Twists of newforms

AbstractLet Sk0(N, ψ) denote the subspace generated by newforms in the space of cuspforms of weight k and character ψ on Σ0(N). In this paper we study decompositions of Sk0(N, ψ) into direct sums of twists (by Dirichlet characters) of other spaces of newforms. Applied to individual newforms, these results immediately yield information on the behavior of newforms under character twists. Most of the results follow from applications of the Eichler Selberg formula for the traces of the Hecke operators. A version of this formula is given in the paper. A sample result is: Let p be a prime and let M be a positive integer prime to p. Let ω be a character mod pv with e = ordpf(ω) > v2 and let φ be a character mod M. Then Sk0(pvM, ωφ) = ⊕χ Sk0(peM, ωχ2φ)χ where the sum ⊕χ is over all primitive characters χ modulo pv − e and where Sk0(N, ψ)χ denotes the twist of Sk0(N,ψ) by χ