  In the spirit of Lehmer's speculation that Ramanujan's tau-function never
vanishes, it is natural to ask whether any given integer $\alpha$ is a value of
$\tau(n)$. For odd $\alpha$, Murty, Murty, and Shorey proved that $\tau(n)\neq
\alpha$ for sufficiently large $n$. Several recent papers have identified
explicit examples of odd $\alpha$ which are not tau-values. Here we apply these
results (most notably the recent work of Bennett, Gherga, Patel, and Siksek) to
offer the first examples of even integers that are not tau-values. Namely, for
primes $\ell$ we find that $$ \tau(n)\not \in \{ \pm 2\ell \ : \ 3\leq \ell<
100\} \cup \{\pm 2\ell^2 \ : \ 3\leq \ell <100\} \cup \{\pm 2\ell^3 \ : \ 3\leq
\ell<100\ {\text {\rm with $\ell\neq 59$}}\}.$$ Moreover, we obtain such
results for infinitely many powers of each prime $3\leq \ell<100$. As an
example, for $\ell=97$ we prove that $$\tau(n)\not \in \{ 2\cdot 97^j \ : \
1\leq j\not \equiv 0\pmod{44}\}\cup \{-2\cdot 97^j \ : \ j\geq 1\}.$$ The
method of proof applies mutatis mutandis to newforms with residually reducible
mod 2 Galois representation and is easily adapted to generic newforms with
integer coefficients.

Number Theory