
// Make newform 4680.2.l.e in Magma, downloaded from the LMFDB on 03 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_4680_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4680_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_4680_2_l_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_4680_2_l_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 := [8, -20, 25, -4, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [323, -121, -4, 10, 25, 2], [-223, -121, 14, -35, -27, -7], [258, -605, 50, -4, -10, -25], [574, -1573, 372, 38, -26, -65], [-774, 1573, -392, 12, 30, 75]];
    Rf_basisdens := [1, 121, 121, 242, 242, 242];
    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_4680_l();"
function MakeCharacter_4680_l()
    N := 4680;
    order := 2;
    char_gens := [3511, 2341, 2081, 937, 1081];
    v := [2, 2, 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_4680_l_Hecke();"
function MakeCharacter_4680_l_Hecke( : Kf := false)
    N := 4680;
    order := 2;
    char_gens := [3511, 2341, 2081, 937, 1081];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [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 := 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, 1, 0], [0, 0, 0, 0, 0, 0], [2, 0, 0, 0, -1, -1], [0, 0, 0, -1, 0, 0], [0, -1, -1, 2, 1, -1], [0, -1, 1, 0, -1, -1], [0, 1, 1, -4, -1, 1], [-2, 2, -2, 0, 0, 0], [4, 0, 0, 0, -2, -2], [0, 2, 2, 2, 0, 0], [4, 1, -1, 0, -2, -2], [0, 1, 1, -2, -1, 1], [0, -2, -2, 6, 1, -1], [0, 0, 0, -2, 2, -2], [-2, 0, 0, 0, -3, -3], [0, -1, 1, 0, 1, 1], [0, 2, 2, 0, 2, -2], [-2, 1, -1, 0, 0, 0], [0, 4, 4, 2, 0, 0], [-6, 1, -1, 0, 1, 1], [0, -2, -2, -2, -1, 1], [-8, -1, 1, 0, 2, 2], [0, 0, 0, -6, 0, 0], [10, 0, 0, 0, -2, -2], [0, 0, 0, 6, 2, -2], [0, 2, 2, 0, 0, 0], [2, -3, 3, 0, 1, 1], [0, 1, 1, 2, 1, -1], [0, -1, -1, -12, -1, 1], [0, 2, -2, 0, 0, 0], [0, -2, -2, 8, 3, -3], [-4, -2, 2, 0, -6, -6], [-4, 0, 0, 0, -3, -3], [4, -2, 2, 0, -4, -4], [0, -5, -5, -12, -1, 1], [0, -2, -2, 4, 0, 0], [0, 2, 2, 10, -1, 1], [0, -4, -4, 2, 4, -4], [8, 2, -2, 0, 2, 2], [2, 2, -2, 0, 6, 6], [0, -4, 4, 0, 0, 0], [0, -2, -2, -2, 4, -4], [0, 6, 6, 8, 3, -3], [2, 5, -5, 0, 1, 1], [-2, 1, -1, 0, 1, 1], [0, -4, -4, 8, 2, -2], [0, -6, -6, -6, 1, -1], [2, -1, 1, 0, -1, -1], [0, 5, 5, 2, 1, -1], [2, 1, -1, 0, 2, 2], [14, -2, 2, 0, 0, 0], [-4, -2, 2, 0, -8, -8], [0, 1, 1, 10, -3, 3], [0, 1, 1, 0, 7, -7], [-18, -4, 4, 0, -6, -6], [4, 6, -6, 0, 2, 2], [0, 3, 3, 0, -1, 1], [-12, -5, 5, 0, -2, -2], [0, -4, -4, 4, 4, -4], [0, 4, 4, -4, 3, -3], [0, 0, 0, -20, 0, 0], [16, 4, -4, 0, 2, 2], [0, -3, -3, 10, 1, -1], [0, -2, -2, 0, 3, -3], [4, -1, 1, 0, -3, -3], [0, 1, 1, -6, 5, -5], [0, 6, 6, -4, 2, -2], [2, 5, -5, 0, 3, 3], [0, -4, -4, -8, -5, 5], [14, 3, -3, 0, 6, 6], [0, -2, -2, -10, -4, 4], [0, 6, 6, 6, -2, 2], [-16, -1, 1, 0, -1, -1], [0, -4, -4, 14, 3, -3], [-14, 0, 0, 0, 0, 0], [0, 4, 4, -6, -6, 6], [0, -3, 3, 0, 8, 8], [-2, 2, -2, 0, -8, -8], [-4, -4, 4, 0, -6, -6], [18, 1, -1, 0, -3, -3], [6, 5, -5, 0, 0, 0], [0, -5, -5, -18, -1, 1], [-16, 4, -4, 0, -4, -4], [0, 8, 8, 4, 2, -2], [8, 7, -7, 0, 0, 0], [0, -2, -2, -18, 2, -2], [0, 0, 0, 0, -1, -1], [0, 0, 0, 16, 0, 0], [0, 0, 0, 12, 2, -2], [18, 1, -1, 0, 2, 2], [0, -6, -6, 4, 4, -4], [-4, -4, 4, 0, 6, 6], [-20, 3, -3, 0, -3, -3], [0, 7, 7, 8, -3, 3], [0, -2, 2, 0, 3, 3], [2, 0, 0, 0, 6, 6], [0, 5, 5, -2, -9, 9], [6, -3, 3, 0, 1, 1], [0, 11, 11, -10, -7, 7], [0, -2, -2, 12, -9, 9], [0, -2, -2, -20, -2, 2], [-14, 2, -2, 0, 8, 8], [-2, 1, -1, 0, 13, 13], [0, 4, 4, 18, 2, -2], [0, -10, -10, -10, -1, 1], [0, 6, 6, -20, -3, 3], [0, 4, -4, 0, -6, -6], [8, -3, 3, 0, 7, 7], [0, -2, -2, -22, 0, 0], [0, -4, -4, -6, -2, 2], [0, 6, 6, 8, -3, 3], [-12, -3, 3, 0, 7, 7], [8, 4, -4, 0, 14, 14], [-6, 0, 0, 0, -4, -4], [0, 6, 6, 0, 2, -2], [0, -1, -1, -8, -5, 5], [0, 0, 0, -2, 0, 0], [24, -6, 6, 0, 2, 2], [-10, -7, 7, 0, -9, -9], [0, 10, 10, -8, 2, -2], [0, 4, 4, 30, 2, -2], [0, -6, -6, 2, 5, -5], [40, 1, -1, 0, -1, -1], [34, 0, 0, 0, -4, -4], [2, -1, 1, 0, 11, 11], [-4, 6, -6, 0, -4, -4], [0, -1, -1, -12, -5, 5], [0, 0, 0, 2, 4, -4], [0, -3, 3, 0, -1, -1], [0, -16, -16, -14, -3, 3], [16, 4, -4, 0, 0, 0], [0, 0, 0, 10, -8, 8], [4, 7, -7, 0, -6, -6], [-2, 4, -4, 0, 0, 0], [0, 8, 8, -4, -7, 7], [0, -8, -8, 20, 4, -4], [0, -10, -10, -10, 0, 0], [-18, -8, 8, 0, 2, 2], [20, -3, 3, 0, 1, 1], [-12, -2, 2, 0, 11, 11], [0, -8, -8, 2, -6, 6], [0, 6, 6, -14, -3, 3], [24, -3, 3, 0, -1, -1], [2, 3, -3, 0, 4, 4], [0, 4, 4, -10, -2, 2], [0, -1, -1, 26, -1, 1], [-18, -7, 7, 0, 5, 5], [0, 2, 2, 10, -5, 5], [0, -2, -2, 10, -4, 4], [10, 6, -6, 0, 16, 16], [0, 2, 2, -4, -6, 6], [0, -3, -3, -36, 1, -1], [0, 6, 6, 12, 2, -2], [24, 4, -4, 0, -12, -12], [-6, 5, -5, 0, 5, 5], [-4, 5, -5, 0, 14, 14], [0, 7, 7, 6, -9, 9], [20, 4, -4, 0, 3, 3], [0, 14, 14, -26, -9, 9], [0, -5, -5, -14, -1, 1], [0, 2, 2, -8, 8, -8], [-12, -2, 2, 0, -2, -2], [0, -4, -4, 4, -7, 7], [0, -4, -4, 18, 1, -1], [-8, -4, 4, 0, 0, 0], [0, 9, 9, -8, 1, -1], [-10, 4, -4, 0, 2, 2], [0, 10, 10, 14, -2, 2], [-2, 2, -2, 0, -1, -1], [6, 5, -5, 0, 3, 3], [32, 4, -4, 0, 2, 2], [0, -12, -12, 22, 4, -4], [-24, -4, 4, 0, 4, 4], [-10, -2, 2, 0, 2, 2], [-4, -7, 7, 0, -5, -5], [20, -8, 8, 0, 3, 3], [0, 2, 2, 10, 0, 0], [-4, -1, 1, 0, 9, 9], [0, -6, -6, -16, -2, 2], [-8, -10, 10, 0, 6, 6], [0, -8, -8, 30, 0, 0], [0, 4, 4, -20, -9, 9], [0, 0, 0, 38, -1, 1], [10, 10, -10, 0, -6, -6], [0, -5, -5, 12, -1, 1], [0, 2, 2, -4, 8, -8], [-6, 0, 0, 0, -6, -6], [-26, -1, 1, 0, -2, -2], [0, -2, -2, -16, 2, -2], [0, 6, 6, 10, -3, 3], [4, -8, 8, 0, 8, 8], [44, 4, -4, 0, -5, -5], [0, -8, -8, 4, 4, -4], [0, -1, -1, -18, -9, 9], [42, 6, -6, 0, 6, 6], [0, -4, -4, 10, 4, -4], [0, 16, 16, 4, -5, 5], [0, -15, -15, -8, -7, 7], [-24, 8, -8, 0, 1, 1], [24, 4, -4, 0, 0, 0], [0, -4, -4, 26, 4, -4], [2, 2, -2, 0, 2, 2], [-10, 0, 0, 0, -3, -3], [0, 14, 14, 26, 4, -4], [-12, 8, -8, 0, -6, -6], [0, 18, 18, 0, -4, 4], [24, 3, -3, 0, -8, -8], [34, 5, -5, 0, -3, -3], [0, -7, -7, 18, -3, 3], [14, -10, 10, 0, 2, 2], [0, -5, -5, -20, 7, -7], [0, 10, 10, -26, -1, 1], [-6, 9, -9, 0, 10, 10], [22, 6, -6, 0, -2, -2], [0, -3, -3, -28, -3, 3], [-38, 4, -4, 0, 4, 4], [0, 0, 0, 6, -1, 1], [0, 14, 14, -12, 1, -1], [-30, 0, 0, 0, -8, -8], [24, 2, -2, 0, 2, 2], [-24, -7, 7, 0, 0, 0], [0, -4, -4, 24, -2, 2], [0, 2, 2, -12, -8, 8], [6, 8, -8, 0, 8, 8], [0, -9, -9, -26, 5, -5], [8, -8, 8, 0, 2, 2], [0, -9, -9, -36, 3, -3], [6, -14, 14, 0, 3, 3], [0, 5, 5, -28, 1, -1], [12, -8, 8, 0, -4, -4], [8, 0, 0, 0, -8, -8], [2, -2, 2, 0, 2, 2], [0, 7, 7, -10, -11, 11], [0, 2, 2, -26, 1, -1], [-14, 4, -4, 0, 10, 10], [0, -12, -12, 12, 5, -5], [-24, -2, 2, 0, 14, 14], [20, -10, 10, 0, -2, -2], [0, 6, 6, 14, -5, 5], [-14, -7, 7, 0, -3, -3], [0, 1, 1, -20, 9, -9], [-18, 9, -9, 0, -9, -9], [0, 14, 14, 44, 3, -3], [-8, -12, 12, 0, 2, 2], [0, -10, -10, 24, -6, 6], [38, 4, -4, 0, 9, 9], [0, -9, 9, 0, 3, 3], [0, 15, 15, 12, 5, -5], [0, -6, -6, -6, 6, -6], [-52, 5, -5, 0, 0, 0], [0, 12, 12, 6, -1, 1], [-24, -9, 9, 0, -3, -3], [0, 4, 4, -22, 0, 0], [46, 4, -4, 0, 1, 1], [8, -13, 13, 0, 9, 9], [0, 0, 0, 24, 6, -6], [0, 10, 10, -10, 4, -4], [0, -6, -6, -34, -8, 8], [0, 3, 3, 20, -5, 5], [0, 12, 12, -4, 12, -12], [-22, -13, 13, 0, -9, -9], [0, -7, -7, 8, -15, 15], [0, -14, -14, 12, 3, -3], [-26, -3, 3, 0, 17, 17], [-16, -4, 4, 0, -17, -17], [-4, 5, -5, 0, -4, -4], [0, 2, 2, -64, -2, 2], [0, -12, -12, 22, 0, 0], [-4, 5, -5, 0, -13, -13], [0, -6, -6, 0, -2, 2], [0, -2, -2, -6, -10, 10], [-8, 12, -12, 0, -8, -8], [0, -5, -5, 34, -1, 1], [0, -2, -2, 36, 2, -2], [0, -2, -2, -42, 3, -3], [2, -1, 1, 0, -19, -19], [66, 2, -2, 0, 2, 2], [-16, 0, 0, 0, 10, 10], [0, 9, 9, 4, -7, 7], [-68, 0, 0, 0, 4, 4], [0, 7, 7, 8, -5, 5], [10, 11, -11, 0, -7, -7], [0, 24, 24, -12, -4, 4], [20, 2, -2, 0, 6, 6], [0, -13, -13, 10, 15, -15], [0, 2, 2, -36, -5, 5], [-8, -2, 2, 0, -12, -12], [-18, -6, 6, 0, -12, -12], [10, 0, 0, 0, 20, 20], [0, 4, 4, -48, -8, 8], [0, 0, 0, -44, -1, 1], [-6, 6, -6, 0, 1, 1], [0, -1, -1, -36, 3, -3], [-42, -6, 6, 0, -2, -2], [-8, -16, 16, 0, -4, -4], [0, 8, 8, 34, -2, 2], [-12, -2, 2, 0, -10, -10], [0, 8, 8, 4, 8, -8], [0, 2, 2, 12, 6, -6], [0, -4, -4, 16, 9, -9], [-32, -4, 4, 0, 24, 24], [0, 0, 0, 20, 0, 0], [60, -2, 2, 0, 2, 2], [0, -6, -6, -6, -6, 6], [0, 12, 12, -32, 0, 0], [-2, 12, -12, 0, -12, -12], [-58, -1, 1, 0, -6, -6], [0, 2, 2, -34, 6, -6], [0, 1, 1, -4, 15, -15], [40, -6, 6, 0, 1, 1], [-2, -4, 4, 0, -22, -22], [0, -4, -4, -20, 12, -12], [0, -14, -14, -14, 3, -3], [-28, -7, 7, 0, -1, -1], [30, 12, -12, 0, 13, 13], [-26, -3, 3, 0, -20, -20], [0, -8, -8, 2, 8, -8], [-18, 8, -8, 0, -8, -8], [-36, -12, 12, 0, -8, -8], [0, -2, -2, 46, -4, 4], [-26, -6, 6, 0, 14, 14], [0, -8, -8, -4, -6, 6], [0, 4, 4, 20, 7, -7], [-16, -5, 5, 0, 25, 25], [-4, -11, 11, 0, -3, -3], [0, 22, 22, 24, 2, -2], [0, 9, 9, -4, 9, -9], [0, 6, 6, -2, -2, 2], [-2, 3, -3, 0, -13, -13], [0, -4, -4, -38, -4, 4], [6, -1, 1, 0, -5, -5], [0, 10, 10, -26, -5, 5], [44, -11, 11, 0, 5, 5], [0, -14, -14, 22, 3, -3], [-10, 7, -7, 0, -11, -11], [0, 0, 0, -24, 3, -3], [18, 14, -14, 0, 16, 16], [0, 2, 2, -34, -12, 12], [0, 8, 8, -54, -10, 10], [0, 3, 3, 34, -1, 1], [12, -2, 2, 0, 13, 13], [-8, 16, -16, 0, -8, -8], [0, -8, -8, -8, -3, 3], [8, 12, -12, 0, -4, -4], [-8, 13, -13, 0, -1, -1], [0, 4, 4, 0, -2, 2], [22, 13, -13, 0, 20, 20], [0, -6, -6, -6, -12, 12], [-8, -15, 15, 0, -13, -13], [0, -12, -12, -10, -4, 4], [-44, -2, 2, 0, -11, -11], [0, 27, 27, 4, -5, 5], [-4, 1, -1, 0, 7, 7], [0, 13, 13, 26, 9, -9], [-6, 3, -3, 0, -8, -8], [-14, 16, -16, 0, -1, -1], [0, 9, 9, -10, 5, -5], [0, 20, 20, 18, 5, -5], [0, -14, -14, 18, 8, -8], [-30, 14, -14, 0, -2, -2], [0, 13, 13, -24, 7, -7], [10, 0, 0, 0, -17, -17], [0, 1, 1, -46, 3, -3], [0, 12, 12, -36, -8, 8], [0, -10, -10, 32, -5, 5], [0, -6, -6, -12, -4, 4], [-34, -6, 6, 0, -10, -10], [-32, -16, 16, 0, -18, -18], [-4, -2, 2, 0, 18, 18], [0, -8, -8, -38, 9, -9], [10, -8, 8, 0, -4, -4], [-14, -3, 3, 0, -19, -19], [0, 1, 1, -4, 13, -13], [-2, -10, 10, 0, -13, -13], [40, 8, -8, 0, -8, -8], [0, -4, -4, 54, -10, 10], [74, 6, -6, 0, 10, 10], [0, -12, -12, 44, 8, -8], [36, -4, 4, 0, -9, -9], [0, -12, -12, -8, 5, -5], [0, 16, 16, -4, -16, 16], [0, 0, 0, -4, -7, 7], [16, 11, -11, 0, -21, -21], [0, 0, 0, -18, -9, 9], [24, 8, -8, 0, 16, 16], [0, 20, 20, 10, 4, -4], [0, -4, -4, 64, -2, 2], [0, 21, 21, 4, -1, 1], [2, 18, -18, 0, 4, 4], [0, -6, -6, 28, 1, -1], [-86, -2, 2, 0, 5, 5], [0, -2, -2, -20, 18, -18], [-30, -7, 7, 0, -6, -6], [0, 23, 23, -6, 3, -3], [-4, -6, 6, 0, 18, 18], [62, 2, -2, 0, -14, -14], [-54, 5, -5, 0, 5, 5], [40, 4, -4, 0, 11, 11], [22, 7, -7, 0, 5, 5], [0, 7, 7, 34, 9, -9], [0, -12, -12, -12, -8, 8], [0, 10, 10, 44, 11, -11], [-40, -2, 2, 0, 3, 3], [30, -5, 5, 0, -9, -9], [0, -18, -18, -26, -6, 6], [16, 21, -21, 0, 16, 16], [0, 10, 10, -32, -18, 18], [2, -10, 10, 0, -15, -15], [0, -4, -4, 4, 8, -8], [0, 20, 20, -30, -14, 14], [0, 18, 18, -24, -2, 2], [4, 12, -12, 0, 20, 20], [0, 26, 26, 16, -2, 2], [26, 2, -2, 0, -2, -2], [-30, -9, 9, 0, -16, -16], [0, -1, -1, -4, 11, -11], [0, 24, 24, 12, 7, -7], [0, -9, -9, -4, 9, -9], [10, 2, -2, 0, 0, 0], [0, 12, 12, 54, -2, 2], [0, 2, 2, -54, -1, 1], [-16, 0, 0, 0, 2, 2], [0, 12, 12, 6, -10, 10], [0, -4, -4, 92, 3, -3], [0, 6, 6, -32, -4, 4], [56, 1, -1, 0, -8, -8], [-56, 7, -7, 0, -1, -1], [0, -8, 8, 0, 4, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4680_l_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4680_2_l_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_4680_2_l_e(:prec:=6)
    chi := MakeCharacter_4680_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_4680_2_l_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_4680_2_l_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4680_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![20, 2, -6, 1]>],Snew);
    return Vf;
end function;