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We develop an algorithm to test whether a non-CM elliptic curve $E/\mathbb{Q}$ gives rise to an isolated point of any degree on any modular curve of the form $X_1(N)$. This builds on prior work of Zywina which gives a method for computing the image of the adelic Galois representation associated to $E$. Running this algorithm on all elliptic curves presently in the $L$-functions and Modular Forms Database and the Stein-Watkins Database gives strong evidence for the conjecture that $E$ gives rise to an isolated point on $X_1(N)$ if and only if $j(E)=-140625/8, -9317,$ $351/4$, or $-162677523113838677$., Comment: With an appendix by Maarten Derickx and Mark van Hoeij, to appear in Math. Comp

For a fixed natural number $h$, we prove meromorphic continuation of the two-variable Dirichlet series $\sum_m r_2(m) \sigma_w(m + h) (m + h)^{-s + w}$ to $\mathbb{C}^2$ and use this to obtain asymptotics for $\sum_{m^2 + n^2 \leq X} \sigma_w(m^2 + n^2 + h)$. We approach this continuation through spectral theory. Our results are comparable to earlier work of Bykovskii, who used different methods to study the sums $\sum_{n^2 \leq X} \sigma_w(n^2 + h)$., Comment: 41 pages, including an appendix by Kuan and Huang

We prove the existence of "murmurations" in the family of holomorphic modular forms of level $1$ and weight $k\to\infty$, that is, correlations between their root numbers and Hecke eigenvalues at primes growing in proportion to the analytic conductor. This is the first demonstration of murmurations in an archimedean family., Comment: 30 pages

Let $f(z) = \sum A(n) n^{(k-1)/2} e(nz)$ be a cusp form of weight $k \geq 3$ on $\Gamma_0(N)$ with character $\chi$. By studying a certain shifted convolution sum, we prove that $\sum_{n \leq X} A(n^2+h) = c_{f,h} X + O_{f,h,\epsilon}(X^{\frac{3}{4}+\epsilon})$ for $\epsilon>0$, which improves a result of Blomer from 2008 with error $X^{\frac{6}{7}+\epsilon}$. This includes an appendix due to Raphael S. Steiner, proving stronger bounds for certain spectral averages., Comment: 22 pages, with a 14 page appendix from Raphael S. Steiner. This version corrects a mistake in the previous, where lifts of holomorphic modular forms to Maass forms were omitted; and it has an updated abstract

We study the number of degree $n$ number fields with discriminant bounded by $X$. In this article, we improve an upper bound due to Schmidt on the number of such fields that was previously the best known upper bound for $6 \leq n \leq 94$., Comment: 19 pages; now includes section 3, clarifying relationships between polynomial coefficients, polynomial roots, and etale algebras

We study the properties of certain graphs involving the sums of primes. Their structure largely turns out to relate to the distribution of prime gaps and can be roughly seen in Cram\'er's model as well. We also discuss generalizations to the Gaussian integers., Comment: Preliminary version. The authors expect to add moderate modifications to the style and content. 18 pages, 11 figures

We study sign changes in the sequence $\{ A(n) : n = c^2 + d^2 \}$, where $A(n)$ are the coefficients of a holomorphic cuspidal Hecke eigenform. After proving a variant of an axiomatization for detecting and quantifying sign changes introduced by Meher and Murty, we show that there are at least $X^{\frac{1}{4} - \epsilon}$ sign changes in each interval $[X, 2X]$ for $X \gg 1$. This improves to $X^{\frac{1}{2} - \epsilon}$ many sign changes assuming the Generalized Lindel\"{o}f Hypothesis., Comment: 14 pages