Prime-power coefficients of a level-one eigenform are often large and positive ($a_{p^e}\ge \frac{1}{2}{p}^{e(k-1)/2}\text{ for infinitely many }e$)

For a normalized level-one cusp form Hecke eigenform of weight $k$, fix a prime number $p$ whose local quadratic has distinct roots. Hecke multiplication relations give $b_e=x{b}_{e-1}-b_{e-2}$ for $b_e=a_{p^e}/p^{e(k-1)/2}$ and real $x=a_p/p^{(k-1)/2}$. If $|x|<2$, write $x=2\cos \theta$; then $b_e=\sin ((e+1)\theta )/\sin \theta$. Periodicity when $\theta /(2\pi )$ is rational, and density of irrational rotations otherwise, give infinitely many $b_e\ge 1/2$. If $|x|>2$, the distinct real reciprocal roots give positive unbounded even-index terms. Reality follows from Hecke operators are self-adjoint for the Petersson inner product.