We study which integers are admissible as Fourier coefficients of even integer weight newforms. In the specific case of the tau-function, we show that for all odd primes $\ell <100$ and all integers $m\ge 1$, we have$\tau (n)\ne \pm \ell ,\pm 5^m.$ For general newforms f with even integer weight 2k and integer coefficients, we prove for most integers j dividing $2k-1$ and all ordinary primes p that $a_f(p^2)$ is never a j-th power. We prove a similar result for $a_f(p^4)$, conditional on the Frey-Mazur Conjecture. Our primary method involves relating questions about values of newforms to the existence of perfect powers in certain binary recurrence sequences, and makes use of bounds from the theory of linear forms in logarithms. The method extends without difficulty to a large family of Lebesgue-Nagell equations with fixed exponent. To prove results about general newforms, we also make use of the modular method and Ribet's level-lowering theorem.

我们研究了哪些整数可以作为偶数整数权重新形式的傅里叶系数。在tau函数的特定情况下，我们表明对于所有奇数素数$\ell <100$ 和所有整数 $米\ge \mathrm{1个}$， 我们有$\tau （ñ）\ne \pm \ell ，\pm 5^{米}。$对于具有偶数权重2 k和整数系数的一般新形式f，我们证明了对于大多数整数j除$\mathrm{2个}ķ-\mathrm{1个}$和所有普通的素数p是${\mathrm{一个}}_F（p^{\mathrm{2个}}）$永远不是第j次方。我们证明了类似的结果${\mathrm{一个}}_F（p^4）$，以Frey-Mazur猜想为条件。我们的主要方法涉及将有关新形式的值的问题与某些二元递归序列中的完美幂的存在相关联，并利用对数线性形式理论的界限。该方法毫不费力地扩展到具有固定指数的一大族Lebesgue-Nagell方程。为了证明有关一般新形式的结果，我们还使用了模块化方法和Ribet的降级定理。