
// Make newform 2240.2.g.n in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2240_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2240_g_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_2240_2_g_n();".
// 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_2240_2_g_n();".
// 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 := [4, 0, 61, 0, 97, 0, 56, 0, 13, 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, 20, 0, 55, 0, 44, 0, 12, 0, 1], [-2, 0, -107, 0, -112, 0, -34, 0, -3, 0], [-12, 0, -111, 0, -112, 0, -34, 0, -3, 0], [0, 16, -97, -93, -110, -110, -34, -34, -3, -3], [0, -13, 0, -82, 0, -78, 0, -23, 0, -2], [0, 16, 97, -93, 110, -110, 34, -34, 3, -3], [0, 29, 0, 137, 0, 122, 0, 35, 0, 3], [20, -14, 203, 151, 198, 178, 58, 56, 5, 5], [-20, -14, -203, 151, -198, 178, -58, 56, -5, 5]];
    Rf_basisdens := [1, 2, 2, 2, 4, 2, 4, 2, 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_2240_g();"
function MakeCharacter_2240_g()
    N := 2240;
    order := 2;
    char_gens := [1471, 1541, 897, 1921];
    v := [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_2240_g_Hecke();"
function MakeCharacter_2240_g_Hecke(Kf)
    N := 2240;
    order := 2;
    char_gens := [1471, 1541, 897, 1921];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [-1, 0, 1, 0, 0, 0, 0, 0, -1, 1], [0, 2, 0, 0, 0, -1, 0, -1, 0, 0], [0, 0, 0, 0, 0, -1, 0, 1, -1, -1], [2, 0, 0, 0, -1, 0, 1, 0, 0, 0], [0, 0, 0, 0, -1, 2, -1, 0, 1, 1], [-2, 0, -2, -1, -1, 0, 1, 0, 0, 0], [2, 0, 2, 2, 2, 0, -2, 0, 0, 0], [0, 0, 0, 0, -1, 2, -1, 0, -1, -1], [0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 2, 2, -2, 2, 2], [0, -1, 0, 0, 0, -2, 0, 0, 1, 1], [0, 2, 0, 0, 1, -2, 1, 0, 1, 1], [4, 0, -2, -2, -1, 0, 1, 0, -2, 2], [2, 0, -2, 0, -1, 0, 1, 0, 0, 0], [0, -2, 0, 0, 2, 0, 2, 0, 0, 0], [0, 0, -2, 2, -1, 0, 1, 0, -1, 1], [0, -2, 0, 0, -1, -2, -1, 0, -1, -1], [-7, 0, 1, 2, 1, 0, -1, 0, 0, 0], [0, -2, 0, 0, -2, 6, -2, 4, -1, -1], [0, 0, -2, -2, -2, 0, 2, 0, 2, -2], [0, 0, 0, 0, -2, 3, -2, 1, 1, 1], [-2, 0, 2, 4, 1, 0, -1, 0, -2, 2], [0, 1, 0, 0, -2, 6, -2, 0, 1, 1], [0, 4, 0, 0, -2, -4, -2, 0, 2, 2], [-2, 0, 2, 1, -1, 0, 1, 0, -2, 2], [0, -4, 0, 0, 1, 2, 1, 0, 1, 1], [0, 4, 0, 0, -1, -4, -1, -2, -1, -1], [0, 0, -2, 4, 1, 0, -1, 0, 2, -2], [0, -2, 0, 0, 4, -4, 4, 0, -2, -2], [12, 0, 2, -2, 1, 0, -1, 0, -2, 2], [-2, 0, 0, 0, 0, 0, 0, 0, 4, -4], [7, 0, 1, 0, 0, 0, 0, 0, -3, 3], [0, 4, 0, 0, -3, -4, -3, 0, 2, 2], [0, 2, 0, 0, 0, 4, 0, -4, 0, 0], [0, -1, 0, 0, 2, -2, 2, 4, -1, -1], [0, 2, 0, 0, 0, 5, 0, -3, 4, 4], [8, 0, 0, 0, 4, 0, -4, 0, 0, 0], [-8, 0, 0, 6, 1, 0, -1, 0, 2, -2], [1, 0, 1, -2, -3, 0, 3, 0, 0, 0], [0, 0, 0, 0, 1, 8, 1, 2, -1, -1], [0, 4, 0, 0, 1, -2, 1, -4, -1, -1], [-10, 0, 2, 0, 0, 0, 0, 0, 0, 0], [-7, 0, -1, -8, 0, 0, 0, 0, 3, -3], [0, -3, 0, 0, 4, 6, 4, 0, -3, -3], [0, -1, 0, 0, -4, 0, -4, 0, 0, 0], [2, 0, -2, -2, -3, 0, 3, 0, 0, 0], [0, 4, 0, 0, -2, -8, -2, 4, 2, 2], [-15, 0, -1, -4, -4, 0, 4, 0, -1, 1], [-4, 0, -2, 0, 2, 0, -2, 0, 2, -2], [-4, 0, -2, 2, 3, 0, -3, 0, -2, 2], [0, -2, 0, 0, -3, 10, -3, 0, -3, -3], [0, 0, 0, 0, 4, 8, 4, 0, 0, 0], [8, 0, 0, 6, 3, 0, -3, 0, 2, -2], [12, 0, -4, -6, 0, 0, 0, 0, 6, -6], [0, 6, 0, 0, 1, 6, 1, 0, 1, 1], [6, 0, -2, 3, 1, 0, -1, 0, 0, 0], [0, -1, 0, 0, 0, 8, 0, 4, -2, -2], [0, -6, 0, 0, -4, -7, -4, 5, -6, -6], [0, 1, 0, 0, 0, 0, 0, 4, -6, -6], [0, 0, 0, 6, -4, 0, 4, 0, -2, 2], [0, 4, 0, 0, 4, -5, 4, -3, -1, -1], [0, 2, 0, 0, 1, -4, 1, 2, -5, -5], [-8, 0, 4, 8, 2, 0, -2, 0, 0, 0], [0, 2, 0, 0, -3, -10, -3, 0, -1, -1], [0, -2, 0, 0, -4, -10, -4, 2, -2, -2], [2, 0, -6, 0, -1, 0, 1, 0, 2, -2], [0, 8, 0, 0, -4, -19, -4, -1, -1, -1], [-4, 0, -4, 4, -4, 0, 4, 0, 4, -4], [0, -5, 0, 0, -2, -6, -2, 4, -7, -7], [0, -4, 0, 0, 3, 0, 3, -2, -5, -5], [4, 0, -4, 4, -4, 0, 4, 0, 2, -2], [0, 4, 0, 0, 2, 8, 2, 0, -4, -4], [2, 0, -2, -7, -5, 0, 5, 0, -6, 6], [0, 2, 0, 0, 0, 3, 0, 3, 0, 0], [6, 0, -4, -1, -6, 0, 6, 0, -3, 3], [10, 0, -4, -2, 0, 0, 0, 0, 8, -8], [8, 0, -6, -10, -1, 0, 1, 0, 2, -2], [2, 0, -6, -5, 1, 0, -1, 0, 0, 0], [-11, 0, 3, 4, 0, 0, 0, 0, -1, 1], [0, 10, 0, 0, -1, 2, -1, 0, 1, 1], [-14, 0, -2, 0, 0, 0, 0, 0, -2, 2], [0, -8, 0, 0, 0, 14, 0, 2, 2, 2], [-2, 0, -2, 5, -3, 0, 3, 0, -2, 2], [0, -8, 0, 0, -1, 2, -1, 4, 3, 3], [-4, 0, 0, -4, 3, 0, -3, 0, 6, -6], [0, -4, 0, 0, -2, 14, -2, -2, -4, -4], [0, -7, 0, 0, 4, -4, 4, -4, 0, 0], [6, 0, -6, -10, 0, 0, 0, 0, 2, -2], [0, -8, 0, 0, -3, 12, -3, 2, -3, -3], [5, 0, -1, -12, -2, 0, 2, 0, 1, -1], [1, 0, 3, 0, -2, 0, 2, 0, 5, -5], [0, -9, 0, 0, 0, 14, 0, 4, 3, 3], [-4, 0, 4, 6, 1, 0, -1, 0, -6, 6], [-4, 0, 6, -2, 4, 0, -4, 0, -2, 2], [0, -6, 0, 0, 6, 10, 6, 4, -3, -3], [2, 0, 2, -7, -5, 0, 5, 0, 0, 0], [0, 4, 0, 0, 2, 6, 2, 2, -6, -6], [0, 2, 0, 0, 7, 4, 7, -2, 5, 5], [0, 2, 0, 0, 2, 18, 2, -8, 3, 3], [2, 0, 6, 2, 5, 0, -5, 0, -1, 1], [-12, 0, -4, -8, -4, 0, 4, 0, -2, 2], [0, -8, 0, 0, -4, 9, -4, 3, -5, -5], [0, 10, 0, 0, -2, -18, -2, 0, 1, 1], [0, 4, 0, 0, -2, -5, -2, 5, -5, -5], [7, 0, 7, 2, 3, 0, -3, 0, -4, 4], [-14, 0, 0, 6, 2, 0, -2, 0, 4, -4], [0, -11, 0, 0, 0, -14, 0, 4, -7, -7], [0, 10, 0, 0, 5, -16, 5, -6, 1, 1], [0, 2, 0, 0, -1, -2, -1, -4, 5, 5], [0, 0, 6, 4, 5, 0, -5, 0, -2, 2], [1, 0, 1, 10, 5, 0, -5, 0, -4, 4], [2, 0, -2, -14, -3, 0, 3, 0, 7, -7], [0, 1, 0, 0, 8, -12, 8, -4, -4, -4], [0, 0, 0, 0, 2, -8, 2, -8, 0, 0], [0, -4, 0, 0, 3, 24, 3, 2, 1, 1], [-7, 0, 3, 0, 6, 0, -6, 0, -5, 5], [8, 0, 0, -8, 3, 0, -3, 0, 0, 0], [0, 2, 0, 0, 0, 16, 0, -4, 2, 2], [0, 10, 0, 0, -2, 1, -2, 1, 4, 4], [0, 12, 0, 0, 0, 10, 0, -10, 6, 6], [-6, 0, 0, 6, 3, 0, -3, 0, 4, -4], [18, 0, 2, -3, 1, 0, -1, 0, 4, -4], [-2, 0, -6, -9, 1, 0, -1, 0, 2, -2], [18, 0, 2, -2, 2, 0, -2, 0, -4, 4], [0, 4, 0, 0, 4, 8, 4, -4, 6, 6], [0, -10, 0, 0, 2, 7, 2, 11, -6, -6], [9, 0, -5, 0, -6, 0, 6, 0, 7, -7], [0, 0, 0, 0, 3, 8, 3, 2, -5, -5], [27, 0, -3, -8, 0, 0, 0, 0, 5, -5], [0, 2, 0, 0, -9, 12, -9, 10, -5, -5], [12, 0, 6, 4, 0, 0, 0, 0, 4, -4], [2, 0, 0, 14, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 27, 0, 7, -6, -6], [0, -1, 0, 0, 0, 12, 0, -8, -2, -2], [0, 2, 0, 0, 2, 3, 2, -5, 0, 0], [-2, 0, -4, 1, -10, 0, 10, 0, 7, -7], [-16, 0, -6, 6, -5, 0, 5, 0, 2, -2], [22, 0, -2, -5, 5, 0, -5, 0, 0, 0], [0, 2, 0, 0, 2, -10, 2, 2, 2, 2], [0, -2, 0, 0, 4, -24, 4, -4, 6, 6], [10, 0, 2, 14, 5, 0, -5, 0, 2, -2], [2, 0, 6, 6, -4, 0, 4, 0, -2, 2], [0, 8, 0, 0, -5, -4, -5, 0, 6, 6], [0, -6, 0, 0, -1, 14, -1, 4, 5, 5], [2, 0, 0, 2, -1, 0, 1, 0, 4, -4], [0, -6, 0, 0, 6, 6, 6, 2, 2, 2], [0, 0, 0, 0, 11, -4, 11, -2, 1, 1], [-26, 0, 0, 2, 0, 0, 0, 0, -2, 2], [0, -16, 0, 0, 0, 10, 0, 6, -2, -2], [0, -4, 0, 0, -6, 8, -6, 8, 0, 0], [0, 6, 0, 0, -8, -2, -8, -10, 4, 4], [-8, 0, 0, 4, -4, 0, 4, 0, -8, 8], [5, 0, -3, 2, -3, 0, 3, 0, -4, 4], [-6, 0, -4, 12, 0, 0, 0, 0, 2, -2], [0, 4, 0, 0, -4, 3, -4, 1, -7, -7], [-14, 0, -2, -8, 1, 0, -1, 0, 4, -4], [0, 10, 0, 0, -4, -6, -4, 6, 6, 6], [0, -10, 0, 0, 0, 18, 0, 2, 0, 0], [0, -2, 0, 0, 3, 16, 3, 6, -7, -7], [20, 0, 6, -4, 11, 0, -11, 0, -10, 10], [0, -2, 0, 0, 0, -12, 0, 0, -6, -6], [0, 11, 0, 0, 0, 6, 0, -8, 9, 9], [4, 0, -2, 10, 1, 0, -1, 0, 5, -5], [0, -2, 0, 0, 2, -25, 2, 3, -2, -2], [10, 0, 0, -3, 10, 0, -10, 0, 3, -3], [0, 12, 0, 0, 5, 4, 5, 0, -2, -2], [-16, 0, 8, 12, 4, 0, -4, 0, -4, 4], [-24, 0, -4, -4, -5, 0, 5, 0, 0, 0], [-23, 0, -3, -6, -1, 0, 1, 0, 2, -2], [0, 0, 0, 0, 8, 6, 8, 2, 2, 2], [-4, 0, 8, -6, 2, 0, -2, 0, -6, 6], [10, 0, -4, -6, 2, 0, -2, 0, 0, 0], [-21, 0, 9, 0, 6, 0, -6, 0, -1, 1], [-6, 0, -6, -5, -9, 0, 9, 0, 0, 0], [0, -8, 0, 0, -2, -4, -2, 0, -6, -6], [2, 0, -2, 4, -1, 0, 1, 0, 4, -4], [0, 10, 0, 0, 3, 2, 3, 4, -5, -5], [2, 0, 0, 4, -1, 0, 1, 0, -8, 8], [0, 0, 0, 0, -5, -28, -5, -2, 5, 5], [0, -6, 0, 0, 9, -6, 9, 0, -3, -3], [0, 10, 0, 0, -3, 18, -3, -8, 9, 9], [-22, 0, -2, -2, -3, 0, 3, 0, 2, -2], [0, -2, 0, 0, 5, 2, 5, -12, 1, 1], [0, 5, 0, 0, -4, -36, -4, -4, 0, 0], [-6, 0, -10, 3, -1, 0, 1, 0, 6, -6], [-4, 0, 12, 10, 4, 0, -4, 0, -4, 4], [0, -6, 0, 0, -3, -2, -3, 4, 5, 5], [0, -4, 0, 0, 2, 2, 2, 10, 0, 0], [1, 0, -9, 0, -8, 0, 8, 0, 7, -7], [18, 0, -2, -2, 3, 0, -3, 0, 6, -6], [0, -4, 0, 0, -8, 20, -8, 4, 4, 4], [0, 8, 0, 0, -2, -37, -2, 5, 3, 3], [-14, 0, 8, 17, 4, 0, -4, 0, -3, 3], [0, 12, 0, 0, 5, 18, 5, -4, 9, 9], [0, 4, 0, 0, 4, -25, 4, -3, 5, 5], [0, -7, 0, 0, 2, -10, 2, 4, -5, -5], [-2, 0, 0, 12, 2, 0, -2, 0, -2, 2], [0, 0, 4, 8, -10, 0, 10, 0, 0, 0], [0, 4, 0, 0, 3, 12, 3, -12, 0, 0], [-8, 0, 2, -2, 8, 0, -8, 0, -8, 8], [-26, 0, 4, 0, -5, 0, 5, 0, -4, 4], [0, 2, 0, 0, -8, -5, -8, 15, -10, -10], [22, 0, 2, -8, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 2, -18, 2, -6, 2, 2], [-2, 0, 2, 10, -1, 0, 1, 0, 5, -5], [-6, 0, -12, -4, -7, 0, 7, 0, 4, -4], [0, -8, 0, 0, 11, -2, 11, 0, -5, -5], [-28, 0, 0, 10, 5, 0, -5, 0, 4, -4], [0, 8, 0, 0, 1, -18, 1, 8, 3, 3], [0, -5, 0, 0, 8, -24, 8, -8, 0, 0], [4, 0, 4, 8, 4, 0, -4, 0, 8, -8], [12, 0, -10, -2, -10, 0, 10, 0, 0, 0], [0, 6, 0, 0, -10, -6, -10, -2, -2, -2], [8, 0, 6, 8, 10, 0, -10, 0, -6, 6], [0, 12, 0, 0, 6, 2, 6, -6, 8, 8], [0, -2, 0, 0, 9, -26, 9, -12, 3, 3], [26, 0, 8, 12, 6, 0, -6, 0, -4, 4], [8, 0, 8, 4, 8, 0, -8, 0, -8, 8], [-30, 0, 2, 7, -5, 0, 5, 0, 6, -6], [0, 2, 0, 0, -13, 8, -13, 6, -3, -3], [0, 3, 0, 0, 12, -20, 12, -4, 0, 0], [-6, 0, 0, 20, 2, 0, -2, 0, 4, -4], [0, 16, 0, 0, 2, -7, 2, -9, 1, 1], [3, 0, 5, 12, 0, 0, 0, 0, 1, -1], [0, -9, 0, 0, 8, 22, 8, -4, -5, -5], [-4, 0, -8, -4, -2, 0, 2, 0, 0, 0], [0, -8, 0, 0, -9, -4, -9, 14, -11, -11], [6, 0, 0, 6, -11, 0, 11, 0, 0, 0], [8, 0, -10, -18, -1, 0, 1, 0, -1, 1], [-22, 0, 10, 13, -3, 0, 3, 0, 2, -2], [0, -7, 0, 0, -8, -4, -8, 4, -4, -4], [0, -13, 0, 0, -4, -10, -4, 4, 7, 7], [-4, 0, 2, 8, -4, 0, 4, 0, 0, 0], [0, -14, 0, 0, -7, 20, -7, 10, 7, 7], [-13, 0, 1, 4, -2, 0, 2, 0, 11, -11], [-14, 0, -2, -2, 12, 0, -12, 0, 4, -4], [0, 0, 0, 0, 4, 4, 4, 0, 10, 10], [-22, 0, 4, -6, -7, 0, 7, 0, -8, 8], [0, 1, 0, 0, 14, -26, 14, -12, 3, 3], [10, 0, -4, 0, -10, 0, 10, 0, -2, 2], [0, 0, 0, 0, 12, 7, 12, -3, 5, 5], [0, 0, -4, -10, 6, 0, -6, 0, -2, 2], [0, 4, 0, 0, -6, -8, -6, 4, -2, -2], [18, 0, -4, 6, -7, 0, 7, 0, 0, 0], [-21, 0, -3, 4, -4, 0, 4, 0, 3, -3], [0, -16, 0, 0, -6, -10, -6, 6, 4, 4], [0, 0, 0, 0, -1, 24, -1, -2, 7, 7], [10, 0, 8, -2, 12, 0, -12, 0, -6, 6], [0, -14, 0, 0, 0, -4, 0, 0, -10, -10], [10, 0, 16, 10, 6, 0, -6, 0, 4, -4], [0, 4, 0, 0, 3, 28, 3, -4, -12, -12], [-11, 0, 7, 0, -6, 0, 6, 0, -3, 3], [-10, 0, 0, 8, 6, 0, -6, 0, 12, -12], [0, -10, 0, 0, -6, -10, -6, 8, -7, -7], [0, -2, 0, 0, 0, 13, 0, -7, 2, 2], [0, -4, 0, 0, 2, 21, 2, -9, 7, 7], [0, 10, 0, 0, 5, -18, 5, -12, 1, 1], [0, 2, 0, 0, 12, 8, 12, 0, 0, 0], [-20, 0, 8, -2, -1, 0, 1, 0, -8, 8], [0, -10, 0, 0, 4, -5, 4, -1, 6, 6], [0, -10, 0, 0, -1, 18, -1, 8, 5, 5], [-8, 0, 2, -4, 11, 0, -11, 0, -6, 6], [6, 0, -2, -7, -5, 0, 5, 0, 8, -8], [-26, 0, 0, 14, 2, 0, -2, 0, 4, -4], [0, 18, 0, 0, 2, -16, 2, -8, 4, 4], [0, 0, 0, 0, 1, -16, 1, -18, 15, 15], [14, 0, -6, 2, -3, 0, 3, 0, 16, -16], [0, -6, 0, 0, 4, 6, 4, 2, -10, -10], [0, 8, 0, 0, -2, -39, -2, -1, 5, 5], [-28, 0, 4, 8, -4, 0, 4, 0, 0, 0], [0, -18, 0, 0, 13, 18, 13, 0, -7, -7], [0, -7, 0, 0, -6, -18, -6, 4, -1, -1], [0, 4, 0, 0, 6, -4, 6, 4, -4, -4], [6, 0, 0, 12, 6, 0, -6, 0, 14, -14], [14, 0, 4, 7, 8, 0, -8, 0, 3, -3], [-6, 0, 4, -10, -7, 0, 7, 0, -4, 4], [0, -7, 0, 0, -12, -10, -12, 8, -7, -7], [11, 0, 3, -6, 3, 0, -3, 0, 8, -8], [0, -8, 0, 0, -6, 0, -6, 16, -8, -8], [-40, 0, 4, 6, -5, 0, 5, 0, -4, 4], [0, -17, 0, 0, 4, -4, 4, 16, -6, -6], [-23, 0, -9, 0, 4, 0, -4, 0, -1, 1], [0, -2, 0, 0, -6, 20, -6, 4, 8, 8], [0, 4, 0, 0, 7, -16, 7, 6, -5, -5], [2, 0, 6, -4, -8, 0, 8, 0, -4, 4], [34, 0, -8, 0, -2, 0, 2, 0, 2, -2], [-22, 0, -2, 3, 3, 0, -3, 0, -14, 14], [0, -3, 0, 0, 8, -4, 8, -4, 8, 8], [0, -16, 0, 0, -9, -14, -9, 12, -13, -13], [32, 0, -6, -16, -5, 0, 5, 0, 10, -10], [0, 8, 0, 0, -15, -12, -15, -2, 1, 1], [6, 0, 18, -2, 11, 0, -11, 0, -4, 4], [42, 0, -2, 4, -4, 0, 4, 0, 2, -2], [0, 6, 0, 0, -12, 3, -12, -5, 4, 4], [-18, 0, -4, 14, -5, 0, 5, 0, 8, -8], [0, -1, 0, 0, -4, 48, -4, 8, -4, -4], [0, 4, 0, 0, 6, 1, 6, 3, 7, 7], [0, 14, 0, 0, -7, -2, -7, -4, 7, 7], [-5, 0, -5, 2, 7, 0, -7, 0, 4, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2240_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2240_2_g_n();".
// 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_2240_2_g_n(:prec:=10)
    chi := MakeCharacter_2240_g();
    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_2240_2_g_n();".
// 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_2240_2_g_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2240_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![256, 0, 708, 0, 532, 0, 165, 0, 22, 0, 1]>,<11,R![-128, -248, -152, -25, 4, 1]>,<19,R![-256, -184, 72, 30, -12, 1]>],Snew);
    return Vf;
end function;