
// Make newform 3920.2.k.c in Magma, downloaded from the LMFDB on 14 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3920_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3920_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_3920_2_k_c();".
// 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_3920_2_k_c();".
// 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 := [1, 0, 0, 0, 0, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 1, 0, 0], [0, 0, 1, 0, 0, 0, 1, 0], [0, 1, 0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0, 0, -1], [0, 0, 1, 0, 0, 0, -1, 0], [0, 0, 0, 1, 0, -1, 0, 0]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 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_3920_k();"
function MakeCharacter_3920_k()
    N := 3920;
    order := 2;
    char_gens := [1471, 981, 3137, 3041];
    v := [1, 2, 2, 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_3920_k_Hecke();"
function MakeCharacter_3920_k_Hecke(Kf)
    N := 3920;
    order := 2;
    char_gens := [1471, 981, 3137, 3041];
    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 := 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], [1, 0, 0, 0, 0, -1, -1, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 2, -1, 0, 0, 0, 0], [0, 1, -1, -1, 0, 0, 0, 0], [0, 3, -2, 0, 1, 0, 0, 0], [-2, 0, 0, 0, 0, 1, 0, -3], [0, 2, 2, -3, 1, 0, 0, 0], [1, 0, 0, 0, 0, 2, -3, 2], [-6, 0, 0, 0, 0, -1, 0, 2], [0, 0, 0, 0, 0, 3, 1, 2], [0, 2, -1, -2, -2, 0, 0, 0], [0, 0, -3, 2, -3, 0, 0, 0], [-3, 0, 0, 0, 0, -1, -3, -2], [4, 0, 0, 0, 0, -2, 1, 3], [-2, 0, 0, 0, 0, 0, -4, 1], [0, 0, -1, -4, 1, 0, 0, 0], [0, 2, 1, 3, 4, 0, 0, 0], [0, -4, -2, -7, 4, 0, 0, 0], [0, 0, 3, 5, -1, 0, 0, 0], [0, 5, 2, -2, 0, 0, 0, 0], [6, 0, 0, 0, 0, -4, 4, 2], [0, -8, -2, -2, -4, 0, 0, 0], [0, 1, 5, 1, -6, 0, 0, 0], [0, 6, -1, 4, -5, 0, 0, 0], [-7, 0, 0, 0, 0, 0, 0, -3], [0, 0, 1, 0, -7, 0, 0, 0], [-1, 0, 0, 0, 0, 2, 0, 6], [-2, 0, 0, 0, 0, 4, -3, -3], [0, 8, 2, 5, -5, 0, 0, 0], [0, 0, 0, 0, 0, 9, 2, -2], [-2, 0, 0, 0, 0, -7, 0, 3], [4, 0, 0, 0, 0, 4, -2, 5], [-2, 0, 0, 0, 0, 2, -1, 8], [0, -11, -2, -2, -8, 0, 0, 0], [0, 8, 0, -4, 0, 0, 0, 0], [0, -8, 1, 5, 0, 0, 0, 0], [3, 0, 0, 0, 0, 9, -1, -6], [0, -3, 7, -1, -6, 0, 0, 0], [0, -10, 2, 3, 4, 0, 0, 0], [0, 6, 5, -10, 0, 0, 0, 0], [0, -5, 0, 3, 8, 0, 0, 0], [-4, 0, 0, 0, 0, 0, 10, 4], [4, 0, 0, 0, 0, 10, 1, -1], [18, 0, 0, 0, 0, -2, 2, 5], [0, -15, -4, -4, 4, 0, 0, 0], [-7, 0, 0, 0, 0, 7, -1, -6], [19, 0, 0, 0, 0, 2, 0, -1], [0, 2, 4, -6, 5, 0, 0, 0], [-6, 0, 0, 0, 0, 1, 4, 5], [0, 9, 6, 1, -4, 0, 0, 0], [0, 4, -10, -2, -3, 0, 0, 0], [-4, 0, 0, 0, 0, -9, 8, -8], [0, -8, -2, -4, 2, 0, 0, 0], [0, -14, -8, 4, 2, 0, 0, 0], [0, 8, -2, -2, -1, 0, 0, 0], [-4, 0, 0, 0, 0, 6, 2, 10], [-4, 0, 0, 0, 0, 7, -5, 4], [11, 0, 0, 0, 0, 0, 0, -8], [3, 0, 0, 0, 0, 10, -2, -1], [0, -13, -3, -5, 2, 0, 0, 0], [9, 0, 0, 0, 0, 6, 8, -3], [-2, 0, 0, 0, 0, -6, 18, 0], [0, 7, 7, -13, 2, 0, 0, 0], [-8, 0, 0, 0, 0, 1, -10, 3], [0, -4, -4, -1, 14, 0, 0, 0], [10, 0, 0, 0, 0, -12, -1, 7], [0, 10, -6, 9, -1, 0, 0, 0], [0, 2, 4, 4, -10, 0, 0, 0], [0, -7, 10, 8, 9, 0, 0, 0], [0, -2, 6, 14, -2, 0, 0, 0], [11, 0, 0, 0, 0, 4, -6, 5], [-2, 0, 0, 0, 0, 1, 16, 5], [0, 6, -2, 5, 10, 0, 0, 0], [4, 0, 0, 0, 0, -10, 2, 4], [-3, 0, 0, 0, 0, 0, -9, -6], [0, 5, -8, 0, -11, 0, 0, 0], [9, 0, 0, 0, 0, 2, -6, -8], [0, -6, 1, -4, 3, 0, 0, 0], [10, 0, 0, 0, 0, 9, 6, -2], [-1, 0, 0, 0, 0, 6, 10, 2], [0, -5, 0, -10, -6, 0, 0, 0], [0, 14, -6, 16, 10, 0, 0, 0], [8, 0, 0, 0, 0, 7, -12, -6], [0, 14, -8, -4, 8, 0, 0, 0], [-9, 0, 0, 0, 0, -2, -7, -8], [10, 0, 0, 0, 0, -1, 3, -14], [0, -4, 3, 0, -9, 0, 0, 0], [0, -4, 2, -4, 8, 0, 0, 0], [-5, 0, 0, 0, 0, 3, -7, -12], [4, 0, 0, 0, 0, 3, -4, 2], [0, -20, 7, 9, 2, 0, 0, 0], [0, 1, -8, -9, -8, 0, 0, 0], [0, 3, 12, -10, -4, 0, 0, 0], [-5, 0, 0, 0, 0, 11, 7, 2], [0, 20, -4, 6, 7, 0, 0, 0], [0, -2, -10, 6, -10, 0, 0, 0], [-4, 0, 0, 0, 0, 8, -12, -6], [-1, 0, 0, 0, 0, 10, -16, -6], [0, 2, -5, -2, -13, 0, 0, 0], [4, 0, 0, 0, 0, 10, 8, -12], [-6, 0, 0, 0, 0, 3, -5, 17], [8, 0, 0, 0, 0, -10, -5, -4], [0, -2, 6, 5, 4, 0, 0, 0], [0, -31, -3, -7, -2, 0, 0, 0], [2, 0, 0, 0, 0, -2, -6, 14], [0, 5, 15, -1, -2, 0, 0, 0], [0, -1, 12, 11, -10, 0, 0, 0], [0, 6, 3, 8, 1, 0, 0, 0], [-9, 0, 0, 0, 0, -6, 10, -9], [4, 0, 0, 0, 0, 11, 8, 1], [6, 0, 0, 0, 0, -12, 9, -7], [0, 0, 0, 0, 0, -20, -4, 1], [0, -7, -10, 2, -8, 0, 0, 0], [2, 0, 0, 0, 0, -8, -11, 0], [3, 0, 0, 0, 0, -8, 12, -9], [-2, 0, 0, 0, 0, 0, -6, 8], [-4, 0, 0, 0, 0, -13, 12, 1], [0, 3, -20, 8, 4, 0, 0, 0], [0, 8, -12, -16, -6, 0, 0, 0], [-14, 0, 0, 0, 0, -21, 2, 3], [0, 1, -17, 15, 6, 0, 0, 0], [0, 14, 7, 14, -7, 0, 0, 0], [-8, 0, 0, 0, 0, 7, -16, -11], [21, 0, 0, 0, 0, -4, 11, 10], [-1, 0, 0, 0, 0, 0, 5, -8], [28, 0, 0, 0, 0, -2, -8, 3], [26, 0, 0, 0, 0, -5, 7, 1], [0, 15, 16, -10, -1, 0, 0, 0], [0, -21, -6, -1, -6, 0, 0, 0], [0, 0, -18, 1, -5, 0, 0, 0], [0, 1, 4, -2, -6, 0, 0, 0], [10, 0, 0, 0, 0, -1, 0, 9], [0, -16, -9, 18, -4, 0, 0, 0], [0, 18, -6, 14, 18, 0, 0, 0], [0, -13, 11, 1, 10, 0, 0, 0], [-19, 0, 0, 0, 0, -1, 1, 16], [0, -11, 9, -1, 16, 0, 0, 0], [11, 0, 0, 0, 0, -10, -5, -14], [-16, 0, 0, 0, 0, 3, -10, 12], [-3, 0, 0, 0, 0, -18, -2, -16], [0, 2, -13, -3, 4, 0, 0, 0], [0, -2, -2, -9, 7, 0, 0, 0], [0, -26, 6, -10, -15, 0, 0, 0], [28, 0, 0, 0, 0, -10, -6, 1], [0, -8, 8, 2, 14, 0, 0, 0], [0, 10, -4, 10, 6, 0, 0, 0], [-24, 0, 0, 0, 0, 11, 0, 5], [0, -22, -1, -13, -14, 0, 0, 0], [-2, 0, 0, 0, 0, 16, 0, -18], [0, -12, 0, 6, -12, 0, 0, 0], [0, 4, -8, -8, -26, 0, 0, 0], [2, 0, 0, 0, 0, 9, 9, -5], [0, -8, 9, -25, -2, 0, 0, 0], [0, -8, 16, 14, 8, 0, 0, 0], [0, -1, 2, 27, 4, 0, 0, 0], [0, -22, 3, -2, -14, 0, 0, 0], [0, 9, 6, 22, -11, 0, 0, 0], [0, -8, -4, -14, -6, 0, 0, 0], [0, -2, 7, 1, 28, 0, 0, 0], [-14, 0, 0, 0, 0, 5, -23, -4], [0, -28, 1, 10, -3, 0, 0, 0], [6, 0, 0, 0, 0, -23, -6, 0], [-30, 0, 0, 0, 0, 2, -12, -2], [23, 0, 0, 0, 0, -1, 7, -4], [0, -2, 0, 5, 12, 0, 0, 0], [0, 7, -20, -14, -5, 0, 0, 0], [7, 0, 0, 0, 0, -24, -3, 12], [0, 4, 13, 15, 5, 0, 0, 0], [0, 0, -14, -19, -14, 0, 0, 0], [0, 34, 3, 2, 1, 0, 0, 0], [0, -21, -10, 2, -14, 0, 0, 0], [16, 0, 0, 0, 0, 5, -1, 8], [-6, 0, 0, 0, 0, 20, 2, -16], [0, -12, -23, 4, -17, 0, 0, 0], [0, 5, 26, 8, -16, 0, 0, 0], [-3, 0, 0, 0, 0, 0, -26, -8], [-14, 0, 0, 0, 0, 15, 13, 5], [0, 4, 0, -24, 10, 0, 0, 0], [0, -16, -18, 2, -2, 0, 0, 0], [8, 0, 0, 0, 0, 19, -6, -9], [-12, 0, 0, 0, 0, -1, -12, 17], [0, 8, 4, 20, 22, 0, 0, 0], [0, -22, -5, -20, 13, 0, 0, 0], [0, 4, -14, 14, 7, 0, 0, 0], [4, 0, 0, 0, 0, 18, -5, 13], [27, 0, 0, 0, 0, 5, 13, 10], [-9, 0, 0, 0, 0, -8, -14, -14], [16, 0, 0, 0, 0, 6, -2, -16], [0, 12, 2, -10, -16, 0, 0, 0], [0, -6, 18, -10, 6, 0, 0, 0], [0, 7, 14, 6, -14, 0, 0, 0], [0, -14, 4, 6, 1, 0, 0, 0], [0, -36, -12, 8, -8, 0, 0, 0], [0, -9, -2, 10, 13, 0, 0, 0], [-7, 0, 0, 0, 0, 24, 12, -2], [0, 0, 0, 0, 0, 18, -1, 15], [0, 11, 23, 9, -12, 0, 0, 0], [-11, 0, 0, 0, 0, 8, 10, -1], [-12, 0, 0, 0, 0, 8, -11, -10], [18, 0, 0, 0, 0, 13, -2, -1], [0, 24, -8, -12, -10, 0, 0, 0], [0, 6, 1, -16, -9, 0, 0, 0], [10, 0, 0, 0, 0, 1, -16, -5], [0, 27, 20, -20, 3, 0, 0, 0], [4, 0, 0, 0, 0, 21, -8, 26], [0, 2, -11, 10, 21, 0, 0, 0], [16, 0, 0, 0, 0, -8, -17, -8], [20, 0, 0, 0, 0, 19, -18, 9], [30, 0, 0, 0, 0, 9, 1, -4], [0, 0, -1, -14, 2, 0, 0, 0], [0, -28, 10, 7, 21, 0, 0, 0], [-19, 0, 0, 0, 0, 10, 2, 29], [18, 0, 0, 0, 0, -4, 14, -6], [0, 8, -20, -8, 18, 0, 0, 0], [0, 8, -17, -9, -10, 0, 0, 0], [0, 2, -28, 4, 1, 0, 0, 0], [0, 6, -8, 20, 8, 0, 0, 0], [-28, 0, 0, 0, 0, 9, 9, -8], [16, 0, 0, 0, 0, 10, 4, 10], [10, 0, 0, 0, 0, 1, -4, -13], [-5, 0, 0, 0, 0, 12, 2, 2], [0, 4, -35, 0, -1, 0, 0, 0], [-15, 0, 0, 0, 0, -22, 10, 13], [-4, 0, 0, 0, 0, 8, 14, 4], [0, -21, 14, -2, -13, 0, 0, 0], [0, -3, 18, 0, -18, 0, 0, 0], [35, 0, 0, 0, 0, 6, 0, 7], [0, 8, -4, -21, -8, 0, 0, 0], [10, 0, 0, 0, 0, 5, 6, 31], [-14, 0, 0, 0, 0, -19, -16, 13], [0, 44, -4, 3, 6, 0, 0, 0], [-5, 0, 0, 0, 0, -10, 6, -22], [-27, 0, 0, 0, 0, -18, -4, 7], [-11, 0, 0, 0, 0, -4, 18, -13], [0, 30, -7, -12, -8, 0, 0, 0], [-28, 0, 0, 0, 0, -14, 14, 2], [0, -9, 16, 1, 30, 0, 0, 0], [0, 0, 0, 0, 0, -2, 22, 5], [0, 34, 9, -12, 13, 0, 0, 0], [-20, 0, 0, 0, 0, -13, -2, -15], [-19, 0, 0, 0, 0, 10, 4, 5], [10, 0, 0, 0, 0, 34, -3, 12], [0, -9, -2, -8, -1, 0, 0, 0], [2, 0, 0, 0, 0, 23, -4, 19], [-24, 0, 0, 0, 0, -16, 2, 0], [38, 0, 0, 0, 0, -5, -8, -13], [0, -29, 20, -9, -8, 0, 0, 0], [0, 0, 3, -18, -9, 0, 0, 0], [-10, 0, 0, 0, 0, 4, -24, 8], [0, 10, -15, -16, 9, 0, 0, 0], [0, -16, -8, -19, -7, 0, 0, 0], [0, 4, 17, 4, 7, 0, 0, 0], [0, -10, -2, 20, 0, 0, 0, 0], [0, 35, -4, -17, -8, 0, 0, 0], [-24, 0, 0, 0, 0, 6, 5, 4], [-22, 0, 0, 0, 0, -7, -1, 3], [0, -25, 6, -10, 11, 0, 0, 0], [0, -27, 11, -5, 2, 0, 0, 0], [0, -22, 2, -12, 38, 0, 0, 0], [0, -4, -13, -23, -16, 0, 0, 0], [0, 30, 9, 14, 6, 0, 0, 0], [0, 47, 4, -2, 1, 0, 0, 0], [0, 0, 10, 20, -20, 0, 0, 0], [12, 0, 0, 0, 0, -5, 2, 4], [-31, 0, 0, 0, 0, 18, 10, 0], [0, 26, 21, -20, 5, 0, 0, 0], [0, -14, 7, -13, -18, 0, 0, 0], [-20, 0, 0, 0, 0, -15, 10, -23], [0, 2, -5, -16, -18, 0, 0, 0], [0, 2, 37, 5, 0, 0, 0, 0], [0, -3, -13, -7, -10, 0, 0, 0], [0, -2, 18, 5, 22, 0, 0, 0], [0, 26, -2, -4, -2, 0, 0, 0], [11, 0, 0, 0, 0, -1, -19, -8], [0, 4, -2, -8, 0, 0, 0, 0], [-18, 0, 0, 0, 0, -4, -11, 40], [-35, 0, 0, 0, 0, -6, 5, -22], [-10, 0, 0, 0, 0, 3, -14, -3], [-23, 0, 0, 0, 0, 16, -20, -13], [0, 15, 2, -14, 22, 0, 0, 0], [20, 0, 0, 0, 0, -18, 8, -10], [0, -20, 1, 0, -32, 0, 0, 0], [11, 0, 0, 0, 0, 13, -11, 2], [0, 15, 6, -18, 8, 0, 0, 0], [30, 0, 0, 0, 0, 2, 12, 24], [46, 0, 0, 0, 0, 0, -10, 0], [-26, 0, 0, 0, 0, 8, -18, -9], [0, -6, -6, -18, -27, 0, 0, 0], [-9, 0, 0, 0, 0, -16, 13, 10], [-39, 0, 0, 0, 0, -10, -6, -1], [18, 0, 0, 0, 0, 3, 7, 34], [2, 0, 0, 0, 0, -14, 8, 15], [0, 0, 0, 0, 0, 13, 2, 3], [0, -14, -16, -20, 29, 0, 0, 0], [-4, 0, 0, 0, 0, 2, -10, -5], [0, -15, 20, 6, -7, 0, 0, 0], [-12, 0, 0, 0, 0, 18, 4, -14], [-23, 0, 0, 0, 0, 4, 12, -7], [0, 3, 5, -1, -2, 0, 0, 0], [-40, 0, 0, 0, 0, 5, -7, 2], [0, 45, -2, -18, 4, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3920_k_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3920_2_k_c();".
// 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_3920_2_k_c(:prec:=8)
    chi := MakeCharacter_3920_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_3920_2_k_c();".
// 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_3920_2_k_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3920_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-17, 20, -2, -4, 1]>],Snew);
    return Vf;
end function;