
// Make newform 2240.2.b.g in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2240_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2240_b_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_b_g();".
// 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_b_g();".
// 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 := [61, -364, 1035, -1822, 2258, -2048, 1463, -794, 362, -120, 35, -6, 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], [-9, 8, -9, 2, -1, 0, 0, 0, 0, 0, 0, 0], [-2, 6, -7, 2, -1, 0, 0, 0, 0, 0, 0, 0], [782, -3168, 5139, -4333, 657, 1376, -2198, 1474, -717, 251, -55, 10], [-3, 28, -68, 92, -77, 40, -18, 4, -1, 0, 0, 0], [29, -90, 152, -138, 105, -46, 20, -4, 1, 0, 0, 0], [-2716, 3421, 10441, -23506, 31600, -27301, 17283, -8844, 3210, -1025, 187, -34], [-1977, 7590, -13106, 16668, -14089, 10310, -5362, 2552, -831, 258, -44, 8], [-1019, 8492, -26920, 45874, -51855, 42112, -24808, 12188, -4155, 1290, -220, 40], [-4401, 22528, -53132, 79210, -80033, 62732, -35532, 17292, -5817, 1806, -308, 56], [82, -323, 639, -748, 682, -421, 229, -82, 28, -5, 1, 0], [-13186, 65395, -150067, 220982, -221274, 173957, -99071, 48532, -16506, 5153, -891, 162]];
    Rf_basisdens := [1, 2, 1, 286, 4, 2, 572, 143, 572, 572, 4, 572];
    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_b();"
function MakeCharacter_2240_b()
    N := 2240;
    order := 2;
    char_gens := [1471, 1541, 897, 1921];
    v := [2, 1, 2, 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_b_Hecke();"
function MakeCharacter_2240_b_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], [-1, 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, 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, 1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1], [0, 0, 0, 0, 0, 0, 1, 0, 1, -1, 0, 1], [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, -1, -1, -1, 0, 0, 0], [0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, -1, 0, 0, 1, -1, 0, -3, 0, 1], [1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0, 1, -1, -1, 0, 0, -2], [0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, -1, 0, 0, -1, 1, 2, 1, 0, 2], [2, -1, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0], [0, 0, 0, -2, 0, 0, 0, -1, 0, 2, 0, 2], [0, 0, 0, 2, 0, 0, -2, -1, -2, 2, 0, 0], [0, 0, 0, 1, 0, 0, 1, -1, 0, 3, 0, 2], [0, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0], [-4, -4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 2, 0], [2, -1, -1, 0, -1, 1, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 2, 2, 0, 6, 0, 0], [2, 0, -1, 0, -1, 0, 0, 0, 0, 0, 1, 0], [3, -1, -1, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 1, 1, 2, 7, 0, 0], [-2, 1, 0, 0, -2, -1, 0, 0, 0, 0, -1, 0], [0, 0, 0, -2, 0, 0, 0, 0, 0, 6, 0, 2], [0, 0, 0, 1, 0, 0, -1, 1, 2, 1, 0, 1], [7, 4, -2, 0, -3, -1, 0, 0, 0, 0, -1, 0], [0, 4, -2, 0, -2, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, -1, 1, -1, -2, 0, -4], [-3, 2, 0, 0, 1, 1, 0, 0, 0, 0, -1, 0], [0, 0, 0, -2, 0, 0, 2, 1, 2, -6, 0, 2], [0, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0], [2, -1, -3, 0, 1, -1, 0, 0, 0, 0, -3, 0], [0, 0, 0, -4, 0, 0, 0, 2, 2, -4, 0, 4], [0, 0, 0, 2, 0, 0, -2, -2, 0, -2, 0, -2], [0, 3, -2, 0, -2, 1, 0, 0, 0, 0, 1, 0], [0, 0, 0, -2, 0, 0, 1, -2, -1, 3, 0, 3], [0, 0, 0, 0, 0, 0, 0, -2, 2, 6, 0, 0], [0, 0, 0, -2, 0, 0, -2, 0, 0, 0, 0, 6], [-8, -1, 3, 0, 5, 3, 0, 0, 0, 0, 1, 0], [3, -2, -4, 0, -1, -1, 0, 0, 0, 0, 1, 0], [0, 0, 0, 3, 0, 0, -1, -3, -1, -6, 0, 0], [-6, -4, 2, 0, 4, 0, 0, 0, 0, 0, 2, 0], [0, 0, 0, -1, 0, 0, -2, 2, -2, 1, 0, 1], [0, -5, 4, 0, 4, 1, 0, 0, 0, 0, 3, 0], [0, 0, 0, -2, 0, 0, 2, 0, 2, 0, 0, 1], [0, 0, 0, -3, 0, 0, -1, 1, -2, 5, 0, -2], [-4, -2, 2, 0, 4, 2, 0, 0, 0, 0, -2, 0], [-6, -1, 1, 0, -3, -1, 0, 0, 0, 0, 1, 0], [2, 6, -3, 0, -3, -2, 0, 0, 0, 0, 1, 0], [0, 0, 0, 2, 0, 0, 2, 3, -2, -6, 0, 2], [6, 0, 0, 0, -4, -1, 0, 0, 0, 0, -2, 0], [8, 6, 2, 0, -2, 2, 0, 0, 0, 0, 2, 0], [0, 0, 0, 0, 0, 0, -2, 0, 2, -12, 0, 4], [5, 0, 2, 0, 1, -1, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 2, 1, -4, -4, 0, 0], [2, -3, -1, 0, -1, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, 2, 0, 0, -10, 0, -3], [0, 0, 0, 2, 0, 0, 1, 0, 1, -7, 0, 3], [0, 0, 0, 2, 0, 0, 2, -2, -2, 4, 0, 3], [-7, -6, 6, 0, 1, 3, 0, 0, 0, 0, 3, 0], [-19, 3, -1, 0, 2, 0, 0, 0, 0, 0, -2, 0], [0, 0, 0, -2, 0, 0, 2, 1, 2, 4, 0, 4], [0, 0, 0, 4, 0, 0, -4, 0, -2, -6, 0, -4], [8, 0, 0, 0, 2, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 1, -5, 2, 1, 0, 2], [0, 0, 0, -5, 0, 0, 5, 3, 2, -9, 0, 4], [7, 3, 1, 0, 2, 0, 0, 0, 0, 0, -2, 0], [12, 0, 0, 0, 4, 1, 0, 0, 0, 0, -4, 0], [-4, 1, 2, 0, 2, -3, 0, 0, 0, 0, 3, 0], [0, 0, 0, -1, 0, 0, -3, 3, -1, -6, 0, -8], [0, 0, 0, 0, 0, 0, 0, 2, -4, 4, 0, -4], [6, 0, 0, 0, -2, 0, 0, 0, 0, 0, -2, 0], [0, 0, 0, -1, 0, 0, -3, -3, 0, 9, 0, -1], [0, 0, 0, 4, 0, 0, 1, 4, 3, 5, 0, 1], [4, 1, -1, 0, 1, 5, 0, 0, 0, 0, 2, 0], [-16, 0, 0, 0, -2, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, -4, -10, 0, 0], [0, 0, 0, 3, 0, 0, 1, 3, -4, 1, 0, 5], [0, 3, 1, 0, -1, -3, 0, 0, 0, 0, 3, 0], [2, 4, 2, 0, 6, -1, 0, 0, 0, 0, 0, 0], [-2, 6, -2, 0, -4, -4, 0, 0, 0, 0, 4, 0], [0, 0, 0, 3, 0, 0, -1, -3, -6, 5, 0, 2], [0, -7, 1, 0, 3, -3, 0, 0, 0, 0, 0, 0], [-9, 2, 4, 0, -5, 1, 0, 0, 0, 0, 3, 0], [0, 0, 0, 0, 0, 0, -2, -2, -2, 0, 0, -6], [-4, -6, -1, 0, -1, -2, 0, 0, 0, 0, 3, 0], [0, 0, 0, -6, 0, 0, -2, 2, 6, 12, 0, 7], [13, -2, 0, 0, -5, 1, 0, 0, 0, 0, 7, 0], [0, -6, 4, 0, 0, -2, 0, 0, 0, 0, 2, 0], [0, 0, 0, 1, 0, 0, 0, 4, 2, -9, 0, -3], [0, 0, 0, -1, 0, 0, 0, -2, 0, -5, 0, -1], [2, 7, 2, 0, 0, -1, 0, 0, 0, 0, 3, 0], [0, 0, 0, 2, 0, 0, -4, 0, -6, -14, 0, -4], [-12, -2, 1, 0, -1, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, -2, 0, 0, 4, -2, -2, -2, 0, -4], [0, 0, 0, 5, 0, 0, -1, 3, 0, 19, 0, -1], [0, 0, 0, 1, 0, 0, -1, 1, 2, -17, 0, 4], [0, 0, 0, -2, 0, 0, -2, 1, 0, 6, 0, -2], [0, 0, 0, 2, 0, 0, 0, 0, -4, 8, 0, 0], [-6, 4, -4, 0, 4, -4, 0, 0, 0, 0, -4, 0], [0, 0, 0, -2, 0, 0, -2, -2, 2, -8, 0, -8], [5, -9, 3, 0, -2, -4, 0, 0, 0, 0, 2, 0], [0, 0, 0, 0, 0, 0, -2, 0, -2, 4, 0, -8], [-1, 1, 1, 0, -2, 2, 0, 0, 0, 0, 2, 0], [6, -7, 1, 0, 1, -5, 0, 0, 0, 0, 1, 0], [-16, 0, 4, 0, 6, 2, 0, 0, 0, 0, 4, 0], [26, 7, -2, 0, 0, -3, 0, 0, 0, 0, -1, 0], [0, 0, 0, 2, 0, 0, 2, 5, 6, 0, 0, 0], [4, -4, 4, 0, -2, 0, 0, 0, 0, 0, 8, 0], [0, 0, 0, 1, 0, 0, 3, 1, -1, -22, 0, 0], [-6, 1, 3, 0, -1, -1, 0, 0, 0, 0, -1, 0], [6, -4, 0, 0, 0, 8, 0, 0, 0, 0, -4, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, -18, 0, 5], [8, -4, -2, 0, -2, 4, 0, 0, 0, 0, -6, 0], [0, 0, 0, 3, 0, 0, -7, 1, 1, 8, 0, -2], [0, 0, 0, 3, 0, 0, 2, -6, 0, 11, 0, 1], [0, 0, 0, 0, 0, 0, 2, 6, 6, -8, 0, -6], [-7, -6, 4, 0, -3, -3, 0, 0, 0, 0, -1, 0], [0, 0, 0, -2, 0, 0, -3, 0, 1, 1, 0, 11], [0, 0, 0, -1, 0, 0, 5, -1, 6, -7, 0, -4], [0, 0, 0, 2, 0, 0, 4, -3, 0, -4, 0, 4], [0, 0, 0, -1, 0, 0, -3, 1, 2, 7, 0, 5], [0, 0, 0, 1, 0, 0, 7, 5, 0, 15, 0, -3], [21, 2, 0, 0, 3, 5, 0, 0, 0, 0, 3, 0], [2, -4, -4, 0, 2, 2, 0, 0, 0, 0, -6, 0], [0, 0, 0, 4, 0, 0, 3, 4, -1, 13, 0, 3], [0, 0, 0, -3, 0, 0, 0, -2, 2, -9, 0, 13], [-4, 8, 2, 0, -2, 2, 0, 0, 0, 0, 4, 0], [-6, -3, 1, 0, -7, -3, 0, 0, 0, 0, 5, 0], [0, 0, 0, -4, 0, 0, -4, -1, 2, -14, 0, 2], [-2, 2, -1, 0, -1, 4, 0, 0, 0, 0, 3, 0], [12, 0, -2, 0, -4, -2, 0, 0, 0, 0, -6, 0], [0, 0, 0, -4, 0, 0, 3, -2, -1, -7, 0, -13], [0, 0, 0, 4, 0, 0, -8, 0, -6, 10, 0, -5], [0, 0, 0, -2, 0, 0, -1, 6, -1, -13, 0, 1], [18, -1, -1, 0, -1, 1, 0, 0, 0, 0, -2, 0], [0, 0, 0, -2, 0, 0, -2, 3, -2, -22, 0, -6], [0, 0, 0, 5, 0, 0, -5, -7, -8, 11, 0, -5], [8, 2, 1, 0, 9, -2, 0, 0, 0, 0, -5, 0], [0, 0, 0, 6, 0, 0, 2, -2, 0, -2, 0, -6], [0, 0, 0, -3, 0, 0, -9, 1, 2, -15, 0, -2], [15, -2, 4, 0, 5, -1, 0, 0, 0, 0, -3, 0], [0, 0, 0, -4, 0, 0, -4, 0, 2, -8, 0, 0], [14, 4, 6, 0, 2, 1, 0, 0, 0, 0, 4, 0], [0, 0, 0, -2, 0, 0, 4, 1, 0, 24, 0, 4], [-20, -6, -1, 0, 3, 2, 0, 0, 0, 0, 5, 0], [0, 0, 0, -3, 0, 0, -5, -3, 3, -4, 0, 6], [10, -2, -4, 0, -12, -4, 0, 0, 0, 0, -2, 0], [0, 0, 0, 9, 0, 0, 7, 5, -2, 3, 0, 4], [-10, 4, 6, 0, -4, 4, 0, 0, 0, 0, 4, 0], [0, 0, 0, -5, 0, 0, -7, -1, 2, 1, 0, -6], [6, 4, 0, 0, 2, 7, 0, 0, 0, 0, 2, 0], [12, 9, -3, 0, -9, 1, 0, 0, 0, 0, -9, 0], [-8, -2, 0, 0, 2, -2, 0, 0, 0, 0, -2, 0], [-15, -1, -5, 0, -2, 0, 0, 0, 0, 0, -2, 0], [0, 0, 0, -1, 0, 0, 3, -1, 0, -3, 0, 6], [0, 0, 0, 7, 0, 0, -1, -3, -2, 9, 0, -4], [7, -6, 4, 0, 15, 3, 0, 0, 0, 0, -1, 0], [0, 2, -2, 0, -10, -6, 0, 0, 0, 0, -2, 0], [0, 0, 0, 1, 0, 0, 3, -1, -5, 10, 0, -2], [5, 0, -2, 0, -5, 3, 0, 0, 0, 0, -1, 0], [-22, 5, 6, 0, 4, 9, 0, 0, 0, 0, 1, 0], [-8, 0, 8, 0, -4, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, 1, 0, 1, -7, 0, 3], [2, 1, -1, 0, -5, -1, 0, 0, 0, 0, 8, 0], [0, 0, 0, 6, 0, 0, 2, -2, 0, 0, 0, 0], [0, 0, 0, -4, 0, 0, -4, -8, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, -2, 2, 4, 12, 0, -4], [16, 5, -9, 0, -3, -3, 0, 0, 0, 0, 1, 0], [6, 4, 0, 0, -4, 4, 0, 0, 0, 0, -6, 0], [-1, -2, 2, 0, 7, -3, 0, 0, 0, 0, 1, 0], [22, 2, 8, 0, 8, -4, 0, 0, 0, 0, -2, 0], [0, 0, 0, -1, 0, 0, -2, 4, -4, 15, 0, 1], [0, 0, 0, -3, 0, 0, 3, -3, 2, 13, 0, 5], [12, 0, -2, 0, -2, -2, 0, 0, 0, 0, 2, 0], [0, 0, 0, 1, 0, 0, 1, 5, 8, -1, 0, 10], [-20, -2, -3, 0, 9, -6, 0, 0, 0, 0, 1, 0], [0, 0, 0, -1, 0, 0, -5, 1, -1, -8, 0, 6], [0, 0, 0, 3, 0, 0, -3, 1, 1, -24, 0, 2], [10, -8, -2, 0, 2, -5, 0, 0, 0, 0, 4, 0], [12, -2, -7, 0, -3, -4, 0, 0, 0, 0, -3, 0], [0, 0, 0, -7, 0, 0, -7, -5, -4, 11, 0, -4], [0, 0, 0, 2, 0, 0, -4, -1, -4, 2, 0, -14], [0, 0, 0, -4, 0, 0, 0, 2, -6, 6, 0, -3], [-16, 7, 1, 0, 3, 3, 0, 0, 0, 0, 2, 0], [3, -4, 2, 0, 3, 7, 0, 0, 0, 0, -5, 0], [12, -4, -4, 0, -6, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0, 3, -3, -6, 21, 0, 2], [0, 0, 0, 5, 0, 0, -2, -4, -4, 5, 0, -5], [0, 0, 0, -7, 0, 0, 3, -5, 2, -11, 0, 14], [0, 0, 0, 8, 0, 0, -4, -4, -4, 8, 0, -4], [-1, 5, -7, 0, -6, -4, 0, 0, 0, 0, -6, 0], [-4, 11, -5, 0, -3, -5, 0, 0, 0, 0, 4, 0], [0, 0, 0, -7, 0, 0, -7, 1, -7, 14, 0, -4], [-19, 1, 1, 0, 8, 2, 0, 0, 0, 0, -4, 0], [-12, 3, -6, 0, 2, -1, 0, 0, 0, 0, -3, 0], [0, 0, 0, -12, 0, 0, 0, 4, 8, -12, 0, -4], [26, -6, -4, 0, -6, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, -6, 0, 0, 2, -2, 2, 18, 0, -4], [24, -2, -5, 0, 5, 2, 0, 0, 0, 0, -1, 0], [0, 0, 0, 7, 0, 0, -5, 1, 3, -16, 0, -12], [0, 0, 0, 4, 0, 0, -3, 2, -1, 5, 0, 9], [4, 2, -6, 0, -6, -4, 0, 0, 0, 0, 10, 0], [0, 0, 0, -1, 0, 0, 5, 7, 6, -3, 0, 8], [30, 8, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 5, 0, 0, 5, -1, 5, 8, 0, 6], [13, -2, 0, 0, -3, 1, 0, 0, 0, 0, -3, 0], [0, 0, 0, 0, 0, 0, -6, 0, 2, 4, 0, 8], [-8, 14, -7, 0, -3, -4, 0, 0, 0, 0, 3, 0], [0, 0, 0, -2, 0, 0, 6, -4, -8, -6, 0, 1], [-2, 6, -4, 0, -2, -2, 0, 0, 0, 0, -4, 0], [-2, 4, -7, 0, -7, -4, 0, 0, 0, 0, 3, 0], [-4, 12, -5, 0, 3, 4, 0, 0, 0, 0, -3, 0], [-20, -8, 5, 0, 7, 2, 0, 0, 0, 0, -1, 0], [4, -12, 5, 0, 1, 0, 0, 0, 0, 0, 5, 0], [0, 0, 0, 2, 0, 0, -4, -1, -2, 36, 0, 0], [0, 0, 0, 6, 0, 0, -2, -2, 0, -30, 0, 0], [2, -2, 4, 0, 2, -6, 0, 0, 0, 0, -4, 0], [-26, -7, 7, 0, -1, -1, 0, 0, 0, 0, 8, 0], [-16, 6, 6, 0, -2, 8, 0, 0, 0, 0, 2, 0], [0, 0, 0, -2, 0, 0, 2, 0, -2, 36, 0, -3], [0, 0, 0, 2, 0, 0, -6, 4, 0, -2, 0, 12], [23, 9, 1, 0, -6, 6, 0, 0, 0, 0, 6, 0], [22, -3, 1, 0, -3, 1, 0, 0, 0, 0, -3, 0], [2, -15, 8, 0, -2, -3, 0, 0, 0, 0, -1, 0], [0, 0, 0, -10, 0, 0, -14, -2, 2, -8, 0, -8], [0, 0, 0, -3, 0, 0, -9, -1, -3, 2, 0, -16], [0, 0, 0, -6, 0, 0, -12, 5, 0, 4, 0, -2], [-10, -12, 12, 0, 2, 7, 0, 0, 0, 0, 2, 0], [14, -7, -9, 0, 7, -5, 0, 0, 0, 0, -4, 0], [0, 0, 0, 2, 0, 0, -2, -4, -8, -2, 0, 5], [-4, 15, -8, 0, -4, 1, 0, 0, 0, 0, -5, 0], [-8, -20, 1, 0, -1, 2, 0, 0, 0, 0, 3, 0], [0, 0, 0, 2, 0, 0, -6, 9, 4, -2, 0, -6], [0, 0, 0, -9, 0, 0, -2, -6, -2, -7, 0, 5], [11, 2, 8, 0, -3, 3, 0, 0, 0, 0, 1, 0], [0, 0, 0, -4, 0, 0, 0, 8, 0, 0, 0, 0], [0, 0, 0, -2, 0, 0, -4, -5, 4, -8, 0, -4], [2, 7, 10, 0, -4, 1, 0, 0, 0, 0, -5, 0], [0, 0, 0, 0, 0, 0, -4, 4, -8, 0, 0, 8], [-1, -9, 9, 0, 10, 10, 0, 0, 0, 0, -2, 0], [-19, 2, 2, 0, 9, 5, 0, 0, 0, 0, 3, 0], [-6, 8, 4, 0, -2, 8, 0, 0, 0, 0, -6, 0], [0, 0, 0, 4, 0, 0, -2, -1, -2, -16, 0, -14], [0, 0, 0, 3, 0, 0, -4, 4, -4, -17, 0, 3], [-24, 0, 1, 0, 1, -4, 0, 0, 0, 0, -11, 0], [0, 0, 0, -11, 0, 0, 3, -3, -1, 4, 0, 14], [-12, -16, 6, 0, 8, 0, 0, 0, 0, 0, 10, 0], [-20, 4, -2, 0, 6, 0, 0, 0, 0, 0, 0, 0], [6, -6, -10, 0, -2, 2, 0, 0, 0, 0, 8, 0], [0, 0, 0, 6, 0, 0, -2, 6, 6, 38, 0, 0], [0, 0, 0, 1, 0, 0, -4, -4, -6, 31, 0, -15], [0, 0, 0, 6, 0, 0, 2, 4, 0, -30, 0, 4], [0, 0, 0, -6, 0, 0, 4, 4, 0, 24, 0, 0], [0, 0, 0, -4, 0, 0, -9, 4, -3, -9, 0, 11], [7, -9, -5, 0, -4, 6, 0, 0, 0, 0, -4, 0], [-12, 4, 1, 0, 1, -4, 0, 0, 0, 0, -3, 0], [0, 0, 0, 2, 0, 0, 2, 10, 4, -10, 0, 2], [0, 0, 0, 12, 0, 0, 0, 0, -4, 18, 0, -18], [0, 0, 0, 10, 0, 0, -5, -2, -5, 23, 0, -3], [-18, 12, -8, 0, -8, 3, 0, 0, 0, 0, -6, 0], [0, 0, 0, 15, 0, 0, -3, -5, -11, -10, 0, -4], [0, 0, 0, 1, 0, 0, -9, 1, -4, -17, 0, 13], [20, 4, -6, 0, 0, 0, 0, 0, 0, 0, 2, 0], [0, 0, 0, 8, 0, 0, 6, 6, 0, 20, 0, 0], [0, 0, 0, -7, 0, 0, -1, 3, -5, 8, 0, 6], [0, 0, 0, 7, 0, 0, -3, -1, -6, 15, 0, 2], [0, 0, 0, 13, 0, 0, 1, -1, -6, 23, 0, -8], [-11, 17, 7, 0, 2, -4, 0, 0, 0, 0, 2, 0], [-28, -16, 4, 0, 8, 3, 0, 0, 0, 0, -8, 0], [-14, 8, 4, 0, -4, 7, 0, 0, 0, 0, 2, 0], [-10, 17, 6, 0, 0, 5, 0, 0, 0, 0, 1, 0], [0, 0, 0, 6, 0, 0, -6, 6, -8, -22, 0, -16], [0, 0, 0, 8, 0, 0, 0, -8, -12, -12, 0, -8], [-4, -11, 3, 0, 5, 1, 0, 0, 0, 0, 6, 0], [0, 0, 0, 6, 0, 0, 8, 7, 4, 40, 0, -2], [18, -5, 0, 0, 6, -1, 0, 0, 0, 0, 1, 0], [0, 13, -1, 0, 5, 1, 0, 0, 0, 0, -7, 0], [2, 12, -2, 0, -8, 8, 0, 0, 0, 0, -4, 0], [0, 0, 0, -8, 0, 0, 2, 2, -2, -12, 0, 4], [0, 0, 0, 6, 0, 0, 6, -2, 0, -26, 0, -1], [-18, 1, -3, 0, 1, -3, 0, 0, 0, 0, -11, 0], [5, 0, 2, 0, 7, 1, 0, 0, 0, 0, -13, 0], [0, 0, 0, -9, 0, 0, 9, 1, 9, -24, 0, -10], [-22, 8, 2, 0, -4, -2, 0, 0, 0, 0, -2, 0], [-16, 14, -2, 0, -4, -6, 0, 0, 0, 0, 8, 0], [0, 0, 0, 5, 0, 0, 11, 3, 10, -11, 0, 7], [0, 0, 0, 6, 0, 0, 6, -2, 0, -10, 0, 1], [-9, -6, 8, 0, 3, 7, 0, 0, 0, 0, -1, 0], [0, 0, 0, -9, 0, 0, -3, 1, 9, -2, 0, 10], [0, 0, 0, 1, 0, 0, -5, -3, -1, -32, 0, -2], [0, 0, 0, 1, 0, 0, -1, 7, -6, 5, 0, 2], [4, -2, -10, 0, -2, -14, 0, 0, 0, 0, -6, 0], [0, 0, 0, 4, 0, 0, -5, -4, -1, 5, 0, 3], [0, 0, 0, 0, 0, 0, 6, 1, 2, 4, 0, 12], [0, 0, 0, -8, 0, 0, 4, -2, -6, 2, 0, 15], [-21, -3, 1, 0, 10, -4, 0, 0, 0, 0, 6, 0], [0, 0, 0, 2, 0, 0, 8, -9, 2, 16, 0, -4], [0, -13, 1, 0, -5, -9, 0, 0, 0, 0, -5, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2240_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2240_2_b_g();".
// 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_b_g(:prec:=12)
    chi := MakeCharacter_2240_b();
    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_b_g();".
// 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_b_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2240_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, 0, 488, 0, 1481, 0, 1044, 0, 270, 0, 28, 0, 1]>,<23,R![1024, -2304, 1504, -160, -80, 4, 1]>,<31,R![1024, -4608, 784, 400, -80, -4, 1]>],Snew);
    return Vf;
end function;