
// Make newform 6032.2.a.u in Magma, downloaded from the LMFDB on 23 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_u();".
// 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_u();".
// 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 := [6, -8, -15, 15, 11, -7, -2, 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], [-2, -1, 1, 0, 0, 0, 0], [1, -4, -1, 1, 0, 0, 0], [4, 2, -5, -1, 1, 0, 0], [-5, 5, 8, -5, -2, 1, 0], [-3, -7, 10, 9, -6, -2, 1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 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_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, -1, 0, 0, 0, 0], [-1, 0, 1, 0, 0, 0, -1], [-1, 0, 0, 0, 0, 1, 0], [1, -1, 0, -2, -1, 0, 1], [-1, 0, 0, 0, 0, 0, 0], [2, 1, 0, 1, 0, -1, 0], [2, -1, -1, 2, 1, -1, -1], [-1, 1, 0, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0, 0], [0, -1, 1, 0, 1, 0, 0], [2, 2, 1, 1, -1, -1, 2], [1, -3, -1, 1, -1, 1, 1], [-5, 4, 0, 0, 1, 1, 0], [0, 3, 1, -2, 0, 1, 1], [-1, 5, 0, 2, -1, 2, 3], [-2, 3, 0, -2, 0, 1, 3], [-1, -3, 1, -1, 3, -1, -1], [2, 0, 1, 0, -2, 1, 1], [-3, -6, 0, 2, 1, -1, -4], [0, 4, -3, -1, 0, 1, 0], [-7, 1, 1, 0, -2, 0, -1], [2, -1, 2, 2, 4, -4, -2], [-3, 0, -2, 3, -1, -5, -3], [1, -4, -2, -3, 1, 0, 0], [-4, 1, 3, 0, -2, 0, -2], [0, 0, 3, -1, -2, -1, 2], [-1, -3, 0, 2, 3, -5, 0], [-2, -5, 2, 1, 0, -3, -2], [-3, -3, -3, -1, 2, 0, 0], [-15, -1, 2, -1, 1, -1, -1], [0, -6, 0, 1, 2, -1, -4], [8, 1, -2, 1, -3, -1, 2], [-3, -2, 2, -5, -2, 5, 3], [0, 0, 0, 0, 0, 5, -1], [-8, 5, 2, 4, 3, 0, -4], [1, 0, 2, 1, 1, 2, 2], [-4, 0, -1, 1, 0, -1, 2], [-1, 3, -1, -2, -2, 8, 5], [1, -2, 2, -1, -1, 6, 2], [5, 2, -6, -1, -3, 1, 5], [1, 5, 0, 2, 0, 1, 2], [-1, 4, 5, 5, 0, 0, 2], [1, -1, -2, -4, 1, 1, 2], [-8, 4, 1, 5, 5, 0, -3], [-4, -7, -2, 3, 1, -3, -2], [-4, -5, -4, -2, 2, -3, -5], [-8, -2, -2, -3, -1, 3, 0], [4, 3, 0, -2, -4, 6, 2], [1, -2, 2, 1, 0, -6, 6], [4, 1, 3, -6, -1, 4, 6], [2, 6, -3, 2, 2, -1, 1], [-10, -1, 5, 4, 7, -4, -6], [0, -5, -5, 7, -2, -1, -6], [-10, -5, 0, -1, 0, -6, -1], [-16, 2, 2, 1, 1, -1, -2], [-1, 0, -4, -5, -3, 5, 1], [-1, 7, -2, 2, -1, -1, -2], [5, -8, 1, -2, 3, 3, -6], [1, -3, -1, 5, -1, 1, 1], [-9, -2, 3, -4, 2, 0, 1], [6, -1, -2, 3, -2, -6, -1], [-8, 2, 2, 0, -5, 3, -1], [-6, -4, -2, 1, 3, -1, 0], [-1, 2, -4, -3, -6, 3, 7], [5, 3, 0, 4, 5, -1, 6], [12, -1, 2, -5, 1, 0, 3], [0, -9, -1, 4, 2, -7, -9], [-3, -2, 9, -4, -3, 7, 0], [-2, 0, -5, 1, -2, 1, 10], [-12, 2, 3, 1, 4, -2, -3], [-1, 4, -7, -2, -2, 2, -5], [-3, 6, 0, 8, -2, -1, 2], [-2, 2, 0, 6, 2, 4, -4], [-1, -1, -1, 7, -1, -5, 3], [6, 2, -7, -2, -3, 4, -4], [-1, 0, -8, -3, -1, -6, -4], [-10, -3, 3, -10, -3, 2, 2], [2, -1, -10, 1, -3, 4, 3], [-10, 3, 1, 2, 3, -10, -8], [-7, 11, 5, 1, -1, -1, -3], [10, -3, 0, -3, -6, 3, 2], [6, -3, -1, 3, -1, 0, -7], [-1, -6, 5, 0, 2, -6, -15], [-10, -2, -2, -2, 1, -2, -4], [-12, -6, -1, 2, -2, -2, -12], [-4, 4, -2, -4, 5, 7, 1], [-7, 9, 0, 0, 2, 3, 12], [-4, 3, 1, -2, 5, -4, 2], [-8, -7, 4, 0, 4, -5, -3], [5, -9, -1, -4, 3, -6, 7], [-5, 3, -3, -5, -5, 2, 6], [-21, 8, 0, 2, 2, 4, -3], [0, -5, 2, 4, 4, -10, -6], [8, -2, -7, -2, -2, -2, 2], [4, -16, -3, 0, 3, -7, -9], [14, 1, -6, -3, -1, 1, 12], [-1, -6, -1, -8, -3, 5, -2], [20, -9, -3, -8, -4, 4, 4], [-2, -9, -9, 0, 1, 0, -4], [-1, -3, -6, -4, -4, 11, 0], [-5, 2, 1, 0, 8, -1, -2], [1, 6, -4, -4, -5, 0, 13], [4, 10, -4, -6, -4, 7, 3], [-10, -3, 6, -3, 6, -3, -4], [3, -10, 2, -3, -2, -4, 0], [3, 0, -9, 15, 2, -9, -7], [-11, 2, 5, -10, 10, 3, 2], [-10, 5, 5, -3, 1, 1, 6], [-2, 8, -3, 9, 2, -1, 0], [2, -6, -7, 0, -2, -6, 4], [19, 1, -2, 2, -1, 6, 5], [-6, -2, 4, -2, -3, -2, 2], [21, 5, 0, 0, -4, 1, 2], [4, 7, 1, -5, -3, 14, 9], [9, -11, 3, -5, 4, 3, -5], [-23, -4, 6, 0, 10, -8, -7], [-7, 10, 10, -3, 4, -4, 0], [14, 10, -3, 5, 2, 3, 0], [-12, -1, 0, 0, 3, -5, -15], [-2, 3, -3, -6, 1, 5, -3], [0, -8, 0, 12, 8, -9, -13], [10, -13, -1, -4, -1, 0, 2], [18, 0, 6, 1, 2, 0, 1], [11, -7, 6, -5, 1, -4, -8], [18, 4, -5, 3, -8, 2, 1], [-6, 4, 9, 1, 4, -6, -1], [1, 3, 2, -10, 2, 11, 4], [-16, 18, 4, -1, -5, 8, 1], [-13, -4, 0, -5, 1, 0, -2], [11, -13, -12, 2, -3, 1, -8], [-11, 8, -5, -6, -2, 8, 7], [-4, 3, -8, 4, 1, 10, -4], [-10, 5, 3, 2, 3, 6, -6], [-12, -6, 7, 7, 8, -3, -10], [8, 13, 6, -7, -2, 11, 4], [19, 9, -9, 5, -1, 10, 4], [-2, 7, 6, 8, 1, -2, -12], [7, 0, -12, -1, 0, 9, -1], [1, 3, -4, -4, -9, 2, 9], [1, 14, 5, 3, 1, -2, -4], [18, -1, -8, 3, -12, 2, -3], [-17, -3, 0, 7, 8, -3, -3], [-24, 7, -3, 6, 8, -5, -9], [5, 6, -2, -1, -5, 1, -5], [17, -10, -10, -1, 0, -5, 7], [-25, 7, 2, -6, -10, 4, 1], [17, 0, 2, 1, -4, 10, 10], [12, 6, -6, 1, -11, 6, -3], [-9, -5, 5, -2, 4, -6, -1], [-3, -3, -4, 4, -5, 0, -9], [11, 9, -5, 7, 4, 7, 1], [10, 1, -2, -3, -1, 8, 21], [-3, -1, 7, -12, 2, 9, 0], [-5, -22, -1, 3, 2, 0, -8], [-3, 6, -7, -1, -6, 11, 5], [-5, 3, -6, 4, -8, 0, 5], [11, 6, 7, -6, -4, -3, 10], [-2, 17, 9, -4, 1, -1, 3], [-13, 2, 5, -14, 3, 10, 13], [-7, -4, -11, -6, 7, 1, 2], [9, 8, 9, 4, -6, -4, -3], [-11, -12, -4, -5, 2, -10, -2], [-6, 4, 4, -7, -1, 1, 6], [5, 6, 13, -2, -5, 11, 4], [10, -12, -8, 6, 2, -16, -6], [-7, -5, 13, -7, 1, 5, -9], [8, 4, -9, -1, -10, 8, -7], [-10, 3, 5, 0, -7, 7, -11], [11, -1, -2, 4, 3, 2, -5], [1, 11, 4, 5, -4, 6, -8], [-9, 0, -4, -9, -7, 6, 10], [-2, -11, 2, 1, 2, 3, -2], [20, 10, -2, 4, -3, -3, -3], [12, -8, 7, 9, -5, -8, -5], [-11, -9, 3, -7, -5, -12, 2], [-1, 0, 0, -6, 5, -4, -1], [-8, 6, -1, -7, -7, 17, 2], [-15, -1, -5, -10, 1, -5, -2], [5, 1, 9, -3, 4, 2, 10], [5, -7, 0, -6, -12, -1, 2], [12, 11, 1, -5, 0, 10, 17], [-10, 19, 2, -3, -3, 14, 13], [-8, 7, -5, -6, -6, 10, 10], [13, -2, 1, 13, 2, -8, -18], [5, -11, 3, -11, 2, 5, 9], [-8, 2, -1, 9, 3, -7, 2], [0, 3, 9, -6, -3, 3, -3], [1, 0, -6, -13, -9, 13, 11], [14, -2, 14, 3, -8, -3, -4], [-3, 10, 6, -9, 1, 6, 18], [5, -5, -9, 11, 15, -8, -2], [-5, 20, 2, 0, 3, -4, -5], [4, 19, -5, 4, -5, -1, 9], [-3, 18, 3, -9, -11, 17, 1], [-13, -6, -6, -7, 0, 2, 2], [-13, -6, -10, -5, 8, -6, -2], [-11, -2, 5, -4, 0, -7, -16], [-2, -17, 6, -9, -4, -3, -8], [7, -2, 2, -1, 5, 5, -5], [29, -6, -4, 3, 0, -6, -8], [-7, 0, 8, -3, 2, 11, 7], [4, 22, -6, 2, 7, 1, 5], [12, -11, -1, -4, -1, -2, 2], [-9, 7, 3, 1, 14, 0, 0], [9, -7, 6, 12, 1, -3, -8], [9, -6, -2, -17, -4, -5, 11], [14, -20, -1, -3, -6, 2, 1], [26, 3, -2, -5, -9, 8, 1], [15, 3, 8, -1, -14, 6, 2], [3, 9, 11, 13, -2, -6, -14], [-9, -17, 3, 1, 4, -1, -23], [1, 15, -10, 1, -3, -1, 17], [-6, -14, -2, 10, 10, -13, -11], [13, -2, -14, 7, -1, 7, 1], [-12, 14, -1, 15, 6, -5, -12], [-15, 5, 5, 1, 5, -2, -20], [2, -3, 4, -5, -1, -9, 8], [-2, 3, 6, -2, -3, 5, 9], [-4, -15, 0, -1, -7, -10, 9], [-6, -3, -6, -5, -8, 13, 4], [-27, -10, 2, 6, 7, -6, -1], [-20, 5, -3, 10, -9, -5, 5], [-1, 16, 8, 11, -7, -10, 6], [11, -17, 2, 4, 3, -16, -1], [-6, -7, -9, -2, 1, -8, -16], [5, 3, 2, 8, 3, 6, 1], [-1, 4, -3, 9, 9, -14, -14], [-23, 10, 3, 3, 12, -2, 0], [-5, -3, 7, -16, 2, 4, -1], [12, -5, 5, -12, -2, 7, 17], [10, -3, -20, 1, -1, 2, 15], [9, 0, -18, -9, -11, -6, 16], [-29, 19, 17, 9, 1, 5, -9], [5, 1, 19, 1, -3, 1, -9], [15, -3, -6, 8, 7, 4, -3], [-25, 5, -6, -4, -11, -1, -2], [15, -28, -5, 12, 3, -1, -6], [20, -12, 7, 11, 6, -3, -4], [-16, 2, -1, 10, 10, -5, -7], [0, -5, 4, -5, 6, -1, 0], [-13, -5, 4, -10, -6, 1, -8], [4, 4, -10, -10, -10, 13, 23], [-1, 2, -15, -10, -6, 3, 12], [-3, 7, -3, 5, -9, 3, -1], [19, -24, -16, -1, 1, 10, -4], [10, -20, -5, -2, 2, -7, 3], [31, 4, 0, 4, 2, 6, 17], [8, -28, -13, 8, 4, -6, -4], [-1, -3, 16, 11, 14, -14, -6], [26, -4, 6, -16, -6, 7, 15], [-19, -15, 4, -2, 3, -4, -15], [-20, 7, 8, -11, -8, -9, 6], [16, 1, 7, 4, 11, -6, 6], [-15, 3, -5, 3, 3, -9, -7], [33, 8, -11, 11, 1, 6, 8], [-11, -19, -2, -4, 10, -3, -6], [-16, 5, 0, 1, -4, -8, -9], [0, -7, 3, 12, -2, -13, -13], [-24, 4, 8, 4, 8, 3, -7], [-9, -11, 0, -11, -6, 9, -5], [-13, -28, -3, 9, 15, -17, -13], [-1, 15, -5, -1, -7, 19, -1], [1, 8, -9, 10, 4, -7, 4], [3, 0, -6, 13, -4, -18, -8], [-13, -11, -5, 10, 0, -5, -14], [32, 2, -11, 5, 5, 11, -2], [-7, -13, -2, -2, 1, -4, -21], [-27, -7, 16, -6, -4, 16, -5], [-13, 7, 2, 2, 15, 0, 3], [7, -10, -17, -4, 0, -3, 14], [-4, 19, 6, -2, 1, 2, 20], [5, 2, -4, 23, -2, 7, 3], [20, -20, -4, -3, -3, -3, -4], [-28, 8, 18, 4, 2, 11, 1], [-9, 13, -6, -4, -2, 13, 28], [-12, 9, -4, 19, -10, -6, 7], [18, 0, -6, 20, -3, -18, -16], [12, 5, -8, -13, -16, 0, 25], [14, 0, -19, -1, -5, 6, 13], [7, 5, -2, 0, -12, 13, -2], [-7, -14, 0, 0, 2, -13, -12], [11, 0, 10, 13, 1, -6, 10], [0, -25, -3, -12, 12, 9, -13], [28, -4, -17, 13, 3, 0, 1], [-21, 0, 11, 5, -3, -10, 2], [33, -18, -3, -12, -6, 9, -2], [20, -4, 1, 1, 2, -8, -7], [4, 6, 4, 3, -14, 3, 8], [11, -2, -3, -8, -13, 6, 15], [-15, 4, 7, -16, -2, 15, 4], [-27, 10, 11, 4, 6, 12, -5], [-20, 16, 25, -11, 7, 7, 12], [13, 5, 4, 0, -11, 10, 1], [-12, -4, -16, 8, 6, -7, 7], [15, -13, -20, 6, -2, 1, -2], [-15, -2, 2, 8, 9, -18, -9], [-11, 5, -13, -9, -1, 2, -12], [-6, -2, -10, 2, -5, 10, 12], [19, 10, -3, 0, -6, 3, -10], [21, -12, -3, 0, -11, 14, 13], [-21, -2, -6, 5, 1, -12, -30], [28, 0, 2, 11, 6, -1, 6], [-3, 13, 1, -16, -6, 8, 19], [9, -9, -4, -3, -2, -19, 5], [10, 8, -2, 14, 8, -25, -11], [-21, 14, 7, -3, 7, 2, 14], [16, 29, -1, 2, -7, 0, 20], [11, 1, 5, 8, 7, -7, -2], [-32, 12, 11, 1, 13, -4, -5], [8, 25, 13, -12, -7, 22, 8], [-13, -6, 0, -9, -5, -6, -2], [35, 20, -1, 12, -6, 4, -3], [-7, 0, -16, -1, 1, -3, 13], [11, 20, 3, 1, -10, 0, 4], [5, -5, 3, -21, -3, 7, -1], [19, -1, -8, 4, 6, -21, -2], [-17, 13, 6, -1, 10, 15, 5], [3, -7, 12, -2, -9, 9, 0], [8, 6, -6, -6, -5, 17, 15], [30, 15, 3, -3, -8, 5, 28], [22, -9, 4, -13, 0, 15, -2], [29, -13, -2, -8, -21, 8, 19], [19, -5, -22, 10, -6, 3, 12], [-14, 25, -4, -13, -8, 9, 16], [-22, -19, -7, 4, -2, -1, -1], [2, 31, 18, 6, -15, 8, 0], [-3, 6, -2, 8, 4, 13, -6], [6, 13, 1, 9, -7, 9, 10], [-6, 12, 2, 8, 9, -1, -25], [-1, -1, 15, -7, -1, -7, 11], [-39, 13, 3, 11, -2, -8, -12], [0, -10, 9, -11, -9, 0, -1], [4, -6, 8, 16, 9, -11, -31], [-43, 22, 17, 6, 0, 0, -13], [1, 10, 14, -8, 5, 12, -1], [1, -6, 1, 8, 6, -20, -23], [7, -11, 1, -7, -2, 10, 4], [-19, -17, -7, 1, 1, 0, 10], [-25, -16, -1, 6, 3, -5, -20], [-10, 10, -2, 10, 0, 13, -1], [-18, -14, 6, -14, 6, 3, 7], [5, -5, 13, -5, -9, 5, 13], [-5, -26, 7, -12, -4, 9, 2], [-23, 12, 9, 4, -3, -16, -1], [11, 1, -9, 14, -8, -1, 2], [19, -19, -10, 6, -3, -13, -22], [-2, 25, -7, -10, -16, 15, 5], [-32, -7, 9, -4, -6, -14, -12], [8, -1, -6, -7, 0, -13, -2], [-15, 8, 1, -7, -7, 15, 7], [6, -12, 12, -8, 3, 17, -1], [-42, 20, 6, 8, 0, -2, 0], [-15, 6, 3, -2, 7, 0, 3], [15, 5, -3, -1, 14, -5, 21], [-13, -2, 1, 10, 12, 6, -33], [-5, 11, 20, 4, 8, -6, -11], [-5, 13, 19, 8, 3, 1, 10], [-45, 21, 13, 7, 5, 2, -6], [2, 11, -12, 10, -11, -8, 0], [-11, -6, -6, 19, 7, -17, -11], [-24, 2, -4, 8, -5, -8, 6], [-35, 10, 14, 10, 9, 7, -10], [12, -3, 16, -4, 0, 5, -5], [-35, -12, 13, -9, 3, -13, -21], [-19, -10, 15, -6, 9, -10, -21], [7, 21, 16, 0, 1, -8, 11], [-26, -6, 3, -4, -15, 14, -8], [7, -20, -7, 18, -13, -13, -18], [-23, -11, 9, -6, 1, 16, -5], [-1, -4, 0, 11, 5, -11, 7], [-35, -2, 15, -1, 10, -16, -24], [-19, 12, 21, -8, 13, -3, 8], [-16, 31, 9, 14, -9, 18, -6], [-43, -29, 4, -8, 15, -13, -14], [-10, -17, 19, 0, 14, -7, -19], [24, 11, -6, 14, 8, -4, -10], [9, 7, 4, -20, -6, 9, -2], [1, -3, 1, 19, 11, -11, -19], [-20, 40, 2, -4, -7, 10, 6], [-13, 11, 17, -5, 8, 2, 2], [-4, -22, -14, -12, 2, 4, -10], [-14, -13, -5, -10, -6, -3, 7], [7, 1, 15, -5, -1, 19, 19], [-25, 23, 19, -4, 7, 3, 8], [-22, 25, 1, -1, 8, 5, -12], [27, -7, 7, -23, -4, 1, 5], [22, 7, -8, -11, -2, -3, 20], [-37, 20, 14, 5, 1, 5, -5], [4, 4, -6, 19, 16, -7, 0], [2, 2, 5, -1, 14, 12, -17], [0, -18, -26, -10, 9, -12, 2], [10, 0, -4, 10, -9, 5, 1], [14, 10, -5, 2, 3, -7, 7], [4, -5, 11, -2, 6, -1, -21], [4, 23, 16, 1, -13, -2, 13], [-7, 17, 12, 1, 9, -7, 17], [6, -2, 7, 3, 1, -22, -17], [-10, -5, -1, 16, 10, -13, 13], [-15, -16, -1, -2, 14, -22, -7], [-4, -22, -2, 18, 6, -22, -6], [30, -17, -13, 6, 7, -23, -3], [-34, 6, 12, -6, 0, 11, 17], [10, 17, 14, -5, -11, 8, 13], [42, 10, 4, -12, -4, 13, 25], [1, 2, -19, 16, 2, -6, 9], [17, -5, -15, 7, -12, 15, 11], [-16, 7, 0, -21, -7, 26, 15], [20, 7, -5, -14, -20, 17, 5], [5, 6, -16, 6, 2, 12, 3], [-34, 15, 20, 5, 0, 5, -4], [-13, -2, -3, 10, 7, 5, -18], [-4, 11, -16, -1, -2, -2, -3], [-1, -7, 2, -4, -6, -11, -10], [9, 5, -17, -3, -12, -13, 17], [45, 12, -16, -3, -8, 4, 18], [23, -21, -15, -3, 3, 4, -2], [-30, -34, 3, -5, 5, -2, -3], [-10, -4, 16, -4, 4, 21, -5], [40, -24, -12, -3, -8, -9, -4], [-3, -12, 1, 12, -2, 4, -15], [11, 12, 10, 3, 1, 14, 10], [22, 15, -5, 14, 10, 10, 0], [-2, 5, 15, -11, -11, 5, 10], [-18, -16, -11, 1, -1, -16, -17], [4, -23, -7, -12, -7, 21, -3], [14, 15, 2, -7, 0, 15, 10], [20, -15, 11, 2, 4, 2, -22], [-13, -7, 6, -11, 2, 0, 0], [-24, 3, 5, 7, 17, -17, -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_u();".
// 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_u(:prec:=7)
    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_u();".
// 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_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6032_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, -6, 33, 30, -16, -11, 2, 1]>,<5,R![36, -8, -111, 106, -7, -18, 2, 1]>],Snew);
    return Vf;
end function;