L-values of certain weight 3 modular forms and transformations of hypergeometric series

Recently, Allen, Grove, Long, and Tu proposed an explicit method to compute modularity for certain hypergeometric series, which gives a concrete link between certain hypergeometric objects and modular forms. The theory is exemplified by a collection of 199 weight 3 modular forms. Among other properties, their process shows that the L -value of such a modular form at 1 is an explicit multiple of a "Equation missing" hypergeometric series. Using the framework of a finite Coxeter group governing the invariance group of normalized "Equation missing" series, this paper fully classifies and describes the possible Hecke eigenforms whose L -values can be obtained using this method. In addition, we determine when these modular forms differ by twist of a finite-order character using the perspective of hypergeometric functions. As one application, we reinterpret a classical identity of hypergeometric series as a formula involving L -values of two Hecke eigenforms that differ by a twist.