NEWFORMS OF LEVEL 4 AND OF TRIVIAL CHARACTER

In this paper, we consider characters of SL 2 (Z) and thenapply them to newforms of integral weight, level 4 and of trivial character.More precisely, we prove that all of them are actually level 1 forms ofsome nontrivial character. As a byproduct, we prove that they all areeigenfunctions of the Fricke involution with eigenvalue −1. IntroductionThe Fricke involution W N of level N, also known as the canonical involution,actsonthe spaceofnewformsoflevelN, integralweightk, and trivialcharacter.Here k is necessarily even and positive. It is well-known that Hecke eigenformsbehave well under the Fricke involution. More specifically, if f is a normalizedHecke eigenform of some level N, weight k and of trivial character, then wehave f| k W N = cg with c ∈ C × and g another normalized Hecke eigenform inthe same space (see Lemma 1.1 below or Theorem 4.6.16 in [6]). The Fouriercoefficients of g can be explicitly determined by that of f but the scalar c isleft mysterious in general.Question 1. Can we explicitly determine c with the information on f?Let f =P