On the parity of coefficients of eta powers

Abstract We study a new notion of mod- p twisted density for a modular form f supported on an arithmetic progression: the proportion of primes $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> for which the mod- p order of infinity of $$U_\ell f$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>ℓ</mml:mi> </mml:msub> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> 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 $$p = 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> 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 En passant we take the opportunity to communicate proofs of two of Bellaïche’s unpublished results on densities of mod-2 modular forms.