- Let $f$ be modular of level $p^nN$, $(p,n) = 1$, $p > 2$ with character $\chi\psi\eta$, where $\chi$ has conductor dividing $N$, $\psi$ conductor power of $p$ and order power of $p$, and $\eta$ conductor $p$ and order dividing $p-1$. Since $p$ is odd, one can write $\psi = \xi^{-2}$. (i) Why is the character of $f \otimes \xi$ equal to $\chi\eta$; (ii) why is the reduction mod $p$ of this equal to the reduction of $f$; (iii) why is for $r \gg n$ $f \otimes \xi$ modular with respect to $\Gamma_0(p^r) \cap \Gamma_1(pN)$? Can this be proven using the converse theorem?

2.a Why is the twisted Eisenstein series $G = a_0 + \sum_{n=1}^\infty\sum_{d \mid n}\eta^{-1}(d)d^{i-1}q^n$ of Nebentypus $\eta^{-1}$ with respect to $\Gamma_0(p)$?

2.b Why is $fG$ modular with respect to $\Gamma_0(p^r) \cap \Gamma_1(N)$, wenn $f \in S_k(\Gamma_0(p^r) \cap\ \Gamma_1(pN), \chi\eta)$ ist?

(The article is here: math.berkeley.edu/~ribet/Articles/motives.pdf )