Conjectures

Conjecture 6.1   Suppose $p$ is a prime and $k\geq 4$ is an even integer. If

$$(p,k) \not\in \{ (2,4),(2,6),(2,8),(2,10),$$

$$(3,4),(3,6), (3,8),$$

$$(5,4), (5,6), (7,4), (11,4)\}$$

then $d_k(\Gamma_0(p))>0$.

Frank Calegari outlined a possible strategy for proving this conjecture.

Conjecture 6.2   Suppose $p>2$ is a prime and $k\geq 3$ is an integer. If

$$(p,k) \not\in \{ (3,3),(3,4),(3,5),(3,6),(3,7),(3,8),$$

$$(5,3),(5,4), (5,5), (5,6), (5,7)$$

$$(7,3), (7,4), (7,5), (11,3), (11,4), (11,5),$$

$$(13,3), (17,3), (19,3)\}$$

then $d_k(\Gamma_1(p))>0$.