Linear relations between $L$-values of newforms and moments of elliptic $K$ integral
Kam Cheong Au
Published: 2025/9/24
Abstract
By systematically translating certain integrals involving moments of the elliptic integral into $L$-values of modular forms on $\Gamma_1(4), \Gamma(4)$ and $\Gamma_1(8)$, and then utilizing relations among the critical $L$-values of cuspidal newforms, we obtain a dimension formula for the space spanned by these integrals. We also report a hypergeometric motive associated to a weight $6$ newform.