Three Primes And Garner

Run the NTT under $10^9+9$, which is NTT-friendly, and adjust by the difference of the two moduli. The true coefficients are bounded by $n \cdot (10^9)^2$, which one modulus near $10^9$ covers, then reduce modulo $10^9+7$ at the very end.

Run the NTT under three different NTT-friendly primes and recombine with the Chinese remainder theorem. The true coefficients are bounded by $n \cdot (10^9)^2$, which three moduli near $10^9$ cover, then reduce modulo $10^9+7$ at the very end.

Run the NTT under one NTT-friendly prime and lift the result with Hensel's lemma. The true coefficients are bounded by $n \cdot (10^9)^2$, which one modulus near $10^9$ pins down, then reduce modulo $10^9+7$ at the very end.

Run a plain double-precision FFT directly on the coefficients and round. The rounding error stays below $0.5$ for any $n$ and any coefficient size, which one transform covers, then reduce modulo $10^9+7$ at the very end.