
// Make newform 6042.2.a.bc in Magma, downloaded from the LMFDB on 24 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_bc();".
// 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_bc();".
// 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, 109, 8, -185, 27, 69, -20, -3, 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, 1, 0, 0, 0, 0, 0, 0, 0], [4020, -11054, -11661, 7311, 6181, -2018, -599, 96, 18], [774, -79830, -19643, 77067, -3014, -16721, 2752, 676, -133], [23946, 94666, -22041, -173817, 27438, 64977, -18066, -2846, 895], [-2730, -43032, 43687, 96198, -39950, -32777, 12805, 1377, -586], [20028, -51710, -37220, 120673, -7016, -49418, 13373, 2181, -695], [25494, -52526, -94575, 227599, 4786, -105613, 26920, 4740, -1449], [83714, 95310, -242785, -206163, 144790, 69901, -35292, -2864, 1541]];
    Rf_basisdens := [1, 1, 2078, 8312, 8312, 4156, 4156, 8312, 8312];
    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, -1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, -1, 1, 0, -1], [-1, 1, 0, -1, -1, 1, -1, 0, 1], [-3, 0, 1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 1, 0, 1, -1, 0, 0], [1, -1, -1, 2, 1, -1, 1, -1, 0], [0, 0, 1, 0, -1, 0, 0, 2, 0], [-4, 1, 1, -1, -1, 3, 0, 1, 0], [0, 0, 0, 0, 0, -1, 1, -2, -1], [-2, 1, 0, 1, 0, -1, 1, 0, 0], [0, 0, -1, 0, -2, 2, -3, 0, 2], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, -3, 0, -1, -2, 1, -1, 2], [-3, 1, 1, -1, 0, 1, -1, 2, 0], [-1, 1, 2, 0, 2, -3, 1, -1, -3], [-1, 1, 1, -2, -1, 1, 2, -2, -2], [-5, 4, 0, -1, 0, 0, 0, 1, -2], [2, 0, 0, 0, 2, -1, 1, -2, -2], [-3, 0, 0, 1, 0, 0, -3, 2, 0], [-5, 2, 2, -3, 0, 0, 2, 1, -1], [-2, 1, 0, -1, 0, -1, -1, 0, 2], [-6, 0, 3, 1, 3, -4, 2, 0, -3], [-8, 0, -2, -1, -1, 1, 0, 0, 3], [-2, 2, 1, -1, -2, 2, -1, 2, 3], [0, -1, 2, -2, 0, 3, -3, -2, 1], [-4, -2, 0, 2, 4, 0, 2, 2, -1], [-2, 3, -1, -1, -3, 3, -2, -1, 1], [3, -2, -2, 2, 0, 0, -4, -1, 3], [-6, -1, 0, -2, -2, -1, 3, 0, 2], [-1, -4, 0, 2, 1, 0, 0, -2, 2], [-7, -1, -2, -1, -1, 1, 2, -1, -1], [8, -2, 2, 1, 1, 0, -2, -1, 2], [-4, 4, -1, 0, -1, 2, -1, 1, -2], [-1, 1, 2, -1, 3, 1, 0, -1, -2], [-6, -2, 0, -1, -1, 2, 2, 3, -1], [-1, -3, -2, 2, 2, -1, -1, -3, 2], [-4, 2, 0, 0, -2, 0, -2, 2, 0], [-6, -1, 2, -2, 0, 1, 3, 0, 0], [2, -2, -4, 0, 0, 2, 2, 0, 1], [-3, -3, -5, 3, 1, -1, 1, -1, 3], [-4, -1, 0, 0, 1, -1, 0, 2, -1], [-1, -4, 1, 2, 2, -2, 2, 2, 1], [-3, 2, -2, -2, 0, -2, 1, 0, -2], [-4, -1, -2, 1, -1, 1, -2, 2, 2], [5, -1, -2, 0, 0, -4, 0, -3, -2], [-1, 1, -4, -3, -3, 3, 0, -3, -1], [1, 0, 4, 0, -2, 2, -2, -1, -1], [2, -2, 4, -1, 1, -2, 1, 0, -5], [-7, 1, 0, 4, 2, -3, 5, 1, -2], [-7, -2, 1, -5, -1, 2, 0, -1, 1], [1, -1, -1, 1, -2, -3, -1, -2, -2], [1, -1, 2, 2, 0, 3, -5, 1, 2], [7, 1, -6, 4, -2, 2, -4, 1, 6], [1, 1, 6, 1, 3, 0, -3, 1, 0], [3, -2, -2, 0, 1, -6, 3, -3, 1], [-4, -1, -2, -1, 0, -1, -1, 0, 0], [0, -4, -1, 1, 0, -4, 5, -4, 1], [-1, -2, -5, -2, -2, 6, -5, -3, 3], [1, 3, 5, 2, 2, -1, -2, 1, -4], [0, 2, -1, 3, 3, -2, 4, 0, -2], [6, -1, 2, -4, -1, 1, 2, -2, -3], [-7, -6, -2, 1, 0, -1, 3, -1, 1], [1, 5, 0, 5, 0, 0, 0, 5, 1], [-5, 2, 4, 0, 0, 0, -5, 2, -3], [-3, -2, 0, -2, -2, 6, -2, 1, 0], [9, 0, 1, -3, 1, 2, -2, 1, 0], [-3, 1, 0, -2, -1, -1, -2, -5, 2], [-5, 0, 2, -1, -4, 2, -2, 3, 7], [-10, 1, 3, -8, -2, 1, 5, -1, 0], [-2, 2, -2, 3, -2, 4, -6, 6, 4], [9, 5, 2, -2, -2, -1, 1, 3, -1], [-4, 0, -4, -4, -4, -1, 5, -2, 0], [3, 2, 2, 3, -1, -4, -2, 4, -2], [-3, -1, 1, 2, 7, -1, 5, -3, -3], [-4, -7, -2, 0, 1, 5, -1, 3, 1], [0, 2, 0, 1, -1, -6, 3, 2, 1], [6, 1, 0, 0, 1, -2, 1, 4, -2], [-8, 2, 2, 0, -3, 4, -5, 4, 2], [-1, 0, 2, 0, 3, 0, -3, 1, -6], [9, -3, 0, 0, 2, -5, 1, -5, -1], [0, 5, 0, -6, -8, 3, -1, 4, -2], [-1, -4, -2, 0, 1, -2, -1, -1, 2], [-6, 5, 0, 0, -5, 5, -4, 0, 3], [5, 3, 0, 5, 3, -7, 4, -3, -5], [-5, 1, 2, 6, 4, -3, 3, 3, -3], [5, -4, -4, 0, 1, 6, -1, -1, 3], [-4, 4, 2, -1, 0, 6, 0, 6, 1], [-1, 2, 0, -1, 2, 2, -4, -3, 0], [-5, 3, -4, -8, -7, 3, 2, -1, 2], [-6, -5, 4, 0, 1, 0, 3, 0, -4], [19, 0, 1, 4, 2, -2, -1, -3, 3], [2, 1, 2, -2, 3, -7, 6, -6, -9], [-2, -1, -6, -1, -2, -3, 1, -8, 3], [-16, 0, 0, 1, 3, 0, 1, 0, 1], [8, -5, -1, 3, 0, 3, 1, -1, 4], [13, -6, 4, 8, 5, -4, -1, 3, -4], [8, 4, -3, 3, 1, 2, 1, 3, 0], [-2, 2, -3, 1, 3, -6, 1, -1, -3], [-9, -1, 6, -4, 0, -5, 9, -3, -3], [6, -1, 1, -1, -1, 5, -11, 4, 4], [2, 0, 0, -7, -1, 1, 0, -4, -3], [4, 5, 0, 3, 0, -7, 2, 3, 0], [6, -2, -4, -1, 2, -2, 2, -2, 0], [-5, -2, 1, -6, -2, 4, -2, -2, 1], [3, 4, -4, -1, -2, -4, 2, 3, 0], [-4, -2, 2, -4, -5, -2, 4, 3, -1], [8, -2, -4, 2, 1, -2, 7, -2, 3], [8, -9, -4, -1, 2, -3, 4, -1, 2], [-4, -3, 0, 3, 1, -5, 6, 4, 0], [8, 0, 8, 2, 6, 4, 0, 2, -1], [9, -7, 4, -3, -1, -1, 6, -3, -1], [-19, -3, -4, -2, 2, 6, -2, 3, 2], [4, -8, -4, 5, 1, 0, 1, 2, 5], [16, -6, 0, 4, -1, -1, -4, -2, 2], [-4, 5, -4, -5, -2, 3, 1, 0, 1], [7, -9, -2, -4, 0, 7, 3, -5, 5], [-6, 7, 2, -4, -1, 1, 4, -2, -5], [-14, 5, 8, 2, 3, -7, 0, 2, -4], [-11, 6, 4, -3, -4, 4, -6, 1, 2], [8, -3, 1, -1, -3, -1, 1, -4, -1], [11, -2, -1, -3, -3, 12, -7, -4, 5], [0, 2, 6, 2, 7, 2, 2, 3, -7], [4, -7, 5, 5, 9, -3, 4, 3, -3], [0, 1, 6, 3, 2, -1, -3, -4, -1], [5, 2, 0, -1, 2, -1, -5, 5, -5], [0, 0, -4, 1, 1, 0, -1, 4, -2], [4, 8, 4, 5, 1, -7, 2, 0, -7], [-2, 2, 0, 1, 5, 0, -5, 2, -2], [11, 2, 2, -4, 0, -2, 4, -5, -9], [3, 5, -2, -2, -10, -1, -5, 1, 5], [8, -8, 2, 3, -4, 8, -8, 2, 3], [8, 5, -4, 5, 0, 1, 1, 6, 1], [7, -1, -6, 3, -3, -1, 2, -1, 4], [-9, 4, 2, 1, 2, -4, 4, -1, -6], [10, -1, 4, 6, 9, -5, 4, -2, -4], [14, -3, -2, 3, 1, -1, -6, 0, 0], [-12, 2, -9, 7, 1, -6, 1, -3, -1], [-2, -5, 1, -1, 6, 7, 1, 1, 4], [3, -1, -3, 1, 0, 5, -2, -3, 0], [2, -11, -2, 5, 8, -1, 3, 2, 3], [-6, -4, 6, -1, 4, -6, 6, -6, -8], [1, -3, 0, 5, 3, -5, 8, -7, 1], [-8, 0, -2, 1, -3, 4, -8, 1, 2], [-8, -8, 2, 3, 3, -6, 3, -6, -5], [-11, -8, 2, 3, 2, 8, -2, 3, 5], [-9, 0, 4, -4, 5, 8, 1, 1, 0], [4, -2, 3, 3, 6, -8, -4, 1, -3], [-1, -2, -2, 7, 7, -4, 11, 3, -9], [-2, -7, 0, -2, 3, -4, 3, 4, 0], [3, -4, -4, 10, 7, -8, 2, -2, -3], [13, 6, 6, -3, -3, 1, -2, 1, -3], [0, 5, 6, 3, 6, -5, -1, 0, -5], [17, -1, -10, 3, -5, 1, -8, -7, 4], [-16, 6, 8, -9, -3, 2, -2, 3, -3], [-5, 1, 5, -2, 7, -3, 5, 1, -5], [-1, 7, 0, 1, 5, 8, -11, -1, 2], [18, -10, -6, -1, 0, 4, -4, -6, 3], [1, 0, -2, -3, -4, 8, -4, -7, 0], [-10, 4, 5, -6, -7, 6, 2, 8, -3], [-6, 3, -6, -5, 2, -1, 4, -1, -2], [-1, 5, 0, 0, 6, -1, -2, -2, -3], [-20, 2, -2, 5, -1, -2, -3, 4, 4], [-15, 10, -2, -3, 3, -2, 3, 3, -1], [4, 7, 2, -1, 8, 3, -3, -4, -5], [1, -6, 2, 0, -2, 12, 0, 1, 3], [-10, -5, 2, 4, 7, -1, 2, 2, -3], [10, 2, -8, 1, -5, 8, -3, 6, 4], [-13, 5, 6, -7, -2, 3, 3, 5, 2], [0, -3, -3, -3, 0, 5, 0, 4, 3], [11, 5, -2, 5, -1, 1, 0, 5, 7], [3, -3, -8, -2, -4, 3, -5, -7, 6], [1, -6, -2, 1, -2, -6, 1, -2, 0], [6, -1, 10, -4, 3, -5, 10, -6, -6], [9, -7, 0, -6, -2, 7, -5, -5, 3], [-5, -6, 1, -4, -4, 0, 3, 1, 4], [11, 6, 4, -2, 1, 6, -1, 5, 2], [28, -2, -2, 3, -1, -2, -9, -4, 3], [11, 3, -2, 1, 3, -7, 6, -1, -3], [-5, -1, 6, -2, -3, -1, -4, -3, 1], [-4, 3, 4, -6, 1, 9, -8, 0, 0], [-3, 2, 1, -7, -6, 2, -4, -2, 10], [5, -6, -2, -6, -3, 2, -7, 3, 4], [6, 4, -7, -1, -2, -2, 3, 2, 5], [3, 4, -4, 4, 4, -3, -3, 3, 0], [-8, 2, -6, 5, 1, -2, 1, -4, 3], [9, -11, -6, 1, -6, 1, 4, 0, 6], [-21, 3, 2, 3, -1, -5, 0, 7, -1], [-7, 0, 4, 5, 6, -4, 3, 2, -2], [1, 4, 2, -3, 0, -6, 4, 1, 6], [-6, 0, 11, 2, -1, 0, 4, 6, -3], [9, 2, 15, 0, 8, -4, 3, 7, -5], [-5, -3, 5, 3, -2, 1, -2, 1, 4], [1, -1, 6, 0, -4, 3, -3, 11, 3], [-6, -2, -2, 9, 3, -4, 1, 4, 0], [0, -9, 6, 4, 4, -9, 5, 0, -10], [6, 1, -8, 1, 3, -12, 5, -8, -3], [-13, 8, 0, 1, 4, -2, 13, 0, -6], [-10, -6, 6, 5, 7, -7, 8, -2, -4], [-7, 9, 8, -11, -8, 5, -7, -1, -1], [13, 3, 2, -1, 2, -9, 11, -7, -10], [4, 1, 8, 5, 7, -9, 6, -4, -11], [4, -10, 1, 0, 0, -2, -7, -6, 6], [-10, -8, 7, 0, 7, -2, 6, -4, -1], [12, 2, 6, -10, -10, 7, -7, 0, -4], [-12, -5, -4, 5, -4, 4, -12, 8, 9], [28, -1, -8, 6, 5, 3, -8, -6, 3], [7, 1, 3, -8, -9, 5, 7, 1, -3], [-6, -3, 4, -5, 8, -1, 5, 0, -13], [11, 0, 0, -7, -4, 0, 6, -1, -4], [-1, 0, -4, -3, -6, 0, 6, -1, 6], [-16, 2, -6, 0, -6, 10, -14, 0, 8], [3, -10, 0, 2, 5, 8, 3, 5, -3], [-7, -5, 2, 0, -2, 7, 3, -3, -1], [6, -2, -8, 4, 6, -14, 8, -8, -9], [-3, 0, 14, -2, 7, 0, 3, 5, -8], [8, 10, -1, -2, 4, -4, 4, -3, -3], [-10, 1, 4, -3, -9, 1, 0, 6, 5], [1, 10, 6, -2, -1, -8, 7, -5, -5], [28, -4, 0, -2, -6, 0, -1, -9, 6], [-11, 8, 2, -6, -11, 16, -5, 3, 0], [0, -3, -4, 7, 6, -7, 11, -4, 6], [9, 3, 10, -3, 5, 5, 6, 3, -8], [-39, 2, 8, -7, 0, -6, 7, 4, -4], [16, 7, 2, 0, 3, 2, -9, -4, -3], [-21, 13, 4, 5, 0, -9, -1, 3, -6], [6, 5, 2, 6, 3, -7, -9, -7, 3], [0, 15, -7, -5, -5, 9, -6, -1, 4], [0, -1, 4, 3, -7, 7, -5, -1, -1], [0, 4, 0, 2, 3, -12, 9, 0, -1], [8, 2, -9, 2, -3, 4, -3, -1, 5], [-9, 6, 4, -7, -2, 12, -8, 5, 8], [-5, -6, -4, 0, -9, 8, -7, 5, 16], [-23, 12, 10, -13, -4, 12, -12, 5, -4], [-8, -4, 6, 3, 9, -10, 15, 4, -8], [6, 0, -2, 3, 4, 4, -12, 6, 4], [-14, 0, 1, 0, 1, 4, 3, 11, -7], [12, -11, 6, -1, 0, -9, 13, -8, -7], [28, 4, 2, -1, -1, 4, -5, -4, -1], [-1, 14, 4, -3, -2, 3, -1, 11, -3], [-15, 7, 0, -7, 2, 3, 9, -1, -1], [-7, -1, -4, 6, 8, -11, 11, 1, -8], [-26, -1, 0, -2, 3, -7, 6, -12, -4], [11, 14, -4, -6, -1, 2, -1, -1, 0], [-2, 2, 2, 0, -2, 4, -14, 4, 6], [16, 0, 0, 4, 9, 0, 2, 1, -9], [18, -6, -10, 7, -2, -4, -5, -11, 5], [15, -1, -8, -4, -4, 9, 1, 1, 3], [-2, 7, -5, 8, 0, -5, -6, 10, -1], [22, -4, 3, -3, -3, 0, 0, 10, 4], [-7, 2, 1, 13, 4, -6, -4, 4, -6], [-9, -1, -12, 11, 4, -10, 10, -5, 3], [11, 0, 10, -2, 4, 4, 2, 5, -4], [7, 6, -8, -2, -11, 3, -8, 1, 9], [2, 14, 0, -4, -7, 6, -4, -5, -3], [12, -11, 0, -3, -10, 5, -4, 3, 11], [29, -7, 10, 1, 1, 1, -8, 1, -3], [-18, 1, 3, -12, -4, -3, 11, -3, -12], [-16, 8, -2, -13, -3, 0, 5, -8, -5], [-19, -1, 0, -1, 7, -10, 5, 3, -11], [-19, 4, 4, -1, 9, 2, 3, -7, -14], [-18, 4, 10, 5, 9, 7, -4, 6, -3], [19, -2, -6, -2, -3, 6, -9, 7, 8], [-23, 6, 8, -1, 2, -2, -1, 0, -2], [-1, 5, 0, 0, 0, -4, 4, -5, -12], [-12, -13, -2, -5, -2, -3, -1, -2, 5], [0, 0, 6, -3, 9, -4, 7, 6, -8], [-12, -7, 12, -3, 1, -7, 12, 2, -3], [-2, 3, -10, -1, -12, 5, -5, 6, 11], [-8, -2, 15, -2, -3, -2, 6, 8, -4], [-7, -15, -3, -6, 5, 15, 2, -2, 6], [-3, 14, -6, 3, -4, -4, -12, 1, 12], [-10, 7, 0, 0, -1, -9, 2, -6, -6], [14, 4, -8, 2, 1, 4, -7, -6, 5], [1, 16, 6, -1, -2, -7, 1, -1, -1], [17, -8, 2, -3, 6, -2, 2, -5, -1], [-5, 0, 0, 10, -4, 2, -6, 5, 2], [-24, -11, 0, -6, -1, 13, 5, 7, 2], [-15, -3, 10, -10, -8, 5, -2, 8, -5], [6, 0, -12, -1, 3, 2, 7, -8, 1], [22, -10, 10, -2, 0, 5, -5, 8, -3], [-13, -5, 2, 1, 11, -7, 8, -3, -7], [11, -9, -6, 7, 8, 3, -2, -8, 0], [1, 10, 4, 2, 1, -6, 1, 7, 4], [11, -8, -10, 15, 8, -4, -10, -3, 3], [-7, 7, -9, -13, -4, -1, 5, -6, 4], [-10, -1, -12, 3, -10, -3, -1, 0, 11], [-5, 9, 20, -9, 1, 9, 0, 15, -9], [-8, 15, -10, -7, -11, 17, -10, 2, 17], [-41, 2, 4, 1, 6, -5, 7, 1, -7], [26, 4, -4, -9, -5, 6, -7, -8, 6], [4, -13, 8, 5, 6, -1, 1, 4, -1], [-4, -3, 2, -9, -7, -7, 13, -7, -1], [-32, 3, 4, -4, 12, 3, 3, 4, -8], [-13, -9, -12, 9, 4, -5, 0, 2, 0], [6, -7, 4, 11, 8, 5, -3, 6, 3], [-4, 3, 2, -9, -12, 22, -14, 14, 14], [-24, -11, 0, 2, 3, -3, 10, -2, 1], [-14, 2, -6, 0, 6, 5, 3, -4, -1], [-29, -11, -2, 3, 9, 0, 7, 3, 6], [-10, 0, 10, 0, -2, 6, -9, 7, 5], [-7, -1, -6, 6, 12, -7, 4, -2, -7], [2, 7, -5, 3, -6, 9, -11, -1, 3], [-4, -6, 1, 1, -8, 16, -4, 5, 8], [-9, 8, -3, 5, -5, -4, -11, 0, 7], [-7, -8, -4, 1, -7, -2, -7, -3, 3], [16, 0, 4, -3, 3, -5, 2, -2, -16], [-27, 16, 12, -15, -7, 2, -7, 5, -5], [-13, 16, -11, 3, -8, 2, -7, 5, 1], [-8, 1, 1, 8, 12, -7, 12, 2, -8], [-16, -2, -10, 0, -11, -1, -4, -4, 9], [-28, 0, -16, -6, -10, -4, 2, -6, 9], [-13, 4, 13, 0, 5, 2, 1, -4, -8], [-33, 0, 3, -8, 0, 10, -2, -2, -7], [2, 1, -8, 6, -3, 3, -8, 0, 5], [-1, 10, -1, -5, -5, -2, 0, 3, 4], [-7, -9, -13, 3, -10, 1, -13, -2, 17], [-21, 7, -1, -4, -6, -11, 2, -5, 7], [6, 0, 10, -1, 5, -7, 0, -8, -3], [0, 6, -4, 0, -4, -10, 4, -8, 5], [-5, -24, -8, 13, 0, 1, 1, 3, 10], [-5, -7, -10, 5, -2, -11, 1, -9, 4], [-29, 0, 13, -5, 6, -2, 5, 1, -8], [12, -10, -14, -1, -4, 0, 9, -7, 11], [-24, -2, 2, 2, 10, -16, 6, 2, -2], [19, 1, -2, 5, -5, 2, -5, -1, -4], [-19, 6, -8, 7, 0, -14, 8, 3, 2], [25, -10, 6, 14, 3, -4, 0, 4, -7], [2, -3, -6, -2, -7, -1, 9, 1, -3], [-25, -18, -4, 7, 10, -6, 4, -5, 6], [-4, 5, 2, 2, 10, 7, -15, 4, 5], [29, 5, 8, 1, 1, 1, 4, -3, -10], [-17, 15, -3, 2, -1, -7, 4, 6, -10], [12, -6, -11, -3, 0, 0, -6, -13, 7], [-4, 1, -12, 10, -3, -3, 2, -4, 5], [12, -18, 0, 0, -4, -14, 16, -16, -6], [35, -9, -2, 8, 2, -11, -9, -1, 2], [-19, -5, -10, 2, -3, -5, 0, -1, -6], [-10, -6, 4, -5, 2, 14, 2, -4, -3], [24, -5, 2, -5, -4, 5, -3, 0, 4], [6, 6, 13, 1, 12, 10, -8, 3, -16], [11, 3, 2, -4, 4, -2, 6, -1, -8], [18, -2, 0, -4, -8, 4, 4, 2, -1], [18, 10, 4, -11, -11, 8, 3, 0, 2], [-1, -7, 11, 1, 7, -9, 15, -1, -12], [-25, 7, -5, 0, 3, 7, 4, 6, 3], [6, -12, -10, 2, -4, 6, -4, -10, 6], [30, -14, -12, 15, 5, -8, 5, -4, 10], [12, -9, -4, -1, -7, 0, 15, -10, -1], [11, 4, 10, -7, -4, 4, 0, 5, -8], [-1, 13, 1, -9, -10, 11, -8, 5, 6], [25, -15, 3, 0, -9, 11, -12, 8, 16], [-20, 4, 10, -7, 9, -10, 10, 1, -8], [21, -2, -2, 2, 1, 2, 1, -7, 8], [0, -1, -15, 9, 7, -23, 8, -5, -2], [13, -14, -6, 3, 9, -6, 9, -5, -3], [-20, 3, 4, -1, 4, -5, -1, 8, -3], [4, -3, -6, 4, -7, 3, -4, 12, 11], [-9, 16, 6, 3, -5, -4, 5, 13, -1], [21, 4, 10, -3, 0, -6, 12, 9, -16], [-5, 19, -2, -10, -6, 1, -1, 1, -7], [0, -4, 14, 2, 4, 2, -2, 2, -5], [-1, 2, -8, -1, -13, 13, -6, 5, 2], [14, -21, 5, 10, 1, -1, 0, -1, -3], [-16, 15, 1, 7, 5, -9, 1, -10, -12], [-11, -3, 9, 2, 0, 21, -6, 15, -3], [22, -9, 4, 3, 5, 17, -10, 10, 9], [6, -14, 10, 0, -8, 16, -6, 10, 4], [0, -19, -10, 0, 2, -13, -1, -8, 8], [-3, 8, 8, 4, 5, -12, 5, -7, -7], [20, 9, -14, 5, 0, -3, -6, -5, -3], [18, -12, -6, -1, -5, -4, 3, 0, 8], [-2, 16, 8, -6, -4, 8, -4, 0, -2], [-26, 12, -3, 1, 7, -6, 9, 1, -4], [-15, -4, 8, -1, 2, 4, -1, -2, -1], [-27, 13, -6, -10, -7, 1, 4, -9, 2], [-11, -12, -3, 16, 7, 0, -6, 3, 4], [-11, 21, 10, -3, 5, 5, -9, 2, -15], [-11, 13, 3, 5, 2, 7, -12, 5, 0], [-1, 9, -12, -13, -7, -5, 6, -5, 5], [23, 10, -3, 8, 0, -2, -5, 7, 1], [25, -1, 8, -7, -6, 6, -8, -3, 3], [12, -1, -4, 2, -5, -7, 6, 4, 1], [31, -5, -4, 2, 2, 3, -3, -3, 4], [7, -7, 10, 4, 8, -18, 6, -7, -14], [24, -2, 5, 4, 1, 6, -9, 9, -3], [5, 7, 0, -2, 8, -5, 17, -5, -5], [-17, 2, 10, -20, 0, 16, -10, -1, 7], [-5, -3, 0, 4, 2, -7, -5, -9, 1], [9, 9, -2, -4, -15, 15, -10, 9, 8], [33, -8, 2, 3, -6, -4, 2, 3, 6], [-7, 5, 10, 7, 3, -6, -1, 5, -1], [-2, 4, -8, 12, -6, -8, -6, 0, -1], [-27, 4, 2, 2, -6, 6, 0, 11, 13], [-17, 11, 1, -2, -13, 11, -14, 4, 2], [-2, 11, 8, -1, -1, 1, -1, 9, -8], [-27, 2, 2, 2, 13, -14, 5, -1, -2], [-8, -2, -2, -15, 1, 10, -5, -4, 6], [-11, 13, 2, -3, 0, 17, -17, 1, 0], [-1, 11, -12, 17, -2, -9, -3, 3, -2], [1, -8, 10, -2, 7, 18, -2, 4, 2], [-5, -1, 4, 1, -1, -7, 2, 13, -5], [12, 3, -6, 1, 6, -11, 1, -6, -7], [7, -4, -8, -2, -7, 5, 6, -7, 9], [-38, 0, 7, -4, 1, -4, 4, -4, -1], [-1, -6, -9, 6, -5, -4, 1, 4, 0], [-18, 15, -16, -7, -7, 3, 2, -4, 4], [-22, 7, 2, 6, 1, -1, -6, 10, 2], [0, 23, 0, -17, -10, 1, 5, 6, 1], [-9, 7, -14, 0, -8, -5, -1, -5, 13], [-15, 11, 20, -6, 6, -1, 13, 9, -14], [5, -15, -4, -4, -8, 1, 9, -1, 9], [34, -9, -6, 6, 7, -19, 14, -12, -1], [29, -22, -6, 6, 5, -14, 1, -13, 0], [21, -4, 0, -6, -2, -8, 20, 3, -13], [20, -6, -8, 5, 0, -2, -18, -4, 7], [-27, 8, 11, 6, -2, 10, -6, 12, 0], [-7, 16, 24, -11, -10, 6, 0, 17, -6], [-4, -12, -8, 10, 7, 4, -5, -4, 2], [8, 0, -18, 5, -1, -6, -3, -6, 6], [-17, 0, -10, 0, -5, 14, -19, 11, 12], [-23, 8, -14, 3, 8, 0, -2, 3, 7], [-22, -7, 6, -7, 4, 22, -6, 2, 0], [8, 4, -8, -6, -6, -2, 4, -10, -2], [-24, 5, -6, -9, 0, -7, 16, -5, 8], [51, 6, -10, 7, 0, 6, -13, -2, -2], [32, 5, -4, 9, -2, 7, -5, 0, -8]];
    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_bc();".
// 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_bc(: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_bc();".
// 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_bc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6042_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![14, -8, -109, 8, 185, 27, -69, -20, 3, 1]>,<7,R![-5968, -15512, -6282, 4969, 3203, -45, -287, -29, 7, 1]>,<11,R![144, 696, -8, -2282, -774, 593, 149, -49, -4, 1]>],Snew);
    return Vf;
end function;