Atlas › Modular forms › Weight 2 › Level 688 › 688.2.a › 688.2.a.f

Newform orbit 688.2.a.f

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 4q^{5} + 4q^{7} - 2q^{9} + 2q^{11} + 2q^{13} - 4q^{15} + 10q^{17} + 4q^{19} - 4q^{21} - 2q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{13}$ , whose minimal polynomial is

$x^{2} - 64x + 1006$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 688$ )
43	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	0	0	4	4	2	2	10	4	−2	0	6	0	−2	−2	−12	22	4	8	−2	12	−24	−4	−18	−12	−2

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.345 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 4, 8, 16, 43, 86, 172, 344, 688]: [0, 1, 2, 3, 4, 4, 13, 25, 48, 92] (certified at ell = 2147483353)
T = [(3, 1), (5, 3), (7, 5), (11, 7), (13, 9)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 6 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(688, 2)[5]   # 688.2.a.f

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 688:2 1000