Lifting $L$-polynomials of genus 3 curves

Let $C$ be a smooth plane quartic curve over $\mathbb{Q}$. Costa, Harvey, and Sutherland provide an algorithm with an implementation, improving Harvey's average polynomial-time algorithm, to compute the $\bmod \ p$ reduction of the numerator of the zeta function of $C$ at all $p\leq B$, where $p$ is