Galois representations attached to eigenforms with nebentypus

A Possible Generalization of Maeda’s Conjecture

“…CM forms of trivial Nebentypus (and therefore even weight). The interested reader should consult [Rib77,§3] Let K be an imaginary quadratic field, O = O K its ring of integers and m an ideal of K. Let us also denote by J m the group of fractional ideals of O K that are coprime to m. A Hecke character ψ of K, of modulus m, is then a group homomorphism ψ : J m −→ C * such that there exists a character…”

“…The only possible group homomorphisms ψ ∞ are then of the form σ u , where σ is one of the two conjugate complex embeddings of K and u is a non negative integer. The eigenform associated with the Hecke character is new if and only if ψ is primitive (see Remark 3.5 in [Rib77]). …”

“…It turns out that one has #(O/m) * ϕ(|D|) different finite types whose values on integers match those of the Kronecker symbol. Out of all these choices, one must consider only those for which ψ f is of conductor m, in order for the corresponding cuspforms to be new (see Remark 3.5 in [Rib77]). …”

Sur la lacunarité des puissances de η

“…II s'agit done d'une lacunarite tres peu marquee, bien moindre que dans les cas r = 1 et r = 3, ou M r (x) est de l'ordre de grandeur de Vx. On a i-(n)^0 pour l*£n=Sl0 15 .…”

“…Soit p le plus petit entier strictement positif s£lO 15 pour lequel r(p) = 0, s'il en existe. C'est un nombre premier ( [6], th.…”