The First Newform on Γ_{0}(N) such that Q(a_{n}) ≠ Q(…, a_{m}, …) for all n

William A. Stein

November 20, 2001

Abstract We compute the minimum N such that there is a newform f = ∑ a_{n}q^{n} ∈ S_{2}(Γ_{0}(N)) with the odd property that Q(a_{n}) ≠ Q(…, a_{m}, …) for all n.^{1}

1 The Newform

Using modular symbols, we computed the first few terms of each newform f = ∑ a_{n}q^{n} on S_{2}(Γ_{0}(N)) for N ≤ 512. For every single newform of level N < 512, we found that there is an integer n such that the field K_{f} = Q(…, a_{m}, …) generated by all Fourier coefficients of f equals the field Q(a_{n}), for some n. However, the characteristic polynomial of T_{3} on S_{2}^{new}(Γ_{0}(512)) is

(x^{2} − 6)^{2}(x^{2} − 2)^{4}(x^{2} + 4x + 2)(x^{2} − 4x + 2),

and if we let f = ∑ a_{n}q^{n} be one of the newforms in the four-dimensional kernel V of (T_{3}^{2} − 6)^{2}, then

f = q − \sqrt{6}q^{3} + 2\sqrt{3}q^{5} + 2\sqrt{2}q^{7} + 3q^{9} − \sqrt{6}q^{11} − 2\sqrt{3}q^{13} − 6\sqrt{2}q^{15} + ⋯

provides an example in which K_{f} ≠ Q(a_{n}) for some n. We use the following theorem of Shimura (see [1, Prop. 3.64]) to prove this assertion about f:

Theorem 1.1 (Shimura). Let N, r, s, k > 0 be integers such that s|N and let M be the least common multiple of N, r^{2}, and rs. Let χ (resp. Ψ) be a primitive character mod r (resp. s). If f = ∑ a_{n}q^{n} ∈ S_{k}(Γ_{0}(N), Ψ) then

f ⊗ χ := ∑ χ(n)a_{n}q^{n} ∈ S_{k}(Γ_{0}(M), Ψχ^{2}).

1Henri Cohen told me that phe discoved this example many years ago, and Elkies “explained” it to him. This example is not in the literature, and I found it indepedently of Cohen and Elkies.

Proposition 1.2. Suppose n ≥ 6 and let ε be a primitive character of conductor dividing 8. Then the map f ↦ f ⊗ ε preserves S_{k}(Γ_{0}(2^{n})).

Proof. Set r = 8 in Theorem 1.1, and note that ε^{2} = 1.

Proposition 1.3. We have K_{f} = Q(\sqrt{2}, \sqrt{3}) and the companions of f coincide with the twists of f by the four Dirichlet characters of conductor dividing 8.

Proof. The newform f and its companions lie inside of V. By “multiplicity one”, V has dimension 4 so [Q(f) : Q] ≤ 4. The characteristic polynomial of T_{7} on V is (x^{2} − 8)^{2} so a_{3} = ±\sqrt{6} and a_{7} = ±2\sqrt{2} so E = Q(\sqrt{2}, \sqrt{3}).

Let χ be a character of conductor dividing 8. By Theorem 1.1, f ⊗ χ lies in S_{2}(Γ_{0}(512)). In fact, it must lie in S_{2}^{new}(Γ_{0}(512)) because if f ⊗ χ lies in S_{2}(Γ_{0}(2^{n})) for n ≥ 6 then Theorem 1.1 implies that f = (f ⊗ χ) ⊗ χ lies in S_{2}(Γ_{0}(2^{n})), and certainly we can take n ≥ 6. So we’re done.

Alternatively, since the image of χ is contained in {±1}, the T_{3} eigenvalue of f equal ±\sqrt{6}. The factor (x^{2} − 6)^{2} exactly divides the characteristic polynomial of T_{3} on S_{2}(Γ_{0}(512)) so f ⊗ χ must be a companion of f.

Remark 1.4. f is the first example of a weight 2 newform having trivial character for which Q(a_{n}) is a proper subfield of E for all n.

Acknowledgment: It is a pleasure to thank Kevin Buzzard and Ken Ribet for helpful comments.

References

[1] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, (1994).