3 problems

Let $k<p-1$, and let $f$ and $g$ be two newforms of level $\Gamma_0(p)$. Write $\varepsilon(f)$ and $\varepsilon(g)$ for their Atkin--Lehner signs, and say that the forms are congr…

Fix $k,r$ and $0\le m<r$, and fix primes $p_1<\dots<p_m$. Let $\mathcal N$ be the set of squarefree levels $N=p_1\dots p_r$ with $r$ prime factors, where the first $m$ primes are f…

Let $(\mathcal N,\mathcal Q)$ be an arithmetically compatible sequence of $(N,Q)$'s of Type I, II or III as above, and fix a weight $k$. Let $\mathcal F$ be the family of newforms…