On L-functions of modular elliptic curves and certain K3 surfaces

Inspired by Lehmer's conjecture on the nonvanishing of Ramanujan $\tau$-function, one may ask whether an odd integer $\alpha$ can be equal to $\tau(n)$ or any coefficient a newform $f(z)$. Balakrishnan, Craig, Ono, and Tsai used theory Lucas sequences Diophantine analysis characterize non-admissible values newforms even weight $k\geq 4$. We use these methods for $2$ $3$ apply our results $L$-functions modular elliptic curves certain $K3$ surfaces with Picard number $\ge 19$. In particular, complete list $f_\lambda(z)=\sum a_\lambda(n)q^n$ that are $\eta$-products, $N_\lambda$ conductor some curve $E_\lambda$, we show if $|a_\lambda(n)|<100$ is $n>1$ $(n,2N_\lambda)=1$, then \begin{align*} a_\lambda(n) \in \,& \{-5,9,\pm 11,25, \pm41, \pm 43, -45,\pm47,49, \pm53,55, \pm59, \pm61, 67\}\\ & \,\,\, \cup \, \{-69,\pm 71, 73,75, \pm79,\pm81, 83, \pm89,\pm 93 97, 99\}. \end{align*} Assuming Generalized Riemann Hypothesis, rule out few more possibilities leaving 11,25,-45,49,55,-69,75,\pm 81,\pm 93,