Sums of Hurwitz Class Numbers and newform of weight 2 and level 49

TL;DR

This paper studies sums of Hurwitz class numbers with a focus on the case where the modulus is 7, revealing connections to modular forms and primes represented by quadratic forms, and completing a related mock modular form.

Contribution

It introduces a new formula for sums of Hurwitz class numbers involving a specific newform of level 49, linking Fourier coefficients to primes represented by quadratic forms.

Findings

Derived a formula connecting Hurwitz sums to modular forms of level 49.

Established a relationship between Fourier coefficients and primes of the form x^2+7y^2.

Completed a mixed mock modular form related to Hurwitz class numbers.

Abstract

We consider sums of Hurwitz class number $H_{m,M}(n)=\sum_{t\equiv m (\text{mod} M)}{H(4n-t^2)}$, where $H(N)$ denotes the Hurwitz class number. In this article, we consider the case of $M=7$. By completing the mixed mock modular form generated by $H_{m,7}(n)$, We obtain the formula of modular forms consist of a computable part and a part from newform 49.2.a.a whose prime terms of Fourier expansion has a connection with with $p=x^2+7y^2$ $(p\equiv 1,2,4 \mod 7)$.