Even Values of Ramanujan’s Tau-Function

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