Odd prime power values of Fourier coefficients of Hecke eigenforms
Abstract
We prove a number of results about odd prime power values of Fourier coefficients of newforms with rational integer coefficients and trivial mod $2$ residual Galois representation. In particular, we show that under mild conditions on such a newform $f(z)$, its Fourier coefficients $\lambda _f(n)$ satisfy $\lambda _f(n) \ne \pm q^\alpha $, $\alpha \ge 0$ integer, unless $n$ is itself a power of $q$, where $q$ belongs to a subset of odd primes less than $100$. For example, this result holds unconditionally for newforms with rational integer coefficients, weight $k = 3a + 1$, $a \ge 7$ odd integer, and level $N = 2^b N_0$, $b \ge 1$ integer, $N_0 \in \{1, 3, 5, 15, 17\}$, for each odd prime $q \lt 100$. We also establish stronger results in the case of level $1$ newforms, extending the results of the first author, Gherga, Patel, and Siksek on odd values of the Ramanujan $\tau $-function.
