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.