Evaluate polynomial with imaginary number as input?

edit

If you want to assume that v^3 + 2 vanishes, perhaps could you work in a quotient ring:

sage: I = R.ideal(v^3 + 2)
sage: I
Principal ideal (v^3 + 2) of Univariate Polynomial Ring in v over Finite Field of size 67
sage: S = R.quotient(I)
sage: S
Univariate Quotient Polynomial Ring in vbar over Finite Field of size 67 with modulus v^3 + 2

Then, you can send your poly in the quotient ring and compute its 67th power:

sage: P = S(poly)
sage: P
19*vbar^2 + 49*vbar + 8
sage: P^67
15*vbar^2 + 4*vbar + 8

Essentially you are trying to compute the image of Frobenius endomorphism on 19*v^2 + 49*v + 8 in the field $\mathrm{GF}(67^3) \cong \mathrm{GF}(67)[v]/\langle v^3+2 \rangle$. You can define it as

R.<v> = PolynomialRing(GF(67))
K.<w> = GF( 67^3, name='w', modulus=v^3 + 2 )
Frob = K.frobenius_endomorphism()
print(Frob)

to get

Frobenius endomorphism w |--> w^67 on Finite Field in w of size 67^3

Then it remains to compute Frob( 19*v^2 + 49*v + 8 ), which gives 15*w^2 + 4*w + 8.