
// Make newform 6040.2.a.n in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6040_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_6040_2_a_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_6040_2_a_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 := [11, -82, 108, 373, -708, -444, 870, 109, -403, 41, 70, -14, -4, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4040, -5889, -19240, 21771, 13459, -28959, 1109, 14165, -3344, -2522, 769, 146, -49], [-8423, 31798, 38301, -154817, -53107, 171713, 8885, -74902, 9026, 12639, -2588, -711, 178], [-15237, 72146, 49407, -370815, -64733, 419143, -14213, -184594, 31378, 31293, -7452, -1765, 486], [-28888, 112017, 120244, -559995, -154375, 640815, 4559, -285965, 43236, 48834, -11097, -2766, 741], [14604, -57175, -58934, 289139, 72987, -329565, 1545, 146093, -22886, -24848, 5723, 1404, -379], [-31106, 122083, 124130, -607915, -157027, 687971, 1023, -304011, 46236, 51640, -11715, -2916, 779], [32784, -123081, -137988, 608887, 176827, -691835, -10663, 307329, -44824, -52362, 11701, 2962, -785], [-756, 2899, 3076, -14373, -3721, 16377, -71, -7271, 1148, 1238, -287, -70, 19], [-42039, 167008, 166041, -845711, -214621, 959715, 127, -424736, 64938, 72203, -16426, -4079, 1092], [75647, -299723, -302861, 1510060, 385094, -1722952, 3104, 766063, -118974, -130545, 29981, 7385, -1991]];
    Rf_basisdens := [1, 1, 1, 212, 212, 212, 212, 106, 212, 212, 4, 212, 212];
    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_6040_a();"
function MakeCharacter_6040_a()
    N := 6040;
    order := 1;
    char_gens := [1511, 3021, 2417, 761];
    v := [1, 1, 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;

function MakeCharacter_6040_a_Hecke(Kf)
    return MakeCharacter_6040_a();
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, -1, 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, 0, -1, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [-1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 2, -1, -1, -1, 1, 0, 0, -1, -1, -1, 1, 1], [-1, -1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, -1], [0, 1, 0, 0, -1, -1, 0, 0, 0, -1, -1, 1, 1], [0, -1, 0, 0, 0, -1, -1, 2, 1, 1, 1, -1, 0], [1, 0, -1, -1, 0, 1, 1, -1, 0, 1, 0, 0, -1], [0, 0, -1, -1, 1, 1, 0, 1, 1, -1, 1, 0, 0], [2, 0, -1, -1, 2, 0, 1, 0, 0, 1, -1, 0, 0], [-2, 2, 0, 0, -1, 0, 0, 1, -1, -1, -2, 0, 1], [-4, -3, 2, 3, 1, -1, -1, 0, 1, 2, 3, -1, -2], [-1, 1, 1, 0, 0, 0, -1, 0, 0, 0, -1, 1, 1], [1, 3, -1, 0, -1, 1, 0, 0, 0, -2, 0, 1, 0], [0, 1, 0, 0, -1, 0, 1, 0, 0, 1, 0, -1, 0], [0, 1, 0, -1, 1, 0, 0, 2, -1, 0, 0, 1, 2], [-1, -2, 0, 0, 2, -2, 1, -1, 1, 2, -1, 0, -1], [3, 0, 2, 1, 1, 0, 2, 0, 1, -1, 0, 0, -1], [-3, -1, 1, 3, 1, -1, -1, 1, 4, 3, 2, -1, -1], [-5, 3, 0, -1, 0, 1, 1, 0, 1, 1, 0, -1, 0], [3, 0, -2, -3, 0, 1, 2, -1, -1, -1, -1, 2, 3], [-3, 1, 1, 0, -1, -1, -1, 0, 0, -1, -3, 0, -1], [1, 3, -1, -2, 1, 1, 1, -1, -2, 0, -1, 1, -1], [-4, -3, 3, 2, 0, -1, 0, 1, 2, 1, 3, -2, -2], [-5, 2, 2, 2, -1, -1, -3, 1, 1, -1, 0, -2, 1], [-4, -2, 0, 0, -1, -1, -1, -1, -3, 0, 0, 2, 0], [-4, 2, -1, -2, -2, 2, -2, 2, -1, -1, 0, -1, 1], [-2, 1, 1, 1, 0, -2, 1, -1, 0, 1, 1, -1, -2], [0, 0, -2, -3, 1, -1, 1, 1, -2, -1, -3, 1, 1], [-5, -4, 3, 3, 1, -1, -2, 0, 2, 4, 5, -2, -4], [-3, 2, 1, 1, 2, 1, 2, 0, 2, -2, 2, -2, -1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -6, 3, 1, 2, -1, 0, 2, 4, 5, 2, -1, 0], [-1, 2, -1, -1, -1, 0, -1, 1, -2, 2, 2, -1, 2], [-4, 1, 2, 1, 0, 0, 2, 0, 4, -1, -2, -1, -1], [1, 0, -1, -1, -1, 0, 3, -1, -1, 0, -1, 1, 0], [0, 1, -1, -2, 0, 0, -1, -2, -3, -1, -2, 3, 1], [-1, -4, 0, 0, 0, -1, -1, 2, -3, 3, 1, 2, 0], [0, 1, 1, -1, -1, -1, 0, -1, 3, -2, -4, 1, 1], [-4, -3, 0, -2, 1, 1, -2, 1, 3, 0, 0, 0, 0], [-1, -5, 0, -1, 2, 0, -1, 0, -1, 2, 0, 2, -1], [-2, 0, 2, 0, -1, -1, 0, 4, 1, 3, -1, -1, 1], [1, 6, 0, 1, -2, 2, 0, -2, 1, -3, -3, 2, 2], [5, 5, -2, -2, -1, 1, 2, -3, -3, -4, -4, 3, 3], [1, 3, -4, -3, 2, 2, 3, 0, -2, -2, -1, 1, 1], [3, 0, 1, 0, 0, -1, -1, -2, -4, -1, -3, 5, 4], [-11, -3, 2, 4, 1, -1, -1, 0, 3, 3, 2, -2, -5], [-8, -8, 5, 6, 2, -4, -3, 2, 5, 4, 5, -3, -5], [-2, 1, -1, 1, -2, 4, -2, -1, -2, -5, 5, 1, -1], [1, 2, -2, -2, 0, 2, 3, -3, -3, 2, 0, 1, 1], [-8, -4, 0, 0, -3, -1, -2, 1, -3, 3, -1, 0, 1], [0, -2, -2, 1, -1, -3, -3, -1, -1, 2, -1, 3, 0], [0, -2, 0, -2, 2, 3, 0, 1, -1, 1, 3, 1, -2], [8, 1, -3, -3, 0, 3, 0, -1, -4, -5, 2, 1, 0], [3, -3, -4, -4, 2, 2, 3, -1, -1, 0, -1, 2, 3], [-5, 2, -4, -3, -2, 0, 0, 4, -1, 1, -3, -2, 2], [4, 0, -2, 1, 2, 1, 1, -2, -1, -2, 0, -1, -1], [-3, 1, 1, -1, -1, 3, -1, -1, 1, 1, 2, 0, -2], [0, -8, 2, -1, 4, -2, 1, -1, -2, 1, 0, 1, -1], [6, 4, -4, -3, 2, 2, 3, 1, -2, 1, -2, 2, 4], [-4, 6, -2, -2, -1, 3, -1, 0, -5, -4, -4, 2, 4], [-3, -2, 1, -2, -1, 3, -3, 2, -1, -1, 1, 0, -2], [2, 0, 1, 2, -1, 3, -1, -1, 3, 0, 5, -1, -1], [2, 9, -6, -1, -4, 3, 4, -6, -4, -7, -5, 5, 3], [-2, -7, 2, 3, 3, -3, 1, 2, 0, 0, -2, 0, 2], [-3, -2, 3, 4, 2, -2, -2, 1, 1, -1, 3, 0, 0], [-5, 2, 2, -1, 0, 2, 1, 0, 1, 0, -2, 2, 3], [0, 0, -6, -5, 1, 4, -2, 3, 0, -2, 0, 0, 3], [2, -5, 3, -1, 1, -1, 2, -1, -2, -1, -1, 1, -1], [-1, 2, 1, -1, -4, -1, 1, -3, -5, -3, -1, 2, -1], [2, 1, -1, -2, -3, 0, -3, 3, 1, -5, 1, -1, 2], [-8, -3, 3, 2, -2, 0, -3, -2, 1, -1, 3, -4, -4], [-4, 0, -1, -1, 3, 0, -1, -1, -2, 1, 0, 0, -1], [0, 1, -2, -3, -2, -1, -1, -1, -4, 0, -2, 5, 0], [-11, -5, 5, 8, -4, -3, -4, 2, 2, 1, 4, -5, -2], [-7, 7, 2, 1, -4, 0, 1, 0, 2, -4, 2, -4, 0], [-1, 5, -2, 0, 0, 3, 4, -2, 0, -2, -2, -4, 2], [-2, -3, -1, -2, 1, -1, -2, 6, 1, 1, -4, 3, 4], [6, -6, 4, 1, -2, -3, -4, -3, -3, -1, -1, -1, -2], [-13, -12, 6, 5, -1, -2, -3, 0, 3, 6, 9, -5, -8], [-9, -5, 4, 8, -2, -6, -4, 4, 4, 3, 2, -4, -2], [2, 1, 1, -2, -1, 0, 0, 2, -2, 2, 2, -2, 0], [-5, 9, -2, -6, -2, 2, 3, 0, -2, -8, -5, 2, 4], [-9, 0, 0, -1, 0, 2, 2, -1, 0, 0, -1, 1, -1], [0, -6, 4, 3, 2, -4, -3, 5, 4, -1, 0, -1, 5], [5, 1, 0, 2, 2, 2, 0, -1, -3, 0, 3, 1, 3], [-7, 2, -3, -2, -4, 4, 2, 0, -2, -1, -1, 1, 0], [-3, -1, 2, 2, 1, -4, 3, -1, 2, -2, 0, -4, -1], [-8, 0, -1, 5, -3, -1, 1, -1, 2, -1, 5, -7, -5], [8, -2, 0, -1, 1, 0, -4, -1, -4, -2, -4, 6, 3], [8, -2, -1, -2, -4, -1, 2, -4, -4, -3, -3, 0, 0], [0, -13, 2, -2, 7, -1, 1, 2, 5, 11, 2, 2, -6], [0, 8, -3, -6, 0, 1, 4, 3, 0, -3, -7, 2, 4], [-2, -7, 7, 3, 4, -7, 0, 2, 5, 2, 0, -5, -2], [8, 0, 1, 2, -1, -1, -4, 4, 0, -3, 4, 1, 1], [11, 4, -10, -7, 0, 3, 2, -1, -8, 2, 1, 6, 5], [-9, -12, 6, 9, 1, -4, -6, 0, 5, 3, 9, -3, -6], [4, 0, -2, -7, 4, 4, 5, 2, 1, 1, -2, 0, 4], [1, -5, 3, -1, 0, -3, -3, 0, -3, 5, -3, 0, -5], [-6, -1, 2, -1, 3, -2, -1, 2, 0, 4, 4, 2, -1], [0, -6, 3, 3, 0, -2, 1, 1, 6, 7, -2, -2, -4], [-14, -4, 3, 3, -3, -3, -2, -3, 0, -2, -4, 0, -4], [0, 5, -2, -6, -5, 2, 0, -3, -9, -6, -5, 7, 2], [-8, -2, 3, 6, -1, -1, -4, 0, -1, 3, 0, 1, 2], [-3, 4, -3, -1, -3, 4, 1, 1, 1, -4, 0, -6, -1], [5, -7, 0, -2, 5, 1, 1, -1, -1, 3, 1, 3, -1], [7, -9, 1, -1, 5, 1, 2, -1, 7, 2, 3, 3, -1], [-4, 0, 5, 5, -1, 0, -1, 1, 6, 5, 3, -7, -2], [-8, 3, 2, 4, 0, 0, -1, 0, 5, -3, 1, 3, 2], [11, -4, 4, 1, 4, -1, 0, 0, 0, -6, 1, 1, 2], [11, -5, -4, -3, 3, 5, 0, 3, 1, 4, 4, 2, 0], [5, -11, -1, -1, 2, -3, 1, -4, 0, 6, 3, 0, -3], [-6, -7, 3, 2, 5, -1, -1, 6, 10, 8, 2, -3, -5], [-1, 1, -1, -4, -1, 5, 0, -3, 0, -5, -2, 0, -4], [5, -1, -5, -2, 0, -2, 1, 2, -4, 0, -3, 2, 0], [9, 1, -4, -8, -3, 1, 3, -6, -1, -5, -8, 2, 2], [12, 0, -2, -3, 2, 0, 1, -1, -1, 3, -2, 9, 0], [-2, 9, -3, -3, -2, 4, 4, 0, 0, -7, -4, -1, 5], [-6, -8, 2, 3, 5, -2, -2, 0, 8, 8, 4, -4, -5], [3, -4, -2, 0, 0, 3, 3, 1, -7, 3, 1, 3, -4], [8, 7, -4, 0, 0, -1, 1, -4, -1, 0, -3, 5, 3], [8, -11, 2, 1, 5, -1, 1, -1, 2, 4, 5, -3, -3], [-5, 14, -1, 4, -3, 1, 2, 0, -2, -2, 1, -4, 2], [-9, -3, 7, 6, 0, -4, -3, 0, 2, 4, 2, -3, 3], [0, -10, 1, 5, 2, -2, -1, -5, -1, 8, 2, 0, -6], [17, 4, -2, -3, 3, 1, 1, -2, -1, -1, -3, 5, 5], [14, -7, -6, -6, 2, -4, 1, 4, 1, 5, -5, -2, 2], [3, -4, -5, -7, 7, 1, 4, 5, 2, 4, 4, 3, 3], [6, 9, -3, -3, -11, 5, -5, -5, -12, -8, -4, 4, 6], [-1, 9, 0, 2, -4, 3, -3, 2, 1, -13, 1, -3, 4], [5, -2, 5, 3, -1, -4, 1, 4, -1, 4, 0, -2, -3], [-2, 1, -4, 0, -2, -1, 3, -1, -2, 8, -1, 3, 2], [5, -2, 5, 3, 2, -4, 4, -4, 7, 3, 0, -3, -5], [-4, 5, 1, 6, -3, 3, -3, -3, 2, -6, 13, -6, -2], [1, 0, -2, -2, 2, 0, 4, 2, -1, -1, -4, -2, 6], [-5, 11, -2, -1, -5, 2, -4, -1, 0, -9, -6, 3, 6], [4, 1, 0, -1, 1, -7, 1, 6, 1, 0, -1, 3, 0], [5, 9, -7, -5, 0, 4, 1, -6, -5, -6, -5, 1, 6], [-2, 7, 5, 6, -3, -1, 0, 0, 3, -10, -1, -5, 1], [-9, 3, -2, 0, -10, 1, -3, -1, -3, -5, -2, 2, 5], [-1, 4, -4, 2, 1, 0, 3, -7, -5, 4, -1, -1, 0], [-3, -7, 1, 2, 1, -1, -2, -4, -2, -1, 9, 1, -1], [-14, -1, 1, 10, -3, -3, -4, 0, 2, 1, 7, -6, -4], [-13, -4, 10, 10, 1, -6, -5, 2, 5, 2, 10, -4, -6], [10, 12, -9, -5, -3, 4, 4, -4, -11, -8, -3, 3, 3], [5, -4, -5, -8, 5, 3, 6, 2, 4, 10, 2, -4, -4], [11, -22, 1, -1, 15, -5, 4, 0, 4, 10, -1, 3, -4], [11, 2, -7, -11, -1, 3, 0, -4, -4, -2, -5, 6, 4], [11, 6, 1, 1, 0, 1, 4, -4, -2, -10, -4, 6, -1], [-1, -8, -2, 3, -1, 0, 0, -2, 1, 5, 2, -1, -1], [4, -6, -1, -4, 5, 1, -2, 4, 1, 5, 1, 5, 3], [-7, -10, 6, 3, 1, -1, 1, 1, 7, 8, 9, -6, -4], [8, -7, -3, 0, 8, 0, 4, 6, 5, 4, -3, 0, -2], [-12, 9, 1, -3, -5, 1, -6, -3, -6, -3, -3, 3, 3], [9, 2, -3, -5, 0, 0, 0, -2, -8, -6, -4, 0, 4], [4, 3, -2, 2, -3, 2, -7, -6, -5, -3, -6, 4, 1], [-14, -10, 5, 5, 2, -4, -2, 6, 10, 6, 4, -6, -3], [2, -5, -3, -5, 8, 2, 6, 1, 4, 5, 1, 4, -4], [8, -2, -4, -5, 7, 2, 9, 0, 4, 5, -5, 2, 0], [0, -11, 10, 6, 5, -7, 2, 2, 8, 5, 5, -4, -6], [-4, -2, 3, 4, 2, -1, -2, 0, 1, 3, -3, 4, 1], [4, 12, -10, -11, -4, 2, 1, -2, -4, -9, -8, 11, 11], [13, 2, -8, -2, -4, 4, 2, -7, 0, 0, -1, 1, -1], [0, 2, -2, -3, -5, -2, -2, -2, -6, -2, 5, 1, 0], [1, 1, 3, -1, -1, -3, -2, 3, -2, 7, 0, 1, 1], [9, 9, -4, -2, -6, 5, -6, -4, -8, -4, -4, 5, 6], [-14, -6, 1, 4, 2, 0, 5, -4, 5, 7, 8, 0, -7], [6, -6, 3, -2, 2, -3, -4, 3, 5, 1, 1, 0, -3], [-1, -3, 5, 7, 0, 0, 3, -13, -2, -4, 2, -3, -7], [-6, 8, -2, -5, 1, 2, 1, 2, -4, -3, 6, -2, -1], [10, -1, 2, -2, 2, 5, 7, 2, -4, 2, -1, 3, -2], [-6, 16, 1, -3, -9, 3, -1, 0, 1, -10, -10, -1, 5], [2, 8, 2, 5, -1, -1, 1, -4, 1, -8, 0, -2, -2], [2, -12, -1, -3, 5, -3, 2, 12, -1, 15, 3, -3, 1], [11, 3, -2, 2, -1, 0, -2, 2, -2, -7, 3, 8, 0], [13, -11, -3, -2, 6, 0, 5, -1, 0, 2, -2, 3, -7], [-5, 7, 3, 0, -3, -2, -1, -2, 0, -2, -9, 0, 4], [-7, -3, -1, -1, 6, -2, -2, 6, 6, 3, 2, -9, -3], [5, 4, -7, -3, -7, 4, -5, 2, -9, -6, 4, 4, 3], [-7, -3, 2, 1, 3, -8, -2, 3, 5, 10, 8, 0, -1], [-10, 0, 2, 3, -4, 1, -7, -7, -5, 2, 2, -1, 1], [7, 11, -3, -5, 3, -3, 4, -3, -4, -2, -7, 1, -2], [11, 6, 3, 4, 4, 0, 5, 0, 4, 2, 0, 0, -3], [6, 20, -9, -2, -7, 2, 5, -4, -10, -5, -9, 4, 3], [-14, 2, 5, 2, 1, 2, 2, -3, 3, 3, 4, 1, -4], [-5, -2, -8, 2, 1, 2, -3, 0, 4, 1, 4, -3, 0], [-13, -4, 0, 5, -3, -3, -6, 6, 2, 2, 8, -2, -1], [0, -1, -5, -6, 1, -4, 3, 2, 1, 3, -13, 1, 4], [8, -1, -3, -7, 5, -1, -3, 16, 5, 2, -2, -2, 3], [11, 8, -8, -5, -5, 5, 3, -7, -2, -11, 5, 2, 2], [-5, -2, -4, 6, -7, 2, -5, -3, -1, -6, 2, 2, 2], [-2, 2, -1, 1, 3, 4, 2, -2, -5, -1, 1, 6, -5], [-2, -1, 0, 3, -1, 4, -5, 4, 2, -4, 4, -3, 6], [-7, -4, -1, 1, 7, 4, 1, 0, 3, 12, 6, -10, -9], [22, 10, -5, -13, -1, 5, 7, -8, -5, -14, -10, 5, 3], [-2, 3, -3, -3, 0, 0, -3, 7, -1, -3, -3, 1, 5], [4, -8, 5, 4, 0, -3, 4, 0, 3, -5, -2, 0, -3], [20, -10, 0, -2, 6, -1, 7, -2, 5, -4, -3, 2, 3], [8, 8, -7, -10, -4, 4, -2, 2, 2, -6, -2, -2, 4], [0, -19, 4, -2, 14, -8, 0, 6, 7, 10, 4, -3, -5], [-11, -1, 2, 7, -3, -4, -6, -11, -5, -9, -4, 9, -1], [-4, -2, 4, 5, 4, -4, 0, 9, 10, 12, 1, -10, -8], [2, -4, 9, 7, -2, 2, -3, 2, 2, -3, 1, 0, -2], [4, -6, 2, -4, 9, -1, 4, 2, 12, 9, 5, 0, -4], [-13, -5, 5, 4, 3, -6, 5, -5, -2, 2, 5, -4, -3], [18, 6, -5, -3, -1, 1, 4, -2, 0, -3, -5, 4, 3], [-14, 2, 4, 4, 2, -5, 2, -5, -1, 1, 0, 2, -4], [3, -8, 8, 3, 3, 1, 0, 9, 1, 8, 2, -7, -2], [21, -11, -2, -7, 5, -1, 1, -6, 1, -1, -6, 10, 0], [-17, -14, 11, 5, 4, -6, -4, 3, 7, 3, 3, -10, -6], [1, -7, 5, -6, 9, -2, 2, 5, 8, 3, 0, -4, 5], [-13, 5, 9, 4, -4, -2, -4, -4, 2, -7, -3, -2, -4], [15, 20, -14, -12, -11, 6, 1, 2, -11, -8, -10, 8, 10], [-8, 12, 2, 4, -10, 2, -4, 2, -4, -8, -2, -7, 1], [2, 2, 10, -1, 3, -4, 2, -6, 2, 0, -5, 1, 4], [0, -8, -4, 5, -5, -2, -9, -2, -6, 3, 9, 4, -4], [9, 8, -3, 6, -3, 2, -4, -1, -3, -9, -2, 2, 3], [19, 11, -10, -9, -4, 6, 6, 2, -11, -12, 2, 5, 5], [-13, -9, 13, 9, 2, -8, -8, 5, 6, 5, 5, -4, 0], [7, -1, 9, -1, 4, -3, 0, 6, 8, 1, -2, -6, 1], [-20, 12, 2, 4, -12, 6, -2, -5, -4, -10, -1, -1, 3], [4, 10, 0, -8, -1, 3, -3, -5, -2, -2, -6, 8, 3], [-9, -22, 15, 11, 13, -9, 9, -3, 20, 15, 7, -12, -14], [-3, 8, -6, -2, -8, 1, -5, -7, -9, -10, -8, 8, 3], [0, 13, -6, -3, -3, 6, 2, -3, -12, -1, 4, 4, 3], [-3, -1, 2, -1, 1, 0, 1, 11, 3, 4, 1, 0, 2], [12, 4, -9, 0, -1, 0, 5, -3, -4, 3, -3, 2, -7], [17, 10, -7, -5, -4, 0, -3, 4, 1, 4, -5, -1, 3], [17, 5, -7, -6, -7, 1, 1, 2, -5, 3, 3, 0, 2], [5, 1, -1, 2, -2, 2, 4, -6, -2, -3, -3, 2, -6], [-22, -5, 8, 10, -1, -10, -8, 11, 1, 4, 4, -6, 3], [-21, 1, 1, 6, 2, 8, 1, -5, 2, 2, 6, -3, -1], [8, 5, -1, 0, 3, 6, 2, 8, -1, -4, 6, 0, -1], [4, 0, -7, 3, 0, -2, -3, -5, 2, -9, -3, -1, 0], [-10, -20, 6, 5, 5, -5, 3, -3, 4, 14, 1, -8, -11], [-4, 2, 9, 10, -9, 0, -8, 0, -7, -1, 3, -6, -9], [6, -15, 2, -2, 7, -4, -4, 5, 0, 8, -1, 8, 1], [4, 12, 5, 0, -6, -1, 4, 1, -1, -5, -9, -9, 10], [13, 18, -11, -7, -6, 6, -4, -7, -15, -7, -2, 10, 11], [0, 0, 9, 6, -4, 6, 3, -9, -2, -9, 0, -3, -1], [-2, -12, -7, -1, 5, -3, 0, -2, 8, 2, -3, -3, -1], [-20, 4, 3, 4, 6, -3, -1, 2, 7, -2, 3, -5, 2], [8, -3, -11, -6, 12, 5, 7, -5, 4, -2, -2, 1, 3], [21, 4, -5, -8, 0, 3, 3, -5, 2, -5, 0, -1, -4], [9, -11, 0, -5, -2, -1, -2, 2, -9, -6, -4, 4, 1], [-11, -3, -3, -2, 2, -1, 6, -3, 3, -6, -5, -2, -4], [-1, 17, -4, 1, -10, 7, -5, 0, 0, -11, -4, 6, 5], [-2, -13, 9, 13, -6, 0, -5, -9, 5, 1, 6, -3, -5], [5, -8, -3, -9, -2, -1, -4, -7, -1, -3, -3, 0, -1], [-16, 1, 0, -3, -1, -5, -1, -5, -8, -2, -1, 0, -4], [11, -9, 7, 6, 13, -2, 2, 5, 8, 11, 16, -6, -7], [-4, -5, -5, -7, 7, -3, 6, -6, 1, 5, -1, -1, -2], [19, -7, -1, 8, -1, -8, 0, 2, 4, -3, -2, 4, 1], [-10, -6, 3, 10, -3, -3, -3, -3, 6, 0, 9, -4, -2], [0, 11, 1, 4, -8, -1, -4, 6, 4, -7, -2, -4, 11], [-6, 4, 7, 7, -4, -1, -9, -1, 7, -2, 5, 3, -8], [21, -19, -2, -8, 11, -6, 2, 2, -1, 14, -1, 0, 1], [-6, 1, -4, -2, -2, 8, -2, 7, -10, 2, 2, 1, -2], [19, -7, 3, -6, 3, -4, 2, 9, 6, 7, -4, -1, -2], [14, 1, -12, -10, 2, 6, 6, -3, -13, -2, 3, 4, 2], [11, 15, -8, -5, 2, 3, 6, -1, 3, -10, -3, 0, 1], [16, -31, 3, 1, 10, -4, 0, 0, 5, 18, 6, 2, -4], [-14, -18, 1, 6, 1, -3, -5, 5, 8, 6, 15, -11, 0], [9, -4, -7, 0, -2, 6, 5, -5, -7, 0, 2, 0, -2], [-4, -6, -2, -2, -6, 0, -8, 2, -8, -8, -4, 4, 6], [6, 8, -12, 1, -12, 2, -1, -8, -6, -5, -5, 4, 2], [4, -2, -3, -7, 1, -1, 2, 8, 5, 1, -3, 4, 12], [0, -9, 4, 1, 1, 2, 4, -4, -1, 16, -1, -2, -1], [-10, 8, -2, 2, -2, 7, -5, -5, -3, -8, 5, -2, 3], [-3, 6, -2, -4, -8, -3, -7, -5, -5, -4, -10, 9, 4], [3, 8, 0, 3, -1, 4, -1, 3, 2, -1, 13, -8, -3], [-10, 0, 0, 7, -5, 0, -1, 2, 2, 3, 11, -8, -14], [1, -1, -2, -7, 0, 6, 3, 2, -1, -4, 6, 5, 5], [6, 19, -15, -10, -13, 6, -1, -5, -13, -19, -11, 7, 14], [2, 14, -1, 5, -12, -1, 0, 4, -7, -7, 6, -8, 3], [5, 4, -5, -5, -3, 4, -9, 5, -4, -8, 0, 2, 4], [9, 18, -9, -10, -3, 5, -1, -1, -6, -5, -11, 7, 11], [18, 2, -9, -3, 5, 6, 8, -3, -13, 9, 0, 5, -2], [-26, 9, 10, 12, -3, 1, 0, -11, 4, -1, 7, -7, -8], [17, -9, -4, -6, 4, 4, 6, -2, 8, -1, -8, 9, -1], [4, -25, 7, 2, 9, -3, -1, -4, 2, 10, 6, 3, -11], [-10, -5, 0, 0, 1, -6, 5, -2, -7, 5, -6, 4, 5], [0, -22, 4, 0, 2, -2, 1, 2, 4, 3, 5, -4, -3], [7, -1, -2, -2, 3, -1, -1, 8, 8, 1, 0, 4, 2], [32, -2, -5, -13, -3, 8, 2, 0, -12, 2, -5, 11, 10], [-9, -5, 12, 6, -3, -8, -1, -3, -4, -6, -6, -4, 1], [4, 2, -9, -8, 3, 5, 6, -2, -3, 1, -14, 4, 8], [-6, -8, -7, -3, 1, -4, -1, 3, -8, 13, -5, 7, 0], [-7, 23, 0, 3, -9, 4, -4, -4, -2, -15, -8, 2, 8], [6, 13, -14, -13, -5, 8, 1, 1, -12, -13, -15, 6, 6], [-13, -8, 4, -1, 3, -2, 9, -4, 11, 4, 0, -2, -1], [-22, 27, -7, 0, -16, 7, -9, 4, -10, -18, 3, -6, 7], [11, -25, -2, -3, 8, -6, 2, -1, 2, -3, -1, 5, -1], [-1, 12, -12, -2, -1, 7, -1, -16, -10, 1, 4, 4, 2], [-10, 7, 5, -2, -10, 6, -1, -5, -5, -15, -2, 2, 8], [-3, 10, -2, -1, -5, 2, 7, -12, -1, 2, -9, -1, -2], [-21, 8, -3, 3, -11, -3, -5, 1, -4, 3, -4, -6, 1], [-22, -8, 4, 7, -3, -8, -2, -1, -6, 12, 12, -4, -7], [-10, 10, 10, 10, -10, 3, -3, -2, 4, -5, 5, -14, -3], [-21, 10, -4, 3, -11, 2, -12, -1, -3, -9, 0, 4, 5], [-7, 9, 5, 8, -5, 1, -2, -7, -5, -16, -10, 1, -1], [-1, -7, -4, 4, -3, -5, 0, -3, 1, 0, -5, -7, -2], [-9, 10, -4, -3, -4, 0, -6, 15, 3, 7, -3, 2, 2], [-6, -10, 4, 5, -4, -2, -4, 4, 1, -6, 11, -10, -9], [-7, 10, -2, 5, -5, 5, 0, -2, 3, -9, 11, -4, 4], [-21, -10, 7, 8, 12, -2, 0, 5, 19, 19, -3, -7, -7], [-1, 12, 5, 2, -2, -2, 0, 2, 2, -8, -2, 1, 1], [7, -5, 7, 5, -2, -5, -1, 0, 11, -5, -3, 0, 1], [12, -8, -2, -7, 3, -2, -5, -9, 0, 1, -11, 7, -4], [0, 2, 6, 2, -7, -3, -4, -5, -2, -5, -13, -1, 2], [-8, 8, -6, -8, 4, 1, 7, -5, -10, -7, -2, 8, 1], [-29, 9, -3, 7, -5, 3, -3, -1, 1, 1, 2, -10, -8], [-1, -24, 6, -6, 5, -8, 0, 5, -5, 13, 0, 0, -7], [1, 10, -11, 5, -9, 7, -2, -6, -11, -13, -3, 4, 6], [-5, 2, 7, 1, -9, -4, 4, -16, -11, -7, -1, 0, 0], [-1, -16, -3, -5, 2, 5, 0, 3, -2, 12, 4, 0, 3], [-10, -3, 3, 13, 0, -2, 3, 0, 11, 9, 19, -13, -3], [-11, 25, 3, -1, -8, 1, 2, 0, 3, -8, -10, -12, 6], [-6, -9, 1, 6, 6, -2, -1, -7, 2, 3, 6, -1, 0], [-7, 2, -4, -3, -2, -3, 9, -8, -1, 2, -5, 4, -1], [8, 13, -10, -8, -8, 12, -3, -11, -10, -2, -8, 4, 6], [-2, -6, 3, -9, 4, -2, -2, 17, 7, 5, -7, 5, 1], [-1, -7, 8, 9, 3, 3, -1, 5, 14, 7, 7, -9, -8], [-10, -9, -9, 0, 9, 0, 3, -14, 7, 5, 11, -9, -6], [-26, 24, 8, 2, -5, 1, 0, 1, 1, -17, -7, 0, 3], [6, -13, 3, -1, 10, 0, 9, -11, 9, 9, -6, 4, -2], [-17, -2, -10, 1, 0, 2, 3, -4, 4, -8, 4, -3, 0], [6, -12, 7, 4, 0, -6, -3, 11, 16, 7, 2, -4, -4], [5, 12, 1, 9, -4, 8, -2, -1, -2, -8, 7, 0, 4], [13, -3, 1, 2, 4, -2, 10, -8, 0, 5, 9, 3, -2], [-6, -15, 7, 6, 11, -2, -4, 9, 17, 10, 10, -12, -15], [-16, 15, 6, 4, -10, 7, 3, 0, 0, -14, -5, -3, 8], [-18, -13, 3, 0, 7, -4, 0, 16, 11, 14, -3, -6, -3], [-3, -11, 12, 10, 6, -12, 0, -5, 1, 6, 3, -9, -9], [-9, -5, 2, 5, -1, -6, -11, 9, 7, 7, 15, -6, -1], [-27, -7, 7, 17, -3, -3, -7, 3, 6, 8, 7, -4, -10], [-3, 4, 1, -1, -6, 2, -4, 6, -4, -3, 12, -2, 8], [-17, 12, 2, 1, -3, -2, 0, 6, 9, 7, 7, -13, 2], [-10, -5, 0, 8, 1, -2, 6, -4, 7, 14, 11, -6, -9], [-25, 4, 12, 14, -11, 0, -6, -2, -3, -5, 7, -10, -9], [-5, 17, 8, 0, -6, -7, -8, 1, -1, -10, 1, -3, 0], [-12, -14, 0, 7, -1, -6, -6, -3, -2, 14, -2, -4, -3], [-15, 1, 2, -5, -9, -1, 0, 5, -5, 4, -4, -5, -3], [-2, 1, 1, -3, 7, 0, 9, -7, -2, 3, 0, 0, -5], [-18, 1, -2, -1, 1, 5, -3, 10, -5, 9, 0, -1, -5], [5, 5, 2, -2, -6, -1, 2, -3, -9, -2, -4, 7, -1], [16, -10, -5, -8, 7, 0, -6, 6, 1, 10, 12, -3, -1], [-14, 4, 7, 13, 7, -4, 3, 0, 2, 1, 6, -13, -3], [-4, 1, 8, 1, 3, 8, 3, 0, -5, 1, 4, 2, -1], [-8, 2, -4, 3, 0, 2, -1, 8, -9, 2, 3, 1, 3], [11, -8, 3, -10, -2, 0, 6, -2, -1, 5, -6, 3, 3], [-7, 9, -9, -7, -4, 2, 3, 0, -11, -1, -9, 6, 6], [-5, 7, 1, 0, 11, 0, 8, -5, 8, 1, 0, 4, 1], [-7, 13, -6, -6, -13, 5, -9, 2, -10, -9, 0, 5, 4], [-13, 16, 2, 1, 8, 2, -1, 12, 3, -8, 3, 0, 0], [-28, 2, 17, 12, -11, -8, -8, -3, 10, -10, -2, -2, -4], [16, 13, -2, -3, -3, 3, 7, -1, -4, -10, -15, 7, 9], [-3, -1, 1, 2, 5, -3, -2, 2, 6, 7, -7, 5, 3], [-21, 16, 12, 2, -3, 0, 1, 0, 0, -13, 0, -6, 5], [-18, -1, 2, -4, 3, 5, -5, 3, -2, 0, 3, 1, 5], [-4, 40, -11, -10, -17, 3, 4, -2, -10, -19, -18, -4, 11], [-29, 15, -3, -4, -8, 0, -13, -7, -10, -14, -2, 10, 3], [-5, -12, -2, -1, -6, -3, -9, -7, -13, 3, 0, 7, 2], [15, 3, -11, 3, 6, -7, 4, -2, 1, 10, 0, 0, -6], [2, -10, -2, 4, -1, -6, 5, -1, 3, 12, -9, -4, -7], [18, -11, 3, -1, 6, -7, -1, -1, 5, 8, 7, 10, -6], [15, 1, -2, -4, -9, 8, 2, -13, -10, -11, -10, 1, 0], [-9, 7, 2, -7, -3, 6, 1, -2, -9, 7, 0, 6, 1], [-11, -7, 6, -7, 18, -4, 6, 3, 15, -1, -7, -5, -2], [3, 1, -1, 8, 1, -7, 5, 0, 7, 5, -11, -6, 0], [-11, 3, 9, 10, -4, 0, -5, 0, -7, 0, 0, -2, 0], [7, -21, 0, 3, 6, -2, 1, 4, 9, 4, 14, -4, -7], [-16, -8, 4, 4, 7, -6, 0, 12, 6, -2, -5, -6, 9], [-13, 24, 2, 2, 2, 1, 1, 5, 0, -11, -15, -4, 9], [6, 14, -18, -17, 1, 5, 13, -2, -9, -2, -14, 8, 12], [-5, -14, 8, 5, -1, -5, -1, 4, 9, 12, 14, -6, -8], [1, 7, -1, 7, -8, -6, 2, 4, -6, -17, -6, -2, 8], [-9, -4, 16, 12, 12, -6, 4, -8, 13, 7, 10, -9, -12], [-15, 6, 3, -2, 0, -3, 1, 11, -1, -6, 2, -2, 5], [-12, 2, 1, -2, -10, 8, 2, -9, -5, -9, 3, 4, -2], [-5, 7, 10, 6, -1, -1, 1, 2, 14, -13, -2, -9, -2], [2, 26, -8, -1, -10, 6, -3, -14, -16, -5, 0, 7, -5], [3, -3, -9, -4, -2, 6, 2, 4, 6, 11, -1, -2, -2], [-3, 7, 12, 9, 6, 0, 0, -9, -2, 7, 9, 7, -5], [-19, 10, 5, 7, 7, 3, 8, -8, 2, -8, 13, -10, -4], [-15, -9, 9, 7, 11, -6, -4, 4, 10, 17, 13, -13, -11], [-22, 14, -9, 4, -13, 8, -8, 3, -6, -14, -4, 7, 12], [12, 30, -9, -6, -9, 13, 6, -5, -4, -15, -6, -1, 3], [19, 7, -16, -20, -8, 9, 8, 2, -14, -4, -20, 14, 16], [11, -16, -2, 6, 6, 6, -2, 3, -4, 9, 12, 6, 3], [0, -8, 4, 9, 3, -6, -1, 2, 0, 3, 10, 5, -10], [-7, -4, 14, 3, 1, -5, -3, 10, 2, -5, -13, 0, 6], [22, 16, -8, -9, -1, 5, 2, 12, 3, -5, 1, -1, 3], [-14, -29, 2, 17, 19, -11, 4, 3, 14, 17, 16, -11, -9], [-21, 10, 4, 11, -7, 1, -2, 6, -3, 10, 8, -11, -6], [-19, 30, -1, -2, -15, 12, 2, -5, -4, -7, -8, -9, 1], [-5, -11, 4, 14, -1, -8, -5, 8, 8, 6, 20, -11, -2], [-17, 3, 8, 1, 8, -5, 5, -4, 1, 5, -4, 10, 2], [-35, 3, 3, 5, -4, -7, -14, 16, -1, 2, 6, -5, 2], [3, -5, -10, -4, 8, 1, 4, 5, 4, 5, -1, -6, -1], [-2, -5, 16, 7, 9, 0, 3, 6, 4, 3, 3, 0, 2], [-1, -6, 11, 4, -5, -5, -7, -2, 4, -3, -1, -7, -3], [-15, 10, 8, 6, -3, 2, -1, -5, 10, -8, 4, -13, -5], [6, 8, -3, -14, 0, 7, 3, -7, -9, -6, -8, 7, 10], [3, 9, 2, 4, -2, 10, 1, -7, -2, -8, 6, -1, 1], [-31, 24, -5, 10, -9, 5, 3, -5, 1, -11, 5, -17, -6], [14, 16, -5, -12, -1, -1, 10, 0, -7, -9, -16, 7, 4], [-10, 11, 1, 6, -3, 2, 0, 7, 6, 2, 2, -5, -10], [-39, -11, 2, 7, 1, -4, 3, -5, 2, 8, 5, -10, -11], [-16, 22, -4, -2, -22, 9, -10, -5, -20, -23, 0, 5, 7], [10, 17, -6, 0, 0, 4, 13, -8, 0, -4, -7, 7, -1], [-10, -2, -5, -11, 6, -1, 12, -3, 8, 9, -16, -2, 4], [-5, 22, 0, -3, 1, 3, 0, -2, 1, -7, -5, 6, 10], [1, 7, 5, 0, -7, -3, 5, 4, 5, -1, 4, -13, -11], [-7, -1, 3, 3, -4, -2, -5, -2, -2, -4, -7, 6, 3], [-30, 8, -11, -3, -2, -2, 2, 4, 0, -4, -2, -9, 5], [-26, -5, 11, 7, -3, -3, -9, -8, 4, -4, 8, -5, -1], [18, 22, -18, -20, -6, 10, 1, 7, -15, -17, -9, 2, 18], [1, 17, -10, -7, 11, 5, 10, -5, -7, -8, -8, 0, 10], [-20, -11, 9, 12, 0, -3, -8, 2, 4, 9, 5, 1, -4], [-2, 15, 2, 2, -9, 5, -4, -6, -7, -6, -1, 11, 9], [-7, 9, -3, 1, -4, -4, -4, -5, -4, 7, 1, 13, -9], [-11, 9, -1, 10, -2, -3, -13, 3, 3, -9, -2, 4, 11], [-2, 2, -17, -12, -6, 4, -6, -4, -15, -9, -4, 12, 6], [-38, -13, 21, 18, 3, -11, -10, 5, 2, 13, 2, -7, -7], [20, 7, -11, -2, 0, 0, -5, -3, -11, 13, -2, 3, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6040_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6040_2_a_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_6040_2_a_n(:prec:=13)
    chi := MakeCharacter_6040_a();
    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_6040_2_a_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_6040_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6040_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-11, -82, -108, 373, 708, -444, -870, 109, 403, 41, -70, -14, 4, 1]>,<7,R![-1024, 9472, -5520, -24336, 6980, 19284, -1233, -5580, 31, 738, 3, -45, 0, 1]>,<11,R![20408, 69112, 57333, -28324, -56109, -10247, 15599, 6947, -1089, -1140, -124, 51, 14, 1]>],Snew);
    return Vf;
end function;