
// Make newform 6042.2.a.be 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_be();".
// 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_be();".
// 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 := [1496, 2072, -10086, -10858, 7140, 6443, -2668, -1287, 465, 105, -36, -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, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [66361412, 56806586, -436949806, -208168678, 291739913, 64541040, -62240161, -7829211, 5289977, 429814, -153659, -8627], [-138362652, -82642718, 945017570, 276730430, -669511535, -26871376, 144429857, -7457875, -12276379, 1189180, 357911, -44133], [-168446612, -86183202, 1277688810, 93884262, -1021791637, 153700554, 228244687, -49855477, -19687587, 4906370, 580219, -153659], [26647108, 95475990, -122170844, -686539810, -171960525, 408639469, 58062170, -81842389, -5889366, 6681248, 192809, -189500], [-130878392, -141548970, 853386646, 789802582, -395937951, -362487140, 69405275, 62549233, -5213807, -4654614, 136741, 123929], [484423404, 632768662, -3143496692, -2983182366, 1479064815, 1330188249, -260559368, -225402809, 19588198, 16556874, -514389, -436706], [348290980, 537284210, -2297594618, -2788215142, 849811789, 1346156238, -121976379, -240145831, 7660263, 18247374, -163655, -492933], [-295289888, -546585460, 1862852610, 2889298104, -457273178, -1483888803, 32314681, 275258286, 99389, -21463244, -65830, 590365], [-339214300, -689742890, 2015976264, 3800163318, -239539453, -2037578341, -39850156, 387502119, 7336736, -30694620, -302055, 853346], [-669738660, -1162833098, 4086959558, 6075452558, -1083764241, -3089711460, 92646333, 569886495, -1709593, -44279466, -86525, 1215147]];
    Rf_basisdens := [1, 1, 2955272, 2955272, 2955272, 2955272, 1477636, 2955272, 2955272, 2955272, 2955272, 2955272];
    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, 0, 0, 0], [-1, 0, 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, 0, 1, 0], [-1, 0, 0, 0, 0, 0, 1, 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, -1, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0], [1, 0, -1, -1, 0, 0, 0, 1, 0, 0, 0, 0], [1, 1, 0, 0, 1, 0, -1, 0, 1, 0, -1, 0], [2, 0, 0, -1, 0, 0, 0, 0, -1, -1, 1, 0], [0, 0, 0, -1, 0, 1, 0, 0, -1, 0, 0, 1], [1, 0, -2, -1, 0, 1, -2, -1, -1, -1, 0, -2], [0, 0, -1, -1, 0, 0, -1, -1, 0, -1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 1, 1, 0, 0, 0, 0, 0, 0, -1, 0], [-2, 1, 2, 1, 0, -2, 1, 1, 1, 2, -1, 1], [1, -2, 0, 0, -1, -1, 0, -1, 1, 0, 1, -1], [0, 0, -1, -1, -1, 0, 0, 0, 0, -1, -1, -1], [-3, 2, 0, 0, 1, 1, 0, 1, -1, 1, -1, 1], [0, 0, 1, -1, 0, -1, 0, 0, 0, 2, 0, -1], [-5, 2, 1, 1, 1, 1, 0, 1, 0, -1, -1, 1], [1, 1, 0, -1, -1, 1, 1, 0, -1, -2, 0, 0], [2, 1, 0, 1, 2, 0, -1, 1, 0, 0, -1, 1], [-3, 2, 1, 1, 1, 1, 1, 2, -1, -1, -1, 1], [-1, 2, 1, 1, -1, 0, 0, 1, 1, 2, -1, 2], [0, 0, -1, -1, -2, -1, 0, -2, -1, -1, 1, -2], [2, -1, -2, -1, 2, 1, 0, 0, -1, 0, 0, -1], [-2, 1, 2, 2, 0, 0, 1, -1, 2, 0, 0, 1], [-1, 1, 1, 0, 0, -1, 3, 0, -1, -1, 1, 1], [-2, -3, -2, -1, -1, 1, 0, -2, -1, -1, 2, -2], [-1, 3, -2, -1, 0, 0, -2, 1, 0, -1, -1, 1], [-4, 1, 1, 1, 2, 0, 1, -1, 1, 1, 0, -1], [-1, 0, 2, 2, 0, -3, 0, -1, 3, 1, -3, 0], [5, 2, -2, -1, -3, 1, -2, -1, 1, 0, -1, -1], [1, -3, 1, -1, -2, -1, 1, 0, -1, -1, 2, 1], [-2, 3, 1, 0, 0, -2, 1, 1, 1, 0, -2, -1], [-2, 1, -1, 0, 1, 2, -1, 1, 1, 2, 1, 0], [-3, -3, 2, 1, 1, 1, -1, -2, -1, 1, 0, -2], [-5, -2, -1, 1, 0, -1, 0, -1, 2, 0, 0, 0], [0, 1, -1, 0, 0, 1, 0, 2, 0, 1, 1, -1], [-7, 0, 2, 2, 0, -1, 0, -1, 1, 1, 1, 0], [0, -2, 0, -1, 0, 0, -1, -1, -1, -1, 2, -2], [-3, 0, 1, 2, -1, -1, 0, -1, 2, 3, -1, 1], [3, 1, 1, 1, 0, 1, -1, -2, 1, 0, 0, 1], [2, -3, 2, 1, -1, -1, 2, 0, 1, 0, 1, 2], [1, -1, 0, 1, -3, -2, 2, 1, 0, 2, -1, 0], [-1, -2, 2, -1, -1, 0, 0, -1, -2, 2, 1, 0], [1, -1, 1, 2, 2, 1, -1, 0, 2, -1, -1, 1], [-8, 1, 0, 1, 1, -1, 0, 2, 1, 3, 0, 0], [-3, -1, -1, -1, -3, 1, 3, -2, -2, -1, 2, -2], [2, -1, 1, 1, 3, 0, -1, -1, 1, 0, -1, -2], [-4, 4, -2, -2, 0, 1, -2, 0, -2, -3, 2, 0], [-2, 1, 2, 3, 2, -2, 1, 1, 3, 0, -1, 5], [-8, 0, 2, -2, -1, 2, 0, 0, 2, 1, 0, 1], [-4, 2, -1, -1, 1, 3, -2, 0, 0, -3, 0, 0], [-4, -1, -2, 0, 0, 1, -1, -1, -1, -1, 1, 0], [3, 1, -2, -2, 2, 1, -1, 2, -3, -2, -3, 1], [-2, 2, 3, 3, 2, -2, 2, 2, 1, -2, -1, 2], [-4, -1, -2, 1, -2, 1, -1, -1, 2, -1, -3, -1], [-8, -1, 1, 2, 1, -3, 1, 1, 1, 2, -4, 0], [-3, 1, -1, -3, -1, 1, 0, -1, -5, -1, 3, 0], [-1, -1, 1, 0, -1, 1, 0, 3, -1, 0, 0, 0], [0, -2, -4, -1, 1, 0, -2, -2, 0, -1, -1, 1], [-1, 2, -2, -3, -1, 0, 0, 1, -4, 3, 1, 0], [1, -4, 2, 0, 1, 1, -1, 0, 1, -1, -1, 0], [-1, -4, 3, 0, 1, 0, 1, 0, 1, 2, -1, -1], [-5, 1, -2, -2, -1, 0, 0, 3, -2, 1, 2, 0], [0, 0, 1, 3, 4, -3, -2, 2, 1, 2, -5, 0], [-5, 5, -3, 0, 0, 2, -2, 1, 1, 1, -2, 1], [-11, -1, 2, 2, -1, -3, 1, 0, -1, -1, 1, 2], [-6, 1, 3, 1, 5, 0, 1, 1, 0, 2, -1, -2], [-1, 2, 2, 3, 2, -2, -1, 2, 2, 2, -3, 4], [1, -1, -1, 0, 1, -1, -1, -4, 2, -2, -1, -2], [-6, -2, -1, 1, -2, -1, 0, 0, 2, 0, 0, -3], [0, 1, 2, -1, 2, 1, 0, 2, 1, 3, -5, -1], [-9, 1, -1, -1, 0, 0, 1, 2, 4, 0, -4, 3], [-1, 2, -2, 2, 1, -1, 0, -1, 3, 0, -2, 1], [-3, 3, 5, 4, 3, -3, 1, 4, 2, 2, -1, 0], [-10, 0, 2, 2, -3, -3, 4, -2, -1, 1, 1, 0], [-2, 1, 0, 0, -3, -1, 0, 2, 1, -1, 1, 2], [1, 6, 0, 1, 2, -1, -2, 3, -1, 3, 0, 2], [1, -3, 1, 0, -1, -1, 3, -2, -2, 0, -1, -2], [4, 0, -6, 0, 0, 1, -3, -1, -1, -5, -1, 0], [0, -2, 1, -1, 0, 0, -2, -2, 3, -2, -3, 0], [6, 2, 2, 0, 3, 2, 0, 2, 0, 4, 0, -1], [-5, 0, 0, 1, 0, -3, 0, 1, 1, 2, 2, 4], [1, -1, -5, -1, -1, -1, -5, 0, 0, -1, 0, -2], [-1, -1, 0, 0, -6, 1, 3, -4, 1, -2, 3, 1], [-9, -1, 4, 1, 2, 2, 2, -1, 0, -2, 1, 1], [0, 1, -4, -1, -1, 0, -3, -5, 0, -1, 1, -2], [4, -1, -2, -3, -2, 3, -2, -2, -1, 1, 2, -7], [-6, 3, -7, -4, -1, 3, -3, 1, -2, -3, 1, -1], [-11, 5, 2, 2, 1, 2, 1, 0, 1, -1, -1, 2], [-11, 3, -2, -4, 1, -1, 1, 0, -1, -3, -1, 0], [-10, -1, 1, 0, -4, 0, 5, -1, -1, -1, 4, -1], [4, -3, 0, -1, 3, 0, -1, 1, 0, 0, 1, 2], [-4, 0, -1, 1, 1, -1, 3, -1, 1, 0, -2, 3], [2, -4, 5, -1, -2, -4, 4, 0, 1, 0, -1, 0], [3, 0, -3, -3, -4, 0, 3, 0, -6, -1, 1, 0], [-4, -1, 2, 1, -2, -1, 6, -2, -2, -2, 2, 1], [-6, 1, -2, 1, 1, 0, 1, -1, -2, -2, 3, 2], [-1, 1, -7, -6, 1, 6, -3, 0, -4, -5, 5, 0], [-10, -3, -3, -1, 1, 1, -5, -1, -4, 1, 2, -3], [1, 5, 0, 0, 0, 3, -1, 0, 1, 1, 2, 3], [-17, 3, 1, 0, 0, 0, -4, -3, 1, -1, -2, -1], [-1, 6, 0, -3, 0, -1, -4, 3, 0, 2, 0, 0], [-3, 2, 2, 2, -1, -1, 6, 1, -1, 3, 3, 3], [-2, 5, -4, -1, -1, 1, -2, -2, 1, -1, -3, 0], [-4, -4, -4, -3, -1, 1, -5, -1, -1, 0, 6, -3], [-16, 1, 5, 2, -2, -2, 5, 1, 3, 4, 1, 1], [-2, 1, -1, 0, -1, -5, 4, 4, 2, 2, -1, 2], [-8, -5, 10, 5, 0, -6, 5, -1, 6, 7, -3, -1], [-1, -2, 2, 0, 0, -1, -2, -3, -1, -1, -1, -4], [6, -5, -4, -3, -3, 3, -5, -3, 1, -3, 0, -3], [-9, 2, 0, 0, 7, 1, -2, 3, -1, -2, -3, 1], [-4, 2, -2, 0, -5, 0, -3, -1, 2, -2, 0, 0], [-3, -1, 4, 3, 4, 1, -1, 2, 5, -1, -6, 1], [0, -5, 0, 3, 1, -2, -3, 1, 2, 0, 1, 0], [7, -4, 1, 2, 1, 1, 0, -3, 2, 1, 1, 1], [-2, -2, 1, 2, 3, 4, -2, 0, 3, 0, 0, -1], [-2, -2, -1, -2, 1, 2, -4, -2, -3, -2, 0, -7], [-7, 3, 2, 3, -5, -8, -1, 2, 5, 2, -6, 6], [1, -5, -3, 0, 1, 1, -3, 0, -3, 2, 1, -8], [-4, 3, 0, -2, 3, 1, 2, -2, -1, -4, 2, 2], [-3, -2, 0, -3, -2, 1, 0, 5, -2, 6, 0, -2], [1, -2, -1, 3, -6, 1, 2, -3, 0, -3, 2, 0], [-12, 2, 1, 3, -1, -1, -2, -2, 6, -2, 0, -2], [-8, -2, -2, -4, -1, -2, 2, 6, 2, -1, -4, -1], [1, -3, -4, 0, 3, -2, -3, 0, 4, 1, 0, -1], [-5, 9, 0, 1, 1, -1, 1, 4, -1, -1, -2, 4], [5, 1, 1, 0, 5, 4, -2, 5, -3, -3, 0, 4], [3, 4, 0, 2, -3, -3, 4, -5, -1, -4, 1, 1], [-1, 1, -4, -1, 2, 6, -2, 1, 0, 0, 1, -3], [8, -3, -1, -6, -2, 0, -1, -1, -5, -1, 0, -3], [6, 2, -2, -4, 0, 4, -4, -4, -2, -2, 2, 0], [-4, 5, 0, 2, 1, 0, -1, -1, 0, -2, -2, 4], [-5, 0, 2, 2, 1, 1, 2, 1, -1, 5, 1, 1], [0, 4, -2, -2, -1, -1, 1, 5, -3, 0, 1, -1], [-2, 4, 3, 4, 0, -1, 1, -3, 3, 3, -1, 2], [1, 3, 1, -4, -3, 2, 2, 1, -4, -2, -2, -2], [2, 0, -2, 3, 1, -1, -4, 2, 0, -2, -1, 3], [-8, -1, 1, 1, 2, 0, 3, 1, 1, -2, 7, 3], [11, 2, -4, -3, -4, -2, -1, -2, 0, -4, 3, 2], [-10, -7, 4, 1, 6, -2, 3, 3, 2, 3, 1, -1], [-4, -5, -4, 0, -1, 2, -5, 1, -2, 0, -1, -4], [20, 0, -1, 1, 1, -4, 0, 0, 3, -2, -1, -1], [-8, -2, 2, 3, 5, 0, 2, 0, 0, 5, 1, -3], [5, 6, -6, -5, 0, 5, -2, -1, -3, -1, -2, -4], [-5, 0, 5, 4, 1, -3, 2, -3, 3, -1, -3, 7], [10, 1, -2, -4, -4, 1, -2, 0, -5, -1, 3, -5], [-1, 0, -7, -5, 1, 6, 0, -1, -5, -1, 1, -2], [-4, -5, 0, 4, 4, 2, -5, 1, 6, 2, 1, -3], [-2, -4, 4, 8, 0, -2, 3, -1, 2, 2, -2, 3], [4, 2, 0, 0, 2, 0, 0, 0, 0, 1, 0, -2], [4, 2, 4, -2, -4, 2, 6, 2, -6, -8, 0, 2], [10, -5, -5, -5, -3, 2, -1, -1, -1, -1, 4, 0], [8, 2, -3, -3, 2, 3, -4, 0, 3, -1, 1, 2], [12, 1, -2, -5, 0, 5, -1, 1, -1, -2, 6, -6], [-3, 1, 4, 1, 4, 0, 0, 3, 2, 0, -3, 5], [-2, 1, 2, 0, -3, 0, 1, 1, 2, -2, -2, 4], [-14, -2, 1, -4, 1, 1, 2, 0, -5, 1, 4, -3], [16, -4, -6, -2, 2, 0, -2, -2, -6, -2, -1, 0], [3, -9, 2, 1, -3, -1, 1, 2, 3, -2, -5, -2], [5, -2, 1, -5, -3, 1, 2, 1, -2, 1, 0, 5], [5, 6, 2, 3, 4, 2, -1, 2, 2, 1, -7, 6], [3, -2, 1, 3, 0, -1, 2, -1, 4, 1, -2, -2], [26, 3, 2, 1, 1, -3, -4, 0, 3, 1, -2, 2], [-7, 4, 1, 6, 1, -3, -2, 1, 6, 5, -7, -1], [4, -1, -3, 1, 3, 2, 1, 1, 1, 0, -3, -2], [13, -10, -8, -8, -5, 4, -7, -4, 0, 0, 2, -5], [3, -5, 1, -1, 4, 5, -4, -3, 3, -1, 5, -3], [6, 2, 1, 1, 0, 0, 2, -2, -3, -1, -3, 2], [6, 0, 0, 2, 1, 4, 0, 2, 0, 3, 6, 1], [-3, 0, 7, 3, -1, -5, 4, 1, 6, 9, -3, 1], [-4, 6, 3, 4, 1, -4, 4, 2, -1, -2, -6, 9], [3, 1, 1, 3, -1, 3, 1, 0, 0, 3, 8, 2], [-15, 0, 0, 5, 1, -2, 2, 1, 5, 6, -2, -1], [4, -4, -1, 0, 4, 0, 2, 0, 5, 5, -2, 2], [-1, 3, -5, -3, -3, 1, -3, -4, 0, -1, -4, 0], [-2, -4, -3, -4, -6, -2, -2, -4, 1, -4, 4, -2], [3, 0, -1, -4, -4, -2, 1, 0, 1, 7, 2, -2], [-5, -6, 0, 3, -3, -1, 0, -5, 5, 2, -2, -5], [0, -3, 6, 1, 6, -3, 0, 2, 6, 7, -6, 1], [12, 3, -4, 2, 4, 2, -1, -3, 0, -5, 0, 1], [4, -1, 6, -2, 4, -3, 3, -1, 0, 2, -6, -1], [5, -3, 5, 2, 1, 1, 8, -1, -4, -4, 7, 1], [-5, 5, -3, 0, -1, -5, -3, 2, -2, 2, -7, 2], [-8, -6, 8, 9, 1, -3, 3, -1, 3, -1, -1, 5], [15, 0, -8, -5, 2, -3, -2, -1, -5, -2, -2, -4], [1, 1, -4, -3, -1, 3, -5, 0, -5, -3, 1, 4], [-10, 3, 4, 4, 1, 0, 3, 5, 4, -2, -4, 8], [16, -7, 6, 2, -3, -5, 7, 3, 2, 1, -6, 4], [2, -10, 0, 3, -1, 3, 0, -6, 0, 3, 1, -5], [-10, 0, 4, 5, 5, -1, 2, -2, 4, -2, -3, 1], [0, -1, 0, 2, 5, -2, 1, 3, -4, -5, -2, -2], [18, -1, -3, -6, -5, -2, 1, -3, -7, -3, 2, -4], [-17, 1, 0, 5, 6, 0, 0, 3, 4, 3, 1, -3], [-11, 1, -4, -2, 3, 1, -6, -1, -1, -2, 1, -3], [13, 3, 4, -1, 1, -3, 5, 0, -3, -1, 1, 2], [8, 6, 5, 2, 3, -5, 8, 10, 2, 2, -5, 8], [7, -7, -2, -7, 1, 2, 0, -1, -4, -1, 7, -6], [-4, -5, -2, -1, 1, 5, -2, 0, -7, 2, 2, -4], [0, 7, 0, 1, -3, 5, 2, 2, 1, 1, 4, 4], [-7, 1, 7, 1, 6, -3, 4, 5, 1, -2, -1, 1], [7, 3, -1, -4, -3, -2, 3, 0, 0, 2, -1, -2], [1, -7, 5, 0, 8, 6, -5, -4, -1, -2, 0, -4], [25, -5, 1, 2, -2, 2, -1, -2, -1, 1, 4, -8], [-7, 9, 2, 1, -5, 1, 1, -2, 1, -2, 0, 4], [0, 0, 1, -5, 0, 3, 4, 4, -7, -1, 5, 2], [5, 1, 4, 5, 4, 1, 3, 2, 1, 5, -2, 7], [-2, 2, 1, 1, -3, -2, 2, 2, -3, -1, -5, 9], [12, -2, 3, 0, -3, 2, -6, 0, 1, 4, -4, 3], [4, 11, 5, 0, 2, -2, 3, 5, -1, 2, -4, 3], [9, 7, 2, 4, -4, -3, -5, -4, 3, 0, -3, 3], [-1, 7, 1, 2, 9, -3, 3, 6, -2, -2, -3, 8], [11, 3, -1, -1, 0, 1, -3, 4, -4, 8, -4, 1], [0, 0, 1, 1, -2, 0, -4, 0, 3, 4, -5, 2], [5, 3, -5, 0, -5, -2, 2, 1, -6, -5, 0, -2], [10, -6, -5, 0, -3, -1, 1, 1, 2, 0, 1, -1], [3, 2, 7, 3, -4, -3, 8, 1, 2, 7, 6, 0], [1, -8, 4, -3, -8, -3, -2, -7, 3, 5, 1, -2], [4, 1, 4, 1, -2, -4, 5, 3, 4, 2, -7, 5], [12, 6, -7, -9, -4, 10, 0, 0, -11, -6, 11, -4], [3, 4, 1, 2, -8, 1, 2, -9, 0, -3, 3, 2], [5, 0, 3, 1, 1, 1, 2, -1, -2, 1, 3, 3], [19, 2, 4, 7, 1, -4, 0, -5, 4, -1, -7, 0], [2, -4, -3, -10, -3, 3, 3, -3, -8, -3, 9, -7], [-16, 3, 10, 6, 3, -6, 3, 3, 10, 3, -5, 8], [-4, -8, 4, 6, 2, -6, 2, -2, 5, 8, -6, -2], [3, -3, -2, -2, -1, 0, -4, 3, 0, -2, -1, -4], [8, 4, 0, -7, -3, 1, -5, -1, -1, -2, -4, -1], [-9, -3, 2, 4, 2, 1, 1, -2, -6, 2, 3, -1], [-20, 3, 2, 0, 7, -4, -3, 5, 4, 2, -8, 2], [20, 2, -3, -2, 0, -2, -8, -2, -3, -3, -4, 0], [11, -4, -6, 4, 5, -1, -10, -3, 5, 3, -7, -5], [18, -1, -11, -10, -1, 10, -7, -1, -7, -7, 4, -6], [11, -1, -7, 0, 0, -2, -6, -1, -3, 5, -2, -5], [-7, 9, 4, 6, 2, 1, -1, 4, 6, 6, -5, 5], [-5, -3, 2, 4, 0, -1, 3, -4, 1, 2, -1, -1], [10, 0, 1, -4, -1, 2, 0, 0, -3, 1, 6, -3], [4, -5, -8, -7, -7, 1, 1, -5, -3, 2, 0, -5], [15, -2, -1, -1, -5, 0, -1, -4, 1, -1, -1, -5], [-2, 0, 2, 1, -3, -2, 0, -2, 6, 4, -3, -3], [0, 1, -6, -1, -1, 0, -5, 5, 0, 1, 3, -2], [-8, -1, 0, 3, 4, 0, 1, 3, -2, -4, -1, 5], [2, -2, 10, 6, 2, -4, 6, 4, 0, 6, 2, 6], [22, -4, -5, 3, -1, 1, -5, -1, 0, 1, 5, -3], [-10, 1, 0, -1, 0, 1, 4, -2, 3, -2, 5, -5], [-15, 1, -7, -8, 1, 1, -5, 2, 0, 0, 1, -4], [16, -2, 4, -2, 2, 3, -2, 2, 0, 1, 2, 10], [10, -5, -8, 0, 3, 4, -6, -4, -1, -3, 7, -2], [11, 2, 0, -4, -4, 0, 7, -2, -1, -1, 6, 0], [-30, 2, 1, 3, 3, -2, 3, 1, 1, -1, -5, 4], [-19, -1, 12, 10, 1, -2, 2, -5, 8, 3, -4, 0], [-6, -1, 14, 8, 4, -1, 7, 3, 5, 6, 3, 4], [-11, 10, 0, 7, -1, -6, 5, 0, 0, 2, -2, 1], [0, 4, 3, 9, 4, -4, -4, -2, 5, 3, -4, 0], [-6, -6, 3, 3, 4, 4, 6, 6, -1, -3, 5, 10], [8, 9, 5, -4, -5, 0, 5, 5, 2, -5, -6, 4], [-11, -2, 2, 2, -1, -1, -2, 1, 1, 0, 1, 1], [3, 0, 1, 3, 2, 8, -8, -5, 3, -1, -3, -5], [12, 0, 4, -2, -4, -1, -7, -3, 1, 3, 3, 2], [17, 3, -6, -4, -5, 4, -7, -6, -6, -9, 6, 1], [2, -5, 2, 1, 4, 0, -3, -3, 2, -4, 3, -3], [-5, -7, -2, -2, -3, 1, 1, -2, -7, 4, 0, -8], [22, -3, 0, 5, 3, 1, -3, -1, -1, -9, 0, 1], [-11, 2, -3, -4, -4, 3, 0, -3, -2, -6, -3, -6], [-14, -5, -1, 0, -1, -2, -3, -5, -1, -1, -4, -4], [4, -1, 0, 0, -1, 1, 0, -4, -9, -2, 5, 0], [13, -6, -1, -4, -6, 5, -2, -5, -6, -4, 13, -4], [-14, 3, -3, -6, 1, 7, 1, 7, -9, -3, 4, 2], [9, 0, 0, 8, 3, 5, 0, 1, 4, -1, 1, 1], [-24, -4, -1, 4, 1, -2, 2, -2, 3, -4, 4, -1], [-14, 2, 5, 9, 5, -6, 6, 6, 3, 5, -3, 3], [-8, -10, -4, -2, -9, 3, -1, -3, -1, 1, 5, -5], [3, 0, 4, -2, 3, 6, -2, -3, 2, 6, -2, -2], [-5, 4, 1, 7, 3, 1, -2, 5, 6, 6, 0, 5], [-2, -7, 7, 6, 3, 3, 2, 4, 0, 6, -1, 0], [-13, -6, 1, 3, -5, 5, 0, -7, 2, -7, -3, -7], [6, -16, -4, -3, -7, -2, 4, -4, 2, -3, 4, -7], [-3, -3, -1, -9, -1, 3, 3, 6, -4, -3, 4, -4], [-3, 6, -9, -6, 2, 0, 6, 5, -9, -6, 6, 1], [0, -6, -3, -4, -8, 2, 4, 4, -3, -4, 10, 2], [0, 6, -8, -7, -3, 1, -1, 3, 3, -7, 5, -3], [-2, -3, 4, 3, 0, -8, 7, 1, -2, 2, -5, 7], [2, -6, 2, -3, -5, -2, 6, -4, -2, -3, 7, -1], [-12, -1, 4, 7, 6, 0, -7, -3, 1, 2, 3, -3], [-2, -5, 3, 2, -2, -6, 5, -1, -3, 5, 2, -3], [3, 8, -2, -2, 4, 1, -2, 5, 5, 5, 1, 10], [-5, 3, -5, -9, -1, 4, -4, 3, -5, 6, 1, -4], [2, 8, -3, 5, -1, -3, 4, 4, 0, -2, -8, 6], [-34, -3, -1, -1, 3, 4, 1, 3, 1, -4, -1, 0], [11, -11, 2, 3, 6, -3, -3, 2, 3, 1, 2, -1], [0, -8, -2, -3, -7, -3, -1, -7, 1, -4, 9, 1], [13, -2, 1, 0, -10, -6, 9, 0, -3, 0, 0, 0], [12, -9, 11, 3, 1, -5, 0, 0, 0, 5, 5, 0], [-21, -3, -4, 4, 0, -3, 5, 2, -1, -2, -1, -3], [-1, -9, -4, -4, -4, -8, -1, 2, 0, 3, 0, -6], [1, 4, -11, -7, -3, 3, -2, -1, -8, -8, 8, 3], [-3, 2, 5, 5, 0, -2, -2, -5, 5, 1, -11, -5], [10, 2, -5, 0, -4, -5, 2, 6, 0, 1, -5, 1], [-10, 1, -5, -4, -4, -2, 3, 3, -3, -5, 1, -5], [-18, -3, -1, -5, 3, 0, -1, 1, -7, -5, -4, -4], [-10, 4, 6, 6, -3, 3, 3, -1, 6, 5, -1, 3], [-1, -7, 13, 8, 1, -3, -1, -8, 9, 5, -9, 0], [9, -3, -1, 5, 5, -5, -5, 6, 7, 5, -6, 0], [14, 0, 3, 5, 4, -1, -5, 3, 4, 2, -8, 8], [-8, 1, -2, 1, -9, -5, -1, -5, 1, -3, -6, 1], [5, 6, 2, -3, -4, -2, 11, 6, 0, -5, 7, 4], [5, 8, -2, -2, -3, -1, 1, 2, -4, -5, 8, 1], [1, 2, -6, -1, 10, 2, -6, -1, -1, -1, 4, 2], [-14, 12, -7, 0, -3, 2, 5, 3, -1, -4, 0, 2], [-17, 1, -9, -8, 4, 7, -5, 2, -8, -9, 3, -1], [-8, -5, -2, -3, 1, 1, 2, -2, -2, -2, 4, 0], [19, -7, -3, -3, 4, 3, -3, 4, -5, -2, 4, 5], [16, -2, -6, -3, -9, 4, -2, -8, 0, -13, -1, -5], [-18, -3, 4, 2, 4, -5, 3, 3, 6, 3, 2, 3], [3, 7, 12, 2, 2, -3, 7, 4, 12, 4, -7, 7], [3, 3, -8, -8, -3, 9, 2, 3, -6, -8, 8, 0], [-7, 8, 3, 12, 5, -2, 0, 7, 7, 7, -6, 4], [15, 8, 0, 3, 6, 5, 2, -1, -1, 4, 2, 2], [-12, 1, 5, 5, 9, 7, -5, 3, 2, 6, -8, 1], [16, -10, 1, 3, -1, 1, -1, -5, -2, -4, 14, 3], [-21, -15, 6, 4, -1, -3, 6, -3, 2, 1, 2, -4], [16, -15, 2, -4, 4, -1, 0, -2, -3, -1, 0, -5], [-20, 3, -3, -8, -4, 0, -7, -5, 1, 0, 0, -1], [-20, 0, 3, 4, -2, -5, 10, 4, 8, 9, -1, 5], [-6, 4, 5, 9, 3, -6, 4, 4, 9, 3, -1, 3], [-14, -5, -6, -3, -2, 0, -9, -3, -2, -3, -2, -1], [2, 4, -1, -2, -1, 6, -8, -2, 5, 1, -5, -3], [-2, 3, 2, 4, 1, 2, -5, 1, 6, 0, -8, 6], [8, -8, -9, -6, -4, 3, -1, -5, -6, -3, 3, -2], [26, -13, 3, -2, 2, -2, 1, -3, -3, -2, 2, -3], [-16, -5, 0, 0, -7, 0, 1, -5, 5, 0, -10, -2], [1, 0, 5, 8, -1, -9, 2, -9, 7, 5, -1, -3], [-5, -7, -4, -5, -3, -3, 1, 2, -9, 2, 6, -2], [-15, -8, 1, -2, -8, -3, 6, 1, 0, 1, 5, -6], [20, 5, 5, 6, -7, -6, 5, -7, 3, 7, 4, -2], [23, -7, 7, -5, 4, -2, 2, -1, -1, 2, -3, -1], [-7, -7, 8, 5, 0, -4, 2, -1, 8, 10, -5, -1], [8, 9, -8, -5, 2, 0, 1, 9, -2, -4, 5, 1], [-2, 11, 6, 7, -3, -8, 5, -5, 3, 1, -9, 2], [18, -8, 1, -2, -3, 1, 0, -4, 2, -4, 13, -6], [14, -5, -2, 7, 2, 0, -3, -7, 4, 7, -1, -5], [14, -8, -4, 1, -1, 0, -2, -4, 4, -2, -5, -7], [-11, 6, 2, -2, -4, 1, 9, 4, 2, 1, -4, 10], [15, -4, -7, -7, -1, -7, 0, 3, -7, 0, 2, -3], [22, -6, -2, -7, -2, 6, 4, -2, -8, -11, 9, -6], [2, -4, -8, -3, 3, -4, -6, -2, 0, -2, -3, -3], [22, -12, -6, 1, 3, 2, -3, 1, -8, 4, 5, -8], [-4, 4, -7, 3, -1, 0, -2, 0, -7, -5, 1, -1], [9, 3, 0, -8, -1, -2, 1, 2, -7, 0, 3, 8], [14, 2, -7, -7, -5, 6, -4, -2, -12, -7, -1, 5], [-5, -11, -5, -2, 1, -3, 1, 2, 2, 3, 8, -8], [20, -2, 3, 6, -6, 4, -4, -6, 7, 1, 2, -4], [0, -11, -1, -2, 0, -2, 1, -3, 2, 2, 0, -1], [-23, -6, -2, -2, 4, 4, -3, -4, 2, 6, -2, -2], [19, -6, -11, -8, -4, 9, -6, -3, -6, -8, 3, -4], [-11, -2, 1, -4, -1, 1, 6, 3, -6, 1, 11, -1], [17, 8, -4, -4, 4, 6, 1, 8, -6, 0, -2, 4], [21, 1, -1, 4, 3, -3, -1, 4, -6, -1, -3, 6], [7, 5, -3, -2, -3, 5, -3, 8, -2, 4, 3, 4], [0, -11, -2, -3, -2, 2, 3, -3, -4, -1, -5, 5], [-11, -4, 10, 9, 0, -3, 3, -8, 7, 6, -8, 1], [12, -6, -2, 0, 0, 1, -1, 1, -4, 2, 9, -2], [29, -12, -2, 3, 1, 2, 0, -1, 1, -4, 6, -1], [-20, -2, -2, 0, 5, 0, -1, 9, 3, 3, 3, -3], [4, -5, -8, -9, 1, 1, -6, 2, 1, 3, 2, -8], [-4, 2, 10, 8, 0, -4, 2, 2, 4, 14, -6, 8], [2, -10, -8, -12, 0, -3, -1, -1, -5, -3, 5, -10], [-18, -5, 1, 0, -6, 0, 3, -3, -1, 5, -2, -7], [3, -11, -5, 1, 2, 5, -7, -2, 0, -4, 1, -1], [12, -5, -11, -11, -5, 2, -5, 5, -5, -6, 3, -8], [-9, -2, -1, 3, 10, 7, 0, 3, 2, -2, 2, 0], [-11, 4, -3, 6, 8, 4, -1, 0, 4, 0, 0, -8], [-1, -13, 7, 1, -3, -3, 1, -2, 6, 5, -3, 0], [18, -2, -1, -6, 0, 5, 1, -3, -12, -8, 5, 0], [-12, -1, 1, -3, 2, -6, 4, -2, 0, 0, -2, 5], [10, 0, -8, -5, -10, -8, -6, -2, 6, -4, -4, 4], [-7, -12, -3, 3, -1, 4, -2, -7, -4, -3, -2, -5], [-17, 10, -6, -3, -2, -1, -6, 5, 1, -4, -6, 8], [16, 3, -1, 7, 3, -7, 3, 7, -1, 3, -9, 4], [8, 0, 3, -6, -3, -2, 3, 7, -3, 0, 2, 6], [-29, 1, 0, -1, 3, 7, -3, 4, 1, 2, -10, 6], [20, -7, 3, -4, 3, 8, 1, 1, -3, 3, 8, -6], [-25, -1, -5, -3, -5, -2, -1, 2, -5, -8, -3, 1], [-16, -9, 7, -1, -4, 0, 7, -1, -4, 2, 1, -3], [-17, 6, -4, 4, 1, -7, 10, 3, -3, -4, -1, 5], [-10, -6, 1, -7, 1, -1, 1, -1, -2, 5, -4, 3], [1, -4, 4, -7, -1, 3, 10, 7, -3, -1, 8, 3], [10, 3, -6, -12, 2, 5, 0, 4, -7, -6, 7, -5], [4, 1, 4, 4, -3, 4, -1, -1, 4, -2, -2, -4], [33, -6, -5, 1, -2, -4, 4, -1, -5, 3, 1, -1], [21, -6, -4, -7, -6, -1, 4, 3, -7, 4, 4, 0], [11, -7, 9, 8, -1, -2, 10, -5, 3, 1, 12, 0], [-9, 4, 0, 6, 7, 3, 2, -1, 4, -7, 3, 3], [-13, 1, 3, -1, -5, 2, -2, -7, 3, -3, -14, 0], [26, -3, -2, 0, 2, -2, -7, 1, -2, 1, -2, -3], [20, -3, -7, -8, -1, 8, -11, -3, -1, -1, 4, -10], [-10, -3, 2, 7, 7, -3, -8, 4, 1, 6, -2, -4], [18, -2, 3, 4, -1, -3, 5, -1, 0, 1, 1, 7], [-13, -3, 6, 6, -6, -7, 1, -8, 5, -1, 0, 1], [18, 10, 1, -5, -6, 2, -8, 0, 1, 5, 3, 6], [-3, 3, -6, -1, -12, 3, -1, -2, -1, -13, -2, 3], [1, -5, -5, -4, -10, 2, -7, -4, -1, 3, -2, -4], [-14, 6, 3, 8, 3, -1, 4, 0, 3, 3, 8, 1], [4, -2, -9, -9, -5, -1, -7, -3, -5, -6, 8, -9], [7, 5, -2, 4, 7, 2, 8, 3, -2, -1, 2, 2], [-32, 0, 3, 2, -1, 0, 6, 6, 7, 8, 2, -3], [-5, -1, 8, -1, 6, 7, -3, 6, 7, 7, -2, 3], [10, -3, 0, -13, -5, 6, 1, 3, -2, -4, 9, 0], [-10, 2, -2, -1, 3, 8, -2, -2, -8, -10, 7, 7], [-15, -5, 3, 10, 5, 1, -5, -2, 10, 0, -1, -4], [-23, 8, 5, 2, -3, -3, -2, 5, 6, 5, 3, 9], [3, -5, 7, 5, 4, 1, 1, 2, -6, 5, 6, -1], [-9, 1, -5, -8, 5, 2, -8, 5, 3, 5, -6, 4], [25, -6, 3, -3, -6, -2, 3, 2, -5, -6, 7, 0], [-28, 2, 4, 13, 4, -2, 12, 6, -1, 3, 5, 6], [16, -7, -4, -5, -7, -6, 5, -5, -2, 3, 7, -8], [-17, 10, 2, 8, 9, 4, 5, 2, 2, 8, -4, 1], [8, 5, -1, 4, 11, 4, -9, -3, 5, -3, -4, 0], [5, 6, 0, -5, -6, -3, -12, 1, 5, 7, -6, 0], [3, 10, 9, 1, 5, -1, 0, 7, 4, 0, 0, 7], [-22, 1, -1, 0, -2, 3, 13, 5, -8, -8, 7, 4], [-21, 10, -1, -5, 4, 3, 2, 5, -14, 0, 2, 4], [3, 5, -2, -4, 2, -1, 7, 6, 1, 7, 0, 1], [-25, 10, 13, 9, 7, -5, 8, -3, 4, 5, -3, 7], [-6, -18, 5, 8, 2, -4, -2, -6, 9, 5, 8, -12]];
    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_be();".
// 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_be(:prec:=12)
    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_be();".
// 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_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6042_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1496, -2072, -10086, 10858, 7140, -6443, -2668, 1287, 465, -105, -36, 3, 1]>,<7,R![15488, -13176, -67908, 4330, 47405, -1053, -12348, 438, 1306, -40, -60, 1, 1]>,<11,R![333824, -173568, -490016, 293664, 144616, -88708, -19302, 9848, 1548, -445, -64, 7, 1]>],Snew);
    return Vf;
end function;