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2 clarifications

Joel -- it's difficult to work out what you wantyou're asking. Of course both possibilities can occur, as Wanax said. Furthermore both possibilities can occur even for the same modular form. For example, if you consider the 2-adic representation attached to Ramanujan $\Delta$ then there is a lattice for which $\overline{\rho}(c)=1$ and another lattice for which it is not 1 (in fact for any quadratic extension of the rationals with discriminant a power of two there is some lattice which gives rise to the reducible non-semisimple representation with kernel corresponding to this field). And already in Serre's 1987 paper on his conjecture he observes that for any $S_3$ extension of $\mathbf{Q}$, totally real or not, it will be modular, and for $A_5$ extensions, totally real or not, computationally they seem to work too (thanks to Mestre).

If you gave me a form in practice, I think that I would do a global calculation to try and locate the kernel of the mod 2 representation (you know where it's ramified, you know an upper bound for the degree and you know lots about how primes split so you can use tables to find the kernel), and then ask if it's totally real or CMnot. Certainly I don't know a local method..method (which uses only information at 2 and infinity) and I'm not sure it would be reasonable to expect one...

1

Joel -- it's difficult to work out what you want. Of course both possibilities can occur, as Wanax said. Furthermore both possibilities can occur even for the same modular form. For example, if you consider the 2-adic representation attached to Ramanujan $\Delta$ then there is a lattice for which $\overline{\rho}(c)=1$ and another lattice for which it is not 1 (in fact for any quadratic extension of the rationals with discriminant a power of two there is some lattice which gives rise to the reducible non-semisimple representation with kernel corresponding to this field). And already in Serre's 1987 paper on his conjecture he observes that for any $S_3$ extension of $\mathbf{Q}$, totally real or not, it will be modular, and for $A_5$ extensions, totally real or not, computationally they seem to work too (thanks to Mestre).

If you gave me a form in practice, I think that I would do a global calculation to try and locate the kernel of the mod 2 representation (you know where it's ramified, you know an upper bound for the degree and you know lots about how primes split so you can use tables to find the kernel), and then ask if it's totally real or CM. Certainly I don't know a local method...