
// Make newform 6042.2.a.bg 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_bg();".
// 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_bg();".
// 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 := [7848, 18296, -19370, -26766, 11014, 11545, -3056, -2095, 453, 169, -34, -5, 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], [-5802453752, -16849170666, 26161780884, 13238233488, -13476233233, -1959913603, 2214031276, 23174311, -137165906, 7340808, 2608739, -204446], [-13115755176, -16773810608, 28087836042, 15577520444, -14675891754, -4306515439, 3363307163, 325470868, -340995353, 10822604, 12257282, -1345873], [12411594952, 25762553016, -18686256534, -39438600788, -1970474342, 12914192153, 2590798999, -1847667036, -375232369, 135031392, 15259486, -4094329], [71971042128, 120758335624, -208346946326, -112755982232, 109575504662, 24690558217, -22214565039, -1283733404, 1961538359, -89204962, -62872796, 7020169], [-49602823488, -64977367004, 173526880946, 58800907072, -97762562352, -9521117757, 20509140469, -448056866, -1833840689, 170893902, 58777566, -8194199], [20499466684, 38591078772, -81138473578, -24245652146, 48288656186, 1783597851, -10137638892, 648769138, 886007227, -105119461, -27548113, 4234532], [-6270409172, -12568245276, 40111343774, 564837718, -31912312038, 3349514017, 8194525736, -1119132754, -829014291, 131490613, 28749929, -5101356], [-2493542228, -328369874, 9465077226, -2801209518, -7130254537, 1716947858, 1727123514, -359968771, -164915409, 32741387, 5435946, -1085494], [1555699728, 194392374, -9331099726, 2801209518, 7130254537, -1716947858, -1727123514, 359968771, 164915409, -32741387, -5435946, 1085494], [61388109096, 98517025968, -214479866182, -48828499424, 127093683434, -3108483931, -26085348123, 2858059572, 2197952363, -329920034, -65451972, 11405033]];
    Rf_basisdens := [1, 1, 669887500, 1339775000, 1339775000, 1339775000, 1339775000, 669887500, 669887500, 133977500, 133977500, 1339775000];
    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], [1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [2, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, -1, 0, -1, 0, 0, 0, 0, -1, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [2, -1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0], [0, -1, -1, 1, 0, 1, 1, 0, 0, 0, 0, 0], [-2, 0, 0, 0, -1, 0, 0, -1, 1, 0, 0, 0], [1, -1, -1, 1, 0, 0, 0, 0, -1, 0, -1, 0], [1, 0, 0, 0, 0, -1, -1, 0, 0, -1, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 1, -1, -1, 0, 0, -1, -1, 0], [2, -1, 1, 0, 1, 1, 1, 1, 0, 2, 1, 0], [-1, 0, -1, 0, -1, -1, -1, -1, 1, -1, -1, 0], [0, -1, 0, 0, 0, 0, 0, 1, 2, -1, -1, 1], [3, -1, 0, 0, -1, 1, 0, -1, 0, 1, 1, 0], [-2, 1, 0, 0, 0, -2, -1, -1, 2, -1, 1, 0], [4, 0, 0, 0, 0, 0, 1, 1, -1, 0, 0, 0], [1, -1, 0, 1, 0, 0, 0, 0, 1, -1, 0, 0], [-2, 0, -1, -1, 1, 0, -1, -1, -1, -1, -1, -1], [0, 0, -1, 0, -1, 0, 0, -1, 1, 0, -1, 0], [-1, -1, 1, 0, 1, -1, 0, 0, 1, 2, 2, 0], [3, 0, -1, -1, 0, 0, 0, 0, 1, -1, -1, 0], [0, -1, 0, -2, -1, 0, 0, -1, 1, 0, 0, 0], [-4, -1, -1, -1, 1, 0, 1, 1, 1, 0, -1, 0], [-1, 0, 2, 1, 0, -2, 0, -1, 1, 0, 1, 1], [0, -1, 0, -1, 0, 1, 1, 0, -1, 2, 0, -1], [3, 0, 1, 1, 1, 1, 0, 3, 1, 2, 1, 0], [-1, -1, -2, -1, 0, -1, 1, 1, -1, -1, -2, 0], [1, 0, 1, -1, 0, 0, -2, 0, -1, 1, -1, -1], [2, -1, -1, -2, -1, 1, -1, -1, 0, 0, -1, -1], [3, 0, 0, -1, 0, -1, 0, 1, 2, 0, 1, 1], [4, -4, -1, -1, 0, 2, 0, -1, -4, 1, -2, 0], [1, 2, -2, 2, -1, 0, 0, -2, -1, -2, 0, 0], [-3, 3, 1, -1, 0, -1, -1, -2, 1, 0, 0, 1], [-3, 0, 0, 1, 1, 1, 2, 0, 0, 2, 2, -1], [6, 0, -1, 0, 0, -2, 0, 1, 1, -2, -1, 1], [-1, 2, -1, 1, -2, 0, 0, 0, 1, -3, -1, 1], [3, -1, 2, 2, 3, 1, 0, 2, -1, 2, 1, 0], [1, 0, 2, 0, 2, 1, -1, 3, -1, 0, 0, 0], [1, 0, -1, -1, -2, 0, 1, 1, 0, -1, 1, 0], [3, 0, -1, 1, -2, 1, 0, 2, -3, 0, 0, -1], [-1, -4, -1, -2, 0, 2, 1, -2, -1, 0, -3, -1], [-3, 3, -1, 4, 0, 0, -1, 0, 3, -2, 1, 1], [3, -3, 2, 0, 2, 1, 2, -1, -2, 3, 1, 0], [-6, 0, 2, -1, 2, 0, 1, 0, 4, 1, 2, 2], [-1, -4, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1], [2, -4, 2, 2, 1, 1, 2, 0, -1, 2, 0, 0], [-5, -2, -1, -1, 1, 0, 1, -2, 0, 1, 0, 1], [0, 2, -1, 0, 1, 0, 1, -1, 1, -1, -1, -1], [4, 4, 0, 1, -1, -1, -1, 1, 2, -2, 0, 1], [0, -1, 2, -3, -1, -1, 0, 0, 0, 1, 1, 1], [6, 3, 0, 1, 0, -4, -2, -2, 1, -1, 1, 1], [1, -3, 3, 0, 3, 1, 0, 1, 2, 2, 2, 0], [-8, 1, 0, 0, -1, 1, 1, 0, 0, -1, -1, 0], [1, -3, 0, 0, 0, -3, 0, -1, 0, 1, 3, -1], [-1, 0, 2, 1, 2, 1, 0, 2, 2, 0, 2, -2], [0, -1, -2, 1, 0, 0, 1, 4, 0, -2, -2, 0], [4, 1, -3, 1, -4, 0, -2, -2, -3, -3, -2, -1], [6, 1, 4, -1, 1, 1, -1, -1, 0, 3, 2, 0], [-3, -1, 0, 0, 2, -1, 0, 0, -3, 1, -1, 0], [-5, 3, 1, 3, 2, -3, 1, 2, 1, 1, 4, 0], [8, -2, 0, 1, 0, -1, -1, 1, -3, 1, 1, -1], [-1, 3, 0, 0, 0, -1, 0, 0, 1, 0, 1, -1], [1, -3, 0, 1, 0, -2, 1, 0, 0, 1, -1, 0], [5, 0, -1, -1, 0, 1, -1, 1, -2, 0, 0, 1], [3, 0, -1, -1, -2, 3, -1, 0, -3, -1, 0, 0], [-6, 2, 4, -1, -1, -1, 0, -2, 1, 2, 2, 2], [7, -3, 3, 0, -2, 0, 0, -2, 2, 1, 2, 1], [-5, 2, -2, -1, -3, -1, -1, -4, -1, -2, -1, 0], [0, 4, 2, 2, 2, -3, -1, -1, -1, 2, 3, -1], [-3, -2, 1, -2, 1, 0, 1, -1, -3, 2, 0, 0], [-10, 1, 1, 1, -1, -2, 1, 3, 1, 0, 1, 1], [4, 4, -1, 0, -3, 1, -1, 1, 0, -5, -3, 2], [1, -2, -2, 3, -1, 1, 3, 0, -1, 0, 1, 0], [-10, 2, 0, 1, 0, -2, -1, 0, 0, 0, -2, 1], [-8, 2, 0, 1, 0, -3, -1, -4, 7, -1, 2, 2], [-1, -1, -3, 1, 0, -2, 0, 0, 1, -4, -1, 0], [1, 2, 4, 1, 1, -3, 1, 2, 1, 2, 3, 0], [-2, 3, 2, 1, 2, -3, 2, 2, 2, 2, 3, 2], [-2, -1, -2, -4, -1, 1, -1, -2, -2, -1, -3, 0], [-8, 3, -1, -1, 1, 3, 0, -1, -1, 0, 0, -1], [5, 2, 0, -1, 0, 0, -1, -2, 2, 0, -3, 0], [-1, 0, -3, 1, -1, 1, 0, 2, 0, -3, -1, -1], [-3, -3, 1, 0, 1, 1, 0, -3, -4, 2, 0, 0], [3, -2, -4, 1, 1, -2, -1, -1, -3, -3, -3, -1], [2, 2, -5, 2, -2, -1, 0, -2, -3, -3, -1, 0], [-16, -1, 3, 2, 1, -5, -1, -1, 1, -1, 2, -1], [-10, 4, 5, 0, 4, 0, 1, 1, 0, 5, 4, 1], [-8, -3, 1, -2, 3, -1, 1, 0, 0, 4, -2, 1], [-5, -2, -3, 1, 0, -3, 3, -2, -1, -1, 0, 0], [-4, 3, 2, -1, 0, -4, -1, 3, -3, 1, 0, 2], [6, 1, 0, 0, -2, 1, -3, 0, -4, -3, -3, 0], [8, 0, -2, 1, 2, 1, 0, 1, -2, 1, 2, -3], [0, 0, 2, -2, 1, 2, -1, 0, 5, 4, 3, -1], [-3, 3, -2, 1, -1, 0, -2, -2, 0, 0, 1, 1], [0, 1, -2, 1, -1, -2, 0, 1, 1, 0, 2, 0], [6, 2, 4, 0, -4, 0, 1, 2, -1, 0, 3, 1], [-6, 3, -1, 2, 2, 2, 0, 2, 0, 2, 3, 0], [-4, 3, 1, 0, 4, -2, 1, 5, 4, 0, 2, 2], [-10, 1, 0, 0, 3, 0, 3, 3, 3, 1, 0, 1], [-9, -1, 1, 1, 0, -4, 1, -1, 1, 1, 2, 0], [-5, 2, 2, -4, 0, -2, -1, -4, -2, 0, 1, 0], [3, -3, 2, -6, 3, -1, 2, 1, 0, 3, 1, 0], [5, -1, -2, -1, 0, -1, -1, -4, 6, 0, 2, -1], [4, 2, -2, 4, 1, -2, -1, 5, 0, 1, 1, 0], [-3, 1, -1, -5, 0, -1, 0, -5, 4, 0, -2, 1], [-4, -3, 1, 1, -1, -2, -1, -2, 0, 1, 1, -1], [2, 1, -2, 1, -1, 0, 0, -3, 2, -3, -1, 1], [-1, 2, 1, -3, 0, 4, 4, 2, -3, 3, 1, -1], [-8, 1, 1, 0, 1, -5, 0, 1, 4, 2, 3, 3], [-1, 0, 1, -2, -3, -1, -3, -1, 3, -1, -1, 2], [2, 1, 0, -2, -2, -3, -1, -4, 0, 0, 3, -1], [-1, -4, 4, 0, 1, 0, 1, -1, -2, 2, 0, 0], [0, 2, 2, -6, -4, 0, -3, -2, -1, 0, 3, -1], [1, 1, 0, -2, 2, 2, -3, 1, -2, -1, -4, -1], [-10, 0, -3, -1, -2, 1, -1, -5, -2, -5, -5, 1], [-4, 4, 1, 1, 2, 1, -1, 5, -2, 1, -1, -1], [-6, 3, 0, 2, 1, 1, 0, 2, 3, 3, 2, 0], [-3, -6, 0, -2, -3, 3, 1, -2, 3, 2, 1, -2], [-3, 3, 1, 1, 0, 1, 2, 4, -3, 1, 0, -2], [11, 5, -2, 0, -2, 1, -1, 2, -1, -1, -1, 1], [3, 0, -1, -5, 1, 2, -1, -2, -2, 1, 0, -2], [10, 0, 1, 1, -1, 5, 1, 4, -1, -1, -1, 1], [4, 1, 6, 1, 3, 0, 1, -1, -1, 5, 2, -2], [-9, 1, -7, 3, 1, 3, 1, -1, -2, 0, -2, -1], [5, -3, 3, 1, 2, 1, 1, 6, -1, 2, 0, 1], [3, 3, -2, 2, -3, -2, 1, -4, 5, -4, -2, 0], [-11, 0, -1, -2, -1, 5, 3, 1, 3, 1, -3, -2], [1, 1, 2, 1, -1, -5, -1, 1, 0, 0, 1, 2], [6, -1, 2, -3, 0, -4, -3, 1, 3, -3, -2, 0], [-4, -2, 2, -4, 2, 0, -4, 0, 2, 2, -2, 0], [-16, 2, 2, 2, -4, 5, 1, 0, -2, 4, 3, -1], [-5, -3, -4, 2, -1, -3, 2, -3, 4, -1, -1, 0], [-8, -1, -2, -2, -1, -2, 3, -1, 0, 2, 1, 1], [0, 5, 0, 0, 1, 2, -1, 0, 6, 0, 2, 1], [-7, -3, 3, 3, 5, 1, 0, 6, -1, 1, 0, -1], [-1, 0, -3, 0, 2, -6, -2, -1, -1, -3, -2, 0], [-4, 4, -3, 5, -2, -3, -2, -3, 5, -3, 0, 2], [-1, -5, 3, -2, -2, -4, 0, 0, 6, -1, 0, 3], [0, -1, 4, -1, 1, 2, 0, -1, 2, 1, 3, 1], [-1, -3, -4, 2, -4, 4, 1, 2, 0, -1, -3, 0], [0, 5, 1, 3, 2, -1, -1, 3, 4, -4, -1, 0], [-6, 7, 4, 3, -1, 1, -1, 1, 2, 1, 4, -2], [-3, -7, -1, 1, -2, -2, -2, -2, 5, 0, 1, 0], [1, -5, 2, 1, -2, 0, 3, -1, 2, 1, 2, 3], [12, 0, -1, -2, -5, 1, -1, -1, -1, -4, 0, 0], [1, 3, -2, 0, -3, 4, -1, 0, -3, -5, -6, 1], [7, -2, 3, 5, 1, -1, 1, 3, -1, 3, 3, 1], [-2, 4, -1, 3, 5, -2, 1, 3, -1, 0, 1, 2], [-12, 1, -1, 4, 4, -2, 1, 6, 1, 0, 2, -2], [4, -7, -2, 2, 2, -1, 0, 2, 1, -1, -4, 1], [7, -1, -5, -2, -7, 1, 1, 0, -1, -4, 0, 0], [-1, 1, 3, 3, 0, 1, 2, 6, 4, 2, 3, -1], [10, -2, 3, 1, -2, -3, 1, 0, 1, 1, 1, -1], [-13, 0, 2, 3, 5, -2, 1, 3, -1, 1, 1, 1], [-14, -1, -2, 0, 0, -4, -3, 2, 1, -5, -1, 0], [-11, 3, 0, -6, 1, -4, -2, -1, 5, -1, 1, 3], [3, -1, 4, 0, 0, 1, 0, -4, 4, 1, 4, 1], [-6, -4, -4, 1, -4, -3, 2, 0, 1, -2, -2, 1], [-9, -1, 0, -6, -2, 3, -1, -1, 5, 1, -2, -1], [1, 1, 8, -6, 4, 1, 1, 2, 4, 4, 4, 0], [11, 1, 3, 1, 5, -3, 1, 6, 1, 2, -2, 3], [4, -1, 0, 3, 4, 1, -1, 4, -1, 2, -2, 1], [13, 2, -6, -4, -4, -1, -1, -5, -1, -4, -6, 0], [-12, 4, 2, -4, -1, -1, -2, -2, -1, 4, 0, 2], [13, -7, 2, -1, 1, 3, -2, -1, -7, 2, -2, -1], [0, -6, -1, -2, 5, 1, -1, -1, 1, 2, 0, -1], [-4, 1, 5, 2, 1, -2, 0, -4, -2, 6, 5, -1], [-10, 0, -4, 3, -3, 3, 3, -3, -6, -2, -2, -1], [2, -3, -4, 0, 2, 3, -1, -1, -4, 3, 2, -1], [-4, 6, -1, -1, -2, -1, 1, 3, 0, 1, 1, -1], [11, -2, -3, -3, -1, 5, 0, 2, -6, -1, -3, -3], [8, -7, -5, -5, 2, 0, 1, -3, -1, 0, -5, 1], [2, 3, -1, -2, 1, 1, -2, -6, -3, 4, -1, -3], [3, 0, 1, 3, -1, 5, 2, -4, -4, 3, 1, -1], [-8, -2, -2, 2, -1, -3, -1, -2, -2, -4, 1, -1], [23, -8, 4, 4, 3, 0, 0, 6, -2, 1, 1, 0], [-3, -5, -1, -2, 3, 1, 3, -3, 3, 2, 3, 1], [8, 0, -3, 3, -3, 4, 1, 1, -6, -4, 0, -2], [-10, 2, 0, 5, 1, 0, 2, -1, -2, 2, 5, 0], [15, 0, 1, 2, 0, -2, 3, -3, -6, 3, 3, -1], [-15, -2, 1, 0, -2, -3, 0, 1, -4, 2, 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2, 1], [-16, -10, -6, -2, -6, -2, 4, -4, 0, -4, -4, 0], [2, -4, 8, 5, 2, 2, 6, 4, -1, 7, 5, 3], [-6, 1, 8, 3, -8, -4, 3, 5, 1, 3, 6, 2], [26, -10, -3, 4, -2, 6, -1, -1, -3, 1, -7, -2], [10, 12, -2, 8, 5, -5, 0, -2, 1, -2, 2, 0], [9, 18, 2, 1, -5, -1, -4, 6, 6, -10, -6, 3], [5, -1, -3, -7, -1, -2, -5, -2, 3, -5, -5, 2], [14, -5, -3, 1, 5, 6, -4, -4, 0, 2, -1, -4], [20, -1, 3, -7, 1, -1, 0, 2, 2, 7, 0, -2], [15, 0, 4, -1, 3, 3, 3, 3, 2, 3, 1, 0], [-3, -2, 4, -6, -2, -3, -4, -8, -6, 4, -4, 0], [26, -13, 8, -3, 5, 6, -1, 3, 3, 7, 2, 1], [7, -11, -6, -1, -6, 13, 5, 0, -9, -3, -5, -1], [3, 5, 3, -9, -12, 6, -3, -4, -2, -2, 2, 0], [-4, -2, 8, -2, 5, 3, -2, 2, 1, 9, 2, -3], [-29, 2, 4, -5, 6, 1, 0, 0, 6, 2, -2, -1], [10, -8, 9, -8, 3, 7, -3, 4, 0, 7, 0, -3], [-5, -1, -2, -5, -2, 6, -1, 1, -3, -1, -5, -3], [22, -6, -5, 4, 4, -13, 0, 1, -5, -3, -4, 4], [-15, 0, -3, 6, -3, -1, -1, 3, 9, -5, 3, 4], [18, 3, 7, 0, -4, 5, 3, 5, -11, 3, -4, -1], [-3, -3, 6, 0, 3, -4, 0, -3, 11, 1, -1, 2], [2, 6, -2, -4, 3, 4, 6, 5, -7, -1, -4, -1], [2, 6, -8, 0, 4, 3, 1, -2, -6, -6, -5, -1], [-1, -15, -9, 3, -2, 4, 5, -1, -5, 0, -4, -1], [1, -10, 2, -5, 4, 8, 5, 6, -2, 4, -1, 4], [-1, -1, 8, 2, -5, 0, -1, 0, -5, 2, 0, 0], [-13, -7, -2, 7, -1, -5, 2, -5, 6, 1, 7, 0], [14, 1, 4, 0, 5, -13, -1, 4, 5, 0, 0, 6], [4, 14, 4, 1, 1, 0, 2, -3, -8, 6, 3, -6], [24, -1, 5, 8, 1, 0, 3, 6, -5, -2, -4, 0], [-4, -3, 1, -3, -2, 12, 0, -2, -3, -1, 0, -4], [-26, -7, 1, -4, -2, -4, -5, -3, 6, -1, 0, 1], [4, -5, -2, 6, -1, 6, 1, -7, -5, 3, 4, -3], [18, 1, -5, 0, 9, 0, 2, 8, -4, 2, -3, 3], [19, -8, 0, 2, -3, 0, 1, -1, -4, 6, 2, 2], [5, 8, 4, -2, 1, -3, 2, 4, -3, -2, -9, -1], [15, 3, 4, -8, 5, 2, 2, 7, -1, 1, 1, 1], [-8, -8, -6, -3, -6, -1, 0, -9, -4, -7, -4, -3], [-29, 5, 2, -2, -2, -1, -2, -2, 9, 0, 5, 3], [2, 0, 3, 5, 2, -7, -1, -1, -8, 3, 3, 5], [15, 2, 6, 8, 9, 0, -3, -1, 4, 0, 4, 2], [8, -4, 6, 0, 6, 6, 5, 2, 3, 12, 9, -1], [0, 4, -3, 1, 1, 11, 2, -3, -1, 1, 0, 0], [5, -4, 6, -9, 7, 7, -1, 4, 5, 8, 5, 2], [-7, 6, 2, -4, -10, 1, -3, -9, 13, 4, 6, 4], [19, 3, 6, 0, 2, -10, -2, 1, 7, -1, 1, 2], [-11, -1, -10, -2, -7, -2, -1, -6, -9, -10, -8, -2], [1, 1, 11, 2, 3, -9, -1, 8, 10, -3, 1, 4], [22, 5, -3, 1, 0, -3, -5, 2, -2, -2, 0, -4], [-6, 3, 5, 4, -2, -10, -2, 4, 12, 0, 5, 3], [21, -5, 0, 3, 7, 3, -2, 1, -3, 2, -6, 3], [0, 16, 5, -2, 1, -9, -3, -2, 0, -3, -2, 3], [7, 0, 1, 5, 0, 7, 8, 2, -8, 3, 5, 1], [-3, 4, 5, 2, -6, 3, 4, 1, -4, 7, 6, 1], [8, 4, 5, -1, -3, -7, 0, 4, -8, 1, 0, 4], [-1, 8, 2, 3, -3, -3, -6, -6, -4, -6, -4, -5], [8, 3, -6, 12, -3, -10, -1, 1, 3, -11, 0, 3], [-10, 5, 4, 10, -6, -3, 7, 7, 6, -3, 8, 1], [-6, 0, 3, -4, -5, 2, -2, 4, -4, -1, -7, 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_bg();".
// 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_bg(: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_bg();".
// 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_bg( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6042_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![7848, 18296, -19370, -26766, 11014, 11545, -3056, -2095, 453, 169, -34, -5, 1]>,<7,R![-1536, -17696, 67480, -55900, -17126, 30013, -1304, -4596, 548, 280, -42, -6, 1]>,<11,R![-475136, -316928, 1954464, -1180960, -244184, 277620, -9570, -21152, 2126, 653, -84, -7, 1]>],Snew);
    return Vf;
end function;