8], [22, 13, 0, -10, 0, 0], [12, -14, 0, -4, 0, 0], [-12, 0, -3, 0, 12, -21], [0, 9, -6, -6, -36, -9], [-40, -2, 0, 1, 0, 0], [-17, -9, 0, 8, 0, 0], [-15, 0, -11, 0, 15, 13], [-13, 0, 6, 0, 13, 3], [1, 0, 13, 0, -1, 7], [0, 11, -15, -15, 6, -11], [0, 8, -17, -17, 17, -8], [-31, -13, 0, -11, 0, 0], [5, 0, 2, 0, -5, -17], [28, 0, 3, 0, -28, -18], [21, 12, 0, -12, 0, 0], [0, 27, -12, -12, 5, -27], [-4, 0, -10, 0, 4, -16], [4, 0, 7, 0, -4, 7], [-17, 0, -9, 0, 17, -6], [-10, 0, 0, -23, 0, 0], [-27, 0, 3, 0, 27, -20], [0, 12, -9, -9, 16, -12], [-12, -6, 0, 15, 0, 0], [18, -7, 0, 12, 0, 0], [-9, 0, 4, 0, 9, 16], [6, -7, 0, 25, 0, 0], [-13, 0, 11, 0, 13, 4], [-7, 0, -2, 0, 7, -11], [13, 14, 0, -2, 0, 0], [-4, 0, -4, 0, 4, -4], [0, 0, -6, -6, 4, 0], [-58, -10, 0, 7, 0, 0], [12, 0, -12, 0, -12, 13], [0, -7, -2, -2, 37, 7], [-30, 0, 3, 0, 30, 15], [-21, 18, 0, 15, 0, 0], [0, -22, 1, 1, 18, 22], [19, -8, 0, 14, 0, 0], [3, 10, 0, 5, 0, 0], [0, 1, 5, 5, 47, -1], [-7, 0, 8, 0, 7, 8], [14, -5, 0, -5, 0, 0], [34, 10, 0, -13, 0, 0], [-32, 0, -10, 0, 32, -16], [0, -1, -2, -2, 38, 1], [21, 0, 0, 0, -21, 30], [-7, 0, -4, 0, 7, 17], [30, 0, 9, 0, -30, 9], [-29, 0, 5, 0, 29, -13], [-1, -10, 0, 7, 0, 0], [10, 0, 1, 0, -10, 26], [0, -26, 17, 17, 4, 26], [0, -15, 13, 13, 35, 15], [-7, 1, 0, -13, 0, 0], [-5, 22, 0, 11, 0, 0], [0, 15, 0, 9, 0, 0], [0, 20, 4, 4, 2, -20], [36, -12, 0, 3, 0, 0], [0, -16, -1, -1, 16, 16], [-10, 0, -5, 0, 10, 31], [28, 0, -8, 0, -28, -17], [0, 15, -3, -3, -19, -15], [-22, 0, 26, 0, 22, -10], [48, 0, -12, 0, -48, 9], [-23, 0, 4, 0, 23, 10], [-23, -4, 0, 4, 0, 0], [6, 0, 3, 0, -6, 3], [0, 33, -3, -3, 5, -33], [0, -4, 7, 7, 14, 4], [11, -6, 0, 6, 0, 0], [23, -15, 0, 15, 0, 0], [-24, 7, 0, -3, 0, 0], [0, -13, -5, -5, 1, 13], [11, 10, 0, 10, 0, 0], [7, 0, 6, 0, -7, 27], [0, 27, -18, -18, 0, -27], [0, -6, -13, -13, -17, 6], [0, -19, 6, 6, -30, 19], [3, 0, 17, 0, -3, 20], [31, 8, 0, 23, 0, 0], [0, -5, 11, 11, 9, 5], [-28, 0, 8, 0, 28, 14], [0, 0, 0, 0, 0, 11], [67, 0, -16, 0, -67, -4], [0, 8, -8, -8, 26, -8], [2, 31, 0, -1, 0, 0], [-18, -22, 0, -17, 0, 0], [-25, -19, 0, -5, 0, 0], [-21, 8, 0, -26, 0, 0], [34, -7, 0, 5, 0, 0], [-7, 0, 21, 0, 7, -24], [-6, 0, -1, 0, 6, -1], [0, 32, -11, -11, 3, -32], [0, 9, 4, 4, 29, -9], [4, 27, 0, -1, 0, 0], [0, -8, 26, 26, 50, 8], [51, 0, -6, 0, -51, -8], [11, 0, -28, 0, -11, 14], [-16, 30, 0, 19, 0, 0], [0, -17, -10, -10, -6, 17], [-2, -9, 0, 5, 0, 0], [-38, -11, 0, 14, 0, 0], [-4, -2, 0, 1, 0, 0], [0, 4, 11, 11, -15, -4], [28, 0, 9, 0, -28, -18], [5, 6, 0, 12, 0, 0], [-6, 0, 12, 0, 6, 15], [0, 15, 6, 6, -42, -15], [-37, 0, -7, 0, 37, -1], [0, -25, 16, 16, -16, 25], [-21, 0, 0, 15, 0, 0], [0, -1, -8, -8, -5, 1], [0, 14, -2, -2, -54, -14], [25, 0, -8, 0, -25, 4], [-12, 0, -9, 0, 12, -30], [36, 26, 0, 1, 0, 0], [0, 26, -26, -26, -12, -26], [0, -23, 35, 35, 16, 23], [50, -1, 0, 1, 0, 0], [26, -11, 0, 35, 0, 0], [33, 1, 0, 9, 0, 0], [-30, -19, 0, -15, 0, 0], [0, 35, -20, -20, 35, -35], [22, -6, 0, 26, 0, 0], [12, 0, -3, 0, -12, 10], [20, 0, -24, 0, -20, -6], [0, -7, -33, -33, -15, 7], [0, 41, -12, -12, 48, -41], [0, -16, 5, 5, -68, 16], [0, 18, -6, -6, 28, -18], [0, -36, 6, 6, -26, 36], [-32, 0, 13, 0, 32, -5], [0, -1, 7, 7, 62, 1], [-6, 29, 0, 7, 0, 0], [9, 8, 0, 15, 0, 0], [3, 0, 11, 0, -3, 5], [-15, -3, 0, 21, 0, 0], [40, 0, -14, 0, -40, -6], [-48, 0, 11, 0, 48, 14], [-52, 0, -6, 0, 52, -21], [12, 0, 0, 0, -12, -21], [-25, -11, 0, 4, 0, 0], [0, 14, 13, 13, 36, -14], [-8, 0, -8, 0, 8, 15], [-26, 0, -10, 0, 26, -10], [-61, -7, 0, -11, 0, 0], [0, -1, 24, 24, 36, 1], [16, -3, 0, 33, 0, 0], [9, -27, 0, 9, 0, 0], [0, -20, -4, -4, -55, 20], [0, -8, 0, 0, 36, 8], [21, -5, 0, 11, 0, 0], [-22, 0, 10, 0, 22, 22], [-1, 0, 15, 0, 1, 6], [0, 11, 0, 0, -9, -11], [-23, 0, 30, 0, 23, 6], [0, -5, 11, 11, -41, 5], [0, -20, -15, -15, 9, 20], [0, 5, 19, 19, -17, -5], [0, -5, -26, -26, -4, 5], [45, 0, -6, 0, -45, 24], [0, 11, 10, 10, -24, -11], [-12, 0, -21, 0, 12, 26], [0, 25, 5, 5, 13, -25], [-18, 0, 30, 0, 18, 10], [24, 23, 0, -14, 0, 0], [-16, 0, -16, 0, 16, 35], [17, 7, 0, -25, 0, 0], [3, 23, 0, -21, 0, 0], [0, -7, -14, -14, -22, 7], [-9, 0, -11, 0, 9, -29], [19, -27, 0, 12, 0, 0], [32, 0, 17, 0, -32, 35], [31, 0, -19, 0, -31, -4], [0, 9, -7, -7, -23, -9], [0, 28, -4, -4, 26, -28], [0, 29, 14, 14, 19, -29], [5, 25, 0, 1, 0, 0], [0, 14, -29, -29, 2, -14], [0, -28, 1, 1, 15, 28], [-24, 0, 24, 0, 24, 14], [8, 0, 8, 0, -8, 33], [-83, 2, 0, 1, 0, 0], [43, 1, 0, -4, 0, 0], [-15, -1, 0, 9, 0, 0], [-9, 23, 0, -36, 0, 0], [-49, 0, -3, 0, 49, -3], [4, 0, 7, 0, -4, 5], [0, -29, 20, 20, 4, 29], [-66, -25, 0, 10, 0, 0], [-18, 0, 24, 0, 18, -14], [0, -22, -12, -12, 0, 22], [-51, 9, 0, 0, 0, 0], [0, -18, 15, 15, -39, 18], [-44, -34, 0, 8, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5292_l_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5292_2_l_f();".
// 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_5292_2_l_f(:prec:=6)
    chi := MakeCharacter_5292_l();
    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_5292_2_l_f();".
// 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_5292_2_l_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5292_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, -4, -1, 1]>],Snew);
    return Vf;
end function;