A Conjecture About Nonvanishing of Twists

Fix $E$ and suppose $p$ is rigid for $E$. For every $\ell\equiv 1\pmod{p}$, fix

$$\chi_{p,\ell} : (\mathbb{Z}/\ell\mathbb{Z})^* \rightarrow\!\!\!\!\rightarrow \boldsymbol{\mu}_p$$

of order $p$ and conductor $\ell$.

Conjecture 3.1 (-)   There exists a prime $\ell\nmid N_E$ such that

$$L(E,\chi_{p,\ell},1)\neq 0$$

and

$$a_\ell(E)\not\equiv \ell+1\pmod{p}.$$

Evidence: The conjecture is true for every pair $(E,p)$ I've tried, e.g., for all rigid $p<50$ for the first $20$ rank $1$ optimal quotients of $J_0(N)$ and the first two rank $2$ quotients.

The following ``Density Conjecture'' will not be needed for our application:

Conjecture 3.2 (-)   The set of primes $\ell\equiv 1\pmod{p}$ such that $L(E,\chi_{p,\ell},1)=0$ has Dirichlet density 0 amongst all primes.