Atlas › Modular forms › Weight 2 › Level 686 › 686.2.a › 686.2.a.b

Newform orbit 686.2.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 3q^{2} + q^{3} + 3q^{4} - q^{6} - 3q^{8} - 4q^{9} + q^{12} + 9q^{13} + 3q^{16} + 4q^{17} + 4q^{18} + 13q^{19} + 7q^{23} - q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{3} - x^{2} - 65x + 169$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
7	— ( $7^2 \mid 686$ )

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
$\operatorname{tr} a_p$	−3	1	0	0	0	9	4	13	7	3	16	6	11	−11	−8	−3	−4	31	23	7	27	−32	−11	10	5

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.354 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, 7, 14, 49, 98, 343, 686]: [0, 1, 1, 4, 5, 16, 33, 100] (certified at ell = 2147479489)
T = [(3, 1), (5, 3)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 4 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(686, 2)[1]   # 686.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 686:2 1000