
// Make newform 8034.2.a.r in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8034_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_8034_2_a_r();".
// 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_8034_2_a_r();".
// 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, -54, 87, 37, -123, 7, 45, -9, -4, 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], [-2412, 4811, 3935, -6789, -1267, 2869, -307, -272, 55], [-3444, 7065, -223, -11785, 2815, 4781, -1331, -442, 129], [1964, -5995, 1175, 12931, -2377, -5381, 1273, 498, -131], [-1977, 5938, 1539, -9287, 774, 3825, -819, -357, 92], [5692, -29201, 11433, 38965, -12675, -13551, 4171, 1176, -359], [10534, -39183, 3219, 49497, -10555, -17037, 4449, 1476, -415], [-5619, 20133, 28, -28776, 4656, 10779, -2575, -968, 261]];
    Rf_basisdens := [1, 1, 634, 634, 634, 317, 634, 634, 317];
    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_8034_a();"
function MakeCharacter_8034_a()
    N := 8034;
    order := 1;
    char_gens := [5357, 1237, 5773];
    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_8034_a_Hecke(Kf)
    return MakeCharacter_8034_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], [-1, 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], [-1, 0, 0, 0, 0, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 2, -1, 0, -1, 1, 1, 0, 0], [-1, -1, 2, 1, 1, -1, 0, 0, 0], [1, 0, -1, 0, 1, 1, -1, 3, 1], [-4, 1, 1, 0, -1, -1, 0, -2, -1], [-1, 1, -2, 1, -1, 1, 0, 0, 0], [-1, 1, 0, -1, 0, 0, -1, 2, -1], [-2, -1, 0, 1, 1, 0, 0, 0, 0], [-4, 1, -2, 2, -1, -2, 1, -2, 0], [2, 1, 0, 0, 0, 0, -3, 3, 1], [-1, -1, 1, 2, 2, -2, -1, 1, 0], [-2, 1, 2, -1, 0, -1, 1, 0, -3], [-2, 2, -1, -1, -1, 2, 0, 1, 1], [-1, 0, 1, 1, 0, 3, 1, 0, -2], [-3, 2, 3, 0, 0, 0, 3, -2, -1], [-3, 0, 1, 0, 0, -1, -1, -1, 3], [1, -1, -1, -1, -1, 2, 0, 1, 4], [2, -2, -1, 2, -1, 0, 3, -3, 1], [2, 2, -2, -1, -2, -1, 1, 2, 3], [-4, 0, 1, 0, 0, 1, 0, -3, -2], [0, 1, 1, -1, 1, -3, 2, -2, 1], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 1, -3, -3, 3, -4, 0], [-1, -1, -5, 2, -1, -1, 2, -2, 2], [3, 0, 2, -1, 5, 4, -2, 4, -1], [0, 2, 1, -2, -3, 2, 4, -3, -3], [-1, -3, 1, -1, -1, -2, -3, 0, -1], [-4, 3, -1, -4, 0, 1, -5, 2, 1], [-1, 0, -3, 0, 2, 1, -2, 4, -2], [0, -1, -2, 2, 5, -2, -4, 4, 0], [3, -1, 0, 2, 2, -2, -3, 3, -2], [-6, -5, 4, 0, 3, -3, 0, 0, -3], [-2, -6, 2, -1, -1, -2, -5, -1, 1], [-1, 1, 0, 2, 3, -2, -1, -1, -4], [-2, -5, 4, 1, 5, -3, -1, 0, -3], [4, -4, -1, -1, -3, -2, 0, -4, 2], [-4, -2, -1, 0, 0, 0, -5, -1, -1], [5, -1, 1, -3, -1, 4, 1, 5, -2], [-4, -1, -4, -1, -1, -2, 5, -3, 1], [-7, -1, 0, -2, -3, 2, -1, -2, 3], [-6, 3, -2, -3, -1, 4, -3, -2, 0], [-2, 3, 0, -2, -1, 3, 5, 4, 1], [-1, 0, -6, 8, 0, -3, 3, -2, 4], [-1, 7, -1, -3, 0, 1, 2, 3, 1], [-4, -1, -1, 0, -5, 2, 2, 0, 0], [-4, 0, 1, -2, -3, 4, 3, -3, 1], [1, 5, -5, -1, -5, 7, 4, -4, 2], [-4, 2, -2, 0, 0, -3, 4, -5, 2], [-5, -2, 0, -3, -1, -3, -2, -2, 0], [6, 0, 2, -5, -1, 4, -2, -2, -2], [3, -7, -1, 4, 2, 0, 0, 2, -1], [3, -3, -6, 0, -4, 2, 1, -3, 4], [-3, 0, 5, -3, -1, 0, -2, -1, 2], [-13, -3, 3, 0, -1, -2, 5, -5, -2], [-1, 2, -1, 0, 0, -3, 1, 6, 0], [6, -3, 4, -3, 2, -5, -5, 2, 3], [-7, -1, 8, 0, 1, -3, 1, -6, -6], [-3, -2, 1, 2, -1, 1, 5, -8, -2], [-2, -2, -1, 0, 3, -4, 1, 1, -3], [-8, -4, 2, -1, 2, 0, -3, -2, 0], [-8, 3, 0, 3, 3, 0, 4, -1, -1], [-3, 0, 6, 5, 6, 0, -1, 1, 0], [-7, -2, 5, 0, 2, -1, 2, -2, -2], [3, -2, -5, 0, -5, 2, -1, -1, 8], [-3, 1, -4, 1, 5, 6, -2, 4, 1], [8, -1, -2, -2, 3, 3, -2, 14, 5], [-11, 7, 0, 1, 3, 3, 4, -5, -5], [0, 9, -4, -4, -2, 5, -2, 4, -3], [-9, -3, 5, 5, 4, 1, 3, -3, -1], [-4, -2, 1, 3, -5, -5, 2, -2, -1], [10, -4, 1, 3, -3, 2, 1, 0, 6], [-1, 0, 7, -2, 3, 2, 0, -3, 0], [-1, 1, 5, -4, 6, 8, -2, 1, -4], [2, 0, 1, -7, -6, 4, -4, 4, 0], [-1, 6, -4, -1, 0, 0, -4, 11, 4], [-5, -3, 5, 1, 6, 0, 1, -6, 3], [-5, -6, 2, 5, -1, -7, 9, -6, -2], [-9, 2, -2, 3, 3, -3, -3, 3, 3], [-15, 3, 4, 4, -2, -4, 7, -5, -2], [4, -1, 0, 4, 3, -6, 6, 0, -2], [1, -3, 3, 3, 4, 3, -6, 8, -2], [-5, -1, -1, 3, 0, -5, 1, -6, 2], [-8, 3, 2, -2, -4, 2, 3, -3, -3], [4, 0, 0, 0, 2, 3, -5, 5, -4], [8, 1, -5, -2, 2, 1, -2, -1, 0], [10, -3, -4, 2, -2, -2, -3, 2, -2], [1, 2, 1, -1, 1, 0, -4, 1, 0], [2, 1, 4, 1, 9, -4, -4, 4, -4], [-4, -2, 7, -6, -1, -3, 1, -10, -5], [-1, -2, -1, -4, 1, -1, -9, 9, 1], [-1, 8, 4, -2, 1, -4, -2, -1, -2], [-9, 6, -1, 1, 1, 2, 6, -2, -1], [5, 11, -5, 0, -5, 2, 7, -3, -2], [-3, 3, -6, 0, 3, 3, -4, 4, -2], [2, 0, -7, 3, 4, -5, -2, 13, 2], [2, 5, -8, -3, -2, 2, -5, 6, 6], [7, 1, 4, -1, -2, 4, 0, 1, 0], [3, -3, -3, 6, 0, -1, 9, -8, 0], [8, -2, -3, 2, 4, -8, -2, -1, 3], [2, -4, 1, -2, 3, -2, -4, 8, 8], [-4, -7, -2, 0, 3, -3, -2, -6, -1], [-3, 4, 4, -5, -4, 1, -2, 1, -5], [4, 2, -4, 1, -8, -6, 3, -1, 3], [13, -2, -10, -2, 2, 8, -6, 11, 8], [7, 1, 4, -5, 0, -2, -6, 9, 8], [1, -10, 2, 6, 4, -1, 8, -6, 0], [-2, 0, -1, -7, 0, 4, -2, -5, 1], [5, 7, -11, -5, -8, 4, -1, 1, 2], [7, -2, 0, -7, -9, 5, -2, 0, 6], [1, 10, 4, -7, -2, 3, 7, 1, -3], [-1, 0, 4, -3, 2, 7, 0, -4, -6], [-5, -4, -2, 6, -3, 1, -3, -6, 0], [3, 2, 1, 3, 0, -3, 2, -10, 0], [-6, -6, 6, 2, -4, -10, -7, -5, 7], [-17, -1, -4, 5, 0, -5, 3, -11, -5], [0, 14, -2, -3, 0, 2, 4, 7, -1], [2, -3, 3, 4, -2, 0, -2, 2, -6], [2, -8, 5, 6, -2, -12, -1, -8, 4], [2, -5, 5, -3, 1, 2, -1, 1, -3], [1, 8, 1, -3, 0, 10, 2, 6, -7], [0, -5, 5, 4, 2, -3, -7, -6, -3], [-3, 1, -7, 4, -4, 3, 7, -6, 6], [-9, 9, 0, -7, -3, 8, 3, 4, -7], [18, 7, -2, -3, -7, 8, 3, -2, 4], [16, -5, -4, 2, -1, -2, -1, 10, 0], [-2, -4, -1, -5, 1, 3, -2, 5, 4], [-1, -3, -1, 4, -8, -6, 5, -6, -5], [-1, -5, 1, -3, 5, 4, -7, 4, -3], [-15, -5, 0, -4, -3, -6, -7, -4, -1], [0, 3, -15, 9, -2, 1, 8, -4, 1], [-1, 0, -2, 2, -2, -2, -4, -5, -4], [9, -6, 2, 4, 1, 3, -5, 3, -3], [9, -2, 2, 2, 4, -8, -2, 4, -1], [-4, -1, -1, 3, 0, -7, 1, 11, 4], [-1, -4, -4, -6, 3, -1, 4, 1, -3], [6, -12, 7, 1, 8, 3, -2, -4, 3], [-8, -10, -3, 1, -6, 2, 6, -5, 5], [-3, 5, -6, -2, 1, -2, 0, 5, 4], [16, 1, -8, -3, 2, 7, -5, 10, 3], [1, 4, 1, -1, -8, -5, 5, -3, 7], [6, 6, -1, -3, -8, 4, 8, -6, 4], [-2, 0, 6, 3, -1, -4, 3, 2, -2], [14, -8, -8, 6, -4, 2, 2, -3, 1], [5, -4, -1, -1, 5, 1, 3, 4, -8], [3, -5, 8, 8, 5, -9, -9, -2, 0], [-1, 2, -3, -8, -13, 8, 6, -6, 7], [5, 12, 2, -14, -2, 10, -3, 1, -2], [-1, 0, -13, 9, 3, -4, 3, -9, 6], [22, -3, -7, -1, 3, 2, -2, 9, 11], [11, -3, 9, 7, 8, -3, 8, 1, -7], [4, 7, 2, -1, 2, -2, -5, 10, -4], [-2, -7, 2, 4, 3, -9, -4, -10, -1], [7, -7, -14, 3, 0, -4, 2, 1, 6], [10, -6, 1, -6, -1, -7, -2, -4, 5], [14, 0, -4, 4, -5, 0, 9, -5, 1], [7, -10, 4, 7, 6, 0, 9, -4, -5], [-3, 0, 1, -3, 1, 13, -8, 2, 0], [4, 0, -6, -10, -7, 4, -10, 11, 7], [13, 1, -1, -2, 0, 11, -2, 1, -7], [14, -1, 0, 4, 0, 3, -7, 17, -6], [3, 12, 6, -8, 3, 8, 3, 9, -4], [1, 1, 15, -2, 2, -5, 1, -4, -6], [6, 4, 3, -6, 2, 12, -5, 2, 2], [6, 11, -6, -9, -5, 18, -2, 4, 2], [7, 2, -3, 2, 1, -1, -9, 2, 4], [8, -12, 1, 1, 1, -3, 9, -4, -3], [-3, -7, 7, 6, -2, -4, -4, -7, -2], [-5, -7, -1, 5, -2, 0, 0, 12, 9], [0, 7, 2, 3, 2, -8, 13, -5, -3], [-9, 2, 9, -8, -3, -3, 5, -9, -5], [2, 0, 5, 0, -6, -3, 4, 4, 1], [-17, 3, -5, 6, 4, -5, -5, 4, 6], [-5, -1, 1, -4, 12, 9, -17, 14, -4], [-8, 8, 0, -2, 13, -6, -2, 9, -3], [11, 6, -8, -6, 3, 10, -1, 9, 8], [-3, 5, 5, -1, 2, -6, 0, -4, -9], [33, -13, -8, 2, 2, -5, -8, 11, 7], [-11, -2, 6, -3, 10, 2, -7, 12, -5], [22, -2, -8, 2, -3, 1, -7, 14, 9], [7, -11, 11, 3, 1, -17, 2, -13, 3], [-9, -13, -4, 6, -5, 5, 3, 2, 2], [10, -2, -14, 0, -9, -4, 1, -3, 7], [2, -8, 3, 8, 0, -1, -2, -13, -6], [-21, -10, -1, 9, 1, -4, 12, -14, -9], [-5, -9, 8, -3, 1, -4, 2, -7, 8], [-5, 7, 3, 1, 1, 0, 2, 3, 4], [-11, 0, -3, 7, 6, -10, 6, -2, 3], [-19, -4, 12, -1, -3, -8, 8, -10, -11], [1, 11, -8, -9, -9, 11, 7, -3, 1], [-12, 1, 1, -5, -1, 4, 3, -14, 2], [0, -6, -2, -5, -1, -2, 2, -9, -3], [-13, -11, 9, 1, 3, 7, 0, -3, -3], [-14, -6, 1, 4, 0, -2, 4, -2, 0], [4, -5, 10, -3, -10, 2, 11, -10, 0], [11, -1, 4, -5, 4, 10, -3, 15, -4], [-4, -10, 7, 4, -1, -8, -4, 0, -12], [8, 2, 16, -12, 0, 4, -8, -2, -2], [-8, 3, 0, -5, -8, 6, 12, 0, -4], [18, -7, 2, -8, 4, 16, -3, 8, 4], [-4, -8, 6, -4, 13, -7, -5, -2, -1], [-3, 5, 2, -5, -9, 1, 9, -7, -3], [15, 3, 7, -2, -2, 3, -3, 10, 0], [-14, 10, 4, -6, -1, 5, 4, -7, -2], [-5, -5, 9, 3, 5, 7, 6, 5, -1], [-26, -14, 7, 10, 5, -18, 7, -11, -1], [9, -14, -2, 8, 5, -9, 5, 3, 5], [3, 11, -2, 5, -3, -2, 4, -4, -7], [3, 4, -6, -4, -6, 6, -2, 16, -1], [18, -6, 5, -10, -2, 2, -18, 15, 3], [3, -8, 6, 1, 1, -6, -5, 2, 5], [-8, -1, 11, 5, 9, -6, -15, 18, -4], [4, -8, -4, -7, 2, -10, -16, 7, 7], [0, 6, -10, -5, -4, 8, -2, -7, -3], [-3, 12, 8, -5, 6, 7, -2, -5, -13], [-6, 6, 6, -3, 9, 2, -2, 14, 0], [-25, 9, -5, -5, -14, 0, 4, -1, 0], [-3, 6, 6, -12, -3, 10, 0, -7, -8], [25, -12, -3, -3, 4, 10, -12, 13, 0], [6, 6, 4, -1, 7, 12, -8, 14, -10], [5, 7, -9, -6, -18, 0, 12, -11, 6], [14, -4, -2, -1, -2, -2, 3, 5, -7], [0, 9, -11, 3, 5, 8, -4, 9, 1], [-2, -7, -10, 4, -1, 2, 5, -6, 2], [-15, 5, -6, 2, -3, -3, 1, -13, 5], [-26, -2, -3, -4, -8, 0, 1, -8, 10], [-6, 1, 0, 9, 2, 1, -4, 3, 6], [3, 5, 4, -2, 1, -3, -6, 0, -10], [-9, 11, -3, 6, -1, -3, -5, -4, 10], [5, -6, -1, 6, 1, -2, 17, -9, 0], [-2, -1, -14, 5, 2, 11, 5, 13, 0], [14, -1, 1, 6, -5, -10, 4, -16, -2], [7, 5, -1, 7, -6, 0, -2, 7, -2], [4, -5, -4, -7, -4, -2, 6, -9, -1], [-5, -9, -4, 9, -1, -2, 3, 9, 10], [15, -6, -12, 7, 7, -2, -4, -1, 0], [20, -11, 1, 0, -1, -5, -11, 9, 2], [3, -12, 0, -4, -1, 1, -1, -2, 12], [24, -5, -6, 2, -2, -1, 0, 5, 6], [12, 5, -8, -13, -3, 13, -6, 14, 1], [-4, 4, 18, -6, 4, -2, -12, 4, 0], [16, 8, -20, -8, -12, 8, 6, -2, 2], [19, 7, -5, -4, 2, -8, -6, -1, 4], [-9, -5, -5, 8, -14, -17, 5, -10, 6], [-1, 13, -10, -5, 1, 15, 4, 4, -8], [-10, 9, 1, -3, 2, 0, 7, 1, -11], [-7, -5, 7, 8, 15, 5, -5, 20, -2], [-5, 1, -16, -2, -9, 4, -4, -7, 6], [-5, -6, 3, 0, -7, -1, -5, -21, -5], [28, -3, -12, 2, -2, -2, -9, 11, 17], [4, 1, -2, -1, -6, -6, 2, 11, -3], [11, 4, -4, 0, 0, 13, 4, -5, 1], [-20, 11, -9, 0, -6, 16, 21, -10, 0], [10, 9, -10, -1, 4, 4, 2, 3, -1], [22, -14, -8, 2, 8, -1, -6, 10, 1], [-17, -12, 5, -3, -9, -13, -3, -22, -4], [-3, 1, 17, -3, -1, -1, 0, -11, 1], [4, -8, 2, 5, 8, -8, -12, 6, -6], [-25, -21, 7, 14, 8, -5, -7, 0, 4], [-5, 3, -9, 1, 10, 10, -4, 7, -10], [16, 22, -7, 2, 1, -2, 12, -2, -6], [15, 1, 7, -4, -9, 4, 6, -22, -1], [-7, -8, 16, -10, -1, 5, -9, -1, -1], [7, -8, 11, -8, 5, -12, -5, -16, -3], [23, -5, 5, -9, 6, 9, -5, 21, 7], [-8, -12, 17, 2, 3, -22, 4, -17, -3], [-9, -1, 8, 2, -3, -10, -3, -6, -9], [12, -9, 0, 7, 4, -3, -13, 15, -12], [-18, 4, 10, 4, -5, 3, 1, -6, -3], [-22, 12, 1, -1, -3, -3, -1, 0, -1], [-16, -9, 10, 3, 6, -1, -9, -3, -4], [33, 7, 2, -8, 1, 11, 2, 16, 2], [20, 2, -19, 0, -1, 11, 11, -1, -8], [-9, -11, 2, 9, -10, -9, 7, -7, 5], [1, -1, 2, -10, 14, -9, -5, 9, 3], [-4, -20, 4, -1, 4, -4, -2, -7, -9], [-6, -11, 5, 14, 1, 0, 8, 2, 8], [-30, 1, 2, 0, -7, -10, 0, -17, -1], [25, 6, -4, -10, -5, 3, 10, -4, 6], [-6, -8, 2, 15, 10, -8, -5, 12, 4], [18, 8, -13, -2, 0, 5, 7, -2, 5], [25, 9, 10, -2, 2, 11, 8, 16, -6], [3, -9, -1, 2, -1, 8, -2, 7, 16], [10, 2, -1, -5, -9, -3, 17, -9, 2], [22, 9, -15, -3, -4, -4, 13, -14, 8], [7, 6, -5, 7, 6, -6, -13, 11, 2], [-16, -5, -3, 9, -15, -1, 8, -15, -8], [-15, 9, -3, -2, 9, 11, -12, 6, 4], [10, -12, 4, 5, 5, 10, -4, 10, 4], [-11, -4, 11, -5, 1, 3, -11, -7, -9], [-12, -4, 0, 8, 4, -3, -5, -9, -4], [23, -5, 8, 4, 9, -8, 1, 4, 3], [9, -6, 4, 4, 7, -15, -1, 2, -10], [2, 16, -10, -1, -3, 9, -12, 14, -1], [11, -2, 9, -11, -3, 0, 7, -6, -1], [-3, -9, 8, -4, 10, 0, -10, 10, 13], [-12, -10, 8, 3, 17, 8, -14, 15, -5], [-21, 5, 1, 6, 6, -14, 11, -12, -5], [-1, -14, -1, 6, 2, -2, -6, -4, -13], [24, 6, -16, 6, -3, 5, 2, 3, 2], [-16, -13, -2, 5, -1, -2, 6, 4, -2], [19, 1, -7, -3, 4, 0, 5, 9, 18], [25, 9, 3, -13, -5, 12, 2, 7, 4], [-7, -12, 1, -5, 1, -5, -12, 9, -5], [13, -6, -3, -2, 3, -7, -21, 2, -2], [-5, -3, -9, 2, -2, -17, -1, -16, 10], [27, 3, -12, -7, 3, 7, -6, 20, 12], [-31, -2, -13, 4, -6, -4, 0, -4, 9], [8, 8, -8, -5, -10, 20, 3, -2, 8], [-22, 4, 6, 3, -3, -5, 0, 8, -3], [7, -9, -1, -5, -5, -5, 1, -4, 6], [-18, 1, -3, -4, 9, 6, -3, 4, -10], [-3, 14, -1, -2, -2, 14, 2, 8, -5], [25, 4, -9, -11, -13, -9, -12, 9, 11], [-8, 4, -7, 8, -10, -3, 10, -6, 3], [42, 2, 18, -11, 2, 4, -11, 10, 6], [7, 4, 13, 8, 18, 1, 3, 12, -10], [-5, 7, -6, 3, -6, 6, 6, 2, 5], [-11, -3, 0, -3, 10, 15, 0, 1, -5], [3, 13, -1, -1, -8, 0, 20, -17, 10], [3, 2, 0, -14, -14, 6, -5, -3, -4], [-9, -11, -4, -3, -2, -14, 0, -11, 12], [-32, 8, 4, 3, 8, -3, 2, -5, 6], [-6, 6, -14, -5, -3, 11, 2, 5, 2], [20, 8, -11, -9, 4, -9, -8, 7, 12], [-19, 9, -8, 11, -4, -6, -8, -11, 14], [-46, -5, 12, 1, -6, -2, -2, -8, -6], [10, -4, -20, 11, -7, -14, 8, 0, 14], [10, -4, 7, 2, 4, -16, 5, -12, -2], [19, 5, -11, 8, 13, 2, 4, 18, -1], [1, 21, -6, 2, 0, -3, 4, 16, -2], [5, 15, -15, 8, 6, 10, 7, 8, 5], [20, 1, -8, -10, -2, 12, -11, 9, 11], [-4, -4, 0, -5, 13, -8, 4, -2, -4], [-15, 6, -18, 5, -4, 7, 3, -2, 2], [-10, 15, 6, 8, -8, 5, 23, -18, -15], [-18, 11, -2, -1, -16, -5, 12, -19, 0], [-7, 3, -3, 11, 1, 3, 3, 4, 8], [0, 11, -5, -4, -2, 18, -5, 8, -6], [9, -4, -13, 3, -26, 0, 4, -13, 4], [-9, -8, 0, -2, -8, -3, 12, -10, 4], [5, -5, 9, -11, -6, -5, 9, -19, -3], [-35, -2, -1, -2, -3, -6, 7, -9, -8], [11, -13, 13, -9, 9, -9, -5, 2, -2], [-15, -2, 22, 16, 16, -10, 10, -10, -15], [-14, 4, -2, 6, 3, -16, 15, -2, -2], [-10, -8, 3, 16, 6, 0, -7, 8, -2], [-11, 18, 9, -2, 5, -2, 2, -2, -15], [19, -3, 3, 11, 14, 5, -7, 21, 5], [-37, 13, -2, 3, 3, 0, -2, -6, 3], [-16, -6, 8, 8, 5, -9, -14, -3, -4], [8, -1, 1, 17, 9, -1, 13, -8, 5], [17, -2, 8, -10, 2, 14, -9, 19, -6], [-28, -12, 9, 13, 2, -12, 0, -7, -7], [23, -2, 10, 8, -3, -7, 0, -15, -1], [24, -9, 4, 0, 6, 7, -7, 3, 14], [1, 2, 11, -12, 7, 8, -11, 9, -8], [-9, 12, 12, -4, 4, 6, 2, 3, -14], [18, 14, 6, -13, -7, 25, 6, 2, -3], [-1, 11, -16, -10, -2, 8, -13, 16, 15], [7, -30, 5, -5, -2, -12, -14, -8, 5], [-42, -11, 15, -5, 9, 2, 6, -2, -8], [20, 13, -14, -8, -16, 17, 2, 4, -7], [1, 12, -10, -15, -14, 12, 7, -10, 11], [-5, -4, 17, -6, 3, -15, -16, 8, 8], [-3, -19, 3, 6, 1, 3, 14, -1, -1], [-4, -9, 14, 2, 4, -5, -16, 24, 3], [-3, 4, -3, 2, 15, -7, 3, -9, -1], [22, -11, 10, 12, 7, -20, -10, -3, -5], [8, -11, 3, 1, 5, 12, 10, -9, 5], [-30, 7, 10, -2, -1, 3, -6, 9, -14], [23, -13, 3, 7, 12, -15, -6, -2, -4], [-9, 11, -5, 11, -7, -2, 15, 10, 9], [-19, 12, -2, -3, -9, -1, 18, -18, -2], [-4, 9, 22, -6, 11, 3, -8, 12, -15], [-33, 4, 2, -11, -2, 5, 8, -15, -1], [-7, 12, 0, -1, -19, 2, 10, 3, 4], [0, 17, -4, 2, 1, 11, 9, 4, 9], [12, -3, -16, -4, 0, 6, -2, 7, 7], [7, 21, -2, -15, -3, -2, 3, -11, -12], [-3, 11, 4, -3, -7, -2, 1, 15, -8], [5, -10, -10, 3, 1, -10, -9, 1, 2], [1, 6, 0, -20, -11, 15, -19, 14, -2], [-10, 23, -8, -17, -16, 24, 18, -13, -3], [3, 8, -22, 7, 13, -9, -12, 23, 7], [-17, -11, 16, 7, -2, -5, 1, -8, 0], [-40, -2, -3, 12, 6, -24, 5, 7, 5], [-20, -13, 15, -2, 9, -8, 8, -11, -27], [-23, 7, -13, -8, -4, 5, -13, 1, 9], [-34, 5, -10, -11, -14, 12, -4, -6, 12], [1, 13, -6, -14, 14, 9, 2, 14, -2], [-23, -3, 0, 11, -6, 3, 4, 9, 7], [-10, -18, 26, -7, 3, -7, -14, -6, 5], [31, -2, 5, 2, 4, -6, 5, -20, -7], [-7, -13, -5, 11, 16, -12, -21, 13, -10], [-7, 0, -2, 19, 8, 2, 16, -1, 0], [-6, -3, -15, 3, -15, -8, 5, 2, 20], [14, -8, 5, 4, -2, 5, 1, -6, -5], [14, 2, 25, -2, 6, 6, 5, 10, -12], [22, -22, 8, 7, 17, -13, -19, 8, -3], [-4, 16, 13, -4, -8, 3, 23, -9, -6], [14, 14, -15, -11, -15, 0, 4, -3, 17], [-22, 3, -7, 12, 3, -11, -3, -2, 11], [-39, -1, -2, 21, -2, -21, 3, -9, -9], [7, 8, -11, 13, -8, -2, 18, -10, -1], [5, 2, 13, -2, 14, 5, -1, 20, -8], [0, -7, 2, 10, 7, -4, 11, -11, 1], [-11, 5, -6, 1, -2, -8, -11, 10, -1], [-31, 16, 4, 1, -6, -16, -10, 5, -6], [-26, 8, 1, 2, -6, -11, 13, -14, -15], [-40, 17, 11, -5, -7, -6, 25, -25, -11], [-26, 5, 13, -6, -1, 7, -12, 1, -14], [-40, 0, 0, 9, -5, -19, -4, -11, 4], [-14, 1, 15, -5, 8, 5, -2, -7, -6], [-10, 17, -19, 9, -23, 3, 23, -18, -3], [-27, -8, 19, 10, 6, 3, -5, -4, -22], [-3, 9, -7, -8, -8, 16, 3, -6, -1], [-29, 14, 4, -11, 1, -4, 6, -11, -2], [15, -14, -4, 4, -4, -7, -11, 7, 17], [-10, 7, 11, 2, -6, 7, 0, -4, 5], [9, -3, 3, -3, -5, 9, 0, -15, -5], [-31, -13, 10, 0, 3, -17, -1, -10, -12], [-17, 21, -18, -4, -13, 19, 16, 6, -6], [13, 29, -13, -16, -1, 17, -14, 13, -1], [26, -19, 3, 0, 10, 13, 1, 16, -3], [24, -15, 7, 3, 22, 4, -10, 21, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8034_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8034_2_a_r();".
// 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_8034_2_a_r(:prec:=9)
    chi := MakeCharacter_8034_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_8034_2_a_r();".
// 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_8034_2_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8034_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-8, -54, -87, 37, 123, 7, -45, -9, 4, 1]>,<7,R![200, -199, -929, -210, 456, 139, -74, -22, 4, 1]>],Snew);
    return Vf;
end function;