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How to compute the sums of squares of elements of a quotient ring?


I'm new to Sage, and I would like to be able to test, given some $q$, whether $$\sum_{p(t) \in \mathbb{F}_{q}[t]/(f)}^{}{p^2(t)}=k \bmod (f)$$ for some fixed $f \in \mathbb{F}_q [t]$ and $k \in \mathbb{F}_q$.

I can get as far as (for $q=3$ and $f=x^2+1$):

sage:    R = PolynomialRing(GF(3),'x'); x = R.gen()
sage:    S = R.quotient(x^2 + 1)

But I'm not sure how to sum over all the elements of the quotient ring, let alone their squares.

Any hints?