
// Make newform 6042.2.a.y 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_y();".
// 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_y();".
// 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 := [-10, -35, -14, 48, 29, -14, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [-445, -480, 682, 517, -216, -51, 16], [-405, -695, 548, 888, -274, -89, 24], [335, 915, -456, -1086, 303, 108, -28], [115, 260, -163, -338, 99, 34, -9], [1285, 2665, -1886, -3466, 1043, 348, -93]];
    Rf_basisdens := [1, 1, 25, 25, 25, 5, 25];
    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], [1, 0, 0, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0, 0], [-1, 0, 0, -1, 0, 0, 0], [-1, 0, 0, 1, 0, 0, 1], [-1, 1, 0, 0, 0, 0, -1], [-1, 0, 1, -1, 1, 0, -1], [-1, 0, 0, 0, 0, 0, 0], [0, 2, -1, -1, 0, -1, 0], [-1, 0, 0, 2, -1, 1, -1], [0, -1, 0, 1, 0, 1, 1], [-1, 1, 1, 0, -1, 1, 0], [-2, 0, -1, 0, -1, 1, 1], [-4, 1, -2, 0, 1, 0, 0], [2, 2, -2, 1, -1, -2, 0], [1, 0, 0, 0, 0, 0, 0], [1, 1, 2, -3, 1, 0, 0], [-3, -1, 3, 1, -2, 0, 1], [-1, 0, -1, 3, -1, -3, 1], [0, 3, -1, -1, 1, 1, -2], [1, 0, 0, -1, 0, 2, -2], [-2, -4, 1, -1, 2, 1, 2], [-2, 3, 1, -2, 0, 0, -3], [-1, -3, 1, 1, -2, -1, 1], [-2, -1, 0, 0, 3, 0, -2], [-3, 0, 0, 2, 3, -1, 0], [-5, 2, -1, 0, 2, -4, 2], [1, 2, -1, -1, -1, -1, 1], [-7, 3, -1, 0, -1, -1, 1], [0, 0, -2, -1, 3, -2, 1], [-3, 0, 0, -2, 1, -2, 1], [-6, -1, 0, 2, -3, 4, 3], [-3, 1, 3, -1, -2, 1, 0], [-3, -1, 3, 0, -3, 5, -2], [0, 0, 3, -3, 4, -2, 2], [-6, 1, -1, -1, 1, -3, 0], [-3, 1, 4, -3, -1, 4, -4], [-6, 0, 1, 0, -1, 2, -1], [4, -4, 2, 3, -1, 4, -1], [-1, 1, 0, 1, -3, 4, -2], [4, -1, 1, -1, -3, -1, 2], [1, 0, -6, 6, 1, -4, 4], [0, -2, -2, 3, 1, -2, 1], [2, -2, -5, 0, 5, -4, 0], [-3, 0, 1, -2, 0, 2, -6], [1, -3, -4, 3, -1, 1, 2], [-3, 1, 4, -3, 3, 1, -1], [-8, -5, 1, 1, -3, 0, 7], [-7, 0, -2, -1, 0, -2, 4], [-8, 0, 1, -1, 0, 0, 2], [6, -3, 0, -2, 1, -2, -1], [1, -3, 1, 0, 1, 0, 2], [-1, 3, -2, -1, 1, 0, -6], [-1, 1, 0, -1, 1, 3, -4], [-5, -1, 3, -7, 4, 2, -1], [-5, 0, -3, 2, -2, 3, 4], [9, -4, 2, -5, -4, 4, -2], [0, -9, 1, 4, 0, 2, 3], [-10, 2, 2, -2, -4, 3, -6], [4, -3, -2, 4, -3, 0, 2], [-13, 2, -3, -3, -3, -1, -3], [6, -1, -4, 1, 2, 2, -3], [1, -8, 5, -3, 1, 1, -4], [5, 7, 1, -4, 3, -2, -5], [5, -5, 0, 5, -5, 3, 2], [-1, -2, 1, -1, 5, 5, -1], [0, 3, 2, -2, -5, 4, -3], [-9, -2, -3, 1, -1, 0, 0], [0, -2, 1, 0, -1, -3, 4], [3, 0, 3, -1, 5, -6, -1], [11, 0, 4, 1, -2, -1, 1], [-3, 0, -2, 8, -3, -2, 7], [-4, -3, -1, 0, 0, -4, 5], [-1, -1, 4, -2, -2, 4, -2], [-5, 4, -3, 5, -1, 1, 1], [2, 2, 5, -3, 2, -1, 0], [10, -5, 2, 3, -4, -2, 2], [-5, 0, -4, -5, 8, -3, -2], [14, 0, -1, -2, 1, -3, -1], [-1, -6, 4, 1, -2, 3, 0], [1, -9, 1, 3, -4, 2, 1], [0, -2, -1, 5, -2, 2, 3], [4, -9, 1, 2, -8, 4, 3], [3, 2, 7, -7, 1, 1, -1], [-7, 0, -4, 0, 7, -10, 4], [-14, -3, 3, 2, -4, 6, -3], [3, 4, -3, 4, 2, -2, 4], [-6, 3, -8, 4, 5, -7, 4], [1, 3, 0, 0, 0, 0, 7], [10, 2, 4, 0, 6, -1, -2], [3, 3, -6, 1, 1, 0, 1], [-5, 5, -5, -2, 5, -9, 2], [7, -5, 9, -6, 3, 0, -5], [-1, 5, -7, 3, 4, -4, -1], [-6, 5, -4, 3, -2, -4, 5], [-5, -2, -1, 4, 2, -8, 2], [12, -2, 3, 6, -3, -1, 3], [-1, 5, -5, -2, -5, -4, -4], [3, 1, 0, -7, -1, -3, 3], [8, -5, -1, 4, 0, 4, 3], [-10, 9, -7, -4, 0, -3, -2], [-4, 1, -7, -1, 5, 1, -1], [-9, 1, 4, 0, 0, 4, -5], [-2, -1, 9, -8, 0, 0, -3], [-21, 8, -7, 4, -4, -2, 2], [11, -10, 5, -1, -5, 2, 1], [-9, 2, 1, -1, -3, 7, -4], [1, 1, 2, -3, 1, 3, -2], [7, -6, -2, 5, -6, 4, -4], [-1, 1, -2, -3, 5, -6, 1], [0, -3, 4, -1, -2, 7, -4], [3, 4, 1, -7, 1, 4, -9], [12, 6, -2, -4, 0, -1, -7], [-8, 7, -5, -4, 2, -5, 1], [-16, 12, 1, -5, 6, -6, -2], [-1, -6, 2, -7, 6, -4, 4], [-11, 4, -6, 3, 0, -3, 4], [-6, 3, -3, -9, -1, -2, -11], [14, -1, 2, -3, -4, 2, 3], [-7, 5, -2, -4, 4, -6, 5], [-5, 4, 3, 2, -8, 0, 6], [-5, -2, 3, 3, 1, 2, 3], [-9, -6, 6, 3, -4, 10, 4], [8, 4, -8, -2, 6, -3, -4], [14, -6, -1, 0, -3, 7, 1], [1, 0, -7, 8, 0, 1, 1], [-1, -1, 3, 5, 2, 7, -7], [14, 5, 6, -3, 0, 4, -1], [-7, 0, -1, -1, 3, -1, 1], [-11, -4, 0, 8, 3, -7, 7], [-23, 3, 6, -3, -3, 1, 0], [-1, -1, -3, 5, -6, 0, -5], [16, -6, -1, -1, 0, 3, -3], [-15, 0, 1, 1, 3, 5, -5], [-9, -5, -6, 1, 3, -4, -1], [6, 4, -1, 0, -1, -1, 3], [-4, -10, -3, 6, -1, -4, 11], [-7, 0, 2, 5, 4, -4, -4], [-3, -6, 1, -3, -7, 2, 5], [6, 2, 4, -2, 6, -1, -1], [8, 4, -2, -8, 8, -8, -9], [-13, 8, 3, -2, -6, 2, -5], [6, -8, 4, 12, -8, 0, 12], [-2, -5, 0, 11, -8, 8, -7], [5, 5, 4, -2, 2, 10, -12], [-6, 7, 2, -6, 5, 7, -16], [-6, -6, 4, -2, -6, 6, 6], [-21, -5, 1, -4, -3, 0, 4], [-7, 5, 3, 7, 0, -3, 1], [12, -9, 7, 4, 0, 0, 3], [2, 2, -7, 9, -2, -6, 10], [17, -12, 9, 1, -1, 10, -6], [-5, 4, 0, -2, -5, 1, 1], [-1, 2, 0, -6, 1, -2, 5], [7, -11, -5, 5, -6, 0, 4], [-4, 6, -7, 0, 3, -1, 7], [-4, 4, -5, -4, -5, -2, -1], [8, 3, 7, -10, 8, -1, 1], [-1, 0, -4, -1, 2, 7, -4], [4, 8, 2, -5, 7, -5, -11], [-13, 3, 0, 3, 1, 4, -3], [1, -6, 2, 5, 4, -2, 4], [-12, 13, -2, 2, 7, -5, 0], [9, -10, 3, -5, 11, -6, 3], [10, 5, 6, -10, 1, -1, -2], [7, 7, -5, -11, 8, -4, -13], [9, -1, 2, 12, -10, 3, 10], [-2, 7, 4, -3, 2, 2, -15], [-4, -9, 4, 8, -1, -4, 7], [-1, 8, 1, -8, -8, 2, -8], [7, 13, -3, -1, 0, -2, -3], [10, -2, -2, 3, -5, -4, 4], [14, 0, -9, 3, 6, -11, 1], [0, -3, 0, -8, -1, 1, 0], [13, 8, -5, 4, -4, 1, -2], [14, 1, -7, 12, -2, -4, 7], [-3, 0, -7, 13, -5, -8, 5], [3, 3, 4, 4, -2, -2, 5], [0, -8, -5, 1, 0, -5, 8], [12, 10, -2, -13, 5, -3, -11], [3, 3, -1, -4, -9, 2, -6], [-6, -8, 1, 5, 4, 0, 10], [19, 4, 6, 2, 1, 1, 0], [-9, -3, -2, 4, 2, -1, -3], [10, -1, 1, -3, -3, -4, -3], [2, 0, -2, -3, 11, -7, 7], [-7, 5, -4, 12, -6, -6, -1], [16, -1, 1, 14, -10, 5, 0], [9, -10, -4, 7, 2, 1, 10], [3, 1, -7, 6, -1, -4, 1], [10, -5, 6, -4, -5, 11, -4], [-10, 5, 1, 0, 8, -2, 1], [1, 4, -6, -7, 4, -8, 6], [6, -7, -10, 8, -3, -5, 6], [-6, -8, 7, -10, 5, 3, 3], [2, -2, 4, -4, 8, 6, 4], [-3, 12, -7, 7, -1, -7, -1], [9, -8, -6, -4, 9, -1, 9], [-23, -5, 3, 3, -8, 5, 8], [4, 5, -2, 3, 10, -9, 2], [10, -1, -3, -4, 0, 6, -5], [1, 5, 6, -16, 4, 3, -5], [2, 5, -3, 1, 11, 7, -13], [10, -13, -1, -10, 14, -5, 2], [10, -5, 7, -6, -4, 4, -6], [-6, -2, 0, 7, -5, 4, 2], [-8, -7, 2, 0, 11, 1, -6], [6, -1, -6, 12, -11, 0, 10], [-2, 3, -1, 4, 4, -11, 3], [1, -7, 4, -11, 9, -3, -4], [9, -8, 0, -3, -6, 5, 6], [12, -3, 2, 3, -2, 8, 1], [-1, -1, -3, 2, 1, 11, -4], [11, -6, -8, 3, -4, 0, -2], [10, 0, -6, -4, 8, -4, -6], [14, -6, 6, -12, 2, 17, -6], [13, -6, -3, 7, -3, 3, 1], [-2, 9, 11, -14, -8, -3, -7], [6, -1, 9, -8, -14, 10, -3], [7, -6, 6, -6, 7, -1, 8], [10, -6, 8, -5, 11, -3, 8], [12, 3, 1, -3, -9, 4, 6], [21, 5, -3, -11, -2, 12, -7], [2, 9, -3, 0, 8, -4, -9], [0, -15, 10, 0, -5, -2, 2], [22, 6, 11, 0, -1, 8, -13], [-8, -1, 1, 2, 12, 6, 1], [21, -20, 2, -3, -6, 1, 2], [4, -10, 14, -5, 1, -4, 2], [20, 0, 7, -6, -9, 15, -7], [25, 2, -4, -3, 6, -6, 4], [7, 14, -10, -5, 0, -2, -13], [12, 9, -10, 9, 6, -7, -4], [3, 8, 3, -1, 1, 1, 1], [5, 2, 6, -5, -4, -4, -6], [-30, -8, -6, 1, 5, -3, 5], [23, 0, -2, -9, 4, 6, -10], [15, -12, 10, 2, 1, -1, 7], [1, 2, -9, -5, 9, 3, -2], [21, 0, 3, -5, 5, -5, -1], [20, 4, -9, -4, 1, 1, -11], [-5, 0, 0, -11, 0, -7, 0], [7, -8, 3, -13, -3, 11, -1], [8, 1, -1, -17, -7, -1, -7], [15, -9, -4, 1, -3, 6, -4], [-19, -15, 4, 6, -4, -1, 7], [18, -7, -5, -4, -2, -4, -5], [28, 8, -2, 8, 2, -2, -10], [7, -3, -2, 9, 7, -6, 5], [-6, -9, 6, -1, 2, 5, -4], [-11, -12, 9, 9, -9, 11, -5], [5, -3, 11, -15, 2, 5, -9], [15, 17, -7, -6, -1, 0, 1], [10, -10, 6, -2, -2, 5, -7], [22, 6, -5, -5, 6, -15, -3], [-4, -6, -3, -8, -1, 5, 4], [8, 1, 0, -6, -3, 9, -2], [-5, 9, 2, -7, 5, 4, -11], [11, 3, 0, 1, -5, -3, 10], [6, 15, 6, -18, 3, -1, -18], [4, 0, -6, -9, 13, -5, 7], [-1, -10, 2, 11, -8, -3, 6], [6, 4, 14, -6, -6, 5, -8], [15, 8, -3, 6, -6, 2, 13], [-21, -9, 0, 1, -5, 11, 8], [-40, -11, 9, -2, -8, 8, 1], [-6, 6, -10, -1, 3, -7, 7], [-1, 22, -8, 7, -10, -6, -4], [34, -4, 3, 6, 1, 1, -1], [-1, 4, -4, 6, 11, 1, -11], [8, 6, 8, -7, 7, -7, -8], [8, 6, 3, 0, 3, -5, -11], [36, -9, 16, -7, 4, -3, -5], [-2, -8, 2, 16, -6, -4, 4], [15, 12, -2, -7, -4, 6, -6], [-5, 3, -8, 0, -10, 1, 1], [-22, -22, 9, -4, 7, 4, 8], [17, 17, -2, -16, 0, -2, -13], [11, 13, -13, 5, -8, 1, -5], [-42, 2, 1, 8, -1, -5, 0], [-6, -5, 10, 0, -9, 10, 0], [-2, -1, 12, -2, 1, 1, 4], [-33, 3, -3, -4, 1, -4, 10], [22, 5, -4, -17, 6, 0, -17], [16, 0, -3, -9, 4, 4, 2], [21, -11, 7, 8, 1, 14, -9], [-20, -1, -1, -2, 6, -10, -3], [21, 17, 7, -11, 0, 11, -7], [-9, -5, -1, 3, 4, -2, 9], [4, 8, 1, -12, -1, -7, -9], [-32, -8, 11, 3, -2, 0, 10], [-12, -7, 1, 6, -4, -8, 12], [-1, -6, 1, 21, -7, -3, 9], [13, -7, 9, 11, -6, 10, -11], [-24, 7, 0, 3, 10, -10, 5], [11, -1, 9, -18, 3, 11, -7], [-13, 6, 2, 2, -7, -4, -8], [-25, 5, -3, 22, 1, -2, 3], [-16, 4, -7, -2, 5, -15, 3], [12, -1, -3, -11, 3, -5, -2], [-3, 11, 0, -19, 5, -13, 10], [4, 6, -1, -12, 9, -11, -11], [-6, 1, 9, -4, 0, 4, -5], [19, -12, -8, 3, 0, 3, 4], [-6, -7, -6, 22, -7, -7, 22], [7, -2, 3, 5, 1, 4, 14], [-8, 2, 7, 4, -13, 0, 3], [10, -17, 6, -5, 2, 4, -7], [-25, 1, -6, 22, 2, 3, -2], [-3, 8, -13, 13, -1, -4, 17], [5, 11, -4, 14, -2, -3, -8], [5, -1, -3, 0, -15, 3, -1], [-13, 18, -2, 9, 6, 0, -8], [44, 11, -3, -2, 4, -4, -2], [8, 10, -4, -13, 11, -6, -5], [1, 11, -7, 5, -2, 5, -16], [9, 1, -2, -3, 7, -9, 0], [-13, 15, -10, 5, -7, -1, 16], [1, 3, -2, -4, 10, -7, -3], [-25, -16, 0, 3, 10, -3, 2], [-45, 1, -6, 7, -1, 4, 1], [-7, 11, -2, 11, -7, 5, 14], [8, 9, -9, 20, -12, -5, -5], [1, -10, 11, -4, 14, 3, 4], [-18, 7, 3, 8, -4, -19, -2], [-17, -3, 9, -1, -12, -3, -4], [22, -12, 3, -6, 11, 2, 2], [2, 0, 2, -14, -2, 6, -6], [-14, 9, 1, -4, 0, -2, 13], [9, -7, -9, 10, -7, -11, 0], [7, -7, 1, 3, -20, 21, 3], [4, -14, 3, -2, 7, 13, 9], [-5, 2, -16, 13, 2, -2, 6], [-37, 0, 2, -1, -2, 2, -1], [-28, -2, 5, -12, 7, 12, 1], [-16, -4, -8, 8, -4, -8, 10], [-2, 0, -3, -17, 14, 0, 0], [-21, -1, 0, 21, -9, -1, 20], [-4, 16, 4, 0, 4, 8, -2], [-31, -2, -7, -4, 16, -3, 0], [27, 9, -9, 12, 5, -4, 1], [1, -7, 8, -3, -3, 8, -13], [10, 19, -8, -2, 1, 2, 2], [-8, -11, 8, -3, -14, 19, -5], [7, 11, 9, -10, -1, 15, -16], [18, -3, -3, -14, 14, -13, 11], [-9, 17, -13, 5, -6, 4, -11], [-8, -21, 18, 0, -9, 11, -1], [24, -11, -1, -8, 6, -7, -3], [-40, -3, 5, 11, -1, 11, 4], [7, -19, 11, 3, -2, 8, -3], [8, -2, -7, 20, -5, -6, 14], [-35, 7, -9, 6, -5, -17, 14], [-30, 7, 7, -11, 1, -7, 6], [-18, 13, 1, -11, 3, -7, 2], [-4, 2, 2, 16, 2, -8, -2], [18, 10, -12, 11, -5, 3, -7], [-21, 14, -2, -9, 10, -4, -4], [-15, -4, 9, 17, -7, -8, 10], [4, 8, -1, 4, 5, -9, 1], [23, 14, 11, -18, 2, 12, -16], [-29, 2, 7, 11, -11, 12, 2], [5, 10, -10, -11, 0, 10, -18], [-11, -6, 3, -9, 19, -7, -5], [-12, -9, -9, 18, 0, -3, 7], [-9, 24, 4, -1, -12, 1, -13], [-2, 8, 5, -13, 2, 21, -17], [-26, -12, 0, 1, -1, 1, 5], [7, -3, -6, -1, 11, -6, 13], [-30, -3, 7, 0, -4, 6, -4], [-2, -6, -8, -6, 18, -10, 0], [19, 8, 10, -14, -3, 0, -4], [-6, 12, -8, 0, 20, -7, -2], [21, 13, -2, -5, -9, 3, -2], [-41, 1, 1, -5, 2, 8, -3], [12, -19, 7, -15, 3, 3, 2], [-4, -24, 18, 1, -5, 4, -4], [26, -7, 3, -3, -7, 12, 2], [-31, -4, 2, 17, 8, -15, 9], [4, 2, 2, 26, -12, 1, -1], [-19, -7, 8, 0, -10, 8, -3], [-8, 0, 2, 10, -2, 12, -12], [-12, -11, -6, -16, 5, -1, 10], [-4, -2, -5, 16, 5, -13, 9], [-8, 9, 8, -2, -15, -2, -9], [-3, 5, 8, 11, -1, 1, -10], [-3, 2, 9, -4, -8, -1, 12], [25, -4, 2, -9, 10, 0, 10], [-9, 1, 0, 10, -14, 0, 11], [11, -4, 7, 1, -3, 5, -3], [16, 9, -10, 20, -9, -12, 5], [13, 9, -8, 14, -16, -10, 3], [-1, 14, 8, -5, 2, 5, -25], [-23, -6, -12, -5, 6, -2, 8], [-2, -22, 4, -6, 12, -1, 10], [24, -9, -13, 4, 18, -7, 11], [2, -10, -5, 7, 2, -5, 19], [12, -14, 2, 18, -14, 6, -4], [-13, 11, -1, -23, 16, -5, -24], [10, -16, 10, -5, -7, 9, -7], [33, -12, -4, 10, -15, -9, 13], [22, -7, -3, 12, -4, -3, 0], [-2, 6, -2, 15, -1, 2, -12], [-27, 4, 8, 7, 10, -14, 8], [12, 9, -14, 8, 5, -9, -4], [-2, 1, 0, 17, -2, 10, -19], [-24, -3, 4, 0, 13, 5, -4], [-26, -7, 8, 16, -9, 9, 18], [-3, -1, 5, -3, 0, -7, 2], [-4, 2, 4, -4, -10, 15, -17], [-11, 10, 5, -3, 7, -6, -21], [-34, 13, -16, 7, -2, -10, 9], [5, 5, -4, 0, 14, -2, -3], [-9, -19, 20, -3, -3, 2, -2], [-7, -2, 13, -13, -1, -1, 7], [3, -6, 21, -20, 2, 10, -16], [-8, -25, 3, 20, -12, 12, 3], [-22, -8, 11, -1, 4, -1, -7], [21, 17, 2, -5, 5, -10, -15], [-3, -5, 4, -2, -2, -5, 9], [5, -21, -15, 16, -5, 9, 20], [6, -4, -5, 21, -12, 0, 15], [-53, -10, -2, 2, 9, -1, 5], [0, 13, -9, -4, 2, 12, -19], [-27, 5, 5, 3, -6, 9, -7], [24, -9, -11, -7, 15, -17, 14], [-16, 7, -7, 3, 5, 1, 12], [-15, 2, -7, 15, 1, -8, 11], [-34, 15, 3, -12, 0, -10, -11], [-2, 6, -7, 9, -10, -1, -6]];
    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_y();".
// 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_y(:prec:=7)
    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_y();".
// 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_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6042_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 116, -140, -1, 51, -11, -4, 1]>,<7,R![-16, 48, 56, -80, -39, 16, 9, 1]>,<11,R![190, -330, -14, 193, -11, -29, 2, 1]>],Snew);
    return Vf;
end function;