
// Make newform 4140.2.f.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4140_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4140_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_4140_2_f_b();".
// 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_4140_2_f_b();".
// 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, 80, 0, 508, 0, 530, 0, 188, 0, 24, 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], [159, -1064, 121, -752, -261, -144, -157, -8, -23, 0, -1, 0], [83, 0, 869, 0, 1215, 0, 503, 0, 69, 0, 3, 0], [415, 0, 3689, 0, 3779, 0, 1331, 0, 169, 0, 7, 0], [497, 0, 3895, 0, 4493, 0, 1661, 0, 215, 0, 9, 0], [0, -663, 0, -3913, 0, -3875, 0, -1339, 0, -169, 0, -7], [0, 58, 0, 496, 0, 529, 0, 188, 0, 24, 0, 1], [159, -1419, 121, -12597, -261, -13815, -157, -5007, -23, -645, -1, -27], [-143, 1515, 231, 12597, 453, 13815, 173, 5007, 23, 645, 1, 27], [143, 1515, -231, 12597, -453, 13815, -173, 5007, -23, 645, -1, 27], [159, 1419, 121, 12597, -261, 13815, -157, 5007, -23, 645, -1, 27], [0, -3245, 0, -19547, 0, -19993, 0, -7017, 0, -891, 0, -37]];
    Rf_basisdens := [1, 96, 48, 96, 96, 48, 6, 96, 96, 96, 96, 96];
    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_4140_f();"
function MakeCharacter_4140_f()
    N := 4140;
    order := 2;
    char_gens := [2071, 461, 1657, 3961];
    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_4140_f_Hecke();"
function MakeCharacter_4140_f_Hecke(Kf)
    N := 4140;
    order := 2;
    char_gens := [2071, 461, 1657, 3961];
    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, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0], [-1, 0, -1, 1, 1, 0, 0, 0, 1, -1, 0, 0], [0, 0, 0, 0, 0, 1, -1, 1, 1, 1, -1, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1], [-1, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [2, 0, -2, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 1, 0, 0, -1, 1, -1, -1, 0], [0, -1, 0, 0, 0, 0, 5, 2, 0, 0, -1, -2], [0, 0, 2, 0, -1, 0, 0, -1, -1, 1, -1, 0], [0, 3, 0, 0, 0, -1, -5, -2, 0, 0, -1, 1], [0, 0, 0, 0, 0, -1, -3, 1, -1, -1, -1, 0], [0, 2, 0, 0, 0, -1, -2, 0, 1, 1, -2, 1], [-3, 0, -1, 2, 1, 0, 0, 2, 3, -3, 2, 0], [1, 0, 1, -1, -3, 0, 0, 0, -2, 2, 0, 0], [0, 1, 0, 0, 0, -2, 3, -2, 0, 0, 1, 0], [4, 0, -2, -2, 1, 0, 0, 2, 0, 0, 2, 0], [0, 0, 0, 0, 0, -1, 1, 0, -2, -2, 0, 2], [-1, 0, 1, -1, -1, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, -3, 2, 0, -2, -2, 2, 3], [-5, 0, 3, -1, 1, 0, 0, 2, -2, 2, 2, 0], [0, -1, 0, 0, 0, 1, 1, 2, 3, 3, -1, 1], [1, 0, -1, -2, -1, 0, 0, 3, 0, 0, 3, 0], [0, -2, 0, 0, 0, 2, -4, 2, 1, 1, 0, -4], [0, -1, 0, 0, 0, 2, -5, 2, 1, 1, -1, 2], [-1, 0, -1, 3, 1, 0, 0, -2, 0, 0, -2, 0], [0, -1, 0, 0, 0, -2, -5, 0, 0, 0, 1, 2], [0, 0, 0, 0, 0, -3, 3, -2, 0, 0, 2, 2], [3, 0, -3, 0, 0, 0, 0, 1, -2, 2, 1, 0], [0, 0, 0, 0, 0, -2, -4, 0, 0, 0, 0, 2], [4, 0, -6, 0, 3, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 2, 4, -2, 0, 0, 2, 2, -2, 2, 0], [-7, 0, 1, 0, 4, 0, 0, 2, -1, 1, 2, 0], [0, 1, 0, 0, 0, 6, 7, 0, -1, -1, -1, 0], [0, 2, 0, 0, 0, 1, -5, 0, -2, -2, -2, 0], [0, 2, 0, 0, 0, 0, 2, -1, -3, -3, -1, -2], [0, 0, 0, 0, 0, 2, -6, 3, 1, 1, -3, -2], [-3, 0, 1, -2, 2, 0, 0, 2, -1, 1, 2, 0], [-1, 0, 1, 1, 1, 0, 0, 2, 5, -5, 2, 0], [4, 0, -6, 2, 4, 0, 0, 0, 1, -1, 0, 0], [0, 2, 0, 0, 0, -3, -1, 0, 0, 0, -2, 0], [0, 2, 0, 0, 0, 1, -5, -2, -2, -2, 0, 4], [-3, 0, -3, 1, 3, 0, 0, 4, 3, -3, 4, 0], [7, 0, 3, 2, -3, 0, 0, -1, 0, 0, -1, 0], [0, -2, 0, 0, 0, 2, -4, 2, -4, -4, 0, 0], [0, 3, 0, 0, 0, 3, 1, -2, 3, 3, -1, -1], [5, 0, 1, -1, -1, 0, 0, -2, 0, 0, -2, 0], [0, 0, 0, 0, 0, -1, -1, 2, -2, -2, -2, -4], [-2, 0, -6, 4, 3, 0, 0, -1, 5, -5, -1, 0], [2, 0, 6, 0, 2, 0, 0, 2, -1, 1, 2, 0], [2, 0, 0, -4, 0, 0, 0, 0, -1, 1, 0, 0], [0, -4, 0, 0, 0, -1, 13, 5, 1, 1, -1, -4], [0, -5, 0, 0, 0, -2, 3, 4, 3, 3, 1, -2], [0, 0, 2, 0, 5, 0, 0, -1, -1, 1, -1, 0], [-3, 0, -3, 4, 1, 0, 0, 1, 4, -4, 1, 0], [0, 0, 0, 0, 0, 7, 5, 1, 3, 3, -1, 0], [13, 0, -1, -1, -3, 0, 0, 0, 3, -3, 0, 0], [0, -2, 0, 0, 0, -1, -16, 0, -2, -2, 2, -1], [0, 6, 0, 0, 0, -1, -8, -2, -1, -1, -4, 3], [0, -4, 0, 0, 0, 0, 8, 5, -3, -3, -1, -2], [-3, 0, 1, 4, -4, 0, 0, -4, 1, -1, -4, 0], [0, -4, 0, 0, 0, 7, 10, 6, 1, 1, -2, -1], [0, -4, 0, 0, 0, -8, 2, 5, -1, -1, -1, 0], [-6, 0, -4, -2, 1, 0, 0, 4, 0, 0, 4, 0], [0, -3, 0, 0, 0, -5, -7, -2, -4, -4, 5, 5], [0, -2, 0, 0, 0, 0, 4, 0, 0, 0, 2, 6], [-2, 0, 4, 2, -1, 0, 0, -3, 3, -3, -3, 0], [0, 4, 0, 0, 0, -3, -9, -2, -4, -4, -2, 2], [9, 0, -9, -3, -3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 5, 4, 2, -2, -2, -2, -1], [0, 0, 0, 0, 0, 0, 0, -4, 5, 5, 4, 0], [-2, 0, 2, 0, -2, 0, 0, -2, 0, 0, -2, 0], [0, -3, 0, 0, 0, 2, 3, 4, -1, -1, -1, 0], [-4, 0, 6, 8, 0, 0, 0, 0, 2, -2, 0, 0], [0, -2, 0, 0, 0, 5, 5, 3, 3, 3, -1, -8], [-12, 0, -4, 2, 4, 0, 0, 4, -1, 1, 4, 0], [0, 0, 2, -2, 1, 0, 0, -1, -3, 3, -1, 0], [1, 0, 3, -1, -5, 0, 0, -2, 0, 0, -2, 0], [-5, 0, -3, -5, -1, 0, 0, 0, -1, 1, 0, 0], [7, 0, -3, -1, 1, 0, 0, 2, 0, 0, 2, 0], [0, 2, 0, 0, 0, 1, -8, 4, 2, 2, -6, -3], [13, 0, 7, -6, -6, 0, 0, -1, -2, 2, -1, 0], [0, -4, 0, 0, 0, 5, -7, 3, -1, -1, 1, -2], [3, 0, -1, 4, 3, 0, 0, 0, 5, -5, 0, 0], [0, 0, 0, 0, 0, -3, -24, -2, -4, -4, 2, -1], [-3, 0, 1, 6, -2, 0, 0, 3, 4, -4, 3, 0], [0, -4, 0, 0, 0, -2, -2, 0, -4, -4, 4, -4], [0, -4, 0, 0, 0, -5, 4, -2, 3, 3, 6, 1], [8, 0, -6, -8, 2, 0, 0, 0, -5, 5, 0, 0], [0, -4, 0, 0, 0, 11, 15, 1, 1, 1, 3, -6], [-18, 0, 0, 6, -3, 0, 0, -4, 0, 0, -4, 0], [6, 0, 2, 4, -1, 0, 0, -3, 7, -7, -3, 0], [0, 0, 0, 0, 0, -9, 0, -6, -1, -1, 6, -3], [11, 0, -1, -2, -2, 0, 0, 3, 2, -2, 3, 0], [14, 0, 0, -8, 2, 0, 0, -2, -2, 2, -2, 0], [0, 1, 0, 0, 0, 9, -5, -2, 3, 3, 1, -3], [15, 0, 1, 0, -4, 0, 0, -2, -3, 3, -2, 0], [0, 4, 0, 0, 0, 5, 5, 1, -1, -1, -5, 6], [0, 7, 0, 0, 0, 8, 5, 0, -1, -1, -7, -2], [0, 2, 0, 0, 0, 3, 8, -4, 1, 1, 2, -5], [-4, 0, -4, 4, 4, 0, 0, 0, 7, -7, 0, 0], [-14, 0, 4, 0, -4, 0, 0, -8, -3, 3, -8, 0], [0, 2, 0, 0, 0, 5, -9, 3, -3, -3, -5, -2], [0, 4, 0, 0, 0, -5, 1, -10, -6, -6, 6, 6], [0, 4, 0, 0, 0, 4, -10, -4, -2, -2, 0, 6], [4, 0, 2, -4, 0, 0, 0, -3, 3, -3, -3, 0], [6, 0, -2, 4, 3, 0, 0, 6, 4, -4, 6, 0], [0, 6, 0, 0, 0, 8, 16, -3, -3, -3, -3, -4], [0, -4, 0, 0, 0, 2, -6, 2, -3, -3, 2, 0], [0, 3, 0, 0, 0, -12, 3, -4, 3, 3, 1, 4], [8, 0, -8, -2, 8, 0, 0, 2, 0, 0, 2, 0], [3, 0, 5, 11, -3, 0, 0, 0, 2, -2, 0, 0], [-1, 0, -9, 1, 5, 0, 0, 0, -2, 2, 0, 0], [0, -2, 0, 0, 0, 1, -12, -2, 0, 0, 4, 5], [0, -6, 0, 0, 0, 1, -3, -2, 0, 0, 8, -2], [0, 0, 0, 0, 0, 5, -1, 3, 1, 1, -3, -8], [-14, 0, 4, 4, 4, 0, 0, -4, 1, -1, -4, 0], [-6, 0, 8, -6, 4, 0, 0, -2, -6, 6, -2, 0], [0, 0, 0, 0, 0, -1, 1, 6, 6, 6, -6, -2], [0, 0, 0, 0, 0, 9, 12, 2, 5, 5, -2, -5], [0, -2, 0, 0, 0, 1, 5, 4, 6, 6, -2, -2], [-22, 0, 6, 6, 6, 0, 0, 0, 0, 0, 0, 0], [-9, 0, -3, -3, 3, 0, 0, -6, 5, -5, -6, 0], [19, 0, -7, -3, -5, 0, 0, 2, -3, 3, 2, 0], [-9, 0, 3, 4, -5, 0, 0, 1, 2, -2, 1, 0], [0, -8, 0, 0, 0, -9, 10, 8, -2, -2, 0, -1], [0, 8, 0, 0, 0, -7, -24, -10, -4, -4, 2, 3], [12, 0, -2, -6, -9, 0, 0, -4, -8, 8, -4, 0], [0, 4, 0, 0, 0, 4, -6, -6, 3, 3, 2, 2], [-1, 0, -11, 1, -3, 0, 0, -4, -2, 2, -4, 0], [0, -5, 0, 0, 0, 2, -7, 6, 8, 8, -1, -4], [16, 0, -2, 4, 7, 0, 0, -1, 3, -3, -1, 0], [-2, 0, -6, -4, -6, 0, 0, 4, -3, 3, 4, 0], [0, -7, 0, 0, 0, -1, 5, 2, -6, -6, 5, -7], [0, 4, 0, 0, 0, -5, 18, -6, -3, -3, 2, -1], [0, -7, 0, 0, 0, 2, -11, -4, -5, -5, 11, 2], [7, 0, -7, 2, 9, 0, 0, 3, 4, -4, 3, 0], [6, 0, -14, 0, 3, 0, 0, -1, -5, 5, -1, 0], [2, 0, 8, -6, -6, 0, 0, 0, -6, 6, 0, 0], [0, 4, 0, 0, 0, -1, -13, -8, -4, -4, 4, 0], [0, -10, 0, 0, 0, -3, 0, 0, -9, -9, 10, -3], [5, 0, 7, -8, -5, 0, 0, 5, -4, 4, 5, 0], [13, 0, -3, -1, 1, 0, 0, -4, -1, 1, -4, 0], [0, -2, 0, 0, 0, -8, -18, 1, -9, -9, 1, 2], [0, -8, 0, 0, 0, -3, 7, 2, 4, 4, 6, -2], [-8, 0, -8, -4, 1, 0, 0, -3, 5, -5, -3, 0], [0, 8, 0, 0, 0, -11, -23, -6, 0, 0, -2, 8], [0, -10, 0, 0, 0, -2, 16, 1, -5, -5, 9, -2], [16, 0, 4, -8, -2, 0, 0, 6, -1, 1, 6, 0], [0, -4, 0, 0, 0, 4, -14, -1, -1, -1, 5, 0], [0, 10, 0, 0, 0, 5, -13, -3, 1, 1, -7, -2], [0, 1, 0, 0, 0, -14, -3, -6, 1, 1, 5, 2], [5, 0, 7, 7, -3, 0, 0, -6, 0, 0, -6, 0], [-7, 0, 1, 1, 1, 0, 0, 2, 5, -5, 2, 0], [2, 0, 8, 4, -5, 0, 0, -1, -1, 1, -1, 0], [0, -9, 0, 0, 0, -9, 3, 4, 3, 3, 5, 1], [-16, 0, 0, 8, 6, 0, 0, -6, 1, -1, -6, 0], [0, -2, 0, 0, 0, -7, 5, -2, -4, -4, 4, -6], [0, 6, 0, 0, 0, -2, -8, -4, 6, 6, -2, 0], [0, 0, 0, 0, 0, 5, 1, -8, 8, 8, 8, -4], [18, 0, -6, 6, 0, 0, 0, 8, 6, -6, 8, 0], [0, 3, 0, 0, 0, -8, -23, 0, 0, 0, -3, 2], [0, 4, 0, 0, 0, -11, -12, -8, 7, 7, 4, -5], [-27, 0, 13, -8, -1, 0, 0, -9, -6, 6, -9, 0], [0, 4, 0, 0, 0, -6, 16, 1, 1, 1, -5, 4], [-17, 0, -1, 9, 7, 0, 0, 6, 12, -12, 6, 0], [0, 6, 0, 0, 0, -9, 5, -8, -2, -2, 2, 8], [7, 0, -5, 5, 5, 0, 0, 0, 3, -3, 0, 0], [14, 0, -4, -2, -9, 0, 0, -5, -1, 1, -5, 0], [2, 0, 16, 10, 0, 0, 0, -4, -1, 1, -4, 0], [0, 6, 0, 0, 0, 3, -24, -4, 7, 7, -2, -5], [-13, 0, -3, 6, 6, 0, 0, -7, 0, 0, -7, 0], [-23, 0, -9, 1, 9, 0, 0, 8, 5, -5, 8, 0], [3, 0, -3, 2, -10, 0, 0, -5, 2, -2, -5, 0], [-10, 0, -2, -4, 1, 0, 0, -8, 0, 0, -8, 0], [0, 5, 0, 0, 0, -6, -31, -2, -5, -5, -3, 2], [-14, 0, 2, -4, 8, 0, 0, 0, 1, -1, 0, 0], [0, -6, 0, 0, 0, 7, 29, 8, 8, 8, -2, -4], [-5, 0, 1, -5, 1, 0, 0, -10, -2, 2, -10, 0], [0, -8, 0, 0, 0, 7, 3, 2, -4, -4, 6, 0], [0, 0, 0, 0, 0, 8, 4, 7, -3, -3, -7, -6], [0, 6, 0, 0, 0, -3, -12, 4, 2, 2, -10, 5], [-2, 0, -2, -8, 4, 0, 0, -4, 1, -1, -4, 0], [0, 0, 0, 0, 0, -2, 14, 0, -2, -2, 0, 8], [0, -7, 0, 0, 0, -10, -3, 4, 1, 1, 3, 2], [2, 0, 8, -8, -1, 0, 0, 1, 1, -1, 1, 0], [9, 0, 5, -4, -5, 0, 0, -3, -4, 4, -3, 0], [0, 10, 0, 0, 0, 1, -3, -9, -3, -3, -1, 8], [0, -6, 0, 0, 0, 4, 0, -1, 3, 3, 7, 0], [18, 0, -16, -6, 2, 0, 0, 4, -5, 5, 4, 0], [-17, 0, -15, -4, 7, 0, 0, 1, 8, -8, 1, 0], [0, -4, 0, 0, 0, 6, -4, 0, -5, -5, 4, 6], [0, 1, 0, 0, 0, 2, 7, -2, -1, -1, 1, 0], [-25, 0, -7, 7, 1, 0, 0, -6, 3, -3, -6, 0], [0, 2, 0, 0, 0, -4, -18, 2, 6, 6, -4, 0], [0, 0, 0, 0, 0, 2, 16, -2, -2, -2, 2, 6], [0, 12, 0, 0, 0, 13, 1, -5, -5, -5, -7, -2], [0, 0, -6, -2, -8, 0, 0, 2, -6, 6, 2, 0], [4, 0, -6, 0, 4, 0, 0, 13, 7, -7, 13, 0], [0, -4, 0, 0, 0, -1, 9, -3, 11, 11, 7, 8], [17, 0, 3, 5, -1, 0, 0, 4, -1, 1, 4, 0], [-14, 0, -10, 4, 2, 0, 0, -2, -4, 4, -2, 0], [0, 0, 0, 0, 0, 9, -13, 2, 0, 0, -2, -2], [-9, 0, 3, -1, 9, 0, 0, -4, 3, -3, -4, 0], [0, -12, 0, 0, 0, -5, 3, -1, -3, -3, 13, -6], [-6, 0, 8, 2, -5, 0, 0, -5, -9, 9, -5, 0], [7, 0, -1, 8, 8, 0, 0, 2, 7, -7, 2, 0], [0, -2, 0, 0, 0, 13, 21, 3, -7, -7, -1, -8], [-15, 0, 3, 8, 8, 0, 0, -2, -1, 1, -2, 0], [0, -3, 0, 0, 0, -11, -7, 2, -4, -4, 1, 7], [0, 10, 0, 0, 0, -4, 8, 4, 2, 2, -14, -6], [7, 0, 17, 2, -3, 0, 0, -2, -5, 5, -2, 0], [31, 0, -9, -7, 5, 0, 0, 4, -4, 4, 4, 0], [0, -12, 0, 0, 0, 1, 23, 9, -3, -3, 3, -4], [-28, 0, -4, -8, -5, 0, 0, 2, -6, 6, 2, 0], [0, 8, 0, 0, 0, 9, 2, -10, 2, 2, 2, 1], [0, 0, 0, 0, 0, -7, -27, -5, -1, -1, 5, -6], [2, 0, -6, -2, -9, 0, 0, -4, 0, 0, -4, 0], [7, 0, -5, 1, 5, 0, 0, 6, -3, 3, 6, 0], [0, 0, 2, -6, 5, 0, 0, 7, -7, 7, 7, 0], [0, -15, 0, 0, 0, 0, 21, 8, 2, 2, 7, -4], [0, -2, 0, 0, 0, -1, -13, -1, 7, 7, 3, -4], [6, 0, 6, -2, -9, 0, 0, -4, 0, 0, -4, 0], [0, -3, 0, 0, 0, -7, -11, 14, 1, 1, -11, -1], [0, 0, 8, 8, -5, 0, 0, -1, -3, 3, -1, 0], [0, 4, 0, 0, 0, 6, 22, 6, 2, 2, -10, 0], [21, 0, 3, -4, -1, 0, 0, 8, 5, -5, 8, 0], [0, 4, 0, 0, 0, -2, -24, -6, -4, -4, 2, 2], [-5, 0, 7, -11, -1, 0, 0, -2, -15, 15, -2, 0], [0, 0, -6, -4, -4, 0, 0, 2, 5, -5, 2, 0], [5, 0, -7, -2, 7, 0, 0, 8, 5, -5, 8, 0], [0, 8, 0, 0, 0, 13, -14, 6, -7, -7, -14, -7], [0, -11, 0, 0, 0, 5, 7, 4, -3, -3, 7, -3], [-4, 0, -22, 8, 8, 0, 0, 0, 4, -4, 0, 0], [0, -9, 0, 0, 0, -5, -9, 4, 5, 5, 5, 1], [38, 0, -10, -8, -5, 0, 0, 3, 1, -1, 3, 0], [16, 0, -2, -2, 1, 0, 0, 4, 10, -10, 4, 0], [0, -4, 0, 0, 0, 7, 2, 6, 1, 1, -2, -13], [16, 0, -8, -12, -5, 0, 0, 5, -1, 1, 5, 0], [0, -6, 0, 0, 0, 7, 11, -3, -3, -3, 9, -4], [-1, 0, -11, 6, 6, 0, 0, -4, 7, -7, -4, 0], [0, -14, 0, 0, 0, 6, 16, 11, 7, 7, 3, -2], [-9, 0, -1, -8, 0, 0, 0, 0, -11, 11, 0, 0], [0, 6, 0, 0, 0, -1, -11, -5, -5, -5, -1, 6], [14, 0, 2, 10, 4, 0, 0, -8, 4, -4, -8, 0], [-8, 0, 0, 10, 2, 0, 0, -12, 2, -2, -12, 0], [0, -14, 0, 0, 0, 19, 12, 16, -2, -2, -2, -5], [0, -2, 0, 0, 0, 8, 20, -2, 9, 9, 4, -6], [20, 0, 0, 6, -8, 0, 0, 2, -3, 3, 2, 0], [0, -6, 0, 0, 0, 3, 22, 4, -9, -9, 2, -7], [-1, 0, 5, -9, -9, 0, 0, 4, -1, 1, 4, 0], [0, 10, 0, 0, 0, -8, 0, -9, 1, 1, -1, -4], [-32, 0, -2, 4, 7, 0, 0, 2, 12, -12, 2, 0], [31, 0, 13, -3, -3, 0, 0, 2, -5, 5, 2, 0], [0, -2, 0, 0, 0, -8, -6, -2, 1, 1, 4, 4], [0, 4, 0, 0, 0, 1, -51, -1, 3, 3, -3, 6], [0, -4, 0, 0, 0, -5, -17, 1, -13, -13, 3, 2], [0, 8, 0, 0, 0, 3, -10, 4, 5, 5, -12, -5], [0, 3, 0, 0, 0, 2, 61, -2, 1, 1, -1, 0], [3, 0, 11, 6, -9, 0, 0, -4, -1, 1, -4, 0], [0, -3, 0, 0, 0, 4, -19, 0, -2, -2, 3, -8], [0, 0, 0, 0, 0, 8, -2, 14, 4, 4, -14, 2], [-21, 0, -11, 5, 7, 0, 0, 2, -4, 4, 2, 0], [-1, 0, -3, 9, -11, 0, 0, -6, 9, -9, -6, 0], [13, 0, 9, 7, 3, 0, 0, 10, 8, -8, 10, 0], [0, -11, 0, 0, 0, -6, 1, 8, 13, 13, 3, -2], [0, -16, 0, 0, 0, 19, 33, 11, 1, 1, 5, -12], [-22, 0, -20, 12, 17, 0, 0, 3, 9, -9, 3, 0], [0, -8, 0, 0, 0, -7, 20, -2, -7, -7, 10, -9], [0, 5, 0, 0, 0, 2, 15, -4, -5, -5, -1, -8], [-7, 0, -3, -7, -7, 0, 0, 4, -1, 1, 4, 0], [0, 8, 0, 0, 0, 13, -17, -8, 0, 0, 0, 2], [0, 6, 0, 0, 0, 7, -19, 1, 1, 1, -7, 2], [0, 4, 0, 0, 0, -10, -12, 0, -4, -4, -4, 10], [-15, 0, 3, 8, 3, 0, 0, 1, -6, 6, 1, 0], [-16, 0, -8, -6, 0, 0, 0, -4, -5, 5, -4, 0], [12, 0, 8, 2, 4, 0, 0, -2, -10, 10, -2, 0], [0, -2, 0, 0, 0, -1, -17, 3, -1, -1, -1, 16], [-16, 0, 2, -12, 4, 0, 0, 0, -3, 3, 0, 0], [0, -7, 0, 0, 0, -9, 5, 0, 1, 1, 7, -3], [-6, 0, 12, 6, 0, 0, 0, 0, -6, 6, 0, 0], [0, -8, 0, 0, 0, 0, 4, 0, -6, -6, 8, -4], [-27, 0, 9, -14, 1, 0, 0, -3, -8, 8, -3, 0], [0, -2, 0, 0, 0, -3, 22, 10, 9, 9, -8, -13], [0, 11, 0, 0, 0, -9, -15, -8, 0, 0, -3, -1], [38, 0, -2, -12, 4, 0, 0, 4, -4, 4, 4, 0], [24, 0, 18, -14, -2, 0, 0, 6, -4, 4, 6, 0], [-4, 0, -6, 18, 10, 0, 0, 4, 16, -16, 4, 0], [0, 12, 0, 0, 0, -9, -8, -6, 4, 4, -6, -3], [0, 0, 0, 0, 0, 2, 16, 14, 0, 0, -14, 2], [-23, 0, 3, 7, -3, 0, 0, 10, 10, -10, 10, 0], [0, -6, 0, 0, 0, 7, -1, 10, -12, -12, -4, 2], [-2, 0, 8, 0, 4, 0, 0, -10, 0, 0, -10, 0], [-42, 0, 0, 12, 4, 0, 0, -2, -5, 5, -2, 0], [0, -8, 0, 0, 0, 11, -1, 5, -1, -1, 3, -10], [-8, 0, 2, -2, -1, 0, 0, 8, -8, 8, 8, 0], [0, 10, 0, 0, 0, 1, -9, 7, 7, 7, -17, 10], [0, 17, 0, 0, 0, -12, -19, -8, 2, 2, -9, 12], [0, 5, 0, 0, 0, 0, 13, 4, -5, -5, -9, -16], [8, 0, 14, -4, -10, 0, 0, 6, -11, 11, 6, 0], [0, -4, 0, 0, 0, 3, -11, -4, -18, -18, 8, -2], [32, 0, 8, -8, 8, 0, 0, -2, -14, 14, -2, 0], [0, 4, 0, 0, 0, 7, -27, 2, 4, 4, -6, 6], [0, -16, 0, 0, 0, -2, 0, 12, 2, 2, 4, 2], [30, 0, -4, -4, 2, 0, 0, 0, 4, -4, 0, 0], [16, 0, -10, -8, -4, 0, 0, -6, -1, 1, -6, 0], [0, 12, 0, 0, 0, -12, -36, -10, 6, 6, -2, 8], [0, -4, 0, 0, 0, 19, -5, 10, -8, -8, -6, -8], [-21, 0, -9, 7, -5, 0, 0, 0, 4, -4, 0, 0], [-30, 0, 6, 12, -12, 0, 0, -12, -3, 3, -12, 0], [0, -10, 0, 0, 0, -7, 9, 1, -11, -11, 9, -12], [0, 1, 0, 0, 0, -10, -11, -2, 5, 5, 1, 6], [21, 0, 7, 1, 5, 0, 0, -6, -15, 15, -6, 0], [-2, 0, -16, 6, 1, 0, 0, 2, -4, 4, 2, 0], [10, 0, -6, -2, 3, 0, 0, 11, -3, 3, 11, 0], [0, -10, 0, 0, 0, -9, -4, -2, -5, -5, 12, 7], [11, 0, -25, 0, 12, 0, 0, 9, -2, 2, 9, 0], [7, 0, -7, -15, -7, 0, 0, 16, -2, 2, 16, 0], [0, 3, 0, 0, 0, -12, -33, -8, 2, 2, 5, 4], [7, 0, 21, 16, -12, 0, 0, -8, -1, 1, -8, 0], [0, -2, 0, 0, 0, -4, 26, 1, 7, 7, 1, 2], [0, 6, 0, 0, 0, -13, -4, -8, 2, 2, 2, 3], [-30, 0, 8, 4, -6, 0, 0, 8, 8, -8, 8, 0], [-20, 0, 10, 18, 4, 0, 0, 10, 11, -11, 10, 0], [0, -4, 0, 0, 0, 7, 19, 8, 4, 4, -4, 0], [0, -21, 0, 0, 0, 17, 31, 12, 5, 5, 9, -5], [0, -12, 0, 0, 0, 5, 16, -2, 4, 4, 14, 3], [-14, 0, -4, 12, 3, 0, 0, -7, 5, -5, -7, 0], [0, -4, 0, 0, 0, 7, -11, -6, 10, 10, 10, -10], [-30, 0, 2, 6, 1, 0, 0, 3, 15, -15, 3, 0], [0, -6, 0, 0, 0, 3, -43, 11, 7, 7, -5, -2], [15, 0, -3, 1, -1, 0, 0, 10, -10, 10, 10, 0], [0, 6, 0, 0, 0, -9, 27, 7, 5, 5, -13, -4], [-16, 0, 8, 4, -4, 0, 0, 4, 7, -7, 4, 0], [0, -5, 0, 0, 0, -3, -47, 0, 11, 11, 5, -3], [27, 0, 17, 0, -16, 0, 0, -2, -5, 5, -2, 0], [0, 3, 0, 0, 0, -10, 1, -8, 11, 11, 5, 12], [0, -4, 0, 0, 0, 7, 1, 2, 6, 6, 2, 2], [0, -1, 0, 0, 0, -3, 7, 8, -2, -2, -7, -5], [4, 0, -18, 10, 6, 0, 0, 4, 10, -10, 4, 0], [30, 0, 6, -4, -12, 0, 0, 12, -6, 6, 12, 0], [0, 8, 0, 0, 0, 3, -6, -4, -2, -2, -4, 7], [7, 0, -1, -22, -9, 0, 0, -5, -16, 16, -5, 0], [-14, 0, -4, 22, 3, 0, 0, -1, -1, 1, -1, 0], [0, -8, 0, 0, 0, -6, -32, 5, 1, 1, 3, 8], [6, 0, 18, 0, 0, 0, 0, 0, -12, 12, 0, 0], [0, -2, 0, 0, 0, -5, -32, -6, 2, 2, 8, 11], [-8, 0, -26, 8, 1, 0, 0, -2, 14, -14, -2, 0], [0, 16, 0, 0, 0, 15, -23, 0, 6, 6, -16, -6], [-7, 0, 13, -6, -13, 0, 0, -18, -13, 13, -18, 0], [0, 2, 0, 0, 0, -9, -46, 0, 7, 7, -2, 17], [12, 0, -12, 0, 6, 0, 0, 0, 3, -3, 0, 0], [0, -8, 0, 0, 0, -8, 36, 9, -1, -1, -1, -10], [28, 0, 6, 2, 10, 0, 0, 2, 5, -5, 2, 0], [38, 0, -8, 0, -10, 0, 0, 10, -1, 1, 10, 0], [0, 12, 0, 0, 0, -21, -15, -14, -4, -4, 2, 14], [0, -6, 0, 0, 0, 2, -8, 7, -7, -7, -1, -6], [0, 6, 0, 0, 0, 7, -4, -2, -6, -6, -4, -5], [-30, 0, 2, -2, -3, 0, 0, 0, 0, 0, 0, 0], [0, 12, 0, 0, 0, 9, -3, 4, 6, 6, -16, 8], [-14, 0, 0, -2, -6, 0, 0, 16, 3, -3, 16, 0], [0, 0, 0, 0, 0, 3, -15, -1, 11, 11, 1, -12], [0, -2, 0, 0, 0, 7, 39, 8, 0, 0, -6, -6], [0, 8, 0, 0, 0, 9, -5, 0, -4, -4, -8, -6], [0, -5, 0, 0, 0, 18, 25, 10, 14, 14, -5, -16], [-2, 0, -4, 2, 8, 0, 0, -8, 4, -4, -8, 0], [-2, 0, -4, -2, 17, 0, 0, 6, 2, -2, 6, 0], [-17, 0, 3, 18, 11, 0, 0, 3, 14, -14, 3, 0], [0, 14, 0, 0, 0, -5, -31, -8, 4, 4, -6, -8], [8, 0, 8, -6, -6, 0, 0, -20, -15, 15, -20, 0], [-26, 0, 6, 2, -10, 0, 0, 0, -10, 10, 0, 0], [0, 4, 0, 0, 0, 1, -57, 1, 5, 5, -5, 4], [60, 0, 0, -8, -17, 0, 0, -5, -7, 7, -5, 0], [9, 0, 5, 7, -9, 0, 0, 4, -5, 5, 4, 0], [0, -10, 0, 0, 0, -15, 26, 6, -7, -7, 4, 3], [-24, 0, 14, -6, -4, 0, 0, -4, -8, 8, -4, 0], [0, 8, 0, 0, 0, 1, -17, -4, 12, 12, -4, 14], [16, 0, -4, 6, 16, 0, 0, -4, 15, -15, -4, 0], [0, -3, 0, 0, 0, 1, 23, 2, -16, -16, 1, -3], [0, 0, 0, 0, 0, -8, 14, 2, 0, 0, -2, 2], [0, 10, 0, 0, 0, -15, 25, -2, 4, 4, -8, 2], [3, 0, -1, 11, 7, 0, 0, -4, 15, -15, -4, 0], [0, -8, 0, 0, 0, -29, 21, 1, -3, -3, 7, 4], [18, 0, 10, 16, -5, 0, 0, -11, 11, -11, -11, 0], [0, -2, 0, 0, 0, -18, -32, -5, -5, -5, 7, 2], [0, 4, 0, 0, 0, -5, -6, -4, 0, 0, 0, -9], [0, 10, 0, 0, 0, -1, -14, 0, -4, -4, -10, 13], [20, 0, -8, -16, -2, 0, 0, -10, -19, 19, -10, 0], [0, -6, 0, 0, 0, -6, 12, -5, -9, -9, 11, -10], [34, 0, 2, 2, 15, 0, 0, -3, 3, -3, -3, 0], [0, 14, 0, 0, 0, 2, 6, -17, -7, -7, 3, 12], [-15, 0, -3, 7, 1, 0, 0, -8, 5, -5, -8, 0], [0, -4, 0, 0, 0, 3, 14, 6, -16, -16, -2, 3], [13, 0, -17, 3, -9, 0, 0, -6, 10, -10, -6, 0], [3, 0, 1, -7, 3, 0, 0, 6, 9, -9, 6, 0], [18, 0, -12, -8, -4, 0, 0, -14, -7, 7, -14, 0], [-35, 0, -9, 8, -5, 0, 0, -5, 2, -2, -5, 0], [-4, 0, -20, -12, -3, 0, 0, -2, -2, 2, -2, 0], [0, 8, 0, 0, 0, -11, 1, -4, -12, -12, -4, 12], [0, 1, 0, 0, 0, -6, -23, 2, -2, -2, -3, -6], [0, -20, 0, 0, 0, -1, 4, -4, -6, -6, 24, -7], [14, 0, -14, -2, 9, 0, 0, -14, -2, 2, -14, 0], [16, 0, -2, 0, 5, 0, 0, 2, 12, -12, 2, 0], [0, 1, 0, 0, 0, 0, 25, -12, -3, -3, 11, 8], [13, 0, -11, -2, -2, 0, 0, 5, -4, 4, 5, 0], [0, 15, 0, 0, 0, 7, 29, -4, 9, 9, -11, -9], [32, 0, -10, 14, 2, 0, 0, 5, 5, -5, 5, 0], [0, -18, 0, 0, 0, 5, 19, 6, -6, -6, 12, 4], [0, 24, 0, 0, 0, -7, -17, -8, -2, -2, -16, 6], [0, 1, 0, 0, 0, 3, 13, 0, -15, -15, -1, 9], [12, 0, -4, 6, -10, 0, 0, 6, -2, 2, 6, 0], [0, -1, 0, 0, 0, 22, -5, 8, -3, -3, -7, -2], [9, 0, -9, -12, 0, 0, 0, 2, -5, 5, 2, 0], [20, 0, 14, 10, 10, 0, 0, 5, 17, -17, 5, 0], [0, 0, 0, 0, 0, 2, 28, -3, -1, -1, 3, -20], [0, 1, 0, 0, 0, 26, -5, 2, -3, -3, -3, -4], [0, 5, 0, 0, 0, -19, 9, -4, 2, 2, -1, 11], [1, 0, 31, -15, -15, 0, 0, -2, -15, 15, -2, 0], [0, 11, 0, 0, 0, -31, -9, -16, 4, 4, 5, 11], [0, 6, 0, 0, 0, -18, 4, -16, -6, -6, 10, 16], [4, 0, 0, 10, -6, 0, 0, -8, 2, -2, -8, 0], [0, -2, 0, 0, 0, 23, 11, 10, -4, -4, -8, -6], [0, 2, 0, 0, 0, 3, 11, 8, 8, 8, -10, -26], [0, 5, 0, 0, 0, -13, -17, -16, -2, -2, 11, 11], [27, 0, -21, -14, -11, 0, 0, 10, 3, -3, 10, 0], [17, 0, 1, -10, 10, 0, 0, 4, 5, -5, 4, 0], [-12, 0, 8, -22, -5, 0, 0, -7, -11, 11, -7, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4140_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4140_2_f_b();".
// 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_4140_2_f_b(:prec:=12)
    chi := MakeCharacter_4140_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(2999) - 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_4140_2_f_b();".
// 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_4140_2_f_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4140_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![82944, 0, 139536, 0, 68992, 0, 14312, 0, 1380, 0, 61, 0, 1]>],Snew);
    return Vf;
end function;