In the spirit of Lehmers unresolved speculation on the nonvanishing of Ramanujans tau-function, it is natural to ask whether a fixed integer is a value 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 coefficients and trivial residual mod 2 Galois representation, that answers this question for odd integers. We determine infinitely many spaces for which the primes $3leq ellleq 37$ are not absolute values of coefficients of newforms with integer coefficients. For $tau(n)$ with $n>1$, we prove that $$tau(n) ot in {pm 1, pm 3, pm 5, pm 7, pm 13, pm 17, -19, pm 23, pm 37, pm 691},$$ and assuming GRH we show for primes $ell$ that $$tau(n) ot in left { pm ell : 41leq ellleq 97 {textrm{with}} left(frac{ell}{5}right)=-1right} cup left { -11, -29, -31, -41, -59, -61, -71, -79, -89right}. $$ We also obtain sharp lower bounds for the number of prime factors of such newform coefficients. In the weight aspect, for powers of odd primes $ell$, we prove that $pm ell^m$ is not a coefficient of any such newform $f$ with weight $2k>M^{pm}(ell,m)=O_{ell}(m)$ and even level coprime to $ell,$ where $M^{pm}(ell,m)$ is effectively computable.
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