
// Make newform 3380.2.f.i in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3380_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3380_f_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_3380_2_f_i();".
// 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_3380_2_f_i();".
// 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, -32, 20, 4, -11, 2, 5, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [2, -10, 6, 1, -4, 1, 2, -1], [-24, 32, -8, -13, 10, 3, -8, 3], [40, -64, 28, 19, -18, -5, 12, -5], [40, -54, 13, 21, -13, -5, 9, -3], [48, -62, 15, 24, -17, -6, 11, -4], [-30, 40, -11, -15, 11, 3, -7, 3], [-80, 90, -11, -35, 19, 11, -15, 5]];
    Rf_basisdens := [1, 2, 8, 8, 4, 4, 2, 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_3380_f();"
function MakeCharacter_3380_f()
    N := 3380;
    order := 2;
    char_gens := [1691, 677, 1861];
    v := [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_3380_f_Hecke();"
function MakeCharacter_3380_f_Hecke(Kf)
    N := 3380;
    order := 2;
    char_gens := [1691, 677, 1861];
    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]];
    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, 1, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, -1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, -1, 0, -1, 0, 0, 0], [0, 0, 0, -2, 0, -1, 1, 1], [1, -1, 1, 0, -1, 0, 0, 0], [0, 1, 0, 0, -2, 0, 0, 0], [0, 0, 0, 4, 0, 2, -2, 0], [0, 0, 0, -3, 0, 3, 2, 0], [0, 0, 0, 0, 0, -6, 1, 0], [-2, 2, -1, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0], [5, -3, -4, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, 2, 3, 2], [2, 1, -2, 0, 4, 0, 0, 0], [0, 0, 0, -1, 0, -5, -4, 0], [0, 0, 0, 0, 0, -3, 3, -3], [0, 0, 0, 2, 0, 7, -2, 1], [-3, -1, 2, 0, -6, 0, 0, 0], [0, 0, 0, 2, 0, 10, 0, -2], [0, 0, 0, -2, 0, 2, -1, 2], [0, 0, 0, -3, 0, -3, 4, -2], [2, 1, 2, 0, -2, 0, 0, 0], [3, 1, 4, 0, -6, 0, 0, 0], [-2, 2, 1, 0, -7, 0, 0, 0], [0, 0, 0, -4, 0, -5, 2, 3], [-3, -1, -3, 0, 5, 0, 0, 0], [-9, -1, -1, 0, -3, 0, 0, 0], [1, 3, -2, 0, 0, 0, 0, 0], [0, 0, 0, 7, 0, 2, -6, -1], [-6, 1, 4, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, -10, -3, 2], [0, 0, 0, 0, 0, 3, 0, -1], [3, 1, -2, 0, -4, 0, 0, 0], [0, 0, 0, 1, 0, 8, 2, 1], [0, 0, 0, 7, 0, 2, -8, -1], [4, -2, -5, 0, 1, 0, 0, 0], [5, 4, 2, 0, 4, 0, 0, 0], [-3, 1, 10, 0, -4, 0, 0, 0], [-12, 1, 6, 0, -2, 0, 0, 0], [0, 0, 0, 3, 0, 15, -4, -2], [0, 0, 0, -3, 0, -9, -6, 0], [-3, 2, 2, 0, -10, 0, 0, 0], [2, 1, -2, 0, 0, 0, 0, 0], [0, 0, 0, -5, 0, -4, 0, 3], [0, 0, 0, 1, 0, 2, 2, 5], [0, 0, 0, 4, 0, 8, -4, 4], [-15, -1, 6, 0, -4, 0, 0, 0], [0, 0, 0, -8, 0, -4, 6, 8], [0, 0, 0, 0, 0, 3, -7, 5], [-11, 4, 4, 0, -2, 0, 0, 0], [-20, 0, 1, 0, -3, 0, 0, 0], [2, 2, -7, 0, -1, 0, 0, 0], [-15, 4, 6, 0, -2, 0, 0, 0], [0, 0, 0, 4, 0, 11, -5, -1], [-4, 0, -3, 0, 3, 0, 0, 0], [0, 0, 0, 8, 0, 10, -4, -2], [4, 6, 3, 0, -1, 0, 0, 0], [0, 0, 0, -1, 0, -2, 2, 1], [0, 0, 0, 0, 0, -3, -6, 1], [-5, 5, -2, 0, -4, 0, 0, 0], [-6, 2, 2, 0, -4, 0, 0, 0], [0, 0, 0, -2, 0, 2, 2, -4], [0, 0, 0, 4, 0, 17, -1, -1], [9, 5, -4, 0, 8, 0, 0, 0], [11, 3, -7, 0, 9, 0, 0, 0], [0, 0, 0, 6, 0, -3, 5, 3], [0, 0, 0, 1, 0, -4, 10, -1], [0, 0, 0, -4, 0, -8, 6, 4], [12, 4, 1, 0, 1, 0, 0, 0], [-17, 3, 3, 0, -3, 0, 0, 0], [0, 0, 0, 6, 0, 3, 9, 1], [0, 0, 0, -13, 0, 1, 4, -2], [-9, -1, -6, 0, 14, 0, 0, 0], [0, 0, 0, -3, 0, 6, -8, -5], [0, 0, 0, 0, 0, -3, 3, 3], [0, 0, 0, -2, 0, 2, -3, 0], [-11, -4, -2, 0, -4, 0, 0, 0], [0, 0, 0, -8, 0, -22, 4, 2], [0, 0, 0, -4, 0, 22, 1, 4], [-6, -4, -1, 0, 5, 0, 0, 0], [15, -2, -2, 0, -4, 0, 0, 0], [-3, -3, -6, 0, 6, 0, 0, 0], [0, 0, 0, -2, 0, -10, 5, 2], [0, 0, 0, -1, 0, -8, -12, 1], [0, 0, 0, 4, 0, 2, -1, -4], [0, 0, 0, 2, 0, -5, 0, -5], [10, 0, 10, 0, -6, 0, 0, 0], [0, 0, 0, -10, 0, -8, 3, -2], [0, 0, 0, 3, 0, 3, 8, 2], [1, -10, -8, 0, 8, 0, 0, 0], [0, 0, 0, 0, 0, -12, 2, -2], [-2, 0, 1, 0, -9, 0, 0, 0], [0, 0, 0, 14, 0, 16, -1, -2], [-4, -6, -16, 0, 12, 0, 0, 0], [-16, 0, 9, 0, 7, 0, 0, 0], [0, 0, 0, -4, 0, 19, -2, -3], [29, -3, -6, 0, -6, 0, 0, 0], [0, 0, 0, 1, 0, -4, 2, -7], [23, 1, -1, 0, 7, 0, 0, 0], [26, 7, -4, 0, 10, 0, 0, 0], [-8, 0, 0, 0, 2, 0, 0, 0], [0, 0, 0, 10, 0, -19, -14, -5], [0, 0, 0, -1, 0, 13, -16, -2], [0, 0, 0, -2, 0, -1, 2, 5], [5, -5, 2, 0, 10, 0, 0, 0], [-20, 3, 6, 0, 12, 0, 0, 0], [2, 0, -3, 0, -15, 0, 0, 0], [0, 0, 0, 7, 0, -1, 0, 2], [0, 0, 0, -13, 0, -2, 10, 7], [0, 0, 0, 0, 0, -12, 2, -8], [0, 0, 0, 16, 0, 11, -5, -3], [-15, -2, 12, 0, -8, 0, 0, 0], [0, 0, 0, -5, 0, 26, -4, -1], [-20, -2, 7, 0, 1, 0, 0, 0], [3, -3, 9, 0, 3, 0, 0, 0], [-1, 0, 2, 0, 12, 0, 0, 0], [0, 0, 0, -12, 0, 6, 9, 0], [2, 0, 15, 0, -9, 0, 0, 0], [7, -3, 10, 0, 0, 0, 0, 0], [0, 0, 0, 5, 0, -8, -6, 1], [0, 0, 0, 8, 0, 13, 7, -5], [24, 2, 0, 0, 8, 0, 0, 0], [0, 0, 0, -2, 0, 8, 17, -2], [15, -8, -18, 0, 4, 0, 0, 0], [-7, 1, -2, 0, -14, 0, 0, 0], [0, 0, 0, -2, 0, -28, 6, 2], [0, 0, 0, 10, 0, 2, -5, 2], [0, 0, 0, 1, 0, 26, 2, 5], [0, 7, -2, 0, 6, 0, 0, 0], [4, -6, -9, 0, 17, 0, 0, 0], [0, 0, 0, -12, 0, -27, 5, -3], [0, 0, 0, -2, 0, -13, 9, 7], [0, 0, 0, 13, 0, -4, -16, -1], [0, 0, 0, 1, 0, -19, -10, 0], [-19, -3, -13, 0, 15, 0, 0, 0], [-11, 4, -2, 0, 10, 0, 0, 0], [0, 0, 0, -12, 0, -18, 14, 2], [0, 0, 0, 0, 0, 3, -9, 3], [-11, 5, 5, 0, -13, 0, 0, 0], [0, 0, 0, -4, 0, -17, 6, -5], [-3, 8, 14, 0, -18, 0, 0, 0], [0, 0, 0, 14, 0, 4, -9, 4], [0, 0, 0, 4, 0, 5, 8, 7], [8, 0, 2, 0, 12, 0, 0, 0], [-11, 3, -6, 0, -12, 0, 0, 0], [0, 0, 0, -16, 0, -17, 6, -5], [0, 0, 0, 1, 0, 2, 4, 7], [11, 4, 8, 0, 4, 0, 0, 0], [8, 4, -8, 0, 10, 0, 0, 0], [3, -3, 9, 0, -15, 0, 0, 0], [-15, -7, 11, 0, -15, 0, 0, 0], [-10, -2, -4, 0, 10, 0, 0, 0], [-16, 7, -8, 0, -6, 0, 0, 0], [0, 0, 0, 10, 0, 5, -5, -7], [-9, 5, 14, 0, -16, 0, 0, 0], [0, 0, 0, -4, 0, -5, 2, 13], [0, 0, 0, -7, 0, -32, 6, 1], [-5, 13, 13, 0, 1, 0, 0, 0], [0, 0, 0, -6, 0, -24, -4, 0], [-17, -2, 4, 0, -2, 0, 0, 0], [0, 0, 0, -1, 0, 7, -16, -8], [0, 0, 0, 6, 0, 3, -20, -9], [24, -5, -14, 0, 4, 0, 0, 0], [3, 13, 13, 0, -9, 0, 0, 0], [0, 0, 0, -4, 0, 16, 0, 6], [-35, -1, -2, 0, 8, 0, 0, 0], [0, 0, 0, 0, 0, 33, 8, -3], [0, 0, 0, -16, 0, 7, 1, -1], [4, -3, 10, 0, 6, 0, 0, 0], [0, 0, 0, -3, 0, 0, -4, 11], [-8, -4, 8, 0, -16, 0, 0, 0], [3, -2, 6, 0, -2, 0, 0, 0], [0, 0, 0, 12, 0, 9, -9, -11], [0, 0, 0, 0, 0, 15, -10, -9], [29, -3, -21, 0, 3, 0, 0, 0], [-6, -13, -16, 0, 10, 0, 0, 0], [0, 0, 0, 20, 0, 22, -2, -6], [10, -14, -8, 0, 4, 0, 0, 0], [-8, 0, 6, 0, -16, 0, 0, 0], [0, 0, 0, 4, 0, 17, 0, -1], [0, 0, 0, 11, 0, -8, -8, 7], [-2, -1, -8, 0, -4, 0, 0, 0], [8, 0, 6, 0, 0, 0, 0, 0], [0, 0, 0, -6, 0, -18, 8, 14], [0, 0, 0, 2, 0, 4, 3, -10], [0, 0, 0, -4, 0, 19, 7, 1], [-8, 2, 11, 0, -3, 0, 0, 0], [0, 0, 0, -1, 0, 4, -4, 13], [19, 5, 14, 0, -18, 0, 0, 0], [0, 0, 0, -4, 0, -20, 13, 10], [-20, -2, -11, 0, 1, 0, 0, 0], [-25, 5, 5, 0, -13, 0, 0, 0], [0, 0, 0, 4, 0, -31, -14, -1], [-6, -4, -1, 0, 9, 0, 0, 0], [0, 0, 0, -8, 0, -25, -12, 5], [-13, 5, -4, 0, 14, 0, 0, 0], [0, 0, 0, -6, 0, 3, 7, -9], [17, 0, 0, 0, -12, 0, 0, 0], [0, 0, 0, -1, 0, -14, -4, -5], [-3, 1, -2, 0, -12, 0, 0, 0], [0, 0, 0, 2, 0, -5, 5, 1], [0, 4, -15, 0, -11, 0, 0, 0], [0, 0, 0, 0, 0, 9, 8, -1], [17, -3, 5, 0, 15, 0, 0, 0], [0, 0, 0, -10, 0, -29, 9, 13], [9, -4, -4, 0, -14, 0, 0, 0], [-25, 1, 13, 0, -3, 0, 0, 0], [-5, 7, 10, 0, 10, 0, 0, 0], [-23, 3, 9, 0, -13, 0, 0, 0], [0, 0, 0, -7, 0, -23, -2, 4], [0, 0, 0, 10, 0, -19, -23, -1], [0, 0, 0, 8, 0, 22, -20, -2], [6, 2, 8, 0, -22, 0, 0, 0], [4, -5, -14, 0, 22, 0, 0, 0], [0, 0, 0, -3, 0, -15, -2, 6], [0, 0, 0, -22, 0, -5, 14, 13], [11, 0, 0, 0, 6, 0, 0, 0], [0, 0, 0, 0, 0, 24, 6, 4], [0, 0, 0, 16, 0, 23, -14, -7], [0, 0, 0, -7, 0, -35, 10, -2], [18, -4, -9, 0, 5, 0, 0, 0], [-12, -2, -8, 0, -8, 0, 0, 0], [9, 11, 9, 0, -13, 0, 0, 0], [5, 2, -22, 0, 2, 0, 0, 0], [2, 4, -11, 0, -15, 0, 0, 0], [0, 0, 0, -12, 0, -3, 20, 3], [1, -1, -7, 0, -1, 0, 0, 0], [-12, 1, -8, 0, 6, 0, 0, 0], [0, 0, 0, 0, 0, -9, 11, -5], [17, -7, -10, 0, -10, 0, 0, 0], [-13, -3, 12, 0, -30, 0, 0, 0], [0, 0, 0, 4, 0, 23, -10, -1], [0, 0, 0, 14, 0, 22, -19, 10], [0, 0, 0, -11, 0, -7, 2, -4], [0, 3, 0, 0, 6, 0, 0, 0], [-6, 1, -6, 0, -14, 0, 0, 0], [0, 0, 0, -21, 0, -39, 12, 0], [-1, -8, -8, 0, -8, 0, 0, 0], [3, 11, 17, 0, -13, 0, 0, 0], [0, 0, 0, 6, 0, -15, -7, -11], [0, 0, 0, -19, 0, -17, -8, -2], [8, 0, 8, 0, 0, 0, 0, 0], [0, 0, 0, -15, 0, 6, 14, 13], [0, 0, 0, -12, 0, 3, 11, -9], [0, 0, 0, -14, 0, 23, 5, 3], [21, -5, -15, 0, 7, 0, 0, 0], [0, 0, 0, 1, 0, -13, -6, 8], [0, 0, 0, -4, 0, 19, -21, -5], [0, 0, 0, -20, 0, 5, 14, 17], [15, 2, -22, 0, 8, 0, 0, 0], [29, 3, 14, 0, -24, 0, 0, 0], [0, 0, 0, -2, 0, 17, 13, -1], [21, -10, 8, 0, 2, 0, 0, 0], [0, 0, 0, 5, 0, -17, -10, 4], [15, -5, -24, 0, 4, 0, 0, 0], [0, 0, 0, 23, 0, -5, -10, 4], [33, -13, -4, 0, 2, 0, 0, 0], [16, -6, -17, 0, 9, 0, 0, 0], [0, 0, 0, 4, 0, 8, -12, -4], [20, 12, 9, 0, 3, 0, 0, 0], [0, 0, 0, -1, 0, 1, 10, -14], [10, -15, 0, 0, 6, 0, 0, 0], [0, 0, 0, -6, 0, -33, 11, 9], [0, 0, 0, -20, 0, 20, 16, 8], [0, 0, 0, -13, 0, 1, -16, -2], [16, -4, -11, 0, -7, 0, 0, 0], [-2, 0, 0, 0, -18, 0, 0, 0], [0, 0, 0, -4, 0, 7, 6, 7], [0, 0, 0, -15, 0, -6, 30, -3], [1, 2, 2, 0, 2, 0, 0, 0], [12, -10, -1, 0, -1, 0, 0, 0], [0, 0, 0, 9, 0, 0, -18, -1], [0, 0, 0, 5, 0, -14, -10, 1], [0, 0, 0, -16, 0, -2, 16, -10], [0, 0, 0, 0, 0, 39, 5, -7], [-31, 4, 8, 0, -14, 0, 0, 0], [0, -4, -3, 0, -13, 0, 0, 0], [0, 0, 0, 8, 0, 43, -13, 7], [-4, 2, -10, 0, 14, 0, 0, 0], [0, 0, 0, 4, 0, -34, 5, -2], [0, 0, 0, -32, 0, -13, 26, 3], [22, 6, 16, 0, -24, 0, 0, 0], [37, 3, -30, 0, 12, 0, 0, 0], [0, 0, 0, 6, 0, 15, 14, -3], [0, 0, 0, -26, 0, -19, 21, 5], [-14, -9, 4, 0, -6, 0, 0, 0], [-32, -5, -2, 0, -14, 0, 0, 0], [-24, -6, 9, 0, -3, 0, 0, 0], [0, 0, 0, 7, 0, 11, 24, -10], [0, 0, 0, 16, 0, 17, -9, -1], [-9, -5, -17, 0, 1, 0, 0, 0], [-14, 4, -20, 0, 4, 0, 0, 0], [35, -3, -18, 0, -8, 0, 0, 0], [-26, -8, 7, 0, 7, 0, 0, 0], [-8, 13, 4, 0, -2, 0, 0, 0], [0, 0, 0, 11, 0, 1, -8, -2], [-6, -2, 1, 0, -3, 0, 0, 0], [0, 0, 0, 26, 0, 22, -18, 4], [-13, 6, 6, 0, -18, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3380_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3380_2_f_i();".
// 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_3380_2_f_i(:prec:=8)
    chi := MakeCharacter_3380_f();
    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_3380_2_f_i();".
// 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_3380_2_f_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3380_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 10, -6, -2, 1]>,<19,R![33, 0, 30, 0, 1]>],Snew);
    return Vf;
end function;