
// Make newform 6032.2.a.z in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6032_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_6032_2_a_z();".
// 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_6032_2_a_z();".
// 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 := [-8, 12, 330, 364, -283, -235, 104, 47, -17, -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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-524, 5022, 12100, -1999, -6935, 828, 1243, -185, -71, 13], [-1132, -6898, -5804, 4997, 5005, -1808, -1257, 359, 97, -27], [1040, -5242, -6374, 6317, 4459, -2966, -1017, 539, 73, -33], [56, -2290, -5882, 5825, 4459, -2966, -1017, 539, 73, -33], [-2024, 1798, 6374, -5825, -4459, 2966, 1017, -539, -73, 33], [136, 8402, 12453, -2031, -6678, -239, 1103, 79, -57, -4], [-1612, 14066, 17272, -13795, -9599, 5200, 1767, -817, -107, 45], [1684, 2330, -3210, -11955, -945, 6262, 551, -1073, -51, 59]];
    Rf_basisdens := [1, 1, 492, 492, 492, 492, 492, 246, 492, 492];
    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_6032_a();"
function MakeCharacter_6032_a()
    N := 6032;
    order := 1;
    char_gens := [5279, 1509, 1393, 1249];
    v := [1, 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_6032_a_Hecke(Kf)
    return MakeCharacter_6032_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, 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], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-1, 0, 0, 0, 0, -1, 0, 1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, -1, 0, 0, -1, -1], [-1, 0, 0, 0, -1, 0, 1, 0, 0, 1], [0, 0, 1, 1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 1, 0, 0, 1, 0, -1, 1], [2, -1, 1, -1, -2, -1, 1, 0, 0, 0], [0, 1, -1, 0, 1, 2, -2, -1, 2, 1], [-3, 0, 1, 0, 0, 0, 1, 0, -1, -1], [-2, 1, 0, -1, 1, 3, 0, -1, 0, 1], [1, -1, 2, 0, -1, -2, 0, 0, 0, -1], [-2, 0, 0, 1, -1, -1, 0, -1, 0, 1], [-2, 1, -1, 0, 1, 0, 0, -1, 0, 1], [-3, 1, 0, -1, 3, 2, 0, 0, 3, 1], [1, 0, 1, 0, -2, 0, 3, 0, -1, -1], [2, 1, 2, 1, 1, -2, -1, -1, -1, -3], [-1, 2, -1, 0, 0, -1, 2, 1, 1, 2], [-5, 1, 1, -1, 0, 2, 1, 0, 0, -1], [4, -1, 2, 1, 0, -1, 1, 0, 0, 0], [0, -1, 0, -2, 0, -2, -1, 1, -1, -1], [1, -2, 0, 0, -1, 0, -2, 0, 1, -3], [-2, 0, 2, -1, 0, -1, -1, 0, 1, -2], [-2, 1, -3, -2, 1, 2, -1, -2, 2, 0], [2, -2, 2, -1, 2, 0, 1, -1, 0, -1], [0, -2, -1, 0, -2, 1, -1, -2, 1, 0], [0, -1, 2, -3, 1, -1, -1, 0, -1, -2], [-2, 1, -2, 1, -2, -3, -1, 2, 1, 0], [2, 3, 1, -1, 0, 4, 1, -1, 1, 1], [-6, 4, 1, -3, 2, 3, -1, 0, 1, 0], [0, -1, -2, 1, 3, 0, 0, 0, -1, -2], [-1, 3, -4, 3, -1, -1, 1, 3, -1, 2], [0, -3, 0, -2, 2, 0, -3, -3, 1, -3], [-5, 0, -1, 2, 2, 0, -2, -1, 0, -2], [-3, 0, 1, 0, -2, -1, 0, 3, -1, 2], [1, 1, 1, -1, 2, 1, 0, 0, 0, 3], [-7, 1, -2, -3, 1, 2, 1, 1, 2, 4], [0, 2, -3, 1, 2, 4, 4, 2, 2, 2], [-1, 2, 0, 3, 1, 2, 0, -1, 1, 0], [0, -1, -3, 4, -3, -2, -1, 2, 0, 0], [0, -4, 3, 0, -1, -1, -1, -2, -1, -2], [-7, 5, 0, 0, 2, 1, 2, 2, 0, 1], [-4, 1, -3, 2, -1, -2, -1, 0, -1, 0], [4, -4, -2, 0, -1, 5, 0, -4, 3, 0], [-3, -3, -1, -1, 0, 4, -3, -4, 0, -3], [1, -2, 0, -1, -2, 3, 2, 0, 1, -1], [-2, -2, -2, 1, -2, 2, 3, 0, -1, 2], [-5, 1, -1, 2, 2, 0, -3, 0, -2, -3], [1, 5, 4, -1, 2, 0, -3, 0, -1, -3], [-3, 1, -3, 4, 0, 3, 2, 2, -1, -1], [-2, 0, 0, -2, 2, -1, -1, 0, -1, 0], [-4, 8, -1, -3, 2, 5, 1, 0, 2, 6], [6, -3, 2, 1, 2, -3, 3, 0, 0, 0], [-2, 3, 0, -1, 4, 2, 2, 2, -1, -2], [-1, -3, -3, 2, -2, 0, -3, 2, 3, -1], [1, 5, 2, -5, 1, 3, 5, 2, 1, 3], [5, -8, 1, 2, -3, -3, 2, 1, -4, -2], [-4, 1, -2, 0, -1, 0, -3, -1, -1, 1], [-3, -4, 5, -3, 0, -1, -2, -3, -2, -6], [-7, 4, 2, 0, 4, 6, -4, -1, 1, 2], [-6, -2, -3, -1, 0, -3, -1, 0, 1, 2], [2, -4, -3, 2, -2, -3, 1, 1, 0, 5], [-2, -1, 1, -4, 4, 3, 3, -2, 2, 0], [-10, -3, 0, 3, -1, 1, 2, 1, -2, 1], [-8, -4, 3, 2, -2, -4, -5, -4, -3, -2], [-4, -5, -3, -2, 4, 2, -2, -1, 2, 1], [1, 1, 3, -3, 3, 2, 0, 1, 3, 0], [-4, 0, -4, 3, 1, -2, -7, 0, 1, -4], [5, -4, 2, 4, -3, -5, 3, 1, -2, -4], [-10, -6, 2, 0, 0, 0, 0, 0, -4, -2], [3, -3, 4, 3, -3, -3, 1, 2, -3, -1], [4, -3, 2, -3, -2, -5, 5, -1, 0, 1], [5, -4, -1, -3, 3, 4, -4, -3, 4, 0], [0, -5, 5, 4, -6, -7, -2, 1, -4, -7], [4, 2, 3, 0, -5, -4, 1, 1, -2, 5], [-11, 7, -6, -1, 2, 4, 3, 4, 3, 5], [-7, 1, -3, 2, 4, 1, 0, 0, -2, -3], [1, -2, 5, -4, 2, 7, 2, -4, 3, -1], [3, -5, 0, -1, -3, -7, 6, 4, -5, -3], [2, -3, -3, -2, 1, -5, -2, 1, 0, 1], [8, -6, 0, 1, 2, -3, -1, 1, 0, 3], [-10, 1, -4, 4, 4, 6, -4, 0, 0, 2], [15, -1, 3, 5, -1, -4, 4, 4, -2, -1], [-8, 0, -1, -3, 2, 0, 0, -4, 0, 2], [7, 2, -3, -2, 4, 3, -4, -1, 4, -4], [-3, -8, -1, -4, 1, 4, -1, 0, 5, 1], [5, 3, -1, 3, -1, -5, 1, 1, -1, 4], [-4, -3, 2, 0, 0, -3, 7, 0, 0, 2], [4, 1, 3, 4, -4, -4, 2, 2, -2, -4], [-9, -6, -2, 0, 4, 5, -4, -3, 1, 0], [-2, 2, -1, -3, 2, 3, -1, 3, 1, -3], [-2, 5, -3, -5, -1, 4, 2, 0, 2, 0], [-2, 3, 1, -7, 6, 6, -3, -1, 5, 1], [6, 0, 3, 0, -6, 0, 3, 2, -3, -4], [0, -1, -3, 1, -4, 2, 1, 0, 2, 0], [-5, 12, -5, 4, 2, 7, 0, 1, -2, 4], [-1, 5, 1, 3, 0, 0, 0, -4, 2, 3], [3, -4, 4, 0, -2, -4, -3, 4, -2, 1], [-2, 1, -6, 0, -1, 3, 2, 1, 1, 1], [-1, -2, 2, 7, -7, -6, 2, 4, -5, -3], [3, 5, 1, 6, -5, -2, 10, 3, -7, 0], [3, 5, -3, 4, -2, -2, 5, 3, -4, 2], [-2, 6, -1, -7, 7, 4, -1, -3, 5, 7], [3, 5, 7, 3, 0, -1, 2, 1, -1, 2], [2, 5, 4, -6, -2, 1, 2, 2, 0, 2], [-12, 5, -5, 8, -4, 4, -2, 3, -4, 1], [1, -4, -4, 0, 0, 4, -8, 0, 0, -3], [-13, 4, 2, 0, 4, 3, 6, 1, -2, -2], [2, 3, 2, 1, -1, 4, 5, -2, -4, 0], [-8, 2, -1, 1, 0, 0, -4, -2, 2, -2], [16, -4, 1, -2, 0, 1, -1, -4, 0, -4], [11, 3, -2, 0, 3, -4, 0, 1, 1, -2], [-2, -4, 1, 4, 3, 1, -4, -1, 4, 1], [11, -5, 7, 0, -4, -6, 5, 3, -6, -2], [0, 3, 0, -7, 1, 4, -1, -3, 5, 5], [6, 4, 6, 0, 1, -4, 5, 0, -4, 0], [0, -5, 5, -1, -6, -8, 2, -2, 1, 0], [-10, -2, 3, 0, -3, -3, -5, 3, 2, 1], [-7, 3, 0, 1, 0, -2, 3, 4, 3, 9], [-9, -5, 0, -4, 1, -1, -7, -2, 0, -3], [6, 0, 0, 4, 0, 5, 5, -2, 0, 2], [1, 1, 3, 1, -3, -4, -8, -1, -3, 2], [-5, 4, -1, -1, 4, 1, -6, -2, 1, 1], [0, 0, 5, 1, -10, -12, 2, 2, -4, 2], [9, 8, 4, -1, 4, -1, -6, -2, 2, -1], [-1, -6, 2, 2, 0, 0, -1, 1, 3, 0], [-11, 2, 6, 3, -4, -4, -2, 0, 1, -5], [14, -1, 3, -2, 3, -7, 2, 3, -4, -3], [15, -7, -3, 0, -4, 1, 2, 3, 0, 4], [-17, 2, 4, -3, 3, 3, -3, -5, -1, -8], [4, 2, -2, 5, 2, -3, 1, -2, -3, 0], [1, -9, 0, 1, -4, -8, 3, 1, -4, 0], [16, 5, 2, 0, -2, -1, -1, -4, -4, -6], [5, 3, 1, -6, 4, 5, -2, -1, 4, 2], [13, -2, -5, -3, 3, 0, -3, 1, 1, 0], [-5, 10, -5, -7, 5, 9, -1, 1, 1, 4], [-19, 5, -2, 2, 2, 2, -3, 3, 1, 2], [-2, 8, 0, -2, -2, -7, 1, 4, -5, 2], [-7, 10, -9, 1, 2, 8, -1, 3, 4, 10], [-14, -4, -3, -2, 1, -4, -3, 0, -1, -2], [4, -4, 6, -3, 3, 10, 3, -5, 6, -5], [2, 8, 6, 2, -1, -5, 2, 0, -4, 2], [11, -4, 1, -2, -1, 0, -8, -4, -1, -5], [-5, -4, 5, 4, 1, -7, 1, -2, -8, -5], [-3, 5, 2, -5, 5, 4, 4, 1, -2, -4], [0, -8, -7, 2, 1, 0, 2, 2, 1, 2], [-5, 1, 1, 5, 2, -6, 5, 4, -1, -3], [-1, -6, -3, -5, -1, 3, -7, -2, 8, -1], [-2, -4, 8, -5, -2, 2, -8, -1, 1, -3], [13, 2, -1, -1, 2, 6, -6, -4, 2, -3], [1, 0, 3, -2, 8, 4, -1, -1, 2, 0], [0, 2, -5, 1, 3, 0, -3, -5, 3, 3], [-15, -1, -8, 3, -1, -1, 3, 1, -2, 2], [7, -5, -3, 7, -6, -7, 0, 3, -7, 0], [-10, 11, -5, -5, 2, 4, -4, -2, 7, 4], [0, -1, -3, 3, 3, 2, -11, 0, 8, -4], [-14, -1, -6, -1, 6, 5, -4, -3, 6, 1], [12, 0, 9, -4, 8, 6, -6, -4, 6, 0], [-3, 16, -8, -3, 2, 3, -6, 2, 5, 3], [-2, -5, 2, 0, -4, -4, 0, -3, 5, 1], [13, -5, -5, 10, -8, -4, 1, 2, -7, -1], [-2, -1, 9, -1, 8, 3, -1, -7, 3, -5], [5, -5, 2, 1, -3, -13, -1, 2, -7, -3], [-19, 12, -1, -3, 14, 15, -2, 0, 5, 5], [8, -6, 3, -5, -3, -3, 2, -1, -1, -1], [-14, 7, -3, 3, 3, -1, -5, 7, -5, 1], [6, -2, 4, 1, -8, -2, 14, 1, -6, 7], [-6, 10, 1, 3, 0, 3, 2, 2, 0, 12], [-9, 1, -4, 1, 4, -4, -3, 1, -2, 0], [-1, 4, 7, 3, 2, -1, 10, 3, -8, -2], [13, -7, 1, 2, -1, -11, -5, 0, -1, -5], [9, -3, -3, 5, 2, 0, -2, 5, -1, 0], [-4, -6, -5, 1, -3, 2, 3, 2, 0, -4], [0, 6, 0, 0, 0, 8, 0, 1, 3, 3], [-4, 1, -2, -1, -3, -1, 0, 1, 1, 5], [-16, 4, -7, 0, 6, 9, -3, 2, 7, 6], [-12, -2, -7, 1, -4, 0, 8, 4, -2, -4], [-4, 4, -12, -2, 3, 8, -5, -5, 5, 3], [10, 0, 7, -6, -1, 2, 9, -1, 4, 5], [-5, 7, 2, -4, 2, 13, 8, -3, 1, 6], [-1, 4, 6, 1, -1, 0, 6, -1, -1, 4], [13, 3, 2, -4, -5, -10, 8, 3, -1, 0], [-5, 10, 5, -1, -2, -7, 9, 4, -6, 3], [-6, -7, -7, -1, -2, 2, 0, -6, 1, 0], [1, -16, -1, -4, -5, -3, -2, -2, 5, 1], [-7, 8, -6, -3, 8, 9, -1, 4, 3, 5], [-2, 0, -1, 9, -6, 1, -4, 4, -4, -4], [-2, -15, -1, -9, 3, -3, -10, -4, 5, -2], [-10, 9, -8, 4, -7, 7, 2, 5, -1, 5], [22, -4, 1, -3, 3, -3, -6, 3, 3, -7], [-22, 15, -6, -3, 7, 8, -3, 1, 0, 1], [-11, 8, -9, -4, 3, -1, 3, 2, 2, 3], [0, -1, -2, 5, -4, -7, 0, 0, -5, 2], [21, 7, -1, 1, 0, -7, -4, 1, 3, -6], [5, -5, 0, -3, 2, -10, -3, -3, -2, 0], [0, 0, -4, 3, 3, 0, -7, 7, 4, 1], [15, -10, 4, -5, -4, -11, 0, 0, 1, -7], [2, 8, -2, -6, 7, 1, 6, 4, 2, -2], [-9, 6, -10, 8, -7, -3, -1, 3, -7, 2], [7, -4, 3, 8, -4, 1, 4, -1, 0, 2], [-12, 0, -7, -2, 4, 9, 5, -2, -1, 2], [-6, -2, -3, -4, -4, 5, 3, -5, 4, -3], [10, -3, -7, 5, -1, -4, -3, 5, -5, 7], [-7, -1, -3, 9, 3, 0, 6, 5, -3, -2], [6, -4, 1, -2, -3, -6, 3, -1, -2, 1], [8, -13, 3, -2, 0, 5, -7, -10, 5, -6], [2, -2, 11, 2, -2, -3, 7, 6, -9, -2], [11, -9, 2, 2, -13, -6, 8, -1, -3, -6], [7, -11, 5, 0, -13, -6, 8, 3, -6, -4], [14, -3, 4, 3, -3, -2, 2, 6, -1, -4], [0, 5, -5, 0, 5, 13, 0, -2, 7, 0], [-4, 8, 0, -5, 10, 11, -3, -4, 9, 2], [8, -2, -5, 1, 6, 6, 0, -1, 1, 1], [-11, 8, 1, -7, 8, 0, -1, -1, -1, 0], [12, -15, -2, 4, 0, -3, 1, 1, -2, -3], [16, 5, -4, 1, 7, 9, 4, 2, 1, 0], [-2, 15, 9, 0, -2, -5, 2, -2, -1, -2], [16, 0, -2, 4, -8, -8, -2, 0, 5, 0], [-1, -2, 4, -4, 3, -2, -7, -6, 0, 3], [1, -14, -2, -6, 0, 2, -9, -7, 7, -4], [19, -3, 3, -1, 2, 0, -1, -4, -1, -5], [16, -7, 5, 2, -8, -7, -8, -3, -4, 1], [5, 0, -1, 3, -3, 5, 11, 3, -9, 0], [2, -9, -1, 1, 0, 1, 6, 0, 0, -8], [11, -3, 6, -7, 4, -1, -6, -8, 7, -5], [-15, -3, 1, -2, -7, -2, -6, 4, 1, -7], [-26, 7, -1, 0, 2, 5, 5, -1, -3, 5], [-13, 6, -8, -3, 7, 15, -4, 0, 5, 9], [13, -7, 3, 5, 6, 5, -10, -8, 6, -1], [15, -10, -7, 4, -9, -6, 6, 11, -4, 2], [1, -12, -9, 6, -5, -2, 8, 5, 0, 4], [17, 0, 11, 3, 5, -5, -1, -5, -1, -2], [20, 3, 9, 4, -9, -10, 5, 10, -6, 0], [31, -14, 4, 3, -4, -7, 4, -1, 0, -2], [-29, -2, -7, 2, 8, 10, -2, -1, 0, -2], [8, -5, -1, -4, -11, -3, -5, 3, 2, -1], [4, -11, 3, 0, 0, -11, -8, 2, 1, -10], [-2, 10, 4, 2, 5, 2, -5, 0, -5, 4], [-3, 3, -3, 6, 1, 7, 5, -1, 2, 0], [9, 10, -2, -2, 8, -4, 3, 6, 0, 3], [11, 9, 2, -3, 3, -7, 9, 4, -11, -1], [3, -19, 11, -7, 2, -9, -4, -8, 2, -11], [11, 10, -1, -6, 1, 1, 1, 1, 2, 4], [4, -14, 13, -2, -1, -7, 6, -7, -6, -7], [8, -16, 7, -5, 0, -11, -5, -5, -5, -13], [-7, -2, -5, 4, 0, -5, 1, 8, -3, 5], [13, -6, -2, 1, -3, 10, -2, -8, 3, 1], [-7, 5, 4, -8, 9, 4, 2, 2, 4, 3], [-21, -4, 3, 4, -6, -3, 0, -4, -5, 3], [7, -4, -3, -2, -6, -3, 2, 3, 6, 10], [-7, 16, -12, 1, 10, 2, -5, 3, 5, 2], [-3, 15, 0, -2, 2, -6, 11, 5, -1, 12], [3, 6, 6, 2, 0, -5, 3, -5, 9, 0], [10, 2, 11, 1, -1, 1, 8, 6, -4, 0], [-18, 10, 11, -2, 1, 0, -7, -2, 3, -4], [-29, 1, -5, -2, 3, 1, -11, 1, 0, 6], [-9, 4, -3, 5, -6, 0, 5, -3, -6, 2], [-12, 5, -4, -5, 1, 8, 9, -7, -2, 7], [7, 1, 1, 8, 2, -13, -2, 10, -4, -1], [-17, 11, -5, 5, -2, 3, 6, 1, -3, 2], [27, 4, -7, 0, -1, 5, 9, 6, 2, 5], [-2, -1, 9, 3, -6, 0, 6, 2, -2, 4], [36, -11, -5, 3, -6, -5, -3, 4, 2, 4], [-15, 14, -10, 0, 4, 15, 2, -3, 10, 12], [-5, 6, -6, 5, 4, -6, 1, 8, -2, 5], [3, 9, 4, -8, 5, 4, 4, 0, 2, 3], [-23, -1, -5, -1, -2, 1, -6, 5, -3, -6], [1, 5, -5, 2, -2, 7, -12, -5, 0, -2], [10, -2, -10, 8, 1, 9, 8, 2, 0, -2], [-7, -8, 12, 2, -2, 0, -6, -1, -7, -6], [13, 3, 13, -2, -5, -7, -3, -5, -2, 2], [6, -6, 9, -1, -2, 4, 6, -2, 2, 4], [-12, 1, -9, -1, 12, 4, -16, -5, 10, 1], [-6, -5, -4, 9, 1, -2, 7, 0, -2, 8], [-24, -3, -5, -9, 0, -4, -8, -1, 4, 5], [-8, 6, -1, 2, -5, 3, -7, 0, -1, 2], [-6, -12, 1, -1, 4, 2, -16, -6, 6, -8], [4, -6, 8, -2, -4, 4, -13, 0, 2, 0], [-3, -3, -5, 1, -8, -11, 6, 4, -2, 11], [9, 0, -6, -1, 3, -5, -4, -1, -4, -10], [-8, 9, -2, -9, 7, 8, 6, -3, 12, 7], [-33, 6, -10, 2, -9, 7, 1, 4, 3, 15], [-11, 0, 10, -8, 4, 10, 5, -4, 4, 3], [-7, -10, 2, 4, 1, -15, -9, 4, -4, -5], [3, -5, 1, -4, -4, -12, -2, -3, 4, 4], [24, -13, 6, 0, -6, -6, 0, 4, -1, -2], [-15, 3, -9, -3, 4, -3, -4, -5, 3, 10], [0, -16, 3, -6, -2, -4, -6, 4, -2, -6], [8, -18, -2, 2, -6, -8, 8, -3, 1, 1], [1, -8, 8, 8, -2, -13, -4, 3, -8, -12], [3, 2, 5, -4, 2, 4, -3, -4, -6, 3], [2, 12, -1, 15, -10, -6, 16, 10, -8, 2], [13, -2, 12, -6, 3, -11, 10, 1, -7, -8], [-16, -12, -11, 9, -2, -2, 2, -1, -3, 3], [6, -3, 1, 0, -4, 0, 2, -4, 5, 0], [-22, 5, -14, -4, -2, 0, 1, 2, -2, 8], [-2, 1, -3, 7, -9, -14, 5, 10, -10, 8], [2, 6, 3, 5, -2, 3, 18, 2, 0, 12], [-2, -10, 1, 0, 8, -10, -12, -7, 2, -5], [5, -9, -3, -4, -1, -6, -16, -4, 3, 1], [-17, 0, 4, 2, -5, -12, 6, 3, -5, -2], [9, -21, 1, 0, 3, 5, -13, -11, 6, -12], [11, -4, -5, 11, -10, -13, 12, 8, -14, 1], [5, -7, -2, 5, 2, -10, 3, 10, -3, -3], [-15, -4, 7, -6, -4, 9, -5, -5, 0, -6], [11, 7, -3, 2, 1, 5, -9, 4, 9, 1], [3, -7, 2, -9, 4, 6, 9, 6, 3, 7], [11, -4, -4, 6, -8, -6, -3, -5, -5, -4], [-4, 6, -5, -4, 6, 10, 4, -6, 10, 0], [-27, 12, -7, 0, -1, 3, 0, 6, 3, 9], [-7, 16, 3, 0, 2, 20, 5, -7, 6, -2], [-20, 8, 0, 11, -5, 7, 1, -2, -7, 4], [6, 9, -3, 11, -1, -1, -2, 1, -10, -3], [-19, 2, -1, 1, -6, 2, -9, 1, -2, 0], [-8, 0, -5, 8, -4, -13, 3, 5, -6, -3], [21, -3, 1, 1, 5, -2, 6, 8, 0, 1], [-27, -7, -7, 0, 8, 1, -8, -4, 5, 1], [-3, -1, 10, -5, -8, -4, -1, 1, 0, 0], [-5, 19, 0, -2, 1, 2, 8, 4, 0, 7], [14, 1, -10, 6, -6, -1, 1, 11, 3, 1], [-9, -15, -8, 1, -2, -4, 5, 3, -4, -8], [-16, 2, -1, 0, -1, 0, -5, 10, 5, 0], [-13, -5, -13, 12, -4, 1, -6, -1, 2, 0], [-21, -13, 4, -2, -6, -2, 1, -4, 0, -9], [17, 6, -1, -3, 1, 3, 5, -4, 13, 1], [-7, 17, 7, 7, -1, -6, 8, 8, -12, -1], [9, -13, -2, 7, -9, -3, 11, 2, -17, -3], [-8, -6, 7, -2, -18, -15, 9, 2, -9, -4], [-3, 11, 5, -6, 9, 9, -9, 0, 5, 7], [6, -1, 12, -8, 1, -13, -9, -3, -7, -3], [1, -2, 5, 0, 1, -11, -2, 3, -6, 0], [32, 4, 5, 13, -1, -2, 1, 0, -2, -6], [-1, 6, -9, 0, -1, 3, 0, 3, 6, 16], [-17, -2, -8, -7, 6, 8, 0, -5, 8, 8], [24, 5, 7, 3, -1, -17, -6, 4, 5, -6], [-23, 3, -1, 0, -10, -6, -3, 0, -1, 15], [-1, 6, 10, -3, -1, 6, -6, 2, -1, 1], [-9, -5, 10, -1, 0, 5, 4, -3, 3, -6], [3, -2, -5, 4, 7, 4, -6, 7, 0, -2], [-9, 11, 3, 15, -2, -1, 10, 9, -15, 0], [-4, 8, 9, 1, 0, 1, 7, 3, 0, 5], [12, -3, -13, -5, 10, -2, -8, -2, 2, -10], [-1, 0, 14, 1, 0, 4, 12, -4, -5, 3], [-22, 1, -11, -2, -2, 7, 11, 3, -1, 3], [-1, 4, 11, 5, -2, -10, 15, 10, -12, 5], [7, 11, -2, 1, 8, -10, 3, 5, 2, -2], [16, -14, -15, -4, -7, -11, -2, 6, 3, 4], [-11, -1, -15, 3, 7, 14, -2, 4, 6, 9], [-6, 11, 3, -11, 18, 13, -17, -9, 15, -5], [-14, -2, -20, 2, 5, 17, -4, -3, 5, 9], [-35, -1, -4, -6, 13, 4, 2, -3, 9, 4], [8, -2, 10, -9, 1, -8, -5, -2, 3, -4], [1, -10, -1, 0, -7, -14, -14, -2, 9, -1], [22, -4, -10, 11, 0, -6, 2, 4, -3, 8], [-18, -1, -8, -6, -4, 3, 3, -4, -6, 2], [3, 5, -6, 5, -3, 17, 13, -1, 1, 2], [-16, 7, 0, -3, -8, -7, 5, 4, -2, 10], [11, -2, -3, -2, 12, 12, 6, -2, 1, -5], [33, -8, 6, -3, -5, -6, -2, -6, -4, -9], [-16, 6, -12, 5, -7, 5, 0, -3, 2, 5], [22, 6, -9, -14, 7, 13, 0, -6, 13, 6], [-16, 3, -7, -11, -1, 4, -17, -8, 10, 8], [23, 8, 9, 3, 3, 13, 1, -7, 1, 0], [-8, 2, -13, 4, 4, 14, -8, -12, 7, -2], [-17, 6, 6, -5, -2, -3, 20, 1, -2, 12], [3, 4, 2, 2, -1, 6, 20, 6, -6, 3], [-5, 5, 10, 3, -9, 8, 19, -1, -2, 12], [-31, -11, 5, -18, 3, 1, -13, -4, 8, -1], [-16, -3, -12, 1, 0, -1, 8, 1, 2, 1], [-1, -3, 4, -1, -14, 10, 3, 4, -1, -3], [-20, 2, -3, -8, 2, 5, -15, -5, 4, -9], [4, -11, 5, -2, 2, 5, -1, -13, 7, -7], [-16, -5, 2, -12, -2, 13, 2, -5, 9, 3], [1, 12, 2, -16, 4, 15, 9, -3, 7, 10], [-9, 6, 2, 1, -2, 12, 21, 3, -7, 6], [-19, 2, -9, -8, -4, 6, -4, 2, 3, 7], [-24, 8, -5, 12, -2, 5, 7, 6, -5, 0], [-10, -12, 8, -6, -2, 4, 0, -10, -2, -6], [-26, 14, -5, -6, 11, 10, -9, 6, 1, 4], [-18, -1, -25, 4, 7, 4, -10, 3, 2, -5], [-4, -8, 18, 9, -8, -18, 2, 0, -11, -10], [16, -2, 3, 2, 0, -9, 12, -6, -7, 2], [9, 0, -7, -15, 5, 7, -1, -2, 16, -1], [-15, 19, -10, -3, 4, 10, -7, -11, 0, -6], [-5, -2, 19, 12, -9, -13, 6, 6, -13, -9], [-11, 14, -12, -9, 6, 17, -3, 2, 7, 11], [27, -1, -11, -5, -7, -12, -4, 5, -3, 0], [17, -7, 2, 5, -14, 5, 1, -1, -6, -4], [-20, 25, 4, 8, 1, 9, 13, 1, -7, 3], [-23, 2, 5, -9, 3, 4, -10, -6, 4, -3], [6, -7, -6, -3, 1, 13, 4, 1, 8, -5], [-36, 8, -8, 1, -2, 4, -4, 3, 7, 11], [17, 10, 4, -11, 5, 5, 13, 6, -5, 5], [-21, 6, -14, 13, -7, 8, 4, 3, -4, 10], [-15, -10, 2, -13, -13, -5, 2, -1, 4, 4], [-10, 13, -13, 0, -3, -9, -10, 5, -6, 7], [10, 20, 9, 5, 1, 2, -1, 4, -9, 2], [2, 12, -13, 9, 3, 7, 0, 3, 5, -3], [9, -2, 0, 7, -7, -4, 8, 8, -7, 5], [-18, 2, 7, -4, -3, 0, -9, -1, 8, -3], [26, -2, 3, -8, 7, 15, -5, -13, 8, -1], [8, -6, 8, 7, -15, -6, 3, 2, -11, -6], [-12, 18, -6, -4, 11, 10, -2, -11, 3, -1], [-2, -5, 1, -2, 14, -3, -13, -2, 10, -2], [-34, 1, -1, -12, 8, 9, 3, -15, -1, 1], [12, 2, -7, 14, -17, 3, 2, 1, -8, -3], [-6, -4, -12, 7, -2, 4, 1, -9, 8, 1], [-35, 17, -9, -4, 12, 22, 5, -4, 11, 15], [28, -6, 8, 2, -14, -9, -3, 0, -11, -10], [-25, 13, 7, -3, -6, -8, 16, 4, -2, 15], [-13, -16, -11, 5, 3, 1, -11, -14, 6, -5], [-7, -3, -3, 0, 2, 10, 15, 7, 2, 6], [10, -11, 4, 4, -14, 3, -5, -4, -8, -8], [-3, -7, -13, 6, -15, 1, 11, 2, -5, 1], [4, -11, 8, 3, -9, -8, 4, 10, -9, -8], [29, -12, 2, 9, -8, 3, -3, -8, -3, -3], [-2, 4, -10, -9, -7, -4, -5, -2, -1, 2], [-19, -9, 7, 15, -8, -21, 6, 4, -12, -1], [-9, -1, -10, 2, 14, 15, -6, 1, 7, 4], [9, -2, -2, 8, -6, -1, 0, 6, -2, -11], [-7, -13, -3, 6, 9, -1, -3, -4, -1, 1], [-13, 0, 6, 8, -11, -4, -1, 2, -14, -11], [23, 0, 7, -10, 0, -3, 6, 2, 3, -15], [-23, 5, 6, 0, 6, 5, -14, -5, -5, -16], [22, 6, -10, 5, 10, -2, -16, 5, 6, 3], [-23, -1, 1, -1, 4, -4, 3, 1, -1, 12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6032_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6032_2_a_z();".
// 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_6032_2_a_z(:prec:=10)
    chi := MakeCharacter_6032_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_6032_2_a_z();".
// 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_6032_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6032_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-8, -12, 330, -364, -283, 235, 104, -47, -17, 3, 1]>,<5,R![-1124, -1716, 1251, 1894, -639, -657, 174, 88, -22, -4, 1]>],Snew);
    return Vf;
end function;