
// Make newform 2400.2.k.e in Magma, downloaded from the LMFDB on 17 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2400_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2400_k_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_2400_2_k_e();".
// 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_2400_2_k_e();".
// 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 := [16, -16, 0, -4, 9, -2, 0, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [-12, -4, 1, 7, -2, -2, -1, 1], [56, -20, -12, -21, 18, 4, 2, -5], [64, -20, -18, -25, 18, 8, 4, -5], [64, -12, -8, -29, 18, 4, 6, -5], [-88, 12, 30, 35, -22, -8, -4, 7], [-88, 24, 17, 37, -22, -6, -5, 7], [-84, 28, 15, 34, -24, -6, -5, 8]];
    Rf_basisdens := [1, 4, 8, 8, 8, 8, 4, 4];
    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_2400_k();"
function MakeCharacter_2400_k()
    N := 2400;
    order := 2;
    char_gens := [1951, 901, 1601, 577];
    v := [2, 1, 2, 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_2400_k_Hecke();"
function MakeCharacter_2400_k_Hecke( : Kf := false)
    N := 2400;
    order := 2;
    char_gens := [1951, 901, 1601, 577];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 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, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 1, -1], [0, -1, 0, 0, 1, 0, 0, 0], [0, 0, -1, 0, 0, 1, -2, 0], [1, -2, 0, 1, -2, 0, 0, 0], [0, 0, 3, 0, 0, -2, 1, -2], [-1, -1, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, 0, 2], [0, 1, 0, 2, -1, 0, 0, 0], [0, 0, -1, 0, 0, 3, 0, 2], [0, 0, 0, 2, -2, 0, 0, 0], [0, 0, -1, 0, 0, 0, 3, 0], [0, 0, 0, 0, 0, 0, -2, 4], [0, 0, 4, 0, 0, 2, 1, 1], [0, 0, -3, 0, 0, -1, 2, -4], [5, -2, 0, -1, 0, 0, 0, 0], [2, 0, 0, 4, -2, 0, 0, 0], [2, 2, 0, 0, 4, 0, 0, 0], [0, 0, -4, 0, 0, 2, 2, 0], [0, 2, 0, -4, 2, 0, 0, 0], [1, -2, 0, 0, -4, 0, 0, 0], [0, 0, -5, 0, 0, 0, 3, 0], [8, -2, 0, 2, 0, 0, 0, 0], [0, 0, 4, 0, 0, 1, 2, 4], [0, 0, 10, 0, 0, 0, -3, 1], [4, -2, 0, 0, -2, 0, 0, 0], [-4, -4, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 2, 0], [8, 2, 0, -2, 2, 0, 0, 0], [0, 0, 6, 0, 0, 0, -2, -2], [0, 0, -12, 0, 0, -2, -2, -2], [3, 1, 0, 4, -4, 0, 0, 0], [0, 0, -6, 0, 0, -4, 3, 3], [0, 0, 3, 0, 0, 3, 0, -2], [-5, 0, 0, 5, -2, 0, 0, 0], [0, 0, 1, 0, 0, 4, -1, -2], [0, 0, -2, 0, 0, -4, 4, -4], [0, 0, -6, 0, 0, 2, -1, 1], [-7, 0, 0, -3, 0, 0, 0, 0], [1, 0, 0, 4, 2, 0, 0, 0], [0, 0, 8, 0, 0, -2, 2, -6], [-1, 5, 0, 2, 4, 0, 0, 0], [0, 0, 3, 0, 0, 1, 0, 2], [-7, 3, 0, 4, 0, 0, 0, 0], [0, 0, -6, 0, 0, 3, -4, 4], [0, 0, -12, 0, 0, 6, -5, 3], [0, -7, 0, -2, -1, 0, 0, 0], [9, 2, 0, -1, -4, 0, 0, 0], [5, 6, 0, 0, 4, 0, 0, 0], [0, 0, -6, 0, 0, -4, 2, -4], [2, 0, 0, -2, 4, 0, 0, 0], [5, 6, 0, -1, 0, 0, 0, 0], [0, 0, 15, 0, 0, 4, 1, 6], [0, 2, 0, -2, 4, 0, 0, 0], [0, 0, 12, 0, 0, -4, 5, -5], [6, 3, 0, -2, -3, 0, 0, 0], [0, 0, 7, 0, 0, -3, 0, -6], [0, 0, -14, 0, 0, -4, 0, -2], [0, 0, 1, 0, 0, -1, 0, 2], [10, 2, 0, 0, 4, 0, 0, 0], [3, 2, 0, -4, 6, 0, 0, 0], [0, 0, -8, 0, 0, 2, 0, 0], [0, 0, 16, 0, 0, 2, 0, 4], [-7, 2, 0, 0, 2, 0, 0, 0], [0, 0, -8, 0, 0, 2, -6, -4], [0, 0, 16, 0, 0, 4, -4, 0], [-2, 9, 0, -2, 3, 0, 0, 0], [-11, 4, 0, 3, -2, 0, 0, 0], [-11, 3, 0, -2, 0, 0, 0, 0], [0, 0, -6, 0, 0, 2, -5, 5], [0, 0, -3, 0, 0, 7, 0, 2], [-4, -2, 0, -8, -2, 0, 0, 0], [0, 0, -19, 0, 0, -2, 5, 0], [0, 0, -18, 0, 0, -6, 1, -5], [-2, 1, 0, 0, 7, 0, 0, 0], [-3, 4, 0, 4, 6, 0, 0, 0], [0, 0, 0, 0, 0, 5, 2, 4], [0, 0, -4, 0, 0, -6, -4, -6], [-17, -2, 0, 3, -2, 0, 0, 0], [-5, 2, 0, 0, -2, 0, 0, 0], [11, 3, 0, 2, 4, 0, 0, 0], [0, 0, 6, 0, 0, 2, -6, 0], [6, -2, 0, -2, -6, 0, 0, 0], [10, -2, 0, -4, 0, 0, 0, 0], [0, 0, 5, 0, 0, -2, 1, 0], [-12, 2, 0, -2, 4, 0, 0, 0], [0, 0, 18, 0, 0, -3, 2, 0], [10, -12, 0, 0, -2, 0, 0, 0], [9, 7, 0, -4, 4, 0, 0, 0], [0, 0, 16, 0, 0, 2, -8, 4], [0, 0, 5, 0, 0, 9, -2, 4], [-9, -8, 0, -3, 2, 0, 0, 0], [0, 0, 16, 0, 0, -2, 2, -10], [0, 9, 0, -8, 7, 0, 0, 0], [0, 0, -5, 0, 0, -1, -6, -4], [0, 0, 12, 0, 0, -4, -3, 3], [0, 0, -2, 0, 0, 0, 2, 10], [0, 0, -20, 0, 0, -4, -2, 2], [0, 0, 2, 0, 0, -1, -2, 4], [-6, -5, 0, 6, -11, 0, 0, 0], [0, 0, -9, 0, 0, -3, -6, 4], [7, 6, 0, -4, 0, 0, 0, 0], [0, 0, 10, 0, 0, -9, 8, -12], [2, 5, 0, -4, 3, 0, 0, 0], [-14, -10, 0, -6, 6, 0, 0, 0], [3, 0, 0, -4, 6, 0, 0, 0], [-14, 2, 0, 0, -4, 0, 0, 0], [0, 0, -14, 0, 0, -6, 2, -2], [12, 3, 0, 6, -3, 0, 0, 0], [0, 0, -3, 0, 0, 7, -4, 2], [-1, -7, 0, -4, 4, 0, 0, 0], [-14, -2, 0, -10, 2, 0, 0, 0], [0, 0, 10, 0, 0, 8, -6, 6], [-20, -4, 0, -8, 4, 0, 0, 0], [0, 0, 14, 0, 0, -12, 8, -10], [0, 0, -4, 0, 0, 1, -2, 8], [0, 0, -2, 0, 0, -2, 6, 2], [14, 0, 0, -8, 10, 0, 0, 0], [0, 0, 15, 0, 0, -2, 1, 2], [0, 0, -16, 0, 0, -6, 0, -8], [0, 0, 4, 0, 0, -6, 0, -4], [0, 0, 17, 0, 0, -6, 7, -2], [0, 0, -22, 0, 0, -4, 3, 3], [19, 0, 0, -1, 4, 0, 0, 0], [1, -3, 0, 8, -4, 0, 0, 0], [0, 0, 20, 0, 0, 0, 0, 4], [0, 0, -4, 0, 0, -4, 0, 8], [-10, 0, 0, -6, 4, 0, 0, 0], [4, 4, 0, -2, 0, 0, 0, 0], [0, 0, 24, 0, 0, -2, 3, 3], [12, 0, 0, 8, 0, 0, 0, 0], [3, -6, 0, -4, 2, 0, 0, 0], [0, 0, -9, 0, 0, 10, -9, 8], [0, 0, 15, 0, 0, -5, 0, -2], [0, 0, -6, 0, 0, 2, 2, 8], [8, 5, 0, -6, 3, 0, 0, 0], [0, 0, 17, 0, 0, -3, 6, -12], [0, 0, -18, 0, 0, -8, -4, -2], [9, 3, 0, -4, -4, 0, 0, 0], [0, 0, -20, 0, 0, 11, -10, 8], [0, 0, 22, 0, 0, 2, 2, -6], [3, 6, 0, -5, -6, 0, 0, 0], [0, 0, -12, 0, 0, -6, 7, -9], [-26, -4, 0, -2, -8, 0, 0, 0], [0, 0, -12, 0, 0, -10, 0, -4], [16, 2, 0, 8, 2, 0, 0, 0], [0, 0, -4, 0, 0, 2, 9, 1], [26, -10, 0, -2, -6, 0, 0, 0], [0, 0, 3, 0, 0, -9, 10, -12], [-18, 16, 0, 0, 6, 0, 0, 0], [0, 0, 8, 0, 0, 0, -4, 0], [-10, -4, 0, -4, -2, 0, 0, 0], [9, -1, 0, 6, -12, 0, 0, 0], [8, 1, 0, 4, -1, 0, 0, 0], [11, -6, 0, 8, 2, 0, 0, 0], [0, 0, 10, 0, 0, 4, -8, -10], [0, 0, -12, 0, 0, 5, 4, 0], [0, -4, 0, 4, -8, 0, 0, 0], [14, -2, 0, 8, 0, 0, 0, 0], [0, 0, 34, 0, 0, -4, 4, -4], [0, 12, 0, 6, 4, 0, 0, 0], [-18, 4, 0, 6, 0, 0, 0, 0], [25, 7, 0, -4, 4, 0, 0, 0], [0, 0, 20, 0, 0, -12, 8, -8], [-13, -8, 0, -12, 4, 0, 0, 0], [0, 0, -14, 0, 0, 2, 8, -2], [0, 0, 2, 0, 0, -5, 6, -8], [0, 0, 14, 0, 0, 2, -7, 11], [22, 6, 0, 16, 0, 0, 0, 0], [9, -18, 0, -8, 2, 0, 0, 0], [-23, 3, 0, -4, 0, 0, 0, 0], [-10, 6, 0, 6, 6, 0, 0, 0], [0, 0, -49, 0, 0, 3, -2, 4], [0, 0, -13, 0, 0, 6, 9, -2], [-3, -15, 0, -10, -4, 0, 0, 0], [0, 0, 16, 0, 0, 6, -5, 15], [-9, -3, 0, 12, -4, 0, 0, 0], [0, 0, -4, 0, 0, -10, 8, 4], [0, 0, -26, 0, 0, 2, 1, -5], [14, 2, 0, 0, -2, 0, 0, 0], [30, -4, 0, -4, -10, 0, 0, 0], [0, 0, -22, 0, 0, 4, -12, -2], [0, 0, 6, 0, 0, 10, -3, 11], [0, 0, 9, 0, 0, -3, 16, 2], [15, -10, 0, 0, -10, 0, 0, 0], [-7, -4, 0, 5, -4, 0, 0, 0], [3, -8, 0, 4, -14, 0, 0, 0], [0, 0, -6, 0, 0, 3, 8, -8], [0, 0, 17, 0, 0, -11, 0, -2], [0, 0, -3, 0, 0, 10, 7, -2], [0, 0, -8, 0, 0, -1, 4, -4], [14, 0, 0, -4, 0, 0, 0, 0], [-2, -8, 0, 4, -2, 0, 0, 0], [0, 0, -4, 0, 0, 0, 8, -12], [4, 2, 0, -2, -2, 0, 0, 0], [-11, 8, 0, 9, -4, 0, 0, 0], [0, 0, 15, 0, 0, 4, -9, -4], [16, -6, 0, 10, -8, 0, 0, 0], [0, 0, -2, 0, 0, 6, -14, -6], [-10, 16, 0, -8, 6, 0, 0, 0], [0, 0, 4, 0, 0, -8, 0, 0], [0, 0, -19, 0, 0, -12, 3, -14], [-27, -1, 0, 2, -8, 0, 0, 0], [0, 0, 6, 0, 0, -5, 14, 0], [-2, 0, 0, -6, 8, 0, 0, 0], [0, 0, 33, 0, 0, 5, -4, 2], [-11, 2, 0, 12, -12, 0, 0, 0], [0, 0, -4, 0, 0, 12, -10, 14], [-25, -5, 0, 16, -16, 0, 0, 0], [0, 0, -16, 0, 0, -10, 8, 4], [2, 16, 0, 0, 18, 0, 0, 0], [-10, -16, 0, -8, 2, 0, 0, 0], [-11, 5, 0, 2, 12, 0, 0, 0], [12, -5, 0, 4, -15, 0, 0, 0], [-33, 0, 0, 5, -10, 0, 0, 0], [0, 0, -17, 0, 0, 6, -7, 6], [0, 0, -16, 0, 0, 0, 7, 1], [7, 17, 0, -10, 4, 0, 0, 0], [34, 6, 0, -4, 10, 0, 0, 0], [12, -2, 0, -10, 4, 0, 0, 0], [0, 0, 12, 0, 0, 9, -2, -4], [0, 0, -26, 0, 0, -6, 13, 7], [-20, 3, 0, 4, 9, 0, 0, 0], [-4, -10, 0, 6, -4, 0, 0, 0], [17, 1, 0, -10, -4, 0, 0, 0], [0, 0, 16, 0, 0, -7, 4, -16], [0, 0, 24, 0, 0, -4, 15, 5], [0, 0, 20, 0, 0, -2, -4, 4], [3, -15, 0, -8, 0, 0, 0, 0], [10, -8, 0, 12, 0, 0, 0, 0], [0, 0, -24, 0, 0, -6, 8, 0], [20, -4, 0, 0, -16, 0, 0, 0], [-2, -4, 0, -8, -6, 0, 0, 0], [0, 0, -7, 0, 0, 8, 9, -4], [0, 0, -8, 0, 0, 4, -14, 12], [25, 10, 0, -9, 12, 0, 0, 0], [0, 0, 10, 0, 0, 5, -6, -4], [0, 0, -27, 0, 0, -1, 18, -8], [11, 1, 0, -6, 16, 0, 0, 0], [0, 0, -4, 0, 0, -6, 1, -15], [10, 0, 0, 0, -4, 0, 0, 0], [-42, 0, 0, 4, -2, 0, 0, 0], [1, 13, 0, -2, 12, 0, 0, 0], [0, 0, -16, 0, 0, 9, -16, -4], [0, 0, -7, 0, 0, 9, 2, 16], [30, 8, 0, -4, 2, 0, 0, 0], [0, 0, -10, 0, 0, -10, 14, -18], [-16, 7, 0, 8, -11, 0, 0, 0], [-52, -2, 0, 4, -6, 0, 0, 0], [-21, -8, 0, 8, -14, 0, 0, 0], [0, 0, 19, 0, 0, 8, -11, -10], [0, 0, -12, 0, 0, 6, 0, -4], [0, 0, 8, 0, 0, -18, 3, -13], [0, 0, -16, 0, 0, 2, -8, 4], [0, 0, -5, 0, 0, 10, -1, 12], [-22, 4, 0, 8, 4, 0, 0, 0], [9, -3, 0, 0, -20, 0, 0, 0], [0, 0, -16, 0, 0, -13, 8, 4], [0, 0, -6, 0, 0, -6, 10, -18], [0, 0, 12, 0, 0, -8, 7, -3], [2, 14, 0, -12, 18, 0, 0, 0], [0, 0, -28, 0, 0, -6, -8, -8], [0, 0, 18, 0, 0, -10, 12, -22], [-18, -10, 0, -14, -10, 0, 0, 0], [0, 0, 32, 0, 0, -16, 12, -16], [0, 0, -4, 0, 0, -10, 6, -2], [0, 0, -8, 0, 0, -8, -3, 3], [0, 0, 24, 0, 0, -2, -4, 4], [-10, 6, 0, -16, 12, 0, 0, 0], [-1, -9, 0, -10, 0, 0, 0, 0], [-5, -10, 0, 8, -8, 0, 0, 0], [5, 3, 0, -6, 12, 0, 0, 0], [0, 0, -28, 0, 0, -6, 10, -4], [0, 0, -8, 0, 0, -12, 5, -1], [-9, 10, 0, 0, -6, 0, 0, 0], [0, 0, 18, 0, 0, -12, 14, -8], [7, 12, 0, 5, -2, 0, 0, 0], [10, -18, 0, 0, -16, 0, 0, 0], [-41, -4, 0, 3, -12, 0, 0, 0], [0, 0, -6, 0, 0, 0, -9, -9], [0, 0, 27, 0, 0, -15, -2, -4], [-18, 16, 0, -16, 10, 0, 0, 0], [-53, -8, 0, -4, 0, 0, 0, 0], [0, 0, 7, 0, 0, 0, -1, -20], [44, -4, 0, -6, 0, 0, 0, 0], [22, 9, 0, 0, 11, 0, 0, 0], [0, 0, -6, 0, 0, -10, 12, 6], [0, 0, -6, 0, 0, -10, -2, 4], [6, 8, 0, -10, 0, 0, 0, 0], [0, 0, 58, 0, 0, 6, -2, 12], [0, 0, 4, 0, 0, 6, -9, 15], [0, 0, 15, 0, 0, -2, -5, 16], [9, 9, 0, 2, 0, 0, 0, 0], [0, 0, 33, 0, 0, -10, -7, -4], [0, 0, 8, 0, 0, 5, 8, 16], [0, 0, 12, 0, 0, 8, 0, -4], [-6, 12, 0, 4, 12, 0, 0, 0], [0, 0, 6, 0, 0, 8, 2, -4], [-27, 1, 0, -8, 4, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2400_k_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2400_2_k_e();".
// 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_2400_2_k_e(:prec:=8)
    chi := MakeCharacter_2400_k();
    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_2400_2_k_e();".
// 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_2400_2_k_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2400_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 12, -6, -4, 1]>],Snew);
    return Vf;
end function;