
// Make newform 2240.2.e.f in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2240_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2240_e_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_e_f();".
// 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_e_f();".
// 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 := [1444, 0, -524, 0, 1625, 0, 610, 0, -40, 0, 16, 0, 28, 0, -6, 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, 0, 0, 0, 0], [-140017061, 0, -1954893928, 0, -1220553906, 0, 259974236, 0, -108971684, 0, 10863016, 0, -2074564, 0, -84844, 0], [0, 81935469158, 0, 38410007337, 0, -3062111390, 0, 3306399632, 0, 1131088376, 0, -191415108, 0, 39930036, 0, 2967085], [0, -93332422574, 0, -12446679765, 0, -5606760042, 0, -24668656, 0, -2071902984, 0, 576987844, 0, -106481348, 0, 4207655], [-3209825990, 0, -124066909, 0, 137240250, 0, -84357892, 0, -23555944, 0, 13430352, 0, -2939696, 0, 274551, 0], [-13312648, 0, -2426116, 0, 3678039, 0, -805342, 0, -360443, 0, 225632, 0, -44446, 0, 4034, 0], [-1386720738, 0, -311886279, 0, 613266210, 0, 6532248, 0, -25125688, 0, 27045620, 0, -5214580, 0, 778741, 0], [0, -4522635990, 0, -5600946403, 0, -2721018508, 0, -8711350, 0, -73573152, 0, -139846998, 0, 30566954, 0, -5290343], [0, 6038703550, 0, 11275740801, 0, 2173230995, 0, -132623279, 0, 248960394, 0, 75636915, 0, -18581871, 0, 4087383], [0, -53518138, 0, -81508947, 0, -26060814, 0, 645500, 0, -1111276, 0, -1062732, 0, 221960, 0, -40619], [0, -8462940866, 0, 101267919077, 0, 28747923386, 0, -4344262192, 0, 2436041896, 0, 1118627340, 0, -271267084, 0, 51214505], [-774884638, 0, 8291019412, 0, 2170744901, 0, -56159602, 0, 92658541, 0, 130546916, 0, -28350288, 0, 4924818, 0], [-198617650, 0, 2016951625, 0, 628515030, 0, -149350828, 0, 74001744, 0, 20915688, 0, -5994576, 0, 1160909, 0], [-371424458, 0, 5299245757, 0, 1528314658, 0, 12504832, 0, 16571240, 0, 78559372, 0, -16104476, 0, 2650265, 0], [0, -3111082718, 0, 2269259769, 0, 974013580, 0, -99229442, 0, 8859636, 0, 49775130, 0, -10635114, 0, 1659481], [0, 7322587014, 0, -5301496115, 0, -2577780150, 0, 324888696, 0, -54427560, 0, -110686692, 0, 24179876, 0, -3799279]];
    Rf_basisdens := [1, 2498768831, 39980301296, 39980301296, 768851948, 10116473, 1537703896, 9995075324, 4997537662, 31933148, 39980301296, 2498768831, 526056596, 1052113192, 768851948, 1537703896];
    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_e();"
function MakeCharacter_2240_e()
    N := 2240;
    order := 2;
    char_gens := [1471, 1541, 897, 1921];
    v := [1, 2, 1, 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_2240_e_Hecke();"
function MakeCharacter_2240_e_Hecke(Kf)
    N := 2240;
    order := 2;
    char_gens := [1471, 1541, 897, 1921];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, -1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, 0, 0, 1, -1, 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, 0, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0], [0, 0, 0, -2, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, -2], [0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 2, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 5, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, -3, 0], [0, 0, 3, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, -4, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -2, 0, 0], [0, 0, 0, -2, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 2], [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, -3, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, -3, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 2, -1, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0], [0, 0, 0, 4, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 7, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 7, -4, 0, 0, 0, 0, 0, 0, 0], [0, 0, -3, 0, 0, 0, 0, 0, 0, -1, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 6, 0, 0, 0, 0, 0, 0, 1, 6, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0], [0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, 0, 0, -2, 3, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0], [-6, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 8, 0, 0, 0, 0, 0, 0, 0], [0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 11, -8, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -3], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -3, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 2, 0, 0], [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, 0, 0, 4, -2, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -3, 0, 0, 0, 0, 0, 0, 9, -3, 0, 0, 0, 0, 0], [0, 0, 5, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, -3, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 16, -9, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -4, 0, 0], [0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 12, 0, 0, 0, 0, 0, 0, 0], [-6, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 4, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 5, 0], [16, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 6, 0, 0, 0, 0, 0, 0, 6, 6, 0, 0, 0, 0, 0], [0, 0, 0, 6, 0, 0, 0, -3, 3, 0, 0, 0, 0, 0, 0, -2], [-10, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0], [0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 6, 0], [0, 0, 0, 8, 0, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0, -2], [0, 0, 0, 0, 1, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [14, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, -4, 0, 0], [0, 0, -7, 0, 0, 0, 0, 0, 0, -2, -7, 0, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -3, -14, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -6, 0, 0, 0, 3, -3, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 8, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 6, 5, 0, 0, 0, 0, 0, 0, 0], [0, 0, 5, 0, 0, 0, 0, 0, 0, 6, 5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 7, 5, 0, 0, 0, 0, 0, 0, 0], [22, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 11, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, -3, 11, 0, 0, 0, 0, 0, 0, 0], [-8, 0, 0, 0, 0, -14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0], [0, 0, 9, 0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, -3, 3, 0, 0, 0, 0, 0, 0, 0], [0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 5, 0], [0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0], [0, 0, -14, 0, 0, 0, 0, 0, 0, -4, -14, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -16, -9, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0], [0, 0, 0, 4, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, 3], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0], [4, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 15, 4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 16, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0], [0, -12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0], [0, 0, 7, 0, 0, 0, 0, 0, 0, -6, 7, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 0, 0], [0, 0, -10, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, -7, 0], [0, 0, 0, 0, -1, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 6, 0, 0, 0, -3, 3, 0, 0, 0, 0, 0, 0, 4], [-18, 0, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-18, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 7], [0, 0, 0, 0, 0, 0, 0, 12, 6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 6, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 1, 0], [0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0], [0, 0, -8, 0, 0, 0, 0, 0, 0, -9, -8, 0, 0, 0, 0, 0], [0, 0, -4, 0, 0, 0, 0, 0, 0, -2, -4, 0, 0, 0, 0, 0], [0, 0, 6, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, -7, 0], [0, 0, 0, 0, 0, 0, 0, 21, -10, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 9, 0], [-4, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0], [18, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, 0, 0, 13, 3, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, -9], [0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 8, 0], [0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0], [0, 0, 0, -2, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -9, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, -4, 0, 0, 0, 0, 0, 0, 14, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -16, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -5, 0, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0], [0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, -8, 0, 0, 0, 0, 0, 0, -4, -8, 0, 0, 0, 0, 0], [0, 0, -13, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 1, 0], [0, 0, -5, 0, 0, 0, 0, 0, 0, 7, -5, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, -24, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 6, 0, 0], [0, 0, 0, 0, 2, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -4, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, -7], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, -26, 9, 0, 0, 0, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, -14, 0, 0, 0, 0], [0, 0, 12, 0, 0, 0, 0, 0, 0, 0, -12, 0, 0, 0, 1, 0], [16, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 4, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0], [0, 0, -13, 0, 0, 0, 0, 0, 0, -8, -13, 0, 0, 0, 0, 0], [0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 4, 0, 0], [0, 0, 0, 6, 0, 0, 0, -3, 3, 0, 0, 0, 0, 0, 0, 1], [0, 0, 6, 0, 0, 0, 0, 0, 0, 16, 6, 0, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0], [14, 0, 0, 0, 0, -17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 8, -6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 9, 0, 0, 0, 0, 0, 0, -3, 9, 0, 0, 0, 0, 0], [0, 0, 0, 0, -9, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -16, 0, 0, 0, 8, -8, 0, 0, 0, 0, 0, 0, 7], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, -8, 0, 0], [0, 0, 2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, -6, 0], [0, 0, 0, 0, 9, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 11, 0, 0, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, -7, -2, 0, 0, 0, 0, 0, 0, 0], [-18, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -10, 0, 0, 0, 5, -5, 0, 0, 0, 0, 0, 0, 1], [0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, -2, 0], [0, 0, 0, 0, -3, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 3, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -26, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 4, 0, 0], [0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, -7, 0], [0, 0, 0, 0, 0, 0, 0, -14, 5, 0, 0, 0, 0, 0, 0, 0], [-10, 0, 0, 0, 0, -12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 8, 0, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0, 4], [0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, -8, 0], [0, 0, 10, 0, 0, 0, 0, 0, 0, 7, 10, 0, 0, 0, 0, 0], [0, 0, 0, 4, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, -3], [0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, -3, 0], [0, 0, 0, -8, 0, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0, -8], [0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-8, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -4, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 9], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 4, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, -6, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, -14, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -23, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10], [0, 0, 16, 0, 0, 0, 0, 0, 0, 3, 16, 0, 0, 0, 0, 0], [0, 0, 0, 0, 7, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 14, 0, 0, 0, 0, 0, 0, -1, 14, 0, 0, 0, 0, 0], [0, 0, 0, 0, -9, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 6, 0, 0], [38, 0, 0, 0, 0, -4, 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, 6], [-14, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 14, 0, -12, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5, -16, 0, 0, 0, 0, 0, 0, 0], [-18, 0, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 13, 0], [0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 21, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -9, -4, 0, 0], [0, 0, 0, -2, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, -8], [0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0], [14, 0, 0, 0, 0, -23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -21, -24, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 24, -21, 0, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 0, 0, 0, -3, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 10, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0], [0, 0, 0, -2, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 3], [0, 0, 0, 0, -14, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, -2, -23, 0, 0, 0, 0, 0, 0, 0], [-30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 5, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 9, 0], [0, 0, 0, -14, 0, 0, 0, 7, -7, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 10, -4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0], [0, -16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0], [0, 0, 0, 0, 7, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 4, 0, 0], [0, 0, -12, 0, 0, 0, 0, 0, 0, 2, -12, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -10, 0, 0, 0, 0, 0, 0, 2, -10, 0, 0, 0, 0, 0], [0, 0, -13, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, -4, 0], [0, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, -13, 0, 0, 0, 0], [6, 0, 0, 0, 0, -20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 17, 23, 0, 0, 0, 0, 0, 0, 0], [0, 0, 8, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 5, 0], [0, 0, 3, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, -3, 0], [0, 0, 0, 0, -9, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -16, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 9, 0, 0, 0, 0, 0, 0, -10, 9, 0, 0, 0, 0, 0], [0, 0, -12, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, -15, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 0], [0, 0, 0, 12, 0, 0, 0, -6, 6, 0, 0, 0, 0, 0, 0, 1], [22, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 0, 0, 0, -16, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 10, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -4, 0, 0], [0, 0, 15, 0, 0, 0, 0, 0, 0, 17, 15, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -13, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, -3, 0], [0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, -12, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0], [0, 0, 0, 0, 0, 0, 0, -2, -19, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 22, 0, -14, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-14, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [52, 0, 0, 0, 0, -13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -6, 0, 0, 0, 3, -3, 0, 0, 0, 0, 0, 0, -8], [0, 0, 0, 0, 0, 0, 0, 16, -5, 0, 0, 0, 0, 0, 0, 0], [0, -14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -16, 18, 0, 0, 0, 0, 0, 0, 0], [0, 0, 9, 0, 0, 0, 0, 0, 0, 6, 9, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -13, 20, 0, 0, 0, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, -6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 8, 0, 0], [0, 0, 0, 4, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, -8], [0, 0, 12, 0, 0, 0, 0, 0, 0, -14, 12, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 5, -18, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 8, 0, 0], [0, 0, 0, 8, 0, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0, -5], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0], [0, 0, -15, 0, 0, 0, 0, 0, 0, -1, -15, 0, 0, 0, 0, 0], [0, 0, 0, -4, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 14], [0, 0, 10, 0, 0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 1, 0], [0, 0, 0, 10, 0, 0, 0, -5, 5, 0, 0, 0, 0, 0, 0, 10], [0, 0, 0, 0, 0, 0, 0, -15, -6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 23, 0, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, -3, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, -6, 0, 0], [0, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2240_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2240_2_e_f();".
// 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_e_f(:prec:=16)
    chi := MakeCharacter_2240_e();
    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_e_f();".
// 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_e_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2240_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 0, 5, 0, 1]>,<11,R![52, 0, 15, 0, 1]>],Snew);
    return Vf;
end function;