We study a new notion of mod-p twisted density for a modular form f supported on an arithmetic progression: the proportion of primes \(\ell \) for which the mod-p order of infinity of \(U_\ell f\) is minimal. We show that for classical modular forms, twisted density is always defined and rational, connecting it with an earlier notion of density studied by Bellaïche. Finally, we specialize to \(p = 2\) and f a positive power of the Dedekind eta function, studying densities via Galois-theoretic techniques developed by Bellaïche in level 1 and extending them to level 9. In particular, we explicitly calculate twisted densities for certain eta powers corresponding to CM/dihedral mod-2 modular forms in the sense of Nicolas and Serre. En passant we take the opportunity to communicate proofs of two of Bellaïche’s unpublished results on densities of mod-2 modular forms.
