Serre mod pq table

In Table 1.6 we tabulate computations that address Mazur's letter (see Section 1.1). We analyzed each pair f, g of newforms of levels N, M, with $NM\leq 671$. Each pair of rows in the table corresponds to a pair of forms f, g. Let $h_1,\ldots, h_n$ be representatives for the Galois conjugacy classes of newforms at level NM. The ``Congruences'' column is a block of columns that contain, in order, the primes p,q such that $f\equiv h_i\pmod{p}$ and $g\equiv h_i\pmod{q}$. The ``Irreducible nonsuccess'' column lists the pairs p,q, with p and q both non-Eisenstein primes listed in column two, such that $\rho_{f,p}\times\rho_{g,q}$ does not arise from a modular form of level NM. The third column is empty exactly when we have maximal success; this occurs for only 4 of the 23 pairs f,g. The table gives only a small part of the full computation, which covers all pairs f,g such that NM<2000. [[A summary of computation up to level 2000 will go here.]]

  

Table: Serre mod pq data for $NM\leq 671$