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

By systematically translating certain integrals involving moments of the elliptic integral into $L$-values of modular forms on ${\mathrm{\Gamma }}_1(4),\mathrm{\Gamma }(4)$ and ${\mathrm{\Gamma }}_1(8)$, and then utilizing relations among the critical $L$-values of cuspidal newforms, we obtain a dimension formula for the space spanned by these integrals.