Inspired by Lehmer’s conjecture on the non-vanishing of the Ramanujan \(\tau \)-function, one may ask whether an odd integer \(\alpha \) can be equal to \(\tau (n)\) or any coefficient of a newform f(z). Balakrishnan, Craig, Ono and Tsai used the theory of Lucas sequences and Diophantine analysis to characterize non-admissible values of newforms of even weight \(k\ge 4\). We use these methods for weight 2 and 3 newforms and apply our results to L-functions of modular elliptic curves and certain K3 surfaces with Picard number \(\ge 19\). In particular, for the complete list of weight 3 newforms \(f_\lambda (z)=\sum a_\lambda (n)q^n\) that are \(\eta \)-products, and for \(N_\lambda \) the conductor of some elliptic curve \(E_\lambda \), we show that if \(|a_\lambda (n)|<100\) is odd with \(n>1\) and \((n,2N_\lambda )=1\), then

Assuming the Generalized Riemann Hypothesis, we can rule out a few more possibilities leaving

受雷曼关于Ramanujan \（\ tau \）函数不消失的猜想的启发，人们可能会问奇数整数\（\ alpha \）是否可以等于\（\ tau（n）\）或任何系数的newform ˚F（ž）。Balakrishnan，Craig，Ono和Tsai使用Lucas序列理论和Diophantine分析来表征重量为（k \ ge 4 \）的新形式的不可接受值。我们将这些方法用于权重2和3的新形式，并将我们的结果应用于模块化椭圆曲线的L-函数以及皮卡德数\（\ ge 19 \）的某些K 3曲面。特别是对于权重为3的新表格的完整列表\（f_ \ lambda（z）= \ sum a_ \ lambda（n）q ^ n \）是\（\ eta \）-乘积，对于\（N_ \ lambda \）则是某些椭圆曲线的导体\ { E_ \ lambda \），我们证明如果\（| a_ \ lambda（n）| <100 \）为\（n> 1 \）和\（（n，2N_ \ lambda）= 1 \）为奇数，则

假设广义黎曼假设，我们可以排除更多的可能性