[THOUGHT] I'll use Python (sympy) to:

1. Compute the symmetric sums from the given power sums to find the cubic $Q(x)$.
2. Use complex analysis: the integral equals $2\pi i$ times the residue at the 4th order pole $z_0 = \frac{205}{69}i$ in the upper half-plane.
3. The residue of order 4: $\text{Res} = \frac{1}{3!}\left.\frac{d^3}{dz^3}[Q(z)e^{iz}]\right|_{z=z_0}$.
4. Then $I = 2\pi i \cdot \text{Res} = \frac{\pi i}{3}\left.\frac{d^3}{dz^3}[Q(z)e^{iz}]\right|_{z=z_0}$.
5. Numerically evaluate $I$, extract $\alpha=\Re(I)$ and $\beta=\Im(I)$, then compute $\lfloor 10\alpha \rfloor + \lfloor \beta \rfloor$.

I'll do everything symbolically then evaluate to high precision to be safe. [/THOUGHT] [PYTHON] import sympy as sp

Define symbols

x, z = sp.symbols('x z', complex=True) I = sp.I pi = sp.pi

Given power sums

s1 = 151 s2 = 10939 s3 = 957871

Compute elementary symmetric sums: p1 = s1, p2 = ab+bc+ca, p3 = abc

p1 = s1 p2 = (s1**2 - s2) // 2 # (a+b+c)^2 = a^2+b^2+c^2 + 2(ab+bc+ca)

Use formula: a^3 + b^3 + c^3 = s1^3 - 3s1p2 + 3*p3

p3 = (s3 - (s1**3 - 3s1p2)) // 3

p1, p2, p3

Construct Q(x) = (x-a)(x-b)(x-c) = x^3 - p1 x^2 + p2 x - p3

Q = x3 - p1*x2 + p2*x - p3 Q_simplified = sp.expand(Q) Q_simplified

Factor to verify roots are natural numbers

Q_factored = sp.factor(Q_simplified) Q_factored

Now define Q(z) as polynomial in z

Qz = Q_simplified.subs(x, z)

Define the pole

z0 = I * sp.Rational(205, 69)

Compute the 3rd derivative of Q(z)*exp(i z)

f = Qz * sp.exp(I*z) D3 = sp.diff(f, z, 3)

Residue at z0 for order 4 pole

Res = D3.subs(z, z0) / sp.factorial(3)

Integral I = 2pii * Res

Integral = 2piI*Res

Simplify Integral

Integral_simplified = sp.simplify(Integral) Integral_simplified

Get numerical approximation with high precision

Integral_num = sp.N(Integral_simplified, 50) Integral_num

Extract real and imaginary parts

alpha = sp.re(Integral_num) beta = sp.im(Integral_num) alpha, beta

Compute the desired value: floor(10*alpha) + floor(beta)

result = sp.floor(10*alpha) + sp.floor(beta) result [/PYTHON]

[THOUGHT] Using the Python result as ground truth, the value of $\lfloor 10\alpha \rfloor + \lfloor \beta \rfloor$ is $-36330$.