Linear relations between $L$-values of newforms and moments of elliptic $K$ integral

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.