Modular forms and certain ₂𝐹₁(1) hypergeometric series

Using the framework introduced recently by Allen, Grove, Long, and Tu [Res. Math. Sci. 12 (2025), DOI 10.1007/s40687-025-00566-y] relating certain hypergeometric motives to modular forms, we define an explicit family of weight 2 Hecke eigenforms with complex multiplication. We use the theory of 2 F 1 ( 1 ) {}_2F_1(1) hypergeometric series and Ramanujan’s theory of alternative bases to compute the exact central L L -value of these Hecke eigenforms in terms of special beta values. We also show the integral Fourier coefficients can be written in terms of Jacobi sums, reflecting a motivic relation between the hypergeometric series and the modular forms.