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In the spirit of Lehmer's unresolved speculation on the nonvanishing ofRamanujan's tau-function, it is natural to ask whether a fixed integer is avalue of $\tau(n)$ or is a Fourier coefficient $a_f(n)$ of any given newform$f(z)$. We offer a method, which applies to newforms with integer coefficientsand trivial residual mod 2 Galois representation, that answers this questionfor odd integers. We determine infinitely many spaces for which the primes$3\leq \ell\leq 37$ are not absolute values of coefficients of newforms withinteger coefficients. For $\tau(n)$ with $n>1$, we prove that $$\tau(n)\not \in\{\pm 1, \pm 3, \pm 5, \pm 7, \pm 13, \pm 17, -19, \pm 23, \pm 37, \pm 691\},$$and assuming GRH we show for primes $\ell$ that $$\tau(n)\not \in \left \{ \pm\ell\ : \ 41\leq \ell\leq 97 \ {\textrm{with}}\\left(\frac{\ell}{5}\right)=-1\right\} \cup \left \{ -11, -29, -31, -41, -59,-61, -71, -79, -89\right\}. $$ We also obtain sharp lower bounds for the numberof prime factors of such newform coefficients. In the weight aspect, for powersof odd primes $\ell$, we prove that $\pm \ell^m$ is not a coefficient of anysuch newform $f$ with weight $2k>M^{\pm}(\ell,m)=O_{\ell}(m)$ and even levelcoprime to $\ell,$ where $M^{\pm}(\ell,m)$ is effectively computable.

Variants of Lehmer's speculation for newforms