Suggestion for database of modular symbols

[chrisxu3]

I would like to suggest storing a database of modular symbols that encode a basis for the Hecke decomposition of S_k(\Gamma(N)), for various values of k and N. They are needed in order to compute q-expansions on arbitrary congruence subgroups: note that if H is a subgroup of GL_2(Z/NZ), it is generally not the case that the q-expansion coefficients of eigenforms on X_H are the Hecke eigenvalues (although it is true e.g. for H = GL_2(Z/NZ)). So one instead has to start with \Gamma(N) and then take H-invariants. (See Assaf's paper on computing modular forms on arbitrary subgroups, as well as Box's thesis which Assaf's paper links to.)

It takes a long time to perform the Hecke decomposition since S_k(\Gamma(N)) gets large quickly. But if the user is able to simply grab them off the LMFDB, the decomposition will only need to be performed once.

[assaferan]

I'm not sure this is really a CMF issue, as we have decided to tabulate CMFs with respect to $\Gamma_0(N)$. These are relevant for the Modular Curves pages, as these are used for the equations of modular curves. However, we have decided against including the actual $q$-expansions around cusps of the modular forms for each congruence subgroup (that would mean an exceptional abuse of storage).

On the other hand, I'm not sure we would like to tabulate these modular symbols at all, as computing them is linear in the dimension, which implies that for values of $k,N$ for which computation is too difficult, storage will also be an issue.