
// Make newform 8034.2.a.o in Magma, downloaded from the LMFDB on 23 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_o();".
// 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_o();".
// 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 := [1, -3, -12, 3, 14, -4, -3, 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], [3, 3, -5, -1, 1, 0, 0], [0, 6, 1, -12, 3, 3, -1], [-2, 5, 2, -12, 3, 3, -1], [0, -9, -2, 13, -3, -3, 1], [2, -10, -9, 17, -1, -4, 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_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], [-1, 0, 0, 0, 0, 0, 0], [1, -1, 1, 0, 1, 0, -1], [-2, 0, 0, -1, 0, -1, 0], [0, 1, -1, 1, -1, 1, -1], [1, 0, 0, 0, 0, 0, 0], [-1, 1, 1, -3, 1, 0, -1], [-3, 0, 0, -1, 0, -1, 0], [1, -1, -1, 1, -2, 0, 1], [-1, 2, 0, 1, 1, -1, -1], [-1, 0, 0, 1, 0, 3, 0], [4, -1, 1, 1, 2, 0, 0], [1, 0, 1, -2, 2, -2, 4], [-4, 3, 2, 0, 1, 1, -2], [2, 0, -4, 3, -1, 2, 1], [2, -4, 0, 2, 0, -3, 4], [-2, -1, 0, 0, -4, -1, 2], [0, -2, -1, 0, 1, -2, -1], [1, -2, 1, 3, -4, 2, 0], [-1, 3, 1, -2, 0, -3, 1], [3, 2, -1, 0, 1, 3, -2], [-5, 0, 3, -1, 0, -1, 1], [1, 3, -2, 0, 0, 1, 3], [-3, 0, 3, 2, -3, -3, 3], [-2, 6, -1, 2, 0, 0, -1], [3, -4, 2, -4, 3, 2, 3], [1, 0, 0, 0, 0, 0, 0], [1, 4, -2, 2, -2, 7, -5], [5, -1, -3, 2, -3, 6, -4], [-2, 6, -3, -1, 1, 1, 0], [-6, -4, 1, 5, -3, -1, 5], [6, -1, -4, -2, -2, 1, 1], [4, 0, -5, 1, -1, 0, -4], [-8, 9, -4, -1, -3, 3, -2], [-3, 0, 1, -3, 0, 1, 4], [-8, -3, -1, 4, -1, -3, 6], [-3, -3, 1, 7, -4, -3, 5], [5, 2, -1, -5, 3, 2, 5], [-2, 2, -1, -5, 6, 0, -1], [2, -6, -4, 5, 1, 2, -1], [-2, 2, -1, -2, 6, -2, 1], [-1, 4, -4, 3, -1, 6, 6], [-10, 5, -2, 0, -5, -3, 3], [4, -3, 4, -1, 2, -4, 4], [6, 0, 3, -5, -1, 0, 3], [-8, -1, 0, -3, 3, -4, -2], [0, -6, 2, -2, 3, -3, 5], [-3, 1, 0, -4, 3, -7, -2], [1, 6, 1, -6, -1, -4, 1], [9, -7, -6, 3, 5, -1, 2], [-7, 1, 0, 4, -2, 3, -1], [-1, -3, -1, -1, 3, 3, 0], [-6, 1, 0, 2, -7, -6, 3], [-4, -5, 5, 1, -5, -5, -1], [2, -5, 8, 4, 1, 4, -1], [1, -9, 1, 11, 0, -1, -4], [0, 1, -1, -2, -1, 1, -8], [4, 3, 2, 5, -1, 1, -8], [2, -7, -4, -3, 1, 2, -3], [-5, -2, 1, 4, -2, 5, -4], [-6, 1, 0, -10, 4, -3, 2], [10, -1, 3, 1, 1, 2, 2], [-7, 5, 7, -3, 5, -8, 5], [-3, 7, 4, 0, -2, -1, -5], [3, 5, -7, 0, -1, 7, 4], [-5, 10, -1, -8, 3, 2, -4], [-7, -8, -4, -3, -2, 2, 5], [4, -7, -2, -5, 9, 1, 0], [-5, -8, -3, -5, 0, 2, 0], [-7, -5, 2, 1, 0, -1, -3], [5, 1, 3, 1, -2, 1, -1], [-9, -2, 6, -3, -1, -5, 2], [1, -2, -3, -2, 3, 0, -10], [-12, 1, 0, 3, -4, 0, 3], [-4, 5, -3, 0, 2, 10, -6], [13, 3, -2, 7, -5, 12, -3], [2, 1, 4, -4, 5, -8, 2], [3, 1, -5, 11, -7, -1, -6], [-2, -1, -3, -3, 5, 1, 2], [-18, 3, 1, -8, 1, -2, -3], [6, -2, 4, -3, 5, -4, 6], [5, -4, -4, -6, 10, 3, 1], [-1, 0, 2, 2, 3, -3, -3], [0, -5, 1, 6, 1, 1, -4], [-13, 8, 1, -17, 8, 1, -6], [-19, 8, 7, -8, -2, -8, -5], [0, -11, 8, -3, -3, -2, 3], [2, 6, 0, -9, 3, 0, 11], [-1, -4, -3, -1, 0, -2, -5], [9, -3, 4, 0, 7, 6, -10], [16, -2, -2, 13, -12, -4, 3], [4, -7, -2, 1, -1, 9, 2], [-17, 8, -1, 4, -4, -8, 0], [7, -8, 0, 14, -7, 9, -2], [-11, -1, 7, -9, 0, -8, -1], [-4, 0, -7, -2, -8, 2, 4], [6, -5, -2, 7, -4, -3, 4], [-8, -1, -4, 7, -1, -2, 1], [-11, 6, -4, 6, -14, 2, 0], [-6, -10, 7, 1, -4, -3, 5], [-16, 8, -4, -8, -4, -3, -3], [-7, -6, -6, -2, 8, -2, 6], [10, -15, -4, 2, 5, 5, 1], [2, -9, 1, 2, 4, -2, -2], [-4, -2, -1, 3, 4, 1, -7], [11, -15, -3, 11, -4, 1, 9], [-8, 0, 13, -4, 9, -14, 4], [1, 1, -3, -2, 2, 9, -8], [-16, 5, 3, 5, -3, -5, 7], [-9, 2, -2, 10, -12, -4, -2], [-14, 14, 1, -3, -4, -13, 2], [-8, 13, -1, 3, -12, -1, 12], [-20, 2, -2, 5, -10, -2, 8], [-28, 13, 1, -9, 7, 5, -1], [-17, 9, 0, 1, -4, 6, 0], [-8, 10, 5, -6, -8, -6, 4], [-5, -4, -4, 9, -8, 8, -1], [-10, -1, -2, 12, -7, 2, -2], [11, 11, -1, -11, 2, 2, -4], [-20, 18, -5, -9, -3, 2, -3], [14, 0, -6, 15, -5, 13, 4], [7, -4, 2, 8, -8, -2, 1], [-19, 6, -6, 3, -16, 6, 4], [8, 8, 1, -1, -3, 4, 4], [8, 10, -14, 2, -5, 4, -5], [4, -8, 7, -11, 5, -6, 8], [-20, 3, 0, 6, -9, -1, -13], [4, -6, 5, 4, 13, -1, -3], [-6, -5, 6, -3, -9, -6, 12], [2, 15, -2, -4, 5, -5, -8], [-12, -15, 5, -2, -9, -6, 10], [4, -2, 2, -10, 1, -1, -4], [0, 5, 2, 4, 4, 3, -5], [0, -6, -1, 7, 1, -4, 7], [-3, 7, 0, 11, -12, -5, -4], [-22, 3, 1, -5, -3, -13, 7], [-3, -14, 10, -7, -1, -10, -5], [4, -3, -1, -1, 6, -9, 9], [2, -18, 7, -2, 0, -5, 18], [-2, -9, -5, -6, 14, -4, 3], [-2, 2, 5, 5, -1, 3, 9], [6, -5, -9, 1, 8, -3, -6], [1, -5, -4, 2, 1, 12, -7], [-10, -13, -4, -2, 4, -1, 0], [7, 4, -5, 1, -1, -2, 0], [-4, -13, 3, -1, 5, -1, -12], [-21, 8, 3, -17, 4, -8, -5], [0, -8, 2, -4, 5, -4, 8], [-3, -13, -1, -8, 15, -3, -9], [6, -3, -3, 15, -5, 0, 4], [-16, 15, -2, -10, 4, 6, 0], [-5, 4, 12, 6, 6, 5, -13], [-11, 2, -3, -2, -8, 5, 8], [1, -1, 6, -6, 10, 0, -2], [-7, -14, 11, -4, -3, -6, 9], [6, 0, -10, 8, -2, 13, 7], [-27, 1, -5, -11, 4, -5, 3], [15, 2, -12, 10, 12, 14, 0], [-6, 1, 3, -7, 0, -7, 0], [-17, 6, -7, 4, -5, -11, 13], [-13, 14, -2, -5, -9, 12, -1], [16, -16, 4, -1, 14, -1, -7], [2, 3, 6, -1, -2, -1, -11], [1, 6, -3, -15, 11, 13, -9], [2, 7, 8, -5, 1, 0, -6], [7, -15, 2, 13, -5, 0, 8], [-6, 1, 4, -6, 17, 1, -5], [-13, -6, -6, 9, -4, 14, 2], [-3, 6, -9, -2, 0, 6, -2], [29, -25, 1, 3, -5, 0, 3], [-25, 4, -2, -3, -1, -8, 12], [13, -1, 5, -11, 9, -13, 0], [-7, -5, 7, -4, 10, 6, -16], [-8, 8, -12, 8, -19, 19, -9], [-15, -7, 7, 9, -10, -2, 3], [-2, 8, -9, -2, 1, 10, 1], [-12, 9, -2, 2, 13, -3, 1], [-6, -3, -4, 4, 7, 13, -3], [16, 5, -2, -1, 14, 1, -13], [3, 7, 2, 5, -3, 6, -2], [-7, -7, 5, -8, 8, 4, 0], [-7, 11, -2, 9, 2, -1, -5], [-11, 10, 12, -9, -1, -10, 3], [7, -11, 9, 4, 2, -9, 0], [-3, -7, 7, -7, 2, -21, -2], [-12, 9, 11, 1, -7, -2, -8], [15, -18, -3, 2, 7, -9, 9], [0, -12, -3, -10, 2, -6, 1], [-10, 7, -12, -3, 2, -1, 2], [1, 9, -2, -11, 23, 9, -11], [-31, 10, -1, -7, -6, -7, 3], [8, 8, 0, 0, -7, -4, 5], [-12, -6, 3, -11, -9, -3, 14], [4, 7, -7, 6, 0, -3, -8], [-8, 4, 13, -11, 18, -2, 0], [22, -9, -6, 6, 4, 8, -5], [3, 1, 7, -5, 8, 7, -20], [6, 13, 7, -6, 5, 0, -16], [24, -4, -10, 16, -10, 9, -5], [12, 5, -11, -15, 11, 8, 6], [32, 6, 5, -9, 8, -3, -1], [-10, -10, -6, 4, 0, 0, 2], [13, 1, -9, 21, -11, -5, 5], [4, 0, -14, 21, -10, 18, 9], [-1, 5, -9, 4, -12, -5, 8], [2, 7, 1, 10, -16, -3, 8], [24, -11, 6, 0, 1, 5, -5], [0, 8, 12, 11, 6, -5, 8], [-5, 5, 7, -7, -2, 3, -24], [15, -15, -4, 14, -12, 17, -5], [-17, 10, 6, 3, -5, 4, 3], [7, -1, 12, 5, 10, 3, 5], [14, -1, 3, 11, -6, -3, 13], [2, 7, 5, -8, 10, 2, 6], [-5, -1, 6, 4, 2, -11, 0], [-2, -6, -5, 21, -12, -6, 28], [-4, -8, 16, 3, 11, -7, -2], [-4, 16, -2, -7, -5, -9, 4], [24, -22, 15, -6, 15, 8, -6], [4, -1, -8, 2, -3, 12, -13], [-5, -17, 7, -14, 11, -14, 10], [26, -32, 5, 16, -6, 8, -2], [-6, 12, -8, 8, -10, 10, -9], [28, 1, -12, -8, 13, 4, -1], [3, 1, 1, -3, -7, -9, 22], [5, 3, -11, 14, -16, -3, -8], [21, -11, 0, 12, -2, 10, 6], [-10, -11, 14, 0, -5, -13, 6], [-2, -7, 7, 15, -14, -4, 12], [15, -12, -14, 23, -4, 16, 0], [23, 2, 1, -1, 8, 3, 0], [1, -10, -8, 22, -22, 12, -8], [32, 6, -6, 3, -4, 2, 6], [25, -10, -1, -1, 2, 2, -2], [12, 9, -2, 6, -11, 1, 2], [-17, 5, 14, -21, 13, -7, -8], [19, -11, 13, -6, -7, -2, -6], [19, 1, 7, -11, -2, -5, -5], [-16, 8, 1, -4, 10, 1, -5], [20, -6, 11, 3, 12, -1, -8], [-12, -3, 3, -2, -9, -7, 11], [-1, 8, -8, -1, 8, 4, 9], [9, -28, 5, 24, 0, 4, -8], [-5, 9, -15, 0, -6, 2, 19], [10, -22, -4, 0, 6, -4, 12], [14, 6, -10, -6, 4, 0, -4], [7, -13, 7, -9, 9, -1, -6], [6, -11, 4, 20, -7, 5, 7], [3, -10, 6, 13, 2, -11, 2], [-18, -3, 3, -4, 14, -21, 13], [-18, 8, 8, -11, -2, 4, -8], [-3, -3, 14, 2, 9, -4, 11], [3, 9, 7, 7, -14, -4, 1], [4, 0, -2, 7, 13, 16, -13], [21, -15, -3, 3, 4, 3, -9], [-32, 0, 9, -1, -11, -17, -2], [6, -9, -13, -7, -7, 8, 5], [15, -13, -1, 11, -8, -9, 3], [-14, 17, 2, -2, 0, 6, -25], [-15, 7, -7, 14, -17, 11, 9], [-17, -5, -1, -3, 4, -1, -14], [-14, 12, -6, 9, 4, 1, 22], [-2, -5, 12, -6, 7, -7, -15], [-11, -5, -7, -4, -3, -4, 2], [48, -24, 3, 7, 6, -13, 11], [19, -28, -13, 2, 15, -6, 9], [-21, 10, 2, -7, -11, -18, 5], [13, 12, -7, -3, -11, 8, -4], [24, -9, 14, -9, 24, -18, 7], [-20, 6, 9, 3, 1, -17, -1], [-12, -16, 13, -9, 11, -16, 1], [6, 11, 6, -14, 11, 5, -6], [-3, 13, 9, -18, 22, -13, 18], [7, -24, -2, 3, -1, 4, 8], [4, 7, 4, -14, -5, 1, -10], [1, 10, -12, 12, -12, 30, -28], [9, 14, -6, 12, 2, 1, -16], [0, 8, 13, -29, 9, -10, -9], [6, -6, 7, -6, -9, 5, -9], [20, 6, 8, -11, 15, -3, -6], [31, -11, -6, 5, -3, 13, 2], [41, -20, 7, 5, 5, 7, 2], [22, 9, 5, -10, 5, -2, 5], [15, 15, -13, -14, -5, 5, 8], [9, -1, 4, 5, -10, 6, 15], [-11, -7, -12, 21, -13, 29, -3], [41, -6, -11, 16, -6, 6, -1], [-17, 14, -12, -5, -12, -4, -2], [24, -25, -3, 20, -3, 7, 19], [5, -9, -1, 21, 1, 0, -7], [35, -8, 2, 6, 16, 8, 3], [3, 15, -11, -4, -11, 18, -18], [-10, 15, -10, -14, 5, -2, -17], [13, -3, -3, -8, 4, 1, -1], [35, -14, 9, 11, 14, 4, -13], [-6, 16, -9, -3, 1, 18, 5], [15, -8, -20, -3, -2, 0, -1], [8, 2, 6, 12, -21, 4, 2], [-19, 6, 3, -4, -13, -9, 4], [31, -32, -8, 23, -9, -1, 20], [-6, 21, 18, -28, 12, -20, -5], [-38, 8, 0, -6, 1, 9, -10], [18, -12, 18, 3, 5, 5, -31], [-6, 28, 8, -11, -2, 1, -8], [25, 6, -3, 8, -6, 30, -16], [42, 12, -14, -5, 8, 12, -2], [27, 12, -5, 7, 8, 3, -17], [2, 16, -2, -9, 9, 11, -25], [35, 7, -7, -9, 17, -8, -13], [-14, -1, 4, 14, -27, -11, 21], [-9, 14, -8, 17, -16, -11, 20], [-42, -8, 4, -11, 13, -23, 15], [7, -15, -3, -2, 3, -23, 26], [-6, 16, 12, -20, 9, -8, 6], [8, -8, -20, 0, -24, 15, -6], [-16, 0, 5, -16, -4, -29, 16], [36, 8, -16, 11, 1, 33, -9], [6, 0, 0, 21, -13, 1, -15], [12, -30, 15, -4, -7, -10, 1], [-17, -7, 17, -12, 17, -3, -6], [-9, 16, -19, 16, 2, 7, -10], [-17, 4, 6, -4, 17, 16, -4], [3, -5, -2, 18, -10, 2, 7], [3, 27, -9, -2, 13, 16, -22], [19, 6, -4, -8, 8, 11, 19], [-19, 26, 4, -4, -16, 16, -16], [24, 1, 2, 9, 25, 7, -15], [2, 10, 2, 10, -6, 2, -28], [-30, 4, 23, -1, 4, 7, -5], [-9, 6, 0, -16, 10, -10, 10], [19, -17, 19, -3, 1, -3, -11], [2, 9, -8, -6, 3, 4, -11], [-11, 0, -14, 9, -18, -4, 34], [25, -2, 11, -20, 15, -18, 15], [1, 7, 12, 19, -1, 13, -35], [16, 6, -2, -4, 9, 19, -28], [-12, -4, 12, -5, 1, -12, -1], [10, 7, 4, -2, -19, 10, -1], [-3, -9, 14, -21, -5, -16, 0], [-12, -15, -6, -17, 8, -4, 18], [19, 2, 13, 1, 12, -13, -4], [-6, -12, -12, 0, -20, 5, 6], [6, 15, 5, 13, -1, 24, -29], [-4, 2, -12, 18, -2, 26, 3], [-8, 12, -11, 7, 12, 5, -2], [4, 13, 2, -5, -10, -14, 15], [-3, -6, 11, 3, -16, 14, 6], [-10, 8, -12, -2, 24, -3, -4], [38, -21, 5, 7, 12, 3, -7], [-5, 2, -3, -4, 0, 21, 13], [-29, 38, -4, -9, 2, 8, -16], [28, -13, -6, -4, 24, 0, 8], [-2, 3, -6, -7, 6, -12, -17], [-21, 5, 26, -3, 10, -5, 15], [-8, -2, 20, -21, 16, -11, 12], [24, 13, -11, -1, -14, 28, -9], [1, 2, 4, -2, -12, 5, 13], [18, -23, 7, 21, 18, -25, 2], [4, 3, -17, -2, 0, -8, -15], [0, -5, 14, -21, 5, -6, -14], [-25, -14, 11, 3, -2, 2, -4], [-10, 13, -9, 9, 5, -5, 11], [-36, -10, -6, 2, -5, 4, 2], [-2, -9, -3, -10, 8, -3, -2], [28, -20, -6, 10, -5, -6, -7], [1, 13, 0, -28, 9, -10, 4], [31, -32, -7, 4, 7, -12, 18], [1, 12, 4, -23, 11, -10, -25], [20, -30, -4, 10, -4, 20, 5], [-31, 27, -3, 4, -14, 8, 8], [-5, -2, -13, -16, 13, -12, -4], [-27, 27, -3, -27, 4, -2, -3], [-3, 0, 7, -13, -1, -15, 12], [-31, -1, -2, -14, 26, -4, 0], [-15, 11, -13, -25, 7, -11, 1], [-3, 34, -17, -24, 0, 8, -1], [-14, 13, 16, 12, -12, -7, -23], [6, 3, -13, -15, 1, -11, 13], [21, -5, -15, 13, -4, 21, -15], [8, 6, -13, 0, -15, 6, 12], [-14, -10, 17, 19, 4, -15, 4], [16, -16, -2, 4, 3, -19, 37], [-9, -33, 21, 10, 4, -18, -19], [-14, -6, 15, -14, 6, -27, -11], [-10, 10, -21, 22, -20, -5, 27], [-21, -17, 24, -22, 1, -25, -10], [-31, 2, 0, 11, -16, 4, -7], [3, -1, -9, -7, 20, 15, 3], [36, -23, 11, 1, -10, -11, -3], [-6, 4, 1, 5, -10, 2, -31], [25, 6, -18, 25, -13, 26, 2], [-9, 9, 0, 1, -2, 9, -30], [-26, -5, 24, 0, 4, -9, 5], [-23, -13, -3, -15, -7, -5, 19], [11, 9, 8, -7, -9, 11, 1], [-39, 21, 4, -18, 6, 2, -19], [-6, -6, 10, -8, -7, -20, 28], [49, -31, 5, -2, 16, 7, 3], [38, -36, 10, 3, 22, 8, -12], [36, -3, -23, 12, 9, 18, 7], [-2, -2, -3, -15, 8, 28, -12], [3, -17, 2, -1, -4, 20, 1], [-21, 16, 4, -15, 10, 12, -32], [-19, -19, 3, -1, 0, 0, 22], [1, 3, -10, -1, 3, 26, -9], [-22, 0, 11, 4, -5, -8, 13], [8, -25, 11, 6, 6, -35, 25], [-30, 4, -15, 3, -25, 8, -1], [28, -12, 0, 20, -9, 26, -11], [-3, -22, 9, -18, 28, -7, -14], [-40, 29, -20, -4, -10, -1, 12], [6, 5, -1, 15, -24, 18, -22], [20, 5, -15, 2, 15, 28, 1], [-25, 19, 8, -11, -2, -4, -11], [-20, 22, -11, 1, -11, -24, 2], [-11, 0, 0, 19, -20, -27, 35], [-24, -20, 4, 4, -6, -6, -2], [-5, 5, -4, -5, 13, 11, -18], [18, -3, 3, -3, -10, -16, -13], [-15, 24, 5, -20, 14, 7, -12], [-36, 14, -12, -24, 3, -27, 6], [8, -16, 9, 27, -2, 5, 8], [39, 15, -2, 26, -6, 15, -6], [-3, 11, -12, -2, 14, 18, -14], [15, -23, 11, 7, -6, -5, 26], [-32, 11, -5, -7, 15, -12, 6], [15, -22, 6, 16, -8, -20, 48], [-17, 29, 11, 0, -16, 8, -20], [18, 2, -3, 23, -23, 10, -8], [-31, -28, 16, -1, 1, -15, 15]];
    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_o();".
// 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_o(:prec:=7)
    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_o();".
// 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_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8034_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![657, -417, -279, 173, 41, -23, -2, 1]>,<7,R![183, 270, -32, -178, -51, 17, 9, 1]>],Snew);
    return Vf;
end function;