
// Make newform 6021.2.a.l in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6021_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_6021_2_a_l();".
// 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_6021_2_a_l();".
// 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 := [-63, 0, 331, 0, -384, 0, 139, 0, -20, 0, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, -30, 0, 52, 0, -14, 0, 1], [2, 0, -95, 0, 116, 0, -29, 0, 2, 0], [0, 173, 0, -675, 0, 539, 0, -121, 0, 8], [0, -238, 0, 840, 0, -799, 0, 191, 0, -13], [51, 0, -325, 0, 353, 0, -87, 0, 6, 0], [0, -154, 0, 1110, 0, -1147, 0, 278, 0, -19], [-73, 0, 425, 0, -469, 0, 116, 0, -8, 0], [38, 0, -185, 0, 183, 0, -44, 0, 3, 0]];
    Rf_basisdens := [1, 1, 3, 5, 15, 15, 5, 15, 5, 1];
    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_6021_a();"
function MakeCharacter_6021_a()
    N := 6021;
    order := 1;
    char_gens := [893, 4240];
    v := [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_6021_a_Hecke(Kf)
    return MakeCharacter_6021_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 := [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, -1, 1, 0, 0, 0, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 1], [0, 0, 1, 0, 0, 1, 0, 0, 0, 0], [3, 0, 0, 1, 0, 0, 0, 0, 1, 0], [0, -1, -2, 0, 1, 1, 0, 2, 0, 0], [0, 3, -1, 0, -1, 0, 0, -1, 0, 0], [-1, 0, 0, 2, 0, 0, -1, 0, 1, 0], [3, 0, 0, 2, 0, 0, 1, 0, 0, -1], [0, -3, 1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, -2, 0, -1, 1], [0, -1, -4, 0, 0, 2, 0, 2, 0, 0], [0, -1, -2, 0, 1, 1, 0, 0, 0, 0], [0, 3, -3, 0, 1, -2, 0, -1, 0, 0], [4, 0, 0, 1, 0, 0, 0, 0, 2, 0], [3, 0, 0, 0, 0, 0, -3, 0, -1, -1], [0, 4, 1, 0, 1, -3, 0, 0, 0, 0], [5, 0, 0, 0, 0, 0, -2, 0, -1, -3], [-3, 0, 0, -1, 0, 0, 1, 0, 2, 1], [0, -2, 2, 0, 0, 0, 0, 2, 0, 0], [0, -1, 1, 0, -4, 0, 0, -1, 0, 0], [-1, 0, 0, 0, 0, 0, 2, 0, 0, -3], [0, 0, 1, 0, 4, -3, 0, 2, 0, 0], [5, 0, 0, -4, 0, 0, 2, 0, -1, 2], [0, 0, 2, 0, -2, -2, 0, -1, 0, 0], [3, 0, 0, 4, 0, 0, 3, 0, 4, 0], [0, -4, 1, 0, -1, -2, 0, 2, 0, 0], [-1, 0, 0, -7, 0, 0, 5, 0, -4, -1], [0, 5, -3, 0, -2, -5, 0, -1, 0, 0], [0, 4, -6, 0, 2, -4, 0, 1, 0, 0], [6, 0, 0, 1, 0, 0, 2, 0, 3, -3], [0, -2, -3, 0, -2, 4, 0, -3, 0, 0], [10, 0, 0, -1, 0, 0, 1, 0, 1, -2], [8, 0, 0, 3, 0, 0, -3, 0, -1, -1], [0, 0, 0, 2, 0, 0, -4, 0, 2, -3], [0, 1, 2, 0, -4, -5, 0, -4, 0, 0], [0, -4, -1, 0, 2, 2, 0, 1, 0, 0], [0, -1, 3, 0, 3, -6, 0, 4, 0, 0], [7, 0, 0, 2, 0, 0, 1, 0, 5, 2], [0, 2, -2, 0, -2, 0, 0, -2, 0, 0], [-1, 0, 0, -1, 0, 0, -1, 0, 4, 3], [0, -2, 3, 0, 0, 1, 0, 5, 0, 0], [2, 0, 0, 4, 0, 0, -6, 0, 1, 3], [13, 0, 0, 2, 0, 0, 2, 0, -3, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, -1, 4, 0, 2, 0, 0], [-14, 0, 0, -3, 0, 0, 4, 0, 1, 3], [0, 2, -5, 0, -4, 3, 0, -4, 0, 0], [0, -3, 8, 0, 5, 0, 0, -4, 0, 0], [3, 0, 0, -6, 0, 0, -4, 0, -6, 2], [0, 5, 0, 0, -2, 6, 0, -2, 0, 0], [0, -5, 2, 0, 2, 1, 0, 1, 0, 0], [0, 4, -5, 0, -2, -3, 0, -2, 0, 0], [0, 1, -1, 0, -6, 6, 0, -4, 0, 0], [3, 0, 0, -1, 0, 0, -1, 0, 7, 1], [14, 0, 0, 1, 0, 0, -1, 0, -1, 1], [0, 6, -7, 0, -5, 1, 0, -3, 0, 0], [-2, 0, 0, 3, 0, 0, 4, 0, 6, -8], [0, 3, -6, 0, 3, -4, 0, -2, 0, 0], [5, 0, 0, 1, 0, 0, -4, 0, -4, 1], [0, -12, 7, 0, 6, 5, 0, 3, 0, 0], [-1, 0, 0, -2, 0, 0, -7, 0, 6, -1], [0, -2, -3, 0, -4, 3, 0, 4, 0, 0], [1, 0, 0, -2, 0, 0, -3, 0, -10, 5], [-9, 0, 0, 1, 0, 0, 4, 0, 1, 0], [0, 0, -4, 0, 4, 1, 0, 10, 0, 0], [4, 0, 0, -11, 0, 0, 0, 0, -8, 3], [0, 4, 5, 0, -6, 5, 0, -2, 0, 0], [0, 4, -3, 0, -5, 4, 0, 1, 0, 0], [1, 0, 0, 7, 0, 0, -1, 0, 0, -3], [2, 0, 0, 1, 0, 0, -2, 0, -5, 3], [6, 0, 0, 3, 0, 0, -13, 0, 0, 2], [0, 2, -9, 0, 0, 3, 0, 2, 0, 0], [0, -8, 8, 0, -4, 0, 0, 1, 0, 0], [7, 0, 0, 0, 0, 0, -5, 0, 2, 7], [0, -3, -2, 0, 1, 11, 0, 1, 0, 0], [10, 0, 0, -7, 0, 0, -1, 0, 4, -2], [0, 10, -7, 0, -8, -4, 0, -2, 0, 0], [1, 0, 0, 4, 0, 0, 6, 0, 6, -11], [0, -9, 6, 0, 1, -1, 0, 0, 0, 0], [-5, 0, 0, -4, 0, 0, 11, 0, 5, -6], [0, 0, 0, -4, 0, 0, -2, 0, -11, 0], [0, -3, 0, 0, 3, 0, 0, 3, 0, 0], [0, -2, -7, 0, 1, -3, 0, -3, 0, 0], [4, 0, 0, -5, 0, 0, -6, 0, -7, 1], [0, 8, -8, 0, 8, 2, 0, 8, 0, 0], [3, 0, 0, -4, 0, 0, -2, 0, -11, 2], [0, 13, 0, 0, -8, 1, 0, -5, 0, 0], [0, 6, -1, 0, 4, -6, 0, 3, 0, 0], [12, 0, 0, 11, 0, 0, -1, 0, 1, 1], [0, -14, -2, 0, -2, 6, 0, -2, 0, 0], [8, 0, 0, 1, 0, 0, 4, 0, -1, -5], [0, 1, -5, 0, 5, 2, 0, 13, 0, 0], [0, -10, 4, 0, 0, 14, 0, 3, 0, 0], [0, 3, 3, 0, 7, -7, 0, 0, 0, 0], [12, 0, 0, -6, 0, 0, -2, 0, -3, -2], [23, 0, 0, 3, 0, 0, 2, 0, 2, 5], [-5, 0, 0, 10, 0, 0, -1, 0, 8, -9], [0, -5, 1, 0, 2, -9, 0, 10, 0, 0], [0, -5, 2, 0, 7, -7, 0, 6, 0, 0], [0, 2, 3, 0, 5, -8, 0, 1, 0, 0], [-1, 0, 0, 2, 0, 0, -3, 0, 12, -7], [5, 0, 0, 5, 0, 0, 4, 0, 14, -2], [0, -12, 13, 0, 8, 3, 0, 5, 0, 0], [0, 7, -7, 0, -3, -1, 0, -8, 0, 0], [0, -10, 2, 0, -5, -3, 0, 1, 0, 0], [15, 0, 0, -3, 0, 0, -8, 0, 5, -6], [7, 0, 0, -1, 0, 0, 5, 0, 1, -5], [6, 0, 0, 9, 0, 0, 8, 0, 8, -8], [0, -17, 3, 0, 0, 0, 0, -1, 0, 0], [7, 0, 0, -8, 0, 0, 3, 0, 7, -3], [18, 0, 0, 0, 0, 0, -1, 0, -4, 7], [0, 2, -1, 0, 1, -5, 0, 4, 0, 0], [8, 0, 0, -2, 0, 0, -3, 0, 2, 0], [0, -5, -2, 0, -2, 6, 0, 8, 0, 0], [0, -5, -1, 0, -12, 3, 0, 0, 0, 0], [0, -5, -2, 0, 2, -2, 0, 2, 0, 0], [20, 0, 0, 0, 0, 0, 7, 0, 3, -7], [-7, 0, 0, -3, 0, 0, -5, 0, -4, 8], [0, -10, -4, 0, 1, 2, 0, 2, 0, 0], [0, -1, 5, 0, 11, 0, 0, 0, 0, 0], [-3, 0, 0, -7, 0, 0, 4, 0, -7, 12], [0, -4, 8, 0, 6, 0, 0, -5, 0, 0], [-3, 0, 0, -1, 0, 0, -14, 0, -1, 6], [0, 2, -13, 0, -3, 10, 0, -2, 0, 0], [23, 0, 0, 8, 0, 0, 9, 0, -5, 0], [10, 0, 0, 8, 0, 0, -3, 0, -5, 0], [12, 0, 0, 8, 0, 0, 8, 0, 0, 3], [0, -5, -2, 0, 2, -4, 0, 4, 0, 0], [-11, 0, 0, -15, 0, 0, 1, 0, -2, 9], [-22, 0, 0, 7, 0, 0, 8, 0, 1, 11], [0, 4, 3, 0, -4, 1, 0, -3, 0, 0], [1, 0, 0, 1, 0, 0, -5, 0, -6, 6], [0, 15, -12, 0, -8, -3, 0, -2, 0, 0], [-8, 0, 0, 9, 0, 0, 7, 0, 1, -4], [0, -5, -3, 0, -1, -2, 0, 4, 0, 0], [0, -11, -5, 0, -2, 6, 0, 1, 0, 0], [-15, 0, 0, 10, 0, 0, 8, 0, 1, 7], [0, 1, -4, 0, -2, -3, 0, 0, 0, 0], [14, 0, 0, -18, 0, 0, 4, 0, -5, -4], [0, -4, -2, 0, -5, 0, 0, 1, 0, 0], [-19, 0, 0, -3, 0, 0, -5, 0, 0, -1], [0, 7, -2, 0, -2, 9, 0, -7, 0, 0], [-8, 0, 0, -4, 0, 0, -1, 0, -4, 10], [0, -8, 5, 0, 1, 21, 0, 2, 0, 0], [-1, 0, 0, 2, 0, 0, -20, 0, -5, 17], [0, -19, 7, 0, 0, 2, 0, -3, 0, 0], [24, 0, 0, 2, 0, 0, -4, 0, 1, -7], [0, 0, -5, 0, -4, 17, 0, -2, 0, 0], [28, 0, 0, 5, 0, 0, -3, 0, -6, 3], [0, -8, 3, 0, 10, -5, 0, 4, 0, 0], [7, 0, 0, -13, 0, 0, -1, 0, -2, 11], [0, 7, 12, 0, 4, -1, 0, -4, 0, 0], [-28, 0, 0, -10, 0, 0, -4, 0, 3, 1], [0, 3, 6, 0, 0, -3, 0, -9, 0, 0], [4, 0, 0, 14, 0, 0, -6, 0, -1, -1], [0, -16, 2, 0, -8, 16, 0, 0, 0, 0], [0, 13, -1, 0, -12, -1, 0, -10, 0, 0], [0, 3, 12, 0, -6, -7, 0, -11, 0, 0], [-14, 0, 0, 8, 0, 0, -6, 0, 4, 3], [0, -11, -2, 0, -13, 11, 0, -3, 0, 0], [0, -12, 4, 0, -11, 3, 0, -3, 0, 0], [0, -6, 9, 0, 9, 7, 0, 5, 0, 0], [-3, 0, 0, -12, 0, 0, 5, 0, -2, 12], [-5, 0, 0, -7, 0, 0, -7, 0, -6, 20], [-17, 0, 0, -15, 0, 0, -13, 0, -4, 5], [0, 10, 5, 0, -3, 1, 0, -4, 0, 0], [0, -18, 15, 0, 5, 8, 0, 2, 0, 0], [-19, 0, 0, 2, 0, 0, 1, 0, 3, 0], [0, 2, -4, 0, -3, -3, 0, 7, 0, 0], [6, 0, 0, -17, 0, 0, 6, 0, 3, -7], [8, 0, 0, 9, 0, 0, 4, 0, 2, -17], [0, -1, 11, 0, -5, -1, 0, -11, 0, 0], [24, 0, 0, 5, 0, 0, -5, 0, -2, -5], [0, 4, 15, 0, -4, 11, 0, -7, 0, 0], [20, 0, 0, -3, 0, 0, -10, 0, 1, -5], [-13, 0, 0, -3, 0, 0, -8, 0, -12, 4], [-14, 0, 0, 4, 0, 0, 8, 0, -7, 9], [0, 4, 3, 0, 7, -14, 0, -9, 0, 0], [-6, 0, 0, -12, 0, 0, -11, 0, -16, 9], [0, -7, 9, 0, -1, 0, 0, -1, 0, 0], [0, -9, -2, 0, 1, 0, 0, -5, 0, 0], [0, 11, -5, 0, -3, 0, 0, -10, 0, 0], [-8, 0, 0, -4, 0, 0, 11, 0, -7, -6], [23, 0, 0, 4, 0, 0, -6, 0, -7, 8], [-8, 0, 0, -11, 0, 0, 7, 0, 0, 0], [0, -5, -8, 0, -22, 14, 0, -1, 0, 0], [15, 0, 0, 12, 0, 0, 2, 0, -1, -15], [0, 11, 4, 0, 7, -9, 0, -2, 0, 0], [7, 0, 0, -3, 0, 0, 8, 0, -5, 2], [0, -5, 1, 0, -3, 16, 0, 3, 0, 0], [0, 15, 3, 0, -6, 5, 0, 0, 0, 0], [0, 3, 5, 0, -1, 6, 0, 2, 0, 0], [0, 0, 0, -8, 0, 0, 18, 0, 4, -14], [-16, 0, 0, -7, 0, 0, 14, 0, -2, -9], [0, 7, 5, 0, 2, -7, 0, -3, 0, 0], [0, 1, 6, 0, -14, 7, 0, -5, 0, 0], [0, 16, -10, 0, 3, 1, 0, 10, 0, 0], [24, 0, 0, 0, 0, 0, 7, 0, 13, -15], [-17, 0, 0, -5, 0, 0, 7, 0, 0, 13], [-19, 0, 0, 17, 0, 0, -3, 0, 3, -2], [0, 6, 10, 0, 2, -7, 0, 0, 0, 0], [0, 2, 6, 0, 2, 0, 0, -4, 0, 0], [-18, 0, 0, -9, 0, 0, -4, 0, -2, 14], [0, -10, 11, 0, 7, -1, 0, -4, 0, 0], [0, 16, -16, 0, -9, -9, 0, 6, 0, 0], [-15, 0, 0, -16, 0, 0, -5, 0, -6, 2], [11, 0, 0, -1, 0, 0, -5, 0, -4, 9], [0, -3, 17, 0, 6, -3, 0, 6, 0, 0], [9, 0, 0, 7, 0, 0, 1, 0, 2, -15], [0, 13, -8, 0, -5, 3, 0, -13, 0, 0], [0, -3, 5, 0, 15, -12, 0, -2, 0, 0], [-13, 0, 0, -5, 0, 0, -5, 0, 11, 4], [-1, 0, 0, 3, 0, 0, -11, 0, 4, -1], [0, 16, -13, 0, 8, 2, 0, 8, 0, 0], [0, -8, 8, 0, 19, -9, 0, 11, 0, 0], [0, 1, -1, 0, 13, -9, 0, 10, 0, 0], [0, 0, 0, -3, 0, 0, -1, 0, 13, 4], [17, 0, 0, 8, 0, 0, 2, 0, 1, 7], [0, 17, -11, 0, -19, -5, 0, -7, 0, 0], [9, 0, 0, 1, 0, 0, -12, 0, 11, 5], [0, 8, -10, 0, -11, 1, 0, -13, 0, 0], [-19, 0, 0, -4, 0, 0, -2, 0, 2, 13], [0, -6, 9, 0, 13, -15, 0, 14, 0, 0], [0, 1, 0, 0, 13, -16, 0, 0, 0, 0], [16, 0, 0, 2, 0, 0, 11, 0, -1, -11], [0, -3, 8, 0, -2, 11, 0, 7, 0, 0], [-27, 0, 0, -7, 0, 0, -19, 0, 4, 4], [19, 0, 0, 3, 0, 0, 6, 0, -8, 3], [17, 0, 0, 0, 0, 0, -11, 0, 15, -6], [0, 3, 7, 0, -8, -6, 0, -12, 0, 0], [-1, 0, 0, -5, 0, 0, 25, 0, 2, -18], [0, 7, 3, 0, -2, -22, 0, -1, 0, 0], [36, 0, 0, 13, 0, 0, 13, 0, 7, -15], [0, -1, 16, 0, 0, -2, 0, 7, 0, 0], [0, 8, -16, 0, -10, -6, 0, 4, 0, 0], [0, -6, 14, 0, -11, -2, 0, -13, 0, 0], [0, -10, -1, 0, -10, 21, 0, -1, 0, 0], [-3, 0, 0, -8, 0, 0, 9, 0, 4, 4], [-9, 0, 0, 0, 0, 0, 11, 0, -11, 11], [-5, 0, 0, -13, 0, 0, 2, 0, -23, 5], [0, -12, 20, 0, 5, 7, 0, 1, 0, 0], [0, -2, 0, 0, -2, 8, 0, -17, 0, 0], [-1, 0, 0, -6, 0, 0, 0, 0, -9, 8], [0, 14, -13, 0, 10, -5, 0, 10, 0, 0], [22, 0, 0, -2, 0, 0, -11, 0, 15, -5], [0, -3, 17, 0, 8, 2, 0, -2, 0, 0], [27, 0, 0, -3, 0, 0, -1, 0, 11, -12], [0, -29, 15, 0, 1, 11, 0, -5, 0, 0], [0, 1, -4, 0, 13, -21, 0, -1, 0, 0], [9, 0, 0, 0, 0, 0, -8, 0, -5, 8], [0, -14, 13, 0, -16, 5, 0, -11, 0, 0], [0, -10, 4, 0, 2, -1, 0, 8, 0, 0], [-9, 0, 0, -5, 0, 0, 1, 0, 8, 4], [14, 0, 0, 10, 0, 0, -14, 0, 2, 8], [0, 13, 3, 0, -7, -6, 0, -1, 0, 0], [-17, 0, 0, -13, 0, 0, 3, 0, -5, 20], [-7, 0, 0, -7, 0, 0, -13, 0, 6, 8], [0, -1, -20, 0, -7, 11, 0, 3, 0, 0], [19, 0, 0, -20, 0, 0, 9, 0, -8, 9], [7, 0, 0, -13, 0, 0, 11, 0, -3, -1], [0, -6, 2, 0, 7, -6, 0, 11, 0, 0], [22, 0, 0, -11, 0, 0, -3, 0, -4, -8], [0, 15, -21, 0, -3, 0, 0, 12, 0, 0], [0, -4, 0, 0, 11, -19, 0, 11, 0, 0], [16, 0, 0, 0, 0, 0, 13, 0, 4, 0], [0, 21, -6, 0, -8, 7, 0, -4, 0, 0], [-10, 0, 0, -19, 0, 0, -1, 0, -9, -7], [35, 0, 0, 15, 0, 0, -10, 0, 2, 6], [-24, 0, 0, -11, 0, 0, 1, 0, -3, 11], [18, 0, 0, 10, 0, 0, -6, 0, 2, -9], [1, 0, 0, 9, 0, 0, -4, 0, -9, -14], [-6, 0, 0, -5, 0, 0, 4, 0, -18, 8], [0, 10, 5, 0, -13, 15, 0, -8, 0, 0], [-15, 0, 0, -4, 0, 0, -27, 0, 8, 3], [-1, 0, 0, 14, 0, 0, 2, 0, 20, -10], [0, 12, -13, 0, 0, -7, 0, -4, 0, 0], [0, 3, -16, 0, -4, -1, 0, 3, 0, 0], [10, 0, 0, 4, 0, 0, -14, 0, 5, 17], [0, 14, -4, 0, 9, 1, 0, 0, 0, 0], [-23, 0, 0, -4, 0, 0, 13, 0, 13, 0], [45, 0, 0, -6, 0, 0, 8, 0, -5, 10], [0, -15, -9, 0, 7, 14, 0, 12, 0, 0], [6, 0, 0, -6, 0, 0, 8, 0, 2, 3], [0, 12, -23, 0, 13, -6, 0, 7, 0, 0], [29, 0, 0, 16, 0, 0, 13, 0, 14, -8], [0, -2, -6, 0, 10, -8, 0, 9, 0, 0], [0, 6, 11, 0, 11, -25, 0, 7, 0, 0], [0, 10, 14, 0, -6, 12, 0, -6, 0, 0], [0, 2, -7, 0, -6, 5, 0, 11, 0, 0], [0, 5, -3, 0, -4, 18, 0, 0, 0, 0], [-20, 0, 0, -16, 0, 0, -10, 0, -3, 5], [0, 3, 5, 0, 7, -9, 0, -13, 0, 0], [9, 0, 0, -18, 0, 0, -1, 0, -16, 3], [0, 31, -16, 0, -10, -14, 0, 2, 0, 0], [0, 2, -8, 0, -9, -2, 0, 1, 0, 0], [24, 0, 0, 16, 0, 0, 12, 0, 23, -5], [-29, 0, 0, 10, 0, 0, -2, 0, -1, -2], [0, -19, 1, 0, 9, 0, 0, 9, 0, 0], [-5, 0, 0, -3, 0, 0, 6, 0, -11, 6], [0, 27, -11, 0, 1, -12, 0, -4, 0, 0], [-19, 0, 0, 15, 0, 0, -2, 0, 1, -4], [-26, 0, 0, 0, 0, 0, 1, 0, -20, 10], [0, 6, 20, 0, -7, -8, 0, -13, 0, 0], [24, 0, 0, 19, 0, 0, 7, 0, 14, -4], [0, -3, -5, 0, 17, 8, 0, 18, 0, 0], [-5, 0, 0, 17, 0, 0, 8, 0, 9, 4], [0, -21, 22, 0, 10, 6, 0, -3, 0, 0], [0, 7, 1, 0, 2, 6, 0, 7, 0, 0], [0, 4, -1, 0, -15, 11, 0, 3, 0, 0], [-28, 0, 0, 1, 0, 0, -1, 0, -11, 5], [0, 28, -12, 0, 0, -2, 0, 1, 0, 0], [2, 0, 0, 3, 0, 0, 1, 0, 6, -20], [0, 3, -1, 0, -2, 12, 0, 15, 0, 0], [0, -8, -5, 0, 11, -5, 0, 7, 0, 0], [9, 0, 0, -11, 0, 0, -24, 0, -19, 13], [0, 9, -7, 0, 8, 11, 0, -5, 0, 0], [-2, 0, 0, 9, 0, 0, -4, 0, -19, 13], [29, 0, 0, 12, 0, 0, 24, 0, -5, 1], [0, 10, -3, 0, -24, 13, 0, -18, 0, 0], [2, 0, 0, -10, 0, 0, -8, 0, -14, -4], [0, -4, -4, 0, 9, -6, 0, 13, 0, 0], [25, 0, 0, -5, 0, 0, -3, 0, 1, -10], [15, 0, 0, 2, 0, 0, -7, 0, 4, -17], [-2, 0, 0, -16, 0, 0, -9, 0, -6, 26], [0, 25, -18, 0, -15, -4, 0, 6, 0, 0], [0, 7, -18, 0, -2, 12, 0, -1, 0, 0], [-29, 0, 0, 7, 0, 0, -4, 0, -6, 1], [0, 24, 0, 0, 3, -22, 0, 7, 0, 0], [-26, 0, 0, 0, 0, 0, 25, 0, -3, 1], [0, 20, -20, 0, -9, -4, 0, -2, 0, 0], [30, 0, 0, 1, 0, 0, -8, 0, 13, -2], [0, 2, 16, 0, -10, -6, 0, -13, 0, 0], [-9, 0, 0, -2, 0, 0, 5, 0, -22, 10], [0, 5, -23, 0, -17, 6, 0, 7, 0, 0], [-36, 0, 0, -22, 0, 0, 0, 0, -10, -6], [2, 0, 0, 16, 0, 0, -8, 0, 14, -3], [-14, 0, 0, -22, 0, 0, -4, 0, -15, 4], [0, -6, 18, 0, 2, 7, 0, -18, 0, 0], [0, 17, 19, 0, 10, -22, 0, 5, 0, 0], [13, 0, 0, 14, 0, 0, -15, 0, 12, 5], [0, -2, 17, 0, 11, -7, 0, 6, 0, 0], [0, 13, -1, 0, -2, 3, 0, 0, 0, 0], [4, 0, 0, -5, 0, 0, -2, 0, -2, 8], [14, 0, 0, 2, 0, 0, 2, 0, 30, -7], [0, 3, -2, 0, -16, -8, 0, -9, 0, 0], [0, 3, 10, 0, 7, -3, 0, -20, 0, 0], [40, 0, 0, 8, 0, 0, 1, 0, -6, -3], [15, 0, 0, 13, 0, 0, -2, 0, 3, -14], [0, 7, -25, 0, 10, -17, 0, 5, 0, 0], [41, 0, 0, -1, 0, 0, -7, 0, -2, 1], [-41, 0, 0, 11, 0, 0, -6, 0, 7, 0], [0, 18, -5, 0, -7, -9, 0, -5, 0, 0], [0, 31, -2, 0, -11, 2, 0, -7, 0, 0], [0, -20, 25, 0, 9, 5, 0, -10, 0, 0], [0, -1, -8, 0, 13, 1, 0, 15, 0, 0], [0, 16, -6, 0, -5, 2, 0, -11, 0, 0], [-6, 0, 0, -35, 0, 0, 12, 0, -12, 0], [0, 27, -26, 0, -3, -13, 0, 4, 0, 0], [0, -8, 17, 0, 3, 14, 0, -5, 0, 0], [0, -11, 11, 0, -1, 10, 0, 10, 0, 0], [13, 0, 0, -6, 0, 0, -6, 0, -20, 17], [20, 0, 0, 18, 0, 0, 12, 0, 14, 0], [0, 1, 1, 0, -16, 3, 0, -9, 0, 0], [7, 0, 0, 12, 0, 0, 7, 0, 12, 9], [1, 0, 0, 0, 0, 0, -20, 0, -4, 0], [0, -30, 26, 0, 2, 17, 0, -6, 0, 0], [31, 0, 0, 12, 0, 0, -5, 0, 15, 18], [0, -23, -4, 0, 0, 7, 0, 1, 0, 0], [0, -12, -9, 0, -17, 16, 0, -14, 0, 0], [21, 0, 0, 13, 0, 0, -8, 0, 1, 14], [3, 0, 0, 10, 0, 0, 1, 0, 6, -11], [0, 6, -1, 0, -7, -8, 0, 4, 0, 0], [0, -2, 4, 0, 32, -10, 0, 18, 0, 0], [2, 0, 0, 20, 0, 0, -5, 0, -6, 6], [0, 20, -4, 0, -3, -24, 0, -2, 0, 0], [25, 0, 0, 8, 0, 0, -22, 0, -7, 15], [0, 1, -14, 0, -15, -6, 0, 9, 0, 0], [0, -4, -30, 0, -13, 15, 0, 7, 0, 0], [22, 0, 0, -18, 0, 0, 8, 0, -7, -8], [0, 22, -2, 0, 6, -13, 0, -6, 0, 0], [20, 0, 0, -13, 0, 0, 9, 0, -4, -10], [0, -25, -1, 0, -18, 13, 0, -7, 0, 0], [19, 0, 0, -5, 0, 0, -8, 0, -13, 0], [-39, 0, 0, 7, 0, 0, 4, 0, 10, 6], [9, 0, 0, 11, 0, 0, 18, 0, -8, -3], [0, 17, 12, 0, -5, -1, 0, -10, 0, 0], [1, 0, 0, -6, 0, 0, -12, 0, 0, 33], [0, 18, -12, 0, 16, -4, 0, 24, 0, 0], [0, 3, 5, 0, -5, 20, 0, 10, 0, 0], [-18, 0, 0, -14, 0, 0, 2, 0, -24, -2], [0, -19, 0, 0, 24, -3, 0, 12, 0, 0], [-18, 0, 0, -26, 0, 0, 19, 0, -19, -6], [-5, 0, 0, -27, 0, 0, 13, 0, -4, 7], [0, -3, 19, 0, -10, 16, 0, -10, 0, 0], [-26, 0, 0, -3, 0, 0, 15, 0, -8, -14], [0, -25, 20, 0, -10, 19, 0, 4, 0, 0], [26, 0, 0, 1, 0, 0, -14, 0, 12, 16], [0, -23, 2, 0, 10, 4, 0, 11, 0, 0], [-9, 0, 0, 34, 0, 0, -14, 0, 4, 8], [0, 5, -10, 0, -4, 5, 0, 21, 0, 0], [0, -17, 14, 0, 7, 10, 0, -1, 0, 0], [15, 0, 0, 0, 0, 0, -7, 0, 8, -11], [-11, 0, 0, 16, 0, 0, 12, 0, 14, 10], [0, 12, -5, 0, -13, -3, 0, -3, 0, 0], [-29, 0, 0, 1, 0, 0, 31, 0, 12, -17], [0, 8, -5, 0, -20, 15, 0, -16, 0, 0], [-13, 0, 0, -11, 0, 0, -11, 0, 19, -2], [0, 5, -30, 0, 3, 5, 0, 10, 0, 0], [0, 5, 12, 0, 8, -7, 0, -2, 0, 0], [-52, 0, 0, -9, 0, 0, 3, 0, 2, -10], [-50, 0, 0, -8, 0, 0, -15, 0, -8, 29], [0, -20, 7, 0, -2, 14, 0, -9, 0, 0], [0, 8, 10, 0, -3, 6, 0, -1, 0, 0], [3, 0, 0, 22, 0, 0, 2, 0, 16, -1], [0, 5, -34, 0, 15, 2, 0, 23, 0, 0], [0, -31, 27, 0, -2, 11, 0, 4, 0, 0], [0, 0, 14, 0, 8, -8, 0, -20, 0, 0], [-11, 0, 0, 3, 0, 0, -5, 0, -11, -7], [0, 2, 27, 0, 5, 1, 0, -20, 0, 0], [0, 8, 13, 0, -7, 11, 0, -11, 0, 0], [24, 0, 0, -11, 0, 0, -3, 0, 1, 9], [0, 4, -13, 0, 0, -13, 0, -1, 0, 0], [0, 19, -12, 0, -14, -4, 0, -3, 0, 0], [0, 1, 0, 0, -6, 8, 0, 0, 0, 0], [-28, 0, 0, 16, 0, 0, 1, 0, 22, -10], [0, -26, -7, 0, 16, 0, 0, 12, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6021_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6021_2_a_l();".
// 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_6021_2_a_l(:prec:=10)
    chi := MakeCharacter_6021_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_6021_2_a_l();".
// 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_6021_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6021_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-63, 0, 331, 0, -384, 0, 139, 0, -20, 0, 1]>,<5,R![-63, 0, 4078, 0, -2404, 0, 460, 0, -36, 0, 1]>],Snew);
    return Vf;
end function;