
// Make newform 6042.2.a.z in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6042_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_6042_2_a_z();".
// 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_6042_2_a_z();".
// 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 := [-14, -8, 119, 29, -180, 25, 51, -11, -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], [-862, -3266, -755, 1732, -96, -119, 34, -9, 1], [-4894, -6482, 14685, 1984, -4444, 361, 306, -81, 9], [3390, 5146, -8937, -14420, 7192, 4455, -1682, -355, 119], [906, -3714, 7505, 14420, -7192, -4455, 1682, 355, -119], [3654, 4794, 775, -14884, 3560, 4935, -1410, -427, 127], [1478, -2806, -5701, 8343, 273, -2520, 362, 199, -42], [402, 4476, -1305, -8192, 2736, 2407, -790, -191, 61], [244, -3070, -1102, 6026, -840, -1802, 387, 145, -36]];
    Rf_basisdens := [1, 716, 1432, 1432, 1432, 1432, 358, 358, 179];
    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_6042_a();"
function MakeCharacter_6042_a()
    N := 6042;
    order := 1;
    char_gens := [2015, 4771, 2281];
    v := [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_6042_a_Hecke(Kf)
    return MakeCharacter_6042_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 := [[-1, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, -1, 1, -1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [2, 1, -1, 0, 1, 0, 0, 0, 0], [0, 0, -1, -1, -1, 0, -1, -1, 0], [-2, 0, 0, 0, -1, 1, 1, 0, 0], [0, 0, 1, 0, 1, -2, 1, 1, -1], [0, 1, 1, 0, 1, 0, -1, 1, 1], [2, 0, 1, 0, 0, 1, 0, 1, 0], [5, 1, 1, 1, 1, -1, -1, 0, 1], [1, 0, 0, 0, 0, 0, 0, 0, 0], [2, 1, 0, -2, 0, -2, -1, 1, 1], [2, 1, 0, 0, 1, 1, 0, 1, 1], [-2, -2, 1, 1, -2, 1, 1, 0, 1], [2, 2, -2, 0, 2, 0, 0, 0, 0], [1, -1, 2, 1, 1, -2, 2, 1, 1], [-1, 0, -1, 2, 0, 0, 2, -3, -1], [3, 0, 2, 1, 0, 1, 0, -2, 0], [-2, 0, 1, -2, 1, -2, 2, 2, -2], [1, 0, 1, 2, 2, 0, 0, -1, 1], [-2, -2, -2, 0, -1, 5, -1, -3, -1], [-4, -2, -2, -1, -3, 1, 0, -3, -1], [3, 0, 0, 0, -1, 0, 0, 1, 3], [-2, 1, -1, 0, 3, 0, 0, -1, 1], [3, -2, 1, -1, 1, -3, 2, 2, -1], [-1, 3, -2, 1, 1, 0, 1, 2, 0], [9, 1, -2, 1, 1, 0, 0, 0, 0], [2, -1, 0, 1, -2, 0, 4, 1, -1], [3, 0, 0, -1, 2, -5, 0, 4, 3], [0, -1, 2, 0, -2, 0, 0, 0, 1], [-1, 1, 2, 2, 1, -2, 2, -1, 1], [5, 0, -2, 2, 3, 0, -1, -2, 0], [0, 2, -2, 0, 1, 3, -3, 0, 4], [7, 0, -2, 4, 2, -1, 3, 1, 0], [2, -2, -1, -1, 0, 1, -2, -1, 0], [-2, -1, 4, 0, 3, -1, 0, 1, -4], [-5, 2, 1, 2, 0, 2, -3, 1, 1], [-1, -1, 2, 0, 1, 0, 1, 0, -2], [1, 1, 1, 3, 0, -2, -1, -1, 2], [7, 3, 0, 1, 4, -1, -2, 2, 2], [-4, 2, -2, -1, -1, 3, -2, -3, -3], [-3, -2, 1, 0, 1, -1, 0, 1, -3], [2, -2, 3, 0, 2, -5, 3, 1, -3], [7, -1, 3, 2, 2, -2, -1, 0, -2], [3, 2, 2, 2, -1, -2, -4, 1, 2], [-2, 0, 3, 2, -3, 2, -1, -2, 0], [-2, -1, -5, 1, 0, 5, 2, -5, -3], [-4, -1, -1, 1, 1, 0, 0, -4, -4], [10, 0, 3, 4, 4, -7, 5, 2, 2], [-5, 0, -1, 3, -1, 5, -3, -3, -1], [7, 0, 3, -1, -2, -2, 0, 3, 3], [0, -2, -2, 1, -2, 2, -2, -2, -2], [5, 0, 1, -2, -1, -5, 1, 5, 3], [4, 1, 0, -1, -1, -1, 1, 4, 4], [-1, -2, 2, 3, -1, -2, 1, 2, 2], [5, -2, -1, -1, -2, 2, -3, 0, -2], [8, 0, -1, -2, 2, -3, -3, 0, 1], [9, 1, 0, 4, -1, 2, 1, -3, -3], [8, 0, 2, 2, -3, 1, -2, 2, 4], [-1, -3, 2, 1, 0, 1, -4, -2, 0], [-3, -2, 5, 1, 0, 0, -1, -2, 2], [1, 1, 2, -5, -1, 2, -4, 2, 4], [5, 0, -1, 2, -1, -1, -5, 0, 2], [12, -2, -2, 3, 2, -2, 5, 0, -2], [-5, -1, -3, 0, 3, -1, 2, -2, -6], [1, -1, 1, -5, -3, -3, 3, 6, 2], [9, 1, 0, 5, 3, -2, 1, 2, 0], [6, 1, 1, -4, 2, -5, 3, -2, -2], [2, 2, 0, -4, 0, -2, 5, 2, -6], [-1, 1, -3, -5, 1, 5, -3, -1, -4], [15, 0, 3, 4, 4, -2, -2, 3, 1], [7, 1, 0, 3, 1, -4, -3, 1, 7], [11, 0, 0, 0, 2, -3, -3, 5, 3], [4, -1, 5, 4, -1, -2, 1, -3, 3], [4, -1, 3, 7, 4, -7, 2, 3, 5], [6, -3, 2, 2, -5, 3, 0, 1, 3], [1, 2, -5, -1, -1, 1, -2, 1, 5], [1, -2, 5, 4, 1, 1, 5, 1, -1], [-1, 0, -3, -5, 0, 6, -1, -4, -6], [5, 0, 2, -3, -1, -4, -2, 8, 4], [0, 1, -4, -3, -1, 7, -2, -7, -7], [0, 3, 3, 2, -1, -2, 3, 3, 0], [10, 0, 2, 2, 5, -3, 0, 5, -3], [5, 3, -5, 1, 5, 3, 1, -5, -2], [2, -3, 2, 1, 5, -5, 3, 6, 0], [10, -2, -4, 4, -4, 2, 4, -5, -1], [4, 2, -6, -1, -2, 4, 4, 0, 0], [3, 1, 3, -1, -1, -3, -1, 6, 4], [7, -2, -2, -6, 0, 1, 2, 6, 0], [2, 2, -1, 0, 2, 1, -5, 4, -1], [8, 0, 4, 3, -1, -3, -2, 1, -1], [-4, 1, -9, -4, -3, 2, 3, -1, 1], [-1, -1, -4, -10, 1, 2, 1, -2, -9], [1, -2, -5, 2, 3, 3, 0, -9, -7], [-1, 2, 3, -6, -3, -5, 1, 7, -1], [1, 2, 3, -1, 0, -4, 5, -1, -1], [7, 2, 3, 3, -2, -8, 0, 4, 6], [9, 3, -1, -3, -3, -1, -1, -1, 4], [-2, 3, -5, -6, 0, 1, -1, 0, 2], [1, -2, 0, 3, -5, 4, 0, -7, -6], [6, 1, 0, 5, -2, 2, -5, -1, 10], [3, -1, 0, -4, 3, -4, 3, 1, -7], [13, 1, -2, -1, 2, 1, 0, -2, -2], [-11, 1, -4, 5, -1, 8, -6, -7, -7], [3, -6, -2, 4, -2, 5, 0, -5, 2], [4, 3, -4, -2, -3, 1, -4, -2, 4], [-1, 1, -1, -4, -3, 3, 2, 2, -2], [-6, -4, 5, 0, 2, -1, 1, 1, -5], [4, -4, 8, 0, -2, -4, 5, 5, 3], [1, 2, -5, -1, -2, -4, 1, -2, 5], [-5, 5, -2, -5, -3, 4, -2, -5, -3], [7, 1, -1, 1, -6, 4, -4, -1, 2], [21, 0, 0, 7, 5, -4, 6, 2, -1], [12, -2, -3, -6, -6, 1, 3, 0, -3], [1, 3, -1, 1, 1, -5, 3, 3, 6], [6, 2, 3, -2, 8, -9, 5, 8, 0], [4, 0, 0, 1, 1, 3, -5, -8, 2], [-5, -2, 4, 3, 2, -5, 4, 0, -3], [14, -1, 2, -7, -1, 1, -5, 6, -4], [-2, -3, -1, -2, -5, 4, -1, -5, -3], [11, 3, 0, 2, -3, 2, 4, 2, 0], [-9, -1, 7, 6, 2, -2, 6, 1, -3], [13, 2, 5, 8, 6, -8, 1, 0, 2], [5, 0, -5, -1, -2, 6, 1, -6, -4], [0, 4, 3, -7, -1, 4, -5, -1, -2], [-1, -3, 2, -1, 0, -1, -6, -4, 4], [3, 0, -3, -5, 1, -1, 3, -1, 5], [8, -4, 1, 4, -5, -6, 5, -4, 0], [-6, -3, -7, 3, 1, 2, 5, -7, -7], [9, -3, 1, 7, 4, -6, 1, 5, -3], [0, -1, 5, 8, 3, -2, -5, 1, 9], [-7, 0, -4, 6, 3, 6, 1, -8, -1], [12, -3, -4, 3, 3, 3, -4, -1, 0], [12, 5, -14, 1, 4, 4, -1, -3, 0], [-1, -3, -3, 0, 3, 1, 7, -4, -8], [6, -1, 5, 1, 3, -8, -6, 9, 1], [2, -1, 5, 6, 4, 1, -2, -6, 0], [6, 0, -8, 4, 4, 2, -2, -4, 4], [-10, 2, -3, 6, 0, 5, -1, -8, 3], [-20, -2, 10, 0, -8, 6, -3, 0, 2], [1, -1, 4, -4, 0, -9, 2, 10, 6], [11, 3, -3, 3, 6, -2, 7, 4, 2], [-2, -2, 6, 8, -4, 6, 5, -1, 4], [6, -3, -8, -5, -3, 5, -3, -8, -7], [11, 4, -1, 2, -1, 3, -5, 3, 1], [-1, 1, 7, 7, -3, 1, -1, -3, -3], [2, 3, 4, -5, 3, -5, 0, 2, 0], [-7, -3, -3, 3, -6, 0, -3, 0, 1], [-6, -2, 0, -1, -4, 2, -5, -5, -3], [-3, -3, -2, 6, -2, 5, -9, -2, 0], [11, 0, 7, 0, 0, -2, -1, 8, 6], [-14, 0, 5, -1, 0, 5, -2, 1, -1], [-17, -3, 6, 2, -1, 4, -3, 2, -4], [-3, 0, 3, 3, 3, 5, 4, 0, -4], [-13, -3, 5, -2, 6, -4, -1, 4, -5], [14, 2, 4, -4, 2, -8, -8, 9, 1], [-1, -5, -9, -4, -4, 6, -5, -4, -4], [-4, -2, 4, 0, -7, 3, 6, 1, -5], [8, 4, -1, 0, -2, 7, -2, 1, 5], [-6, 6, -6, 0, 9, -3, 4, 1, 7], [4, -3, 5, -3, -4, 3, -5, -4, -2], [-11, 2, 2, -9, -6, 9, -6, 1, 1], [-3, -1, 1, 0, 3, -3, -9, 1, 0], [-3, 2, 4, 5, 8, -13, 4, 5, 2], [-7, 9, 2, -3, 6, -7, -6, 4, 2], [1, -8, 1, 3, -1, 7, -5, -3, 1], [5, 2, -3, 1, 0, 0, -5, -3, 7], [-13, 1, 2, -5, -3, 10, -6, -4, 2], [-8, 2, -6, 0, 5, -5, 0, 1, -1], [-13, -4, 7, -5, -4, 0, -4, -5, -3], [0, -3, 3, -8, -6, -3, 7, 6, -2], [-2, 1, -2, -9, 2, -2, 9, 3, -4], [-9, -2, 7, -5, -4, 4, -1, 0, -4], [8, 7, 1, -6, 5, -4, 0, 0, 2], [15, -1, 1, 4, 2, -12, 3, 4, 3], [-9, -3, -10, 8, -3, 10, 7, -12, -7], [-16, 1, 2, 6, -2, -4, 2, -4, -3], [-2, 4, -4, -4, 7, -1, -3, 1, 1], [-15, -2, 1, -4, -10, 6, 4, -3, 0], [-5, 1, -2, 1, -2, 9, 8, -1, -5], [-2, 1, -5, -4, -2, -5, -7, -4, 6], [-28, 1, -6, -6, 0, 2, 0, 0, -4], [-13, -4, 1, 5, -4, 12, 3, -11, -7], [4, -3, 5, 5, 2, -7, 4, 1, -5], [8, -5, 8, 2, 4, -10, 2, 9, 10], [-8, 2, 0, -2, 1, 1, 0, 9, 1], [2, 3, 0, 5, 3, -3, 2, 7, 5], [4, 3, 0, 3, 3, -5, 6, 1, 4], [-3, -3, 0, 2, 0, 1, -12, -4, 4], [2, 1, 2, 1, -4, -6, -4, 6, 2], [-2, 2, 8, -6, 2, -2, -6, 2, -2], [-5, 2, -3, 5, 2, 4, -5, -2, -4], [-9, 1, -7, -5, 6, -4, 1, -2, -9], [-8, -1, 2, 1, -2, 4, -8, -1, 1], [-3, 2, -1, -1, 9, -9, -3, 0, 0], [-7, 0, -3, -6, 3, -7, 5, -1, -5], [3, 1, -4, -7, 2, 5, 5, -5, 1], [7, 10, 5, 3, 6, -6, -2, 12, 8], [-18, 3, 1, -7, 1, 6, -11, -2, -2], [4, -4, 8, 3, -1, -5, -1, -1, -3], [6, -1, 1, -3, 1, -2, 11, 6, -2], [-5, 1, 0, -2, -4, 9, -8, 2, 8], [3, -2, 4, 11, -3, 0, -9, 0, 8], [1, -1, 2, 10, 3, 6, 5, -4, 2], [12, 2, 2, 11, -1, 3, 8, -7, -7], [19, 1, -7, 3, -4, -2, 0, -3, 6], [2, 5, -11, 6, 1, -2, 3, 1, 7], [0, 3, 1, 0, -4, -3, -7, 2, 7], [-7, -1, 8, -5, 1, -6, 2, -1, -7], [-3, 7, -8, 4, -3, 14, -6, -7, -1], [-3, -2, 1, -2, -1, 7, -2, 4, 0], [-6, 5, 2, -6, -4, 2, -6, 8, 4], [3, 3, 3, 13, -2, -2, 4, 3, 11], [7, -2, -5, -4, -5, 3, -3, -9, -1], [-1, 3, 8, 2, 2, -1, 1, -2, -2], [-5, 3, -2, 1, 4, -1, 13, 2, -2], [12, -1, 3, 4, 0, 3, 10, 3, -1], [-23, -6, -5, -7, -5, 7, -5, -3, -3], [23, 6, 2, 3, 4, -3, 0, 2, 2], [-3, 0, 12, -9, 1, -2, 1, 2, -6], [8, 0, -1, -1, -6, 1, 2, -7, 3], [4, 7, 5, 4, 6, -5, -9, 4, 10], [10, -1, 3, 3, 2, -1, 0, 7, 5], [4, 7, 10, -2, 4, 0, -6, 13, 5], [-20, -1, 2, -1, -6, 10, -6, -6, 6], [-18, -4, 3, 0, -3, 6, -9, -1, 1], [-15, 6, 5, -4, -2, 6, -4, 2, 8], [2, 1, 2, -2, 11, -9, 8, 6, -6], [-3, -6, 3, 3, -4, -6, 1, 8, 6], [22, -3, 2, -1, 3, -13, 6, 3, 0], [-15, 0, -3, -8, -5, 7, -7, -7, -9], [-9, 7, -4, 5, 5, -6, 8, 1, -1], [35, -4, 5, 5, 7, -13, 15, 5, -1], [3, 1, -2, 0, -1, 6, -1, -3, -11], [-23, 9, 3, -5, -3, 1, -5, 7, 4], [-12, 1, -9, -14, -7, 4, -5, -7, -9], [-5, -4, -7, -3, -3, 5, -10, 0, -2], [0, 1, 5, -3, -5, -8, 5, 9, 11], [10, -9, -1, 1, -3, -2, 10, -7, 1], [-6, 3, 6, 2, 12, -10, 1, 3, -3], [7, -5, -4, 3, 2, -3, -7, -5, -5], [-21, 0, -2, -1, 2, -5, 0, 0, -2], [-2, 6, -2, -8, 6, -2, -14, -2, 0], [-6, -2, 0, 4, 2, -8, -4, -1, 3], [-5, 4, -1, -3, 1, 13, -1, -2, -2], [9, -2, 7, -3, -10, 0, -7, -4, -6], [-16, 0, 0, -5, -10, 14, -1, 1, -5], [14, -8, 6, -2, -5, 1, 2, 0, 4], [-22, 1, 4, -5, -10, 4, -4, -3, -4], [7, 1, 2, 13, -1, -6, 12, -8, 0], [4, 6, 4, -3, 8, -2, -1, 6, 4], [-17, 5, -9, -1, 3, 11, -3, -3, -7], [-2, -2, -2, -6, -10, 2, 10, -1, 5], [14, -4, -4, 1, -1, 1, -10, -11, 5], [18, -10, -2, 10, 5, 1, 2, -1, 3], [-32, -3, 5, -8, 0, 7, -3, -8, -4], [16, 1, -6, -6, 7, -5, -5, 2, 0], [-10, -5, 6, 9, -5, 7, 5, 0, -10], [-13, 0, 1, -8, -3, 7, -10, -1, 7], [9, 3, 4, -3, 4, -3, -10, -4, 4], [4, 6, 1, -8, -5, -6, 5, 5, -8], [7, 1, 5, -8, 0, -6, 1, 12, 11], [5, -1, 2, 5, -3, -2, 12, -5, -11], [-25, -5, -7, 2, -3, 1, 9, -13, -8], [25, 8, 5, 7, 1, -5, 7, 1, 5], [28, 4, 4, 11, 5, -1, -5, 3, 9], [-5, -4, -1, -4, 4, -2, 3, 4, -2], [-10, -3, -6, 8, 0, 8, -9, -3, 4], [-2, -2, -4, 0, -2, 4, 2, -8, -8], [9, -7, -2, 9, 5, 6, 8, -9, -7], [-9, 0, 5, -6, 2, -6, 2, -3, 0], [33, 0, 5, 5, 7, -17, 7, 8, 8], [-12, -5, -10, -10, -10, 8, 4, -6, -16], [-6, 0, 2, -3, 8, 4, -7, 5, -5], [8, -4, 2, -3, -4, -12, 1, 12, -4], [1, 9, 7, -12, -1, -7, -4, 2, -5], [1, 3, 4, 16, 0, 3, -6, -4, -4], [4, -5, 3, 1, 0, 7, 8, 1, -9], [-9, -2, 2, 3, -1, 4, 8, 1, -9], [38, 1, 4, -2, 6, -8, 12, 8, -5], [-8, 0, 4, 4, -1, -3, -8, 4, 4], [-14, -3, -1, -4, -10, 3, -3, -2, 0], [2, -6, -6, -3, -6, 10, 3, 1, 1], [-9, -2, -1, -7, -1, 3, 7, 1, 3], [6, -1, -5, 4, -11, 8, 3, -11, -1], [-10, 7, -12, 0, -6, 8, -8, -10, 3], [-24, -4, -8, -7, -13, 19, -15, -1, -3], [-7, 2, -1, 3, -2, 8, -7, 0, 6], [-17, -8, 7, 7, -9, 11, 9, -3, -5], [17, -10, 2, 1, -7, 8, 10, -1, 1], [-2, -1, -3, 12, 9, 0, 4, 1, 3], [9, -4, -12, -1, -3, 4, 10, 4, 4], [5, 5, 6, 2, -4, 3, -2, -1, 5], [18, 5, 5, 8, 7, -2, -5, 1, -3], [-18, -7, -6, 2, -7, 15, -14, -1, 4], [27, 1, 8, -3, 8, -5, 4, 10, -2], [-14, -1, -3, 0, -3, 6, 15, -1, -3], [-12, -3, -5, 4, -8, 5, -13, -12, 4], [2, -1, -8, 8, -1, 7, 7, -16, -10], [-2, 12, -6, -6, 13, -1, -20, -5, -1], [-4, -11, 3, 0, -16, 7, 5, -5, 1], [-21, 1, 5, -9, 11, -7, 0, 8, -2], [-3, 2, 0, 11, -3, -6, -2, -1, 1], [40, -1, 5, 10, 4, -9, -5, -3, 9], [9, 1, -5, -12, -1, 7, -2, -6, -10], [14, 10, -9, -2, 1, 0, -13, -2, 3], [15, 2, 3, 8, 8, -10, -2, 0, 6], [-19, -3, -6, 3, -7, -2, -2, -1, -5], [-16, 0, -8, -5, 5, 9, -5, -9, -11], [-20, -8, -2, -6, -14, 2, 3, -5, -5], [11, 10, -1, 2, 10, -10, 5, 1, 9], [-19, -1, 1, 6, -5, -7, -9, -3, 3], [3, 3, -10, -1, -6, -3, 2, 4, 4], [11, -5, 2, -11, 0, -7, 1, 1, -7], [22, 5, -1, -5, 10, -7, 8, 1, -5], [20, 2, -2, 2, 4, -6, -10, -1, 5], [-16, -7, -6, -7, -4, 10, -6, -2, -1], [-13, 2, -12, -2, 4, 7, 1, -1, -4], [-13, 4, 3, -9, 0, 0, -6, -5, -1], [-16, -1, 2, -6, -2, 4, -14, -1, -3], [-9, 4, -3, 0, 0, 2, -3, -3, -11], [-11, 2, 13, 11, 5, -3, -5, 10, 4], [-19, 3, 2, -8, -3, 4, -2, 1, -1], [20, 0, -11, 4, 1, 4, -1, -7, -2], [-25, -6, 10, 7, -2, 1, 2, 2, -7], [1, -5, 15, -10, -4, -6, 4, 7, -1], [6, 7, 5, 4, 11, -8, 3, -7, 3], [6, -2, 4, 1, 6, -8, -4, 2, 2], [17, 0, 7, -2, 4, 2, 12, 0, -4], [12, 9, -4, -2, 8, 4, -1, 1, 0], [-2, 4, -11, 7, -5, 4, -3, -13, 7], [-1, -1, 17, -11, 3, -3, -5, 11, 3], [19, -5, -1, -10, 5, -15, -1, 13, -5], [36, 4, -4, 4, 2, 0, -4, 4, 4], [-3, -1, 0, -4, -8, -1, -5, -3, 0], [-10, -2, -2, 2, -3, 5, -2, 0, 12], [25, 7, -9, -3, 0, -4, 1, -1, 3], [-29, -3, 13, -2, -13, 3, 0, -4, 1], [18, -8, -6, -6, -7, 17, -7, -6, 4], [-2, 3, 6, 5, 3, -9, 8, 5, -12], [3, 1, -13, -1, 2, 4, -2, 3, -3], [32, -3, -7, 1, -6, 3, 5, -10, 0], [10, 7, 3, -13, 2, -1, -2, 0, -4], [3, 3, -12, -8, 1, 10, -21, -4, -4], [18, 0, -10, -2, -2, 4, -12, -8, 6], [6, 0, 1, 11, -7, 6, -1, -19, -2], [-16, -3, 5, 1, -4, -3, -1, 3, 1], [13, 1, -10, -7, 5, -2, -2, 7, -1], [9, 3, 0, 6, 7, -4, 4, -7, -3], [-7, -1, -1, -5, 2, -6, 3, -4, -6], [2, -6, 4, 4, -8, 4, -12, -14, 4], [1, 4, 3, -2, 5, -5, 3, 3, 5], [7, -7, -3, 16, 0, 14, 7, -12, -4], [15, 6, 8, -7, 6, -9, -4, 1, -7], [-21, 6, -11, -6, 2, 10, -5, -8, -6], [9, 12, 0, 3, 3, 2, -8, -5, 10], [-21, -2, -2, 11, 2, 1, 0, -7, -3], [18, 8, 4, -1, 8, -2, -2, 10, 8], [5, 2, -13, 3, -6, 10, -11, -4, 14], [-16, -9, -4, 12, -7, 11, -1, -4, 7], [2, -7, 6, -8, -4, 8, -4, -7, 4], [-20, 0, -4, -1, -2, 0, 2, -1, 5], [7, 8, -11, -2, -1, 1, -7, 1, 7], [-29, -7, 0, 0, 13, -6, 12, 0, -4], [6, 10, 0, 3, 19, -15, 5, 1, 3], [-34, -5, -12, -12, -11, 19, 6, -13, -18], [-26, -11, -3, 2, -9, 2, 1, -4, -6], [5, -5, 5, 6, -4, 0, -4, -7, 12], [9, 5, -14, -11, 6, 7, -6, -2, -10], [-11, 0, -9, -7, -1, -1, 3, 3, -7], [14, 3, -2, 0, 9, 3, 4, 5, -2], [5, 0, -15, 5, 4, 6, -3, -15, -9], [-8, -3, -5, -14, 4, -1, 4, 3, -15], [6, 6, -9, 9, 4, -1, -3, 1, -5], [-5, -9, 4, 5, -10, 1, 0, 3, 5], [-19, -1, 3, 3, -3, 1, 3, -7, -14], [-8, 0, 0, 6, 0, 10, 12, 6, -6], [9, 1, 1, -9, -9, 5, -17, -7, 1], [-12, 3, -5, -2, 7, 4, -1, -11, -5], [49, 5, -6, 5, 17, -20, 12, 16, -2], [8, -4, -4, 15, 1, -1, 20, -1, -3], [17, -5, 4, 6, 4, -9, -3, 3, 8], [2, -8, -3, -12, -9, 14, -7, -1, -5], [-11, 1, -1, 9, -5, 5, -1, -11, -6], [4, 5, 15, 3, 5, 4, -3, 9, 11], [30, 4, 2, -3, -2, -4, -4, -1, 5], [24, -6, -7, 9, 4, -3, 9, 0, 5], [10, 2, 2, -2, -5, 5, 2, -2, -2], [6, -2, -2, 11, -10, -2, -4, -2, 10], [17, 8, -1, 0, 7, -9, -22, 2, 6], [-9, 5, -6, 4, -7, -6, 1, -6, 4], [-22, -4, 1, -2, 3, -8, -17, -2, 12], [-10, 0, 4, -8, -8, 8, -2, 14, -2], [-6, -1, 11, -14, 1, -14, 7, 6, -8], [-8, 1, -5, 0, 2, -3, 3, 10, 2], [14, 3, -10, -8, 5, 3, -8, -5, 3], [11, 4, -8, 4, 3, -6, 0, 4, 8], [14, 3, -2, -10, 4, -18, 2, 5, 5], [17, 7, 8, -5, 3, 6, -4, 11, -1], [-12, -1, 22, -2, 12, -10, -4, 18, -1], [21, 7, -6, -5, 18, -15, 4, 10, -5], [5, 7, 1, 7, 9, -7, -9, -1, 11], [12, -8, 10, -4, -1, -7, -4, 9, -5], [14, -3, -3, -4, -7, -4, 1, -4, -4], [5, 7, -10, -15, 4, -7, -20, 5, 3], [2, 0, 0, -1, -3, 1, -13, 6, 0], [-32, -10, 5, -8, -16, 15, -9, -8, -3], [17, 1, -3, -13, 1, 1, -3, 3, 0], [11, 1, 7, -1, -11, 5, -10, 2, 7], [21, 8, -3, 3, 12, -4, -3, 12, 0], [25, 0, 6, 1, -1, -6, -2, 9, 10], [4, -7, -13, 0, -2, 3, 11, 2, 2], [36, -2, 11, 10, -1, -12, 13, 0, 2], [-14, 4, 1, -1, 13, -4, 3, 10, 4], [-4, 7, 6, -11, 8, -8, 3, 3, -10], [-16, -3, -6, 1, -11, 11, -18, -11, 6], [6, -9, 3, -3, -4, -1, 12, 4, -6], [35, 4, -9, 7, 5, 1, 1, 13, 13], [-26, -9, 8, 7, -9, -3, -2, 5, -1], [4, -12, 4, 13, -8, 0, 13, -1, 3], [0, -5, 0, 0, -7, -3, 10, -1, -12], [14, -3, 2, 0, 9, -9, -2, 7, 2], [8, 2, -16, -1, -11, 3, -7, -16, 8], [-20, -1, 5, 8, -4, 11, 3, -10, 0], [-32, -17, -1, 2, -15, 14, -2, -6, -10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6042_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6042_2_a_z();".
// 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_6042_2_a_z(:prec:=9)
    chi := MakeCharacter_6042_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_6042_2_a_z();".
// 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_6042_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6042_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![8, 40, -66, -250, 157, 186, -22, -27, 1, 1]>,<7,R![256, 384, -592, -1184, -274, 265, 71, -25, -4, 1]>,<11,R![-64, -272, 384, 1402, -232, -447, 107, 31, -12, 1]>],Snew);
    return Vf;
end function;