
// Make newform 2340.2.u.g in Magma, downloaded from the LMFDB on 13 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2340_u();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2340_u_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_2340_2_u_g();".
// 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_2340_2_u_g();".
// 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 := [4, 8, 8, -20, 32, 8, 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], [-6130, -1300, -1148, 4010, 316, 220, -173, 97], [696, 738, -1530, 5280, 1116, 216, -237, 140], [654, 2922, -8760, 4452, 1176, 585, -488, 174], [-1266, -5566, 22288, -8076, -2412, -1539, 1227, -382], [4076, -5144, -4588, 15640, 4201, 416, -730, 452], [2656, 12070, -11272, 19300, 4471, 1461, -1244, 605]];
    Rf_basisdens := [1, 1, 2462, 2462, 2462, 2462, 2462, 2462];
    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_2340_u();"
function MakeCharacter_2340_u()
    N := 2340;
    order := 4;
    char_gens := [1171, 2081, 937, 1081];
    v := [4, 4, 3, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2340_u_Hecke();"
function MakeCharacter_2340_u_Hecke( : Kf := false)
    N := 2340;
    order := 4;
    char_gens := [1171, 2081, 937, 1081];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 1, 0], [0, 0, 0, 0, 1, 1, 0, 0], [-2, -1, 0, 0, 2, 0, -1, 1], [2, -1, 0, 0, 0, 0, 0, -1], [0, 0, 0, -2, 0, 0, 1, 1], [0, 1, 1, 0, 0, -1, 0, 0], [3, 1, 0, 0, -3, 0, 1, -1], [0, -1, 0, -1, 5, 1, 0, 2], [0, 0, 0, 1, 0, 0, 1, 1], [0, 1, 0, 1, 3, -1, 0, 0], [1, 0, -1, 2, 1, -1, -1, -1], [-1, 3, 0, 0, 1, 0, 3, -3], [0, 0, 0, 0, 1, 3, 0, -2], [2, 0, -1, 0, 2, -1, 2, 2], [0, 0, -1, 1, 0, -1, 0, 0], [1, -1, -1, 1, 0, 0, 4, 0], [3, -1, 1, 1, 0, 0, 0, 0], [1, 0, 0, 1, 1, 0, -4, -4], [-5, -1, -3, 1, 0, 0, -4, 0], [0, -1, 0, -1, 2, -2, 0, -2], [0, 2, 0, 2, 1, 1, 0, -2], [-2, 0, -1, -2, -2, -1, -2, -2], [0, 3, -2, -3, 0, 0, 0, 0], [0, 4, 0, 4, 4, 0, 0, -2], [-1, 5, 1, 0, 1, -1, 2, -2], [-6, 5, 1, 0, 6, -1, 1, -1], [0, -2, 0, 0, 0, 0, 1, -1], [-1, 0, 1, 6, -1, 1, -3, -3], [5, 0, 1, -3, 5, 1, 2, 2], [0, 2, 4, -2, 0, 0, 4, 0], [0, 1, 0, 1, 8, 2, 0, 0], [0, 1, 0, 1, 2, 0, 0, 0], [-5, 4, 1, 0, 5, -1, 3, -3], [1, -5, 0, 0, -1, 0, -4, 4], [-4, -4, 0, 0, 4, 0, -1, 1], [11, -7, -3, 7, 0, 0, -2, 0], [0, 0, 0, 0, 5, -1, 0, 4], [5, 0, -2, 0, -5, 2, -2, 2], [-8, 0, 0, 0, 0, 0, 6, 0], [0, -3, 0, -3, 7, 1, 0, 0], [-2, 2, -4, -2, 0, 0, -2, 0], [4, -1, 2, 1, 0, 0, 6, 0], [10, -1, 0, 1, 0, 0, -2, 0], [4, 4, -2, -4, 0, 0, 4, 0], [-6, 6, 0, -6, 0, 0, -4, 0], [0, -2, 0, -2, -3, 1, 0, 4], [-5, -5, 1, 5, 0, 0, -8, 0], [4, 0, 1, -8, 4, 1, 4, 4], [7, 4, 4, 0, -7, -4, 2, -2], [7, -3, 1, 0, -7, -1, 3, -3], [-13, 0, -1, 0, 13, 1, -1, 1], [0, -5, 0, -5, 2, 2, 0, 4], [4, 0, -1, -6, 4, -1, 0, 0], [-2, 0, 0, -1, -2, 0, 2, 2], [0, -2, 0, -2, 4, 2, 0, -4], [-3, 5, 2, 0, 3, -2, -2, 2], [1, 0, -2, 2, 1, -2, -2, -2], [5, 4, -1, 0, -5, 1, 1, -1], [7, 7, 0, 0, -7, 0, 1, -1], [-15, -3, -1, 3, 0, 0, -4, 0], [0, 0, 0, 0, -7, 3, 0, 0], [0, -1, 0, -1, 6, -2, 0, 4], [-8, 0, 0, 2, -8, 0, 1, 1], [5, -1, 1, 1, 0, 0, -6, 0], [0, 0, 0, 9, 0, 0, -3, -3], [1, 0, -1, 2, 1, -1, -5, -5], [-8, 7, 4, 0, 8, -4, 0, 0], [-3, 0, 0, 0, -3, 0, 2, 2], [0, -5, 0, -5, 3, -1, 0, 0], [-2, 0, 3, -11, -2, 3, 4, 4], [8, 1, 2, 0, -8, -2, -8, 8], [0, 0, 4, 4, 0, 4, -5, -5], [-3, -7, 1, 0, 3, -1, -5, 5], [0, -6, 0, -6, -17, 5, 0, 4], [-10, -2, 4, 2, 0, 0, 0, 0], [0, 5, 0, 5, -15, 3, 0, -6], [-10, -6, 3, 0, 10, -3, 0, 0], [9, -12, -3, 0, -9, 3, -3, 3], [0, 3, 0, 3, -4, -4, 0, -8], [2, 0, 2, -4, 2, 2, -1, -1], [-6, 0, -2, -1, -6, -2, -1, -1], [-11, 6, 2, 0, 11, -2, -4, 4], [-14, 2, 4, -2, 0, 0, 0, 0], [14, 0, 0, 3, 14, 0, -2, -2], [-2, 0, -2, 8, -2, -2, -7, -7], [16, 1, 0, -1, 0, 0, -2, 0], [11, 0, -2, -8, 11, -2, 4, 4], [-3, 5, 3, -5, 0, 0, 6, 0], [-7, 0, -3, 9, -7, -3, -10, -10], [-4, 0, 1, -7, -4, 1, 4, 4], [3, 1, 3, -1, 0, 0, -4, 0], [0, 7, 0, 7, 4, 2, 0, 4], [-2, 5, 5, 0, 2, -5, -2, 2], [4, 0, 5, 3, 4, 5, 5, 5], [-1, -2, -6, 0, 1, 6, 0, 0], [-9, 9, -1, -9, 0, 0, 0, 0], [2, 0, -2, -9, 2, -2, 10, 10], [-2, 4, 3, 0, 2, -3, 2, -2], [-14, 3, 1, 0, 14, -1, -1, 1], [0, 3, 0, 3, -11, 1, 0, -8], [13, -3, 0, 0, -13, 0, 3, -3], [-7, -1, 9, 1, 0, 0, -6, 0], [0, -5, 0, -5, 14, 6, 0, 10], [-21, -1, -3, 1, 0, 0, -4, 0], [35, -1, -1, 1, 0, 0, 0, 0], [14, -5, -2, 5, 0, 0, -10, 0], [0, -1, 0, -1, -24, -2, 0, -2], [6, 6, 8, -6, 0, 0, 8, 0], [0, -7, -1, 0, 0, 1, -3, 3], [0, -5, 0, -5, -30, 2, 0, -4], [-10, 13, 4, -13, 0, 0, 10, 0], [-8, 0, -1, -9, -8, -1, 8, 8], [8, -1, 2, 0, -8, -2, 7, -7], [0, 10, 0, 10, 2, -4, 0, -14], [0, 2, 0, 2, -1, -1, 0, -12], [7, 0, 3, -9, 7, 3, 10, 10], [-17, 0, 1, -2, -17, 1, 1, 1], [0, 1, 0, 1, -4, 4, 0, -4], [6, -10, 0, 0, -6, 0, -7, 7], [-10, 12, 0, 0, 10, 0, 9, -9], [16, -2, 1, 0, -16, -1, -2, 2], [-23, 1, 1, -1, 0, 0, 6, 0], [2, 0, -2, 3, 2, -2, -3, -3], [0, -2, 0, -2, 10, 0, 0, -4], [-12, 0, 1, -10, -12, 1, -8, -8], [0, -6, 0, 6, 0, 0, 4, 0], [-1, 0, 2, 15, -1, 2, -11, -11], [0, 7, 0, 7, 6, -6, 0, 2], [11, 0, -1, 1, 11, -1, -1, -1], [0, 8, 0, 8, -1, 3, 0, 6], [0, 3, 0, 3, -18, 4, 0, -2], [5, -2, -6, 0, -5, 6, 0, 0], [-1, -10, 4, 0, 1, -4, -2, 2], [-17, -12, 0, 0, 17, 0, -4, 4], [0, -5, 0, -5, 11, 5, 0, 14], [0, -8, 0, -8, -1, -1, 0, -10], [16, 0, 2, 8, 16, 2, 3, 3], [0, -13, 0, -13, 3, -1, 0, 10], [-2, 0, -2, 15, -2, -2, -11, -11], [-17, 0, -3, -2, -17, -3, -7, -7], [-3, -5, 1, 0, 3, -1, -4, 4], [0, -2, 0, -2, 3, 5, 0, 8], [9, 13, 5, -13, 0, 0, 18, 0], [13, 0, -1, -1, 13, -1, -5, -5], [9, 7, -3, -7, 0, 0, 6, 0], [-2, 0, 5, -8, -2, 5, -2, -2], [0, 3, 0, 3, 8, 0, 0, -10], [0, 0, 0, 0, 13, -3, 0, 0], [-2, -9, 2, 9, 0, 0, -14, 0], [0, -9, 0, -9, -17, 1, 0, -6], [-17, -7, -6, 0, 17, 6, -9, 9], [-10, 1, 6, 0, 10, -6, -2, 2], [13, 0, 5, -13, 13, 5, 10, 10], [-8, -12, -4, 12, 0, 0, -18, 0], [0, 5, 0, 5, -6, 6, 0, -12], [10, -10, -4, 0, -10, 4, 3, -3], [-23, 6, 2, 0, 23, -2, 0, 0], [2, 0, 9, -6, 2, 9, 2, 2], [0, 4, 0, 4, -15, 9, 0, -2], [14, 0, -3, 0, -14, 3, 6, -6], [-1, 5, 5, -5, 0, 0, 0, 0], [26, -14, -6, 14, 0, 0, 2, 0], [-3, 3, 7, -3, 0, 0, -8, 0], [1, 5, 9, -5, 0, 0, 4, 0], [0, 12, 4, -12, 0, 0, -2, 0], [10, -2, 7, 0, -10, -7, 6, -6], [5, 0, -2, 12, 5, -2, 6, 6], [-6, 10, -3, 0, 6, 3, -4, 4], [11, -23, -1, 0, -11, 1, 1, -1], [-22, -6, 7, 0, 22, -7, -2, 2], [0, 5, 0, 5, 8, -10, 0, -4], [0, 1, 0, 1, 29, 1, 0, 8], [24, -4, 0, 4, 0, 0, 10, 0], [0, 2, 0, 2, -4, -4, 0, -8], [-9, -3, -2, 0, 9, 2, 10, -10], [5, -6, 2, 0, -5, -2, 2, -2], [13, 13, 6, 0, -13, -6, 5, -5], [0, -14, 0, -14, 8, 2, 0, 10], [0, 6, 0, 6, 13, -5, 0, -10], [0, -1, 0, -1, -4, 12, 0, -4], [-11, 0, 0, 2, -11, 0, 4, 4], [20, 1, 0, -1, 0, 0, 0, 0], [5, -13, 1, 13, 0, 0, 10, 0], [-17, 7, 9, -7, 0, 0, 2, 0], [14, 0, -2, 0, 14, -2, 7, 7], [0, -4, 0, -4, -17, -1, 0, -4], [9, 0, -1, -2, 9, -1, -3, -3], [13, 9, 6, 0, -13, -6, 0, 0], [19, 0, -3, 20, 19, -3, 1, 1], [0, -2, 0, -2, -19, -5, 0, -12], [-20, -14, -6, 14, 0, 0, -24, 0], [-5, 0, -5, 0, 5, 5, 5, -5], [1, 0, 0, -11, 1, 0, 7, 7], [-15, 18, -2, 0, 15, 2, -2, 2], [27, 0, -1, 0, 27, -1, 3, 3], [-8, 8, -4, 0, 8, 4, 9, -9], [0, 11, 0, 11, 19, 1, 0, -2], [-2, 0, -5, 5, -2, -5, -3, -3], [7, 0, -6, 8, 7, -6, 2, 2], [10, 0, -8, 0, 0, 0, 6, 0], [20, 1, 8, -1, 0, 0, -14, 0], [0, 11, 0, 11, -3, -11, 0, -14], [3, 0, -5, -9, 3, -5, -1, -1], [17, -22, -2, 0, -17, 2, -8, 8], [-7, 15, 1, 0, 7, -1, -7, 7], [-20, 0, -6, -1, -20, -6, -10, -10], [-15, -6, -4, 0, 15, 4, -8, 8], [0, 9, 0, 9, 0, 4, 0, -14], [1, 0, 4, 2, 1, 4, 8, 8], [-15, -3, -5, 3, 0, 0, 12, 0], [-4, 0, -7, -1, -4, -7, -9, -9], [0, -6, 0, -6, 13, 5, 0, -4], [-8, 3, 5, 0, 8, -5, -2, 2], [6, -14, 0, 0, -6, 0, -9, 9], [0, 5, -8, 0, 0, 8, 6, -6], [-16, -6, -10, 6, 0, 0, -2, 0], [7, 3, -1, -3, 0, 0, -12, 0], [6, 3, -6, -3, 0, 0, 8, 0], [-9, -9, -11, 9, 0, 0, -16, 0], [-19, 0, 5, 9, -19, 5, -5, -5], [12, 16, 7, 0, -12, -7, 8, -8], [0, -4, 0, -4, 5, -3, 0, 14], [15, 0, 1, 9, 15, 1, 6, 6], [-1, 3, 3, -3, 0, 0, -10, 0], [-2, 0, 3, -30, -2, 3, 12, 12], [0, -5, 0, -5, -28, 6, 0, -6], [25, 0, -6, 13, 25, -6, 3, 3], [15, 9, -8, 0, -15, 8, 8, -8], [-3, -4, -7, 0, 3, 7, 1, -1], [0, -5, 0, -5, 16, -8, 0, 14], [11, 0, -2, -15, 11, -2, 14, 14], [0, -8, 0, -8, -6, 4, 0, 0], [6, 0, -2, -1, 6, -2, -10, -10], [3, -15, -7, 15, 0, 0, -14, 0], [17, 0, 1, -8, 17, 1, -3, -3], [-29, -1, -9, 1, 0, 0, -22, 0], [8, 0, -2, 0, 0, 0, -4, 0], [28, 2, -6, -2, 0, 0, 4, 0], [0, -6, 0, -6, 7, 11, 0, 2], [0, 9, 0, 9, 6, -8, 0, 18], [-2, 0, 9, -11, -2, 9, 5, 5], [2, -6, -1, 0, -2, 1, -2, 2], [0, -1, 0, -1, -2, -10, 0, -10], [8, -8, -6, 8, 0, 0, -12, 0], [0, 16, 0, 16, -3, -5, 0, -18], [8, -19, 4, 0, -8, -4, -15, 15], [-9, 3, -5, 0, 9, 5, -1, 1], [21, -5, 7, 0, -21, -7, 2, -2], [0, -5, 0, -5, -11, -9, 0, -6], [-13, 0, 0, -14, -13, 0, -6, -6], [0, 14, 0, 14, -1, -3, 0, -18], [-6, 20, 0, -20, 0, 0, 20, 0], [-15, 0, -5, -6, -15, -5, 5, 5], [12, 0, -1, 15, 12, -1, 4, 4], [-13, -9, -5, 9, 0, 0, 0, 0], [-17, 11, 11, -11, 0, 0, -8, 0], [-5, 0, 11, -6, -5, 11, 1, 1], [15, 3, -1, -3, 0, 0, 10, 0], [7, -11, 6, 0, -7, -6, 13, -13], [-14, -9, -2, 0, 14, 2, 0, 0], [5, 2, 12, 0, -5, -12, -10, 10], [0, 0, 1, 6, 0, 1, 0, 0], [0, 9, 0, 9, -27, -9, 0, -14], [0, 11, 0, 11, 16, 6, 0, -2], [-14, 4, -5, 0, 14, 5, -4, 4], [-1, 0, -4, 12, -1, -4, -8, -8], [-1, -11, 5, 11, 0, 0, -12, 0], [14, 14, -3, 0, -14, 3, -4, 4], [0, 2, 0, 2, -38, 6, 0, -2], [39, -13, -7, 13, 0, 0, 2, 0], [-10, 15, -4, -15, 0, 0, -6, 0], [-14, 14, 6, -14, 0, 0, 10, 0], [-8, 8, -8, 0, 8, 8, 15, -15], [0, -14, 0, -14, 19, 7, 0, 18], [-3, -3, -9, 3, 0, 0, 12, 0], [-8, 0, -9, 4, -8, -9, -4, -4], [2, -6, -8, 0, -2, 8, 1, -1], [0, -1, 0, -1, -2, -8, 0, 16], [4, 0, -1, -11, 4, -1, 11, 11], [0, 19, 8, 0, 0, -8, 1, -1], [26, -11, 5, 0, -26, -5, 7, -7], [-28, 0, -1, 6, -28, -1, -2, -2], [0, -10, 0, -10, 9, 5, 0, 18], [0, 5, 0, 5, 12, 0, 0, 4], [1, 0, 2, -2, 1, 2, -6, -6], [-4, -9, 2, 9, 0, 0, 12, 0], [-20, 0, 3, 9, -20, 3, -4, -4], [-13, -7, -1, 7, 0, 0, -22, 0], [-17, 15, 5, -15, 0, 0, 12, 0], [-34, -11, 5, 0, 34, -5, -1, 1], [0, 1, 0, 1, -10, 4, 0, -2], [-28, -11, 2, 0, 28, -2, -1, 1], [9, 0, -3, -18, 9, -3, -1, -1], [16, -8, 6, 8, 0, 0, -28, 0], [-6, -8, 14, 8, 0, 0, -2, 0], [12, 16, 0, 0, -12, 0, 7, -7], [0, 1, 0, 1, -18, -2, 0, 20], [0, 4, 0, 4, -17, 11, 0, -10], [11, 28, 2, 0, -11, -2, 12, -12], [0, -5, 0, -5, 6, -8, 0, 22], [-12, 4, -8, -4, 0, 0, 2, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2340_u_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2340_2_u_g();".
// 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_2340_2_u_g(:prec:=8)
    chi := MakeCharacter_2340_u();
    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_2340_2_u_g();".
// 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_2340_2_u_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2340_u();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1936, 0, 1840, 0, 456, 0, 40, 0, 1]>,<11,R![4, -8, 8, 124, 580, 332, 98, 14, 1]>,<17,R![11664, 864, 32, -192, 360, -8, 0, 0, 1]>],Snew);
    return Vf;
end function;