
// Make newform 8034.2.a.p in Magma, downloaded from the LMFDB on 25 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_p();".
// 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_p();".
// 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 := [20, 23, -49, -38, 43, 12, -12, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-17, -5, 32, 1, -11, 0, 1, 0], [-11, -53, 7, 92, -6, -32, 1, 3], [-25, -37, 43, 60, -18, -21, 2, 2], [-28, -37, 44, 60, -18, -21, 2, 2], [46, -2, -95, 29, 33, -11, -3, 1], [-45, -47, 76, 62, -29, -21, 3, 2]];
    Rf_basisdens := [1, 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_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], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 1, 0, 1], [-1, 0, 0, 0, 1, -1, 0, -1], [0, -1, -1, -1, 0, 0, 2, 1], [1, 0, 0, 0, 0, 0, 0, 0], [-3, 1, 0, 0, -1, 0, -1, -1], [-2, 0, 0, 2, -1, -1, -2, -1], [-1, 1, 2, -1, -2, 0, -1, 0], [-4, -1, -1, 0, 2, -1, -1, -1], [-2, 0, 4, 2, -3, 1, -2, -3], [-2, 2, 1, 0, 0, -2, -2, -1], [-3, 1, -1, -1, -1, 2, 0, 0], [-2, -2, -1, 1, 2, 1, 0, 0], [-1, 1, -4, 1, 1, -1, 2, 3], [-2, -2, -1, 1, -1, -2, 1, 0], [1, 1, 0, -1, -2, -1, 2, 2], [1, -1, -1, -4, 1, 2, 3, 3], [-1, 0, -1, 1, 2, -1, -1, 2], [-3, 0, -1, -2, 2, 3, 1, 0], [0, -2, 5, 4, -4, -2, -1, -6], [0, 4, 4, 4, -2, -1, -1, -2], [0, -4, -1, -6, 4, 3, 3, 3], [-2, -1, 2, -3, 0, 0, -2, -1], [2, 6, 4, 4, -1, -2, -3, -1], [1, -3, -3, 0, 1, 0, 3, 6], [1, 0, 0, 0, 0, 0, 0, 0], [-5, 3, 0, 0, -1, -2, 0, -3], [-4, 2, -2, 2, -1, 3, 1, -1], [-8, 2, 6, 3, -2, 3, -2, -1], [0, -5, -7, -2, 2, 5, 5, 2], [-4, 1, 4, 3, -1, -1, 1, -2], [3, -1, 2, -5, 2, 3, 1, 6], [-4, 1, -7, -1, 9, 0, 3, 4], [-2, -1, -5, -3, 0, 3, 4, 3], [-1, 6, 5, 1, -2, -3, -4, -1], [-1, -1, -5, 0, 1, 0, 4, 4], [-2, 0, -7, -6, 4, 3, 4, 5], [-4, 1, -4, -6, 6, 3, 4, 1], [-6, -4, -1, 0, 2, 1, 2, 2], [-5, 5, 1, 1, -2, 0, -5, 2], [5, -7, -2, -1, 8, -5, 1, 1], [0, -3, -2, -2, -2, 7, 5, 5], [-3, -2, 3, -2, 2, 4, -4, -4], [-6, -3, -2, 1, -3, 5, 0, 2], [-1, 2, 1, 3, 1, 2, -2, -3], [-7, -2, 1, 8, -3, -4, -4, -5], [1, 2, -1, -3, 4, 1, -2, 6], [0, -1, -1, 0, 7, -4, 3, 1], [1, -3, -5, 1, 1, -2, 3, -2], [2, 0, 7, 4, -6, -3, -3, -5], [-1, 2, 8, 8, -6, -4, -7, -2], [-5, -2, -6, -6, 8, 5, 6, 4], [-5, -4, 2, 8, -1, 4, -2, 2], [-4, -3, -5, -1, 5, 1, 2, 5], [4, 1, -3, -4, -4, 10, 7, 6], [-5, 2, 9, -3, -2, -3, -6, -1], [-3, -6, -2, 0, 8, -6, -3, -5], [-5, 7, 0, 0, -1, -2, -7, 0], [0, 0, 1, 9, -1, -4, -3, -7], [-3, 7, 6, 1, -6, -1, -4, -4], [1, -9, 0, -4, 0, 0, 4, -7], [7, -1, 8, -2, -7, -2, -1, -5], [2, 4, -5, 2, 4, -7, -1, 1], [-5, 3, 4, 6, 1, -7, 6, -6], [-1, -5, -3, -4, 6, 2, 8, 9], [7, -2, 5, 3, -5, -5, -4, -5], [4, 3, -1, 1, 3, -2, 1, -6], [-6, 7, -2, -4, -3, 1, -7, 5], [5, 4, 4, 5, 2, -14, -2, -4], [-3, -1, -1, 2, 3, 0, -4, -4], [6, 4, 0, -3, -2, 3, 4, 6], [-6, -6, -4, -9, 3, 10, 2, -2], [0, 2, 9, 10, -9, -4, -3, -8], [-10, 1, 5, 6, -1, -7, -11, -5], [1, -2, -9, -7, 6, 4, 7, 7], [8, 0, 4, -4, -9, 5, -3, 0], [8, -5, 5, 5, -7, -6, -1, -1], [-8, 2, 6, 4, -2, 0, -5, -10], [5, 5, -1, -5, 5, -3, 0, 4], [-3, 0, -3, 0, 3, -3, -5, 0], [9, -3, 8, 4, -15, -2, -10, -5], [5, -3, -4, 1, -6, 4, -6, 8], [-1, 6, -9, -11, 6, 7, 4, 9], [-2, 5, -3, -5, 0, -1, 2, -5], [-6, -7, -1, -3, 6, 4, -7, 0], [-6, 12, 5, 1, -4, -2, -5, -1], [-13, 7, -4, 5, -5, 1, -4, -1], [7, 1, -6, -2, -4, -1, 2, 1], [1, 4, 0, 6, -9, 7, -1, 0], [5, 1, 12, 8, -16, -5, -12, -14], [1, 4, 2, 14, -10, -6, -9, -13], [11, -3, 9, -3, -7, 6, -2, -4], [-7, 0, 5, 2, 8, 4, -11, -1], [3, 5, -2, -13, -4, 8, 9, 10], [13, -4, 8, 6, -12, 0, 0, -5], [1, 8, 6, 11, -5, -6, -1, -10], [-9, -1, -2, 0, 5, -6, -3, 8], [5, 0, 8, 0, -16, -2, -12, -2], [3, 0, -12, -4, 6, -3, 7, 0], [-8, -15, -6, -4, 11, 0, 4, -9], [-10, 9, 5, -5, -1, -2, -8, -11], [7, 2, 1, -3, 1, 3, 7, 6], [1, -6, -4, -11, 4, -3, 11, 2], [10, 7, 9, -3, -9, 5, -9, 1], [3, 5, 5, 4, -5, -4, -8, 0], [0, -7, -9, -2, 8, 0, 8, 3], [7, -5, 14, 13, -8, 3, -6, -3], [-8, -3, -1, 1, 4, 0, 8, -3], [4, -9, -7, -7, 7, 4, 12, -1], [-3, -9, -7, 3, -4, 5, 4, 5], [6, 6, -2, 1, -3, -6, -3, -5], [-3, 2, 3, 11, -6, -3, -5, -1], [1, -10, -12, -4, 13, 10, 14, 6], [6, 2, 5, 10, -8, -8, -8, -4], [-10, 7, -5, -7, -3, 0, 6, 2], [30, -1, 6, 2, -8, -3, 0, 0], [3, -1, -11, -5, 4, -3, 3, 9], [-4, -1, -6, -5, 3, -2, 10, 4], [5, -10, -1, 0, 10, 1, 8, -3], [15, 2, 0, 3, 8, -7, -4, -2], [6, 0, 16, 2, -5, -4, -3, -8], [-17, 1, 10, 12, -11, -9, -5, -16], [12, 0, -11, -2, -3, -1, 7, 15], [2, 5, 15, -4, -11, 6, 5, 0], [-5, 7, 2, -1, 4, -7, -7, -6], [-1, -2, 7, -7, 3, 5, 5, -6], [1, 1, -2, -17, 2, 6, 10, 11], [9, 6, -9, -11, 5, 10, 2, 19], [10, -12, -13, 1, 2, 5, 14, 8], [6, 1, -7, -5, 8, 3, 11, -2], [4, 0, 15, -4, -2, -10, -7, -7], [14, 7, -2, 1, 7, -3, 1, 8], [2, 1, -2, 15, -11, 1, 0, -8], [-2, -10, -16, -5, 9, -4, 11, -4], [6, 5, -11, 3, -6, -10, 6, 3], [14, -11, -8, -5, 5, -5, 6, -7], [10, -1, -9, -12, 9, 16, 6, 18], [-4, -7, 0, -14, 4, -4, -1, 2], [0, -6, -14, -3, 13, -3, 12, 12], [-5, -6, -16, 1, 5, 11, 11, 5], [6, -12, -12, -9, 9, -6, 5, 3], [8, -10, 4, 8, 0, -5, -6, 0], [5, -7, 2, -5, 8, 1, -10, -2], [3, 8, -5, 8, -7, -17, -5, -5], [2, 4, 2, -12, -2, -2, 7, -8], [2, 2, 15, 3, -7, -9, -14, -12], [2, -14, -2, -3, 11, -6, 6, -7], [10, 7, 10, -10, -8, -11, -9, -2], [-11, -9, -6, 14, 4, -2, 0, -7], [14, 6, 4, 11, -4, -5, -7, -5], [0, 0, 7, 1, -12, 18, 0, 6], [1, -9, -14, 2, -2, 2, 5, 8], [-26, 5, 5, -4, 7, -5, -5, -16], [-12, -13, 3, 8, 1, 0, -7, -1], [0, 9, -8, -6, 11, 2, 10, 3], [-3, -1, 3, 8, -1, -14, -10, -14], [-4, 5, 7, 7, 1, -8, -14, -17], [-11, -7, -3, -2, -2, 6, -3, -8], [7, 10, 9, -4, -5, 2, -6, 14], [-13, -2, -9, 2, 10, -11, -2, 8], [-6, -8, -12, 2, 0, -3, -1, 1], [13, -5, -7, 0, 13, 0, 0, 9], [17, -1, 7, -8, -6, 7, -3, 0], [12, -19, -2, -8, 12, 2, 16, -2], [-5, 1, -8, 6, 8, 0, -4, -2], [-5, 6, 1, -3, 7, 1, -11, 8], [-13, -5, -2, 2, -5, 5, 1, -8], [-11, 7, -13, -11, 7, 0, -6, 6], [-13, 7, -10, -2, 1, 6, 3, 3], [-12, 6, -3, 4, 3, -11, -13, -8], [8, 7, 7, 9, -5, 2, -3, -11], [1, -3, -3, -2, 5, -13, 9, -17], [12, 5, 16, 1, -12, 8, -8, -6], [15, 15, 14, 3, -22, 0, -11, -4], [4, 8, 12, 5, 8, -10, -3, -6], [-3, -12, -13, -9, 13, 5, 13, 10], [7, 4, 9, 13, -13, -5, -19, -18], [-7, 3, -1, 0, 13, -8, -2, -7], [13, -3, -6, 0, 8, 2, 6, 12], [-9, -5, 9, 20, -1, -15, -11, -9], [2, -1, 9, 4, -13, -10, -8, -13], [-11, 0, -29, -6, 20, -3, 16, 8], [-22, 12, -1, -4, 7, 3, -4, -6], [19, -8, 6, 7, 1, -2, 3, 1], [7, -3, 4, -12, 10, -14, 8, -11], [7, 10, 15, -4, -1, -4, 0, -2], [8, 8, -12, 0, -5, 8, 7, 21], [-3, 12, 16, 7, -9, 0, -15, -10], [15, -6, 4, 6, -9, -4, -6, -13], [-8, -6, -18, 4, 18, 11, 11, 6], [18, 5, -15, 0, 7, 0, 11, 6], [-10, 0, 20, 9, -6, -11, -18, -27], [2, 5, 0, -21, 2, -3, 4, -1], [-20, 12, 9, 1, -6, -3, 1, -10], [-10, 9, -5, -8, -6, 5, 4, 16], [19, -8, -16, -11, 15, 8, 7, 21], [-3, 4, -12, 4, 7, 3, 11, 19], [-4, -5, -4, -6, 11, -6, 16, 1], [-20, 0, -7, -11, 9, 10, 2, 2], [18, 1, 15, 21, -13, -5, -13, -5], [2, 8, 4, -2, -10, -2, 4, -4], [14, 6, 14, -2, -7, -10, -8, -6], [-9, 5, -22, 8, 6, -15, 0, 2], [8, -10, -1, -8, 5, 17, 8, 4], [6, 13, 18, -1, -12, 1, -6, 5], [1, 2, 19, 19, -13, -9, -11, -14], [6, -13, -15, 1, 10, -11, 4, 14], [5, -4, -18, -5, 11, 6, 15, 27], [-8, 16, 17, 0, -2, 1, -5, -5], [-8, -1, 8, -7, -1, 7, -4, -15], [-25, 10, 22, 13, -24, -6, -12, -20], [-10, -3, -13, -2, 19, -10, -4, 2], [11, 6, -10, -5, -2, -17, 5, -4], [-2, -5, 22, -1, 0, -11, -18, -22], [-4, -2, 21, 3, -12, 15, -5, -10], [9, -4, 12, 1, -20, -1, -2, -10], [1, 4, -18, -5, 14, -9, 2, 12], [2, 3, 7, 14, -6, -4, 8, -1], [-8, 11, -7, -3, -3, -3, -10, 5], [-3, -5, -1, -25, 12, 21, 19, 16], [-25, 14, -4, 2, -14, -2, -8, -5], [-26, -7, 3, 11, 0, 10, -3, 1], [4, -7, 1, 3, -1, -5, 8, 5], [6, -5, 13, -11, -1, 8, 11, 9], [-1, -9, -2, 1, 18, 3, -2, 13], [-8, 11, 9, 6, 3, -5, 1, -18], [-8, -4, -19, -7, 34, -7, 6, 10], [-1, -7, -4, -19, 5, 20, 10, 22], [-15, -9, -20, -8, -5, 17, 7, 6], [-9, -4, -2, -12, -9, 21, 10, 8], [-9, -10, -4, -4, 8, -2, 14, 8], [0, 1, -4, 4, 11, -5, -7, -12], [-10, 17, 0, 0, -1, 9, -9, 13], [15, -1, -16, -6, -1, 13, 8, 19], [-5, -9, 3, 1, -9, 4, -7, 8], [9, 1, -11, -7, -2, 13, 7, 14], [-12, 4, 6, 13, -12, 8, -4, 5], [-5, 1, 8, -8, -11, 8, 6, 14], [-14, 6, 12, 14, -9, -9, -8, -6], [-12, 0, 7, 7, 5, -11, -5, -27], [-22, -9, -14, 18, -2, -1, 0, -10], [-14, -2, 20, 14, -16, 14, -6, -7], [8, -11, -11, -8, 6, 3, 30, 23], [0, -2, 2, -2, -12, 18, 10, 10], [16, 2, -12, 2, 10, 8, 8, -8], [-2, 11, 15, 15, -29, -2, -19, -15], [13, -4, -3, -3, -7, 9, 17, 8], [-2, -8, 16, 30, -11, -21, -18, -25], [21, 5, -14, -13, -5, 19, 21, 21], [10, 7, 10, 6, -3, -17, -3, -6], [16, -10, 0, 8, -4, 3, -4, 14], [-5, 1, -4, -15, 12, 6, 17, 12], [-29, 13, -6, 7, -1, -3, 6, 3], [-20, 0, 4, 3, -4, 18, 8, 17], [1, -1, 11, 10, 3, 11, -12, 4], [2, -1, 16, 4, -7, -4, -9, -5], [6, 2, 28, 33, -18, -14, -22, -17], [17, 10, 10, 17, -7, -9, -4, -7], [14, 9, -22, -16, 26, 5, 1, 30], [15, 10, 36, 15, -23, 2, -25, -13], [-11, -6, 6, 8, 5, 13, -2, -3], [3, -14, -9, 5, 23, -5, 5, -4], [-17, -1, 15, 9, -16, -5, -13, -14], [4, -5, 1, -11, -5, 23, -1, 9], [-17, -6, -12, -5, 2, 4, 3, 6], [-16, 7, -18, -9, 15, 7, -4, -3], [20, 1, 4, -11, -4, 13, 13, 12], [14, -2, 7, -12, -3, 8, 7, 12], [4, -5, -15, -16, 12, 23, 9, 12], [16, 7, 8, 25, -17, 7, -16, 0], [5, -19, -2, -8, 3, 7, 14, -9], [8, 10, -21, 6, 13, 5, 6, 8], [-20, -6, -5, -2, 11, 13, -5, 0], [-3, 1, -16, -6, 13, -15, 0, 9], [-15, 4, 6, -12, 6, 16, 6, -2], [1, 18, 1, 13, -21, -14, -1, -6], [-9, -2, 13, 4, -4, 17, 0, 9], [-4, -4, -13, -4, -5, 28, 6, 24], [-5, -16, -6, 7, -4, 1, 10, 12], [-21, -1, -13, -3, 17, 14, -5, 0], [14, -7, 9, -8, 9, 7, -6, -4], [-4, 4, 12, 4, -18, 5, -20, -3], [19, -5, -2, -10, -1, 19, 8, 18], [12, 1, -8, -5, 6, -22, 13, 3], [-10, -13, -9, 5, 10, 0, 4, -6], [11, -6, -16, -4, 19, -10, -5, 3], [-14, 2, 1, 15, 8, -11, -5, 11], [29, 9, -12, -16, 16, -5, -3, 16], [6, 18, 7, -13, -8, 10, -5, -4], [-13, -21, 7, -15, 0, 8, 1, 5], [-14, 12, 0, 8, -15, 15, 5, 5], [-6, 5, -13, -25, 7, -3, 16, 1], [4, -3, 6, 3, 3, -3, -23, -15], [7, -1, 8, 18, -14, -27, -6, -27], [22, 1, 26, 5, -5, -9, -16, -16], [16, -5, -8, -6, 8, 11, 8, 16], [-21, 7, 5, 0, 2, 4, 14, 12], [-7, -5, -24, -13, 17, 10, 21, 23], [15, -13, -37, -26, 29, 17, 38, 35], [-2, 17, 1, 6, 2, -11, -26, -12], [0, -6, 1, -18, 2, 14, 13, 9], [9, -7, -13, -4, 4, -11, 9, -11], [-8, 15, 27, 5, -26, -5, -18, -4], [-5, 17, 3, 3, -15, -12, 2, -4], [-17, 0, 5, -7, -6, 5, -13, 15], [-8, 22, 10, 8, -10, 9, -7, 12], [22, 8, -6, 1, -9, 1, -5, 4], [23, 1, 14, -6, 3, -2, 3, 3], [23, 6, 13, 5, -7, -19, -5, -10], [24, -6, 12, 22, -19, -5, -8, -7], [25, 9, 5, 8, -2, 9, -5, -3], [-33, -13, -6, 10, 9, -11, 14, -12], [-5, 27, 13, 20, -22, -4, -12, -4], [17, 9, -6, 4, 0, 8, 1, 3], [-19, 10, 25, 17, -7, -22, -27, -37], [3, -5, 7, -16, 8, -11, -3, -15], [-16, 2, 32, 15, -21, -11, -29, -24], [18, 16, 8, 5, -10, 4, -3, 10], [-7, -1, 0, -2, -1, 3, 12, 24], [6, 14, 11, 15, -11, 6, -15, -7], [4, -9, 5, 0, -10, -2, -10, -16], [-25, 3, 29, 22, -14, -3, -22, -28], [-1, 7, -1, -9, -2, 13, 13, 6], [6, 14, 27, 23, -14, -22, -12, -12], [-16, 7, 7, 3, -3, -2, 12, -1], [-6, -16, 0, -16, 9, -4, 20, -3], [11, 15, 23, -7, -17, 10, -12, -15], [1, -8, -26, -8, 28, -5, 17, 10], [2, -23, 1, -1, 13, 2, 6, 3], [-18, 0, -14, -20, 13, 24, 4, 18], [-12, -11, -9, -13, 11, 25, 24, 19], [3, -17, 6, -8, -13, 17, 3, 6], [-13, 6, -2, 17, -6, 12, -1, 2], [-16, -3, -25, -27, 18, 14, 18, 8], [20, -8, 11, 0, -10, -26, -7, -13], [3, 1, -12, 7, -1, -5, -6, 7], [-38, -5, 1, 9, 1, -7, -8, -23], [-14, -10, -3, 13, 1, 6, -6, -11], [12, -5, 30, 5, -11, 6, -24, -13], [13, -24, -24, -14, 27, 9, 26, 16], [-7, -15, -14, -6, 16, 10, 3, 3], [-16, 1, -4, -14, -3, 16, 7, 9], [-38, 19, 13, -1, -6, -8, -7, -27], [-1, -19, -37, -21, 24, -11, 24, 19], [34, 12, -15, -10, 15, -24, 7, 4], [8, 21, 10, 8, -7, 7, -21, 1], [25, 5, 10, -12, -6, -12, -3, -5], [2, 9, 25, 12, -15, -12, -16, -22], [5, -22, 3, 19, -2, -22, -16, -13], [39, 3, 22, 14, -19, -9, -5, -6], [-30, 4, 14, 9, 4, 13, -9, 17], [-10, -6, -3, 10, 13, -14, 19, -6], [-1, 10, 0, 17, -4, -10, -26, -16], [10, 15, 5, -3, -12, 1, -10, 8], [-15, -8, -11, 2, 15, -2, 11, 7], [26, -16, -27, -17, 18, 2, 4, 6], [2, 28, 12, 3, -19, -10, -15, 5], [-14, -20, -28, -19, 39, -11, 22, 8], [6, 5, -8, -14, 16, 2, 0, 16], [10, 5, 2, -14, -25, 27, 13, 15], [-14, 11, -11, -3, 2, 4, 8, 13], [-3, 9, 9, -2, 10, -24, -14, -16], [-45, 6, -5, -14, 25, 3, -4, 8], [24, -13, -5, 4, 15, -12, 9, 24], [0, 17, 29, 9, -8, -9, -21, -19], [2, -30, -26, -23, 17, 0, 30, 8], [11, -12, 1, 6, 2, 13, 6, 18], [15, 4, 16, 32, -19, -12, -9, -6], [-6, 22, -14, 7, 4, -11, -13, 4], [-24, 2, -18, 5, 7, -6, -4, 4], [13, 9, 8, 1, -8, -26, -1, -20], [4, -1, 5, 4, 7, 9, 4, 14], [28, -1, 1, -2, 3, 5, 7, -16], [-11, -9, -5, 17, -4, -8, 6, 1], [-18, 21, 11, 19, -18, 0, -17, -12], [-26, 27, 10, 3, -1, -15, -17, -18], [-1, 4, 34, 30, -17, -24, -34, -40], [13, -27, 1, 10, -5, 2, 2, 2], [-6, 13, 5, 0, -26, 22, -2, 5], [-26, -8, 2, -2, -3, 15, 0, 18], [13, 6, -15, 8, -1, -2, 8, -11], [7, -28, -18, -11, 9, -1, 8, 16], [-13, -3, -16, 12, -3, 0, 10, -8], [5, -17, 4, -12, 7, 0, 10, -6], [24, 1, 0, -4, -17, -7, 12, -9], [-4, 5, 6, -8, -6, 7, -2, 20], [20, -14, -30, -17, 4, 14, 18, 11], [9, -8, -20, -19, 14, 4, 14, 13], [15, 8, 9, 9, -11, 3, -30, -16], [13, -5, -24, -3, 8, 1, 23, 25], [-19, 9, 25, 10, -20, -10, -15, -35], [-5, 12, -17, -4, 18, -5, 21, -3], [2, 4, 22, 4, -1, 4, 0, -16], [-2, -17, 3, -9, 18, 26, 10, 4], [-9, 7, -9, 0, -6, 11, 4, 6], [5, -19, 11, 2, -12, -2, 0, -16], [5, 2, -23, -24, 20, -1, 5, 18], [23, -10, 7, -1, -18, -1, -5, -13], [-23, -7, -9, -13, 13, 7, 18, 26], [4, 18, 27, 7, -6, 7, -5, -10], [-2, 1, -4, -7, -8, 6, 17, 27], [35, 3, -4, -2, -7, 1, 3, 24], [-1, -28, -15, -22, 14, -4, 20, -1], [12, 7, 24, 10, -19, -14, -11, -18], [-5, -19, -15, -22, 25, 12, 13, 21], [-24, -8, 5, 6, 6, 11, -9, 12], [-33, -4, -5, 8, 0, 9, 0, 5], [10, 3, -11, -20, 19, 6, 29, 20], [26, -28, -7, -3, -1, -2, 7, 5], [-12, 6, -15, -10, 15, -11, -10, -9], [-30, 22, 13, 6, -26, 6, 2, -11], [9, 3, 3, 1, -9, 9, 6, 34], [-16, 1, -12, 1, -18, 24, 1, 15], [20, 0, -1, 16, 7, -1, -1, -9], [-14, -1, -9, -13, 5, 29, 13, 28], [-7, -11, 3, -1, 7, 6, -2, -19], [-14, 12, -14, 2, 16, -8, -1, -3], [15, -18, 3, 12, -13, -12, -1, 1], [16, 10, -11, -15, 29, -10, 1, 11], [-28, 8, 41, 16, -34, 4, -19, -25], [-16, 34, -2, 4, -9, -15, -2, -20], [-3, -10, -3, 6, -15, 23, -10, 15], [3, -4, -12, -7, 16, -17, 3, 13], [-11, -12, -8, -23, 3, 10, 1, 19], [11, -9, 22, 20, -28, 4, 6, -21], [-11, -2, 26, 38, -26, -18, -30, -38], [12, 12, -20, -27, 28, -1, 5, 22], [-11, 5, -4, 5, -16, -17, 7, -24], [-21, 6, 28, 9, -11, 15, 1, 6]];
    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_p();".
// 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_p(:prec:=8)
    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_p();".
// 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_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8034_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, -27, -125, -237, -189, -47, 13, 8, 1]>,<7,R![-199, -81, 408, 228, -117, -94, -6, 6, 1]>],Snew);
    return Vf;
end function;