
// Make newform 6042.2.a.bb in Magma, downloaded from the LMFDB on 25 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_bb();".
// 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_bb();".
// 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 := [16, -106, 101, 300, -366, 30, 76, -16, -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, 1, 0, 0, 0, 0, 0, 0, 0], [-19336, -146543, -71814, 285404, -46158, -57016, 11484, 2940, -615], [44587, -90525, -276671, 336392, -12713, -70426, 11855, 3724, -779], [17350, 70569, 48122, -371346, 148714, 78330, -33928, -4158, 1831], [120443, -320194, -350096, 678224, -110149, -138099, 34561, 7273, -2002], [-166196, 451057, 371756, -1097896, 299972, 227530, -74202, -12078, 4037], [63294, -309497, -279212, 684937, -141607, -140186, 38702, 7391, -2164], [224837, -539153, -727251, 1236964, -167719, -249600, 58897, 12997, -3474]];
    Rf_basisdens := [1, 1, 26914, 13457, 26914, 13457, 26914, 13457, 13457];
    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], [-1, 0, -1, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, -1, -1, -1], [0, -1, 0, 1, 0, 1, 1, 0, 0], [-2, -1, 0, -1, 1, -1, 0, 1, 1], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [2, -1, 1, 0, -1, 0, -1, 1, 0], [-1, 0, 0, 0, 0, -1, 0, -1, 1], [-2, 0, 0, -1, 0, 0, 2, 1, 2], [0, 1, 2, -1, -1, 1, 0, 1, -1], [-2, 1, 2, -1, 1, 1, -1, -1, -1], [0, -1, -1, 2, 1, 0, -1, 1, 0], [-1, 0, 1, 1, 0, 0, 0, -2, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 1, -1, 0, 1, 1, 0, -1], [0, 1, 3, 2, -1, 0, -3, -4, -1], [2, 0, -1, 1, 0, -1, 2, 3, 1], [-1, -1, -1, -1, -1, -1, 1, 2, 0], [-3, 0, -1, -4, 0, 0, 3, 3, 1], [1, -2, 1, 0, 0, 4, 1, -2, 1], [-4, -2, -2, -2, 1, 0, 2, 2, 3], [-4, 0, -2, 4, 2, -4, -5, -2, 1], [-2, 4, 1, 1, -2, -1, -3, 1, -2], [-3, 0, -2, 0, 0, -1, 4, 1, -1], [1, 0, -2, 3, 0, 1, -1, -3, -2], [-2, 1, -1, -3, -1, 2, 2, 0, -1], [-1, 3, -1, -1, -1, -3, -1, 1, -1], [-5, -2, -1, 0, 0, 0, 2, -1, -2], [-3, 1, -3, 1, 1, 1, 0, -2, -1], [0, 4, 3, -1, 0, 1, -5, -3, -4], [0, 2, 2, 0, 0, 2, -1, 2, -1], [3, 0, 0, 3, -1, -1, -2, 0, 0], [-1, 3, -1, 1, -1, 1, 0, -5, 0], [-6, -1, -1, 0, 2, -2, 1, -3, -1], [-10, 1, 1, 2, 3, 0, -3, -2, -1], [-8, 1, -4, -1, 1, -3, 0, 4, 4], [-3, 1, -2, 2, 0, -4, -3, -4, 1], [-5, 4, 0, -3, -2, 1, 0, 2, 1], [2, -2, -2, -2, 2, -2, 2, 2, 2], [0, 2, 2, 0, -4, 2, 1, 0, -1], [-5, 1, -3, 1, 1, -1, 6, 1, 0], [-2, 3, 0, 0, -1, -3, 2, 1, 2], [-2, -3, -2, -5, -1, 3, 7, 2, 1], [-7, 1, -3, -1, -1, 1, 0, 4, 3], [-2, 1, -1, -4, 0, -4, 0, 5, 6], [-5, 7, 0, 4, -1, -2, -9, -6, -2], [-10, -1, 1, 2, -1, -2, -1, -2, 2], [-5, -4, 1, 2, 2, 0, -2, 1, 4], [-12, 2, -1, 5, 2, -3, -5, -5, -2], [-2, -1, -3, 2, -3, 0, 2, -1, -1], [-3, -2, -2, -1, 4, -1, 0, 0, -1], [4, -4, -4, 1, 0, -2, 5, 1, -1], [-4, 1, 1, 2, -1, 2, -5, -4, -3], [1, -3, -2, 4, 3, 0, 0, 2, 1], [-10, -3, -1, -4, 1, 4, 3, 0, 2], [-5, 3, 1, -5, -3, 3, 1, 0, 1], [5, -3, 8, -2, -3, 8, -1, -2, 0], [0, -3, 1, -4, -3, 2, 1, 3, -2], [-4, 4, 3, 0, 0, -1, -4, 0, -1], [-3, 0, -4, 0, 4, -3, -3, -1, 2], [-2, 0, -3, -2, 0, -5, 2, 4, -1], [-10, 0, -5, 0, 0, -3, -2, 0, 3], [-8, -1, -2, 1, 1, 3, 7, 4, -1], [-5, -5, -7, 1, -1, -3, 4, 4, 4], [11, 0, -3, 0, 0, 0, -1, 2, -3], [-1, 3, 2, -4, 2, -2, -2, -4, -4], [-3, -1, -2, 0, 3, -2, 3, -2, -2], [1, -3, -1, -4, -3, 3, 9, 4, -1], [-12, 1, -5, -2, -1, 0, 4, 5, 5], [8, 0, 4, 4, 0, 4, -5, -6, -5], [-5, -5, -2, -4, 3, 0, 0, 3, 2], [6, 4, 3, 7, 0, -7, -10, -5, -3], [2, 1, 1, 2, 3, -6, -8, 1, -1], [-1, -4, -3, 6, 2, -4, 1, 2, -3], [-11, 2, -3, 2, 1, 0, 6, 3, -3], [-6, -2, -3, 10, 6, -5, -6, -4, -3], [1, 7, 5, 3, 2, 1, -10, -3, -3], [8, -1, 5, -2, 1, 0, 0, 1, 1], [0, 2, 4, -2, -4, 4, 2, -1, 4], [13, 0, 5, 0, 2, 2, -6, -1, -2], [6, -1, 3, 0, -3, 6, 3, 1, -2], [-6, -1, -5, 8, 1, 0, -2, -3, 3], [0, 1, -2, -5, -3, 3, -1, 0, 3], [8, -1, 9, -2, 1, 6, 1, 1, -2], [-2, -5, 2, -3, 1, 5, 7, -2, -3], [3, -5, 1, 7, 1, -1, 0, -3, 2], [-8, 1, -1, 0, -3, 4, 6, -1, 5], [3, -11, -2, -6, -1, 6, 13, 4, 2], [-3, 2, 0, -3, 4, 3, 4, 2, 1], [-9, -3, -5, 5, 3, 1, 1, -5, -1], [4, 1, -3, 2, 1, -4, 0, 7, -1], [10, 9, 8, -3, -1, 1, -4, -7, -4], [3, 0, -4, -2, -2, -3, 3, 3, -2], [2, -2, 4, -6, -6, 8, 5, 4, -1], [5, -7, 6, -2, 1, 2, 1, 6, 0], [0, -1, 3, -4, 1, -2, 8, 5, -1], [5, 2, 5, 0, -8, 6, 0, 0, -1], [2, -1, -7, -2, -1, -2, 7, 11, 4], [2, -4, 2, 0, 4, 8, 1, -3, -2], [9, 0, 9, 6, -2, -2, -7, -8, 2], [-2, 7, -3, -2, 1, -4, 6, 1, -1], [-14, 3, -1, 3, -1, -6, -5, -4, 4], [19, 3, 6, -4, 1, -2, -5, -1, -2], [-3, 0, 4, 7, 3, -1, -14, -10, -2], [-10, 3, -5, -4, -1, -6, 0, -1, 1], [9, -8, 5, 2, 0, 6, 0, 2, 3], [5, 4, 3, 0, 2, -2, -9, -5, 3], [11, -7, 7, -5, -2, 7, 3, 3, 6], [0, 4, 8, -10, -8, 8, 4, 2, -2], [-1, 0, -6, -7, -1, 7, 14, 2, 0], [-5, 1, 0, -4, 5, -2, -3, 2, 0], [-1, 4, 5, 6, -6, 2, -8, -11, -8], [-5, -3, -7, 5, 5, -5, -2, -5, 2], [12, -2, -1, -7, -4, 3, 3, 2, 0], [18, 0, 13, 3, -2, 5, -2, -5, -7], [-2, 5, 0, -3, 1, -7, -1, 9, 5], [-5, -11, -6, 6, 1, -4, -1, -2, 4], [2, -1, 7, 2, 5, -6, -4, 3, 5], [8, 3, 4, -5, -1, 5, 3, 4, 3], [7, 6, -3, 2, -2, -2, -5, -7, 3], [-5, 14, 3, 4, -4, 0, -3, 1, -1], [7, 6, 2, -6, -2, 1, 5, -1, -6], [6, 2, -7, 7, -2, -1, -4, -4, 0], [3, -1, -1, 9, 4, -7, -11, -7, 0], [15, -2, -3, -2, 4, -2, -2, 6, 5], [13, 0, 0, -1, -2, -3, 5, 8, -2], [11, -3, 3, 3, -1, -1, -6, 2, -2], [-5, -1, -4, 2, -3, 4, 8, 6, 1], [9, -7, -6, -4, 3, 2, 7, -3, 4], [11, -6, 8, -1, -2, 5, 0, -4, -9], [-14, 6, -9, 5, 2, -3, 7, 5, 4], [-2, 0, -5, 1, 2, -3, -6, -3, 3], [5, 2, 6, -5, -4, 7, -4, 2, -5], [5, -1, 6, 8, 1, 0, -13, -2, 2], [-6, 0, 4, 0, -2, -4, -6, 6, 4], [-3, 5, 5, -1, -1, 5, -1, -7, 1], [-11, -2, -7, -4, 2, 4, 9, 9, 3], [-13, 2, -2, 1, 6, 1, -3, 4, 4], [19, 1, -11, 7, 3, -5, -9, 2, 0], [-1, 1, 3, -3, 3, 3, -2, 4, 3], [-16, 7, -2, 8, 7, -5, -10, -7, -4], [9, -5, 4, -4, 1, 0, -3, -2, 2], [8, -8, 1, 3, 0, 9, 8, -3, -1], [-9, -4, 1, 12, 4, -2, -4, -5, -2], [-4, 2, -7, 3, 1, -1, -3, 5, 5], [2, -6, -1, -5, -4, 3, 12, 7, 3], [10, 5, 4, -1, -5, 7, 7, 0, -7], [12, -5, 1, -2, 5, 6, 3, -3, 2], [4, 12, 0, 2, 2, 2, -7, -9, 0], [11, 5, 7, 1, -5, 7, -5, 1, -9], [-2, 4, 2, 2, -2, -6, -10, -3, 2], [12, -1, 11, 4, 2, 4, -4, -7, -10], [-2, 2, 0, 4, 8, -2, -6, -5, -2], [-5, -8, -6, -5, 0, 9, 17, 4, 4], [0, 12, 6, -8, -2, 2, -2, 2, 0], [-7, 3, 2, 4, 2, -4, -11, -10, -7], [-4, -5, -3, -3, -3, 2, 4, -4, -5], [-5, -3, 3, 1, 1, 9, -7, 2, 1], [-5, 12, 0, -3, 0, -5, 2, 6, -1], [-17, -3, 1, 5, 3, 1, -5, -3, -1], [-13, 4, -1, 1, 2, -2, -2, 0, 6], [13, 5, 12, -2, 6, -2, -10, -6, -6], [11, -1, 0, -4, -4, 4, 9, 2, 1], [-7, -9, -6, -4, 3, -2, 4, 4, 7], [1, 2, -11, 2, 2, 2, 3, -3, -5], [5, -12, 0, -1, 2, -1, 1, 10, -2], [2, 0, -1, -9, -2, 5, 7, -3, -2], [10, -1, 6, 13, 1, 9, -5, -4, -5], [-1, -4, 8, 3, -2, 11, 0, -16, -1], [-3, -9, -5, 1, 3, -5, -6, 3, 6], [-2, -1, -5, -15, 3, -4, 19, 12, 2], [1, -5, -1, 3, -3, 3, 4, -5, -8], [19, -1, -4, -4, -5, -2, -6, 0, -3], [-14, 6, 12, 0, -1, -6, -12, -2, -5], [19, -12, 3, -2, 2, 12, 11, 1, 5], [18, 5, 1, 0, -7, 4, 0, -11, -3], [-16, -6, 3, -8, 8, 9, 11, 0, 8], [-12, -1, -9, 2, -7, 2, 3, 9, 2], [-4, -1, -9, -8, -5, 4, 18, 7, 5], [-11, -7, -1, -7, 7, -1, 0, -1, -4], [-9, 7, 0, -4, 3, 0, -2, 4, 3], [-11, -4, -7, 4, 4, -6, -1, 5, 7], [-4, -2, 5, -7, 4, -3, 2, 0, 2], [-10, 8, -1, 1, -6, -3, -3, 1, 2], [-2, -10, -3, 2, 2, 7, 12, 8, -3], [10, -7, -11, -2, 1, -4, 2, 8, 5], [12, -3, -11, -2, -3, 6, 6, -1, 1], [2, 3, 0, 3, 0, -7, -5, 2, 10], [11, -7, -11, 5, 1, -3, 0, -3, -2], [-24, 1, -4, 1, 4, -7, -6, -10, -3], [21, 3, -7, -5, 3, -9, -9, 7, 5], [-5, -1, -9, 8, -3, 7, 12, 8, 2], [-3, 1, 2, 3, 1, -6, -7, -1, 6], [-14, -1, -2, -16, -3, -1, 14, 11, 10], [4, -3, 7, -8, 1, 8, 4, 1, -5], [7, 0, -4, -5, 4, 1, 12, 12, 11], [-14, 0, -2, 12, -2, 6, -5, -12, -1], [-19, -6, -4, 3, 6, 5, -6, -9, 1], [10, 1, -6, 13, 0, 3, -10, -6, -3], [22, 3, 3, 10, -3, -4, -7, 2, -8], [6, -1, -6, 5, -5, -3, 8, -2, -2], [7, 6, 11, 14, 0, -8, -16, -15, -2], [-2, 0, 5, -9, 0, 5, -5, -3, -8], [9, 3, 1, -4, 1, -9, 2, -2, 6], [9, 9, 5, -6, -5, -1, 2, 6, -6], [-1, -5, -9, -7, 3, -7, 5, 10, -1], [2, -17, -7, -6, 5, 0, 12, 10, 9], [6, 0, 2, -2, 2, -4, -9, 10, 3], [0, 11, -1, 13, 1, 0, -12, -16, 1], [5, 5, 12, -4, 1, 0, -1, 4, 6], [-17, 6, -9, -2, -4, -2, -1, 1, 3], [3, -10, -7, -4, 2, 0, 13, -3, -3], [12, -5, 2, 1, 7, 1, 2, -2, -4], [5, 7, 0, -2, -1, -10, -11, -2, -2], [-2, 3, -1, 4, 9, -10, -8, -7, 3], [-19, -2, -7, 0, 6, 0, 2, 1, -8], [8, -3, 5, 2, 5, 4, -1, -9, -8], [-24, 1, -3, 0, 3, -2, 1, -4, 1], [16, -3, -7, -6, 1, 2, 3, 7, 0], [-15, 8, 4, 1, -4, -3, 1, 4, 0], [-19, -4, -3, 6, 3, 2, 3, -3, -8], [2, -2, 5, 7, 0, 7, 5, 1, 2], [6, 1, 5, -6, -1, -6, 1, 9, 8], [0, 4, -2, -8, 0, 0, -4, 6, -4], [0, 2, -2, -12, -1, 2, 2, 14, 3], [-9, -1, 5, -1, -3, -5, -10, 0, 8], [19, 1, -5, 9, -3, 13, 8, -3, 0], [19, -2, 1, 8, 5, 4, -12, -9, -1], [-16, 3, 0, 1, 3, 1, 2, -15, -5], [9, -12, -6, -9, 1, -1, 12, 4, 10], [12, 0, -8, 6, -6, 0, 8, 2, -4], [19, -8, 2, -1, -8, -1, -3, 7, 7], [5, -7, -3, -7, 1, 1, 15, 9, -1], [-14, -1, -7, 8, -1, 4, 11, 5, -10], [7, 10, 13, -10, -6, 6, -5, -9, -9], [-15, 0, -12, 11, -4, 3, -4, -12, -1], [0, 5, 3, -8, -5, -10, 2, 17, 1], [-3, 4, -10, 3, 6, -17, -5, 11, 1], [-5, 1, 4, 6, -1, 0, -1, 12, -4], [0, -6, -1, 1, 4, -5, 12, 3, 5], [-29, 10, -3, 0, -4, 2, 16, 10, -2], [12, -10, 10, -8, 2, 2, 2, 4, 14], [-7, 1, 6, -4, -5, -6, -10, 8, 1], [7, 1, 1, -7, -9, 3, -5, 7, 7], [-10, -8, 11, -7, 4, 3, 3, 1, 4], [14, -2, 6, 6, 6, 14, 0, -18, -6], [5, -8, -1, 8, 5, -2, -5, 11, 0], [6, -6, -1, 11, 9, 1, -5, -1, 3], [-10, 3, 1, 8, 3, -14, -12, -5, 3], [14, 0, -4, 5, -10, -6, 5, 5, -5], [0, 0, 9, 0, -4, 1, -14, -2, -5], [8, -15, -7, -10, -1, 2, 6, 4, 2], [-3, -2, 9, -6, -2, 2, 3, -12, -3], [-13, -9, 8, -16, 7, -6, 5, 10, 6], [6, 7, 17, -12, -5, 4, 2, 4, 1], [9, 4, 13, -10, 1, 0, -5, 9, 2], [-4, 22, 7, 3, -7, 1, -12, -11, -14], [13, -1, 1, 17, 7, 1, -20, -11, -8], [15, 5, 18, -8, -1, 14, -6, -3, -4], [12, 4, -9, 3, 2, 1, -6, -5, 3], [-2, -1, -4, -9, -3, 1, 7, 11, 1], [-26, 7, -12, 13, 9, -21, -17, -6, -1], [13, -3, 0, -6, -7, -6, -5, 13, 6], [-8, 4, -7, 3, 4, -5, -3, -3, -2], [-6, 1, 2, -3, -4, 11, 14, 10, 1], [-24, 6, -7, 1, 0, -19, -13, -6, 0], [-7, -6, -6, 19, 8, -3, -15, -8, -4], [-17, 6, 6, 11, -6, 5, 0, -20, -7], [0, -4, 11, -5, 4, 3, 3, -9, -4], [1, 2, 8, 1, -10, -1, 5, 2, 0], [-2, -6, -2, 11, -6, 10, 2, -3, 6], [-19, -5, -7, -11, 5, 3, 11, 4, 10], [-5, 10, 7, 2, 0, 0, -11, -17, -9], [9, -8, -17, 8, 6, -6, 5, 13, 7], [30, -10, 8, -10, -2, 4, 10, 12, 6], [19, 1, -1, -5, -7, 1, 14, 16, 10], [14, -1, 16, -3, 1, 11, -9, -14, 5], [-11, 9, 14, 8, 1, 0, -21, -12, -8], [-10, -4, -14, 8, -5, 8, 12, -2, -1], [1, -2, 4, 1, -5, 3, 3, 8, 1], [-14, 5, 11, -8, -3, 12, 10, -9, -7], [0, -15, -12, -7, -9, -3, 19, 19, 9], [-19, 5, -1, 7, 3, -1, 2, -9, 8], [2, -11, 4, 1, -4, 7, 13, 2, -6], [-4, 14, 1, -3, -2, -9, -9, 15, -2], [10, -1, 12, 1, 5, 11, 1, -4, -9], [8, 3, -7, 12, 5, -6, -9, -6, -3], [-11, 7, -4, -4, 3, -14, -3, 6, 2], [-49, 9, -9, -1, -3, -5, -4, -1, 2], [10, -4, 11, -11, -4, 7, -1, -11, -8], [5, 11, -1, 9, -3, 1, -14, -2, -6], [-15, 4, 6, 5, -2, 3, 0, 2, -3], [-19, 1, 6, 0, 5, 10, -1, -4, -10], [-21, -3, -2, -2, 10, -4, 6, 2, 6], [14, 11, 0, -5, -7, -15, -9, 0, 7], [-12, -4, -8, -10, -1, -2, 3, 6, -6], [-21, -5, -2, 0, 5, 8, 3, -6, -2], [-31, 8, -6, -7, -2, 1, 6, -11, -7], [-6, -9, 2, -13, 1, 5, 7, 12, 3], [-10, 11, -14, 1, 1, -13, -7, -1, 5], [-3, -8, 5, 8, 6, -2, -1, 10, 9], [6, 4, 10, 2, -3, 12, -5, -14, -20], [31, -9, -12, 0, 2, 6, 15, 8, 7], [8, 9, -2, 7, -7, 1, -8, -5, -11], [21, 18, 5, 8, -4, 2, -3, -8, -10], [-20, 12, -13, -7, 4, -5, 12, 4, 2], [17, -18, 1, -8, 0, 2, 15, 9, 9], [-18, -4, 13, -13, 4, 5, 1, 0, -6], [-39, 2, 1, -4, 0, 6, 7, 5, -3], [-5, -5, -5, 4, -3, -9, 3, 2, 5], [-9, -7, -1, 8, 1, -3, -5, -4, 5], [-11, 10, 1, 0, -2, 8, 1, -6, -11], [15, 2, 7, 2, -2, 2, 2, 1, -2], [21, 5, 9, -5, -5, -1, 2, -4, 5], [19, -9, -1, -11, 3, -5, 0, 10, -7], [-10, -1, -2, -9, 1, -9, 7, 14, 11], [-25, 0, -1, 11, -6, 0, -11, -10, -1], [-9, -10, 0, 5, 8, -5, 3, 13, -3], [33, -11, 4, 17, 7, 0, -14, -7, -5], [-15, 2, -6, -15, -2, -7, 11, 19, 6], [-17, -3, -13, -15, -1, -5, 6, 13, 10], [-26, 23, -6, -3, -1, -3, 0, 7, -6], [-15, -7, -17, -1, 5, -11, 0, -1, 0], [22, 9, 14, 12, 1, -3, -15, -15, -11], [14, -2, 7, 11, 3, 1, -11, -7, -5], [16, -3, 6, 9, -7, 15, 2, -8, 2], [-15, 2, -13, 4, -4, 2, 5, -2, -11], [-17, 10, -3, 6, 0, -14, -17, -9, -5], [26, 1, 5, -14, 2, 0, 18, 9, -4], [-15, 1, -9, 1, 8, -3, 14, -3, 5], [15, 17, 3, 9, -1, -3, -5, 1, 5], [3, -6, 9, -6, 2, 0, 11, -3, -3], [-4, 9, 2, -7, -9, 3, 10, 4, 12], [29, 10, 1, 6, 2, -10, -5, 4, 2], [-2, -6, -2, -8, 0, 20, 19, -5, 8], [-14, 14, -4, 4, -4, -2, 3, -8, 7], [-26, 12, -4, -10, 0, 0, 2, 12, -8], [5, -6, 8, -7, -8, 7, 2, -8, -11], [-2, -5, 5, -14, -1, -2, 2, 5, 13], [11, -14, -8, -5, 2, 3, 4, 2, 11], [-16, -6, -17, 3, -8, -3, 13, 5, 0], [15, -10, -3, 6, 2, -8, -9, -6, -2], [-14, 7, -9, -6, -1, 2, 5, 16, 4], [1, -12, 7, 0, -2, 2, 4, 1, 14], [-13, -1, -8, -16, -9, -4, 24, 14, 5], [-5, 4, -1, -7, 0, -10, -14, -4, -2], [-32, -6, -22, -8, 0, -8, 13, 19, 12], [-6, 0, 10, -8, 8, 10, 0, -12, -14], [-9, -9, 4, 16, -1, -6, -8, -14, 9], [-9, 1, 4, 16, 9, 0, -12, -7, -1], [-11, 2, -5, -12, 8, -2, -1, 15, 3], [-30, 6, -10, -15, -6, -8, 12, 9, 2], [-23, 5, -21, 9, 9, -19, -1, 12, 14], [-16, 0, 11, 3, -3, 7, 1, -3, -11], [-14, 7, 3, -10, -5, -18, -11, 7, 8], [2, 6, -8, 10, 0, 0, -18, -9, 1], [7, 2, 9, -4, 0, -10, -11, -1, -7], [13, 10, 6, 7, 0, -11, -15, 8, -6], [-18, 3, -18, 3, 3, -19, 9, 14, -3], [-7, 6, 17, 10, -8, 12, -7, -21, -13], [17, 9, -11, 11, -5, 5, 11, 7, -3], [24, -9, 7, 8, -3, 16, -8, -5, -1], [13, 0, 3, 8, -2, -2, 2, -9, 6], [-31, -7, -3, -13, 7, 5, 12, 6, 4], [-3, 8, 20, -15, -4, 11, 6, -7, -2], [10, -10, -3, -2, 0, -7, -1, 16, 4], [29, -12, 4, -10, 0, 5, 12, -3, -5], [18, -4, 9, 11, -8, -7, -5, 5, -12], [18, 0, 0, 4, 0, -2, 10, 12, 0], [10, 3, -4, -1, 7, 5, -3, -8, 5], [-13, 3, 6, 0, -1, -2, -3, 2, 0], [18, -5, 5, -6, -6, 0, 0, -13, -12], [9, 16, 4, -3, -10, -13, 0, 12, -5], [-16, 6, 10, 2, 4, 8, -4, -12, -2], [-12, 8, -10, 0, 4, -6, 15, 9, 6], [-13, 14, -5, 10, -1, 0, -6, -7, -3], [-18, -10, -1, -11, 8, -5, 10, 3, 9], [1, 2, -2, -16, 2, -7, 1, 11, 8], [-8, 3, 6, 1, 0, -17, -17, -16, -8], [-20, 7, 10, 6, -7, 9, -8, -9, 8], [11, -7, -13, 5, -7, -3, 16, 11, 12], [-29, -6, 4, -10, 16, 1, -3, 5, 0], [-9, 8, 1, -8, 0, 0, 13, 14, 3], [-22, -10, 4, -12, 4, -8, 13, 14, 9], [-21, 10, 18, 1, -4, 7, -6, -10, -1], [2, 2, 3, -9, -6, 9, 19, -2, -10], [-1, 6, 0, 11, -8, -3, -21, 5, 5], [14, -10, 7, 9, 0, 1, -20, 1, 1], [-2, 3, -9, 0, -7, -4, 9, 19, 4], [-18, 18, -11, 7, 0, -7, -12, -15, -1], [-2, 1, -9, -2, -3, -4, 0, 7, -3], [-15, 19, 5, -7, -5, -1, -7, -3, -3], [-6, 3, -10, 23, 11, -19, -27, -13, 3], [5, -5, -11, -1, -1, -7, 6, 11, -4], [-14, 2, -7, 1, 10, -1, 7, -3, -4], [27, -11, -9, 3, -5, 3, 13, -4, 8], [22, -1, 1, 16, -14, 8, -12, -13, 0], [38, -10, 17, 3, 2, 21, -3, -9, -4], [-21, -4, 10, -1, 0, -9, -18, 4, 5], [9, 2, -13, -6, -2, 10, 14, -7, 2], [15, -1, 3, 19, -1, 11, 0, -13, 0], [-27, 0, -12, 9, 9, -13, -4, -2, 0], [-9, -9, -11, 9, 13, 3, -6, 1, 6], [9, -5, -8, -8, -9, 0, 15, 6, 18], [42, -9, 13, 4, -1, 4, -12, 0, -1], [19, 4, 15, 13, -4, 4, -6, -4, -2], [-9, 10, 7, 4, -12, 6, -2, 12, -5], [-9, 3, -18, -2, -1, 0, 23, 2, 6], [-41, 0, -17, -10, 6, -2, 9, 17, 3], [11, -3, 4, -2, -7, 14, -9, -4, -8], [-2, 9, -9, 8, -1, 0, -8, 15, 5], [3, 1, 4, 24, -1, 12, -6, -16, -7], [-8, 11, 19, 12, -1, 2, -10, -9, -11], [10, 11, 14, 15, -7, -3, -9, -6, -15], [18, 4, 7, 5, -20, -1, 7, -1, -2], [6, -2, 11, -5, 0, 1, 9, 11, 4], [-15, 17, 10, 18, -3, 2, -16, -18, -15], [-6, 3, 2, -15, 5, -5, 6, 5, 11], [25, -13, -9, 3, 1, -1, 17, 13, 11], [1, -5, 3, 15, -1, 15, -14, -21, -4], [7, -18, -2, 1, 8, -1, -4, 6, 1], [10, 7, -7, -12, -3, 0, 21, 1, -4], [14, -5, -2, 3, -3, 5, 9, 2, -3], [3, 6, -8, 1, 0, -19, -14, 13, 7], [-10, -4, -6, 2, -10, -8, 0, 6, 8], [-23, -8, 2, -1, -5, -7, 16, 10, 2], [-4, -2, 8, -2, -5, 2, 20, 10, -3], [4, -3, 3, -6, 1, -10, -7, 3, -2]];
    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_bb();".
// 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_bb(: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_bb();".
// 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_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6042_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![68, 428, -647, -238, 389, 71, -79, -14, 5, 1]>,<7,R![32, 3, -167, -44, 210, 52, -64, -12, 5, 1]>,<11,R![-1024, 1408, 2624, -1320, -1052, 424, 103, -38, -3, 1]>],Snew);
    return Vf;
end function;