Narrow Results

Let $f(z)$ be the normalized primitive holomorphic Hecke eigenforms of even integral weight $k$ for the full modular group SL$(2,\mathbb{Z})$. In this paper, we investigate the second moment of Fourier coefficients of symmetric power $L$-function $L(s,{\mathrm{\text{sym}}}^{j}f)$ for $j\ge 2$ and improve on previous error estimates. We also consider the second moment of ${\lambda }_f(n^2)$.

Given a fixed quadratic extension K of ℚ, we consider the distribution of elements in K of norm one (denoted ). When K is an imaginary quadratic extension, is naturally embedded in the unit circle in ℂ and we show that it is equidistributed with respect to inclusion as ordered by the absolute Weil height. By Hilbert's Theorem 90, an element in can be written as for some , which yields another ordering of given by the minimal norm of the associated algebraic integers. When K is imaginary we also show that is equidistributed in the unit circle under this norm ordering. When K is a real quadratic extension, we show that is equidistributed with respect to norm, under the map β ↦ log|β|(mod log|ϵ²|) where ϵ is a fundamental unit of .