
// Make newform 1120.2.bw.c in Magma, downloaded from the LMFDB on 01 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1120_bw();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1120_bw_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_1120_2_bw_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_1120_2_bw_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, -3, 0, 8, 0, -3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [-3, 4, 8, 0, -4, 0, 1, 0], [0, -13, 0, 0, 0, 0, 0, -1], [3, 4, -8, 0, 4, 0, -1, 0], [9, 0, -24, 0, 8, 0, -3, 0], [-13, -1, 0, 24, 0, -8, -1, 3], [1, 9, -16, -24, 8, 8, -3, -3], [0, 15, 0, -40, 0, 16, 0, -5]];
    Rf_basisdens := [1, 4, 4, 4, 8, 8, 8, 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_1120_bw();"
function MakeCharacter_1120_bw()
    N := 1120;
    order := 6;
    char_gens := [351, 421, 897, 801];
    v := [6, 6, 3, 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_1120_bw_Hecke();"
function MakeCharacter_1120_bw_Hecke(Kf)
    N := 1120;
    order := 6;
    char_gens := [351, 421, 897, 801];
    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], [0, 0, 0, 0, -1, 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, 1, 0, -1, -1, 0, 1, 0], [0, -1, 0, 1, 1, 0, 0, 0], [2, -1, 0, 0, -1, 1, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, 0, 1, 3, 3, 0], [-5, 4, -1, 2, 0, -2, 2, 1], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, -1, 0, 1, 1, -3], [-4, -1, 2, 1, 10, 1, 0, 2], [0, -3, -2, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 2, 3, 3, 0, 0, 4], [-4, 0, 0, 5, 1, -5, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-7, -3, -2, 0, 14, 0, 3, -2], [-8, 4, 2, -4, -3, -8, 8, 4], [0, 0, 0, 0, 0, 0, 0, 0], [-11, -2, 2, -4, 9, -2, 2, 1], [-9, -5, -1, -5, 7, 5, 5, 0], [-13, 3, -4, 3, 5, -3, -3, 0], [0, 3, -8, 0, -3, -3, -3, -4], [0, 0, 0, 0, 0, 0, 0, 0], [7, 0, -2, 7, 0, 7, -7, -2], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 1, -6, -1, -1, 0, 0, -12], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -7, 8, -7, 0, 0, 0, 0], [7, -3, 5, 0, -14, 0, 3, 5], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [7, -12, 3, -6, 0, 6, -6, -3], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -10, -1, 0, -1, 1, -10], [9, -9, 0, 0, 0, 9, -9, -8], [0, 0, 0, 0, -14, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, -6, -6, -6, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-10, 0, 0, 9, -1, -9, 0, 5], [0, -5, -12, 0, 5, 5, 5, -6], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -14, 0, 14, 14, 0], [5, -5, 0, 0, 0, 5, -5, 8], [0, 0, 0, 0, 0, 0, 0, 0], [-7, -11, 2, 0, 14, 0, 11, 2], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [4, 0, 0, 9, 13, -9, 0, -2], [7, 12, -3, 6, 0, -6, 6, 3], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [13, 1, 0, -1, 13, 0, 1, 7], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-19, 0, 7, 0, 5, -9, -9, 0], [0, -14, 0, 0, 14, 14, 14, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-16, 8, 4, -8, 9, -16, 16, 8], [0, 9, 2, 0, -9, -9, -9, 1], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 2, 9, 0, -9, -9, -2], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [29, 0, -8, 0, -13, 3, 3, 0], [0, 0, 9, -5, 0, 5, 5, -9], [0, 0, 0, 0, 0, 0, 0, 0], [-42, 0, 0, 0, 0, 0, 0, 0], [11, 13, -1, 0, -22, 0, -13, -1], [23, -9, 8, -9, -7, 9, 9, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-7, 7, 0, 0, 0, -7, 7, -10], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [14, 3, -7, -3, -34, -3, 0, -7], [-20, 10, 5, -10, 3, -20, 20, 10], [-42, 0, 0, 0, 42, 0, 0, 0], [0, -11, 16, -11, 0, 0, 0, 0], [-12, 6, 3, -6, 19, -12, 12, 6], [17, 1, 8, 0, -34, 0, -1, 8], [0, 0, 0, 0, 0, 0, 0, 0], [8, 0, 0, -15, -7, 15, 0, -4], [0, -14, 0, 14, 14, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [15, 0, -4, 15, 0, 15, -15, -4], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 18, 0, -18, -18, 0], [33, 0, -7, 0, -19, -5, -5, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -11, 6, 0, 11, 11, 11, 3], [-13, 0, -4, -13, 0, -13, 13, -4], [1, -15, 0, 15, 1, 0, -15, -7], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -9, -16, 0, 9, 9, 9, -8], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [11, 9, 0, -9, 11, 0, 9, 10], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 12, 5, 0, -5, -5, -12], [12, -6, -3, 6, 23, 12, -12, -6], [0, 0, 0, 0, 0, 0, 0, 0], [7, 17, -5, 0, -14, 0, -17, -5], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-17, -15, -1, 0, 34, 0, 15, -1], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -42, 0, 0, 0], [-14, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-20, 0, 0, 13, -7, -13, 0, 10], [0, 0, 0, 0, 0, 0, 0, 0], [-23, 4, -4, 8, 27, 4, -4, -2], [0, 0, 0, 0, 0, 0, 0, 0], [0, 14, 0, -14, -14, 0, 0, 0], [17, 5, 0, -5, 17, 0, 5, 11], [4, 15, -2, -15, -38, -15, 0, -2], [0, -6, 0, 0, 6, 6, 6, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-41, 3, -11, 3, 19, -3, -3, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -3, -13, 0, 13, 13, 3], [7, 0, 16, 7, 0, 7, -7, 16], [37, 0, -7, 0, -23, -9, -9, 0], [11, -19, 0, 19, 11, 0, -19, -4], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 11, 9, -11, -11, 0, 0, 18], [0, 0, 0, 0, 0, 0, 0, 0], [42, 0, 0, 0, -42, 0, 0, 0], [43, 3, 10, 3, -23, -3, -3, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-10, -13, 5, 13, 46, 13, 0, 5], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [22, 0, 0, -15, 7, 15, 0, -11], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-5, -24, 6, -12, 0, 12, -12, -6], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [35, -12, 3, -6, 0, 6, -6, -3], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [35, -16, 4, -8, 0, 8, -8, -4], [0, 0, 0, 0, 0, 0, 0, 0], [18, 0, 0, 0, -18, 0, 0, 0], [14, 11, -7, -11, -50, -11, 0, -7], [0, -3, 16, 3, 3, 0, 0, 32], [27, -17, 11, -17, -5, 17, 17, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [21, -21, 0, 0, 0, 21, -21, -10], [-31, 2, -2, 4, 33, 2, -2, -1], [0, 0, 0, 0, 0, 0, 0, 0], [-19, 19, 0, 0, 0, -19, 19, 20], [0, 15, 1, -15, -15, 0, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-21, 0, 16, -21, 0, -21, 21, 16], [0, 0, 0, 0, 0, 0, 0, 0], [0, -11, 12, 11, 11, 0, 0, 24], [21, -21, 0, 0, 0, 21, -21, -4], [0, 0, 0, 0, 0, 0, 0, 0], [0, -3, -34, 0, 3, 3, 3, -17], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 26, -3, 0, 0, 0, 0], [37, 10, -10, 20, -27, 10, -10, -5], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-11, 12, -12, 24, 23, 12, -12, -6], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 21, -20, 21, 0, 0, 0, 0], [19, -32, 8, -16, 0, 16, -16, -8], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 18, 1, 0, -1, -1, -18], [-25, 1, -13, 0, 50, 0, -1, -13], [-47, 0, 14, 0, 19, -9, -9, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -30, 0, 30, 30, 0, 0, 0], [10, 0, 0, -23, -13, 23, 0, -5], [25, 16, -16, 32, -9, 16, -16, -8], [0, 21, -22, 21, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -15, -8, 15, 15, 0, 0, -16], [0, 0, 0, 0, 0, 0, 0, 0], [16, -8, -4, 8, 33, 16, -16, -8], [0, -7, -22, -7, 0, 0, 0, 0], [-28, 3, 14, -3, 50, -3, 0, 14], [0, 0, 16, 9, 0, -9, -9, -16], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-33, 19, -13, 19, 7, -19, -19, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 17, 6, 0, -17, -17, -17, 3], [28, -17, -14, 17, -22, 17, 0, -14], [29, -15, 0, 15, 29, 0, -15, 7], [0, 3, 19, -3, -3, 0, 0, 38], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [21, -21, 0, 0, 0, 21, -21, 4], [0, 0, 0, 0, 0, 0, 0, 0], [-23, 23, 0, 0, 0, -23, 23, 22], [-12, 6, 3, -6, -37, -12, 12, 6], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [14, -25, -7, 25, 22, 25, 0, -7], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-21, 0, 26, -21, 0, -21, 21, 26], [0, 0, 0, 0, 0, 0, 0, 0], [8, -4, -2, 4, -39, 8, -8, -4], [-31, 9, 0, -9, -31, 0, 9, -11], [0, 0, 11, 15, 0, -15, -15, -11], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-19, -18, 18, -36, 1, -18, 18, 9], [-29, 1, 0, -1, -29, 0, 1, -14], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-10, 0, 0, -19, -29, 19, 0, 5], [-31, -15, -8, 0, 62, 0, 15, -8], [-14, 0, 0, 0, 14, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [54, 0, 0, 0, 0, 0, 0, 0], [42, 0, 0, 0, 0, 0, 0, 0], [32, -13, -16, 13, -38, 13, 0, -16], [0, 0, 0, 0, 0, 0, 0, 0], [-36, 18, 9, -18, 11, -36, 36, 18], [-7, 7, 0, 0, 0, -7, 7, 32], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 25, -2, 25, 0, 0, 0, 0], [0, 21, 8, 21, 0, 0, 0, 0], [5, 18, -18, 36, 13, 18, -18, -9], [0, 15, -13, -15, -15, 0, 0, -26], [0, 0, 0, 0, 0, 0, 0, 0], [13, -27, 0, 27, 13, 0, -27, -7], [0, 0, 0, 0, 0, 0, 0, 0], [27, 0, -10, 27, 0, 27, -27, -10], [35, 12, -3, 6, 0, -6, 6, 3], [33, 0, -14, 0, -5, 23, 23, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 10, 0, -10, -10, 0], [0, 11, 36, 0, -11, -11, -11, 18], [20, 0, 0, -27, -7, 27, 0, -10], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 19, 12, 0, -19, -19, -19, 6], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 32, -1, 0, -1, 1, 32], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1120_bw_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1120_2_bw_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_1120_2_bw_c(:prec:=8)
    chi := MakeCharacter_1120_bw();
    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_1120_2_bw_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_1120_2_bw_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1120_bw();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![121, 462, 654, 252, -37, -36, 6, 6, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;