Swinnerton-Dyer type congruences for certain Eisenstein series (Q2782407)

A theorem of Swinnerton-Dyer states that a normalized Eisenstein series \(E_k(z)\), which is a modular form of weight \(k\) with respect to \(SL(2,\mathbb{Z})\), satisfies the congruence \(E_k(z)\equiv 1\pmod p\) for prime \(p\geq 5\) if and only if \(k\equiv 0\pmod {p-1}\). The author generalizes this result to certain Eisenstein series in spaces of modular forms of weight \(k\) on a congruence subgroup of type \(\Gamma_0(N)\) with Nebentypus character \(\chi\).NEWLINENEWLINEFor the entire collection see [Zbl 0980.00024].