4 problems

Let $N_1$ and $N_2$ be coprime squarefree positive integers, with $\omega(N_1)$ odd, and let $E_4$ be the weight-$4$ Eisenstein series on $\operatorname{SL}_2(\mathbb Z)$. Squarefr…

Let $k$ be the weight and let the preceding result give a congruence between a weight-$2k$ Eisenstein series and a newform at prime level. Let $N$ be a squarefree level, with the l…

Let $k \ge 2$ be an integer, let $E$ be an imaginary quadratic field of discriminant $D$, and let $\ell=1$ or a prime with $\ell\nmid D$. Let $\lambda_q=\lambda_q(k)$ be the quanti…

Let $\tilde{\Pi}$ denote an irreducible, cuspidal, tempered automorphic representation of $G({\mathbb A})$, and let $T_{\mu}$ be the normalized Hecke operator associated with…