Problems

Let $f$ be a newform in $S_2(\Gamma_0(N))$ (or, more generally, in $S_k(\Gamma_1(N))$ ). Let $\lambda$ be a prime ideal in the ring generated by the Fourier coefficients of $f$ . Let $n\geq 1$ be a positive integer.

Problem 7.1.1   Let $E$ be the elliptic curve 11a given by the equation

$$y^2 + y = x^3 - x^2 - 10x - 20,$$

and let $f=f_E=q - 2q^{2} - q^{3} + \cdots \in S_2(\Gamma_0(11))$ be the corresponding newform. For

$$r = 9, \quad 27,\quad 7^2$$

compute all newforms $g \in S_2(\Gamma_0(11q))$ with $q<500$ prime and $g\equiv f\pmod{r}$ . Is there a pattern?

Problem 7.1.2   Formulate a level raising conjecture modulo $\lambda^n$ . Provide computational and theoretical evidence.

Problem 7.1.3   Formulate a level lowering conjecture modulo $\lambda^n$ . Provide computational and theoretical evidence.