Mod 2 eigensystems not defined over Z/2--looking for simple examples Consider the weight 2 newform 67.2.1 b in the LMFDB table. It is defined over Q(root 5), and reducing modulo the inert prime (2) we get a mod 2 eigensystem defined over an extension of Z/2 but not over Z/2 itself. Are there N smaller than 67 for which such mod 2 eigensystems of level Gamma_0 (N) exist?
Edit--(a)---It seems that any one of the 4 conjugate weight 2 newforms of level Gamma_0 (47) will work. (2) is inert in the field of definition, with residue class field GF(16). The a_n for n odd all appear to have reduction lying in GF(4), and the resulting mod 2 eigensystem takes as values just the elements of GF(4). But can one do still better?
(b)---Here's a proof that the weight 2 newform f of level Gamma_0 (23) doesn't work. Let u--> u* be the automorphism taking root (5) to -root (5). Then 
(1/root 5)(f-f*) is of weight 2 for Gamma_0 (23), has coefficients in Z, and is divisible by q^2. It follows that it is an integer multiple of the square of the expansion of eta(z)*eta(23z); since the coefficient of any q^n, n odd, in this square is an even integer, we find that the coefficient of such a q^n in f is 0 or 1 mod 2. The same argument should apply in level 31, but now we replace eta(z)*eta(23z) by the weight 1 newform for Gamma_1 (31).
(c)---Thanks, Kimball, for your suggestions. But sometimes one has to go to weights bigger than 2 to get all the mod 2 eigensystems. I wonder if (apart from the eigensystem where every a_p is 0, and weight 12 is needed) weights 2,4 and 6 always suffice.
 A: The prime levels $N < 67$ such that there is a newform of weight 2 for $\Gamma_0(N)$ with non-rational Fourier coefficients are 23, 29, 31, 41, 43, 47, 53 and 61.  You can just examine their Fourier coefficients to find such examples.  Note $N=29$ (coefficient field $\mathbb Q(\sqrt 2)$) gives you an example where the odd prime eigenvalues are not integers mod 2.  Namely, looking at the Fourier expansion of this form, we see $a_3 = -\alpha$ and $a_{31} = -5\alpha + 2$, where $\alpha = -1 \pm \sqrt{2}$.
In levels 23 (as you observed) and 31, it appears that the odd prime eigenvalues are congruent to 0 or 1 mod 2.  I didn't check the other levels up to 67.
A: A bit surprisingly, the 47 mentioned in my edit can be lowered to 37; the construction of the mod 2 eigensystem is close to the construction I describe in my comments on Kimball's answer. I doubt that further improvement is possible.
This time O is the ring of integers in K=Q(root(-74)). The class group is Z/10 and psi is an ideal-class character of order 5. Now we get a weight 1 newform  f with character the character chi attached to K; when chi(p)=1, so that p=PP', then a_p= psi(P) + psi(P'). Once again T_2(f^2) is a modular form giving rise by reduction to the desired mod 2 eigensystem.
The hitch is that chi has conductor 296=8*37, so that our form T_2(f^2) is of level Gamma_0 (296). But there are arguments going back to Serre, that allow one to replace our form by a form with the same mod 2 reduction but of level Gamma_0 (37), at the expense of a possible increase in weight.
The final description of the eigensystem is this. Let GF(4) be {0,1,d,d^2}. If chi(p)= -1, then a_p is 0. Suppose (chi)p=1, so that p is represented by a reduced positive definite form of discriminant -296. If the form is (1,0,74) or (2,0,37) then a_p is 0. If the form is (3,2,25), or (6,4,13) a_p is d. If the form is (5,2,15) or (9,8,10), a_p is d^2.
It's not obvious what the weight of the newform giving our eigensystem is, and the LMFDB database is not at first sight helpful. But indeed our eigensystem comes from the horrible looking newform 37.4.1a defined over Q(a) where a is a root of x^4+6*x^3-x^2-16*x+6 = 0. If tau, lying in the degree 2 unramified extension of the field of 2-adic numbers satisfies tau^2 + tau -1 = 0, then our fourth degree equation has a root a lying in this unramified extension that is
congruent to 3+ 8*tau mod 16. Using the formulas for the coefficients of the newform given in the database, together with the Sturm bound, one can show that our eigensystem comes from this wight 4 newform.
EDIT____Alex Ghitza has recently calculated all the mod 2 eigensystems of level Gamma_0 (N) for N odd and < 100. (In particular his calculations show that no lowering of my 37 is possible). Perhaps he'll say something about his methods and results here?
