Hello, I've found that the modular symbols code in Sage may detect interesting examples of torsion modular forms. For example:
G=DirichletGroup(13); e=G.gen()^6; # e has order 2 M=ModularSymbols(e,2,-1); # M is 0 ep=e.change_ring(GF(3)); Mp=ModularSymbols(ep,2,-1); # Mp is one-dimensional I think this is coming from the famous counterexample to the naive version of Serre's conjecture: there is a mod-3 modular form of weight 2 and level 13 with quadratic character such that any lift to characteristic zero has non-quadratic character. This is something peculiar to mod-p forms for p=2 or 3. But it also sometimes gives confusing results: M=ModularSymbols(7,8,1).cuspidal_subspace(); # M has dimension 3 Mp=ModularSymbols(7,8,1,GF(5)).cuspidal_subspace(); # M has dimension 4 I don't think this should happen: mod-5 modular forms of weight 8 and level Gamma0(7) should lift to characteristic 0 forms of the same type. I assume that what is happening is that Mp is not really computing H^1(X_0(7),\F_5)^+, which is I thing I think modular symbols should be. Math question: Where is this extra dimension coming from? Does it have any interesting number theoretic meaning? Sage question: Is this a bug? Should Sage warn us that, with finite-field coefficients, ModularSymbols might not be computing the thing I think it is computing? Best wishes, Preston -- You received this message because you are subscribed to the Google Groups "sage-nt" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-nt/18758bbb-b636-4db9-815f-ebe79354fbb4%40googlegroups.com.
