
// Make newform 8046.2.a.h in Magma, downloaded from the LMFDB on 14 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8046_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_8046_2_a_h();".
// 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_8046_2_a_h();".
// 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 := [2, 25, 34, -43, -58, 29, 25, -9, -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, 1, 0, 0, 0, 0, 0, 0, 0], [-3, -1, 1, 0, 0, 0, 0, 0, 0], [-347, 355, 992, -975, -329, 516, -40, -69, 15], [34, -116, -235, 850, -35, -515, 102, 72, -19], [-30, -1030, -232, 1714, -105, -764, 141, 95, -24], [107, 722, 155, -1637, 105, 764, -141, -95, 24], [-194, -1168, 589, 1926, -679, -839, 265, 101, -32], [232, 1600, -59, -2978, 427, 1355, -305, -170, 47]];
    Rf_basisdens := [1, 1, 1, 77, 77, 77, 77, 77, 77];
    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_8046_a();"
function MakeCharacter_8046_a()
    N := 8046;
    order := 1;
    char_gens := [299, 3727];
    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_8046_a_Hecke(Kf)
    return MakeCharacter_8046_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, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 1, 0, -1, 0], [0, -1, 0, 1, 0, 0, 0, 0, 0], [-1, 1, 0, -1, -1, -1, 0, 0, 0], [-1, 0, 0, 0, 0, 1, -1, 0, 0], [-1, 2, 0, 0, 0, -1, 1, 1, 1], [-2, 2, 1, -1, -1, -1, 0, 1, 0], [-1, -1, -1, 1, 1, 1, 0, 1, -1], [-1, 2, 0, -1, 1, -2, -1, 2, 1], [-2, -1, -2, 0, 1, 1, -1, 0, -1], [-2, 2, -1, -2, 1, -1, 1, 0, 0], [0, 2, 0, 1, 2, -1, 0, 1, -1], [-2, 2, 2, -1, -1, -1, 3, -1, 0], [0, 1, 2, -1, -1, -2, -1, -1, 0], [1, -5, 0, 0, 2, 0, 0, 0, -1], [-2, 0, 1, -1, -2, 1, 0, 2, 0], [-1, -1, 0, 1, 1, -1, 2, 1, -1], [-1, -1, 1, 0, 1, 2, 0, 0, -3], [-3, -3, -2, 4, -1, 3, -2, -1, 0], [0, 0, 0, 2, 1, -1, -2, -1, 2], [-5, 1, -1, -1, -3, 0, -3, -1, 2], [-3, 5, 1, 0, 0, -1, 0, 4, 1], [-1, -3, -1, 1, 1, 1, 0, -2, 1], [-3, 0, -1, 0, 1, 0, 0, 2, 2], [2, 0, -2, 0, -1, 1, 0, 1, -2], [-2, 1, 0, 0, -3, 2, 1, -1, -1], [-2, -2, 0, 3, 0, 2, -1, -2, -1], [-5, -1, 0, 2, 0, 1, -4, 0, 1], [1, -1, 1, -1, -2, 0, 1, -4, 3], [-2, -5, 0, 2, -2, 4, 0, -1, -2], [4, -3, 3, 2, 5, -3, 1, 0, -1], [5, -2, 1, 0, 1, -3, 3, 2, -2], [3, -3, 3, 0, -1, 2, 1, -5, 1], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [-3, -5, 0, 2, 1, 3, -1, -4, -3], [3, -5, -2, -2, 5, 0, 0, -1, 1], [-4, -3, 0, -4, -3, -2, 1, 2, 2], [-6, -1, -3, -1, -5, 1, 0, -4, 2], [1, 4, -3, 0, 3, -3, -1, -1, -4], [2, 1, 4, -4, -1, -4, 7, -2, 3], [-3, -6, 1, 3, 2, 3, 0, -3, -5], [2, 0, 0, 4, -3, 4, 0, 0, -1], [1, 1, 1, -1, -1, -1, 1, -6, 1], [-7, 7, 1, -1, -2, 2, -1, -3, -1], [-2, -1, 2, 0, 4, -2, 2, -1, 1], [-4, 0, -5, 3, 2, 4, -5, 1, 1], [-5, 6, 0, -1, -2, -3, 2, -4, -1], [4, 1, 2, 2, 1, -4, 2, 0, 0], [3, -4, 2, 1, 2, 2, 0, -2, 1], [-3, 5, 2, 1, -3, 2, -5, 1, 2], [2, -6, 3, 2, 5, 3, 2, 0, -7], [-6, -3, -7, 6, 1, 5, -5, 0, -2], [5, -3, 1, 0, -2, 2, 7, -7, 0], [-3, -3, -5, -1, 1, 3, 0, 0, -2], [1, -5, -1, -1, -8, 3, -5, -1, 2], [-1, -1, -5, -1, 2, 3, 0, 3, 2], [-2, -2, 1, 1, 3, -6, 4, 3, 1], [-1, -10, 2, 3, -1, 2, -1, 1, -2], [6, 1, -2, 0, 2, 1, -4, 0, 4], [-4, 6, -3, -3, 2, -2, -1, 8, 0], [2, -3, -1, -1, 1, 2, 1, -1, 1], [-7, 6, -2, -5, -3, -6, 0, -2, 1], [2, 2, 2, -2, 4, -5, 3, -1, -2], [-7, 9, -1, -4, 0, -6, 4, 8, 4], [5, -10, 0, 3, 2, 6, -4, 0, -4], [-5, 4, 2, -7, -8, -5, 0, 0, 3], [-4, 0, -8, 7, 2, 5, -6, 5, -1], [-5, 3, 4, -3, -5, 2, -3, -6, 1], [4, 1, 8, -3, 5, -2, 6, -2, 0], [9, -3, -2, 2, 6, -1, 5, 3, -2], [-3, -5, -4, 4, 5, 5, -1, 4, 0], [0, -3, -4, 1, 4, -1, 0, 1, 3], [1, -8, 1, -3, -7, 2, -2, -7, 0], [-11, 10, 0, 1, -3, -3, 2, 6, 0], [-4, -2, -3, -3, 5, -5, 2, -1, 5], [6, -1, 7, 3, -4, 6, -2, -3, -2], [-5, -5, -6, -3, -7, 8, 0, -1, 2], [7, 3, 4, 2, 3, -4, 1, -3, 3], [-3, 11, 4, -3, -6, 2, 2, -1, 0], [-9, 1, 1, 2, -9, 2, -1, -3, -1], [-5, 3, 0, -2, 5, -4, 1, 1, 5], [4, -1, 1, 4, 0, 4, 5, -3, -2], [-2, -2, 4, -2, 1, -5, 2, -5, -3], [-7, 5, 2, -5, 1, -4, -2, 9, 0], [7, 4, 6, -1, 4, -5, 11, 1, 3], [5, -4, -1, 5, 3, 3, -8, -4, 0], [3, -3, -3, 6, 2, 5, -2, -4, -1], [5, 2, -2, -2, -3, 1, -5, 2, 6], [8, -3, 3, -1, 6, -3, 1, 10, -5], [-9, 1, -1, 2, -2, 4, -6, 7, 0], [6, -5, -5, 10, 2, 5, -4, -2, -3], [-13, 7, -7, -2, 1, 2, 0, 4, 0], [-11, 4, -3, -6, -6, 5, 4, 4, 2], [-5, 7, -4, -1, -5, 1, 1, -4, 10], [11, -1, 5, -1, 2, -5, 5, -3, -1], [6, -9, 2, 3, -4, 7, -3, -4, -3], [5, -5, 2, -5, 1, -4, 6, 4, -5], [-5, 9, 2, -1, 3, -11, 0, 5, 3], [-7, -2, -3, 0, 11, -5, 4, 1, -4], [-5, 4, -1, -4, -3, -6, -10, -1, 3], [-7, 11, 10, -5, -3, -1, 5, 2, 0], [-5, 9, 0, -2, -1, -6, -10, 3, 7], [10, 7, -8, 0, 2, 1, -1, 0, 2], [1, -7, -3, 0, 3, -3, 0, -1, 1], [2, -2, 2, -1, 1, 10, 8, 2, -4], [0, -4, 0, -7, -6, 9, -9, -1, -2], [20, -13, 0, 8, -2, 6, -7, -3, 4], [-14, 8, 6, -6, -6, -4, 6, -2, 5], [-1, -2, 0, 5, -5, 2, -3, 9, -2], [4, 4, 8, 2, -1, 2, -8, 0, -2], [-11, 7, 0, 1, -3, 1, -4, -3, 8], [-4, -6, -2, -5, -2, 3, -7, 3, -5], [9, -11, 0, -4, -2, 2, -1, -2, -2], [-8, 6, 2, -6, -14, 0, 4, -6, 7], [5, 5, -2, -5, -3, 1, 0, -11, 0], [0, 14, 3, -3, -9, -9, 0, 2, 10], [4, -13, 0, 9, 1, 9, -8, -1, -9], [10, -4, 1, 0, 2, -3, 5, -3, 4], [5, -10, 0, 9, 10, 7, -3, -5, -7], [-6, -1, -6, -4, -9, 1, 3, -10, 6], [3, 3, -1, -2, 5, -1, 6, -3, 0], [15, -1, 2, -4, 0, 1, -7, -9, -3], [-11, -1, -6, -2, -13, 8, -7, -6, 1], [2, 4, 1, 5, 5, -1, 6, -1, -2], [2, 9, -11, -3, -2, -2, -4, 5, 4], [-5, 0, 1, -8, -7, -1, -6, 1, -7], [2, -2, -3, 0, 1, 0, -3, 2, -6], [9, 12, 12, -6, -8, -4, 1, -1, 4], [2, 15, 6, 0, -5, -8, 5, 1, 2], [-16, 7, 0, -1, 6, -7, 1, 7, 4], [-15, 6, -8, -2, 0, 3, -2, 4, 1], [-3, -15, -4, 7, 0, 9, -1, -3, -10], [-5, 6, 9, -8, 2, -4, 1, 7, -1], [10, -3, -7, 4, 10, 4, -3, 2, -4], [7, -3, -3, -7, -2, -2, 7, 5, 0], [-4, -8, -6, 8, 4, 10, -9, 0, -12], [12, -5, 5, -6, 6, -5, 1, 0, 0], [-2, 4, -7, -5, -1, -5, 0, 12, 0], [1, 5, 7, 0, 3, 4, 0, 2, -3], [-9, 9, 6, -1, -9, -7, 0, 4, 11], [6, 11, 4, -8, 3, -4, 11, 8, 0], [-1, 5, -3, 0, 5, -4, -2, 4, 1], [2, 8, 1, -5, -1, -6, 4, 2, 11], [-16, -7, -7, 4, -1, 20, -6, -1, -3], [1, -9, 0, 7, -5, 11, -3, -2, -3], [2, -2, 9, -4, -8, -6, -1, -6, -3], [9, 14, 9, -13, -19, -6, 3, -11, 11], [-5, 5, -4, 8, -8, 2, -9, 5, 1], [10, -3, 4, 8, 5, -2, -4, 1, 2], [-8, 11, -3, -9, -7, 2, 8, 1, 2], [-4, 3, -5, -3, 4, 0, -9, 10, 3], [-1, 4, 7, 3, 7, -12, -1, 12, -2], [-5, -4, 0, -7, 3, -2, 10, -3, -9], [-3, -3, 2, -11, -12, 0, 0, -14, 6], [7, -29, -5, 16, 3, 25, -10, -12, -11], [-14, 13, -3, 5, 3, 1, -5, 12, 7], [10, 5, -1, -2, 1, -7, 11, 9, -1], [-15, 11, 9, -11, -9, -5, -1, -4, 6], [-1, -1, 0, 8, -8, 3, -8, 1, 3], [-2, 10, -7, -3, 2, -2, -7, 4, -2], [10, -12, -8, 11, 6, 2, -4, -2, -11], [23, -12, 1, 13, 7, 7, -1, -1, -1], [10, 22, 5, -15, -5, -18, 8, 6, 9], [1, 9, 8, 0, 4, -7, 11, 2, 0], [-7, 4, -4, 2, 7, -7, 6, 4, 5], [-16, 3, -6, 1, 1, -2, 9, 13, -4], [18, 3, -9, -1, 2, -1, 6, 1, -2], [-8, -1, -4, 3, 7, 8, -9, 6, -2], [3, 1, 6, 2, -1, -1, -7, -1, 8], [3, 19, 4, -10, 0, -20, 6, 8, 11], [16, -32, -2, 10, -2, 15, -12, -12, -8], [11, -1, 7, 10, 0, 11, -1, 1, -9], [15, -1, 0, -8, 6, -16, -3, -4, 3], [4, 5, -4, -1, 7, -4, 10, 9, 3], [19, 1, 8, 13, 0, -3, -6, -1, 9], [8, -4, 4, 4, -6, 1, 9, -4, -4], [-19, 8, -6, -5, -3, -5, 10, 0, 11], [-19, 8, -3, 2, 0, 2, 5, 12, -3], [-8, 14, 5, -1, -5, -7, -6, -4, -2], [18, -7, 1, 11, 5, 2, 2, 4, -8], [-1, 4, -6, -8, 3, 0, -9, 9, -1], [-3, -8, -3, -2, -6, 10, -11, -5, 2], [11, 0, 8, -10, -5, -10, 17, -2, -1], [-2, -11, -4, 0, 1, -3, 11, -7, 3], [10, 7, 2, 7, -4, -4, -11, 4, 6], [0, 10, 2, 0, 3, -7, 5, 10, -5], [16, -16, -3, 17, -1, 17, 0, -15, -4], [14, -26, -7, 18, 11, 14, -12, -7, -17], [-10, 15, -4, -4, 2, -11, 6, 2, 8], [-2, -10, -11, 8, -1, 4, -8, 3, 1], [-5, 11, 1, 2, 9, 0, 3, -3, -1], [-18, 4, -3, 0, 2, -2, -6, 9, -5], [13, -9, -2, -1, 2, 2, 8, -9, -7], [-6, 8, 2, -5, -4, -6, 2, 5, -4], [5, -8, -4, 4, 4, 16, -1, -3, -13], [8, 4, 10, -5, 9, -18, 8, 7, 0], [18, -17, 3, -1, 5, 8, 1, 1, 0], [7, -14, -3, 4, 4, 8, -8, 6, -1], [0, -4, 1, -1, -5, 7, -6, 3, 1], [9, 3, 3, 3, -2, 0, 8, 5, -6], [4, -8, 5, -1, 7, -5, 8, 4, 5], [15, 15, 0, -6, -6, -18, 3, 0, 11], [-5, 16, -5, -10, -1, -16, 14, 8, 9], [-6, 3, -2, 0, 12, 4, -1, 7, -3], [-19, 4, 1, -3, -4, -1, 2, 1, -2], [6, -3, 1, 5, 3, 5, 2, -18, -2], [3, 3, -1, -5, -7, -1, 3, -8, 7], [-5, 7, 0, -17, 0, -6, 3, 4, 2], [4, 0, -4, 9, 11, 5, 4, 1, -6], [-6, 17, 2, 6, 8, -8, -8, 3, 9], [12, 0, 1, -9, 3, -6, 4, -5, 0], [-7, 4, 0, 8, 9, 5, -5, 15, -4], [-14, -4, -7, 0, -12, 10, -13, 0, -7], [-18, 12, 6, -9, -15, -6, 11, 3, 8], [0, 11, -5, 2, -6, -2, -11, 7, 4], [-3, -18, -8, 14, -6, 22, -17, -8, -4], [20, -22, -5, 9, 8, 5, -12, -1, -7], [-8, -6, -1, 1, 1, 11, 4, -4, -10], [12, -26, 1, 12, 10, 4, -13, -6, -6], [-4, -13, -2, -7, -4, 11, 4, 4, 3], [8, -16, -8, -1, 1, 5, 5, -1, -5], [15, -22, -3, 5, 6, -8, 3, 0, 5], [11, 8, 14, -11, -4, -7, 7, -6, 0], [-11, 15, 5, -13, -3, 3, 2, 4, 0], [8, 10, 13, -4, -7, -4, -1, -9, 7], [1, 3, 9, 1, -10, 1, -6, -6, 0], [-4, 7, 1, -8, 13, -22, 14, 1, 0], [3, -7, -2, -4, 4, -5, 5, -2, 1], [-9, -2, 0, -10, -6, 1, 2, -9, 8], [1, 9, 5, -9, -1, 2, -3, -1, -5], [-7, 10, -4, -5, -10, -8, 2, -4, 14], [2, -3, -2, 14, 1, 14, -2, 6, -8], [24, -25, -8, 12, -2, 23, -10, -13, -2], [0, 2, -7, -1, -5, 15, 6, 0, -1], [15, -12, 3, 5, -9, 2, -1, -11, 1], [17, -20, -2, 9, 7, 10, -8, -9, -3], [-13, 17, 0, -1, 1, -12, 10, 4, 9], [-7, -13, -9, 5, 3, 8, -13, -1, -4], [2, 4, 7, 4, -6, 12, -7, -15, 1], [-25, 19, -9, -9, 6, -6, -1, 2, 3], [5, 0, -2, -6, 11, 6, 4, 7, -13], [12, 6, -10, 4, -2, 3, -7, 6, 1], [-18, -21, -5, 6, -8, 14, 8, -4, 0], [1, 4, 1, -2, -15, 11, 6, -15, 8], [-3, -11, -3, -5, -13, 3, -9, 4, 6], [7, -16, -1, -2, -3, 12, -4, -22, -7], [-6, 8, -3, -2, -6, 11, -16, 14, 6], [1, 10, -6, 6, 1, 2, -3, 5, 14], [-10, -1, 11, -5, 0, 4, 11, -3, 0], [16, 10, -6, -4, 4, -4, -6, 12, -8], [-5, 0, 7, -17, -23, -5, -4, -16, 11], [-11, 26, 12, -4, 3, -17, -5, 7, -2], [1, -30, 5, 12, 1, 14, -11, -19, -12], [-9, 7, 15, -2, -12, -5, 15, 5, 0], [20, -4, 3, 7, 16, 6, 6, 6, -8], [-8, -2, -7, -6, -10, 0, 11, -14, 5], [7, -21, 5, 8, 5, 13, -7, 1, -9], [-6, -11, -2, 1, 4, 11, 9, -5, -3], [15, 3, 9, 0, 3, -5, 6, -1, 6], [5, -16, -13, 10, -1, 27, -5, -12, 1], [-22, 10, -11, 2, 6, -2, -1, 2, 7], [0, 16, 0, -4, -9, -11, -4, 8, 16], [6, -12, -6, 13, 2, 3, -1, 3, -17], [-9, 4, -13, 3, 0, -4, -10, 9, -4], [-5, 2, -2, 1, 0, 12, -19, 5, 2], [-8, -5, -2, 7, 8, -1, -11, 13, 4], [8, -6, -1, -2, 7, 5, -3, -15, 0], [10, 11, -4, -5, 11, -13, 14, -6, 14], [1, 3, 3, -11, 4, -12, 13, -7, 3], [4, 3, 2, 7, 4, -5, -9, -8, -6], [3, 15, -5, -5, -12, -15, -10, 9, 7], [22, -17, 4, -2, -15, 6, -3, -16, -6], [5, 10, -8, -3, 3, 4, -13, -2, 8], [18, -3, 8, 5, -7, -1, 3, -10, -5], [7, -15, -3, 6, 10, 2, 3, 2, -3], [-14, -7, -9, -7, 11, 5, -2, -3, 2], [-19, 0, -12, -2, 10, 9, -3, 5, -2], [5, 13, 1, -10, 0, -9, 0, 8, 8], [17, 2, -2, -9, 20, -13, 10, 2, 2], [-17, 21, 8, -11, 2, -30, -2, 11, 2], [2, -2, 13, 2, -6, 2, -7, -5, -10], [9, -1, -3, 12, 10, 6, -3, -3, 1], [25, -1, -6, -2, -2, 5, 11, -6, -5], [2, 1, -2, -17, -13, -1, -5, 2, 1], [-9, 15, 19, -5, 6, -9, 3, 9, -2], [7, 0, -7, 2, 1, -8, 2, -3, -1], [-5, 7, 8, -5, -9, 5, 10, -7, 5], [-5, -3, 3, 7, 5, 1, 9, 1, 2], [1, -6, 0, 14, -16, 4, -13, 3, 2], [3, 3, 9, -4, 8, -13, 15, 4, -7], [-22, 8, -2, -13, 3, 8, 0, 1, 2], [14, 11, 14, -2, 9, -8, 4, 15, -1], [16, -1, 0, -8, -4, 0, 7, 1, 1], [8, -11, 8, 12, 4, 17, 6, -16, -6], [3, 4, 9, 2, 6, -11, -2, -3, -2], [17, -27, -2, 9, -5, 22, -6, -24, 1], [12, -26, -8, 9, -10, 19, 2, -20, -1], [-23, 9, 2, -4, -9, -6, -9, -6, -8], [-7, 12, 4, -10, -18, -5, 10, -6, -2], [15, 0, -2, -12, 5, -14, 9, 11, -4], [6, -9, 7, 8, 10, -3, 5, 1, -7], [26, -8, 14, -1, 9, -4, 3, -7, -3], [-14, 4, -8, -12, -4, 4, 6, -2, -13], [21, -3, 14, 10, 2, -7, 9, -2, -5], [-8, 4, -18, 2, 5, -1, -5, 14, 4], [4, -4, 4, 14, 18, -12, 10, 9, 3], [28, 8, 3, -9, 14, -16, 5, 1, 7], [-8, -1, -10, -1, -2, 8, -3, -13, 7], [-1, 5, 7, 1, -2, 3, 5, -1, 0], [-12, 17, 7, 7, -9, -11, 3, 14, 5], [-28, -8, -9, -5, -7, 22, -3, -3, -4], [18, 6, -13, -3, -8, 8, -18, 2, 11], [3, 1, -8, -12, 7, -2, -4, 9, 2], [-10, 0, 10, -3, 5, -6, -5, -1, 14], [-4, -11, -11, 10, 21, 7, 3, 11, -10], [-8, -21, 10, 5, -4, -1, 7, -7, 4], [14, -23, -2, 13, 4, -8, 1, -9, 7], [20, -8, -2, 8, 1, -8, -16, 9, -7], [22, -25, 11, 3, -4, 10, 7, -2, -3], [14, 23, 3, -5, 4, -17, -14, 4, 9], [25, -19, 0, 12, 0, 15, 7, -16, 0], [-8, 2, 0, -11, -12, -8, 3, 1, 12], [-16, 9, -12, -3, -13, 1, -7, -10, 9], [-15, 2, -15, 0, -10, 9, -16, 12, 7], [-3, -1, -5, -3, -9, 15, -15, -18, 4], [5, 18, -3, -12, 4, -20, 1, 21, -6], [6, -34, -15, 9, 7, 26, -7, -13, -12], [-17, 18, -4, -7, 0, -16, 20, 8, 6], [9, -8, -3, 12, 8, -3, -1, 9, -13], [1, -18, 9, 6, 7, 3, -3, 5, -12], [-35, 14, -13, -13, -1, 4, -9, -5, 12], [-1, -3, 11, 1, -8, 6, -9, -15, 2], [-21, 17, -1, -1, 11, -21, 7, 17, -5], [-31, 7, 3, -7, -6, -8, -3, -4, 9], [18, -7, -3, 6, -2, -1, 1, -11, 10], [-2, -8, 6, -1, -1, 6, 20, -12, -6], [-2, -10, 4, 6, -12, 0, -5, 9, -5], [-13, -6, 9, 5, -6, 9, 4, -1, -1], [11, -7, 4, -9, -4, 0, -10, 2, -12], [3, -1, -3, 8, -7, 11, -10, 2, -8], [-10, 24, -9, -5, -11, 3, -8, 3, 9], [4, 1, 5, -1, -8, -8, 6, -10, 0], [-17, -17, -6, 9, -5, 9, -16, -13, -4], [7, 28, 1, -16, -15, -30, 2, 4, 10], [-12, 10, 14, -17, -6, -3, 15, -7, 12], [-11, -6, -3, 4, -14, 8, -11, 7, 2], [8, -15, 0, 0, 4, 14, -12, -5, -4], [20, 3, 6, 6, 29, -15, 16, 18, -12], [4, 2, 5, 18, 33, -2, 1, 5, -2], [-9, -1, 0, -3, -6, 1, 5, 13, 6], [2, -7, 2, 12, -12, 8, 3, 8, -8], [18, -12, -3, 2, 6, 14, 9, 3, -3], [4, -22, -4, 0, -6, 9, -6, -16, -5], [2, -28, -1, 14, -5, 13, -7, -8, -12], [-11, 5, 4, -5, -15, 16, -18, -9, 3], [-17, -1, -15, 16, 10, -1, -6, 12, -4], [-11, 33, 5, -14, 0, -34, 8, 14, 8], [7, -1, -8, -6, 2, -2, 15, -1, -9], [-26, 18, 5, -2, -5, -16, 15, 18, 6], [-5, -5, 2, 15, 24, -2, 10, 6, -5], [3, 24, -9, -15, -8, -14, -7, 2, 8], [-28, 2, -14, 1, 12, 0, -3, 20, -8], [11, -3, 13, 3, 13, -2, 5, 14, -3], [1, 20, 7, -18, -13, -10, 14, -4, 24], [11, 15, 19, -2, 16, -22, 12, 12, 4], [-11, -10, 6, 9, 5, 12, -16, 1, -4], [12, -23, 5, 24, 16, 16, -4, -14, -6], [-11, 24, 6, -11, -5, -19, 8, 16, 0], [-6, 17, 3, -13, -8, -2, 8, -7, 14], [-19, 5, -9, -9, -17, 7, -7, 1, 7], [-13, -4, 4, 11, -15, 7, -5, 2, 7], [-10, -5, 4, 2, -7, -6, -13, 4, -6], [-14, 16, 1, 4, -2, -15, -9, 4, 2], [8, 23, -3, -10, -13, -22, -11, 7, 20], [-31, 7, -2, 2, -2, 3, -8, -7, 5], [8, -11, 15, -2, 11, -13, 12, 1, -15], [-25, 23, -7, 9, 18, 5, -9, 29, -9], [-14, 3, 6, 0, -5, -1, -8, -8, 9], [-23, 7, 7, -17, -13, -1, -3, -12, -9], [1, -9, 1, -6, -5, 2, 9, -9, -8], [-14, 34, 13, -25, -10, -10, 0, 5, 10], [12, -1, -3, -9, -6, -8, -5, -7, -3], [-9, 18, -13, -7, 2, 7, -1, 2, -11], [8, -28, -3, 20, 27, -4, 4, 16, -7], [9, -10, 10, 2, 17, -1, 3, 11, -7], [7, 17, 4, -5, 25, -16, 16, 7, 14], [-9, -17, 6, 10, -11, 5, -9, -8, -1], [-28, 26, -8, 7, 0, -9, -5, 13, 15], [-20, -21, -11, 6, 10, 18, -9, 7, -10], [-31, 17, 16, -9, -11, -23, 17, 12, 6], [5, -11, -4, 3, 21, -17, 14, 5, -1], [-4, -17, -6, 0, -7, -1, 15, -20, 5], [13, -12, 5, -1, 0, 1, 7, -10, -14], [-1, -8, -11, 16, -10, 0, -26, 12, -5], [-14, 20, -11, 4, 7, -2, -14, 12, 15], [-7, 8, 10, -4, -15, -4, -6, -18, -5], [-12, 1, -9, -9, -18, 21, -16, -2, 3], [-14, -12, -8, 9, 5, 21, 1, 3, -6], [22, -8, 17, -10, -12, -8, 9, -13, -9], [5, 20, 15, 9, 10, -12, 11, 18, -1], [3, 1, 6, 8, 8, -17, 16, 5, 9], [-20, -10, 2, -10, -7, 5, 4, 4, -3], [-12, -2, -1, -10, -4, -5, 9, -14, -1], [-35, 7, -4, 0, 12, -10, 7, 9, 5], [-13, 19, 6, -5, 4, 0, -2, 25, -3], [28, -21, -2, 12, 27, 5, -8, 11, 2], [-2, 30, 14, 2, 0, -3, -12, 1, -4], [-12, 11, 11, 2, 3, -3, 17, 8, -2], [-39, 26, 9, -12, -2, -10, 8, 4, -5], [7, -19, 8, 1, 2, 2, -12, -3, -20], [11, -2, -6, 15, 5, -7, -3, 23, -16], [3, 3, -12, 9, 12, -1, -2, 18, 2], [-33, 15, -5, -8, 0, -8, 2, -2, 8], [-16, -9, -13, 12, 3, 13, -20, 7, -13], [-7, 2, 3, 10, -4, 5, 5, -12, -2], [-17, 23, 1, -16, -18, -17, 21, -5, 6], [18, -14, -2, 8, 15, -8, 14, 0, -10], [0, 9, 19, 4, -9, 0, 12, 4, 8], [-28, -2, -1, -4, -1, 3, 5, 8, -8], [-45, 11, -5, 2, 17, -23, 2, 15, 7], [26, -9, -5, -4, 13, -3, 7, 17, -7], [-15, -2, -13, -8, 4, -4, 4, 2, 5], [13, -12, -15, 5, 29, 9, 5, -5, -7], [-5, -8, 7, 16, 17, -4, -4, 19, 10], [13, 3, 5, -17, -15, -15, 13, 12, 4], [3, -18, -8, 20, 16, 3, -17, -3, -10], [13, 12, 7, 4, 13, -14, 16, 9, -3], [-6, 13, 2, 6, 1, -3, 4, -8, -3], [-18, -5, -14, -8, -6, -7, -12, -8, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8046_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8046_2_a_h();".
// 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_8046_2_a_h(:prec:=9)
    chi := MakeCharacter_8046_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_8046_2_a_h();".
// 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_8046_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8046_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![18, -143, -330, 9, 240, 43, -56, -13, 4, 1]>,<11,R![114, 585, -138, -871, 137, 338, -68, -31, 4, 1]>],Snew);
    return Vf;
end function;